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abstract: 'We have studied the optical properties of four (LaNiO$_3$)$_n$/(LaMnO$_3$)$_2$ superlattices (SL) ($n$=2, 3, 4, 5) on SrTiO$_3$ substrates. We have measured the reflectivity at temperatures from 20 K to 400 K, and extracted the optical conductivity through a fitting procedure based on a Kramers-Kronig consistent Lorentz-Drude model. With increasing LaNiO$_3$ thickness, the SLs undergo an insulator-to-metal transition (IMT) that is accompanied by the transfer of spectral weight from high to low frequency. The presence of a broad mid-infrared band, however, shows that the optical conductivity of the (LaNiO$_3$)$_n$/(LaMnO$_3$)$_2$ SLs is not a linear combination of the LaMnO$_3$ and LaNiO$_3$ conductivities. Our observations suggest that interfacial charge transfer leads to an IMT due to a change in valence at the Mn and Ni sites.'
author:
- 'P. Di Pietro$^1$, J. Hoffman$^2$, A. Bhattacharya$^2$, S. Lupi$^3$, A. Perucchi$^1$'
date:
-
-
title: 'Spectral weight redistribution in (LaNiO$_3$)$_n$/(LaMnO$_3$)$_2$ superlattices from optical spectroscopy'
---
Recent progress in the growth of correlated oxide heterostructures has sparked interest in understanding and exploiting the novel electronic and magnetic properties that emerge at the interfaces between different material systems [@mannhart]. In transition metal oxides, electronic orbitals tend to overlap less than the $s$- and $p$- orbitals in semiconductors, resulting in strong electronic correlations [@imada], higher values of the effective mass of charge carriers, and larger coupling to the polarized lattice. Correlation effects can be further enhanced at oxide interfaces, since the rupture of the periodicity of the ion lattice at the interface may reduce the electronic screening and thus increase the on-site Coulomb interaction. Interfacial charge redistribution in these systems results in the reconstruction of orbital and spin degrees of freedom (electronic reconstruction) producing correlated two-dimensional (2D) states with novel electronic and magnetic behaviors [@bhatta].
We address here (LaNiO$_3$)$_n$/(LaMnO$_3$)$_2$ superlattices [@hoffman13], where the subscript denotes the layer thickness in terms of pseudocubic unit cells. X-ray spectroscopy of this system shows that the Mn oxidation state is converted from 3+ to 4+, while Ni is intermediate between 2+ and 3+ [@hoffman13]. A metal to insulator transition is observed as $n$ is decreased from 5 to 2. Charge transfer at the LNO/LMO interface therefore provides the opportunity to control the interplay between magnetism and charge transport in a Ni based oxide, as highlighted by several recent theoretical [@lee; @dong] and experimental [@gibert; @hoffman13] studies.
LaNiO$_3$ (LNO) is the only member of the perovskite rare-earth nickelate series that does not undergo a metal to insulator transition at low temperatures (T) [@torrance92]. Due to the low-spin 3$d^7$ configuration of Ni$^{3+}$, it has been proposed that the combination of LNO with other oxides in some superlattice (SL) structure may provide the possibility to mimic the CuO$_2$ planes of high-temperature superconductors [@chaloupka08; @hansmann09]. LaMnO$_3$ (LMO) on the other hand, is an antiferromagnetic insulator, known for being the parent compound of colossal magnetoresistance manganites [@tokura99; @tokura06].
In this work we have measured the reflectivity of four (LNO)$_n$/(LMO)$_2$ SLs ($n$=2, 3, 4, 5) together with a LNO thin film, from 100 to 16000 cm$^{-1}$ at nearly normal incidence and between 20 and 400 K. The measurements were performed by means of a BRUKER 70v interferometer, using a gold mirror as a reference and various beamsplitters, detectors and thermal sources to cover the whole infrared range. The samples were grown by molecular beam epitaxy, as previously described in Ref. [@hoffman13]. LNO and LMO thin films prepared under the same conditions as the superlattices were found to possess electronic and magnetic properties comparable to high-quality bulk single-crystal samples, suggesting the samples in this study are stoichiometric. By varying the number of bilayers within each superlattice, the total thickness of the samples was kept constant at about 64 nm, as confirmed by x-ray reflectivity measurements. Our reflectivity measurements elucidate the mechanism behind the metal insulator transition with decreasing $n$. In particular, we observe a systematic shift in the spectral weight away from the Drude peak, and also from higher frequencies (near-infrared and above) into the mid-infrared, with the onset of insulating behavior as $n$ decreases from 5 to 2. Our observations are consistent with a localization mechanism that is driven by interfacial transfer of electrons from LMO into LNO.\
In Fig. 1 we report the reflectivities of the four SLs, together with that of LNO, from 100 to 8000 cm$^{-1}$. Due to the small thickness of the SLs, the reflectivity is dominated by the STO substrate as shown by the gray line in Fig. 1a. Below 800 cm$^{-1}$, i.e., in the far-infrared region characterized by the large STO phonon band, the reflectivity decreases with increasing $n$. On the other hand, the opposite occurs in the mid-infrared (800-8000 cm$^{-1}$), where the reflectivity of the SL is enhanced with respect to the substrate. A similar effect was previously observed in the (LaMnO$_3$)/(SrMnO$_3$) series [@perucchi], where an increase (decrease) in reflectivity in the mid- (far-) infrared range is attributed to higher conductivity properties. The reflectivity of the $n$=2 superlattice does not vary significantly with temperature. As shown in Fig. 1b, the temperature dependence of the $n$=3 compound is essentially confined to frequencies above 1000 cm$^{-1}$. As $n$ increases, the observed T-dependence progressively moves towards the far-infrared range, with the $n$=5 and pure LNO samples being independent of temperature above 2000 cm$^{-1}$.
![Reflectivity of the LNO/LMO superlattices. a) Room temperature reflectivities of the $n$=2,3,4,5 samples together with the reflectivity of pure LNO and that of the STO substrate. b) Temperature dependence of the reflectivity of the $n$=3 compound. c-e) Same for the $n$=4,5 and for pure LNO. Dashed lines correspond to intermediate temperatures 100, 150, 200, 250, and 350 K. In the inset of panel c) the data at 20 K and the relative fit are reported in the whole range.[]{data-label="Fig.1"}](fig1.eps){width="9cm"}
The optical conductivity of the LNO/LMO superlattices can be extracted through a fitting procedure based on the Kramers-Kronig consistent Lorentz-Drude model [@kuzmenko05], by taking into account the finite thickness of the sample and the substrate’s contribution to the reflectivity [@perucchi]. An overview of all the room temperature optical conductivities is provided in Fig. 2a, after subtracting the phonon modes of the SL. This helps to single out the electronic contribution to the optical conductivity.
For the $n$=2 compound, the optical conductivity smoothly tends to zero in the dc limit, thus indicating insulating behavior. However, the onset of a mid-infrared band is clearly present. We note that in pure LMO the first absorption band is centered at about 2.5 eV (20000 cm$^{-1}$), while no spectral weight is found in the mid-infrared region [@okimoto97]. At larger values of $n$, the optical conductivity increases in the whole infrared range, as a signature for the onset of metallization. For the $n$=5 compound, a clear Drude term is present, superimposed on an almost flat background infrared conductivity. Interestingly, the $n$=5 SL differs from pure LNO, which displays a sharper Drude peak and lower mid-infrared spectral weight.
In panels b to e in Fig. 2, we report the T-dependence of the optical conductivities in the various compounds. For low $n$, the T-dependence is more pronounced in the mid-infrared range, while with increasing $n$, the T-induced changes gradually shift towards the far-infrared. We observe a non-negligible increase of the mid-infrared spectral weight, with increasing temperature for the low $n$ SLs. For larger $n$, as well as for pure LNO, the Drude peak lowers and slightly broadens, with increasing temperature.
![Optical conductivity of the LNO/LMO superlattices as extracted from a Lorentz-Drude fitting. a) Room temperature optical conductivities of the $n$=2,3,4,5 samples and pure LNO and LMO. b) Temperature dependence of the optical conductivity of the $n$=3 compound. c-e) Same for the $n$=4,5 and for pure LNO. []{data-label="Fig2"}](fig2_v3.eps){width="9cm"}
The LNO film displays metallic properties, with dc conductivity values reaching up to 10$^{4}$ ($\Omega$.cm)$^{-1}$, in good agreement with previously reported values of resistivity [@hoffman13]. A clear Drude peak coexists with a broad electronic background associated with interband transitions within the Ni $3d$ manifold [@basov]. The plasma frequency ($\omega_p$), calculated as the spectral weight underlying the Drude peak, is about 9000 cm$^{-1}$. As displayed in Fig. \[Fig2b\], the temperature dependence of the spectral weight $W=\int_0^{\Omega}\sigma_1(\omega)d\omega$, integrated up to $\Omega=\omega_p$, follows the quadratic relation $W(T)=W_0-BT^2$ [@ortolani05; @toschi05]. The $T^2$ dependence of $W(T)$ is a general feature of metallic systems, that finds a natural explanation within a tight-binding approach, through the temperature dependence of the kinetic energy. In conventional metals, the energy scale set by $\omega_p$ should be large enough to fully recover the conservation of the spectral weight, resulting in a vanishingly small $B$ [@ortolani05]. The parameter $b=B/W_0$ gauges the presence of correlation effects [@baldassarre08], which extend the temperature dependence of the spectral weight to energy scales larger than $\omega_p$. For LNO $b=3.2\times$10$^{-7}$K$^{-2}$, a value which is sizably larger than that found in gold (1.3$\times$10$^{-8}$K$^{-2}$) [@ortolani05], and comparable to La$_{2-x}$Sr$_2$CuO$_4$ (2.5$\times$10$^{-7}$K$^{-2}$) [@ortolani05]. An alternative method to estimate correlations is to calculate the ratio of kinetic energies obtained from experiment and from band theory $K_{exp}/K_{band}$, which ranges from 0 for a Mott insulator, to 1 for conventional metals [@qazilbash09]. $K_{exp}$ is proportional to the spectral weight due to the Drude term. By integrating the optical conductivity up to 500 cm$^{-1}$ (which roughly corresponds to a cut-off for the Drude’s spectral weight [@ouellette10]) and employing the LDA (Local Density Approximation) data from Ref. [@ouellette10] we find $K_{exp}/K_{band}\sim0.2$ for LNO, similar to underdoped and optimally doped cuprates [@qazilbash09].
![Temperature dependence of the spectral weight $W(T)$ integrated up to $\omega_p$, for the $n$=3,4,5 and LNO compounds. For $n$=3 and $n$=4, $W(T)$ increases with increasing temperature, thus signaling the presence of incoherent excitations at low energies. For both $n$=5 and LNO the spectral weight can be fitted (continuous line) according to the quadratic relation $W(T)=W_0-BT^2$.[]{data-label="Fig2b"}](fig2_b_v3.eps){width="9cm"}
The optical conductivity of the $n$=5 SL presents many analogies with pure LNO. Since a Drude peak is also well distinguishable in the data, we can again estimate the plasma frequency to be about 8500 cm$^{-1}$, roughly the same as for LNO. A $T^2$ behavior of $W(T)$ is found at temperatures between 150 and 400 K, with $b=1.6\times$10$^{-7}$K$^{-2}$. Below 150 K, which notably corresponds to the magnetic ordering temperature of bulk LMO and LNO/LMO SLs [@hoffman13], a clear change is observed in the $T^2$ slope of $W(T)$, thus suggesting a coupling between magnetism and carrier density. The overall spectral weight is significantly larger in the $n$=5 compound, due to a larger mid-infrared conductivity.
In the compounds with $n<$5, the separation of the Drude term from the incoherent electronic background is more arbitrary. We can roughly estimate $\omega_p$ of about 5000 and 4000 cm$^{-1}$ for the $n$=4 and $n$=3 compounds respectively. $W(T)$ does not follow a quadratic law, and even increases with increasing temperature (panels a and b in Fig. \[Fig2b\]), thus highlighting the increased role of incoherent excitations in shaping the infrared conductivity.
![Comparison between the room temperature $\sigma_{1,n}$, and $\sigma^*_{1,n}$ (see text) for $n$=2,3,4,5. The blue (red) area corresponds to spectral weight lost (gained) by the SL in the far- (mid-) infrared range due to the presence of interfacial effects. The green area highlights the transfer of spectral weight from higher energies.[]{data-label="Fig3"}](fig3_v3.eps){width="9cm"}
In order to better understand the present phenomenology, we introduce the average optical conductivity $\sigma^*_1$, defined as a weighted linear combination of the optical conductivity of the single constituents of a given SL: $$\sigma^*_{1,n}=\frac{n\cdot\sigma_{1,LNO}+2\cdot\sigma_{1,LMO}}{n+2}.$$ Since $\sigma^*_1$ disregards interfacial effects, the comparison between the measured $\sigma_{1,n}$ and $\sigma^*_{1,n}$ allows us to single out the features directly induced by the presence of the interfaces (Fig. \[Fig3\]).
The presence of the interface induces a loss of spectral weight $\Delta SW$ in the low frequency side, for all SL compounds (blue area in Fig. \[Fig3\]). From this quantity, we can calculate the charge density ($\Delta N^{FIR}$) involved in such spectral weight redistribution by assuming $\Delta SW=\sqrt{\frac{4\pi \Delta N^{FIR}e^2}{m}}$, where $m=m_e$. As shown in the inset of Fig. \[Fig3\]c, $\Delta N^{FIR}$ increases monotonically with decreasing $n$, i.e., by increasing the number of interfaces in the SL.
The interfaces are also responsible for the piling up of new spectral weight at mid-infrared frequencies (red area in Fig. \[Fig3\]). The mid-infrared spectral weight redistribution can not be systematically estimated for all compounds, since for $n$=2 and $n$=5, the red area extends above our maximum frequency limit (16000 cm$^{-1}$). However, we can safely conclude that the red area is always larger than the blue one (note log scale on x axis in Fig. \[Fig3\]). This indicates that the additional mid-infrared spectral weight is only in part due to the loss of spectral weight at low frequencies, and that a redistribution from the high frequency side (green area in Fig. \[Fig3\]) is also at play.
![Schematics of the spectral weight redistribution in LNO/LMO. a) Transfer of spectral weight from coherent (blue) to in-coherent (red) excitations in LNO, due to the reduction in the Ni oxidation state. b) Spectral weight transfer in LMO from the high energy Jahn-Teller band (green) to the mid and far infrared (red), due to the increased Mn valence. c) Sketch of the spectral weight transfer in LNO/LMO SLs, due to the valence mixing associated with interfacial effects.[]{data-label="Fig4"}](fig4_v3.eps){width="9cm"}
The optical data can be understood within the qualitative model depicted in Fig. \[Fig4\]. In the superlattice compound, with decreasing $n$, the valence of Ni decreases from its nominal Ni$^{3+}$ value typical of stoichiometric LaNiO$_3$. The reduction in the Ni oxidation state is known [@sanchez96] to drive a metal to insulator transition already at values about +2.75. From transport measurements on LNO reduced in oxygen [@sanchez96], it is known that the semiconducting behavior is characterized by infrared activation energies (250 cm$^{-1}$ for Ni$^{2.75}$, and 1000 cm$^{-1}$ for Ni$^{2.5}$). We thus expect that the introduction of LMO layers in the SL, by reducing Ni valence, progressively depletes the low energy, coherent spectral weight of LNO, which piles up in the infrared, as schematically shown in Fig. \[Fig4\]a.
On the other hand, LMO is an insulator, with the first optical absorption occurring at about 2.5 eV [@okimoto97], corresponding to the Jahn-Teller splitting of the $e_g$ band. When the Mn valence increases from 3+ to a higher oxidation state (as in La$_{1-x}$(Sr/Ca)$_x$MnO$_3$), polaronic states appear in the mid-infrared, whose spectral weight is taken from the Jahn-Teller band at 2.5 eV (see Fig. \[Fig4\]b). At intermediate values of the Mn oxidation state ($x=0.3\div0.5$) a metallic state may also set in, due to double-exchange physics [@zener51; @millis95]. While we know from x-ray spectroscopy that Mn has mainly 4+ character in the SLs, the spectral weight increase observed below the magnetic ordering temperature in the inset of Fig. \[Fig2\]d, may be a hint at the survival of the double-exchange physics in the LNO/LMO series. Panel \[Fig4\]c, pictorially describes the effect of the charge transfer between Mn and Ni sites on the overall optical conductivity, thus providing a qualitative explanation for the piling up of spectral weight at mid-infrared frequencies, at the expense of the far-infrared, and high energy (visible/UV) sides. Interfacial charge redistribution was previously identified as a possible origin of changes to the mid-infrared spectral response in manganite [@perucchi; @choi11] and vanadate superlattices [@jeong11]. This is in contrast to what is observed in LNO/LaAlO$_3$ superlattices [@liu12; @benckiser11] and in ultrathin LNO films [@scherwitzl11], where localization is believed to occur due to dimensional confinement and enhanced correlations.
Infrared data show that the LNO/LMO superlattices display the presence of significant mid-infrared excitations that are not present in LNO or LMO alone. Since SLs are intrinsically ordered structures, we can rule out disorder as a possible source for the presence of such incoherent excitations. On the other hand, both electronic correlation and polaronic effects (or even more likely, a mixture of the two) can be invoked as the mechanisms underlying the piling up of spectral weight in the mid infrared range. By increasing $n$, it is possible to progressively enhance the coherent excitations, thereby tuning the degree of metallicity of the SL. For $n$=5 the low energy electrodynamics is dominated by coherent excitations, while some coupling with magnetic ordering takes place below 150 K. Ab initio calculations could be of help in better elucidating these points [@baldassarre08]. The LNO/LMO systems thus provide an exciting new platform to manipulate and control the interplay of electronic, magnetic and vibrational degrees of freedom in a disorder-free 2D oxide material.
This work was partially supported by Italian Ministry of Research (MIUR) program FIRB Futuro in Ricerca grant no. RBFR10PSK4. J.D.H and A.B. acknowledge support from Department of Energy, Office of Basic Energy Science, Materials Science and Engineering Division. Work at Argonne National Laboratory, including the use of the Center for Nanoscale Materials and Advanced Photon Source, was supported by the U.S. Department of Energy, Office of Basic Energy Sciences under contract number DE-AC02-06CH11357.
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abstract: |
Untraceable communication is about hiding the identity of the sender or the recipient of a message. Currently most systems used in practice (e.g., TOR) rely on the principle that a message is routed via several relays to obfuscate its path through the network. However, as this increases the end-to-end latency it is not ideal for applications like Voice-over-IP (VoIP) communication, where participants will notice annoying delays if the data does not arrive fast enough.
We propose an approach based on the paradigm of Dining Cryptographer networks (DC-nets) that can be used to realize untraceable communication within small groups. The main features of our approach are low latency and resilience to packet-loss and fault packets sent by malicious players. We consider the special case of VoIP communication and propose techniques for a P2P implementation. We expose existing problems and sketch possible future large-scale systems composed of multiple groups.
author:
- Christian Franck and Uli Sorger
bibliography:
- 'references.bib'
title: 'Untraceable VoIP Communication based on DC-nets'
---
Introduction
============
A few decades ago telecommunications were mainly realized using circuit switching, such that an electrical signal carrying the audio information could be sent from one correspondent to another. Nowadays it is more common that speech is digitized and then sent in small packets over a private network or over the internet.
In such real-time VoIP communication the quality of the user experience is dependent on the latency of the network. While some package-loss can be tolerated, the one-way end-to-end latency should ideally not exceed 150ms [@itu_g114]. To this effect a stream of small packets is sent at a fast pace, for instance around 50 packets per second [@BandwidthVOIP].
We are interested to have a system in which two users can communicate with each other via VoIP anonymously to the others. That means these two users know that they are speaking to each other, but nobody else can infer who is communicating with whom.
One approach would be to use an anonymisation system like the well-known TOR network [@dingledine2004tsg], or alternatively one of the more recently proposed systems like Drac [@danezis2010text] or Herd [@le2015herd]. However, in all these systems the security is based on the principle that the message is relayed several times on its way through the network, so that it is hard to distinguish who is the sender or the recipient of a message. One problem of this is that users have to trust the operators of intermediary relays and mixes that they are honest and do not disclose any information. Another problem is that this relaying increases the latency and the jitter, which reduces the quality of phone calls or video-conferences.
Another approach would be to use Dining Cryptographer Networks (DC-nets). Here, messages are not relayed but multiple players simultaneously send ciphertexts to the recipient, and then the recipient combines these ciphertexts to obtain a message. The recipient knows that one of the players must have sent the message, but he is not be able to distinguish which one. As opposed to relay based systems like TOR and mixing, the latency is lower, no central authority is required and might therefore be interesting for applications which require a low latency and for P2P scenarios.
However, in classical DC-nets the recipient can only recover the message once he has obtained all the ciphertexts. Furthermore, all these ciphertexts must be correct; if a malicious player provides faulty data the recipient cannot recover the message. Thus, if packets are late, lost or faulty, then no useful information is transmitted.
In this paper, we propose a protocol that is based on the paradigm of DC-nets but adapted for VoIP streaming. It enables two players out of a group of $n$ players to communicate without disclosing to anybody else that they are communicating. Our protocol is resilient to packet-loss and to faulty packets, and can be implemented using lightweight cryptography. We study the performance, discuss practical implementations and propose possible future work.
The paper is organized as follows. In Section 2 we describe our model and our security requirements. In Section 3, we derive our new protocol. In Section 4, we discuss practical considerations, e.g. an implementation as a P2P protocol. In Section 5 we consider the performance. In Section 6 we sketch possible future work, and in Section 7 we review related work. In Section 8 we conclude with some remarks.
Model and Definitions \[sec:Model-and-Definitions\]
===================================================
We consider the case where two players want to communicate and stay anonymous w.r.t. to the others, so our requirements are different from those of the original DC-net protocol. In the original DC protocol the recipient does not know the identity of the sender and the sender does not know the identity of the recipient, but in our model we assume that the two correspondent know each other’s identity.
Model
-----
We assume a communication network with $n$ players $P_{1},...,P_{n}$ and an aggregator $A$, and we consider the problem where two of these players, $P_{a}$ and $P_{b}$, want to communicate with each other such that nobody else can see that they are communicating.
After an appropriate setup, the messages are exchanged in successive transmission rounds which consist of two phases:
1. A collection phase where each player send data to the aggregator.
2. A broadcast phase during which the aggregator sends the aggregated data back to the participants.
This star topology is a well-known scheme for implementing a DC-net in networks without physical broadcast.
Security Properties
-------------------
In this section we define the properties of correctness and privacy we want to achieve.
#### Correctness {#correctness .unnumbered}
We say that a protocol is correct if, assuming the players $P_{a}$, $P_{b}$ and the aggregator $A$ participate correctly, it holds that:
- if the message of $P_{a}$ encoding $m_{a}$ reaches the aggregator $A$ in time and the result forwarded by the aggregator $A$ reaches $P_{b}$, then $P_{b}$ can compute $m_{a}$; and
- if the message of $P_{b}$ encoding $m_{b}$ reaches the aggregator $A$ in time and the result forwarded by the aggregator reaches $P_{a}$, then $P_{a}$ can compute $m_{b}$.
Thus, if the players $P_{a}$, $P_{b}$ and the aggregator $A$ behave correctly and their packets arrive timely, then $P_{b}$ and $P_{a}$ can respectively recover $m_{a}$ and $m_{b}$.
#### Privacy {#privacy .unnumbered}
We say that a protocol is private if a computationally bounded adversary controlling up to $n-3$ players (other than $P_{a}$ and $P_{b}$) and the aggregator $A$ cannot determine who are the two correspondents with a probability better than random guessing.
For a set of honest participants $H\subset\{P_{1},...,P_{n}\}$ there are $c=\left|H\right|\cdot(\left|H\right|-1)\cdot2^{-1}$ different pairs of correspondents. Thus a protocol is private if the adversary has a probability of guessing correctly the pair of correspondents which is not better than a random guess, i.e., with probability $1/c$.
A Protocol for Untraceable Streaming
====================================
To start we consider a simple protocol that meets the requirements from the previous section in an ideal situation and we progressively adapt it to end up with a protocol that can be used in a real-world scenario.
In the description we assume that the setups for the protocols have already been performed. Typically one of the correspondents will anonymously provide the other players with the required data using an anonymous channel akin to [@FranckMscThesis; @corrigan2013proactively].
A Simple Protocol
------------------
1. **Setup**: $P_{1},...,P_{n}$ respectively have secret keys $k_{1},...,k_{n}$ such that $k_{1}+...+k_{n}=0$. $P_{a}$ additionally has $m_{a}$ and $P_{b}$ additionally has $m_{b}$.
2. **Collection phase**: Every $P_{i}$ for $i\ne a,b$ sends $O_{i}=k_{i}$ to $A$. $P_{a}$ sends $O_{a}=k_{a}+m_{a}$ to $A$. $P_{b}$ sends $O_{b}=k_{b}+m_{b}$ to $A$.
3. **Broadcast phase**: $A$ broadcasts $X=O_{1}+...+O_{n}$ to $P_{1},...,P_{n}$. $P_{a}$ computes $m_{b}=X-m_{a}$ and $P_{b}$ computes $m_{a}=X-m_{b}$.
We assume that during the execution of the protocol
- the number of players remains constant,
- no messages are lost (or arrive too late), and
- all players are honest.
In this situation two players $P_{a}$ and $P_{b}$ can communicate with each other in full-duplex using the classic DC-net principle. This approach is described in detail in Protocol \[algo:1\]. All players are initially provided with secret keys $k_{1},...,k_{n}$ that sum up to $0$, so that the keys will vanish when all ciphertexts are aggregated. So if both $P_{a}$ and $P_{b}$ send during the same round, the result forwarded by the aggregator is the sum of their messages. By subtracting their own message from this result they can recover each other’s message.
The previous system obviously fails, if only one user does not send anything or if his packet is lost. This seems to be a very strong restriction of the protocol.
A Packet-Loss Resilient Protocol\[sub:A-Packet-Loss-Resilient\]
---------------------------------------------------------------
The problem of packet-loss is that if the aggregator does not receive all the packets, he can only make the sum $X$ over a strict subset of $O_{1},...,O_{n}$. This means that the keys will not cancel and $P_{a}$ and $P_{b}$ cannot recover the messages like in the previous scenario.
To overcome this problem we can modify the previous protocol as shown in Protocol \[protocol:2\]. During the initialization $P_{a}$ and $P_{b}$ are provided with the keys $k_{1},...,k_{n}$ of all players. Further the aggregator does not only broadcast the sum $X$, but also a list $L$ informing which packets he received. Said list $L$ informs the players $P_{a}$ and $P_{b}$ about which keys are included in the partial sum and since they know all the keys, they can subtract them from $X$ and recover the messages.
1. **Setup**: $P_{1},...,P_{n}$ respectively have secret keys $k_{1},...,k_{n}$. $P_{a}$ additionally has $m_{a}$ and $P_{b}$ additionally has $m_{b}$, and both know all $k_{1},...,k_{n}$.
2. **Collection phase**: Each $P_{i}$ for $i\ne a,b$ sends $O_{i}=k_{i}$ to $A$. $P_{a}$ sends $O_{a}=k_{a}+m_{a}$ to $A$. $P_{b}$ sends $O_{b}=k_{b}+m_{b}$ to $A$. $A$ receives $O_{i}$ for $i\in L\subseteq\{1,...,n\}$.
3. **Broadcast phase**: $A$ broadcasts $(L,X)$ to $P_{1},...,P_{n}$, where $X=\sum_{i\in L}O_{i}$. If $b\in L$ then $P_{a}$ computes $m_{b}$; with $m_{b}=X-\sum_{i\in L}k_{i}-m_{a}$ in case $a\in L$ or otherwise with $m_{b}=X-\sum_{i\in L}k_{i}$. If $a\in L$ then $P_{b}$ computes $m_{a}$; with $m_{a}=X-\sum_{i\in L}k_{i}-m_{b}$ in case $b\in L$ or otherwise with $m_{a}=X-\sum_{i\in L}k_{i}$.
This leaves us with the problem of users who deliberately send faulty packets to disrupt the communication. Such a case should be caught and the corresponding packets should be dropped. To identify such packets, we propose the following protocol.
A Protocol Resilient to Lost and Faulty Packets
-----------------------------------------------
The problem is that if a malicious player $P_{i}$ sends a random value instead of $k_{i}$ then $P_{a}$ and $P_{b}$ who expect $k_{i}$ will not be able to properly extract the messages $m_{a}$ and $m_{b}$ from $X$ anymore.
In order to protect against such malicious players it is obvious the aggregator must be able to distinguish if a received packet is correct. However the aggregator should not be able to distinguish which packets contain messages. Thus every player must include a proof that the submitted data is correct, and the aggregator must be able to verify this proof without gaining any other information from it. This means that a player $P_{i}\notin\{P_{a},P_{b}\}$ must be able to prove that $O_{i}=k_{i}$, and $P_{a}$ and $P_{b}$ must keep the freedom to send $O_{a}=k_{a}+m_{a}$ and $O_{b}=k_{b}+m_{b}$.
An elegant way to achieve this is to bind each player $P_{i}$ to his key $k_{i}$ using a trapdoor commitment, where the secret trapdoor information $\alpha$ is only known to $P_{a}$ and $P_{b}$. Then each player $P_{i}\notin\{P_{a},P_{b}\}$ can only open the commitment to the value $k_{i}$, but $P_{a}$ and $P_{b}$ who know $\alpha$ can open their commitments to any value they like, that is to $k_{a}+m_{a}$ and $k_{b}+m_{b}$.
In our description we use Pedersen commitments [@Pedersen91] which are of the form $c=g^{r}h^{k}$, and we assume that the secret $\alpha=\log_{g}h$ is only known to $P_{a}$ and $P_{b}$.
As shown in Protocol \[protocol:3\], during the setup the aggregator is provided with a commitment for each expected $O_{i}$, and each player $P_{i}$ is provided with the corresponding secret pairs $(k_{i},r_{i})$. Then, during the collection phase, each participant $P_{i}\notin\{P_{a},P_{b}\}$ must send $(k_{i},r_{i})$ to the aggregator, since without $\alpha$ he cannot find any other pair $(k'_{i},r'_{i})$ that corresponds to the commitment. $P_{a}$ and $P_{b}$ can use $\alpha$ to compute valid pairs $(k_{a}+m_{a},r'_{a})$ and $(k_{b}+m_{b},r'_{b})$. The aggregator verifies for each received pair if it corresponds to the commitment and rejects pairs that do not. Thus only valid $k_{i}$s are used to compute $X$.
1. **Setup**: $P_{1},...,P_{n}$ respectively have secret value pairs $(k_{1},r_{1}),...,(k_{n},r_{n})$. $P_{a}$ additionally has $m_{a}$ and $P_{b}$ additionally has $m_{b}$, and both know all $(k_{1},r_{1}),...,(k_{n},r_{n})$. $A$ is provided with $c_{1},...,c_{n}$ where $c_{i}=g^{r_{i}}h^{k_{i}}$. Only $P_{a}$ and $P_{b}$ know $\log_{g}h$.
2. **Collection phase**: Each $P_{i}$ sends $(O_{i},s_{i})$ to $A$. Each $P_{i}$ for $i\ne a,b$ uses $O_{i}=k_{i}$ and $s_{i}=r_{i}$. $P_{a}$ uses $O_{a}=k_{a}+m_{a}$ and $s_{i}=r_{i}-m_{a}\cdot\log_{g}h$. $P_{b}$ uses $O_{b}=k_{b}+m_{b}$ and $s_{i}=r_{i}-m_{b}\cdot\log_{g}h$. $A$ receives $(O_{i},s_{i})$ where additionally $g^{s_{i}}h^{O_{i}}=c_{i}$ holds for $i\in L\subseteq\{1,...,n\}$.
3. **Broadcast phase**: $A$ broadcasts $(L,X)$ to $P_{1},...,P_{n}$, where $X=\sum_{i\in L}O_{i}$. If $b\in L$ then $P_{a}$ computes $m_{b}$; with $m_{b}=X-\sum_{i\in L}k_{i}-m_{a}$ in case $a\in L$ or otherwise with $m_{b}=X-\sum_{i\in L}k_{i}$. If $a\in L$ then $P_{b}$ computes $m_{a}$; with $m_{a}=X-\sum_{i\in L}k_{i}-m_{b}$ in case $b\in L$ or otherwise with $m_{a}=X-\sum_{i\in L}k_{i}$.
This scenario is good if there is only one transmission round, but the anonymous sending of $c_{1},...,c_{n}$ to $A$ is expensive and does not scale well to multiple rounds. Therefore we need a more efficient way to provide the aggregator $A$ with means for verifying the data from the participants when there are multiple transmission rounds.
A Protocol Resilient to Packet-Loss and Malicious Players for Multiple Rounds\[sub:A-Protocol-Resilient\]
---------------------------------------------------------------------------------------------------------
In order to extend the previous protocol to multiple transmission rounds, we propose to use Merkle trees [@merkle1989certified]. For each player $P_{i}$ we use a Merkle tree $T_{i}$ that allows to verify that a given commitment is valid for a given round. It is then not necessary anymore to provide the aggregator with a commitment for each round, but it is sufficient to provide the aggregator with the roots of the Merkle trees. As illustrated in Figure \[Fig:Merkle\], such a Merkle tree $T_{i}$ can be constructed from a sequence $(k_{i}^{(1)},r_{i}^{(1)}),...,(k_{i}^{(J)},r_{i}^{(J)})$, which can be derived from a secret seed $S_{i}$.
So what changes compared to the preceding protocol is that the aggregator is provided with the roots of the Merkel trees instead of the commitments, and each player $P_{i}$ is provided with a secret seed $S_{i}$ that corresponds to a pseudorandom sequence. Then, during the transmission phase each player $P_{i}$ does not only send $(k_{i}^{(j)},r_{i}^{(j)})$ but $(k_{i}^{(j)},r_{i}^{(j)},Z_{i}^{(j)})$ where $Z_{i}^{(j)}$ is a proof the commitment corresponding to $(k_{i}^{(j)},r_{i}^{(j)})$ is the right one for round $j$ (i.e., that it is at position $j$ in the sequence). The aggregator computes a commitment and verifies using $Z_{i}^{(j)}$ that it is correct. A detailed description of the protocol is shown in Protocol \[protocol:4\].
It is easy to see that this protocol satisfies the properties of correctness and privacy defined in Section \[sec:Model-and-Definitions\].
1. **Setup**: For each round $j\in\{1,...,J\}$, $P_{1},...,P_{n}$ respectively have secret value pairs $(k_{1}^{(j)},r_{1}^{(j)}),...,(k_{n}^{(j)},r_{n}^{(j)})$. $P_{a}$ additionally has $m_{a}^{(j)}$ and $P_{b}$ additionally has $m_{b}^{(j)}$, and both know all $(k_{1}^{(j)},r_{1}^{(j)}),...,(k_{n}^{(j)},r_{n}^{(j)})$. $A$ is provided with $R_{1},...,R_{n}$ the roots of a merkle trees $T_{1},...,T_{2}$ constructed from $(c_{1}^{(1)},...,c_{1}^{(J)}),...,(c_{n}^{(1)},...,c_{n}^{(J)})$ where $c_{i}^{(j)}=g^{r_{i}^{(j)}}h^{k_{i}^{(j)}}$. Only $P_{a}$ and $P_{b}$ know $\log_{g}h$.
2. **Collection phase** (round $j$): Each $P_{i}$ sends $(O_{i}^{(j)},s_{i}^{(j)},z_{i}^{(j)})$ to $A$. Each $P_{i}$ for $i\ne a,b$ uses $O_{i}^{(j)}=k_{i}^{(j)}$ and $s_{i}^{(j)}=r_{i}^{(j)}$. $P_{a}$ uses $O_{a}^{(j)}=k_{a}^{(j)}+m_{a}^{(j)}$ and $s_{i}^{(j)}=r_{i}^{(j)}-m_{a}^{(j)}\cdot\log_{g}h$. $P_{b}$ uses $O_{b}^{(j)}=k_{b}^{(j)}+m_{b}^{(j)}$ and $s_{i}^{(j)}=r_{i}^{(j)}-m_{b}^{(j)}\cdot\log_{g}h$. Further, $z_{i}^{(j)}$ is a proof that $c_{i}^{(j)}=g^{r_{i}^{(j)}}h^{k_{i}^{(j)}}$ is in the Merkle tree $T_{i}$ at position $j$. $A$ receives $(O_{i}^{(j)},s_{i}^{(j)},z_{i}^{(j)})$ where $z_{i}^{(j)}$ proves that $c_{i}^{(j)}=g^{s_{i}^{(j)}}h^{O_{i}^{(j)}}$ is at position $j$ in $T_{i}$, for $i\in L^{(j)}\subseteq\{1,...,n\}$.
3. **Broadcast phase** (round $j$): $A$ broadcasts $(L^{(j)},X^{(j)})$ to $P_{1},...,P_{n}$, where $X^{(j)}=\sum_{i\in L^{(j)}}O_{i}^{(j)}$. If $b\in L^{(j)}$ then $P_{a}$ computes $m_{b}^{(j)}$; with $m_{b}^{(j)}=X^{(j)}-\sum_{i\in L^{(j)}}k_{i}^{(j)}-m_{a}^{(j)}$ in case $a\in L^{(j)}$ or otherwise with $m_{b}^{(j)}=X^{(j)}-\sum_{i\in L^{(j)}}k_{i}^{(j)}$. If $a\in L^{(j)}$ then $P_{b}$ computes $m_{a}^{(j)}$; with $m_{a}^{(j)}=X^{(j)}-\sum_{i\in L^{(j)}}k_{i}^{(j)}-m_{b}^{(j)}$ in case $b\in L^{(j)}$ or otherwise with $m_{a}^{(j)}=X^{(j)}-\sum_{i\in L^{(j)}}k_{i}^{(j)}$.
Variants of the Protocol
------------------------
In order to recover the messages in presence of packet-loss we proposed in Section \[sub:A-Packet-Loss-Resilient\] that the aggregator should send the list of packets that have been received along with the sum. This allows the receiver to directly compute the message. If one can assume that only a few (e.g., 1 or 2) packets are lost per round, one can also omit to send this list. The recipient can then still recover the message by trying all the possible combinations of missing packets. This way the packet length is reduced, however at the cost of a more expensive computation at the recipient.
Similarly one could also completely omit the cryptographic proof and go for a completely different mechanism. The players $P_{a}$ and $P_{b}$ could, upon detection of problems, use the anonymous channel from the setup and ask the aggregator to publish all the packets he received during a problematic round. Since $P_{a}$ and $P_{b}$ know all the keys, they would directly distinguish who sent a faulty packet, and they could anonymously ban those players from the group. This optimistic approach would lead to shorter packets, but as the latency of the anonymous channel is expected to be high, the stream would be interrupted for a non negligible amount of time.
Practical considerations
========================
In this section we discuss some aspects to consider in a real implementation of the protocol.
#### Channel Setup
Concerning the setup of the channel, we assumed so far that the initialization is performed anonymously by one of the correspondents $P_{a}$ or $P_{b}$. This correspondent will provide all other players and the aggregator with the the required data via an anonymous communication channel.
One way to implement such an anonymous channel is to use a DC-net. However, such a DC-net must then be run periodically, since in general it is not known in advance when a correspondent will want to talk with another. The higher the frequency with which such a DC-net is run, the better the user experience. But as each run consumes bandwidth, one does not want to do this more often than necessary either. So there is a tradeoff to be made between bandwidth and user experience with this approach.
Another way to implement an anonymous channel is to use a relay based approach like onion routing (e.g. TOR). Here the problem is that the overall security provided by the system is only as strong as the weakest link in the chain. The use of such a relay based approach would weaken the overall security of the system.
#### Channel Termination
In the protocol of Section \[sub:A-Protocol-Resilient\] the number of rounds (and thus the length of the call) is fixed during the setup of the channel. If a call ends earlier the correspondents can actively terminate the call by notifying the other players via the same anonymous channel that they used to do the initialization.
#### Load Distribution with P2P
Especially for the aggregator the computational costs and the bandwidth requirements and can rise to non-negligible levels, since they are proportional to the number of players. For instance if all packets are around 100 bytes and if 100 players send 50 packet per second, the aggregator must aggregate $5000$ packets per second and has a corresponding incoming and outgoing traffic of $4\,\mbox{Mbps}$. Each participant would have an incoming an outgoing traffic of $40\,\mbox{kbps}$.
In a P2P system one is not obliged to have only one aggregator as illustrated in Figure \[Fig:p2p\_a\], but the players can distribute this load between all of them by successively have each one of them play the role of the aggregator in a round robin fashion as illustrated in Figure \[Fig:p2p\_b\]. This way the load is more evenly distributed and for the same setup as in the preceding example each player would have of around $80\,\mbox{kbps}$ of ignoring and outgoing traffic.
#### Synchronization
All players should send their packets such that they arrive at the aggregator practically at then same time, in order to minimize the overall latency of a transmission. The aggregator will only wait for a certain period of time before aggregating the received data and sending the result to the players. It is therefore important that the clocks of the players are properly synchronized.
#### Cryptography
The cryptographic assumption for the commitments and the hashtables only needs to hold for a short time. It is therefore possible to use a lower security parameter than for digital signatures that have to be secure for decades.
Performance
===========
In this section we consider the latency, the bandwidth and the computing complexity for some setups.
#### Latency
Latency of packets in computer networks is often assumed to follow a log-normal distribution, e.g. in [@huberman1997social; @Sorger2011]. This distribution defined by $$Pr(t=x)=\frac{1}{\sqrt{2\pi s^{2}}}\cdot\exp\left(\frac{(\ln x-u)^{2}}{2s^{2}}\right)$$ has a characteristic heavy tail, as shown shown in Figure \[fig:latency\_log\_normal\]. If we assume an average $u=0.97$ and a standard error $s=0.06$, then the average time the aggregator has to wait until all $n$ independent ciphertexts have arrived is given by $$Pr(t<x)^{n}=\left(\int_{l=-\infty}^{x}Pr(t=l)\right)^{n}.$$ where $n$ is the number of players. As shown in Figure \[fig:cumul\_latency\] the cumulated latency increases with the number of players. In our case, we see that for $n=100$ players we already have a latency increase of more than $30$ms.
#### Packet Loss
Packet loss typically occurs in bursts and can be modeled using the well known Gilbert–Elliott (GE) channel [@gilbert1960capacity; @elliott1963estimates]. We estimate the number of rounds during which no packet is lost on its way to the aggregator, based on the probability $p$ that a packet is lost. The probability that a packet is not lost is then $1-p$, and the probability that no packet is lost is then $$q=(1-p)^{n}.$$ Figure \[packetloss\] illustrates the number of rounds during which at least one packet does not make it to the aggregator for various values of $p$.
#### Bandwidth
During one transmission round each of the $n$ players sends a packet to the aggregator, and the aggregator sends a packet back to each player. The total number of packets per second $b(n)$ is thus proportional to the number of players $n$. That is $$b(n)=2\cdot\frac{n-1}{f},$$ where $f$ is the number of rounds per second.
The load of the aggregator increases with the number of players.
In a P2P scenario where all players successively play the role of the aggregator, the bandwidth usage is distributed evenly amongst all players. Each player will perform like a normal player for n-1 rounds, and in one out of $n$ rounds he will not have to send anything, but he will have to broadcast the aggregated data to the $n-1$ other players. The bandwidth per player $p(n)$ is shown in Figure \[Fig\_bandwidth\_per\_participant\]. It can be computed with $$p(n)=\frac{b}{n}=\frac{n-1}{n}\cdot\frac{2}{f}\sim\frac{2}{f}.$$
#### Packet size
Packets in our protocol are composed of two parts, the audio payload on one hand and the cryptographic overhead on the other hand.
The amount of audio data depends on the frequency of the packets, on the quality (sampling frequency, compression rate) of the sound and the number of sound channels (e.g., mono or stereo). For voice transmission in mono this could be 50 packets with 60 bytes per second per packet, but for high-end music in stereo it will be significantly more.
The amount of cryptographic data depends on the strength of the cipher that is used. Since the cryptographic assumption only has to hold during the communication one can use lightweight cryptographic primitives.
Future Work
===========
There are basically two directions for future work, the improvement of the protocol itself and the building of larger systems composed of multiple groups.
![In an untraceable communication system with groups, there is a whole group of potential correspondents on either side of the line. Nobody except the real correspondents – neither an external observer, nor any other group member – can distinguish who is communicating with whom. []{data-label="Figure:GlobalVision-1"}](Groups)
#### Detection of Malicious Aggregators
In this paper we assume that the aggregator is honest but curious. This means that he will not drop packets, nor omit to send packets. While he cannot send wrong results and remain undetected, he can just disrupt a transmission round by just dropping data. A more powerful malicious aggregator could however just ignore some of the packets he receives, or he could deliberately not broadcast the aggregated result to anybody. In a P2P setting the effect of such an aggregator can be mitigated using the rotation principle proposed in section X, but ideally one would like to detect and to ban such aggregators from the group.
#### Larger Systems with Multiple Groups
Protocols based on DC-net do not scale to a very large number of participants, as the bandwidth and the computational power used by the aggregator are proportional to the number of participants. So the idea which was already proposed in [@goel2003hsa] is to realize systems composed of many small groups, as illustrated in Figure \[Figure:GlobalVision-1\]. Only the correspondents will know that they are communicating, all other players or observers cannot distinguish who is communicating.
For example one could have on one side a group of 500 politicians and on the other side a group of 500 journalists. When a politician then talks to a journalist, it would only be possible to see that one of the 500 politicians is talking to one of the 500 journalists, but it would not be possible for anybody to distinguish which politician is talking to which journalist. As there would be $500\cdot500=250000$ different possibilities, such a system would offer a fairly good protection.
There are many open questions, such as: How can we locate a given participant within the system, if we do not know in which group he is? How can we handle participants joining and leaving the system? How can we ban malicious participants from the entire system?
Related Work
============
The Dining Cryptographers protocol was proposed in [@chaum1988dcp] and further studied in [@bos1989ddd; @BWPW91; @waidner1990usa; @waidner1989dcd]. A first system composed of multiple DC-nets was proposed in [@goel2003hsa].
Computationally secure DC-net protocols with zero-knowledge verification of the data have been proposed in [@golle2004dcr] and further studied in [@FranckMscThesis; @wolinsky2012dissent; @corrigan2013proactively; @Franck_DC_0924; @corrigan2015riposte].
Recent group based communication systems include [@corrigan2015riposte; @kwon2015riffle; @le2015herd; @danezis2010text]. There have also been TOR [@dingledine2004tsg] extensions for VoIP [@gegelonionphone] and for group communication [@yang2015mtor].
Concluding Remarks
==================
Starting from the classic DC-net paradigm we derived a protocol for untraceable VoIP telephony that is resilient against packet-loss and faulty packets. It enables two players within a larger group to communicate with VoIP without anybody else being able to distinguish that they are communicating. Further we discussed practical issues and showed how to distribute the load in a P2P network.
We consider this work a first step towards larger systems composed of multiple groups so that can scale to a larger number of participants.
|
---
abstract: 'The structural properties of static, jammed packings of monodisperse spheres in the vicinity of the jamming transition are investigated using large-scale computer simulations. At small wavenumber $k$, we argue that the anomalous behavior in the static structure factor, $S(k) \sim k$, is consequential of an excess of low-frequency, collective excitations seen in the vibrational spectrum. This anomalous feature becomes more pronounced closest to the jamming transition, such that $S(0) \rightarrow 0$ at the transition point. We introduce an appropriate dispersion relation that accounts for these phenomena that leads us to relate these structural features to characteristic length scales associated with the low-frequency vibrational modes of these systems. When the particles are frictional, this anomalous behavior is suppressed providing yet more evidence that jamming transitions of frictional spheres lie at lower packing fractions that that for frictionless spheres. These results suggest that the mechanical properties of jammed and glassy media may therefore be inferred from measurements of both the static and dynamical structure factors.'
author:
- 'Leonardo E. Silbert$^{1}$'
- 'Moises Silbert$^{2,3}$'
title: Long wavelength structural anomalies in jammed systems
---
Introduction
============
The emergence of similarities between the properties of molecular glasses, dense colloidal suspensions, foams, and granular materials, have led to the notion of jamming [@liu1] - the transition between solid-like and fluid-like phases in disordered systems - as a manner through which one can gain a deeper understanding of the traditional liquid-glass transition and the fascinating, complex phenomena observed in amorphous materials in general [@coniglio1]. Although there has recently been some works highlighting the differences between the onset of rigidity in jammed matter and glassiness [@kurchan8; @zamponi2], the emphasis in this work is on the commonalities between the two [@ohern1; @ohern3; @leo14; @leo17]. Here, we aim to provide a heuristic physical picture that accounts for specific, long wavelength, structural features that emerge in the jammed state and relate these features to their dynamical properties.
Donev et al. [@torquato6] found that in the [*hard sphere*]{} jamming transition, the structure factor, $S(k)$, exhibits a linear dependence on wavenumber $k \equiv |\bf{k}|$, $$S(k) \propto k~~\textrm{ as }~~k \rightarrow 0
\label{eq1}$$ This behavior of $S(k)$ suggests that the total correlation function $h(r)$, decays as $|r^{-4}|$, at large separations $r$, as deduced from the asymptotic estimates of Fourier transforms [@lighthill1; @stell1; @silbert5]. Similarly it suggests a long range behavior for the direct correlation function $c(r)$. This is indeed in contrast to standard liquid state theory for liquids whose constituents interact via a finite range potential - hence $c(r)$ is short ranged - which predicts, $S(k) \propto k^{2}$. This anomalous low-$k$ behavior can be interpreted as being indicative of the suppression of long wavelength density fluctuations due to hyperuniformity [@torquato7]. Here, we propose an alternative interpretation related to large length-scale collective dynamics.
This paper is arranged as follows. We provide a brief overview of the Molecular Dynamics (MD) simulations used here to generate liquid and jammed states. We then review previous results from studies of the jamming transition pertinent to the discussion here. This is followed by a discussion on the relevant concepts from liquid state theories that indicate that our results and those of Ref. for frictionless particles are indeed rather unusual. We then present our results for the static structure factor, $S(k)$, at small values of the wavenumber $k$, in our jammed, model glassy system. We then put forward a conjecture that relates the asymptotic behavior of $S(k)$ to an excess of vibrational modes relative to the Debye model. We end with results from ongoing work on frictional systems and conclusions.
Simulation Model
================
The computer simulations performed here are for monodisperse, [*soft spheres*]{} of diameter $d$ and mass $m$, interacting through a finite range, purely repulsive, one-sided, harmonic potential, $$V(r) = \left\{
\begin{array}{cc}
\frac{\varepsilon}{2d^{2}}(d-r)^{2} & r<d,\\
0 & r>d
\end{array}
\right\}
\label{eq2}$$ where $r=|{\bf r}_{i} - {\bf r}_{j}|$ is the center-to-center separation between particles $i$ and $j$ located at ${\bf r}_{i}$. The strength of the interaction is parameterized by $\varepsilon$, which is set to unity in this study. Most of the results presented below are for frictionless particles, with friction coefficient $\mu = 0$. We also present preliminary results for frictional packings using a static friction model [@leo7] to compare between frictionless and frictional systems. In the frictional packings the particle friction coefficient was varied, $0.01 \leq \mu \leq 1.0$. We simulated systems ranging in size, $1024\leq N \leq 256000$ particles, in cubic simulation cells of size $10d \lesssim L \lesssim 50d$, with periodic boundary conditions, over a range of packing fractions, $\phi = N\pi
d^{3}/L^{3}$.
The jamming protocol implemented here to generate zero-temperature jammed packings is similar to other soft-sphere protocols [@makse1; @makse2]. Initially, starting from a collection of spheres randomly placed in the simulation cell at low packing fraction $\phi_{i} = 0.30$, we compressed the system to a specified over-compressed state, $\phi = 0.74$, minimizing the energy of the system in a steepest descent manner. At this value of $\phi =
0.74$, all the particles experience overlaps with several other particles - their contact neighbors. The packings are mechanically stable and disordered with no signs of long range order. To generate packings at $\phi < 0.74$, we then incrementally reduced $\phi$ in steps of, $10^{-6} \leq \delta \phi \leq
10^{-2}$, minimizing the energy after each step. This allowed us to accurately determine the location of the jamming transition where the system unjams at a packing fraction $\phi_{c}$ (see below), for *each independent realization*, down to an accuracy of $10^{-6}$ in $\phi$ for $N \leq 10000$ and an accuracy of $10^{-3}$ for $N > 10000$. For the largest system, $N=256000$, we generated four independent configurations at each value of $\phi$.
We also studied equilibrated liquids of frictionless spheres at a dimensionless temperatures $T = 0.01$. This value of $T$ was chosen to compare and contrast our results between liquid and jammed states.
Our main tool of analysis is the static structure factor $S(k)$, which we obtain in the standard way as a direct Fourier transform of the particle positions, $$S(k) = \frac{1}{N}\left|\sum_{i=1}^{N} \exp(\imath {\bf k} \cdot {\bf r}_{i})\right|^{2} .
\label{eq3}$$ Here, ${\bf k}$ is the reciprocal lattice vector for the periodic simulation cell, which is restricted to $k \geq 2\pi / L$. Although the data is noisy at the lowest wavenumbers, our largest packings ($N=256000$) display the anomalous low-$k$ behavior, as described below, well above the noise level. Throughout this study, $d=m=\varepsilon=1$, and all quantities have been appropriately non-dimensionalized.
Review
======
Jamming Transition
------------------
Frictionless, purely repulsive, soft sphere system have been shown to undergo a zero-temperature transition between jammed and unjammed phases as a function of density or packing fraction [@durian7; @makse1; @makse2; @ohern2; @ohern3; @hecke6]. In the infinite-size limit, this transition occurs at a packing fraction coincident with the value often quoted for random close packing, $\phi_{rcp}=0.64$ [@ohern3]. Above the jamming threshold $\phi > \phi_{c}$, the jammed state is mechanically stable to perturbations with non-zero bulk and shear moduli, whereas below the transition $\phi < \phi_{c}$, it costs no energy to disturb the system [@ohern2]. The relevant parameter here is not the absolute value of the packing fraction, rather the *distance* to the jamming transition defined through, $\Delta\phi \equiv \phi - \phi_{\rm c}$. Thus, as a jammed state is brought closer to the transition point, $\Delta\phi \rightarrow 0$, it gradually loses its mechanical rigidity and becomes increasingly soft. Intriguingly, this loss of mechanical stability as the jamming transition is approached can be related to diverging length scales [@leo14] that characterize the extent of soft regions that determine the macroscopic behavior of system [@wyart2]. Experiments [@dauchot3] have also identified growing length scales in the vicinity of the jamming transition thus promoting the idea that the jamming transition can be considered in the context of critical phenomena.
In this paper, we connect the long wavelength structural features observed in $S(k)$ to correlation lengths characterizing the typical length scale of collective, low-frequency vibrational modes in jammed, zero-temperature, disordered packings [@leo14]. In Ref. it was shown that the approach of the jamming transition in a soft sphere packing is accompanied by a dramatic increase in the number of low-frequency vibrational modes over the expected Debye behavior. In traditional glasses, these excess low-frequency modes in the vibrational density of states, $\mathcal{D}(\omega)$, are often referred to as the *boson peak* [@phillips1] in reference to the peak observed when plotting $\mathcal{D}(\omega)/\omega^{2}$. For convenience, we also employ this language here. A detailed study of $D(\omega)$ for jammed sphere-packings and the appearance of the so-called boson peak can be found in [@leo14].
In Fig. \[fig1\] we compare the vibrational density of states for dense, soft sphere liquids (Fig. \[fig1\](a)) and amorphous jammed solids, identifying the location of the boson peak, $\omega_{\rm B}$, for our jammed system (Fig. \[fig1\](b)). The location of the boson peak tends to zero at the jamming transition point, i.e. $\omega_{\rm B} \rightarrow 0$, as $\Delta\phi \rightarrow 0$ [@leo14]. The two values of $\Delta\phi = 1
\times 10^{-4}$ and $1 \times 10^{-1}$ correspond to actual packing fraction values of $\phi = 0.6405$ and $0.74$ respectively, for sample configurations used to generate this data.
![The vibrational density of states, ${\mathcal D}(\omega)$, for two systems of $N=1024$ soft, frictionless spheres, characterized by the distance $\Delta\phi \equiv \phi - \phi_{c}$, from their zero-temperature, jamming transition packing fraction. Solid line: $\Delta\phi=1\times 10^{-1}$. Dashed line: $\Delta\phi=1\times
10^{-4}$. (a) Equilibrated liquids at $T=0.01$, well above their respective freezing or apparent glass transition temperatures. Data obtained from Fourier transforms of velocity autocorrelation functions, and has been rescaled such that the $\omega=0$ intercepts coincide. A Lorentzian function (dotted) corresponding to Langevin diffusion [@mcquarrie1] is also shown. (b) Jammed packings at zero temperature. The location of the *boson peak* is identified as $\omega_{\rm B}$, for $\Delta\phi=1\times 10^{-4}$. In (b), the Debye result (dotted), corresponding to the system $\Delta\phi = 0.1$, was generated using values for the bulk and shear moduli [@ohern3], and is shown for comparison.[]{data-label="fig1"}](fig1a.eps "fig:"){width="7cm"} ![The vibrational density of states, ${\mathcal D}(\omega)$, for two systems of $N=1024$ soft, frictionless spheres, characterized by the distance $\Delta\phi \equiv \phi - \phi_{c}$, from their zero-temperature, jamming transition packing fraction. Solid line: $\Delta\phi=1\times 10^{-1}$. Dashed line: $\Delta\phi=1\times
10^{-4}$. (a) Equilibrated liquids at $T=0.01$, well above their respective freezing or apparent glass transition temperatures. Data obtained from Fourier transforms of velocity autocorrelation functions, and has been rescaled such that the $\omega=0$ intercepts coincide. A Lorentzian function (dotted) corresponding to Langevin diffusion [@mcquarrie1] is also shown. (b) Jammed packings at zero temperature. The location of the *boson peak* is identified as $\omega_{\rm B}$, for $\Delta\phi=1\times 10^{-4}$. In (b), the Debye result (dotted), corresponding to the system $\Delta\phi = 0.1$, was generated using values for the bulk and shear moduli [@ohern3], and is shown for comparison.[]{data-label="fig1"}](fig1b.eps "fig:"){width="7cm"}
Associated with this excess in the vibrational density of states are two diverging length scales: the longitudinal correlation length, $\xi_{L}$, characterizing the scale of collective excitations of longitudinal modes contributing to the boson peak, while $\xi_{T}$, characterizes transverse excitations. These correlation lengths scale with the boson peak position as [@leo14; @wyart2], $$\begin{aligned}
\omega_{\rm B} \propto \xi_{L}^{-1}~~~{\textrm{(L: longitudinal modes)}}
\label{eq4}\\
\omega_{\rm B} \propto \xi_{T}^{-2}~~~{\textrm{(T: transverse modes)}}.
\label{eq5}\end{aligned}$$ Thus, when the Debye contribution, $\omega_{D}$, is included the corresponding dispersion relations become, $$\begin{aligned}
\omega_{L}(k) - \omega_{D,L}\cong \beta k
\label{eq6}\\
\omega_{T}(k) - \omega_{D,T} \cong \alpha k^{2} .
\label{eq7}\end{aligned}$$ The preceeding relations are only approximate, valid for low frequencies. They both contribute to the boson peak: The longitudinal term by modifying the slope in the Debye relation, whereas the transverse contribution contains an anharmonic term. We discuss the relevance of these results further below.
Analyticity of $S(k)$
---------------------
In order to avoid any misunderstanding we define the concepts of regular and singular in the following sense [@lighthill1]. Assuming $k$ to be a complex variable, for any three dimensional system interacting with tempered pairwise, additive potentials, namely when $$\phi (r): \frac{1}{r^{3+\eta}}~~{\textrm as}~~r\rightarrow \infty~;~\eta>0
\label{eq8}$$ and therefore integrable, $| \int d{\bf r} \phi (r)| < \infty$. Then, the structure factor, in the long wavelength limit reads [@silbert5], $$S(k): A \rho |k|^{\eta} + F(k^{2})~~{\textrm as}~~ k \rightarrow 0
\label{eq9}$$ where $\rho = \frac{N}{V}$ is the number density, $A$ is a constant that depends on the thermodynamic properties of the equilibrium system, and $F$ is an analytic function of $k$, regular in the neighborhood of $k=0$. Thus, the second term on the right hand side of Eq. \[eq9\] is the regular contribution to $S(k)$, whereas the first term - originating from the potential of interaction - is the singular term.
For instance, when $\eta = 3$, as in the attractive part of the Lennard-Jones potential, $$S(k) = A\rho |k|^{3} + F(k^{2})
\label{eq10}$$ a result originally derived by Enderby, Gaskill, and March [@enderby1]. Thus, a linear behavior of the structure factor at small values of $k$ corresponds to $\eta=1$. Physically, this would correspond to a charge-dipole interaction, which Chan [*et al.*]{} [@chan1] have shown can only be present if *long-ranged* dipole-dipole interactions are also included. Thus, in the present work, the linear behavior of $S(k)$, at small $k$, does not come from the potential of interaction, as the finite range harmonic potential used here will only give the regular contribution to $S(k)$. However, this does suggest that the anomalous low-$k$ behavior originates in long-ranged correlations associated with the system.
Results
=======
Frictionless Spheres
--------------------
We initially generated a number of over-compressed, $\phi > \phi_{c}$, zero-temperature packings at various distances, $\Delta\phi \equiv \phi -
\phi_{c}$, from the jamming transition point. We also point out that the main distinction between this work and that of Ref. , is that we study soft-spheres and approach the jamming transition from above, $\phi
\rightarrow \phi_{c}^{+}$, whereas, hard spheres necessarily approach the transition from below. In Fig. \[fig2\] we show $S(k)$ for two soft-sphere systems at $T = 0.01$, well above their respective apparent glass transition temperatures $T_{g}$. (The corresponding density of states are shown in Fig. \[fig1\](a).) For $\Delta\phi = 10^{-1}$, $T_{g}<0.001$, while for smaller $\Delta\phi$, $T_{g}$ lies below this. Qualitatively the curves are similar. We find the usual primary peak corresponding to nearest neighbors, and, as expected from standard liquid state theory [@mcquarrie1; @silbert5], at lower $k$, $S(k) \propto k^{2}$, with a non-zero intercept at $k=0$, which corresponds to the finite compressibility sum rule, Fig. \[fig2\] inset. Thus, for equilibrated liquids the structure factor and the long-wavelength limit are quite insensitive to the location of the zero-temperature jamming transition, at these values of the number density, $\rho \equiv N/L^{3} =
6\phi/\pi d^{3}= 1.22, 1.44$ for $\phi = 0.6405~(\Delta \phi = 1 \times
10^{-4}), 0.74~(\Delta \phi = 1 \times 10^{-1})$ respectively.
![Two $N=10000$ soft-sphere systems in the liquid state, $T=0.01$, for $\Delta \phi = 1 \times 10^{-1}$ and $1 \times 10^{-4}$ (see legend in panel (b)). This temperature is well above the respective glass transition temperatures for the two systems, $T>>T_{g}$. (a) Static structure factor $S(k)$, as a function of wavenumber $k$. The inset is a zoom of the region near the origin, where the arrows indicate the non-zero intercepts as $k \rightarrow 0$. (b) Radial distribution function, $g(r)$ shows a prominent nearest neighbor peak at $r \approx
1$ and correlations that die off rapidly.[]{data-label="fig2"}](fig2a.eps "fig:"){width="7cm"} ![Two $N=10000$ soft-sphere systems in the liquid state, $T=0.01$, for $\Delta \phi = 1 \times 10^{-1}$ and $1 \times 10^{-4}$ (see legend in panel (b)). This temperature is well above the respective glass transition temperatures for the two systems, $T>>T_{g}$. (a) Static structure factor $S(k)$, as a function of wavenumber $k$. The inset is a zoom of the region near the origin, where the arrows indicate the non-zero intercepts as $k \rightarrow 0$. (b) Radial distribution function, $g(r)$ shows a prominent nearest neighbor peak at $r \approx
1$ and correlations that die off rapidly.[]{data-label="fig2"}](fig2b.eps "fig:"){width="7cm"}
Turning our attention to the jammed phases at zero temperature [@footnote23], in Fig. \[fig3\] we show the radial distribution function $g(r)$ and the low-$k$ region of $S(k)$, for $N=256000$ soft-spheres systems at three values of $\Delta\phi = 1 \times 10^{-1}$, $1 \times 10^{-2}$, and $3
\times 10^{-3}$. Note that in Fig. \[fig3\](a) we use log-log scales to clearly demonstrate the linear region in $S(k)$. Far from the jamming transition, $\Delta\phi = 1 \times 10^{-1}$, $S(k)$ plateaus near $k\approx 1$ and tends to constant as $k \rightarrow 0$. For the systems closer to the jamming transition, $\Delta\phi = 1 \times 10^{-2}$ and $\Delta\phi = 3 \times
10^{-3}$, there is a *qualitative change* in the low-$k$ behavior of $S(k)$. We find a linear region, $S(k) \sim k$, extending over almost an order of magnitude at low-$k$. We also point out that on closer inspection the linear region extends to lower $k$ for the system closest to the jamming transition point $(\Delta\phi = 3 \times 10^{-3})$. At the smallest $k$ attainable, $S(k)$, flattens out again because the system is not exactly at $\Delta\phi=0$. The radial distribution function shown in Fig. \[fig3\](b) exhibits typical features characterisitc of a glassy phase; namely a split second peak an additional shoulder on the third peak.
![Zero-temperature, jammed packings containing $N=256000$ purely-repulsive, frictionless, soft-spheres at three different values of $\Delta\phi = 1 \times 10^{-1}$ (dotted line), $1 \times 10^{-2}$ (dashed), and $3 \times 10^{-3}$ (solid). (a) Static structure factor $S(k)$. Far from the jamming transition, $\Delta\phi = 1 \times
10^{-1},~S(k)$ plateaus nears $k\approx 1$. Closer to jamming, $\Delta\phi = 1\times 10^{-2}$ and $\Delta\phi = 3 \times 10^{-2},~S(k)$ exhibits approximately linear dependence on $k$ over almost an order of magnitude in $k$, extending down to low-$k$. A linear curve on this log-log plot is shown for comparison. (b) Radial distribution function. The jamming transition is characterized by a diverging nearest neighbor peak at $r\approx 1$, a clear splitting of the second peak [@leo17], and oscillations that persist out to larger $r$ than for the liquid state.[]{data-label="fig3"}](fig3a.eps "fig:"){width="7cm"} ![Zero-temperature, jammed packings containing $N=256000$ purely-repulsive, frictionless, soft-spheres at three different values of $\Delta\phi = 1 \times 10^{-1}$ (dotted line), $1 \times 10^{-2}$ (dashed), and $3 \times 10^{-3}$ (solid). (a) Static structure factor $S(k)$. Far from the jamming transition, $\Delta\phi = 1 \times
10^{-1},~S(k)$ plateaus nears $k\approx 1$. Closer to jamming, $\Delta\phi = 1\times 10^{-2}$ and $\Delta\phi = 3 \times 10^{-2},~S(k)$ exhibits approximately linear dependence on $k$ over almost an order of magnitude in $k$, extending down to low-$k$. A linear curve on this log-log plot is shown for comparison. (b) Radial distribution function. The jamming transition is characterized by a diverging nearest neighbor peak at $r\approx 1$, a clear splitting of the second peak [@leo17], and oscillations that persist out to larger $r$ than for the liquid state.[]{data-label="fig3"}](fig3b.eps "fig:"){width="7cm"}
For completeness, we examine the influence of system size on the results presented here. In Fig. \[fig4\], we show $S(k)$ at $\Delta\phi \approx 3
\times 10^{-3}$ for three different systems sizes, $N = 1000, ~ 10000, ~
256000$, corresponding to $L \approx 10, ~ 20, ~ 60$. The main panel of Fig. \[fig4\] shows that the gross properties of $S(k)$ do not depend on system size. Oscillations in $S(k)$ persist out to the largest $k$ and is a consequence of the diverging nearest neighbor peak in $g(r)$ [@leo17]. The inset to Fig. \[fig4\] indicates that the linear portion of $S(k)$ at small $k$ becomes resolvable for $N \geq 10000$.
![$S(k)$ at $\Delta\phi \approx 3 \times 10^{-3}$ for different systems sizes, $N = 1000$ (line), $N = 10000$ ($\circ$), and $N =
256000$ ($+$). Oscillations persist out to large $k$, reflecting the dominant nearest neighbor peak in $g(r)$ shown in Fig. \[fig3\]. The inset indicates that the linear behavior at low-$k$ becomes better resolved with increasing system size.[]{data-label="fig4"}](fig4.eps){width="7cm"}
Thus the first aim of this work shows that, similarly to the findings in Ref. for hard spheres just below the jamming transition point, for our soft sphere packings above $\phi_{\rm c}$, $S(k)$ exhibits a linear behavior at small values of momentum transfer. Hence, this unusual behavior not only pertains to hard spheres, but also to soft spheres, both above and below the jamming transition point, and possibly, to finite range repulsive potentials in general. We will attempt below to produce what we believe is a reasonable explanation for the origin of this behavior which, as stated at the beginning, does appeal to a conjecture.
Since, in the long wavelength limit, the linear behavior of $S(k)$ cannot arise from the singular contribution of the potential used in this work [@silbert5; @enderby1], it can only be due to the collective excitations present in the jammed or glassy state [@march2]. The collective excitations we have in mind are the same that are responsible for the boson peak. The range of $k$, over which $S(k)\sim k$, characterizes the length scale over which these excitations may be considered collective. As one approaches the jamming transition, this length scale diverges, a result consistent with work on the density of states of jammed packings [@leo14].
Recent work by Chumakov [*et al.*]{} [@schirmacher2] does indeed show that the excitations leading to the boson peak are predominantly collective; in agreement with inelastic neutron scattering experiments [@ruocco1]. Although there have been a number of theories put forward [@elliott2; @parisi1; @wyart1], there is, at present, no agreed explanation as to the origins of the boson peak. For the purposes of this work this is not necessary, except insofar as there is agreement that these excitations are predominantly collective, and that the vibrational modes are likely to be kinetically driven. The arguments that follow are somewhat oversimplified, but we believe they are along the correct lines.
The following comments are in order: (i) In a system that undergoes the liquid-glass transition, the long wavelength limit of the static structure factor, $S(k\rightarrow 0)$, is not the thermodynamic compressibility $\kappa_{T}$, but a different value, say $\kappa$, which can actually be extracted from experiments [@gotze2]. (ii) Independently of the model or approximation used, the density fluctuations in the glass phase are kinetically, not thermodynamically, driven [@gotze2]. (iii) In spite of their different origins, the kinetic temperature, say $T_{\textrm{dyn}}$, defined for glasses, matches the Edwards temperature, $T_{\textrm{Edw}}$, used in granular matter. They happen to coincide within mean field theories when the glass is in contact with an almost zero temperature heat bath, and the athermal grains are jammed [@kurchan5].
The connection between the dispersion relations and the structure factor, follows from the second moment of the dynamical structure factor, $S(k,\omega)$ [@egelstaff1], $$2\int_{0}^{\infty} d \omega \omega^{2} S(k,\omega) = \nu^{2}_{0} k^{2},
\label{eq11}$$ where $\nu_{0}= \nu_{0}(T)$ (for a system in thermodynamic equilibrium $\nu_{0}^{2} = \frac{k_{B} T}{m}$, where $k_{B}$ is Boltzmann’s constant). The static structure factor is the zero moment of $S(k,\omega)$, $$S(k) = \int d\omega S(k,\omega).
\label{eq12}$$
We now put forward the following conjecture: assume that there is only one very well defined collective mode with dispersion relation $\omega_{B}(k)$, the boson peak. This gives us a relation between the asymptotic long wavelength behavior of $S(k)$ and the boson peak, $$S(k,\omega) = S(k)\delta(\omega - \omega_{B}(k))
\label{eq13}$$ whence, $$S(k) = \nu_{0}^{2} \frac{k{^2}}{2\omega^{2}_{B}(k)}.
\label{eq14}$$ This collective mode is associated with vibrational excitations with a wavelength of the order of the correlation length of the jammed state. We expand the dispersion relation at these small values of momentum transfer [@march2] $$\omega_{B}(k) \cong c k + a k^{2} + ... .
\label{eq15}$$ In Eq. \[eq15\], $c$ is the speed of sound, and the second term on the rhs denotes departures from the usual Debye behavior, such that both $c$ and $a$ are independent of $k$. Replacing Eq. \[eq15\] into \[eq14\], in the limit of $k \rightarrow 0$, we find, $$S(k) = \frac{\nu_{0}^{2}}{2c^{2}} \left(1 - 2\frac{a}{c}k \right) + \mathcal{O}(k^2)
\label{eq16}$$ Thus, on comparing Eqs. \[eq15\] and \[eq16\] to Eqs. \[eq6\] and \[eq7\], it transpires that it is the transverse modes that contribute to the linear behavior of $S(k)$. On the other hand, the longitudinal components, Eq. \[eq6\], only contribute a constant term to $S(k)$, modifying the slope of the Debye relation.
We illustrate the differences in the dispersion behavior at two extreme values of $\Delta\phi = 10^{-6}$ and $10^{-1}$ in Fig. \[fig5\]. These results were obtained from diagonalization of the dynamical matrix for systems with $N =
10000$ and then locating the peaks in the transverse components of the Fourier transforms of the eigenmodes - transverse mode structure factors [@nagel11; @leo23] - and then averaging over a small range of frequencies. The data shown in Fig. \[fig5\] distinguishes between the regular, linear dispersion relation that dominates the dispersion behavior far from the jamming transition at $\Delta\phi = 10^{-1}$, while the quadratic contribution to the dispersion behavior is significant closer to the jamming transition, $\Delta\phi = 10^{-6}$.
![Transverse dispersion behavior for $\Delta\phi= 10^{-1} (\circ)$ and $10^{-6} (\square)$. The solid lines correspond to quadratic fits to the data as in Eq. \[eq16\]. Data obtained from the low-frequency portion of the transverse structure factors of the vibrational modes for $N=10000$.[]{data-label="fig5"}](fig5.eps){width="7cm"}
From the analysis we obtain the transverse speed of sound, $c_{t}$, as a fitting parameter. At these two compressions: $c_{t} \approx 0.26
(\Delta\phi=10^{-1})$ and $c_{t} \approx 0.023 (\Delta\phi=10^{-6})$. The actual values calculated from the bulk and shear moduli data [@ohern3] give: $c_{t} \approx 0.28 (\Delta\phi=10^{-1})$ and $c_{t} \approx 0.018
(\Delta\phi=10^{-6})$. Therefore, we find reasonable agreement within this approximation. If we use only a linear dispersion relation we find that the values of $c_{t}$ computed here and the actual values [@ohern3] vary by as much as an order of magnitude. This can be realized by the observation that a linear fit to data for $\Delta \phi = 10^{-6}$ does not pass through the origin as required in the hydrodynamic limit.
The linear feature of $S(k)$ in the small-$k$ regime appears here as a consequence of assuming an “excess” relative to the Debye model for the dispersion relation, and only one collective mode. Although the “excess” model used here appears as an approximation appropriate for the low-$k$ regime, we have reconciled this with existing results on the emergence of characteristic length scales associated with these collective excitations. More generally, however, it does suggest that the boson peak and the linear behavior of $S(k)$, as $k \rightarrow 0$, are two sides of the same coin. Therefore, this anomalous suppression of long-wavelength density fluctuations emerges as a consequence of large length-scale correlated dynamics in the low-frequency modes of the jammed solid. Furthermore, although the boson peak has been traditionally associated with the glassy phase, recent low frequency Raman spectroscopic studies of glassy, supercooled, and molten silica reveal that the boson peak persists into the liquid phase [@papatheodorou1]. This also appears to be the case with other network systems (see references in Ref. ) which, in the molten state, show a distinctive prepeak at small $k$ in $S(k)$. This prepeak is indicative of intermediate range order in those melts, representing a characteristic length in those melts that measures the correlations between the centers of “clusters” present in the liquid state. This intermediate-range clustering may in fact promote longer wavelength correlated dynamics in the liquid state. It may therefore be interesting to investigate experimentally, and by means of MD simulations, whether the low $k$-behavior of $S(k)$ in those systems is also linear, both in their glassy and liquid phases.
Frictional Packings
-------------------
Studies on jammed packings of *frictional* particles are less well-developed. The picture that is emerging is that frictional packings undergo a similar zero-temperature jamming transition at packing fractions which now become friction dependent [@hecke7; @hecke8; @makse7; @leo21]. In the limit of high friction coefficient the frictional jamming transition coincides with the value associated with random loose packing, $\phi_{rlp} \approx
0.55$ [@liniger1; @schroter3; @leo9]. Thus the relevant parameter that measures the distance to the jamming transition now becomes a friction-dependent quantity, $\Delta\phi(\mu)$ [@leo21].
![Log-log plot of the static structure factor $S(k)$, focusing on the low-$k$ region. All packings at the same $\phi = 0.64$, contain $N=256000$ purely-repulsive, monodisperse, soft-spheres, with different friction coefficients $\mu = 0~(\rm{solid-line}), ~0.01~(\rm{dash}),
~0.1 (\rm{dot}), ~1.0 (\rm{dot-dash})$.[]{data-label="fig6"}](fig6.eps){width="7cm"}
We test these concepts through $S(k)$ for packings over a range of friction coefficients at the *same* $\phi = 0.64$, which translates to different values of $\Delta\phi(\mu)$. Our preliminary results in Fig. \[fig6\] are data for $S(k)$ for jammed packings with $0 \leq \mu \leq 1$. At this fixed value of $\phi = 0.64$, $\Delta\phi(\mu)$ increases with increasing friction coefficient. Therefore, we expect that the anomalous linear behavior in the low-$k$ region of $S(k)$ should become less prominent with increasing $\mu$. Indeed this trend is observed in Fig. \[fig6\], thus providing yet further evidence that the jamming transition point shifts to lower $\phi$ with increasing friction. To recover the linear behavior at low-$k$ in frictional packings we would therefore need to study mechanically stable packings at lower $\phi$. This work in ongoing.
Conclusions
===========
In conclusion, we have provided a physical, albeit naive, explanation for the observation of the apparent suppression of long wavelength density fluctuations in jammed model glassy materials. We submit that it is the presence of low-frequency, collective excitations, mainly of transverse character, contributing to the excess of low-frequency modes, that are responsible for the linear behavior at low-$k$ in $S(k)$. The relevant length scale here is the transversal correlation length that contributes to the diverging boson peak in the jammed phase. This low-$k$ feature is most pronounced in the model system studied here for soft particles at packing fractions above the zero-temperature jamming transition matching the behavior on the other side of the jamming transition for hard spheres. This connection between the long-wavelength behavior on either side of the transition reinforces the view that the jamming transition is critical in nature. Although we also point out that the jamming transition is very different from a thermodynamic liquid-gas critical point in that we do not observe a divergence in $S(k=0)$, but rather it becomes zero.
Moreover, we expect that as the transition is approached, $$\nu_{0}(\mathcal{T}) \rightarrow 0 \Rightarrow S(0) \rightarrow 0 ~~~~\textrm{as}~~~~\Delta\phi \rightarrow 0.
\label{eq18}$$ The dependence of $\nu_{0}(\mathcal{T})$ on a generalized temperature-like quantity, $\mathcal{T}$, in jammed packings is consistent with the concept of the *angoricity* [@blumenfeld4; @henkes3]. This plays the role of temperature in packings of elastic particles and goes to zero when the particles fall out of contact which occurs at the jamming transition. Future studies may also provide a way to understand the connection between different temperature definitions relevant to the study of thermal and jammed systems. Our preliminary data for frictional packings also provide further evidence that the location of the jamming transition occurs at lower packing fractions with increasing friction coefficient. The underlying nature of the low-frequency modes in frictional materials has yet to be investigated in three dimensional systems and forms part of ongoing work. When friction is present the modes will contain not only the translational character as seen in frictionless systems, but also rotational character due to the additional degrees of freedom.
This anomalous low-$k$ behavior is also likely to be present in other, finite range, model repulsive systems. These features may be detected, using the appropriate spectroscopy, in (hard-sphere) colloidal glasses, and granular packings, which can be prepared close to their respective jamming transitions.
We are grateful to Gary Barker for insightful discussions on some aspects of this work. LES is especially grateful to Jane and Gary McIntyre for support during the course of this work and also gratefully acknowledges the support of the National Science Foundation CBET-0828359.
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---
abstract: 'Is the large influence that mutual funds assert on the U.S. financial system spread across many funds, or is it is concentrated in only a few? We argue that the dominant economic factor that determines this is market efficiency, which dictates that fund performance is size independent and fund growth is essentially random. The random process is characterized by entry, exit and growth. We present a new time-dependent solution for the standard equations used in the industrial organization literature and show that relaxation to the steady-state solution is extremely slow. Thus, even if these processes were stationary (which they are not), the steady-state solution, which is a very heavy-tailed power law, is not relevant. The distribution is instead well-approximated by a less heavy-tailed log-normal. We perform an empirical analysis of the growth of mutual funds, propose a new, more accurate size-dependent model, and show that it makes a good prediction of the empirically observed size distribution. While mutual funds are in many respects like other firms, market efficiency introduces effects that make their growth process distinctly different. Our work shows that a simple model based on market efficiency provides a good explanation of the concentration of assets, suggesting that other effects, such as transaction costs or the behavioral aspects of investor choice, play a smaller role.'
author:
- 'Yonathan Schwarzkopf[^1] and J. Doyne Farmer[^2]'
title: 'What drives mutual fund asset concentration? '
---
Introduction {#section_introduction}
============
In the past decade the mutual fund industry has grown rapidly, moving from $3\%$ of taxable household financial assets in 1980, to $8\%$ in 1990, to $23\%$ in 2007[^3]. In absolute terms, in 2007 this corresponded to 4.4 trillion USD and 24% of U.S. corporate equity holdings. Mutual funds account for a significant fraction of trading volume in financial markets and have a substantial influence on prices. This raises the question of who has this influence: Are mutual fund investments concentrated in a few dominant large funds, or spread across many funds of similar size? Do we need to worry that a few funds might become so large that they are “too big to fail"? What are the economic mechanisms that determine the concentration of investment capital in mutual funds?
Large institutional investors are known to play an important role in the market [@corsetti-2001]. Gabaix et al. recently hypothesized that the fund size distribution plays a central role in explaining the heavy tails in the distribution of both trading volume and price returns[^4]. If their theory is true this would imply that the heavy tails in the distribution of mutual fund size play an important role in determining market risk.
While it is standard in economics to describe distributional inequalities in terms of statistics such as the Gini or Herfindahl indices, as we show in Appendix A, this approach is inadequate to describe the concentration in the tail. Instead, the best way to describe the concentration of assets is in terms of the functional form of the tail. As is well-known in extreme value theory [@Embrechts97], the key distinction is whether all the moments of the distribution are finite. If the tail is truly concentrated, the tail is a power law, and all the moments above a given threshold, called the tail exponent, are infinite. So, for example, if the tail of the mutual fund size distribution follows Zipf’s law as hypothesized by Gabaix et al., i.e. if it were a power law with tail exponent one, this would imply nonexistence of the mean. In this case the sample estimator fails to converge because the tails are so heavy that with significant probability a single fund can be larger than the rest of the sample combined. This is true even in the limit as the sample size goes to infinity. Thus power law tails imply a very high degree of concentration.
Instead, empirical analysis shows that the tail of the mutual fund size distribution is not a power law, and is well-approximated by a lognormal [@Schwarzkopf10a]. Thus, while the distribution is heavy tailed, it is not as heavy tailed as it would be if the distribution were a power law. The key difference is that for a log-normal all of the moments exist.
This naturally leads to the question of what economic factors determine the tail properties of the mutual fund distribution. There are two basic types of explanation. One type of explanation is based on a detailed description of investor choice, and another is based on efficient markets, which predicts that growth should be random, and that the causes can be understood in terms of a simple random process description of entry, exit and growth. Of course market efficiency depends on investor choice, but the key distinction is that the random process approach does not depend on any of the details, but rather only requires that no one can make superior investments based on simple criteria, such as size.
Explanations based on investor choice can in turn be divided into two types: Rational and behavioral. For example, Berk and Green \[[-@Berk04]\] have proposed that investors are rational, making investments based on past performance. Their theory implies that the distribution of fund size is determined by the skill of mutual fund managers and the dependence of transaction costs on size. If we assume, for example, that the transaction cost is a power law (which includes linearity) if the distribution of fund size is log-normal, then it is possible to show that the distribution of mutual fund skill must also be log-normal. Unfortunately, without a method of measuring skill this is difficult to test.
Another type of explanation is behavioral, i.e. that investors are strongly influenced by factors such as advertising, fees, and investment fads[^5]. We strongly suspect that this is true, and that they play an important role in determining the size of individual funds. The question we investigate here is not whether such effects exist, but whether they are essential to explain the form of the distribution.
The alternative is that the details of investor choice don’t matter, and that the distribution of fund size is driven by market efficiency, which dictates an approach based on the random process of entry, exit and growth. The random process approach was originally pioneered as an explanation for firm size by Gibrat, Simon and Mandelbrot, and is popular in the industrial organization literature[^6]. The basic idea is that while details of investor choice are surely important in determining the size of [*individual*]{} funds, the details may average out or be treatable as noise, so that in aggregate they do not matter in shaping the overall size distribution.
On the face of it, however, there seems to be a serious problem with this approach. Under simple assumptions about the entry, exit and growth of fund size, [@gabaix-2003-mit] showed that the steady state solution is a power law; a similar argument is described in [@Montroll82] and [@Reed01][^7]. As already mentioned, however, the upper tail of the empirical distribution is a log-normal, not a power law. Thus there would seem to be a contradiction. Apparently either the correct random process is more complicated, or this whole line of attack fails.
We show here that the central problem comes from considering only the steady state (i.e. infinite time) solution. We study the same equations considered by Gabaix et al. and Reed, but we find a more general time-dependent solution, and show that the time required to reach steady state is very long. The mutual fund industry is rapidly growing and, even if the growth process had been stationary over the last few decades, not enough time has elapsed to reach the stationary solution for the fund size distribution. In the meantime the solution is well approximated by a log-normal. This qualitative conclusion is very robust under variations of the assumptions. In contrast to the hypothesis of Berk and Green, it does not depend on details such as the distribution of investor skill – the log-normal property emerges automatically from market efficiency and the random multiplicative nature of fund growth.
To test our conjectures more quantitatively we study the empirical properties of entry, exit and growth of mutual funds, propose a more accurate model than those previously studied, and show it makes a good prediction of the empirically observed fund size distribution. The model differs from previous models in that it incorporates the fact that the relative growth rate of funds slows down as they get bigger[^8]. This makes the time needed to approach the steady state solution even longer: Whereas the relaxation time for the size-independent diffusion model is several decades, for the more accurate size-dependent model it is more than a century.
Market efficiency is the key economic principle that makes the random process model work, and dictates many of its properties. It enters the story in several ways. (1) The fact that stock market returns are essentially random implies that growth fluctuations are random, for two reasons: (a) Without inflows and outflows, under the principle that past returns are not indicative of future returns, fund growth is random. (b) Although investors chase past returns, since what they are chasing is random, fund growth due to inflow and outflow is random on sufficiently long time scales. (2) Efficiency dictates that mutual fund performance must be independent of size. Thus as mutual funds randomly diffuse through the size space, there is no pressure pushing them toward a particular size. (3) Efficiency, together with the empirical fact that the relative importance of fund inflows and outflows diminishes as funds get bigger, implies that the mean growth rate and the growth diffusion approach a constant in the large size limit. As we show, this shapes the long-term properties of the size distribution. All of these points are explained in more detail in Section \[sec:size\_change\].
Market efficiency makes mutual funds unusual relative to most other types of firms. For most firms, in the large size limit the mean and standard deviation of the growth rate are empirically observed to decay to zero. For mutual funds, in contrast, due to market efficiency they both approach a positive limit. This potentially affects the long-term behavior: Most firms approach a solution that is thinner than a log-normal, i.e under stationary growth conditions their tails are getting thinner with time, whereas mutual funds approach a power law, so their tails are getting fatter with time. Nonetheless, as we have already mentioned, even under stationary growth conditions the approach to steady-state takes so long that this is a moot point.
At a broader level our work here shows how the non-stationarity of market conditions can prevent convergence to an “equilibrium" solution. Nonetheless, even under stationary conditions the random process model usefully describes the time-dependent relationships between entry, exit and growth phenomena on one hand and size on the other hand. While we cannot show that the random process model is the only possible explanation, we do show that it provides a good explanation[^9]. The conditions for this are robust, depending only on market efficiency, without the stronger requirements of perfect rationality, or the complications of mapping out the idiosyncrasies of human behavior.
The paper is organized as follows. In Section \[section\_model\] we develop the standard exit and entry model. Section \[section\_N\] presents the time-dependent solution for the number of funds, Section \[section\_solution\] presents the time-dependent solution for the size distribution, and Section 2.3 introduces a size-dependent model. Section \[data\_set\] describes our data. In Section \[section\_empiric\_justification\] we perform an empirical analysis to justify our assumptions and to calibrate the model. In Section \[sec:comparison\] we present simulation results of the proposed model and compare them to the empirical data. Finally Section \[conclusions\] presents our conclusions.
Model {#section_model}
=====
Our central thesis in this paper is that due to market efficiency the mutual fund size distribution can be explained by a stochastic process governed by three key underlying processes: the size change of existing mutual funds, the entry of new funds and the exit of existing funds. In this section we introduce the standard diffusion model and derive a time-dependent solution for the special case when the diffusion process has constant mean and variance. We then make a proposal for how to model the more general case where the mean and variance depend on size.
The aim of the model we develop here is to describe the time evolution of the size distribution, that is, to solve for the probability density function $p({\omega},t)$ of funds with size $s$ at time $t$, where ${\omega}= \log s$. The size distribution can be written as $$p({\omega},t)=\frac{n({\omega},t)}{N(t)},$$ where $n({\omega},t)$ is the number of funds at time with logarithmic size ${\omega}$ and $N(t)=\int n({\omega},t){\mathrm{d}}{\omega}$ is the total number of funds at time $t$. To simplify the analysis we solve separately for the total number of funds $N(t)$ and for the number density $n({\omega},t)$.
Dynamics of the total number of funds {#section_N}
-------------------------------------
As we will argue in Section \[section\_empiric\_justification\], the total number of funds as a function of time can be modeled as $$\frac{dN}{dt} = \nu - \lambda N$$ where $\nu$ is the rate of creating new funds and $\lambda$ is the exit rate of existing funds. Under the assumption that $\nu$ and $\lambda$ are constants this has the solution $$\label{eq_Nt}
N(t)=\frac{\nu}{\lambda}\left(1-{\mathrm{e}}^{-\lambda t}\right)\theta(t),$$ where $\theta(t)$ is a unit step function at $t = 0$, the year in which the first funds enter. This solution has the surprising property that the dynamics only depend on the fund exit rate $\lambda$, with a characteristic timescale $1/\lambda$. For example, for $\lambda \approx 0.09$, as estimated in Section \[section\_empiric\_justification\], the timescale for $N(t)$ to reach its steady state is only roughly a decade. An examination of Table \[table\] makes it clear, however, that $\nu = \mbox{constant}$ is not a very good approximation. Nonetheless, if we crudely use the mean creation rate $\nu \approx 900$ from Table \[table\] and the fund exit rate $\lambda \approx 0.09$ estimated in Section \[section\_empiric\_justification\], the steady state number of funds should be about $N \approx 10,000$, compared to the $8,845$ funds that actually existed in 2005. Thus this gives an estimate with the right order of magnitude.
The important point to stress is that the dynamics for $N(t)$ operate on a different timescale than that of $n(\omega, t)$. As we will show in the next section the characteristic timescale for $n(\omega, t)$ is much longer than that for $N(t)$.
Solution for the number density $n({\omega},t)$ {#section_solution}
-----------------------------------------------
We define and solve the time evolution equation for the number density $n({\omega},t)$. The empirical justification for the hypotheses of the model will be given in Section \[section\_empiric\_justification\]. The hypotheses are:
- The entry process is a Poisson process with rate $\nu$, such that at time $t$ a new fund enters the industry with a probability $\nu\mathrm{d}t$ and (log) size ${\omega}$ drawn from a distribution $f({\omega},t)$. We approximate the entry size distribution as a log-normal distribution in the fund size $s$, that is a normal distribution in ${\omega}$ given by $$\label{entry_distribution}
f({\omega},t)=\frac{1}{\sqrt{\pi\sigma_{{\omega}}^2}}\exp\left(-\frac{({\omega}-{\omega}_0)^2}{\sigma_{{\omega}}^2}\right)\theta(t-t_0),$$ where ${\omega}_0$ is the mean log size of new funds and $\sigma_{\omega}^2$ is its variance. $\theta(t-t_0)$ is a unit step function ensuring no funds funds enter the industry before the initial time $t_0$.
- The exit process is a Poisson process such that at any time time $t$ a fund exits the industry with a size independent probability $\lambda{\mathrm{d}}t$.
- The size change is approximated as a (log) Brownian motion with a size dependent drift and diffusion term $$\label{geometric_random_walk}
\mathrm{d}{\omega}=\mu({\omega})\mathrm{d}t+\sigma({\omega})\mathrm{d}W,$$ where $\mathrm{d}W$ is an i.i.d random variable drawn from a zero mean and unit variance normal distribution.
Under these assumptions the forward Kolmogorov equation (also known as the Fokker-Plank equation) defining the time evolution of the number density [@Gardiner04] is given by $$\label{fokker-plank}
\frac{\partial }{\partial t} n({\omega},t)= \nu f({\omega},t) -\lambda n({\omega},t) - \frac{\partial}{\partial {\omega}}[\mu({\omega})n({\omega},t)]+\frac{\partial^2}{\partial {\omega}^2}[D({\omega}) n({\omega},t)],$$ where $D({\omega})=\sigma({\omega})^2/2$ is the size diffusion coefficient. The first term on the right describes the entry process, the second describes the fund exit process and the third and fourth terms describe the change in size of a existing funds.
### Approximate solution for large funds
To finish the model it is necessary to specify the functions $\mu({\omega})$ and $D({\omega})$. It is convenient to define the relative change in a fund’s size $\Delta_s(t)$ as $$\label{ds}
\Delta_s(t) =\frac{s(t+1)-s(t)}{s(t)},$$ such that drift and diffusion parameters in our model are given by $$\mu({\omega})=\mathrm{E}[\log(1+\Delta_s)] \,\,\,\,\,\,\,\,\,\, D({\omega})=\frac{1}{2} \mathrm{Var}[\log(1+\Delta_s)].$$ The relative change can be decomposed into two parts: the return $\Delta_r$ and the fractional investor money flux $\Delta_f (t)$, which are simply related as $$\label{ds_decomposed}
\Delta_s(t)=\Delta_f(t)+\Delta_r(t).$$ The return $\Delta_r$ represents the return of the fund to its investors, defined as $$\label{dr}
\Delta_r(t)=\frac{NAV(t+1)-NAV(t)}{NAV(t)},$$ where $NAV(t)$ is the Net Asset Value at time $t$. The fractional money flux $\Delta_f (t)$ is the change in the fund size by investor deposits or withdrawals, defined as $$\label{df}
\Delta_f(t)=\frac{s(t+1)-[1+\Delta_r(t)]s(t)}{s(t)}.$$
In Section \[section\_empiric\_justification\] we will demonstrate empirically that the returns $\Delta_r$ are independent of size, as they must be for market efficiency. In contrast the money flux $\Delta_f$ decreases monotonically with size. In the large size limit the returns $\Delta_r$ dominate, and thus it is reasonable to treat $\mu(s)$ as a constant, $\mu= \mu_\infty$. Market efficiency also implies that in the large size limit the standard deviation $\sigma(s)$ is a constant, i.e. $\sigma= \sigma_\infty$. Otherwise investors would be able to improve their risk adjusted returns by simply investing in larger funds.
With these approximations the evolution equation becomes $$\frac{\partial }{\partial t} n({\omega},t)= \nu f({\omega},t) -\lambda n({\omega},t) -\mu\frac{\partial}{\partial {\omega}}n({\omega},t)+D\frac{\partial^2}{\partial {\omega}^2}n({\omega},t),
\label{fokker-plank_large_size}$$ In this and subsequent equations, to keep things simple we use the notation $D=\sigma_\infty^2/2$ and $\mu = \mu_\infty$.
The exit process is particularly important, since it is responsible for thickening the upper tail of the distribution. The intuition is as follows: Since each fund exits the industry with the same probability, and since there are more small funds than large funds, more small funds exit the industry. This results in relatively more large funds, making the distribution heavy-tailed. As we will now show this results in the distribution evolving from a log-normal upper tail to a power law upper tail. In contrast, the entry process is not important for determining the shape of the distribution, and influences only the total number of funds $N$. This is true as long as the entry size distribution $f({\omega},t)$ is not heavier-tailed than a lognormal, which is supported by the empirical data.
In the large size limit the solution for an arbitrary entry size distribution $f$ is given by $$n({\omega},t)=\nu \int_{-\infty}^{\infty}\int_0^{t}\exp^{-\lambda t'}\frac{1}{\sqrt{4 \pi D t'}}\exp\left[-\frac{({\omega}-{\omega}' -\mu t')^2}{4 D t'}\right] f({\omega}',t-t')\, {\mathrm{d}}t'{\mathrm{d}}{\omega}'.$$ Stated in words, a fund of size ${\omega}'$ enters at time $t-\tau$ with probability $f({\omega}',t-\tau)$. The fund will survive to time $t$ with a probability $\exp(-\lambda \tau)$ and will have a size ${\omega}$ at time $t$ with a probability according to (\[solution\_no\_exit\_or\_entry\]).
If funds enter the industry with a constant rate $\nu$ beginning at $t=0$, with a log-normal entry size distribution $f({\omega},t)$ centered around ${\omega}_0$ with width $\sigma_{\omega}$ as given by (\[entry\_distribution\]), the size density can be shown to be $$\begin{aligned}
n({\omega},t)&=&\frac{\nu\mu}{4 \sqrt{\gamma } D} \exp\left[(\gamma +\frac{1}{4})\frac{ \sigma_{\omega}^2}{2}-\sqrt{\gamma } \left|\frac{\sigma_{\omega}^2}{2}+ \frac{\mu}{D}\left({\omega}-{\omega}_0\right)\right|+\frac{\mu}{2D} \left({\omega}-{\omega}_0\right)\right] \nonumber\\ &\,&\times\left( A +\exp\left[\sqrt{\gamma }|\sigma_{\omega}^2+2 \frac{\mu}{D} \left({\omega}-{\omega}_0\right)|\right] B\right).\end{aligned}$$ The parameters A, B and $\gamma$ are defined as $$\label{eq_gamma}
\gamma=\sqrt{\frac{1}{4}+\frac{\lambda D}{\mu^2}},$$ $$\begin{aligned}
A&=&\mathrm{Erf}\left[\frac{\left|\frac{\sigma_{\omega}^2}{2}+ \frac{\mu}{D}\left({\omega}-{\omega}_0\right)\right|- \sqrt{\gamma } \sigma_{\omega}^2}{ \sqrt{2} \sigma_{\omega}}\right]\\&& -\mathrm{Erf}\left[\frac{\left|\frac{\sigma_{\omega}^2}{2}+ \frac{\mu}{D}\left({\omega}-{\omega}_0\right)\right|- \sqrt{\gamma } \left(\sigma_{\omega}^2+2\frac{\mu^2}{D}t\right)}{\sqrt{2} \sqrt{\sigma_{\omega}^2+2\frac{\mu^2}{D} t}}\right]\nonumber\end{aligned}$$ and $$\begin{aligned}
B&=& \mathrm{Erf}\left[\frac{ \sqrt{\gamma } \left(\frac{\sigma_{\omega}^2}{2}+\frac{\mu^2}{D}t\right)+|\frac{\sigma_{\omega}^2}{2}+ \frac{\mu}{D}\left({\omega}-{\omega}_0\right)|}{\sqrt{2} \sqrt{\sigma_{\omega}^2+2 \frac{\mu^2}{D}t}}\right]\\
&&-\mathrm{Erf}\left[\frac{ \sqrt{\gamma } \sigma_{\omega}^2+\left|\frac{\sigma_{\omega}^2}{2}+ \frac{\mu}{D}\left({\omega}-{\omega}_0\right)\right|}{ \sqrt{2} \sigma_{\omega}}\right],\nonumber\end{aligned}$$ where $\mathrm{Erf}$ is the error function, i.e. the integral of the normal distribution.
Approximating the distribution of entering funds as having zero width simplifies the solution. Let us define a large fund as one with ${\omega}\gg{\omega}_0$, where ${\omega}_0$ is the logarithm of the typical entry size of one million USD. For large funds we can approximate the lognormal distribution as having zero width, i.e. all new funds have the same size ${\omega}_0$. The number density is then given by $$\begin{aligned}
\label{time_dependent_solution}
n({\omega},t)&=&\frac{ \nu D}{4 \sqrt{\gamma }\mu ^2}e^{\frac{1}{2} \frac{\mu}{D}\left({\omega}-{\omega}_0\right)}\Bigg[e^{-\sqrt{\gamma } \frac{\mu}{D}|{\omega}-{\omega}_0|}\left(1+ \mathrm{erf}\left[\sqrt{\frac{\gamma\mu^2 t}{D}}-\frac{|{\omega}-{\omega}_0|}{2 \sqrt{D t}} \right]\right)\nonumber\\&&- e^{\sqrt{\gamma }\frac{\mu}{D} |{\omega}-{\omega}_0|} \left(1-\mathrm{erf}\left[\mu\sqrt{\frac{t}{D}}(\frac{1}{2}+\sqrt{\gamma})\right]\right)\Bigg].
\label{solution_constant_size}\end{aligned}$$ Since $\gamma > 1/4$ (\[eq\_gamma\]), the density vanishes for both ${\omega}\to \infty$ and ${\omega}\to -\infty$.
### Steady state solution for large funds
The steady state solution for large times is achieved by taking the $t\to\infty$ limit of (\[solution\_constant\_size\]), which gives $$\label{n_delta_s}
n({\omega})=\frac{\nu}{2\mu\sqrt{\gamma}}
\exp\frac{\mu}{D}\left(\frac{{\omega}-{\omega}_0}{2}-\sqrt{\gamma}|{\omega}-{\omega}_0|\right).$$ Since the log size density (\[n\_delta\_s\]) has an exponential upper tail and the CDF for $s$ has a power law tail with an exponent[^10] $\zeta_s$, i.e. $$P(s>X)\sim X^{-\zeta_s}.$$ Substituting for the parameter $\gamma$ using Eq. (\[eq\_gamma\]) for the upper tail exponent yields $$\label{eq_zeta}
\zeta_s=\frac{-\mu+\sqrt{\mu^2+4D\lambda}}{2D}.$$ Note that this does not depend on the creation rate $\nu$. Using the average parameter values in Table \[table\_fit\] the asymptotic exponent has the value $$\zeta_s=1.2\pm0.6.$$ This suggests that if the distribution reaches steady state it will follow Zipf’s law, which is just the statement that it will be a power law with $\zeta_s\approx1$. As discussed in the introduction, this creates a puzzle, as the empirical distribution is clearly log-normal [@Schwarzkopf10a].
### Timescale to reach steady state
Since we have a time dependent solution we can easily estimate of the timescale to reach steady state. The time dependence in Eq. \[time\_dependent\_solution\] is contained in the arguments of the error function terms on the right. When these arguments become large, say larger than 3, the solution is roughly time independent, and can be written as $$t > \frac{9 D}{4 \gamma \mu ^2}\left(1+\sqrt{1+\frac{2}{9} \frac{ \sqrt{\gamma\mu^2} }{D} \left|{\omega}-{\omega}_0\right| }\right)^2.
\label{timeScale1}$$ Using the average values in Table \[table\_fit\] in units of months $\mu=\mu_\infty \approx 0.005$, $D=\sigma_\infty^2/2$ and $\sigma_{\infty} \approx 0.05$. This gives $$t >180 \left(1+\sqrt{1+0.7\left|{\omega}-{\omega}_0\right| }\right)^2,$$ where the time is in months. Plugging in some numbers from Table \[table\_fit\] makes it clear that the time scale to reach steady state is very long. For instance, for funds of a billion dollars it will take about 170 years for their distribution to come within 1 percent of its steady state. This agrees with the empirical observation that there seems to be no significant fattening of the tail in the fifteen years from 1991 - 2005. Note that the time required for the distribution $n({\omega}, t)$ to reach steady state for large values of ${\omega}$ is much greater than that for the total number of funds $N(t)$ to become constant.
During the transient phase the solution remains approximately log-normal for a long time. If funds only change in size and no funds enter or exit, then the resulting distribution is normal $$\label{solution_no_exit_or_entry}
\tilde{n}(w,t)=\frac{1}{\sqrt{4 \pi D t}}\exp\left[-\frac{({\omega}-\mu t)^2}{4 D t}\right],$$ which corresponds to a size distribution $p(s)$ with a lognormal upper tail. While the exit process acts quickly in changing the total number of funds, it acts slowly in changing the shape. This is the key reason why the distribution remains approximately log-normal for so long.
A better model of size dependence \[muAndSigma\]
------------------------------------------------
The mean rate of growth and diffusion are in general size dependent. We hypothesize that the mean growth rate $\mu(s)$ and the standard deviation $\sigma(s)$ are the sum of a power law and a constant, of the form $$\begin{aligned}
\label{eq_mu_sigma_power_law}
\sigma_s(s)&=&\sigma_0s^{-\beta} +\sigma_{\infty} \\
\mu_s(s)&=&\mu_0s^{-\alpha}+\mu_{\infty}.\nonumber\end{aligned}$$ The constant terms come from mutual fund returns (neglecting inflow or outflow of funds), and must be constant due to market efficiency, as explained in more detail in Section \[sec:size\_change\]. The power law terms, in contrast, are due to the flow of funds in and out of the market. There is a substantial literature of proposed theories for this, including ours[^11]. We present the empirical evidence for the power law hypothesis and explain the role of efficiency in more detail in Section \[sizeDependence\].
The functional form given above for the size dependence can be used to make a more accurate diffusion model. The non vanishing drift $\mu_\infty>0$ and diffusion terms $\sigma_\infty>0$ are essential for the distribution to evolve towards a power law. As already mentioned, due to market efficiency $E[\Delta_r(s)]$ must be independent of $s$, and since $E[\Delta_f(s)]$ is a decreasing function of $s$, for large $s$ $ \mu(s) = E[\Delta_r(s)] + E[\Delta_f(s)] = \mu_\infty > 0$. This distinguishes mutual funds from other types of firms, which are typically observed empirically to have $\mu_\infty = \sigma_\infty = 0$ [@stanley-1996; @Matia04]. Assuming that other types of firms obey similar diffusion equations to those used here, it can be shown that the resulting distribution has a stretched exponential upper tail, which is much thinner than a power law[^12].
Data Set {#data_set}
========
We test our model against the CRSP Survivor-Bias-Free US Mutual Fund Database. Because we have daily data for each mutual fund, this database enables us to investigate the mechanism of fund entry, exit and growth to calibrate and test our model[^13]. We study the data from 1991 to 2005[^14]. We define an equity fund as one whose portfolio consists of at least $80\%$ stocks. The results are not qualitatively sensitive to this, e.g. we get essentially the same results even if we use all funds. The data set has monthly values for the Total Assets Managed (TASM) by the fund and the Net Asset Value (NAV). We define the size $s$ of a fund to be the value of the TASM, measured in millions of US dollars and corrected for inflation relative to July 2007. Inflation adjustments are based on the Consumer Price Index, published by the BLS. In Table \[table\] we provide summary statistics of the data set and as seen there the total number of equity funds increases roughly linearly in time, and the number of funds in the upper tail $N_{tail}$ also increases.
variable 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 mean std
-------------------- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
$N$ 372 1069 1509 2194 2699 3300 4253 4885 5363 5914 6607 7102 7794 8457 8845 - -
E$[s]$ (mn) 810 385 480 398 448 527 559 619 748 635 481 335 425 458 474 519 134
Std$[s]$ (bn) 1.98 0.99 1.7 1.66 1.68 2.41 2.82 3.38 4.05 3.37 2.69 1.87 2.45 2.64 2.65 2.42 0.8
E$[{\omega}]$ 5.58 4.40 4.40 3.86 3.86 3.91 3.84 3.85 4.06 3.97 3.60 3.37 3.55 3.51 3.59 3.96 0.54
Std$[{\omega}]$ 1.51 1.98 2.09 2.43 2.50 2.46 2.50 2.51 2.46 2.45 2.63 2.42 2.49 2.59 2.50 2.34 0.29
$\mu (10^{-3})$ 26 42 81 46 72 67 58 39 39 20 -3 -10 50 18 30.5 38.5 26
$\sigma (10^{-1})$ 0.78 2.0 2.6 3.0 2.3 2.8 2.9 3.0 2.8 2.8 2.5 2.4 2.4 2.4 2.5 2.5 0.55
$N_{\tt exit}$ 0 41 45 61 139 115 169 269 308 482 427 660 703 675 626 - -
$N_{\tt enter}$ 185 338 581 783 759 885 1216 1342 1182 1363 1088 1063 1056 796 732 891 346
\[default\]
Empirical investigation of size dynamics {#section_empiric_justification}
========================================
In this section we empirically investigate the processes of entry, exit and growth, providing empirical justification and calibration of the model described in Section 2.
Fund entry {#sec:entry}
----------
We begin by examining the entry of new funds. We investigate both the number $N_{\tt enter}(t)$ and size $s$ of funds entering each year. We perform a linear regression of $N_{\tt enter}(t)$ against the number of existing funds $N(t-1)$, yielding slope $\alpha=0.04 \pm 0.05$ and intercept $\beta =750 \pm 300$. The slope is not statistically significant, justifying the approximation of entry as a Poisson process with a constant rate $\nu$, independent of $N(t)$.
![\[Winit\] The probability density for the size $s$ of entering funds in millions of dollars (solid line) compared to that of all funds (dashed line) including all data for the years 1991 to 2005. The densities were estimated using a gaussian kernel smoothing technique.](figures/Winit){width="10cm"}
The size of entering funds is more complicated. In Figure \[Winit\] we compare the distribution of the size of entering funds $f(s)$ to that of all existing funds. The distribution is somewhat irregular, with peaks at round figures such as ten thousand, a hundred thousand, and a million dollars. The average size[^15] of entering funds is almost three orders of magnitude smaller than that of existing funds, making it clear that the typical surviving fund grows significantly after it enters. It is clear that the distribution of entering funds is not important in determining the upper tails[^16]. The value of the mean log size and its variance are calculated from the data for each period as summarized in Table \[table\_fit\].
Thus the empirical data justifies the approximation of entry as a Poisson process in which an average of $\nu$ funds enter per month, with the size of each fund drawn from a distribution $f({\omega},t)$.
Fund exit
---------
Unlike entry, fund exit is of critical importance in determining the long-run properties of the fund size distribution. In Figure \[Nfinal\_vs\_N\] we plot the number of exiting funds $N_{\tt exit}(t)$ as a function of the total number of funds existing in the previous year, $N(t-1)$. There is a good fit to a line of slope $\lambda$, which on an annual time scale is $\lambda=0.092 \pm 0.030$.
![\[Nfinal\_vs\_N\] The number of equity funds exiting the industry $N_{\tt exit}(t)$ in the year $t$ as a function of the total number of funds existing in the previous year, $N(t-1)$. The plot is compared to a linear regression (full line). The error bars are calculated for each bin under a Poisson process assumption, and correspond to the square root of the average number of funds exiting the industry in that year. ](figures/Nfinal_vs_N){width="10cm"}
This justifies our assumption that fund exit is a Poisson process with constant rate $\lambda$.
Fund growth {#sec:size_change}
-----------
We first test the i.i.d and normality assumptions of the diffusion growth model, and then test to demonstrate the size dependence of the growth process that we proposed in Section \[muAndSigma\]. We also discuss the diverse roles that efficiency plays in shaping the random process for firm growth in more detail.
### Justification for the diffusion model\[justification\]
In the absence of entry or exit we have approximated the growth of existing funds as a multiplicative Gibrat-like process[^17] satisfying a random walk in the log size ${\omega}$. This implicitly assumes that $\Delta_s$ is an i.i.d normal random variable.
The assumption of independence is justified by market efficiency, which requires that the returns $\Delta_r$ of a given fund should be random [@bollen05; @carhart97]. Under the decomposition of the total growth as $\Delta_s = \Delta_r + \Delta_f$, as demonstrated in the next sub-section, in the large size limit the returns $\Delta_r$ dominate, so under market efficiency the i.i.d. assumption is automatically valid.
This is not so obvious for smaller size firms, where the money flux $\Delta_f$ dominates the total growth $\Delta_s$. It is well known that investors chase past performance[^18]. Even though the past performance they are chasing is random, if they track a sufficiently long history of past returns, this can induce correlations. This causes correlations in the money flux $\Delta_f$, which in turn induces correlations in the total size change $\Delta_s$.
To test whether such correlations are strong enough to cause problems with the random process hypothesis, we perform cross-sectional regressions of the form $$\Delta_f(t)=\beta+\beta_1\Delta_r(t-1)+\beta_2\Delta_r(t-2)+\ldots +\beta_6\Delta_r(t-6) + \xi(t),
\label{performanceRegression}$$ where $\xi(t)$ is a noise term. The results are extremely noisy; for example, when we perform separate regressions in five different periods, eight of the thirty possible coefficients $\beta_i$ shown in Table \[corr\_table\] are negative and only two of them are significant at the two standard deviation level. We also perform direct tests of the correlations in $\Delta_f$ and we find that they are small. This justifies our use of the i.i.d. hypothesis.
The normality assumption is also not strictly true. Here we are saved by the fact that the money flux $\Delta_f$ is defined in terms of a logarithm, and while it has heavy tails, they are not sufficiently heavy to prevent it from converging to a normal. We have explicitly verified this by tracking a group of funds in a given size range over time and demonstrating that normality is reached within 5 months. Thus even though the normality assumption is not true on short timescales it rapidly becomes valid on longer timescales.
date 12/2005 9/2005 6/2005 3/2005 12/2004
----------- ---------------- ---------------- --------------- ---------------- ----------------
$\beta_1$ $0.10\pm0.16$ $0.40\pm0.98$ $0.27\pm0.68$ $1.17\pm4.68$ $-0.23\pm1.24$
$\beta_2$ $0.14\pm0.27$ $0.36\pm1.20$ $0.48\pm0.83$ $-0.79\pm3.13$ $-0.65\pm2.31$
$\beta_3$ $0.28\pm0.45$ $0.01\pm1.07$ $0.33\pm0.83$ $1.79\pm3.24$ $0.60\pm2.57$
$\beta_4$ $0.56\pm0.40$ $-0.28\pm0.85$ $0.24\pm1.27$ $-0.28\pm1.65$ $0.44\pm2.32$
$\beta_5$ $0.24\pm0.43$ $-0.25\pm1.13$ $0.21\pm0.90$ $-0.24\pm2.95$ $0.43\pm2.49$
$\beta_6$ $0.48\pm0.38$ $-0.02\pm1.03$ $0.30\pm0.92$ $1.27\pm3.50$ $0.31\pm2.09$
$\beta$ $-0.02\pm0.02$ $0.03\pm0.05$ $0.01\pm0.05$ $0.14\pm0.21$ $0.06\pm0.15$
: \[corr\_table\] Cross-sectional regression coefficients of the monthly fund flow, computed for several months, against the performance in past months, as indicated in Eq. \[performanceRegression\]. The regression was computed cross-sectionally using data for 6189 equity funds. For example the entry for $\beta_1$ in the first (from the left) column represents the linear regression coefficient of the money flux at the end of 2005 on the previous month’s return. The errors are 95% confidence intervals.
### Size dependence of the growth process \[sizeDependence\]
We now test the model for the size dependence of the growth process proposed in Section \[muAndSigma\]. We also discuss the crucial role of the decomposition into returns and money flux in determining the size dependence.
![\[delta\_vs\_s\] A summary of the size dependence of mutual fund growth. The average mean $\mu_r$ and volatility $\sigma_r$ of fund returns, as well as the average $\mu_f$ and volatility $\sigma_f$ of money flux (i.e. the flow of money in and out of funds), are plotted as a function of the fund size (in millions) for the year 2005 (see Eqs. (\[ds\] - \[df\])). The data are binned based on size, using bins with exponentially increasing size; we use monthly units. The average monthly return $\mu_r$ is compared to a constant return of 0.008 and the monthly volatility $\sigma_r$ is compared to 0.03. The average monthly flux $\mu_f$ is compared to a line of slope of -0.5 and the money flux volatility $\sigma_f$ is compared to a line of slope -0.35. Thus absent any flow of money in or out of funds, performance is independent of size, as dictated by market efficiency. In contrast, both the mean and the standard deviation of the money flows of funds decrease roughly as a power law as a function of size.](figures/mu_and_sigma_vs_s_05){width="12cm"}
Figure \[delta\_vs\_s\] gives an overview of the size dependence for both the returns $\Delta_r$ and the money flux $\Delta_f$. The two behave very differently. The returns $\Delta_r$ are essentially independent of size[^19]. This is expected based on market efficiency, as otherwise one could obtain superior performance simply by investing in larger or smaller funds [@malkiel-1995]. This implies that equity mutual funds can be viewed as a constant return to scale industry [@gabaix-2006]. Both the mean $\mu_r = E[\Delta_r]$ and the standard deviation $\sigma_r = \mbox{Var}[\Delta_r]^{1/2}$ are constant; the latter is also expected from market efficiency, as otherwise it would be possible to lower one’s risk by simply investing in funds of a different size.
In contrast, the money flux $\Delta_f$ decreases with size. Both the mean money flux $\mu_f=E[\Delta_f]$ and its standard deviation $\sigma_f = \mbox{Var}[ \Delta_f]^{1/2}$ roughly follow a power law over five orders of magnitude in the size $s$. This is similar to the behavior that has been observed for the growth rates of other types of firms [@stanley-1995; @stanley-1996; @Amaral97; @Bottazzi03a]. As already discussed in footnote 9, there is a large body of theory attempting to explain this (and we believe our own theory presented elsewhere provides the correct explanation [@Schwarzkopf10b]).
As explained in Section \[muAndSigma\], the steady state solution is qualitatively different depending on whether the parameters $\mu_\infty$ and $\sigma_\infty$ in Eq. \[eq\_mu\_sigma\_power\_law\] are positive. As can be seen from the fit parameters in Table \[table\_fit\], based on data for $\Delta_s$ alone, we cannot strongly reject the hypothesis that the drift and diffusion rates vanish for large sizes, i.e. $\mu_{\infty}\to 0$ and $\sigma_{\infty} \to 0$. However, because the size change $\Delta_s$ can be decomposed as $\Delta_s = \Delta_r +\Delta_f$, efficiency dictates that $\Delta_r$ is independent of size, and since $E[\Delta_r] > 0$, we are confident that neither $\mu_{\infty}$ nor $\sigma_{\infty}$ are zero.
variable 1991- 1998 1991- 2005
------------------- ----------------------- -------------------
${\omega}_0$ $0.14$ $-0.37$
$\sigma_{\omega}$ $3.02$ $3.16$
$\sigma_0$ $0.35 \pm 0.02 $ $0.30 \pm 0.02$
$\beta$ $0.31 \pm 0.03$ $0.27 \pm 0.02$
$\sigma_{\infty}$ $0.05 \pm 0.01 $ $0.05 \pm 0.01$
$R^2$ 0.93 0.96
$\mu_0$ $0.15 \pm 0.01 $ $0.08 \pm 0.05$
$\alpha$ $0.48 \pm 0.03 $ $0.52 \pm 0.04$
$\mu_{\infty}$ $0.002 \pm 0.008$ $0.004 \pm 0.001$
$R^2$ 0.98 0.97
: \[table\_fit\] Model parameters as measured from the data in different time periods. ${\omega}_0$ and $\sigma_{\omega}^2$ are the mean and variance of the average (log) size of new funds described in (\[entry\_distribution\]). $\sigma_0$, $\beta$ and $\sigma_\infty$ are the parameters for the size dependent diffusion and $\mu_0$, $\alpha$ and $\mu_{\infty}$ are the parameters of the average growth rate (\[eq\_mu\_sigma\_power\_law\]). The confidence intervals are 95$\%$ under the assumption of standard errors. The adjusted $R^2$ is given for the fits for each period. The time intervals were chosen to match the results shown in Fig. \[simV2\].
\[default\]
As we showed in Table \[corr\_table\], the correlation between the returns $\Delta_r$ and the money flux $\Delta_f$ is small. This implies that the standard deviations can be written as a simple sum. Since $\Delta_r$ is independent of size and both the mean and standard deviation of $\Delta_f$ are power laws, this indicates that Eq. \[muAndSigma\] is a good approximation, and that $\mu_\infty$ and $\sigma_\infty$ are both greater than zero. As illustrated in Figure \[fit\], these functional forms fit the data reasonably well, with only slight variations of parameters in different periods, as shown in Table \[table\_fit\].
![\[fit\] An illustration that the empirical power law-based model provides a good fit to the distribution of mutual funds. (a) The standard deviation $\sigma$ of the logarithmic size change $\Delta_s = \Delta(\log s)$ of an equity fund as a function of the fund size $s$ (in millions of dollars). (b) The mean $\mu$ of $\Delta_s = \Delta(\log s)$ of an equity fund as a function of the fund size $s$ (in millions of dollars). The data for all the funds were divided into 100 equally occupied bins. $\mu$ is the mean in each bin and $\sigma$ is the square root of the variance in each bin for the years 1991 to 2005. The data are compared to a fit according to (\[eq\_mu\_sigma\_power\_law\]) in Figures (a) and (b) respectively. ](figures/fit91-05){width="12cm"}
Testing the predictions of the model {#sec:comparison}
====================================
In this section we use our calibrated model of the entry, exit and size-dependent growth processes to simulate the evolution of the firm size distribution through time. We are forced to use a simulation since, once we include the size dependence of the diffusion and drift terms as given in equation (\[eq\_mu\_sigma\_power\_law\]), we are unable to find an analytic solution for the general diffusion equation (Eq. (\[fokker-plank\])). The analytic solution of the size independent case (Eq. (\[time\_dependent\_solution\])) gives the correct qualitative behavior, but the match is much better once one includes the size dependence.
The simulation was done on a monthly time scale, averaging over 1000 different runs to estimate the final distribution. As we have emphasized in the previous discussion the time scales for relaxation to the steady state distribution are long. It is therefore necessary to take the huge increase in the number of new funds seriously. We begin the simulation in 1991 and simulate the process for varying periods of time, making our target the empirical distribution for fund size at the end of each period. In each case we assume the size distribution for injecting funds is log-normal, as discussed in Section \[sec:entry\].
To compare our predictions to the empirical data we measure the parameters for fund entry, exit and growth using data from the same period as the simulation, summarized in Table \[table\_fit\]. A key point is that we are not fitting these parameters on the target data for fund size[^20], but rather are fitting them on the individual entry, exit and diffusion processes and then simulating the corresponding model to predict fund size. One of our main predictions is that the time dependence of the solution is important. In Figure \[simV2\] we compare the predictions of the simulation to the empirical data at two different ending times.
![ \[simV2\] The model is compared to the empirical distribution at different time horizons. The left column compares CDFs from the simulation (full line) to the empirical data (dashed line). The right column is a QQ-plot comparing the two distributions. In each case the simulation begins in 1991 and is based on the parameters in Table \[table\_fit\]. The first row corresponds to the years 1991-1998 and the second row to the years 1991-2005 (in each case we use the data at the end of the quoted year). ](figures/simV2_98_05){width="12cm"}
The model fits quite well at all time horizons, though the fit in the tail is somewhat less good at the longest time horizon. Note, that our simulations make it clear that the fluctuations in the tail are substantial. The deviations between the prediction and the data are thus very likely due to chance – many of the individual runs of the simulation deviate from the mean of the 1000 simulations more than the empirical data does.
Conclusions\[conclusions\]
==========================
We have argued that the mutual fund size distribution is driven by market efficiency, which gives rise to a random growth process. The essential elements of the growth process are multiplicative random changes in the size of existing funds, entry of new funds, and exit of existing funds as they go out of business. We find, however, that entry plays no role at all other than setting the scale; exit plays a small role in thickening the tails of the distribution, but this acts only on a very slow timescale. The log-normality comes about because the industry is young and still in a transient state, and the exit process has not had a sufficient time to act. In the future, if the conditions for fund growth and exit were to remain stationary for more than a century, the distribution would become a power law. The thickening of the tails happens from the body of the distribution outward, as the power law tail extends to successively larger funds. We suspect that the conditions are highly unlikely to remain this stationary, and that the fund size distribution will remain indefinitely in its current log-normal, out of equilibrium state. There is also an interesting size dependence in the growth rate of mutual fund size, which is both like and unlike that of other types of firms. Mutual funds are distinctive in that their overall growth rates can be decomposed as a sum of two terms, $\Delta_s = \Delta_f + \Delta_r$, where $\Delta_f$ represents the flow of money in and out of funds, and $\Delta_r$ the returns on money that is already in the fund. The money flow $\Delta_f$ decreases as a power law as a function of size, similar to what is widely observed in the overall growth rates for other types of firms. Furthermore the exponents are similar to those observed elsewhere. The returns $\Delta_r$, in contrast, are essentially independent of fund size, as they must be under market efficiency. As a result, for large sizes the mean and variance of the overall growth are constant – this is unlike other firms, for which the mean and variance appear to go to zero in the limit. As we discuss here, this makes a difference in the long-term evolution: While the exit process is driving mutual funds to evolve toward a heavier-tailed distribution, other firms are evolving toward a thinner-tailed distribution. Again, though, due to the extremely slow relaxation times, we suspect this makes little or no difference.
Our analysis here suggests that the details of investor preference have a negligible influence on the upper tail of the mutual fund size distribution, except insofar as investors choose funds so as to enforce market efficiency. Investor preference enters our analysis only through $\Delta_f$, the flow of money in and out of the fund. Since $\Delta_f$ becomes relatively small in the large size limit, the growth of large funds is dominated by the returns $\Delta_r$, whose mean and variance are constant. Thus the upper tail of the size distribution is determined by market efficiency, which dictates both that returns are essentially random, and thus diffusive, and that there is no dependence on size. As a result, for large fund size investor preference doesn’t have much influence on the growth process. This is reinforced by the fact that the statistical properties of the money flux $\Delta_f$ are essentially like those of the growth of other firms.
How can size-dependent transaction costs be compatible with our results here? We have performed an empirical study, which we will report elsewhere, that demonstrates that as size increases fund managers maintain constant after-transaction cost performance by lowering fees, reducing trading and diversifying investments. This is in contrast to the theory proposed by Berk and Green ([-@Berk04]) that fund size is determined by the skill of fund managers, i.e. that better managers attract more investment until increased transaction cost causes excess returns to disappear. Both our theory and that of Berk and Green are based on market efficiency. The key difference is that we find that the flatness of performance vs. size is enforced by simple actions taken by fund managers that do not influence the diffusion of fund size. In contrast, the Berk and Green theory requires choices by investors that directly influence fund size, and thus is not compatible with the free diffusion that we have prevented empirical evidence for here. In their theory transaction costs and investor skill determine fund size; in our theory, neither plays a role.
We would like to stress that, while we are fitting econometric models to the entry, exit and growth processes, and calibrating these models against the data, we are not fitting any parameters on the size data itself. This makes it challenging to get a model that fits as well as the model shown in Fig. \[simV2\]. Of course, we have only demonstrated that the random process model is sufficient to explain fund size; we cannot demonstrate that other explanations might not also be able to explain it. However, the assumptions that we make here are simple and natural. The stochastic nature of fund growth is not surprising: It is well known that past returns do not predict future returns. Thus even if investors chase returns, they are chasing something that is inherently random. We believe that this is at the core of why our model works so well. Our demonstration that a good explanation can be obtained based on market efficiency alone, which requires weaker assumptions than full rationality, provides a theory that is robust and largely independent of the details of human choice.
Acknowledgements {#acknowledgements .unnumbered}
================
[We would like to thank Brad Barber, Giovani Dosi, Fabrizio Lilo, Terry Odean, Eric Smith and especially Rob Axtell and Andrew Lo for useful comments. YS would like to thank Mark B. Wise. We gratefully acknowledge financial support from NSF grant HSD-0624351. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.]{}
Inadequacy of Gini coefficients to characterize tail behavior {#gini}
=============================================================
![The Gini coefficients as described in equation (\[eq:gini\]) are calculated numerically for a lognormal distribution and a Pareto distribution. The Gini Coefficients were calculated for different parameter values and are plotted as a function of the resulting standard deviation. For the Pareto distribution (footnote 20) we used $s_0=0.01$ and different exponents $\alpha$ in the range $(2,5]$, i.e. a finite second moment. The lower standard deviation $\sigma=0.0033$ corresponds to $\alpha=5$ and $\sigma=1916.17$ corresponds to $\alpha\to2$. For the lognormal we used $a=0$ and different $b$ in the range $[0.1,2.8]$ where $b=0.1$ corresponds to $\sigma=0.101$ and $b=2.757$ corresponds to $\sigma=2000$. []{data-label="fig:gini"}](figures/GiniCoeffsLN "fig:"){width="6cm"} ![The Gini coefficients as described in equation (\[eq:gini\]) are calculated numerically for a lognormal distribution and a Pareto distribution. The Gini Coefficients were calculated for different parameter values and are plotted as a function of the resulting standard deviation. For the Pareto distribution (footnote 20) we used $s_0=0.01$ and different exponents $\alpha$ in the range $(2,5]$, i.e. a finite second moment. The lower standard deviation $\sigma=0.0033$ corresponds to $\alpha=5$ and $\sigma=1916.17$ corresponds to $\alpha\to2$. For the lognormal we used $a=0$ and different $b$ in the range $[0.1,2.8]$ where $b=0.1$ corresponds to $\sigma=0.101$ and $b=2.757$ corresponds to $\sigma=2000$. []{data-label="fig:gini"}](figures/GiniCoeffsPL "fig:"){width="6cm"}
The Gini coefficient [@Gini1912] is commonly used as a measure of inequality but as we show here it is not suitable for distinguishing between highly skewed distributions when one wishes to focus on tail behavior. For a non negative size $s$ with a CDF $F(s)$, the Gini coefficient $G$ is given by $$\label{eq:gini}
G=\frac{1}{E[s]}\int_0^{\infty} F(s)(1-F(s))\mathrm{d}s,$$ where $E[s]$ is the mean [@Dorfman79]. To illustrate the problem we compare the Gini coefficients of a Pareto distribution to those of a lognormal[^21]. For a Pareto distribution with tail parameter $\alpha$ the $m>\alpha$ moments do not exist. This is in contrast to the lognormal distribution, for which all moments exist. Naively one would therefore expect that the Gini coefficient of the Pareto distribution (see footnote 18) to be larger than that of a lognormal since it has a heavier upper tail. This is true for a Pareto distribution with $\alpha < 2$, for which the Gini coefficient is one due to the fact that the standard deviation does not exist. However, when $\alpha < 2$, for large standard deviations the Gini coefficient of the log-normal is greater than that of the Pareto, as shown in Figure \[fig:gini\]. In order to compare apples to apples in Figure \[fig:gini\] we plot the Gini coefficient as a function of the standard deviation (which is a function of the distribution parameters). For a Pareto distribution with a finite second moment ($\alpha>2$) the lognormal has a higher coefficient.
Thus, even though the Gini coefficient is frequently used as measure for inequality, it is not a good measure when one seeks to study tail properties, particularly for comparisons of distributions with different functional representations. The reason is that the Gini coefficient is a property of the whole distribution, and depends on the shape of the body as well as the tail. Similar remarks apply to the Herfindahl index.
Simulation model {#appendix_simulation_model}
================
We simulate a model with three independent stochastic processes. These processes are modeled as Poisson process and as such are modeled as having at each time step a probability for an event to occur. The simulation uses asynchronous updating to mimic continuous time. At each simulation time step we perform one of three events with an appropriate probability. These probabilities will determine the rates in which that process occurs. The probability ratio between any pair of events should be equal to the ratio of the rates of the corresponding processes. Thus, if we want to simulate this model for given rates our probabilities are determined.
These processes we simulate are:
1. The rate of size change taken to be 1 for each fund and $N$ for the entire population.Thus, each fund changes size with a rate taken to be unity.
2. The fund exit rate $\lambda$ which can depend on the fund size.
3. The rate of creation of new funds $\nu$.Each new fund enters with a size ${\omega}$ with a probability density $f({\omega})$.
Since some of these processes are defined per firm as opposed to the creation process, the simulation is not straightforward. We offer a brief description of our simulation procedure.
1. At every simulation time step, with a probability $\frac{\nu}{1+\lambda +\nu}$ a new fund enters and we proceed to the next simulation time step.
2. If a fund did not enter then the following is repeated $(1+\lambda)N$ times.
a.
: We pick a fund at random.
b
: . With a probability of $\frac{\lambda}{1+\lambda}$ the fund enters.
c.
: If it is not annihilated, which happens with a probability of $\frac{1}{1+\lambda}$, we change the fund size.
We are interested in comparing the simulations to both numerical and empirical results. The comparisons with analytical results are done for specific times and for specific years when comparing to empirical data. In order to do so, we need to convert simulation time to “real” time. The simulation time can be compared to ’real’ time if every time a fund does not enter we add a time step. Because of the way we defined the probabilities each simulation time step is comparable to $1/(1+\lambda)$ in “real” time units. The resulting “real” time is then measured in what ever units our rates were measured in. In our simulation we use monthly rates and as such a unit time step corresponds to one month.
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[^1]: California Institute of Technology, Pasadena, CA 91125. Santa Fe Institute, Santa Fe, NM 87501. yoni@caltech.edu.
[^2]: Santa Fe Institute, Santa Fe, NM 87501.Luiss Guido Carli, ROMA Italy. jdf@santafe.edu.
[^3]: Data is taken from the Investment Company Institute’s 2007 fact book available at www.ici.org.
[^4]: The equity fund size distribution was argued to be responsible for the observed distribution of trading volume [@levy-1996; @solomon-2001], and Gabaix et al. have argued that it is important for explaining the distribution of price returns [@gabaix-2003-nature; @gabaix-2006].
[^5]: @Barber05 have found that investors flows are correlated to marketing and advertising while they are not correlated to the expense ratio.
[^6]: For past stochastic models see [@gibrat-1931; @simon-1955; @simon-1958; @mandelbrot-1963; @ijiry-1977; @sutton-1997; @gabaix-2003-mit; @gabaix-2003-nature].
[^7]: For a review on similar generative models see [@Mitzenmacher03].
[^8]: For work on the size dependence of firm growth rate fluctuations see[@stanley-1995; @stanley-1996; @Amaral97; @Bottazzi01; @Bottazzi03a; @Bottazzi05; @dosi-2005; @Defabritiis03].
[^9]: While variations in the assumptions about the random process preserve certain qualitative conclusions, such as the log-normal character of the upper tail, we found that getting a good fit to the data requires a reasonable degree of fidelity in the modeling process. The size-dependent nature of the diffusion process, for example, is quite important.
[^10]: To calculate the tail exponent of the density correctly one must change variables through $p(s)=p({\omega})\frac{\mathrm{d}{\omega}}{\mathrm{d}s}\sim s^{-\zeta_s-1}$. This results in a CDF with a tail exponent of $\zeta_s$.
[^11]: There has been a significant body of work attempting to explain the heavy tails in the growth rate of firms and the associated size dependence in the diffusion rate. See [@Amaral97; @Buldyrev97; @Amaral98; @Defabritiis03; @Matia04; @Bottazzi01; @Sutton01; @Wyart03; @Bottazzi03b; @Bottazzi05; @Fu05; @Riccaboni08; @Podobnik08]. Our theory argues for an additive replication model, and produces predictions that fit the data extremely well for a diverse set of different phenomena, including mutual funds [@Schwarzkopf10b]. We argue that the fundamental reason for the power tails is the influence network of investors.
[^12]: A stretched exponential is of the form $p(x) \sim \exp(a x^{-b})$, where $a$ and $b$ are positive constants. There is some evidence in the empirical data that the death rate $\lambda$ also decays with size. However, in our simulations we found that this makes very little difference for the size distribution as long as it decays slower than the distribution of entering funds, and so in the interest of keeping the model parsimonious we have not included this effect.
[^13]: Note that we treat mergers as the dissolution of both original firms followed by the creation of a new (generally larger) firm. This increases the size of entering firms but does not make a significant difference in our conclusions.
[^14]: There is data on mutual funds starting in 1961, but prior to 1991 there are very few entries. There is a sharp increase in 1991, suggesting incomplete data collection prior to 1991.
[^15]: When discussing the average size one must account for the difference between the average log size and the average size: Due to the heavy tails the difference is striking. The average entry log size E$[{\omega}_c]\approx 0$, corresponding to a fund of size one million, while if we average over the entry sizes E$[s_c]=\mathrm{E}[{\mathrm{e}}^{{\omega}_c}]$, we get an average entry size of approximately 30 million. For comparison, both the average size and the average log size of existing funds are quoted in Table \[table\].
[^16]: In Section \[section\_solution\] we showed that the entry process is not important as long as the tails of the entry distribution $f$ are sufficiently thin. We compared the empirical $f$ to a log-normal and found that the tails are substantially thinner.
[^17]: A Gibrat-like process is a multiplicative process in which the size of the fund at any given time is given as a multiplicative factor times the size of the fund at a previous time. In Gibrat’s law of proportionate effect [@gibrat-1931] the multiplicative term depends linearly on size while here we allow it to have any size dependence.
[^18]: For empirical evidence that investors react to past performance see [@remolona-1997; @busse-2001; @chevalier-1997; @sirri-1998; @delguercio-2002; @bollen-2007].
[^19]: The independence of the return $\Delta_r$ on size is verified by performing a linear regression of $\mu_r$ vs. $s$ for the year 2005, which results in an intercept $\beta=6.7\pm 0.2 \times10^{-3}$ and a slope $\alpha=0.5\pm 8.5 \times10^{-8}$. This result implies a size independent average monthly return of 0.67%.
[^20]: It is not our intention to claim that the processes describing fund size are constant or even stationary. Thus, we would not necessarily expect that parameters measured on periods outside of the sample period will be a good approximation for those in the sample period. Rather, our purpose is to show that the random model for the entry, exit and growth processes can explain the distribution of fund sizes.
[^21]: The CDF of the Pareto distribution is defined as $$\label{eq:cdf_pareto}
F_{p}(s)=1-\left(\frac{s}{s_0}\right)^{-\alpha},$$ where $s_0$ is the minimum size and $\alpha$ is the tail exponent. The CDF of a lognormal is given by $$\label{eq:cdf_ln}
F_{ln}(s)=\frac{1}{2} \left(1+\mathrm{Erf}\left[\frac{\log(s)-a}{\sqrt{2} b}\right]\right),$$ where $a$ is a location parameter and $b$ is the scale parameter.
|
---
abstract: 'We investigate the Gamow-Teller strength distributions in the electron-capture direction in nuclei having mass $A=90-97$, assuming a $^{88}$Sr core and using a realistic interaction that reasonably reproduces nuclear excitation spectra for a wide range of nuclei in the region as well as experimental data on Gamow-Teller strength distributions. We discuss the systematics of the distributions and their centroids. We also predict the strength distributions for several nuclei involving stable isotopes that should be experimentally accessible for one-particle exchange reactions in the near future.'
author:
- 'A. Juodagalvis$^{1,2}$ and D.J. Dean$^1$'
date: 'June 17, 2004'
title: 'Gamow-Teller $GT_+$ distributions in nuclei with mass $A=90-97$'
---
Introduction {#sect-Intro}
============
New frontiers of nuclear structure experiments to probe the Gamow-Teller distributions in medium-mass nuclei are currently being pursued. These experiments will be able to measure Gamow-Teller data in the mass 90-100 region. Extensive theoretical studies have been devoted to Gamow-Teller total strengths and strength distributions in $1s$-$0d$ shell nuclei (mass $A=16$-40 nuclei) [@BW-88] and the $0f$-$1p$ shell (mass $A=40$-80 nuclei) [@Langanke95; @Martinez96; @Caurier99GT]. Due to an excellent agreement between shell model results and the available experimental data, the calculated results have been used extensively to predict numerous Gamow-Teller strength distributions in nuclei that have not yet become experimentally accessible [@Langanke00].
In addition to their nuclear structure interests, an appropriate description of Gamow-Teller transitions in nuclei directly affects the early phases of type II supernova core collapse since electron capture rates are partly determined by them. The effects of the improved rate estimates are rather dramatic, as was recently discussed in Refs. [@Langanke03; @Hix03]. In addition to the standard Gamow-Teller transitions, first- and second-forbidden transitions contribute to the electron capture rates in the supernova environment. For terrestrial experiments, the primary focus is on the Gamow-Teller transitions.
Recently, Zegers [*et al*]{} [@Zegers04] proposed to measure the Gamow-Teller distributions using stable Zr and Mo isotopes as targets in $(t,{}^3{\rm He})$ reactions [@Mo-Zegers]. Estimates indicate that the Gamow-Teller strength is sufficiently large to be measured. In this paper, we will investigate these transitions using standard shell-model diagonalization techniques for 36 nuclei with the mass number $90\leq A\leq97$ ($Z=40$-47, $N=50$-57). To validate the interaction, we also studied excitation spectra in those and other nuclei in the region. Since our model space does not contain all spin-orbit partners, i.e. it is not a complete $0\hbar\omega$ calculation, the total Gamow-Teller strength will be overestimated in our calculations. We adopt a single quenching factor similar to the one discussed in Ref.[@N50GT-Brown]. We estimated this factor based on recent experimental data on 97Ag [@Ag97GT-Hu]. We used this measurement to gauge our calculation for two reasons. First, it used the total absorption spectrometry which accounts also for the weak $\gamma$-ray cascades that follow the $\beta^+$ decay. Second, almost all total Gamow-Teller strength is inside the $Q$-window. We note that this factor need not be universal as it is simply a phenomenological tool at this point.
The remainder of this paper is organized as follows. In Section \[sect-Model\], we present results on the nuclear spectra, generated with an effective interaction that uses $^{88}$Sr as a core, and compare them to experiment. In section \[sect-GT\], we present our shell-model diagonalization results for the Gamow-Teller strength distributions and compare to experiment when available. We also present systematics of the Gamow-Teller centroids. In section \[sect-SMCode\], we discuss the distributed-memory shell-model computer code that we developed and used for these calculations. Finally, we conclude and give a perspective in Section \[sect-Summary\].
Calculated spectra using the $^{\bm8\bm8}$Sr core {#sect-Model}
=================================================
We perform our shell-model diagonalization calculations in a model space taking $^{88}$Sr as the core nucleus and allowing excitations within the valence space of $1p_{1/2}$ and $0g_{9/2}$ proton shells and $1d_{5/2}$, $2s_{1/2}$, $1d_{3/2}$, $0g_{7/2}$, and $0h_{11/2}$ neutron shells. While it cannot be used for calculations of $\beta$-decays, it appears suitable for Gamow-Teller distributions in the electron-capture direction. The effective interaction [@In102-Lipoglavsek] was derived from a CD-Bonn potential [@CDBonn] using the machinery of many-body perturbation theory [@HjorthJensen95]. We use the following single-particle energies: $\varepsilon(p_{1/2})=0.0$ MeV and $\varepsilon(g_{9/2})=0.9$ MeV for protons; and $\varepsilon(d_{5/2})=0.0$ MeV, $\varepsilon(s_{1/2})=1.26$ MeV, $\varepsilon(d_{3/2})=2.23$ MeV, $\varepsilon(g_{7/2})=2.90$ MeV, and $\varepsilon(h_{11/2})=3.50$ MeV for neutrons. A slightly different version of this interaction was used to describe Sr and Zr isotopes [@SrZr-Holt]. We have not attempted to adjust the interaction to obtain a better fit to experimental data [@NDSOnline].
We calculated low-energy spectra of more than 50 nuclei with masses $90\leq A\leq 98$, $38\leq Z\leq48$, and $50\leq N\leq 58$. General agreement between the calculated lowest states and experimentally observed states is satisfactory. We judged the agreement based on reproduction of low-lying states up to a chosen excitation energy. For odd-odd isotopes, which have higher density of states, the upper limit was chosen to be 1 MeV. For even-even nuclei the limit was up to 3 MeV. Not all observed states were found in the model space, as would be expected from a restricted calculation, and for some nuclei our calculations suggested some low-lying states that have not yet been observed.
The interaction generally reproduces the correct spin for the lowest states of both parities as well as their ordering, though there are cases where some levels are interchanged. The energy splitting between the lowest states with different parities is reproduced with varying success, although this is difficult to judge for some nuclei because of the lack of experimental information.
For representative spectra which indicate the overall quality of the interaction, we show nuclei having mass $A=96$ in Figs.\[fig-A96Spectra\]-\[fig-A96Spectra2\]. The maximum energy range shown in the plots is varied following the density of states. From these figures we observe that the spectra of even-even nuclei (96Pd, 96Ru, 96Mo, 96Zr, and 96Sr) are reproduced well. For 96Pd there are more calculated states than experimentally known. In some nuclei the model space is insufficient to describe all observed states. Odd-odd nuclei (96Rh,96Tc, 96Nb, and 96Y), having more states, are also more difficult to describe although, even here, the interaction performs reasonably well. The position of 8[$^+$]{} isomer in 96Y is not known experimentally [@A96DataSheets]. This state appears in the calculation at a relatively high excitation energy, 1.1 MeV above the lowest positive parity state which was calculated to be 5[$^+$]{}. The nucleus 96Tc reflects a situation where the lowest states are experimentally very close (in this nucleus there are 6 states in the energy range of 50 keV), while the calculation reproduces the states but not their energies (the calculated range is 310 keV). A similar situation occurs in $^{92,94}$Nb.
These $A=96$ nuclei reflect the situation in other cases as well, with a general conclusion that the interaction reproduces excitation spectra reasonably well, though fine tuning might increase the accuracy. We do not discuss them in a greater detail, since the focus of our paper is Gamow-Teller distributions.
Gamow-Teller strength distributions {#sect-GT}
===================================
Our study focuses on the Gamow-Teller transitions from the lowest positive parity states which is natural for the most nuclei in the region above 88Sr, with the exception of the Y isotopes where the odd proton in the $p_{1/2}$ shell is responsible for low-lying negative parity states. Since our model space is not sufficient to reasonably reproduce negative parity states in Sr isotopes where no valence protons are available, we do not calculate the transitions between these two isotope chains. Among the calculated nuclei, there are three cases where we chose the lowest experimental state to be the initial state for $GT$ excitations rather than using our calculated lowest energy state. This affected two $N=51$ nuclei, 92Nb and 94Tc, where the calculation places 2[$^+$]{} to be the lowest state, and the nucleus 96Tc.
The Gamow-Teller strength was calculated using the formula $$GT_+\, =\, \langle \bm{\sigma } \bm{\tau} \rangle^2\,
=\,
\frac1{2J_i+1} \sum\limits_f
\left| \langle \Psi_f ||
\sum_k \bm{\sigma}(k) \bm{\tau_+}(k) || \Psi_i \rangle \right|^2.$$ To obtain the strength distribution, we used the method of moments [@StrengthFunction]. We performed 33 iterations for each $J_f$ in all nuclei except for the decays of 97Mo, where we did 24 iterations per final state, and 97Ag, where a complete convergence was achieved. The $GT_+$ strength inside the experimental $Q_{EC}$ window [@Audi03] is marked as $B_{Q_{EC}}$. This value is only an estimate, since we did not strive to achieve the convergence of states inside the $Q$-window.
As discussed above, our calculation overpredicts the Gamow-Teller strength; thus we include a hindrance factor, $h$, so that $S(GT_+)=GT_+/h$. This factor is found by comparing experimental data to the calculated Gamow-Teller total strength. For nuclei around 100Sn, the single-particle estimate of the Gamow-Teller strength is commonly used, since the main contribution comes from a transition of a $g_{9/2}$ proton into a $g_{7/2}$ neutron. The estimate is given by a formula (see e.g. [@Towner85]): $$\sum GT_+=
\frac{N_{9/2}}{10}\,
\left(1-\frac{N_{7/2}}8\right)\,
GT_+(^{100}{\rm Sn}),
\label{eq-GTocc}$$ where $N_{9/2}$ is the occupation of the $g_{9/2}$ shell by protons, and $N_{7/2}$ is the occupation of $g_{7/2}$ shell by neutrons in the initial state of a parent nucleus, and $GT_+(^{100}{\rm Sn})=17.78$ is the single particle estimate of the total Gamow-Teller strength for 100Sn. In the simplest, non-interacting shell model, the occupation numbers are replaced by the numbers of valence particles in the corresponding shells. This simplest estimate does not exactly reproduce our calculated strength even though the values are close. It underestimates the strength for isotopes below the mass $A=96$, which is not surprising because our model space allows excitations out of the $p_{1/2}$ shell. On the other hand, it overestimates the strength for some isotopes with $A=97$, which is related to the partial occupation of the $g_{7/2}$ shell by neutrons reducing the total strength as compared to the non-interacting picture.
Since our calculated total strength is reasonably close to the single-particle estimates, we could use the experimental hindrance factor quoted relative to the single-particle estimate: $h^{exp}=GT_{sp}/GT_{exp}$. Unfortunately, in this region, experimental information on $h^{exp}$ is limited. Some of the calculated nuclei naturally decay from the ground state by $\beta^-$-decay instead of electron capture, while in other nuclei, the $Q$-window contains only a small fraction of the total strength. Thus the total $GT_+$ strength could be obtained only by $(n,p)$ or similar one-particle exchange reactions. An additional uncertainty in deriving the hindrance factor, even for nuclei where the $Q$-window is large, comes from the fact that $\gamma$-ray spectroscopy misses a significant fraction of the Gamow-Teller decay strength due to sensitivity limits of detectors, a low population of nuclear levels close to the $Q$-limit as well as weak intensity of their decays [@N50GT-Brown; @In102GT-Gierlik]. This limitation can be largely overcome combining a high-resolution $\gamma$-ray detector with total absorption spectrometry (TAS), as was done in a number of recent experiments on nuclei in the 100Sn region. For example, a study of 97Ag decay [@Ag97GT-Hu] showed that only 2/3 of the total Gamow-Teller strength is obtained by high-resolution $\gamma$-ray spectrometry, while the same number for 102In was about 1/8 [@In102GT-Gierlik].
One nucleus, 97Ag, has almost 98% of the total $GT$ strength inside the $Q_{EC}$ window. We can use this nucleus to estimate the experimental hindrance factor. Hu [*et al*]{} [@Ag97GT-Hu] reported $\sum
B(GT)=3.00(40)$ based on TAS measurements, which leads to the hindrance factor $h^{exp}=4.24_{-0.50}^{+0.65}$. Another nucleus where this window is large, and there is a TAS measurement available is 98Ag [@Ag98GT-Hu]. Hu [*et al*]{} reported the total strength in 98Ag to be 2.7(4), giving the hindrance factor $h^{exp}=4.27_{-0.55}^{+0.74}$, since the calculated $B_{Q_{EC}}$ is 11.53 (this is 92% of the total Gamow-Teller strength inside the $Q_{EC}=8.24$ MeV window). Thus the hindrance in two Ag isotopes, 97Ag and 98Ag, are of the order of 4.25, and we adopt this value for the total hindrance factor $h$. We did not consider heavier nuclei for the hindrance estimate, because they are further away from our region of interest, and the possible $Z$-dependence of this factor is not clear [@Ag97GT-Hu].
In this region, the only available total Gamow-Teller strength measured using $(n,p)$ reaction is for 90Zr. Raywood [*et al*]{} [@Zr90GT-Raywood] deduced a value of $1.0\pm0.3$ for the total strength. Our calculated total strength for this isotope is $S(GT_+)=0.34$. We should note, however, that in our restricted model space there is only one 1[$^+$]{} state in 90Y. These values can also be compared to recently reported measurements of $3.0\pm1.9$ for the total strength by Sakai and Yako [@Sakai04].
We turn now to strength distributions. Our calculated Gamow-Teller distributions in the decay of nuclear systems with a few valence protons, like Zr or Mo isotopes, have the strength concentrated in a narrow energy range (less than 0.5 MeV) or sometimes in only one transition. The strength in systems with $Z_v^p>4$ (Tc and above) is distributed over the energy range of about 4 MeV. The Nb isotope chain is intermediate in this respect, because in the lowest configuration Nb has only one valence proton in the $g_{9/2}$ shell, while decays to Zr isotopes are distributed over several states. We show these systematics using a few examples below.
In Figs. \[fig-N50GTExpDistr\]-\[fig-Rh96GTcompar\], we compare the calculated Gamow-Teller distributions with available data collected from several sources. All measurements were done using $\gamma$-ray spectroscopy. Data for $N=50$ and 51 isotopes were obtained from Refs.[@N50GT-Brown; @N50GT-Johnstone; @N51GT-Johnstone] (see also references there in). The $GT$ distribution in 97Ag [@Ag97GT-Hu] was obtained with a TAS measurement. We note that in comparisons to experimental data, we do not include the sensitivity limits of experimental detectors. This sensitivity artificially cuts off Gamow-Teller strength near the $Q$-window so that calculated states are often not observed even 2 MeV below the $Q$-window (see, e.g., [@N50GT-Brown; @Ag97GT-Hu; @In102GT-Gierlik]). For this reason, comparisons to experiment are somewhat difficult, and one should focus on unambigous regions of low-lying strength. In Table \[tbl-GTQec\] we listed fractions of the calculated Gamow-Teller strength that lie inside the $Q$-window. A comparison to the calculation by Brown and Rykaczewski [@N50GT-Brown] reveals some interaction dependence in the values. For example, they estimated $f_{Ec}=29\%$ and 99% in the decays of 94Ru and 96Pd, while our estimates are only 19% and 72%, respectively. Johnstone [@N50GT-Johnstone; @N51GT-Johnstone], following a different approach, estimated significantly higher fractions of the strength inside the $Q$-window for most cases except 95Ru.
Reaction %
--------------------- ------- ------- ---- ---
93Tc$(\beta^+)$93Mo 6.12 0.13 2
94Tc$(\beta^+)$94Mo 5.74 0.13 2
94Ru$(\beta^+)$94Tc 7.89 1.53 19
95Tc$(\beta^+)$95Mo 5.43 0.08 1
95Ru$(\beta^+)$95Tc 7.49 0.59 8
95Rh$(\beta^+)$95Ru 9.41 6.35 67
96Tc$(\beta^+)$96Mo 5.36 0.04 1
96Rh$(\beta^+)$96Ru 9.05 5.14 57
96Pd$(\beta^+)$96Rh 11.18 8.08 72
97Rh$(\beta^+)$97Ru 8.53 2.60 30
97Pd$(\beta^+)$97Rh 10.90 7.38 68
97Ag$(\beta^+)$97Pd 12.71 12.49 98
98Ag$(\beta^+)$98Pd 12.47 11.53 92
: \[tbl-GTQec\] Fraction of the calculated Gamow-Teller strength inside $Q_{EC}$ window, $f_{Ec}=B_{Q_{EC}}/GT_+$
From Figs. \[fig-N50GTExpDistr\]-\[fig-Rh96GTcompar\] we note that the calculated strength distributions follow the trend observed in experiments. Most odd-$Z$ $N=50$ isotones and $N=51$ isotones have little strength at low excitation energies ($E_x{\; \raisebox{-0.4ex}{\tiny$\stackrel
{{\textstyle<}}{\sim}$}\;}2$-3 MeV), with the strength distributed among many states at a higher excitation energy, some of which are above the $Q$-window. The strength in even-even nuclei, represented here by even-$Z$ $N=50$ isotones, is concentrated in a few states.
The Gamow-Teller distribution in 97Ag shown in Fig.\[fig-N50GTExpDistr\] is converged (around 60 iterations per $J_f$ was required); thus the calculated shape is as good as it can be for the interaction. The centroid of experimental Gamow-Teller strength distribution in 97Ag is lower than the calculation predicts: $E_{centr}^{exp}=4.3$ MeV versus $E_{centr}^{calc}=4.7$ MeV. This is one of the indicators that the interaction may require some fine-tuning.
We already showed and discussed parts of the calculated Gamow-Teller distributions in Figs.\[fig-N50GTExpDistr\]-\[fig-N51GTExpDistr\]. Having in mind upcoming experiments [@Mo-Zegers] on Mo isotopes, we show the calculated distributions for the decays of Mo isotopes with masses $A=93$-97 in Fig. \[fig-Mo9497\]. The main contributions to the total strength are located within 1 MeV energy range around the centroid. The decays of Tc isotopes (see Fig.\[fig-Tc9497\]) have the strength distributed within 4 MeV range. These two isotope chains display the difference in the decays of even-even and odd-odd nuclei that we discussed above.
We also show the distributions for Zr isotopes, see Fig.\[fig-Nb9296\]. There we observed the migration of the strength from higher energies to lower energies as the neutron number increases. At 96Zr, where the lowest configuration is a completely occupied proton $p_{1/2}$ shell and a similar situation occurs in the neutron $d_{5/2}$ shell, the entire strength is peaked in one transition. The trend remains also in 97Zr.
We turn now to a discussion of the calculated total Gamow-Teller strength and the centroids. The isotopic dependence of the total strength is smooth with the strength decreasing together with the increasing number of neutrons and/or the decreasing number of valence protons in the $g_{9/2}$ shell. Assuming no mass dependence, an approximate formula can be derived: $GT_+=0.086(Z_v-1.5)(20-N_v)$. (The factor $(20-N_v)$ is due to the relative unimportance of the $h_{11/2}$ shell because of its negative parity.) This form is somewhat similar to the dependence $Z_v(20-N_v)/A$ observed in the $pf$ shell nuclei (see e.g. [@SMMC97]). The difference may be related to the active $j$ shells. In the $pf$ shell nuclei, discussed in Ref. [@SMMC97], protons predominantly occupy the $f_{7/2}$ shell; thus its occupation is proportional to the number of valence protons. While in our model space, the occupancy of the $g_{9/2}$ proton shell increases due to excitations out of the $p_{1/2}$ shell via configuration mixing. This increase is greater for isotopes closer to the core (around 0.6), and is 0.2 for $A=97$ nuclei with $Z_v>2$. The formula’s $\chi^2$ per degree of freedom is 0.05.
Another systematic relates to the centroids of the $GT_+$ distribution. If plotted with respect to the lowest positive parity state of the daughter nucleus, the centroids of Gamow-Teller distributions show a characteristic odd-even staggering, see Fig.\[fig-GT-CentroidEx\]. They are low in even-even nuclei, high in odd-odd nuclei, and average in odd-$A$ nuclei. A similar trend was observed in the mid-$pf$ shell nuclei [@Koonin94].
Langanke and Mart[í]{}nez-Pinedo [@Langanke00] interpreted this odd-even staggering as a result of pairing energy contributions to the mass splitting between the parent and daughter nuclei (see also [@Sutaria95]). The pairing structure goes away if the centroids are measured with respect to the parent nucleus. We plotted the centroid energies calculated in this way in Fig.\[fig-GT-CentroidEx-TzMass\]. We also included the Coulomb energy difference, calculated using a formula [@Myers66] $E_c=0.72(Z^2/A^{1/3})(1-1.69/A^{2/3})$, but ignored the proton-neutron mass difference and the splitting between the proton and neutron single-particle orbits which would be present if the lowest single-particle energies would be taken with respect to the core nucleus, 88Sr. The figure shows that centroid energies indeed lose information about the pairing structure. It is interesting to note that there seems to be a cross-over behavior, which we highlighted by connecting the transitions corresponding to the decays of Zr isotopes. These centroid energies follow a linear dependence as well, but the inclination is different from that in other nuclei. This behavior is probably related to the fact that $GT_+$ strength in Zr is due to proton excitations out of the $p_{1/2}$ shell.
Distributed-memory shell-model code {#sect-SMCode}
===================================
Our calculations were performed using a new parallel shell-model code [orpah]{} (Oak Ridge PArallel shell model code), which is still under development. The basic ideas are similar to those employed in the serial $m$-scheme computer code [antoine]{} [@Antoine; @Caurier99]. However, there are differences in the approach, since the code was developed targeting the distributed-memory computational paradigm. While the distributed-memory approach sets no limits on the available memory or the number of processors involved in the computation, a natural limitation occurs due to the need to communicate data from one processor to another, a process which for collective operations scale as $N_p^2$, where $N_p$ is the number of processors. However, even in cases when the communication becomes unfavorable, there is still a possible trade-off because of a greater amount of available memory.
The most time-consuming part of the shell-model problem is the operation of the Hamiltonian on a vector to produce a new vector which occurs during the Lanczos procedure. This affects load-balancing since each processor needs a similar workload for an effective use of computational time. We consider parallelization at two levels: the vector amplitudes are distributed among the processors, and each processor produces a portion of the final vector in a time-balanced way. We then redistribute the final partial vectors to the appropriate positions after each iteration.
Similar to [antoine]{}, the code numerically builds “blocks” of identical-particle Slater determinants having the same quantum numbers and sets up tables allowing construction of the elements of the Hamiltonian matrix [@Caurier99]. Differences arise from the parallel implementation. If the dimension of the model space is $D$, then the part of amplitudes which reside on a particular processor has size $D/N_p$. For a sufficiently large number of processors, $D/N_p$ can get smaller than the size of the largest block. This would place the limit for the maximum reasonable number of processors. Operations involving this block are also the most time-consuming. We decided to split the blocks in order to have smaller pieces of tasks, which could be distributed among the processors more efficiently. During the operation of the Hamiltonian acting on a vector, some of the amplitudes are prefetched and others are requested during the calculation. There is no predefined communication pattern, since some amplitudes can be delivered with some delay while the process would still be able to employ the ones already present in the memory. To enable this disconnection of computation and communication, each processor consists of two threads responsible for those two tasks. Those threads communicate via shared variables. The need to deliver amplitudes creates a communication overhead on top of the time needed to produce the final Lanczos vector. The latter operation is well balanced (i.e., is inversely proportional to the number of processors), while the overhead depends on the number of processors involved in the calculation. The final Lanczos vector is reorthogonalized to previous vectors. Due to orthogonality of the basis, each processor can produce the partial sum of a scalar product, and a global communication is needed only to obtain the total sum.
There is no coded-in restriction on the number of processors, with the exception that the minimum number of processors is two, because of the manager-worker algorithm employed in the Hamiltonian table set-up procedure. The current version of the code can calculate eigenvalues and eigenvectors of the Hamiltonian, the total angular momentum and isospin, as well as $GT$ properties. Some computations were done on a 2-4 CPU computer; others were done at the NERSC computer Seaborg using up to 80 processors. The largest problem that we tried to solve was the ground state energy of 52Fe ($D=110\times10^6$) on 48 processors, but the current set-up did not allow us to reach such dimensions in the region of our study. In addition, a further improvement in the performance is required before the code assessment is done, though the distributed-memory computation is a venue to solve larger interacting shell-model problems.
Summary {#sect-Summary}
=======
We calculated nuclei above 88Sr having masses $A=90$-97 using a realistic effective interaction derived from the CD-Bonn potential. The agreement between the calculated and measured spectra is satisfactory. Improvements to the interaction through fine tuning of the matrix elements could be useful to obtain the finer spectroscopic details, including the level ordering or the placement of negative-parity states in several nuclei, but this is beyond the scope of this exploratory work.
We also calculated the total Gamow-Teller strength and strength distributions for the decay in the electron capture direction. We found that the total strength follows the single-particle estimate based on the $\pi g_{9/2}$ and $\nu g_{7/2}$ occupation numbers obtained from the ground state wave functions of the parent nucleus, although the values slightly differ from a naive single-particle shell-model picture. Calculated strength distributions appear to reasonably recover experimental distributions in regions that are unaffected by detector sensitivity limits. From TAS data on $^{97}$Ag, we were able to obtain an estimate of the phenomenological quenching factor relative to single-particle estimates. Furthermore, our $GT$ distribution for 97Ag reproduces the measured data, see Fig. \[fig-Ag97GTcompar\]. By analyzing the centroids of the Gamow-Teller distributions, we found that the odd-even staggering behavior disappears if the centroids are measured from the parent ground state, as was suggested by Langanke and Mart[í]{}nez-Pinedo [@Langanke00]. We also observed that the centroids measured in this way have a quasi-linear dependency on the parameterization $(N-Z)/A$, with a different inclination for nuclei where no protons are present in the $g_{9/2}$ shell in the non-interacting picture. Finally, we made several predictions of the strength distributions for measurements that may soon be available in mass $A=92$-97 nuclei.
The description of low-lying Gamow-Teller strength distributions in a large range of nuclei (from roughly mass $A=50$ to mass $A=150$) is one important ingredient in understanding type-II supernova explosions, since electrons get captured by nuclei through these levels. Of course, this is not the whole story since first- and second-forbidden transitions (typically difficult to access in the laboratory) also play an important role in the cross section and rate calculations relevant for supernova. For low-energy capture, the Gamow-Teller transitions will dominate. They also dominate the neutrino-nucleus scattering processes that may occur at later times in the supernova event. While theory can produce many things, measurements are necessary to validate the estimates. Such is the case in the nuclear region discussed in this paper. We eagerly await experimental comparisons to our calculations.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are pleased to acknowledge useful discussions with T. Engeland, M. Hjorth-Jensen, K. Langanke, K. Rykaczewski, R.G.T. Zegers, B.A. Brown, and I.P. Johnstone. We are also grateful to M. Karny and L. Batist for providing us TAS data on 97Ag. Oak Ridge National Laboratory (ORNL) is managed by UT-Battelle, LLC, for the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. A.J. work was partially supported by the Department of Energy through the Scientific Discovery through Advanced Computing (SciDAC) program.
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|
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abstract: |
Coalescing massive black hole binaries are powerful emitters of gravitational waves, in the [*LISA*]{} sensitivity range for masses $M_{\rm BH} \approx 10^{4-7}$M$_{\odot}$. According to hierarchical galaxy merger models, binary black holes should form frequently, and should be common in the cores of galaxies. The presence of massive black hole binaries has been invoked to explain a number of class properties of different types of galaxies, and in triggering various forms of activity. The search for such binary black holes is therefore of great interest for key topics in astrophysics ranging from galaxy formation to activity in galaxies.
A number of phenomena were attributed to the presence of supermassive binary black holes, including X-shaped radio galaxies and double-double radio galaxies, helical radio-jets, periodicities in the lightcurves of blazars, (double-horned emission-line profiles), binary galaxies with radio-jet cores, binary quasars, and the X-ray active binary black hole at the center of the galaxy NGC 6240. Here, I review the observational evidence for the presence of supermassive binary black holes in galaxies, and the scenarios which have been discussed to explain these observations.
author:
- Stefanie Komossa
title: Observational evidence for supermassive black hole binaries
---
[ address=[Max-Planck-Institut für extraterrestrische Physik, Giessenbachstr. 1, 85748 Garching, Germany; skomossa@xray.mpe.mpg.de]{} ]{}
Introduction
============
The major route of forming binary black holes is via mergers of galaxies. If both merging galaxies harbor supermassive black holes at their centers (e.g., Richstone 2002), these two black holes will finally sink to the center of the potential and will eventually merge. Hierarchical merger models of galaxy formation predict that binary black holes should be common in galaxies (e.g., Haehnelt & Kauffman 2002, Volonteri et al. 2003). The merging process will be enhanced in clusters and groups of galaxies. Under some circumstances, black holes may already form as binaries at the centers of the first galaxies (Bromm & Loeb 2003).
Mergers of massive binary black holes produce strong gravitational wave signals (e.g., Thorne & Braginsky 1976, Centrella 2003, Baker 2003) which will be detectable for the first time with the future space-borne gravitational wave interferometer mission [*LISA*]{} (e.g., Bender et al. 1998, Danzmann 1996, Haehnelt 1994) for masses in the range $\sim$10$^{4-7}\,M_{\odot}$.
Supermassive binary black holes (BBHs) are expected to be of wide astrophysical relevance. Their detection and number estimates provide important constraints on models for galaxy formation and evolution. Their detection allows to study one possible route of BH growth, that by merging of BHs with each other, the frequency of these events, and their relevance for BH growth. BBHs were suggested to play a role in increasing AGN activity (e.g, Gaskell 1985, Gould & Rix 2000), in triggering starburst activity (Taniguchi & Wada 1996), and in the formation of molecular tori (Zier & Bierman 2001, 2002) which are believed to be an important ingredient in unified models of AGN. The possible relevance of BBHs in explaining different classes of radio-loud AGN was variously addressed (e.g., Basu et al. 1993, Wilson & Colbert 1995, Villata & Raiteri 1999, Britzen et al. 2001), and it was suggested that the misalignments between the direction of radio jets and disks in active galaxies (AGN) are due to past black hole merges in these galaxies (Merritt 2002, 2003). Once [*LISA*]{} results become available, the gravitational waves from coalescing binary black holes will be used to infer merger rates and the merger history of BHs/galaxies (e.g., Haehnelt 1994, Menou et al. 2001, Hughes 2002, Menou 2003), and possibly black hole masses (Hughes 2002, Menou 2003). In the future, gravitational waves of coalescing BHs may be used as cosmological standard candles, if the optical counterparts of these systems can be identified (Holz & Hughes 2003).
Given the importance of BBHs, the wide role they may play in explaining observations of classes of activity, and given the expectations that they should be common, based on galaxy formation scenarios, one key question is: what is the actual observational evidence that BBHs do exist ? This is the topic of this review. An overview is given of the different types of observations which point to the presence of BBHs, and of the scenarios invoked to explain these observations. Remaining uncertainties in both, observations and models are discussed. Due to space limitations, referencing will be incomplete. My apologies in advance.
Binary black hole formation and evolution
-----------------------------------------
BBHs will be formed in the course of merging of two galaxies (e.g., Begelman et al. 1980, Valtaoja et al. 89, Milosavljevic & Merritt 2001, Yu 2002, and references therein). The merging of the two black holes basically proceeds in three stages (e.g., Fig. 1 of Begelman et al. 1980). In a first stage, the cores of the merging galaxies reach closer towards each other by dynamical friction (e.g. Valtaoja et al. 1989). The third stage is the actual merging of the two black holes by emission of gravitational waves (GWs). Which processes are effective in the intermediate second stage and how efficiently they operate, i.e., how quickly the separation between the BHs shrinks to a distance where GW emission becomes significant, is still a subject of intense theoretical study (see Merritt 2003 for a review).
A number of processes which could lead to a hardening of the black hole binary were discussed, including stellar slingshot effects and re-filling of the loss cone (Saslaw et al. 1974, Quinlan & Hernquist 1997, Milosavljevich & Merritt 2001, Zier & Biermann 2001, and references therein), black hole wandering (e.g. Merritt 2001, Hemsendorf et al. 2002, Chatterjee et al. 2003), interaction with surrounding gas and the accretion disk (e.g. Ivanov et al. 1999, Gould & Rix 2000, Haehnelt & Kauffmann 2002, Armitage & Natarajan 2002), and/or the Kozai mechanism (Blaes et al. 2002). Theoretical calculations show, that under some circumstances none of these mechanisms may operate in sufficient strength, with the consequence that the BBH is expected to stall at separations of 0.01-1 pc (e.g., Milosavljevich & Merritt 2001).
There is circumstantial evidence that most BBHs do actually merge in less than a Hubble time. Haehnelt & Kauffman (2002) argued, if the binaries lasted too long, some of them would be ejected from the nucleus in the course of a three-body interaction with a third black hole, once a new merger occurs. That would lead to the prediction of the existence of galaxies without BHs at their center, as opposed to observations that most galaxies do harbor SMBHs.
Theoretical issues of BBH formation and evolution are covered in much greater detail by Haehnelt, Volonteri et al., and Milosavljevich & Merritt in these proceedings, and in the review by Merritt (2003).
Most of the BBH evidence described below is based on observations of [*active*]{} black holes, i.e. phenomena like presence of radio-jets, presence of hard and luminous X-ray emission, and optical emission-line ratios typical of Seyfert galaxies.
BBHs as explanations for different classes of astrophysical objects or specific components of the AGN core
----------------------------------------------------------------------------------------------------------
It is generally expected, that merging between galaxies may trigger all kinds of activity in the center of the merger remnant (e.g., Gaskell 1985), for instance by driving gas to the central region (Barnes & Hernquist 1996) which then triggers starburst and AGN activity.
In particular, the presence of a [*binary*]{} black hole was explicitly invoked to explain different facets of activity in galaxies: Taniguchi & Wada (1996) proposed that the BBH which was formed in the course of a major merger triggers starburst activity near the nucleus, while BBHs in minor mergers lead to formation of hot-spot nuclei.
According to Zier & Bierman (2001, 2002), the influence of the BBH on the surrounding stellar distribution will lead to a torus-like structure. Winds of these stars would then form the tori of molecular gas, thought to be ubiquitous in AGN and thought to play an important role in unification scenarios of AGN (Antonucci et al. 1993).
The possible relevance of BBHs in explaining different classes of radio-loud AGN was addressed repeatedly (e.g., Basu et al. 1993, Wilson & Colbert 1995, Villata & Raiteri 1999, Valtonen & Heinämäki 2000, Britzen et al. 2001, Merritt 2002). Villata & Raiteri (1999) speculated that all blazars “owe their origin to the presence of binary black holes”, and that differences between different classes of AGN are due to an evolutionary sequence: BL Lacs and FRI radio galaxies would represent the advanced stages with close binary pairs and low mass accretion, while FRII galaxies would harbor the wide pairs with long orbital periods (observationally less easily recognizable).
Wilson & Colbert (1995) suggested that the difference between radio-loud and radio-quiet AGN arises because the former, in elliptical galaxies, posses rapidly spinning BHs, spun up (not by accretion but) as a result of the coalescence of the two original BHs, while the latter, residing in spiral galaxies, have non-spinning BHs. The jets are considered to be powered by the spin of the BHs in the radio-loud galaxies.
Merritt (2002, 2003) pointed out that the spin re-orientation of the primary black hole caused by repeated mergers would explain the fact that the orientations of radio jets of AGN are almost random with respect to the plane of the stellar disk.
Finally, different types of jet structures in individual galaxies were interpreted as observational evidence for the presence of BBHs. This topic is addressed in more detail in the next sections.
Observational evidence for binary black holes: spatially unresolved systems
===========================================================================
BBH merger remnants
-------------------
### X-shaped radio galaxies
Some radio galaxies show jets with very peculiar morphology; abrupt changes in jet direction, forming X-shaped patterns (e.g., Fig. 2 of Leahy & Williams 1984, Fig. 1 of Parma et al. 1985, Leahy & Parma 1992, Capetti et al. 2002, Wang et al. 2003). The changes are more abrupt than in S-shaped radio galaxies. While the latter shapes are usually explained in terms of precession effects (e.g. Ekers et al. 1978), Parma et al. (1985) noted that the cross-shaped morphology is likely linked to an abrupt change in jet ejection axis, or is consistent with a precession phenomenon for a specific viewing geometry.
About 15 X-shaped or ‘winged’ radio sources are known, most of them associated with low-luminosity FRII sources. Except one (Wang et al. 2003), none of the X-shaped radio galaxies shows optical quasar activity in form of [*broad*]{} emission lines at their center. The host galaxies mostly exhibit high ellipticities (Capetti et al. 2002). A number of them have companion galaxies (Tab. 1 of Merritt & Ekers 2002), and the host galaxy of one (3C293) shows clear signs of interaction (Evans et al. 1999).
Several different models to explain the X-shaped patterns were addressed in the literature, which either link the characteristic X-shape to a re-orientation of the jet axis, or to effects of backflow from the active lobes into the wings (see, e.g., Sect. 5,6 of Dennett-Thorpe et al. 2002 for a summary)[[^1]]{}.
According to Merritt & Ekers (2002) and Zier & Biermann (2002), the X-shaped patterns reflect changes in the orientation of the black hole’s spin axis, caused by the merger with a second SMBH. This is also one of the two explanations preferred by Dennett-Thorpe et al. (2002). Merritt & Ekers favored minor mergers and showed that these can significantly change the spin axis of the primary black hole.
### Double-double radio galaxies with interrupted jet activity
Secondly, so called double-double radio galaxies (Schoenmakers et al. 2000) were suggested to be remnants of merged BBHs (Liu et al. 2003). These are sources which exhibit pairs of symmetric double-lobed radio-structures, aligned along the same axis. Inner and outer radio lobes have a common center (Fig. 1-3 of Schoenmakers et al. 2000) and there is a lack of radio emission between inner and outer lobes. The most likely origin of these structures is an interruption and re-starting of the jet formation. The interruption time scale is about 1 Myr.
Several ideas were proposed to explain this phenomenon (see Sect. 5 of Schoenmakers et al. 2000). Liu et al. (2003) favor the presence of binary black holes which have already merged in these systems. According to Liu et al. the inward spiralling secondary black hole temporarily leads to the removal of the inner parts of the accretion disk around the primary black hole, thus to an interruption of jet formation. Jet activity restarts after the outer parts of the accretion disk refill the inner parts of the disk.
0.4cm
These two important classes of candidates for BBH merger remnants – the X-shaped radio galaxies and the double-double radio galaxies – will certainly receive intense observational interest in the next few years, in order to address questions like: are there any double-double radio galaxies with [*changes*]{} in the jet direction (Liu et al. assumed the merging BHs to be coplanar but this does not always have to be the case) ? Do we see X-shaped radio galaxies with an interruption of jet-activity between the core and the wings (if the wings reflect an earlier period of activity they would no longer be powered today, which could be confirmed by very high-resolution radio imaging) ?
Helically distorted radio jets
------------------------------
A third phenomenon exhibited by radio jets which was suggested to be linked to the presence of binary black holes (Begelman et al. 1980) is the presence of (semi-periodic) deviations of the jet directions from a straight line. Helical distortions and bendings of jets have been seen in a number of sources (including 3C273, 3C449, BL Lac, Mrk 501, 4C73.18 and PKS0420-014; see e.g., Fig. 1 of Roos et al. 1993, Fig. 3 of Tateyama et al. 1998), interpreted as manifestation of BBHs in these systems.
While there is general agreement that the wiggles in radio jets are most plausibly caused by the presence of BBHs, different mechanisms to explain the observations have been favored. These either link the observations to [*orbital motion*]{} of the jet-emitting black hole (e.g., Kaastra & Roos 1992, Roos et al. 1993, Hardee et al. 1994), or to [*precession effects*]{}, either precession of the accretion disk around the jet-emitting black hole under gravitational torque (on shorter time scales; e.g. Katz 1997, Romero et al. 2000), or to geodetic precession (acting on longer time scales; e.g. Begelman et al. 1980). Correspondingly, black hole mass estimates in these systems are still subject to uncertainties by a factor $\sim$10-1000.
Semi-periodic signals in lightcurves
------------------------------------
### OJ 287
Another periodic phenomenon very often attributed to the presence of BBHs is (semi)periodic changes in lightcurves. The best-studied candidate for harboring a BBH, inferred from the characteristics of its optical lightcurve, is the BL Lac Object OJ 287, which exhibits optical variability with quite a strict period of 11.86 years (Silanpää et al. 1988, 1996, Valtaoja et al. 2000, Pursimo et al. 2000, and references therein). Optical observations of this source can be followed back to 1890 (e.g., Fig. 1 of Pursimo et al. 2000). How can the BBH model explain periodic optical variability ? Basically, two different classes of models were discussed: (i) accretion(disk)-related variations in the luminosity (e.g., Silanpää et al. 1988, Lehto & Valtonen 1996), or (ii) jet-related variability due to Doppler-boosting of varying strength (Katz 1997, Villata et al. 1998). Variants of them have also been invoked to explain apparent periods in the data of other BL Lac objects. (i) According to the original idea of Silanpää et al. (1988), the tidal perturbation, when the secondary approaches closest to the accretion disk of the primary leads to increased accretion activity, thus a peak in the optical lightcurve. (ii) Katz et al. (1997) studied a model in which precession of the accretion disk, driven by the gravitational torque of a companion mass (a second black hole), causes the jet to sweep periodically close to or across our line of sight. This then leads to a modulation of the observed intensity of light due to Doppler-boosting.
Combining radio and optical observations of OJ 287, Valtaoja et al. (2000) recently favored a scenario in which the first optical peak is due to a thermal flare, when the secondary BH plunges into the accretion disk of the primary. Following the major optical flare, a second optical flare is observed about a year after the first peak, and (only) that second peak is accompanied by enhanced radio emission (Valtaoja et al. 2000). This second peak is traced back to the tidal perturbation, excerted by the secondary black hole upon closest approach to the accretion disk of the primary which leads to increased accretion activity. The observed $\sim$12 yr period ($\sim$9 yr in the rest frame) then corresponds to the orbital period of the BBH.
A possible alternative to the BBH models, a well-defined duty-cycle model of accretion, is briefly mentioned by Valtaoja et al. (2000), but is considered as an unlikely explanation for the case of OJ 287, due to its quite strict periodicity.
The next optical maximum of OJ 287 is expected in March 2006. No doubt, this BL Lac will then be the target of intense multi-wavelength monitoring campaigns.
### Other cases
Optical variability with indications for periodicity ($N$ = a few, where $N$ is the number of periodic oscillations) and periods on long (order 10-20 yrs) and intermediate (order 20 days - 1 yr) scales have been observed in other blazars (see Sect. 1 of Fan et al. 1998 and Tab. 1 and 2 of Xie 2003 for summaries). Among them a 336 day period of minima in the optical lightcurve of PKS1510-089 (Xie et al. 2002), a 14 yr period in optical data of BL Lac (Fan et al. 1998), a 5.7 yr period of AO 0235+16 at radio frequencies (Raiteri et al. 2001), and a 23-26 day period of Mrk 501 at TeV energies (Hayashida et al. 1998). No other data set brackets as long a time interval as OJ 287, though, and periodicities are generally less conspicuous and less persistent as in OJ 287[[^2]]{}. In some, but not all cases, the variability in the lightcurves was traced back to the presence of a BBH, but different types of models were involved, which either relate the variability to real changes in the luminosity, or apparent changes due to Doppler-boosting effects in those lightcurves which show periodic brightness peaks.
Rieger & Mannheim (2000, see DePaolis et al. 2002 for a generalization of their approach to other orbits) showed that the periodic variability of the BL Lac object Mrk 501 at TeV energies with a period of about 23 days and $N$=6 is possibly related to the presence of a BBH. According to their model, the observed flux modulation arises from a varying Doppler-factor due to slight variations in inclination angle of (a moving blob in) the jet, caused by the orbital motion of the less massive, jet-emitting black hole. The presence of a BBH in Mrk 501 was also discussed with respect to the complex radio jet morphology of this source (Conway & Wrobel 1985, Villata & Raiteri 1999), and in relation to peculiarities in its spectral energy distribution (Fig. 2 of Villata & Raiteri 1999).
Doule-peaked BLR line profiles
------------------------------
If binary black holes exist in active galaxies, it can plausibly be expected that several are at nuclear separations such that their orbital motion causes observable effects on the profiles of the broad emission lines (e.g. Stockton & Farnham 1991, Gaskell 1996).
A number of AGN were observed which show double-horned emission-line profiles (Arp 102B and 3C390.3 are prominent examples). These line profiles were interpreted as evidence for two physically distinct broad line regions (BLRs) of two black holes at the centers of the two galaxies.
However, in those cases examined closely, so far, the BBH interpretation was disfavored, mostly because the predicted temporal variations in the line profiles, as the two BHs orbit each other, were not detected in optical spectroscopic monitoring programs (Halpern & Filippenko 1988, Eracleous et al. 1997). In one case, the apparent double-horned profile of the Balmer lines turned out to be an artifact of other spectral peculiarities (Halpern & Eracleous 2000).
Yu (2002) pointed out that orbital timescales of binary BHs with BLRs in AGN may generally be expected to be as large as 100-1000 yr, in which case no detectable variability in the red and blue peak of the emission-line profile is expected on the timescale of years. A further complication in the search for BBHs via characteristic line profiles is the existence of a number of other mechanisms which are also known to produce double-peaked line profiles, like bipolar outflows and accretion disks, not related to BBHs.
Galaxies which lack central cusps
---------------------------------
Essentially all evidence presented for the presence of BBHs is linked to some activity of at least one of the two black holes which form the pair (like radio jets, AGN-like emission-lines, etc.).
There is one exception, which is some rather indirect hint for the presence of binary black holes. Lauer et al. (2002) used [*HST*]{} to identify several early-type galaxies with inward-decreasing surface-brightness profiles. The presence of galaxies with such central ‘holes’ rather than cusps is expected in some models of BBH evolution (see Sect. 6 of Merritt 2003 for a summary), reflecting the ejection of stars from the core in the course of the hardening of the black hole binary.
Observational evidence for binary black holes: spatially resolved systems
=========================================================================
If a large number of BBHs exist in very close orbits, we also expect some pairs of larger separations, directly resolvable as individual SMBHs in nearby galaxies. Below, I discuss the few available observations which fall in this category.
X-ray active black hole pair: NGC 6240
--------------------------------------
Recent observations of the [*Chandra*]{} X-ray observatory let to the discovery of a pair of active black holes[[^3]]{} in the center of the (ultra)luminous infrared galaxy NGC 6240 (Komossa et al. 2003). The BBH in NGC 6240 is special in that it is presently the [*only*]{} BH pair I am aware of at the center of one galaxy, and spatially resolved such that both (active) BHs can be separately identified.
The galaxy NGC6240 belongs to the class of (ultra)luminous infrared galaxies (ULIRGs) which are characterized by an IR luminosity exceeding $\sim$10$^{12}$L$_{\odot}$ (see Sanders & Mirabel 1996 for a review). NGC6240 is one the nearest (U)LIRGs and is considered a key representative of its class. NGC6240 is the result of the merger of two galaxies, expected to form an elliptical galaxy in the future. It harbors two optical nuclei (Fried & Schulz 1983). Their nature remained unclear. In particular, no optical signs of AGN activity showed up in ground-based optical spectra. An active search for obscured AGN activity in NGC6240 was carried out during the last two decades (see Sect. 1 of Komossa et al. 2003 for a summary) which let to the detection of absorbed, intrinsically luminous X-ray emission (Schulz et al. 1998) extending up to $\sim$100 keV (Vignati et al. 1999) from the direction of NGC6240. Spatial resolution was insufficient to locate the source of the hard X-ray emission, though.
Employing the superb spatial imaging spectroscopy capabilities of the X-ray observatory [*Chandra*]{}, it was found that [*both*]{} nuclei of NGC6240 are active (Fig. 1), i.e. harbor accreting supermassive black holes (Komossa et al. 2003). The southern and northern nucleus show very similar X-ray spectra which are flat, heavily absorbed, and superposed on each is the presence of a strong neutral (or low-ionization) iron line. These kind of spectra have only been observed in AGN.
The projected separation of the X-ray cores is 1.5 arcsec, corresponding to a physical separation of 1.4 kpc. Over the course of the next few hundred million years, the two black holes in NGC6240 are expected to merger with each other.
{height=".36\textheight"} {height=".35\textheight"}
Binary quasars
--------------
A small fraction of (high-redshift) quasars were observed to be accompanied by a second nearby quasar (image) with the same redshift. These may either be gravitational lenses, true quasar pairs, or chance alignments. The majority of them are in fact the result of gravitational lensing.
Several pairs with separations 3-10$^{\prime\prime}$ were argued to be real pairs, i.e., physically distinct binary quasars (see, e.g., Tab. 1 of Mortlock et al. 1999 and of Kochanek et al. 1999, Sect. 2.5.8 of Schneider et al. 1992). The distinction between a real pair and a lensed quasar is mostly based on: (i) whether or not a lens is detected and (ii) whether or not the quasar spectra differ significantly, and/or whether or not there are other significant differences, like when one image is radio-loud, the other radio quiet. Lack of a detectable lens, and very different quasar spectra/properties strongly argue for real pairs. There is a growing number of quasar images identified as true pairs or as excellent candidates for true pairs, among them Q1343.4+2640 (with substantial spectral differences; Crampton et al. 1988), LBQS0103-2753 (the smallest-separation pair known, with very different optical spectra; Junkkarinen et al. 2001), LBQS0015+0239 (interpreted as probable binary system due to lack of an optically bright lens which then makes it the highest-redshift binary known, at $z$=2.45; Impey et al. 2002), LBQS1429-0053 (similar optical spectra but lack of a lens candidate; Faure et al. 2003), UM425 (similar optical/UV spectra but different amounts of absorption in X-rays and lack of a bright lens; Mathur & Williams 2003, Aldcroft & Green 2003), and Q2345+007 (Green et al. 2002). Q2345+007 at $z$=2.15 is an interesting case: Despite essentially identical optical spectra, no lens could be found in the optical and X-ray band (Green et al. 2002). The X-ray spectra of the quasars do differ, and Green et al. favor an interpretation in terms of a real quasar pair.
The smallest-separation known binary quasar is LBQS0103-2753 at redshift $z$=0.848. Its projected separation of 0.3$^{\prime\prime}$ corresponds to $\sim$ 2.5 kpc. It remains uncertain, whether the projected separation reflects the true separation. For instance, the emission-line shifts in both quasar spectra ($v$=3900 km/s) indicate a much larger separation, but emission lines are not always good indicators of systemic redshift (Junkkarinen et al. 2001). Junkkarinen et al. point out that if the systemic redshift difference is much smaller than indicated by the emission lines, LBQS0103-2753 would most likely be a galaxy merger with two quasar cores.
It is interesting to note that there appears to be a lack of small pair separations. Apart from LBQS0103-2753 other pairs have separations $>$ 2$^{\prime\prime}$, and typically 3-10$^{\prime\prime}$. These projected separations convert to distances of $\sim$10-80 kpc, given the redshifts of the quasars (see Fig. 7 of Mortlock et al. 1999 for an attempt to derive physical separations by a random deprojection method).
In any case, a chance projection of the quasar pairs on average is very unlikely, because the pairs are in excess of what is expected from chance projections of single, unrelated sources. Open questions then are: are the host galaxies of the quasars interacting, are they bound to each other and in the process of merging ? At low redshifts, this question can be addressed by searching for morphological and kinematical distortions in the host galaxies. However, most binary quasars are at large redshifts ($z \approx 1-2$), and sometimes the host galaxies are not even detected[[^4]]{}.
If the quasar cores reside in galaxies which are already interacting with each other, then the quasar activity might have plausibly been triggered in the course of the interacting/merging process (e.g., Kochanek et al. 1999).
Assuming the observed binary quasars are bound pairs, Mortlock et al. (1999) estimated that dynamical friction will drive them closer together on a timescale comparable to a Hubble timescale. Since this is longer than the typical activity time scale of quasars, these authors pointed out that closer pairs may already be beyond the phase of quasar activity (no fuel left in the center) and therefore not easily detectable.
Pairs of galaxies which both harbor radio-jet sources: 3C75
-----------------------------------------------------------
In the course of a VLA radio survey of Abell clusters, the very unusual morphology of the radio source 3C75 close to the center of the cluster of galaxies Abell 400 was discovered. It showed [*two*]{} pairs of radio jets (Fig. 1 of Owen et al. 1985) and was first considered an apparently single galaxy with two cores.
However, optical follow-up observations showed two elliptical galaxies which each posses a radio-jet emitting core. Such a configuration is quite unusual, and an important question became: are these two galaxies interacting or merging with each other ? Generally, this can be decided upon examination of the isophotes and kinematics of the galaxies. For instance, asymmetric isophote distortions and tidal tails indicate an ongoing (advanced) merger (e.g. Lauer 1988). As regards 3C75, no distortions of the inner isophotes and no kinematic disturbances were found (Balcells et al. 1995, Govoni et al. 2000). However, the outer isophotes show an off-centering and twist (Lauer et al. 1988, DeJuan et al. 1994, Balcells et al. 1995, Govoni et al. 2000). These observations indicate, that the two galaxies are presently not bound to each other, but do interact with each other (Balcells et al. 1985). Balcells et al. cautioned, that dust may play a role as well in explaining the non-concentric isophotes. Independently, though, it was argued that the galaxies likely are not spatially very distant from each other, because the jets show similar bends, i.e. should have passed similar regions of the ICM (Owen et al. 1985).
In any case the two elliptical galaxies are close to the cluster center, and are expected to sink to the center of the potential and merge, ultimately.
Concluding remarks, LISA rates, uncertainties, future observations
==================================================================
In summary, various lines of evidence point to the existence of supermassive binary black holes at the centers of galaxies. According to BBH evolution models, the longest timescales in the evolution of the binary up to coalescence are likely those in which the binary is closely bound ($\sim$ 0.01-10 pc; see, e.g., Fig. 1 of Begelman et al. 1980); most BH pairs would therefore not be spatially resolved. These pairs likely manifest their presence in imprinting periodic phenomena on observed galaxy properties, like wiggling radio jets and quasi-periodic variations in lightcurves (most prominently seen in OJ 287).
However, we do expect to see some wider pairs which can be spatially resolved. It should be noted, though, that these can only be easily recognized, [*if both black holes are active*]{}. Likely, a number of them escape detection if the second BH is not in an active state at the epoch of observation. The pair with the smallest physical separation spatially resolved ($\sim$1.5 kpc), and the only one we are aware of at the center of a single galaxy, is the X-ray active pair of black holes in NGC 6240. There are wider pairs, in form of binary galaxies with active centers, like, the two galaxies which make the radio source 3C75 or several binary quasars; those with closest spatial separations may already be in an advanced stage of merging.
Some post-merger candidates for BBHs were proposed, especially the X-shaped radio galaxies, interpreted as result of minor mergers between two black holes, leading to a spin-flip in the resulting black hole and a corresponding change in jet direction. Based on this scenario, Merritt & Ekers (2002) predict a merger event rate for this type of galaxies of $\sim$1/year detectable by gravitational wave interferometers.
Within the framework of the BBH models, uncertainties arise from questions like, which scenario is the correct one (in particular, orbital motion versus precession effects) to describe the data. This, in particular, leads to large uncertainties in predictions of the BH masses of the observed systems, which range from 10$^6$ - 10$^{10}$ M$_\odot$. The high BH mass range would be outside the [*LISA*]{} sensitivity range.
Future search for (active) binary BHs will likely concentrate on topics like (i) an extended X-ray search for active binary black holes at the centers of (ultraluminous IR) galaxies using [*Chandra*]{}, (ii) deep [*HST*]{} imaging of the host galaxies of quasar pairs at high redshift, to search for interaction-induced morphological distortions, (iii) high-resolution radio imaging of X-shaped radio galaxies to search for interruption of the radio emission between core and wings, and (iv) multi-wavelength monitoring of blazars with (suspected) periodical variability, in particular an intense coverage of the next expected maximum of OJ 287.
Once [*LISA*]{} operates, it will provide us with valuable complementary information on the rate and merger history of binary black holes. Ideally, if [*LISA*]{} sources could be identified with optical counterparts, gravitational wave signals and observations across the electromagnetic spectrum could be combined to study BBH merger systems in detail.
It is a pleasure to thank Joan Centrella for organizing this very interesting workshop, Bill Keel for providing the optical image of NGC6240, and Günther Hasinger and David Merritt for pleasant discussions.
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[^1]: Leahy & Williams (1984) studied a backflow model to explain the X-shaped radio structures (their Fig. 6; see Capetti et al. 2002 for another variant of these type of models). In their model of a light jet, post-jet-shock material is re-accelerated towards the center, causing backflows which, back at the source, expand laterally. In favor of backflow type of models, as opposed to jet-reorientation models, Capetti et al. (2002) noted that a number of the winged radio galaxies show wings preferentially aligned with the minor axis of the host galaxies. On the other hand, Dennett-Thorpe et al. (2002) argued that backflow models are untenable for sources like 3C223.1 and 3C403 which show wings that are longer than the active lobes. Other types of models, argued to be unlikely by Merrit & Ekers (2002), involve accretion of dwarf galaxies and warping instabilities of the accretion disk in order to explain a re-orientation of the jet axis.
[^2]: For instance, the lightcurves of BL Lac also indicate periods of 0.6 yr, 0.88 yr, 7.8 yr and 34 yr, and it is unclear whether all of them are real, and what causes them.
[^3]: Actually, what is observed is the characteristic signs of AGN activity. The step that this is equivalent to SMBH detection is based on the widespread belief, that AGN are indeed powered by SMBHs. This statement basically holds for all other BBH evidence: it is always the activity associated with the black hole(s) which is detected.
[^4]: Kochanek et al. (1999b) used HST to find that the host galaxies of the binary quasar MGC 2214+3550 are undisturbed, implying their physical separation is greater than their projected separation of $\sim$20 kpc
|
---
abstract: 'In this work we develop an analytic approach to study pulsar spindown. We use the monopolar spindown model by Alvarez and Carramiñana (2004), which assumes an inverse linear law of magnetic field decay of the pulsar, to extract an all-order formula for the spindown parameters which are expressed in terms of modified Bessel functions. We further extend the analytic model to incorporate the quadrupole term that accounts for the emission of gravitational radiation, and obtain expressions for the period $P$ and frequency $f$ in terms of transcendental equations. We derive the period of the pulsar evolution as an approximate first order solution in the small parameter present in the full solution. We find that the first three spindown parameters of the Crab, PSR B1509-58, PSR B0540-69 and Vela pulsars are within their known bounds providing a consistency check on our approach. After the four detections of gravitational waves from binary black hole coalescence and a binary neutron merger 170814, which was a novel joint gravitational and electromagnetic detection, a detection of gravitational waves from pulsars will be the next landmark in the field of multi-messenger gravitational wave astronomy.'
author:
- 'F.A. Chishtie'
- 'S.R. Valluri'
title: An analytic approach for the study of pulsar spindown
---
10.0in 9.0in -0.60in
PACS No.: 97.60.Gb, 95.85.Sz\
Key Words: Pulsars; Spindown; Gravitational Waves and Detectors
email: fchishti@uwo.ca, valluri@uwo.ca
Introduction
============
Pulsars are highly magnetized neutron stars which are known to rotate rapidly due to various physical mechanisms [@LK2005]. These mechanisms involve electromagnetic and gravitational emissions which result in the spindown of the GW frequency of the emitted pulsar signal [@AC2004].
In their paper on monopolar pulsar spindown, Alvarez and Carramiñana have considered a general multipole spindown to study pulsar evolution [@AC2004]. In this model, a monopolar term was introduced to ensure that the braking indices defined in terms of the frequency and higher derivatives are in agreement with the trajectories in the $P\neg\dot{P}$ diagram for pulsar evolution, where $P$ and $\dot{P}$ denote the pulsar period and its time derivative. Their detailed analysis of the stationary multipole model ruled out the possibility of a time independent evolutionary equation for the pulsar frequency. Their time dependent multipole model included dynamics of the pulsar magnetic moment and thereby the decay of the magnetic field, $B(t)$. They show in their analysis that an inverse linear decay proposed by Chanmugham and Sang [@CG89], in contrast to an exponential decay law for pulsar magnetic field [@Ostriker], did a better fit of the evolutionary trajectories of the four pulsars studied namely the Crab, PSR B1509-58, PSR B0540-69 and Vela. Following their approach, we use an inverse linear decay law for $B(t)$. The spindown of pulsars due to the intense magnetic fields that surround them is a phenomenon that will significantly impact on the younger pulsars which can lose a large amount of rotational energy due to physical processes such as electromagnetic and gravitational multipole radiation. The evolution of pulsars has been studied in great detail [@Camilo; @Chanmugam1; @Chanmugam2; @Colpi; @ATNF; @Lyne1; @Lyne2; @Manchester1; @Manchester2; @Manchester3; @Manchester4; @Manchester5; @ES2011; @Taylor; @Ostriker; @Alvarez3; @Damour; @Lyne2015].
The first observations of gravitational waves by Advanced LIGO are identified as black hole mergers, namely GW150914 [@GW1], GW151226 [@GW2] and a less significant candidate LVT151012 [@GW3]. Recently, GWs from another such merger, GW170104 has been reported [@GW4]. GW170814 was coherently observed by the advanced Virgo and two advanced LIGO detectors, produced by the coalescence of two stellar mass black holes. Recently, a neutron merger has been detected by both GW and electromagnetic observations [@GW5]. With increasing detector sensitivity of LIGO, VIRGO and the upcoming KAGRA, SKA and IndIGO detectors, the next wave of optimism and discovery should be on the detection of GW from pulsars. Pulsars are remarkably stable objects and although their GW amplitude is weaker compared to black holes, the fact that they can be tracked over a long period of time and that they emit continuous GW signals will be invaluable in detecting their GW signals.
We extend the mathematical analysis of the monopolar pulsar spindown model introduced by Alvarez and Carramiñana [@AC2004; @Alvarez1]. We further present a means of analytically finding the pulsar spindown parameters as defined by Brady et al. [@BC2000; @BC1998] and Jaranowski et al. [@JK1999] for the GW frequency and the phase measured at the ground based detectors.
Our paper is organized as follows. In the next section, we present a model for pulsar spindown along with a new solution that includes the quadrupole term due to gravitational radiation emission, which is an extension relevant for younger pulsars. In section 3, we connect this model to the form of frequency spindown shifts by Brady et al. [@BC2000], via the orthogonal properties of the Chebyshev polynomials and determine an analytical form of the spindown parameters. We also compute the first three spindown parameters via this approach for the Crab, PSR 1509-58, PSR 0540-69 and Vela pulsars and find that these are within the stated bounds provided by [@BC2000]. These bounds provide a consistency check to the model. Section 4 summarizes our conclusions.
A model for pulsar spindown
===========================
We consider the spindown model derived from a general spindown law [@AC2004], $$\dot{f}=-F(f,t)$$
This law expresses the change of pulsar frequency with respect to time and assumes that the frequency is a positive, antisymmetric and continuous function of time. Following these properties, the even powers of $f$ are excluded and the first three non-zero terms in the Taylor expansion of Eq.(1) are,
$$\dot{f}=-s(t)f-r(t)f^{3}-g(t)f^5$$
The first term, namely the monopolar term was introduced by Alvarez and Carramiñana (2004) to take into account particle acceleration mass loss or pulsar winds. This term further enables the braking indices to be in accord with the trajectories in the $P\neg\dot{P}$ diagrams for pulsar evolution. The second and third terms incorporate the normal spindown mechanisms of magnetic dipole radiation and gravitational radiation. For the frequencies measured in known, isolated pulsars, higher order terms can be neglected. In contrast, the spindown of millisecond pulsars has shown features strongly in contrast with typical pulsars. Hence, this model does not describe binary pulsars. A thorough analyses of binary pulsar spindown has been done in several excellent works [@Damour; @Taylor], to cite but a few.
Since $f=\frac{1}{P}$ and $\dot{f}=\frac{-\dot{P}}{P^2}$
$$\dot{P}=s(t)P +\frac{r(t)}{P}+\frac{g(t)}{P^3}$$
For a simpler analysis for older pulsars, the quadrupole term $g(t)$ is dropped, as it does not give an appreciable contribution:
$$\dot{P} \simeq s(t)P+\frac{r(t)}{P}$$
The assumption is that the frequency/period evolution changes as the magnetic field $B(t)=B(0)/ \left(1+t/t_c \right)$ decays. Moreover, the $\{s(t),r(t)\}\propto B(0)^{2} \psi (\frac{t}{t_c})$; thereby $r_0$, $s_0$ are obtained from pulsar evolution studies[@AC2004].\
From the use of the inverse linear law in Eq.(4), we find,
$$\dot{P}=\left({\frac{1}{1+t/t_c}} \right)^{2}\left(\frac{r_0}{P}+s_{0}P \right)$$
where $r_0 \geqslant 0, s_0 \geqslant 0$, and $t_c$ is the characteristic time of the magnetic field decay.\
If we have the initial condition $P(t_0)=P_0$, then the solution of this equation is given below:
$$\begin{aligned}
P(t)=\frac{1}{s_0} \sqrt{s_{0} \exp \left \{\frac{2t_{c}^{2} s_0(t-t_0)}{(t+t_c)(t_c+t_0)} \right\} (P_{0}^{2}s_{0}+r_0)-r_0s_0} \\ \intertext{\indent For $s_0<0, r_0\ll |s_0| \ll1, t_c\gg t, t_0=0, g_0=0$, we have}
P(t)=\sqrt{\frac{e^{2s_0t}\left(P_0^2s_0^2+r_0s_0\right)}{s_0^2}-\frac{r_0}{s_0}}\sim P_0e^{s_0t}
\end{aligned}$$
In the most general form, for $s_0>0, r_0>0$,
$$\begin{split}
f(t) &=s_0\left[\sqrt{s_0\exp\left(\frac{2t_c^2s_0(t-t_0)}{(t+t_c)(t_c+t_0)}\right)\left(\frac{1}{f_0^2}s_0+r_0\right)-r_0s_0}~\right]^{-1}\\
&=f_0s_0^{1/2}\left[\sqrt{\lambda_0\exp\left(\frac{2t_c^2s_0(t-t_0)}{(t+t_c)(t_c+t_0)}\right)-\lambda_1}\right]^{-1}
\end{split}$$
where $\lambda_0 = f_{0}^{2}r_0+s_0$, $\lambda_1 = r_{0}f_0^2$, and $f(0)=f_0$,
For $t_c\gg t $ and $t_0=0$,
$$\begin{aligned}
f(t)=\frac{f_{0}s_{0}^{1/2}}{\sqrt{\lambda_0\exp(2s_0t)-\lambda_1}} \\ \intertext{\indent It should be noted that Eq.(8a) can also be expressed to account for a possible negative sign in $s_0$ as}
f(t)=\frac{f_0|s_0|}{\sqrt{s_0\lambda_0 \exp (2s_0t)-\lambda_1s_0}}
\end{aligned}$$
It should be observed that the inclusion of a quadratic term in $f$ for a case in which $f$ is not antisymmetric would not present undue problems in the solution of the nonlinear differential Eq.(2).
For pulsars such as PSR B1509-58, there is an initial phase of strong gravitational spindown with the quadrupole parameters being at least two orders of magnitude higher than that of the other pulsars. We extend this model to further include the quadrupole term. Inclusion of the quadrupole term to consider the effects of spindown to GW emission, gives
$$\frac{dP}{dt}=\left(s_0P+\frac{r_0}{P}+\frac{g_0}{P^3}\right)\left(1+\frac{t}{t_c}\right)^{-2}$$
Multiplying by $P^3$ we obtain
$$P^3dP=\left(s_0P^4+r_0P^2+g_0\right)\left(1+\frac{t}{t_c}\right)^{-2}dt$$
$$\frac{P^3dP}{s_0P^4+r_0P^2+g_0}=\left(1+\frac{t}{t_c}\right)^{-2}dt$$
Let $Q=P^2$, in the above differential equation. Thereby, $dQ=2PdP$, and
$$\label{2}
\frac{QdQ}{2(s_0Q^2+r_0Q+g_0)}=\left(1+\frac{t}{t_c}\right)^{-2}dt$$
$$\frac{QdQ}{2s_0(Q^2+\frac{r_0Q}{s_0}+\frac{g_0}{s_0})}=\frac{QdQ}{2s_0(Q+a)(Q+b)}$$
Since $$\label{general}
\int \frac{QdQ}{(Q+a)(Q+b)}=\frac{1}{a-b} \{a \ln(Q+a)-b \ln(Q+b)\}$$
Integrating both sides of Eq.(9), we obtain
$$\label{3}
\frac{1}{a-b} \left \{a \ln(Q+a)-b \ln(Q+b) \right\}=\frac{-2t_cs_0}{1+t/t_c}+C$$
The initial condition $P(t_0)=P_0$ can be used to fix $C$ ,\
which can be expressed in the form: $$C=\frac{1}{a-b} \left\{a \ln(P_0^2+a)-b \ln(P_0^2+b) \right \}+\frac{2t_cs_0}{1+t_0/t_c}$$
Therefore the full solution to Eq.(10) can be expressed as $$\label{4}
\frac{1}{a-b} \left \{a \ln \left(\frac{P^2+a}{P_0^2+a}\right)-b \ln \left(\frac{P^2+b}{P_0^2+b}\right) \right\} =2t_cs_0 \left\{1-\frac{1}{1+t/t_c}\right\} \approx 2s_0t (t_c\gg t)$$
Eq.(\[4\]) is symmetric under interchange of $a$ and $b$, suggesting a very interesting pattern of the generalized Lambert W function [@AT2017], which we explore in the next section.\
We can solve for $a$ and $b$ by finding the roots of the quadratic expression of $Q^2+\frac{r_0}{s_0}Q+\frac{g_0}{s_0}$. Here $a+b=\frac{r_0}{s_0}$, $ab=\frac{g_0}{s_0}$, $a-b=\frac{\sqrt{r_0^2-4s_0g_0}}{s_0}$.\
Hence,
$$\begin{aligned}
a=\frac{1}{2}\left (\frac{r_0+\sqrt{r_0^2-4s_0g_0}}{s_0}\right)\\
b=\frac{1}{2} \left(\frac{r_0-\sqrt{r_0^2-4s_0g_0}}{s_0}\right)\end{aligned}$$
Here $s_0\geq 0$, $r_0 \geq 0$ and $g_0 \geq 0$ where $s_0 > r_0 > g_0$. By letting $b=a\epsilon $ for small $b$, we obtain
$$a \ln \left ( \frac{Q+a}{Q_0+a} \right ) + a\epsilon \ln \left ( \frac{Q_0+a\epsilon}{Q+a\epsilon} \right) = 2s_0ta-2s_0t a\epsilon$$
which simplifies to $$\label{13}
\left [ \ln\left( \frac{Q+a}{Q_0+a}\right) - 2s_0t \right] + \epsilon \left [ \ln \left(\frac{Q_0+a\epsilon}{Q+a\epsilon}\right) +2s_0t \right] = 0$$
It is worthwile to keep in consideration the possibility that the coefficients $s_0$, $r_0$,$g_0$ can change sign to indicate the occurence of glitches in some time segments of pulsar data. Then there is a spinup in those time segments. Although Alvarez and Carramiñana [@AC2004] considered the coefficients $s_0, r_0$ and $g_0$ to be $\geq 0$, to restrict the study to spindown of pulsars, it is worth to keep open the cases where one or more of $s_0, r_0$ and $g_0$ could be $<0$, giving rise to the possibility that $a$ and/or $b$ could be $<0$. Such cases of negative values can indicate pulsar spinups, also known as glitches. Glitches are discrete changes in the pulsar rotation rate that is often followed by a relaxation [@ES2011; @Lyne2015]. The cumulative effect of glitches is to reduce the regular long-term spindown rate $|\dot{f}|$ of the pulsar.
The Lambert W Solution
----------------------
Under the special case of $a=b$, which may be a rare situation when the expressions in the radical sign in Eqs.(18a) and (18b) vanish, the left hand side of Eq.(\[general\]) can be integrated as
$$\int \frac{Q}{(Q+a)^2}dQ =\log (Q+a) + \frac{a}{Q+a} = -\frac{2t_cs_0}{1+ t/t_c} + C$$
When $t=0$ and $Q=Q_0$,
$$\log \left (Q_0 + a \right) + \frac{a}{Q_0+a} = C-2t_cs_0$$
The solution can be concisely written as , $$-\log\left( \frac{a}{Q+a}\right) + \frac{a}{Q+a} + \log a = 2s_0t + \log \left ( Q_0+a \right ) + \frac{a}{Q_0+a}$$
Substituting $\frac{a}{Q+a}=z$ and $\frac{a}{Q_0+a}=z_0$, we obtain by exponentiation on both sides
$$\begin{aligned}
e^{-\log z+z} = e^{2s_0t + z_0- \log z_0}\nonumber \\\end{aligned}$$
Rearranging gives
$$-ze^{-z} = -z_0e^{-z_0}e^{2s_0t}$$
This transcendental equation can be solved to yield
$$-z=W\left (-z_0e^{-z_0}e^{-2s_0t} \right)$$
where $W$ is the Lambert W function [@Lambert]. Thus, again for $t_c \gg t$
$$-\frac{a}{Q+a} = W \left (-\frac{a}{Q_0+a} e^{-a/(Q_0+a)} e^{-2s_0t} \right )$$
where $Q$ and $Q_0$ as defined previously are $Q=P^2$ and $Q_0=P_0^2$.\
$W(z)$ has the series expansion
$$\begin{split}
W(z)&=\sum_{n\geqslant 1}\frac{(-n)^{n-1}}{n!}z^n\\
&\approx z-z^2+\frac{3z^3}{2}+- - - -
\end{split}$$
Spindown as a function of frequency
-----------------------------------
From Eq.(19), we give an equivalent expression in terms of the frequency $$\ln \left ( \frac{1}{f^2} +a \right ) - \epsilon \ln \left ( \frac{1}{f^2}+a\epsilon \right ) = (1-\epsilon) 2s_0t + \ln \left ( \frac{1}{f_0^2}+a \right ) - \epsilon \ln \left ( \frac{1}{f^2_0}+a\epsilon \right )$$
$$\ln\left[\frac{\frac{1+af^2}{f^2}}{\left(\frac{1+a\epsilon f^2}{f^2}\right)^{\epsilon}}\right]=\ln\left[\frac{\frac{1+af_0^2}{f_0^2}}{\left(\frac{1+a\epsilon f_0^2}{f_0^2}\right)^{\epsilon}}\right]+ (1-\epsilon)2s_0t$$
For $g_0 \ll r_0 < s_0$, $a$ given by Eq.(18a), with $b=a\epsilon$ ($\epsilon \approx 10^{-2}$) and $s_c=(1-\epsilon)s_0$, we obtain,
$$\left (\frac{1+af^2}{1+af_0^2} \right ) \left (\frac{f_0^2}{f^2} \right )= e^{2s_ct}$$
which can be written as
$$\frac{1}{f^2}=\left(\frac{1+af_0^2}{f_0^2}\right)e^{2s_ct}-a$$
For the Crab pulsar, for this case of using the inverse magnetic law, $r_0 = 7.5\times10^{-12}$ Hz$^{-1}$, $s_0=9.4 \times 10^{-15}$ Hz was determined in \[2\], however $g_0$ was not found as this parameter was assumed to be zero in their model.\
There are three possible cases for the roots in Eqs.(18a) and (18b):
1. $r_0^2-4s_0g_0 > 0, a > b$, e.g. in the case of the Crab pulsar.
2. $r_0^2-4s_0g_0 = 0, a = b$, there are no data available for this case, but this case can arise from pulsars that have certain values for $s_0, r_0$ and $g_0$.
3. $r_0^2-4s_0g_0 < 0$, $\sqrt{r_0^2-4s_0g_0}\pm i\sqrt{-r_0^2+4s_0g_0}$, where $a$ and $b$ are complex. The Vela is a fine example of where this could be possible.
For $b=a\epsilon \ll a$, $ g_0 \neq 0$, and $s_c = s_0(1-\epsilon)$ we approximately obtain,
$$f(t)=\frac{1}{\left[\left(\frac{1+af_0^2}{f_0^2}\right)e^{2s_ct}-a\right]^{1/2}}$$
Comparing the expression of $f$ here with that for $g_0=0$,
$$f(t)=\frac{f_0s_0^{1/2}}{\sqrt{\lambda_0e^{2s_0t}-\lambda_1}}$$
where $\lambda_0=f_0^2r_0+s_0, \lambda_1=r_0f_0^2, a=\frac{r_0}{s_0}$, we find that when $\epsilon=0$, the above two equations coincide exactly.\
For $\epsilon \neq 0$, Eq.(32) can be written as,
$$f(t) = \frac{f_0}{\left\{\left(1+af_0^2\right)e^{2s_ct}-af_0^2 \right\}^{1/2}}$$
$$f(t)=\frac{f_0}{\left[(1+af_0^2)e^{2s_ct}\left\{1-\left(\frac{af_0^2}{1+af_0^2}\right)e^{-2s_ct}\right\}\right]^{1/2}}$$
This can be written approximately, if one ignores the higher order terms in the curly brackets as $$f(t) =f_0\left(1+af_0^2\right)^{-1/2}e^{-s_ct}$$
It should be noted that for $g_0\neq 0$, $$1+af_0^2=1+\frac{r_0}{s_0}f_0^2+Cf_0^2$$
where $C=\frac{1}{2} \left(-\frac{2s_0g_0}{r_0^2}+\ldots\right)$ in the expansion of the radical $\sqrt{1-\frac{4s_0g_0}{r_0^2}}$. Consequently, $\lambda_0$ will not be the same as given previously ($\lambda_0=s_0+r_0f_0^2$), but will be
$$\lambda_c=\lambda_0+s_0Cf_0^2+\ldots$$
Hence,
$$f_{pulsar}=f_0\sqrt{\frac{s_0}{\lambda_c}}e^{-s_ct}\left[1-\frac{\lambda_1}{\lambda_c}e^{-2s_ct}\right]^{-1/2}$$
For $g_0 \neq 0$, simplification of Eq.(17a) in terms of $\lambda_0=r_0f_0^2+s_0$ gives
$$a=\frac{r_0}{2\left(\lambda_0-f_0^2r_0\right)}\left\{1+\frac{|r_0|}{r_0}\sqrt{1-4g\left(\frac{\lambda_0}{r_0^2}-\frac{f_0^2}{r_0}\right)}\right\}$$
where the $|r_0|$ addresses the situation where $r_0<0$.
This expression can be more concisely written as $$a=\frac{r_0}{2\Lambda}\left\{1+\frac{|r_0|}{r_0}\sqrt{1-\frac{4g}{r_0^2}\Lambda}\right\}$$
where $\Lambda=\lambda_0-f_0^2r_0$.
Gravitational Wave Signal with Spindown Corrections
===================================================
Based on the derivations done in section 2, we have two cases: $g_0=0 \ (\epsilon=0)$ and $g_0\neq 0 \ (\epsilon \neq 0)$. We explore both cases below.
It should be noted that time-dependent $r_0, s_0$ are taken from Table 3. in Alvarez and Carramiñana [@AC2004], assuming an inverse linear magnetic field decay timescale consistent with $r_0\geq0$ and $s_0\geq 0$. $g_0$ is taken from the stationary multipole model in Table 1.[@AC2004], which also contains values of $r_0$ and $s_0$, which differ little from those in Table 3. [@AC2004]. Better values of $r_0$, $s_0$ including the presently undetermined $g_0$ should be available from pulsar data in the coming years.
a. $g_0=0 ~ (\epsilon=0)$ {#a.-g_00-epsilon0 .unnumbered}
-------------------------
$$f_{pulsar}(t)=f_0 \sqrt{\frac{s_0}{\lambda_0}} \exp \left[\frac{-s_{0}t}{1+t/t_c}\right]{\left[1-\frac{\lambda_1}{\lambda_0} \exp \left\{\frac{-2s_{0}t}{1+t/t_c}\right\}\right]}^{-1/2}$$
Using the binomial expansion we can rewrite this as $(1-x)^{-1/2}=1+\frac{1}{2}x+\frac{3}{8}x^2+...$, thus
$$f_{pulsar}(t)=f_{0}\sqrt{\frac{s_0}{\lambda_0}} \exp \left\{\frac{-s_0t}{1+t/t_c}\right\} \left[1+\frac{\lambda_1}{2\lambda_0} \exp \left\{\frac{-2s_0t}{1+t/t_c}\right\} + \frac{3\lambda_1^2}{8x_0^2} \exp \left\{\frac{-4s_0t}{1+t/t_c}\right\} + ...\right]$$
Higher order terms were dropped. Since $t/t_c \ll 1$, we have
$$f_{pulsar}(t)=f_0\sqrt{\frac{s_0}{\lambda_0}} \exp \{-s_0t\} \left[1+\frac{\lambda_1}{2\lambda_0} \exp \{-2s_0t\} + ...\right]$$
The spindown of GW signal from pulsars has been studied in pioneering works by a parametrized model for the gravitational wave frequency [@BC2000; @JK1999]:
$$f=f_0\left(1+\frac{\vec{v}}{c} \cdot \hat{n}\right)\left(1+\sum_{k=1} f_k\left[t+\frac{\vec{x}}{c}\cdot \hat{n}\right]^k \right)$$
where the terms $\frac{\vec{v}}{c} \cdot \hat{n}$ and $\frac{\vec{x}}{c}\cdot \hat{n}$ account for the Doppler shift. By ignoring the Doppler shift (which we investigate in a later paper), we can further express the GW signal in terms of the parametrized series for pulsars as a linear combination of Chebyshev polynomials:
$$f_{GW}(t)=\sum_{k} f_k\left(\frac{t}{\tau_{min}}\right)^k = \sum_{k} f_k T_k \left(\frac{t}{\tau_{min}}\right)$$
Let $x = t/ \tau_{min}$, where $T_k(x)$ are the Chebyshev polynomials of the first kind.
Therefore,
$$f_{GW}(t)=\sum_{k} f_k T_k(x)$$
and $f_k$ are the spindown parameters.\
We can evaluate the spin-down parameter by using the orthogonal properties of the Chebyshev polynomials, by using the equation
$$f_{GW}(t) \approx f_{pulsar}(t)$$
Multiplying by $\frac{T_l(x)}{\sqrt{1-x^2}}$ on both sides, we obtain
$$\frac{T_l(x)f_{GW}(t)}{\sqrt{1-x^2}} = \frac{T_l f_{pulsar}(t)}{\sqrt{1-x^2}}$$
Integration over the domain $[-1,1]$ gives,
$$\int_{-1}^{1} \sum_{k} f_k \frac{T_l(x)T_k(x)}{\sqrt{1-x^2}} dt = \int^{1}_{-1} \frac{T_l f_{pulsar}(t)}{\sqrt{1-x^2}} dt.$$
From the orthogonality conditions of Chebyshev polynomials,
$$\begin{aligned}
\int_{-1}^{1} \frac{T_j(x)T_k(x)}{\sqrt{1-x^2}} dx = \left\{ \begin{array}{cc}
\pi & \hspace{5mm} j=k=0 \\
\frac{\pi}{2} & \hspace{5mm} j=k\neq 0 \\
0 & \hspace{5mm} j\neq k\\
\end{array} \right.\end{aligned}$$
we obtain the following expression for the spindown parameters,
$$f_k=\frac{2}{\pi} \int^{1}_{-1} \frac{T_k(x)f_{pulsar}(t)}{\sqrt{1-z^2}} dt$$
Evaluating the integral enables us to find $f_k$,
$$\int_{-1}^{1} \frac{1}{\sqrt{1-z^2}} e^{-pz} T_k(z) dz = (-1)^{k} I_k(p)$$
where $I_k(p)$ is the modified Bessel function.\
For the Crab pulsar with $\tau_{min}\approx 962$ years, we have
$$\frac{1}{\tau_{min}}\approx 3.296 \times 10^{-11} s^{-1}$$
$$f_1 \approx -5.1804 \times 10^{-12} s^{-1} < \frac{1}{\tau_{min}}$$
$$f_2 \approx 9.6924 \times 10^{-24} s^{-2} < \frac{1}{\tau^2_{min}}=1.0865 \times 10^{-21} s^{-2}$$
$$f_3 \approx -1.2103 \times 10^{-35} s^{-3} < \frac{1}{\tau^3_{min}} = 3.581 \times 10^{-32} s^{-3}$$
The period of the Crab pulsar is $33.5 \times 10^{-3} s$, hence the frequency is $29.8508~Hz$.\
---------------- ------------- ------------ -------------------------- ------------------------- ---------------------------
Pulsar Age (years) $f_0$ (Hz) $f_1 (s^{-2})$ $f_2 (s^{-3})$ $f_3 (s^{-4})$
\[0.5ex\] Crab 962 29.937 $-5.180 \times 10^{-12}$ $9.692 \times 10^{-24}$ $-1.210 \times 10^{-35}$
PSR1509-58 1553 6.627 $-1.657 \times 10^{-12}$ $1.988\times 10^{-24}$ $-1.591\times 10^{-36}$
PSR0540-69 1664 19.881 $-4.405\times 10^{-12}$ $1.101\times 10^{-23}$ $-1.835\times 10^{-35}$
Vela 11000 11.198 $-8.499 \times 10^{-13}$ $3.612 \times 10^{-25}$ $ -1.023 \times 10^{-37}$
---------------- ------------- ------------ -------------------------- ------------------------- ---------------------------
: First three spindown parameters for the Crab, PSR1509-58, PSR0540-69 and Vela pulsars[]{data-label="table:1"}
The values of the calculated spindown parameters are shown in Table 1 above and within the prescribed limit $|{f_k}| \leq \frac{1}{\tau^k_{min}}$, which is consistent with the parameterization of Brady & Creighton and Jaranowski et al.[@BC2000; @JK1999]. This is also a good indication of the robustness of the model by Alvarez and Carramiñana [@AC2004] where $r_0$ and $s_0$ were determined independently of $f_1, f_2$ and $f_3$.
b. $g_0\neq 0~(\epsilon \neq 0)$ {#b.-g_0neq-0epsilon-neq-0 .unnumbered}
--------------------------------
$f_{pulsar}(t)$ now assumes the form
$$f_{pulsar}(t) \approx f_0 \sqrt{\frac{s_0}{\lambda_c}} e^{-s_ct}$$
where higher order terms are dropped, but can be included if a more accurate approximation is warranted.\
Hence,
$$f_k=\ \sqrt{\frac{s_0}{\lambda_c}} (-1)^{k} I_k (s_c)$$
For $s_0>0, g_0\neq 0, \lambda_c <\lambda_0$, the spindown coefficient $|f_k|$ will be having slightly higher values in comparison to the case when $g_0=0$. Also when $g_0<0$, as can occur for spinups, $C$ will be positive and $|f_k|$ will be lower during such time segments. For $s_0<0$, the expression for $f(t)$ is modified accordingly as demonstrated in the limiting case of $g_0 = 0$ in Eq.(8b) above, whereby higher values of the spindown parameters are obtained.\
For a relatively old pulsar, such as the Vela pulsar, only three spindown parameters may be adequate. For a young pulsar, $g_0$ could be more significant and more spindown parameters need to be evaluated. Eq.(52) provides the analytic expression for $f_k$, which could be used for all spindown parameters.
Conclusions
===========
In earlier works [@JVD96; @CQG2002; @VCV2005; @CVRSW; @NO2008] we have implemented the Fourier transform (FT) of the Doppler shifted GW signal from a pulsar with the Plane Wave Expansion in Spherical Harmonics (PWESH). It turns out that the consequent analysis of the Fourier Transform (FT) of the gravitational wave (GW) signal from a pulsar has a very interesting and convenient development in terms of the resulting spherical Bessel, generalized hypergeometric, Gamma and Legendre functions. These works considered frequency modulation of a GW signal due to rotational and circular orbital motions of the detector on the Earth. In later analysis, rotational and orbital eccentric motions of the Earth, as well as perturbations due to Jupiter and the Moon were considered [@VCV2005]. The numerical analysis of this analytic expression for the signal offers a challenge for efficient and fast numerical and parallel computation.
The recent detection of gravitational waves from black hole mergers is an outstanding success of theoretical physics and experimental gravitation. Gravitational wave detectors like the LIGO, VIRGO, LISA, KAGRA and GEO 600 are opening a new window for the study of a great variety of nonlinear curvature phenomena. Detection of GW necessitates sufficiently long observation periods to attain an adequate Signal to Noise ratio (S/N). The data analysis for continuous GW, for example from rapidly spinning neutron stars, is an important problem for ground based detectors that demands analytic, computational and experimental ingenuity. The Crab and Vela pulsars are among the iconic sources of GW emissions. Abbott et al. have presented direct upper limits on GW emissions from the Crab pulsar [@Abbott2008]. The searches use the known frequency and position of the Crab pulsar. They find that, under the assumption that GW and the electromagnetic signals are phase locked, their single template search results constrain the GW luminosity to be less than 6% of the observed spindown luminosity, and beats the indirect limits obtained from all electromagnetic observations of the Crab pulsar and nebula. Similarly, Abadie et al. have given the direct upper limits on GW emissions from the Vela pulsar using data from the VIRGO detector’s second science run [@Abadie]. The Square Kilometer Array (SKA) will soon be in operation, and along with Advanced LIGO, VIRGO, KAGRA and the upcoming IndIGO, direct detection of GW from pulsars may become a reality and serve as a landmark in the field of multi-messenger grativational wave astronomy in the very near future.
In this work, we have presented an analytic formulation for determining spindown parameters in the Brady and Creighton approach [@BC2000] using a pulsar model which assumes an inverse linear law decay of the magnetic field [@AC2004]. We were able to extract these parameters using the exact solution involving the monopolar, dipolar and quadrupolar terms in the model and found these to be proportional to the modified Bessel functions. For the Crab, PSR 1509-58, PSR 0540-69 and Vela pulsars, we obtained the first three spindown parameters which were within the limit $|{f_k}| \leq \frac{1}{\tau^k_{min}}$. This is consistent with the parameterization of Brady & Creighton [@BC2000] and Jaranowski et al. [@JK1999], and a good indication of the robustness of the model by Alvarez and Carramiñana [@AC2004]. Further, we were able to find the full solution for the period (frequency) evolution. With the determination of the quadrupole coefficient $g_0$ from data, our solution can be incorporated for further improvement in accuracy of the spindown parameters.The study of pulsar spindown and evolution of its braking index will lead to further interesting explorations of the anomalies present in the timing structure, not only in connection with gravitational waves, but also in the fundamental aspects of quark deconfinement in pulsar cores [@Weber]. The study of pulsar spindown also implicitly involves the role of spinups. The physics behind glitches is an active ongoing area of research that presents challenging studies such as the interior of neutron stars and the properties of matter at ultra high nuclear densities [@Baym]. We are presently working on utilizing the available data on isolated pulsars towards finding fits for the $P\neg\dot{P}$ diagram that would also include the quadrupole term $g_0$ using the analytic expression derived in this work to study gravitational wave data mining for the Crab Pulsar [@FVK2]. We hope to further improve the accuracy of the spindown parameters for GW signal detection and extend the applicability of our approach to younger pulsars. In forthcoming work, we plan to develop the analytic Fourier Transform of the pulsar GW signal to include spindown [@FV1].
Acknowledgments
===============
Firstly, we thank the anonymous reviewers of our earlier paper in CQG (2006) for their thorough and inspiring critique that has stimulated our continued efforts to tackle this problem. SRV would like to thank the Natural Sciences Engineering Research Council (NSERC) Canada for funding support through an NSERC Discovery Grant, and King’s University College for their continued support in our research endeavors. We would like to also thank Vladimir Dergachev, Ken Roberts, Xingyi Wang and Justin Tonner for useful discussions.
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---
abstract: 'We investigate a level-set type method for solving ill-posed problems, with the assumption that the solutions are piecewise, but not necessarily constant functions with unknown level sets and unknown level values. In order to get stable approximate solutions of the inverse problem we propose a Tikhonov-type regularization approach coupled with a level set framework. We prove the existence of generalized minimizers for the Tikhonov functional. Moreover, we prove convergence and stability for regularized solutions with respect to the noise level, characterizing the level-set approach as a regularization method for inverse problems. We also show the applicability of the proposed level set method in some interesting inverse problems arising in elliptic PDE models.'
author:
- 'A. De Cezaro [^1]'
bibliography:
- 'Final\_v\_123643\_lset-const.bib'
title: 'On a level-set method for ill-posed problems with piecewise non-constant coefficients'
---
#### Keywords:
Level Set Methods, Regularization, Ill-Posed Problems, Piecewise Non-Constant Coefficients.
Introduction {#sec:1}
============
Since the seminal paper of Santosa [@San95], level set techniques have been successfully developed and have recently become a standard technique for solving inverse problems with interfaces (e.g., [@BurOsher05; @CBGI09; @CVK10; @DCMYVO10; @DL09; @MS08; @NLT07; @TO05; @DA06]).
In many applications, interfaces represent interesting physical parameters (inhomogeneities, heat conductivity between materials with different heat capacity, interface diffusion problems) across which one or more of these physical parameters change value in a discontinuous manner. The interfaces divide the domain $\Omega\subset \mathbb{R}^n$ in subdomains $\Omega_j$, with $j=1,
\cdots, k$, of different regions with specific internal parameter profiles. Due to the different physical structures of each of these regions, different mathematical models might be the most appropriate for describing them. Solutions of such models represent a free boundary problem, i.e., one in which interfaces are also unknown and must be determined in addition to the solution of the governing partial differential equation. In general such solutions are determined by a set of data obtained by indirect measurements [@BurOsher05; @CBGI09; @ChungChanTai05; @ChungVese03; @CVK10; @LieLysTai06; @TaiYao; @VeseOsher04]. Applications include image segmentation problems [@ChungVese03; @LieLysTai06; @TaiYao; @VeseOsher04], optimal shape designer problems [@TaiChan04; @BurOsher05], Stefan’s type problems [@BurOsher05], inverse potential problems [@CLT08; @DCLT2010; @DCLT09], inverse conductivity/resistivity problems [@ChungChanTai05; @CVK10; @DCMYVO10; @Isa90; @DA06] among others [@BurOsher05; @CBGI09; @CVK10; @DL09; @TaiChan04].
There is often a large variety of priors information available for determining the unknown physical parameter, whose characteristic depends on the given application. In this article, we are interested in inverse problems that consist in the identification of an unknown quantity $u \in D(F) \subset X$ that represents all parameter profiles inside the individual subregions of $\Omega$, from data $y
\in Y$, where $X$ and $Y$ are Banach spaces and $D(F)$ will be adequately specified in Section \[sec:min-concept\]. In this particular case, only the interfaces between the different regions and, possibly, the unknown parameter values need to be reconstructed from the gathered data. This process can be formally described by the operator equation $$\label{eq:inv-probl}
F(u) \ = \ y \, ,$$ where $F: D(F) \subset X \to Y$ is the forward operator.
Neither existence nor uniqueness of a solution to are guarantee. For simplicity, we assume that for exact data $y \in Y$, the operator equation admit a solution and we do not strive to obtain results on uniqueness. However, in practical applications, data are obtained only by indirect measurements of the parameter. Hence, in general, exact data $y \in Y$ are not known and we have only access to noise data $y^\delta \in Y$, whose level of noise $\delta
> 0$ are assumed be known *a priori* and satisfies $$\label{eq:noisy-data}
\| y^\delta - y \|_Y \ \le \ \delta \, .$$
We assume that the inverse problem associated with the operator equation is ill-posed. Indeed, it is the case in many interesting problems [@CVK10; @DL09; @EngHanNeu96; @HR96; @Isa90; @TaiChan04; @DA06]. Therefore, accuracy of an approximated solution call for a regularization method [@EngHanNeu96]. In this article we propose a Tikhonov-type regularization method coupled with a level-set approach to obtain a stable approximation of the unknown level sets and values of the piecewise (not necessarily constant) solution of .
Many approaches, in particular level set type approaches, have previously been suggested for such problems . In [@ChanTai03; @ChanTai04; @ChungChanTai05; @FSL05; @LS03; @San95], level set approaches for identification of the unknown parameter $u$ with distinct, but known, piecewise constant values were investigated.
In [@ChanTai04; @ChungVese03; @CLT08], level set approaches were derived to solve inverse problems, assuming that $u$ is defined by several distinct constant values. In both cases, one needs only to identify the level sets of $u$, i.e. the inverse problem reduces to a shape identification problem. On the other hand, when the level values of $u$ are also unknown, the inverse problem becomes harder, since, we have to identify both the level sets and the level values of the unknown parameter $u$. In this situation, the dimension of the parameter space increases by the number of unknown level values. Level set approaches to ill- posed problems with unknown constant level values appeared before in [@DCLT09; @DCLT2010; @TaiChan04; @TaiHongwei07; @TaiYao]. Level set regularization properties of the approximated solution for inverse problems are described in [@Burger01; @CLT08; @DCLT2010; @DCLT09; @FSL05].
However, regularization theory for inverse problems where the components of the parameter $u$ are variable and have discontinuities have not been well investigated. Indeed, level set regularization theory applied to inverse problems [@DCLT2010; @CLT08; @DCLT09] that recover the shape and the values of variable discontinuous coefficients are unknown to the author. Some early results in the numerical implementation of level set type methods were previously used to obtain solutions of elliptic problems with discontinuous and variable coefficients in [@CVK10].
In this article, we propose a level set type regularization method to ill-posed problems whose solution is composed by piecewise components which are not necessarily constants. In other words, we introduce a level set type regularization method to recover the shape and the values of variable discontinuous coefficients. In this framework a level set function is used to parameterized the solution $u$ of . We obtain a regularized solution using a Tikhonov-type regularization method. Since the level values of $u$ are not-constant and also unknown.
In the theoretical point of view, the advantage of our approach in relation to [@BurOsher05; @DCLT09; @CLT08; @DCLT2010; @FSL05; @DAL09] is that we are able to obtain regularized solutions to inverse problems with piecewise solutions that are more general than those covered by the regularization methods proposed before. We still prove regularization properties for the approximated solution of the inverse problem model , where the parameter is a non-constant piecewise solution. The topologies needed to guarantee the existence of a minimizer (in a generalized sense) of the Tikhonov functional (define below in ) is quite complicated and differ in some key points from [@DCLT2010; @DCLT09; @FSL05]. In this particular approach, the definition of generalized minimizers are quite different from other works [@DCLT09; @CLT08; @FSL05] (see Definition \[def:quadruple\]). As a consequence, the arguments used to prove the well-posedeness of the Tikhonov functional, the stability and convergence of the regularized solutions of the inverse problem are quite complicated and need significant improvements (see Section \[sec:min-concept\]).
The main applicability advantage of the proposed level set type method compared to those in the literature is that we are able to apply this method to problems whose solutions depend of non-constant parameters. This implies that we are able to handle more general and interesting physical problems, where de components of the desired parameter is not necessarily homogeneous, as those presented before in the literature [@Ber04; @CVK10; @DL09; @DCLT09; @DCLT2010; @TaiChan04; @TaiHongwei07; @TaiYao; @TL07; @DAP11]. Examples of such interesting physical problems are heat conduction between materials of different heat capacity and conductivity, interface diffusion processes and many other types of physical problems where modeling components are related with embedded boundaries. See for example [@Ber04; @DCLT09; @CBGI09; @CVK10; @DL09; @DAP11] and references therein. As a benchmark problem we analyze two inverse problems modeled by elliptic PDE’s with discontinuous and variable coefficients.
In contrast, the non-constant characteristics of the level values impose different types of theoretical problems, since the topologies where we are able to provide regularization properties of the approximated solution are more complicated than the ones presented before [@DCLT09; @DCLT2010; @TaiChan04; @TaiHongwei07; @TaiYao]. As a consequence, the numerical implementations becomes harder than the others approaches in the literature [@DCLT09; @DCLT2010; @DAP11; @DAL09].
The paper is outlined as follows: In Section \[sec:level-set-formularion\], we formulate the Tikhonov functional based on the level-set framework. In Section \[sec:min-concept\], we present the general assumptions needed in this article and the definition of the set of admissible solutions. We prove relevant properties about the admissible set of solutions, in particular, convergence in suitable topologies. We also present relevant properties of the penalization functional. In Section \[sec:conv-an\], we prove that the proposed method is a regularization method to inverse problems, i.e., we prove that the minimizers of the proposed Tikhonov functional are stable and convergent with respect to the noise level in the data. In Section \[sec:num-sol\], a smooth functional is proposed to approximate minimizers of the Tikhonov functional defined in the admissible set of solutions. We provide approximation properties and the optimality condition for the minimizers of the smooth Tikhonov functional. In Section \[sec:numeric\], we present an application of the proposed framework to solve some interesting inverse elliptic problems with variable coefficients. Conclusions and future directions are presented in Section \[sec:conclusions\].
The Level-set Formulation {#sec:level-set-formularion}
=========================
Our starting point is the assumption that the parameter $u$ in assumes two unknown functional values, i.e., $u(x) \in \{ \psi^1(x), \psi^2(x) \}$ a.e. in $\Omega \subset
\mathbb R^d$, where $\Omega$ is a bounded set. More specifically, we assume the existence of a mensurable set $D \subset\subset \Omega$, with $0<|D|< |\Omega|$, such that $u(x) = \psi^1(x)$ if $x \in D$ and $u(x) = \psi^2(x)$ if $x \in \Omega/D$. With this framework, the inverse problem that we are interested in this article is the stable identification of both the shape of $D$ and the value function $\psi^j(x)$ for $x$ belonging to $D$ and to $\Omega/D$, respectively, from observation of the data $y^\delta \in Y$.
We remark that, if $\psi^1(x)=c^1$ and $\psi^2(x)=c^2$ with $c^1$ and $c^2$ unknown constants values, the problem of identifying $u$ was rigorously studied before in [@DCLT09]. Moreover, many other approaches to this case appear in the literature, see [@DCLT09; @ChanTai04; @ChanTai03; @BurOsher05] and references therein. Recently, in [@DCLT2010], a $L^2$ level set approach to identify the level and constant contrast was investigated.
Our approach differs from the level set methods proposed in [@DCLT2010; @DCLT09], by considering also the identification of variable unknown levels of the parameter $u$. In this situation, many topological difficulties appear in order to have a tractable definition of an admissible set of parameters (see Definition \[def:quadruple\] below). Generalization to problems with more than two levels are possible applying this approach and following the techniques derived in [@CLT08]. As observed before, the present level set approach is a rigorous derivation of a regularization strategy for identification of the shape and non-constant levels of discontinuous parameters. Therefore, it can be applied to physical problems modeled by embedded boundaries whose components are not necessarily piecewise constant [@FSL05; @BurOsher05; @CLT08; @DCLT2010; @DCLT09].
In many interesting applications, the inverse problem modeled by equation is ill-posed. Therefore a regularization method must be applied in order to obtain a stable approximate solution. We propose a regularization method by: First, introduce a parametrization on the parameter space, using a level set function $\phi$ that belongs to $H^1(\Omega)$. Note that, we can identify the distinct level sets of the function $\phi\in
H^1(\Omega)$ with the definition of the Heaviside projector $$\begin{aligned}
H:\, & H^1(\Omega) \longrightarrow L^\infty(\Omega)\\
& \,\, \phi \,\, \, \, \longmapsto \,\, H(\phi):=\begin{cases}
1 \,\, \mbox{ if } \,\,
\phi(x) > 0\,,\\
0 \,\, \mbox{ other else}\,.
\end{cases}\end{aligned}$$ Now, from the framework introduced above, a solution $u$ of , can be represented as $$\begin{aligned}
\label{eq:def-P}
u(x) \ = \ \psi^1(x) H(\phi) + \psi^2(x) (1 - H(\phi)) \ =: \
P(\phi,\psi^1,\psi^2)(x) \, .\end{aligned}$$ With this notation, we are able to determine the shapes of $D$ as $
\{x \in \Omega\, ;\ \phi(x) > 0 \}$ and $\Omega / D$ as $ \{x \in
\Omega\, ;\ \phi(x) < 0 \}$.
The functional level values $\psi^1(x)$, $\psi^2(x)$ are also assumed be unknown and they should be determined as well.
\[ass:1\] We assume that $\psi^1, \psi^2\, \in {\mathbb{B}}:=\{f: f \mbox{ is measurable
and}\, f(x) \in [m, M]\,, a.e. \,\, in\, \Omega\}$, for some constant values $m, M$.
We remark that, $f \in {\mathbb{B}}$ implies that $f\in L^\infty(\Omega)$. Since $\Omega$ is bounded $f\in L^1(\Omega)$. Moreover, $$\begin{aligned}
\int_\Omega f(x) \nabla\cdot \varphi(x) dx \leq |M| \int_\Omega
|\nabla\cdot (\varphi)(x)|dx \leq |M| {{\left\lVert\nabla
\cdot\varphi\right\rVert_{}}}_{L^1(\Omega)}\,,\, \forall \varphi \in C_0^1(\Omega,
\mathbb{R}^n)\,.\end{aligned}$$ Hence $f \in \bv(\Omega)$.
Note that, in the case that $\psi^1$ and $\psi^2$ assumes two distinct constant values (as covered by the analysis done in [@BurOsher05; @DCLT09; @DCLT2010] and references therein) the assumptions above are satisfied. Hence, the level set approach proposed here generalizes the regularization theory developed in [@DCLT09; @DCLT2010].
From , the inverse problem in , with data given as in , can be abstractly written as the operator equation $$\label{eq:inv-probl-mls}
F( P(\phi, \psi^1,\psi^2) ) \ = \ y^\delta \, .$$ Once an approximate solution $(\phi, \psi^1,\psi^2)$ of is obtained, a corresponding solution of can be computed using equation .
Therefore, to obtain a regularized approximated solution to , we shall consider the least square approach combined with a regularization term i.e., minimizing the Tikhonov functional
$$\label{eq:mc}
\hat{\Ga}(\phi,\psi^1,\psi^2):= \| F(P(\phi,\psi^1,\psi^2)) -
y^\delta \|^2_{Y}
+ \alpha \Big\{ \beta_1 |H(\phi)|_\bv
+\beta_2 \| \phi - \phi_0\|^2_{H^1(\Omega)}
+ \beta_3 {\textstyle\sum\limits_{j=1}^2 |\psi^j-\psi^j_0|_\bv} \Big\} \, ,$$
where, $\phi_0$ and $\psi^j_0$ represent some *a priori* information about the true solution $u^*$ of . The parameter $\alpha > 0$ plays the role of a regularization parameter and the values of $\beta_i\,, i=1,2,3$ act as scaling factors. In other words, $\beta_i\,, i=1,2,3$ need to be chosen *a priori*, but independent of the noise level $\delta$. In practical, $\beta_i\,, i=1,2,3$ can be chosen in order to represent *a priori* knowledge of features the of the parameter solution $u$ and/or to improve the numerical algorithm. A more complete discussion about how to choose $\beta_i\,, i=1,2,3$ are provided in [@DCLT2010; @DCLT09; @CLT08].
The regularization strategy in this context is based on $TV-H^1-TV$ penalization. The term on $H^1$-norm acts simultaneously as a control on the size of the norm of the level set function and a regularization on the space $H^1$. The term on $\bv$ is a variational measure of $H(\phi)$. It is well known that the $BV$-semi-norm acts as a penalizing for the length of the Hausdorff measure of the boundary of the set $\{x \,:\,\phi(x) > 0\}$ (see [@EG92 Chapter 5] for details). Finally, the last term on $\bv$ is a variational measure of $\psi^j$ that acts as a regularization term on the set ${\mathbb{B}}$. This Tikhonov functional extends the ones proposed in [@DCLT09; @CLT08; @ChanTai03; @ChanTai04; @TaiChan04] (based on $TV$-$H^1$ penalization).
Existence of minimizers for the functional , in the $H^1\times {\mathbb{B}}^2$ topology does not follow by direct arguments, since, the operator $P$ is not necessarily continuous in this topology. Indeed, if $\psi^1 = \psi^2 =\psi$ is a continuous function at the contact region, then $P(\phi^1,\psi^2, \psi)= \psi$ is continuous and the standard Tikhonov regularization theory to the inverse problem holds true [@EngHanNeu96]. On the other hand, in the interesting case where $\psi^1$ and $\psi^2$ represents the level of discontinuities of the parameter $u$, the analysis becames more complicated and we need a definition of generalized minimizers (see Definition \[def:quadruple\]) in order to handle with these difficulties.
Generalized Minimizers {#sec:min-concept}
======================
As already observed in [@FSL05], if $D \subset \Omega$ with $\mathcal{H}^{n-1}(\partial D) < \infty$ where $\mathcal{H}^{n-1}(S)$ denotes the (n-1)-dimensional Hausdorff-measure of the set $S$, then the Heaviside operator $H$ maps $H^1(\Omega)$ into the set $$\mathcal{V} := \{ \chi_{D} \, ; \
D \subset \Omega \mbox{ measurable},\,:\, \mathcal{H}^{n-1}(\partial
D) < \infty \}\,.$$ Therefore, the operator $P$ in maps $H^1(\Omega) \times {\mathbb{B}}^2 $ into the admissible parameter set $$D(F) := \{ u = q(v,\psi^1,\psi^2)\,;\, v \in \mathcal V
\mbox{ and } \psi^1, \psi^2 \in {\mathbb{B}}\}\,,$$ where $$q: \mathcal{V}
\times {\mathbb{B}}^2 \ni (v,\psi^1,\psi^2) \mapsto \psi^1 v+\psi^2(1-v) \in
BV(\Omega)\,.$$
Consider the model problem described in the introduction. In this article, we assume that:
[**(A1)**]{} $\Omega \subseteq {\mathbb{R}}^n$ is bounded with piecewise $C^1$ boundary $\partial \Omega$.
[**(A2)**]{} The operator $F:D(F)\,\subset L^1(\Omega) \to
Y$ is continuous on $D(F)$ with respect to the $L^1(\Omega)$-topology.
[**(A3)**]{} $\ve$, $\alpha$ and $\beta_j\,,j=1,2,3$ denote positive parameters.
[**(A4)**]{} \[ass:5\] Equation (\[eq:inv-probl\]) has a solution, i.e. there exists $u_* \in{D(F)}$ satisfying $F(u_*)=y$ and a function $\phi_* \in H^1(\Omega)$ satisfying $|\nabla \phi_*|
\neq 0$, in the neighborhood of $\{\phi_* = 0\}$ such that $H(\phi_*)
= z_*$, for some $z_* \in \mathcal{V}$. Moreover, there exist functional values $\psi^1_*, \psi^2_* \in {\mathbb{B}}$ such that $q(z_*,\psi^1_*,\psi^2_*) = u_*$.
For each $\ve > 0$, we define a smooth approximation to the operator $P$ by $$\label{eq:def-Pve}
P_\ve(\phi,\psi^1,\psi^2) \ := \ \psi^1 H_\ve(\phi) + \psi^2(1 -
H_\ve(\phi))\; ,$$ where $H_\ve$ is the smooth approximation to $H$ described by $$H_\ve(t) := \left\{
\begin{array}{rl}
1 + t/\ve & \mbox{ for \ } t \in \left[-\ve,0\right] \\
H(t) & \mbox{ for \ } t \in \mathbb{R} / \left[-\ve,0\right] \\
\end{array} \right. .$$
It is worth noting that, for any $\phi_k \in H^1(\Omega)$, $H_\ve(\phi_k)$ belongs to $L^\infty(\Omega)$ and satisfies $0\leq
H_\ve(\phi_k)\leq 1$ a.e. in $\Omega$, for all $\ve
> 0$. Moreover, taking into account that $\psi^j \in \mathbb{B}$, follows that the operators $q$ and $P_\ve$, as above, are well defined.
In order to guarantee the existence of a minimizer of ${\cal G}_\alpha$ defined in in the space $H^1(\Omega) \times {\mathbb{B}}^2$, we need to introduce a suitable topology such that the functional ${\cal
G}_\alpha$ has a closed graphic. Therefore, the concept of generalized minimizers (compare with [@CLT08; @FSL05]) in this paper is:
\[def:quadruple\] Let the operators $H$, $P$, $H_\ve$ and $P_\ve$ be defined as above and the positive parameters $\alpha, \beta_j$ and $\ve$ satisfying the Assumption **(A3)**.
A [**quadruple**]{} $(z,\phi,\psi^1,\psi^2) \in
L^\infty(\Omega) \times H^1(\Omega) \times \bv(\Omega)^2$ is called [**admissible**]{} when: a) There exists a sequence $\{
\phi_k \}$ of $H^1(\Omega)$-functions satisfying $\lim\limits_{k\to\infty} \| \phi_k - \phi \|_{L^2(\Omega)} = 0$.
b\) There exists a sequence $\{ \ve_k \} \in \mathbb R^+$ converging to zero such that $\lim\limits_{k\to\infty} \|
H_{\ve_k}(\phi_k)-z \|_{L^1(\Omega)} = 0$.
c\) There exist sequences $\{ \psi^1_k \}_{k\in\mathbb{N}}
\mbox{ and } \{ \psi^2_k \}_{k\in\mathbb{N}}$ belonging to $\bv \cap
C^\infty(\Omega)$ such that $$|\psi^j_k|_\bv \longrightarrow |\psi^j|_\bv\,,\quad j=1,2\,.$$
d\) A [**generalized minimizer**]{} of $\hat{\Ga}$ is considered to be any admissible quadruple $(z,\phi,\psi^1,\psi^2)$ minimizing $$\label{eq:gzphi}
{\cal{G}}_\alpha(z,\phi,\psi^1,\psi^2) :=
{{\left\lVert F(q(z,\psi^1,\psi^2)) - y^\delta \right\rVert_{}}}_Y^2 + \alpha
R(z,\phi,\psi^1,\psi^2)$$ on the set of admissible quadruples. Here the functional $R$ is defined by $$\begin{aligned}
\label{def:R}
R(z,\phi,\psi^1,\psi^2) \ = \ \rho(z,\phi) +
\beta_3 {\textstyle\sum\limits_{j=1}^2 |\psi^j-\psi^j_0|_\bv} \,
$$ and the functional $\rho$ is defined as $$\begin{aligned}
\label{eq:rho}
\rho(z,\phi) := \inf \Big\{ \liminf\limits_{k\to\infty}
\Big[ \beta_1 |H_{\ve_k} (\phi_k)|_\bv +
\beta_2 \| \phi_k - \phi_0 \|_{H^1(\Omega)}^2 \Big] \Big\} \, .
\end{aligned}$$ The infimum in is taken over all sequences $\{\ve_k\}$ and $\{\phi_k \}$ characterizing $(z,\phi,\psi^1,\psi^2)$ as an admissible quadruple.
The convergence $ |\psi^j_k|_\bv \longrightarrow |\psi^j|_\bv$ in Item c) in Definition \[def:quadruple\] is in the sense of variation measure [@EG92 Chapter 5]. The incorporation of item c) in the Definition \[def:quadruple\] implies the existence of the $\Gamma$-limit of sequences of admissible quadruples [@FSL05; @AcaVog94]. This appears in the proof of Lemmas \[lemma:auxil\], \[lemma:limit-ad\] and \[lemma:rho-lsc\], where we proove that the set of admissible quadruples are closed in the defined topology (see Lemmas \[lemma:auxil\] and \[lemma:limit-ad\]) and in the weak lower semi-continuity of the regularization functional $R$ (see Lemma \[lemma:rho-lsc\]). The identification of non-constant level values $\psi^j$ imply in a different definition of admissible quadruples.
As a consequence, the arguments in the proof of regularization properties of the level set approach are the principal theoretical novelty and the difference between our definition of admissible quadruples and the ones in [@DCLT2010; @DCLT09; @FSL05].
\[remark-ad\] For $j=1,2$ let $\psi^j\in {\mathbb{B}}\cap C^\infty(\Omega)$, $\phi\in
H^1(\Omega)$ be such that $|\nabla \phi| \neq 0 $ in the neighborhood of the level set $\{\phi(x)=0 \}$ and $H(\phi)=z \in \mathcal{V}$. For each $k\in {\mathbb{N}}$ set $\psi_k^j=\psi^j$ and $\phi_k =\phi$. Then, for all sequences of $\{\ve_k\}_{k\in {\mathbb{N}}}$ of positive numbers converging to zero, we have $$\begin{aligned}
{{\left\lVertH_{\ve_k}(\phi_k) - z\right\rVert_{}}}_{L^1(\Omega)} & =
{{\left\lVertH_{\ve_k}(\phi_k) - H(\phi)\right\rVert_{}}}_{L^1(\Omega)} =
\int_{(\phi)^{-1}[-\ve_k,0]}\left| 1-\frac{\phi}{\ve_k}\right|dx \\
& \leq \int_{-\ve_k}^0\int_{(\phi)^{-1}(\tau)}1d\tau \leq
meas\{(\phi)^{-1}(\tau)\}\int_{-\ve_k}^01dt \longrightarrow 0\,.\end{aligned}$$ Here, we use the fact that $|\nabla \phi|\neq 0$ in the neighborhood of $\{\phi =0\}$ implies that $\phi$ is a local diffeomorphism together with a co-area formula [@EG92 Chapter 4]. Moreover, $\{\psi_k^j\}_{k\in {\mathbb{N}}}$ in ${\mathbb{B}}\cap C^\infty(\Omega)$ satisfyes Definition \[def:quadruple\], item c).
Hence, $(z,\phi,\psi^1,\psi^2)$ is an admissible quadruple. In particular, we conclude from the general assumption above that the set of admissible quadruple satisfying $F(u)=y$ is not empty.
Relevant Properties of Admissible Quadruples {#subsec:relevant-propoerties}
--------------------------------------------
Our first result is the proof of the continuity properties of operators $P_\ve$, $H_\ve$ and $q$ in suitable topologies. Such result will be necessary in the subsequent analysis.
We start with an auxiliary lemma that is well known (see for example [@DZ94]). We present it here for the sake of completeness.
\[lemma:0\] Let $\Omega$ be a measurable subset of $\mathbb{R}^n$ with finite measure.
If $(f_k) \in {\mathbb{B}}$ is a convergent sequence in $L^p(\Omega)$ for some $p$, $1\leq p<\infty$, then it is a convergent sequence in $L^p(\Omega)$ for all $1\leq p < \infty$.
In particular Lemma \[lemma:0\] holds for the sequence $z_k: =
H_\varepsilon(\phi_k)$.
See [@DZ94 Lemma 2.1].
The next two lemmas are auxiliary results in order to understand the definition of the set of admissible quadruples.
\[lemma:q-cont\] Let $\Omega$ as in assumption **(A1)** and $j=1,2$.
1. Let $\{z_k\}_{k\in \mathbb{N}}$ be a sequence in $L^{\infty}(\Omega)$ with $z_k \in [m,M]$ a.e. converging in the $L^1(\Omega)$-norm to some element $z$ and $\{\psi_k^j\}_{k\in
\mathbb{N}}$ be a sequence in ${\mathbb{B}}$ converging in the $\bv$-norm to some $\psi^j\in {\mathbb{B}}$. Then $ q(z_k,\psi_k^1,\psi_k^2)$ converges to $ q(z,\psi^1,\psi^2)$ in $L^1(\Omega)$.
2. Let $(z,\phi) \in L^1(\Omega) \times H^1(\Omega)$, be such that $H_\ve(\phi)\rightarrow z$ in $L^1(\Omega)$ as $\ve \to 0$ and let $\psi^1,\psi^2\in {\mathbb{B}}$. Then $P_\ve(\phi,\psi^1,\psi^2) \rightarrow
q(z,\psi^1,\psi^2)$ in $L^1(\Omega)$ as $\ve \to 0$.
3. Given $\ve >0$, let $\{\phi_k\}_{k\in {\mathbb{N}}}$ be a sequence in $H^1(\Omega)$ converging to $\phi \in H^1(\Omega)$ in the $L^2$-norm. Then $H_\ve(\phi_k) \to H_\ve(\phi)$ in $L^1(\Omega)$, as $k \to \infty$. Moreover, if $\{\psi_k^j\}_{k\in\mathbb{N}}$ are sequences in ${\mathbb{B}}$, converging to some $\psi^j$ in ${\mathbb{B}}$, with respect to the $L^1(\Omega)$-norm, then $q(H_\ve(\phi_k),\psi^1_k,\psi^2_k)
\to q(H_\ve(\phi),\psi^1,\psi^2)$ in $L^1(\Omega)$, as $k \to
\infty$.
Since $\Omega$ is assumed to be bounded, we have $L^{\infty}(\Omega)
\subset L^{1}(\Omega)$ and $BV(\Omega)$ is continuous embedding in $L^2(\Omega)$ [@EG92]. To prove [*(i)*]{}, notice that $$\begin{aligned}
\| q(z_k,\psi^1_k,\psi^2_k) - q(z,\psi^1,\psi^2)\|_{L^1(\Omega)}
& = {{\left\lVert\psi^1_k z_k + \psi^2_k(1-z_k) - \psi^1z -
\psi^2(1-z)\right\rVert_{}}}_{L^1(\Omega)}\\
&
\leq {{\left\lVertz_k\right\rVert_{}}}_{L^\infty(\Omega)}
{{\left\lVert\psi^1_k-\psi^1\right\rVert_{}}}_{L^1(\Omega)} + {{\left\lVert\psi^1\right\rVert_{}}}_{L^2(\Omega)}
{{\left\lVertz_k-z\right\rVert_{}}}_{L^2(\Omega)} \\
& \, + {{\left\lVert1-z_k\right\rVert_{}}}_{L^\infty(\Omega)}
{{\left\lVert\psi^2_k-\psi^2\right\rVert_{}}}_{L^1(\Omega)}+ {{\left\lVert\psi^2\right\rVert_{}}}_{L^2(\Omega)}
{{\left\lVertz_k-z\right\rVert_{}}}_{L^2(\Omega)} \stackrel{k\to\infty}{\longrightarrow} 0
\,.\end{aligned}$$ Here we use Lemma \[lemma:0\] in order to guarantee the convergence of $z_k$ to $z$ in $L^2(\Omega)$.
Assertion [*(ii)*]{} follows with similar arguments and the fact that $H_\ve(\phi) \in L^\infty(\Omega)$ for all $\ve >0$.
As $ \| H_\ve(\phi_k) - H_\ve(\phi) \|_{L^1(\Omega)} \leq \ve^{-1}
\sqrt{ {\rm meas}(\Omega) } \| \phi_k - \phi \|_{L^2(\Omega)}$ the first part of assertion [*(iii)*]{} follows. The second part of the assertion [*(iii)*]{} holds by a combination of the inequality above and inequalities in the proof of assertion [*(i)*]{}.
\[lemma:auxil\] Let $\{\psi^j_k\}_{k\in \mathbb{N}}$ be a sequence of functions satisfying Definition \[def:quadruple\] converging in $
L^1(\Omega)$ to some $\psi^j$, for $j=1,2$. Then $\psi^j$ also satisfies Definition \[def:quadruple\].
[*Sketch of the proof.*]{}
Let $k\in {\mathbb{N}}$ and $j=1,2$. Since $\psi^j_k$ satisfies Definition \[def:quadruple\], $\psi_k^j \in \bv$. From [@EG92 Theorem 2, pg 172] there exist sequences $\{\psi_{k,l}^j\}_{l\in{\mathbb{N}}}$ in $\bv\times C^\infty(\Omega)$ such that $$\psi_{k,l}^j \stackrel{l\to\infty}{\longrightarrow} \psi_k^j \,\,
\mbox{ in }\, \, L^1(\Omega) \,\, \mbox{ and } \, \,
|\psi_{k,l}^j|_\bv \stackrel{l\to\infty}{\longrightarrow}
|\psi_k^j|_\bv\; .$$ In particular, for the subsequence $\{\psi_{k,l(k)}^j\}_{k\in{\mathbb{N}}}$ follows that $$\begin{aligned}
\label{eq:limit}
\psi_{k,l(k)}^j \stackrel{k\to\infty}{\longrightarrow} \psi^j \,\, \mbox{ in }\,
\, L^1(\Omega) \,\, \mbox{ and } \, \, |\psi_{k,l(k)}^j|_\bv
\stackrel{k\to\infty}{\longrightarrow} |\psi^j|_\bv\; .\end{aligned}$$ Moreover, by assumption $\psi^j \in L^1(\Omega)$. From the lower semi-continuity of variational measure (see [@EG92 Theorem 1 pg. 172]), equation and the definition of $\bv$ space, it follows that $\psi^j \in \bv$. $\square$
In the next lemma we prove that the set of admissible quadruples is closed with respect the $ L^1(\Omega) \times L^2(\Omega) \times
(L^1(\Omega))^2$ topology.
\[lemma:limit-ad\] Let $(z_k,\phi_k,\psi^1_k,\psi^2_k)$ be a sequence of admissible quadruples converging in $ L^1(\Omega) \times L^2(\Omega) \times
(L^1(\Omega))^2$ to some $(z,\phi,\psi^1,\psi^2)$, with $\phi \in
H^1(\Omega)$. Then, $(z,\phi,\psi^1,\psi^2)$ is also an admissible quadruple.
[*Sketch of the proof.*]{} Let $k\in {\mathbb{N}}$. Since $(z^1_k,\phi^1_k,\psi^1_k,\psi^2_k)$ is an admissible quadruple, it follows from Definition \[def:quadruple\] that there exist sequences $\{\phi_{k,l}\}_{l\in{\mathbb{N}}}$, in $H^1(\Omega)$, $\{\psi_{k,l}^1\}_{l\in{\mathbb{N}}}$, $\{\psi_{k,l}^2\}_{l\in{\mathbb{N}}}$ in $\bv\times C^\infty(\Omega)$ and a correspondent sequence $\{\ve^l_{k}\}_{l\in{\mathbb{N}}}$ converging to zero such that $$\phi_{k,l} \stackrel{l\to\infty}{\longrightarrow} \phi_k \ \mbox{ in
} \ L^2(\Omega)\,, \quad H_{\ve^l_{k}}(\phi_{k,l})
\stackrel{l\to\infty}{\longrightarrow} z_k \ \mbox{ in } \
L^1(\Omega)\, \quad \mbox{ and } \quad |\psi_{k,l}^j|_\bv
\stackrel{l\to\infty}{\longrightarrow} |\psi_k^j|_\bv\,, j=1,2\, .$$
Define the monotone increasing function $\tau : {\mathbb{N}}\to {\mathbb{N}}$ such that, for every $k \in {\mathbb{N}}$ it holds $$\label{eq:diag-argument}
\ve_k^{\tau(k)} \leq \frac{1}{2} \ve_{k-1}^{\tau(k-1)} \, , \ \big\|
\phi_{k,\tau(k)} - \phi_k \big\|_{ L^2(\Omega)} \leq \frac{1}{k} \,
, \ \big\| H_{\ve_k^{\tau(k)}}(\phi_{k,\tau(k)}) - z_k
\big\|_{L^1(\Omega)} \leq \frac{1}{k} \, , \,
|\psi_{k,\tau(k)}^j|_\bv \longrightarrow |\psi_k^j|_\bv\,, j=1,2\, .$$ Hence, for each $k\in {\mathbb{N}}$ $$\begin{aligned}
\big\| \phi - \phi_{k,\tau(k)} \big\|_{ L^2(\Omega)} & \leq &
\| \phi - \phi_k \|_{L^2(\Omega)}
+ \big\| \phi_{k,\tau(k)}- \phi_k \big\|_{L^2(\Omega)} \\
\big\| z - H_{\ve_k^{\tau(k)}}(\phi_{k,\tau(k)})
\big\|_{L^1(\Omega)} & \leq &
\| z - z_k\|_{L^1(\Omega)}
+ \big\| H_{\ve_k^{\tau(k)}}(\phi_{k,\tau(k)}) - z_k
\big\|_{L^1(\Omega)}\,.\end{aligned}$$ From , $$\label{eq:zj-conv}
\lim_{k\to\infty} \big\| \phi - \phi_{k,\tau(k)} \big\|_{
L^2(\Omega)} = 0 \,\, , \,\,\lim_{k\to\infty} \big\| z -
H_{\ve_k^{\tau(k)}}(\phi_{k,\tau(k)})
\big\|_{L^1(\Omega)} = 0\,.$$ Moreover, with the same arguments as Lemma \[lemma:auxil\], it follows that $$|\psi_{k,\tau(k)}^j|_\bv \to |\psi^j|_\bv\,,\qquad j=1,2\,,$$ and $\psi^j\in \bv(\Omega)$. Therefore, it remains to prove that $(z,\phi,\psi^1,\psi^2)$ is an admissible quadruple. From Definition \[def:quadruple\] and Lemma \[lemma:auxil\], it is enough to prove that $z \in L^\infty(\Omega)$. If this is not the case, there would exist a $\Omega' \subset \Omega$ with $|\Omega'| >
0$ and $\gamma > 0$ such that $z(x) > 1 + \gamma$ in $\Omega'$ (the other case: $z(x) < - \gamma$ is analogous). Since $(H_{\ve_k^{\tau(k)}}(\phi_{k,\tau(k)}))(x) \in [0,1]$ a.e. in $\Omega$ for $k \in {\mathbb{N}}$ (see remark after Definition \[def:quadruple\]), we would have $$\| z - H_{\ve_k^{\tau(k)}}(\phi_{k,\tau(k)}) \|_{L^1(\Omega)} \ge \|
z - H_{\ve_k^{\tau(k)}}(\phi_{k,\tau(k)}) \|_{L^1(\Omega')} \ge
\gamma |\Omega'| \, , \ k \in {\mathbb{N}}\, ,$$ contradicting the second limit in . $\square$
Relevant Properties of the Penalization Functional {#subsec:relevant-propoerties-penalization}
--------------------------------------------------
In next lemmas, we verify properties of the functional $R$ which are fundamental for the convergence analysis outlined in Section \[sec:conv-an\]. In particular, these properties implies that the level sets of $\Ga$ are compact in the set of admissible quadruple, i.e., $\Ga$ assume a minimizer on this set. First, we prove a lemma that simplify the functional $R$ in . Here we present the sketch of the proof. For more details, see the arguments in [@DCLT09 Lemma 3].
\[lemma:R-auxl\] Let $(z,\phi,\psi^1,\psi^2)$ be an admissible quadruple. Then, there exists sequences $\{\ve_k\}_{k\in {\mathbb{N}}}$, $\{\phi_k\}_{k\in {\mathbb{N}}}$ and $\{\psi^j_k\}_{k\in {\mathbb{N}}}$ as in the Definition \[def:quadruple\], such that $$\begin{aligned}
R(z,\phi,\psi^1,\psi^2) = \lim_{k \to \infty} \left\{\beta_1
|H_{\ve_k}(\phi_k)|_\bv + \beta_2{{\left\lVert\phi_k -
\phi_0\right\rVert_{}}}^2_{H^1(\Omega)} + \beta_3
\sum_{j=1}^2|\psi^j_k-\psi^j_0|_\bv \right\}\,.\end{aligned}$$
[*Sketch of the proof.*]{} For each $l \in {\mathbb{N}}$, the definition of $R$ (see Definition \[def:quadruple\]) guaranties the existence of sequences $\ve_k^l$, $\{\phi_{k,l}^j\}\in
H^1(\Omega)$ and $\{\psi_{k,l}^j\}\in {\mathbb{B}}$ such that $$\begin{aligned}
R(z,\phi,\psi^1,\psi^2) = \lim_{l\to \infty}\left\{ \liminf_{k\to
\infty}\left\{ \beta_1 |H_{\ve_k^l}(\phi_{k,l})|_\bv +
\beta_2{{\left\lVert\phi_{k,l} - \phi_0\right\rVert_{}}}^2_{H^1(\Omega)} \right\} +\beta_3
\sum_{j=1}^2|\psi^j_{k,l}-\psi^j_0|_\bv \right\}\,.\end{aligned}$$ Now a similar extraction of subsequences as in Lemma \[lemma:limit-ad\] complete the proof. $\square$
In the following, we prove two lemmas that are essential to the proof of well posedness of the Tikhonov functional .
\[lemma:rho-coer\] The functional $R$ in is coercive on the set of admissible quadruples. In other words, given any admissible quadruple $(z, \phi, \psi^1, \psi^2)$ we have $$\begin{aligned}
R(z, \phi, \psi^1, \psi^2) \geq \left(\beta_1|z|_{\bv}
+\beta_2{{\left\lVert\phi - \phi_0\right\rVert_{}}}^2_{H^1(\Omega)} +\beta_2 \sum_{j=1}^2
|\psi^j - \psi_0^j|_{\bv}\right)\,.\end{aligned}$$
[*Sketch of the proof.*]{} Let $(z,\phi,\psi^1,\psi^2)$ be an admissible quadruple. From [@CLT08 Lemma 4], it follows that $$\begin{aligned}
\label{eq:rho-coercivo}
\rho(z,\phi) \ \geq \
\big( \beta_1 |z|_\bv + \beta_2 \|\phi-\phi_0\|_{H^1(\Omega)}^2 \big) \, .\end{aligned}$$ Now, from and the definition of $R$ in , we have $$\big( \beta_1 |z|_\bv + \beta_2 \| \phi - \phi_0 \|^2_{H^1(\Omega)}
+ \beta_3 \sum_{j=1}^2|\psi^j-\psi_0^j|_\bv\big)
\, \leq \, \rho(z,\phi) + \beta_3 \sum_{j=1}^2|\psi^j-\psi_0^j|_\bv
\, = \, R(z,\phi,\psi^1,\psi^2) \, ,$$ concluding the proof. $\square$
\[lemma:rho-lsc\] The functional $R$ in is weak lower semi-continuous on the set of admissible quadruples, i.e. given a sequence $\{(z_k,\phi_k,\psi^1_k,\psi^2_k)\}$ of admissible quadruples such that $z_k \to z$ in $L^1(\Omega)$, $\phi_k \rightharpoonup \phi$ in $H^1(\Omega)$, $\psi^j_k \to \psi^j$ in $L^1(\Omega)$, for some admissible quadruple $(z,\phi,\psi^1,\psi^2)$, then $$R(z,\phi,\psi^1,\psi^2) \ \leq \ \liminf_{k \in {\mathbb{N}}}
R(z_k,\phi_k,\psi^1_k,\psi^2_k) \, .$$
The functional $\rho(z,\phi)$ is weak lower semi-continuous cf. [@CLT08 Lemma 5]. As $\psi_k^j \in \bv$ follows from [@EG92 Theorem 2 pg 172] that there exist sequences $\{\psi_{k,l}^j\} \in \bv \cap C^\infty(\Omega)$ such that ${{\left\lVert\psi_{k,l}^j - \psi_k^j\right\rVert_{}}}_{L^1(\Omega)} \leq \frac{1}{l}$. From a diagonal argument, we can extract a subsequence $\{\psi_{k,l(k)}^j\}$ of $\{\psi_{k,l}^j\}$ such that $\{\psi_{k,l(k)}^j\} \to \psi^j$ in $L^1(\Omega)$ as $k \to \infty$. Let $\xi \in C^1_c(\Omega, \mathbb{R}^n)\,,\,|\xi|\leq 1$. Then, from [@EG92 Theorem 1 pg 167], it follows that $$\begin{aligned}
\int_\Omega \psi^j\; \nabla \cdot \xi dx & = \lim_{k\to
\infty}\int_\Omega
\psi_{k,l(k)}^j \; \nabla \cdot \xi dx =
\lim_{k\to \infty}\left[\int_\Omega \left(\psi_{k,l(k)}^j -
\psi_k^j\right) \; \nabla \cdot \xi dx + \int_\Omega \psi_k^j \; \nabla \cdot \xi dx
\right]\\
&
\leq
\lim_{k\to \infty}\left[{{\left\lVert\psi_{k,l(k)}^j -
\psi_k^j\right\rVert_{}}}_{L^1(\Omega)} {{\left\lVert\nabla \cdot \xi\right\rVert_{}}}_{L^\infty(\Omega)}|\Omega| -
\int_\Omega \xi\cdot \sigma_k d|\psi_k^j|_\bv \right]
\leq
\liminf_{k\to \infty} |\psi_k^j|_\bv\,.\end{aligned}$$ Thus, form the definition of $|\cdot|_\bv$ (see [@EG92]), we have $$\begin{aligned}
|\psi^j|_\bv = \sup\left\{ \int_\Omega \psi^j\; \nabla \cdot \xi
dx\,; \xi \in C^1_c(\Omega, \mathbb{R}^n)\,,\,|\xi|\leq 1 \right\}
\leq \liminf_{k\to \infty} |\psi_k^j|_\bv\,.\end{aligned}$$ Now, the lemma follows from the fact that the functional $R$ in is a linear combination of lower semi-continuous functionals.
Convergence Analysis {#sec:conv-an}
====================
In the following, we consider any positive parameter $\alpha$, $\beta_j\,, j=1,2,3$ as in the general assumption to this article. First, we prove that the functional $\Ga$ in is well posed.
\[th:admissible\] The functional $\Ga$ in attains minimizers on the set of admissible quadruples.
Notice that, the set of admissible quadruples is not empty, since $(0,0,0,0)$ is admissible. Let $\{(z_k,\phi_k,\psi^1_k,\psi^2_k) \}$ be a minimizing sequence for $\Ga$, i.e. a sequence of admissible quadruples satisfying ${\cal{G}}_\alpha(z_k,\phi_k,
\psi^1_k,\psi^2_k) \to \inf {\cal{G}}_\alpha \leq
{\cal{G}}_\alpha(0,0,0,0) < \infty$. Then, $\{ \Ga(z_k,\phi_k,
\psi^1_k,\psi^2_k) \}$ is a bounded sequence of real numbers. Therefore, $\{(z_k,\phi_k,\psi^1_k,\psi^2_k)\}$ is uniformly bounded in $\bv \times H^1(\Omega)\times \bv^2$. Thus, from the Sobolev Embedding Theorem [@Ada75; @EG92], we guarantee the existence of a subsequence (denoted again by $\{ (z_k, \phi_k,\psi^1_k,\psi^2_k)
\}$) and the existence of $(z,\phi,\psi^1,\psi^2) \in L^1(\Omega)
\times H^1(\Omega)\times \bv^2$ such that $\phi_k \to \phi$ in $L^2(\Omega)$, $\phi_k \rightharpoonup \phi$ in $H^1(\Omega)$, $z_k
\to z$ in $L^1(\Omega)$ and $\psi_k^j \to \psi^j \mbox{ in }
L^1(\Omega)$. Moreover, $z, \psi^1 \mbox{ and } \psi^2 \in \bv$. See [@EG92 Theorem 4, pp. 176].
From Lemma \[lemma:limit-ad\], we conclude that $(z,\phi,\psi^1,\psi^2)$ is an admissible quadruple. Moreover, from the weak lower semi-continuity of $R$ (Lemma \[lemma:rho-lsc\]), together with the continuity of $q$ (Lemma \[lemma:q-cont\]) and continuity of $F$ (see the general assumption), we obtain $$\begin{aligned}
\inf \ \Ga & = & \lim_{k \to \infty} {\cal{G}}_\alpha(z_k,
\phi_k,\psi^1_k,\psi^2_k) \ = \ \lim_{k \to \infty} \big\{
\|F(q(z_k,\psi^1_k,\psi^2_k))-y^\delta\|^2_Y
+ \alpha R(z_k,\phi_k,\psi^1_k,\psi^2_k) \big \} \nonumber \\
& \geq & \| F(q(z,\psi^1,\psi^2)) - y^\delta \|^2_Y + \alpha
R(z,\phi,\psi^1,\psi^2) \ = \ \Ga(z,\phi,\psi^1,\psi^2) \, ,
\label{eq:no}\end{aligned}$$ proving that $(z,\phi,\psi^1,\psi^2)$ minimizes $\Ga$.
In that follows, we shall denote a minimizer of $\Ga$ by $(z_\alpha,
\phi_\alpha,\psi^1_\alpha,\psi^2_\alpha)$. In particular the functional $\hat{\Ga}$ in attain a generalized minimizer in the sense of Definition \[def:quadruple\]. In the next theorem, we summarize some convergence results for the regularized minimizers. These results are based on the existence of a generalized *minimum norm solutions*.
\[def:R-min-norm-solution\] An admissible quadruple $(z^{\dag}, \phi^{\dag}, \psi^{1,\dag},
\psi^{2,\dag})$ is called a *$R$-minimizing solution* if satisfies $$\begin{aligned}
(i) \, & \, F( q(z^{\dag},\psi^{1,\dag},\psi^{2,\dag}) ) = y \, , \\
(ii)\, & \, R(z^{\dag}, \phi^{\dag},\psi^{1,\dag},\psi^{2,\dag}) =
\mbox{ms}
:= \inf \big\{ R(z, \phi,\psi^1,\psi^2) \, ;\
(z, \phi,\psi^1,\psi^2) \ \mbox{\rm is an} \\
& \hskip5cm \mbox{\rm admissible quadruple and} \
F(q(z,\psi^1,\psi^2)) = y \big\} \; .\end{aligned}$$
\[th:min-norm\] Under the general assumptions of this paper there exists a $R$-minimizing solution.
From the general assumption on this paper and Remark \[remark-ad\], we conclude that the set of admissible quadruple satisfying $F(q(z,\psi^1,\psi^2)) = y$ is not empty. Thus, $ms$ in ([*ii*]{}) is finite and there exists a sequence $\{(z_k,\phi_k,\psi^1_k,\psi^2_k)\}_{k\in{\mathbb{N}}}$ of admissible quadruple satisfying $$F(q(z_k,\psi^1_k,\psi^2_k)) = y \quad \mbox{and} \quad
R(z_k,\phi_k,\psi^1_k,\psi^2_k) \to ms < \infty \; .$$ Now, form the definition of $R$, it follows that the sequences $\{
\phi_k \}_{k\in{\mathbb{N}}}$, $\{ z_k \}_{k\in{\mathbb{N}}}$ and $\{ \psi^j_k
\}_{k\in{\mathbb{N}}}^{j=1,2}$ are uniformly bounded in $H^1(\Omega)$ and $\bv(\Omega)$, respectively. Then, from the Sobolev Compact Embedding Theorem [@Ada75; @EG92], we have (up to subsequences) that $$\phi_k \to \phi^{\dag} \ \mbox{ in } \ L^2(\Omega)\,, \quad z_k \to
z^{\dag} \ \mbox{ in } \ L^1(\Omega) \quad \mbox{ and } \quad
\psi^j_k \to \psi^{j,\dag} \ \mbox{ in } \ L^1(\Omega) \, , \ j =
1,2 \; .$$ Lemma \[lemma:limit-ad\] implies that $(z^{\dag},
\phi^{\dag},\psi^{1,\dag},\psi^{2,\dag})$ is an admissible quadruple. Since $R$ is weakly lower semi-continuous (cf. Lemma \[lemma:rho-lsc\]), it follows $$ms = \liminf_{k\to\infty} R(z_k,\phi_k,\psi^1_k,\psi^2_k)
\geq R(z^{\dag},\phi^{\dag},\psi^{1,\dag},\psi^{2,\dag}) \; .$$ Moreover, we conclude from Lemma \[lemma:q-cont\] that $$q(z^{\dag},\psi^{1,\dag},\psi^{1,\dag}) = \lim\limits_{k\to\infty}
q(z_k,\psi^1_k,\psi^2_k) \quad \mbox{ and } \quad F(
q(z^{\dag},\psi^{1,\dag},\psi^{2,\dag}) ) = \lim\limits_{k\to\infty}
F(q(z_k,\psi^1_k,\psi^2_k)) = y\,.$$ Thus, $(z^{\dag},\phi^{\dag},\psi^{1,\dag},\psi^{2,\dag})$ is a $R$- minimizing solution.
Using classical techniques from the analysis of Tikhonov regularization methods (see [@EngKunNeu89; @EngHanNeu96]), we present below the main convergence and stability theorems of this paper. The arguments in the proof are somewhat different of that presented in [@DCLT09; @DCLT2010]. But, for sake of completeness, we present the proof.
\[th:converg\] Assume that we have exact data, i.e. $y^\delta=y$. For every $\alpha
> 0$ let $(z_\alpha,\phi_\alpha,\psi^1_\alpha,\psi^2_\alpha)$ denote a minimizer of $\Ga$ on the set of admissible quadruples. Then, for every sequence of positive numbers $\{\alpha_k\}_{k\in{\mathbb{N}}}$ converging to zero there exists a subsequence, denoted again by $\{\alpha_k\}_{l\in{\mathbb{N}}}$, such that $(z_{\alpha_k},\phi_{\alpha_k},\psi^1_{\alpha_k},\psi^2_{\alpha_k})$ is strongly convergent in $L^1(\Omega) \times L^2(\Omega)\times
(L^1(\Omega))^2$. Moreover, the limit is a solution of .
Let $(z^{\dag}, \phi^{\dag},\psi^{1,\dag},\psi^{2,\dag})$ be a $R$-minimizing solution of – its existence is guaranteed by Theorem \[th:min-norm\]. Let $\{\alpha_k\}_{k\in{\mathbb{N}}}$ be a sequence of positive numbers converging to zero. For each $k
\in {\mathbb{N}}$, denote $(z_k,\phi_k,\psi^1_k,\psi^2_k) :=
(z_{\alpha_k},\phi_{\alpha_k}, \psi^1_{\alpha_k},\psi^2_{\alpha_k})$ be a minimizer of $G_{\alpha_k}$. Then, for each $k \in {\mathbb{N}}$, we have $$\label{eq:0}
G_{\alpha_k}(z_k,\phi_k,\psi^1_k,\psi^2_k)
\leq \big\| F(q(z^{\dag},\psi^{1,\dag},\psi^{2,\dag})) - y \big\|
+ \alpha_k R(z^{\dag}, \phi^{\dag},\psi^{1,\dag},\psi^{2,\dag})
= \alpha_k R(z^{\dag}, \phi^{\dag},\psi^{1,\dag},\psi^{2,\dag}) .$$ Since $\alpha_k R(z_k,,\phi_k,\psi^1_k,\psi^2_k) \le
G_{\alpha_k}(z_k, \phi_k,\psi^1_k,\psi^2_k)$, it follows from that $$\label{eq:1}
R(z_k,\phi_k,\psi^1_k,\psi^2_k) \ \leq \
R(z^{\dag},\phi^{\dag},\psi^{1,\dag},\psi^{2,\dag}) \ < \ \infty \;
.$$ Moreover, from the assumption on the sequence $\{ \alpha_k \}$, it follows that $$\label{eq:2}
\lim_{k\to\infty} \ \alpha_k R
(z^{\dag}, \phi^{\dag},\psi^{1,\dag},\psi^{1,\dag}) \ = \ 0 \; .$$ From and Lemma \[lemma:rho-coer\], we conclude that sequences $\{ \phi_k \}$, $\{ z_k \}$ and $\{\psi^j_k\}$ are bounded in $H^1(\Omega)$and $\bv$, respectively, for $j=1,2$. Using an argument of extraction of diagonal subsequences (see proof of Lemma \[lemma:limit-ad\]), we can guarantee the existence of an admissible quadruple $(\tilde{z},\tilde{\phi},
\tilde{\psi}^1,\tilde{\psi}^2)$ such that $$(z_k, \phi_k,\psi^1_k,\psi^2_k) \to (\tilde{z},\tilde{\phi},
\tilde{\psi}^1,\tilde{\psi}^2) \ \mbox{ in } \ L^1(\Omega) \times
L^2(\Omega)\times (L^1(\Omega))^2 \, .$$ Now, from Lemma \[lemma:q-cont\] (i), it follows that $q(\tilde{z},\tilde{\psi}^1,\tilde{\psi}^2) =
\lim\limits_{k\to\infty} q(z_k,\psi^1_k,\psi^2_k)$ in $L^1(\Omega)$. Using the continuity of the operator $F$ together with and , we conclude that $$y \ = \ \lim_{k\to\infty} F( q(z_k,\psi^1_k,\psi^2_k)) \ = \
F(q(\tilde{z},\tilde{\psi}^1,\tilde{\psi}^2)) \; .$$ On the other hand, from the lower semi-continuity of $R$ and it follows that $$R(\tilde{z}, \tilde{\phi},\tilde{\psi}^1,\tilde{\psi}^2) \ \leq \
\liminf_{k\to\infty} R(z_k,\phi_k,\psi^1_k,\psi^2_k) \ \leq \
\limsup_{k\to\infty} R(z_k,\phi_k,\psi^1_k,\psi^2_k)) \ \leq \
R(z^{\dag},\phi^{\dag},\tilde{\psi}^1,\tilde{\psi}^2) \, ,$$ concluding the proof.
\[th:stabil\] Let $\alpha = \alpha(\delta)$ be a function satisfying $\lim\limits_{\delta \to 0} \alpha (\delta)$ $= 0$ and $\lim\limits_{\delta \to 0} \delta^2 \alpha(\delta)^{-1} = 0$. Moreover, let $\{ \delta_k \}_{k\in{\mathbb{N}}}$ be a sequence of positive numbers converging to zero and $y^{\delta_k} \in Y$ be corresponding noisy data satisfying . Then, there exist a subsequence, denoted again by $\{ \delta_k \}$, and a sequence $\{
\alpha_k := \alpha(\delta_k) \}$ such that $(z_{\alpha_k},\phi_{\alpha_k},
\psi^1_{\alpha_k},\psi^2_{\alpha_k})$ converges in $L^1(\Omega)
\times L^2(\Omega)\times (L^1(\Omega))^2$ to solution of .
Let $(z^{\dag},\phi^{\dag}, \psi^{1,\dag},\psi^{1,\dag})$ be a $R$-minimizer solution of (such existence is guaranteed by Theorem \[th:min-norm\]). For each $k \in {\mathbb{N}}$, let $(z_k,\phi_k, \psi^1_k,\psi^2_k) :=
(z_{\alpha(\delta_k)},\phi_{\alpha(\delta_k)},
\psi^1_{\alpha(\delta_k)},\psi^2_{\alpha(\delta_k)})$ be a minimizer of $G_{\alpha(\delta_k)}$. Then, for each $k \in {\mathbb{N}}$ we have $$\begin{aligned}
\label{eq:0a}
G_{\alpha_k} (z_k,\phi_k,\psi^1_k,\psi^2_k)
& \leq \big\| F( q(z^{\dag},\psi^{1,\dag},\psi^{1,\dag}) ) - y^{\delta_k} \big\|_Y^2
+ \alpha(\delta_k) R(z^{\dag},\phi^{\dag},\psi^{1,\dag},\psi^{2,\dag})
\nonumber \\
& \leq \ \delta_k^2 + \alpha(\delta_k)
R(z^{\dag},\phi^{\dag}, \psi^{1,\dag},\psi^{2,\dag}) \, .\end{aligned}$$ From and the definition of $G_{\alpha_k}$, it follows that $$\label{eq:1a}
R(z_k,\phi_k,\psi^1_k,\psi^2_k) \ \leq \
\frac{\delta_k^2}{\alpha(\delta_k)} +
R(z^{\dag}, \phi^{\dag}, \psi^{1,\dag},\psi^{2,\dag}) \; .$$ Taking the limit as $k\to\infty$ in , it follows from theorem assumptions on $\alpha(\delta_k)$, that $$\lim_{k\to\infty} {{\left\lVert F( q(z_k,\psi^1_k,\psi^2_k)) - y^{\delta_k}
\right\rVert_{}}} \ \leq \ \lim_{k\to\infty} \left(\delta_k^2 + \alpha(\delta_k)
R(z^{\dag},\phi^{\dag}, \psi^{1,\dag},\psi^{2,\dag}) \right) \ = \ 0 \,
,$$ and $$\label{eq:1b}
\limsup_{k\to\infty} R(z_k,\phi_k,\psi^1_k,\psi^2_k) \ \leq \
R(z^{\dag},\phi^{\dag},\psi^{1,\dag},\psi^{2,\dag}) \; .$$ With the same arguments as in the proof of Theorem \[th:converg\], we conclude that, at least a subsequence that we denote again by $(z_k,\phi_k,\psi^1_k,\psi^2_k)$, converge in $L^1(\Omega)\times L^2(\Omega) \times (L^1(\Omega))^2$ to some admissible quadruple $(z,\phi,\psi^1,\psi^2)$. Moreover, by taking the limit as $k\to\infty$ in , it follows from the assumption on $F$ and Lemma \[lemma:q-cont\] that $$F(q(z,\phi,\psi^1,\psi^2))=\lim\limits_{k\to\infty}
F(q(z_k,\psi^1_k,\psi^2_k)) = y\,.$$
The functional $\Ga$ defined in is not easy to handled numerically, i.e., we are not able to derive a suitable optimality condition to the minimizers of $\Ga$. In the next section, we work in sight to surpass such difficulty.
Numerical Solution {#sec:num-sol}
==================
In this section, we introduce a functional which can be handled numerically, and whose minimizers are ’near’ to the minimizers of $\Ga$. Let $\Ger$ be the functional defined by $$\label{eq:m-reg}
\Ger(\phi,\psi^1,\psi^2) := \| F(P_\ve(\phi,\psi^1,\psi^2)) -
y^\delta \|_Y^2 + \alpha \big( \beta_1 |H_\ve(\phi)|_{\bv} + \beta_2
\| \phi - \phi_0 \|_{H^1}^2 + \beta_3 {\textstyle
\sum\limits_{j=1}^2|\psi^j-\psi^j_0|_\bv} \big) ,$$ where $P_\ve(\phi,\psi^1,\psi^2) := q(H_\ve(\phi), \psi^1,\psi^2)$ is defined in . The functional $\Ger$ is well-posed as the following lemma shows:
\[lemma:Gae\] Given positive constants $\alpha$, $\ve$, $\beta_j$ as in the general assumption of this article, $\phi_0 \in H^1(\Omega)$ and $\psi_0^j \in {\mathbb{B}}$, $j = 1,2$. Then, the functional $\Ger$ in attains a minimizer on $H^1(\Omega) \times
(\bv)^2$.
Since, $\inf \{\Ger(\phi,\psi^1,\psi^2) \, : \ (\phi,\psi^1,\psi^2)
\in H^1(\Omega) \times (\bv)^2 \} \leq \Ger(0,0,0) < \infty$, there exists a minimizing sequence $\{ (\phi_k,\psi^1_k,\psi^2_k) \}$ in $H^1(\Omega)\times {\mathbb{B}}^2$ satisfying $$\lim_{k\to\infty} \Ger(\phi_k, \psi^1_k,\psi^2_k) \ = \ \inf \{
\Ger(\phi,\psi^1,\psi^2) \; : \ (\phi,\psi^1,\psi^2) \in H^1(\Omega)
\times
{\mathbb{B}}^2 \} \, .$$ Then, for fixed $\alpha > 0$, the definition of $\Ger$ in implies that the sequences $\{ \phi_k \}$ and $\{
\psi^j_k\}^{j=1,2}$ are bounded in $H^1(\Omega)$ and $(\bv)^2$, respectively. Therefore, from Banach-Alaoglu-Bourbaki Theorem [@Yosida95] $\phi_k \rightharpoonup \phi$ in $H^1(\Omega)$ and from [@EG92 Theorem 4 pg. 176], $\psi_k^j \rightarrow \psi^j$ in $L^1(\Omega)$, $j=1,2$. Now, a similar argument as in Lemma \[lemma:auxil\] implies that $\psi^j\in {\mathbb{B}}$, for $j=1,2$. Moreover, by the weak lower semi-continuity of the $H^1$–norm [@Yosida95] and $|\cdot|_\bv$ measure (see [@EG92 Theorem 1 pg. 172]), it follows that $$\| \phi - \phi_0\|^2_{H^1} \leq
\liminf\limits_{k\to\infty} \| \phi_k - \phi_0 \|^2_{H^1} \, \mbox{
and }\; \ |\psi^j-\psi^j_0|_\bv \leq \liminf\limits_{k\to\infty} |
\psi^j_k - \psi^j_0 |_\bv\;.$$
The compact embedding of $H^1(\Omega)$ into $L^2(\Omega)$ [@Ada75] implies in the existence of a subsequence of $\{\phi_k\}$, (that we denote with the same index) such that $\phi_k
\to \phi$ in $L^2(\Omega)$. Follows from Lemma \[lemma:q-cont\] and [@EG92 Theorem 1, pg 172] that $| H_\ve(\phi) |_{\bv} \leq
\liminf\limits_{k\to\infty} | H_\ve(\phi_k) |_{\bv}$. Hence, from continuity of $F$ in $L^1$, continuity of $q$ (see Lemma \[lemma:q-cont\]), together with the estimates above, we conclude that $$\begin{aligned}
\Ger(\phi,\psi^1,\psi^2)
& \leq \lim_{k\to\infty} \| F( P_\ve(\phi_k, \psi^1_k,\psi^2_k)) - y^\delta \|_{Y}^2 \\
& \quad + \alpha \, \Big( \beta_1 \liminf_{k\to\infty} |H_\ve(\phi_k)|_{\bv}
+ \beta_2 \liminf_{k\to\infty} \|\phi_k - \phi_0\|_{H^1(\Omega)}^2
+ \beta_3 \liminf_{k\to\infty} {\textstyle\sum\limits_{j=1}^2 |\psi^j_k - \psi^j_0|_\bv}
\Big) \\
& \leq \liminf_{k\to\infty} \Ger(\phi_k, \psi^1_k,\psi^2_k) \ = \ \inf \Ger \, ,\end{aligned}$$ Therefore, $(\phi,\psi^1,\psi^2)$ is a minimizer of $\Ger$.
In the sequel, we prove that, when $\ve \to 0$, the minimizers of $\Ger$ approximate a minimizer of the functional $\Ga$. Hence, numerically, the minimizer of $\Ger$ can be used as a suitable approximation for the minimizers of $\Ga$.
\[th:just\] Let $\alpha$ and $\beta_j$ be given as in the general assumption of this article. For each $\ve > 0$, denote by $(\phi_{\ve,\alpha},
\psi^1_{\ve,\alpha}, \psi^2_{\ve,\alpha})$ a minimizer of $\Ger$ (that there exist form Lemma \[lemma:Gae\]). Then, there exists a sequence of positive numbers $\ve_k \to 0$ such that $(H_{\ve_k}(\phi_{{\ve_k},\alpha}), \phi_{{\ve_k},\alpha},
\psi^1_{{\ve_k},\alpha}, \psi^2_{{\ve_k},\alpha})$ converges strongly in $L^1(\Omega) \times \lzo \times (L^1(\Omega))^2$ and the limit minimizes $\Ga$ on the set of admissible quadruples.
Let $(z_\alpha,\phi_{\alpha}, \psi^1_{\alpha},\psi^2_{\alpha})$ be a minimizer of the functional $\Ga$ on the set of admissible quadruples (cf. Theorem \[th:admissible\]). From Definition \[def:quadruple\], there exists a sequence $\{ \ve_k
\}$ of positive numbers converging to zero and corresponding sequences $\{ \phi_k \}$ in $H^1(\Omega)$ satisfying $\phi_k \to
\phi_{\alpha}$ in $L^2(\Omega)$, $H_{\ve_k}(\phi_k) \to z_\alpha$ in $L^1(\Omega)$ and, finally, sequences $\{ \psi_k^j \}$ in $\bv\times
C_c^\infty(\Omega)$ such that $|\psi_k^j|_\bv \longrightarrow
|\psi^j|_\bv$ . Moreover, we can further assume (see Lemma \[lemma:R-auxl\]) that $$R(z_\alpha, \phi_{\alpha},\psi^1_{\alpha},\psi^2_{\alpha}) \ = \
\lim_{k\to\infty}
\big( \beta_1 |H_{\ve_k} (\phi_k)|_\bv + \beta_2 \|\phi_k -\phi_0\|^2_{H^1(\Omega)}
+ \beta_3 {\textstyle\sum\limits_{j=1}^2|\psi_k^j-\psi^j_0|_\bv} \big) \, .$$ Let $(\phi_{\ve_k},\psi^1_{\ve_k},\psi^2_{\ve_k})$ be a minimizer of ${\cal{G}}_{\ve_k,\alpha}$. Hence, $(\phi_{\ve_k},\psi^1_{\ve_k},\psi^2_{\ve_k})$ belongs to $H^1(\Omega)\times {\mathbb{B}}^2$ (see Lemma \[lemma:Gae\]). The sequences $\{ H_{\ve_k}(\phi_{\ve_k}) \}$,$\{ \phi_{\ve_k} \}$ and $\{\psi_{\ve_k}^j\}$ are uniformly bounded in $\bv(\Omega)$, $H^1(\Omega)$ and $\bv(\Omega)$, for $j=1,2$, respectively. Form compact embedding (see Theorems [@Ada75] and [@EG92 Theorem 4 pg. 176]), there exist convergent subsequences whose limits are denoted by $\tilde{z}$, $\tilde{\phi}$ and $\tilde{\psi}^j$ belong to $\bv(\Omega)$,$H^1(\Omega)$ and $\bv(\Omega)$, for $j=1,2$, respectively.
Summarizing, we have $\phi_{\ve_k} \to \tilde{\phi}$ in $L^2(\Omega)$, $H_{\ve_k}(\phi_{\ve_k}) \to \tilde{z}$ in $L^1(\Omega)$, and $\psi^j_{\ve_k} \to \tilde{\psi}^j$ in $L^1(\Omega)$, $j = 1,2$. Thus, $(\tilde{z},\tilde{\phi},
\tilde{\psi}^1,\tilde{\psi}^2) \in L^1(\Omega) \times H^1(\Omega)
\times \L^1(\Omega)$ is an admissible quadruple (cf. Lemma \[lemma:limit-ad\]).
From the definition of $R$, Lemma \[lemma:q-cont\] and the continuity of $F$, it follows that $$\begin{aligned}
& \| F(q(\tilde{z},\tilde{\psi}^1,\tilde{\psi}^2)) - y^\delta
\|^2_Y \ = \
\lim_{k\to\infty} \| F( P_{\ve_k}(\phi_{\ve_k},\psi^1_{\ve_k},\psi^2_{\ve_k}) )
- y^\delta \|_Y^2 \, , & \\
& R(\tilde{z},\tilde{\phi}, \tilde{\psi}^1,\tilde{\psi}^2) \ \leq \
\liminf_{k\to\infty} \big( \beta_1 | H_{\ve_k}(\phi_{\ve_k}) |_\bv +
\beta_2 \| \phi_{\ve_k}-\phi_0 \|^2_{H^1(\Omega)} +
\beta_3 {\textstyle\sum_{j=1}^2 |\psi^j_{\ve_k} - \psi_0^j|_\bv} \Big) \, . &\end{aligned}$$ Therefore, $$\begin{aligned}
\Ga(\tilde{z}, \tilde{\phi},\tilde{\psi}^1,\tilde{\psi}^2) & = &
\| F(q(\tilde{z},\tilde{\psi}^1,\tilde{\psi}^2)) - y^\delta \|_Y^2
+ \alpha R(\tilde{z}, \tilde{\phi},\tilde{\psi}^1,\tilde{\psi}^2) \\
& \le & \liminf_{k\to\infty} \ {\cal{G}}_{\ve_k,\alpha}
(\phi_{\ve_k}, \psi^1_{\ve_k},\psi^2_{\ve_k})
\ \le \ \liminf_{k\to\infty} {\cal{G}}_{\ve_k,\alpha}(\phi_k, \psi^1_k,\psi^2_k) \\
& \le & \limsup_{k\to\infty}
\| F( P_{\ve_k}(\phi_k, \psi^1_k,\psi^2_k) ) - y^\delta \|_Y^2 \\
& & + \ \alpha \limsup_{k\to\infty} \big( \beta_1 | H_{\ve_k}(\phi_k) |_\bv
+ \beta_2 \| \phi_k - \phi_0 \|^2_{H^1(\Omega)}
+ \beta_3 {\textstyle\sum_{j=1}^2 |\psi^j_k-\psi^j_0|_\bv} \big) \\
& = & \| F(q(z_\alpha, \psi^1_\alpha,\psi^2_\alpha)) - y^\delta \|_Y^2
+ \alpha R(z_\alpha, \phi_{\alpha},\psi^1_\alpha,\psi^2_\alpha)
\ = \ \Ga(z_\alpha, \phi^1_{\alpha},\psi^1_\alpha,\psi^2_\alpha) \, ,\end{aligned}$$ characterizing $(\tilde{z},
\tilde{\phi},\psi^1_\alpha,\psi^2_\alpha)$ as a minimizer of $\Ga$.
Optimality Conditions for the Stabilized Functional
---------------------------------------------------
For numerical purposes it is convenient to derive first order optimality conditions for minimizers of the functional $\Ga$. Since $P$ is a discontinuous operator, it is not possible. However, thanks to the Theorem \[th:stabil\], the minimizers of the stabilized functionals $\Ger$ can be used for approximate minimizers of the functional $\Ga$. Therefore, we consider $\Ger$ in ,with $Y$ a Hilbert space, and we look for the Gâteaux directional derivatives with respect to $\phi$ and the unknown $\psi^j$ for $j=1,2$.
Since $H'_\ve(\phi)$ is self-adjoint[^2], we can write the optimality conditions for the functional $\Ger$ in the form of the system
\[eq:formal0\] $$\begin{aligned}
& \alpha (\Delta-I)(\phi - \phi_0) \ = \
L_{\ve,\alpha,\beta}(\phi,\psi^1,\psi^2) \,,\qquad & \ {\rm in } \
\Omega \, \\
& (\phi - \phi_0) \cdot \nu \ = \ 0 \, , \qquad\qquad \quad \quad & \ {\rm at } \ \partial\Omega \\
& \alpha \, \nabla \cdot \big[\nabla(\psi^j-\psi^j_0)/|\nabla
(\psi^j-\psi^j_0)|\big] \ = \
L^j_{\ve,\alpha,\beta}(\phi,\psi^1,\psi^2) \, , & \ j=1,2 \, .\end{aligned}$$
Here $\nu(x)$ represents the external unit normal quadruple at $x
\in \partial\Omega$, and
\[eq:formal1\] $$\begin{aligned}
L_{\ve,\alpha,\beta}(\phi,\psi^1,\psi^2) & = &
(\psi^1-\psi^2)\beta_2^{-1}
H'_\ve(\phi)^* F'( P_\ve(\phi,\psi^1,\psi^2) )^*
( F(P_\ve(\phi,\psi^1,\psi^2)) - y^\delta ) \nonumber \\
& & \quad - \beta_1(2\beta_2)^{-1} H'_\ve(\phi) \,
\nabla \cdot \big[ \nabla H_\ve(\phi) / |\nabla H_\ve (\phi)| \big] \, , \\
L^1_{\ve,\alpha,\beta}(\phi,\psi^1,\psi^2) & = & (2\beta_3)^{-1}
\big( F'(P_\ve(\phi,\psi^1,\psi^2) )\, H_\ve(\phi) \big)^*
( F(P_\ve(\phi,\psi^1,\psi^2)) - y^\delta ) \\
L^2_{\ve,\alpha,\beta}(\phi,\psi^1,\psi^2) & = & (2\beta_3)^{-1}
\big( F'(P_\ve(\phi,\psi^1,\psi^2) )\, (1 - H_\ve(\phi)) \big)^*
( F(P_\ve(\phi,\psi^1,\psi^2)) - y^\delta ) \, .\end{aligned}$$
It is worth noticing that the derivation of is purely formal, since the $\bv$ seminorm is not differentiable. Moreover the terms $|\nabla H_\ve (\phi)|$ and $|\nabla
(\psi^j-\psi^j_0)|$ appearing in the denominators of and , respectively.
In Section \[sec:numeric\], system and is used as starting point for the derivation of a level set type method.
Inverse Elliptic Problems {#sec:numeric}
=========================
In this section, we discuss the proposed level set approach and their application in some physical problems model by elliptic PDE’s. We also discuss briefly the numerical implementations of the iterative method based on the level set approach. We remark that, in the case of noise data the iterative algorithm derived by the level set approach need an early stooping criteria [@EngHanNeu96].
The Inverse Potential Problem
-----------------------------
In this subsection, we apply the level set regularization framework in an inverse potential problem [@DAP11; @DCLT2010; @HR96]. Differently from [@DAP08; @DAP11; @DCLT2010; @DCLT09; @FSL05; @DA06; @DAL09], we assume that the source $u$ is not necessarily piecewise constant. For relevant applications of the inverse potential problem see [@HR96; @Isa90; @DAP11; @DAP08] and references therein.
The forward problem consists of solving the Poisson boundary value problem $$\label{eq:ipp}
-\nabla \cdot (\sigma \nabla w) \ = \ u \, ,\ {\rm in} \ \Omega \,
, \quad \gamma_1 w + \gamma_2 w_\nu \ = \ g \, \ {\rm on} \
\partial\Omega\, ,$$ on a given domain $\Omega \subset \mathbb R^n$ with $\partial
\Omega$ Lipschitz, for a given source function $u \in L^2(\Omega)$ and a boundary function $g \in L^2(\partial \Omega)$. In , $\nu$ represent the outer normal vector to $\partial \Omega$, $\sigma$ is a known sufficient smooth function. Note that, depending of $\gamma_1, \gamma_2 \in \{0,1\}$, we have Dirichlet, Neumann or Robin boundary condition. In this paper, we only consider the case of Dirichlet boundary condition, that corresponds to $\gamma_1 = 1$ and $\gamma_2 =0$ in . Therefore, it is well known that there exists a unique solution $w
\in H^1(\Omega)$ of with $w- g \in H_0^1(\Omega)$, [@DLions].
Assuming homogeneous Dirichlet boundary condition in , the problem can be modeled by the operator equation $$\begin{aligned}
\label{eq:operator-ipp}
F_{1}\, :\, L^2(\Omega) & \to L^2(\partial\Omega)\nonumber\\
\quad \quad \quad & u \longmapsto F_{1}(u) := w_{\nu}
|_{\partial\Omega}\,.\end{aligned}$$
The corresponding inverse problem consists in recover the $L^2$ source function $u$, from measurements of the Cauchy data of its corresponding potential $w$ on the boundary of $\Omega$.
Using this notation, the inverse potential problem can be written in the abbreviated form $F_1(u) = y^\delta$, where the available noisy data $y^\delta \in L^2(\partial\Omega)$ have the same meaning as in . It is worth noticing that this inverse problem has, in general, non unique solution [@HR96]. Therefore, we restrict our attention to minimum-norm solutions [@EngHanNeu96]. Sufficient conditions for identifiability are given in [@Isa90]. Moreover, we restrict our attention to solve the inverse problem in $D(F)$, i.e., we assume that the unknown parameter $u \in D(F)$, as defined in Section \[sec:min-concept\]. Note that, in this situation, the operator $F_1$ is linear. However, the inverse potential problem is well known to be exponentially ill-posed [@Isa90]. Therefore, the solution call for a regularization strategy [@EngHanNeu96; @HR96; @Isa90].
The following lemma implies that the operator $F_1$ satisfies the Assumption [**(A2)**]{}.
The operator $F_1\,:\, D(F) \subset L^1(\Omega) \longrightarrow
L^2(\partial \Omega)$ is continuous with the respect to the $L^1(\Omega)$ topology.
It is well known form the elliptic regularity theory [@DLions] that ${{\left\lVertw\right\rVert_{}}}_{H^1(\Omega)} \leq c_1 {{\left\lVertu\right\rVert_{}}}_{L^2(\Omega)}$. Let $u_n, u_0 \in D(F)$ and $w_n, w_0$ the respective solution of . Then, the linearity and continuity of the trace operator from $H^1(\Omega)$ to $L^2(\partial \Omega)$ [@DLions], we have that $${{\left\lVertF_1(u_n) - F_1(u_0)\right\rVert_{}}}_{L^2(\partial \Omega)} \leq C {{\left\lVertw_n -
w_0\right\rVert_{}}}_{H^1(\Omega)} \leq \tilde{C} {{\left\lVertu_n -
u_0\right\rVert_{}}}_{L^2(\Omega)}\leq \tilde{C_1} {{\left\lVertu_n -
u_0\right\rVert_{}}}_{L^1(\Omega)}\,,$$ where we use Lemma \[lemma:0\] to obtain the last inequality. Therefore, $F_1$ is sequentially continuous on the $L^1(\Omega)$ topology. Since $L^1(\Omega)$ is a metrizable spaces [@Yosida95], the proof is complete.
### A level set algorithm for the inverse potential problem
We propose an explicit iterative algorithm derived from the optimality conditions and for the Tikhonov functional $\Ger$.
For the inverse potential problem with Dirichlet boundary condition ($\gamma_1=1 \mbox{ and } \gamma_2=0$) the algorithm reads as:
Each step of this iterative method consists of three parts (see Table \[tab:sls\]): 1 - The residual $r_k \in L^2(\partial\Omega)$ of the iterate $(\phi_k, \psi_k^j)$ is evaluated (this requires solving one elliptic BVP of Dirichlet type); 2 - The $L^2$–solution $h_k$ of the adjoint problem for the residual is evaluated (this corresponds to solving one elliptic BVP of Dirichlet type); 3 - The update $\delta\phi_k$ for the level-set function and the updates $\delta \psi_k^j$ for the level values are evaluated (this corresponds to multiplying two functions).
In [@DAL09], a level set method was proposed, where the iteration is based on an inexact Newton type method. The inner iteration is implemented using the conjugate gradient method. Moreover, the regularization parameter $\alpha > 0$ is kept fixed. In contrast to [@DAL09], in Table \[tab:sls\], we define $\delta t = 1/\alpha$ (as a time increment) in order to derive an evolution equation for the levelset function. Therefore, we are looking for a fixed point equation related to the system of optimality conditions for the Tikhonov functional. Moreover, the iteration is based on a gradient type method as in [@DCLT2010].
The Inverse Problem in Nonlinear Electromagnetism
-------------------------------------------------
Many interesting physical problems are model by quasi-linear elliptic equations. Examples of applications include the identification of inhomogeneity inside nonlinear magnetic materials form indirect or local measurements. Electromagnetic non-destructive tests aim to localize cracks or inhomogeneities in the steel production process, where impurities can be described by a piecewise smooth function, [@CBGI09; @ChungChanTai05; @BurOsher05; @CVK10].
In this section, we assume that $D \subset \subset \Omega$ is measurable and $$\begin{aligned}
\label{eq:u}
u = \begin{cases}
\psi_1 \,, x \in D\,,\\
\psi_2 \,, x \in \Omega \ D\,,
\end{cases}\end{aligned}$$ where $\psi_1, \psi_2 \in {\mathbb{B}}$ and $m>0$.
The forward problem consists of solving the Poisson boundary value problem $$\label{eq:iep}
-\nabla \cdot (u \nabla w) \ = \ f \, ,\ {\rm in} \ \Omega \, ,
\quad w \ = \ g \, \ {\rm on} \
\partial\Omega\,,$$ where $\Omega \subset \mathbb R^n$ with $\partial \Omega$ Lipschitz, the source $f \in H^{-1}(\Omega)$ and boundary condition $g \in
H^{1/2}(\partial \Omega)$. It is well known that there exists a unique solution $w \in
H^1(\Omega)$ such that $w-g \in H_0^1(\Omega)$ for the PDE , [@DLions].
Assuming that during the production process the workpiece is contaminated by impurities and that such impurities are described by piecewise smooth function, the inverse electromagnetic problem consist in the identification and the localization of the inhomogeneities as well as the function values of the impurities. The localization of support and the tabulation of the inhomogeneities values can indicate possible sources of contamination in the magnetic material.
In other words, the inverse problem in electromagnetism consists in the identification of the support (shape) and the function values of $\psi^1, \psi^2$, of the coefficient function $u(x)$ defined in . The voltage potential $g$ is chosen such that its corresponding the current measurement $h := (w)_\nu
|_{\partial\Omega}$, available as a set of continuous measurement in $\partial \Omega$. This problem is known in the literature as the inverse problem for the Dirichlet-to-Neumann map [@Isa90].
With this framework, the problem can be modeled by the operator equation $$\begin{aligned}
\label{eq:operator-iep}
F_{2}\,:\, D(F) \subset L^1(\Omega) \to H^{1/2}(\partial\Omega)\nonumber\\
\quad \quad \quad u \longmapsto F_{2}(u) := w|_{\partial\Omega}\,,\end{aligned}$$ where the potential profile $g = w|_{\partial\Omega} \in
H^{1/2}(\Omega)$ is given.
The authors in [@CVK10] investigated a level set approach for solve an inverse problems of identification of inhomogeneities inside a nonlinear material, from local measurements of the magnetic induction. The assumption in [@CVK10] is that part of the inhomogeneities are given by a crack localized inside the workpiece and that outside the crack region, magnetic conductivities are nonlinear and they depends on the magnetic induction. In other words, that $\psi_1 = \mu_1$ and $\psi_2 = \mu_2(|\nabla w|^2)$, where $\mu_1$ is the (constant) air conductivity and $\mu_2=\mu_2(|\nabla w|^2)$ is a nonlinear conductivity of the workpiece material, whose values are assumed be known. In [@CVK10], they also present a successful iterative algorithm and numerical experiment. However, in [@CVK10], the measurements and therefore the data are given in the whole $\Omega$. Such amount of measurements are not reasonable in applications. Moreover, the proposed level set algorithm is based on an optimality condition of a least square functional with $H^1(\Omega)$-semi-norm regularization. Therefore, there is no guarantee of existence of minimum for the proposed functional.
\[remark:1\] Note that $F_2(u) = T_D(w)$, where $T_D$ is the Dirichlet trace operator. Moreover, since $T_D: H^1(\Omega) \to H^{1/2}(\partial
\Omega)$ is linear and continuous [@DLions], we have ${{\left\lVertT_D(w)\right\rVert_{}}}_{H^{1/2}(\partial \Omega)} \leq c
{{\left\lVertw\right\rVert_{}}}_{H^1(\Omega)}$.
In the following lemma, we prove that the operator $F_2$ satisfies the Assumption [**(A2)**]{}.
Let the operator $F_2\,:\, D(F) \subset L^1(\Omega) \longrightarrow
H^{1/2}(\partial \Omega)$ as defined in . Then, $F_2$ is continuous with the respect to the $L^1(\Omega)$ topology.
Let $u_n, u_0 \in D(F)$ and $w_n, w_0$ denoting the respective solution of . The linearity of equation implies that $w_n - w_0 \in H_0^1(\Omega)$ and it satisfies $$\begin{aligned}
\label{eq:wf}
\nabla \cdot ( u_n \nabla w_n) - \nabla \cdot ( u_0 \nabla w_0) =
0\,,\end{aligned}$$ with homogeneous boundary condition. Therefore, using the weak formulation for we have $$\begin{aligned}
\int_\Omega \left( \nabla \cdot ( u_n \nabla w_n) - \nabla \cdot (
u_0 \nabla w_0) \right) \varphi dx = 0\,, \quad \forall \varphi \in
H^1_0(\Omega)\,.\end{aligned}$$ In particular, the weak formulation holds true for $\varphi = w_n -
w_0$. From the Green formula [@DLions] and the assumption that $m>0$ (that guarantee elipticity of ), follows that $$\begin{aligned}
\label{eq:est1}
m {{\left\lVert\nabla w_n - \nabla w_0\right\rVert_{}}}^2_{L^2(\Omega)} \leq \int_\Omega u_n |\nabla w_n - \nabla w_0|^2 dx \leq \int_\Omega |(u_n - u_0)| |\nabla w_0| |(\nabla w_n - \nabla
w_0)| dx\,.\end{aligned}$$ From [@M63 Theorem 1], there exist $\varepsilon > 0$ (small enough) such that $ w_0 \in W^{1,p}(\Omega)$ for $p = 2+
\varepsilon$. Using the Hölder inequality [@DLions] with $1/p + 1/q = 1/2$ (note that $q> 2$ in the equation , follows that $$\begin{aligned}
\label{eq:a}
m {{\left\lVert\nabla w_n - \nabla w_0\right\rVert_{}}}^2_{L^2(\Omega)} \leq {{\left\lVertu_n -
u_0\right\rVert_{}}}_{L^q(\Omega)} {{\left\lVert\nabla w_0\right\rVert_{}}}_{L^p(\Omega)} {{\left\lVert\nabla w_n
- \nabla w_0\right\rVert_{}}}_{L^2(\Omega)}\,.\end{aligned}$$ Therefore, using the Poincaré inequality [@DLions] and equation , we have $$\begin{aligned}
{{\left\lVert w_n - w_0\right\rVert_{}}}_{H^1(\Omega)} \leq C {{\left\lVertu_n -
u_0\right\rVert_{}}}_{L^q(\Omega)}\,,\end{aligned}$$ where the constant $C$ depends only of $m, \Omega,{{\left\lVert\nabla w_0\right\rVert_{}}}$ and the Poincaré constant. Now, the assertion follows from Lemma \[lemma:0\] and Remark \[remark:1\].
### A level set algorithm for inverse problem in nonlinear electromagnetism
We propose an explicit iterative algorithm derived from the optimality conditions and for the Tikhonov functional $\Ger$. Each iteration of this algorithm consists in the following steps: In the first step the residual vector $r\in L^2(\partial\Omega)$ corresponding to the iterate $(\phi_n,\psi_n^1, \psi_n^2)$ is evaluated. This requires the solution of one elliptic BVP’s of Dirichlet type. In the second step the solutions $v \in H^1(\Omega)$ of the adjoint problems for the residual components $r$ are evaluated. This corresponds to solving one elliptic BVP of Neumann type and to computing the inner-product $ \nabla w \cdot \nabla v$ in $L^2(\Omega)$. Next, the computation of $L_{\varepsilon,\alpha, \beta}(\phi_n,\psi_n^1, \psi_n^2)$ and $L^j_{\varepsilon,\alpha, \beta}(\phi_n,\psi_n^1, \psi_n^2)$ as in . The four step is the updates of the level-set function $\delta\phi_n \in H^1(\Omega)$ and the level function values $\delta\psi_n^j \in \bv(\Omega)$ by solve .
The algorithm is summarized in Table \[tab:ls\].
Conclusions and Future Directions {#sec:conclusions}
=================================
In this article, we generalize the results of convergence and stability of the level set regularization approach proposed in [@DCLT09; @DCLT2010], where the level values of discontinuities are not piecewise constant inside of each region. We analyze the particular case, where the set $\Omega$ is divide in two regions. However, it is easy to extend the analysis for the case of multiple regions adapting the multiple level set approach in [@CLT08; @DCLT09].
We apply the level set framework for two problems: the inverse potential problem and in an inverse problem in nonlinear electromagnetism with piecewise non-constant solution. In both case, we prove that the parameter-to-solution map satisfies the Assumption **(A1)**. The inverse potential problem application is a natural generalization of the problem computed in [@CLT08; @DCLT2010; @DCLT09]. We also investigate the applicability of an inverse problem in nonlinear electromagnetism in the identification of inhomogeneities inside a nonlinear magnetic workpiece. Moreover, we propose iterative algorithm based on the optimality condition of the smooth Tikhonov functional $\Ger$.
A natural continuation of this paper is the numerical implementation. Level set numerical implementations for the inverse potential problem was done before in [@CLT08; @DCLT09; @DCLT2010], where the level values are assumed to be constant. Implementations of level set methods for resistivity/conductivity problem in elliptic equation have been intensively implemented recently e.g., [@ChungChanTai05; @DL09; @SLD06; @DA07; @DA06; @CVK10; @BurOsher05].
Acknowledgments {#acknowledgments .unnumbered}
===============
A.D.C. acknowledges the partial support from IMEF - FURG.
[^1]: Institute of Mathematics Statistics and Physics, Federal University of Rio Grande, Av. Italia km 8, 96201-900 Rio Grande, Brazil (<adrianocezaro@furg.br>).
[^2]: Note that $H'_\ve(t)=\begin{cases} \frac{1}{\ve} \,\,
t\in (-\ve,0)\\
0\,\, \mbox{ other else}\,.
\end{cases}$
|
---
abstract: 'Rumour is a collective emergent phenomenon with a potential for provoking a crisis. Modelling approaches have been deployed since five decades ago; however, the focus was mostly on epidemic behaviour of the rumours which does not take into account the differences of the agents. We use social practice theory to model agent decision making in organizational rumourmongering. Such an approach provides us with an opportunity to model rumourmongering agents with a layer of cognitive realism and study the impacts of various intervention strategies for prevention and control of rumours in organizations.'
author:
- Amir Ebrahimi Fard
- Rijk Mercuur
- Virginia Dignum
- 'Catholijn M. Jonker'
- Bartel van de Walle
bibliography:
- 'Mendeley\_Rijk-Amir\_ESSA2018.bib'
title: 'Towards Agent-based Models of Rumours in Organizations: A Social Practice Theory Approach[^1]'
---
Introduction
============
The phenomenon of rumourmongering has malicious impacts on societies. Rumours make people nervous, create stress, shake financial markets and disrupt aid operations [@Vosoughi2018]. In organizations, rumours lead to unpleasant consequences such as, breaking the workplace harmony, reduction of profit, drain of productivity and damaging the reputation of a company [@DiFonzo1994; @Michelson2000]. Recent work on the McDonald’s wormburger rumour and the P&G Satan rumour confirm the negative impact of rumours on the productivity of firms [@DiFonzo1994].
For 120 years, scholars from a wide range of disciplines are trying to understand different dimensions of this phenomenon. Research in rumour studies can be classified according to the approach followed: a case-based approach and a model-based approach. In the case-based approach, results are based on case studies, not on models, making it hard to generalize their conclusions. The model-based approach tries to explain the phenomenon of rumours by model-based based simulations. The model-based approaches, so far, focus only on the dynamic of the spread, while rumour is a collective phenomenon and the acts of individuals can influence the whole system. Rumours in organizations have been mainly approached with case-based studied and dynamic spreading model. To our knowledge there are no studies where the cognition of the individual is taken into account.
In our agent-based approach, we study the dynamics of the spread of rumours in organizations as an emergent (collective) behavior resulting from the behavior of individual agents using social practice theory. We use the proposed model to study the impact of change in organizational layout on control of organizational rumour.
The concept of social practices stems from sociology, and aims to depict our ‘doings and sayings’ [@Schatzki1996 p. 86], such as dining, commuting and rumourmongering. This paper uses the semantics of the social practice agent (SoPrA) model [@MercuurSoPrA] to gain insights in rumourmongering in organizations.[^2] SoPrA provides an unique tool to combine habitual behavior, social intelligence and interconnected practices in one model. This makes SoPrA especially well-suited for studying the spread of rumours in organizations as this practice is largely habitual, social [@Hackman1990] and interconnected with practices as working and moving around. To build the model with SoPrA, owing to lack of available empirical dataset, we give a proof-of-concept on how to collect data by doing eight semi-structured interviews.
This paper is organized as follows. The next section provides an overview of the research on rumours with an emphasis on studies of organizational rumour. Section \[context\] describes the context for our experiment, and the methodology of data collection and data preprocessing. The model is introduced in Section \[modelsection\]. One possible experiment is described in Section \[experiments\] and Section \[concl\] presents our conclusions, discussion and ideas for future work.
Background & Related Work {#sec:literature}
=========================
Rumours are unverified propositions or allegations which are not accompanied by corroborative evidence [@DiFonzo1994]. Rumours take different forms such as exaggerations, fabrications, explanations [@Prasad1934], wishes and fears [@Knapp1944]. Rumours have a lifecycle and change over the time. Allport and Postman in their seminal work “psychology of rumour” concluded that, “as a rumour travels, it grows shorter, more concise, more easily grasped and told.” [@Allport1965]. Buckner considers rumour a collective behaviour which is becoming more or less accurate while being passed on as they are subjected to the individuals’ interpretations which depends on the structure of the situation in which the rumour originates and spreads subsequently [@Buckner1965]. Rumours are conceived to be unpleasant phenomena that should be curtailed. Therefore, a number of strategies have been proposed to prevent and control them [@Buckner1965; @Oh2013; @Knopf1975].
One of the rumour contexts that has received attention from researchers for almost four decades is organizations. Like rumour in general context which is explained in above paragraph, rumour in organizations has different types and follow its own life-cycle [@DiFonzo1994; @DiFonzo1998; @Bordia2003; @Kimmel2004; @Bordia2006; @Bordia2014]. Also, to quell credible and non-credible organizational rumours, a number of different techniques and strategies have been suggested [@DiFonzo1994; @Kimmel2004]. The research approach also follow the same pattern, with a slight difference which to best of our knowledge is qualitative without adopting any modelling approach.
The related literature reported above are based on case studies or experiments in the wild. This pertains to the types of rumour, dynamics of rumour and strategies to control rumours, either in general or in organizational contexts. These case studies and experiments are to inform the construction of theories and models underlying the phenomenon of rumourmongering. Theories and models, in turn, should be tested in case studies and simulations. Model-based approaches do just that. However, the current state-of-the-art in model-based simulations of rumourmongering focus only on the dynamics of the rumourmongering, comparable to the epidemic modelling and spread of viruses [@Daley1964; @Zanette2002; @Nekovee2007; @Wang2017; @Turenne2018]. These models do not consider the complexities of the agents that participate in rumourmongering.
The research area of agent-based social simulations (ABSS) specializes on simulating the social phenomena as phenomena that emerge from the behaviour of individual agents. ABSS is a powerful tool for empirical research. It offers a natural environment for the study of connectionist phenomena in social science. This approach permits one to study how individual behaviour give rise to macroscopic phenomenon [@Epstein1999]. Such an approach is an ideal way to study the macro effects of various social practices, because it can capture routines which are practiced by individuals on a regular basis in micro level and see their collective influence in a macro level.
Domain {#context}
======
This research investigates the daily routine of rumourmongering in a faculty building on the campus of a Dutch University. In this faculty, students, researchers and staff work in offices with capacity of one to ten people. Aside from the actual work going on in the building, filling a bottle with water, getting coffee from the coffee machine, having lunch at the canteen and going to the toilet are among the most obvious practices that every employee in this faculty does on a daily basis.
Nevertheless, there are other daily routines in the organization which are not that obvious. One of these latent routines is rumourmongering. Rumours or unverified information are transferred between students, researchers and staffs on a daily basis, during lunch, while queuing for coffee, when seeing each other in the hallways, and when meeting in classrooms and offices. All these situations are potential contexts for casual talks and information communication without solid evidence.
For data collection we conducted semi-structured explorative interviews with people from the above-mentioned faculty. Semi-structured interviews allows us to ask questions that are specifically aimed at acquiring the content needed for the SoPrA model, while still giving the freedom to ask follow up questions on unclear answers. The data collection can be improved in future works by increasing the number of interviewees and diversifying them (Not only asking from students). For demographic information, the reader is referred to Table \[demographics\]. We prepared following question set to ask from each interviewee based on the meta-model which will be explained in the next section:
1. What are the essential competencies for rumourmongering?
2. What are the associated values with rumourmongering?
3. What kind of physical setting is associated with rumourmongering?
\[demographics\]
[|X|X|l|X|X|]{} **Number of Interviews** & **Number of Different Countries** & **Lowest Educational Level** & **Mean age** & **Female %**\
8 & 6 & MSc & 28 & 50\
Given the thin line between personality traits and competences, we used the Big Five model [@Goldberg1993] to differentiate between personality traits and competences. For Question 2, we asked the interviewees to choose the relevant values from Schwartz’s Basic Human Values model [@Schwartz2012]. We asked the same set of questions about fact-based talk.
We processed the collected data in two ways before using it in the model. Firstly, we clustered answers that point to the same concept. For example, in Question 3, interviewees gave answers such as cafeteria, coffee shop and cafe to point to a place where people can get together and drink coffee. In the coffee example, we clustered answers under the term of “coffee place”.
Secondly, we classified the answers to Question 2. As mentioned, for that question, we asked interviewees to pick associated values from Schwartz’s Basic Human Values model. We used the third abstraction level of the model which is more fine-grained and compared to other levels, and gave the interviewees a better idea of what they point to. However, a model based on level three, would not allow us to compare the agents effectively. Therefore, we decided to wrap the answers and classify them based on second abstraction level. Using a classification based on the first abstraction level would have been too homogeneous in the sense that the agents would behave too similar, which would loose the effectiveness of the simulation.
Model {#modelsection}
=====
The model has two main parts: (i) static part and (ii) dynamic part. In the static part, the components of the model and their properties are described, and in the dynamic part we explain the interaction of those components.
Static Part
-----------
This section describes the SoPrA meta-model which is used as the groundwork for our agent-based model, how we use empirical data to initiate the model, the model choices we make and how we tailor the model to the context of organization.
The SoPrA meta-model was introduced by @MercuurSoPrA and describes how the macro concept of social practices can be connected to micro level agent concepts. Figure \[fig:uml\] shows SoPrA in a UML-diagram. The main objects in a SoPrA model are activities (e.g., fact talk, rumourmongering), agents (e.g., PhD students, supervisors), competences (e.g., networking, listening), context elements (e.g., office, cafetaria) and values. Values here refer to human values as found by the earlier stated Schwartz model, such as, power or conformity. The social practice is an interconnection of (1) activities and (2) related associations as depicted by the grey box in Figure \[fig:uml\]. For example, the practice of talking consists of two possible activities fact talk or rumourmongering. The social practice connects these different activities with the `Implementation` association. If activity $A$ implements activity $B$ this means that $A$ is a way of or a part of doing $B$.
![The social practice meta-model captured in the Unified Modelling Language, including classes (yellow boxes), associations (lines), association classes (transparent boxes), navigability (arrow-ends) and multiplicity (numbers).[]{data-label="fig:uml"}](UMLSimplified){height="0.4\textheight"}
The `Implementation` association is the first of several associations that are related to an activity (see Table \[table:assocations\]). Most associations are fairly self-explanatory, however the `Trigger` and `Strategy` association are a bit more complex. Following @Wood2007, triggers are the basis for habitual behaviour. If an agent is near a context element that has a trigger association with an activity, then it will do that activity automatically (without for example considering its values). Following @Ostrom2007, strategies are related to norms and signify that something is the normal way to do something.. If an agent believes that activity $A$ is a strategy for activity $B$, then it believes that other agents usually implement activity $B$ by doing activity $A$.
The SoPrA meta-model does not only relate the activities to other classes, but the agent itself also has two types of associations: `HasCompetence` and `ValueAdherence` which plays a role in choosing the activities it will do:. The `HasCompetence` association links possible skills to the agent who masters those. The `ValueAdherence` association captures if an agent finds that value important.
[|l|X|]{} **Association** & **Specification**\
Implementation & which activities are a way of or a part of doing the activity\
Affordance & which context elements are needed to do the activity\
RequiredCompetence & which competences are needed to do the activity\
Knowledge & which activities an agent knows about\
Belief & which personal beliefs an agent has about the activity\
RelatedValue & which values are promoted or demoted by the activity\
Trigger & which context elements habitually start the activity\
Strategy & which activities usually implement the activity\
The model can be initiated using empirical data. Note that in this study we focussed on a small set of explorative interviews. We show with this initial data a proof-of-concept of how the model can be initiated. To properly ground the model a larger and more rigorous empirical study is necessary.
The activity class has three instances: talking, rumourmongering and fact talk. The number of instances of agent can vary in the different experiments (see Section \[experiments\]). The instances of the context element, competence and values class are based on the gathered data and can be found in Table \[table:rumourelements\] and \[table:facttalkelements\].[^3] The complete static model consists both of object instances and associations between these instances. An example focusing on one agent (i.e., `Bob`) and one activity (i.e., `rumourmongering`) is shown in Figure \[fig:umlinstance\]. Bob beliefs that the activity of rumourmongering is related to the value of privacy, curiosity and social power. He thinks it requires the competence of networking and noticing juicy details and thinks the activity is triggered (to some extent) by the hallway, restaurant and another agent named Alice. Furthermore, he himself has the competence of networking and adheres strongest to the value of ambition and weakest to the value of pleasure.
[| c | c | c |]{}\
**Context Elements** & **Meaning** & **Competence**\
Friend & Self-Direction &Sneaky Skills\
Coffee place & Power & Network Skills\
Hallway & Hedonism & Talking Skills\
Restaurant & Achievement & Observing Skills\
Office & Benevolence &\
Phone & &\
Computer & &\
[| c | c | c |]{}\
**Context Elements** & **Meaning** & **Competence**\
Colleague & Universalism & Being knowledgeable\
Academic Staff & Self-Direction & Listening Skills\
Office & Benevolence & Critical Thinking Skills\
Conference & Achievement & Communication Skills\
Meeting room & Tradition &\
Classroom & &\
Restaurant & &\
Phone & &\
Computer & &\
Pen & &\
Coffee & &\
![An instance of the SoPrA meta-model for the activity of rumourmongering and one agent. For illustration purposes the assocations related to the activity ’talking and the agent ’alice’ are omitted.[]{data-label="fig:umlinstance"}](UMLModelOneAgentRumour){width="0.9\paperwidth"}
The agents differ in which activity they associate with which element. In other words, the SoPrA meta-model does not initiate one social practice that all agents share, but one social practice *per agent*. The chance that an agent relates an activity to a competence is based on the empirical data we gathered in the interviews. For example, if 50% of the interviewees linked critical thinking skills to fact talk the chance an agent makes this association depends on a binomial distribution with $p=0.5$. For `relatedValue` association and `HabitualTrigger` association all agents make the associations as mentioned in Table \[table:rumourelements\] and \[table:facttalkelements\]. However, the weights differ per agent. The weights for the `relatedValue` association are picked from a normal distribution between $0$ and $1$. Given the lack of empirical data on the relation between activities and human values, we follow the related finding of the World Value Survey that people adhere to values with roughly a normal distribution [@worldvaluesurvey]. The weights for `HabitualTrigger` are picked on a logarithmic distribution based on the empirical work of [@Lally2010]. One interesting modelling choice we made was to drop the `Affordance` assocations in the conceptual model. The SoPrA meta-model conceptualizes two associations with context elements. The `HabitualTrigger` association representing that some context element can automatically lead to a reactive action and the `Affordance` association representing that some context elements are a pre-condition to enact a certain behaviour. None of our interviewees mentioned a possible context element that affords rumourmongering fact talk. As such this association seemed irrelevant for our model.
The associations related to the agents themselves are based on random distributions. Each competence has a 50% chance to be related to an agent. Each value is associated to each agent, but the weights differ. The weights for the `hasValue` association strength is based on a correlated normal distribution. @Schwartz2012 shows that the strength to which people adhere to values is correlated. For example, people who positively value universalism usually negatively value achievement. We use the correlations found by @Schwartz2012 to simulate intercorrelated normal distribution from which we pick the weights. In future work, we aim to extent our interviews to also gather data that can inform these weights.
For our modelling context, we need to extend the SoPrA model with a spatial component. We do this by adding two attributes to the `ContextElement` class called `x-coordinate` and `y-coordinate`. These coordinates can be used by the agent to sense which objects are near. Note that every agent is also a context element as indicated with the ’generalization’ association in the UML-diagram.
Dynamic Part
------------
This section describes the dynamic part of the model which on each tick comprises:
1. An agent decides on its location using the moving submodel and updates its coordinate attributes.
2. An agent decides if it will engage in fact talk or rumourmongering based on the choose-activity submodel.
The moving submodel has four components that agents can transfer between. As it is shown in Figure \[fig:movingDiagram\] the initial state is offices and from that state agents can leave their offices and pass the hallway to either have lunch at the restaurant or grab a cup of coffee at the coffee place. During the interviews, we discovered most of the people do those daily routines around the same period of time and only a few people do not follow this pattern and leave their offices out of usual time periods, so we concluded the transition of agents between different locations is a random phenomenon which follows a normal probability distribution.
![The moving model for agents[]{data-label="fig:movingDiagram"}](mvdiagram){width="\linewidth"}
The choose-activity submodel is based on @Mercuur2017a and has three stages. The submodel is depicted in Figure \[fig:choose-activity\]. The agent starts by considering both rumourmongering and fact talk. At each stage the agent makes a decision on one cognitive aspect. If this aspect is not conclusive it will prolong the decision to the next stage. In the first stage, the agent compares its own competences to the competences that it beliefs to be required for the activity. In our example model depicted in Figure \[fig:umlinstance\], Bob would decide it cannot do the activity of `rumourmongering`, because it requires a competence he does not have: noticing juicy details. As such, Bob will engage in fact talk. (Note that if Bob does not have the skill to do either activity, then the decision is also prolonged to the next stage.) In the second stage, an agent tries to make a decision based on its habits. It will survey its context and decide which context elements are near, i.e., resources, places or other agents. If it has a habitual trigger association with a particular strong strength between one of those context elements and either rumourmongering or fact talk it will automatically do that action. In the last stage, the agent will consider how strongly it relates certain values to both activities and how strongly it adheres itself to these values. Consequently, it makes a comparison between the two activities and decides which best suits its values. For the complete implementation of the habitual model and value model we refer to [@Mercuur2015].
![The choose-activity submodel and the three stages the agent uses to decide on its activity: competences, habits and values.[]{data-label="fig:choose-activity"}](DynamicModel){width="\linewidth"}
Experiment {#experiments}
==========
The proposed rumour model with elements associated with physical settings, individuals’ values and competencies enables us to investigate impacts of a variation of settings and interventions on the spread of rumours in organizations.
One of the open questions in organizational rumour literature is the effectiveness of different prevention and control strategies. In our approach we only need to extend the model with the specific elements and characteristics of the case that we would like to study. In this paper we study the effect of organizational layout on rumour dynamics. In our case, we take the size of offices and number of coffee places as the proxies for organizational layout and juxtapose two organizational layouts cases (Figure 4) to understand the impact of layout on rumourmongering dynamic.
To setup the model, we determine the number of agents, then initialize the context and agents. In the organization that we studied each section has on average 50 people, therefore, we pick 50 as the number of the agents. For context initialization, we design the layouts and assign agents to different locations, then we initialize agents with probability distributions for routines such as grab a cup of coffee or having lunch. After the model setup, it can be executed.
![(a) In this case, we study the impact of office size on dynamics of rumourmongering (b) In this case we study the impact of number of coffee places on the dynamics of rumourmongering[]{data-label="fig:lay"}](cases){width="\linewidth"}
Discussion & Future Research {#concl}
============================
Modelling rumourmongering has been studied since 1964. So far, the modelling did not consider the complexities of individual agents, and mostly focused on the spreading behaviour of the phenomenon. In the model proposed in this paper, agents have a cognitive layer that deploys social practice theory and views rumour as a routine with associated competencies, values and a physical setting.
In this research, we narrowed our study to the context of organization and after introducing the generic model, we tailored our model to the context of organization via empirical data collected though interviews conducted in a Dutch University. Based on explorative interviews we established that social practice theory are likely to be applicable as people shared a view on rumour, and their habits regarding rumour and rumours seem to be intertwined with other activities.
Our model can be used to study a wide range of topics in organizational rumour studies, in particular for testing the effectiveness of interventions for prevention and control of rumours in organizations.
Future work is to extend the questionnaire by asking about associations, conduct more and more rigorous interviews, implement the model and run the proposed experiments that explore different organization layouts. Furthermore, we aim to validate our model by looking at how rumours travel from person to person in the organization during a pre-selected time period.
Contributions & Acknowledgements {#contributions-acknowledgements .unnumbered}
================================
Fard & Mercuur wrote the first draft. Fard provided the domain knowledge and collected most data, whereas Mercuur provided the meta-model and methodological knowledge. Dignum, Jonker and van der Walle. supervised the process and contributed to the draft by providing comments, feedback and rewriting. This research was supported by the Engineering Social Technologies for a Responsible Digital Future project at TU Delft and ETH Zurich.
[^1]: This paper has been peer-reviewed and accepted for the Social Simulation Conference 2018 in Stockholm. The final authenticated version will be available online at Springer LNCS. The DOI will be provided when available.
[^2]: @MercuurSoPrA provides a static model of SoPrA based on literature and argued modelling choices. This paper applies this model to the domain and extends it by including competences and affordances and modelling a dynamic component based on [@Mercuur2015]. Note that @MercuurSoPrA is still under review and only available as pre-print at the moment of writing.
[^3]: The context-element ‘Friend’ and ‘Colleague’ are special cases; these are rather attributes of context-elements (i.e., agents) than context-elements themselves. In our model these are to some extent implicitly captured, because the agents who one sees most often (i.e., friends, colleagues) are mostly likely to be habitually associated with an action.
|
---
abstract: 'With increasing ubiquity of artificial intelligence (AI) in modern societies, individual countries and the international community are working hard to create an innovation-friendly, yet safe, regulatory environment. Adequate regulation is key to maximize the benefits and minimize the risks stemming from AI technologies. Developing regulatory frameworks is, however, challenging due to AI’s global reach and the existence of widespread misconceptions about the notion of regulation. We argue that AI-related challenges cannot be tackled effectively without sincere international coordination supported by robust, consistent domestic and international governance arrangements. Against this backdrop, we propose the establishment of an international AI governance framework organized around a new AI regulatory agency that — drawing on interdisciplinary expertise — could help creating uniform standards for the regulation of AI technologies and inform the development of AI policies around the world. We also believe that a fundamental change of mindset on what constitutes regulation is necessary to remove existing barriers that hamper contemporary efforts to develop AI regulatory regimes, and put forward some recommendations on how to achieve this, and what opportunities doing so would present.'
author:
- |
Olivia J. Erdélyi\
School of Law\
University of Canterbury\
Christchurch 8140, New Zealand\
olivia.erdelyi@canterbury.ac.nz
- |
Judy Goldsmith\
Department of Computer Science\
University of Kentucky\
Lexington, KY 40506, USA\
goldsmit@cs.uky.edu
bibliography:
- 'AIRegs.bib'
title: |
Regulating Artificial Intelligence\
Proposal for a Global Solution
---
Introduction {#Intro}
============
Emerging technologies commonly described by the generic term *artificial intelligence (AI)* are becoming increasingly pervasive in human society. They are extremely transformative, advance rapidly, and affect virtually all aspects of our existence: Self-driving cars are being released on the roads. AI-driven medical diagnosis tools sometimes outperform humans in catching rare diagnoses. Product recommendation systems analyze our needs and optimize our shopping experience. Automated surveillance techniques, killer bots, and other weaponized AI technologies shore up the defenses of our countries. Powerful data mining applications allow us to sift through a wealth of information within a short period of time. AI is revolutionizing financial services with applications reaching from detecting fraud, tax evasion, or money laundering to regulatory technology (RegTech), enhancing regulatory processes like monitoring, reporting, and compliance. The justice system increasingly relies on AI-enabled decision-making systems for predictive policing and sentencing. And the list of examples could go on.
Undoubtedly, some of these technologies can make life a lot easier by providing previously unimaginable benefits. But, given their highly disruptive nature, they also present substantial challenges. Sometimes these problems result from the imperfection of AI applications, as when AI systems produce discriminatory biases, or inappropriate inferences due to biases in training data [@cabitza2017unintended; @chouldechova2017fair]. At other times, issues arise because AI is doing its job far too perfectly, as evidenced by the increasing privacy threat posed by pattern recognition applications [@kosinski2017deep]. Some AI applications lead to human de-skilling [@cabitza2017unintended]. Some instantiations of AI are ethically questionable (e.g., child-like sex bots [@strikwerda]) or potentially dangerous (e.g., autonomous weapons systems). Less obviously, AI innovation may also raise broader systemic challenges in the economic [@AGG2019], legal [@EE2020], and many other domains, likely forcing us to reevaluate many of our most fundamental ethical, legal, and social paradigms. The overall destabilization of the international community through what is commonly referred to as the *AI race* — a dangerous competition for technological superiority between AI developers, countries, and regional groupings, encouraging safety and governance corner-cutting and potentially exacerbate existing or even create new conflict situations [@CO2018] — is yet another danger we face in relation to AI. For a good overview of the contemporary AI landscape and policy environment, see [@RC2017].
In order to optimally harness AI’s benefits and address its potential risks — preferably proactively rather than retroactively and in a manner beneficial to all humanity — it is indispensable to develop adequate policies in relation to AI technologies at the earliest possible stage. Society’s growing interest in and anxiety over AI — fueled by incessant hype and shocking scandals, respectively — are putting additional pressure on policymakers. The AI community has long been calling for policy action, with criticism getting louder on the growing legal vacuum in virtually every domain affected by technological advancement [@wadhwa2014laws; @EM2017; @TW2017]. Policymakers around the world have begun to address AI policy challenges. International organizations and groupings have put forward and/or are in the process of developing ethical principles, policy guidelines, and reports concerning AI to provide guidance and assist policymakers’ efforts to tackle AI challenges [@Asilomar2017; @GGAR2018; @EC2019; @OECD2019; @G202019; @UNESCO2019; @ECAIWP2020]. Countries have started to launch ambitious strategies to promote the development and commercialization of AI with a view to maintain sustained economic competitiveness after the inevitable global transition to an AI-driven economy [@JN2017; @RV2017; @OECDPSI; @OECDAIS; @FoLI2020; @ECAIW]. There are also many academic, joint public-private sector, and other venues that support governments in promoting AI R$\&$D. Examples include the International Association for Artificial Intelligence and Law, the Partnership on AI, The Future Society, the Artificial Intelligence Forum of New Zealand, and SPARC in the EU. More indirectly, cutting-edge tech firms like Amazon, Apple, Baidu, Google, Facebook, IBM, Microsoft, and others are also instrumental in shaping AI policies. Courts, faced with first AI-related disputes, contribute to clarifying situations, although some decisions reflect a complete lack of technological expertise [@CJEU:160/15; @LGH]. Such decisions add to, rather than decrease, confusion and underscore the need for interdisciplinary cooperation and improving policymakers’ AI expertise [@RC2017].
These diverse societal stakeholders enthusiastically engage in shaping our future with AI. Yet the question remains: Do these applications really make human society more efficient, better, or safer? Or is AI rather a looming menace that will ultimately destroy mankind [@SK2015]? In any case, both the AI revolution and the challenges it presents to society are very real and policymakers’ efforts to do something about it must continue.
The importance of *adequately* regulating AI cannot be overstated. History and economic research shows that societal benefits from technological innovation — including AI — cannot be taken for granted, but are largely determined by the quality of the market-structuring regulatory environment [@KS2019]: Appropriate regulation corrects market failures by incentivizing socially optimal behavior, ensuring that the benefits of innovation are equally distributed across society. Poorly designed or inappropriately implemented policy measures may, however, impair the status quo by aggravating inequalities and generating tensions between the winners and losers of innovation. In some (perhaps many) cases, some or all segments of society are unnecessarily disadvantaged by uses of AI-driven technology. Inadequate regulatory interventions and protracted periods of uncertainty during regulatory adjustments may also irreversibly destroy society’s trust in new technologies. This, in turn, may thwart their societal adoption or even annihilate entire emerging markets, withholding potentially substantial benefits from society [@TP2018].
Once an issue — such as the emergence of AI technologies — is constructed as a problem and the need for regulation is correctly identified, the next question is how we come up with a regulatory regime that stimulates trust, enables innovation yet also provides safety, and yields socially optimal outcomes? What factors impact on the efficiency of regulation? Addressing them all would not be possible in one paper (but see [@BCL2012] for a comprehensive treatise). Thus, we will concentrate on three interrelated issues that are paramount in the AI context: (1) Why the nature of lawmaking commands international coordination, (2) the importance of institutional architecture in determining the quality of regulation, and (3) practical communication challenges hindering contemporary AI policy development and related regulatory design questions.
Regarding the nature of lawmaking, a number of peculiarities should be considered so any newly conceived legal norms governing AI become truly authoritative, that is, accepted as legitimate and institutionalized. Otherwise we risk creating merely formal or symbolic rules without any impact on normative orientations and behavior [@H2008]. While regulatory initiatives are predominantly propelled by nation states to address problems at a domestic level, caution is advised with purely national approaches. Whenever the regulation of an issue has externalities that transcend national boundaries — as does AI regulation — differing domestic approaches tend to conflict, raising significant difficulties for those affected by more than one regime. An additional problem in such cases is that domestic policymakers’ ability and willingness to control the effects of their actions in foreign jurisdictions are limited. These problems are then typically perceived as transnational in scope, with the consequence that actors increasingly deem national rules inapt to provide suitable solutions. This discrepancy between the transnational nature of a problem and the national character of the law governing it creates pressures for transnational regulation. Aligning the scope of regulatory coordination with the reach of externalities has a number of benefits. First, it is the only way to effectively control those externalities. Second, it also facilitates welfare enhancing AI adoption instead of aggravating already pronounced worldwide inequalities, which, in turn, could help increase social and political support [@KS2019; @BCL2012]. Transnational legal ordering, however, is characterized by a set of complex, recursive, multi-directional processes, which follow their own logic and crucially affect norms’ authority [@HS2015]. Although the legitimacy of legal norms has predominantly social rather than moral roots, their acceptance also depends on the degree to which they conform to prevalent moral values of a given society and — in the transnational context — are able to bridge the gaps between conflicting morals of different cultures. Ethical considerations are thus an essential part of any regulatory endeavor.
On the matter of institutional architecture, generally, people immediately associate regulation with its most visible aspect: the actual rules produced by regulators. As the tools of regulation, rules are admittedly important. But such a narrow conception of regulation ignores that developing rules to address a given problem is a small segment of the full regulatory process — a series of tasks from detecting an anomaly and devising an adequate regulatory response to effective supervision, enforcement, as well as continuous assessment and adaptation of regulatory regimes to ensure optimal performance, all of which must seamlessly complement each other so that rules can be successful [@BCL2012]. Throughout this paper, when we speak of the efficiency or quality of *regulation*, or use the terms *regulatory regime* and *regulatory environment*, we intend to refer to this comprehensive regulatory process. One of those less palpable components of creating an enabling regulatory environment — for AI as much as any other issue — is *institutional architecture* (sometimes also referred to as *governance* framework or arrangements), which structures the collaboration of all parties involved in policymaking. In part because the significant workload associated with the urgency to issue tangible pieces of regulation typically exhausts regulators’ capacities, architectural design questions are often neglected or even overlooked. This is unfortunate, as they strongly affect the quality of both domestic and international AI regulation [@BCL2012; @AS2000]. We argue that it makes a huge difference whether international AI policy coordination occurs in an ad-hoc, voluntary manner, or is streamlined by robust, consistent national and international governance frameworks.
Turning to communication challenges and related regulatory design questions, another prevalent misconception in relation to regulation is that it is some necessary evil that the state imposes upon society to safeguard order and influence behavior in desired directions. Again, while not inherently wrong, this view overlooks the modern reality of regulation, which is much more a decentered, dynamic process of co-creation than a purely state-driven enterprise [@JB2001]. Due to the immense complexity of most modern regulatory domains, regulators generally lack both expertise and resources to face regulatory challenges by themselves. As a consequence, innovative, hybrid regulatory settings and strategies leveraging the collaboration and expertise of multiple societal stakeholders are typically superior to their traditional counterparts, and increasingly taking over as the default mode of regulation [@BCL2012]. From this follows that regulation presents vital opportunities for diverse societal stakeholders to instill their interests into regulatory processes early on. This improves the odds of creating rules and regulatory frameworks — which may be binding or not depending on parties’ preferences — that more adequately reflect aggregate collective preferences. We posit that this imperfect understanding of the notion of regulation is currently prevailing among stakeholders. It clouds the aforementioned opportunities from their view and evokes an overly cautious or even hostile attitude in them towards regulation, significantly impeding AI regulatory efforts worldwide. We believe that targeted emphasis on educating parties and clear communication about regulatory intentions, expectations, and opportunities could significantly alleviate these problems. In light of the fact that in AI regulation both expertise and resource problems are heightened due to the field’s complexity and rapid development, we urge for using these insights and see regulation as an opportunity rather than an obstacle.
Against this background, we hold the view that efforts to develop AI policies, should, from the very beginning, be coordinated and supported by adequate national, regional, and international governance frameworks to avoid risks and inefficiencies stemming from the imperfect interaction of fragmented domestic regulatory approaches. To date, such frameworks are missing. AI policies are developed with limited levels of coordination between governments and various academic and industry groupings. The regulatory purviews of agencies involved in policymaking processes are not clearly delineated, and discussions on issues of institutional architecture design are, if at all, in preliminary stages both within governments and across various regional and international fora. In the preliminary version of this paper [@EG2018], we proposed the establishment of a new intergovernmental organization — potentially named International Artificial Intelligence Organization (IAIO) — to serve as an international forum for discussion and engage in standard setting activities. Marchant and Wallach [@MW2018] cultivate a similar idea proposing to set up what they term governance coordination committees (GCCs) either at the national or international level, as appropriate, depending on the issue area addressed. We suggested the IAIO should unite a diverse group of stakeholders from public sector, industry, and academic organizations, whose interdisciplinary expertise can support policymakers in the overwhelming and crucially important task of regulating this novel, immensely complex, and largely uncharted area. Our hope was that such a wide-scale, in-depth cooperation among all interested stakeholders at this early stage would put national and international policymakers in the position to take proactive action instead of lagging behind technological innovation with potentially devastating implications. Yet, establishing a new body is not necessarily the best and surely not the only option to achieve those goals: Since then, a number of new initiatives led by existing and new groupings have emerged that could step up to assume this function or spearhead discussions considering the pros and cons of repurposing other existing bodies or establishing a new agency. The French-Canadian initiative to establish an International Panel on Artificial Intelligence (IPAI) — renamed Global Partnership on AI, (GPAI) — the Organisation for Economic Cooperation and Development’s (OECD) new AI Policy Observatory (OECD.AI), the Global Governance on AI Roundtable (GGAR) hosted by the World Government Summit (WGS), and a new work stream on AI within the United Nations Educational, Scientific and Cultural Organization (UNESCO) are among the most prominent examples. Thus, in this paper we will also consider how these new developments may tie in with our proposal.
The paper will proceed as follows: Section \[TLO\] will present a brief analysis of transnational normmaking processes. This will provide the foundations for our proposal — introduced in Section \[IAIO\] — to establish a global AI governance framework organized around either a new or an existing but repurposed intergovernmental organization as the lead global AI policy body. Section \[CRC\] will look into the above mentioned communication challenges that currently hamper regulatory efforts in the AI space and discuss related regulatory design issues in a hope to improve the situation. Finally, we summarize our thoughts in a short conclusion.
Dynamics of Transnational Lawmaking {#TLO}
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In response to economic and cultural globalization, legal, political science, and sociology scholarship have made many attempts to capture processes of various forms of transnational social ordering. Examples include the traditional, purely state-centric legal notion of *international law* with a dichotomous view towards national and international law; *global law*, which refers to legal norms of universal scope while also acknowledging the role of non-state actors in normmaking; *transnational law*, which can have several connotations in reference to norms with a more than national but less than global purview; the concept *regime theory* developed by international relations scholars, which is likewise state-centric and has a sole focus on international political processes without any regard to the impact of domestic politics or law’s normativity; the sociological *world polity theory*, which studies the diffusion of legal norms and assumes that global conceptual models frame national societies in one-dimensional top-down processes; and *transnational or global legal pluralism*, which emphasizes the coexistence of different legal orders and normative contestations among them. Giving a comprehensive overview (including references) of the respective merits and limitations of existing theories, Shaffer [@S2010] and Halliday and Shaffer [@HS2015] introduce a further, socio-legal notion termed *transnational legal order (TLO)*. It builds on these approaches and is defined as a social order of transnational scope consisting of ”a collection of formalized legal norms and associated organizations and actors that authoritatively order the understanding and practice of law across national jurisdictions.” We explain the determinants influencing transnational legal processes through which legal norms are constructed, conveyed, and institutionalized based on the concept of TLO, owing to its ability to highlight both the legal and institutional aspects justifying the proposed governance framework.
Disaggregating the above definition into two parts — (1) formalized legal norms produced solely or partially by some type of transnational legal organization or network, which are (2) aimed at inducing changes within nation-states — we will first give account of the bewildering variety of governance arrangements characterizing modern international relations, and then illuminate the complex ways in which legal norms interact and institutionalize. The terms *international* and *transnational* shall be used interchangeably, referring to norms and institutions spanning national boundaries (whether global or geographically more restricted in scope).
Turning to the first element of our TLO definition, both the norms and the institutions issuing them come in a diverse array of forms. Norms are contained in various formal texts of softer or harder legal character such as treaties, conventions, codes, model laws, standards, administrative rules and guidelines, and judicial judgments. International institutionalization displays a similar diversity featuring public intergovernmental (also called international) organizations (IGOs or IOs) and private non-governmental organizations (NGOs) of varying levels of formality [@JK2015; @SB2011; @VS2013]. The widespread use of both hard and soft legalization in international governance begs the questions of what *hard* and *soft law* are and what drives actors’ choices between disparate legal and institutional settings.
Note that the existing literature is divided on what constitutes hard and soft law. Some authors concentrate on legal rules’ binding quality either in binary terms or along a continuum between fully binding legal instruments and purely political, non-binding arrangements, while others focus on their ability to impact behavior. For a good overview see [@GM2010; @SP2010; @AS2000]. Because it includes both the legal norms and the institutional arrangements responsible for their development within the scope of its analysis, the most interesting definition for our purposes is the one adopted by Abbott and Snidal. They distinguish hard from soft law along three dimensions, namely (1) the extent of rules’ precision, (2) the degree of legal obligation they establish, and (3) whether or not they delegate authority to a third-party decision-maker for interpreting and implementing the law. Hard law refers to legally binding obligations that are either precise or can be made such by adjudication or further clarifying regulation, and that empower a third party to oversee their interpretation and enforcement. Soft law, on the other hand, embodies legal instruments that exhibit some degree of softness along any of these three dimensions.
Guzman and Meyer [@GM2010] and Abbott and Snidal [@AS2000] provide a very instructive comparison of the relative advantages and disadvantages of hard and soft legalization and the various factors that determine actors’ preferences towards different types of international governance arrangements.
Hard legalization is typically characterized by a coherent, established, and formalized institutional and procedural framework to ensure smooth implementation, elaboration, and enforcement of commitments. These arrangements are generally perceived as legitimate, resulting in a concomitant enhanced compliance-pull, and backed up by international law that provides international actors readily available mechanisms (e.g., for recognition or enforcement) to order their relations. The combination of these factors enhances the credibility of commitments by constraining opportunistic behavior and increasing the costs of reneging; reduces post-contracting transaction costs by restricting/constraining attempts to alter the status quo by way of frequent renegotiation, persuasion, or coercive behavior; allows parties to pursue political strategies through legal rather than political channels at low political cost; and solves problems of incomplete contracting by vesting an administrative or judicial institution with power for interpreting and clarifying rules intentionally left imprecise in anticipation of unforeseeable future contingencies. Yet, hard legalization comes at certain costs: it restricts actors’ behavioral freedom, entails potentially severe sovereignty implications, and is less effective in accommodating diversity or adapting to changing circumstances by reason of its relative rigidity.
Thus, in many instances, softer forms of legalization, which offer some of hard law’s perks yet alleviate its intrinsic disadvantages through their flexible, more or less informal cooperation mechanisms, may better serve parties’ purposes. By relaxing the level of formality along one or more of the dimensions precision, obligation, and delegation, soft legalization minimizes initial contracting costs and facilitates speedy conclusion of agreements. Bargaining problems become less pronounced, negotiation and drafting requires less scrutiny, and there is no need for potentially challenging approval and ratification processes. Thanks to soft legal commitments’ malleable cooperation frameworks, parties retain more control over the overall design and organization of their cooperation, incur lower sovereignty costs, and have an easily adjustable system at their disposal to deal with change and uncertainty. Soft law also has a way of evening out power asymmetries by securing and perpetuating powerful actors’ interests at lower sovereignty costs, while at the same time shielding the weak from their pressure. Furthermore, soft law is the only directly available instrument to non-state actors for ordering their interactions. Due to their conciliatory properties, softer forms of legalization leave actors time to acquire sufficient information and expertise to gradually test and develop solutions to problems, encouraging collective learning processes and ever deeper cooperation between them — benefits that plentifully compensate soft law’s central weakness: diminished compliance pull.
In conclusion, the choice between harder and softer types of legalization involves a context-dependent tradeoff, which actors should carefully consider on a case-by-case basis. Vabulas and Snidal [@VS2013] describe the pros and cons of institutional formality and the tradeoffs actors face when moving along a broad spectrum of intergovernmental organizational formality — especially between formal and informal intergovernmental institutions (FIGOs and IIGOs) — in an analogous fashion.
These three analyses show that, in general, actors opt for hard law/higher institutional formality when they (1) wish to enter into a binding commitment in issue areas subject to a high degree of consensus, because violations are hard to detect, or parties wish to signalize their intention to engage in sincere cooperation; (2) are willing to accept sovereignty costs stemming from delegating decision-making authority to a central body in order to establish stronger collective oversight over issue areas where the probability of violations is high and monitoring and enforcement is important; (3) put more value on collective control of information, for instance, to unveil violations and increase peer pressure to induce universal compliance; (4) aim for lower long-term transaction costs to effectively tackle recurring or clear-cut issues in standard operating procedures; (5) intend to set up a sophisticated centralized administration to provide legitimacy and stability for supporting complex work processes such as the design and elaboration of norms, coordination involving multiple parties, or judiciary and/or enforcement procedures; (6) are faced with the task of managing routine problems, which is more easily done with established administrative and implementing systems.
Conversely, soft law/lower institutional formality is preferable when actors (1) want to maintain flexibility to deal with uncertainty, distribution problems, diversity, and changing circumstances; (2) prefer to preserve state autonomy and avoid sovereignty intrusions because welfare gains of cooperation outweigh the potential for defection and opportunism so that agreements are self-enforcing once any focal point for discussions has been established, or when external effects elicited by domestic actions are negligible; (3) insist on avoiding formal transparency mechanisms to maintain closer control of information typically among a more homogeneous group; (4) need lower initial contracting costs to speed up negotiations to be able to act fast (e.g., in crisis situations) or because hard law is not available for lack of consensus; (5) find that minimalist administrative functions are sufficient to support their purposes; (6) must manage high uncertainty (e.g., in initial stages of cooperation or in new/complex issue area) and want to allow themselves time for coordination and establishing common ground without making strong commitments.
Sometimes soft law eventually paves the way towards harder forms of legalization and cooperation becomes increasingly formalized, but in many cases soft legalization and institutional informality have their own justification. In practice, both highly institutionalized FIGOs, such as the United Nations (UN) or World Trade Organization (WTO), IIGOs allowing for laxer cooperation, like the Basel Committee on Banking Supervision (BCBS), private NGOs, for instance the International Chamber of Commerce (ICC), and hybrid forms can be fairly successful and instrumental actors in international lawmaking.
Table \[T1\] gives an overview of the above outlined six tradeoffs actors have to weigh when choosing their desired level of legalization/institutional formalization.
Hard Law/High Institutional Formality Soft Law/Low Institutional Formality
--------------------------------------- --------------------------------------
binding commitment flexible cooperation arrangements
delegation/high sovereignty costs state autonomy/low sovereignty costs
collective control of information close control of information
low long-term transaction costs low initial contracting costs
complex centralized administration minimalist administrative functions
routine management crisis/uncertainty management
: Tradeoffs in legalization/institutional formality.[]{data-label="T1"}
Moving on to the second part of our TLO definition, transnational legal norms directly or indirectly pursue the ultimate goal to induce shifts in countries’ policies and individuals’ normative preferences through various formal or informal channels. This generates convoluted, recursive cycles of international lawmaking processes across diverse transnational and national fora, until norms eventually settle and institutionalize [@S2010]. Halliday and Shaffer [@HS2015] note that transnational norm-making may encounter difficulties in the following situations: First, actors may find themselves caught up in diagnostic struggles over the framing of problems, which favors particular alliances and antagonisms, supporting diagnoses reflecting the respective interests of these groupings. Second, domestic implementation of transnationally agreed rules is frequently thwarted and a new cycle of lawmaking is triggered by parties who are influential at the national level, but are not represented or are unsuccessful in international negotiations and therefore refuse to accept such norms as legitimate — a situation referred to as *actor mismatch*. Third, in their endeavor to reach widely accepted compromises, parties often resort to vague language or leave delicate issues unresolved in their agreements. The resulting ambiguity and built-in contradictions of transnational norms open avenues for nationally fragmented, likely conflicting implementation, again calling for further transnational lawmaking to eliminate related problems.
Inspired by Shaffer [@S2010], we now describe the recursive processes of international lawmaking, which encompass mutual interactions both vertically between transnational and domestic venues, as well as horizontally among various TLOs. Vertically, transnational norms impact states in a process referred to as *state change*. Their impact can encompass the whole or parts of the state (*location of change*), it may occur in a slow, progressive process or abruptly owing to unexpected circumstances (*timing of change*), and across five interrelated dimensions. The most obvious aspect of state change is the dynamic evolution of domestic legal systems elicited by the formal national enactment of transnational law. Formal enactment may or may not have a substantial effect on rules’ practical implementation depending on the extent to which the transnationally induced change is viewed as legitimate. In more subtle ways, however, these primary legal changes set much broader systemic transformations in motion with potentially heavy social repercussions. For one thing, they continuously reshape established governance models by altering the allocation of functions between the state, the market, and other forms of social ordering. At times, this prompts more state intervention, giving birth to new public and public-private hybrid agencies, while at other times it propels deregulatory tendencies resulting in a retreat of state administration and simultaneous engagement in self-regulation by the private sector. Moreover, transnational legal processes are often responsible for revamping states’ institutional architecture. They shift power between different branches of government and upset the division of responsibilities among existing state institutions, sometimes giving rise to new additions to the institutional landscape. It is not hard to see that domestic systems may starkly differ, and such fragmentation often entails devastating consequences in issue areas with cross-border effects. These legal, governance, and institutional changes directly affect individuals by reconfiguring markets for professional expertise, which, in turn, feeds back into the adaptation of governance models by, e.g., a move towards more technocratic forms of governance. This highlights an important, yet admittedly somewhat elusive point, namely that not only institutions but also individuals — acting as conduits facilitating the diffusion of transnational norms — play a crucial role in domestic and transnational lawmaking. The fifth domain of state change concerns the modifications in patterns of association and mechanisms of accountability across various national and international sites of governance. These shifts ultimately shape individuals’ legal culture and consciousness, as well as their expectations towards the state, triggering new processes of state change where these views conflict with the prevailing state of affairs.
The extent, location, and timing of state change hinges on three clusters of factors pertaining to the TLO’s nature, its relation to the receiving state, and the receiving state’s peculiarities. First, TLOs are generally better received if perceived legitimate, i.e., norms are adopted by respected actors with preferably similar interests, in a fair (especially non-coercive) process, and they effectively tackle designated target problems. Myriads of international and national, state and non-state actors interact in complementary or conflicting ways in shaping every aspect of transnational lawmaking. They seek to legitimize rules that serve their purposes and delegitimize those running against them. Powerful players typically dictate the outcome of such struggles. TLOs are more likely to have real behavioral impact if they consist of accepted, clear, and well-understood norms. As discussed above, binding hard law does not necessarily score better in this respect. In a large part, TLOs’ coherence is a function of the quality and quantity of their horizontal interaction, and can be threatened where significantly overlapping TLOs interact in an antagonistic rather than complementary fashion [@SP2010].
As far as TLOs’ relation to the receiving state is concerned, powerful actors sometimes resort to coercive measures to impose their will on weaker countries. However, because coercion irrevocably destroys norms’ legitimacy, changes forced on states in this manner are at best symbolic and short-lived before they are successfully blocked at the stage of domestic implementation. Another essential prerequisite for the sustainability of transnationally triggered change is the support of intermediaries, who link transnational and national lawmaking processes and are deeply familiar with the interests of both sides. Whether government representatives, industry specialists, academics, social movement leaders, or professionals employed with various public or private organizations on the national and international platform, these intermediaries are instrumental in coordinating communication, easing tensions, and conveying norms between the national and transnational levels.
Finally, the single most important condition for transnational legal norms’ national acceptance is their conformity with the target country’s existing cultural and institutional settings and pursued reform initiatives. It strongly depends on the receiving country’s prevailing power configurations, institutional capacities, path dependencies, and cultural disposition, and tends to decrease as the distance between the transnational and national contexts and interests and/or the extent of state change increases.
This concludes our analysis of transnational legal ordering, highlighting the main factors instrumental in determining transnational legal norms’ efficiency in influencing the behavior of states and their various constituencies. Our aim was to show that the choice of governance arrangements for an issue area in question — in our case AI — crucially determines the overall efficiency of the regulatory regime governing it. We have also seen that legal norms and lawmaking processes interact in complex ways between the transnational and national levels. This means we must strive to design AI governance frameworks that are consistent across these levels, ensuring that all actors affected by regulation are appropriately represented at some point in the process, and hence willing to accept any rules the frameworks produce. We now turn to our proposal on how an international AI governance framework could look like.
Proposal for a New International Artificial Intelligence Organization {#IAIO}
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International institutions are the prevalent vehicles of international cooperation in our interconnected world. When a critical mass of states and/or non-state actors feel that transnational cooperation is necessary to solve a problem that is impossible to tackle by isolated national measures, they establish a new IGO or NGO for that particular purpose. Based on legal and international relations definitions in circulation — see [@JK2015; @SB2011; @VS2013] — we define an IGO as a formal entity (1) established by an international agreement governed by international law; (2) with at least three (sometimes two) members — typically states but increasingly also IGOs; and (3) having at least one organ with a will distinct from that of its members. FIGOs’ organizational purpose is laid down in a binding international agreement such as a treaty or a formal legal act of another IGO, their membership is clearly defined in the founding legal act, and they have a permanent and significant institutionalization in place. By contrast, IIGOs operate based on an explicitly shared, but informal expectation about purpose, their membership is not always clear, as members are explicitly associated but only by non-legal mutual acknowledgment, and they do not possess any significant institutionalization. NGOs differ from IGOs in that they are not created by treaty — meaning they are governed by national rather than international law — and their membership is made up of non-state actors.
Given the severity and global nature of AI’s anticipated impact on humanity, we expect it to join the long line of issue areas requiring interstate cooperation, raising the question of establishing an IGO at some point in the future. Against this background, we propose the creation of the IAIO as a new IGO, which could initially serve as a focal point of policy debates on AI-related matters and — given sufficient international support — acquire increasing role in their regulation over time. We start by determining the degree of desired institutional formalization by examining, in turn, the six above elaborated tradeoffs in relation to AI.
*Binding commitment vs. flexible cooperation arrangements*: As pointed out earlier, AI will fundamentally transform human society worldwide. Since this process of transformation is likely to be inescapable for any single state, states will probably wish to cooperate sincerely. Also, violations will be difficult to detect as keeping pace with technological innovation will require considerable technical expertise and capacities, presumably exceeding especially weaker countries’ capabilities and evoking severe power asymmetries. While apart from this latter circumstance, these factors speak for hard legal commitments, it must be kept in mind that AI research and AI-human interactions are relatively young phenomena and their novelty severely restricts our ability to anticipate the spectrum and extent of the impending changes, let alone the dimension of the problems they will raise. Many AI instantiations encroach on our most basic rights, pose an existential threat, or bring up profound ethical and social questions, not to mention that they will utterly and completely upset our legal system. So, we are looking at heated debates among radically diverse parties over a variety of uncertain issues, which may change in rapid and currently unimaginable ways — conditions that, based on past experience, do not exactly favor international consensus. Therefore, we need all the flexibility we can get to acquire familiarity with the issues at hand, sort out differences, and establish common ground, before we can contemplate drawing up a more binding framework for cooperation.
*Delegation/high sovereignty costs vs. state autonomy/low sovereignty costs*: Weaponized AI technologies and certain data mining practices are clearly relevant for national security. As this is a sensitive issue area involving particularly high sovereignty costs, at least initially, states will show reluctance to give up and delegate decision-making authority to the IAIO. In the long run, however, powerful collective oversight and enforcement mechanisms will probably be indispensable in order to curb incentives for violations and opportunistic behavior, which should otherwise be high in light of the major shifts in international power constellations triggered by changes in countries’ competitive positions. Also, domestic AI policies will produce significant externalities, affecting other countries. Based on this analysis, it is hard to escape the conclusion that a highly institutionalized organization with binding legislative, dispute resolution, and enforcement authority would be better suited as new international AI regulator. Nevertheless, the political reality remains that until sufficient clarity is reached on the IAIO’s precise purpose, membership, the issues to regulate, and the broad directions to follow, international consensus supporting such a high degree of institutionalization is off the table.
*Collective control of information vs. close control of information*: History shows that states are generally cautious about sharing information on fate-changing technologies, which speaks for close control of information with respect to AI. However, if and when we manage to gather consensus for hard legal commitments (e.g., treaty on certain AI applications), we will probably need to be more forthcoming with certain information to ensure compliance with those instruments. This is again a strong argument in favor of starting cooperation on AI regulation in a softer institutional framework and using soft law instruments, although a move towards harder legalization seems to be desirable over time.
*Low long-term transaction costs vs. low initial contracting costs*: International discussions on AI are just beginning and powerful states will likely have divergent preferences with respect to the regulation of this high-impact field. Compounded with the difficulties discussed in the context of previous tradeoffs, this makes the prospect of reaching a workable international consensus in the short term rather remote. Yet crucially, swift regulatory response is imperative to prevent proliferating unregulated AI applications from causing social harm and to ensure that the opportunity presented by the rise of AI is harvested to humanity’s benefit rather than detriment — an aim best facilitated by lowering initial contracting costs with soft legalization and low institutional formalization. This is not to say that the idea of setting up a more robust governance framework with standard operating procedures should be abandoned. On the contrary, such a step has merit, but only at a later stage, in possession of sufficient expertise and political consensus to better assess the implications of various policy options and formulate informed policy recommendations.
*Complex centralized administration vs. minimalist administrative functions*: Similar considerations apply as far as the level of administrative sophistication of the IAIO is concerned. In the initial stage of determining the purpose of the organization, its membership, the issues that need to be regulated, and the backbone of its regulatory agenda, less is probably more. Later, with perhaps binding legal instruments governing selected aspects of AI for a wide membership, work will get more complex, requiring stronger oversight, dispute resolution, and enforcement mechanisms as well as more powerful bureaucratic functions to service them.
*Routine management vs. crisis/uncertainty management*: In view of AI’s novelty, extreme complexity, unforeseeable evolution, and the controversies it is expected to elicit among a very heterogeneous circle of members, we are up against managing an extraordinarily uncertain issue area. Consequently, we need time and soft legalization’s flexibility to establish commonly shared ideas, interests, cooperation mechanisms, and solutions, which can then form the basis of more formalized cooperation arrangements in the future.
In summary, at least initially, the IAIO should start out as an IIGO displaying a relatively low level of institutional formality. It should use soft law instruments, such as non-binding recommendations, guidelines, and standards, to support national policymakers in the conception and design of AI-related regulatory policies. Its interim goal should be to galvanize international cooperation, fostering internationally consistent AI policy approaches by directly engaging governments in this domain as early as possible, before states develop their own, diverging policies, which may be hard to rescind without political damage. Like many other key IGOs, the IAIO should be hosted by a neutral country to provide for a safe environment, limit avenues for political conflict, and build a climate of mutual tolerance and appreciation. Whether the international community wishes to move towards more formalized cooperation at some point in the future remains to be seen. Diverse institutional choices in other areas of international cooperation suggest that many different settings can be successful. Sometimes informality turns out to be the key to an organization’s success. This seems to be the case with the Bank of International Settlements especially during its initial years of operation and World War II, or the BCBS and the different G-Groups at present [@BT2006]. Another common trajectory is when initially informal arrangements turn into formal frameworks of cooperation. A case in point is the General Agreement on Tariffs and Trade’s (GATT) gradual transformation into the WTO [@AS2000]. Finally, there are examples for remarkably successful, sustained, complementary, and mutually beneficial cooperation between several organizations of varying institutional formality in the same issue area. This sort of relationship is characteristic for the IMF and various G-Groups in financial regulation, or the Australia Group (AG), an IIGO, and the Organization for the Prohibition of Chemical Weapons (OPCW), a FIGO, in the regulation of chemical and biological weapons [@VS2013]. So, even though the prospect of a formal global AI agency with regulatory and perhaps also conflict resolution powers is rather remote at the moment, higher or even full institutional formality might become the best option one day.
As mentioned in the Introduction, when we first came up with the idea to propose the establishment of a global AI governance framework, there was virtually no discussion — let alone one involving governments — addressing institutional architectural design questions. Instead, many newly-formed non-governmental stakeholders were just starting to work on selected high-priority topics in an uncoordinated manner. This led us to suggest the formation of a new intergovernmental organization, which we dubbed IAIO. However, since then work on AI has intensified in a number of existing organizations, and new actors and initiatives have also appeared on the horizon. This presents a new situation: Instead of a void, we now have a rudimentary institutional structure. Our goal should be to build on these foundations and find an institutional configuration that maximizes incentives for cooperation and minimizes competition between these new international AI policy actors. As long as one of them can take on the role we envisage for the IAIO — an issue we will now examine — it is counterproductive to add yet another institution to the existing landscape of potential AI regulators.
In February 2020, capitalizing on previous AI work done by the organization, the OECD has established a new AI Policy Observatory (OECD.AI) [@OECD.AI]. This inclusive platform for public policy on AI has the purpose to facilitate international dialogue and collaboration between a wide range of stakeholders representing governments, domestic and international regulators, the private sector, academia, the technical community, and civil society. It provides multidisciplinary, evidence-based policy analysis in areas most strongly affected by AI. Since its inception in 1961, the OECD has proven to be a successful global standard-setter in multiple public policy areas — the most recent case in point are the OECD AI Principles adopted in 2019 [@OECD2019], which constitute the first intergovernmental standard on AI. The organization’s global reach and multi-stakeholder approach facilitates the gradual development of widely accepted best practices by encouraging open, international dialogue, comparison of each other’s policy responses, and mutual learning. As an intergovernmental, yet informal forum for AI policymaking, OECD.AI provides the necessary flexibility that is required in the initial stages of global AI policy coordination. At the same time, it is backed by the OECD’s established, formal institutional arrangements, which — if the international community so desires in the future — may help it transition into a more formal vehicle of cooperation. Should the grouping have aspirations to become the undisputed focal point of global AI policymaking, it will have to find a way to officially represent the entire international community rather than just OECD countries. Ultimately, OECD.AI’s success and the trajectory of its evolution will depend on the global political climate.
According to the decision of its 40th General Conference in November 2019, UNESCO has also embarked on an ambitious two-year project with a view to draft the first global standards on AI ethics [@UNESCOAI; @UNESCOAHEG2020]. This mission statement seems narrower than that of OECD.AI at first sight. However, it has to be seen within UNESCO’s broader mandate to build peace through international cooperation in education, science, and culture, and contribute to achieving the Sustainable Development Goals adopted by the UN General Assembly in 2015 as part of the organization’s 2030 Agenda for Sustainable Development. Moreover, it has to be noted that UNESCO’s mode of operation is also characterized by a highly inclusive multi-stakeholder approach, engaging multifaceted, interdisciplinary expertise and powerful parties from around the globe. Again, political considerations will play a central role in determining the direction in which UNESCO’s AI work-stream will develop over time. In any case, this is another informal grouping embedded into and supported by a — this time truly global — FIGO, which could decide to broaden its mandate and become the sort of intergovernmental AI policy body we propose to put in place.
Another potentially interesting new initiative is the Global Governance on AI Roundtable (GGAR) [@GGAR]. The first two editions of the Roundtable in 2018 and 2019 have been hosted by the World Government Summit (WGS) held in Dubai. Likewise employing an international, interdisciplinary, and multi-stakeholder approach, this endeavor’s primary objective is to assist the UAE State Minister for AI in developing the UAE’s AI strategy, which was announced in 2017. However, by moving the venue of discussion away from developed western countries, which traditionally lead global dialogue in virtually all issue areas, GGAR offers a radically novel and different political context. This may resonate better with certain countries and provide an opportunity to bring parties with previously irreconcilable political positions on board. This hope is reflected in GGAR’s other key aim: To serve as a neutral forum that coordinates with a wide range of existing stakeholders vested in global AI policymaking, enabling the international community to shape globally accepted and culturally adaptable norms for AI governance. GGAR is a brand new informal grouping, which — if it awakens the international communities’ interest — could be morphed into a global intergovernmental AI agency and perhaps evolve into a more formal organization without being constrained by existing path dependencies.
In June 2018, France and Canada announced an initiative to establish an International Panel on Artificial Intelligence (IPAI), later renamed Global Partnership on AI (GPAI) [@MIPAI2018; @AG2019]. Once taking up work, the GPAI’s mission will be to support and guide responsible, human-centric AI adoption, respecting human rights and ensuring inclusion, diversity, innovation, and economic growth. It is also envisaged to rely on interdisciplinary, multi-stakeholder mechanisms to harness leading global expertise, and collaborate with other international AI policy bodies to facilitate AI research, information sharing, and the development of widely accepted, international best practices. At the present juncture, there is little clarity on the GPAI’s organizational structure, relationship to other existing AI policy actors, specific mandate, and political reception, so it is hard to predict if it could grow into a universally accepted, intergovernmental AI policymaking institution.
A last organization we would like to mention here is the World Economic Forum (WEF), in particular its Centre for the Fourth Industrial Revolution (4IR) launched in 2017 [@WEF]. The WEF is an NGO founded on the principle of public-private cooperation, which also relies on a global network of expertise, partnering with a wide range of stakeholders from both the public and private sectors to contribute to shaping local, regional, and global policy agendas. 4IR is intended to function as a hub for global, multi-stakeholder cooperation that takes the lead in co-designing and piloting innovative policy frameworks and governance protocols related to emerging technologies, including but not limited to AI. The interesting thing about the WEF is that as a non-governmental entity, it is led by the private sector and only involves governments indirectly. Yet it essentially works with the same methods as the other intergovernmental initiatives listed above. What is more, by focusing on well delineated pilot projects implemented jointly with selected government partners, it has substantial real impact on AI policies and industry practices at the national, regional, and global levels. While as an NGO, the 4IR cannot directly assume the role of a global intergovernmental AI regulatory body, it is definitely an actor with major influence that needs to be taken into account in the design of the suggested global AI governance framework.
There are also several other intergovernmental bodies that do valuable work on AI and are instrumental in steering global AI policies. Examples include the European Commission [@ECworkAI; @ECAIW], the Council of Europe [@CAHAI], and the G7 and G20 groups. However, as European institutions, the first two represent the EU rather than the international community. As for the informal G-Groups, these have a broader focus than just AI and there are good reasons to preserve them in their current roles as quickly mobilizable, flexible vehicles that devise informal solutions supporting the work of FIGOs in various policy domains. Hence, neither of them constitute adequate fora to serve the purpose we envisage for the IAIO. That said, these are also highly influential stakeholders that need to be involved in any international AI governance arrangements.
To sum up, OECD.AI, UNESCO’s AI group, GGAR, and the GPAI are potentially viable vehicles to take on the role of an intergovernmental AI regulatory agency as an alternative to setting up an entirely new organization in the form of the IAIO. This ends our excursus in the domain of international lawmaking, which also shows that beyond the optimal level of legalization and institutional formality, the proposed new intergovernmental AI policy body — be it one of the above introduced new players or a newly established IAIO — must fulfill a number of more subtle requirements to be perceived as a *fair and legitimate* regulator. While leaving the elaboration of details to political decisions and future research, we would like to stress three points: (1) It is necessary to put an IGO in charge of leading international AI regulatory efforts to ensure sufficient government involvement and impact on domestic AI policies, as well as to keep the option open to move towards more formal institutional arrangements. (2) We must maintain an inclusive, interdisciplinary, and multi-stakeholder approach in all aspects global AI policy design, including the initial deliberations related to the IAIO’s or its equivalent’s establishment, modus operandi, and regulatory agenda. This is the only way to ensure the availability of expertise necessary to effectively tackle AI-related challenges. (3) Finally, it is paramount to choose an institutional setting that adequately reflects all relevant AI actors’ interests and existing power constellations to guarantee the widespread acceptance and legitimacy of the proposed global AI governance framework. There is already some collaboration between the policy bodies and other stakeholders introduced in this paper. However, the status quo is much less effective compared to what would be possible if a well-designed and universally accepted global AI governance framework organized around a single agency was introduced. Such a framework could streamline all these stakeholders’ efforts ensuring an appropriate division of labor by optimally exploiting synergies in a clear and transparent manner. We are not proposing a complete overhaul of the current system, merely to eliminate inefficiencies by introducing an institutional architecture that maximizes collaboration and minimizes the duplication of tasks and competition between all actors involved. As mentioned earlier, the quality of a governance framework crucially determines the efficiency of the rules it produces: We cannot have efficient and widely accepted transnational AI norms if the adequacy and legitimacy of the governance framework producing them is put into question. Hence, in an ideal world, figuring out the right governance arrangements logically precedes the creation of any rules.
Communication: An Insidious Regulatory Challenge {#CRC}
================================================
Like all emerging technologies, AI’s successful societal adoption hinges on trust, which, in turn, flows from an agile, transparent, and sustainable regulatory environment. As we have seen, governance arrangements are just a part of that, and this paper has only provided some preliminary thoughts on the international setting. Ideally, the proposed global AI governance framework needs to be complemented by robust national AI regulatory regimes, which duly represent national interests and reflect the domestic stakeholder landscape. We respect that — being shaped by each country’s unique political situation as well as cultural and other path dependencies — national regimes will inevitably display differences. Still, some international coordination in setting up domestic regimes is desirable to prevent discrepancies which may lead to clashes between countries. Leaving these questions aside for now, we would like to direct attention to two instances of what we believe to be ultimately communication challenges: (1) issues around the general perception of the notion of regulation, and (2) interdisciplinary and inter-stakeholder communication barriers. Both are less obvious but highly pernicious problems, which currently heavily stifle efforts to establish AI regulatory regimes at all levels. Thus, both the international community and individual countries may find these insights helpful when designing the fundamental elements of their respective regulatory regimes.
As noted in the Introduction, people usually think of regulation as binding rules forced on society by the state either in the form of purely domestic regulatory measures or implementing international legal commitments. Especially businesses see regulation as an obstacle designed by the state — the *enemy* — to restrict their activities, which they consequently somehow have to work around. Admittedly, there is some truth in this view, as regulation ideally aims to incentivize socially rather than individually optimal behavior, and hence unavoidably restrains activities that threaten to decrease society’s welfare. However — apart from the aforementioned fact that creating rules is just a small part of regulation — this conception raises at least two additional problems by assuming an adversarial relationship between the state, businesses, and civil society.
First, it does not necessarily reflect the genuine preferences of these three stakeholder groups — which may or may not contradict depending on the particular scenario in question. Take AI innovation for instance. At first sight, innovators’ interests misalign with those of the state and civil society when it comes to safety considerations. The former may see an opportunity for cost saving by engaging in corner-cuttings, while the latter definitely value safety very highly. Yet this only holds true on the short run, if at all, as safety issues destroy society’s trust and hence markets in AI technologies. The resulting situation is against everybody’s interests and potentially even welfare decreasing: Innovators are no longer able to derive any financial gains from developing AI, while state and society are deprived from any benefits of innovation.
Second, it is not in line with modern regulatory reality [@BCL2012; @JB2001]. Regulation is no longer a responsibility reserved solely to the state but a decentered process of co-creation, involving multiple stakeholders. The main driver behind this regulatory paradigm shift is the increasing complexity and rapid pace of change of modern regulatory domains, which makes them a prohibitively big challenge for the state — or any other stakeholder for that matter — to tackle alone. The recognition that no single party has the knowledge, power, or capacity to effectively control and regulate all segments of society has led modern regulatory theory to move away from the state’s regulatory monopoly and advocate a shared responsibility of different actors for regulation instead. According to current regulatory best practices, regulation should be a series of convoluted multi-stakeholder interactions, in which autonomous social actors and government stakeholders are mutually interdependent co-producers of regulation, jointly constructing knowledge and exercising power. The core aim is to create dynamically adaptive patterns of interaction between a multitude of regulatory actors and strategies that best serve the public interest. In widespread opinion, hybrid regulatory mixes and networks deliver the best results in today’s globalized and interconnected world. Yet, finding the right blend of institutions and instruments remains challenging. This fundamentally different understanding of regulation dictates the use of diverse regulatory strategies — incorporating both more state-driven and self-regulatory elements — and coordination between multiple regulatory actors drawing on wide interdisciplinary and multi-stakeholder expertise. So yes, regulation will always remain state-driven to a certain extent. But it is also an indirect, flexible, and sensitive process of steering, coordinating, balancing, and influencing that not only gives affected societal stakeholders an opportunity to stand up for their interests but cannot be efficiently done without their participation.
Unfortunately, all this is easier in theory than in practice. It is hard enough to build a regulatory regime that is sufficiently inclusive to ensure that all stakeholders’ interests are duly taken into account, really putting public interest first. The trickiest part, however, is to maintain high levels of engagement, satisfaction, and performance in the face of changing conditions. A potential danger stakeholders currently active in AI policymaking may encounter is loss of drive. In the past few years, a large number of public and private bodies around the world have engaged individuals of diverse background into developing solutions to various AI-related challenges. However, lacking clear objectives and adequate coordination within and across such groupings, very little of that energy has been actually transformed into concrete, implementable actions. Low productivity levels have already led to a tangible decline of enthusiasm among participants in some venues, and we expect this trend to continue, especially as the AI-hype continues to wane. This is undesirable, seeing as governments depend on these stakeholders both in terms of expertise and regulatory capacity, and it is also in the latter’s best interest to work with governments to implant their preferences into policy initiatives. Hence, there is a sense of urgency in developing regulatory regimes — perhaps also relying on self-regulatory organizations or otherwise incorporating self-regulatory features — that set well-defined regulatory objectives and more efficiently coordinate all stakeholders involved.
To link back to the point of communication challenges raised earlier: In our experience, the true nature of regulation — including the above explained relationship of co-dependence between different regulatory actors and the benefits that stem from participating in regulatory processes — are generally not well understood or met by strong skepticism by stakeholders involved in AI policymaking across various fora. Our recommendations to alleviate these problems are twofold: (1) Acknowledging those national and transnational AI policy actors that already comply with modern regulatory best practices, we urge those not yet on this path to embrace and apply these insights in practice when designing their respective AI regulatory regimes. (2) Governments and international policymaking bodies should approach actors they wish to involve in regulatory and policymaking processes with a clear and realistically implementable agenda. They should unambiguously communicate — better yet, educate — them about the nature of regulation, current regulatory best practices, and their intention to follow them. We need to elicit a change of mindset about regulation — not state-imposed restrictions but an opportunity to co-create regimes and rules that serve aggregate collective preferences — and give stakeholders reason to trust that these new expectations will be fulfilled in practice.
This leads us to the last point we would like to tackle in this paper, namely interdisciplinary and inter-stakeholder communication barriers. As explained above, all modern regulatory domains have their fair share of complexity. This holds all the more true for AI regulation, as we are dealing with a diverse set of unusually fast-developing technologies, which penetrate virtually all domains of human existence with far-reaching consequences. To make matters worse, due to their complexity, the workings of AI technologies are very hardly accessible for individuals without some technical background. Unfortunately, the vast majority of people — including numerous policymakers and other stakeholders involved in regulatory and policymaking processes — grapple with this problem. Owing to the technological intricacies and AI’s widespread societal impacts on a global scale, developing sound regulatory approaches requires deep understanding of a wide spectrum of multidisciplinary concepts and internationally concerted, collaborative efforts between multiple stakeholders: policymakers, the AI and other affected industries, academic institutions, and civil society.
As previously noted, many AI policy actors are aware of these problems and aim to gather the right bundle of expertise to the table. So at least on paper, we seem to be doing just fine. The problem is that the interests, ways of thinking, and modus operandi of different disciplines and stakeholder groups starkly deviate, inducing massive coordination and communication challenges, not to mention frustration. An illustrative example for such tensions are circles of frustration between government, industry, and academic stakeholders. As the drivers of innovation and economic growth, businesses come up with cutting-edge solutions and products to harness AI’s benefits. They are much more flexible and faster than academia or the public sector, but do not necessarily have public interest at their heart. Some also lead the way in R$\&$D to underpin their business activity, but many operate on less sound theoretic foundations. As a consequence, businesses tend to be annoyed by governments’ lack of expertise, and the slow pace with which both governments and academia operate. Academic stakeholders excel in research, heavily contributing to developing the theoretical foundations that allow for introducing new technologies into society. Often, however, businesses and governments are not aware of existing research results that would solve problems they wrestle with — a fact of life that understandably upsets academics. Governments, in turn, ideally serve public interest, lack funding and resources comparable to the private sector, and are often also short on expertise. Hence, they are frequently overwhelmed when it comes to assessing the risks and benefits of new technologies, deciding the fate of those technologies, and keeping pace with industry.
As regards interdisciplinary communication, knowledge transfer between disciplines is far beneath the desired levels due to people’s inability to find a way to explain and understand each others’ problems and needs. This is counterproductive, as their pieces of knowledge are complementary and cumulatively necessary to successfully tackle AI-related challenges. In our view, a currently severe problem is that many research contributions — e.g., dealing with various societal impacts of AI — and policy decisions are made without due consultation of technical experts, even though the researchers or decision-makers obviously lack the necessary technical background to make meaningful contributions to the intersection of AI and their respective fields or to make informed policy decisions. This not only upsets technically literate individuals, but also results in incorrect and technically infeasible scientific recommendations and polices, both of which are very problematic. The former provide flawed foundations for further research and policy action, and also confuse technically illiterate readers. The latter promote the mis-assessment of AI, providing wrong behavioral incentives and tricking society into believing themselves to be protected from potential negative effects of these technologies.
These problems adversely affect the levels of engagement in regulatory processes and in societal dialogues on AI, society’s trust in regulator’s ability to design and maintain adequate AI regulatory regimes, and ultimately the acceptance and legitimacy of emerging regimes and norms governing AI. We need to get better at collaborating with and actually listening to each other, and make more responsible judgments about the limits of our own expertise if we are serious about developing adequate AI regulatory regimes and technically feasible rules and policy solutions.
Conclusion {#Conclusion}
==========
Given the intensifying worldwide activism in AI regulation and AI’s substantial and global impact on human society, we have highlighted some key regulatory considerations and problems to assist domestic and international AI policymakers. We have also proposed a consistent international regulatory framework — with either a new or a repurposed existing IGO as its focal point — to streamline and coordinate national policymaking efforts. Learning from past experience in other regulatory fields, our objective is to offer a viable framework for international regulatory cooperation in the issue area of AI to avoid the development of nationally fragmented AI policies, which may lead to international tensions. Should our proposal find sufficient support in the international community, more concrete steps towards setting up the here advocated regulatory framework, and regulatory policies on specific AI issues can be elaborated.
|
---
abstract: 'We present a generative model that is defined on *finite sets* of exchangeable, potentially high dimensional, data. As the architecture is an extension of Real NVPs, it inherits all its favorable properties, such as being invertible and allowing for exact log-likelihood evaluation. We show that this architecture is able to learn finite non-i.i.d. set data distributions, learn statistical dependencies between entities of the set and is able to train and sample with variable set sizes in a computationally efficient manner. Experiments on 3D point clouds show state-of-the art likelihoods.'
author:
- |
Kashif Rasul\
Zalando Research\
Mühlenstraße 25, 10243 Berlin, Germany\
`kashif.rasul@zalando.de` Ingmar Schuster\
Zalando Research\
Mühlenstraße 25, 10243 Berlin, Germany\
`ingmar.schuster@zalando.de` Roland Vollgraf\
Zalando Research\
Mühlenstraße 25, 10243 Berlin, Germany\
`roland.vollgraf@zalando.de` Urs Bergmann\
Zalando Research\
Mühlenstraße 25, 10243 Berlin, Germany\
`urs.bergmann@zalando.de`
bibliography:
- 'references.bib'
title: 'Set Flow: A Permutation Invariant Normalizing Flow'
---
Introduction
============
Most of machine learning research concerns itself with either modeling independent and identically distributed (i.i.d.) data, or a full joint probability over a number of variables is modeled, i.e. $p(\mathbf{x}_1, ..., \mathbf{x}_s)$. However, some data is conceptually best represented as a finite unordered set: e.g. point clouds of objects, voice data from a given speaker, or documents as bag-of-words. This is why there has been growing interest in set modeling typically via composition of elementwise operations with permutation invariant reduction operations such as averaging as in [@NIPS2017_6931] or taking the maximum as in [@Qi2017PointNetDL] which introduces a bottleneck in what information about the set can be extracted. Formally, any finite joint probability distribution over $s$ exchangeable random variables $\mathbf{x}_i$ (called “entities” from now on) must fulfill the following requirement for all $s!$ permutations $\pi$: $$p(\mathbf{x}_1, \ldots, \mathbf{x}_s) = p(\mathbf{x}_{\pi(1)}, \ldots, \mathbf{x}_{\pi(s)}).$$ It has been shown that finite exchangeable distributions can be written as a *signed mixture* of i.i.d. distributions [@Kerns2006]. Note that this differs from de Finetti’s theorem, which is a similar statement for exchangeable processes, i.e. infinite sequences of random variables, and states that in this case the distribution is a mixture of i.i.d. processes: $p(\mathbf{x}_1, .., \mathbf{x}_s) = \int p(\theta) \prod_{i=1}^s p(\mathbf{x}_i|\theta)\, d\theta$, with $p(\theta)$ a probability distribution. For illustration of this difference, take a distribution of two exchangeable random variables that sum up to a fixed number $N$ (here commutativity ensures exchangeability)—in this case sampling of the two numbers cannot be written as conditionally independent given a $\theta$. Recent generative models build on top of de Finetti’s results [@NIPS2018_7949; @DBLP:journals/corr/abs-1902-01967], and hence assume an underlying infinite sequence of exchangeable variables. In this work we present an architecture to explicitly model finite exchangeable data. In other words, we are concerned with data comprised of sets $X$ which are i.i.d. sampled from our underlying data, but each set $X = \{\mathbf{x}_i\}_{i=1}^{s}$ is a finite exchangeable set, which can be non-i.i.d. data samples in some arbitrary order. We develop a density estimation model that is permutation invariant and is able to model dependencies between the entities in this setting. We call the resulting architecture *Set Flow*, as it builds on ideas of normalizing flows, in particular compositions of bijections like Real NVP [@45819], and combines these ideas with set models [@NIPS2017_6931].
The paper is structured as follows. In Section 2 we review background concepts and Section 3 has related work. Section 4 describes our model and how it is trained. In Section 5 we present experiments on synthetic and real data, and finally conclude in Section 6 with a brief summary and discussion of future directions.
Background
==========
Sets
----
One straight-forward approach to generate a set function is to treat the input as a sequence and train an RNN, but augmented with all possible input permutations, in the hopes that the RNN will become invariant to the input order. This approach might be robust to small sequences but for set sizes in the thousands it becomes hard to scale. Also, as described in [@44871], the order of the sequences does matter and cannot be discarded.
A recently proposed neural network method, which is invariant to the order if its inputs, is the Deep Set architecture [@NIPS2017_6931]. The key idea of this approach is to map each input to a learned feature representation, on which a pooling operation is performed (e.g. a sum), which then is passed through another function. With $\mathcal{X}$ being the set of all sets, $X \in \mathcal{X}$ being a set, the deep set function $f: \mathcal{X} \mapsto \mathbb{R}^S$ can be written as $f(X) = \rho \left( \sum_{\mathbf{x} \in X} \phi(\mathbf{x}) \right)$, where $\phi \colon \mathbb{R}^D \mapsto \mathbb{R}^K$ and $\rho \colon \mathbb{R}^K \mapsto \mathbb{R}^S$ are chosen as a neural networks.
Recent methods like Janossy pooling [@murphy2018janossy] expresses a permutation invariant function as the average of a permutation variant function applied to all reorderings of the input sequence which allows the layer to leverage complicated permutation variant functions to construct permutation invariant ones. This is computationally demanding, but can be done in a tractable fashion via approximation of the ordering or via random permutations. One can also train a permutation optimization module that learns a canonical ordering [@zhang2019permoptim] to permute a set and then use it in a permutation invariant fashion, typically by processing it via an RNN.
Density Estimation via Normalizing Flows
----------------------------------------
Real NVP [@45819] is a type of normalizing flow [@tabak] where densities in the input space $\mathcal{X}=\mathbb{R}^{D}$ are transformed into some simple distribution space $\mathcal{Z}=\mathbb{R}^{D}$, like an isotropic Gaussian, via $f \colon \mathcal{X} \mapsto \mathcal{Z}$, which is composed of stacks of bijections or invertible mappings, with the property that the inverse $\mathbf{x} = f^{-1}(\mathbf{z})$ is easy to evaluate and computing the Jacobian determinant takes $O(D)$ time. Due to the change of variables formula we can evaluate $p_{\mathcal{X}}(\mathbf{x})$ via the Gaussian by $$p_{\mathcal{X}}(\mathbf{x}) = p_{\mathcal{Z}}(\mathbf{z}) \left| \det \left( \frac{\partial f(\mathbf{x})}{\partial \mathbf{x}} \right)\right|.
\label{eq:variable_change}$$
The bijection introduced by Real NVP called the *Coupling Layer* satisfies the above two properties. It leaves part of its inputs unchanged and transforms the other part via functions of the un-transformed variables $$\begin{cases}
\mathbf{y}^{1:d} = \mathbf{x}^{1:d} \\
\mathbf{y}^{d+1:D} = \mathbf{x}^{d+1:D} \odot \exp(s( \mathbf{x}^{1:d})) + t( \mathbf{x}^{1:d}),
\end{cases}$$ where $\odot$ is an element wise product, $s$ is a scaling and $t$ a translation function from $\mathbb{R}^{d} \mapsto \mathbb{R}^{D-d}$, given by neural networks. To model a complex nonlinear density map $f(\mathbf{x})$, a number of coupling layers $\mathcal{X} \mapsto \mathcal{Y}_1 \mapsto \cdots \mapsto \mathcal{Y}_{K-1} \mapsto \mathcal{Z}$ are composed together, while alternating the dimensions which are unchanged and transformed. Via the change of variables formula the probability density function (PDF) of the flow given a data point can be written as $$\label{Real NVP-logp}
\log p_{\mathcal{X}}(\mathbf{x}) = \log p_{\mathcal{Z}}(\mathbf{z}) + \log | \det(\partial \mathbf{z}/ \partial\mathbf{x})| = \log p_{\mathcal{Z}}(\mathbf{z}) + \sum_{i=1}^{K} \log | \det(\partial \mathbf{y}_{i}/ \partial\mathbf{y}_{i-1})|.$$
Note that the Jacobian for the Real NVP is a block-triangular matrix and thus the log-determinant simply becomes $$\label{Real NVP-logdet}
\log | \det(\partial \mathbf{y}_{i}/ \partial\mathbf{y}_{i-1})| = \mathtt{sum}(\log|\mathtt{diag}( \exp (s(\mathbf{y}_{i-1}))|),$$ where $\mathtt{sum}()$ is the sum over all the vector elements, $\log()$ is the element-wise logarithm and $\mathtt{diag}()$ is the diagonal of the Jacobian. This model, parameterized by the weights of the scaling and translation neural networks $\theta$, is then trained via stochastic gradient descent (SGD) on training data points where for each batch $\mathcal{D}$ the log likelihood (\[Real NVP-logp\]) as given by $$\mathcal{L} = \frac{1}{|\mathcal{D}|} \sum_{\mathbf{x}\in\mathcal{D}} \log p_{\mathcal{X}}(\mathbf{x}; \theta),$$ is maximized. One can trivially condition the PDF on some additional information $\mathbf{h} \in \mathbb{R}^H$ to model $p_{\mathcal{X}}(\mathbf{x} | \mathbf{h})$ by concatenating $\mathbf{h}$ to the inputs of the scaling and translation function approximators, i.e. $s(\mathtt{concat}(\mathbf{x}^{1:d}, \mathbf{h}))$ and $t(\mathtt{concat}(\mathbf{x}^{1:d}, \mathbf{h}))$ which are modified to map $\mathbb{R}^{d+H} \mapsto \mathbb{R}^{D-d}$. This does not change the log-determinant of the coupling layers given by (\[Real NVP-logdet\]).
In practice Batch Normalization [@Ioffe:2015:BNA:3045118.3045167] is applied, as a bijection, to outputs of coupling layers to stabilize training of normalizing flow. This bijection implements the normalization procedure using a weighted average of a moving average of the layer’s mean and standard deviation values, which are different depending if we are training or doing inference.
Related Work
============
The Real NVP approach can be generalized as in the Masked Autoregressive Flow (MAF) [@Papamakarios:2017:maf] which models the random numbers used in each stack to generate data. Glow [@NIPS2018_8224] augments Real NVP by the addition of a reversible $1 \times 1$ convolution, as well as removing other components and thus simplifying the overall architecture to obtain qualitatively better samples for high dimensional data like images.
The BRUNO model [@NIPS2018_7949] performs exact Bayesian inference on sets of data such that the joint distribution over observations is permutation invariant in an autoregressive fashion, in that new samples can be generated conditional on previous ones and a stream of new data points can be easily incorporated at test time. This is easily possible for our method as well, where the network architecture is considerably simple as it only draws upon ideas from normalizing flows. BRUNO, on the other hand, makes use of Student-$t$ processes, i.e. Bayesian models of real-valued functions admitting closed form marginal likelihood and posterior predictive expressions [@shah2014student]. The main issue with this building block is that inference typically scales cubically in the number of data points, although the Woodbury matrix inversion lemma can be used to alleviate this issue for the streaming data setting.
Similar to BRUNO, the PILET model [@DBLP:journals/corr/abs-1902-01967] utilizes an autoregressive model, build upon normalizing flow ideas instead of Student-$t$-processes [@pmlr-v80-oliva18a]. This is combined with a permutation equivariant function to capture interdependence of entities in a set while maintaining exchangeability. They extend their method to make use of a latent code in an exchangeable variational autoencoder framework called PILET-VAE. Note both BRUNO and PILET transform base distributions by applying bijections to entity dimension. Bayesian Sets [@NIPS2005_2817] also models exchangeable sets of binary features but it is not reversible so does not allow sampling from it.
Set Flow
========
In order to make a model invariant to input permutations, one can try to sort the input into some canonical order. While sorting is a very simple solution, for high dimensional points the ordering is in general not stable with respect to the point perturbations and thus does not fully resolve the issue. This makes it hard for a model to learn a consistent mapping even if we constrain the model to have the same set size.
We propose a normalizing flow architecture called Set Flow that in each stack transforms each entity of the set via a shared global Gaussian noise vector, and then this noise vector gets transformed via a symmetric function of all the transformed elements of the set, for example via a Deep Set [@NIPS2017_6931] layer.
![Schematic of a single Set Flow stack where a set of entities $\{ \mathbf{x}_1, \ldots \mathbf{x}_s\}$, where $\mathbf{x}_i \in \mathbb{R}^D$ and a global Gaussian noise vector $\mathbf{z}_0 \sim \mathcal{N}(0, I) \in \mathbb{R}^G$, are transformed $ (\mathbf{z}_0, \{\mathbf{x}_1, \ldots, \mathbf{x}_s\}) \mapsto (\mathbf{z}_1, \{\mathbf{y}_1, \ldots, \mathbf{y}_s\})$ via (\[eq:set-flow\]). See text for detailed description.[]{data-label="fig:set-flow"}](images/set-flow-simple.pdf){width="0.7\linewidth"}
Figure \[fig:set-flow\] shows a single Set Flow stack, which takes its input from layer $k=0$ to the next stack $k=1$. The block takes a set of entities $\{ \mathbf{x}_1, \ldots \mathbf{x}_s\}$ where $\mathbf{x}_i \in \mathbb{R}^D$, and a global Gaussian noise vector $\mathbf{z}_0 \sim \mathcal{N}(0, I) \in \mathbb{R}^G$ and transforms it to $ (\mathbf{z}_0, \{\mathbf{x}_1, \ldots, \mathbf{x}_s\}) \mapsto (\mathbf{z}_1, \{\mathbf{y}_1, \ldots, \mathbf{y}_s\})$ given by: $$\begin{cases}
\hat{\mathbf{y}}_i = \mathbf{x}_i \odot \exp(s_1^{(0)}( \mathbf{z}_0)) + t_1^{(0)}( \mathbf{z}_0) \quad \mathrm{for} \quad i=1,\ldots,s\\
\hat{\mathbf{y}}_i \mapsto \mathbf{y}_i \quad \mathrm{via\ RealNVP}^{(0)} \mathrm{\ for} \quad i=1,\ldots,s \quad \mathrm{if} \quad D > 1 \\
\mathbf{z}_1 = \mathbf{z}_0 \odot \exp(s_2^{(0)}( f^{(0)}(\mathbf{y}_1, \ldots,\mathbf{y}_s ))) + t_2^{(0)}(f^{(0)}(\mathbf{y}_1, \ldots, \mathbf{y}_s))
\end{cases}
\label{eq:set-flow}$$ where $f^{(k)}$ is a permutation invariant function given via a Deep Set, $t_i^{(k)}$ and $s_i^{(k)}$ are deep neural networks function approximators and $\mathrm{RealNVP}^{(k)}$ is a standard Real NVP—all of these functions are layer $k$ specific and do not share weights across layers. By stacking $K$ such Set-Coupling layers we arrive at our *Set Flow* model. As one can see from the construction this mapping is permutation equivariant due to the Deep Set layer and invertable via the bijections.
The inverse transformation starts by sampling a global noise vector $\mathbf{z}_{K-1} \sim \mathcal{N}(0, I) \in \mathbb{R}^G$ as well as a set of the desired number of Gaussian sample entities and going through the flow model in reverse (or from the top to bottom in Figure \[fig:set-flow\]).
As in the Real NVP we can also condition this model via some $\mathbf{h} \in \mathbb{R}^H$ for each set of entities $\{\mathbf{x}_i\}_{i=1}^{s}$ by the following modification in (\[eq:set-flow\]): $$\hat{\mathbf{y}}_i = \mathbf{x}_i \odot \exp(s_1^{(0)}( \mathtt{concat}(\mathbf{z}_0, \mathbf{h}))) + t_1^{(0)}( \mathtt{concat}(\mathbf{z}_0, \mathbf{h}))$$ to obtain a set-conditioned model, for example when the entities of a set come from a particular category.
Training
--------
We train the model by sampling batches where for each batch $\mathcal{D}$ the size of the set $s$ is fixed, and construct $|\mathcal{D}|$ sets where each set has $s$ entities as well as a global noise vector. We use Adam [@kingma:adam] with standard parameters, to maximize the log likelihood: $$\begin{aligned}
\mathcal{L} &= \mathcal{L}_{\mathcal{X}} + \mathcal{L}_{\mathcal{N}} \nonumber \\
&= \sum_{i=1}^{s} \log p_{\mathcal{X}} (\mathbf{x}_i; \theta) + \log p_{\mathcal{N}}(\mathbf{z}_0; \theta),
\label{eq:SetFlowLikelihood}\end{aligned}$$ where for each term above (\[eq:variable\_change\]) is employed to explicitly evaluate the likelihoods and calculate their derivatives, with respect to $\theta$ which denotes all parameters of the Set Flow model.
Note that we choose $\mathbf{z}_0 \sim \mathcal{N}(0, I) \in \mathbb{R}^G$ in all our experiments. As we’re interested in the likelihoods of the sets we hence subtract $G$ times the entropy of a Gaussian (the maximum likelihood solution of the global variables) with variance $1$ from the calculated likelihoods of (\[eq:SetFlowLikelihood\]) when reporting the test set likelihoods.
Experiments
===========
Our first goal in the experiments is to demonstrate and analyze the ability of the proposed model to capture non-i.i.d. dependencies within finite sets. In a second set of experiments, we show that the model scales to much larger and complex datasets by learning 3D point clouds.
Generation of Non-i.i.d. Exchangeable Data Sets
-----------------------------------------------
In order to best understand the ability of the model to capture dependencies of entities, we generate a toy dataset of finite sets with a non-i.i.d. structure: equidistant 2D points on circles with varying radius and position. The generative process of each set is given as follows: first, the center position $x,y$, radius $r$ and a rotation offset $\phi$ is sampled uniformly as $x,y \sim \mathcal{U}(-10,10)$, $r \sim \mathcal{U}(0.5,3)$ and $\phi \sim \mathcal{U}(0,2\pi)$. Then $N$ points are generated with coordinates $x_i = x + (r + \Delta r_i) \; \mathrm{cos}(\psi_i)$ and $y_i = y + (r + \Delta r_i) \; \mathrm{sin}(\psi_i)$, where $\psi_i = \phi + 2\pi i/N + \Delta \psi_i$, with independent radial noise $\Delta r_i \sim \mathcal{N}(0,0.1)$ and angular noise $\Delta \psi_i \sim \mathcal{N}(0,0.3)$.
Figure \[fig:noniid\] (left) shows sample sets with a size $N=5$ drawn from this generative model—colors indicate set membership. For the experiment, we trained a model on uniformly random sampled set sizes in $\{3,4,5,6\}$, where each minibatch of $16$ sets contained the same set sizes. The second subfigure from left shows that after $10^5$ set samples, the model groups elements of sets together in clusters, but fails to produce discernible circles with equidistant points on them. After $3\times10^6$ set samples, the model reproduces the dataset more faithfully, as can be seen in the second to right Figure. The rightmost subfigure in Figure \[fig:noniid\] (top) shows the distribution of inferred phases from fitted circles to sets of size $N=3$ (the mean phase across the set is subtracted for alignment). It can be seen that the model (green) nicely captures the equidistant peaks, similar to the original data (blue). Note that this implies that the model captured the generative process of the finite set—otherwise there would be more mass in between the peaks. The variance, however, is larger than in the ground truth phases. Similarly, the model has a bias towards smaller circles—as can be seen in the distribution of inferred radii.
![Non-i.i.d. analysis. The leftmost subfigure shows samples of sets with $N=5$ 2D equidistant entities drawn on circles with random positions and radii. The model initially captures the global position variance in the data (second from left at $t=10^5$ samples) and later in learning captures the non-i.i.d. equidistance property on the circles (at $t=3\times10^6$, second from right). The rightmost subfigure shows that the model nicely captures equidistant phases on the sampled circles, but has a bias towards smaller circles than in the original dataset.[]{data-label="fig:noniid"}](images/noniid_training_data.pdf "fig:"){width="0.261\linewidth"} ![Non-i.i.d. analysis. The leftmost subfigure shows samples of sets with $N=5$ 2D equidistant entities drawn on circles with random positions and radii. The model initially captures the global position variance in the data (second from left at $t=10^5$ samples) and later in learning captures the non-i.i.d. equidistance property on the circles (at $t=3\times10^6$, second from right). The rightmost subfigure shows that the model nicely captures equidistant phases on the sampled circles, but has a bias towards smaller circles than in the original dataset.[]{data-label="fig:noniid"}](images/noniid_generated_citer_10528.pdf "fig:"){width="0.261\linewidth"} ![Non-i.i.d. analysis. The leftmost subfigure shows samples of sets with $N=5$ 2D equidistant entities drawn on circles with random positions and radii. The model initially captures the global position variance in the data (second from left at $t=10^5$ samples) and later in learning captures the non-i.i.d. equidistance property on the circles (at $t=3\times10^6$, second from right). The rightmost subfigure shows that the model nicely captures equidistant phases on the sampled circles, but has a bias towards smaller circles than in the original dataset.[]{data-label="fig:noniid"}](images/noniid_generated_citer_3077776_2.pdf "fig:"){width="0.261\linewidth"} ![Non-i.i.d. analysis. The leftmost subfigure shows samples of sets with $N=5$ 2D equidistant entities drawn on circles with random positions and radii. The model initially captures the global position variance in the data (second from left at $t=10^5$ samples) and later in learning captures the non-i.i.d. equidistance property on the circles (at $t=3\times10^6$, second from right). The rightmost subfigure shows that the model nicely captures equidistant phases on the sampled circles, but has a bias towards smaller circles than in the original dataset.[]{data-label="fig:noniid"}](images/noniid_phases_3ents_citer_3077776.pdf "fig:"){width="0.261\linewidth"}
3D Point Cloud Experiments
--------------------------
We train Set Flow from point clouds of Airplane and Chair classes of the ModelNet40 [@CVPR15_Wu] dataset, where we sample $s=1,000$ random points from a point cloud of 10,000 for each model to construct a set for the chosen class. We split the model files into a training and test set via an 80% split. We train the model on two individual classes: airplane and chair separately and report the mean test likelihoods in Table \[tb:modelnet40\]. We also show some sample generated point clouds in Figure \[fig:chair-plane-samples\] for different set sizes.
We also train the model on three classes (airplane, chair and lamp) together and then given two sets we obtain the noise vectors by passing the sets through our model. We can then linearly interpolate between these two sets and generate samples by passing the interpolated noise, both global and for the entities of the set, backwards. Figure \[fig:chairs-inter\] shows the results of this experiment for a chair to another chair, chair to a lamp and chair to an airplane.
Finally, we train the model on all 40 classes both without supplying the class labels and with class labels via a set class embedding vector $\mathbf{h}\in \mathbb{R}^H$. We report the mean test log-likelihoods over each entity in the set in Table \[tb:modelnet40\] together with results from other methods.
We have implemented all the experiments in PyTorch [@paszke2017automatic] and will make the code available after the review process here[^1]. We used the following hyperparameters: batch size $|\mathcal{D}|=16$, global noise vector dimension $G=90$, size of Deep Set pooling output $S =100$, size of conditioning embedding vector $H=10$, number of Set Flow stacks $K=6$, number of random entities in a set $s=1,000$ and a learning rate of $5 \times 10^{-4}$ for all our experiments.
Model Airplane Chair All All with labels
----------- -------------------- ---------------------- ----------------------- -----------------------
PILET-VAE $\textbf{4.08}$ $\textbf{2.03}$ **2.13** **2.29**
PILET $3.75$ $1.47$ $1.58$ $1.94$
BRUNO $2.6$ $0.79$ $0.75$ $0.64$
Set Flow **4.1** $\pm 0.35$ **2.045** $\pm 0.25$ **2.143** $\pm 0.318$ **2.311** $\pm 0.298$
: Mean test log-likelihoods for ModelNet40 [@CVPR15_Wu] dataset models with two times standard error for our method.[]{data-label="tb:modelnet40"}
\
\
Discussion and Conclusions
==========================
We have introduced a simple generative architecture for learning and sampling from exchangeable data of finite sets via a normalizing flow architecture using permutation invariant functions like, for example, Deep Sets. As shown in the experiments our model captures dependencies between entities of a set in a computationally feasible manner. We demonstrated the capability of the model to capture finite exchange invariant generative processes on toy data. We also demonstrated state-of-the art performance for generative modeling of 3D point clouds. In principle, the propose model can be applied to higher dimensional data points, like for example sets of images e.g. in an outfit.
In future work we will further explore alternative architectures of these models, utilize them to learn on sets of images and experiment to see if these methods can be used to learn correlations in time series data across a large number of entities.
[^1]: <https://www.github.com/xxx/xxx>
|
---
abstract: 'Electron-positron pair production in space- and time-dependent electromagnetic fields is investigated. Especially, the influence of a time-dependent, inhomogeneous magnetic field on the particle momenta and the total particle yield is analyzed for the first time. The role of the Lorentz invariant [$\mathbf{E}^2 - \mathbf{B}^2$]{}, including its sign and local values, in the pair creation process is emphasized.'
address:
- |
Theoretisch-Physikalisches Institut, Abbe Center of Photonics,\
Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, D-07743 Jena, Germany\
Helmholtz-Institut Jena, Fröbelstieg 3, D-07743 Jena, Germany
- 'Institute of Physics, NAWI Graz, University of Graz, A-8010 Graz, Austria'
author:
- Christian Kohlfürst
- Reinhard Alkofer
title: |
On the effect of time-dependent inhomogeneous magnetic fields\
in electron-positron pair production
---
Electron-positron pair production, QED in strong fields, Kinetic theory, Wigner formalism
#### Introduction
Although already predicted in the first half of the last century [@Sauter:1931zz] electron-positron pair production attracted renewed attention over the last decade. This interest is strengthened by experiments verifying the possibility of creating matter by light-light scattering [@Burke:1997ew]. Upcoming laser facilities, [*e.g.*]{}, ELI [@eli; @Heinzl:2008an] and XFEL [@xfel; @Ringwald:2001ib], as well as newly proposed experiments [@Marklund:2008gj] are expected to deepen our understanding of matter creation from fields.
Note that in the very special case of constant and homogeneous fields the Lorentz invariants $$\mathcal F = \frac{1}{2} \left( \mathbf{E}^2 - \mathbf{B}^2 \right) ,\quad \mathcal G = \mathbf{E} \cdot \mathbf{B}$$ determine the particle production rate [@Dunne:2004nc]. In [constant crossed fields]{} $\mathcal G$ vanishes which highlights then the role of the action density $\mathcal F$ in pair production.
Although electric and magnetic fields appear in equal magnitude in the quantity $\mathcal F$ magnetic fields are usually ignored in theoretical investigations of pair production. This may be [motivated]{} by the fact that for perfect settings the magnetic field vanishes in the overlapping region of two colliding laser beams [@Alkofer:2001ik]. Hence, the majority of studies on pair production have examined this process for time-dependent electric fields only [@PhysRevLett.101.130404; @abc]. (NB: Configurations with an additional constant magnetic field have been investigated in [@Dunne:1997kw].)
But in studies of pair production by electric fields it turns out that exactly the time-dependence of the fields is most influential, and depending on it one observes different mechanisms behind pair production [@Nousch:2012xe; @Diss]. In a first, almost superficial, way one can distinguish multi-photon pair production [@Kohlfurst:2013ura; @Ruf:2009zz] from the Schwinger effect [@Hebenstreit:2011pm; @Cohen:2008wz]. [Employing]{} multi-timescale fields, [however]{}, a rich phenomenology opens up. Hereby, [*e.g.*]{}, the dynamically-assisted Schwinger effect [@PhysRevLett.101.130404; @Orthaber:2011cm; @Linder:2015vta; @Panferov:2015yda] is only one, although the most prominent, example.
Given this situation, and in view of realistic possibilities of an experimental verification, it is an unsatisfactory situation that so little is known about pair production in non-constant magnetic fields [@Piazza; @Ruf:2009zz]. The clarification of potential, currently unknown phenomena associated with time-dependent magnetic fields is one of the required next steps if theoretical results on Schwinger pair production shall be put to the scrutiny of experiment.
Among worldline [@Dunne:2005sx; @Schneider:2014mla] and WKB-like formalism [@Strobel:2013vza], the introduction of quantum kinetic theory [@Smolyansky:1997fc] has helped to understand pair production in homogeneous, but time-dependent electric fields. (NB: For recent developments concerning quantum kinetic theory see, [*e.g.*]{}, refs. [@Dabrowski:2014ica; @Kohlfurst:2012rb; @Hebenstreit:2014lra; @Diss; @Li:2014nua; @Hebenstreit2015; @Blinne2015]). However, to accurately describe pair production in laser fields one has to take into account spatial inhomogeneities [@Dunne:2005sx; @Berenyi:2013eia; @Hebenstreit:2011wk; @Han:2010rg; @Harvey:2012ie] as well as magnetic fields [@Ruf:2009zz]. In this letter, we will discuss the results of our exploratory study on the influence of time-dependent, spatially inhomogeneous magnetic fields on the particle production rate using still a relatively simple model for the gauge potential. To put these results into perspective, we will also compare the outcome of these calculations with a field configuration not fulfilling the homogeneous Maxwell equations. Our results are based upon the Dirac-Heisenberg-Wigner (DHW) approach [@Vasak:1987um], which was successfully employed for spatially inhomogeneous electric fields only recently [@Berenyi:2013eia; @Hebenstreit:2011wk].
#### Formalism
Throughout this article the convention $c = \hbar = m = 1$ will be used. The theoretical approach employed here is based on the [fundament]{} laid by [refs]{}. [@Vasak:1987um].
[ The fundamental quantity in the DHW approach is the covariant Wigner operator $$\hat{\mathcal W}_{\alpha \beta} \left( r , p \right) = \frac{1}{2} \int d^4 s \ \mathrm{e}^{\mathrm{i} ps} \ \mathcal U \left(A,r,s \right) \ \mathcal C_{\alpha \beta} \left( r , s \right),$$ where we have introduced the density operator $$\mathcal C_{\alpha \beta} \left( r , s \right) = \left[ \bar \psi_\beta \left( r - s/2 \right), \psi_\alpha \left( r + s/2 \right) \right]$$ and the Wilson line factor $$\mathcal U \left(A,r,s \right) = \exp \left( \mathrm{ie} \int_{-1/2}^{1/2} d \psi \ A \left(r+ \psi s \right) \ s \right).$$ The vector potential $A$ is given in mean-field approximation, $r$ and $s$ denote center-of-mass and relative coordinates, respectively. Taking the vacuum expectation value of the Wigner operator and projecting on equal time ([*i.e.*]{}, performing an integral $\int dp_0$) yields the single-time Wigner function $\mathcal W \left( \mathbf x , \mathbf p , t\right)$. ]{}
[ The simplest way to incorporate inhomogeneous magnetic fields is to investigate pair production in the $xz$-plane. However, there are in total three different ways of defining the basis matrices for a DHW calculation with only two spatial dimensions: one representation using 4-spinors and two representations using 2-spinors. Generally, the 4-spinor formulation contains all information on the pair production process, while the results from a 2-spinor formulation are spin-dependent (one describes electrons with spin up and positrons with spin down [@deJesusAnguianoGalicia:2005ta] and the other describes the spin-reversed particles). ]{}
[ To simplify the calculations we use a 2-spinor representation. Hence, we decompose the Wigner function into Dirac bilinears: $$\mathcal W \left( \mathbf x, \mathbf p, t \right) = \frac{1}{2} \left( \mathbbm 1 \ \mathbbm s + \gamma_{\mu} \mathbbm v^{\mu} \right).$$ Following refs. [@Vasak:1987um] we are able to identify $\mathbbm s$ as mass density and $\mathbbm v^{\mu}$ as charge and current densities. ]{}
We can reduce the corresponding equations of motions for the Wigner [coefficients]{} $\mathbbm{s}$ and [$\mathbbm{v}^\mu =
(\mathbbm{v}_0,\mathbbm{v}^1,\mathbbm{v}^3)$]{} to the form (see, [*e.g.*]{}, [ref]{}. [@Diss]): $$\begin{aligned}
{4}
& D_t \mathbbm{v}_0 && +\mathbf{D} \cdot \mathbbm{v} && &&= 0,
\label{eqn1_1} \\
& D_t \mathbbm{s} && && -2 \left( \Pi_x \cdot \mathbbm{v}^3 -
\Pi_z \cdot \mathbbm{v}^1 \right) &&= 0, \label{eqn1_2} \\
& D_t \mathbbm{v}^1 && +D_x \cdot \mathbbm{v}_0 && -2 \Pi_z
\cdot \mathbbm{s} && = -2 \mathbbm{v}^3, \label{eqn1_3} \\
& D_t \mathbbm{v}^3 && +D_z \cdot \mathbbm{v}_0 && +2\Pi_x
\cdot \mathbbm{s} &&= 2 \mathbbm{v}^1, \label{eqn1_4} \end{aligned}$$ with the pseudo-differential operators $$\begin{aligned}
{8}
& D_t && = \quad && \partial_{t} && + e && \int d\xi \, && \mathbf{E} \left(
\mathbf{x}+ \textrm {i} \xi \boldsymbol{\nabla}_p,t \right) && \cdot && \boldsymbol{\nabla}_p,
\label{eqn2_1} \\
& \mathbf{D} && = \quad && \boldsymbol{\nabla}_x && + e && \int d\xi \, && \mathbf{B}
\left( \mathbf{x}+\textrm {i} \xi \boldsymbol{\nabla}_p,t \right) && \times && \boldsymbol{\nabla}_p,
\label{eqn2_2} \\
& \boldsymbol{\Pi} && = \quad && \mathbf{p} && - \textrm {i} e && \int d\xi \,
\xi \, && \mathbf{B} \left( \mathbf{x}+\textrm {i} \xi \boldsymbol{\nabla}_p,t \right) &&
\times && \boldsymbol{\nabla}_p. \label{eqn2_3} \end{aligned}$$ The vacuum initial conditions are given by $$\begin{aligned}
{3}
\mathbbm{s}_{vac} \left(\boldsymbol{p} \right) = -\frac{2}{\sqrt{1 +
\boldsymbol{p}^2}}, \quad &&
\mathbbm{v}_{vac}^{1,3} \left(\boldsymbol{p} \right) = -\frac{2
\boldsymbol{p}}{\sqrt{1 + \boldsymbol{p}^2}}. \label{eqn3}\end{aligned}$$ For later use [we]{} explicitly subtract the vacuum terms by defining $$\begin{aligned}
\mathbbm{w}^v = \mathbbm{w} - \mathbbm{w}_{vac},\end{aligned}$$ [with $\mathbbm{w} = \mathbbm{v}_0 ,~\mathbbm{s} ,~\mathbbm{v}^1$ and $\mathbbm{v}^3$, respectively [@Hebenstreit:2011pm].]{} The particle number density in momentum space is given by $$\begin{aligned}
n \left( p_x, p_z \right) = \int dz \, \frac{\mathbbm{s}^v + p_x
\mathbbm{v}^{v,1} + p_z \mathbbm{v}^{v,3}}{\sqrt{1+\boldsymbol{p}^2}}.
\label{eqn5}\end{aligned}$$ When evaluated at asymptotic times, this quantity gives the particle momentum spectrum. Subsequently, the particle yield per unit volume element is obtained via $N = \int dp_x \, dp_z \, n \left( p_x ,p_z
\right)$.
[ In the following we will discuss pair production for one specific 2-spinor representation. The results for particles with opposite spin can be obtained performing $p_z \to - p_z$.]{}
#### Solution strategies
As momentum derivatives appear as arguments of $\mathbf{E}$ and $\mathbf{B}$ we Taylor-expand the pseudo-differential operators in - up to [fourth order [@Diss]]{}. To increase numerical stability canonical momenta are used: $$\boldsymbol{q} = \boldsymbol{p} + e\boldsymbol{A} \left( \boldsymbol{x}, t \right) .$$
In order to solve eqs. - numerically, spatial and momentum directions are equidistantly discretized, and [additionally we set $\mathbbm{w}^v \left(\mathbf x_0 \right) = \mathbbm{w}^v \left(\mathbf x_{N_\mathbf{x}} \right) $ as well as $\mathbbm{w}^v \left(\mathbf p_0 \right) = \mathbbm{w}^v \left(\mathbf p_{N_\mathbf p} \right) $.]{} [We further demand Dirichlet boundary conditions $$\begin{aligned}
\mathbbm{w}^v \left( \mathbf x_{k_i}, \mathbf p_{k_j} \right) = 0 \quad \textrm{if} \quad
k_i = 0 \ \textrm{or} \ k_j = 0.\end{aligned}$$ ]{} The derivatives are then calculated using pseudospectral methods in Fourier basis [@Boyd]. The time integration was performed using a Dormand-Prince Runge-Kutta integrator of order 8(5,3) [@NR].
#### Model for the fields
For our studies of pair production in electromagnetic fields, we choose a vector potential of the form $$\begin{aligned}
\boldsymbol{A} (z,t) &= \varepsilon \ \tau \left( \tanh \left(
\frac{t+\tau}{\tau} \right) - \tanh \left( \frac{t-\tau}{\tau} \right) \right)
\notag \\
&\times \exp \left( -\frac{z^2}{2 \lambda^2} \right) \ \boldsymbol{e}_x.
\label{AA}\end{aligned}$$ If not stated otherwise, the electric and magnetic field are derived from this expression. [Note that the field configuration obeys $\mathcal G = \boldsymbol{E} \cdot \boldsymbol{B} = 0$. Moreover,]{} the homogeneous Maxwell equations are automatically fulfilled and additionally $\boldsymbol{\nabla} \cdot \boldsymbol{E} = 0$ holds.
The electric field is antisymmetric in time exhibiting a double peak structure with $\varepsilon$ denoting the field strength. The field strength of the magnetic field, however, is [suppressed relative to the electric field strength]{} by a term $\tau/
\lambda^2$, where $\tau$ and $\lambda$ also determine the scale for temporal and spatial variations, respectively. Hence, for $\tau/ \lambda \ll 1$ the field energy is stored almost exclusively in the electric field. For $\tau/ \lambda \gtrsim 1$, however, the energy stored in the magnetic field exceeds the energy fraction coming from the electric part.
In ref. [@Dunne:2004nc] it was argued that pair production is only possible in regions where $\mathbf{E} \left( z,t \right)^2 - \mathbf{B} \left( z,t \right)^2 > 0$. To analyze our results in view of this conjecture we therefore define an “effective field amplitude” $$\tilde E \left( z,t \right)^2 = \mathbf{E} \left( z,t \right)^2- \mathbf{B} \left( z,t \right)^2$$ and a “modified effective field energy” $$\mathcal{E} \left( \mathbf{E}, \mathbf{B} \right) = \int \ dz \ dt \ \tilde E \left( z,t
\right)^2 \ \Theta \left( \tilde E \left( z,t
\right)^2 \right),
\label{EE}$$ with the Heaviside function $\Theta \left(x \right)$.
![Qualitative comparison of the electric field energy (dotted grey line) and the effective field energy using the proposed vector potential (dot-dashed red line) for $\tau = 10/m$. [In addition, we show for later comparison the difference in energy ($\Delta \varepsilon$) added to the electric field energy (blue line).]{} []{data-label="fig:Inv"}](Inv5){width="49.00000%"}
This effective energy for different values of $\lambda$ is displayed in Fig. \[fig:Inv\]. We find that the magnetic field can significantly reduce the effective field strength. While the electric part linearly depends on $\lambda$, the calculation for a combined electric and magnetic field shows a rapid drop off for $\tau/\lambda
\gtrsim 1$.
#### Particle distribution
It is useful to define the reduced particle density $n(p_x)/ \lambda$ to scale out the trivial linear dependence on $\lambda$. As can be seen in Fig. \[fig:distr\_x\], $n(p_x)/ \lambda$ displays a peaked structure superimposed by an oscillating function. This is characteristic for electric fields with peaks of the same absolute value but opposite sign [@Hebenstreit:2009km].
It should be pointed out that especially the peaks in the reduced particle distribution $n(p_x)/ \lambda$ decrease with decreasing $\lambda$. A possible interpretation is that the presence of the magnetic field prevents the particles, created at the different field oscillations, to interfere. In case of $\tau/\lambda \gtrsim 1$ particles created around the first electric field oscillation at $z \neq 0$ are accelerated in $x$ and also $z$ direction. However, particles created at the second oscillation acquire a completely different momentum signature and therefore both wave packages become distinguishable. Moreover, an analysis of our data indicates, that the particle distribution is slowly shifted to lower momenta for small $\lambda$. The reason for this phenomenon seems to be directly linked with the increase in the magnetic field strength. For a configuration of the form , a decrease of the parameter $\lambda$ causes the region with maximal effective field amplitude to be shifted away from $t=0$. Therefore this shift has a different origin compared to the previously discovered particle self-bunching [@Hebenstreit:2011wk].
The reduced particle density $n(p_z)/ \lambda$, [*cf.*]{} Fig. \[fig:distr\_z\], does not show any interference pattern. For $\lambda = 100/m$ the distribution in $p_z$ is symmetric around the origin, in agreement with homogeneous calculations. However, in case of $\tau/\lambda
\gtrsim 1$ the particle peak is shifted towards positive $p_z$.
As noted above, using the second $2$-spinor basis, one obtains a particle density mirrored at $p_z=0$. Therefore, this result is an indicator for interactions between the magnetic field and the electron spin.
![Reduced particle density $n(p_x)/ \lambda$ for various values of the spatial inhomogeneity $\lambda$, a field strength of $e\varepsilon = 0.707 \, m^2$ and a pulse length $\tau = 5 /m$. For $\lambda \gg \tau$ the reduced particle density converges.[]{data-label="fig:distr_x"}](npx_pic){width="49.00000%"}
![Reduced particle density $n(p_z)/ \lambda$ for various values of the spatial inhomogeneity $\lambda$. The particle density is symmetric for $\lambda \gg \tau$ only. The vertical grey line is there to guide the eye: the peak of the particle density is shifted to positive $p_z$. Parameters: $e\varepsilon = 0.707 \, m^2$ and $\tau = 5 /m$.[]{data-label="fig:distr_z"}](npz_pic){width="49.00000%"}
#### Particle yield
The magnetic field is not independent of the electric field, because both stem from the same vector potential . The effect of fixing $\mathbf{B} = 0$ and therefore violating the homogeneous Maxwell equation $\boldsymbol{\nabla} \times \boldsymbol{E} = -
\dot{\boldsymbol{B}}$ shows up in the particle density and subsequently in the particle yield. In order to draw a general conclusion between effective field energy and particles created, we will focus on the particle yield in the following.
![Double-log plot of the reduced particle yield as a function of the parameter $\lambda$. For $\lambda \gg \tau$ the reduced particle yield converges to the homogeneous result (dashed black line). In case of a sizable magnetic field the calculation for $B = 0$ (blue line) leads to an overestimation compared to the correct result (dashed red line). Parameters: $\tau = 10/m$ and $e\varepsilon = 0.707 \, m^2$.[]{data-label="fig:N1"}](Log_Jena.eps){width="49.00000%"}
The effective field energy without a magnetic field $\mathcal{E} \left(E,0
\right)$ is a linear function of $\lambda$, see Fig. \[fig:Inv\]. Hence, it is reasonable to introduce an approximation for the particle yield $$\begin{aligned}
\tilde{N} = \lambda \, N_{hom},\end{aligned}$$ where $N_{hom}$ is the yield obtained from a calculation with a spatially homogeneous field. The figures Fig. \[fig:N1\] and Fig. \[fig:N2\] show, that there is good agreement between the approximation and the full solution for $\lambda \gg \tau$. Reasons are, that in this case the electric field can be considered as quasi-homogeneous. Furthermore, the magnetic field energy is by orders of magnitude smaller than its electric counterpart and therefore negligible.
The effect of spatial restrictions on the electric field has already been investigated in Ref. [@Dunne:2005sx; @Hebenstreit:2011wk]. In our case, also the effect of a magnetic field growing in strength for decreasing $\lambda$, has to be taken into account. The corresponding computation of the effective field amplitude is depicted in Fig. \[fig:Inv\] as $\mathcal{E}
\left(E, B \right)$. One observes a faster than linear decrease. This is in qualitative agreement with the particle yield, as illustrated in Fig. \[fig:N1\]. We have to admit, however, that for calculations with $\lambda < 5/m$ the results are not reliable anymore due to a breakdown of the used Taylor expansion. [(NB: The calculation with $B=0$ does not display this numerical problem.)]{}
Eventually, the configuration $\boldsymbol{E} = - \dot{\boldsymbol{A}}$ and $\boldsymbol{B} = 0$ is analyzed. Contrary to the previous case, simply calculating the effective field energy of the applied field, which would be $\mathcal{E} \left( E,0 \right)$ in Fig. \[fig:Inv\], is not sufficient. We have to consider, that the homogeneous Maxwell equations are not fulfilled. Hence, we suggest to add the missing part of the effective field energy to the electric field energy, [illustrated as $\mathcal{E} \left(E,0 \right)
+ \Delta \mathcal{E}$ in Fig. \[fig:Inv\].]{} In this way, the increase in the particle yield in Fig. \[fig:N1\] and Fig. \[fig:N2\] can be understood in terms of the magnetic field. We assume, that a magnetic field hinders matter creation. Fixing $B$ to zero and ignoring the term in the equations - therefore inevitably leads to an overestimation of the effective field amplitude and consequently to an overestimation of the total particle number. (NB: Comparison of Fig. \[fig:Inv\] with Fig. \[fig:N1\] corroborates this argument.)
The sharp drop off on the left side of Fig. \[fig:N2\] is connected to the fact, that for $\lambda \to 0$ the energy stored in the background field is not sufficient anymore to overcome the particle rest mass [@Hebenstreit:2011pm; @Dunne:2005sx].
![Particle yield drawn in a lin-lin plot for a configuration with $\tau = 10/m$ and $e\varepsilon = 0.707 \, m^2$ (the same set of data as in Fig. \[fig:N1\] is used). At $\lambda = 5/m$ the particle yield obtained from a calculation with $B=0$ exceeds the correct result by a factor of $50$. []{data-label="fig:N2"}](norm_Jena.eps){width="49.00000%"}
#### Conclusions
Based on the DHW formalism, Sauter-Schwinger electron-positron pair production in time-dependent, spatially inhomogeneous electric and magnetic fields has been investigated. For the first time the equations of motion for an effectively $2+1$ dimensional system has been solved numerically. We have focused on the influence of the magnetic field on the pair production process for a special class of vector potentials. We have found that for this kind of potentials the magnetic field is of minor importance for a wide range of parameter sets thereby validating studies which have been performed so far. Additionally, there is perfect agreement in the results when comparing to quantum kinetic theory in the limit of spatially homogeneous fields. However, and as most important result presented here, we have verified in a quantitative manner that in the case of spatially strongly localized fields the results can be explained assuming that pair production is only possible in regions where the electric field exceeds the magnetic field. In this parameter region the correct treatment of the magnetic field is of utter importance.
#### Outlook
In order to investigate pair production in background fields with more realistic length and time scales, improvements in the employed numerical methods will be necessary. [Such work is in progress, and it will allow to investigate more general electromagnetic fields. A possible extension would be, for example, the study of multi-photon pair production. Therefore we are optimistic that the investigation presented here will soon serve as a basis for studies employing fields closer to experimentally feasible conditions.]{}
#### Acknowledgements
We are grateful to Florian Hebenstreit and Daniel Berényi for helpful discussions, especially those about numerical methods used in this study. We thank Holger Gies and Alexander Blinne for many interesting discussions and a critical reading of this manuscript.\
C. K. acknowledges funding by the Austrian Science Fund, FWF, through the Doctoral Program “On Hadrons in Vacuum, Nuclei and Stars”(FWF DK W1203-N16) and by BMBF under grant No. 05P15SJFAA (FAIR-APPA-SPARC). We thank the research core area “Modeling and Simulation” for support.
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|
---
abstract: 'In this note, we investigate Goppa codes which are constructed by means of Elementary Abelian $p$-Extensions of $\mathbb{F}_{p^{s}}(x)$, where $p$ is a prime number and $s$ is a positive integer. We give a simple criterion for self-duality of these codes and list the second generalized Hamming weight of these codes.'
address:
- ' Indian Institute of Science Education and Research, Bhopal'
- 'Indian Institute of Science Education and Research, Bhopal'
author:
- Nupur Patanker
- Sanjay Kumar Singh
title: 'A short note on Goppa codes over Elementary Abelian $p$-Extensions of $\mathbb{F}_{p^{s}}(x)$'
---
**Introduction**
================
Let $\mathbb{F}_{p^{s}}$ be the finite field with $p^{s}$ elements of characteristic $p$ (where $s$ is a positive integer). A linear code is a $\mathbb{F}_{p^{s}}$-subspace of $\mathbb{F}_{p^{s}}^{n}$, the $n$-dimensional standard vector space over $\mathbb{F}_{p^{s}}$. Such codes are used for transmission of information. It was observed by Goppa in 1975 that we can use algebraic function field over $\mathbb{F}_{p^{s}}$ to construct a class of linear codes. In Goppa’s construction, we choose a divisor $G$ and $n$ rational places of the algebraic function field to form a linear code of length $n$. In this note, we study Goppa codes over Elementary Abelian $p$-Extensions of $\mathbb{F}_{p^{s}}(x)$.
The properties of Elementary Abelian $p$-Extensions of $\mathbb{F}_{{p}^{s}}(x)$ have been studied in [@elem3], [@elem4], etc. In [@elem], T. Johnsen, S. Manshadi and N. Monzavi determined parameters of Goppa codes $C_{\mathcal{L}}(D,G)$ over plane projective curves $X$ with affine equation $A(y)=B(x)$, where $A(T) \in \mathbb{F}_{{p}^{s}}[T]$ is a separable, additive polynomial of degree $q = p^k$, for some $k$ and the degree $m$ of $B(T) \in \mathbb{F}_{{p}^{s}}[T]$ is not divisible by $p$. They studied codes over $X$ with the assumption that $deg~D \geq 4g-2$. In [@elem2], Garcia also studied Goppa codes over plane projective curves $X$ with affine equation $A(y)=B(x)$ but with a central hypothesis that there exists a subgroup $H$ of $\mathbb{F}_{{p}^{s}} \backslash \{0\}$ such that if $A(\beta)=B(\alpha)$ with $\alpha \in H \cup \{0\}$, then $\beta \in \mathbb{F}_{{p}^{s}}$. In this note, we investigate codes which are constructed by means of Elementary Abelian $p$-Extensions of $\mathbb{F}_{p^{s}}(x)$ without the above assumptions. We determine a simple condition for self-duality of these codes and list their second generalized Hamming weights.
This note is organised as follows. In section $2$, we recall some results about Goppa’s construction of linear codes, Elementary Abelian $p$-Extensions of $\mathbb{F}_{{p}^{s}}(x)$ and generalized Hamming weight of linear codes. In section $3$, we study the properties of one-point codes over this function field. In section $4$, we determine a simple condition for self-duality of these codes. In section $5$, we conclude the note listing the second generalized Hamming weights of these codes.
**Preliminaries**
=================
Goppa code
----------
Goppa’s construction is described as follows:
Let $F / \mathbb{F}_{p^{s}}$ be an algebraic function field of genus $g$. Let $P_{1},\cdots,P_{n}$ be pairwise distinct places of $F/ \mathbb{F}_{p^{s}}$ of degree 1. Let $D:=P_{1}+ \cdots +P_{n}$ and $G$ be a divisor of $F / \mathbb{F}_{p^{s}}$ such that $supp(G) \cap supp(D)=\emptyset$. The Goppa code $C_{\mathcal{L}}(D,G)$ associated with $D$ and $G$ is defined as $$C_{\mathcal{L}}(D,G):=\{(x(P_{1}), \cdots ,x(P_{n})):~ x \in \mathcal{L}(G)\} \subseteq \mathbb{F}_{p^{s}}^{n}.$$ Then, $C_{\mathcal{L}}(D,G)$ is an $[n,k,d]$ code with parameters $k=dim(\mathcal{L}(G))-dim(\mathcal{L}(G-D))$ and $d \geq n-deg(G)$.
We can define another code with the divisors $G$ and $D$ by using local components of Weil differentials. We define the code $C_{\Omega}(D,G) \subseteq \mathbb{F}_{p^{s}}^{n}$ by $$C_{\Omega}(D,G):=\{(\omega_{P_{1}}(1),\cdots,\omega_{P_{n}}(1)):~\omega \in \Omega_{F}(G-D)\}.$$ Then, $C_{\Omega}(D,G)$ is an $[n,k',d']$ code with parameters $k'=i(G-D)-i(G)$ and $d' \geq deg(G)-(2g-2).$
$C_{\Omega}(D,G)$ is the dual code of $C_{\mathcal{L}}(D,G)$ with respect to Euclidean scalar product on $\mathbb{F}_{p^{s}}^{n}$ i.e. $C_{\Omega}(D,G)=C_{\mathcal{L}}(D,G)^{\perp}.$ Let $\eta$ be a Weil differential of $F$ such that $\nu_{P_{i}}(\eta)=-1$ and $\eta_{P_{i}}(1)=1$ for $i=1,\cdots,n$. Then, $C_{\mathcal{L}}(D,G)^{\perp}=C_{\Omega}(D,G)=C_{\mathcal{L}}(D,D-G+(\eta))$.
([@book], p.200) Elementary Abelian $p$-Extensions of rational function field
-----------------------------------------------------------------------------
Let $K$ be a field of characteristic $p>0$. Consider a function field $F=K(x,y)$ with $$y^{q} + \mu y=f(x)\in K[x],$$ where $q=p^{k}>1$ is a power of $p$ and $0 \neq \mu \in K$. Assume that $m:=deg ~f >0$ is coprime to $p$. Also assume that all the roots of the equation $T^{q}+\mu T=0$ are in $K$. Then the following holds:
1. $[F:K(x)]=q$, and $K$ is the full constant field of $F$.
2. $F/K(x)$ is Galois. The set $A:=\{ \gamma \in K:~{\gamma}^{q}+\mu \gamma=0 \}$ is a subgroup of order $q$ of the additive group of $K$.
3. The pole $P_{\infty}\in \mathbb{P}_{K(x)}$ of $x$ in $K(x)$ has a unique extension $Q_{\infty} \in \mathbb{P}_{F}$, and $e(Q_{\infty}|P_{\infty})=q$. Hence $Q_{\infty}$ is a place of $F/K$ of degree one.
4. $P_{\infty}$ is the only place of $K(x)$ which ramifies in $F/K(x)$.
5. The genus of $F/K$ is $g=(q-1)(m-1)/2$.
6. The divisor of the differential $dx$ is $$(dx)=(2g-2)Q_{\infty}=((q-1)(m-1)-2)Q_{\infty}.$$
7. The pole divisor of $x$ is $(x)_{\infty}=qQ_{\infty}$ and the pole divisor of $y$ is $(y)_{\infty}=mQ_{\infty}$.
8. Let $r \geq 0$. Then, the elements $x^{i}y^{j}$ with $$0 \leq i, ~0\leq j \leq q-1, ~qi+mj \leq r$$ form a basis of the space $\mathcal{L}(r Q_{\infty})$ over $K$.
9. For all $\alpha \in K$, one of the following cases holds:\
Case (1). The equation $T^{q} + \mu T=f(\alpha)$ has $q$ distinct roots in $K$. In this case, for each $\beta$ with ${\beta}^{q} + \mu \beta=f(\alpha)$ there exists a unique place $P_{\alpha,\beta} \in \mathbb{P}_{F}$ such that $P_{\alpha,\beta}|P_{\alpha}$ and $y(P_{\alpha,\beta})=\beta$. Hence, $P_{\alpha}$ has $q$ distinct extensions in $F/K(x)$, each of degree one.\
Case (2). The equation $T^{q} + \mu T=f(\alpha)$ has no root in $K$. In this case, all extensions of $P_{\alpha}$ in $F$ have degree $>1$.
The Hermitian function field over $\mathbb{F}_{q^{2}}$ is defined by $$H=\mathbb{F}_{q^{2}}(x,y) ~~\text{with}~~ y^{q}+y=x^{q+1}.$$ This is a special case of Elementary Abelian $p$-Extension with $K=\mathbb{F}_{q^{2}}$, $\mu=1$ and $f(x)=x^{q+1}$.
$min\{q,m\} \geq 2$.
Generalized Hamming weight of Linear codes
------------------------------------------
The support of a $[n,k]$ linear code $C$ over $\mathbb{F}_{{p}^{s}}$ is defined by $$supp(C):=\{i~:~ x_{i} \neq 0 \text{ for some } \mathbf{x}=(x_{1},\cdots,x_{n}) \in C\}.$$ For $1 \leq l \leq k$, the *$l$th generalized Hamming weight* of $C$ is defined by $$d_{l}(C):=min\{~ \mid supp(D) \mid ~:~ D \text{ is a linear subcode of } C \text { with } dim(D)=l\}$$ In particular, the first generalized Hamming weight of $C$ is the usual minimum distance. The *weight hierarchy* of the code $C$ is the set of generalized Hamming weights $\{d_{1}(C), \cdots, d_{k}(C)\}$. These notions of generalized Hamming weights for linear codes were introduced by Wei in his paper [@ghw].
Few properties of generalized Hamming weight of $C$ have been listed in the following theorems.
[[@ghw]$($Monotonicity$)$]{} For an $[n,k]$ linear code $C$ with $k>0$, we have $$1 \leq d_{1}(C) < d_{2}(C)< \cdots <d_{k}(C) \leq n.$$
Let $\mathbf{H}$ be a parity check matrix of $C$, and let $\mathbf{H}_{i}$, $1 \leq i \leq n$, be its column vectors. For $I \subseteq \{1, \cdots, n \}$, let $\langle \mathbf{H}_{i}: i \in I \rangle$ denote the space generated by those vectors. Then
[[@ghw]]{} $d_{l}(C)=min\{~ \mid I \mid~:~ \mid I \mid -rank(\langle \mathbf{H}_{i}: i \in I \rangle)\geq l\}$
For Goppa code $C_{\mathcal{L}}(D,G)$, the $l$th generalized Hamming weight is given by the following theorem.
[[@ghwag]]{} Let $C=C_{\mathcal{L}}(D,G)$ be a code of dimension $k$ and $a:=dim(\mathcal{L}(G-D)) \geq 0$. Then for every $l$, $ 1 \leq l \leq k$, $$\begin{aligned}
d_{l}(C)&=min\{deg(D')~:~0 \leq D' \leq D,~ dim(\mathcal{L}(G-D+D')) \geq l+a\}\\
&=min\{n-deg(D')~:~0 \leq D' \leq D,~ dim(\mathcal{L}(G-D')) \geq l+a\}.\end{aligned}$$
**Goppa code over Elementary Abelian $p$-extensions of $\mathbb{F}_{p^{s}}(x)$**
================================================================================
Let $Q$ be a rational place in $F$. A positive integer $l$ is called pole number at $Q$ is there exists $z \in F$ such that $(z)_{\infty}=lQ$. Let $p_{1}<p_{2}<\cdots$ be the sequence of pole numbers at $Q$ (that is, $p_{r}$ is the $r$th pole number at $Q$); thus $dim (\mathcal{L}(p_{r}Q))=r$, so $p_{1}=0$.\
\
Few properties of Elementary Abelian $p$-Extensions of $\mathbb{F}_{p^{s}}(x)$, observed from [@book] and [@order], are listed in the following theorem.
[(Properties of function field)]{}
- The Weierstrass semigroup $H$ of $Q_{\infty}$ is generated by $m$ and $q$ i.e. $H=\langle q,m \rangle$.
- $H$ is symmetric numerical semigroup.
- $(2g-1)$ is the largest gap number of $Q_{\infty}$.
- $(2g-2)Q_{\infty}$ is a canonical divisor.
- $H^{*} = H^{*}
(D, Q_{\infty}) := \{ l \in \mathbb{N}_{0} : C_{\mathcal{L}}(D, lQ_{\infty}) \neq C_{\mathcal{L}}(D, (l-1)Q_{\infty})\}.$ If $l_{1},\cdots, l_{g}$ denotes the gap numbers of $Q_{\infty}$, then $ \{ l_{1}, \cdots l_{g}, n+2g-1-l_{g}, \cdots, n+2g-1-l_{1} \}=\{0,1, \cdots, n+2g-1 \} \setminus H^{*}$.
- The sequence $(C_{\mathcal{L}}(D,m_{i}Q_{\infty})_{i=0,\cdots, n}$ satisfy the isometry dual condition where $H^{*}=\{m_{1},\cdots,m_{n}\}$.
- The polynomial $T^{q}+\mu T-\prod_{i=1}^{m} (x-\alpha_{i})\in \mathbb{F}_{p^{s}}(x)[T]$, where $\alpha_{i} \in K$, is absolutely irreducible.
- $\{1, y, \cdots, y^{q-1}\}$ is an integral basis of $F/\mathbb{F}_{p^{s}}(x)$ for all $P \in \mathbb{P}_{\mathbb{F}_{p^{s}}(x)} \setminus P_{\infty}$.
In [@book], Stichtenoth has investigated one-point Goppa codes over Hermitian function field. Using similar idea, we define codes over Elementary Abelian $p$-Extension of $\mathbb{F}_{p^{s}}(x)$ and determine its parameters.
Let $K:=\mathbb{F}_{p^{s}}$, $s$ large enough. Let $m>0$ be an integer coprime to $p$. Choose $m$ distinct elements $\alpha_{1},\cdots,\alpha_{m} \in K$ . Let $f(x):=\prod_{i=1}^{m}(x-\alpha_{i})$. Denote by $P_{\alpha_{i}}$ the zero of $(x-\alpha_{i})$ in $K(x)$. Let $\beta_{1},\cdots,\beta_{q}$ denotes the zeroes of $T^{q}+\mu T$ in $K$ (choose $s$ large enough so that all the zeroes are in $K$). Then, for $1 \leq i \leq m$, $1 \leq j \leq q$, $P_{\alpha_{i},\beta_{j}}$ are the places of $F/K(x)$ of degree one (such places exist by section $2$).
For r $\in \mathbb{Z}$ we define\
$$C_{r}:=C_{\mathcal{L}}(D,rQ_{\infty}),$$ where $$D:=\sum_{i=1}^{m} \sum_{j=1}^{q} P_{\alpha_{i},\beta_{j}}$$.
Then, $C_{r}$ is a code of length $n:=qm$ over the field $K$. For $r <0$, $\mathcal{L}(rQ_{\infty})=\{0\}$, therefore $C_{r}=\{(0, \cdots, 0)\}$. For $r > n+(2g-2)=2qm-q-m-1$, $dim (C_{r})=n$, therefore $C_{r}=\mathbb{F}_{p^{s}}^{n}$. It remains to study codes $C_{r}$ with $ 0\leq r \leq 2qm-q-m-1$.
Parameters of $C_{r}$
---------------------
Let $J$ be the set of pole numbers of $Q_{\infty}$. For $b \geq 0$, let $$J(b):=\{u \in J|~u \leq b\}.$$ Then, $|J(b)|=dim(\mathcal{L}(bQ_{\infty}))$. From section 2, we have: $$J(b)=\{u \leq b|~u=iq+jm \text{ with } i \geq 0 \text{ and } 0 \leq j \leq q-1\}.$$ Hence $$|J(b)|=|\{(i,j) \in \mathbb{N}_{0} \times \mathbb{N}_{0};~ j \leq q-1 \text{ and } iq+jm \leq b \}|.$$\
[$($ [@book], $II.2.2 \text{ and } II.2.3)$]{} $C_{\mathcal{L}}(D,G)$ is an $[n,k,d]$ code with parameters $k=dim(\mathcal{L}(G))-dim(\mathcal{L}(G-D))$ and $d \geq n-deg(G)$. If $deg(G) < n$, then $k=dim(\mathcal{L}(G))$.\
Suppose that $0 \leq r \leq 2qm-q-m-1$. Then the following holds:
1. $dim (C_{r})=dim(\mathcal{L}(rQ_{\infty}))-dim(\mathcal{L}((r-qm)Q_{\infty}))$. For $0 \leq r < qm$, $dim (C_{r})=|J(r)|$. For $qm-q-m-1 < r <qm$, we have $dim (C_{r})=r+1-(q-1)(m-1)/2$.
2. The minimum distance $d$ of $C_{r}$ satisfies $$d \geq qm-r.$$ If $r=qb$, where $0 \leq b < m $ or if $r=cm$, where $0 \leq c <q $, then $d=qm-r$. In addition, if $r \geq qm-q-m$ then $C_{r}$ is not MDS code.
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1. By Theorem $3.3$ and as $D \sim qmQ_{\infty}$ we have, $dim (C_{r})=dim(\mathcal{L}(rQ_{\infty}))-dim(\mathcal{L}((r-qm)Q_{\infty}))$. For $0 \leq r < qm$ (i.e. $deg (G) < n$), $$dim (C_{r})= dim(\mathcal{L} (r Q_{\infty}))= |J(r)|.$$ For $qm-q-m-1 < r <qm$ (i.e. $2g-2< deg (G) < n$), Riemann-Roch theorem yields $$dim (C_{r})=dim(\mathcal{L}(rQ_{\infty}))=deg(rQ_{\infty})+1-g=r+1-(q-1)(m-1)/2.$$
2. The inequality $d \geq qm-r$ directly follows from Theorem $3.3$. If $r=qb$, where $0 \leq b < m $, choose $b$ distinct elements from the set $\{\alpha_{1},\cdots, \alpha_{m}\}$ ( where as before $\alpha_{i}$ are such that $f(x)=\prod_{i=1}^{m} (x-\alpha_{i})$). Let us call these elements $\gamma_{1},\cdots,\gamma_{b}$. Then the element $$z_{1}:=\prod_{j=1}^{b} (x-\gamma_{j}) \in \mathcal{L}(rQ_{\infty})$$ has exactly $qb=r$ distinct zeros in $D$. The weight of the corresponding codeword in $C_{r}$ is $qm-r$. Hence, $d=qm-r$.\
\
Similarly, if $r=cm$, where $0 \leq c < q $, choose $c$ distinct elements from the set $\{\beta_{1},\cdots, \beta_{q}\}$. Let us call these elements $\tau_{1},\cdots,\tau_{c}$. Then the element $$z_{2}:=\prod_{j=1}^{c} (y-\tau_{j}) \in \mathcal{L}(rQ_{\infty})$$ has exactly $cm=r$ distinct zeros in $D$. The weight of the corresponding codeword in $C_{r}$ is $qm-r$. Hence, $d=qm-r$.\
\
If $r=qb$ and $C_{r}$ is MDS code then, $d=n-k+1$ implies $g=0$ which is not possible. Similarly for $r=cm$.
Using the idea from [@tru], we have the following result for minimum distance of $C_{r}$.
Assume $m>q$. For $qm \leq r \leq 2qm-q-m-1$ we have $0 \leq r^{\perp}:=2qm-q-m-1-r \leq qm-q-m-1$. Let $t^{\perp} \leq r^{\perp}$ be the largest integer such that $t^{\perp}$ is a pole number at $Q_{\infty}$ i.e. $t^{\perp}=aq+bm$ where $0 \leq a \leq m-2$ and $0 \leq b \leq q-1$. Then, the minimum distance of $C_{r}$ satisfies $$d(C_{r})=a+2.$$
Let $\mathbf{H}$ be a parity check matrix of $C_{r}$. From section $2$, we have $\{1,x,y,\cdots,x^{a}$, $x^{a-1}y, \cdots, y^{b}\}$ is a basis for $\mathcal{L}(t^{\perp}Q_{\infty})$. Choose $\beta \in K$ such that $\beta^{q}+ \mu \beta=0$. Let $\mathbf{H}_{1}$ be a submatrix of $\mathbf{H}$ with columns corresponding to $P_{\alpha_{1},\beta},\cdots,P_{\alpha_{a+2},\beta}$. We write $\mathbf{H}_{1}$ in the following form using row reduction. $$\mathbf{H}_{1}=
\begin{bmatrix}
1 & 1 & 1 & \cdots & 1 \\
\alpha_{1} & \alpha_{2} & \alpha_{3} & \cdots & \alpha_{a+2}\\
\alpha_{1}^{2} & \alpha_{2}^{2} & \alpha_{3}^{2} & \cdots & \alpha_{a+2}^{2}\\
\vdots & \vdots & \vdots & ~ & \vdots \\
\alpha_{1}^{a} & \alpha_{2}^{a} & \alpha_{3}^{a} & \cdots & \alpha_{a+2}^{a}\\
0 & 0 & 0 & \cdots & 0 \\
\vdots & \vdots & \vdots & ~ & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{bmatrix}$$ Here, $rank(\mathbf{H}_{1})=a+1$ and $\mathbf{H}_{1}$ has $a+2$ columns, so the columns of $\mathbf{H}_{1}$ are linearly dependent. Therefore, $d(C_{r}) \leq a+2$.
On the other hand, we choose any $a+1$ distinct columns from $\mathbf{H}$. Let us call this matrix $\mathbf{H}_{2}$. Since each column of $\mathbf{H}$ corresponds to a place $P_{\alpha,\beta}$ of degree $1$, we reorder columns of $\mathbf{H}_{2}$ according to $\alpha's$ as follows. $$\begin{matrix}
P_{\alpha_{1},\beta_{1,1}}, & P_{\alpha_{1},\beta_{1,2}}, & \cdots, & P_{\alpha_{1},\beta_{1,w_{1}}}\\
P_{\alpha_{2},\beta_{2,1}}, & P_{\alpha_{2},\beta_{2,2}}, & \cdots, & P_{\alpha_{1},\beta_{2,w_{2}}}\\
\vdots & \vdots & \vdots & \vdots \\
P_{\alpha_{\gamma},\beta_{\gamma,1}} & P_{\alpha_{\gamma},\beta_{\gamma,2}} & \cdots, & P_{\alpha_{\gamma},\beta_{\gamma,w_{\gamma}}}\\
\end{matrix}$$ where $\alpha_{i}$’s are pairwise distinct and $w_{1}+w_{2}+\cdots+w_{\gamma}=a+1$ with $w_{1} \geq w_{2} \geq \cdots \geq w_{\gamma} \geq 1$. For $0 \leq j_{i} \leq w_{i}-1; ~1 \leq i \leq \gamma$, $x^{i-1}y^{j_{i}}$ belongs to basis of $\mathcal{L}(t^{\perp}Q_{\infty})$. We rewrite these basis elements in the form $$\begin{matrix}
1, & y, & y^{2}, & \cdots, & y^{w_{1}-1}\\
x, & xy, & xy^{2}, & \cdots, & xy^{w_{2}-1}\\
x^{2}, & x^{2}y, & x^{2}y^{2}, & \cdots, & x^{2}y^{w_{3}-1}\\
\vdots & \vdots & \vdots & ~ & \vdots \\
x^{\gamma-1}, & x^{\gamma-1}y, & x^{\gamma-1}y^{2}, & \cdots, & x^{\gamma-1}y^{w_{\gamma}-1}\\
\end{matrix}$$
Then, we extract an $(a+1) \times (a+1)$ submatrix $\mathbf{H'}$ of $\mathbf{H}_{2}$ such that each row corresponds to a function above in the given order. That is, $\mathbf{H'}=[\mathbf{H'}_{i,j}]$, $i,j=1,2,\cdots,\gamma$ where $\mathbf{H'}_{i,j}$ is a $(w_{i} \times w_{j})$ matrix with $\mathbf{H'}_{i,j}=\alpha_{j}^{i-1}\mathbf{B}_{i,j}$ with $$\mathbf{B}_{i,j}=
\begin{bmatrix}
1 & 1 & 1 & \cdots & 1\\
\beta_{j,1} & \beta_{j,2} & \beta_{j,3} & \cdots & \beta_{j,w_{j}}\\
\beta_{j,1}^{2} & \beta_{j,2}^{2} & \beta_{j,3}^{2} & \cdots & \beta_{j,w_{j}}^{2}\\
\vdots & \vdots & \vdots & ~ & \vdots \\
\beta_{j,1}^{w_{i}-1} & \beta_{j,2}^{w_{i}-1} & \beta_{j,3}^{w_{i}-1} & \cdots & \beta_{j,w_{j}}^{w_{i}-1}\\
\end{bmatrix}$$ Then, from [@tru], Lemma $2$ and Lemma $3$, $$det(\mathbf{H'})=(\prod_{i=1}^{\gamma} det(\mathbf{B}_{i,i})).(\prod_{j=2}^{\gamma} \rho_{j}^{w_{j}})$$ where $$\rho_{j}=\prod_{i=1}^{j-1}(\alpha_{j}-\alpha_{i}),~j=2,3,\cdots,\gamma.$$ And any $a+1$ columns of $\mathbf{H}$ are linearly independent over $K$. Hence, $d(C_{r}) \geq a+2$.
**Condition for self-duality of codes**
=======================================
A linear code $C$ is called self-dual if $C=C^{\perp}$, where $C^{\perp}$ is the dual of $C$ with respect to Euclidean scalar product on $\mathbb{F}_{p^{s}}^{n}$. Self-dual codes are an important class of linear codes. We give a simple criterion for self-duality of codes over Elementary Abelian $p$-Extensions of $\mathbb{F}_{p^{s}}^{n}$ by using the following theorem from [@equal].
We call two divisors $G$ and $H$ equivalent with respect to $D$ if there exists $u \in F$ such that $H=G+(u)$ and $u(P_{i})=1$, for all $i=1,\cdots,n.$
Suppose $n > 2g + 2$. Let $G$ and $H$ be two divisors of the same degree $m$ on a function field of genus $g$. If $C_{\mathcal{L}}(D,G)$ is not equal to $0$ nor to $K^{n}$ and $2g - 1 < m < n -1$, then $C_{\mathcal{L}}(D,G)=C_{\mathcal{L}}(D,H)$ if and only if $G$ and $H$ are equivalent with respect to $D$.
Let $f(x)$ and $D$ as before. Clearly, $n=qm >2g+2 $. Let $\eta:=\frac{f'(x) dx}{f(x)}$ be a differential then we get, $\nu_{P}(\eta)=-1$ and $res_{P}(\eta)=1$ for all $P \in supp(D)$. Therefore, for any divisor $G$ on $F$ with $supp(D)~ \cap~ supp(G)= \emptyset$, we have $C_{\mathcal{L}}(D,G)^{\perp}=C_{\mathcal{L}}(D,D+(\eta)-G)$.
The dual code of $C_{r}$ is given by $$C_{r}^{\perp}=\bar{a}~C_{2qm-q-m-1-r}$$ where $(\bar{a})^{-1}=((f'(x))(P_{\alpha_{1},\beta_{1}}),\cdots,(f'(x))(P_{\alpha_{m},\beta_{q}})) \in (\mathbb{F}^{*}_{p^{s}})^{n}$. Hence, if $qm-q-m+1 \leq r \leq qm-2$ then $C_{r}$ is quasi-self-dual if and only if $r=\frac{(2qm-q-m-1)}{2}$.
$$\begin{aligned}
C_{r}^{\perp}&=C_{\mathcal{L}}(D, D+ (\eta)-rQ_{\infty})\\
&=C_{\mathcal{L}}(D, D+(f'(x))-(f(x))+(dx)-rQ_{\infty})\\
&=C_{\mathcal{L}}(D, D+(f'(x))-D+qm Q_{\infty}+(2g-2)Q_{\infty}-r Q_{\infty})\\
&=C_{\mathcal{L}}(D, (f'(x))+(qm+(2g-2-r) Q_{\infty})\\
&=\bar{a} C_{\mathcal{L}}(D,(2qm-q-m-1-r) Q_{\infty})\\
&=\bar{a} C_{2qm-q-m-1-r}\end{aligned}$$
Now it follows directly from Theorem $4.2$ that $C_{r}$ is quasi-self-dual if and only if $r=\frac{(2qm-q-m-1)}{2}$.
For $0 \leq r \leq 2qm-q-m-1$, define $s:=2qm-q-m-1-r$. The dimension of $C_{r}$ is given by
$dim(C_{r})=\left \{
\begin{array}{ccc}
\mid J(r) \mid & \text{ for } & 0 \leq r < qm,\\
qm-\mid J(s) \mid & \text{for } & qm \leq r \leq 2qm-q-m-1. \\
\end{array}
\right.$\
Let $G$ be a divisor of $F$ with $deg(G)=qm-\frac{q}{2}-\frac{m}{2}-\frac{1}{2}$. Clearly, $n >2g+2 \text{~~and~~} 2g-1 < deg(G) < n-1$. Let $H:=D+(\eta)-G$. Then, $deg(G)=deg(H)$. From Theorem $4.2$, it follows that
$C_{\mathcal{L}}(D,G)$ is self-dual if and only if $2G$ is equivalent to $(f'(x))+(2qm-q-m-1)Q_{\infty}$ with respect to $D$.
By Theorem $4.2$, $$\begin{aligned}
& C_{\mathcal{L}}(D,G)=C_{\mathcal{L}}(D,D+(\eta)-G)\\
\Leftrightarrow &~G=D+(\eta)-G+(u) \text{ for some } u \in F \text{ such that } u(P)=1 \text{ for each } P \in supp(D)\\
\Leftrightarrow &~(u)+(\eta)=2G-D\\
\Leftrightarrow &~(u)+(f'(x))+(dx)-(f(x))=2G-D\\
\Leftrightarrow &~(u)+(f'(x))+[(q-1)(m-1)-2]Q_{\infty}-D+ qm Q_{\infty}=2G-D\\
\Leftrightarrow &~(f'(x))=2G-(2qm-q-m-1)Q_{\infty}-(u).\\
\end{aligned}$$
Let $p=2$. Let $K=\mathbb{F}_{4}$. Let $\omega$ be a primitive element of $\mathbb{F}_{4}$. Consider $F=K(x,y)$ with $$y^{2}+y=x(x-1)(x-\omega)$$ Therefore, all roots of $T^{2}+T$ is in $K$. The genus of $F/K$ is $g=1$. Let $$f(x)=x(x-1)(x-\omega).$$ Let $P_{0},P_{1}$ and $P_{\omega}$ denotes zero in $K(x)$ of $x,(x-1)$ and $(x-\omega)$ respectively. Then, each of $P_{0},P_{1}$ and $P_{\omega}$ has exactly two extensions in $F$. Similarly, the zero of $(x-\omega^{2})$ denoted by $P_{\omega^{2}}$ has two extensions in $F$, say, $Q_{1}$ and $Q_{2}$. Let $D=(f(x))_{0}$ and let $G$ be a divisor in $F$ equivalent to $Q_{1}+Q_{2}+Q_{\infty}$ with respect to $D$. Then $C_{\mathcal{L}}(D,G)$ is self dual.\
Conversely, if $C_{\mathcal{L}}(D,G)$ is self-dual code with $D$ as above then $2G$ is equivalent to $2(Q_{1}+Q_{2}+Q_{\infty})$ with respect to $D$.
**Generalized Hamming weight of code $C_{r}$**
==============================================
Using the idea of [@ghwag], we have the following lemma.
Let $r \leq qm$ be a pole number at $Q_{\infty}$. Then, $r=iq+jm$ where $i \geq 0$ and $0 \leq j \leq q-1$. If either $i=0$ or $j=0$ then $\exists$ a divisor $0 \leq D' \leq D$ such that $rQ_{\infty} \sim D'$.
For $i=0$ and $j=0$, $D'=0$ works.\
\
If $i=0$ and $j \neq 0$ then, $r=jm$. With notation as in section $3$, choose $j$ elements from $\beta_{1}, \cdots, \beta_{q}$. Denote these elements by $\tau_{1},\cdots, \tau_{j}$. Define $g:=\prod_{t=1}^{j}(y-\tau_{t})$. Then, $(g)=D'-rQ_{\infty}$. Therefore, $D' \sim rQ_{\infty}$.\
\
Similarly for $j=0$ and $i \neq 0$.
A positive integer $r \leq qm$ is said to have property (\*) if $r$ is a pole number at $Q_{\infty}$, $r=iq+jm$ for $i \geq 0$, $0 \leq j \leq q-1$ and either $i=0$ or $j=0$.
As before, let $C_{r}=C_{\mathcal{L}}(D,rQ_{\infty})$. If for $1 \leq l \leq k$, $r-p_{l}$ or $qm-r+p_{l}$ has the property (\*) then, $d_{l}(C_{r}) \leq qm-r+p_{l}$.
If $r-p_{l}$ has the property (\*), then according to Lemma $5.1$, there exists a divisor $0 \leq D' \leq D$ such that $(r-p_{l})Q_{\infty} \sim D'$. Thus, $dim(\mathcal{L}(rQ_{\infty}-D'))=dim( \mathcal{L}(p_{l}Q_{\infty}))=l$. Hence, from Theorem $2.5$, $d_{l}(C_{r}) \leq qm-r+p_{l}$.
Now, if $qm-r+p_{l}$ has the property (\*) then according to Lemma $5.1$ there exists a divisor $0 \leq D'' \leq D$ such that $(qm-r+p_{l})Q_{\infty} \sim D''$. Also, $qmQ_{\infty} \sim D$. Thus, $D-rQ_{\infty}+p_{l}Q_{\infty} \sim D''$. Therefore, $D':=D-D'' \sim (r-p_{l})Q_{\infty}$. Hence, $d_{l}(C_{r}) \leq qm-r+p_{l}$.\
An immediate corollary of the above theorem is the following.
Assume that $m >q$. If for $1 \leq r <qm $, $r$ or $qm-r$ has the property (\*), then the minimum distance of $C_{r}$ is $d=qm-r$.
For $C_{r}:=C_{\mathcal{L}}(D,rQ_{\infty})$, we have $d_{k}(C_{r})=n=qm$ where $k=dim(C_{r})$.
The following result is stated in Munuera [@ghwag].
Let $C=C_{\mathcal{L}}(D,G)$ be a code of dimension $k$ and $dim(\mathcal{L}(G-D))=:a > 0$. Then, for $1 \leq l \leq k$, we have $d_{l}(C) \leq deg(D')$ for every effective divisor $D' \leq D$ such that $dim(\mathcal{L}(D'))>l$.
Using the above theorem, we get the following result.
Assume that $m >q$. If $qm \leq r \leq 2qm-q-m-1$, then $d_{2}(C_{r}) \leq \text{min} \{2q, m\}$.
Since $p_{3}=\text{min} \{2q, m\}$ therefore, from Lemma $5.1$, $\exists$ $0\leq D' \leq D$ such that $p_{3}Q_{\infty} \sim D'$. Then, $dim(\mathcal{L}(D'))=dim(\mathcal{L}(p_{3}Q_{\infty}))=3$. Hence, $d_{2}(C_{r}) \leq deg~D'=p_{3}=\text{min} \{2q, m\}$.
Assume $m>2q$. Suppose $qm \leq r \leq 2qm-q-m-1$. Then $0 \leq r^{\perp}:=2qm-q-m-1-r \leq qm-q-m-1$. Let $t^{\perp} \leq r^{\perp}$ be the largest integer such that $t^{\perp}$ is a pole number of $Q_{\infty}$ i.e. $t^{\perp}=aq+bm$ where $0 \leq a \leq m-3$ and $0 \leq b \leq q-1$. Then, $$d_{2}(C_{r})=a+3.$$
Let $\mathbf{H}$ be a parity check matrix for $C_{r}$ over $K$. Choose $\beta \in K$ such that $\beta^{q}+\mu \beta=0$. $\{1,x,y,\cdots,x^{a},x^{a-1}y, \cdots, y^{b}\}$ is a basis for $\mathcal{L}(t^{\perp}Q_{\infty})$. Let $\mathbf{H}_{1}$ be a submatrix of $\mathbf{H}$ with columns corresponding to $P_{\alpha_{1},\beta},\cdots,P_{\alpha_{a+3},\beta}$ (possible since $a+3\leq m$). By using row reduction, we make $\mathbf{H}_{1}$ as follows. $$\mathbf{H}_{1}=
\begin{bmatrix}
1 & 1 & 1 & \cdots & 1 \\
\alpha_{1} & \alpha_{2} & \alpha_{3} & \cdots & \alpha_{a+3}\\
\alpha_{1}^{2} & \alpha_{2}^{2} & \alpha_{3}^{2} & \cdots & \alpha_{a+3}^{2}\\
\vdots & \vdots & \vdots & ~ & \vdots \\
\alpha_{1}^{a} & \alpha_{2}^{a} & \alpha_{3}^{a} & \cdots & \alpha_{a+3}^{a}\\
0 & 0 & 0 & \cdots & 0 \\
\vdots & \vdots & \vdots & ~ & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{bmatrix}$$ Here, $rank(\mathbf{H}_{1})=a+1$ and $\mathbf{H}_{1}$ has $a+3$ columns. So, by Theorem $2.4$, $d_{2}(C_{r}) \leq a+3$.\
\
On the other hand from Theorem $2.3$, $d_{2}(C_{r}) \geq d_{1}(C_{r})+1=(a+2)+1=a+3$.
FENG-RAO DISTANCES ON NUMERICAL SEMIGROUPS
------------------------------------------
\
The Feng-Rao distances on numerical semigroups is defined in [@semi]. We will explain it briefly in this subsection.
Let $A$ be a numerical subgroup. If for a set $G \subseteq A$, every $x \in A$ can be written as a linear combination $$x=\sum_{g \in G} \lambda_{g} g,$$ where finitely many $\lambda_{g} \in \mathbb{N} \cup \{0\}$ are non-zero, then we say that $A$ is generated by $G$. It is well known that every numerical semigroup is finitely generated. An element $x \in A$ is said to be irreducible if $x=a+b$ for $a,b \in A$ implies $a.b=0$. Every generator set contains the set of irreducible elements and the set of irreducibles actually generates $A$. The number of irreducible elements is called the *embedding dimension* of $A$. We enumerate the elements of $A$ in increasing order $$A=\{\rho_{1}=0 < \rho_{2}< \cdots\}.$$ For $a,b \in \mathbb{Z}$ given, we say that $a$ *divides* $b$, and write $$a \leq_{A} b, \text{ if } b-a \in A.$$ The binary relation is an order relation.
The set $D(y)$ denotes the set of *divisors* of $y$ in $A$, and for given $M=\{m_{1}, \cdots, m_{r} \} \subseteq A$, we write $D(M)=D(m_{1}, \cdots, m_{r})=\bigcup^{r}_{i=1} D(m_{i})$.
Let $A$ be a numerical subgroup, that is, a submonoid of $\mathbb{N}$ such that $\mid(\mathbb{N}\backslash A)\mid < \infty$ and $0 \in A$. We call $g:=\mid(\mathbb{N}\backslash A)\mid$ the *genus* of $A$. The unique element $c \in A$ such that $c-1 \not \in A$ and $c+l \in A$ for all $l \in \mathbb{N}$ is called the *conductor* of $A$. The (classical) Feng-Rao distance of $A$ is defined by the function\
$
\begin{array}{llll}
\delta_{FR}:&A &\rightarrow &\mathbb{N}\\
&x &\mapsto &\delta_{FR}(x):=min \{ \mid D(m_{1}) \mid ~: ~m_{1} \geq x, ~m_{1} \in A\}\\
\end{array}
$
There are some well-known results about the function $\delta_{FR}$ for an arbitrary semigroup $A$. One of the important result is that $\delta_{FR}(x) \geq x+1-2g$ for all $x \in A$ with $x \geq c$. The proof of the following theorem can be found in [@semi].
Let $A=\{0=\rho_{1}<\rho_{2}< \cdots < \rho_{n}< \cdots \}$ be an embedding dimension two numerical subgroup. Then $$d_{r}(C_{l}) \geq \delta_{FR}(l+1)+\rho_{r}$$ for $r=1, \cdots, k_{l}$, where $C_{l}$ is a code in an array of codes as in [@array] and $k_{l}$ is the dimension of $C_{l}$.
Using the above theorem, we determine the second generalized Hamming weight of code $C_{r}$ (as defined in section $3$).
Assume $m >q$. For $r <qm$, $d_{2}(C_{r}) \geq qm-r+q.$
Assume $m >q$. For $r <qm$, if $r-q$ or $qm-r+q$ satisfies the property (\*) then, $$d_{2}(C_{r})=qm-r+q.$$
Applying Theorem $5.3$, we get $d_{2}(C_{r})\leq qm-r+q.$\
\
On the other hand, as $\rho_{2}=q$ and as the dual code of $C_{r}$ form an array of codes ( for details see [@array]), from Theorem $4.2$ and the Theorem $5.10$, we have $$\begin{aligned}
d_{2}(C_{r})&=d_{2}(C_{2qm-q-m-1-r}^{\perp})\\
&\geq \delta_{FR}(2qm-q-m-r)+q.\end{aligned}$$ Since $r<qm$, we have $2qm-q-m-r \geq 2g=qm-q-m+1$, therefore $$\begin{aligned}
d_{2}(C_{r})&\geq \delta_{FR}(2qm-q-m-r)+q\\
&\geq 2qm-q-m-r+1-(qm-q-m+1)+q\\
&=qm-r+q.\end{aligned}$$ Hence proved.
The following result is stated in Munuera [@ghwag].
Let $C_{\mathcal{L}}(D,G)$ be a code of dimension $k$ and abundance $dim(C_{\mathcal{L}}(D,G-D))=:a \geq 0$. If there is a rational point $Q$ not in $D$ and $C_{\mathcal{L}}(D,G-p_{r}Q) \neq \{0\}$, where $p_{r}$ is rth pole number at $Q$, then for every $l$, $1 \leq l \leq k$, $$d_{l}(C_{\mathcal{L}}(D,G)) \leq d_{1}(C_{\mathcal{L}}(D,G-p_{l}Q)).$$
Using the above theorem and Theorem $5.11$, we get the following result.
Assume $m >q$. For $r< qm$. If $r-q$ and $qm-r+q$ doesn’t satisfy the property (\*), then $$qm-r+q \leq d_{2}(C_{r}) \leq qm-\overline{(r-q)}.$$ where $\overline{(r-q)}$ is the largest pole number less than or equal $(r-q)$ that satisfies the property (\*).
**Concluding remarks**
======================
In this note, we have defined one-point Goppa codes over Elementary Abelian $p$-Extension of $\mathbb{F}_{p^{s}}(x)$ and determined its dimension and exact minimum distance in few cases. We have also given a simple criterion for one-point Goppa codes to be quasi-self-dual and Goppa codes with divisor $G$ (not necessarily one-point) to be self-dual. We have listed exact second generalized Hamming weight of these codes in few cases.
[9]{}
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HENNING STICHTENOTH, Algebraic Function Fields and Codes, Springer-Verlag Berlin Heidelberg, 1993.
KYEONGCHEOL YANG AND P. VIJAY KUMAR, On the true minimum distance of Hermitian codes, Springer, Berlin, Heidelberg, 2006.
MANUEL DELGADO, JOSE I. FARRAN, PEDRO A. GARCIA-SANCHEZ AND DAVID LLENA, On the Weight Hierarchy of Codes Coming From Semigroups With Two Generators, IEEE Transactions on Information Theory, Volume: 60 , Issue: 1, Jan. 2014, 282-295.
OLAV GEIL, CARLOS MUNUERA, DIEGO RUANO AND FERNANDO TORRES, On the order bounds for one-point AG codes, Advances in Mathematics of Communications, 2011, 5 (3), 489-504.
T. JOHNSEN, S. MANSHADI AND N. MONZAVI, A determination of the parameters of a large class of Goppa codes, IEEE Transactions on Information Theory, Volume: 40, Issue: 5, Sep 1994.
V. K. WEI, Generalized Hamming weights for linear codes. IEEE Trans. Inform. Theory, vol. 37, 1991, pp. 1412-1418.
|
---
abstract: 'We report an experimental verification of conformal mapping with kitchen aluminum foil. This experiment can be reproduced in any laboratory by undergraduate students and it is therefore an ideal experiment to introduce the concept of conformal mapping. The original problem was the distribution of the electric potential in a very long plate. The correct theoretical prediction was recently derived by A. Czarnecki [@andrzej].'
author:
- 'S. Haas'
- 'D. A. Cooke'
- 'P. Crivelli'
title: Testing conformal mapping with kitchen aluminum foil
---
Statement of the problem {#sec:Problem}
========================
The initial problem is part of the collection of the Moscow Institute of Physics and Technology [@cahn]. A voltage $U_s$ is applied to the corners A and B of a semi-infinite long metallic ruler as shown in Fig. \[fig:ruler\]).
![A semi-infinite metal ruler to which the voltage $U_s$ is applied between A and B. Between C and D the voltage $U_d$ is being measured.[]{data-label="fig:ruler"}](ruler)
The question is: when one measures the voltage difference $U_d$ between C and D, how does the voltage difference behave further down the ruler? Assuming that the thickness of the ruler can be neglected, this can be reduced to a two-dimensional problem. A. Czarnecki derived the correct solution to this problem conformal mapping [@andrzej]. In this tutorial, we present the experimental verification of these new calculations.
Conformal mapping
=================
Conformal mapping is a mathematical technique which is widely used not only in physics but also in engineering. The main idea behind this technique is to simplify a certain problem by mapping it to a better suited geometry in order to simplify its solution.
Mathematical definition
-----------------------
A complex function $f : U \to \mathbb{C}$ is called holomorphic, if it is complex differentiable at any point in its domain. Or in other words, if the following limit exists: $$f'(z_0) = \lim_{z \to z_0}\frac{f (z) -f (z_0)} {z - z_0}$$ A holomorphic function $g: U \to \mathbb{C}$ is said to be conformal if $g'(z) \neq 0 \, \forall z \in U$. Conformal functions have the properties to preserve locally angles and the shapes of infinitesimally small figures.
Example: the mercator projection
--------------------------------
The Mercator projection is a cylindrical map projection. It is probably the most common way to map the spherical earth surface in two dimension. The corresponding map is: $$x = R (\theta -\theta_0)$$ $$y = R \ln\Big[\tan{\frac{1}{4}\pi-\frac{1}{2}\phi}\Big]$$ Where $R$ is the Earth radius, $\phi$ is the latitude, $\theta$ the longitude and $\theta_0$ an arbitrary central meridian (commonly chosen to be the one of Greenwich). This mapping satisfies the above condition of a conformal map and visualises well its properties. The circles of longitude and latitude are perpendicular on the map and on small scales the shapes of objects are preserved. whereas large objects can change their shape and size depending where they are located on the globe. For example according to Fig. \[fig:mercator\] the size of Greenland and Africa would be of the same order.
![Mercator projection [@mercator].[]{data-label="fig:mercator"}](MercatorProjection_1000.pdf)
Solution
========
The solution of the problem presented in Sec. \[sec:Problem\] as suggested in [@cahn] starts with comparing the potential differences $U_x$ at two nearby pairs of points: $$U_{x+dx}-U_x=\alpha(x)U_xdx$$ It is then argued that since the ruler is semi-finite, the coefficient $\alpha(x)=\alpha$ is constant and does not depend on position. This leads to the differential equation: $$U_x'=\alpha U_x$$ If this equation would hold everywhere one would get the final solution: $$\label{eq:originalSol}
U_x=U_s\left(\frac{U_s}{U_d}\right)^{-x/d},$$ where $U_s$ is the voltage applied at the edge and $U_d$ the measured voltage difference at a given point with a distance $d$ from the edge. However, the ruler is only semi-infinite and not infinite, so the assumption that the voltage does not depend on the position is not fulfilled. In fact, points very close to the beginning of the ruler do not have the same neighbourhood as points further down the ruler.
Czarnecki [@andrzej] derived the correct solution to this problem using conformal mapping. Complex coordinates $z=x+iy$ were introduced such that the corner B corresponds to $z_B=0$ and the upper corner A to $z_A=i$. Looking at the image of the ruler under the mapping $z \to w(z)=e^{\pi z}$, one sees that the corners A and B are mapped on the $x$-axis, $w_A=(-1,0)$ and $w_B=(1,0)$ as shown in Fig. 2.
If an infinite ruler would be mapped with this function, this would cover the whole upper half-plane. The actual advantage of this mapping is that every function that only depends on the distance to the corners is now symmetric to the real axis since both corners lie on that axis. This symmetry leads to the solution: $$\label{eq:AndrzejSol}
U(x)=\frac{U_s}{c-\ln s}\ln \frac{1+e^{-\pi x}}{1-e^{-\pi x}},$$ where $s$ is the length of the contacts and $c$ is a constant depending on their detailed geometry but not on their size if $s$ is small. The expression $(c-\ln s)$ describes physical rather than idealized contacts and we thus refer to it as the reality factor. These contacts parameters are dependent on the width of the ruler.
If one assumes that the distance between the left corner and the position where we measure the voltage difference is sufficiently large (more then a third of the width of the stripe), this formula can be approximated to: $$U(x)=U_{d_1} \left( \frac{U_{d_2}} {U_{d_1}} \right)^{\frac{x-d_1}{d_2-d_1}}$$ where $U_{d_1}$ and $U_{d_2}$ are the voltage differences measured at distances $d_1$ and $d_2$ from the edge.
Experimental verification
=========================
Experimental setup
------------------
To realize this experiment the following basic laboratory equipment is required:
- Standard power supply (30 V, 3 A)
- Standard voltmeter ($\pm0.01$ mV)
- Aluminium foil (10-50 $\mu$m, typical thickness of aluminium kitchen foil)
The experiment consists of applying and measuring the voltage difference at different points. To simulate a semi-infinite metallic ruler, the metal stripe made of aluminium foil was cut much longer than actually needed. The wires, used to apply the voltage difference, were pulled through holes in the contact stripes as shown in Fig. \[setup\]. The experimental configuration can be described by the following parameters: the length of the ruler $L$, its width $W_R$ , its thickness $T$ and the width of the contact stripes $W_C$ (see Fig. \[setup\]).
![Sketch of the experimental setup.[]{data-label="setup"}](sketch2)
Results
-------
The voltage difference was measured in 5 mm steps form 0 to 50 mm. These measurements were performed with different settings in order to investigate the influence of the following factors: $L$, $T$, $W_R$ and $W_C$. The experimental uncertainties to be taken into account are the ones of the voltmeter ($\pm$0.01 mV) and the one of the measuring position ($\pm$0.3 mm).
The results for $L$ and $T$ are presented in Figs. \[VvsL\]-\[VvsT\]. As one can see, the measured points for ruler lengths of 300 and 500 mm are the same within the experimental errors, one can thus conclude that a ruler with more than 300 mm is sufficiently long for our experiment and it is a good approximation of a semi infinite ruler. The results are also unaffected when using two different thicknesses of $T$=0.01 and 0.05 mm. Therefore aluminium kitchen foil is thin enough to approach a 2-dimensional problem as required by this experiment.
![Influence of the foil length $L$ on the voltage difference.[]{data-label="VvsL"}](L)
![Influence of the foil thickness $T$ on the voltage difference[]{data-label="VvsT"}](T)
To study the effect of the contacts geometry and thus the $(c-\ln s)$ parameter of Eq.\[eq:AndrzejSol\], the width $W_C$ was varied. Apart from the first measured point at $x=0$ mm, the obtained values are the same for all the others distances within the experimental errors (see Fig. \[VvsWC\]).
![Dependence of the width of the voltage stripe $W_c$ on the voltage difference.[]{data-label="VvsWC"}](V)
The last parameter to be investigated is the influence of the width $W_R$ of the ruler. According to the original calculation (Eq. \[eq:originalSol\]) one would expect that the measured values are independent on $W_R$. This is in contradiction with the data as shown in Fig. \[WR\] confirming the inadequacy of this solution and correctness of the newer calculations
![Influence of the width $W_R$ on the voltage difference and fit to the data using Eq. \[eq:AndrzejSol\][]{data-label="WR"}](WR)
Conclusions
===========
Our measurements are in very good agreement with the predictions of the new calculations obtained applying conformal mapping by Czarnecki and point out that the original solution in which the problem of calculating the distribution of the electric potential in a very long plate was proposed is not adequate. This problem is a very nice example of the application of conformal mapping and since this experiment is very simple and uses only basic equipment, it can reproduced in any undergraduate laboratory thus providing a very good introduction to students on this subject.
[99]{} A. Czarnecki, Canadian Journal of Physics 92, 1297 (2014) S. B. Cahn and B. E. Nadgorny, “A Guide to Physics Problems”, Part 1: Mechanics, Relativity and Electrodynamics (Plenum, New York, 1994) http://mathworld.wolfram.com/images/eps-gif/MercatorProjection\_1000.gif
|
---
abstract: 'We rewrite the Lagrangian for a dilute Bose gas in terms of auxiliary fields related to the normal and anomalous condensate densities. We derive the loop expansion of the effective action in the composite-field propagators. The lowest-order auxiliary field (LOAF) theory is a conserving mean-field approximation consistent with the Goldstone theorem without some of the difficulties plaguing approximations such as the Hartree and Popov approximations. LOAF predicts a second-order phase transition. We give a set of Feynman rules for improving results to any order in the loop expansion in terms of composite-field propagators. We compare results of the LOAF approximation with those derived using the Popov approximation. LOAF allows us to explore the critical regime for all values of the coupling constant and we determine various parameters in the unitarity limit.'
author:
- Fred Cooper
- Bogdan Mihaila
- 'John F. Dawson'
- 'Chih-Chun Chien'
- Eddy Timmermans
bibliography:
- 'johns.bib'
date: ', EST'
title: Auxiliary field approach to dilute Bose gases with tunable interactions
---
\[s:intro\]Introduction
=======================
In 1911, Kamerlingh Onnes found that liquid $^{4}$He, when cooled below $2.2$ K began to expand rather than contract[@r:Kamerling:1911fk]. ÊThe transition, later named the lambda-transition was recognized in 1938 as the onset of superfluidity[@r:Kapitza:1938kx; @r:Allen:1938vn]. ÊThe connection with Bose-Einstein condensation (BEC), first argued by F. London on the basis of the near identical values of the lambda transition temperature $T_{c}$ and the critical temperature $T_{c}^{0}$ for BEC of noninteracting bosons[@r:London:1938ys; @r:London:1938zr] sparked a series of weakly interacting BEC studies when Bogoliubov[@r:Bogoliubov:1947ys] pointed out that the BEC elementary excitations satisfy the Landau criterion for superfluidity[@r:Landau:1941ly]. ÊIn the weakly interacting limit, the interactions can be characterized by a single parameter[@r:Lee:1957ve] — the scattering length $a$ — giving the results a powerful, general applicability. The hope of studying bosons with short-range inter-particle interactions of a strength that can be tuned all the way from weakly interacting ($\rho^{1/3}a \ll 1$) to universality ($\rho^{1/3}a \gg 1$), appeared thwarted when it was found that the three-body loss-rate in cold atom traps scales as $\propto a^{4}$ near a Feshbach resonance[@r:Fedichev:1996hc; @r:Esry:1991ij]. In cold atom traps, only fermions have been obtained in the strongly interacting, quantum degenerate regime in equilibrium[@r:Shin:2007oq], in which case three-body loss is reduced by virtue of the Pauli exclusion principle. Recently, however, it was pointed out[@r:Daley:2009bs] that three-body losses can be strongly suppressed in an optical lattice when the average number of bosons per lattice site is two or less. The development of novel cold atom technology[@r:Henderson:2006fv; @r:Henderson:2009dz] leads to the prospect of studying finite temperature properties, such as the BEC transition temperature, $T_{c}$, superfluid to normal fluid ratio, depletion, and specific heat, at fixed particle density $\rho$.
At finite temperature the description of BEC’s even in the weakly interacting regime remains a challenge. ÊStandard approximations such as the Hartree-Fock-Bogoliubov (HFB) and the Popov schemes, generally fall within the Hohenberg and Martin classification[@r:Hohenberg:1965fu] of conserving and gapless approximations which imply that they either violate Goldstone’s (or Hugenholz-Pines) theorem or general conservation laws[@r:Hohenberg:1965fu]. Both these approximations predict the BEC-transition to be a first-order transition, whereas we expect the transition to be second-order[@r:Andersen:2004uq]. The calculation of $T_c$, first undertaken by Toyoda[@r:Toyoda:1982pi] to explain the difference between the lambda-transition temperature $T_{c}$ ($2.2$ K) and the $T_{c}^{0}$ ($3.1$ K) of the noninteracting BEC at the same density, exemplifies the difficulties of understanding the theory near $T_c$: whereas Toyoda found a $T_{c}$ -decrease with increasing scattering length, K. Huang later pointed out that the calculation had a sign error, giving an increasing value of $T_{c}$ [@r:Huang:1999qa]. ÊHowever, Baym and collaborators[@r:Baym:1999mi; @r:Baym:2000fk] noted that the Toyoda expansion involves an expansion in a large parameter. ÊTheir calculation found a linear increase of $(T_{c}-T^{0}_{c})/T^{0}_{c}$ with $\rho a^{3}$. ÊThe fact that the helium lambda transition temperature falls below $T^{0}_{c}$ may be explained by quantum Monte-Carlo calculations[@r:Baym:1999mi], which found that the critical temperature of a hard-sphere boson gas increases at low values of $\rho a^{3}$, then turns over and drops below $T^{0}_{c}$ near $\rho a^{3} \approx 0.1$.
In this paper, we discuss in detail a theoretical description that we introduced recently[@r:Cooper:2010fk] to describe a large interval of $\rho^{1/3}a$ values, satisfies Goldstone’s theorem, yields a Bose-Einstein transition that is second-order, gives the same critical temperature variation found in Refs. but at a lower order of the calculation, while also predicting reasonable values for the depletion. ÊThis method then resolves many of the main challenges in describing boson physics over a large temperature and $\rho^{1/3}a$ regime and it’s predictions will be available for experimental testing in the near future. The approach we present here is different from other resummation schemes such as the large-$N$ expansion (which is a special case of this expansion), in that it treats the normal and anomalous densities on an equal footing.
In the following, we will discuss the general features that arise when rewriting the original theory in terms of composite fields. One aspect of this approach is that one can systematically calculate corrections to the mean-field results presented earlier[@r:Cooper:2010fk] in a loop expansion in the composite-field propagators. We derive the Feynman rules for such an expansion using the propagators and vertices of the mean-field approximation. At each level of this loop expansion one maintains the features that the results are both gapless and conserving. The broken $U(1)$ symmetry Ward identities guarantee Goldstone’s theorem order-by-order in the loop expansion in terms of auxiliary-field propagators[@r:Bender:1977bh].
In our auxiliary field formalism, we introduce two auxiliary fields related to the normal and anomalous densities by means of the Hubbard-Stratonovitch transformation[@r:Hubbard:1959kx; @r:Stratonovich:1958vn], utilizing methods discussed in the quantum field theory community [@r:Bender:1977bh; @r:Coleman:1974ve; @r:Root:1974qf]. This transformation has already been shown to be quite useful in discussing the properties of the BCS-BEC crossover in the analogous 4-fermi theory for the BCS phase [@r:Melo:1993vn; @r:Engelbrecht:1997fk; @r:Floerchinger:2008kxx]. The path integral formulation of the grand canonical partition function can be found in Negele and Orland [@r:Negele:1988fk]. The Hubbard Stratonovich transformation is used to replace the original quartic interaction with an interaction quadratic in the original fields. An excellent review of previous use of path integral methods to study dilute Bose gases is found in the review article of Andersen[@r:Andersen:2004uq]. The use of path integral methods to study various topics in dilute gases began with the work of Braaten and Nieto [@r:Braaten:1997uq]. Path integral methods have recently been used to study static and dynamical properties of the dilute Bose gases [@r:ReyHuCalzettaRouraClark03; @r:gasenzer:2005; @r:gasenzer:2006; @r:gasenzer:2007; @PhysRevB.81.235108; @r:Floerchinger:2008kx; @r:Floerchinger:2008kxx]. An excellent summary of this approach and its connection to the more traditional Hamiltonian approach is to be found in the recent book by Calzetta and Hu [@r:Calzetta:2008pb]. We also point out that the 1/N expansion, which is a special case of the method being proposed here, has a long history of use in high-energy and condensed matter physics [@r:brezin:1993; @r:Moshe:2003uq]. It has been used to calculate the critical temperature by Baym, Blaizot and Zinn-Justin [@r:Baym:2000fk]. This calculation gives the same result for $T_c$ as the method we are describing here. However, our approach can be used at all temperatures. Corrections to the 1/N result to calculating $T_c$ were obtained by Arnold and Tomasik [@PhysRevA.62.063604].
The paper is organized as follows: In Sec. \[s:auxfield\] we discuss the auxiliary-field formalism and rewrite the Lagrangian for weakly interacting Bosons in terms of two auxiliary fields. In Sec. \[s:Seff\] we derive the loop expansion by performing the path integral over the original fields $\phi_i$ and then performing the resulting path integral over the auxiliary fields by stationary phase. In Sec. \[s:effpotcon\] we find the leading-order loop expansion in the auxiliary fields (LOAF) for the action. In Sec. \[s:theta\] we set the auxiliary-field parameter $\theta$ and discuss the leading-order effective potential for both the ground state and at finite temperature. In Sec. \[s:meanfield\] we discuss related mean-field approximations. In Sec. \[ss:results\] we discuss numerical results for the theory at finite temperature and varying dimensionless coupling constant $\rho^{1/3}a$. We compare the LOAF approximation to the Popov approximation in detail. We conclude in Sec. \[s:conclusions\]. Finally, in App. \[s:renorm\] we discuss the connection between regularization of the effective potential and renormalization of the parameters. In App. \[s:building\] we give the rules for determining all the Feynman graphs for the expansion using the mean-field propagators and vertices.
\[s:auxfield\]The auxiliary-field formalism
===========================================
The classical action $S[\, \phi,\phi^{\ast} \, ]$ is given by $$\label{af.e:actionI}
S[\, \phi,\phi^{\ast} \, ]
=
\int \! [{\mathrm{d}}x] \>
{\mathcal{L}}[ \, \phi,\phi^{\ast} \, ] \>,$$ where $[{\mathrm{d}}x] \equiv {\mathrm{d}}t \, {\mathrm{d}}^3 x$ and where the Lagrangian density is $$\begin{aligned}
{\mathcal{L}}[ \, \phi,\phi^{\ast} \, ]
&=
\frac{i \hbar}{2} \,
[ \,
\phi^{\ast}(x) \, ( \partial_t \, \phi(x) )
-
( \partial_t \, \phi^{\ast}(x) ) \, \phi(x) \,
]
\label{af.e:LagI} \\
& \quad
-
\phi^{\ast}(x) \,
\Bigl \{ \,
-
\frac{\hbar^2\nabla^2}{2m}
-
\mu_0 \,
\Bigr \} \,
\phi(x)
-
\frac{\lambda_0}{2} \, | \, \phi(x) |^4 \>.
\notag\end{aligned}$$ Here $\mu_0$ and $\lambda_0$ are the bare (unrenormalized) chemical potential and contact interaction strength respectively. We introduce two auxiliary fields, a real field, $\chi(x)$, and a complex field, $A(x)$, by means of the Hubbard-Stratonovitch transformation[@r:Hubbard:1959kx; @r:Stratonovich:1958vn], utilizing methods discussed in Refs. . In our case, the auxiliary-field Lagrangian density takes the form $$\begin{aligned}
&{\mathcal{L}}_{\text{aux}}[\phi,\phi^{\ast},\chi,A,A^{\ast}]
=
\frac{1}{2 \lambda_0} \,
\bigl [ \,
\chi(x) - \lambda_0 \, \cosh\theta \, | \phi(x) |^2 \,
\bigr ]^2
\notag \\
& \qquad
-
\frac{1}{2 \lambda_0} \,
\bigl | \,
A(x)
-
\lambda_0 \, \sinh\theta \, \phi^{2}(x) \,
\bigr |^2
\>,
\label{af.e:Laux}\end{aligned}$$ which we add to Eq. . Here $\theta$ is a parameter which provides a mixing between the normal and anomalous densities. In Sec. \[s:meanfield\], we will see that choosing $\theta = 0$ leads to the usual large-$N$ expansion which has only the auxiliary field $\chi$ [@r:Coleman:1974ve; @r:Root:1974qf]. In lowest order, $\theta = 0$ gives a gapless solution very similar to the free Bose gas in the condensed phase. If instead we choose $\theta$ such that $\sinh \theta = 1$, then in the weak coupling limit our results agree with the Bogoliubov theory[@r:Bogoliubov:1947ys; @r:Andersen:2004uq], which represents the leading-order low-density expansion. Of course all values of $\theta$ lead to a complete resummation of the original theory in terms of different combinations of the composite fields.
For an arbitrary parameter $\theta$, the action is given by $$\begin{aligned}
&S[\Phi,J]
=
\label{af.e:actionII} \\
&
-
\frac{1}{2} \,
\iint [{\mathrm{d}}x] \, [{\mathrm{d}}x'] \,
\phi_a(x) \, {\mathcal{G}}^{-1}{}^a{}_b[\chi](x,x') \, \phi^b(x')
\notag \\
&
+
\int {\mathrm{d}}x \,
\Bigl \{ \,
\frac{\chi_i(x) \, \chi^i(x)}{2\lambda_0}
+
\Phi_{\alpha}(x) \, J^{\alpha}(x) \,
\Bigr \} \>.
\notag\end{aligned}$$ with $$\begin{aligned}
&{\mathcal{G}}^{-1}{}^a{}_b[\chi]
=
\delta(x,x') \,
\bigl \{ \,
G^{-1}_0{}^a{}_b
+
V^{a}{}_{b}[\chi](x) \,
\bigr \} \>,
\label{af.e:G0invV0def} \\
&G^{-1}_0{}^a{}_b
=
\begin{pmatrix}
h - \mu_0 & 0 \\[3pt]
0 & h^{\ast} - \mu_0
\end{pmatrix} \>,
\quad
h
=
-
\frac{\hbar^2 \nabla^2}{2m}
-
i \hbar \frac{\partial}{\partial t} \>,
\notag \\
&V^{a}{}_{b}[\chi](x)
=
\begin{pmatrix}
\chi(x) \cosh\theta & - A(x) \sinh\theta \\
- A^{\ast}(x) \sinh\theta & \chi(x) \cosh\theta
\end{pmatrix} \>.
\notag\end{aligned}$$ Here we have introduced a two-component notation using Roman indices $a,b,c,\dotsb$ for the fields $\phi(x)$ and $\phi^{\ast}(x)$ and currents $j(x)$ and $j^{\ast}(x)$,
\[af.e:notation\] $$\begin{aligned}
{2}
\phi^a(x)
&=
{ \bigl ( \, \phi(x), \phi^{\ast}(x) \, \bigr )} \>,
&\quad
\phi_a(x)
&=
{ \bigl ( \, \phi^{\ast}(x), \phi(x) \, \bigr )} \>,
\\
j^a(x)
&=
{ \bigl ( \, j(x), j^{\ast}(x) \, \bigr )} \>,
&\quad
j_a(x)
&=
{ \bigl ( \, j^{\ast}(x), j(x) \, \bigr )} \>,\end{aligned}$$
for $a=1,2$, and a three-component notation using Roman indices $i,j,k,\dotsb$ for the fields $\chi(x)$, $A(x)$, and $A^{\ast}(x)$, $$\begin{aligned}
\chi^{i}(x)
&=
\bigl ( \,
\chi(x), A(x)/\sqrt{2}, A^{\ast}(x)/\sqrt{2} \,
\bigr ) \>,
\label{af.e:chiSupperdefs} \\
S^{i}(x)
&=
\bigl ( \,
s(x), S(x)/\sqrt{2}, S^{\ast}(x)/\sqrt{2} \,
\bigr ) \>,
\notag\end{aligned}$$ and $$\begin{aligned}
\chi_{i}(x)
&=
\bigl ( \,
\chi(x), - A^{\ast}(x)/\sqrt{2}, - A(x)/\sqrt{2} \,
\bigr ) \>,
\label{af.e:chiSlowerdefs} \\
S_{i}(x)
&=
\bigl ( \,
s(x), -S^{\ast}(x)/\sqrt{2}, -S(x)/\sqrt{2} \,
\bigr ) \>,
\notag\end{aligned}$$ for $i=1,2,3$. For convenience, we also define five-component fields with Greek indices $\Phi^{\alpha}(x) = { \bigl ( \, \phi^a(x),\chi^i(x) \, \bigr )}$ and currents $J^{\alpha}(x) = { \bigl ( \, j^a(x),S^i(x) \, \bigr )}$. These definitions define a metric $\eta_{\alpha,\beta}$ for raising and lowering indices. We use this notation throughout this paper.
The action is invariant under a global $U(1)$ transformation, $\phi(x) \mapsto e^{i\alpha} \phi(x)$, $A(x) \rightarrow e^{2 i \alpha} A(x)$, and $\chi(x) \mapsto \chi(x)$. In components, the equations of motion are $$\begin{gathered}
[ \ h - \mu_0 + \chi(x) \cosh\theta \, ]\, \phi(x)
-
A(x) \, \phi^{\ast}(x) \, \sinh\theta
=
j(x) \>,
\notag \\
\chi(x) / \lambda_0
=
| \, \phi(x) \, |^2 \, \cosh\theta
-
s(x) \>,
\notag \\
A(x) / \lambda_0
=
\phi^2(x) \, \sinh\theta
-
S(x) \>.
\label{af.e:alleom}\end{gathered}$$ We note that substituting $\chi(x)$ and $A(x)$ from the last two lines of Eqs. (for zero currents) into the first line of Eqs. gives the equation of motion for the field $\phi(x)$ with *no* auxiliary fields[@r:Andersen:2004uq].
Parametrizing the Green function ${\mathcal{G}}$ as $$\label{af.e:Green}
{\mathcal{G}}(x,x')
=
\begin{pmatrix}
G(x,x') & K(x,x') \\
K^{\ast}(x,x') & G^{\ast}(x,x')
\end{pmatrix} \>,$$ and using $$\label{af.e:invert}
\int [{\mathrm{d}}x'] \,
{\mathcal{G}}^{-1}(x,x') \, {\mathcal{G}}(x',x'')
=
\delta(x,x'') \>,$$ we obtain the equations
\[af.e:GKeom\] $$\begin{aligned}
&
\bigl [ \,
h_0
- \mu +
\chi(x) \cosh \theta \,
\bigr ] \, G(x,x')
\label{af.e:Geom} \\
& \qquad\qquad
-
A(x) \, K^{\ast}(x,x') \sinh \theta
=
\delta(x,x') \>,
\notag \\
&
\bigl [ \,
h_0 - \mu
+
\chi(x) \cosh \theta \,
\bigr ] \, K(x,x')
\label{af.e:Keom} \\
& \qquad\qquad
-
A(x) \, G^{\ast}(x,x') \sinh \theta
=
0 \>,
\notag\end{aligned}$$
and the complex conjugates. Here, $G(x,x')$ and $K(x,x')$ are the normal and anomalous correlation functions.
\[s:Seff\]Auxiliary-field loop expansion
========================================
The generating functional for connected graphs is $$\label{afle.e:Z}
Z[J]
=
e^{i W[J] / \hbar}
=
{\mathcal{N}}\int {\text{D}}\Phi \>
e^{ i S[\Phi,J] / \hbar } \>,$$ with $S[\Phi,J]$ given by Eq. . Average values of the fields are given by $$\label{afle.e:avePhis}
{\ensuremath{\langle \, \Phi^{\alpha}(x) \, \rangle}}
=
\frac{\hbar}{i} \,
\frac{1}{Z[J]} \, \frac{\delta Z[J]}{\delta J_{\alpha}(x)} \Big |_{J=0}
=
\frac{\delta W[J]}{\delta J_{\alpha}(x)} \Big |_{J=0} \>.$$ If we integrate out the auxiliary fields $A(x)$ and $\chi(x)$, we obtain the path integral for the original Lagrangian of Eq. . The strategy we will use here is to reverse the order of integration and first do the path integral over the fields $\phi^a(x)$ exactly and then perform the path integration over the auxiliary fields by stationary phase to obtain a loop expansion in the auxiliary fields. Performing the path integral over the fields $\phi^a$, we obtain $$Z[J]
=
{\mathcal{N}}' \int {\text{D}}\chi \, e^{i S_{\text{eff}}[ \chi,J ] / (\epsilon \hbar) }
\>,
\label{eq:Z}$$ where the effective action is given by $$\begin{aligned}
S_{\text{eff}}[ \chi,J ]
&=
\frac{1}{2} \iint [{\mathrm{d}}x] \, [{\mathrm{d}}x'] \,
j_{a}(x) \, {\mathcal{G}}[\chi]^a{}_b(x,x') \, j^a(x)
\notag \\
& \quad
+
\int \! [{\mathrm{d}}x] \,
\Bigl \{
\frac{\chi_i(x) \, \chi^{i}(x)}{2\lambda_0}
+
\chi_{i}(x) \, S^{i}(x)
\notag \\
& \qquad\qquad\qquad
-
\frac{\hbar}{2i}
{\mathrm{Tr} [ \, {\ln [ \, {\mathcal{G}}^{-1}(x,x) \, ]} \, ]} \,
\Bigr \} \>.
\label{afle.e:Seffxx}\end{aligned}$$ Here $\chi^i(x)$ is defined in Eq. . As shown in Ref. , the dimensionless parameter $\epsilon$ (which we eventually set equal to one) in Eq. allows us to count loops for the auxiliary-field propagators in the effective theory in analogy with $\hbar$ which counts loops for the $\phi$-propagator for the original Lagrangian. The stationary point $\chi_0^{i}(x)$ of the effective action are defined by $\delta S_{\text{eff}}[ \chi,J ] / \delta \chi_{i}(x) = 0$, i.e $$\begin{aligned}
\frac{\chi_0(x)}{\lambda_0}
&=
\bigl \{ \,
| \phi_0(x) |^2
+
\hbar\, {\ensuremath{\mathcal{R}e \{ \, G(x,x) \, \} }} / i \,
\bigr \} \, \cosh\theta
-
s(x)
\notag \\
\frac{A_0(x)}{\lambda_0}
&=
\bigl \{ \,
\phi^2_0(x)
+
\hbar \, K(x,x) / i \,
\bigr \} \, \sinh\theta
-
S(x) \>,
\label{afle.e:chi0A0eqs}\end{aligned}$$ where $\phi^a_0(x)$ is given by $$\label{afle.e:phi0def}
\phi^a_0[\chi_0,J](x)
=
\int [{\mathrm{d}}x'] \, {\mathcal{G}}[\chi_0]^a{}_b(x,x') \, j^b(x') \>.$$ Both $\chi_0(x)$ and $A_0(x)$ include self consistent fluctuations and are functionals of all the currents $J^{\alpha}(x)$. Expanding the effective action about the stationary point, we find $$\begin{aligned}
\label{afle.e:Seffexpand}
S_{\text{eff}}[ \chi,J ]
&=
S_{\text{eff}}[ \chi_0,J ]
\\ & \qquad
+
\frac{1}{2} \iint [{\mathrm{d}}x] \, [{\mathrm{d}}x'] \,
{\mathcal{D}}^{-1}_{ij}[\chi_0](x,x')
\notag \\
& \qquad
\times
( \chi^i(x) - \chi^i_0(x) ) \,
( \chi^j(x') - \chi^j_0(x') )
+
\dotsb
\notag\end{aligned}$$ where ${\mathcal{D}}_{ij}^{-1}[\chi_0](x,x')$ is given by the second-order derivatives $$\begin{aligned}
{\mathcal{D}}^{-1}_{ij}[\chi_0](x,x')
&=
\frac{ \delta^2 \, S_{\text{eff}}[ \chi^a] }
{ \delta \chi^i(x) \, \delta \chi^j(x') } \, \bigg |_{\chi_0}
\label{afle.e:Dinverse} \\
&=
\frac{1}{\lambda_0} \eta_{ij} \, \delta(x,x')
+
\Pi_{ij}[\chi_0](x,x') \>,
\notag\end{aligned}$$ evaluated at the stationary points. Here $\Pi_{ij}[\chi_0](x,x')$ is the polarization and is calculated in App. \[s:building\]. We perform the remaining gaussian path integral over the fields $\chi_i$ by saddle point methods, obtaining $$\begin{aligned}
\epsilon \, W[J]
&=
S_0
+
S_{\text{eff}} [\chi_0,J]
\label{afle.e:WtoSD} \\
& \qquad
-
\frac{\epsilon \hbar}{2i}
\int \! [{\mathrm{d}}x] \, {\mathrm{Tr} [ \, {\ln [ \, {\mathcal{D}}^{-1} [\chi_i, J](x,x) \, ]} \, ]} \>,
\notag\end{aligned}$$ where $S_0$ is a normalization constant. From this we calculate the order $\epsilon$ corrections to $\phi^a = \delta W / \delta j_a$ and $\chi^i = \delta W / \delta S_i$. Schematically, these one-point functions are shown in Fig. \[f:phi\_chi\].
![\[f:phi\_chi\] Feynman diagrams for $\phi$ and $\chi$ to first order in $\epsilon$. Solid and wavy lines correspond to the propagators of $\phi$ and $\chi$. Dashed lines denote the zeroth-order $\phi^{(0)}$.](phi-chi_BEC.pdf){width="0.9\columnwidth"}
The vertex function $\Gamma[\Phi]$ is constructed by a Legendre tranformation (see for example Ref. ) by $$\label{afle.e:vertexfctdef}
\Gamma[\Phi]
=
\int [{\mathrm{d}}x] \, J_{\alpha}(x) \, \Phi^{\alpha}(x)
-
W[J] \>.$$ Here $\Gamma[\Phi]$ is the generator of the one-particle-irreducible (1-PI) graphs of the theory[@r:LW; @r:Baym62; @r:CJT], with $$\label{afle.e:dGammadphidchi}
\frac{\delta \Gamma[\Phi]}{\delta \Phi_{\alpha}(x)}
=
J^{\alpha}(x) \>.$$ Keeping only the gaussian fluctuations in $W[J]$, we find $$\begin{aligned}
&\epsilon \, \Gamma[\Phi]
=
\frac{1}{2} \iint [{\mathrm{d}}x] \, [{\mathrm{d}}x'] \,
\phi_a(x) \, {\mathcal{G}}^{-1}[\chi]^{a}{}_{b}(x,x') \, \phi^b(x')
\notag \\
& \quad
-
\int [{\mathrm{d}}x] \,
\Bigl \{ \,
\frac{\chi_{i}(x) \, \chi^{i}(x)}{2\lambda_0}
-
\frac{\hbar}{2i} \,
{\mathrm{Tr} [ \, {\ln [ \, {\mathcal{G}}^{-1}[\chi](x,x) \, ]} \, ]}
\notag \\
&\quad
-
\frac{\epsilon \hbar}{2i} \,
{\mathrm{Tr} [ \, {\ln [ \, {\mathcal{D}}^{-1}[\Phi] \, ]}(x,x) \, ]} \,
\Bigr \}
+
\epsilon \, \Gamma_0
+
\dotsb
\>,
\label{gamma} \end{aligned}$$ which is the negative of the classical action plus self consistent one loop corrections in the $\phi^a$ and $\chi^i$ propagators. Here, $\Gamma_0$ is an adjustable constant used to set the minimum of the effective potential to have finite reference energy. The effective potential ${\mathcal{V}_{\text{eff}}}[\Phi]$ is defined for static fields $\Phi$ by $$\begin{aligned}
&{\mathcal{V}_{\text{eff}}}[\Phi]
=
\frac{\epsilon \Gamma[\Phi]}{V T}
=
{\mathcal{V}_0}+
\frac{1}{2} \,
\phi_a \, V[\chi]^{a}{}_{b} \, \phi^b
-
\frac{\chi_{i} \, \chi^{i}}{2\lambda_0}
\label{afle.e:Veff} \\
& \qquad
-
\frac{\hbar}{2i V T} \,
{\mathrm{Tr} [ \, {\ln [ \, {\mathcal{G}}^{-1}[\chi](x,x) \, ]} \, ]}
\notag \\
& \qquad
-
\frac{\epsilon \hbar}{2i V T} \,
{\mathrm{Tr} [ \, {\ln [ \, {\mathcal{D}}^{-1}[\Phi](x,x) \, ]} \, ]}
+
\text{O}(\epsilon^2) \>,
\notag\end{aligned}$$ where $$\label{afle.e:Vabdef}
V[\chi]^a{}_b
=
\begin{pmatrix}
\chi \cosh\theta - \mu &
-A \sinh\theta \\
-A^{\ast} \sinh\theta &
\chi \cosh\theta - \mu
\end{pmatrix} \>.$$ We will see below that for the static case, ${\mathcal{G}}^{-1}[\chi](x,x)$ and ${\mathcal{D}}^{-1}[\Phi](x,x)$ are independent of $x$.
For a system in equilibrium at temperature $T$, we Wick rotate the time variable to Euclidian time $\tau$ according to the Matsubara prescription, $t \rightarrow -i \hbar \tau$. Then the effective potential becomes the grand potential ${\Omega}[\Phi]$ per unit volume, ${\mathcal{V}_{\text{eff}}}[\Phi] \rightarrow {\Omega}[\Phi] / V$. (Details of the Matsubara formalism can be found for example in Ref. .) So to leading order in $\epsilon$, the thermal effective potential is given by $$\begin{aligned}
{\mathcal{V}_{\text{eff}}}[\Phi]
&=
{\mathcal{V}_0}+
\frac{1}{2} \, \phi_a \, V[\chi]^{a}{}_{b} \, \phi^b
-
\frac{\chi_{i} \, \chi^{i}}{2\lambda_0}
\label{afle.e:leading} \\
& \qquad
-
\frac{1}{2 \beta V}
{\mathrm{Tr} [ \, {\ln [ \, {\mathcal{G}}^{-1}[\chi](x,x) \, ]} \, ]} \>,
\notag\end{aligned}$$ and where ${\mathcal{V}_0}$ is a normalization constant. At the next order we have the additional term $${\mathcal{V}_{\text{eff}}}^{(1)}[\Phi]
=
-
\frac{\epsilon}{2 \beta V}
{\mathrm{Tr} [ \, {\ln [ \, {\mathcal{D}}^{-1}[\Phi](x,x) \, ]} \, ]} \>.$$ Here and throughout this section, we suppress the dependence of quantities on $\theta$ and the thermodynamic variables ${ \bigl ( \, T,\mu,V \, \bigr )}$. The thermodynamic effective potential ${\mathcal{V}_{\text{eff}}}[\Phi_0]$ is obtained by evaluating the effective potential at zero currents. From , this is when the fields $\Phi_0$ satisfy $$\label{afle.e:dGammadphidchiII}
\frac{\delta \, {\mathcal{V}_{\text{eff}}}[\Phi_0]}{\delta \Phi_{\alpha}(x)}
=
0 \>,
{\qquad\text{for $\alpha = 1,\dotsb,5$.}\qquad}$$ We call these the “gap equations” in analogy with the corresponding equations in BCS theory.
The Green functions are periodic with Matsubara frequency $\omega_n = 2 \pi n/\beta$ with $\beta = 1 / ( {k_{\text{B}}}T )$, and are expanded in a Fourier series, $${\mathcal{G}}[\chi](x,x')
=
\frac{1}{\beta}
\sum_{{\mathbf{k}},n}
\tilde{{\mathcal{G}}}[\chi]({\mathbf{k}},n) \,
e^{i [ \, k \cdot ({\mathbf{r}}-{\mathbf{r}}') - \omega_n(\tau-\tau') \, ]} \>.
\label{afle.e:Fourierexpand}$$ Writing the Green function equation in ${\mathbf{k}}$-$n$ space as $$\label{afle.e:GG}
\tilde{{\mathcal{G}}}^{-1}[\chi]({\mathbf{k}},n) \,
\tilde{{\mathcal{G}}}[\chi]({\mathbf{k}},n)
=
1 \>,$$ we find $$\begin{aligned}
&\tilde{{\mathcal{G}}}^{-1}[\chi]({\mathbf{k}},n)
\label{afle.e:tGinverse} \\
& \quad
=
\begin{pmatrix}
\xi_k + \chi \cosh\theta - i \omega_n &
-A \sinh\theta \\
-A^{\ast} \sinh\theta &
\xi_k + \chi \cosh\theta + i \omega_n
\end{pmatrix} \>,
\notag\end{aligned}$$ where $\xi_k = \epsilon_k - \mu_0$. So $$\label{afle.e:detGinv}
{\det [ \, \tilde{{\mathcal{G}}}^{-1}[\chi]({\mathbf{k}},n) \, ]}
=
\omega_k^2
+
\omega_n^2 \>.$$ where $$\label{afle.e:omegak}
\omega_k^2
=
[ \,
\xi_k + \chi \cosh\theta \,
]^2
-
| A |^2 \sinh^2\theta \>.$$ Stable solutions are possible for $\omega_k^2 \ge 0$. The trace-log term then becomes $$\begin{aligned}
&\frac{1}{2V \beta} \,
\mathrm{Tr}
\bigl [ \,
{\ln [ \, {\mathcal{G}}^{-1}[\chi](x,x) \, ]} \,
\bigr ]
=
\frac{1}{2V \beta} \,
\sum_{{\mathbf{k}},n}
{\ln [ \, \omega_k^2 + \omega_n^2 \, ]}
\notag \\
&\qquad
=
\frac{1}{V} \,
\sum_{{\mathbf{k}}}
\Bigl \{ \,
\frac{\omega_k}{2}
+
\frac{1}{\beta}
{\ln [ \, 1 - e^{-\beta \omega_k} \, ]} \,
\Bigr \}
\notag \\
&\qquad
=
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\Bigl \{ \,
\frac{\omega_k}{2}
+
\frac{1}{\beta}
{\ln [ \, 1 - e^{-\beta \omega_k} \, ]} \,
\Bigr \} \>.
\label{aux.e:TrLn}\end{aligned}$$ So from the effective potential to leading order in the auxiliary field loop expansion (LOAF) is given by $$\begin{aligned}
&{\mathcal{V}_{\text{eff}}}[\Phi]
=
{\mathcal{V}_0}+
| \phi |^2 \,
\bigl [ \,
\chi \cosh\theta - \mu_0 \,
\bigr ]
\label{afle.e:GammaIII} \\
& \qquad
-
\frac{1}{2} \,
\bigl [ \,
\phi^{\ast\,2} \, A
+
\phi^2 \, A^{\ast} \,
\bigr ] \,
\sinh\theta
\notag \\
& \qquad
-
\frac{\chi^2 - | A |^2}{2\lambda_0}
+
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\Bigl \{ \,
\frac{\omega_k}{2}
+
\frac{1}{\beta}
{\ln [ \, 1 - e^{-\beta \omega_k} \, ]} \,
\Bigr \} \>.
\notag \end{aligned}$$ It is useful to introduce new variables $\chi'$ and $A'$ as $$\label{afle.e:redefineAF}
\chi'
=
\chi \cosh\theta - \mu_0 \>,
{\qquad\text{and}\qquad}
A'
=
A \, \sinh\theta \>.$$ Then the effective potential becomes $$\begin{aligned}
&{\mathcal{V}_{\text{eff}}}[\Phi']
=
{\mathcal{V}_0}+
| \phi |^2 \, \chi'
-
\frac{1}{2} \,
\bigl [ \,
\phi^{\ast\,2} \, A'
+
\phi^2 \, A^{\prime\,\ast} \,
\bigr ] \,
\notag \\
& \qquad
-
\frac{ ( \chi' + \mu_0 )^2 }
{ 2 \lambda_0 \cosh^2\theta }
+
\frac{| A' |^2}{2\lambda_0 \sinh^2\theta }
\label{afle.e:GammaIV} \\
& \qquad
+
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\Bigl \{ \,
\frac{\omega_k}{2}
+
\frac{1}{\beta}
{\ln [ \, 1 - e^{-\beta \omega_k} \, ]} \,
\Bigr \} \>,
\notag \end{aligned}$$ where now $\omega_k^2 = ( \, \epsilon_k + \chi' \, )^2 - | A' |^2$. The gap equations are now written as $$\begin{gathered}
\begin{pmatrix}
\chi'_0 & -A'_0 \\
-A^{\prime\,\ast}_0 & \chi'_0
\end{pmatrix}
\begin{pmatrix} \phi_0 \\ \phi_0^{\ast} \end{pmatrix}
=
0 \>,
\label{afle.e:gapeqs} \\
\frac{\chi'_0 + \mu_0}{\lambda_0 \cosh^2\theta}
=
| \phi |^2
+
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\frac{\epsilon_k + \chi'_0}{2\omega_k} \,
[ \, 2 n(\beta\omega_k) + 1 \, ] \>,
\notag \\
\frac{A'_0}{\lambda_0 \sinh^2 \theta}
=
\phi^2
+
A'_0
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\,
\frac{ [ \, 2 n(\beta\omega_k) + 1 \, ]}{2\omega_k} \>,
\notag\end{gathered}$$ where $n(x) = 1 / ( e^x - 1 )$ is the Bose-Einstein particle distribution. The solutions $\Phi_0^{\prime\,\alpha}$ of Eqs. substituted into Eq. determine the effective potential.
To calculate the finite temperature effective potential to order $\epsilon$ we need to determine ${\mathrm{Tr} [ \, {\ln [ \, {\mathcal{D}}^{-1}(x,x') \, ]} \, ]}$. For the static case in the imaginary time formalism, ${\mathcal{D}}_{ij}[\chi](x,x')$ and $\Pi_{ij}[\Phi](x,x')$ are expanded in Fourier series’ analogous to Eq. . So from Eq. we obtain $$\label{BEC.NLOAF.e:tD}
\tilde{{\mathcal{D}}}_{ij}[\Phi]({\mathbf{k}},n)
=
\frac{ \eta_{ij} }{\lambda_0}
+
\tilde \Pi_{ij}[\Phi]({\mathbf{k}},n) \>.$$
\[s:effpotcon\] The effective potential in the condensate phase to leading order
================================================================================
In the language of broken symmetry, the condensate phase is a phase where the U(1) symmetry of the theory is broken since then ${\ensuremath{\langle \, \phi \, \rangle}} \neq 0$. From Eq. , the minimum of the effective potential is when $$\label{EP.e:brokencase}
\frac{\delta {\mathcal{V}_{\text{eff}}}[\Phi]}{\delta \phi^{\ast}} \Bigl |_{\phi_0}
=
\chi' \, \phi_0
-
A' \, \phi_0^{\ast}
=
0 \>.$$ Because of the $U(1)$ gauge symmetry, we can choose $\phi_0$ to be real, which means that $A$ is also real. Hence, we have the broken symmetry condition $\chi' = A'$, and the dispersion relation reads $$\label{EP.e:disprel}
\omega_k^2
=
\epsilon_k \, ( \epsilon_k + 2 A' ) \>,$$ The latter is a consequence of the the Hugenholz-Pines theorem which assures that the dispersion relation does not exhibit a gap. This is equivalent to the Goldstone theorem for a dilute Bose gas with a spontaneously-broken continuos symmetry. This connection is discussed in detail in Ref. . In the absence of quantum fluctuations in $\chi'=A'$, one obtains the Bogoliubov dispersion, $\omega_k^2 = \epsilon_k ( \epsilon_k + 2 \lambda \, \phi_0^2 ) $, by setting $A' = \lambda \, \phi_0^2 \, \sinh^2 \theta$ and $\sinh \theta = 1$.
In the spontaneously broken phase, the effective potential is $$\begin{aligned}
{\mathcal{V}_{\text{eff}}}[\chi']
&=
{\mathcal{V}_0}-
\frac{ ( \chi' + \mu_0 )^2 }
{ 2 \lambda_0 \cosh^2\theta }
+
\frac{\chi^{\prime\,2}}{2 \lambda_0 \sinh^2\theta }
\label{EP.e:V-i} \\
& \quad
+
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\Bigl \{ \,
\frac{\omega_k}{2}
+
\frac{1}{\beta}
{\ln [ \, 1 - e^{-\beta \omega_k} \, ]} \,
\Bigr \} \>,
\notag \end{aligned}$$ where $\chi'$ is determined by the equation $$\begin{aligned}
\frac{ \partial {\mathcal{V}_{\text{eff}}}[\chi'] }{ \partial \chi' }
&=
\frac{\chi'}{\lambda_0 \sinh^2\theta}
-
\frac{\chi' + \mu}{\lambda_0 \cosh^2\theta}
\label{EP.e:dVdchip} \\
& \qquad
+
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\, \frac{\epsilon_k}{2 \omega_k} \, [ \, 2 n(\beta\omega_k) + 1 \, ]
=
0 \>.
\notag\end{aligned}$$ These equations for ${\mathcal{V}_{\text{eff}}}[\chi']$ and $\chi'$ contain infinite terms that need to be regulated. In order to regulate the effective potential, we first expand $\omega_k$ in a Laurent series in $\epsilon_k$ $$\label{RR.e:omegakexpand}
\omega_k
=
\sqrt{ ( \, \epsilon_k + \chi' \, )^2 - | A' |^2 }
=
\epsilon_k
+
\chi'
-
\frac{ | A' |^2 }{ 2 \epsilon_k }
+
\dotsb \>,$$ around $k \rightarrow \infty$. The first three terms in the series are responsible for the divergences in the integral in Eq. . To regularize the theory, we subtracting these three terms from $\omega_k$ in the integrand, and replace the constant ${\mathcal{V}_0}$ the bare interaction strength $\lambda_0$ and chemical potential $\mu_0$ by regulated ones. This procedure gives the regulated effective potential $$\begin{aligned}
&{\mathcal{V}_{\text{eff}}^{\text{R}}}[\Phi']
=
{\mathcal{V}_{\text{R}}}+
| \phi |^2 \, \chi'
-
\frac{1}{2} \,
\bigl [ \,
\phi^{\ast\,2} \, A'
+
\phi^2 \, A^{\prime\,\ast} \,
\bigr ] \,
\notag \\
& \qquad
-
\frac{ ( \chi' + {\mu_{\text{}}})^2 }
{ 2 {\lambda_{\text{}}}\cosh^2\theta }
+
\frac{| A' |^2}{2{\lambda_{\text{}}}\sinh^2\theta }
\label{RR.e:GammaV} \\
&
+
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\Bigl \{ \,
\frac{1}{2}
\Bigl [ \,
\omega_k
-
\chi'
+
\frac{ | A' |^2 }{ 2 \epsilon_k } \,
\Bigr ]
+
\frac{1}{\beta}
{\ln [ \, 1 - e^{-\beta \omega_k} \, ]} \,
\Bigr \} \>,
\notag \end{aligned}$$ which is now finite. Similarly, the regulated gap equations for $A'$ and $\chi'$ are now give as $$\begin{aligned}
\frac{\chi'+ {\mu_{\text{}}}}{{\lambda_{\text{}}}\cosh^2 \theta}
&=
| \phi |^2 \!
+ \!\!
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\Bigl \{
\frac{\epsilon_k + \chi'}{2\omega_k}
[ 2 n(\beta\omega_k) + 1 ] \!
- \!
\frac{1}{2}
\Bigr \} \,,
\notag \\
\frac{A'}{{\lambda_{\text{}}}\sinh^2 \theta}
&=
\phi^2 \!
+\!
A' \!\!
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\Bigl \{
\frac{2 n(\beta\omega_k) + 1}{2\omega_k}\!
-\!
\frac{1}{2\epsilon_k}
\Bigr \}
\label{RR.e:gapeqsA1}\end{aligned}$$ which are also finite.
This regularization scheme is equivalent to dimensional regularization as done for example in Ref. , or to conventional renormalization of the coupling constant and chemical potential as described in the review article of Andersen and discussed in detail for the LOAF approximation in the App. \[s:renorm\].
\[s:theta\]Setting the parameter $\theta$
=========================================
Up to this point, we considered a one-parameter class of mean-field approximations governed by the parameter $\theta$. The dispersion relation in the condensate phase for the leading-order auxiliary field (LOAF) approximation is given by Eq. $$\omega_k
=
\sqrt{ \epsilon_k ( \epsilon_k + 2 A \, \sinh\theta ) } \>,$$ where $A/\lambda = {\ensuremath{\langle \, \phi^2 \, \rangle}} \sinh \theta = [ \, \phi^2 + \hbar \, K(x,x) / i \, ] \, \sinh \theta$. Now, we will choose $\theta$ by demanding that in the weak coupling limit, when $K(x,x)$ can be ignored, the dispersion relation agrees with the one-loop low-density result obtained by Bogoliubov. Using a Hamiltonian formalism, Bogoliubov assumed $$\phi = \phi_0 + \psi \>,
\label{eq:bog}$$ subject to the constraint ${\ensuremath{\langle \, \psi \, \rangle}} = 0$. Realizing that $\phi_0 \approx \sqrt N$, he then wrote the theory in terms of the classical Hamiltonian plus a quadratic fluctuation Hamiltonian, which he diagonalized. Using Eq. and limiting to at most quadratic fluctuations, one has $$\begin{aligned}
&[(\phi_0^*+ \psi^*) (\phi_0+\psi)]^2
\rightarrow
(\phi_0^\ast \phi_0)^2
+
4 \, \psi^\ast \psi \, (\, \phi_0^\ast\phi_0 \, )
\notag \\
& \qquad
+
\psi \, \psi \, ( \phi_0^\ast \phi_0)
+
\psi_0^\ast \psi_0^\ast \, (\phi_0^\ast \phi_0) \>.
\label{bog.e:Bogaves}\end{aligned}$$ The minimum of the classical Hamiltonian defines $\mu = \lambda \, (\phi_0^\ast \phi_0)$. One can reformulate[@r:Andersen:2004uq] the Bogoliubov theory in path integral language as the classical approximation plus gaussian fluctuations. The inverse Green function in the gaussian fluctuation approximation now has $$V^{a}{}_{b}[\phi](x)
=
\lambda
\begin{pmatrix}
2 \, \phi_0^* \phi_0 & \phi_0 \phi_0 \\
\phi_0^* \phi_0^* & 2 \, \phi_0^* \phi_0
\end{pmatrix} \>$$ where $V^a{}_b[\phi]$ is defined in Eq. . This leads to the dispersion relation at the minimum: $$\label{ST.e:bogdisp}
\omega_k
=
\sqrt{\epsilon_k ( \epsilon_k + 2 \lambda \, \phi_0^2 ) }$$
We will choose $\theta$ such that our result for $\omega_k$ reduces to the Bogoliubov dispersion relation when we ignore quantum fluctuations in the anomalous density. This sets $\sinh\theta = 1$ and $\cosh\theta = \sqrt{2}$. With our choice of $\theta$, the renormalized effective potential can be written as $$\begin{aligned}
&{\mathcal{V}_{\text{eff}}^{\text{R}}}[\Phi]
=
{\mathcal{V}_{\text{R}}}+
\chi' \, | \phi |^2
-
\frac{1}{2} \,
\bigl (
A^{\ast} \, \phi^2
+
A \, \phi^{\ast\,2} \,
\bigr )
-
\frac{ ( \chi' + \mu)^2}{4 {\lambda_{\text{}}}}
\notag \\
& \qquad
+
\frac{| A |^2 }{2 {\lambda_{\text{}}}}
+
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\,
\Bigl \{ \,
\frac{1}{2}
\Bigl [
\omega_k
-
\epsilon_k
-
\chi'
+
\frac{|A|^2}{2 \epsilon_k}
\Bigr ]
\notag \\
& \qquad\qquad
+
\frac{1}{\beta} \, {\ln [ \, 1 - e^{-\beta \omega_k} \, ]} \,
\Bigr \} \>,
\label{BEC.Seff.e:VeffII} \end{aligned}$$ where now $\chi' = \sqrt{2} \, \chi - \mu$ and $$\omega_k^2 = ( \epsilon_k + \chi')^2 - |A|^2 \>.$$ The equations for the auxiliary fields are obtained from $\delta \, {\mathcal{V}_{\text{eff}}^{\text{R}}}[\Phi] / \delta \chi'_{i} = 0$, as
\[BEC.Seff.e:gapeqs\] $$\begin{aligned}
\frac{A}{{\lambda_{\text{}}}}
&=
\phi_0^2
+
A \!
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\,
\Bigl \{
\frac{[ 2 n(\beta\omega_k) + 1 ]}{2\omega_k}
-
\frac{1}{2\epsilon_k}
\Bigr \} \>,
\label{BEC.Seff.e:gapeqsA} \\
\frac{\chi'+ {\mu_{\text{}}}}{2 {\lambda_{\text{}}}}
&=
| \phi_0 |^2 \!
+ \!
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\,
\Bigl \{
\frac{\epsilon_k + \chi'}{2\omega_k}
[ 2 n(\beta\omega_k) + 1 ]
-
\frac{1}{2}
\Bigr \} \>.
\label{BEC.Seff.e:gapeqsB}\end{aligned}$$
From Eq. we know that at the minimum of the effective potential we have $ (\chi' - A )\, \phi_0 = 0$, and we can replace ${\mu_{\text{}}}$ by the physical density using $$\label{BEC.Seff.e:rhodef}
\rho
=
-
\frac{\partial {\mathcal{V}_{\text{eff}}^{\text{R}}}[\Phi_0]}{\partial \mu}
=
\frac{\chi' + {\mu_{\text{}}}}{2 {\lambda_{\text{}}}} \>.$$ In the broken symmetry phase we have $\chi' = A$ in which case Eqs. become
\[BEC.Seff.e:gapeqsII\] $$\begin{aligned}
\frac{\chi'}{{\lambda_{\text{}}}}
&=
\rho_0
+
\chi'
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\,
\Bigl \{
\frac{[ 2 n(\beta\omega_k) + 1 ]}{2\omega_k}
-
\frac{1}{2\epsilon_k}
\Bigr \} \>,
\label{BEC.Seff.e:gapeqsAII} \\
\rho
&=
\rho_0
+
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\,
\Bigl \{
\frac{\epsilon_k + \chi'}{2\omega_k}
[ 2 n(\beta\omega_k) + 1 ]
-
\frac{1}{2}
\Bigr \} \>,
\label{BEC.Seff.e:gapeqsBII}\end{aligned}$$
where $\rho_0 = \phi_0^2$ is the condensate density.
\[s:meanfield\]Related mean-field approximations
================================================
For comparison, we will review next two related mean-field approximations. We will focus on the leading-order large-$N$ approximations, which corresponds to the choice of $\theta=0$ in our formalism, and the Popov approximation that is widely used in the study of BEC condensates.
\[ss:largeN\]Large-$N$ approximation in leading order
------------------------------------------------------
The large-$N$ approximation corresponds to the value $\theta = 0$. In obtaining the large-N approximation, one rewrites $\phi^\ast \phi$ in terms of two real components and extends the theory to $N$ real components. The $O(2)$ \[$U(1)$\] symmetry is then extended to $O(N)$. Here the composite field is $\chi = \lambda \, \phi_i \phi_i / N$. With appropriate rescaling, one can show[@r:Bender:1977bh] that the composite-field propagator is proportional to $1/N$, so counting loops of bound-state propagators yields the $1/N$ expansion. In lowest order we find that this approximation in the BEC phase leads to the free-field dispersion relation. A related large-$N$ expansion for the Bose gas at the critical temperature has been used successfully to characterize the behavior near the critical point[@r:Baym:2000fk]. One simplicity of this expansion is that the noninteracting-like dispersion relation simplifies the integrals present in the theory, and one can obtain analytic results even at finite temperatures. As with the general $\theta$ result, the large-$N$ expansion also provides a complete resummation of the original theory.
The large-$N$ finite-temperature effective potential in leading order is given by $$\label{ln.e:Vln-I}
{\mathcal{V}}_\text{LN}[\Phi]
=
{\mathcal{V}}_0
+
\chi \, | \phi |^2
-
\frac{(\chi + \mu_0)^2}{2 \lambda_0}
-
\frac{1}{2} \, {\mathrm{Tr} [ \, {\ln [ \, {\mathcal{G}}^{-1} \, ]} \, ]} \>.$$ The Matsubara inverse propagator in momentum space is now diagonal: $${\mathcal{G}}^{-1} ({\mathbf{k}}, n)
=
\begin{pmatrix}
i \omega_n - \epsilon_k - \chi & 0 \\
0 & -i\omega_n - \epsilon_k - \chi
\end{pmatrix}$$ We write the temperature-dependent last term in Eq. as $$\begin{aligned}
\frac{1}{2} \, {\mathrm{Tr} [ \, {\ln [ \, {\mathcal{G}}^{-1} \, ]} \, ]}
&=
\frac{1}{2 \beta}
{\int^{\Lambda} \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\sum_n (\omega_n^2 + \omega_k^2 )
\\
&=
{\int^{\Lambda} \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\,
\Bigl \{ \,
\frac{\omega_k}{2}
+
\frac{1}{\beta} \,
{\ln [ \, 1- e^{-\beta \omega_k} \, ]} \,
\Bigr \} \>,
\notag\end{aligned}$$ where $\omega_k = \epsilon_k + \chi$. Inserting this into Eq. , the effective potential for the large-$N$ case is given by $$\begin{aligned}
{\mathcal{V}}_\text{LN}[\Phi]
&=
{\mathcal{V}}_0
+
\chi \, | \phi |^2
-
\frac{(\chi + \mu_0)^2}{2 \lambda_0}
\label{ln.e:Vln-II} \\
& \qquad
+
{\int^{\Lambda} \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\,
\Bigl \{ \,
\frac{\omega_k}{2}
+
\frac{1}{\beta} \,
{\ln [ \, 1- e^{-\beta \omega_k} \, ]} \,
\Bigr \} \>.
\notag\end{aligned}$$ Setting the derivative of the effective potential with respect to $\chi$ equal to zero yields the gap equation, $$\frac{\chi+\mu_0}{\lambda_0}
=
| \phi |^2
+
{\int^{\Lambda} \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\,
\frac{2 \, n(\beta\omega_k) + 1 }{2} \>.$$
The large-$N$ effective potential in leading order is renormalized following the procedure discussed in Ref. [@r:Andersen:2004uq]. We recognize that the infinite constant is related to the renormalization of the chemical potential, i.e $$\frac{\mu_0}{\lambda_0}
=
\frac{{\mu_{\text{}}}}{{\lambda_{\text{}}}}
+
{\int^{\Lambda} \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\, \frac{1}{2} \>.$$ The renormalization is a consequence of the lack of a normal-ordering step in the the path-integral formalism, in contrast with the usual Hamiltonian formalism. Performing the renormalization, one obtains the finite gap equation, $$\label{LN.e:gap}
\rho
=
\frac{\chi+{\mu_{\text{}}}}{{\lambda_{\text{}}}}
=
\rho_0
+
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\,
n(\beta\omega_k) \>,$$ where we have set the condensate density $\rho_0 = |\phi |^2$. Eq. determines $\chi[\phi]$ implicitly, which is then re-inserted into the expression of ${\mathcal{V}}_{\text{LN}}$, so that ${\mathcal{V}}_{\text{LN}}[\Phi]$ becomes solely a function of $\rho_0 = |\phi |^2$. The renormalized potential is now $$\begin{aligned}
{\mathcal{V}}_{\text{LN}}[\phi]
&=
\chi \, | \phi |^2
-
\frac{ (\chi+ {\mu_{\text{}}})^2}{2 {\lambda_{\text{}}}}
\notag \\
& \qquad
+
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\,
\frac{1}{\beta} \, {\ln [ \, 1- e^{-\beta \omega_k} \, ]} \>.\end{aligned}$$ The minimum of the effective potential is when $$\frac{\partial \, {\mathcal{V}}_{\text{LN}}}{\partial \phi^\ast}
=
0 \>,
{\qquad\text{$\Rightarrow$}\qquad}
\phi \, \chi = 0 \>.$$ So, in the large-$N$ mean-field approximation, the broken-symmetry regime, $\phi \neq 0$, corresponds to the condition $$\chi = 0 \>,$$ which gives the dispersion relation, $\omega_k = \epsilon_k$, which is the same as the free-field theory dispersion. At finite temperature, the gap equation at the minimum $$\label{gap_ln}
{\lambda_{\text{}}}\, \rho_0
=
{\mu_{\text{}}}-
{\lambda_{\text{}}}{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\, n(\beta\omega_k)$$ which gives the chemical potential as $${\mu_{\text{}}}=
{\lambda_{\text{}}}\,
\Bigl \{ \,
\rho_0
+
\frac{\sqrt{\pi}}{2} \, T^{3/2} \, \zeta(3/2) \,
\Bigr \} \>.
\label{mu_ln}$$ Correspondingly, the phase transition ($\phi = 0$) takes place at the free-field critical temperature: $$T_c
=
\Bigl [ \,
\frac{2 {\mu_{\text{}}}}{ {\lambda_{\text{}}}\, \zeta(3/2) \, \sqrt{\pi}} \,
\Bigr ]^{2/3}
\>.$$ At the minimum, $\chi = 0$ so that the value of the effective potential at the minimum as a function of temperature for $T< T_c$ is $$\begin{aligned}
{\mathcal{V}}_{\text{LN}}
& =
-
\frac{{\mu_{\text{}}}^2}{2 {\lambda_{\text{}}}}
+
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\, \frac{1}{\beta} \, {\ln [ \, 1- e^{-\beta \epsilon_k} \, ]} \>,
\\ \notag
& =
-
\frac{{\mu_{\text{}}}^2}{2 {\lambda_{\text{}}}}
-
\frac{T^{3/2}}{16 \pi^2} \,
\frac{\sqrt{\pi } \,
\zeta(5/2)}{2} \>.\end{aligned}$$ The total number density is determined from $$\label{rho_ln}
\rho
=
- \frac{\partial \, {\mathcal{V}}_{\text{LN}} }{\partial {\mu_{\text{}}}}
=
\frac{{\mu_{\text{}}}}{{\lambda_{\text{}}}} \>.$$ Hence, by combining Eqs. and , we obtain the density of particles in the condensate as $$\rho_0
=
\rho
-
\frac {\sqrt{\pi}}{2} \, T^{3/2} \, \zeta(3/2) \>.$$
In summary, below $T_c$ the large-$N$ approximation gives essentially the same results as a non-interacting gas since $\chi =0$. Above $T_c$, the large-$N$ approximation gives rise to a self-consistent correction to the dispersion relation. The large-$N$ result above $T_c$ is the same as that of the Popov approximation we review below. We will also find that the large-$N$ approximation is equal to the LOAF approximation we are proposing here at high temperatures in the regime where $A=0$.
\[ss:popov\]Hartree and Popov approximations
--------------------------------------------
The Hartree approximation is a truncation scheme that ignores correlation functions beyond the first two. Technically this is obtained by setting the third derivative of the generating functional of connected graphs with respect to the external currents to zero, i.e. $$\frac {\delta^3 \, W[j]}{ \delta j(x) \, \delta j(y) \, \delta j(z)}
\equiv
0 \>.$$ Then, the vacuum expectation value of the expectation value of the field $\phi[j](x)$ in the presence of external sources is $$\begin{aligned}
&( h - \mu ) \, \phi(x)
+
\lambda_0 \, | \phi(x) |^2 \, \phi(x)
\label{BEC.hartree.e:funceom} \\
& \qquad
+
2 \lambda_0 \hbar \, G(x,x) \, \phi(x) / i
\notag \\
& \qquad
+
\lambda_0 \hbar \, K(x,x) \, \phi^{\ast}(x) / i
=
j(x) ,
\notag\end{aligned}$$ where $h$ was defined in Eq. . Here $\phi(x)$, $G(x,x')$ and $K(x,x')$ are considered functionals of the current $j(x)$. $G(x,x')$ and $K(x,x')$ have the same meaning as the normal and anomalous correlation functions in Eq. . Introducing new auxiliary fields $\chi(x)$ and $A(x)$ by the definition,
\[BEC.hartree.e:chiAdefs\] $$\begin{aligned}
\frac{\chi(x) + \mu_0}{2 \lambda_0}
&=
| \, \phi(x) \, |^2 + \hbar \, G(x,x) / i \>,
\label{BEC.hartree.e:chidef} \\
\frac{A(x)}{\lambda_0}
&=
[ \, \phi(x) \, ]^2 + \hbar \, K(x,x) / i \>.
\label{BEC.hartree.e:Adef}\end{aligned}$$
and setting $j(x) = 0$ in Eq. gives an equation for the average field, $$\label{BEC.hartree.e:eomphiII}
\bigl [ \,
h
+
\chi(x)
-
2 \lambda \, | \, \phi(x) \, |^2 \,
\bigr ] \, \phi(x)
+
A(x) \, \phi^{\ast}(x)
=
0 \>.
\notag$$ and its complex conjugate. Functional differentiation of Eq. with respect to $j(x')$ and $j^{\ast}(x')$, ignoring third-order functional derivatives leads to equations for the Green functions $G(x,x')$ and $K(x,x')$. We find
\[BEC.hartree.e:GKeom\] $$\begin{aligned}
\bigl [ \,
h
+
\chi(x) \,
\bigr ] \, G(x,x')
+
A(x) \, K^{\ast}(x,x')
&=
\delta(x,x') \>,
\label{BEC.hartree.e:Geom} \\
\bigl [ \,
h
+
\chi(x) \,
\bigr ] \, K(x,x')
+
A(x) \, G^{\ast}(x,x')
&=
0 \>,
\label{BEC.hartree.e:Keom}\end{aligned}$$
and the complex conjugates. Eqs. can be written in matrix form as $$\label{BEC.hartree.e:Gmatform}
\int {\mathrm{d}}x' \,
{\mathcal{G}}^{-1}(x,x') \, {\mathcal{G}}(x',x'')
=
\delta(x,x'') \>,$$ where
\[BEC.hartree.e:Gmatdefs\] $$\begin{aligned}
{\mathcal{G}}^{-1}(x,x')
&=
\delta(x,x') \,
\begin{pmatrix}
h + \chi(x) & A(x) \\
A^{\ast}(x) & h^{\ast} + \chi(x)
\end{pmatrix} \>.
\label{BEC.hartree.e:GmatdefsB}\end{aligned}$$
The renormalized effective potential for the Hartree approximation can be written as: $$\begin{aligned}
&{\mathcal{V}}_{\text{H}}[\Phi]
=
{\mathcal{V}}_{\text{R}}
+
\chi \, | \phi |^2
-
{\lambda_{\text{}}}\, | \phi |^4
-
\frac{( \chi + {\mu_{\text{}}})^2}{4 {\lambda_{\text{}}}}
\label{BEC.hartree.e:effVII} \\
& \qquad
-
\frac{| A |^2}{2 {\lambda_{\text{}}}}
+
\frac{1}{2} \,
[ \,
\phi^2 \, A^{\ast}
+
\phi^{\ast\,2} \, A \,
] \,
\notag \\
&
+
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\,
\Bigl \{ \,
\frac{1}{2}
\Bigl [
\omega_k
-
\epsilon_k
-
\chi
+
\frac{|A|^2}{2 \epsilon_k}
\Bigr ]
+
\frac{1}{\beta} \, {\ln [ \, 1 - e^{-\beta \omega_k} \, ]} \,
\Bigr \} \>.
\notag\end{aligned}$$ The minimum of the potential is given by $$\label{BEC.hartree.e:dVdphi}
\frac{\partial V_{\text{H}}}{\partial \phi^{\ast}} \Big |_{\phi_0}
=
\chi_0 \, \phi_0
-
2 {\lambda_{\text{}}}\, | \phi_0 |^2 \, \phi_0
+
A_0 \, \phi^{\ast}_0
=
0 \>.$$ Again, the $U(1)$ gauge symmetry, allows us to choose $\phi_0$ to be real at the minimum. Then according to Eq. , $A_0$ is also real. Hence, Eq. becomes $$\label{BEC.hartree.e:phimincond}
[ \,
\chi_0 - 2 {\lambda_{\text{}}}\, | \phi_0 |^2 + A_0 \,
] \, \phi_0
=
0 \>.$$ In the broken symmetry case, $\phi_0 \ne 0$, we have $$\label{BEC.hartree.e:wehave}
\chi_0 + A_0
=
2 {\lambda_{\text{}}}\, | \phi_0 |^2 \>,$$ and the dispersion relation is $$\label{BEC.hartree.e:omegahartree}
\omega_k^2
=
[ \, \epsilon_k + 2 {\lambda_{\text{}}}\, | \phi_0 |^2 \, ] \,
[ \,
\epsilon_k
-
2 \, ( \, A_0 - {\lambda_{\text{}}}\, | \phi_0 |^2 \, ) \, ] \>.$$
The Hartree approximation has the defect of not being gapless. This can be fixed by hand by ignoring the fluctuation in the anomalous density, that is by arbitrarily setting $$A_0 - {\lambda_{\text{}}}\, | \phi_0 |^2
=
0 \>.$$ This further approximation is known as the “gapless” Popov approximation [@r:Popov:1983kx]. The Popov approximation includes the self-consistent fluctuations of $\chi$, but treats $A$ classically. Below $T_c$, the Popov approximation has the dispersion relation $$\label{disp:Popov}
\omega_{k}^{2}
=
\epsilon_{k} ( \, \epsilon_{k} + 2 {\lambda_{\text{}}}\rho_0 \, ) \>,$$ and the chemical potential is ${\mu_{\text{}}}= 2 {\lambda_{\text{}}}\rho - {\lambda_{\text{}}}\rho_0$. The condensate density $\rho_0$ is given by $$\rho
=
\rho_0
+
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\,
\Bigl \{ \,
\frac{ \epsilon_{k} + {\lambda_{\text{}}}\rho_0}
{ \omega_{k}} \,
[ \, 2n(\beta\omega_{k}) + 1 \, ]
-
\frac{1}{2} \,
\Bigr \} \>.$$ In the Popov approximation the critical temperature is the same as in the free-field case, $T_c=T_0$.
The Popov approximation is the most commonly used mean-field theory of weakly interacting bosons at finite temperatures[@r:Andersen:2004uq]. This approximation has a gapless spectrum but is known to produce an artificial first-order phase transition as we shall see below. Unlike the LOAF and the Hartree approximations, the equations for $\phi, A, \chi$ in the Popov approximation are not derivable from an effective action.
\[ss:results\]Mean-field results and discussions
================================================
We begin by comparing the predictions of the LOAF and Popov approximations for zero-temperature conditions. In the broken-symmetry phase at $T=0$, we have $\chi' = A$, and the effective potential is given by $$\begin{aligned}
{\mathcal{V}}_{\text{eff}} [\chi']
&=
-\frac{1}{2 \lambda} \,
\Bigl [ \,
\frac{ ( \chi'+ \mu )^2 }{2}
-
\chi'{}^2 \,
\Bigr ]
\\ \notag
& \qquad
+
\frac{1}{4 \pi^2}
\int_{0}^{\infty} \!\! k^2 \, {\mathrm{d}}k \,
\Bigl [ \,
\omega_k
-
\epsilon_k
-
\chi'
+
\frac{\chi'{}^2}{2 \epsilon_k} \,
\Bigr ]
\\ \notag
&=
-
\frac{1}{2 \lambda}
\Bigl [ \,
\frac{ ( \chi' + \mu )^2 }{2}
-
\chi'{}^2 \,
\Bigr ]
+
\frac { 2 \sqrt{2}} {15 \pi^2} \, \chi'{}^{5/2} \>.
\label{VNzero}\end{aligned}$$ Setting $\partial {\mathcal{V}}_{\text{eff}}/\partial \chi' = 0$, then gives $\chi'$ as a function of $\mu$ at the minimum. We have $$\label{mu1}
\chi'
=
\mu
-
\lambda \, \frac{ 2 \sqrt{2}}{3 \pi^2} \, \chi'{}^{3/2}
\>.$$ The above cubic equation can be solved explicitly. In particular, in the weak coupling limit we obtain $$\chi'
=
\mu
-
\lambda \, \frac{2 \sqrt{2} }{3 \pi^2} \, \mu^{3/2} \>.
\label{chi_weak}$$ Using Eq. , in weak coupling we obtain $$\rho
=
\frac{\mu}{\lambda}
\Bigl [ \,
1
-
\frac{ \lambda \sqrt{2 \mu}}{ 3 \pi^2} \,
\Bigr ]
\label{eq:rho_T0}
\>,$$ which agrees with the one-loop result corresponding to the original Bogoliubov approximation (see e.g. Eq. 89 in Ref. [@r:Andersen:2004uq]). By inverting Eq. , we derive $\mu(\rho)$ at weak coupling, as $$\begin{aligned}
\mu
& =
\lambda \rho
\Bigl [ \,
1
+
\frac{1}{3 \pi^2} \sqrt{ \frac{\lambda \rho} {2} } \,
\Bigr ]
\label{mu_weak}
\\ \notag
&
=
8\pi \rho \, a \,
\Bigl ( \,
1
+
\frac{32}{3} \, \sqrt{\rho \, a^{3} / \pi } \,
\Bigr ) \>,\end{aligned}$$ where we have set ${\lambda_{\text{}}}= 4\pi \hbar^2 \, a / m$ with $a$ the $s$-wave scattering length.
We can also calculate the condensate depletion, defined as $\rho - \rho_0$. From Eq. at $T=0$, we obtain the exact LOAF result $$\rho - \rho_0 = \frac{1}{6\sqrt{2} \, \pi} \, \chi'^{3/2}
\>.$$ Hence, using Eqs. and we obtain the weak-coupling result for the fractional depletion (see e.g. Eq. 22.14 in Ref. ) $$1 - \frac{\rho_0}{\rho} = \frac{8}{3} \sqrt{ \frac{\rho a^3}{\pi} }
\>,$$ first obtained by Bogoliubov in 1947 [@r:Bogoliubov:1947ys].
![\[f:Tzero\] (Color online) Comparison of the predictions of the Popov and LOAF approximations regarding the zero-temperature values of the densities, $\chi' = A$, condensate fraction, $\rho_0/\rho$, chemical potential, $\mu$, and effective potential, $V_{\mathrm{eff}}$, as a function of dimensionless parameter, $\rho^{1/3}a$. In the case of the LOAF approximation, the normal and anomalous densities are equal, $\chi'= A$, whereas in the Popov approximation we have $A = \lambda \rho_0$. Note that there is no effective potential in the Popov approximation, because this approximation is not derivable from an action. ](zero_temperature){width="0.9\columnwidth"}
![\[f:densities\] (Color online) Normal density, $\chi'$, and anomalous density, $A$, from the LOAF and Popov approximations. The comparison between the LOAF and Popov approximations is carried out for $\rho^{1/3}a = 1$, $\rho^{1/3}a = 0.4$, and $\rho^{1/3}a = 0.05$. $T_c$ and $T^\star$ indicate vanishing condensate density, $\rho_0$, and anomalous density, $A$, respectively. The Popov approximation leads to a first-order phase transition, whereas LOAF predicts a second-order phase transition. We have that $T_c = T^\star$ in the Popov approximation but not in LOAF. In LOAF $\chi'$ and $A$ are equal for $T \le T_c$. ](comp_A-Chi){width="0.9\columnwidth"}
![\[f:rho0\] (Color online) Condensate fraction, $\rho_0/\rho$, from the LOAF and Popov approximations. Similarly to Fig. \[f:densities\]. Because at $T_c$ the Popov approximation and noninteracting dispersion relations are the same, the Popov approximation does not change $T_c$ relative to the noninteracting case. LOAF increases $T_c$. ](comp_rho0){width="0.9\columnwidth"}
![\[f:mu\] (Color online) Chemical potential, $\mu$, from the LOAF and Popov approximations. Similarly to Fig. \[f:densities\]. ](comp_mu){width="0.9\columnwidth"}
![\[f:unity\] (Color online) LOAF predictictions for the critical regime, $T=T_c$, as a function of $\rho^{1/3}a$: (Top panel) Relative change in $T_c$ with respect to the noninteracting critical temperature, $T_0$. (Bottom panel) Critical value of the normal and anomalous densities, $\chi'_c = A_c$. The insets illustrate the $\rho^{1/3}a$ dependence of $\Delta T_c/T_0 = (T_c - T_0)/T_0$ and $\chi'_c$ in the weak-coupling regime. ](unitarity){width="0.9\columnwidth"}
In Fig. \[f:Tzero\] we depict the coupling constant dependence of the zero-temperature values of the normal densities, $\chi'$, condensate fraction, $\rho_0/\rho$, chemical potential, $\mu$, and effective potential, $V_{\mathrm{eff},0}$. The coupling constant depends linearly of the dimensionless parameter, $\rho^{1/3}a$. We note that in the case of the LOAF approximation, the normal and anomalous densities are equal, $\chi' = A$, whereas in the Popov approximation we have $A = \lambda \rho_0$. Also, in the Popov approximation there is no effective potential, because this approximation is not derivable from an action. At zero temperature, LOAF predicts that the condensate fraction in the unitarity limit is 3/4, whereas in the Popov approximation the condensate fraction approaches zero asymptotically.
Turning now to the discussion of results in the finite temperature regime, we note that throughout this section, the temperature is scaled by its noninteracting critical value, $T_0 = (2\pi \hbar^2 / m) [\rho/\zeta(3/2)]^{2/3}$, where $\zeta(x)$ is the Riemann zeta function. In Fig. \[f:densities\] we depict the temperature dependence of the normal density $\chi'$, and anomalous density, $A$, at constant $\rho^{1/3}a$. We compare the results derived using the LOAF and and the Popov approximations. For illustrative purposes, we show results for $\rho^{1/3}a = 1$, $\rho^{1/3}a = 0.4$, and $\rho^{1/3}a = 0.05$. Similarly, in Figs. \[f:rho0\] and \[f:mu\], we depict the temperature dependence of the condensate fraction, $\rho_0/\rho$, and chemical potential, $\mu$, respectively, for different interaction strengths.
We identify two special temperatures, at $T_c$ where the condensate density vanishes, and at $T^\star$ where the anomalous density, $A$, vanishes. These temperatures are the same in the Popov aproximation formalism, but they are different in the LOAF approximation. The existence of a temperature range, $T_c < T < T^\star$, for which the anomalous density, $A$, is nonzero despite a zero condensate fraction, $\rho_0/\rho$, is a fundamental prediction of LOAF. In this temperature range, the dispersion relation departs from the quadratic form predicted by the Popov approximation for $T > T_c$. Above $T_c$ the solution of the Popov-approximation equations becomes multivalued, indicating that the system undergoes a first-order phase transition at $T_c$. In contrast, LOAF predicts a second-order transition. Because at the critical temperature, $T_c$, in the Popov approximations and the noninteracting gas case, the dispersion relations are the same, the Popov approximation does not change $T_c$ relative to the noninteracting case. The LOAF formalism predicts a higher critical temperature than in the noninteracting case, $T_c \ge T_0$. In the weak coupling limit, we wave $T_c \rightarrow T_0$, as $\rho^{1/3}a \rightarrow 0$.
As illustrated in Figs. \[f:densities\], \[f:rho0\], and \[f:mu\], the LOAF and Popov approximations results become qualitatively similar in the weak coupling limit, even though the order of the phase transitions remains different. However, strengthening the interaction between particles in the Bose gas results in enhanced differences between the LOAF and Popov predictions, even for temperatures, $T \ll T_c$. A larger value of $\rho^{1/3}a$ indicates stronger coupling.
We also note for comparison purposes, that below $T_c$ the large-$N$ approximation gives the same results as a non-interacting gas. Above $T_c$, the large-$N$ result above $T_c$ is the same as that of the Popov approximation. Also, above $T^\star$, where $A=0$ in the LOAF approximation, the large-$N$, Popov and LOAF approximation give the same results.
In Fig. \[f:unity\] we depict the relative change in $T_c$ with respect to $T_0$, $\Delta T_c/T_0 = (T_c - T_0)/T_0$, and the critical value of the normal and anomalous densities, $\chi'_c = A_c$, predicted by LOAF, as a function of the interaction strength characterized by the dimensionless parameter $\rho^{1/3}a $. The insets show the weak-coupling limit of LOAF results, emphasizing the departure from the noninteracting result in lowest order.
The leading-order auxiliary formalism, LOAF, produces a more realistic set of observables away from the weak-coupling limit because of its non-perturbative character. In contrast, the Popov approximation is appropriate only in the case of a weakly-interacting gas of bosons. The former is made explicit by studying the LOAF prediction for the relative change, $\Delta T_c/T_0 = (T_c - T_0)/T_0$, as a function of $\rho^{1/3}a $. The inset in the top panel in Fig. \[f:unity\] illustrates that in the weak-coupling regime, $\rho^{1/3}a \ll 1$, LOAF produces the same slope, 2.33, for the linear departure as that derived by Baym *et al.*[@r:Baym:2000fk] using the large-N expansion, but at next-to-leading order (i.e. they include density fluctuations in their calculation). The LOAF corrections to the critical temperature are due to the inclusion of self-consistent fluctuations effects in the mean-field $\chi'$ and $A$ densities. We note that carrying that approach to the next order, the slope is reduced to $1.71$ [@PhysRevA.62.063604], and is approaching the Monte Carlo estimates of $1.32\pm0.02$ [@PhysRevLett.87.120401; @PhysRevE.64.066113; @PhysRevE.68.049902], and $1.29\pm 0.05$ [@PhysRevLett.87.120402]. It will be interesting to see how our next to leading order calculation compares to these results. A summary of other $\Delta T_c/T_0$ theoretical predictions is found in Ref. .
As the system approaches the unitarity limit, LOAF predicts that $\Delta T_c/T_0 \rightarrow 0.396$ and $\chi'_c/T_0 = A_c/T_0 \rightarrow 0.873$ for $\rho^{1/3}a \gg 1$.
\[s:conclusions\]Conclusions
============================
In this paper we discussed in detail a new auxiliary-field formulation for the BEC problem that was first introduced in Ref. . At mean-field level this approach meets three very important criteria [@r:Andersen:2004uq] for a satisfactory mean-field theory for weakly interacting bosons: (1) the excitation spectrum should be gapless (Goldstone theorem), (2) at $T=0$ and weak coupling, it reproduces the known results from Bogoliubov theory, and (3) it has a smooth second-order phase transition. The commonly used theories violate those criteria: the Hartree approximation violates (1), the Bogoliubov and Popov theories violate (3), and the $T$-matrix Popov theory violates (2). Also at mean-field level, we obtain a result for $\Delta T_c/T_0 = (T_c - T_0)/T_0$ which was obtain only at next-to-leading order in a large-$N$ expansion, showing that including the anamolous density in our auxiliary-field formulation is quite important. This approach will be useful to study both the static and dynamic properties of dilute Bose gases. As described above, one can systematically improve upon the LOAF approximation discussed here by calculating the 1-PI action order-by-order in $\epsilon$. The broken $U(1)$ symmetry Ward identities guarantee Goldstone’s theorem order-by-order in $\epsilon$ [@r:Bender:1977bh]. For time-dependent problems, however, this expansion is secular[@r:MCD01], and a further resummation is required. The latter is performed using the two-particle irreducible (2-PI) formalism[@r:Baym62; @r:CJT]. The corresponding Schwinger-Dyson (SD) equations for the scalar field and the two-particle correlation functions are simplified dramatically because all vertices are trilinear. A practical implementation of this approach is the bare-vertex approximation (BVA)[@r:BCDM01]. The BVA is an energy-momentum and particle-number conserving truncation of the SD infinite hierarchy of equations obtained by ignoring the derivatives of the self-energy, similarly to the Migdal’s theorem[@r:Migdal:1958uq] approach in condensed matter physics. The BVA proved effective in the case of classical and quantum $\lambda \phi^4$ field theory problems[@r:CDM02; @r:CDM02ii; @r:Mihaila:2003ys] and can be applied to the BEC case. In this context, we note that a related approximation is the 2PI-1/N expansion which has been used in particle theory to study thermalization of various quantum field theories [@r:AB01; @r:B02; @r:AABBS]. Its use for studying dilute Bose gases was discussed by Calzetta and Hu [@r:Calzetta:2008pb]. The 2-PI approach has been used also to study the quantum dynamics in the Bose-Hubbard model [@r:ReyHuCalzettaRouraClark03; @PhysRevA.75.013613].
This work was performed in part under the auspices of the U. S. Dept. of Energy. The authors would like to thank E. Mottola and P. B. Littlewood for useful discussions.
\[s:renorm\]Regularization and renormalization
==============================================
Unlike the case of an operator formalism where one can remove vacuum energies by normal ordering, in the path integral method we have to subtract an infinite zero-point vacuum energy ${\mathcal{V}_0}$. In addition the interaction strength $\lambda_0$ needs to be renormalized to obtain the physical scattering amplitude, as in the Bogoliubov theory for a $\delta$-function interaction. This is accomplished by summing the Born series to find the physical $s$-wave scattering amplitude. We will find that regularizing by subtracting the leading divergences in the expression for the potential for the broken symmetry case is equivalent to dimensional regularization, which is known to preserve the Ward identities. It is also equivalent to renormalizing the vacuum energy, chemical potential, and coupling constant.
\[ss:DR\]Dimensional regularization
-----------------------------------
Our regularization scheme of subtracting the leading divergence is equivalent to a dimensional regularization procedure, which guarantees that the Ward identities of the unrenormalized theory are preserved. Dimensional regularization consists of evaluating a generalization of the integral in a regime where it is defined and then analytically continuing to the original ill-defined integral.
In the broken symmetry phase, we need to evaluate an integral of the form $$\label{dr.e:dint}
I[M^2]
=
\frac{1}{4 \pi^2} \int_0^\infty \!\! k^2 \, {\mathrm{d}}k \,
\sqrt{ k^2 ( k^2 + M^2 ) }
\>.$$ If we consider instead the integral $$\begin{aligned}
&\frac{1}{4 \pi^2}
\int_0^\infty \!\! {\mathrm{d}}k \, k^{-\alpha} \,
( k^2 + M^2 )^\gamma
\\ \notag & =
\frac{ [ \, M^2 \, ]^{\gamma +1}}{[ \, M^2 \, ]^{ (\alpha+1)/2} }
\frac{
\Gamma [ \, ( \, 1 - \alpha \, ) / 2 \, ] \,
\Gamma [ \, ( \, \alpha - 2 \gamma - 1 \, ) / 2 \, ]
}{ 8 \pi^2 \Gamma (-\gamma) } \>,
\end{aligned}$$ and then analytically continue this expression to $\alpha=-3$ and $\gamma = 1/2$, we obtain the dimensionally-regularized value of the integral in Eq. as $$I[M^2]
=
\frac{1}{30 \pi^2} \, [ \, M^2 \, ]^{5/2} \>.$$ This is exactly what we obtained by regulating the integral by subtracting the leading divergences, i.e. $$\begin{aligned}
I_{\text{R}}[M^2]
&=
\frac{1}{4 \pi^2} \int_0^\infty \!\!\! k^2 \, {\mathrm{d}}k \,
\Bigl \{
\sqrt{ k^2 ( k^2+M^2 )}
\\
& \qquad
-
k^2
-
\frac{M^2}{2}
+
\frac{M^4}{8 k^2}
\Bigr \}
=
\frac{1}{30 \pi^2} \, [ \, M^2 \, ]^{5/2} \>,\end{aligned}$$ because the terms we subtracted are formally zero in the dimensional regularization scheme.
\[ss:R\]Renormalization
-----------------------
In the broken symmetry phase, our regularization scheme of subtracting the leading divergence is also equivalent to renormalizing the vacuum energy, chemical potential, and coupling constant.
Introducing a cutoff $\Lambda$ in the momentum integrals in Eq. , the effective potential in the broken symmetry case is given by $$\begin{aligned}
&{\mathcal{V}_{\text{eff}}}[\chi']
=
{\mathcal{V}_0}-
\frac{ \mu_0^2 }
{ 2 \lambda_0 \cosh^2\theta }
-
\frac{ \mu_0 \chi' }
{ \lambda_0 \cosh^2\theta }
\label{Re.e:V-i} \\
& \quad
+
\frac{2 \chi^{\prime\,2}}{\lambda_0 \sinh 2\theta }
+
{\int^{\Lambda} \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\Bigl \{ \,
\frac{\omega_k}{2}
+
\frac{1}{\beta}
{\ln [ \, 1 - e^{-\beta \omega_k} \, ]} \,
\Bigr \} \>,
\notag \end{aligned}$$ and $\omega_k = \sqrt{ \epsilon_k \, ( \epsilon_k + 2 \chi' ) }$. We first renormalize the interaction strength $\lambda_0$ by setting $$\label{Re.e:lambdaRdef}
\frac{2}{\lambda_0 \sinh 2\theta}
=
\frac{2}{{\lambda_{\text{}}}\sinh 2\theta}
+
{\int^{\Lambda} \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\, \frac{1}{4\epsilon_k} \>.$$ Next we renormalize the chemical potential $\mu_0$ by setting $$\label{Re.e:muRdef}
\frac{\mu_0}{\lambda_0 \cosh^2 \theta}
=
\frac{{\mu_{\text{}}}}{{\lambda_{\text{}}}\cosh^2 \theta}
+
{\int^{\Lambda} \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\, \frac{1}{2} \>.$$ The renormalized vacuum energy is then defined by the equation $$\label{Re.e:VzeroRdef}
{\mathcal{V}_0}-
\frac{ \mu_0^2 }
{ 2 \lambda_0 \cosh^2\theta }
=
{\mathcal{V}_{\text{R}}}-
\frac{ {\mu_{\text{}}}^2 }
{ 2 {\lambda_{\text{}}}\cosh^2\theta }
+
\frac{1}{2} {\int^{\Lambda} \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\, \epsilon_k \>,$$ so that the effective potential becomes $$\begin{aligned}
&{\mathcal{V}_{\text{eff}}^{\text{R}}}[\chi']
=
{\mathcal{V}_{\text{R}}}-
\frac{ ( \chi' + {\mu_{\text{}}})^2 }
{ 2 {\lambda_{\text{}}}\cosh^2\theta }
+
\frac{\chi^{\prime\,2}}{2 {\lambda_{\text{}}}\sinh^2\theta }
\label{Re.e:VeffR} \\
& \quad
+
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\Bigl \{ \,
\frac{1}{2}
\Bigl [
\omega_k
-
\chi'
+
\frac{\chi^{\prime\,2}}{2 \epsilon_k} \,
\Bigr ]
+
\frac{1}{\beta}
{\ln [ \, 1 - e^{-\beta \omega_k} \, ]} \,
\Bigr \} \>,
\notag \end{aligned}$$ where we have taken the limit $\Lambda \rightarrow \infty$ since the integral is now finite. For completeness, we note that the renormalized gap equation for $\chi'$ is $$\begin{aligned}
&\frac{\partial {\mathcal{V}_{\text{eff}}^{\text{R}}}[\chi']}{\partial \chi'}
=
\frac{\chi'}{{\lambda_{\text{}}}\sinh^2 \theta}
-
\frac{\chi' + {\mu_{\text{}}}}{{\lambda_{\text{}}}\cosh^2 \theta}
\label{Re.e:dVdchip} \\
& \qquad
+
{\int \!\!\frac{ \mathrm{d}^3 k }{ (2\pi)^3 } }\,
\Bigl \{ \,
\frac{\epsilon_k}{2 \omega_k} \,
[ \, 2 \, n(\beta\omega_k) + 1 \, ]
-
\frac{1}{2}
+
\frac{\chi'}{2 \, \omega_k} \,
\Bigr \}
=
0 \>.
\notag\end{aligned}$$
\[s:building\]Building blocks for graphs
========================================
Mean-field perturbation theory is an expansion around the stationary point of the effective action and uses the propagators and vertices of the stationary point to construct all the graphs. The propagators that enter into the loop expansion are the mean-field propagators ${\mathcal{G}}^a{}_b[\chi]$, where ${\mathcal{G}}^{-1}{}^a{}_b[\chi]$ is given by Eq. , and ${\mathcal{D}}_{ij}[\Phi]$, where ${\mathcal{D}}_{ij}[\Phi]^{-1}$ is defined by Eq. . The basic local vertices are the three-point vertex ${\mathcal{V}}^i_{ab}$, which connects $\chi_i$ with a $\phi_a$ and $\phi_b$, and the two-point vertex, ${\mathcal{V}}^i_{ab} \phi^b$, which changes a $\phi_a$ into a $\chi^i$. The lowest-order theory also consists of the nonlocal 1-PI vertices for $N$-$\chi$ lines, namely $$\Gamma_N^{i_1,i_2, \ldots i_N}
=
\frac{\delta^N {\mathrm{Tr} [ \, {\ln [ \, {\mathcal{G}}^{-1}[\chi] \, ]} \, ]} }
{\delta \chi_{i_1} \delta \chi_{i _2} \ldots \delta \chi_{i_N}}$$ These nonlocal $N$-$\chi$ vertices are polygons made up of $N$ mean-field propagators ${\mathcal{G}}[\chi]$. Once we have $\Gamma[\phi_a, \chi_i]$ to some order in $\epsilon$, we can determine the equations for $\phi$ and $\chi$ from $\delta \Gamma/ \delta \phi_a = j^a$ and $\delta \Gamma/ \delta \chi_i = s^i$. Subsequently, all higher-order 1-PI vertex functions can be obtained by knowing what happens when we differentiate either ${\mathcal{G}}$ with respect to $\chi_i$ or ${\mathcal{D}}$ with respect to both $\chi_i$ and $\phi_a$. Because we know both ${\mathcal{G}}^{-1}$ and ${\mathcal{D}}^{-1}$ explicitly, one uses the identity $$\frac{\delta A}{\delta \Phi}
=
- A \circ \frac{\delta A^{-1}}{\delta \Phi} \circ A \,$$ to obtain the rules for how to functionally differentiate ${\mathcal{G}}$ and ${\mathcal{D}}$ in a graph. Here the $\circ$ symbol stands for both an integration and a matrix product. Using the notation of Eq. with $\chi_i = \eta_{ij} \, \chi^j$, we note that $$\begin{gathered}
\frac{\delta \chi^i(x)}{\delta \chi^j(x')}
=
\delta^i{}_j \, \delta(x,x') \>,
\quad
\frac{\delta \chi^i(x)}{\delta \chi_j(x')}
=
\eta^{ij} \, \delta(x,x') \>,
\label{BEC.NLOAF.e:chidervs} \\
\frac{\delta [ \, \chi_i(x) \chi^i(x) \, ]}{\delta \chi^j(x')}
=
2 \, \chi_j(x) \, \delta(x,x') \>.
\notag\end{gathered}$$ Functional derivatives of ${\mathcal{G}}^{-1}[\chi]$ with respect to $\chi^i$ are given in terms of $$\label{BEC.Seff.e:dGinvdchi}
\frac{ \delta {\mathcal{G}}^{-1}[\chi] }{ \delta \chi_i }
=
{\mathcal{V}}^{i}(\theta) \>,$$ where
\[BEC.NLOAF.e:Vimats\] $$\begin{aligned}
{\mathcal{V}}^{1}(\theta)
&=
\cosh\theta
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix} \>,
\\
{\mathcal{V}}^{2}(\theta)
&=
\sqrt{2} \, \sinh\theta
\begin{pmatrix}
0 & 0 \\
1 & 0
\end{pmatrix} \>,
\\
{\mathcal{V}}^{3}(\theta)
&=
\sqrt{2} \, \sinh\theta
\begin{pmatrix}
0 & 1 \\
0 & 0
\end{pmatrix} \>. \end{aligned}$$
In addition, we have $$\label{e:dGdchis}
\frac{\delta {\mathcal{G}}[\chi]}{\delta \chi_i}
=
-
{\mathcal{G}}[\chi] \circ {\mathcal{V}}^{i}(\theta) \circ {\mathcal{G}}[\chi] \>.$$ In this notation, the inverse composite-field propagator ${\mathcal{D}}_{ij}^{-1}[\Phi](x,x')$ defined in Eq. is given by $$\label{BEC.NLOAF.e:DinverseII}
{\mathcal{D}}_{ij}^{-1}[\Phi](x,x')
=
\frac{\eta_{ij} }{\lambda} \, \delta(x,x')
+
\Pi_{ij}[\Phi](x,x') \>,$$ where the polarization $\Pi{}^{ij}[ \Phi ]$ is $$\begin{aligned}
\Pi^{ij}[\Phi]
&=
\phi \circ {\mathcal{V}}^{ij}[\chi](\theta) \circ \phi
\label{BEC.NLOAF.e:Sigmadef} \\
& \qquad
-
\frac{\hbar}{2i} \,
{\mathrm{Tr} [ \, {\mathcal{G}}[\chi] \circ {\mathcal{V}}^i(\theta) \circ
{\mathcal{G}}[\chi] \circ {\mathcal{V}}^j(\theta) \, ]} \>,
\notag \end{aligned}$$ with $$\begin{aligned}
&{\mathcal{V}}^{ij}[\chi](\theta)
\label{BEC.NLOAF.e:Vijdef} \\
&=
\frac{1}{2} \,
\bigl [ \,
{\mathcal{V}}^i(\theta) \circ {\mathcal{G}}[\chi] \circ {\mathcal{V}}^j(\theta)
+
{\mathcal{V}}^j(\theta) \circ {\mathcal{G}}[\chi] \circ {\mathcal{V}}^i(\theta) \,
\bigr ] \>.
\notag\end{aligned}$$ Another quantity we will need for obtaining the graphical rules is ${\mathcal{R}}^{ijk}[\chi]$ defined by by $$\begin{aligned}
&{\mathcal{R}}^{ijk}[\chi](\theta)
=
-\frac{\delta {\mathcal{V}}^{ij}[\chi](\theta)}{\delta \chi_k}
\label{BEC.NLOAF.e:Rijkdef} \\
&=
\frac{1}{2} \,
\bigl \{ \,
{\mathcal{V}}^i(\theta) \circ {\mathcal{G}}[\chi] \circ {\mathcal{V}}^k(\theta)
\circ {\mathcal{G}}[\chi] \circ {\mathcal{V}}^j(\theta)
\notag \\
& \qquad
+
{\mathcal{V}}^j(\theta) \circ {\mathcal{G}}[\chi] \circ {\mathcal{V}}^k(\theta)
\circ {\mathcal{G}}[\chi] \circ {\mathcal{V}}^i(\theta) \,
\bigr \} \>.
\notag\end{aligned}$$ Similarly $$\begin{aligned}
&{\mathcal{R}}^{ijkl}[\chi]
=
\frac{\delta {\mathcal{V}}^{ij}[\chi]}{\delta \chi_k \, \delta \chi_l}
\label{BEC.NLOAF.e:Rijkldef} \\
&=
\frac{1}{2} \,
\bigl \{ \,
{\mathcal{V}}^i \circ {\mathcal{G}}[\chi] \circ {\mathcal{V}}^l
\circ {\mathcal{G}}[\chi] \circ {\mathcal{V}}^k \circ {\mathcal{G}}\circ {\mathcal{V}}^j
\notag \\
& \qquad
+
{\mathcal{V}}^i \circ {\mathcal{G}}[\chi] \circ {\mathcal{V}}^k\circ {\mathcal{G}}\circ {\mathcal{V}}^l
\circ {\mathcal{G}}[\chi] \circ {\mathcal{V}}^j \,
\bigr \}
\notag \\
&
+
\frac{1}{2} \,
\bigl \{ \, {\mathcal{V}}^j \circ {\mathcal{G}}[\chi] \circ {\mathcal{V}}^l \circ {\mathcal{G}}\circ {\mathcal{V}}^k
\circ {\mathcal{G}}[\chi] \circ {\mathcal{V}}^i
\notag \\
& \qquad
+
{\mathcal{V}}^j \circ {\mathcal{G}}[\chi] \circ {\mathcal{V}}^k
\circ {\mathcal{G}}[\chi] \circ {\mathcal{V}}^l \circ {\mathcal{G}}[\chi] \circ {\mathcal{V}}^i \,
\bigr \} \>.
\notag\end{aligned}$$ We also define the leading-order 3-$\chi$ 1-PI vertex function, $Q_3^{ijk}[\chi]$, as $$\begin{aligned}
&Q_3^{ijk}[\chi]
=
\frac{\delta {\mathcal{D}}^{-1}{}^{ij}[\chi] }{ \delta \chi_{k} }
=
\frac{\delta \Pi^{ij}[\chi] }{ \delta \chi_{k} }
\label{BEC.NLOAF.e:Qdef1} \\
&=
{}-
\phi \circ {\mathcal{R}}^{ijk}[\chi](\theta) \circ \phi
\notag \\
&
{}+
\frac{\hbar}{2i} \,
{\mathrm{Tr} [ \, {\mathcal{G}}[\chi] \circ {\mathcal{V}}^k(\theta) \circ
{\mathcal{G}}[\chi] \circ {\mathcal{V}}^i(\theta) \circ
{\mathcal{G}}[\chi] \circ {\mathcal{V}}^j(\theta)
\notag \\
& \qquad
+
{\mathcal{G}}[\chi] \circ {\mathcal{V}}^k(\theta) \circ {\mathcal{G}}[\chi]
\circ {\mathcal{V}}^j(\theta) \circ
{\mathcal{G}}[\chi] \circ {\mathcal{V}}^i(\theta) \, ]} \>.
\notag \end{aligned}$$ The 4-$\chi$ vertex is then given by $$\begin{aligned}
&Q_4^{ijkl}
=
\phi \circ {\mathcal{R}}^{ijkl}[\chi](\theta) \circ \phi
-
\frac{\hbar}{2i} \,
{\mathrm{Tr} [ \, {\mathcal{G}}\circ {\mathcal{V}}^i(\theta)
\label{e:Q4} \\
& \qquad
\circ {\mathcal{G}}\circ {\mathcal{V}}^j(\theta) \circ {\mathcal{G}}\circ {\mathcal{V}}^k(\theta)
\circ {\mathcal{G}}\circ {\mathcal{V}}^l(\theta) \, ]} + \text{perms.}
\notag\end{aligned}$$
With the above definitions we can construct the rules for inserting a $\phi$ or $\chi$ vertex into a graph: Inserting a $\chi$ line into ${\mathcal{G}}[\chi]$, we obtain: $$\begin{aligned}
\frac{\delta {\mathcal{G}}^a{}_b[\chi] (x_1,x_2)} {\delta \chi_i(z_1)}
&=
-
{\mathcal{G}}^a{}_c (x_1,z_1) \, {\mathcal{V}}^i{}^c{}_d(\theta) \, {\mathcal{G}}^d{}_b(z_1,x_2)
\label{BB.e:dGdchi} \\
&=
-
{\mathcal{G}}\circ {\mathcal{V}}^i(\theta) \circ {\mathcal{G}}\>.
\notag\end{aligned}$$ Inserting a $\chi$ line into ${\mathcal{D}}[\Phi]$, we obtain $$\begin{aligned}
&\frac{\delta {\mathcal{D}}[\chi \phi]^{i,j}(z_1,z_2)} {\delta \chi_k(z_3 )}
\label{BB.e:dDdchi} \\
&=
-
\int [ {\mathrm{d}}z_4 ] \, [ {\mathrm{d}}z_5 ] \,
{\mathcal{D}}^{im}(z_1,z_4) \,
\frac{\delta {\mathcal{D}}^{-1}_{mn}(z_4,z_5) }
{\delta \chi_k(z_3)} \,
{\mathcal{D}}^{nj}(z_5, z_2)
\notag \\
&=
- {\mathcal{D}}^{im} \circ Q_{mn}{}^k \circ {\mathcal{D}}^{nj}
\notag\end{aligned}$$ We also need to insert a $\phi$ line into ${\mathcal{D}}$. The $2$-$\chi$ $1$-$\phi$ vertex is given by $$\begin{aligned}
\Gamma^3{}^{ij,a}
&=
\frac {\delta {\mathcal{D}}^{-1}{}^{ij} (z_1,z_2)}
{\delta \phi_a(x_1)}
=
\delta (z_1,x_1) \, {\mathcal{V}}^{ij}{}^a{}_d (z_1,z_2) \, \phi^d(z_2)
\notag \\
&
+
\phi^c(z_1) \, {\mathcal{V}}^{ij\,a}{}_c (z_1,z_2) \, \delta (z_2,x_1) \>,
\notag\end{aligned}$$ and for the $2$-$\chi_i$ $2$-$\phi_a$ vertex we find $$\begin{aligned}
\Gamma^{4\,ij,ab}
&=
\frac{\delta^2 {\mathcal{D}}^{-1}{}^{ij} (z_1,z_2)}
{\delta \phi_a(x_1) \, \delta \phi_b(x_2) }
\\
&
=
\delta (z_1,x_1) \, {\mathcal{V}}^{ijab}(z_1,z_2) \, \delta(z_2,x_2)
\notag \\
& \qquad
+
\delta (z_1,x2) \, {\mathcal{V}}^{ijab}(z_1,z_2) \, \delta (z_2,x_1) \>.
\notag\end{aligned}$$ Thus we obtain $$\begin{aligned}
&\frac{\delta {\mathcal{D}}^{i,j}[\Phi](z_1,z_2)}
{\delta \phi_a(x_1)}
\\
&=
-
\int [ {\mathrm{d}}z_3 ] \, [ {\mathrm{d}}z_4 ] \,
{\mathcal{D}}^{im}(z_1,z_3) \, \Gamma^3_{mn}{}^{a}(z_3,z_4,x_1) \,
{\mathcal{D}}^{nj} (z_4, z_2)
\notag \\
&=
-
{\mathcal{D}}^{im} \circ \Gamma^3_{mn}{}^a \circ {\mathcal{D}}^{nj} \>.
\notag\end{aligned}$$
\[ss:invprop\]Inverse propagators to order $\epsilon$
------------------------------------------------------
Using the above rules, we derive the one and two-point vertex function to order $\epsilon$. For the one-point function in the presence of sources we have the following two equations: For $\phi_c^a$ we have $$\begin{aligned}
j^a
&=
\frac{\delta \, \Gamma[\Phi]}{\delta \phi_a}
\\
&=
\frac{1}{2} \,
\bigl [ \,
\phi \circ {\mathcal{G}}^{-1} \,
\bigr ]^a
+
\frac{1}{2} \,
\bigl [ \,
{\mathcal{G}}^{-1} \circ \phi \,
\bigr ]^a
+
\frac{\epsilon}{2} {\mathrm{Tr} [ \, {\mathcal{D}}\, \Gamma^{3\,..a} \, ]}
\>,
\notag\end{aligned}$$ whereas for $\chi_i$ we find $$\begin{aligned}
s^i
&=
\frac{\delta \, \Gamma[\Phi]}{\delta \chi_i}
=
\frac{1}{2} \, \phi \circ {\mathcal{V}}^i \circ \phi
\\
& \qquad
-
\frac{ \chi^i}{\lambda} +\frac{1}{2i} {\mathrm{Tr} [ \, {\mathcal{G}}\circ {\mathcal{V}}^i \, ]}
+
\frac{\epsilon}{2i} {\mathrm{Tr} [ \, {\mathcal{D}}\circ Q^3 {}^{i..} \, ]} \>.
\notag\end{aligned}$$
In turn, for the inverse propagator matrix we have: $$\begin{aligned}
&\frac{\delta^2 \, \Gamma[\Phi]}{ \delta \chi_i \, \delta \chi_j }
=
-
{\mathcal{D}}^{-1}{}^{ij} [\chi, \phi=0]
\\
& \qquad
-
\frac{\epsilon}{2i} {\mathrm{Tr} [ \, {\mathcal{D}}\circ Q_3^{i..} \circ {\mathcal{D}}\circ Q_3^{j..} \, ]}
+
\frac{\epsilon}{2i} {\mathrm{Tr} [ \, {\mathcal{D}}\circ Q_4^{i...} \, ]} \>,
\notag\end{aligned}$$ and $$\begin{aligned}
&\frac{\delta \, \Gamma[\Phi]} {\delta \phi_a \, \delta \phi_b}
=
G_0^{-1}{}^{ab}
\\
& \qquad
-
\frac{\epsilon}{2i} {\mathrm{Tr} [ \, D \circ \Gamma_3^{b..} \circ D \circ \Gamma_3^{a..} \, ]}
+
\frac{\epsilon}{2i} {\mathrm{Tr} [ \, D \circ \Gamma_4^{ab..} \, ]} \>.
\notag\end{aligned}$$ The term that mixes $\phi$ and $\chi$ is $$\begin{aligned}
&\frac{\delta \, \Gamma[\Phi]} {\delta \phi_a \, \delta \chi_i}
=
\frac{1}{2} \left[ \phi \circ {\mathcal{V}}^i \right]^a
+
\frac{1}{2} \left[ {\mathcal{V}}^i \circ \phi \right]^a
\\
& \quad
-
\frac{\epsilon}{2i} {\mathrm{Tr} [ \, {\mathcal{D}}\circ Q_3^{i ..} \circ {\mathcal{D}}\circ \Gamma_3^{a..} \, ]}
-
\frac{\epsilon}{2i} {\mathrm{Tr} [ \, {\mathcal{D}}\circ [R_3^{i..} \circ \phi]^a \, ]} \>.
\notag\end{aligned}$$ The propagators for the theory are obtained by inverting the $5 \times 5$ inverse propagator matrix. Expanding the propagators in a power series in $\epsilon$ and keeping terms to order $\epsilon$ gives the graphs for the propagators that one would have obtained by working directly with $\ln Z$ to order $\epsilon$. The Feynman diagrams for the second derivatives of $\Gamma[\Phi]$ are shown in Fig. \[f:dGamma\].
![\[f:dGamma\] Feynman diagrams for the second derivatives of $\Gamma$. Solid and wavy lines correspond to the propagators of $\phi$ and $\chi$. Dashed lines denote $\phi$.](dGamma_BEC.pdf){width="0.9\columnwidth"}
\[ss:pi0\]$\Pi^{ij}[\Phi](x,x')$
--------------------------------
To conclude this section, we complete the calculation of $\Pi^{ij}[\Phi](x,x')$ introduced first in Eq. and explicitly evaluated in Eq. above. In the imaginary-time formalism, we first introduce $$\begin{aligned}
&{\mathcal{M}}^{ij}[\chi](x,x')
\label{BEC.NLOAF.e:trGVGV} \\
&
=
\frac{1}{2} \,
{\mathrm{Tr} [ \, {\mathcal{G}}[\chi](x',x) \circ {\mathcal{V}}^i(\theta) \circ
{\mathcal{G}}[\chi](x,x') \circ {\mathcal{V}}^j(\theta) \, ]}
\notag \\
&
=
\frac{1}{\beta^2}
\iint \frac{{\mathrm{d}}^3 k_1 \, {\mathrm{d}}^3 k_2}{(2\pi)^6}
\sum_{n_1,n_2=-\infty}^{+\infty}
\notag \\
& \quad \times
\frac{1}{2} \,
{\mathrm{Tr} [ \, \tilde{{\mathcal{G}}}[\chi]({\mathbf{k}}_2,n_2) \circ {\mathcal{V}}^i(\theta) \circ
\tilde{{\mathcal{G}}}[\chi]({\mathbf{k}}_1,n_1) \circ {\mathcal{V}}^j(\theta) \, ]}
\notag \\
& \qquad\qquad
\times
e^{i [ \,
({\mathbf{k}}_1 - {\mathbf{k}}_2 ) \cdot ( {\mathbf{r}}- {\mathbf{r}}' )
-
(\omega_{n_1} - \omega_{n_2} ) ( \tau - \tau' ) \, ] }
\notag \\
&=
\frac{1}{\beta} \int \frac{{\mathrm{d}}^3 k}{(2\pi)^3}
\sum_{n=-\infty}^{+\infty}
e^{i [ \, {\mathbf{k}}\cdot ( {\mathbf{r}}- {\mathbf{r}}' ) - \omega_{n} ( \tau - \tau' ) \, ] }
\notag \\
& \qquad
\frac{1}{\beta}
\iint \frac{{\mathrm{d}}^3 k_1 \, {\mathrm{d}}^3 k_2}{(2\pi)^6} \,
\sum_{n_1,n_2}
(2\pi)^3 \, \delta( {\mathbf{k}}- {\mathbf{k}}_1 + {\mathbf{k}}_2 ) \,
\delta_{n,n_1 - n_2} \,
\notag \\
&\times
\frac{1}{2} \,
{\mathrm{Tr} [ \, \tilde{{\mathcal{G}}}[\chi]({\mathbf{k}}_2,n_2) \circ {\mathcal{V}}^i(\theta) \circ
\tilde{{\mathcal{G}}}[\chi]({\mathbf{k}}_1,n_1) \circ {\mathcal{V}}^j(\theta) \, ]} \>.
\notag \end{aligned}$$ Expanding ${\mathcal{M}}^{ij}[\chi](x,x')$ in a Fourier series, $$\begin{aligned}
& {\mathcal{M}}^{ij}[\chi](x,x')
\label{BEC.NLOAF.e:trGVGVxx} \\
&=
\frac{1}{\beta} \int \frac{{\mathrm{d}}^3 k}{(2\pi)^3}
\sum_{n=-\infty}^{+\infty}
\tilde{{\mathcal{M}}}^{ij}[\chi]({\mathbf{k}},n) \,
e^{i [ \, {\mathbf{k}}\cdot ( {\mathbf{r}}- {\mathbf{r}}' ) - \omega_{n} ( \tau - \tau' ) \, ] } \>,
\notag \end{aligned}$$ where from , $\tilde{{\mathcal{M}}}^{ij}[\chi]({\mathbf{k}},n)$ is given by the convolution integral, $$\begin{aligned}
&\tilde{{\mathcal{M}}}^{ij}[\chi]({\mathbf{k}},n)
=
\frac{1}{2\beta} \int \frac{{\mathrm{d}}^3 k'}{(2\pi)^3}
\sum_{n'=-\infty}^{+\infty}
\label{BEC.NLOAF.e:tMdef} \\
& \times
{\mathrm{Tr} [ \, \tilde{{\mathcal{G}}}[\chi]({\mathbf{k}}- {\mathbf{k}}',n - n') \circ {\mathcal{V}}^i(\theta) \circ
\tilde{{\mathcal{G}}}[\chi]({\mathbf{k}}',n') \circ {\mathcal{V}}^j(\theta) \, ]} \>.
\notag\end{aligned}$$ From Eqs. , , and , we then have $$\label{BEC.NLOAF.e:tSigma}
\tilde{\Pi}^{ij}[\Phi]({\mathbf{k}},n)
=
\phi \circ \tilde{{\mathcal{V}}}^{ij}[\chi]({\mathbf{k}},n) \circ \phi
-
\tilde{{\mathcal{M}}}^{ij}[\chi]({\mathbf{k}},n) \>,$$ with $$\begin{aligned}
&\tilde{{\mathcal{V}}}^{ij}[\chi]({\mathbf{k}},n)
=
\frac{1}{2} \,
\bigl [ \,
{\mathcal{V}}^i(\theta) \circ \tilde{{\mathcal{G}}}[\chi]({\mathbf{k}},n) \circ {\mathcal{V}}^j(\theta)
\label{BEC.NLOAF.e:tcalVij} \\
& \qquad\qquad
+
{\mathcal{V}}^j(\theta) \circ \tilde{{\mathcal{G}}}[\chi](-{\mathbf{k}},-n) \circ {\mathcal{V}}^i(\theta) \,
\bigr ] \>.
\notag\end{aligned}$$
|
---
abstract: 'The $\alpha m_t^2/m_W^2$ order supersymmetric electroweak corrections arising from loops of chargino, neutralino, and squark to top quark pair production by $gg$ fusion at LHC are calculated in the minimal supersymmetric model. We found that the corrections amount about a few percent.'
address:
- |
CCAST (World Laboratory), P. O. Box 8730, Beijing 100080, P.R. China,\
Institute of Modern Physics and Department of Physics,\
Tsinghua University, Beijing 100084, P.R. China$^*$\
- |
CCAST (World Laboratory), P. O. Box 8730, Beijing 100080, P.R. China,\
Physics Department, Peking University, Beijing 100871, P.R. China$^*$
author:
- 'Hong-Yi Zhou'
- 'Chong-Sheng Li'
title: Supersymmetric Electroweak Corrections to Top Quark Production at LHC
---
[**I. Introduction**]{}
The top quark has been found experimentally by the CDF and D0 Collaborations with the mass and production cross section $m_t=176\pm 8(stat)\pm 10(syst)$ GeV $\sigma=6.8^{+3.6}_{-2.4} pb$, and $m_t=199^{+19}_{-20}(stat)\pm 22(syst)$ GeV $\sigma=6.4\pm 2.2 pb$, respectively [@CDFD0]. Although this measured mass is close to the central value predicted by the best fit of the Standard Model (SM ) to the latest LEP data, the central value of the cross section is somewhat larger than the Standard Model prediction $\sigma_{t\bar t}=5.52^{+0.07}_{-0.45} $ pb for $m_t=175$ GeV at $\sqrt{s}=1.8$ TeV $p\bar p$ collider in which the effects of multiple soft-gluon emissions have been properly resummed[@BERGER]. In addition, there are still a number of unsolved theoretical problems in the SM. New physics beyond the SM are still possible. Among various models of new physics so far considered, supersymmetry (SUSY) is a promising one at present. The simplest and interesting SUSY model is the minimal supersymmetric extension of the standard model (MSSM) [@MSSM]. At the future multi-TeV proton colliders such as the CERN Large Hadron Collider (LHC), $t{\bar t}$ production will be enormously larger than the Tevatron rates and the accuracy with which the top quark production cross section can be measured will be much better (the uncertainty is about 5% at LHC [@TOPPHYSICS]). Thus theoretical calculations of the radiative corrections to the production of the top quark at those colliders are of importance.
QCD corrections to $O(\alpha_s^3)$ and electroweak one-loop corrections in the SM to $t\bar t$ production in hadron colliders are carried out in Ref.[@BERGER][@SMQCD] and Ref.[@SMEW]. Yukawa corrections to $t\bar t$ production at the Fermilab Tevatron and LHC in two-Higgs-doublet models are calculated in [@MSSMYK][@MSSMYKLHC]. In the MSSM, electroweak corrections from chargino, neutralino and squark to the top pair production via $q{\bar q}$ annihilation at the Fermilab Tevatron are calculated in Ref.[@MSSMEW]. Recent calculation of the supersymmetric QCD corrections to the top quark pair production at the Tevatron shows that they increase the cross section by about 20%[@MSSMQCD][@kim]. But this is still within the experimental uncertainty 30%. At LHC the main production mechanism of top quark pair is the gluon-gluon fusion process $gg\rightarrow t\bar t$. In this paper we investigate the electroweak corrections of order $ \alpha m_t^2/m_W^2$ arising from chargino, neutralino and squark to the top quark production by the process $gg\rightarrow t{\bar t}$ at LHC. The formalism of the calculation of the corrections to the matrix elements will be given in Sec.II. In Sec III, we present our numerical examples and discussions of the corrections to the cross sections in the MSSM.
[**II. Formalism**]{}
The tree-level Feynman diagrams and the relevant supersymmetric electroweak corrections to $gg\rightarrow t{\bar t}$ are shown in Fig.1 (u-channel diagrams of (b) and (e)–(h) are not explicitly shown) in which the dashed lines in the loop represent the squark $\tilde{t}_i$ or $\tilde{b}_i~(i=1,2)$ and the solid lines represent neutralinos or charginos, respectively.
The supersymmetric partner of left- and right-handed massive quarks mix [@TTMX]. The mass eigenstates $\tilde{q}_1$ and $\tilde{q}_2$ are related to the current eigenstates $\tilde{q}_L$ and $\tilde{q}_R$ $$\tilde{q} _1 = \tilde{q} _L \cos \theta _q + \tilde{q} _R \sin \theta
_q,~~~\tilde{q} _2 = -\tilde{q} _L \sin \theta _q + \tilde{q} _R \cos
\theta _q$$ The mixing angle $\theta_t$ and the masses $m_{\tilde{t}_1}$, $m_{\tilde{t}_2}$ can be calculated by diagonalizing the following mass matrix $$\begin{aligned}
\label{eqnum2}
& & M^2_{\tilde{t}} =\left(
\begin{array} {ll}
M^2_{\tilde{t}_L} & m_tm_{LR} \\
m_tm_{LR} & M^2_{\tilde{t}_R}
\end{array}
\right) \nonumber\\
& & M^2_{\tilde{t}_L}=m^2_{\tilde{t}_L} + m^2_t+(\frac{1}{2}
-\frac{2}{3}\sin^2\theta_W)\cos(2\beta)m_Z^2\nonumber\\
& & M^2_{\tilde{t}_R}=m^2_{\tilde{t}_R} + m^2_t
+\frac{2}{3}\sin^2\theta_W\cos(2\beta)m_Z^2\nonumber\\
& & m_{LR}=\mu\cot\beta+A_t\end{aligned}$$ where $m^2_{\tilde{t}_L},~m^2_{\tilde{t}_R}$ are the soft SUSY-breaking mass terms of left- and right-handed stops, $\mu$ is the coefficient of the $H_1~H_2$ mixing term in the superpotential, $A_t$ is the parameter describing the strength of soft SUSY-breaking trilinear scalar interaction $\tilde{t}_L\tilde{t}_R H_2$, $\tan\beta=v_2/v_1$ is the ratio of the vacuum expectation values of the two Higgs doublets.
From Eq.(\[eqnum2\]), we can get the expressions for $m^2_{\tilde{t}_{1,2}}$ and $\theta_t$ : $$\begin{aligned}
% 3 & 4
& & m^2_{\tilde{t}_{1,2}}=\frac{1}{2}\left[
M^2_{\tilde{t}_{L}}+M^2_{\tilde{t}_{R}}\mp\sqrt{(M^2_{\tilde{t}_{L}}
-M^2_{\tilde{t}_{R}})^2+4m_t^2m_{LR}^2}\right]\\
& &\tan\theta_t=\frac{m^2_{\tilde{t}_1}-M^2_{\tilde{t}_{L}}}{m_tm_{LR}}\end{aligned}$$
For the sbottoms, we neglect the mixing between the left- and right- handed sbottoms($\theta_b=0$) and have $$\begin{aligned}
% 5
m^2_{\tilde{b}_{1,2}}=m^2_{\tilde{t}_{L},\tilde{b}_{R}}
+ m^2_b\pm(T^3_{L,R}-Q_b
\sin\theta_W)\cos(2\beta)m^2_Z, \end{aligned}$$ where $T^3_{L,R}=-\frac{1}{2},~0$, $Q_b=-\frac{1}{3}$ and $m^2_{\tilde{t}_{L},\tilde{b}_{R}}$ are the soft SUSY-breaking mass terms for left- and right-handed sbottoms.
In the presence of squark mixing,the squark-quark-neutralino and squark-quark-chargino interaction Lagrangian of order $gm_t/m_W$ is given by\
$$\begin{aligned}
L_{\tilde{\chi} \tilde{q} \bar{q}} & = & -\frac{gm_t}{\sqrt{2}m_W\sin\beta}
\sum\limits_{j=1}^{4}\bar{t}[(a_{\tilde{t}_1j}-b_{\tilde{t}_1j}
\gamma_5)\tilde{t}_{1}+(a_{\tilde{t}_2j}-b_{\tilde{t}_2j}\gamma_5)
\tilde{t}_{2}]\tilde{\chi}^0_j \nonumber \\
& & +\frac{gm_t}{\sqrt{2}m_W\sin\beta}
\sum\limits_{j=1}^{2}\bar{t}[(a_{\tilde{b}_1j}-b_{\tilde{b}_1j}
\gamma_5)\tilde{b}_{1}+(a_{\tilde{b}_2j}-b_{\tilde{b}_2j}\gamma_5)
\tilde{b}_{2}]\tilde{\chi}^+_j+H.C.\;,\end{aligned}$$ where $g$ is the SU(2) coupling constant , and $a_{\tilde{t}_1j},~b_{\tilde{t}_1j},~a_{\tilde{t}_2j},
~b_{\tilde{t}_2j},~a_{\tilde{b}_1j},~b_{\tilde{b}_1j},
~a_{\tilde{b}_2j},~b_{\tilde{b}_2j} $ are given by $$\begin{aligned}
\label{ab}
& & a_{\tilde{t}_1j}=\frac{1}{2}(N_{j4}^\ast\cos\theta_t+N_{j4}\sin\theta_t),
~~b_{\tilde{t}_1j}=\frac{1}{2}(N_{j4}^\ast\cos\theta_t-N_{j4}\sin\theta_t),
\nonumber\\
& & a_{\tilde{t}_2j}=\frac{1}{2}(-N_{j4}^\ast\sin\theta_t+N_{j4}\cos\theta_t),
~~ b_{\tilde{t}_2j}=\frac{1}{2}(-N_{j4}^\ast\sin\theta_t-N_{j4}\cos\theta_t),
\nonumber \\
& & a_{\tilde{b}_1j}=b_{\tilde{b}_1j}=\frac{1}{2}V_{j2}^\ast\cos\theta_b,
~~ a_{\tilde{b}_2j}=b_{\tilde{b}_2j}=-\frac{1}{2}V_{j2}^\ast\sin\theta_b\end{aligned}$$ $V_{j2}$ are the elements of $2\times 2 $ matrix V and $N_{j4}$ are the elements of $4\times 4 $ matrix N (see the Appendix).
At the tree level, the S-matrix element is composed of three different production channels(s-,t-,u-channel) as follows:
$$\begin{aligned}
%(8)-(10)
M_0^{s} &= & -ig_s^2(if_{abc}T^c)_{ji}{\bar u}(p_2)
\rlap/\Gamma v(p_1)/\hat{s}\nonumber\\
&=&-i(T^aT^b-T^bT^a)_{ji}M_0^{s\prime}, \\
M_0^t& = & -ig_s^2(T^bT^a)_{ji}{\bar u}(p_2)\rlap/\epsilon_4
(\rlap/q+m_t)\rlap/\epsilon_3 v(p_1)/(\hat{t}-m_t^2)\nonumber\\
& =& -i(T^bT^a)_{ji}M_0^{t\prime},\\
M_0^u& = & M_0^t(p_3\leftrightarrow p_4,\;\;T^a\leftrightarrow T^b,\;\;
\hat{t}\rightarrow \hat{u} )\nonumber\\
&=&-i(T^aT^b)_{ji}M_0^{u\prime},\end{aligned}$$
where $q=p_2-p_4$, $\epsilon_4^\mu=\epsilon^\mu(p_4),~~
\epsilon_3^\mu=\epsilon^\mu(p_3)$, $\Gamma^\mu$ is given in the Appendix. Instead of calculating the square of the amplitudes explicitly, we calculate the helicity amplitudes numerically by using the method of Ref. [@ZEPPEN]. This method greatly simplifies our calculations.
The $O(\alpha m_t^2/m_W^2)$ SUSY electroweak corrections to $gg\rightarrow t{\bar t}$ are shown in Fig.1 (c)–Fig.1 (h). The sum of them is QCD gauge invariant without the strong coupling constant renormalization . In our calculation, we use dimensional regularization to regulate the ultraviolet divergences and adopt the on-mass-shell renormalization scheme. We also discard the terms proportional to $\gamma_5$.
We only give the explicitly results of the s- and t-channel contributions to the SUSY electroweak corrections. The u-channel results can be obtained by the following substitutions: $$\begin{aligned}
& & p_3\leftrightarrow p_4,\;\;
T^a\leftrightarrow T^b,\;\;
\hat{t}\leftrightarrow\hat{u}.\end{aligned}$$ Fig.1(c) lead to the s-channel vertex correction $\delta M^{s1}$: $$\begin{aligned}
\delta M^{s1}& =& -ig_s^2 (if_{abc}T^c)_{ji}
{\bar u}(p_2)
(F_0^{s1}+F_1^{s1}\rlap/\Gamma+\rlap/F_6^{s1} )v(p_1)/\hat{s}\nonumber\\
& =&-i(T^aT^b-T^bT^a)\delta M^{s1\prime} \end{aligned}$$ Fig.1(d) gives $\delta M^{s2}$: $$\begin{aligned}
%(13)
\delta M^{s2}& =&-ig_s^2(T^aT^b+T^bT^a)_{ji}
{\bar u}(p_2)F_0^{s2}v(p_1)\nonumber\\
& = &-i(T^bT^a+T^aT^b)_{ji}
\delta M^{s2\prime}\end{aligned}$$
The top quark self-energy $\delta M^{self,t}$ of Fig.1(e) is: $$\begin{aligned}
%(14)
\delta M^{self,t}&=& -ig_s^2(T^bT^a)_{ji}
{\bar u}(p_2)\rlap/\epsilon_4(\rlap/q+m_t)\nonumber\\
& &[F_1^{self,t}+F_2^{self,t}\rlap/q]
(\rlap/q+m_t)\rlap/ \epsilon_3 v(p_1)/(\hat{t}-m_t^2)\nonumber\\
& =&-i(T^bT^a)_{ji}\delta M^{self,t\prime}\end{aligned}$$
Vertex correction $\delta M^{v1,t}$ of Fig.1(f) is: $$\begin{aligned}
%(15)
\delta M^{v1,t}&=& \displaystyle -ig_s^2(T^bT^a)_{ji}
{\bar u}(p_2)(F_0^{v1,t}+F_1^{v1,t}\rlap/\epsilon_4+\rlap/F_6^{v1,t})
(\rlap/q+m_t)\rlap/\epsilon_3 v(p_1)/(\hat{t}-m_t^2)\nonumber\\
&=&-i(T^bT^a)_{ji} \delta M^{v1,t\prime}\end{aligned}$$
Vertex correction $\delta M^{v2,t}$ of Fig.1(g) is: $$\begin{aligned}
%(16)
\delta M^{v2,t}&=& \displaystyle -ig_s^2(T^bT^a)_{ji}
{\bar u}(p_2)\rlap/\epsilon_4(\rlap/q+m_t)
(F_0^{v2,t}+F_1^{v2,t}\rlap/\epsilon_3+\rlap/F_6^{v2,t})
v(p_1)/(\hat{t}-m_t^2)\nonumber\\
& =&-i(T^bT^a)_{ji}\delta M^{v2,t\prime}\end{aligned}$$
$\delta M^{box,t}$ of the box diagram Fig.1(h) is: $$\begin{aligned}
%(17)
\delta M^{box,t}& =&
\displaystyle -ig_s^2(T^bT^a)_{ji}
{\bar u}(p_2)[F_0^{b,t}+\rlap/F_3^{b,t}]v(p_1)\nonumber \\
& =&-i(T^bT^a)_{ji}\delta M^{box,t\prime}\end{aligned}$$
The total amplitude can be written as: $$\begin{aligned}
& & M_{ji}=-i[(M_0^+ +\delta M^+) O^{(+)}_{ji}
+(M_0^- +\delta M^-)O^{(-)}_{ji}~], %(18)\end{aligned}$$ where $$\begin{aligned}
& & O^{(+)}=\frac{T^bT^a+T^aT^b}{2}\;,\;\; O^{(-)}=\frac{T^bT^a-T^aT^b}{2}\\
& & M_0^+=M_0^{t\prime}+M_0^{u\prime}\;,\;\; M_0^-=M_0^{t\prime}-M_0^{u\prime}
-2M_0^{s\prime},\end{aligned}$$ $$\begin{aligned}
\delta M^+&=& 2\delta M^{s2\prime}+\delta M^{self,t\prime}+\delta
M^{self,u\prime}+\delta M^{v1,t\prime}+\delta M^{v1,u\prime}\nonumber\\
& & +\delta M^{v2,t\prime}+\delta M^{v2,u\prime}+\delta M^{box,t\prime}
+\delta M^{box,u\prime}\nonumber\\
\delta M^-&=&-2\delta M^{s1\prime}+\delta M^{self,t\prime}-\delta
M^{self,u\prime}+\delta M^{v1,t\prime}-\delta M^{v1,u\prime}\nonumber\\
& &+\delta M^{v2,t\prime}-\delta M^{v2,u\prime}
+\delta M^{box,t\prime}-\delta M^{box,u\prime}\end{aligned}$$ The color sum of the corrected amplitude square is: $$\begin{aligned}
\sum\limits_{color}|M|^2&=&\frac{7}{3}|M_0^+|^2+3|M_0^-|^2\nonumber\\
& &+\frac{14}{3}Re(M_0^+\delta M^{+\dagger})+6Re(M_0^-\delta M^{-\dagger})
%(20)\end{aligned}$$ The spin sum as well as phase-space integration and parton distribution convolution are done by the VEGAS program. The correction cross section $\Delta \sigma$ is defined as $$\Delta \sigma=\sigma-\sigma_0, %(21)$$ where $\sigma$ is the cross section given by $|M|^2$ and $\sigma_0$ is the tree level QCD cross section given by $|M_0^+|^2$ and $|M_0^-|^2$ .
[**III. Numerical Examples and Discussions**]{}
The production cross section is obtained by convoluting the partonic cross section with certain parton distribution. The relative correction is not sensitive to parton distribution. In this paper, we take the Martin-Roberts-Stirling (MRS) parton distribution set A$^\prime$ [@MRSA] with $Q^2=\hat{s}$. The following parameters are used in our calculation:
$$\begin{aligned}
%(22)
& & \sqrt{s}=14~TeV,\;\;m_t=176~GeV,~m_b=4.9 ~GeV,\nonumber\\
& & m_W=80.22~GeV,~m_Z=91.175~GeV,~\alpha=\frac{1}{128.8} . \end{aligned}$$
Care must be taken in the calculation of the form factors expressed in terms of the standard loop integrals defined in Ref. [@VELTMAN]. As has been discussed in Ref.[@DENNER], the formulae for the form factors given in terms of the tensor loop integrals will be ill-defined when the scattering is forwards or backwards wherein the Gram determinants of some matrices vanish and thus their inverses do not exist. This problem can be avoided by taking the kinematic cuts on the rapidity $y$ and the transverse momentum $p_T$. In this paper, we take $$%(23)
|y|<2.5,\;\;p_T>20\;GeV.\\$$ The cuts will also increase the relative corrections[@SMEW].
We first checked the QCD gauge invariance by the substitution $p_4\rightarrow\epsilon_4$ and $p_3\rightarrow\epsilon_3$ and find that $\delta M^+,~\delta M^-$ are a few order of magnitudes smaller.
In the calculation of the chargino and neutralino masses, we fix $M=200~GeV$, $\mu=-100~GeV$ and use the relation $M^\prime
=\frac{5g^{\prime2}}{3g^2}M$(see the Appendix). We also assume $m_{\tilde{t}_L}=m_{\tilde{t}_R}=m_{\tilde{b}_R}=m_{\tilde{q}}$.
The relative correction to the hadronic cross section as a function of the squark mass parameter $m_{\tilde{q}}$ with $\tan\beta=1$ and $m_{LR}=0$(corresponding to non-mixing case) is presented in Fig.2. For $\tan\beta=1$, the chargino masses $m_{\tilde{\chi}^+_{j}}=
(220,120)~GeV$ and the neutralino masses $m_{\tilde{\chi}^0_{j}}=
(105,221,128,100)~GeV$. The correction is always negative. For $m_{\tilde{q}}<150~GeV$, the correction is very sensitive to $m_{\tilde{q}}$. A sharp dip at about $m_{\tilde{q}}=56~GeV$ is due to the singularity of the top quark wave function renormalization constant at the threshold point $m_t=m_{\tilde{b}_1}+m_{\tilde{\chi}^+_{2}}$ (note that for $\tan\beta=1$, $m_{\tilde{b}_1}=m_{\tilde{q}}$). This singularity will disappear if the finite widths of the top quark and the charginos are taken into account. The correction exceeds -5% only in a small region near the dip. The correction approaches to zero at large $m_{\tilde{q}}$ which shows the decoupling behaviour.
Fig.3 shows the dependence of the relative correction to the hadronic cross section on the stop mixing parameter $m_{LR}$. We set $m_{\tilde{q}}=100~GeV$ and $\tan\beta=1$. $m_{LR}$ affects the mass splitting and mixing angle of $\tilde{t}_1$ and $\tilde{t}_2$. The mass splitting increases as $m_{LR}$ increases. We fix $\tilde{t}_1$ to be the light one(cf. Eq.(3)). Therefore, $\theta_t=\frac{\pi}{4}$ for $m_{LR}<0$, $\theta_t=-\frac{\pi}{4}$ for $m_{LR}>0$. The mixing angle causes the asymmetry of the relative correction between $m_{LR}>0$ and $m_{LR}<0$ although the mass splitting of $\tilde{t}_1$ and $\tilde{t}_2$ is symmetry between $m_{LR}>0$ and $m_{LR}<0$. The dip at about $m_{LR}=-200~GeV$ is due to the threshold effect at $m_t=m_{\tilde{t}_1}+m_{\tilde{\chi}^0_j}$. No dip is found at $m_{LR}=200~GeV$ because when $m_{LR}>0$, $\theta_t=-\pi/4$, the $t\tilde{t}_1\tilde{\chi}^0_j$ coupling is proportional to $\gamma_5$ ($a_{ij}=0$, $b_{ij}\neq 0$). From the expression of the top quark renormalization constant (see the Appendix), one can see that the singularities of $G_0$ and $G_1$ cancel with each other when $a_{ij}=0$, $b_{ij}\neq 0$.
In Fig.4, we present the $\tan\beta$ dependence of the relative correction to the hadronic cross section at given $m_{\tilde{q}}=100$ GeV and $m_{LR}=100$ GeV. $\tan\beta$ slightly affects the stop mass splitting and $m_{\tilde{b}_1}$. The factor $1/\sin^2\beta$ in the coupling constant leads to the rapid increase of the correction in the range $\tan\beta<1$. But the increase is somewhat more quickly than $1/\sin^2\beta$ because $m_{\tilde{b}_1}$ decreases as $\beta$ decreases.
From Fig.2–4, we see that only for $\tan\beta<1$ and a small region near the threshold $t\to \tilde{b}_1 \tilde{\chi}_j^+$ and $t\to \tilde{t}_1 \tilde{\chi}_j^0$ the correction may exceed $-5\%$. Otherwise, the correction amounts only a few percent smaller than $-5\%$. Therefore, we conclude that the supersymmetric electroweak corrections of order $\alpha m_t^2/m_W^2$ to top quark pair production at LHC are potentially observable for $\tan\beta<1$ and small parameter region near the threshold $t\to \tilde{b}_1 \tilde{\chi}_j^+$, $t\to \tilde{t}_1 \tilde{\chi}_j^0$ .
[**ACKNOWLEDGMENTS**]{}
This work is supported in part by the National Natural Science Foundation of China, the Fundamental Research Foundation of Tsinghua University and a grant from the state commission of Science and Technology of China.
[**Appendix** ]{}
We give here the form factors for the matrix element appeared in the text. They are written in terms of the conventional one-, two-, three- and four-point scalar loop integrals defined in Ref.[@VELTMAN].
$$\begin{aligned}
F_0^{s1}&= & -\sum\limits_{j}\sum\limits_{i=\tilde{t}_1,\tilde{t}_2,
\tilde{b}_1,\tilde{b}_2}CPL [m_j(a_{ij}^2-b_{ij}^2)((p2-p1)\cdot\Gamma C_0\\
& & -2C^{10}\cdot\Gamma)](-p_2,k,m_j,m_i,m_i)\\
F_1^{s1}&= & \sum\limits_{j}\sum\limits_{i=\tilde{t}_1,\tilde{t}_2,
\tilde{b}_1,\tilde{b}_2}CPL [\delta Z_{ij}^v]\\
F_6^{s1\mu}&= & -\sum\limits_{j}\sum\limits_{i=\tilde{t}_1,\tilde{t}_2,
\tilde{b}_1,\tilde{b}_2}CPL [(a_{ij}^2+b_{ij}^2)((p2-p1)\cdot\Gamma C^{10\mu}\\
& & -2C^{21\mu}(\Gamma))](-p_2,k,m_j,m_i,m_i)\\
& & \\
F_0^{s2}&= & -\sum\limits_{j}\sum\limits_{i=\tilde{t}_1,\tilde{t}_2,
\tilde{b}_1,\tilde{b}_2}CPL [m_t(a_{ij}^2+b_{ij}^2)C_{11}
-m_j(a_{ij}^2-b_{ij}^2)C_{0}](-p_2,k,m_j,m_i,m_i)\\
& & \\
F_1^{self,t}& =& \sum\limits_{j}\sum\limits_{i=\tilde{t}_1,\tilde{t}_2,
\tilde{b}_1,\tilde{b}_2}CPL [-m_j(a_{ij}^2-b_{ij}^2)B_0+\delta m_{ij}
+m_t\delta Z_{ij}^v](\hat{t},m^2_j,m^2_i)/(\hat{t}-m^2_t)\\
F_2^{self,t}& =& \sum\limits_{j}\sum\limits_{i=\tilde{t}_1,\tilde{t}_2,
\tilde{b}_1,\tilde{b}_2}CPL [(a_{ij}^2+b_{ij}^2)B_1
-\delta Z_{ij}^v](\hat{t},m^2_j,m^2_i)/(\hat{t}-m_t^2)\\
& &\\
F_0^{v1,t}& =& -\sum\limits_{j}\sum\limits_{i=\tilde{t}_1,\tilde{t}_2,
\tilde{b}_1,\tilde{b}_2}CPL [2m_j(a_{ij}^2-b_{ij}^2)(p_2\cdot \epsilon_4C_0
-C^{10}\cdot\epsilon_4)](-p_2,p_4,m_j,m_i,m_i)\\
F_1^{v1,t}&= & \sum\limits_{j}\sum\limits_{i=\tilde{t}_1,\tilde{t}_2,
\tilde{b}_1,\tilde{b}_2}CPL [\delta Z_{ij}^v]\\
F_6^{v1,t\;\mu}&= & -\sum\limits_{j}\sum\limits_{i=\tilde{t}_1,\tilde{t}_2,
\tilde{b}_1,\tilde{b}_2}CPL [2(a_{ij}^2+b_{ij}^2)(p_2\cdot\epsilon_4
C^{10\mu}\\
& & -C^{21\mu}(\epsilon_4))](-p_2,p_4,m_j,m_i,m_i)\\
& &\\
F_0^{v2,t}& =& -\sum\limits_{j}\sum\limits_{i=\tilde{t}_1,\tilde{t}_2,
\tilde{b}_1,\tilde{b}_2}CPL [2m_j(a_{ij}^2-b_{ij}^2)(-p_1\cdot \epsilon_3C_0\\
& & -C^{10}\cdot\epsilon_3)](p_1,-p_3,m_j,m_i,m_i)\\
F_1^{v2,t}&= & \sum\limits_{j}\sum\limits_{i=\tilde{t}_1,\tilde{t}_2,
\tilde{b}_1,\tilde{b}_2}CPL [\delta Z_{ij}^v]\\
F_6^{v2,t\;\mu}&= & -\sum\limits_{j}\sum\limits_{i=\tilde{t}_1,\tilde{t}_2,
\tilde{b}_1,\tilde{b}_2}CPL [2(a_{ij}^2+b_{ij}^2)(-p_1\cdot\epsilon_3
C^{10\mu}\\
& & -C^{21\mu}(\epsilon_3))](p_1,-p_3,m_j,m_i,m_i)\\
& & \\
F_0^{b,t}& =& \sum\limits_{j}\sum\limits_{i=\tilde{t}_1,\tilde{t}_2,
\tilde{b}_1,\tilde{b}_2}CPL
[4m_j(a_{ij}^2-b_{ij}^2)(p_2\cdot \epsilon_4 p_1\cdot\epsilon_3D_0
-p_1\cdot\epsilon_3D^{10}\cdot\epsilon_4\\
& & +p_2\cdot\epsilon_4D^{10}\cdot\epsilon_3
-\epsilon_3\cdot D^{21}(\epsilon_4))] (-p_2,p_4,p_3,m_j,m_i,m_i,m_i)\\
F_3^{b,t\mu}& = & \sum\limits_{j}\sum\limits_{i=\tilde{t}_1,\tilde{t}_2,
\tilde{b}_1,\tilde{b}_2}CPL
[4(a_{ij}^2+b_{ij}^2)(p_2\cdot \epsilon_4 p_1\cdot\epsilon_3D^{10\mu}
-p_1\cdot\epsilon_3D^{21\mu}(\epsilon_4)\\
& & +p_2\cdot\epsilon_4D^{21\mu}(\epsilon_3)
-D^{32\mu}(\epsilon_3,\epsilon_4))] (-p_2,p_4,p_3,m_j,m_i,m_i,m_i)\end{aligned}$$
In the above $i,~j$ summation, $j=1,2,3,4,~m_j=m_{\tilde{\chi}^0_j}$ for $i=\tilde{t}_1,~\tilde{t}_2$ , $j=1,2,~m_j=m_{\tilde{\chi}^+_j}$ for $i=\tilde{b}_1,~\tilde{b}_2$. As $\theta_b=0$, $\tilde{b}_2$ actually does not contribute to the sum(cf. Eq.(\[ab\])). $$\begin{aligned}
& & k=p_1+p_2=p_3+p_4,\;\;\hat{s}=k^2,\;\;
\hat{t}=q^2=(p_2-p_4)^2,\;\;\hat{u}=(p_2-p_3)^2,\\
& & \Gamma^\mu=(-p_4+p_3)^\mu\epsilon_3\cdot\epsilon_4+(2p_4+p_3)
\cdot\epsilon_3
\epsilon_4^\mu-(2p_3+p_4)\cdot\epsilon_4\epsilon_3^\mu \;,\\
& & CPL=\frac{\alpha m_t^2}{8\pi m_W^2\sin^2\theta_W\sin^2\beta}\end{aligned}$$
$$\begin{aligned}
& & C^{10\mu}=C^{\mu}\;, C^{21\mu}(p)=p_\nu C^{\mu\nu}\;,
C^{20}=g_{\mu\nu}C^{\mu\nu}-\displaystyle\frac{1}{2}\;,\\
& & D^{10\mu}=D^{\mu}\;, D^{21\mu}(p)=p_\nu D^{\mu\nu}\;,
D^{20}=g_{\mu\nu}D^{\mu\nu}\;,\\
& & D^{32\mu}(p,l)=p_\nu l_\alpha D^{\mu\nu\alpha}\;,
D^{30\mu}=g_{\nu\alpha}D^{\mu\nu\alpha}\;. \end{aligned}$$
In our calculation, we calculate the tensor loop integrals $C^\mu$, $C^{\mu\nu}$, $D^{\mu}$, $D^{\mu\nu}$ and $D^{\mu\nu\alpha}$ numerically instead of expanding them explicitly.
The renormalization constants are:\
$$\begin{aligned}
& & Z^v_{ij}=[(a_{ij}^2+b_{ij}^2)(B_1+2m_t^2G_1)+2(a_{ij}^2-b_{ij}^2)
m_tm_jG_0](p^2,m^2_j,m^2_i)|_{p^2=m_t^2}\\
& & \delta m_{ij}=[-(a_{ij}^2+b_{ij}^2)m_tB_1+(a_{ij}^2-b_{ij}^2)m_jB_0]
(p^2,m^2_j,m^2_i)|_{p^2=m_t^2}\end{aligned}$$ where $\displaystyle G_0=-\frac{\partial B_0(p^2,m_i^\prime,m_i)}
{\partial p^2}$, $\displaystyle G_1=\frac{\partial B_1(p^2,m_i^\prime,m_i)}{\partial p^2}$ .
The neutralino and chargino masses , the $2\times 2$ matrix $V$ and the $4\times 4$ matrix $N$ are given by the following relations[@MSSM]: $$X =\left(
\begin{array} {cc}
M & m_W\sqrt{2}\sin\beta \\
m_W\sqrt{2}\cos\beta & \mu
\end{array}
\right)~,$$ $$VX^2V^{-1}=M_D^2$$
$$Y =\left(
\begin{array} {cccc}
M^\prime & 0 & -m_Z s_W\cos\beta & m_Z s_W\sin\beta\\
0 & M & m_Zc_W\cos\beta & -m_Zc_W\sin\beta \\
-m_Z s_W\cos\beta & m_Zc_W\cos\beta & 0 & -\mu\\
m_Z s_W\sin\beta & -m_Zc_W\sin\beta & -\mu & 0
\end{array}
\right)~,$$
$$N Y^2 N^{-1}=N_D^2 \;,\;\;N^\ast Y N^{-1}=N_D~,$$ where $M$, $M^\prime$ are the masses of gauginos corresponding to $SU(2)$ and $U(1)$, respectively. With the grand unification assumption, we have the relation $M^\prime=\frac{5}{3}(g^{\prime 2}/g^2)M$. $M_D^2$ is a diagonal matrix and the chargino mass squares correspond to its diagonal elements. $N_D$ is a diagonal matrix with non-negative entries which give the masses of neutralinos. The chargino and neutralino masses depend on the parameters $M$, $\mu$ and $\tan\beta$.
[7]{}
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[Figure Captions]{}
Fig.1 Feynman diagrams of tree level and $O(\alpha m_t^2/m^2_W)$ SUSY electroweak correction.
Fig.2 Relative correction to hadronic cross section versus $m_{\tilde{q}}$ with $\tan\beta=1$ and $m_{LR}=0$.
Fig.3 Relative correction to hadronic cross section versus $m_{LR}$ with $\tan\beta=1$ and $m_{\tilde{q}}=100$ GeV.
Fig.4 Relative correction to hadronic cross section versus $\tan\beta$ with $m_{\tilde{q}}=100$ GeV and $m_{LR}=100$ GeV.
|
---
abstract: 'We establish a nonminimal Einstein-Yang-Mills-Higgs model, which contains six coupling parameters. First three parameters relate to the nonminimal coupling of non-Abelian gauge field and gravity field, two parameters describe the so-called derivative nonminimal coupling of scalar multiplet with gravity field, and the sixth parameter introduces the standard coupling of scalar field with Ricci scalar. The formulated six-parameter nonminimal Einstein-Yang-Mills-Higgs model is applied to cosmology. We show that there exists a unique exact cosmological solution of the de Sitter type for a special choice of the coupling parameters. The nonminimally extended Yang-Mills and Higgs equations are satisfied for arbitrary gauge and scalar fields, when the coupling parameters are specifically related to the curvature constant of the isotropic spacetime. Basing on this special exact solution we discuss the problem of a hidden anisotropy of the Yang-Mills field, and give an explicit example, when the nonminimal coupling effectively screens the anisotropy induced by the Yang-Mills field and thus restores the isotropy of the model.'
address:
- |
Department of General Relativity and Gravitation,\
Kazan State University,\
Kremlevskaya str. 18, Kazan, 420008, Russia\
Alexander.Balakin@ksu.ru
- |
Fachbereich Physik,\
Universität Konstanz,\
Fach M 677, D-78457, Konstanz, Germany\
Heinz.Dehnen@uni-konstanz.de
- |
Department of General Relativity and Gravitation,\
Kazan State University,\
Kremlevskaya str. 18, Kazan, 420008, Russia\
Alexei.Zayats@ksu.ru
author:
- 'ALEXANDER B. BALAKIN'
- HEINZ DEHNEN
- 'ALEXEI E. ZAYATS'
title: 'NONMINIMAL ISOTROPIC COSMOLOGICAL MODEL WITH YANG-MILLS AND HIGGS FIELDS'
---
Introduction {#Intro}
============
The discussion of a nonminimal coupling (NMC) of gravity with fields and media has a long history. The most intensely this topic has been studied in connection with the problem of nonminimal coupling of gravity and scalar field, which has numerous cosmological applications. The details of the investigations of this problem are discussed, e.g., in the review of Faraoni [*et al*]{}.[@FaraR] The development of the theory of NMC of gravity and scalar field $\phi$ has started by the introduction of the term $\xi \phi^2 R$ to the Lagrangian ($R$ is the Ricci scalar). In Ref. the special choice $\xi = 1/6$ has been motivated by the conformal invariance; in Ref. this quantity was considered as an arbitrary parameter of the model. Such a model has been widely used for the cosmological applications, in which $\xi$ played a role of extra parameter of inflation (see, e.g., Refs. –). In Refs. – the gauge-invariant term $\alpha {\bf \Phi}^{+}{\bf \Phi}R $ has been introduced instead of $\xi \phi^2 R$ in the context of non-Abelian gauge theory (${\bf
\Phi}$ is a multiplet of scalar complex Higgs fields interacting with gravity and spinor matter.) Subsequent generalizations have been related to the replacement of $\xi
\phi^2$ by the function $f({\Phi}^2)$ (see, e.g., Refs. –), as well as, to the inserting of the terms of the type $F({\Phi}^2, {\cal R})$ both linear and nonlinear in the Ricci scalar, Ricci and Riemann tensors (see, e.g., Refs. –). The idea of nonminimal derivative coupling introduced in Ref. and developed further in Refs. has enriched the NMC modeling by the terms $\phi_{,ij..}$. Nonminimal cosmological models based on the formalism of derivative coupling are the multi-parameter ones and have supplementary abilities for a fitting of observational data. Let us note that the NMC of gravity and scalar field leads to the modifications of both the Klein-Gordon and the Einstein equations, and such modifications are of interest for various inflation scenarios. Thus, the modeling of nonminimal interactions of scalar and gravitational fields is one of the well established and physically motivated branch of modern cosmology. Natural extension of the nonminimal theory from the models with scalar fields coupled to curvature to the models describing scalar fields interacting with gauge fields has the same sound motivation and can disclose new aspects of cosmological dynamics.
The study of the nonminimal coupling of gravity with electromagnetic field has another motivation and another history. In 1971 Prasanna[@Prasa1] introduced the invariant $R^{ikmn}F_{ik}F_{mn}$ ($R^{ikmn}$ is the Riemann tensor, $F_{ik}$ is the Maxwell tensor) as a possible element of a Lagrangian, and then in Ref. obtained the corresponding nonminimal one-parameter modification of the Einstein-Maxwell equations. In 1979 Novello and Salim[@Novello1] proposed to insert the gauge non-invariant terms $R A^k A_k$ and $R^{ik}A_i A_k$ in the Lagrangian ($A_k$ is an electromagnetic potential four-vector). A qualitatively new step has been made by Drummond and Hathrell in Ref. , where the one-loop corrections to the quantum electrodynamics (QED) are obtained, which take into account the nonminimal coupling of gravity and electromagnetism. The Lagrangian of such a theory happens to contain three fundamental $U(1)$-gauge-invariant scalars $R^{ikmn}F_{ik}F_{mn}$, $R^{ik}g^{mn}F_{im}F_{kn}$ and $RF_{mn}F^{mn}$ with coefficients reciprocal to the square of the electron mass. This Lagrangian had no arbitrary parameters, but curvature induced modifications of the electrodynamic equations gave the impetus to wide discussions about the formal structure of the nonminimal Lagrangian, basic evolutionary equations, breaking the conformal invariance and the properties of the photons, coupled to curvature in different gravitational backgrounds (see, e.g., Refs. –). The last paper revived, as well, the interest to the paradigm: curvature coupling and equivalence principle, various aspects of which are now discussed (see, e.g., Refs. ). The QED-motivation of the use of the generalized Maxwell equations can also be found in the papers of Kostelecký and colleagues.[@Kost1; @Kost2] The effect of birefringence induced by curvature, first discussed in Ref. , and some of its consequences for the electrodynamic systems have been investigated in Refs. – for the case of pp-wave background. The generalization of the idea of nonminimal interactions to the case of torsion coupled to the electromagnetic field has been made in Refs. (see, also, Ref. for a review on the problem). To summarize we stress that the study of electrodynamic systems nonminimally coupled to the gravity field poses a natural question about curvature induced variations of photon velocity in the cosmological background. Since the interpretation of observational data in cosmology depends essentially on the velocity of photon propagation during different cosmological epochs, the modeling of nonminimal electrodynamic phenomena seems to be well motivated and interesting from physical point of view.
Concerning the nonminimal Einstein-Yang-Mills (EYM) theory, we can distinguish between two different ways to establish it. The first way is the direct nonminimal generalization of the Einstein-Yang-Mills (EYM) theory. In the framework of this approach Horndeski[@Horn] and Müller-Hoissen[@MH] obtained the nonminimal one-parameter EYM model from a dimensional reduction of the Gauss-Bonnet action. Now the Gauss-Bonnet models are of great interest in connection with the problem of dark energy (see, e.g., the Gauss-Bonnet model with nonminimal scalar field[@OdinDE]). Thus, the non-Abelian multi-parameter extensions of nonminimal models are also well motivated, since they give a chance to explain the accelerated expansion of the Universe without addressing to exotic substance. We follow the alternative way, which is connected with a non-Abelian generalization of the nonminimal Einstein-Maxwell theory along the lines proposed by Drummond and Hathrell[@Drum] for the linear electrodynamics. Based on the results of Ref. a three-parameter gauge-invariant nonminimal EYM model linear in curvature is considered.[@1BZ06][@BDZ07] Our goal is to formulate a nonminimal Einstein-Yang-Mills-Higgs (EYMH) theory, and this process, of course, also admits different approaches. In fact, the nonminimal EYMH theory should accumulate the ideas and methods both from the nonminimally extended EYM theory and from the nonminimally extended scalar field theory. Initial attempt to develop nonminimal EYMH theory can be found, for instance, in Ref. , where the scalar Higgs field is nonminimally coupled with gravity via $\xi {\Phi}^2 R$ term, and the Higgs field ${\bf \Phi}$ is included into the Lagrangian of the Yang-Mills field in a composition with a square of the Yang-Mills potential: ${\Phi}^2 A_k^{(a)} A^k_{(a)}$. Such a theory is not gauge-invariant.
In this paper we establish a new six-parameter nonminimal Einstein-Yang-Mills-Higgs model. First three coupling parameters, $q_1$, $q_2$ and $q_3$, describe a nonminimal interaction of Yang-Mills field and gravitational field. The fourth and fifth parameters, $q_4$ and $q_5$, describe the so-called gauge-invariant nonminimal derivative coupling of the Higgs field with gravity. Since the gauge-invariant derivative, ${\hat{D}}_m {\Phi}^{(a)}$, contains the potential of the Yang-Mills field, the corresponding nonminimal term is associated with “triple” interaction, namely, gravitational and scalar fields, gauge and scalar fields, and gauge and gravitational fields. The sixth parameter, $\xi$, is the well-known coupling parameter nonminimally connecting gravitational and scalar fields via the term $\xi R {\Phi}^2$. Of course, this model is only one of a wide class of the nonminimal EYMH models. As for its motivation and possible physical applications, one can see that on the one hand, the interest to a six-parameter nonminimal EYMH model is based on the sound results obtained earlier in the framework of partial nonminimal models (Einstein-Maxwell, Einstein-Yang-Mills and scalar field theories), on the other hand, the six-parameter model under discussion shows new specific solutions of cosmological type, which can not appear in more simple models.
The paper is organized as follows. In Sec. \[Formalism\] we formulate the nonminimal EYMH model, which contains six phenomenological coupling parameters, and establish the nonminimally extended Yang-Mills, Higgs and Einstein equations. In Sec. \[IsModel\] we apply the introduced master equations to the spacetime with constant curvature and obtain the specific relationships between coupling constants, which turn the extended equations for the gauge field and scalar field into identities. In Subsec. \[dSsptime\] we discuss the exact solutions to the nonminimal EYMH equations attributed to the isotropic cosmological model with Yang-Mills field, characterized by hidden anisotropy.
The formalism of the nonminimal EYMH theory {#Formalism}
===========================================
Minimal EYMH theory and basic definitions {#MinEYMH}
-----------------------------------------
The minimal Einstein-Yang-Mills-Higgs theory can be formulated in terms of the action functional $$S_{(\rm EYMH)} = \int d^4 x \sqrt{-g} \left\{\frac{R}{\kappa} +
\frac12 F_{mn}^{(a)} F^{mn}_{(a)} - {\hat{D}}_m \Phi^{(a)} {\hat{D}}^m \Phi_{(a)}
+ {V}({\Phi}^2) \right\} , \label{act}$$ where $g = {\rm det}(g_{ik})$ is the determinant of a metric tensor $g_{ik}$, $R$ is the Ricci scalar, Latin indices run from 0 to 3. The symbol ${\Phi}^{(a)}$ denotes the multiplet of the Higgs scalar fields, ${V}({\Phi}^2)$ is a potential of the Higgs field and $\Phi^2\equiv\Phi^{(a)}\Phi_{(a)}$. Let us mention that there are two formal variants to introduce the cosmological constant into the action (\[act\]): first, explicitly as an additional term $\frac{2\Lambda}{\kappa}$, second, as a term $V(0)$ in the decomposition $${V}({\Phi}^2)= \frac{2\Lambda}{\kappa} + \mu {\Phi}^2 + \omega
{\Phi}^4 + \dots \label{V}$$ Below we consider the second variant. Following Ref. , Section 4.3, we consider the Yang-Mills field ${\bf F}_{mn}$ and the Higgs field ${\bf \Phi}$ taking values in the Lie algebra of the gauge group $SU(n)$: $${\bf F}_{mn} = - i {\cal G} {\bf t}_{(a)} F^{(a)}_{mn} \,, \quad
{\bf A}_m = - i {\cal G} {\bf t}_{(a)} A^{(a)}_m \,, \quad {\bf
\Phi} = {\bf t}_{(a)} \Phi^{(a)} \,. \label{represent}$$ Here ${\bf t}_{(a)}$ are the Hermitian traceless generators of $SU(n)$ group, the constant ${\cal G}$ is the strength of the gauge coupling, $F^{(a)}_{mn}$, $A^{(a)}_m$ and $\Phi^{(a)}$ are real fields ($A^{(a)}_m$ represents the Yang-Mills field potential) and the group index $(a)$ runs from $1$ to $n^2-1$. The symmetric tensor $G_{(a)(b)}\equiv 2\, {\rm Tr} \ {\bf t}_{(a)}
{\bf t}_{(b)}$ plays a role of a metric in the group space so that, e.g., $\Phi_{(a)}\equiv G_{(a)(b)}\Phi^{(b)}$. The Yang-Mills fields $F^{(a)}_{mn}$ are connected with the potentials of the gauge field $A^{(a)}_i$ by the well-known formula (see, e.g., Refs. –) $$\begin{aligned}
F^{(a)}_{mn} = \nabla_m A^{(a)}_n - \nabla_n A^{(a)}_m + {\cal G}
f^{(a)}_{\ (b)(c)} A^{(b)}_m A^{(c)}_n \,. \label{Fmn}\end{aligned}$$ Here $\nabla _m$ is a covariant spacetime derivative, the symbols $f^{(a)}_{\ (b)(c)}$ denote the real structure constants of the gauge group $SU(n)$. The gauge-invariant derivative is defined according to the formula (Ref. , Eqs.(4.46, 4.47)) $${\hat{D}}_m \Phi^{(a)} \equiv \nabla_m \Phi^{(a)} + {\cal G} f^{(a)}_{\
(b)(c)} A^{(b)}_m \Phi^{(c)} \,. \label{DPhi}$$ For the derivative of arbitrary tensor defined in the group space we use the following rule[@Akhiezer]: $$\begin{aligned}
{\hat{D}}_m Q^{(a) \cdot \cdot \cdot}_{\cdot \cdot \cdot (d)} \equiv
\nabla_m Q^{(a) \cdot \cdot \cdot}_{\cdot \cdot \cdot (d)} + {\cal
G} f^{(a)}_{\ (b)(c)} A^{(b)}_m Q^{(c) \cdot \cdot \cdot}_{\cdot
\cdot \cdot (d)} - {\cal G} f^{(c)}_{\ (b)(d)} A^{(b)}_m Q^{(a)
\cdot \cdot \cdot}_{\cdot \cdot \cdot (c)} + \dots \label{DQ2}\end{aligned}$$ The commutator and anticommutator of the generators ${\bf t}_{(a)}$ take the form $$\left[ {\bf t}_{(a)} , {\bf t}_{(b)} \right] = i f^{(c)}_{\ (a)(b)}
{\bf t}_{(c)} \,, \label{fabc}$$ $$\left\{ {\bf t}_{(a)} , {\bf t}_{(b)} \right\} \equiv {\bf t}_{(a)}
{\bf t}_{(b)} + {\bf t}_{(b)} {\bf t}_{(a)} = \frac{1}{n}\,
G_{(a)(b)} {\bf I} + d^{\,(c)}_{\ (a)(b)} {\bf t}_{(c)}\,,
\label{dabc}$$ where $d_{(c)(a)(b)}$ are the completely symmetric coefficients and ${\bf I}$ is the unitary matrix. The metric $G_{(a)(b)}$, the structure constants $f^{(c)}_{\ (a)(b)}$ and the coefficients $d^{\,(c)}_{\ (a)(b)}$ are supposed to be constant tensors in standard and covariant manner[@Akhiezer]. This means that $$\begin{gathered}
\partial_m G_{(a)(b)} = 0 \,,\quad
\partial_m f^{(a)}_{\ (b)(c)} = 0 \,, \quad \partial_m d^{\,(a)}_{\ (b)(c)} =
0\,, \nonumber\\ {\hat{D}}_m G_{(a)(b)} = 0 \,,\quad {\hat{D}}_m f^{(a)}_{\
(b)(c)} = 0 \,, \quad {\hat{D}}_m d^{\,(a)}_{\ (b)(c)} = 0\,. \label{DfG}\end{gathered}$$
Nonminimal extension of the Lagrangian {#Nmextension}
--------------------------------------
Any version of nonminimal generalization of the Lagrangian of the EYMH theory is based on the choice of the set of admissible invariants. The classification of the Yang-Mills fields based on the invariant polynomials in the $F^{(a)}_{ik}$ tensor has been made in Ref. . Nonlinear constitutive equations for Yang-Mills field along a line of the Born-Infeld theory has been first discussed in Ref. (see, also, e.g., Ref. ). Possessing the tensorial quantities $F^{(a)}_{ik}$, $\Phi^{(a)}$, ${\hat{D}}_i \Phi^{(a)}$, $R^{ikmn}$, $R^{ik}$, $R$ one can construct a variety of gauge invariant scalars both minimal and nonminimal. This procedure has been discussed in the framework of scalar field theory and electrodynamics (see Introduction and references therein). Einstein-Yang-Mills-Higgs theory possesses an extended set of basic elements for such a representation, thus, a number of candidates to be included into a Lagrangian is much bigger. For instance, in order to couple the group indices $(a)$ and $(b)$ in the product $F^{(a)}_{ik} F^{(b)}_{mn}$, we can use, first, the standard convolution procedure, based on metric $G_{(a)(b)}$, second, the projections onto $\Phi_{(a)} \Phi_{(b)}$, $\Phi_{(a)} {\hat{D}}_j
\Phi_{(b)}$ or ${\hat{D}}_j \Phi_{(a)} {\hat{D}}_s \Phi_{(b)}$, third, the convolution with symmetric tensors $d_{(a)(b)(c)} \Phi^{(c)}$, $d_{(a)(b)(c)} {\hat{D}}_j \Phi^{(c)}$, or with antisymmetric tensors $f^{(c)}_{ \ (a)(b)} \Phi_{(c)}$, $f^{(c)}_{ \ (a)(b)} {\hat{D}}_j
\Phi_{(c)}$. The corresponding examples of the scalar invariants, admissible for including into the nonminimal Lagrangian are $$\frac{1}{2} {\cal R}^{ikmn}_{(I)} F^{(a)}_{ik} F^{(b)}_{mn} \left[
G_{(a)(b)} + d_{(a)(b)(c)} \Phi^{(c)} \Psi_1( {\Phi}^2) +
\Phi_{(a)} \Phi_{(b)} \Psi_2( {\Phi}^2)\right.$$ $$\left. {}+({\hat{D}}_l \Phi_{(a)}) ({\hat{D}}^l \Phi_{(b)}) \Psi_3( {\Phi}^2)+
d_{(a)(b)(c)} \Phi^{(c)} ({\hat{D}}^l \Phi_{(h)}) ({\hat{D}}_l \Phi^{(h)})
\Psi_4( {\Phi}^2) + \dots\right] \,, \label{inv1}$$ $${\cal R}^{ikmn}_{(II)} F^{(a)}_{ik} \left[ f_{(a)(b)(c)}({\hat{D}}_m
\Phi^{(b)}) ({\hat{D}}_n \Phi^{(c)}) \Psi_5( {\Phi}^2) + \dots\right] \,,
\label{in1}$$ $${\cal R}^{ikmn}_{(III)} \left\{ g_{im} ({\hat{D}}_k \Phi^{(a)}) ({\hat{D}}_n
\Phi^{(b)}) \left[ G_{(a)(b)} {+} d_{(a)(b)(c)} \Phi^{(c)} \Psi_6(
{\Phi}^2) {+} \Phi_{(a)} \Phi_{(b)} \Psi_7( {\Phi}^2) {+}
\dots\right] \right.$$ $$\left. {}+({\hat{D}}_i \Phi^{(a)})({\hat{D}}_k \Phi^{(b)})({\hat{D}}_m
\Phi^{(c)})({\hat{D}}_n \Phi^{(d)}) f^{(h)}_{\cdot (a)(b)} f_{(h)(c)(d)}
\Psi_8( {\Phi}^2) + \dots \right\} \,. \label{in2}$$ Here $\Psi_1,\ \Psi_2,\ \dots,\ \Psi_8$ are arbitrary functions of their argument, and the tensors ${\cal R}^{ikmn}_{(I)}$, ${\cal
R}^{ikmn}_{(II)}$ and ${\cal R}^{ikmn}_{(III)}$ are considered to be appropriate linear combinations of the Riemann tensor and its convolutions with phenomenological coupling constants $q_1,\
q_2,\ \dots,\ q_j$. These constants are treated to be independent and have a dimensionality of area. Below the examples of such tensors are presented explicitly.
In this paper we restrict ourselves to the consideration of a Lagrangian, which satisfy the following requirements: the EYMH Lagrangian is a gauge invariant scalar linear in a spacetime curvature, quadratic in the Yang-Mills field strength tensor $F^{(a)}_{ik}$ and depending on the first derivative of the Higgs field only. In addition, in this paper we consider the convolutions of the standard type only, i.e., the terms including $G_{(a)(b)} {\mathstrut F}^{(a)}_{ik}
{\mathstrut F}^{(b)}_{mn}$, $G_{(a)(b)}\Phi^{(a)}\Phi^{(b)}$, etc. We intend to consider more sophisticated models in future papers.
Explicit example of nonminimal gauge-invariant Lagrangian {#NMEYMH}
---------------------------------------------------------
Consider now an action functional $$\begin{aligned}
\label{1act}
S_{({\rm NMEYMH})} = \int d^4 x \sqrt{-g}\ \left\{
\frac{R}{\kappa}+\frac{1}{2}F^{(a)}_{ik} F^{ik}_{(a)}
-{{\hat{D}}}_m\Phi^{(a)}{{\hat{D}}}^m\Phi_{(a)}+V(\Phi^2) \right.
{}\nonumber\\
\left. {}+\frac{1}{2} {\cal R}^{ikmn}F^{(a)}_{ik} F_{mn(a)} -
{\Re}^{\,mn}{{\hat{D}}}_m\Phi^{(a)}{{\hat{D}}}_n\Phi_{(a)}+\xi R\,{\Phi}^2
\right\}\,,\end{aligned}$$ where the tensors ${\cal R}^{ikmn}$ and $\Re^{\,mn}$ are defined as follows: $$\begin{aligned}
{\cal R}^{ikmn} &\equiv&
\frac{q_1}{2}R\,(g^{im}g^{kn}-g^{in}g^{km}) \nonumber\\
{}&+& \frac{q_2}{2}(R^{im}g^{kn} - R^{in}g^{km} + R^{kn}g^{im}
-R^{km}g^{in}) + q_3 R^{ikmn}\,, \label{sus}\end{aligned}$$ $$\label{Re}
\Re^{\,mn}\equiv {q_4}Rg^{mn}+q_5 R^{mn}\,.$$ This action describes a six-parameter nonminimal Einstein-Yang-Mills-Higgs model, and $q_1$, $q_2,\ \dots,\ q_5$, $\xi$ are the constants of nonminimal coupling.
### Nonminimal extension of the Yang-Mills field equations {#Ymequations}
The variation of the action $S_{({\rm NMEYMH})}$ with respect to the Yang-Mills potential $A^{(a)}_i$ yields $${\hat{D}}_k {H}^{ik}_{(a)} = - {\cal G} ({\hat{D}}_k \Phi^{(b)})f_{(a)(b)(c)}
\Phi^{(c)} \left( g^{ik} + \Re^{ik} \right) \,. \label{Heqs}$$ Here the tensor ${H}^{ik}_{(a)}$ is defined as $${H}^{ik}_{(a)} = \left[ \frac{1}{2}( g^{im} g^{kn} - g^{in} g^{km})
+ {\cal R}^{ikmn} \right] G_{(a)(b)} F^{(b)}_{mn} \,. \label{HikR}$$ This equation looks like Maxwell equation for the medium with the susceptibility tensor ${\cal R}^{ikmn}$ and the current vector ${\cal G} ({\hat{D}}_k \Phi^{(b)})f_{(a)(b)(c)} \Phi^{(c)} \left( g^{ik} +
\Re^{ik} \right)$ induced by the Higgs field.
### Nonminimal extension of the Higgs field equations {#Hequations}
The variation of the action $S_{({\rm NMEYMH})}$ with respect to the Higgs scalar field $\Phi^{(a)}$ yields $$\label{Heq}
{{\hat{D}}}_m\left({{\hat{D}}}^m{\Phi}^{(a)} + \Re^{\,mn}{\hat{D}}_n{\Phi^{(a)}}\right) =
- \xi{R\,{\Phi}^{(a)}}-V'(\Phi^2){\Phi^{(a)}} \,.$$ This equation can be rewritten in the form $$\label{21Heq}
{{\hat{D}}}_m {\Psi}^{m(a)} = - \left[ \xi R + V'(\Phi^2) \right]
{\Phi}^{(a)} \,, \quad {\Psi}^{m(a)} \equiv {{\hat{D}}}^m{\Phi}^{(a)} +
\Re^{\,mn}{\hat{D}}_n{\Phi}^{(a)} \,,$$ and can be considered as scalar analog of (\[Heqs\]) and (\[HikR\]).
### Master equations for the gravitational field {#Einequations}
In the nonminimal theory linear in curvature the equations for the gravity field related to the action functional $S_{({\rm NMEYMH})}$ take the form $$\left(R_{ik}-\frac{1}{2}Rg_{ik}\right)\cdot(1+\kappa\xi\Phi^2)=
\kappa\xi\left({\hat{D}}_i{\hat{D}}_k-g_{ik}{\hat{D}}_m{\hat{D}}^m\right){\Phi}^2+
{k}T^{(NMYMH)}_{ik} \,. \label{Eeq}$$ The principal novelty of these equations, in comparison with the well-known equations for nonminimal scalar field, is associated with the third, fourth, etc., terms in the decomposition $$T^{(NMYMH)}_{ik}=T^{(YM)}_{ik} + T^{(H)}_{ik} + q_1 T^{(I)}_{ik} +
q_2 T^{(II)}_{ik} + q_3 T^{(III)}_{ik} + q_4 T^{(IV)}_{ik} + q_5
T^{(V)}_{ik} \,. \label{Tdecomp}$$ The first term $T^{(YM)}_{ik}$: $$T^{(YM)}_{ik} \equiv \frac{1}{4} g_{ik} F^{(a)}_{mn}F^{mn}_{(a)} -
F^{(a)}_{in}F_{k\,(a)}^{\ n} \,, \label{TYM}$$ is a stress-energy tensor of pure Yang-Mills field. The second one, $T^{(H)}_{ik}$, $$T^{(H)}_{ik}={\hat{D}}_i\Phi^{(a)}{\hat{D}}_k\Phi_{(a)}-\frac{1}{2}g_{ik}{\hat{D}}_m\Phi^{(a)}{\hat{D}}^m\Phi_{(a)}+\frac{1}{2}V(\Phi^2)\,g_{ik}$$ is a stress-energy tensor of the Higgs field. The definitions of other five tensors are related to the corresponding coupling constants $q_1$, $q_2,\,\dots,\,q_5$: $$T^{(I)}_{ik} = R\,T^{(YM)}_{ik} - \frac{1}{2} R_{ik}
F^{(a)}_{mn}F^{mn}_{(a)} + \frac{1}{2} \left[ {{\hat{D}}}_{i} {{\hat{D}}}_{k} -
g_{ik} {{\hat{D}}}^l {{\hat{D}}}_l \right] \left[F^{(a)}_{mn}F^{mn}_{(a)}
\right] \,, \label{TI}$$$$T^{(II)}_{ik} = -\frac{1}{2}g_{ik}\biggl[{{\hat{D}}}_{m}
{{\hat{D}}}_{l}\left(F^{mn(a)}F^{l}_{\ n(a)}\right)-R_{lm}F^{mn (a)}
F^{l}_{\ n(a)} \biggr]$$$${}- F^{ln(a)} \left(R_{il}F_{kn(a)} +
R_{kl}F_{in(a)}\right)-R^{mn}F^{(a)}_{im} F_{kn(a)} - \frac{1}{2}
{{\hat{D}}}^m{{\hat{D}}}_m \left(F^{(a)}_{in} F_{k\,(a)}^{ \
n}\right)$$$$\quad{}+\frac{1}{2}{{\hat{D}}}_l \left[ {{\hat{D}}}_i \left(
F^{(a)}_{kn}F^{ln}_{(a)} \right) + {{\hat{D}}}_k
\left(F^{(a)}_{in}F^{ln}_{(a)} \right) \right] \,, \label{TII}$$$$T^{(III)}_{ik} = \frac{1}{4}g_{ik} R^{mnls}F^{(a)}_{mn}F_{ls(a)}-
\frac{3}{4} F^{ls(a)} \left(F_{i\,(a)}^{\ n} R_{knls} +
F_{k\,(a)}^{\ n}R_{inls}\right)$$$$\quad {}-\frac{1}{2}{{\hat{D}}}_{m} {{\hat{D}}}_{n} \left[ F_{i}^{ \ n
(a)}F_{k\,(a)}^{ \ m} + F_{k}^{ \ n(a)} F_{i\,(a)}^{ \ m} \right]
\,, \label{TIII}$$ $$T^{(IV)}_{ik}=\left(R_{ik}-\frac{1}{2}Rg_{ik}\right){\hat{D}}_m\Phi^{(a)}{\hat{D}}^m\Phi_{(a)}+R\,{\hat{D}}_i\Phi^{(a)}{\hat{D}}_k\Phi_{(a)}$$$$\label{TIV}
{}+\left(g_{ik}{\hat{D}}_n{\hat{D}}^n-{\hat{D}}_i{\hat{D}}_k\right)\left[{\hat{D}}_m\Phi^{(a)}{\hat{D}}^m\Phi_{(a)}\right]\,,$$
$$T^{(V)}_{ik}={\hat{D}}_m\Phi^{(a)}\left[R_i^m{\hat{D}}_k\Phi_{(a)}+R_k^m{\hat{D}}_i\Phi_{(a)}\right]-
\frac{1}{2}R_{ik}\,{\hat{D}}_m\Phi^{(a)}{\hat{D}}^m\Phi_{(a)}$$$${}+\frac{1}{2}g_{ik}{\hat{D}}_m{\hat{D}}_n\left[{\hat{D}}^m\Phi^{(a)}{\hat{D}}^n\Phi_{(a)}\right]-
\frac{1}{2}\,{\hat{D}}^m\biggl\{{\hat{D}}_i\left[{\hat{D}}_m\Phi^{(a)}{\hat{D}}_k\Phi_{(a)}\right]$$$$\label{TV}
{}+{\hat{D}}_k\left[{\hat{D}}_m\Phi^{(a)}{\hat{D}}_i\Phi_{(a)}\right]-
{\hat{D}}_m\left[{\hat{D}}_i\Phi^{(a)}{\hat{D}}_k\Phi_{(a)}\right]\biggr\}\,.$$
### Bianchi identities {#Bids}
The Einstein tensor $R_{ik}-\frac{1}{2}g_{ik}R$ is the divergence-free one, thus, the tensor $T^{(NMYMH)}_{ik}$ in the right-hand-side of (\[Eeq\]) has to satisfy the differential condition $$\nabla^k
\left\{\frac{\kappa\xi\left({\hat{D}}_i{\hat{D}}_k-g_{ik}{\hat{D}}_m{\hat{D}}^m\right){\Phi}^2+
{k}T^{(NMYMH)}_{ik}}{(1+\kappa\xi\Phi^2)} \right\} =0 \,.
\label{Eeeq}$$ One can prove that it is valid automatically, when $F^{(a)}_{ik}$ is a solution of the Yang-Mills equations (\[Heqs\]), and $\Phi^{(a)}$ satisfy the Higgs equations (\[Heq\]). In order to check this fact directly, one has to use the Bianchi identities and the properties of the Riemann tensor: $$\nabla_i R_{klmn} + \nabla_l R_{ikmn} + \nabla_k R_{limn} = 0 \,,
\quad R_{klmn} + R_{mkln} + R_{lmkn} = 0 \,, \label{bianchi}$$ as well as the rules for the commutation of covariant derivatives $$(\nabla_l \nabla_k - \nabla_k \nabla_l) {\cal A}^i = {\cal A}^m
R^i_{\cdot mlk} \,, \label{nana}$$ (this rule is written here for the vector only). The procedure of checking is analogous to one, described in Ref. and we omit it.
Isotropic cosmological model associated with six-parameter nonminimal EYMH theory {#IsModel}
=================================================================================
Generally, the application of the EYMH model to cosmological problems requires the spacetime to be considered as anisotropic one. Clearly, when the spacetime is isotropic, the Einstein tensor in the left-hand-side of (\[Eeq\]) is diagonal, while the tensor $T_{ik}^{({\rm NMYMH})}$ in the right-hand-side is generally non-diagonal. This can be also motivated by the analogy with Einstein-Maxwell theory: it is well-known, for instance, that the minimal models with magnetic field are inevitably anisotropic and can be properly described in terms of Bianchi models. Nevertheless, as it was shown in Ref. , the nonminimal extension of the Einstein-Maxwell theory admits the models in which the spacetime is isotropic while the magnetic field is non-vanishing. Below we discuss the first example of analogous problem in the framework of nonminimal EYMH theory. Our goal is to present explicitly an exact solution to the equations of spatially isotropic EYMH model. When the Yang-Mills field is non-vanishing, the stress-energy tensor (\[Tdecomp\]) is non-diagonal, as in the case of Einstein-Maxwell theory, thus, the symmetry of equations for the gravitational field is, generally, broken. Nevertheless, we will indicate a special choice of the coupling parameters $q_1$, $q_2,\ \dots,\ q_5$, $\xi$, for which one can guarantee, that these equations become self-consistent. Since the de Sitter model is associated with the spacetime of constant curvature, we consider a number of properties of desired solution without solving the master equations.
Constant curvature spacetime\
and restrictions on the Yang-Mills-Higgs fields {#ConstCurv}
-----------------------------------------------
We consider isotropic cosmological models with constant curvature $K$.[@MTW] For these spacetimes the Riemann tensor takes the form $$\label{curv}
R_{ikmn}=-K\left(g_{im}g_{kn}-g_{in}g_{km}\right)$$ and the Ricci tensor, the Ricci scalar are $$\label{1curv}
R_{ik}=-3Kg_{ik}\,, \qquad R=-12K\,.$$ The tensors ${\cal R}_{ikmn}$ and $\Re_{ik}$, introduced phenomenologically, can be transformed into $$\label{simpa}
{\cal
R}_{ikmn}=-K(6q_1+3q_2+q_3)\left(g_{im}g_{kn}-g_{in}g_{km}\right)\,,\quad
\Re_{ik}=-3K(4q_4+q_5)g_{ik}\,.$$ Then, the ${H}^{ik}_{(a)}$ tensor and the ${\Psi}^m_{(a)}$ vector simplify significantly, and the equations (\[HikR\]) and (\[21Heq\]) convert, respectively, into $$\label{1simpa}
[1-2K(6q_1+3q_2+q_3)] {\hat{D}}_k {F}^{ik}_{(a)} = -{\cal G}[1-3K(4q_4+q_5)] f_{(a)(b)(c)}{\hat{D}}^i \Phi^{(b)}\Phi^{(c)} \,,$$ $$\label{2simpa}
[1-3K(4q_4+q_5)] {\hat{D}}_m {\hat{D}}^m {\Phi}^{(a)} = \left[ 12 \xi K -
V'(\Phi^2) \right] {\Phi}^{(a)} \,.$$ We focus on the case, when the equation for the Yang-Mills field turns into identity for arbitrary (non-vanishing) ${F}^{(a)}_{ik}$. It is not possible, when the EYMH theory is minimal one. Nevertheless, in the framework of nonminimal EYMH theory with non-vanishing Higgs field, ${\Phi}^{(a)}\neq 0$, the Yang-Mills equations admit an arbitrary non-vanishing solution, when $$\label{AQU}
2(6q_1+3q_2+q_3)= \frac{1}{K} \,, \quad 3(4q_4+q_5) = \frac{1}{K}
\,.$$ If (\[AQU\]) is valid, the Higgs equations are self-consistent, when $$\label{condV}
\left[ 12 \xi K - V'(\Phi^2) \right] {\Phi^{(a)}} =0 \,.$$ In its turn, it is possible in two cases: first, when $V({\Phi}^2)$ is a linear function of its argument, $$\label{1condV}
V({\Phi}^2) = \frac{2\Lambda}{\kappa} + 12 K \xi {\Phi}^2 \,,$$ ${\Phi}^{(a)}$ being arbitrary, second, when ${\Phi}^2$ is constant satisfying the equation (\[condV\]).
One-parameter nonminimal EYMH model {#1parModel}
-----------------------------------
In order to obtain an analytical progress in the searching for the solution to the gravity field equations let us consider the one-parameter model, which is characterized by the following conditions: $$\label{q}
q_1=q_4=\frac{1}{12K} \,, \quad q_2=q_3=q_5=0 \,, \quad
V({\Phi}^2) = \frac{2\Lambda}{\kappa} + \mu {\Phi}^2 \,, \quad \xi
= \frac{\mu}{12K} \,.$$ These conditions guarantee that the Yang-Mills equations (\[Heqs\]) and the Higgs equations (\[Heq\]) are the trivial identities for arbitrary ${F}^{ik}_{(a)}$ and ${\Phi}^{(a)}$. The Einstein equations for this case take the form $$3Kg_{ik} \ (1+\kappa\xi {\Phi}^2)=
\kappa\xi\left({\hat{D}}_i{\hat{D}}_k-g_{ik}{\hat{D}}_p{\hat{D}}^p\right){\Phi}^2+
\frac{\kappa}{2}\,V({\Phi}^2)g_{ik}$$$${}+\frac{\kappa}{8}\,g_{ik}\left(F_{mn}^{(a)}F^{mn}_{(a)}-2{\hat{D}}_m\Phi^{(a)}{\hat{D}}^m\Phi_{(a)}\right)$$$$\label{EinSpec}
{}+\frac{\kappa}{24K}\left({\hat{D}}_i{\hat{D}}_k-g_{ik}{\hat{D}}_p{\hat{D}}^p\right)\left[F_{mn}^{(a)}F^{mn}_{(a)}-2{\hat{D}}_m\Phi^{(a)}{\hat{D}}^m\Phi_{(a)}\right]\,.$$ It can be reduced formally to ten equations for one scalar function $$\label{W}
\nabla_i \nabla_k W = K g_{ik} W \,,$$ where $$\label{W1}
W \equiv
F_{mn}^{(a)}F^{mn}_{(a)}-2{\hat{D}}_m\Phi^{(a)}{\hat{D}}^m\Phi_{(a)}+24K\xi\Phi^2
+ \frac{24}{\kappa} \left(\frac{\Lambda}{3} - K \right) \,.$$ Let us consider the integrability conditions for such system and calculate the commutator of the covariant derivatives $\hat{{\cal
K}}_{ijk} \equiv [\nabla_i \nabla_j - \nabla_j \nabla_i] \nabla_k
W $. On the one hand with (\[W\]) this commutator yields directly $$\label{KK1}
\hat{{\cal K}}_{ijk} = - K\left(g_{ik}\nabla_j W - g_{jk}\nabla_i
W \right) \,.$$ On the other hand due to (\[nana\]) $$\label{KK2}
\hat{{\cal K}}_{ijk} = -{R^p}_{kij}\nabla_p W = - K
\left(g_{ik}\nabla_j W - g_{jk}\nabla_i W\right) \,.$$ Thus, the integrability conditions are satisfied identically, and the equations (\[W\]) are completely integrable.
De Sitter spacetime {#dSsptime}
-------------------
In order to represent the exact solution to (\[W\]) explicitly we consider the model with positive curvature, $K>0$, and reduce the metric to the de Sitter form[@Weinberg] $$\label{deSitter}
ds^2 = dt^2 + \exp\{2 \sqrt{K} t\} (\eta_{\alpha \beta}
dx^{\alpha} dx^{\beta}) \,,$$ where $\alpha, \beta = 1,2,3$ and $\eta_{\alpha \beta}$ is the spatial part of the Minkowski metric with the signature $(-,-,-)$. Then (\[W\]) splits into three subsystems $$\partial^2_t W -K W =0 \,, \quad \partial_{\alpha} [\partial_t W -\sqrt{K}
W]=0 \,,$$ $$\label{EqE}
\partial_{\alpha} \partial_{\beta} W + \eta_{\alpha \beta} \sqrt{K} \exp\{2 \sqrt{K} t\}
[\partial_t W -\sqrt{K} W] = 0 \,,$$ which can be readily solved $$\label{SolEqE}
W = C_1 e^{\sqrt{K}t} + C_2 e^{- \sqrt{K}t} + e^{\sqrt{K}t}
\left[L_{\alpha} x^{\alpha} + C_2 K \eta_{\alpha \beta} x^{\alpha}
x^{\beta} \right] \,.$$ Here $C_1$, $C_2$ and $L_{\alpha}$ are arbitrary constants. Thus, we obtain an exact solution of the total EYMH system of equations for which the Yang-Mills field $F_{mn}^{(a)}$ and the Higgs fields $\Phi^{(a)}$ are connected by unique condition $$F_{mn}^{(a)}F^{mn}_{(a)}-2{\hat{D}}_m\Phi^{(a)}{\hat{D}}^m\Phi_{(a)}+24K\xi\Phi^2
+ \frac{24}{\kappa} \left(\frac{\Lambda}{3} - K \right)$$ $$\label{W11}
\qquad{}= C_1 e^{\sqrt{K}t} + C_2 e^{- \sqrt{K}t} + e^{\sqrt{K}t}
\left[L_{\alpha} x^{\alpha} + C_2 K \eta_{\alpha \beta} x^{\alpha}
x^{\beta} \right] \,.$$ Clearly, there exists a lot of various Yang-Mills-Higgs configurations, which satisfy this condition.
Discussions {#Discussion}
===========
1\. The main mathematical result of the presented paper is the establishing of a new self-consistent nonminimal system of master equations for the coupled Yang-Mills, Higgs and gravity fields from the gauge-invariant nonminimal Lagrangian (\[1act\]). The obtained mathematical model contains six arbitrary parameters, and, thus, admits a wide choice of special sub-models interesting for the applications to the nonminimal cosmology (isotropic and anisotropic) and nonminimal colored spherical symmetric objects. The applications require the phenomenological coupling constants $q_1$, $q_2,\
\dots,\ q_5$ and $\xi$ to be interpreted adequately. Following the idea, discussed in Ref. , we intend not to introduce “new constants of Nature”, but to relate the phenomenological parameters with the constants well-known in the High Energy Particle Physics, on the one hand, and with the constants of cosmological origin, on the other hand. Indeed, in the specific cosmological model, established above, the sixth phenomenological parameter $\xi$ is expressed in terms of the square of the effective mass of the Higgs bosons $\mu$ and constant curvature $K$, $\xi = \frac{\mu}{12K}$. Other parameters are expressed in terms of $K$ (see (\[q\])). Since in the de Sitter model the Hubble constant is $H=\sqrt{K}$, one can say that $q_1$, $q_2,\ \dots,\ q_5$ are connected with $H$. Analogously, one can consider the equality $H^2=K=\frac{\Lambda}{3}$ and thus, one can say that they are connected with the cosmological constant $\Lambda$. In any case the parameters of nonminimal coupling $q_1$, $q_2,\ \dots,\ q_5$ can be expressed in terms of cosmological parameters $K$, $H$ or $\Lambda$, and define a specific radius of curvature coupling, $r_q \equiv \frac{1}{\sqrt{K}}$ and the corresponding time parameter $t_q \equiv r_q/c$.
2\. The curvature coupling modifies the master equations for the Yang-Mills and Higgs fields. According to (\[Heqs\]) a new tensor ${H}^{ik}_{(a)}$ appears (see (\[HikR\])), which is an analog of the induction tensor in the Maxwell theory[@Maugin]. This means that the curvature coupling of the non-Abelian gauge field with gravity acts as a sort of quasi-medium with a nonminimal susceptibility tensor ${\cal R}^{ikmn}$ (see (\[HikR\])). As well, the curvature coupling modifies the master equations for the Higgs field, and the tensor $\Re^{mn}$, according to (\[Heq\]), can be indicated as a simplest nonminimal susceptibility tensor for the Higgs field, and the vector ${\Psi}^{m}_{(a)}$ (see (\[21Heq\])) can be defined as scalar induction. For the specific set of coupling constants (see (\[AQU\]), (\[q\])) the non-Abelian induction $H^{ik}_{(a)}$ and the scalar induction ${\Psi}^{m}_{(a)}$ can turn into zero, despite the fact that the Yang-Mills field strength ${F}^{ik}_{(a)}$ and the Higgs field ${\Phi}^{(a)}$ are non-vanishing. This means that, when (\[AQU\]) holds, the possibility exists to satisfy the nonminimally extended Yang-Mills and Higgs equations for arbitrary ${F}^{ik}_{(a)}$ and ${\Phi}^{(a)}$. This possibility gives, in principle, a new option for modeling physical processes in Early Universe and shows very interesting analogy between this nonminimal model and resonance phenomena in plasma physics. Indeed, when we deal with plasma waves (for instance, with the longitudinal waves) one can see that electric induction $\vec{D}$ is connected with the longitudinal electric field $\vec{E}_{||}$ with the frequency $\omega$ by the relation $\vec{D}= \varepsilon_{||}\vec{E}_{||}$. Here $\varepsilon_{||}$ is the longitudinal dielectric permittivity, the simplest expression for this quantity can be obtained in the limit of long waves and gives $\varepsilon_{||}= 1-\frac{\Omega^2_{p}}{\omega^2}$, where $\Omega_{p}$ is the well-known plasma frequency. When $\omega=\Omega_{p}$, one obtains $\vec{D}=0$ and electrodynamic equations are satisfied for arbitrary $\vec{E}_{||}$. Analogous feature can be found in the nonminimal model described above (see Eq. (\[1simpa\])). Indeed, the quantity $K$ with the dimensionality of squared frequency ($c=1$) can be regarded as an analog of $\Omega^2_{p}$, the quantity $2(6q_1+3q_2+q_3)$ can be indicated as $1/\omega^2$, then the term $1-2K(6q_1+3q_2+q_3)$ plays a role of effective permittivity scalar $\varepsilon_q$. When this effective permittivity scalar vanishes, i.e., when the constants of nonminimal coupling are connected with the constant curvature $K$ according to (\[AQU\]), we obtain the resonance case, for which the Yang-Mills and Higgs equations are satisfied identically for arbitrary strength field tensor $F^{ik}_{(a)}$ and Higgs multiplet $\Phi^{(a)}$, the color induction $H^{ik}_{(a)}$ being equal to zero.
3\. The vector potential of the Yang-Mills field $ A^{(a)}_i$ enters the master equations via the gauge covariant derivative $\hat{D}_k$, thus, the gauge field generates an anisotropy in the spacetime. Such an anisotropy, in general case, breaks down the symmetry of the model and produces the isotropy violation. Nevertheless, as it was shown above, the nonminimal coupling can effectively screen the anisotropy and guarantee the symmetry conservation. In the framework of this model one can speak about hidden anisotropy of the Yang-Mills field, keeping in mind that non-Abelian gauge field enters the master equations for the gravity field in the isotropic combinations only.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was funded by DFG through the project No. 436RUS113/487/0-5.
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|
---
author:
- 'Lokesh Kumar [^1] (for STAR Collaboration)'
title: '$K^{*0}$(892) and $\phi$(1020) resonance production at RHIC'
---
Introduction {#intro}
============
Quantum Chromodynamics (QCD) predicts a phase transition from nuclear matter to a state of deconfined matter, called the quark gluon plasma (QGP), at high temperature and energy density [@Karsch:2001vs]. High energy heavy-ion collisions provide ideal scenario for the formation of the QGP [@Adams:2005dq]. The study of resonance production can be a useful tool in understanding the properties of the system formed in heavy-ion collisions. Since their lifetimes are comparable to that of fireball, resonance particles are expected to decay, rescatter, and regenerate to the kinetic freeze-out state (vanishing elastic collisions). As a result, their characteristic properties may be modified due to in-medium effects [@medium]. The study of resonance particles can provide the time span and hadronic interaction cross-section of the hadronic phase between chemical (vanishing inelastic collisions) and kinetic freeze-out [@time_span]. Comparisons of $K^{*0}$ and $\phi$ mesons are interesting since their lifetimes differ by a factor of 10 [@pdg]. $K^{*0}$ has a lifetime $\sim$4 fm/$c$ [@rapp], comparable with that of the fireball, so it is expected to suffer changes from the in-medium effects. There are two competing processes affecting $K^{*0}$ yield, rescattering that reduces the $K^{*0}$ yield and regeneration that may lead to increase in $K^{*0}$ yield [@rescatter_regen]. As a result, it is expected that $K^{*0}$/$K$ should change as a function of centrality, i.e., increase due to regeneration or decrease due to rescattering. On the other hand, $\phi$ mesons are expected to freeze-out early [@Adams:2005dq] and have a comparatively larger lifetime ($\sim$45 fm/$c$) than $K^{*0}$, so they may not undergo rescattering and regeneration effects. As a result, the $\phi$/$K$ ratio should remain constant as a function of centrality.
Another interesting property of the $\phi$ meson is that being a meson, it has a mass comparable to that of protons and $\Lambda$, which are baryons. Studying the $\phi$ meson (e.g., elliptic flow) along with these baryons and other mesons may give information on quark coalescence or the partonic phase at the top RHIC energy [@quark_coal]. Once the partonic phase is established at higher energies, one would expect the turn-off of partonic phase or decrease of dominance of the partonic interactions when the energy is decreased. This is one of the goals of the RHIC Beam Energy Scan (BES) program [@bes].
Experimental Detail {#sec-1}
===================
The results presented here are mainly from the Solenoidal Tracker At RHIC (STAR) experiment. The STAR detector has a coverage of 2$\pi$ in azimuth and pseudorapidity $|\eta|<$1. The data sets include Au+Au, Cu+Cu, $d$+Au, and $p+p$ collisions for energies $\sqrt{s_{NN}}=$ 62.4 and 200 GeV. Results from the Beam Energy Scan phase-I, that include data from Au+Au collisions at $\sqrt{s_{NN}}=$7.7, 11.5, 19.6, 27, and 39 GeV, are also presented. The STAR Time Projection Chamber (TPC) is the main detector used for particle identification by measuring the particle energy loss [@tpc].
The centrality selection is done using the uncorrected charged track multiplicity measured in the TPC within $|\eta|<$0.5 and comparing with Monte-Carlo Glauber simulations [@centrality]. Both $K^{*0}$ and $\phi$ resonances are reconstructed via their hadronic decay channels: $K^{*0} \rightarrow K\pi$ and $\phi
\rightarrow KK$. These daughter particles are identified using the TPC as mentioned above. $K^{*0}$ and $\phi$ mesons are reconstructed by calculating invariant mass for each unlike-sign $K\pi$ and $KK$, respectively, in an event. The resultant distribution consists of the true signal ($K^{*0}$ or $\phi$) and contributions arising from the random combination of unlike sign $K\pi$ and $KK$ pairs. To extract the $K^{*0}$ or $\phi$ yield, the large random combinatorial background must be subtracted from the unlike sign $K\pi$ or $KK$ pairs. The random combinatorial background distribution is obtained using the mixed-event technique [@mix_event]. In the mixed event technique, the background distribution is built with uncorrelated unlike-sign $K\pi$ or $KK$ pairs from different events. The generated mixed events distribution is then properly normalized to subtract the background from the same event unlike-sign invariant mass spectrum.
Results and Discussions
=======================
Figure \[spectra\] shows the invariant yields versus transverse momentum $p_T$ of $K^{*0}$ (left plot) and $\phi$ (right plot) in Au+Au collisions at 62.4 and 200 GeV, respectively for different collision centralities [@Aggarwal:2010mt; @Abelev:2007rw]. From these distributions, $dN/dy$ and average transverse momentum $\langle p_T \rangle$ can be obtained. These quantities provide important information about the system formed in high energy collisions. It is observed that $dN/dy$ per participating nucleon pair for $K^{*0}$ increases with increasing energy. $\phi$ meson yields per participating nucleon increases with increasing energy and centrality [@Aggarwal:2010mt; @Abelev:2008aa]. $dN/dy$ per participating nucleon pair increases from $pp$, $d$+Au to Au+Au collisions at 200 GeV for both $K^{*0}$ and $\phi$ mesons. Comparing $\langle p_T \rangle$ of $K^{*0}$ with pions, kaons, and protons in Au+Au collisions at 200 GeV, suggests that it is greater than that of pions and kaons but similar to that of protons, reflecting mass dependence or collectivity [@kstar_meanpt]. When $K^{*0}$ $\langle p_T \rangle$ is compared between $pp$ and Au+Au collisions at 200 GeV, it is found to be larger in Au+Au collisions, suggesting larger radial flow. In general, $\langle p_T \rangle$ increases with particle mass, showing a collective behavior [@Abelev:2008aa].
{height="5cm" width="6cm"} {height="5cm" width="5cm"}
![Left: $K^{*0}$/$K^-$ ratio in $p+p$ and various centralities in $d$+Au and Au+Au collisions as a function of $dN_{\rm{ch}}/d\eta$ [@Abelev:2008yz]. Right: $K^{*0}$/$K^-$ ratio in Au+Au, Cu+Cu, and d+Au collisions divided by that in $p+p$ collisions at $\sqrt{s_{NN}}=$ 200 GeV as a function of $\langle N_{\rm{part}} \rangle$ [@Aggarwal:2010mt]. []{data-label="ratio_kstar"}](ratio_kstar_k.eps "fig:"){height="4.2cm" width="6.cm"} ![Left: $K^{*0}$/$K^-$ ratio in $p+p$ and various centralities in $d$+Au and Au+Au collisions as a function of $dN_{\rm{ch}}/d\eta$ [@Abelev:2008yz]. Right: $K^{*0}$/$K^-$ ratio in Au+Au, Cu+Cu, and d+Au collisions divided by that in $p+p$ collisions at $\sqrt{s_{NN}}=$ 200 GeV as a function of $\langle N_{\rm{part}} \rangle$ [@Aggarwal:2010mt]. []{data-label="ratio_kstar"}](fig10b_n.eps "fig:"){width="0.4\linewidth"}
![Left: $\phi/K^{-}$ ratio in $p+p$ and various centralities in $d$+Au and Au+Au collisions as a function of $\langle N_{\rm{part}} \rangle$ [@Abelev:2008aa]. The dashed line shows results from UrQMD model calculations. Right: $\phi/K^{*0}$ ratio in Au+Au, Cu+Cu, and d+Au collisions divided by that in $p+p$ collisions at $\sqrt{s_{NN}}=$ 200 GeV as a function of $\langle N_{\rm{part}} \rangle$ [@Abelev:2008yz].[]{data-label="ratio_phi"}](Fig15copy3.eps "fig:") ![Left: $\phi/K^{-}$ ratio in $p+p$ and various centralities in $d$+Au and Au+Au collisions as a function of $\langle N_{\rm{part}} \rangle$ [@Abelev:2008aa]. The dashed line shows results from UrQMD model calculations. Right: $\phi/K^{*0}$ ratio in Au+Au, Cu+Cu, and d+Au collisions divided by that in $p+p$ collisions at $\sqrt{s_{NN}}=$ 200 GeV as a function of $\langle N_{\rm{part}} \rangle$ [@Abelev:2008yz].[]{data-label="ratio_phi"}](fig11b_n.eps "fig:"){width="0.45\linewidth"}
![Left: $R_{\rm{CP}}$ of $K^{*0}$ as a function of $p_T$ in Au+Au collisions at 200 GeV and 62.4 GeV compared with that of $K^0_S$ and $\Lambda$ at 200 GeV [@Aggarwal:2010mt]. Right: $v_2(\phi)/v_2(p)$ ratio as a function of $p_T$ in Au+Au collisions at two different centralities [@fortheSTAR:2013cda]. []{data-label="rcpkstar_v2phi"}](rcp_kstar.eps "fig:") ![Left: $R_{\rm{CP}}$ of $K^{*0}$ as a function of $p_T$ in Au+Au collisions at 200 GeV and 62.4 GeV compared with that of $K^0_S$ and $\Lambda$ at 200 GeV [@Aggarwal:2010mt]. Right: $v_2(\phi)/v_2(p)$ ratio as a function of $p_T$ in Au+Au collisions at two different centralities [@fortheSTAR:2013cda]. []{data-label="rcpkstar_v2phi"}](phi_pr_ratio_run10.eps "fig:")
![Left: $K^{*0}$ $v_2$ as a function of $p_T$ in minimum bias Au+Au collisions at 200 GeV [@Aggarwal:2010mt]. The dashed lines represent the $v_2$ of hadrons with different numbers of constituent quarks. Right: $p_T$ dependence of $v_2$ of $\phi$, $\Lambda$, and $K^0_S$ in Au+Au collisions (0-80%) at 200 GeV [@Abelev:2007rw]. The magenta curved band represents the $v_2$ of the $\phi$ meson from the AMPT model with a string melting mechanism. The dash and dot curves represent parametrizations for number of constituent quarks, NQ = 2 and NQ=3, respectively. []{data-label="v2kstar_v2phi"}](v2_kstar.eps "fig:") ![Left: $K^{*0}$ $v_2$ as a function of $p_T$ in minimum bias Au+Au collisions at 200 GeV [@Aggarwal:2010mt]. The dashed lines represent the $v_2$ of hadrons with different numbers of constituent quarks. Right: $p_T$ dependence of $v_2$ of $\phi$, $\Lambda$, and $K^0_S$ in Au+Au collisions (0-80%) at 200 GeV [@Abelev:2007rw]. The magenta curved band represents the $v_2$ of the $\phi$ meson from the AMPT model with a string melting mechanism. The dash and dot curves represent parametrizations for number of constituent quarks, NQ = 2 and NQ=3, respectively. []{data-label="v2kstar_v2phi"}](v2_phi.eps "fig:")
![Left: $\Omega/\phi$ ratio versus $p_T$ for three centralities in $\sqrt{s_{NN}} =$ 200 GeV Au+Au collisions [@Abelev:2007rw]. The solid and dashed lines represent recombination model predictions for central collisions for total and thermal contributions, respectively. Right: $p_T$ dependence of the nuclear modification factor $R_{\rm{CP}}$ $\phi$ meson compared with $K^0_S$ and $\Lambda$ in Au+Au 200 GeV collisions [@Abelev:2008aa]. The top and bottom panels present $R_{\rm{CP}}$ from midperipheral and most-peripheral collisions, respectively. []{data-label="omegaphi_rcpphi"}](fig3_OmegaPhi_ratio.eps "fig:") ![Left: $\Omega/\phi$ ratio versus $p_T$ for three centralities in $\sqrt{s_{NN}} =$ 200 GeV Au+Au collisions [@Abelev:2007rw]. The solid and dashed lines represent recombination model predictions for central collisions for total and thermal contributions, respectively. Right: $p_T$ dependence of the nuclear modification factor $R_{\rm{CP}}$ $\phi$ meson compared with $K^0_S$ and $\Lambda$ in Au+Au 200 GeV collisions [@Abelev:2008aa]. The top and bottom panels present $R_{\rm{CP}}$ from midperipheral and most-peripheral collisions, respectively. []{data-label="omegaphi_rcpphi"}](rcp_phi.eps "fig:")
Rescattering effect
-------------------
Figure \[ratio\_kstar\] (left plot) shows ratio of $K^{*0}$/$K^-$ as a function of $dN_{\rm{ch}}/d\eta$, which reflects the centrality [@Abelev:2008yz]. The right plot shows the double ratio i.e. ratio of $K^{*0}$/$K^-$ in heavy-ion collisions over that in $pp$ collisions [@Aggarwal:2010mt]. One can see that the ratio $K^{*0}$/$K^-$ decreases as a function of increasing number of participating nucleons as well as decreases from $pp$, $d$+Au, to central Au+Au collisions. This decrease in the $K^{*0}$/$K^-$ ratio may be attributed to the rescattering of daughter particles of $K^{*0}$.
Figure \[ratio\_phi\] (left plot) shows ratio of $\phi$/$K^-$ as a function of number of participating nucleons [@Abelev:2008aa]. The ratio remains flat as a function of collision centrality, suggesting that there is a negligible rescattering effect for $\phi$. The results are also compared with UrQMD model which assumes kaon coalescence as the dominant mechanism for $\phi$ production. As seen, the data rules out the kaon coalescence as dominant mechanism for the $\phi$-meson production. The right plot shows the double ratio i.e. ratio of $\phi$/$K^{*0}$ in heavy-ion collisions over that in $pp$ collisions. It shows that the ratio increases with increasing $N_{\rm{part}}$ [@Abelev:2008yz]. This increase might be either due to rescattering of daughter of $K^{*0}$ or strangeness enhancement for $\phi$. Since, we already see that $K^{*0}$ shows rescattering effect, this increase is most likely due to the rescattering effect for $K^{*0}$.
Figure \[rcpkstar\_v2phi\] (left plot) shows the nuclear modification factor ($R_{\rm{CP}}$), defined as yields in central collisions to that in peripheral collisions scaled by the number of binary collisions [@Aggarwal:2010mt; @ref_rcp]. At low $p_T$ ($p_T < $1.8 GeV/c), we observe that $R_{\rm{CP}}$ of $K^{*0}$ is less than that of $K^{0}_S$ (also a meson) and $\Lambda$ (having almost similar mass). The lower value of $R_{\rm{CP}}$ of $K^{*0}$ might be due to the rescattering of daughter particles of $K^{*0}$ at low $p_T$ in the medium. The right plot shows the elliptic flow parameter $v_2$ of $\phi$ divided by $v_2$ of proton as a function of $p_T$ for high statistics Au+Au data at 200 GeV for two different centralities [@fortheSTAR:2013cda]. At low $p_T$, we observe that this ratio is not unity. Since the $\phi$ mass is similar to the proton mass, we expect a similar $v_2$ for $\phi$ and proton at low $p_T$ due to mass-ordering. However, data show that the mass ordering is broken at low $p_T$, which might be due to rescattering of protons at low $p_T$ as suggested in Ref. [@hydro_mod].
Quark coalescence and partonic effects
--------------------------------------
Figure \[v2kstar\_v2phi\] (left plot) shows the $v_2$ versus $p_T$ for $K^{*0}$ in Au+Au collisions at 200 GeV [@Aggarwal:2010mt]. The various curves show the number of constituent quarks ($n=$2 for mesons and 3 for baryons). We observe that the $K^{*0}$ $v_2$ follows the $n=$2 parametrization suggesting quark coalescence for their production [@quark_coal]. The right plot shows the $v_2$ versus $p_T$ for the $\phi$ meson compared with $K^0_S$ and $\Lambda$ along with number-of-constituent-quark parameterizations [@Abelev:2007rw]. We see that the $\phi$ meson follows the $K^0_S$ behavior and the $n=$2 curve. Since $\phi$ meson is not formed via kaon coalescence and undergoes less hadronic interaction (as discussed before), the observed $v_2$ of $\phi$ is due to the partonic phase. These results also suggest that heavier quarks flow as strongly as lighter quarks.
Figure \[omegaphi\_rcpphi\] (left plot) shows the $\Omega$/$\phi$ ratio as a function of transverse momentum $p_T$ for three different centralities in Au+Au collisions at 200 GeV [@Abelev:2007rw]. Data are compared with the model calculations which assume that $\Omega$ and $\phi$ are produced from thermal $s$ quarks coalescence in the medium. The results show that the coalescence model reproduces the data at low $p_T$. The right plot shows the nuclear modification factor $R_{\rm{CP}}$ of $\phi$ for 0–5%/40-60% (top panel) and 0–5%/60-80% (bottom panel), compared with $R_{\rm{CP}}$ of different particles [@Abelev:2008aa]. A suppression of $R_{\rm{CP}}$ at high $p_T$ has been suggested to be the signature of dense medium or quark gluon plasma formation in heavy-ion collisions. Together with Fig. \[rcpkstar\_v2phi\] (left plot), above results suggest that both $K^{*0}$ and $\phi$ $R_{\rm{CP}}$ show suppression at high $p_T$, suggesting dense medium formation at the top RHIC energy.
![Left: $v_2/n_q$ as a function of $(m_T-m_0)/n_q$ for different particles in Au+Au collisions at $\sqrt{s_{NN}} =$ 7.7, 11.5, 19.6, 27, 39 and 62.4 GeV [@v2_prl_bes]. Right: The $p_T$ integrated $\phi$ meson and proton $v_2$ for Au+Au minimum bias (0–-80%) collisions at mid-rapidity $|y|<$ 1.0 at RHIC as a function of $\sqrt{s_{NN}}$ [@ref_wp]. The $\phi$ meson $v_2$ values are compared with corresponding AMPT model calculations at various beam energies. []{data-label="v2bes"}](v2_bes.eps "fig:") ![Left: $v_2/n_q$ as a function of $(m_T-m_0)/n_q$ for different particles in Au+Au collisions at $\sqrt{s_{NN}} =$ 7.7, 11.5, 19.6, 27, 39 and 62.4 GeV [@v2_prl_bes]. Right: The $p_T$ integrated $\phi$ meson and proton $v_2$ for Au+Au minimum bias (0–-80%) collisions at mid-rapidity $|y|<$ 1.0 at RHIC as a function of $\sqrt{s_{NN}}$ [@ref_wp]. The $\phi$ meson $v_2$ values are compared with corresponding AMPT model calculations at various beam energies. []{data-label="v2bes"}](avg_phi_pr_v2_comp_model_star_pre.eps "fig:")
![Left: $R_{\rm{CP}}$ of $\phi$ as a function of $p_T$ in Au+Au collisions at various beam energies [@fortheSTAR:2013cda]. Right: Ratio N($\Omega^- + \Omega^+ )/(2N \phi$) as a function of $p_T$ in central Au+Au collisions at $\sqrt{s_{NN}}=$11.5–200 GeV [@ref_wp; @mpla_lok; @fortheSTAR:2013gwa]. The curves represent model calculations by Hwa and Yang for $\sqrt{s_{NN}} =$ 200 GeV.[]{data-label="omegaphi_rcp_bes"}](rcp_phi_energy_dependence.eps "fig:") ![Left: $R_{\rm{CP}}$ of $\phi$ as a function of $p_T$ in Au+Au collisions at various beam energies [@fortheSTAR:2013cda]. Right: Ratio N($\Omega^- + \Omega^+ )/(2N \phi$) as a function of $p_T$ in central Au+Au collisions at $\sqrt{s_{NN}}=$11.5–200 GeV [@ref_wp; @mpla_lok; @fortheSTAR:2013gwa]. The curves represent model calculations by Hwa and Yang for $\sqrt{s_{NN}} =$ 200 GeV.[]{data-label="omegaphi_rcp_bes"}](omega_phi.eps "fig:")
Energy dependence of partonic interactions
------------------------------------------
In the previous subsection, we have established the partonic nature of the system formed at the top RHIC energy. It is interesting to see what happens to this partonic nature when the collision energy is decreased. RHIC Beam Energy Scan (BES) allows to check this energy dependence.
Figure \[v2bes\] (left plot) shows the $v_2$ scaled by the number of constituent quarks (ncq) plotted versus $m_T$-$m$ divided by number of constituent quarks. Results are shown for 7.7, 11.5, 19.6, 27, 39, and 62.4 GeV, and for various particles that include mesons and baryons [@v2_prl_bes]. We observe that all particles follow ncq scaling down to 19.6 GeV. However, at 11.5 GeV and below $\phi$ mesons deviate from this scaling. Since $\phi$ meson has a small hadronic interaction cross-section [@Adams:2005dq], this small $\phi$ $v_2$ may suggest less partonic contributions at lower energies. However, as can been seen, higher statistics are needed at lower energies to make definite conclusions. The right plot shows the the $v_2$ of $\phi$ mesons compared to corresponding AMPT model calculations [@ref_wp]. The $\langle v_2 \rangle$ values from the model is constant for all the energies at a given parton-parton interaction cross-section. This is expected because it is the interactions between minijet partons in the AMPT models that generate $v_2$. The $v_2$ of $\phi$ mesons for $\sqrt{s_{NN}} >$ 19.6 GeV can be explained by the AMPT model with string melting enabled (AMPT-SM). The AMPT-SM model with 10 mb parton-parton cross-section fits the data at $\sqrt{s_{NN}} =$ 62.4 and 200 GeV, whereas a reduced value of parton-parton cross-section of 3 mb is needed to describe the data at $\sqrt{s_{NN}} =$ 27 and 39 GeV. On the other hand, the data at $\sqrt{s_{NN}} =$ 11.5 GeV are explained within the default version of the AMPT model without the partonic interactions. These model results along with data indicate that for $\sqrt{s_{NN}} <$ 11.5 GeV, the hadronic interaction plays a dominant role, whereas above 19.6 GeV contribution from partonic interactions increases.
Figure \[omegaphi\_rcp\_bes\] (left plot) Nuclear modification factor $R_{\rm{CP}}$ (0–10%/40–60%) of $\phi$ meson at different BES energies along with 200 GeV ((0–5%/40–60%) [@fortheSTAR:2013cda]. We observe that $R_{\rm{CP}} \ge $ 1 for beam energies $\sqrt{s_{NN}} \le$ 19.6 GeV. The right plot shows the $\Omega$/$\phi$ ratio versus $p_T$ for different energies from $\sqrt{s_{NN}} =$ 11.5, 19.6 GeV, up to 200 GeV [@ref_wp; @mpla_lok; @fortheSTAR:2013gwa]. We observe that 19.6 , 27 and 39 GeV follow the same behavior as 200 GeV, however, the ratio at 11.5 GeV show different trend i.e. the ratio turns down at lower $p_T$ when compared to higher energies. This may suggest different particle production phenomenon at 11.5 GeV compared to higher energies.
Summary
=======
In summary, $K^{*0}$(892) and $\phi$(1020) resonance production at RHIC is discussed. The $K^{*0}$/$K^-$ ratio decreases as a function of $N_{\rm{part}}$, $R_{\rm{CP}}$ of $K^{*0}$ is less than that of $K^{0}_S$ and $\Lambda$ at low $p_T$, and the $\phi/K$ ratio remains constant as a function of centrality. These results suggest rescattering effect for $K^{*0}$ and may be negligible rescattering effect for $\phi$ mesons. The number-of-constituent-quark scaling is observed for both $K^{*0}$ and $\phi$, possibly indicating partonic nature of system formed at the top RHIC energy. Similarly, the quark coalescence model explains the $\Omega$/$\phi$ ratio at low $p_T$. The energy dependence of various observables suggest that the system formed at lower energies may be hadron dominant. As an example, $\phi$ meson $v_2$ does not follow ncq-scaling for $\sqrt{s_{NN}} \le$ 11.5 GeV. $v_2$ of $\phi$ mesons compared with model results indicate that for $\sqrt{s_{NN}} <$ 11.5 GeV the hadronic interaction plays a dominant role. Energy dependence of $\Omega$/$\phi$ ratio versus $p_T$ suggests a change of particle production at $\sqrt{s_{NN}} =$ 11.5 GeV and the $\phi$ meson $R_{\rm{CP}} \ge $ 1 for beam energies $\sqrt{s_{NN}} \le$ 19.6 GeV.
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---
abstract: 'In opinion polls, the public frequently claim to value their privacy. However, individuals often seem to overlook the principle, contributing to a disparity labelled the ‘Privacy Paradox’. The growth of the Internet-of-Things (IoT) is frequently claimed to place privacy at risk. However, the Paradox remains underexplored in the IoT. In addressing this, we first conduct an online survey (N = 170) to compare public opinions of IoT and less-novel devices. Although we find users perceive privacy risks, many still decide to purchase smart devices. With the IoT rated less usable/familiar, we assert that it constrains protective behaviour. To explore this hypothesis, we perform contextualised interviews (N = 40) with the public. In these dialogues, owners discuss their opinions and actions with a personal device. We find the Paradox is significantly more prevalent in the IoT, frequently justified by a lack of awareness. We finish by highlighting the qualitative comments of users, and suggesting practical solutions to their issues. This is the first work, to our knowledge, to evaluate the Privacy Paradox over a broad range of technologies.'
author:
-
bibliography:
- 'bib.bib'
title: |
“*Privacy is the Boring Bit*”: User Perceptions\
and Behaviour in the Internet-of-Things
---
Privacy, Internet-of-Things, Privacy Paradox
Introduction {#sec:one}
============
Opinion polls and surveys suggest that the public value their privacy [@Rainie2013; @TRUSTe2015]. However, research indicates that individuals often act to the contrary [@Carrascal2013; @Beresford2012]. This apparent disparity between opinions and actions has been labelled the ‘Privacy Paradox’ [@Barnes2006]. While some attribute concerns to social norms [@Fazio2005], others believe cognitive biases [@Acquisti2004a] have an influence.
The Internet-of-Things (IoT) refers to the agglomeration of ‘smart devices’ which increasingly pervade our lives. These networks offer great benefits to productivity, being widely predicted to benefit production. However, despite the appeal of these networks, many have highlighted their threats to privacy [@Abomhara2014; @Williams2016c]. As this field rapidly expands, what does this mean for perceptions, behaviour and the Privacy Paradox?
Thus far the Paradox has been studied in less-novel environments. Whereas the issue is explored on smartphones [@Park2015] and social networks [@Acquisti2006], it is rarely examined in the IoT [@Hallam2016]. Furthermore, Paradox studies have been criticised for comparing abstract concepts with practical behaviour [@Trepte2014a]. Student samples are frequently solicited, with little consideration of real-life scenarios. Without practical analysis of the Paradox, the IoT might place user privacy at risk.
To explore the phenomenon across both IoT and less-novel products, we conducted two detailed studies. Firstly, to compare opinions of a range of devices, we undertook an online survey (N = 170). We sought evaluations before requesting the rationale for product ownership. IoT devices were considered significantly less private/usable, suggesting protection might be constrained. Although most users recognised the risks, many still decided to purchase IoT products.
Intrigued by this potential disparity between opinion and action, we conducted contextualised interviews with the public (N = 40). Rather than comparing the abstract and the practical, we grounded discussions around each participant’s device. 1/3 of our respondents displayed an opinion-action disparity, suggesting the presence of the Paradox. While some non-IoT owners acted in this manner, the disparity was significantly more prevalent in IoT users [@Williams2017]. We hypothesise this to be due to reduced awareness, with this rationale predominantly given by participants. We finally proposed solutions aligned with these justifications, such as IoT educational campaigns.
Our work is the first to analyse the Privacy Paradox across such a range of devices. We are also the first to compare privacy perceptions between the IoT and less-novel products. Rather than studying student-composed convenience samples, we dissect the privacy rationale of the general public. Our work offers novel insights into the Privacy Paradox, and provides practical solutions to reduce its prevalence.
This paper is structured as follows. Section \[sec:two\] reviews literature on the Paradox and privacy decision-making. Section \[sec:three\] details our methodology, before Section \[sec:four\] reflects on the findings. Section \[sec:five\] introduces our contextualised interviews, followed by the discussion in Section \[sec:six\]. We conclude the paper in Section \[sec:seven\], highlighting limitations and further work.
Background and Related Work {#sec:two}
===========================
The Privacy Paradox
-------------------
While the principle of privacy is widespread, it is also cultural and subjective. With the concept being highly contextual [@Nissenbaum2009], people might value privacy in one situation but not another. Clarke [@Clarke1999] defined information privacy as, “*the interest an individual has in controlling, or at least significantly influencing, the handling of data about themselves*” [@Clarke1999]. While we concern this domain in our work, people might have varying views of privacy in other contexts.
Opinion polls suggest that the public care about privacy. 86% of US respondents reported taking steps to protect themselves [@Rainie2013], while 88% in a UK study claimed to value the principle [@TRUSTe2015]. Despite these assertions, individuals often express behaviour to the contrary. Carrascal et al. [@Carrascal2013] used an auction to assess the value placed on personal data. They found users would sell their browsing history for €7, suggesting a lack of concern. Beresford et al. [@Beresford2012] varied the prices of two online stores to explore privacy valuation. They discovered that when the intrusive store was €1 cheaper, almost every user selected it. Although people might claim to be concerned about privacy, their behaviour can often appear misaligned.
This disparity between opinion and action has been labelled the ‘Privacy Paradox’ [@Barnes2006]. Decision-making is also dissected through ‘Privacy Calculus’ [@Dinev2006], where disclosure benefits and risks are logically compared. However, Acquisti [@Acquisti2004a] prefers ‘bounded rationality’ to explain behaviour, noting that decisions are constrained by cognitive biases. In a 2017 review, Barth and Jong [@Barth2017] also concluded that irrationality is present. With the phrase ‘Privacy Paradox’ under dispute, we prefer ‘disparity’ to describe a discrepancy between privacy opinions and actions. This is similar to the concept of the ‘attitude-behaviour gap’ found in psychological research [@Fazio2005].
Privacy Decision-Making
-----------------------
Privacy decision-making has been analysed through many studies. Acquisti and Grossklags [@Acquisti2005b] rejected perfect rationality, instead considering the role of cognitive biases. They conducted a 119-person survey, identifying a disparity between concerns and behaviour. As most users could not assess their risk, they concluded that a lack of awareness was influential.
Dinev and Hart [@Dinev2006] developed the Extended Privacy Calculus Model, analysing the balance between risks and incentives. Through surveying 369 participants, they confirmed that perceived risks led to a reluctance to disclose. Xu et al. [@Xu2009] investigated location-based services and the factors which influence privacy decisions. Through their survey, they found compensation increased perceived benefits while regulation reduced perceived risks. While these works are purely quantitative, we follow a mixed-methods approach. Furthermore, while they have relevance to our research, they do not concern the IoT. As this field differs in terms of usability [@Foster2016] and ubiquity, decisions might differ from those on familiar systems.
Although the Privacy Paradox is rarely studied in the IoT, Hallam and Zanella [@Hallam2016] did consider wearable devices. They constructed a self-disclosure model before validating it through an online survey. They found that behaviour was more driven by short-term incentives than long-term risk avoidance. Li et al. [@Li2016] studied Privacy Calculus in wearable healthcare products. Through surveying 333 users, they found that adoption increased as functionality outweighed sensitivity.
While the Paradox is not considered, other work explores IoT privacy. Wieneke et al. [@Wieneke2016] studied wearable devices and how privacy affects decisions. Through 22 interviews, they found individuals had little awareness of data sharing. Most also claimed risk did not impact their choices, which might suggest an opinion-action disparity. Lee et al. [@Lee2016b] surveyed 1,682 users on their wearable perceptions. Participants indicated their concern following privacy infractions by a hypothetical product. They found preferences correlated with reactions, even in unfamiliar situations. While these studies analyse wearable devices, we explore a variety of technologies.
Kowatsch and Maass [@Kowatsch2012] developed a model to predict IoT disclosure intention. They conducted surveys with 31 experts, finding usefulness the only factor to consistently encourage usage. Yang et al. [@Yang2017] also considered how concerns affect smart home adoption. The authors developed a theoretical model and validated it through a 216-person survey. They found that while privacy risks limit adoption, trust can counteract the effect. Whereas these studies explore a few scenarios, we examine a range of IoT and less-novel products. This enables analysis of how technology influences behaviour. With the IoT proliferating, it is crucial we ascertain its influence on privacy.
Online Survey Methodology {#sec:three}
=========================
Research Hypotheses
-------------------
Before we describe our methodology, we must outline our research hypotheses. These can be found below in Table \[tbl:hypotheses1\]. They were based on our research goal: to explore the Paradox across IoT and less-novel environments. To ascertain high-level opinions, we designed a public online survey. Rather than solely analysing privacy, we explored other factors which could have an influence. For example, less usable or (less) familiar devices might constrain protective behaviour [@Whitten1999]. Therefore, as outlined in Section \[sec:evalfact\], we asked respondents to evaluate four factors: *privacy*, *familiarity*, *usability* and *utility*. This enabled us to compare opinions of IoT and non-IoT products, with device selection described in Section \[sec:devsec\].
\# Research Hypothesis
-------------- ---------------------
H1
\[2.5ex\] H2
\[2.5ex\] H3
\[2.5ex\] H4
: Online Survey Research Hypotheses
\[tbl:hypotheses1\]
Since studies suggest smart devices could impact privacy [@Abomhara2014; @Williams2016c], we believed non-experts would share this opinion. We therefore asserted that IoT products would be rated less privacy-respecting than non-IoT technologies (**H1**). With smart devices being heterogeneous [@Bandyopadhyay2011] and novel, we posited these technologies would also be less familiar (**H2**). This has particular risk for privacy behaviour, as users might be less able to use protection [@Whitten1999]. As IoT interfaces are often criticised [@Foster2016], we asserted they would be rated less usable (**H3**).
Following the factor ratings, we queried participants on whether they owned the device and why. This qualitative justification sought to identify factors influencing ownership decisions. While we believed the IoT would be considered less private, we doubted this would reduce its popularity. Therefore, we posited that this disparity between opinion and purchasing action would be more prevalent in the IoT (**H4**).
Survey Design
-------------
We chose to begin with an online survey, enabling the analysis of public opinion. Being directed by our high-level findings, we then explored in depth through qualitative interviews. The questionnaire was advertised via Twitter and national/international message boards. Such boards included DailyInfo, GumTree and The Student Room. These fora were selected as we wished to canvas non-expert opinions. No screening criteria were applied, other than the participants being adults. The questionnaire was iteratively refined, with face validation received from privacy and psychometric experts. We sought to disguise an IoT/non-IoT comparison, framing the theme as general technology. We then performed a small pilot test, before the survey was undertaken from Sept to Nov 2016. The form was composed of demographics and factor ratings, with these components discussed in the following subsection.
Demographics and Factor Ratings {#sec:evalfact}
-------------------------------
We solicited gender, age and highest education level. As research [@Sheehan1999] suggests women possess larger privacy concerns than men, we explored whether privacy ratings varied similarly. It is also reported that older people care more about privacy [@Han2002], and this could be reflected in conservative evaluations. Previous work found that education correlates with privacy concern [@ONeil2001], and this could influence our ratings.
Ratings were made from 0 (low) to 5 (high) on an ordinal scale. This scale was selected for simplicity to aid our non-expert audience. As previously mentioned, these factors were *privacy*, *familiarity*, *usability* and *utility*. We chose these non-privacy attributes both to disguise survey purpose and for their aforementioned interest to the study. We chose against including factor definitions, as we wished to explore the unbiased opinions of our non-expert participants. We did substitute ‘utility’ for ‘usefulness’ on the form, as we believed this synonym to be more comprehensible.
Device Selection {#sec:devsec}
----------------
Through our above factors, we compare smart devices with less-novel alternatives. However, with the IoT being nebulous, we constrained our scope. We chose to select six technologies: three IoT and three non-IoT. These labels are not a strict dichotomy; there is a spectrum ranging from novel mobile products to familiar desktop computers. However, to enable a comparison between groups, we selected archetypal products.
Since we sought public opinion, we constrained our focus to consumer devices. We then specified three criteria to aid selection: novelty, ubiquity and autonomy. These were chosen as IoT products are typically modern, ubiquitous and autonomous. Whereas PCs are well-established, the IoT has flowered in the past decade (novelty). Although laptops reside in many houses, they do not pervade like ‘smart homes’ (ubiquity). Finally, older products are typically user-dependent, while the IoT often interacts with its surroundings (autonomy).
By plotting products against novelty, ubiquity and autonomy, we identified which devices fell into which group. Desktops and laptops appeared non-IoT: both are over 20 years old; both require input; and neither would be considered a Ubicomp device. While tablets do have greater portability, they possess similarities to a keyboardless laptop. Since technology research firms [@Rivera2013; @Duffy2014] also judge these products as distinct from the IoT, we are confident in our categorisation. Furthermore, smart products often require human-free interaction [@Rouse2016], which is rarely supported by desktops, laptops or tablets.
Wearables (e.g., Fitbit) have achieved recent success, are highly mobile and use autonomous sensors. Smart appliances, such as connected fridges, are also novel and communicate through online interaction. Home automation systems (e.g., Google Nest), while static, are highly pervasive and react to their environments. We therefore compared (*desktops, laptops, tablets*) with (*wearables, smart appliances, home automation*). Although definitions were not provided (as a means of disguising the IoT/non-IoT comparison) we included images of relevant devices. These products originated from a range of manufacturers to reduce bias from brand predilections. While some products sit between categories, such as the Microsoft Surface, such examples are rare. Furthermore, although diversity exists within groups, the distinction across categories is generally greater. While a Fitbit differs from an Apple Watch, they largely support similar functionality.
Response Bias Mitigation
------------------------
Since self-reporting surveys face a number of risks, we sought to mitigate response biases. Privacy concerns can become inflated if the topic is salient [@Rajivan2016]. Therefore, we disguised the subject through a generic survey with a range of factors. As acquiescence bias can lead participants to agree with researchers, we avoided yes/no questions. While later ratings might be made relative to earlier scores, we shuffled categories to mitigate the effect. To both allay concerns and reduce non-response bias, we treated data anonymously and received ethical approval. This was important, as otherwise those most concerned about privacy might avoid the study.
Survey Results and Discussion {#sec:four}
=============================
Participants and Techniques
---------------------------
We collected 170 responses with 57% male and 43% female. 50% came from the 26-35 age group, reflected in our estimated mean of 32. 36% of participants possessed a Master’s degree, implying a well-educated respondent group.
For correlation, we analysed the Spearman’s Rank-Order Correlation Coefficient ($r_s$). We used this technique as our variables were ordinal and varied monotonically. To perform significance testing on two independent samples, we used the Mann-Whitney U Test. We selected this as our dependent variables were ordinal and our independent variables were nominal. When comparing two related samples, we chose the Wilcoxon Signed-Rank Test. This was used as our dependent variables were ordinal and our independent variables had related groups. In all cases, we required *p*-values <0.05 for significance. We discuss three opinion variables: the mean *privacy* rating, the mean *familiarity* rating and the mean *usability* rating. We use $\bar{x}$ to denote means, $r_s$ for correlation coefficients and include *p* when differences are significant.
Demographic Analysis
--------------------
While women rated technologies less privacy-respecting than men, this difference was not significant. This might have been due to our sample size, or because opinions are unformed for unfamiliar products. We found age was significantly negatively correlated with privacy ratings ($r_s$ = -0.232, *p* = 0.002), implying older people express greater concern. This might be a generational issue, as older individuals did not grow up with new devices. While we found the highly-educated did rate products less privately, the correlation was not significant. Again, this could be due to the unfamiliarity of IoT products.
Factor Comparisons
------------------
To understand how opinions differ, we compared factor ratings across our surveyed devices. We first calculated the mean score of each factor for each product. We then performed significance testing to investigate our hypotheses.
Laptops ($\bar{x}$ = 3.27) were rated most private, with wearables ($\bar{x}$ = 2.31) regarded as most privacy-concerning. Generally, we found IoT devices were rated significantly less privacy-respecting (*Z* = -5.151, *p* <0.001), confirming our hypothesis (**H1: Accept**). This might be due to fear of the unknown, or because IoT products collect a range of data. If the public recognise the risk, it implies security awareness is increasing.
Laptops ($\bar{x}$ = 4.72) were also found most familiar, with home automation receiving the lowest score ($\bar{x}$ = 1.45). Individuals were significantly less-accustomed to IoT devices (*Z* = -11.103, *p* <0.001), which we imagine to be due to their current novelty (**H2: Accept**). Low familiarity could lead to privacy risks, with users less aware of protective techniques [@Whitten1999]. If these technologies are considered alien, this implies that the IoT market is still nascent. We would expect such devices to become better-known as their sector expands.
Laptops were considered most usable ($\bar{x}$ = 4.55), with wearables faring the worst ($\bar{x}$ = 2.52). IoT products were considered significantly less-usable than their counterparts (*Z* = -10.332, *p* <0.001), confirming our hypothesis (**H3: Accept**). This could be because the gadgets are less-understood, or because they often possess small screens. With the IoT rated less usable and (less) familiar, protection might be impeded [@Whitten1999]. The mean factor ratings are presented below in Figure \[fig:chart\].
![Device mean factor ratings: Privacy (PRV), usability (USB), familiarity (FAM) and utility (UTY).[]{data-label="fig:chart"}](chart.png){width="46.00000%"}
Ownership Decisions
-------------------
With IoT products considered less-private (H1), we investigated whether this affected purchasing decisions. For each device, we compared privacy ratings to ownership frequency and user justifications. If a person recognises the risks but still purchases the product, then a disparity might be present.
Laptops were most popular, with 93% claiming ownership. Mobility was the most liked feature, with no privacy concerns expressed. Tablets were owned by 66%, with ownership mainly justified on mobility. Only one person criticised privacy, with them denouncing a lack of settings customisation. Desktops were owned by 52%, with 62% of those praising the product’s functionality. Again, not a single person criticised privacy. This implies that the topic is rarely influential when users purchase computers.
Smart appliances were owned by 42%, with functionality the most popular feature. 9% rejected because of privacy, with these participants worried about monitoring. Despite this, more than 3 times as many were deterred by price (28%). Only 21% owned wearables, with functionality again the main attraction (63%). Although they received the lowest privacy ratings, only 3% of rejections cited this reason. Again, far more were deterred by the cost (20%). Of those 12% with home automation, only 8% of them criticised privacy. While they disliked remote infiltration, again, far more blamed the price (23%). Since prices decrease as products mature, this suggests the IoT will grow in popularity. Therefore even if privacy becomes salient, cheap gadgets might remain attractive.
The Opinion-Action Disparity {#sec:opacsurv}
----------------------------
We now move forward to compare privacy ratings with purchasing decisions. If individuals buy a device despite recognising the risks then the disparity might be present. We take ratings of 1/5 or below to indicate criticism, as 2/5 (or 3/5) could be deemed a cautious evaluation. In this manner, we seek to place a minimum bound on disparity prevalence.
In the case of non-IoT technologies, the disparity was far from common. Of the 89 who bought a laptop, only 7 gave a low privacy rating (7.87%). For tablets the figure was 10.71%, with desktop disparities even less common (8.23%). On average, 8.91% purchased a non-IoT product despite perceiving a risk. These individuals may have felt constrained by the PC market, with operating systems developed by a small number of vendors. Therefore, even if users object to a brand’s privacy practices, they have few alternatives to choose from.
The disparity was more prevalent for IoT devices. Of those who purchased home automation systems, almost 10% rated privacy poorly. 9/71 smart appliance owners criticised privacy (12.68%), with wearables performing the worst (17.14%). Across all IoT owners, this resulted in an average of 14.96%. These purchases might be made because consumers value functionality over data privacy. Alternatively, owners might have inconsistent preferences and detach their opinions from their actions. In either case, this implies that a subset of individuals are willing to sacrifice their privacy. This is concerning as while employees might require a desktop/laptop for work, IoT purchases are largely voluntary. In this manner, privacy is sacrificed for entertainment rather than necessity. Although IoT users were 68% more likely to present this disparity, there was no significant difference between our groups (**H4: Reject**). With a *p*-value of 0.056, our sample may have been too small to confirm the difference.
Findings and Implications
-------------------------
We found the IoT is regarded as less privacy-respecting (H1), less familiar (H2) and less usable (H3). Since confusing and unfamiliar interfaces are harder to use, data protection might be impeded [@Whitten1999]. Ownership justifications imply privacy is rarely considered, and this could contribute to unwise purchases. If the topic is not salient, users might place themselves at risk. While our findings hinted at constrained action, this could not be confirmed without additional data. As decision-making was opaque in these quantitative results, we required detailed discussions. We therefore undertook qualitative analyses to dissect the decision-making rationale. In this manner, we can compare opinion and action to explore the disparity.
Contextualised Interview Methodology {#sec:five}
====================================
Research Hypotheses
-------------------
Again, we begin this section by introducing our hypotheses. These can be found below in Table \[tbl:hypotheses2\]. As described in the following sections, we conducted contextualised interviews with a non-expert public. In these discussions, we solicited both participants’ opinions and their reported behaviour. These responses were then codified and quantified, resulting in the metrics described in Section \[sec:interviewdataanalysis\]. This enabled direct comparison between individuals’ privacy opinions and their actions. In this manner, the prevalence of the disparity could be evaluated across both IoT and less-novel environments.
\# Research Hypothesis
-------------- ---------------------
H5
\[2.5ex\] H6
\[2.5ex\] H7
: Contextualised Interview Research Hypotheses
\[tbl:hypotheses2\]
As more IoT users bought ‘risky’ devices, we posited such owners would care less about their data. Furthermore, since privacy rarely featured in ownership decisions, this suggests the topic is infrequently considered. We asserted that this would result in lower quantified opinion scores (**H5**), with individuals showing less concern. This comprised the opinion component of the opinion-action disparity. As our survey suggested the IoT is less familiar (H2) and (less) usable (H3), we posited users would also be less able to protect themselves. Smart devices are heterogeneous and novel, potentially challenging mental models. This might result in lower quantified action scores (**H6**), with users doing less to protect themselves. This comprised the action component of the disparity.
As IoT owners displayed the disparity frequently in the survey, we asserted that it would be prevalent in the interviews. While their privacy expectations might be lower, we contend that their behaviour is disproportionately constrained. If true, this would lead to an increased discrepancy between perceptions and behaviour. We therefore posit that the opinion-action disparity (the Privacy Paradox) will be more likely in the IoT (**H7**), with metrics outlined in Section \[sec:interviewdataanalysis\].
Interview Design
----------------
With our survey suggesting a potential disparity, rich data was required for further investigation. Therefore we designed interviews to discuss privacy rationale. As we wished to explore the wider applicability of the disparity, we approached a distinct sample of the general public. If a subset of these also display the disparity, then the phenomenon might be common.
To compare IoT and non-IoT owners, we recruited two distinct groups. These were divided based on the survey categories, enabling analysis of whether the same dichotomy exists. Both groups faced the same questions, with only the device name customised in our between-subjects format. Participants were screened for adults who owned a device in one of our six categories. Recruitment was undertaken via Twitter and a local messaging board, ensuring our respondents did not comprise a student sample. Interviews were conducted one-on-one in a seminar room, with informed consent received at the start. Monetary compensation was offered to incentivise participation, and the study was approved by our IRB board.
If participants believe their privacy perceptions are being evaluated, they might adjust their responses [@Rajivan2016]. Therefore, our interview was framed as concerning general opinions. More-overt questions were also placed near the end to ensure earlier responses were not primed. To minimise any deception, the true purpose was revealed at the end of each interview.
We sought to overcome the criticisms of previous Privacy Paradox studies [@Trepte2014a]. Firstly, rather than comparing abstract concepts against practical actions, we grounded our interviews around owned devices. Participants were then able to draw on their personal experiences to answer in a more-informed manner. With privacy being highly contextual, this enabled opinions and actions to be fairly compared. Secondly, instead of considering ‘privacy’, we solicited qualitative reactions to specific incidents. Rather than discussing the nebulous principle, as has been criticised by previous work [@Dienlin2015], we constrained our focus to informational privacy. Thirdly, our interviews were conducted with the public, as opposed to student-composed samples. This should lead our findings to be more-representative of non-expert users.
Fourthly, we discussed protective actions (described below) that were both practical and feasible. While few non-experts use Tor, passwords and settings can help ordinary users. Finally, we considered the rationale behind decisions, rather than just the decisions themselves. If a password is neglected because the data is thought trivial, then the user is not necessarily careless. If despite these controls, they express concern but take no action, we argue a disparity is present.
Interview Questions
-------------------
We first received face validation from a privacy and psychometric expert. We then conducted a pilot study, granting an opportunity to test our questions. Following interviews with 10 individuals, we found our sequence primed privacy. After moving our action queries to later in the session, the topic appeared better disguised. While our interview questions were broad, they were chosen to solicit open-ended comments from non-expert users. A more prescriptive approach might have channelled responses, but also constrained the diversity of replies. Privacy Paradox studies have been criticised for comparing abstract opinions with specific circumstances [@Trepte2014a]. For example, while a person might value privacy, this may bear no relation to their Facebook usage. To ensure that opinions and actions are comparable, we contextualised our questions around a participant’s device. For example, if they owned a Fitbit, all queries concerned their use of that Fitbit.
Questions were of four types: *General (G)*, *Opinion (O)*, *Action (A)* and *Disparity (D)*. These queries can be found below in Table \[tbl:questions\]. *General* questions had two roles: to solicit broad opinions and to disguise the topic of privacy. Although our *General* questions led to intriguing findings, in the interest of brevity, we scope to our other results. *Opinion* queries were used to investigate privacy perceptions. Incidents were selected from the archetypal privacy violations found in Solove’s taxonomy [@Solove2008]. Disclosure and surveillance are both comprehensible ways in which privacy can be violated. Data selling encapsulates the secondary use violation, while unauthorised deletion represents an intrusion into solitude.
\# Interview Question
------------ --------------------
O1
\[2ex\] O2
\[2ex\] O3
\[2ex\] O4
\[2ex\] A1
\[2ex\] A2
\[1ex\] A3
\[2ex\] D1
\[2ex\] D2
\[2ex\] D3
: Contextualised Interview Questions (General Questions excluded for brevity)
\[tbl:questions\]
*Action* questions queried how participants actually use their devices. Protective measures were selected based on three criteria: simplicity, utility and applicability. Techniques must be easy to apply, as we should not expect non-experts to install complex software. Measures must also be beneficial by granting an opportunity for greater knowledge or control. Finally, techniques must apply to both IoT and non-IoT devices to enable a fair comparison. Passwords, privacy policies and privacy settings are all of use, widespread and well-known. Therefore, we avoid comparing opinions against impractical actions. While opaque policies frequently lack usability, they still offer an opportunity to discover device practices.
In addition to assessing the disparity’s existence, we explored privacy rationale. To avoid priming the topic, the *Disparity* questions were placed at the end of the interview. We believed disparity-prone individuals might respond defensively if directly queried on the topic. Therefore, we phrased questions in terms of why other people might act in this manner. While answers were likely to still correspond with their rationale, we avoided antagonising our respondents.
Interview Results and Discussion {#sec:six}
================================
Participants
------------
We conducted 40 contextualised interviews between January and February 2017. 60% were male and 40% were female, closely corresponding with the 57%/43% split in our survey. Respondent ages were also similar, with 45% in the 26-35 group and an estimated mean of 31.6. Educational levels were again relatively high as 53% possessed a Master’s degree.
Data Analysis {#sec:interviewdataanalysis}
-------------
We manually transcribed our recordings, resulting in a transcript for each discussion. We then conducted thematic analysis, labelling responses under a range of codes. Rather than simply noting the answer, these codes also encapsulated the justification for the decision. After all transcripts were reviewed, categories were developed to ensure consistency. For example, privacy policy codes ‘*Did Not Read, Jargon*’ and ‘*Did Not Read, Legalese*’ were categorised under ‘*Did Not Read, Complex*’. Where justifications were clearly distinct, they were preserved to ensure a diversity of views.
In terms of opinions, the intensity of reaction was grouped under ‘*Indifference*’, ‘*Slight Dislike*’, ‘*Dislike*’ or ‘*Strong Dislike*’. While a ‘*Like*’ category was envisaged, none of the participants expressed this reaction. These groups were used to distinguish between those who felt inconvenienced and those who showed strong opposition. Actions were split between ‘*Did*’ and ‘*Did Not*’ unless a ordinal scale appeared necessary. For example, as a sizeable proportion of participants skimmed their policies, responses were divided ordinally between ‘*Did*’, ‘*Briefly*’ and ‘*Did Not*’. In seeking to minimise subjectivity, categorisation was refined to ensure group consistency.
To assess the disparity at an individual level, we quantified opinions and actions. Whereas a comparison could be made qualitatively, we believed this approach would be too subjective. Opinions were scaled from 1/5 (low) to 5/5 (high) based on concern intensity and justification. For example, a person with ‘*Strong Dislike*’ towards deletion, surveillance and selling would receive 5/5. If concerns were contingent on a particular factor, such as high sensitivity, the score was reduced.
We used a similar scale for privacy actions, assessing whether participants set passwords, read policies and configured settings. Their rationale was also considered, as a person might reject a password for trivial data. In these cases their action score was increased. These adjustments sought to place a minimum bound on disparity prevalence.
To identify disparities, we judged whether the opinion and action scores were commensurate. As both question sets were contextualised around the same device, we ensured a correspondence between scores. Furthermore, the actions could be used to directly address the hypothetical violations. For example, passwords can reduce deletion risk, while policies outline how devices are monitored. If users claim concern but take little protective action, a disparity might exist.
Considering the 5-point scale, we defined a disparity as when the action score was at least 2 points less than the opinion score. We did not believe 1-point differences signified a dissonance, but thought a 3-point definition was too extreme. If a respondent strongly objects to threats (5/5) but merely glances at policies and settings (3/5), then their behaviour might be deemed unwise. Similarly, if a person exhibits reasonable concern (3/5) but takes no action (1/5), then they might also be at risk. As we controlled for contingent concerns, we placed a minimum bound on disparity prevalence.
We continued to use the Mann-Whitney U Test to compare ordinal variables between our participant groups. When responses were binary (nominal), such as ‘*Set Password*’ and ‘*Did Not*’, the Chi-Square Test was used instead. When analysing the correlation between ordinal data, we continued to study the Spearman’s Rank-Order Correlation Coefficient ($r_s$). In all cases we required *p*-values <0.05 for significance. As we compare distinct variables once each, we do not expect to be affected by the Multiple Comparisons Problem.
Participant Opinions
--------------------
Opinion questions concerned data deletion, unauthorised sharing, surveillance and data selling. Most participants objected to deletion, with 73% expressing a dislike for the scenario. This implies that individuals generally feel some sense of ownership over their data. However, we discovered IoT product owners cared significantly less about the issue (*U* = 121, *p* = 0.033). Smart device data was often perceived as low in value (expressed by participants including \#18, \#31 and \#34), as shown below. This is concerning, as while some data is trivial, home occupancy metrics can be revealing. Furthermore, GPS data from wearables might reveal where a person lives or works.
“*I wouldn’t be too fussed, there isn’t a whole lot on there that I’m particularly dear to. It’s just settings and stuff like that, nothing to worry about*” (\#34, IoT)
In terms of unauthorised sharing, 78% either disliked or strongly disliked the practice. This implies that despite the popularity of sharing content, people want agency over this process. IoT owners cared significantly less about unauthorised sharing (*U* = 81, *p* = 0.001), suggesting a dichotomy in privacy opinions. Smart device users often cited a lack of data sensitivity (\#4, \#21, \#35), whereas non-IoT owners were troubled by an absence of control (\#12, \#17, \#37). While IoT metrics might not appear sensitive, users may not have knowledge of advanced inference techniques [@Creese2012].
“*Just because it’s only activity, it’s only what I get up to, I don’t see it as a secret*” (\#35, IoT)
Both groups strongly rejected surveillance, with 85% of IoT device users objecting to monitoring. This implies that consumers still criticise the notion of supervision. This is in conflict with modern wearables, as many of these track GPS. Whereas many non-IoT respondents rejected surveillance on principle (\#2, \#9, \#14), IoT users expressed some concern over tracking (\#15, \#23, \#35). With many smart devices offering location services, digital stalking can be a real possibility.
“*I’d feel like, like someone would maybe be stalking me which would be a bit unnerving*” (\#35, IoT)
Data selling was also met with widespread condemnation. 83% at least disliked the practice, with 30% expressing strong objections. Despite the prevalence of data markets, this implies consumers still reject this custom. With information frequently sold by technology firms, users might be unaware how common this practice is. Whereas non-IoT participants were concerned by a lack of consent (\#8, \#19, \#32), smart device users wanted money from the transaction (\#5, \#18, \#36). This suggests IoT owners have a greater understanding of how data is monetised.
“*I would also be angry because I should get part of the share of the money*” (\#36, IoT)
We found that 60% expressed strong privacy opinions, being scaled to either 4/5 or 5/5. This implies that the public still claim to value this threatened principle. However, IoT users were found to have significantly lower privacy concerns (*U* = 127.5, *p* = 0.049) (**H5: Accept**). This confirms our hypothesis that IoT owners appear to care less about their data. From our qualitative justifications, this often appears due to the data being considered less important. Although data can appear trivial, users might not understand the inferences that can be made [@Creese2012]. Therefore, non-expert owners might unwittingly place their privacy at risk.
Participant Actions
-------------------
Since our survey suggests the IoT is less familiar (H2) and (less) usable (H3), user protection might be constrained. Privacy behaviour was gauged on whether users set passwords, read their devices’ privacy policies and configured their devices’ privacy settings. If an individual reads their policies and adjusts their settings, they arguably behave more privately than someone who ignores these opportunities.
Password protection was far from perfect, with only 58% securing their products. We found passwords were used significantly less often on smart devices ($X^2$(1) = 14.11, *p* <0.001). With these products usually connected to the Internet, users might place their data at risk. Inconvenience played a large role, with PINs often reducing usability (\#13, \#20, \#21). This justification indicates that while users might know about passwords, they opt for convenience. This presents a direct trade-off between utility and privacy/security. Modern wearables also face an increasing theft risk, and unsecured interfaces will only encourage this threat.
“*I just want to swipe it, yeah. It just takes too much time to get in there*” (\#21, IoT)
In general, only 13% studied privacy policies in detail, with 65% avoiding the text. This implies a large number of consumers are held to terms of which they have no knowledge. Users might criticise practices as unconsented, but actually agree to them through opaque policies. Again, we found the IoT group was significantly less interested in the documents (*U* = 117.5, *p* = 0.041). Smart device owners often found functionality more exciting, deeming policies a low priority (\#23, \#28, \#40). Such an attitude contributes to users being unaware of data collection. If consumers are preoccupied with novel features, they might treat privacy as an afterthought.
“*I think I was more in a hurry to get it out of the box and set up and start using it*” (\#40, IoT)
Settings adjustment was varied, with 35% fully configuring and 33% taking no action. This implies that while some are eager to adjust their devices, many rely on defaults. Displaying the contrast between groups, IoT users also configured their settings significantly less often (*U* = 127.5, *p* = 0.049). This was frequently justified through a lack of awareness (\#21, \#23, \#25), or because product functionality was considered more exciting (\#27, \#29, \#36). Once again, this displays a trade-off between privacy and utility. Since default settings are often permissive, IoT users might be leaking data. If individuals perceive privacy as boring, they may avoid protection and place themselves at risk.
“*I just want to explore the functions and interesting bits not the privacy bit, privacy is the boring bit*” (\#36, IoT)
Of the 40 participants, 68% had their actions scaled to 3/5 or above. This was greater than the 60% for opinions, suggesting that some users are more private than they claim. However, the IoT mean was again significantly lower than that of the non-IoT group (*U* = 79.5, *p* = 0.001) (**H6: Accept**). This confirms our hypothesis that IoT owners do less to protect their data. Our justifications suggest this is due to both a preoccupation with functionality and a lack of awareness. As highlighted in Büchi et al.’s 2017 work [@Buchi2017], if users do not understand protection, then they cannot guard their data.
The Opinion-Action Disparity {#the-opinion-action-disparity}
----------------------------
While the IoT group does less on average, we must consider individual cases to identify disparities. 13/40 participants displayed a 2-point difference between opinion and action (33%). With these individuals recruited from a non-expert general public, this implies that the disparity might be prevalent. Furthermore, with our sample disproportionately-educated, this may be a minimum bound. While 23% of these owned less-novel devices, 77% possessed IoT products. Accordingly, we found IoT owners are significantly more likely to display the disparity ($X^2$(1) = 5.584, *p* = 0.041) (**H7: Accept**). This confirms our hypothesis and suggests that IoT products might exacerbate the Privacy Paradox. If so, privacy might be placed at risk as smart devices proliferate. The distribution of opinions and actions are displayed below in Figure \[fig:map\]. As the figure suggests, IoT users are more likely to have strong concerns but take little action. With the most protected participants using non-IoT products, smart devices may be a constraint.
![Participant privacy opinion-action distribution: The shaded red area highlights where there is a disparity between privacy opinion and action.[]{data-label="fig:map"}](distribution2.png){width="46.00000%"}
If these technologies are more likely to support the disparity, why is this the case? With both concern and protective behaviour reduced in the IoT, one would expect a similar disparity prevalence. However, although IoT users often regarded their data as trivial, they still objected to privacy violations. In seeking to mitigate the issue, we explored rationale in greater detail. Through our final three *Disparity* questions, we triangulated why individuals might act in this manner. Although these queries referred to other people, 77% of disparity-prone respondents made reference to themselves.
If individuals are not aware their privacy is at risk, they cannot protect themselves [@Acquisti2005b]. Even if they have some knowledge of the threat, they cannot guard their data if they cannot use the system. Disparity-prone participants cited lack of awareness six times (\#19, \#21, \#27, \#32, \#33, \#35), as did 28/40 respondents. Representative quotes are presented below:
“*If that was me, I wouldn’t realise until somebody said ‘you do realise that this is open to everybody’, I’d be like ‘oh no’ and I would change it*” (\#21, IoT)
“*And certainly just undertaking this interview highlighted to me in ways which I may be risky*” (\#27, Non-IoT)
It is often considered a social norm to value privacy, whether or not one’s actions match their claimed concern [@Fazio2005]. Even if one does not care about their data, there is social pressure to desire privacy. When participants were asked why the disparity exists, social norms were mentioned most frequently (12/40). This reason was salient with disparity-prone users, mentioned by the 33% who referred to themselves (\#1, \#23, \#25).
“*There’s certainly a cultural norm of saying privacy is important, which maybe doesn’t always translate into reality or action*” (\#23, IoT)
“*Well its socially unacceptable to say ‘oh I don’t care about privacy at all’, and therefore you want to say that you do care about privacy, but in fact you’re not doing very much*” (\#25, IoT)
Individuals might understand the risks of an action, but do it anyway due to short-term necessity [@Hallam2016]. For example, although public Wi-Fi can be insecure, a person might still use it to send an urgent email. Security fatigue [@Furnell2009] describes the cognitive load users face in following security, and a similar concept might exist for privacy. While privacy can still be aspired to as a principle, it is often sacrificed due to practical necessity. This justification was offered frequently by our respondents (\#11, \#18, \#40).
“*If you need a service and you’re in a rush and you need to get something done really quickly, you don’t really give a s\*\*t about the privacy bit*” (\#18, IoT)
“*If you’re travelling, sometimes you might have to use a laptop like in a café or in a hotel or something like that which I always try not to do, but I think that’s just what makes people do it, the need to do it*” (\#40, IoT)
Participant-Informed Solutions
------------------------------
We have identified several justifications for disparity-prone behaviour. If actions are not commensurate with opinions, users might place themselves at risk. With a third of our sample acting in this manner, further work is required. Therefore, we suggest approaches directly informed by disparity-prone individuals. We are cognisant that technology firms might resist change, as they profit from data monetisation [@Robinson2015]. With this in mind, we give balanced feasible suggestions.
Many respondents (\#35, \#36, \#37, \#38, \#39) recommended awareness campaigns as a means of increasing understanding. While initiatives have frequently concerned security [@Bada2015], few have specifically targeted smart devices. Sessions could be held for school pupils, as they will mature in a connected world. For an effective initiative, topics including default settings and data markets must be addressed. Practical advice would be essential, such as how to disable GPS tracking. If users can understand why their data is collected, they can make decisions in an informed manner. To ascertain whether such initiatives are successful, attendees could be evaluated through a longitudinal process. Whereas education far from guarantees action [@Bada2015], it would give people the tools to guard their data.
With privacy policies often long and complex, respondents appealed for simplification (\#8, \#30, \#40). If individuals could understand how their data was used, perhaps they would make prudent decisions. While attempts have been made to simplify policies, vendors are keen to resist these efforts. As an accommodation, graphical icons could be introduced to highlight functionality. A Wi-Fi symbol could denote wireless, while a padlock could represent password protection. IoT vendors could subscribe to this scheme and compete based on their functionality. Whereas consumers would still favour exciting features, privacy would not be hidden. To assess whether standards improve, icon distribution could be observed over time. Although this approach might hamper IoT innovation, it would reduce the risk of insecure infrastructure.
Several participants believed companies should do more to protect their customers (\#36, \#21). Some complained that privacy is hidden (\#37), while others argued for clearer settings (\#35). To increase salience, privacy options could be embedded in the installation process. However, many vendors are funded through data collection [@Robinson2015], and therefore might resist alterations. As an accommodation, private settings could be default with alternatives highlighted during installation. Therefore, those who desire functionality can opt-in, while ignorance and apathy would not impede privacy. To monitor the success of such an approach, empirical studies would assess the popularity of different settings. We believe such measures are necessary to reduce the opinion-action disparity.
Conclusions and Further Work {#sec:seven}
============================
In this paper, we explored the Privacy Paradox and the influence of the Internet-of-Things. This is of importance as those who display the disparity might place themselves at risk. Through our 170-person online survey, we discovered that IoT devices are considered significantly less private than non-IoT products. We also found smart devices are regarded as less familiar and (less) usable, with this potentially challenging effective protection. Although the IoT was rated poorly, many who recognised the risks still purchased the products.
To examine this potential disparity between opinion and action, we conducted contextualised interviews with 40 members of the public. Rather than comparing abstract concepts with practical behaviour, our discussions concerned respondents’ devices. We found IoT owners both cared significantly less about their data and were significantly less able to protect it. As supported by our survey results, justifications suggest unfamiliarity and complexity led users to neglect protection.
Directly comparing opinions and actions, we found IoT users were significantly more likely to display the disparity. Seeking to deconstruct the issue, we explored the qualitative rationale of disparity-prone users. Social norms, lack of awareness and short-term necessity were all cited as factors. We concluded by proposing mitigative measures, including IoT awareness campaigns and graphical privacy policies. With a third of our interviewees prone to the disparity, we believe further work is required to mitigate the Privacy Paradox.
We accept our current research possesses several limitations. Our surveys and interviews capture an educated demographic, with a large number of Master’s graduates. Although privacy research is often conducted with college-age students, further work will extend these studies with broader demographics. With even these individuals neglecting their data, protection might be rarer for less-educated users. As we phrased our rationale queries in terms of other people, this might have biased responses. While 77% of disparity-prone respondents referred to themselves, participants might state what they consider to be common replies. In future work, we will use a range of scenarios to dissect why decisions are made.
As mentioned, there is no strict dichotomy between IoT and non-IoT products. However, to explore the influence of smart devices, we selected examples of archetypal products. Future work would extend the range of devices and consider technologies, such as mobile phones, nearer the intersection.
Devices within categories are also diverse, with a Mac desktop differing from a Windows computer. Similar products might offer different privacy settings and collect different pieces of data. By contextualising discussions, we sought to compare each device’s concern with its usage. Through identifying disparities at an individual level, we looked to minimise the effect of product diversity. Future work could offer a stricter control by comparing devices from the same vendor. Finally, surveys and interviews are inherently prone to response biases. Through disguising privacy and requesting non-normative opinions, we hope to have minimised their influence. In future work we wish to explore behaviour empirically, comparing actions across a broad range of technologies.
|
---
abstract: 'Köhler’s method is a useful multi-thresholding technique based on boundary contrast. However, the direct algorithm has a too high complexity - $\mathcal{O} (N^2)$ i.e. quadratic with the pixel numbers $N$ - to process images at a sufficient speed for practical applications. In this paper, a new algorithm to speed up Köhler’s method is introduced with a complexity in $\mathcal{O}(NM)$, $M$ is the number of grey levels. The proposed algorithm is designed for parallelisation and vector processing, which are available in current processors, using OpenMP (Open Multi-Processing) and SIMD instructions (Single Instruction on Multiple Data). A fast implementation allows a gain factor of 405 in an image of 18 million pixels and a video processing in real time (gain factor of 96).'
address: |
International Prevention Research Institute\
95 cours Lafayette\
69006 Lyon, France
bibliography:
- 'refs.bib'
title: 'Speeding up the Köhler’s method of contrast thresholding'
---
Köhler multi-thresholding, boundary contrast, fast image segmentation, parallelisation, pattern recognition
Introduction {#sec:intro}
============
Adaptive thresholding is one of the most used technique in many applications because it is fast to compute and when combined with previous filters, it gives robust decision rules for pattern recognition. Among many other techniques of thresholding [@Cai2014]; Köhler’s method computes a curve of contrast of the region boundaries in an image [@Kohler1981]. The contrast steps correspond to the local maxima of the curve and can be extracted for (multi-)thresholding of the image. This is useful for many applications: industrial, biomedical, video, etc [@Hautiere2006; @Jourlin2012; @Jourlin2016_chap3]. However, computing Köhler’s method is time consuming; almost 1 minute using a C++ implementation on a current computer with an image acquired by a recent camera (18 million pixels, fig. \[fig:pre:Kolher\_tulips\]). The purpose of this paper, is to introduce and implement a new algorithm for Köhler’s thresholding method faster than the existing algorithms and making it useful for applications requiring fast or real-time processing (e.g. video thresholding, large datasets) [@Thomee2016].
Previously, two attempts were made to speed up the computation of Kölher’s method. Zeboudj [@Zeboudj1988; @Coster1989] used mathematical morphology operations to give a similar version of Köhler’s method. However, his approach was efficient on specific devices which are not available any more. The other one, from Hautière [@Hautiere2005; @Hautiere2006], consists of making the computation on a reduced part of the neighbourhood and to pre-calculate some intermediate images of minimum and maximum between the image translated horizontally and vertically in order to compute the contrast. However, this algorithm does not introduce any parallelisation.
In this paper, after a reminder on Köhler’s method and on parallel computing in Mathematical Morphology [@Matheron1967; @Serra1982; @Soille2003]; we will introduce a parallel algorithm for Köhler’s method using line translations of the image. We will also propose to compute the contrast on a reduced neighbourhood as in [@Hautiere2005]. Eventually, we will compare an implementation of our algorithm, using vectorisation with SIMD instructions [@Cockshott2010] and multi-core (i.e. parallel) processing with OpenMP [@Chapman2008], to other implementations of Köhler’s method.
Prerequisites {#sec:pre}
=============
Let us remind Köhler’s method and the acceleration of Mathematical Morphology operations. An image is a function $f$ defined on a discrete domain $D \subset {\mathbb Z}^n$ with values in $[0, M[$, $M \in {\mathbb R}$ and $M=256$ for 8 bits images. We denote $x$ the location of a pixel and $f_x$ its value. In the sequel, we will use the 4-neighbourhood $N_4$ of pixels. For bi-dimensional images, we can also use the 8 or the 6-neighbourhood [@Soille2003] with insignificant differences [@Hautiere2006].
Köhler’s method {#ssec:Kohler}
---------------
Let us remind Köhler’s method [@Kohler1981; @Jourlin2012]. Given a grey-level image $f$ and a threshold $t \in [0,M[$, two classes are generated by $t$: $C_0^t(f) = \{ x \in D, f_x \leq t \}$ and $C_1^t(f) = \{ x \in D, f_x > t \}$. A boundary $B(t)$ is also generated: $$\label{eq:pre:Kohler_boundary}
\begin{array}{@{}ccl@{}}
B(t) &=& \left\{ (x_0,x_1) \in D^2, x_0 \in C_0^t(f), x_1 \in C_1^t(f) \right.\\
&&\left. \text{ and } x_1 \in N_4(x_0)\right\}.
\end{array}$$ For each couple of pixels $(x_0,x_1)$ of $B(t)$, Köhler associates a contrast $C_K^t(x_0,x_1)$ defined as: $$\label{eq:pre:Kohler_contrast}
C_K^t(x_0,x_1) = \min\left( f_{x_1}-t , t-f_{x_0} \right)$$ which is the minimum of the two steps (in terms of contrast) generated by the threshold $t$ between $f_{x_0}$ and $f_{x_1}$. The average contrast of the boundary $B(t)$ is defined as: $$\label{eq:pre:Kohler_boundary_contrast}
C_K(B(t)) = \frac{1}{\# B(t)}\times \sum_{(x_0,x_1) \in B(t)} C_K^t(x_0,x_1).$$ $\# B(t)$ is the cardinality (number of elements) of $B(t)$ and the summation is made on the couples of pixels $(x_0,x_1)$ belonging to $B(t)$. This generates a curve of contrasts $C_K(B(t))$ for all the possible thresholds $t \in [0,M[$. The optimal threshold $t_0$ is selected as: $$\label{eq:pre:Kohler_threshold}
C_K(B(t_0)) = \max_{t\in [0,M[} \left( C_K(B(t) \right).$$
In figure \[fig:pre:Kolher\_tulips\], we have extracted the 6 most significant thresholds (i.e. the local maxima) from the contrast curve. These multiple thresholds give an efficient simplification (i.e. compression) of the image grey levels: passing from 256 to 7 grey levels. The 7 grey levels corresponds to the mean value of the pixels for each class of the segmentation.
[@c@[ ]{}c@]{} ![Multiple thresholding by Köhler’s method of the (a) original image into a (b) segmented image (the class value is the mean values of the class pixels) by (c) the seven thresholds selected on the contrast curve.[]{data-label="fig:pre:Kolher_tulips"}](fig/tulips_grey_small.jpg "fig:"){width="0.5\columnwidth"}& ![Multiple thresholding by Köhler’s method of the (a) original image into a (b) segmented image (the class value is the mean values of the class pixels) by (c) the seven thresholds selected on the contrast curve.[]{data-label="fig:pre:Kolher_tulips"}](fig/tulips_multithresholded_Mean_Kolher_small.png "fig:"){width="0.5\columnwidth"}\
&\
\
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A direct C++ implementation consists of computing for each threshold $t \in[0,M[$, the contrast $C_K^t(B(t))$ of the boundary $B(t)$. It has a duration of 53 s using an image of size $3672 \times 4096$ pixels and a processor IntelCore^TM^ i7 CPU 4702HQ, 2.20 GHz, 4 cores, 8 threads. As the algorithm is not parallel, a single thread is used. For real-time applications, or big datasets, a faster algorithm is needed.
Accelerating operations on a neighbourhood {#ssec:accel}
------------------------------------------
In Mathematical Morphology, for operations on a neighbourhood some acceleration methods exists. With a symmetric structuring element $A$ (such as the one associated to the 4-neighbours), the morphological dilation corresponds to the Minkowski addition [@Minkowski1903; @Matheron1967; @Serra1982; @Soille2003; @Najman2013]: $$\label{eq:fast:Minkowski_plus}
X \oplus A = \bigcup_{a \in A} X_a = \{ x + a : x \in X , a\in A\}.$$ $X_a = \{ x + a : x \in X \}$ is the set $X \subset D$ translated by the vector $a$. A direct implementation of a dilation, by computing the union on the neighbourhood of each pixel (fig. \[fig:fast:morpho\_neigh\] (a)), will lead to an algorithm of complexity of $\mathcal{O}(N \times |A|)$ [@Geraud2010]. $N$ is the number of pixels in the image and $|A|$ is the cardinality of $|A|$. However, the Minkowksi addition (i.e. the dilation) has the property to distribute the union [@Hadwiger1957; @Serra1982]: $X \oplus (A \cup A') = (X \oplus A) \cup (X \oplus A')$. This property has important technological consequences as a dilation can be computed elements by elements of the structuring element $A$, before combining the intermediate results by union. Therefore, $$\label{eq:pre:dil_line}
\begin{array}{ccl}
X \oplus A &=& \{ x \in D \> | \> \exists a \in A, x - a \in X \}\\
&=& \bigcup_{a \in A} \{ x \in D \> | \> x - a \in X \}.
\end{array}$$ An implementation by translating all pixel $x$ by the vectors $a \in A$ and checking if they belong to the set $X$ will have a complexity of $\mathcal{O}(|X \oplus A|)$ [@Geraud2010]. As images are stored in computer memory as unidimensional arrays, an efficient implementation [@Faessel_smil2013] consists of translating the lines instead of the pixels (fig. \[fig:fast:morpho\_neigh\] (b)).
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![Computation of a dilation (a) using a neighbourhood iterator or (b) by line translation.[]{data-label="fig:fast:morpho_neigh"}](fig/Fw_4-Neighbourhood_processing.pdf "fig:"){width="0.5\columnwidth"} ![Computation of a dilation (a) using a neighbourhood iterator or (b) by line translation.[]{data-label="fig:fast:morpho_neigh"}](fig/Fw_Line_processing_hor_N4.pdf "fig:"){width="0.5\columnwidth"}
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As the operations in each line are independent from the other lines (eq. \[eq:pre:dil\_line\]), a parallel programming method is used, namely OpenMP [@Chapman2008], which is designed for multi-processor/core, shared memory computers. Each thread is computed by a single core and all cores share a common memory. In addition, vectorised data are used with SIMD instructions [@Cockshott2010] in each thread. Vector operations using SIMD instructions allow multiple data to be processed with a single instruction of the processor while scalar operations use one instruction to process each individual data. The size of the SIMD registers being of 128 bits, 16 operations on integers of 8 bits are performed at the same time instead of a single one. In recent compilers, the vectorisation is performed automatically after activation of the right option. In [@Faessel_smil2013], a dilation with a square structuring element of size 3 pixels in a 8 bit image (of size $1024 \times 1024$ pixels) is accelerated by a factor 136 with a neighbourhood implementation by comparison to a line implementation with parallelisation and vectorisation (45 ms versus 0.33 ms, IntelCore^TM^ i3 CPU M330, 2.13 GHz, 2 cores, 4 threads). Let us introduce an efficient method to compute Köhler’s method.
A fast algorithm for Köhler’s method {#sec:fast}
====================================
Accelerating the computation of Köhler’s contrast {#ssec:accel_Kohler}
-------------------------------------------------
A direct implementation of Köhler’s approach (section \[ssec:Kohler\]) is not designed for parallel and vector processing. Kölher’s contrast (eq. \[eq:pre:Kohler\_contrast\]) is summed on boundaries $B(t)$ which are computed by the set difference between a morphological dilation of the set $C_1^t(f)$ and this same set: $\{x_0 \in D, (x_0,x_1) \in B(t)\} = (C_1^t(f) \oplus A) \setminus C_1^t(f)$. The structuring element $A$ corresponds to the 4-neighbours. The direct implementation has a complexity of $\mathcal{O}(N^2)$ (i.e. the complexity of a dilation, $\mathcal{O}(N)$, multiplied by the complexity of scanning the pixel pairs of the boundary $B(t)$, $\mathcal{O}(M \times \#B(t)) = \mathcal{O}(N)$). A direct acceleration would consist of computing the boundary with an accelerated morphological dilation, as presented above. However, such approach does not reduce the complexity of the algorithm. For this purpose, we propose first to perform the translation of the image lines, which is suited for parallel processing, and then to compute the contribution to the contrast of each pixel pairs, for each threshold. The complexity decreases to $\mathcal{O}(NM)$, i.e. the product of the number of pixels by the number of grey levels. The algorithm \[alg:fast\_Kohler\] presents our approach.
\[alg:fast\_Kohler:begin\_parallel\] \[alg:fast\_Kohler:line\_multicore\] \[alg:fast\_Kohler:translate\_line\] \[alg:fast\_Kohler:autov\_1\] $mini \leftarrow \min( curLine(j) , nLine_{a}(j) )$ \[alg:fast\_Kohler:mini\] $maxi \leftarrow \max( curLine(j) , nLine_{a}(j) )$ \[alg:fast\_Kohler:maxi\] \[alg:fast\_Kohler:contrast\] \[alg:fast\_Kohler:counter\] \[alg:fast\_Kohler:end\_parallel\]
\[alg:fast\_Kohler:norm\_contrast\]
Reduction of the neighbourhood size {#ssec:reduc_neigh}
-----------------------------------
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![(a) The 4-neighbourhood $N_4$ and its (b) “half” $N_4^*$[]{data-label="fig:reduc:neigh"}](fig/4-connexity.pdf "fig:"){width="0.2\columnwidth"} ![(a) The 4-neighbourhood $N_4$ and its (b) “half” $N_4^*$[]{data-label="fig:reduc:neigh"}](fig/4-connexity_half.pdf "fig:"){width="0.2\columnwidth"}
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In order to gain an additional factor 2, let us show that it is the same to make the computation on a “half” neighbourhood $N_4*$ as on the 4-neighbourhood $N_4$ (fig. \[fig:reduc:neigh\]). The idea has been introduced in [@Hautiere2005; @Hautiere2006]. Here, the equality of the two approaches is demonstrated.
When scanning the boundary $B(t)$ with the 4-neighbourhood $N_4$, only the pixels $x_1$, such as $f_{x_0} < f_{x_1}$, are contributing to the boundary contrast. However, both pixels $x_0$ and $x_1$ are neighbours: $x_0 \in N_4(x_1) \Leftrightarrow x_1 \in N_4(x_0)$. Therefore, both (ordered) pairs $(x_0,x_1)$ and $(x_1,x_0)$ are scanned and only one pair is contributing to the contrast between the (unordered) set of points $\{x_0,x_1\}$ $$\label{eq:dem_cont2}
\begin{array}{@{}c@{ }c@{ }l@{}}
\Gamma_{K}^t(\{x_0,x_1\}_{N_4}) &=&
\left\{
\begin{array}{@{}l@{}}
\min\left( f_{x_0}-t , t-f_{x_1} \right), \text{if } f_{x_1} \leq t < f_{x_0}\\
\min\left( f_{x_1}-t , t-f_{x_0} \right), \text{else }f_{x_0} \leq t < f_{x_1}
\end{array} \right.\\
&=& \min\left( |f_{x_0}-t| , |t-f_{x_1}| \right).\\
\end{array}$$ Let us remove the order condition between the grey levels, $f_{x_0} \leq t < f_{x_1}$ and define the absolute pair contrast $\overline{C}_{K}^t(x_0,x_1) = \min\left( |f_{x_0}-t| ,\right.$ $\left.|t-f_{x_1}| \right)$. When scanning both pairs without the grey level order, the contrast between the set of points $\{x_0,x_1\}$, is: $$\label{eq:dem_cont3}
\begin{array}{@{}c@{ }c@{ }l@{}}
\overline{\Gamma}_{K}^t(\{x_0,x_1\}_{N_4}) &=& \overline{C}_{K}^t(x_0,x_1) + \overline{C}_{K}^t(x_1,x_0)\\
&=& 2 \min\left( |f_{x_0}-t| , |t-f_{x_1}| \right)\\
&=& 2 \Gamma_{K}^t(\{x_0,x_1\}_{N_4}), \quad \text{(eq. \ref{eq:dem_cont2})}\\
\end{array}.$$ Using the “half”-neighbourhood $N_4^*$ allows to scan only one pair of pixels. Therefore, we obtain our result: $$\label{eq:cont_equal}
\overline{\Gamma}_{K}^t(\{x_0,x_1\}_{N_4^*}) = \Gamma_{K}^t(\{x_0,x_1\}_{N_4}).$$
Implementation: parallelisation and vectorisation {#ssec:impl}
-------------------------------------------------
In order to make parallel the algorithm, the computation of the contrast can be performed independently line by line. For each parallel thread $k$, two arrays of length $M$, $C_k$ (contrast) and $Card_k$ (counter), need to be created at the beginning of the parallel process (line \[alg:fast\_Kohler:begin\_parallel\], alg. \[alg:fast\_Kohler\]). At the end of the parallel process (line \[alg:fast\_Kohler:end\_parallel\]), they are grouped by summation in two arrays $C$ (contrast) and $Card$ (counter). The parallel programming language used is OpenMP in C++. Instead of being performed between single numbers, several operations can be performed using arrays, allowing the vectorisation of the data. The following operations are vectorised and processed using SIMD instructions: $i)$ the line translation (line \[alg:fast\_Kohler:translate\_line\]), $ii)$ the computation of the minimum (line \[alg:fast\_Kohler:mini\]) and the maximum (line \[alg:fast\_Kohler:maxi\]) between the arrays $curLine$ and $nLine_a$, $iii)$ the computation of the contrast $C_k$ (line \[alg:fast\_Kohler:contrast\]) and of the counter $card_k$ (line \[alg:fast\_Kohler:counter\]) and $iv)$ the normalisation of the contrast (line \[alg:fast\_Kohler:norm\_contrast\]).
Results {#Results}
=======
We now compare the duration of the direct implementation to the fast algorithm. We have used a processor IntelCore^TM^ i7 CPU 4702HQ, 2.20 GHz, 4 cores, 8 threads with 16Gb RAM. Using the image “Tulips” (fig. \[fig:pre:Kolher\_tulips\]) with a current camera resolution of $3672 \times 4896$ pixels and the fast algorithm with parallelisation, the computation of Köhler’s method is made in 0.13 s (tab. \[tab:res:Duration\]) instead of 53 s, with a gain factor of 405. With images of former resolution ($512 \times 512$ pixels), such as Lenna image; the direct method takes 0.69s and the fast method 0.005s with a gain factor of 126. Therefore, the necessity of using a faster algorithm instead of direct implementation becomes essential to process images with current resolution. Other experiments have confirmed this result.
Name Lenna Tulips
---------------- ------------------ --------------------
Size (pixels) $512 \times 512$ $3672 \times 4896$
Direct (D) 6.90e-01 s 5.29e+01 s
Fast (B) 1.62e-02 s 4.86e-01 s
Fast (A) 5.46e-03 s 1.30e-01 s
Gain (B vs. D) 42 109
Gain (A vs. D) 126 405
: Comparison of the duration of the different implementations for images of different sizes: direct (D), fast without parallelisation (B) and fast with parallelisation (A). The gain factors are computed between the implementations B and D and between the implementations A and D.[]{data-label="tab:res:Duration"}
Let us try Köhler’s method with a video of a car from the dataset YFCC100M (Yahoo Flickr Creative Commons 100M) [@Thomee2016; @YFCC100M_video_car2009]. In figure \[fig:res:frames\_video\], two frames and their segmentations in two classes are shown. The direct implementation segments the video at a rate of 1 frame per second while the fast implementation (with parallelisation) processes 97 frames per second, which is faster than the 25 frames/s of the video (tab. \[tab:res:Duration\_video\]). Therefore, the fast algorithm for Köhler’s method is suited for real time video processing.
Name YFCC100M (car)
------------------ ------------------ --
Size (pixels) $502 \times 480$
Number of frames 640
Direct (D) 0.99 frames/s
Fast (A) 96.96 frames/s
Gain (A vs. D) 98
: Frame per seconds segmented by Köhler’s method with different implementations applied on a video: direct (D), and fast with parallelisation (A). The gain factors between the implementations A and D have been computed.[]{data-label="tab:res:Duration_video"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(a), (b) Two frames of a video from the dataset YFCC100M and (c), (d) their segmentations in two classes by Köhler’s method.[]{data-label="fig:res:frames_video"}](fig/4078577031_sd_0101.png "fig:"){width="0.5\columnwidth"} ![(a), (b) Two frames of a video from the dataset YFCC100M and (c), (d) their segmentations in two classes by Köhler’s method.[]{data-label="fig:res:frames_video"}](fig/4078577031_sd_0251.png "fig:"){width="0.5\columnwidth"}
![(a), (b) Two frames of a video from the dataset YFCC100M and (c), (d) their segmentations in two classes by Köhler’s method.[]{data-label="fig:res:frames_video"}](fig/4078577031_sd_bin_0101.png "fig:"){width="0.5\columnwidth"} ![(a), (b) Two frames of a video from the dataset YFCC100M and (c), (d) their segmentations in two classes by Köhler’s method.[]{data-label="fig:res:frames_video"}](fig/4078577031_sd_bin_0251.png "fig:"){width="0.5\columnwidth"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Conclusion and perspectives {#sec:concl}
===========================
A faster algorithm for Köhler’s thresholding has been introduced with a lower complexity, $\mathcal{O}(NM)$, than the direct approach, $\mathcal{O}(N^2)$. It is designed to benefit from the capacities of processors: multi-core processing with OpenMP and vector processing using SIMD instructions. Results show that with an image of 18 million pixels the duration is reduced by a factor 405 (from 53 s to 0.13 s) and that a video can be processed at a rate of 97 frames/s instead of 1 frame/s. Importantly, this algorithm is suited for applications requiring real-time or fast processing: video, industrial, large databases, etc. Its practical interest is to be combined with previous transforms: a low-pass filter, a mathematical morphology transform [@Serra1982; @Serra1988; @Geraud2010; @Noyel2010] or a map of colour distances [@Noyel2015; @Noyel2016]. In future works, the influence on the method of different contrasts will be presented (already studied), such as the contrasts defined in the Logarithmic Image Processing framework [@Jourlin2012; @Jourlin2016_chap3].
|
---
abstract: 'For a classical superconformal gauge theory in a conformal supergravity background, its chiral R-symmetry anomaly, Weyl anomaly and super-Weyl anomaly constitute a supermultiplet. We review how these anomalies arise from the five-dimensional gauged supergravity in terms of the AdS/CFT correspondence at the gravity level. The holographic production of this full superconformal anomaly multiplet provides a support and test to the celebrated AdS/CFT conjecture.'
---
16.0 true cm 22.0 true cm 0 cm 0 cm 0 true in 0.05 true in
[**Superconformal Anomaly from AdS/CFT Correspondence**]{}
[^1]
[M. Chaichian]{} and [W.F. Chen]{}
High Energy Physics Division, Department of Physical Sciences, University of Helsinki\
and Helsinki Institute of Physics, FIN-00014, Helsinki, Finland
Introduction
============
The discovery of a $D$-brane as a fundamental dynamical object carrying $R-R$ charge has played a crucial role in establishing a web of dualities among five superstring theories and unifying them into a single M-theory [@pol]. On one hand, combining other types of branes such as the Neveu-Schwarz (NS) solitonic five-brane and orientifold plane, a supersymmetric gauge theory defined on the world volume of $Dp$-branes can be constructed from an elaborated setting of a brane configuration in a weakly type II string theory [@giku]. On the other hand, a stack of $Dp$-branes can modify the space-time background of the strong coupled type II string theory and arise as the brane solution to the low-energy effective theory of type II string, i.e., type II ($A$ or $B$) supergravity. These two distinct features of a D-brane at both strong and weakly superstring theory have led to the celebrated AdS/CFT correspondence conjecture proposed by Maldacena [@mald].
The original AdS/CFT correspondence conjecture [@mald] states that the type IIB string theory compactified on $AdS_5\times S^5$ theory with $N$ units of $R-R$ flux on $S^5$ describes the same physics as ${\cal N}=4$ $SU(N)$ supersymmetric Yang-Mills theory. The explicit definition was further clarified and generalized as the following [@gkp; @witt1]. Given the type II superstring theory in the background $AdS_{d+1}\times X^{9-d}$, with $X^{9-d}$ being a compact Einstein manifold, there should exist a one-to-one correspondence between a string state a supergravity field in the $AdS_{d+1}$ bulk and a gauge invariant operator of the conformal field theory defined on the boundary of $AdS_{d+1}$ space-time. Concretely, the partition function $Z_{\rm String}\left[\phi_0\right]$ of the type II superstring theory as a functional of the boundary value $\phi_0(x)$ of a bulk field $\phi (x,r)$ should equal the generating functional of the correlation function $Z_{\rm CFT}$ of the gauge invariant operator of the conformal conformal field theory with the external source $\phi_0(x)$ provided by the boundary value of the bulk field $\phi (x,r)$, $$\begin{aligned}
Z_{\rm String}\left[\phi_0\right]
&=& Z_{\rm CFT}\left[\phi_{0}\right], \nonumber\\
\left.Z_{\rm String}\left[\phi_0\right]\right| &=& \int_{\phi
(x,0)=\phi_{0}(x)}{\cal D}\phi (x,r)
\exp\left(-S[\phi (x,r)]\right), \nonumber\\
Z_{\rm CFT}\left[\phi_{0}\right]&=& \left\langle \exp\int_{M^d}
d^dx
{\cal O} (x) \phi_{0}(x)\right\rangle \nonumber\\
&=&\sum_n\frac{1}{n!}\int \prod_{i=1}^n d^d x_i \left\langle {\cal
O}_1 (x_1)\cdots {\cal O}_n (x_n) \right\rangle \phi_{0} (x_1)
\cdots \phi_{0} (x_n)\nonumber\\
&{\equiv}& \exp\left(-\Gamma_{\rm CFT} [\phi_{0}]\right).
\label{acc1}\end{aligned}$$ In above equation, ${\cal O}$ represent certain composite operator in the superconformal field theory such as the energy-momentum tensor and chiral $R$-symmetry current etc., $\phi_0 $ are the corresponding background fields such as the gravitational and gauge fields etc. coming from the boundary value of the corresponding bulk field, and $\Gamma_{\rm CFT} [\phi_{0}]$ is the quantum effective action describing the composite operators interacting with background field $\phi_{0}$.
At low-energy the string effect can be neglected. The partition function of the type IIB superstring can be evaluated as the exponential of the type IIB supergravity action in a on-shell field configuration $\phi^{\rm cl}[\phi^{0}]$ with the boundary value $\phi_{0}(x)$, i.e., $$\begin{aligned}
Z_{\rm String}\left[\phi_0\right]
=\exp\left(-S_{\rm SUGRA}[\phi^{\rm cl}[\phi_{0}]]\right) \,.
\label{acc3}\end{aligned}$$ A comparison between Eq.(\[acc1\]) and Eqs.(\[acc3\]) immediately shows that the large-$N$ quantum effective action of the $d$-dimensional conformal field theory in the background provided by $\phi_0$ can be approximately equal to the on-shell classical action of $AdS_{d+1}$ supergravity with non-empty boundary, $$\begin{aligned}
\Gamma_{\rm CFT}[\phi_{0}]=S_{\rm SUGRA}[\phi^{\rm
cl}[\phi_{0}]]=\int d^d x \phi_0 (x) \left\langle {\cal
O}\right\rangle. \label{acc4}\end{aligned}$$
Let us see how the five-dimensional gauged supergravities [@gst; @awada; @guna2] arises in the AdS/CFT correspondence [@ferr]. The $AdS_5\times S^5$ background comes from the near horizon limit of three-brane solution of type IIB supergravity [@agmo]. Therefore, in the $AdS_5\times S^5$ background, the spontaneous compactification on $S^5$ of the type IIB supergravity occurs [@freu]. With the assumption that there exists a consistent nonlinear truncation of the massless modes from the whole Kluza-Klein spectrum of the type IIB supergravity compactified on $S^5$ [@marcus; @kim], the resultant theory should be the $SO(6)(\cong SU(4))$ gauged ${\cal N}=8$ $AdS_5$ supergravity since the isometry group $SO(6)$ of $S^5$ becomes the gauge group of the compactified theory and the $AdS_5\times S^5$ background preserves all the supersymmetries of type IIB supergravity [@guna2]. Furthermore, if the background for the type IIB supergravity is $AdS_5 \times X^5$ with $X^5$ being an Einstein manifold rather than $S^5$ such as $T^{1,1}=(SU(2)\times
SU(2))/U(1)$ or certain orbifold, then the number of preserved supersymmetries in the compactified $AdS_5$ supergravity is reduced [@roman2; @kw2]. One can thus obtain the gauged ${\cal
N}=2,4$ $AdS_5$ supergravity in five dimensions, and their dual field theories should be $N=1,2$ supersymmetric gauge theories [@ferr]. A supersymmetric gauge theory with lower supersymmetries is not a conformal invariant theory since its beta function usually does not vanish. However, it was shown that the beta function a supersymmetric gauge theory has the zero point, at which the (super)conformal invariance can arise [@shif; @seib]. The $AdS/CFT$ correspondence between the ${\cal N}=2,4$ gauged supergravities in five dimensions and ${\cal N}=1,2$ supersymmertric gauge theories can thus be established [@ferr].
Eq.(\[acc4\]) shows clearly that according the AdS/CFT correspondence a quantum effective action describing a superconformal gauge theory in an external supergravity background can be identified with the on-shell action of the gauged supergravity with non-trivial boundary data. Thus the various anomalies at the leading-order of large-$N$ expansion of a four-dimensional supersymetric gauge theory should be extracted from the $AdS_5$ gauged supergravity. Specifically, the AdS/CFT correspondence exchange strong and weak couplings and vice versa, and the anomalies are independent to the couplings, the production of the superconformal anomaly from the gauged supergravity will provide a supportand and test to the $AdS/CFT$ correspondence at the supergravity level.
This short review is outlined as the following. In Sect.II, we shall briefly introduce a classical superconformal gauge theory in a conformal supergravity background and explain how a superconformal anomaly multiplet arises. In Sect.III we shall explain the structure of the gauged supergravity in five dimensions and emphasize why the superconformal anomaly multiplet can be reproduced from the gauged supergravity in terms of the AdS/CFT correspondence. Sect.IV contains a systematics review on how chiral-, Weyl- and super-Weyl anomalies arise from the classical five-dimensional gauged supergravity. Sect.V is a brief summary.
Superconformal Anomaly Multiplet in External Conformal Supergravity Background
==============================================================================
We focus on a general ${\cal N}=1$ four-dimensional supersymmetric $SU(N)$ gauge theory, its conserved currents including the energy-momentum tensor $\theta^{\mu\nu}$, the supersymmetry current $s^{\mu}$ and the chiral (or equivalently axial vector) R-current $j^{\mu}$ constitute a supermultiplet due to the supersymmetry [@ilio; @anm], $$\begin{aligned}
\partial_\mu T^{\mu\nu}=\partial_\mu s^\mu=\partial_\mu
j^{\mu}=0.\end{aligned}$$ If these currents satisfy further algebraic constraints, $T^{\mu}_{~\mu}=\gamma_\mu s^\mu=0$, the Poincaré supersymmetry will be promoted to a superconformal symmetry since one can construct three more conserved currents, $$\begin{aligned}
d^\mu {\equiv} x_\nu T^{\nu\mu}, ~ k_{\mu\nu}{\equiv} 2 x_\nu
x^\rho T_{\rho\mu}-x^2T_{\mu\nu}, ~ l_\mu{\equiv} ix^\nu
\gamma_\nu s_\mu.\end{aligned}$$ These three new conserved currents lead to the generators for dilatation, conformal boost and conformal supersymmetry transformation. However, the superconformal symmetry may become anomalous at quantum level. If all of them, the trace of energy-momentum tensor, $T^\mu_{~\mu}$, the $\gamma$-trace of supersymmetry current, $\gamma^\mu s_\mu$ and the divergence of the chiral $R$-current, $\partial_\mu j^{\mu}$, get contribution from quantum effects, they will form a chiral supermultiplet with the $\partial_\mu
j^{\mu}$ playing the role of the lowest component of the corresponding composite chiral superfield [@anm; @sibold; @grisaru].
When considering the ${\cal N}=1$ supersymmetric gauge theory in a ${\cal N}=1$ conformal supergravity background, the energy-momentum tensor $T_{\mu\nu}$, the supersymmetry current $s_\mu$, and the chiral (or axial vector) $R$-symmetry current $j_\mu$ will couple to the gravitational field $g_{\mu\nu}$, chiral (or axial) vector field $A_\mu$ and vector-spinor gravitino field $\psi_{\mu}$ in the multiplet of conformal supergravity, respectively [@fra], $$\begin{aligned}
{\cal L}_{\rm ext}=\int d^4x \sqrt{-g}\left( g_{\mu\nu}T^{\mu\nu}
+ A_\mu j^{\mu}+\overline{\psi}_\mu s^{\mu}\right). \label{clag}\end{aligned}$$ The action (\[clag\]) shows that the covariant conservations of the currents, $\nabla_\mu \theta^{\mu\nu}=D_\mu s^\mu=0$, are equivalent to the local gauge transformation invariance of the external supergravity system, $$\begin{aligned}
\delta g_{\mu\nu} (x) = \nabla_\mu \xi_\nu +\nabla_\nu \xi_\mu
,~~~ \delta\psi_\mu (x)= D_\mu \chi (x).
\label{egt1}\end{aligned}$$ Furthermore, the covariant conservation of the chiral (or axial) vector current $j_\mu $ and the vanishing of both the $\gamma$-trace of supersymmetry current and the trace of energy-momentum tensor at classical level, $$\begin{aligned}
\nabla_\mu j^{\mu}=\gamma^\mu s_\mu =T^{\mu}_{~\mu}=0,
\label{supercon}\end{aligned}$$ mean the Weyl transformation invariance of $g_{\mu\nu}$, the super-Weyl symmetry and the chiral gauge symmetry of the external conformal supergravity system, $$\begin{aligned}
\delta g_{\mu\nu}=g_{\mu\nu} \sigma (x),~~~ \delta \psi_\mu =
\gamma_\mu \eta (x),~~~ \delta A_\mu (x) = \partial_\mu \Lambda
(x). \label{busy}\end{aligned}$$ This means that the classical superconformal symmetry of the supersymmetric gauge theory is equivalent to that of the external conformal supergravity. Therefore, in the context of the $AdS/CFT$ (or more generally gravity/gauge) correspondence the superconformal anomaly in ${\cal N}=1$ supersymmetric gauge theory due to the supergravity external sources will be reflected in the explicit violations of the bulk symmetries of ${\cal N}=2$ gauged $AdS_5$ supergravity on the boundary [@witt1; @hesk; @bian1; @chch].
With no consideration on the quantum correction from the dynamics of the supersymmetric gauge theory, the external superconformal anomaly is exhausted at one-loop level. As given in Ref.[@grisaru], for a general ${\cal N}=1$ supersymmetric gauge theory with $N_v$ vector and $N_\chi$ chiral multiplets in an external supergravity background, the chiral R-symmetry and the Weyl anomalies read $$\begin{aligned}
\nabla_\mu j^{\mu} &=&\frac{c-a}{24\pi^2} R_{\mu\nu\lambda\rho}
\widetilde{R}^{\mu\nu\lambda\rho}+\frac{5a-3c}{9\pi^2}F_{\mu\nu}
\widetilde{F}_{\mu\nu}, \nonumber\\
T^\mu_{~\mu} &=& \frac{c}{16\pi^2}C_{\mu\nu\lambda\rho}
C^{\mu\nu\lambda\rho}
-\frac{a}{16\pi^2}\widetilde{R}_{\mu\nu\lambda\rho}
\widetilde{R}^{\mu\nu\lambda\rho}+\frac{c}{6\pi^2}
F_{\mu\nu}F^{\mu\nu},\nonumber\\
\gamma_\mu s^\mu &=&\left(A
R_{\mu\nu\lambda\rho}\gamma^{\lambda\rho}
+B F_{\mu\nu}\right)D^\mu \psi^\nu.
\label{sca}\end{aligned}$$ The coefficients $a$ and $c$ are purely determined by the field contents. For a ${\cal N}=1$ supersymmetric theory in the weak coupling limit, they are, respectively, [@grisaru] $$\begin{aligned}
c=\frac{1}{24}\left(3 N_v+N_{\chi}\right), ~~~
a=\frac{1}{48}\left(9 N_v+N_{\chi}\right).
\label{cc}
\end{aligned}$$ The coefficients $A$ and $B$ in the super-Weyl anomaly $\gamma_\mu
s^\mu$ are relevant to $a$ and $c$. In above equations, $\gamma_{\mu\nu}=i/2\,[\gamma_\mu,\gamma_\nu]$; $F_{\mu\nu}=\partial_\mu A_\nu- \partial_\nu A_\mu$ is the field strength corresponding to the external $U_R(1)$ vector field $A_\mu$; $R_{\mu\nu\lambda\rho}$ and $C_{\mu\nu\lambda\rho}$ are the Riemannian and Weyl tensors corresponding to the gravitational background field $g_{\mu\nu}$; $\widetilde{R}_{\mu\nu\lambda\rho}$ and $\widetilde{F}_{\mu\nu}$ are the Hodge duals of the Riemannian tensor and gauge field strength; $D_\mu$ is the covariant derivative with respect to both the external gravitational and gauge fields.
Five-dimensional gauged supergravity and $AdS_5$ boundary Reduction
=====================================================================
In this section we explain why the superconformal anomaly of a four-dimensional supersymmetric gauge theory in a conformal supergravity background can be extracted from a gauged supergravity in five dimensions [@gst; @awada; @guna2].
First, all the ${\cal N}=2,4,8$ five-dimensional gauged supergravities admit an $AdS_5$ classical solution [@gst; @awada; @guna2], $$\begin{aligned}
ds^2 &=& \frac{l^2}{r^2}\left(\eta_{\mu\nu} dx^\mu dx^\nu
- dr^2\right)
\label{metrican}
\end{aligned}$$ all the other fields vanishing. The cosmological term leading above solution comes from the value of the scalar potential at the critical point. Further, checking the supersymmetric transformation of the fermionic field in this background, one can find the non-vanishing Killing spinor [@gst; @awada; @guna2]. Thus the $AdS_5$ solution preserve the full supersymmetry of the gauged supergravity.
Second, we choose this $AdS_5$ classical solution as the vacuum configuration of the five-dimensional gauged supergravity and investigate the corresponding dynamical features around such a vacuum background. For the ${\cal N}=2$ $U(1)$ gauged supergravity, the Lagrangian density near the $AdS_5$ vacuum up to the quadratic terms in spinor fields is of the form [@bala] $$\begin{aligned}
8\pi G^{(5)}E^{-1} {\cal L} &=& -\frac{1}{2}{\cal R}
-\frac{1}{2}\overline{\Psi}_M^i \Gamma^{MNP}D_N \Psi_{P
i}-\frac{3l^2}{32} {\cal F}_{MN}{\cal F}^{MN}
-\frac{6}{l^2}\nonumber\\
&&-\frac{il^3}{64}E^{-1}\epsilon^{MNPQR} {\cal F}_{MN}{\cal
F}_{PQ}{\cal A}_R
-\frac{3i}{4l}\overline{\Psi}_M^i\Gamma^{MN}\Psi^{N j}
\delta_{ij}\nonumber\\
&&
-\frac{3il}{32}\left( \overline{\Psi}_M^i\Gamma^{MNPQ}
\Psi_{N i}{\cal F}_{PQ}+2\overline{\Psi}^{M i}\Psi^N_i {\cal
F}_{MN}\right), \label{gaugedfm}\end{aligned}$$ where ${\cal R}$ is the five-dimensional Riemannian scalar curvature; $\Psi_M^i$ are the gravitini, $i=1,2$ are the $SU(2)$ $R$ symmetry group indices; ${\cal A}_M$ and ${\cal F}_{MN}$ are the $U(1)$ gauge field and field strength; $D_M$ is the covariant derivative with respect to the (modified) spin connection, the Christoffel and ${\cal A}_M$; $G^{(5)}$ is the five-dimensional gravitational constant. The supersymmetry transformations at the leading order in spinor fields read [@bala] $$\begin{aligned}
\delta E_M^{~A}&=&\frac{1}{2}\overline{\cal E}^i\Gamma^A\Psi_{M
i}, ~~~\delta {\cal A}_M = \frac{i}{l}\overline{\Psi}_M^i{\cal
E}_i,
\nonumber\\
\delta \Psi_{M }^i &=& D_{M}{\cal E}^i+ \frac{il}{16}
\left(\Gamma_{M}^{~NP}-4\delta_{M}^{~N} \Gamma^P \right) {\cal
F}_{NP}{\cal E}^i+\frac{i}{2l}\Gamma_M \delta^{ij}{\cal E}_j.
\label{twostm}
\end{aligned}$$
The investigation on the classical dynamics of the gauged supergravity around the $AdS_5$ vacuum configuration means looking for the solution to the classical equation of motion which should asymptotically approach the $AdS_5$ solution (\[metrican\]). Geometrically, this is actually a process of revealing the asymptotic dynamical behaviour of the bulk fields near the boundary of $AdS_5$ space-time. The procedure of doing this is straightforward [@nish]. As the first step, one should perform fix the local symmetries such as the local Lorentz symmetry, gauge symmetry and supersymmetry in the radial (fifth) direction in a way consistent with the $AdS_5$ classical solution. Then one should reveal the radial coordinate dependence near the boundary of $AdS_5$ space-time of the solutions to the classical equations of motion of the fields. There are some delicate points for the fermionic fields like the gravitino. The fermionic fields are the symplectic Majorana spinors in five dimensions, one should show how they reduce to chiral spinors in four dimensions. Specifically, most of the fields in the five-dimensional gauged supergravity belong to certain representation of $R$-symmetry, while the gauged supergravity in the bulk and the conformal supergravity on the boundary have different $R$-symmetries. Thus one should exhibit how the $R$-symmetry in the bulk converts into the one on the boundary. For the ${\cal N}=2$ gauged supergravity, the solutions to the classical equations of motion display the follow leading-order dependence on the radial coordinate near the $AdS_5$ boundary [@bala], $$\begin{aligned}
{\cal A}_\mu (x,r) &=& {A}_\mu (x)+{\cal O}(r),~~ E_\mu^{~a}
(x,r)=\frac{l}{r} {e}_\mu^{~a}
(x)+{\cal O}(r ),
~~E_r^{~\overline{r}}=\frac{l}{r},\nonumber\\
\Psi_\mu^R &=& \left(\frac{2l}{r}\right)^{1/2}{\psi}_\mu^R(x),~~
\Psi_\mu ^L = \left(2lr\right)^{1/2}{\chi}_\mu^L(x), \nonumber\\
\chi_\mu^L &=& \frac{1}{3}\gamma^\nu\left({D}_\mu\psi_\nu^R
-{D}_\nu\psi_\mu^R\right)-\frac{i}{12}
\epsilon_{\mu\nu}^{~~\lambda\rho} \gamma_5\gamma^\nu
\left({D}_\lambda\psi_\rho^R -{D}_\rho\psi_\lambda^R\right),
\label{refield}\end{aligned}$$ and all other fields vanish. In above equations, $\mu,a=0,\cdots,3$ are the Riemannian and local Lorentz indices on the boundary, respectively, and $\overline{r}$ is the Lorentz index in the radial direction. The various quantities including the $\gamma$-matrices and the covariant derivative reduced from the five-dimensional case are the following [@bala], $$\begin{aligned}
{\gamma}_{a} &=& \Gamma_{a},~~ \Gamma_\mu=
E_\mu^{~a}\Gamma_{a}=\frac{l}{r}
{\gamma}_{\mu},
~~\Gamma^\mu = \overline{E}^\mu_{~a}\Gamma^{a}
=\frac{r}{l} {\gamma}^{\mu}, \nonumber\\
\gamma_5 &=& i\Gamma^{\overline{r}}
=-i\Gamma_{\overline{r}},~~\gamma_5^2=1; ~~
{D}_\mu (x) {\equiv} \nabla_\mu
+\frac{1}{4}{\omega}_\mu^{~ab}
{\gamma}_{~ab}-\frac{3}{4}{A}_\mu\gamma_5, \nonumber\\
\Psi_\mu &\equiv & \Psi_{\mu 1}+i\Psi_{\mu 2}, ~~~
\Psi^R_\mu\equiv \frac{1}{2} (1-\gamma_5)\Psi_\mu, ~~
\Psi^L_\mu \equiv \frac{1}{2} (1+\gamma_5).
\label{rega}
\end{aligned}$$ Redefining the bulk supersymmetry transformation parameter, ${\cal
E} (x,r)={\cal E}_1(x,r)+i{\cal E}_2(x,r)$, decomposing it into the chiral components, and further choosing radial coordinate dependence of ${\cal E}^{L,R}$ in the same way as the bulk gravitino, $$\begin{aligned}
{\cal E}^R(x,r)=\left(\frac{2l}{r}\right)^{1/2}{\epsilon}^R(x),~~~
{\cal E}^L(x,r)= \left(2lr\right)^{1/2} {\eta}^L(x), \label{strp}\end{aligned}$$ one can find that the bulk supersymmetry transformation reduces to that for ${\cal N}=1$ conformal supergravity in four dimensions with $\epsilon$ and $\eta$ playing the roles of parameters for supersymmetry and special supersymmetry transformations, respectively [@fra; @bala], $$\begin{aligned}
\delta {e}_\mu^{~a} &=&-\frac{1}{2}
\overline{\psi}_\mu\gamma^{a}{\epsilon},~~
\delta {\psi}_\mu
= {\nabla}_\mu {\epsilon} -\frac{3}{4}A_\mu \gamma_5
{\epsilon}-\gamma_\mu {\eta},~~ \delta {A}_\mu =
i\left(\overline{\psi}_\mu\gamma_5{\eta} -\overline{\chi}_\mu
\gamma_5\epsilon \right), \label{desut}\end{aligned}$$ where all the spinorial quantities, $\psi_\mu (x)$, ${\chi}_\mu
(x)$ ${\epsilon} (x)$ and ${\eta}(x)$ are Majorana spinors constructed from their chiral components $\psi^R_\mu (x)$, $\chi^L_\mu (x)$, ${\epsilon}^R (x)$ and ${\eta}^L(x)$.
As for other local symmetries of five-dimensional gauged supergravity, it has been proved that for any domain wall solution of the following form which asymptotically approaches the $AdS_5$ solution (\[metrican\]), $$\begin{aligned}
ds^2&=& G_{MN} dX^M d X^N=\frac{l^2}{r^2} \left[g_{\mu\nu}(x,r)
dx^\mu dx^\nu - dr^2\right],
\label{dw1}
\end{aligned}$$ the diffeormorphism symmetry preserving its above form must be a combination of the four-dimensional diffeomorphism symmetry and the Weyl symmetry [@imbi], $$\begin{aligned}
\delta g_{\mu\nu} (x,r) = 2\sigma (x) \left(
1-\frac{1}{2}r\partial_r\right) g_{\mu\nu}(x,r)+ \nabla_\mu
\xi_\nu (x,r) +\nabla_\nu \xi_\mu (x,r). \label{dedw}\end{aligned}$$ The $U(1)$ bulk gauge symmetry under the transformation $\delta
{\cal A}_M (x,r)=\partial_M \Lambda (x,r)$, automatically reduces to the $U(1)$ chiral (or equivalently axial) vector gauge symmetry on the $AdS_5$ boundary.
The above fact indicates that the on-shell five-dimensional gauged supergravity near the $AdS_5$ vacuum configuration leads to the off-shell conformal supergravity on the $AdS_5$ boundary. Therefore, this has provided the justification that the superconfromal anomaly of a supersymmetric gauge theory in a conformal supergravity background can be extracted from the five-dimensional gauged supergravity.
Holographic Superconformal Anomaly
==================================
Holographic Chiral Anomaly
--------------------------
The holographic origin of the R-symmetry anomaly is the Chern-Simons (CS) five-form term in the gauge supergravity [@witt1; @chu]. For the ${\cal N}=8$ $SO (6)\cong SU(4)$ gauged supergravity in five dimensions, the CS term is [@gst] $$\begin{aligned}
S_{\rm CS}[{\cal A}] &=& \frac{l^3}{48 \pi G^{(5)}}\int \mbox{Tr}
\left[ {\cal A}
(d {\cal A})^2+\frac{3}{2}{\cal A}^3d {\cal A} +\frac{3}{5} {\cal
A}^5\right]
\nonumber\\
&=&\frac{l^3}{48 \pi G^{(5)}}\int \mbox{Tr}\left( {\cal A} {\cal
F}^2-\frac{1}{2}{\cal A}^3 {\cal F}
+\frac{1}{10} {\cal A}^5\right)\nonumber\\
&=& \frac{l^3}{192 \pi G^{(5)}}\int d^5x \epsilon^{MNPQR}
d^{abc}\left({\cal A}_M^a {\cal F}_{NP}^b {\cal F}_{QR}^c- f^{ade}
{\cal A}_M^d {\cal A}_N^e {\cal A}_P^b {\cal F}_{QR}^c
\right.\nonumber\\
&& \left. +\frac{2}{5} f^{ade}f^{efg} {\cal A}_M^d {\cal A}_N^f
{\cal A}_P^g {\cal A}_Q^b {\cal A}_R^c \right),\end{aligned}$$ where ${\cal A}_M$ and ${\cal F}_{MN}$ are the $SU(4)$ gauge field and the field strength. A CS term has a particular feature: its gauge variation is a total derivative. Therefore, under the bulk gauge transformation, $$\begin{aligned}
\delta {\cal A}_M^a (x,r)=\left[D_M V(x,r)\right]^a,\end{aligned}$$ $V(x,r)=V^a(x,r) t^a$ being a gauge transformation parameter, the other gauge field relevant terms are gauge invariant, but the gauge transformation of CS term leaves a total derivative term, $$\begin{aligned}
\delta_V S_{\rm SUGRA}[{\cal A},\cdots]&=&
\delta_V S_{\rm CS}[{\cal A}]=\int d Q_4^1 (V,{\cal A})\nonumber\\
&=&\frac{l^3}{48\pi G^{(5)}}\int
d\mbox{Tr}\left[{V}\,d\left(AdA+\frac{1}{2} A^3\right)\right].
\label{csgv}\end{aligned}$$ Choosing the boundary behaviour of the bulk gauge transformation parameter as the bulk gauge field, $\left.V(x,r)\right|_{r\to 0}=
v(x)$ and making use of the AdS/CFT correspondence (\[acc4\]), $$\begin{aligned}
&& \left.\delta_V S_{\rm SUGRA}[{\cal A},\cdots]\right|_{{\cal
A}_M\to A_\mu,\,V\to v } = \delta_v \Gamma_{\rm SYM}
[A_\mu^a,\cdots]= \int d^4x \frac{\delta \Gamma} {\delta
A_\mu^a (x) }\delta A_\mu^a (x)\nonumber\\
&=& \int d^4x j^{a\mu}\delta A_\mu^a (x)
=\int d^4x j^{a\mu}(x)[D_\mu v(x)]^a
=-\int d^4x v^a (x)[D_\mu j^{\mu} (x)]^a. \label{gtr}\end{aligned}$$ one can obtain from Eqs.(\[csgv\]) and (\[gtr\]) $$\begin{aligned}
[D{}^\star j (x)]^a =-\frac{l^3}{48\pi G^{(5)}}\mbox{Tr}t^a\left[
F^2-\frac{1}{2}\left(A^2F+FA^2+AFA\right)+\frac{1}{2} A^4\right].\end{aligned}$$ Considering the following relations among the $AdS_5$ radius $l$, string coupling $g_s$, the number $N$ of $D3$-branes, the five- and ten-dimensional gravitational constants related by the compactification of the type IIB supergravity on $S^5$ of radius $l$ [@agmo], $$\begin{aligned}
G^{(5)}=\frac{G^{(10)}}{\mbox{Volume}\,
(S^5)}=\frac{G^{(10)}}{l^5\pi^3}, ~~~ G^{(10)}=8\pi^6g^2_s,
~~~l=\left(4\pi N g_s\right)^{1/4},\end{aligned}$$ one immediately recognizes the holographic Bardeen (consistent) anomaly, $$\begin{aligned}
[D_\mu j^{\mu} (x)]^a =-\frac{N^2}{24\pi^2}e^{-1}
\epsilon^{\mu\nu\lambda\rho}
\partial_\mu \mbox{Tr}t^a\left(A_\nu \partial_\lambda A_\rho+
\frac{1}{2}A_\nu A_\lambda A_\rho \right). \label{nfour}\end{aligned}$$
For the ${\cal N}=2$ supersymmetric Yang-Mills theory, its $R$-symmetry group is $U(2)_R{\cong} SU(2)_R\times U(1)_R$. It is the $U(1)_R$ that becomes anomalous. The dual gravitational theory is the five-dimensional $SU(2)\times U(1)$ gauged ${\cal N}=4$ supergravity. The holographic chiral $U(1)_R$ anomaly comes from the $SU(2)\times U(1)$-mixed CS term in the gauged supergravity [@awada; @ferr], $$\begin{aligned}
S_{\rm CS}[{\cal W}, {\cal A},\cdots] &=& \frac{l^3}{64\pi
G^{(5)}} \int \mbox{Tr}\left({\cal G}\wedge {\cal G}\right)\wedge
{\cal A},\end{aligned}$$ where ${\cal W}$ and ${\cal A}$ are the $SU(2)$ and $U(1)$ gauge fields, and ${\cal G}$ the $SU(2)$ field strength. The reduction of the bulk $U(1)$ gauge transformation $\delta {\cal A}=d V$ to the $AdS_5$ boundary leads to the holographic $U_R(1)$ anomaly, $$\begin{aligned}
\partial_\mu \left(e j^\mu\right)=-\frac{N^2}{32\pi^2}
\epsilon^{\mu\nu\lambda\rho} \mbox{Tr}\left( G_{\mu\nu}
G_{\lambda\rho}\right). \label{ntwo}\end{aligned}$$ The justification that the boundary value $A_\mu(x)$ of the bulk gauge field ${\cal A}_{M}$ is considered as the external chiral (or axial) gauge field in four dimensions is implied from the boundary reductions of the bulk covariant derivative and of the supersymmetric transformation listed in Eqs.(\[rega\]) and (\[desut\]).
However, Eqs.(\[nfour\]) and (\[ntwo\]) do not contain the gravitational background contribution shown in the general expression (\[sca\]). In fact, for ${\cal N}=4$ supersymmetric Yang-Mills theory, there exists no gravitational contribution. The reason is that the field contents of ${\cal N}=4$ SYM can be considered as one ${\cal N}=1$ vector multiplet plus three chiral multiplets in the adjoint representation of $SU(N)$ and according to Eq.(\[cc\]) this yields $c=a=(N^2-1)/4$ [@blau]. Thus the CS term composed of the $SU_R(4)$ gauge field is fully responsible for the holographic source of the chiral R-symmetry anomaly. For the general ${\cal N}=1,2$ supersymmetric gauge theories, Eq.(\[cc\]) shows that usually $a\neq c$, hence the gravitational background contribution to the chiral anomaly should arise. Its absence implies that the five-dimensional gauged supergravity (or the type IIB supergravity in $AdS_5\times X^5$ background) is only the lowest order approximation to type IIB superstring theory) and corresponds only to the leading order of the large-$N$ expansion of supersymmetric gauge theory. In Ref.[@ahar] it was shown that for an ${\cal N}=2$ supersymmetric $USp(2N)$ gauge theory coupled to two hypermultiplets in the fundamental and antisymmetric tensor representations of the gauge group, respectively, the gravitational background part of the holographic chiral anomaly does originate from a mixed CS term. However, this CS terms is obtained from the compactification on $S^3$ of the Wess-Zumino term describing the interaction of the R-R 4-form field with eight $D7$-branes and one orientifold 7-plane system. Specifically, this gravitational background term is at the subleading $N$-order rather than the leading $N^2$-order in the large-$N$ expansion of the ${\cal N}=2$ supersymmetric $USp(2N)$ gauge theory. This fact exposes the limitation of the gauged supergravity in providing an equivalent physical description to the supersymmetric gauge theory.
Holographic Weyl Anomaly
------------------------
The holographic origin of the Weyl anomaly of a supersymmetric gauge theory lies in the $AdS_5$ boundary behaviour of the on-shell action of the gauged supergravity. Due to the infinity of the boundary, the on-shell action of the five-dimensional gauged supergravity in $AdS_5$ background suffers from the infrared divergences when approaching the boundary. Therefore, one must perform a so-called “ holomorphic renormalization” [@bian1]. That is, first introducing an IR cut-off when one integrate over the radial (fifth) coordinate to evaluate the on-shell action, then similar to dealing with the UV divergence in a renormalizable quantum field theory, defining a counterterm according to a preferred renormalization condition to cancel the IR divergence, finally removing the cut-off to get the renormalized on-shell action for the gauged supergravity. Specifically, Eq.(\[dedw\]) shows that the bulk diffeomorphism symmetry of the gauged supergravity decomposes into the diffeomorphism symmetry and the Weyl symmetry on the boundary [@imbi]. These two symmetries cannot be preserved simultaneously in implementing the holomorphic renormalization. Thus if one requires the diffeomorphism symmetry preserved, the holographic Weyl anomaly of a supersymmetric gauge theory will arise.
We take the ${\cal N}=8$ $SO(6)$ gauged supergravity in five dimensions as an example, and choose the truncated action consisting only of the Einstein-Hilbert action and the non-vanishing scalar potential [@hesk; @bian1], $$\begin{aligned}
8\pi G^{(5)}E^{-1} {\cal L}_{\rm trunc} =-\frac{1}{2}{\cal
R}-P[\phi]. \label{truaction}\end{aligned}$$ The corresponding Einstein equation is $$\begin{aligned}
{\cal R}_{MN}-\frac{1}{2} {\cal R} G_{MN}=P[\phi=0] G_{MN}.
\label{ee}\end{aligned}$$ The solution to the Einstein equation (\[ee\]) is the domain wall (\[dw1\]). It should be emphasized that the existence of scalar field is necessary for the domain wall solution (\[dw1\]). Otherwise there will be no non-trivial vacuum configurations and the domain wall solution does not exist. Near the $AdS_5$ boundary ($r\rightarrow 0$), the solution $g_{\mu\nu}
(x,r)$ admits the following expansion [@hesk; @bian1], $$\begin{aligned}
g_{\mu\nu}(x,r)&=&
g_{(0)\mu\nu}(x)+g_{(2)\mu\nu}(x)\frac{r^2}{l^2}
\nonumber\\
&&+ \left[g_{(4)\mu\nu}+ h_{1(4)\mu\nu}\ln \frac{r^2}{l^2}
+h_{2(4)\mu\nu}\left(\ln \frac{r^2}{l^2}\right)^2\right]
\left(\frac{r^2}{l^2}\right)^2 +\cdots. \label{expa}\end{aligned}$$ Substituting (\[expa\]) into the Einstein equation (\[ee\]) with the cosmological constant provided by the value of the scalar potential at the critical point $\phi=0$, one can determine the coefficients $g_{(2k)\mu\nu}$, $h_{\mu\nu}$ in terms of $g_{(0)\mu\nu}$ [@hesk; @bian1], $$\begin{aligned}
g_{(2)\mu\nu} &=&\frac{l^2}{2}\left( R_{\mu\nu}-\frac{1}{6}R
g_{(0)\mu\nu}
\right),\nonumber\\
h_{1(4)\mu\nu}
&=&\frac{l^4}{8}\left(R_{\mu\lambda\nu\rho}R^{\lambda\rho}
+\frac{1}{6}\nabla_\mu\nabla_\nu R-
\frac{1}{2}\nabla^2R_{\mu\nu}-\frac{1}{3}R R_{\mu\nu}\right)\nonumber\\
&&+\frac{l^4}{32}g_{(0)\mu\nu}\left(\frac{1}{3}\nabla^2
R+\frac{1}{3} R^2
-R_{\lambda\rho}R^{\lambda\rho}\right), \nonumber\\
h_{2(4)\mu\nu} &=& 0,~~~\mbox{Tr}g_{(4)}
=\frac{1}{4}\mbox{Tr}\left(g_{(2)}\right)^2,
\nonumber\\
\nabla^\nu g_{(4)\mu\nu} &=& \nabla^\nu\left\{
-\frac{1}{8}\left[\mbox{Tr}g_{(2)}^2-\left(\mbox{Tr}g_{(2)}\right)^2
\right] g_{(0)\mu\nu}+\frac{1}{2}\left(g_{(2)}^2\right)_{\mu\nu}
-\frac{1}{4}g_{(2)\mu\nu}\mbox{Tr} g_{(2)}\right\},\end{aligned}$$ where $\mbox{Tr}g_{(2)}=g_{(0)}^{\mu\nu}g_{(2)\mu\nu}$, $R_{\mu\nu\lambda\rho}$, $R_{\mu\nu}$ and $R$ are the Riemannian tensor, Ricci tensor and curvature scalar corresponding to $g_{(0)\mu\nu}$, respectively. Inserting above solution into Eq.(\[truaction\]) to evaluate the on-shell action, one finds that it is divergent when approaching the $AdS_5$ boundary. Thus one must perform the radial integration in a finite domain by introducing a cut-off $r=\epsilon >0$ and get the regularized on-shell action, $$\begin{aligned}
S_{\rm reg} &=& \frac{1}{8\pi G^{(5)}}\int_{r=\epsilon >0} d^5X
E(x,r)\left(-\frac{1}{2}{\cal R}-P[\phi=0]\right)\nonumber\\
&=& \frac{l^5}{8\pi G^{(5)}}\int_{\epsilon} \frac{dr}{r^5}\int
d^4x \sqrt{g (x,r)}\,
\frac{2}{3}P[\phi=0]\nonumber\\
&=& -\frac{l^5}{8\pi G^{(5)}}\int_{\epsilon} \frac{dr}{r^5}\int
d^4x \sqrt{g (x,r)}\,
\frac{4}{3}\Lambda = \frac{l^3}{\pi G^{(5)}}\int_{\epsilon}
\frac{dr}{r^5}\int
d^4x \sqrt{g (x,r)}\nonumber\\
&=&\frac{l^3}{\pi G^{(5)}}\int
d^4x \sqrt{g_0(x)} \int_{\epsilon} \frac{dr}{r^5}
\left\{1+\frac{r^2}{2l^2}\mbox{Tr}g_{(2)}
+\frac{r^4}{4l^4}\left[\left( \mbox{Tr}g_{(2)}\right)^2
-\mbox{Tr}g^2_{(2)}\right]+\cdots\right\}\nonumber\\
&=&\frac{l^3}{\pi G^{(5)}}
\int d^4x\sqrt{g_{(0)}}\left[-\frac{1}{4\epsilon^4}
-\frac{1}{12\epsilon^2}R
-\frac{1}{16}\ln\frac{\epsilon}{l}\,\left(R_{\mu\nu}R^{\mu\nu}
-\frac{1}{3}R^2 \right) +{\cal L}_{\rm
finite}\right]\nonumber\\
&=& S_{\epsilon^{-4}}+S_{\epsilon^{-2}}+S_{\ln\epsilon}+S_{\rm
finite}. \label{regaction}\end{aligned}$$ In writing down (\[regaction\]) the Einstein equation (\[ee\]), the identification $\Lambda=-2P[\phi=0]=-{6}/{l^2}$ and the matrix operation $$\begin{aligned}
\sqrt{\det (1+A)}&=&\exp\left[\frac{1}{2}\mbox{Tr} \ln
\left(1+A\right)\right] =\exp\left[\frac{1}{2}\mbox{Tr} \left(A-
\frac{1}{2}A^2+\cdots \right)\right]
\nonumber\\
&=& 1+\frac{1}{2}\mbox{Tr}A+\frac{1}{4} \left[
\left(\mbox{Tr}A\right)^2 -\mbox{Tr}A^2\right]+\cdots,\end{aligned}$$ have been employed. ${\cal L}_{\rm fin}$ consists of the terms standing the $\epsilon{\rightarrow}0$ limit.
To get the renormalized on-shell action action, one must first define a subtracted action by introducing the counterterms to cancel the IR divergence in the limit $\epsilon\rightarrow 0$, $$\begin{aligned}
S_{\rm sub}[g_{(0)\mu\nu},\epsilon^2/l^2,\cdots]=S_{\rm reg}.
+S_{\rm counter}\end{aligned}$$ A holographically renormalized on-shell action is yielded after removing the regulator, $$\begin{aligned}
S_{\rm
ren}[g_{(0)\mu\nu},\cdots]=\lim_{\epsilon{\rightarrow}0}S_{\rm
sub}[g_{(0)\mu\nu},\epsilon^2/l^2,\cdots]\end{aligned}$$ In adding the counterterm, there arise a finite ambiguity similar to cancelling the UV divergence in a perturbative quantum field theory. This ambiguity can be fixed by the symmetry requirement. According to Eq.(\[dedw\]), the bulk diffeomorphism transformation converts into a four-dimensional Weyl and a diffeormorphism transformations near the $AdS_5$ boundary. Requiring the four-dimensional diffeomorphism symmetry preserved in performing subtraction, one can introduce the following counterterm similar to the minimal subtraction in the dimensional regularization of the perturbative quantum field theory [@hesk; @bian1], $$\begin{aligned}
S_{\rm counter} &=& \frac{l^3}{\pi G^{(5)}}
\int d^4x\sqrt{g_{(0)}}\left[\frac{1}{4\epsilon^4}
+\frac{1}{12\epsilon^2}R
+\frac{1}{16} \left(R_{\mu\nu}R^{\mu\nu} -\frac{1}{3}R^2
\right)\ln\frac{\epsilon}{l}\right],
\end{aligned}$$ and consequently, the renormalized on-shell action is just the finite part of the regularized one, $$\begin{aligned}
S_{\rm ren} &=& \frac{l^3}{\pi G^{(5)}} \int d^4x
\sqrt{g_{(0)}}{\cal L}_{\rm finite}.\end{aligned}$$ There are several ways to extract the Weyl anomaly from the renormalized action [@hesk; @bian1]. The most straightforward way is considering the scale transformation of the regularized action, i.e., choosing the parameter $\sigma (x)$ of the Weyl transformation as a constant $\sigma$. The regularized action is invariant under the combination of two transformations, $\delta
g_{(0)\mu\nu}=2\sigma g_{(0)\mu\nu}$, $\delta \epsilon=2 \sigma
\epsilon$. That is [@hesk], $$\begin{aligned}
\left(\delta_{g_0}+\delta_\epsilon\right) S_{\rm reg}
=\left(\delta_{g_0}+\delta_\epsilon\right) \left(
S_{\epsilon^{-4}}+S_{\epsilon^{-2}}+S_{\ln\epsilon}+S_{\rm
finite}\right) =0.\end{aligned}$$ However, it has been found that [@hesk] $$\begin{aligned}
\delta_{g_0} \left(S_{\epsilon^{-4}}+S_{\epsilon^{-2}}\right)
=0,~~~ \delta_\epsilon \left(
S_{\epsilon^{-4}}+S_{\epsilon^{-2}}\right) =0,~~~ \delta_{g_0}
S_{\ln\epsilon}=0, ~~~ \delta_\epsilon S_{\rm finite} =0.\end{aligned}$$ This leads to $$\begin{aligned}
\delta_{g_0} S_{\rm finite}&=&\delta_{g_0} S_{\rm ren}=\int d^4x
\sqrt{g_{(0)}}\langle T^{\mu}_{~\mu}\rangle\sigma
=-\delta_\epsilon S_{\ln\epsilon}\nonumber\\
&=&\frac{l^3}{8\pi G^{(5)}}\int d^4x \sqrt{g_{(0)}}
\left(R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^2 \right)\sigma,\end{aligned}$$ and yields the Weyl anomaly in the gravitational background [@hesk; @bian1], $$\begin{aligned}
\langle T^\mu_{~\mu}\rangle =
\frac{N^2}{4\pi^2}\left(R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^2\right).
\label{trace}\end{aligned}$$ It can be further rewritten as the combination of the $A$- and $B$-type anomalies, i.e, the sum of the Euler number density and the square of the Weyl tensor [@hesk; @deser], $$\begin{aligned}
\langle T^\mu_{~\mu}\rangle &=& -\frac{N^2}{\pi^2} \left(E_4+W_4\right), \nonumber\\
E_4 &=&
\frac{1}{8}\widetilde{R}_{\mu\nu\lambda\rho}\widetilde{R}^{\mu\nu\lambda\rho}
=\frac{1}{8}\left( R^{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}
-4R^{\mu\nu}R_{\nu\nu}+R^2 \right),\nonumber\\
W_{4} &=& -\frac{1}{8}
{C}_{\mu\nu\lambda\rho}{C}^{\mu\nu\lambda\rho} =-\frac{1}{8}\left(
R^{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}
-2R^{\mu\nu}R_{\mu\nu}+\frac{1}{3}R^2 \right).\end{aligned}$$ It is the trace anomaly of the ${\cal N}=4$ supersymmetric Yang-Mills theory in the external gravitational field at the leading order of large-$N$ expansion [@hesk].
The gauge field contribution to the Weyl anomaly can be similarly calculated when switching on the gauge field sector of the ${\cal
N}=8$ gauged supergravity in five dimensions [@bian1]. If one considers only the bilinear terms in the gauge fields, the $SO(6)$ gauge field can be approximated by some uncoupled Abelian sectors ${\cal A}_M$ [@bian2]. The solution to the gauge field equation near the $AdS_5$ boundary is [@bian1] $$\begin{aligned}
{\cal A}_\mu (x,r)= A_\mu (x)+\frac{r^2}{l^2}\left[ A_{(2)\mu}(x)+
\widetilde{A}_{(2)\mu}(x) \ln \frac{r^2}{l^2}\right]+\cdots,
\label{ge}\end{aligned}$$ where $A_{(2)\mu}(x)$ and $\widetilde{A}_{(2)\mu}(x)$ can be expressed as the functional of $A_\mu (x)$ when inserting Eq.(\[ge\]) into the classical equation of motion for ${\cal
A}_\mu$. The regularized action of the gauge field sector is [@bian1] $$\begin{aligned}
S_{\rm reg}=\frac{l^3}{8\pi G^{(5)}}\int d^4x \sqrt{g_{(0)}}
\left[\left(-\frac{1}{4} F_{\mu\nu}F^{\mu\nu}\right)\ln
\frac{\epsilon}{l} +{\cal L}_{\rm finite}\right]\end{aligned}$$ The gauge field part of the Weyl anomaly given in Eq.(\[sca\]) can be derived in the same way as the gravitational case.
Holographic Supersymmetry Current Anomaly
-----------------------------------------
The production of the super-Weyl anomaly of a four-dimensional supersymmetric gauge theory from the five-dimensional gauged supergravity lies in two aspects. First, as a supersymmetric theory, the supersymmetry transformation of the Lagrangian of the gauged supergravity must be composed of the total derivative terms. These terms cannot be naively ignored due to the existence of the boundary $AdS_5$. Second, near the $AdS_5$ boundary the bulk supersymmetry transformation decomposes into the four-dimensional supersymmetry and super-Weyl transformations as shown in Eq.(\[desut\]). If we require four-dimensional supersymmetry preserved on the boundary, the total derivative terms should yield the anomaly of the supersymmetry current via the AdS/CFT correspondence. Therefore, the key point is to calculate the supersymmetric variation of the gauged supergravity and get the total derivative terms. Then putting these terms on the $AdS_5$ boundary, one should find the holographic supersymmetry current anomaly.
We have worked out these total derivative terms of the simplest case in the five dimensional gauged supergravities, the ${\cal
N}=2$ $U(1)$ gauged supergravity whose Lagrangian is given in Eq.(\[gaugedfm\]). The concrete calculation is very lengthy and has been displayed in a great detail in Ref.[@chch]. The variation of the Lagrangian (\[gaugedfm\]) under the supersymmetric transformation (\[twostm\]) yields $$\begin{aligned}
\delta S &=& \frac{1}{8\pi G^{(5)}}\int d^5x E \nabla_M \left(
-\frac{9il}{16}
\overline{\cal E}^i\Psi_{N i}{\cal F}^{MN}
-\frac{1}{2}\overline{\cal E}^i\Gamma^{MNP}\nabla_N
\Psi_{P i}\right.\nonumber\\
&& + \frac{3}{8}
\overline{\cal E}^i\Gamma^{MNP}
\Psi_{P}^j\delta_{ij}{\cal A}_N
- \frac{3il}{32}E^{-1}\epsilon^{MNPQR}
\overline{\cal E}^i\Gamma_{R}
\Psi_{N i}{\cal F}_{PQ}\nonumber\\
&&\left.+\frac{9}{4}\overline{\cal E}^i\Gamma^{MN}
\Psi_{N}^i\delta_{ij}
+\frac{l^2}{16}E^{-1}
\epsilon^{MNPQR}
\overline{\cal E}^i \Psi_{R}{\cal A}_N {\cal F}_{PQ}\right).
\label{var3}\end{aligned}$$ In deriving above derivative terms, one must bear in mind that in the Lagrangian (\[gaugedfm\]) the $U(1)$ gauge field is imaginary and the gravitino field is an $SU(2)$ symplectic Majorana spinor, $$\begin{aligned}
{\cal A}_M^\star &=& -{\cal A}_M, ~~~ \Psi^i =
C^{-1}\Omega^{ij}\overline{\Psi}_j^T=C^{-1}\overline{\Psi}^{iT},
~~~\overline{\Psi}^i=-\Psi^{iT}C,\nonumber\\
&& \overline{\Psi}^i\Gamma_{M_1\cdots M_n}\Phi_i =
-\Psi^{iT}C\Gamma_{M_1\cdots M_n}C^{-1}\overline{\Phi}_i^T\nonumber\\
&=&\left\{\begin{array}{l}
-\Psi^{iT}\Gamma_{M_1\cdots M_n}\overline{\Phi}^T_i
=\overline{\Phi}_i\Gamma_{M_1\cdots M_n} \Psi^i
=-\overline{\Phi}^i\Gamma_{M_1\cdots M_n} \Psi_i,~~ n=0,1,4,5,\\
\Psi^{iT}\Gamma_{M_1\cdots M_n}\overline{\Phi}^T_i
=-\overline{\Phi}_i\Gamma_{M_1\cdots M_n} \Psi^i
=\overline{\Phi}^i\Gamma_{M_1\cdots M_n} \Psi_i,~~ n=2,3,
\end{array} \right.\, .\end{aligned}$$ The convention for the $\Gamma$-matrix in five dimensions and the frequently used relations are chosen as the following, $$\begin{aligned}
\Gamma_{MN}&=&\frac{1}{2}[\Gamma_M,\Gamma_N], ~~
\Gamma^{MNP}=-\frac{1}{2!}E^{-1}\,
\epsilon^{MNPQR}\Gamma_{QR},\nonumber\\
\Gamma^{MNPQ}&=& E^{-1}\, \epsilon^{MNPQR}\Gamma_{R},
~~\Gamma_{MNPQR}=E\, \epsilon_{MNPQR}.
\nonumber \\
\Gamma_{MN}\Gamma_{PQ}
&=& E\,\epsilon_{MNPQR}\Gamma^R- \left(G_{MP}
G_{NQ}-G_{MQ}G_{NP}\right),
\nonumber\\
\Gamma_{M}\Gamma_{NP} &=& \Gamma_{MNP}
+G_{MN}\Gamma_P-G_{MP}\Gamma_N, \nonumber\\
\Gamma^{MNP}\nabla_N \nabla_P \Psi_i &=&
\frac{1}{2}\Gamma^{MNP}[\nabla_N, \nabla_P] \Psi_i =
\frac{1}{8}\Gamma^{MNP}{\cal R}_{NP AB}\Gamma^{AB}\Psi_i.
\end{aligned}$$ The following Ricci and Bianchi identities for the Riemannian curvature tensor and the $U(1)$ field strength are employed in the calculation, $$\begin{aligned}
\epsilon^{MNPQR} {\cal R}_{SPQR}=0, ~~~
\epsilon^{MNPQR}\nabla_N {\cal R}_{STPQ}=0,~~~
\epsilon^{MNPQR}\nabla_N {\cal F}_{QR}=0.
\end{aligned}$$ Specifically, due to the nocommutativity between $\nabla_M$ and $\Gamma_{M_1\cdots M_n}$, we reiteratively make the operation, $$\begin{aligned}
\Gamma_{M_1\cdots M_n} \nabla _M (\cdots)
&=&\left[\Gamma_{M_1\cdots M_n},\nabla _M\right] (\cdots)
+\nabla_M \left[\Gamma_{M_1\cdots M_n} (\cdots)\right].
\end{aligned}$$ In this case it is convenient to choose the inertial coordinate system, i.e. the Christoffel symbol $\Gamma^{M}_{~NP}=0$. Consequently, the metricity condition leads to $\partial_M E_N^{~A}=0$, and hence the modified spin connection $\Omega_{M AB}$ contains only the quadratic fermionic terms. This simplifies the calculation greatly since we retain only the quadratic fermionic terms in calculating the supersymmetry variation.
The holographic super-Weyl anomaly can be extracted from above total derivative terms. First, we take into account the radial coordinate dependence of bulk fields and of the supersymmetry transformation parameter ${\cal E}^i$ near the $AdS_5$ boundary as well as the connection between five- and four-dimensional $\gamma$-matrices listed in (\[refield\]), (\[rega\]) and (\[strp\]). Second, to avoid the possible IR divergence due to the infinite $AdS_5$ boundary, we must integrate over the radial coordinate to the cut-off $r=\epsilon$ and then take the limit $\epsilon\to 0$. Finally, we use the fact that the metric on the boundary should be the induced metric [@hesk; @bian1] $$\begin{aligned}
{g}_{\mu\nu}(x)=\left.\frac{l^2}{\epsilon^2}{g}_{\mu\nu}(x,\epsilon)
\right|_{\epsilon\to 0}\end{aligned}$$ rather than ${g}_{\mu\nu}(x,\epsilon)$ [@bian1]. With all these considerations together, we have found that the non-vanishing contribution comes only from the term $E^{-1}\epsilon^{MNPQR}\overline{\cal E}^i\Gamma_{R}
\Psi_{N i}{\cal F}_{PQ}$. Therefore, we obtain [@chch] $$\begin{aligned}
\delta S &=& \frac{3il^3}{8\times 32\pi G^{(5)}}
\int d^4 x \epsilon^{\mu\nu\lambda\rho}
F_{\nu\lambda}\overline{\eta}\gamma_\rho\chi_\mu,
\label{fvar}
\end{aligned}$$ where ${\chi}_\mu$ is the Majorana spinor constructed from the left-handed spinor $\chi_\mu^L$ given in (\[refield\]).
Eq.(\[fvar\]) definitely leads to the super-Weyl anomaly in the context of AdS/CFT correspondence (\[acc4\]) since it is proportional to the special supersymmetry transformation parameter $\eta$. Inserting the explicit form $\chi_\mu$ expressed in terms of the gravitino $\psi_\mu$ in four-dimensional ${\cal N}=1$ conformal supergravity, we have $$\begin{aligned}
\delta S &=& \int d^4x \overline{\eta}\gamma^\mu s_\mu \nonumber\\
&=& \frac{3il^3}{8\times 32\pi G^{(5)}}\int d^4 x
\epsilon^{\mu\nu\lambda\rho}
F_{\nu\lambda}\overline{\eta}\gamma_5\gamma_\rho \gamma^\alpha
\left[ \frac{1}{3} \left(D_\mu\psi_\alpha- D_\mu\psi_\alpha\right)
-\frac{i}{6}\epsilon_{\mu\alpha\sigma\delta}\gamma_5 D^\sigma\psi_\delta
\right]\nonumber\\
&=& -\frac{l^3}{8\times 16\pi G^{(5)}}\int d^4x
\left[F^{\mu\nu}D_\mu \psi_\nu
+\epsilon^{\mu\nu\lambda\rho}\gamma_5 F_{\mu\nu} D_\lambda\psi_\rho
\right.\nonumber\\
&&\left.+\frac{1}{2}\sigma^{\mu\nu} F_{\nu\lambda}
\left(D_\mu\psi^\lambda-D^\lambda\psi_\mu\right)\right],
\end{aligned}$$ where we have used the $\gamma$-matrix algebraic relations, $\gamma^\mu\gamma^\nu=g^{\mu\nu}-i\gamma^{\mu\nu}$, $\gamma_5\gamma^{\mu\nu}={i}\epsilon^{\mu\nu\lambda\rho}
\gamma_{\lambda\rho}/{2}$. The gauge field part of the holographic super-Weyl anomaly of the $SU(N)$ supersymmetric gauge theory at the leading-order of the large-$N$ expansion is thus yielded, $$\begin{aligned}
\gamma_\mu s^\mu &=& \frac{N^2}{64\pi^2}\left[F^{\mu\nu}D_\mu
\psi_\nu
+\epsilon^{\mu\nu\lambda\rho}\gamma_5 F_{\mu\nu} D_\lambda\psi_\rho
+\frac{1}{2}\sigma^{\mu\nu} F_{\nu\lambda}
\left(D_\mu\psi^\lambda-D^\lambda\psi_\mu\right)\right].
\label{gtra}\end{aligned}$$
There should also has a contribution from the external gravitational background shown in Eq.(\[sca\]), which was found long time ago [@abb]. The reason for having failed to reproduce the gravitational part is not clear to us yet, we have the following two speculations based on the process of deriving the gravitational background parts in both the holographic Weyl and chiral anomalies [@blau; @ahar].
The first intuitive argument, as mentioned before, is that the five-dimensional gauged supergravity (or the type IIB supergravity in $AdS_5\times X^5$ background) is only the lowest approximation to the type IIB superstring theory in $AdS_5\times X^5$ background. Thus, it is possible that the gravitational part cannot be revealed within the five-dimensional gauged supergravity itself, and one must consider the higher-order gravitational action such as the Gauss-Bonnet term generated from the superstring theory [@berg2]. The supersymmetry variation of the gauged supergravity containing the high-order gravitational term and the corresponding fermionic terms required by supersymmetry may lead to the gravitational contribution to the super-Weyl anomaly.
The other possible reason for the failure of getting the gravitational background contribution is that in Eq.(\[refield\]) only the leading-order of radial coordinate dependence of the bulk fields near the $AdS_5$ boundary is taken into account. As shown above in deriving the holographic Weyl anomaly, when one makes a complete near-boundary analysis and considers the asymptotic expansion of the bulk fields in terms of the radial coordinate beyond the leading-order until the emergence of the logarithmic term [@hesk; @bian1], the higher-order gravitational terms can appear in the on-shell action [@hesk], and they lead to the holographic Weyl anomaly composed of the $R_{\mu\nu} R^{\mu\nu}$ and $R^2$ terms. Therefore, it is also possible that the gravitational background part in the super-Weyl anomaly can arise if one takes into account the logarithmic term in the expansion of the on-shell bulk fields. In this case the on-shell action of the five-dimensional gauged supergravity should have the infrared divergence when approaching the $AdS_5$ boundary. One must perform the holographic renormalization to get the renormalized on-shell action [@bian1].
We have not realized whether there are any physical reasons for the difference between these two holographic contributions to the super-Weyl anomaly. The essence of the holographic anomaly is the anomaly inflow from the bulk theory to the $AdS_5$ boundary [@callan]. Thus the absence of the gravitational part might be relevant to the difference between the anomaly inflows contributed from the gravitational and gauge background fields.
Summary
=======
We have reviewed how the superconfromal anomaly multiplet of a supersymmetric gauge theory in a conformal supergravity background can be produced via the AdS/CFT correspondence. The type IIB supergravity in $AdS_5\times X^5$ background reduce to a gauged supergravity in five dimensions since such a background provides a compactification on $X^5$, thus the AdS/CFT correspondence implies that there should exist a holographic correspondence between the gauged supergravity in five dimensions and a four-dimensional $SU(N)$ supersymmetric gauge theory in certain phase at the large-$N$ limit. Based on this consideration, we make use of the fact that the five-dimensional gauged supergravity admits a classical $AdS_5$ solution preserving the full supersymmetry. Then it is found that around this $AdS_5$ vacuum configuration the supermultiplet of the on-shell five-dimensional gauged supergravity converts into the off-shell conformal supergravity multiplet in four dimensions. Therefore, the holographic relation between the $AdS_5$ gauged supergravity and the four-dimensional supersymmetric $SU(N)$ gauge theory at the large-$N$ limit can be established in the sense that the conformal supergravity on one hand has furnished a background for a classical superconformal gauge theory and on the other hand is a $AdS_5$ boundary theory for the five-dimensional gauged supergravity. Therefore, it is natural to reproduce the superconformal anomaly of a four-dimensional supersymmetric gauge theory from the $AdS_5$ gauged supergravity. But still it is amazing that these three distinct anomalies can be extracted in the framework of a five-dimensional gauged supergravity.
[**Acknowledgments:**]{} This work is supported by the Academy of Finland under the Project No.54023.
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[^1]: Contribution to “ Symmetries in Gravity and Field Theory " conference for Professor A. de Azcarraga’s 60th birthday, June 2003, Salmanca, Spain
|
---
abstract: 'In the present work we present the full treatment of scalar and vector cosmological perturbations in a non-singular bouncing universe in the context of metric $f(R)$ cosmology. Scalar metric perturbations in $f(R)$ cosmology were previously calculated in the Jordan frame, in the present paper we successfully use the Einstein frame to calculate the scalar metric perturbations where the cosmological bounce takes place in the Jordan frame. The Einstein frame picture presented corrects and completes a previous calculation of scalar perturbations and adds new information. Behavior of fluid velocity potential and the pure vector fluid velocity terms are elaborately calculated for the first time in $f(R)$ cosmology for a bouncing universe in presence of exponential gravity. It is shown that the vector perturbations can remain bounded and almost constant during the non-singular bounce in $f(R)$ gravity unlike general relativistic models where we expect the vector perturbations to be growing during the contracting phase and decaying during the expanding phase. The paper shows that the Einstein frame can be used for calculation of scalar and vector metric perturbations in a bouncing universe for most of the cases except the case of asymmetric non-singular bounces.'
author:
- |
Pritha Bari$^\dagger$, Kaushik Bhattacharya$^\ddagger$ [^1]\
Department of Physics, Indian Institute of Technology, Kanpur\
Kanpur 208016, India
title: '**Evolution of scalar and vector cosmological perturbations through a bounce in metric $f(R)$ gravity in flat FLRW spacetime**'
---
Introduction
============
Though inflation[@Starobinsky:1980te; @Guth:1982ec] has been tremendously successful in solving most of the problems of the standard big bang cosmology, the issue of singularity and trans-Planckian problem still remains [@Borde:1996pt; @Martin:2000xs; @Brandenberger:2012aj]. A possible solution to the above mentioned problems is to consider the existence of non-singular bouncing cosmologies [@Martin:2001ue; @Martin:2003sf; @Battefeld:2014uga; @Novello:2008ra; @Cai:2012va], in which the universe goes from a contracting phase to an expanding one through bounce without any singularity. Non-singular bounce also addresses the horizon and flatness problem[@Battefeld:2014uga]. If one does not want to introduce some exotic matter components in a 3-dimensionally flat Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime then a way of realizing bounce is modifying general relativity (GR) [@Abramo:2009qk; @Carloni:2005ii; @Paul:2014cxa; @Bhattacharya:2015nda; @Bamba:2013fha; @Bari:2018aac]. It is conjectured that GR may not be the unique, correct theory of gravity to describe geometry of space-time when the curvature scale is high. Modifications of the Einstein-Hilbert gravitational action by higher order curvature invariants is done in very strong gravity regimes, such as in very early universe. At high curvature limits, when bounce happens, modifications to GR is expected. In this paper we study perturbations in a bouncing cosmology where the theory of gravity is given by metric $f(R)$ theory. The $f(R)$ paradigm is important as various models of inflation and late time acceleration of the universe can be modelled on $f(R)$ gravity theories. Current observation has detected a cosmic acceleration starting after the matter domination. Modified gravity theories [@Myrzakulov:2013hca; @Clifton:2011jh; @Atazadeh:2006re; @Carroll:2003wy] have been used to explain this late time acceleration of the universe. Although $f(R)$ gravity [@Nojiri:2017ncd] [@Nojiri:2010wj] is only one amongst the many modified gravity models, but it is one of the simplest modifications of GR which can tackle various cosmological problems.
It is known that $f(R)$ theory of gravity can be analyzed [@Paul:2014cxa; @Maeda:1988ab; @Sotiriou:2008rp; @DeFelice:2010aj] in two conformal frames; the Jordan frame and the Einstein frame. In Jordan frame the theory is a higher derivative theory, because higher than two order of time derivatives appear in the field equation. The Jordan frame field equation is obtained by varying the $f(R)$ action with respect to the metric tensor [@Sotiriou:2008rp]. In $f(R)$ cosmology is is assumed that the problem of cosmological dynamics is fundamentally posed in the Jordan frame although one can work the cosmological dynamics also in the Einstein frame using the conformal transformation connecting the frames. An advantage of working in the Einstein frame is that the theory of gravity becomes GR and known techniques of GR evolution can be applied in the Einstein frame. One can always transform back to the Jordan frame after calculating cosmological dynamics in the Einstein frame. In previous studies this was the method followed [@Paul:2014cxa]. Einstein frame description of $f(R)$ is GR with an added scalar field that is minimally coupled to gravity and non-minimally coupled to matter. Einstein frame description of $f(R)$ gravity provides easier ways to tackle many problems in $f(R)$ gravity.
There have been many attempts to realize bounce in various $f(R)$ gravity models [@Bamba:2013fha] [@Carloni:2005ii]. It was pointed out in [@Bari:2018aac] that exponential $f(R)$ theory might be a good candidate theory which supports a bounce in flat FLRW metric. We will, hence, deal with exponential $f(R)$ gravity in this paper. The present work deals with scalar and vector perturbations in a bouncing universe guided by exponential gravity, the bounce takes place in the Jordan frame. The perturbations have been calculated in both the Jordan and Einstein frames and then the results are matched to gain insight into the nature of the conformal correspondence of the two frames. It has been shown that the Einstein frame can be used for the calculation of scalar cosmological perturbations for symmetric bounce in the Jordan frame. The conformal correspondence fails for asymmetrical bounces in the Jordan frame. No such difficulties arise for vector perturbations where the Einstein frame can be safely used for all the calculations in a much easier way. A part of scalar metric perturbation calculation in the Einstein frame was incompletely presented in an earlier work Ref. [@Paul:2014cxa] where the authors did not take into account the role of fluid velocity potential. In the present work we specify the complete and correct way of calculation of the scalar metric perturbations in a bouncing universe using the Einstein frame. We present the full nature of the scalar perturbations and the dynamics of the fluid velocity potential during a cosmological bounce. The next part of the paper shows the nature of vector metric perturbations during a non-singular bounce. In $f(R)$ cosmology the vector perturbations remains almost constant during the bounce which differs from GR results where the vector perturbations generally increases during the contraction phase [@Battefeld:2004cd]. Throughout the paper the role of the Einstein frame as an important frame for calculations has been emphasized. The issue about tensor perturbations in a bouncing universe will be addressed in a future work.
The material in the paper is presented in the following manner. The next section presents the background cosmological evolution in both the Jordan frame and Einstein frame. It also specifies the conformal transformation relating these frames. Section \[scalar\] introduces the scalar metric perturbations in both the frames. The Jordan frame result is first calculated and next the Einstein frame results are presented. The results of scalar perturbations are calculated for both a symmetric and asymmetric bounce in the Jordan frame. The topic of vector perturbations is taken up in section \[vp\]. The next and the last section concludes the present work with a brief summary of the results obtained.
Field equations of $f(R)$ gravity {#back}
=================================
In the following we work with the spatially flat maximally symmetric FLRW metric given by $$ds^2=-dt^2+a^2(t)d{\bf x}^2
\label{g}$$ where $t$ is the cosmological time, ${\bf x}$ is the Co-moving spatial coordinates and $a(t)$ is the scale-factor of the universe. In the domain of general relativity it is well known that a bouncing solution is possible only for spatially positively curved FLRW universe, if we do not want to include any exotic matter component in the scenario. But in metric $f(R)$ gravity and as shown in some previous works[@Paul:2014cxa; @Bhattacharya:2015nda], it is possible to have bouncing solution in spatially flat FLRW universe without invoking any exotic matter component for certain $f(R)$ theories, simplest of which being the $R+\alpha R^2$ gravity with a negative $\alpha$. The reason is that the extra curvature induced energy density and pressure terms can indeed produce the bouncing conditions. In this section we present the field equations in Jordan and Einstein frame respectively. We will later apply the formalism to study classical cosmological bounce phenomena [@Novello:2008ra].
The relevant cosmological equations in Jordan and Einstein frames {#jefr}
-----------------------------------------------------------------
The modified Friedmann equations for $f(R)$ theory in the Jordan frame are given as [@Sotiriou:2008rp]: $$\begin{aligned}
3 H^2 &=&\frac{\kappa}{F(R)} \rho_{\rm eff}\,,
\label{fried}\\
3H^{2}+2\dot{H} &=&-\frac{\kappa}{F(R)} P_{\rm eff}\,,
\label{2ndeqn}\end{aligned}$$ where $H$ is the conventional Hubble parameter defined as $H\equiv
\dot{a}/a$ and the constant $\kappa = 8\pi G$ where $G$ is the universal gravitational constant. In the above equations $$\begin{aligned}
F(R) \equiv \frac{df(R)}{dR}\,.
\label{fp}\end{aligned}$$ The dot specifies a derivative with respect to cosmological time $t$. The effective energy density, $\rho_{\rm eff}$, and pressure, $P_{\rm
eff}$, are defined as: $$\begin{aligned}
\rho_{\rm eff} \equiv \rho + \rho _{\rm curv}\,,\,\,\,\,\,
P_{\rm eff} \equiv P + P_{\rm curv}\,,
\label{epeff}\end{aligned}$$ where $\rho_{\rm curv}$ and $P_{\rm curv}$ are given by $$\begin{aligned}
\rho _{\rm curv} &\equiv& \frac{RF-f}{2\kappa}-\frac{3H\dot{R}
F_R(R)}{\kappa}\,,
\label{reff}\\
P_{\rm curv} &\equiv& \frac{\dot{R}^{2}F_{RR} + 2H\dot{R}F_R
+ \ddot{R}F_R }{\kappa} - \frac{RF-f}{2\kappa}\,,
\label{peff}\end{aligned}$$ which are curvature induced energy-density and pressure. In the above equations the subscript $R$ specifies derivatives with respect to the Ricci scalar $R$. The curvature induced thermodynamic variables exists in absence of any hydrodynamic matter. The conventional $\rho$ and $P$ are defined through $$\begin{aligned}
\label{tmunu}
T_{\mu \nu} = (\rho + P)u_\mu u_\nu + P g_{\mu\nu}\,, \end{aligned}$$ which has the information of hydrodynamic matter. In this article we assume the fluid to be barotropic so that its equation of state is $$\begin{aligned}
P=\omega \rho\,,
\label{eqns}\end{aligned}$$ where $\omega$ is a constant and its value is zero for dust and one-third for radiation. It must be noted that $u_\mu$ in Eq. (\[tmunu\]) is the 4-velocity of a fluid element and $u_\mu
u^\mu = -1\,.$
One can make a conformal transformation on the system of equations in the Jordan frame to recast the problem in the Einstein frame. The Einstein frame version of the cosmological dynamics sometimes becomes relatively easy to manage as in this version one deals with the known Einstein equations. The Einstein frame description of $f(R)$ gravity is obtained by the following conformal transformation, $$\begin{aligned}
\tilde{g}_{\mu \nu}=F(R)g_{\mu \nu}\,,
\label{gtilde}\end{aligned}$$ and simultaneously defining a new scalar field $\phi$ as $$\begin{aligned}
\phi = \sqrt{\frac{3}{2\kappa}} \ln F(R)\,.
\label{phidef}\end{aligned}$$ This scalar field plays an important role in the Einstein frame. The conformally transformed line element in the Einstein frame is $$\begin{aligned}
d\tilde{s}^2 = -d\tilde{t}^2 + \tilde{a}^2 d{\bf x}^2\,,
\label{dseins}\end{aligned}$$ where the time coordinate, $\tilde{t}$, and the scale factor, $\tilde{a}$, in the Einstein frame are related to their corresponding Jordan frame terms via the relations $$\begin{aligned}
d\tilde{t}= \sqrt{F(R)} \,dt\,\,\,\,\,{\rm and}\,\,\,\,\,
\tilde{a}(t)=\sqrt{F(R)} \,a(t)\,.
\label{aat}\end{aligned}$$ Using these transformations one can formulate the gravitational dynamics of $f(R)$ gravity in the Einstein frame in presence of matter and the scalar field $\phi$ acting as sources. The energy-momentum tensor in the Einstein frame, which is related to $T^{\mu \nu}$ in the Jordan frame, turns out to be $$\begin{aligned}
\tilde{T}_{\mu \nu} = (\tilde{\rho} +
\tilde{P})\tilde{u}_\mu \tilde{u}_\nu + \tilde{P} \tilde{g}_{\mu\nu}\,,
\label{tmnt}\end{aligned}$$ where $\tilde{\rho}=\rho/F^2(R)$, $\tilde{P}=P/F^2(R)$ and $\tilde{u}_\mu =
\sqrt{F(R)}u_\mu$. In the Einstein frame $\tilde{g}^{\mu\nu}\tilde{u}_\mu \tilde{u}_\nu=-1$. Except $\tilde{T}^{\mu \nu}$, the energy-momentum tensor for the scalar field also acts as source of curvature in the Einstein frame and it is given as $$\begin{aligned}
S^{\mu}_{\nu}=\partial_\alpha \phi \partial_\nu \phi\tilde{g}^{\alpha
\mu} - \delta^\mu_\nu {\mathcal L}(\phi)\,,
\label{smn}\end{aligned}$$ where the scalar field Lagrangian is $$\begin{aligned}
{\mathcal L}(\phi)= \frac12 \partial_\alpha \phi \partial_\beta\phi
\tilde{g}^{\alpha \beta} + V(\phi)\,.
\label{philag}\end{aligned}$$ The scalar field potential in the Einstein frame turns out to be $$\begin{aligned}
V(\phi)=\frac{RF-f}{2\kappa F^2}\,,
\label{potphi}\end{aligned}$$ where one has to express $R=R(\phi)$, from Eq. (\[phidef\]) by inverting it, and then express $V(\phi)$ as an explicit function of $\phi$. From the form of $S^{\mu}_{\nu}$ one can write $$\begin{aligned}
S^0_{0} \equiv \rho_\phi = \frac12 \left(\frac{d\phi}{d\tilde{t}}\right)^2
+ V(\phi)\,,\,\,\,\,
S^i_{i} \equiv P_\phi = \frac12 \left(\frac{d\phi}{d\tilde{t}}\right)^2
- V(\phi)\,,
\label{rps}\end{aligned}$$ where the scalar field $\phi$ is assumed to be a function of time only. The total energy-momentum tensor responsible for gravitational effects in the Einstein frame is $\tilde{T}^\mu_{\,\,\,\,\nu} + S^{\mu}_{\,\,\,\,\nu}$ which is a mixed tensor with only diagonal components.
The time coordinate, $t$, and the Hubble parameter, $H$, in the the Jordan frame are related to the time coordinate, $\tilde{t}$, and Hubble parameter $\tilde{H}(\equiv
\frac{1}{\tilde{a}}\frac{d\tilde{a}}{d\tilde{t}})$, in the Einstein frame via the relations: $$\begin{aligned}
\tilde{t}=\int_{t_0}^t \sqrt{F(R)} dt'\,,\,\,\,\,\,\,\,
H=\sqrt{F}\left(\tilde{H}-\sqrt{\frac{\kappa}{6}}\,\frac{d\phi}{d\tilde{t}}
\right)\,.
\label{th}\end{aligned}$$ As we will be interested mainly in bouncing cosmologies $t_0$ will be set to zero. The instant $t_0=0$ is the bouncing time in the Jordan frame. The Einstein frame description of cosmology can be tackled like FLRW spacetime in presence of a fluid and a scalar field. The presence of the Scalar field potential $V(\phi)$ gives one a pictorial understanding of the physical system which is lacking in the Jordan frame. Seeing the nature of the potential and the initial conditions of the problem one gets a hint about the possible time development of the system. The time evolution of the scalar field in the Einstein frame is dictated by the equation $$\begin{aligned}
\frac{d^2 \phi}{d\tilde{t}^2} + 3\tilde{H}\frac{d\phi}{d\tilde{t}}
+\frac{dV}{d\phi}=\sqrt{\frac{\kappa}{6}}(1-3\omega)\tilde{\rho}\,,
\label{phieqn}\end{aligned}$$ where the equation of state of the fluid in the Jordan frame is $P=\omega \rho$. It is interesting to note that the equation of state of the fluid remains the same in the Einstein frame. The evolution of the energy density in the Einstein frame is given by $$\begin{aligned}
\frac{d\tilde{\rho}}{d\tilde{t}}+\sqrt{\frac{\kappa}{6}}(1-3\omega)\tilde{\rho}
\frac{d \phi}{d\tilde{t}} + 3\tilde{H} \tilde{\rho}(1+\omega)=0\,.
\label{rhotildeq}\end{aligned}$$ The above two equations dictate the time evolution of $\tilde{\rho}$ and $\phi$ in the Einstein frame. To generate proper bouncing solution from the above two equations one requires the values of only two quantities at the bouncing time, they are $\phi(\tilde{t}_0)$, $\left(\frac{d\phi}{d\tilde{t}}\right)_{\tilde{t}_0}$, the other parameters are determined from these two at the bouncing time[@Paul:2014cxa]. In general for a flat FLRW spacetime these two values of the respective quantities are enough to solve the whole system in the Einstein frame. The expression of the Hubble parameter and its rate of change in the Einstein frame are given as $$\begin{aligned}
\tilde{H}^2 &=& \frac{\kappa}{3}(\rho_\phi +
\tilde{\rho})\,,
\label{htilde}\\
\frac{d\tilde{H}}{d\tilde{t}}&=&-\frac{\kappa}{2}
\left[\left(\frac{d\phi}{d\tilde{t}}\right)^2 + (1+\omega) \tilde{\rho}\right]\,.
\label{hprime}\end{aligned}$$
Scalar cosmological perturbations in $f(R)$ gravity {#scalar}
===================================================
Scalar perturbation in the cosmological framework, mainly related to inflation, has been widely studied [@Riotto:2002yw; @Baumann:2009ds; @Mukhanov:1990me; @Xue:2013bva]. In this section we discuss the evolution of scalar cosmological perturbation through a bounce in $f(R)$ gravity both in Jordan frame and Einstein frame. For the sake of comparison, it is better to use the conformal time $\eta$, since by definition it is invariant under a conformal transformation, $d\eta={dt}/{a(t)}={d\tilde{t}}/{\tilde{a}(\tilde{t})}$. The most general form of the line element in the Jordan frame is: $$ds^2= a(\eta)^2[-(1+2 \chi) d\eta^2+2B_{,i} dx^i d\eta + ((1-2\psi)\delta_{ij}+
2E_{,ij})dx^i dx^j]\,.$$ Here $\chi$, $B$, $\psi$ and $E$ are functions of space and time. The subscript(s) preceded by a comma specifies partial derivatives. The corresponding expression in Einstein frame for the perturbed spacetimes is: $$d\tilde{s}^2= (F+\delta F) ds^2 = \tilde{a}(\eta)^2[-(1+2 \tilde{\chi}) d\eta^2 + 2\tilde{B}_{,i} dx^i d\eta + ((1-2\tilde{\psi})\delta_{ij}+2\tilde{E}_{,ij})dx^i dx^j],$$ where henceforth $F$ stands for the unperturbed value of $df/dR$. The scale-factors are still related via the relation specified in Eq. (\[aat\]). The scalar metric perturbation functions in the two frames are related via [@DeFelice:2010aj], $$\begin{aligned}
\tilde{\chi}= \chi + \frac{\delta F}{2F}\,, && {\rm and} && \tilde{\psi}=
\psi - \frac{\delta F}{2F}\,,
\label{chipsi}\end{aligned}$$ $$\begin{aligned}
\tilde{B}= B\,, && {\rm and} && \tilde{E}= E\,.
\label{be}\end{aligned}$$ In Jordan frame the gauge-invariant variables are: $$\Phi= \chi + \frac{1}{a}[(B-E')a)]'\,,\,\,\,\,
\Psi= \psi - \frac{a'}{a}(B-E')\,.$$ The corresponding gauge-invariant variables, in the Einstein frame, are: $$\tilde{\Phi}= \tilde{\chi} + \frac{1}{\tilde{a}}[(\tilde{B}-\tilde{E}')\tilde{a})]'
= \Phi + \frac{\delta F^{(gi)}}{2F}\,,\,\,\,\,\,
\tilde{\Psi}= \tilde{\psi} - \frac{\tilde{a}'}{\tilde{a}}(\tilde{B}-\tilde{E}') =
\Psi - \frac{\delta F^{(gi)}}{2F}\,,$$ where $\delta F^{(gi)}=(\partial F/\partial R)\delta R^{(gi)}$. The gauge invariant perturbation of a scalar quantity $q$ is [@Mukhanov:1990me], $\delta q^{(gi)}\equiv\delta
q+q^\prime(B-E^\prime)\,.$ Here the primes stand for derivatives with respect to conformal time. Assuming the perturbation in the matter sector in Jordan frame to be such that $$\delta P^{(gi)}-c_{s}^{2}\delta\rho^{(gi)}=0\,,$$ where $c_s^2={\delta P}/{\delta\rho}$ one can derive the relation between $\tilde{P}^{(gi)}$ and $\tilde{\rho}^{(gi)}$ in the Einstein frame as $$\begin{aligned}
\delta\tilde{P}^{(gi)}-c_{s}^{2}\delta\tilde{\rho}^{(gi)}&=&\delta \tilde{P}-
c_s^2\delta\tilde{\rho}\nonumber\\
&=&\delta(Pe^{-2\sqrt{\frac{2\kappa}{3}}\phi})-c_s^2\delta(\rho e^{-2\sqrt{\frac{2\kappa}{3}}\phi})
\nonumber\\
&=&e^{-2\sqrt{\frac{2\kappa}{3}}\phi}(\delta P-c_s^2\delta\rho)+2\sqrt{\frac{2\kappa}{3}}
e^{-2\sqrt{\frac{2\kappa}{3}}\phi}(c_s^2\rho-P)\delta\phi\nonumber\\
&=&e^{-2\sqrt{\frac{2\kappa}{3}}\phi}(\delta P^{(gi)}-c_s^2\delta\rho^{(gi)})+
2\sqrt{\frac{2\kappa}{3}}e^{-2\sqrt{\frac{2\kappa}{3}}\phi}(c_s^2\rho-P)\delta\phi
\nonumber\\
&=&2\sqrt{\frac{2\kappa}{3}}\tilde{\rho}(c_{s}^{2}-\omega)\delta\phi\,,\end{aligned}$$ showing that for a single barotropic fluid (where $\omega=c_s^2$) whose equation of state is given by Eq. (\[eqns\]) in the Jordan frame one must have $\delta\tilde{P}^{(gi)}-c_{s}^{2}\delta\tilde{\rho}^{(gi)}=0$ in the Einstein frame.
Before we proceed to formulate the scalar cosmological perturbation in the Jordan and Einstein frames it is pertinent to elucidate the nature of gauge choices and their relationship with conformal transformations. If we apply synchronous gauge in Jordan frame, one should impose $\chi=B=0$. It can be noted that although $\tilde{B}=B=0$ in Einstein frame, $\tilde{\chi}$ does not vanish. So the definition of synchronous gauge is not the same in both frames.
In the spatially-flat gauge, in Jordan frame, one has to impose $\chi=E=0$. From Eq. (\[chipsi\]) it is seen that $\tilde{\chi}$ does not vanish in Einstein frame. Consequently the spatially-flat gauge is also not conformally invariant.
Only in the longitudinal gauge, where one imposes $B=E=0$ in the Jordan frame, one obtains $\tilde{B}=\tilde{E}=0$ in Einstein frame. The longitudinal gauge remains invariant under the conformal transformation connecting the Jordan frame and the Einstein frame. Hence we will use longitudinal gauge in the present article from here on.
Scalar perturbations in the Jordan frame {#jords}
----------------------------------------
In the domain of linear perturbations, the scalar perturbed FLRW metric has two gauge invariant degrees of freedom. In the longitudinal gauge this can be expressed as, $$\begin{aligned}
ds^{2}=a^{2}(\eta)\left[ -(1+2\Phi)d\eta^{2}+(1-2\Psi)\delta_{ij}
dx^{i}dx^{j}\right]
\label{delta_g}\end{aligned}$$ Here $\Phi$ and $\Psi$ are the two gauge invariant perturbation degrees of freedom, also called the Bardeen potentials. The, $00$, $ii$, $ij-th(i\neq j)$ elements of the linearized perturbed Einstein equation in the Fourier space are [@Matsumoto:2013sba]: $$\begin{aligned}
F[-k^2(\Phi+\Psi)-3\mathcal{H}(\Phi^\prime+\Psi^\prime)+(3\mathcal{H}^\prime -6\mathcal{H}^2)
\Phi-3\mathcal{H}^\prime\Psi]\nonumber\\
+F^\prime(-9\mathcal{H}\Phi+3\mathcal{H}\Psi-3\Psi^\prime)=\kappa a^2\delta\rho\,,
\label{jord_ptbd_00}
\\
\nonumber\\
F[\Phi^{\prime\prime}+3\mathcal{H}(\Phi^\prime+\psi^\prime)+3\mathcal{H}^\prime
+(\mathcal{H}^\prime+2\mathcal{H}^2)\Psi]+F^\prime(3\mathcal{H}\Phi-\mathcal{H}\Psi+
3\Phi^\prime)\nonumber\\
+F^{\prime\prime}(3\Phi-\Psi)=c_s^2\kappa a^2\delta P \,,
\label{jord_ptbd_ii}
\\
\nonumber\\
\Phi-\Psi-\frac{2F_R}{a^2 F}[3\Psi^{\prime\prime}+6(\mathcal{H}^\prime+\mathcal{H}^2)\Phi
+3\mathcal{H}(\Phi^\prime+3\Psi^\prime)-k^2(\Phi-2\Psi)]=0 \,,
\label{jord_ptbd_ij}\end{aligned}$$ where $F_R=dF/dR$. If there is only a single matter component present, the perturbation in the matter sector can be assumed to be adiabatic, so that the sound velocity can be defined as done in the beginning of this section. Using Eq. (\[jord\_ptbd\_00\]) and (\[jord\_ptbd\_ii\]), we can write
![Evolution of scale factor with $\eta$ through asymmetric bounce[]{data-label="bp2"}](aasy.pdf)
$$\begin{aligned}
&&(1+F)\left[\Phi^{\prime\prime}+\Psi^{\prime\prime}+3\mathcal{H}(1+c_s^2)
(\Phi^\prime + \Psi^\prime)+c_s^2(k^2+6\mathcal{H}^2)\Phi+3\mathcal{H}^\prime(1-c_s^2)\Phi+
\mathcal{H}^\prime(1+c_s^2)\Psi\right.\nonumber\\
&&+\left.(2\mathcal{H}^2+c_s^2k^2)\Psi\right]+F^\prime[\mathcal{H}(1+3c_s^2)(\Phi-3\Psi)+
3\Phi^\prime + 3c_s^2\Psi^\prime]+F^{\prime\prime}(3\Phi-\Psi)=0\,.
\label{jord_ptbd_ii00}\end{aligned}$$
A form of the above perturbation equations of $f(R)$ gravity in the Jordan frame was also calculated in [@Bean:2006up] where the authors studied the problem of structure formation in late times. In this paper we will follow the equations as written above and obtained from [@Matsumoto:2013sba] which seem more appropriate for our analysis. Eq. (\[jord\_ptbd\_ij\]) and (\[jord\_ptbd\_ii00\]) can be solved numerically to get the solutions $\Phi$ and $\Psi$. In the present article we will exclusively work with exponential gravity [@Bari:2018aac] where the form of $f(R)$ is given as $$f(R)= \frac{1}{\alpha} \exp{(\alpha R)}\,,
\label{frform}$$ where $\alpha= 10^{12}$ for phenomenological reasons, it sets the energy scale of bounce[^2]. In the present work all the values of dimensional constants (as $\alpha$) or variables (as ${\cal
H}$, $\eta$, and others) will be represented in Planck units. To go back to mass units one has to multiply the appropriate variables by suitable power of Planck mass expressed in GeV units. We choose exponential gravity as in this case the gravitational theory does not have any instabilities, as because with $\alpha > 0$ we have $$F>0\,,\,\,\,\,\,F_R>0\,.$$ The issues about instability in this kind of theory is discussed in [@Bari:2018aac].
![Evolution of $\Psi$ with $\eta$ through asymmetric bounce, calculated directly in Jordan frame[]{data-label="pa"}](JAP.pdf)
![Evolution of $\Psi$ with $\eta$ through asymmetric bounce, calculated directly in Jordan frame[]{data-label="pa"}](JASP.pdf)
The blue curves in Fig. \[pn2\] and Fig. \[pn3\] shows the evolution of $\Phi(k,\eta)$ and $\Psi(k,\eta)$ for $k=10^{-10}$ for a particular background evolution. We have used $\Phi(0)=0.0001$, $\Phi'(0)=0, \Psi(0)=0.0002, \Psi'(0)=0$ to produce the plots. The bouncing time is $\eta=0$. For the background we have chosen radiation where $\omega=1/3$. The bounce in the background is shown in Fig. \[bp2\] where the scale-factor is plotted with respect to $\eta$. To obtain the the background solutions we have initially solved the system of equations as specified in the initial part of section \[back\] in Jordan frame. The analysis is done in coordinate time. Later the result is transformed to conformal time. The conditions used to produce the bouncing background solution are, ${\cal H}(0)=0$, ${\cal H}^\prime(0)=6.8 \times 10^{-14}$ and ${\cal
H}^{\prime \prime} = 0$. The background evolution in Fig. \[bp2\] specifies a symmetric bounce. In all the calculations of bounce we have normalized the scale-factor in the Jordan frame in such a way that $a(0)=1$. One can also choose the bounce in the background to be asymmetric. In this case the conditions at $\eta=0$ remains the same as before (as in the symmetric case) except that ${\cal H}^{\prime
\prime}(0) \ne 0$. The asymmetry in the background evolution can be generated from an infinitesimal value of the second time derivative of the Hubble parameter as ${\cal H}^{\prime \prime}(0)=8.2 \times
10^{-21}$. The perturbation evolution in an asymmetric background in the Jordan frame are plotted in Fig. \[pb\] and Fig. \[pa\]. To evaluate the dynamics of the perturbations the values of the perturbations are now not specified at $\eta=0$ but at $\eta=-10^6$. The reason for choosing a separate time instant for plotting the asymmetric bounce results will become clear when we analyze the case of asymmetric bounces in the Einstein frame. In the present case $\Phi(-10^6)=0.7$, $\Psi(-10^6)=0.9$ and $\Phi'(-10^6)=\Psi'(-10^6)=0$ and the plots show perturbation evolution for $k=10^{-10}$. The plots of the perturbation evolution in the Jordan frame are visibly continuous and smooth. We will see later that if we try to get these results from the Einstein frame calculation we will hit localized singularities.
In the next section we will try to recast the problem in the Einstein frame. A part of this calculation was incompletely done in an earlier publication [@Paul:2014cxa] where the authors disregarded the velocity potential of the fluid in the Einstein frame. In the present work we complete and rectify the previous calculation.
Einstein frame
--------------
In Einstein frame the scalar perturbation in the longitudinal gauge is given as $$\begin{aligned}
d\tilde{s}^{2}=\tilde{a}^{2}(\eta)\left[-(1+2\tilde{\Phi})d\eta^{2}
+(1-2\tilde{\Psi})\delta_{ij} dx^{i}dx^{j}\right]\,.
\label{delta_g_tilde}\end{aligned}$$ In Einstein frame the gravitational theory is essentially GR and the energy-momentum tensor of the hydrodynamic matter and the scalar field are both diagonal, it is easy to check from the $ij-{\rm th}(i\neq j)$ component of the perturbed field equation that $$\tilde{\Phi}=\tilde{\Psi}\,.$$ This is different from the behavior of the above quantities in the Jordan frame, because the energy-momentum tensor of the existing matter component being diagonal does not necessarily implies the equality of the two Bardeen potentials in a higher derivative gravity theory. This is a good instance where switching to the Einstein frame makes things easier. The Jordan frame Bardeen potentials can be recovered from the Einstein frame Bardeen potential as follows [@Mukhanov:1990me], $$\begin{aligned}
\Phi= -\frac{2}{3}\left(\frac{F^{2}}{F'a}\right)\left[\left(\frac{a}{F}
\right)\tilde{\Phi}\right]'\,,\,\,\,\,
\Psi = \frac{2}{3}\left(\frac{1}{FF'a}\right)(aF^{2}\tilde{\Phi})'\,.
\label{Phi_Psi}\end{aligned}$$ Later it will be shown that these connecting formulae breaks down in the case of asymmetric bounce in the Jordan frame.
As the velocity potential cannot be in general neglected when treating the scalar metric perturbations we rewrite the perturbation equations using this new information. Perturbed Einstein tensor components are: $$\begin{aligned}
\delta G^0_0&=&\frac{2}{\tilde{a}^2}[-3\tilde{\mathcal{H}}(\tilde{\mathcal{H}}\tilde{\Phi}+\tilde{\Phi}^{\prime})+\nabla^{2}\tilde{\Phi} ]\,,\\
\delta G^0_i&=&\frac{2}{\tilde{a}^2}\left[\tilde{\Phi}^{\prime}+\tilde{\mathcal{H}}\tilde{\Phi} \right]_{,i}\,,\\
\delta G^i_j&=&-\frac{2}{\tilde{a}^2}[(2\tilde{\mathcal{H}}^\prime+\tilde{\mathcal{H}}^2)\tilde{\Phi}+3\tilde{\mathcal{H}}\tilde{\Phi}^\prime+\tilde{\Phi}^{\prime\prime}]\delta^i_j\,.\end{aligned}$$ Here a partial derivative with spatial coordinates is specified with the comma followed by a Latin alphabet. Perturbed energy-momentum tensor components for hydrodynamic matter are [@Mukhanov:1990me]: $$\begin{aligned}
\delta \tilde{T}_0^{0}&=&\delta\tilde{\rho}\,,\\
\delta \tilde{T}_i^{0}&=&(\tilde{\rho}+\tilde{P})\tilde{a}^{-1}\delta \tilde{u}_i=-(\tilde{\rho}+\tilde{P})\tilde{a}^{-1}\partial_i \tilde{U}\,,
\label{tt0i}\\
\delta \tilde{T}_j^{i}&=&-\delta \tilde{P}\delta^i_j\,.\end{aligned}$$ The quantity $\tilde{U}$ is the velocity potential given by $\delta
\tilde{u}_i=-\partial_i \tilde{U} +\tilde{v}_i$. Here $\tilde{v}_i$ is is the pure vector part of the fluid velocity perturbation in the Einstein. The perturbed scalar field energy momentum tensor components are: $$\begin{aligned}
\delta S_0^{0}&=&a^{-2}\left(-{\phi^{\prime}}^{2}\tilde{\Phi}+\phi^{\prime}\delta\phi^{{\prime}}
+\tilde{a}^{2}V_{,\phi}\delta\phi\right)\,,\\
\delta S_i^{0}&=&a^{-2}\phi^{\prime}\delta\phi_{,i}\,,\\
\delta S_j^{i}&=&-a^{-2}\left(-\tilde{\Phi} {\phi^{\prime}}^{2}+\phi^{\prime} \delta\phi^{\prime} -\tilde{a}^{2} V_{,\phi}\delta\phi\right)\delta^i_j\,.\end{aligned}$$ A derivative with respect to $\phi$ is specified by a comma followed by $\phi$ in the subscript. Using the results we can write the perturbed Einstein equations as: $$\begin{aligned}
\label{rho}
-3\tilde{\mathcal{H}}(\tilde{\mathcal{H}}\tilde{\Phi}+\tilde{\Phi}^{\prime})+\nabla^{2}\tilde{\Phi} &=& \frac{\kappa}{2} \left( \tilde{a}^{2}\delta \tilde{\rho} -{\phi^{\prime}}^{2}\tilde{\Phi} +\phi^{\prime}\delta\phi^{\prime}
+\tilde{a}^{2}V_{,\phi} \delta\phi\right)\,,
\label{perte1}\\
(2\tilde{\mathcal{H}}^{\prime}+\tilde{\mathcal{H}}^{2}) \tilde{\Phi} +\tilde{\Phi}^{\prime \prime} +3\tilde{\mathcal{H}}\tilde{\Phi}^{\prime} &=& \frac{\kappa}{2}\left( \tilde{a}^{2}\delta \tilde{P} -\tilde{\Phi} {\phi^{\prime}}^{2} + \phi^{\prime} \delta\phi^{\prime} -\tilde{a}^{2} V_{,\phi}\delta\phi\right)\,,\\
\frac{\kappa}{2}\left[\phi' \delta\phi-\tilde{a}(\tilde{\rho}+\tilde{p}) \tilde{U}\right]&=&\tilde{\Phi}^{\prime}+\tilde{\mathcal{H}}\tilde{\Phi}\,.
\label{vpot}\end{aligned}$$ Multiplying the first of the above set of equations by $c_s^2$ and subtracting from the second one yields, $$\begin{aligned}
\label{b}
\tilde{\Phi}^{\prime \prime}- c_{s}^{2}\nabla^{2}\tilde{\Phi} +
\left( 2\tilde{\mathcal{H}}^{\prime} +
\tilde{\mathcal{H}}^{2}
\right)\tilde{\Phi} +
3 \tilde{\mathcal{H}}\tilde{\Phi}^{\prime} +
3 c_{s}^{2}\tilde{\mathcal{H}}\left( \tilde{\mathcal{H}}\tilde{\Phi}
+ \tilde{\Phi}^{\prime} \right) \nonumber\\
= -\frac{\kappa}{2} \tilde{\Phi}\phi^{\prime 2}
\left( 1- c_{s}^{2} \right)+\frac{\kappa}{2}\phi^{\prime}(1-c_{s}^{2})\delta
{\phi}^\prime - \frac{\kappa \tilde{a}^2}{2} V_{,\phi}
(1 + c_{s}^{2})\delta \phi\,.\end{aligned}$$ This is one equation which gives the dynamics of the perturbed potential $\tilde{\Phi}$ in the Einstein frame. The scalar field perturbation is linked with the above dynamics. We require more equations for uniquely solving the perturbation evolutions.
The other equation comes from perturbing the Klein-Gordon equation as: $$\tilde{\Box} \phi -V_{, \phi} + \frac{1}{\sqrt{-\tilde{g}}} \frac{\partial
\mathcal{L_M}}{\partial \phi}=0\,,
\label{kg1}$$ where $\tilde{\Box} \equiv \tilde{D}^\mu \tilde{D}_\mu$, $\tilde{D}_\mu$ being the covariant derivative in the Einstein frame. Here $\mathcal{L_M}$ specifies the Lagrangian of the hydrodynamic fluid. Using the following fact we can write down the last term on the left hand side of the above equation, $$\frac{\partial \mathcal{L_M}}{\partial \phi} = \frac{\partial
\mathcal{L_M}}{\partial g^{\mu \nu}} \frac{\partial
g^{\mu \nu}}{\partial \phi} = \frac{1}{F(\phi)}\frac{\partial
\mathcal{L_M}}{\partial \tilde{g}^{\mu \nu}} \frac{\partial
(F(\phi)\tilde{g}^{\mu \nu})}{\partial \phi}\,.$$ Using the standard definition of the matter energy momentum tensor $$\tilde{T}_{\mu \nu} = - \frac{2}{\sqrt{-\tilde{g}}} \frac{\partial
\mathcal{L_M}}{\partial \tilde{g}^{\mu \nu}}\,,$$ we can write, $$\frac{\partial \mathcal{L_M}}{\partial \phi} = -\sqrt{-\tilde{g}}
\frac{F_{,\phi}}{2F}\tilde{g}^{\mu \nu} \tilde{T}_{\mu \nu}=-\sqrt{-\tilde{g}}
\frac{F_{,\phi}}{2F} \tilde{T}\,,$$ where $\tilde{T}$ is the trace of the energy-momentum tensor in the Einstein frame. In metric $f(R)$ cosmology it is always, $$\frac{F_{,\phi}}{2F}=\sqrt{\frac{\kappa}{6}}\,,$$ and consequently Eq. (\[kg1\]) becomes $$\tilde{D}^\mu \tilde{D}_\mu \phi -V_{, \phi} -\sqrt{\frac{\kappa}{6}}\tilde{T} =0\,.
\label{kg2}$$ Perturbing the terms in the Klein-Gordon equation, without the matter coupling term, one gets $$\begin{aligned}
\delta(\tilde{D}^\mu \tilde{D}_\mu \phi - V_{, \phi})
&=&-\tilde{a}^{-2}\left[\delta \phi^{\prime \prime} +
2\tilde{\mathcal{H}}\delta \phi^\prime - \nabla^2 \delta \phi -
4\phi^\prime \tilde{\Phi}^\prime + \tilde{a}^2 V_{, \phi \phi}
\delta \phi +2 \tilde{a}^2 V_{, \phi}
\tilde{\Phi}\right.\nonumber\\ &&\left.- 2 \tilde{a}^2
\sqrt{\frac{\kappa}{6}} (1-3c_s^2) \tilde{\Phi} \tilde{\rho}
\right]\,,
\label{dkg}\end{aligned}$$ where we have used the background Klein-Gordon equation for the scalar field in the Einstein frame. The perturbation of the matter coupling term gives $$\begin{aligned}
\delta \tilde{T} = \delta \tilde{g}^{\mu \nu} \tilde{T}_{\mu \nu}
+ \tilde{g}^{\mu \nu} \delta \tilde{T}_{\mu \nu}=-(1-3c_s^2) \delta \tilde{\rho}\,.
\label{dt}\end{aligned}$$ Combining the results the perturbed Klein-Gordon equation gives, $$\begin{aligned}
\delta \phi^{\prime \prime} &+& 2\tilde{\mathcal{H}}\delta \phi^\prime - \nabla^2 \delta \phi
-4\phi^\prime \tilde{\Phi}^\prime + \tilde{a}^2 V_{, \phi \phi} \delta \phi
+2 \tilde{a}^2 V_{, \phi} \tilde{\Phi} -2\tilde{a}^2 \sqrt{\frac{\kappa}{6}}
(1-3c_s^2) \tilde{\Phi} \tilde{\rho}\nonumber\\
&-&\tilde{a}^2 \sqrt{\frac{\kappa}{6}} (1-3c_s^2) \delta \tilde{\rho}=0\,.
\label{dkgc}\end{aligned}$$
![Evolution of $\delta\phi$ with $\eta$ through a symmetric bounce in the background.[]{data-label="endens1"}](PhiEin1.pdf)
![Evolution of $\delta\phi$ with $\eta$ through a symmetric bounce in the background.[]{data-label="endens1"}](dphi1.pdf)
Using the expression of $\delta\tilde{\rho}$ from Eq. (\[perte1\]) we can write the above equation as $$\begin{gathered}
\label{perkg}
\delta \phi^{\prime \prime} -\nabla^2 \phi + 2 \tilde{\Phi}
\tilde{a}^2 V_{, \phi} + \sqrt{\frac{\kappa}{6}} (1- 3c_{s}^{2}) \Big(
-\phi^{\prime 2} -2 \tilde{a}^2 \tilde{\rho} + \frac{6
\tilde{\mathcal{H}}^2}{\kappa} \Big)\tilde{\Phi} \\
-2 \sqrt{\frac{\kappa}{6}} (1- 3c_{s}^{2})\frac{\nabla^2
\tilde{\Phi}}{\kappa}-\left[4 \phi^{\prime} - \sqrt{\frac{\kappa}{6}} (1-
3c_{s}^{2}) \frac{6 \tilde{\mathcal{H}}^2}{\kappa}\right] \tilde{\Phi}^{\prime}\\
+\left[V_{, \phi \phi} + \sqrt{\frac{\kappa}{6}} (1- 3c_{s}^{2})V_{, \phi}\right]
\tilde{a}^2 \delta \phi+ \left[2 \tilde{\mathcal{H}} + \sqrt{\frac{\kappa}{6}} (1-
3c_{s}^{2})\phi^{\prime}\right] \delta \phi^{\prime} =0\,.\end{gathered}$$ One can now solve solve Eq. (\[perkg\]) and Eq. (\[b\]) simultaneously and obtain the evolution of $\tilde{\Phi}$ and $\delta\phi$. These are the general results related to evolution of scalar metric perturbations in the longitudinal gauge worked out in the the Einstein frame which are appropriate for early universe cosmological processes as bounce. Previous authors have worked the Einstein frame perturbation equations in the synchronous gauge [@Bean:2006up] to study the problem of structure formation in the late time universe. The results obtained in the cited work cannot in general be applied to study the problem of scalar metric perturbation evolution through a non-singular bounce and to our knowledge the appropriate longitudinal gauge results for scalar perturbation growth which we present in this paper are reported for the first time. Next we will apply the formalism in the case of a background exponential bounce. For the particular $f(R)$ model as chosen in Eq. (\[frform\]) the scalar field potential $V(\phi)$ is given by: $$V(\phi)= \frac{1}{2 \kappa \alpha} \Big(\sqrt{\frac{2 \kappa}{3}} \phi- 1 \Big) e^{- \sqrt{\frac{2 \kappa}{3}}\phi}\,,
\label{potf}$$ where $\alpha= 10^{12}$. The above equations are important results reported for the first time in this paper.
### Using the Einstein frame to model symmetric bounces in the Jordan frame {#sse}
We can choose the conditions in the Einstein frame variables in such a way so that the dynamics produces a symmetric bounce in the Jordan frame. We will choose the conditions in the Einstein frame so that we can reproduce the results of the symmetric bounce in the Jordan frame as presented in the previous subsection, modulo some small numerical error which arises in the program to convert the result from Einstein frame to Jordan frame. In the Einstein frame we choose $\phi(0)=0.1$, $\phi^\prime(0)=0$ for the symmetric bounce background. For the perturbations we use $\tilde{\Phi}(0)=0.00015$, $\tilde{\Phi}^\prime(0)=0$, $\delta \phi(0)=.00002$ and $\delta
\phi^\prime(0)=0$. The above set of values are not chosen randomly, they are chosen in such a way such that they reproduce the analogous conditions at $\eta=0$ imposed in Jordan frame perturbation calculations for the symmetric bounce, presented in the previous subsection. The background cosmological evolution is guided by the equations given at the last part of subsection \[jefr\]. The background evolution was specified in terms of coordinate time in the Einstein frame and we use the same equations to produce the background dynamics. After the background dynamics is done in coordinate time we map the results to conformal time so that the background calculation matches with perturbation dynamics results presented in this subsection. Fig. \[an1\] and Fig. \[endens1\] shows the evolution of $\tilde{\Phi}$ and $\delta \phi$ in the Einstein frame for a symmetric cosmological bounce in the background Jordan frame. One can now use the relations given in Eq. (\[Phi\_Psi\]) to convert the perturbations from the Einstein frame to the Jordan frame. We expect that the results so obtained will closely match with the results obtained by the perturbation dynamics calculations done in the Jordan frame, plotted in blue in Fig. \[pn2\] and Fig. \[pn3\].
![Evolution of $\Psi$ with $\eta$ through symmetric bounce. The blue curve represents Jordan frame result and the orange curve represents the result obtained from the Einstein frame. The details regarding the plots are given in text.[]{data-label="pn3"}](FinalPhiSymPlot.pdf)
![Evolution of $\Psi$ with $\eta$ through symmetric bounce. The blue curve represents Jordan frame result and the orange curve represents the result obtained from the Einstein frame. The details regarding the plots are given in text.[]{data-label="pn3"}](FinalPsiSymPlot.pdf)
The present results are plotted in orange, in Fig. \[pn2\] and Fig. \[pn3\]. We see that our results do match to a great extent, the slight mismatch results from the numerical techniques utilized in calculating the background and perturbation dynamics in the two separate frames.
### Using the Einstein frame to model asymmetric bounces in the Jordan frame
If we consider an asymmetric bounce, we see that the nature of the scalar perturbations calculated from the two frames do not match with each other. This fact was partially noted in Ref. [@Paul:2014cxa], in the present article we fully specify the gravity of the situation. At first we point out the most important difference between perturbation dynamics for the symmetric and asymmetric bounce case. In the case of an asymmetric bounce in the Jordan frame we can smoothly plot the perturbations $\Phi$, $\Psi$ in the Jordan frame as shown in Fig. \[pb\] and Fig. \[pa\] or $\tilde{\Phi}$, $\delta \phi$ in the Einstein frame as shown in Fig. \[panb\] and Fig. \[pana\]. In this paper all the calculations of the perturbation dynamics is done for the Fourier mode $k=10^{-10}$. The asymmetric bounce in the Jordan frame can be obtained from an Einstein frame cosmology where $\phi(0)=0.1$ and $\phi^\prime(0)=10^{-8}$. These values corresponds to the values for $\cal{H}^\prime$ and $\cal{H}^{\prime \prime}$ applied in the Jordan frame to produce an asymmetric bounce. To plot the perturbations in the Einstein frame we have used the conditions $\tilde{\Phi}(-10^6)=0.8, \tilde{\Phi}'(-10^6)= 0$ and $\delta
\phi(-10^6)= 0.05, \delta \phi'(-10^6)=0$. These values match with the analogous conditions used to plot the scalar perturbations in the case of an asymmetric bounce in the Jordan frame as shown in Fig. \[pb\] and Fig. \[pa\]. In the Einstein frame $\tilde{\Phi}$ becomes marginally non-perturbative after for $\eta>0$ but the actual Jordan frame scalar perturbations remain perturbative within the time period of our interest. The marginal non-perturbative behavior in the Einstein frame do not posit any cosmological problem as far as the bounce in the Jordan frame is considered. The interesting feature of the asymmetric bounce appears when one tries to calculate the Jordan frame perturbation evolution using the Einstein frame results via the use of the relations in Eq. (\[Phi\_Psi\]). Using the relations in Eq. (\[Phi\_Psi\]) one obtains the Jordan frame results shown in Fig. \[pnc\] and Fig. \[pnew\]. The results do not match with the expected result as obtained in Fig. \[pb\] and Fig. \[pa\]. The transformation from the Einstein frame to the Jordan frame produces unavoidable singularities. The singularities arise in the case of an asymmetric bounce due to the reason that
![Evolution of $\delta{\phi}$ with $\eta$, in the case of an asymmetric bounce in Jordan frame, calculated directly in Einstein frame.[]{data-label="pana"}](AsyPhiEin.pdf)
![Evolution of $\delta{\phi}$ with $\eta$, in the case of an asymmetric bounce in Jordan frame, calculated directly in Einstein frame.[]{data-label="pana"}](AsydPhi.pdf)
$F^\prime=0$ for some $\eta \ne 0$ in the time period of our interest, making the relations in Eq. (\[Phi\_Psi\]) singular. For symmetric bounces one has $F^\prime(0)=a^\prime(0)=\tilde{\Phi}(0)=0$ and the singularity disappears in the $\eta \to 0$ limit in Eq. (\[Phi\_Psi\]) as both the numerator and denominator tends to vanish at the same time instant. The point was partially discussed in [@Paul:2014cxa]. In the present case the singularities lie near $\eta=0$ and so the Jordan frame potentials blow up near $\eta=0$ when we apply the transformations in Eq. (\[Phi\_Psi\]) to the Einstein frame results. If the conditions used to plot the perturbations were applied at $\eta=0$ (in both the frames) then the blowing up of the potentials near $\eta=0$ distorts the values of $\Phi$ and $\Psi$ obtained far from $\eta=0$. In this case it is better to use initial conditions at $\eta=-10^6$, which is much further from the singularities encountered in the transformations. Before we finish this discussion we must point out that the singularities shown in the perturbation evolutions in the asymmetric case are not real singularities but an artefact of using a conformal frame which is not suitable to tackle asymmetric bounces. As a consequence our results show that the Einstein frame can be used to calculate most of the properties of a symmetric cosmological bounce in the Jordan frame including the scalar perturbation evolution. In this case the Einstein frame actually serves as an auxiliary conformal frame where the calculations can be done and the results can be converted back to the Jordan frame. On the other hand for an asymmetric bounce the Einstein frame can act as a true auxiliary frame for the background evolution but fails to reproduce the scalar perturbation dynamics in the Jordan frame. This fact is of paramount importance showing that the Jordan frame is the natural choice for scalar metric perturbation dynamics.
![Evolution of $\Psi$ with $\eta$ through an asymmetric bounce in the Jordan frame obtained from the results in the Einstein frame.[]{data-label="pnew"}](PPAE.pdf)
![Evolution of $\Psi$ with $\eta$ through an asymmetric bounce in the Jordan frame obtained from the results in the Einstein frame.[]{data-label="pnew"}](PPAEA.pdf)
### Evolution of velocity potential in both the frames
We omitted the equation involving fluid velocity potential in the Jordan frame as the potential can be calculated from the Einstein frame itself. After showing that the scalar perturbations can be correctly calculated from the Einstein frame we directly use the Einstein frame to predict the nature of the fluid velocity potential in the Jordan frame. In the Einstein frame Eq. (\[vpot\]) can be used to predict the evolution of the velocity potential $\tilde{U}$. Once the evolution $\tilde{U}$ is known one can convert the result to the Jordan frame to opine on the behavior of $U$. We have $\tilde{T}^\mu_\nu=T^\mu_\nu/F^2$ and consequently $$\delta \tilde{T}^0_i= \frac{\delta T^0_i}{F^2} -2 \frac{\delta T^0_i}{F^3}\,.$$ As $ T^0_i=0$ and the form of $\delta \tilde{T}^0_i$ (or $\delta T^0_i$) can be obtained from the form of Eq. (\[tt0i\]) we can write $$(\tilde{\rho} + \tilde{P})\tilde{a}^{-1} \partial_i \tilde{U}
=\frac{(\rho + P)a^{-1} \partial_i U}{F^2}\,,$$ which yields $$\begin{aligned}
\tilde{U} = \sqrt{F} U\,.
\label{uut}\end{aligned}$$ The above equation shows how the velocity potentials in the two conformal frames are related to each other. The plots of the velocity potentials, when $k=10^{-10}$, for a symmetric bounce in the background is shown in Fig. \[utf\]. The blue curve represents the Jordan frame result and the orange one represents the result obtained from the Einstein frame. The results reasonably match with each other.
Vector perturbations in $f(R)$ cosmology {#vp}
========================================
In this section we will specialize on vector metric perturbations and try to see how these perturbations evolve in $f(R)$ cosmology. In GR based bouncing models of cosmology the metric vector perturbation is bound to grow in the contracting phase of the universe [@Battefeld:2004cd]. But such a behavior is not in general true in $f(R)$ bouncing models as we will see below. The behavior of vector perturbations in $f(R)$ cosmology is modulated by the behavior of $F(R)$ which can keep the vector perturbations under tight control.
In the Jordan frame the metric is written as $$\renewcommand\arraystretch{1.3}
g_{\mu\nu}=a^2(\eta)
\mleft[
\begin{array}{c|c}
-1 & S_{i} \\
\hline
S_{i} & \delta_{ij}-Q_{i,j}-Q_{j,i}
\end{array}
\mright]\,,$$ and $$\renewcommand\arraystretch{1.3}
g^{\mu\nu}=\frac{1}{a^2(\eta)}
\mleft[
\begin{array}{c|c}
-1 & S^{i} \\
\hline
S^{i} & \delta^{ij}+Q^{i,j}+Q^{j,i}
\end{array}
\mright]\,,$$
![Evolution of fluid velocity potential $U$ and $\tilde{U}$ for a symmetric bounce in the Jordan frame. The blue curve represents the Jordan frame result and the orange curve represents the result obtained from the Einstein frame.[]{data-label="utf"}](vel_pot_plots.pdf)
where $S^i$ and $Q^i$ ate 3-vectors satisfying the constraint $S^i_{\,\,,i}=Q^i_{\,\,,i}=0$. The metric in the Einstein frame is given in an identical way except that $\tilde{a}$, $\tilde{S_i}$ and $\tilde{Q_{i,j}}$ appear instead of $a$, $S_i$ and $Q_{i,j}$. More over the metric perturbations remain the same in both the conformal frames. In the Jordan frame the relevant quantities calculated from the metric given above are: $$\begin{aligned}
R_{00}= -3 \mathcal{H}^\prime\,,\,\,\,\,\,
R_{0i}=(\mathcal{H}^\prime + 2 \mathcal{H}^2) S_{i} -\frac{1}{2} \nabla S_i\,,\end{aligned}$$ where $\nabla\equiv \partial_j \partial^j$, and $$\begin{gathered}
R_{ij}= (\mathcal{H}^\prime + 2 \mathcal{H}^2) \delta_{ij}-\frac{1}{2}(S^\prime_{i,j}+
S^\prime_{j,i})- \mathcal{H}(S_{i,j}+ S_{j,i})\\
-(\mathcal{H}^\prime + 2 \mathcal{H}^2)(Q_{i,j}+ Q_{j,i})-\mathcal {H} (Q^\prime_{i,j} +
Q^\prime_{j,i})- \frac{1}{2} (Q^{\prime\prime}_{i,j} + Q^{\prime\prime}_{j,i})\,.\end{gathered}$$ The Ricci tensor remains unchanged: $$R= 6\frac{(\mathcal{\dot{H}}+\mathcal{H}^2)}{a^2}\,,$$ as $\delta R=0$. The perturbed Einstein tensor components are $$\delta G^{0}_{0}=0 \,,\,\,\,\,
\delta G^{0}_{i}= \frac{1}{2a^2} \nabla S_i\,,
\label{einv1}$$ and $$\begin{aligned}
\delta G^{i}_{j} = -\frac{1}{2a^2}(S^{\prime i}_{\,\,\,,j}+ S^{\prime \,\,i}_{j,})- \frac{\mathcal{H}(S^{i}_{\,\,,j}+ S^{\,\,\,i}_{j,})}{a^2} -\frac{1}{2a^2}(Q^{\prime \prime i}_{\,\,\,\,\,\,,j}+ Q^{\prime \prime \,\,i}_{j,})- \frac{\mathcal{H}(Q^{\prime i}_{\,\,\,,j}+ Q^{\prime \,\,i}_{j,})}{a^2}\,,\end{aligned}$$ where in our convention, $G_{\mu \nu} = R_{\mu \nu} - (1/2) g_{\mu
\nu} R$, and, $\delta G^{\mu}_{\nu}= \delta (g^{\mu \alpha}
G_{\alpha \nu}) = \delta g^{\mu \alpha} G_{\alpha \nu}+ g^{\mu \alpha}
\delta G_{\alpha \nu}\,.$ The field equation in $f(R)$ theory is [@Sotiriou:2008rp]: $$G_{\mu \nu}= \frac{\kappa T_{\mu \nu}}{F(R)} + \frac{g_{\mu
\nu}[f(R)-RF(R)]}{2F(R)}+\frac{D_{\mu} D_{\nu} F(R)- g_{\mu
\nu}\Box F(R)}{F(R)}\,,$$ where $D_\mu A^\nu \equiv \partial_\mu A^\nu + \Gamma^\nu_{\mu
\lambda} A^\lambda$ represents a covariant derivative of a contravariant 4-vector $A^\mu$ and $\Box \equiv g^{\mu \nu} D_\mu
D_\nu$. Perturbing the above field equation one obtains $$\begin{aligned}
\label{2nd}
\delta G^{\mu}_{ \nu} &=& \frac{\kappa \delta T^{\mu}_{\nu}}{F} -
\frac{\kappa T^{\mu}_{ \nu }
F_R \delta R}{F^2} - \frac{\delta ^{\mu }_{\nu} f F_R \delta R}{2 F^2}-
\frac{\left[D^{\mu}D_{\nu} F- \delta ^{\mu }_{\nu} \Box F\right]F_R \delta R}
{F^2(R)} \nonumber\\
&+& \frac{1}{F} \left[\delta g^{\mu \alpha}( {\partial_{\alpha} \partial_{\nu} F -
\Gamma^{\lambda}_{\alpha \nu} \partial_{\lambda} F} )
+g^{\mu \alpha }\left({ \partial_{\alpha} \partial_{\nu}(F_R \delta R) -
\delta \Gamma^{\lambda}_{\alpha \nu} \partial_{\lambda} F -
\Gamma^{\lambda}_{\alpha \nu}\partial_{\lambda}(F_R \delta R)}\right)
\right.\nonumber\\
&-& \delta ^{\mu }_{\nu} ( \partial^{\lambda} \partial_{\lambda}(F_R \delta R) + \delta \Gamma^{\lambda}_{\rho \lambda} g^{\rho \kappa }(\partial_{\kappa} F)+ \Gamma^{\lambda}_{\rho \lambda} \delta g^{\rho \kappa }(\partial_{\kappa} F) + \Gamma^{\lambda}_{\rho \lambda} g^{\rho \kappa } \partial_{\kappa}(F_R \delta R))]\,.\end{aligned}$$ For further progress we require the perturbed fluid energy-momentum tensor in the Jordan frame, whose background value is specified in Eq. (\[tmunu\]). In our convention we specify $u_\mu = -a(1,v_i)$ where the scale-factor is expressed in conformal time and $\delta u_i \equiv -a v_i$. In such a case one can easily see that $u^\mu = a^{-1}(1, -S^i-v^i)$. By perturbing Eq. (\[tmunu\]) one gets the non-zero components of $T^\mu_\nu$ in the absence of any anisotropic stress: $$\delta T^{0}_{i} = - (\rho + P)v_i \,,\,\,\,\,
\delta T^{i}_{0} = (\rho + P)(S^i + v^i)\,,
\label{texp}$$ where $\rho$ and $P$ are the background values of the thermodynamic variables. From the above set of equations one can write the dynamical equations for the perturbations as $$\begin{aligned}
\frac{1}{2a^2} \nabla P^{i}&= -\frac{\kappa }{F} (\rho + P)v^{i}\,,
\label{fv1}\\
\frac{1}{2a^2}\partial_{\eta}\left[a^2(P^{i}_{\,\,,j}+ P^{\,\,\,i}_{j,})\right]&= -\frac{F^\prime}{2F} (P^{i}_{\,\,,j}+ P^{\,\,\,i}_{j,})\,,
\label{fv2}\end{aligned}$$ where\
$$\label{118}
S^{i}+ Q^{\prime i} \equiv P^i$$ is a gauge-invariant quantity. Henceforth we will work in the Newtonian gauge where $Q^i=0$. One can easily check that in the limit when $f(R)=R$ the above equations become identical to the equations for vector perturbations obtained in [@Battefeld:2004cd].
![Evolution of velocity perturbation $v^i_k$ with $\eta$ through symmetric bounce. $S^i_k(0)$’s are the same as in the previous plot.[]{data-label="vs2"}](PertSym.pdf)
![Evolution of velocity perturbation $v^i_k$ with $\eta$ through symmetric bounce. $S^i_k(0)$’s are the same as in the previous plot.[]{data-label="vs2"}](VelSym.pdf)
In the Newtonian gauge the solutions of the above equations, in the Fourier space where the subscript $k$ specifies the $k$th mode, are $$\label{vik}
v^i_k = \frac{k^2 F}{2a^2 \kappa (\rho + P)} S^i_k$$ and\
$$\label{sik}
S^i_k = \frac{C^i_k}{a^2 F}$$ where $C^{i}_{k} $ is a constant 3-vector. Combining the above equations we get an expression for the Fourier mode of velocity perturbation as: $$\label{vpertf}
v^{i}_{k} \sim \frac{k^2 C^{i}_{k}}{a^{1-3\omega}}\,.$$ In GR, in contracting phase of the universe, the vector metric perturbation increases as the scale-factor decreases. This growth of perturbations could pose a fundamental threat to the validity of perturbation theory. But, in $f(R)$ theory, the evolution of vector perturbation depends on the term $a^2F(R)$. The behavior of $F(R)$ in general affects the evolution of the vector perturbations in metric $f(R)$ cosmology.
In this paper we work with exponential gravity where the form of $f(R)$ is given in Eq. (\[frform\]) where $\alpha= 10^{12}$. The metric perturbation $S^i_k$ and velocity perturbation $v^i_k$ for $k=10^{-10}$, for a radiation dominated universe where $\omega=1/3$, have been plotted for a symmetric cosmological bounce in the background in Fig. \[vs1\] and Fig. \[vs2\]. The conditions used for the background evolution for the symmetric bounce remains the same as specified in subsection \[jords\]. The plots clearly show that in $f(R)$ cosmology the vector perturbations remains practically the same through the non-singular bounce. More over the metric vector perturbation slightly diminishes during the contracting phase where as in GR the vector perturbation only increases during the contracting phase. Consequently $f(R)$ cosmology can moderate the growth of vector perturbations, a fact perhaps noted for the first time in this paper.
In the Einstein frame analysis of vector perturbations, the field equation is: $$\tilde{R}^{\mu}_{\nu}-\frac{1}{2}\delta^{\mu}_{\nu} \tilde{R}=\kappa (\tilde{T}^{\mu}_{\nu}+ S^{\mu}_{\nu})\,.$$ In the Einstein frame the effective energy-momentum tensor is $\tilde{T}^{\mu}_{\nu}+ S^{\mu}_{\nu}$ where $S^{\mu}_{\nu}$ is the energy momentum tensor of a scalar field. $\tilde{T}^{\mu}_{\nu}$ is the Einstein frame counterpart of ${T}^{\mu}_{\nu}$. As the scalar field does not produce any vector perturbations only the fluid perturbations, coming from $\tilde{T}^{\mu}_{\nu}$ contribute on the right hand side of the perturbed Einstein equation. Consequently the vector perturbation equations in Einstein frame are: $$\begin{aligned}
\frac{1}{2\tilde{a}^2} \nabla \tilde{S}^{i}=-\kappa (\tilde{\rho} +\tilde{P})\tilde{v}^{i}\,,\,\,\,\,{\rm and}\,\,\,\,\,\,
\partial_{\eta}\left[\tilde{a}^2(\tilde{S}^{i}_{\,\,,j}+ \tilde{S}^{\,\,\,i}_{j,})\right]=0\,,\end{aligned}$$ where we have used the results from [@Battefeld:2004cd]. Solving these equations we get $$\tilde{S}^{i}_{k} = \frac{\tilde{C}^{i}_{k}}{\tilde{a}^2}\,,$$ which shows that $\tilde{S}^{i}_{k}=S^{i}_{k}$ when $\tilde{C}^i_k=C^i_k$. In the present case we have chosen $C^i_k=0.00015$. As in the present case the vector perturbations remain equal in both the conformal frames we do not separately present the Einstein frame result. Both the frames produce the same plots and no singularities are encountered in converting the result from Einstein frame to the Jordan frame. The present calculation shows that one can always use the Einstein frame for the calculation for vector perturbations as the calculations are much easily handled in the Einstein frame.
Discussion and conclusion
=========================
The present paper deals with metric perturbations in a bouncing cosmology guided by $f(R)$ theory of gravity. Scalar and vector metric perturbations during bounce in exponential gravity have been presented in this paper. The cosmological bounce takes place in the Jordan frame. The scalar metric perturbations have been calculated in the longitudinal gauge as this gauge choice remains invariant under a conformal transformation relating the Jordan frame and the Einstein frame. In the case of the vector perturbations the metric perturbations remain the same in both the frames and one can work with any gauge one wishes to. In the present paper we have worked with the Newtonian gauge. While the background evaluation of bouncing cosmologies can efficiently be calculated in the Einstein frame the perturbations on the metric and fluid variables can also be calculated in the Einstein frame where we expect the gravitational theory to be like GR with an extra scalar field. In a previous calculation the authors tried to calculate the scalar perturbations in a bouncing scenario using the Einstein frame [@Paul:2014cxa] but did not take into account the fluid perturbations. Hence the earlier calculations were incomplete. In this paper the full calculation of the scalar perturbations in the Einstein frame are presented for the first time. The perturbation calculations are general although in the present case the results are applied for the particular case of a cosmological bounce in the Jordan frame. All the results presented in this paper is for an exponential $f(R)$ gravity theory which satisfies the basic stability conditions[^3]. The matter content of the universe was always assumed to be radiation fluid as these is the most general fluid which plays an important role in the very early universe.
In the present paper the critical role of the Einstein frame in aiding the calculations of Jordan frame phenomena in $f(R)$ gravity is studied in full detail. The paper establishes that the Einstein frame can be a used for the background evolutions for all the cases in the Jordan frame and also for perturbation evolution of many cases in $f(R)$ gravity except the particular case of asymmetrical bounce in the Jordan frame where one cannot transform the scalar metric perturbation back to the Jordan frame. Although the correspondence of the background cosmological evolution in the conformal frames, for $f(R)$ based cosmology, was noted much earlier [@Maeda:1988ab; @Mukhanov:1990me] here one must note that in [@Paul:2014cxa] it was pointed out that for the case of a cosmological bounce in presence of matter, which satisfies $\rho+P \ge 0$ in the Jordan frame, there is no corresponding cosmological bounce in the conformally related Einstein frame when one works with spatially flat FLRW spacetimes. As far as cosmological bounce in flat FLRW spacetime is concerned, the Einstein frame acts like a true auxiliary frame where one can do all the calculations and then convert the results appropriately in the Jordan frame. For cosmological perturbations one can also use the Einstein frame as the auxiliary frame but as far as scalar perturbations are concerned the conformal correspondence fails for asymmetric bounces. This particular failure is not a physical problem as shown in the present paper. The perturbations evolve smoothly in both the Jordan frame calculation and Einstein frame calculation. The problem arises in the connecting formulae which connect the Jordan frame perturbations with the Einstein frame perturbation. The paper presents the calculations of perturbations separately in both the conformal frames and then connects the results to show the validity of the Einstein frame results. The nature of the variation of the fluid velocity potential in the Jordan frame is also presented in the paper.
The next part of the present paper deals with the evolution of vector perturbations during a non-singular bounce in $f(R)$ gravity. This calculation shows that for vector perturbations one can actually use the Einstein frame for all the cases and the calculations do become much easier in the Einstein frame. The vector perturbations remains the same in both the frames and no singularity appears anywhere in the description of vector perturbations from the Einstein frame. In GR based cosmological models it was shown that one expects the vector perturbations to be increasing during the contracting phase. For a singular bounce the vector perturbations can diverge during the bounce. In the present paper we have only studied non-singular bounce in exponential gravity and the results regarding the evolution of vector perturbations during such a phase are interesting. In this case the vector perturbations practically remains constant during the bouncing phase showing stability of the perturbation modes. One can expect that in the deep expansion phase of the universe this vector modes do get diluted. There has been studies on magnetic field generation in the early universe and the nature of vector perturbations, we expect the specific nature of the evolution of vector perturbations in the present case can have interesting consequences for magnetogenesis.
The question of tensor perturbation in general $f(R)$ gravity and in particular related to bouncing scenarios has not been presented in the paper. Tensor perturbations will be taken up in a future publication as there are various interesting issues related to primordial tensor perturbations which require separate and thorough investigation. The present work presupposes that the earlier universe (much before the bouncing time) and the later universe (much later than the bouncing time) were guided by a theory of gravity like GR which gets effectively modified to exponential gravity in the high curvature limit near the bounce. The change over from GR to $f(R)$ is non-trivial and will require new physics. The perturbation modes earlier and later than the bounce are expected to follow standard cosmological dynamics obtained from GR. In this process our main aim is to show how the perturbation modes evolve during the bouncing time. It is shown that the perturbation modes can solely be tackled from the accompanying, conformally connected Einstein frame for most of the cases. The perturbation modes which have been studied remain perturbative throughout the bounce, but this does not comprehensively specify that all the perturbations are stable. The vector perturbations should be stable for all initial values but it is very difficult to generalize the statement for scalar perturbations for all initial values of the perturbations. There may remain some modes with specific initial conditions which tend to become non-perturbative near the bounce. In such cases one has to bring in new physics to settle the issue of perturbative instability. While our work does not prove perturbative stability in the most general way it definitely shows how the stable perturbations evolve near the bounce.
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[^1]: $^\dagger$ pribari@iitk.ac.in, $^\ddagger$kaushikb@iitk.ac.in
[^2]: The word exponential gravity in the context of $f(R)$ theories may be a bit confusing as previously many authors have used exactly the same name to work a completely different problem. The authors of the works [@Bean:2006up; @Linder:2009jz; @Odintsov:2017qif; @Bamba:2010ws] have used exponential gravity to tackle late time universe problems as structure formation or the dark energy problem. In all of these cases the form of $f(R)$ is given as $R + g(R)$ where the function $g(R)$ contains an exponential function of the Ricci scalar. Compared to those models our model is much simpler and more over our model of exponential gravity tackles an early universe problem. As the form of $f(R)$ we are using is purely exponential we do not modify the name of the theory.
[^3]: Some general remarks on stability of $f(R)$ theories were presented in Ref. [@Barrow:1983rx]. In the referred work the authors find out the stability of $f(R)$ cosmology by slightly perturbing the scale-factor and the issue of metric perturbations was not studied.
|
---
abstract: 'A coupled Camassa-Holm type equation is linked to the first negative flow of a modified Drinfeld-Sokolov III hierarchy by a transformation of reciprocal type. Meanwhile the Lax pair and bi-Hamiltonian structure behaviors of this coupled Camassa-Holm type equation under change of variables are analyzed.'
address: 'School of Mathematics, Huaqiao University, Quanzhou, Fujian 362021, People’s Republic of China'
author:
- 'Nianhua Li, Jinshun Zhang, and Lihua Wu'
title: 'Reciprocal link for a coupled Camassa-Holm type equation'
---
Introduction
============
The Camassa-Holm (CH) equation $$\label{ch}
m_t+um_x+2u_xm=0,\quad m=u-u_{xx}$$ was proposed as a model for long waves in shallow water by the asymptotic approximation of Hamiltonian for Green-Naghdi equations in 1993 [@Holm]. It is completely integrable with a Lax pair and associated bi-Hamiltonian structure [@Holm; @hyman], and is shown to be solvable by inverse scattering transformation [@beal1; @consta1; @consta2]. Meanwhile the CH equation is linked to the first negative flow of the KdV hierarchy by a transformation of reciprocal type [@fuch; @wang1; @lenel]. Furthermore, different from KdV equation, the CH equation admits peakon solutions [@Holm; @hyman; @Beals], and we call the integrable equation possessing peakon solutions CH type equation.
Degasperis and Procesi [@Degasperis], applying the method of asymptotic integrability, discover a new CH type equation $$m_t+um_x+3u_xm=0,\quad m=u-u_{xx},$$which is also integrable admitting a Lax representation as well as a bi-Hamiltonian structure, and is reciprocal linked to a negative flow of the Kaup-Kupershmidt hierarchy [@Deg2]. Besides the multi-peakon solutions of it are studied by inverse scattering approach [@Lun; @lu2].
Applying the tri-Hamiltonian duality approach [@fuch; @olver], two new CH type systems are proposed and the corresponding Lax pairs are given by Schiff [@schiff]. The first one is the modified CH equation (MCH) [@fuch; @olver] $$m_t+[(u^2-u_x^2)m]_x=0, \quad m=u-u_{xx},$$which is later rediscovered by Qiao from two dimensional Euler equation [@qiao]. Furthermore, the MCH equation is related to the (modified) KdV hierarchy via a reciprocal transformation [@Hone]. The peakon and multi-component generalization of it are also researched (see e.g. [@Gui; @xia]). The second one is a two-component CH equation [@olver] $$\begin{aligned}
&& m_t+um_x+2u_xm-\rho\rho_x = 0,\quad m=u-u_{xx}, \\
&& \rho_t+(\rho u)_x = 0,\end{aligned}$$which is rediscovered from the Green-Naghdi equations and the peakon solutions of it in the short waves limit are constructed by Constantin [@constantin]. Moreover it is reciprocal linked to the AKNS hierarchy [@Ming].
Subsequently, the Novikov’s equation [@Novikov] $$\label{novikov}
m_{t}+u^{2}m_{x}+3uu_{x}m=0,\quad m=u-u_{xx}.$$is obtained in the symmetry classification of CH type equation, a Lax pair and bi-Hamiltonian structure are given by Hone and Wang [@Hone], and the explicit formulas for the multi-peakon solutions of the Novikov’s equation are calculated [@honeJ]. Especially the Novikov’s equation is reciprocal connected to the first negative flow of Sawaka-Kotera hierarchy. The Geng-Xue equation [@Geng] $$\begin{aligned}
\label{novikov2}
&&m_{t}+3u_{x}vm+uvm_{x} =0, \nonumber\\
&&n_{t}+3v_{x}un+uvn_{x} =0, \\
&&m=u-u_{xx},\quad n=v-v_{xx}. \nonumber\end{aligned}$$is proposed as a generalization of the Novikov’s equation admitting a Lax pair and a Hamiltonian structure later. Furthermore, the Geng-Xue equation is reciprocal connected to the first negative flow of the modified Boussinesq hierarchy [@Li], and the peakon solutions of it are discussed [@ldk].
Recently, Geng and Wang [@geng1] propose a new coupled CH type equation with cubic nonlinearity $$\begin{aligned}
\label{cubicCH}
&& v_t=2v_x(qr_x-q_xr)+2v(3qr_{xx}-q_{xx}r-q_xr_x-qr), \nonumber\\
&& w_t=2w_x(qr_x-q_xr)-2w(3q_{xx}r-qr_{xx}-q_xr_x-qr),\\
&& v=r_{xxx}-r_x,\quad w=q_{xxx}-q_x,\nonumber\end{aligned}$$associated with a $4\times4$ matrix spectral problem $$\label{lax}
\varphi_x=U\varphi,\quad \varphi=\left(
\begin{array}{c}
\varphi_1 \\
\varphi_2 \\
\varphi_3 \\
\varphi_4 \\
\end{array}
\right),\quad U=\left(
\begin{array}{cccc}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\frac{1}{4} & \lambda v & 0 & 0 \\
\lambda w & \frac{1}{4} & 0 & 0 \\
\end{array}
\right).$$ Bi-Hamiltonian structure and infinite sequences of conserved quantities as well as N-peakon solutions for the new system are also considered by them.
The outline of this paper is as follows. In section 2, we construct a reciprocal transformation for the CH type system (\[cubicCH\]), it shows that the transformed system is a constraint of the first negative flow of a modified Drinfeld-Sokolov (DS) III hierarchy. In section 3, the Hamiltonian structures of the coupled CH type hierarchy under the reciprocal transformation are given. In appendix, a reciprocal transformation for another system related to the problem (\[lax\]) is also studied.
Reciprocal transformation
=========================
Since infinite sequences of conserved quantities for the integrable hierarchy of the spectral problem (\[lax\]) are obtained [@geng1], we may construct reciprocal transformations for the coupled CH type system (\[cubicCH\]) which possesses the Lax representation $$\label{lax1}
\varphi_{x}=U\varphi, \quad \varphi_{t}=V\varphi,$$ where $$V=\left(
\begin{array}{cccc}
2q_{xx}r-k_1 & \frac{r_x}{\lambda}&4k_2 & -\frac{2r}{\lambda} \\
-\frac{q_x}{\lambda} & k_1-2qr_{xx} & \frac{2q}{\lambda} & 4k_2 \\
q_{xx}r_x-q_xr_{xx}-k_2 & 4\lambda vk_2+\frac{2r_{xx}-r}{2\lambda} & 2qr_{xx}-k_1 & -\frac{r_x}{\lambda} \\
4\lambda wk_2+\frac{q-2q_{xx}}{2\lambda} & q_{xx}r_x-q_xr_{xx}-k_2 & \frac{q_x}{\lambda} & k_1-2q_{xx}r\\
\end{array}
\right),$$herein $k_1=\frac{1}{2\lambda^2}+q_xr_x+qr, k_2=\frac{1}{2}(qr_x-q_xr)$.
Especially, notice that the equation (\[cubicCH\]) implies a conservation law $$((wv)^{\frac{1}{4}})_t=(2(wv)^{\frac{1}{4}}(qr_x-q_xr))_x,$$which defines a reciprocal transformation via the relation $$\label{reci1}
dy=udx+2u(qr_x-q_xr)dt,\quad d\tau=dt,$$where $u=(wv)^{\frac{1}{4}}$.
Setting $h =(\frac{v}{w})^{\frac{1}{4}}$, then after a gauge transformation $\varphi_1=hu^{-\frac{1}{2}}\phi$, the scalar form of spectral problem (\[lax\]) can be transformed to $$\label{lax3}
(\partial_y^4+\partial_y m\partial_y+n)\phi=\lambda^2\phi,$$where $$\begin{aligned}
&& m=-\frac{1}{2u^2}+\frac{u_y^2}{2u^2}-\frac{u_{yy}}{u}-6\frac{h_y^2}{h^2}+4\frac{h_{yy}}{h}, \\
&& n=-\frac{h_y}{h}m_{y}+\frac{h_y^2}{h^2}m+\frac{1}{2}m_{yy}
-\frac{h_{yyyy}}{h}+36\frac{h_y^4}{h^4}-48\frac{h_y^2h_{yy}}{h^3}+6\frac{h_{yy}^2}{h^2}+10\frac{h_yh_{yyy}}{h^2}\\
&&\hspace{0.8cm}+\frac{u_{yy}}{4u^3}+\frac{u_{yy}^2}{4u^2}+\frac{1}{16u^4}-\frac{u_y^2u_{yy}}{4u^3}+\frac{u_y^4}{16u^4}-\frac{u_y^2}{8u^4}.\end{aligned}$$
The relation between $m,n$ and $u,h$ may be related to factorization of Lax operator [@Fordy]. To begin with the factorization in [@guha], $$L=\partial_y^4+\partial_y m\partial_y+n=(\partial_y^2+\partial_y i -j)(\partial_y^2-i\partial_y-j),$$where $m$ and $n$ satisfy a pair of generalized Miura system $$m=-(i_y+i^2+2j), \quad n=j^2-j_{yy}-(ij)_y.$$ In fact $L$ may be further decomposed as $$L=(\partial_y-b_1+a_1)(\partial_y-b_1-a_1)(\partial_y+b_1+a_1)(\partial_y+b_1-a_1),$$ with $i,j$ satisfying $$\begin{aligned}
i=-2b_1,\quad j=a_{1y}+a_1^{2}-b_{1y}-b_1^{2},\end{aligned}$$ where $a_1=\frac{u_{y}+1}{2u},\ b_1=\frac{h_y}{h}$.
It is not difficult to find that the spectral problem (\[lax3\]) is which of the DS III system [@metin; @sokolov] and the system (\[cubicCH\]) in the new variable is nothing but a constraint of the first negative flow of the DS III hierarchy. However, detailed calculation shows that we should rewrite the spectral problem (\[lax3\]) as matrix form in order to obtain the transformed system of (\[cubicCH\]) and its Lax pair.
To this end, on the one hand, eliminating $\varphi_3,\varphi_4$ and denoting $s=\frac{u}{h^2}$, the spectral problem (\[lax\]) is transformed to $$\begin{aligned}
&&\phi_{yy}+(\frac{u_y}{u}-\frac{s_y}{s})\phi_y+(\frac{3s_y^2}{4s^2}-\frac{s_{yy}}{2s}-\frac{u_ys_y}{2us}-\frac{1}{4u^2})\phi=\lambda\psi,\\
&&\psi_{yy}+(\frac{s_y}{s}-\frac{u_y}{u})\psi_y+(\frac{s_{yy}}{2s}-\frac{s_y^2}{4s^2}-\frac{u_ys_y}{2us}-\frac{u_{yy}}{u}+\frac{u_y^2}{u^2}-\frac{1}{4u^2})\psi=\lambda\phi,\end{aligned}$$where $\psi=us^{-\frac{1}{2}}\varphi_2$ (here and in the sequel, we use $u,s$ instead of $u,h$ for convenience). Then the two new potentials $i,j$ are obtained $$i=\frac{s_y}{s}-\frac{u_y}{u}, \quad j=\frac{1}{4u^2}+\frac{u_ys_y}{2us}-\frac{3s_y^2}{4s^2}+\frac{s_{yy}}{2s}.$$ Setting $\Phi=(\psi,
\psi_y,
\phi,
\phi_y)^{T}$, then the Lax pair (\[lax1\]) of the system (\[cubicCH\]) is transformed to $$\label{tran1}
\Phi_y= \left(
\begin{array}{cccc}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
j & \lambda & i & 0 \\
\lambda & j-i_y & 0 & -i \\
\end{array}
\right)\Phi,\quad
\Phi_\tau=\left(
\begin{array}{cccc}
-\frac{1}{2\lambda^2} & \frac{g_y-2gi}{\lambda} & 0 & -\frac{2g}{\lambda} \\
-\frac{f_y+2fi}{\lambda} & \frac{1}{2\lambda^2}& \frac{2f}{\lambda} & 0 \\
-2g & \frac{A_2}{\lambda}& -\frac{1}{2\lambda^2} & -\frac{g_y}{\lambda} \\
\frac{A_3}{\lambda} & 2f & \frac{f_y}{\lambda} & \frac{1}{2\lambda^2} \\
\end{array}
\right)\Phi,$$where $f=q\frac{u^2}{s},g=rs$ with $$\begin{aligned}
A_2 &=& g_{yy}-2g_yi-2gj, \\
A_3 &=& -f_{yy}-2(fi)_y+2fj.\end{aligned}$$
On the other hand, under the transformation (\[reci1\]), the system (\[cubicCH\]) is transformed to $$\begin{aligned}
i_\tau &=& -2(rs+q\frac{u^2}{s}),\quad \quad \quad \quad \quad \quad \ u^2s^{-1}=\partial_y u\partial_y u\partial_y r-\partial_y r, \\
j_\tau &=& -3(rs)_y+2rsi+\frac{u^2}{s}(q_y+q\frac{s_y}{s}), \quad s=\partial_y u\partial_y u\partial_y q-\partial_y q.\end{aligned}$$ Furthermore, above system may be reformed as $$\begin{aligned}
\label{md1}
i_\tau &=& -2(f+g),\quad \quad \quad \quad \quad \ F_1=-1, \\\label{md2}
j_\tau &=& f_y-3g_y+2i(f+g),\quad F_2=-1,\end{aligned}$$ where $$\begin{aligned}
&&F_1=3ig_{yy}-g_{yyy}+(i_y-2i^2+4j)g_y+(2j_y-4ij)g, \\
&&F_2=(4ij+2j_y-2i_{yy}-4ii_y)f+(4j-5i_y-2i^2)f_y-3if_{yy}-f_{yyy}.\end{aligned}$$ Then it is easy to verify that the compatibility of the transformed Lax pair (\[tran1\]) is just the transformed system (\[md1\]-\[md2\]), so the system (\[cubicCH\]) and its Lax pair (\[lax1\]) are transformed to (\[md1\]-\[md2\]) and (\[tran1\]) respectively.
Now, we will show that the transformed system (\[md1\]-\[md2\]) is a constraint of the first negative flow of the modified DS III hierarchy. The first negative flow of the modified DS III hierarchy may be formed as (see Appendix) $$\begin{aligned}
\label{neg1}
i_\tau &=& -2(f+g), \quad \quad \quad \quad \quad \ G_1=0,\\\label{neg2}
j_\tau &=& f_y-3g_y+2i(f+g),\quad G_2=0,\end{aligned}$$ where $$\begin{aligned}
G_1 &=& \frac{1}{4}(2(F_1+F_2)_y+\partial_y i\partial_y^{-1}(F_1-F_2)),\\
G_2 &=& \frac{1}{4}((\partial_y^2-i\partial_y)(F_1+F_2)+(-\frac{1}{2}\partial_y^3+\frac{i}{2}\partial_y^2+j\partial_y+\partial_y j)\partial_y^{-1}(F_1-F_2)).\end{aligned}$$ To see the connection between (\[md1\]-\[md2\]) and (\[neg1\]-\[neg2\]), it is clear that the following identity holds: $$\left(
\begin{array}{c}
G_1 \\
G_2 \\
\end{array}
\right)=\frac{1}{2}\left(
\begin{array}{cc}
\partial_y+\frac{i}{2}+\frac{i_y}{2}\partial_y^{-1} & \partial_y-\frac{i}{2}-\frac{i_y}{2}\partial_y^{-1} \\
\frac{1}{4}\partial_y^2-\frac{i}{4}\partial_y+j+\frac{j_y}{2}\partial_y^{-1} & \frac{3}{4}\partial_y^2-\frac{3i}{4}\partial_y-j-\frac{j_y}{2}\partial_y^{-1} \\
\end{array}
\right)\left(
\begin{array}{c}
F_1 \\
F_2\\
\end{array}
\right),$$(here all integration constants are assumed to be zero). This leads to the fact that the system (\[md1\]-\[md2\]) may be regarded as a reduction of the first negative flow of the modified DS III hierarchy (\[neg1\]-\[neg2\]).
The Hamiltonian structure behavior under the transformation
===========================================================
Let us define ${\cal E}=\partial^3-\partial,\ \theta=(v,w)^{T}$ and $v^{(n)}=\frac{\partial^{n}v}{\partial x^n}$, then the coupled system (\[cubicCH\]) can be written as a bi-Hamiltonian structure $$\left(
\begin{array}{c}
v \\
w \\
\end{array}
\right)_t={\cal J}_2\left(
\begin{array}{c}
\frac{\delta}{\delta v} \\
\frac{\delta}{\delta w} \\
\end{array}
\right)H_0={\cal J}_1\left(
\begin{array}{c}
\frac{\delta}{\delta v} \\
\frac{\delta}{\delta w} \\
\end{array}
\right)H_1$$where $$\begin{aligned}
&&{\cal J}_2=-{\cal E}\sigma_1,\\
&&{\cal J}_1=-2\sigma_3\theta\partial^{-1}(\sigma_3\theta)^{T}-2(\theta\partial+\partial \theta){\cal E}^{-1}(\theta\partial+\partial \theta)^{T},\end{aligned}$$and $$\begin{aligned}
H_0 &=& \frac{1}{2}\int (v[2q^2r_{xx}-2qq_xr_x+q_x^2r-q^2r]+w[2rq_xr_x-2q_{xx}r^2-r_x^2q+r^2q]) dx, \\
H_1 &=& \frac{1}{2}\int (wr-qv) dx,\end{aligned}$$ herein $\sigma_1=\left(
\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}
\right)$ and $\sigma_3=\left(
\begin{array}{cc}
1 & 0 \\
0 & -1 \\
\end{array}
\right)
$ are the standard Pauli matrices.
The corresponding structures such as the recursion operator, bi-Hamiltonian structure, conserved quantities between the two connected equations can be generated by the reciprocal transformation (\[reci1\]), and in the following, the bi-Hamiltonian structure for the coupled CH type hierarchy under the change of variables $(x,v,w)\rightarrow (y,i,j)$ $$\begin{aligned}
\left\{\begin{array}{rl}
&y=P(x,v^{(n)},w^{(n)})=\int^x_{-\infty} (wv)^{\frac{1}{4}}dx=\partial^{-1} (wv)^{\frac{1}{4}}, \\
&i(y)=Q_1(x,v^{(n)},w^{(n)})=-\frac{1}{2}(wv)^{-\frac{5}{4}}(wv_x-w_xv),\\
&j(y)=Q_2(x,v^{(n)},w^{(n)})\\
&\hspace{0.8cm}=\frac{1}{64}(wv)^{-\frac{5}{4}}(16w^2v^2-33v^2w_x^2+6ww_xvv_x+7w^2v_x^2+24v^2ww_{xx}-8w^2vv_{xx})\end{array} \right.\end{aligned}$$may be given.
Let $\vartheta=(i,j)^{T}$, following the ideal in [@Bru], an implicit function $B(\theta,\vartheta)=0$ may be defined, then $$B_{\theta}\theta_t+B_{\vartheta}\vartheta_t= 0,$$ where $B_{\theta},B_{\vartheta}$ are corresponding Frechét derivatives for the vector variables, so we get $$\vartheta_t=-T_1\theta_t,\quad T_1=B_{\vartheta}^{-1}B_{\theta}.$$ It is easy to find that $B_{\vartheta}$ is the identity matrix, therefore we may obtain $$\label{T1}
T_1=\left(
\begin{array}{cc}
i_yP'[v]-Q_1'[v] &i_yP'[w]-Q_1'[w] \\
j_yP'[v]-Q_2'[v] &j_yP'[w]-Q_2'[w] \\
\end{array}
\right).$$ Then as we know that $$\begin{aligned}
\label{HH1}
&&\theta_t={\cal J}(v,w)\frac{\delta H}{\delta\theta}={\cal J}(v,w)E_{\theta}h,\\\label{HH2}
&&\vartheta_t=\tilde{{\cal J}}(i,j)\frac{\delta \tilde{H}}{\delta\vartheta}=\tilde{{\cal J}}(i,j)E_{\vartheta}\tilde{h},\end{aligned}$$where $$H=\int h(x,v^{(n)},w^{(n)})dx,\quad \tilde{H}=\int \tilde{h}(y,i^{(n)},j^{(n)})dy,$$ and $E_{\theta}, E_{\vartheta}$ are the Euler operators. In order to connect the Hamiltonian structures of the two evolution equations, we need the action on Euler operator under a change of variables which is related by (see [@olver2], Exercise 5.49) $$E_{\theta}h=T_2E_{\vartheta}\tilde{h},$$ herein $$\label{T2}
T_2=\left(
\begin{array}{cc}
Q'^{*}_{1,v}(D_x P)-P'^{*}_{v}(D_x Q_1) & Q'^{*}_{2,v}(D_x P)-P'^{*}_{v}(D_x Q_2) \\
Q'^{*}_{1,w}(D_x P)-P'^{*}_{w}(D_x Q_1) & Q'^{*}_{2,w}(D_x P)-P'^{*}_{w}(D_x Q_2) \\
\end{array}
\right).$$ We are now in a position to state our main results:
[@Bru] The Hamiltonian structures of two evolution equations (\[HH1\]) and (\[HH2\]) which are related by $B(\theta,\vartheta)$ are linked as $$ \tilde{{\cal J}}_{k}=-T_1{\cal J}_{k}T_2\ (k=1,2), \quad \left(
\begin{array}{c}
\frac{\delta }{\delta v} \\
\frac{\delta}{\delta w} \\
\end{array}
\right)H(v,w)=T_2\left(
\begin{array}{c}
\frac{\delta}{\delta i} \\
\frac{\delta}{\delta j} \\
\end{array}
\right)\tilde{H}(i,j),$$where $T_1$ and $T_2$ are given by (\[T1\]) and (\[T2\]) accordingly.
In the following, we will indicate the explicit formula for the transformed Hamiltonian operators. It is easy to find that $$\begin{aligned}
&&P'_{v}=\frac{1}{4}\partial^{-1}(wv)^{-\frac{3}{4}}w=\frac{1}{4}\partial_y^{-1}\frac{1}{v},\quad \quad \quad \quad \ P'_{w}=\frac{1}{4}\partial_y^{-1}\frac{1}{w},\\
&&P'^{*}_{v}=-\frac{1}{4}(wv)^{-\frac{3}{4}}w\partial^{-1}=-\frac{u}{4v}\partial_y^{-1}\frac{1}{u},\quad \quad P'^{*}_{w}=-\frac{u}{4w}\partial_y^{-1}\frac{1}{u}.\end{aligned}$$ Then through tedious calculations and change of variables, we obtain $$\begin{aligned}
&&Q'_{1,v}=-\partial_y\frac{1}{2v}-\frac{i}{4v},\quad \quad \quad \quad \quad \quad \quad \ Q'_{1,w}=\partial_y\frac{1}{2w}-\frac{i}{4w},\\
&&Q'_{2,v}=(-\frac{j}{2}-\frac{1}{8}\partial_y^2+\frac{i}{8}\partial_y)\frac{1}{v},\quad \quad \quad \quad Q'_{2,w}=(-\frac{j}{2}+\frac{3}{8}\partial_y^2-\frac{3i}{8}\partial_y)\frac{1}{w},\\\end{aligned}$$and $$\begin{aligned}
&&Q'^{*}_{1,v}(D_x P)=\frac{u}{4v}(2\partial_y-i),\quad \quad \quad \quad \quad \quad \ Q'^{*}_{1,w}(D_x P)=-\frac{u}{4w}(2\partial_y+i),\\
&&Q'^{*}_{2,v}(D_x P)=-\frac{u}{v}(\frac{j}{2}+\frac{1}{8}\partial_y^2+\frac{1}{8}\partial_y i),\quad \quad Q'^{*}_{2,w}(D_x P)=\frac{u}{w}(-\frac{j}{2}+\frac{3}{8}\partial_y^2+\frac{3}{8}\partial_y i).\\\end{aligned}$$ A direct computation shows $$\begin{aligned}
T_1 &=& \frac{1}{4}\left(
\begin{array}{cc}
(\partial_y i \partial_y^{-1}+2\partial_{y})\frac{1}{v}& (\partial_y i \partial_y^{-1}-2\partial_{y})\frac{1}{w} \\
(j_y\partial_{y}^{-1}+2j+\frac{1}{2}\partial_y^2-\frac{i}{2}\partial_y)\frac{1}{v}& (j_y\partial_{y}^{-1}+2j-\frac{3}{2}\partial_y^2+\frac{3i}{2}\partial_y)\frac{1}{w} \\
\end{array}
\right), \\ \\
T_2 &=& \frac{1}{4}\left(
\begin{array}{cc}
\frac{u}{v}(2\partial_y-i+\partial_y^{-1}i_y)& \frac{u}{v}(-\frac{1}{2}\partial_y^2-\frac{1}{2}\partial_y i-2j+\partial_y^{-1}j_y) \\
\frac{u}{w}(-2\partial_y-i+\partial_y^{-1}i_y) & \frac{u}{w}(\frac{3}{2}\partial_y^2+\frac{3}{2}\partial_y i-2j+\partial_y^{-1}j_y) \\
\end{array}
\right).\end{aligned}$$ We are now in a position to obtain the Hamiltonian operators $\tilde{{\cal J}}_1,\tilde{{\cal J}}_2$.
Under change of variables, $$ \frac{1}{v}{\cal E}\frac{u}{w}=\Theta_1=\partial_y^3-3i\partial_y^2+(2i^2-i_y-4j)\partial_y+4ij-2j_y.$$
It follows from the proposition 1 that $$\frac{1}{w}{\cal E}\frac{u}{v}=-\Theta_1^{*}=\partial_y^3+3i\partial_y^2+(2i^2+5i_y-4j)\partial_y+2i_{yy}-2j_y+4ii_y-4ij,$$ Hence we deduce that $$\tilde{{\cal J}}_1=-\frac{1}{16}\Lambda\left(
\begin{array}{cc}
0 & \Theta_1 \\
-\Theta_1^{*} & 0 \\
\end{array}
\right)\Lambda^{*},$$where $$\Lambda=\left(
\begin{array}{cc}
\partial_y i\partial_y^{-1}+2\partial_y & \partial_y i\partial_y^{-1}-2\partial_y \\
j_y\partial_{y}^{-1}+2j+\frac{1}{2}\partial_y^2-\frac{i}{2}\partial_y& j_y\partial_{y}^{-1}+2j-\frac{3}{2}\partial_y^2+\frac{3i}{2}\partial_y \\
\end{array}
\right).$$
Under change of variables, $$ \frac{1}{u^2}{\cal E}\frac{1}{u}=\Theta_2=\partial_y^3+(i_y-\frac{1}{2}i^2-2j)\partial_y+\partial_y(i_y-\frac{1}{2}i^2-2j).$$
Using the proposition 2, a direct calculation shows $$\begin{aligned}
&&\tilde{{\cal J}}_2=\frac{1}{2}\left(
\begin{array}{c}
2 \\
\partial_y-i \\
\end{array}
\right)\partial_y\left(
\begin{array}{c}
2 \\
\partial_y-i \\
\end{array}
\right)^{*}-\frac{1}{2}\Theta_2\left(
\begin{array}{cc}
0 & 0 \\
0 & 1 \\
\end{array}
\right)\\
&&\hspace{0.6cm}=\left(
\begin{array}{cc}
2\partial_y & -(\partial_y^2+\partial_y i) \\
\partial_y^2-i\partial_y & j\partial+\partial j-(\partial_y-i)\partial_y(\partial_y+i) \\
\end{array}
\right) .\end{aligned}$$ Since the compatible Hamiltonian operators are gotten, a recursion operator $\tilde{{\cal J}}_2\tilde{{\cal J}}_1^{-1}$ is obtained which is also the recursion operator of the modified DS III hierarchy through complicated calculation (compare with the recursion operator of modified DS III hierarchy in appendix), besides the $n$th equation of the coupled CH type hierarchy may be mapped similarly.
[**ACKNOWLEDGMENTS**]{}
This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11401572 and 11401230) and the Initial Founding of Scientific Research for the introduction of talents of Huaqiao University (Project No. 14BS314).
THE FIRST NEGATIVE FLOW OF THE MODIFIED DS III HIERARCHY
========================================================
Firstly, we will derive the recursion operator of the modified DS III hierarchy. Actually, the bi-Hamiltonian operators of the DS III hierarchy [@metin; @mihaklov] are $$\begin{aligned}
&&P^1=\left(
\begin{array}{cc}
0 & 4\partial_y \\
4\partial_y & 3\partial_y^3+\partial_y m+m\partial_y \\
\end{array}
\right),\\
&&P^2=\left(
\begin{array}{cc}
5\partial_y^3+m\partial_y+\partial_y m & \frac{3}{2}\partial_y^5+\frac{3}{2}\partial_y^2m\partial_y+4n\partial_y+3n_y \\
\frac{3}{2}\partial_y^5+\frac{3}{2}\partial_y m\partial_y^2+4n\partial_y+n_y & P^{2}_{22} \\
\end{array}
\right),\end{aligned}$$ where $P^{2}_{22}=\frac{1}{2}(\partial_y^7+\partial_y(\partial_y^3m+m\partial_y^3+m\partial_y m+n\partial_y+\partial_y n)\partial_y)+\partial_y^3n+n\partial_y^3+\partial_y mn+mn\partial_y$, then the recursion operator of the DS III hierarchy is $R=P^2(P^1)^{-1}$. Notice that $$\left(
\begin{array}{c}
m \\
n \\
\end{array}
\right)=\Omega(i,j)=\left(
\begin{array}{c}
-i_y-i^2-2j\\
j^2-j_{yy}-(ij)_y \\
\end{array}
\right),$$ therefore the recursion operator of the modified DS III hierarchy is $\tilde{R}=\Omega'^{-1}R\Omega'$.
Secondly, let us introduce the first negative flow of the modified DS III hierarchy from the formal Lax pair $$\Phi_y=\left(
\begin{array}{cc}
0 & I\\
A & B \\
\end{array}
\right)\Phi, \quad \Phi_\tau=\left(
\begin{array}{cc}
K^{1} & K^{2}\\
K^{3} & K^{4} \\
\end{array}
\right)\Phi,$$ where $$\begin{aligned}
A=\left(
\begin{array}{cc}
j & \lambda \\
\lambda & j-i_y \\
\end{array}
\right),\quad B=\left(
\begin{array}{cc}
i & 0 \\
0 & -i \\
\end{array}
\right),\end{aligned}$$$I$ is the identity matrix, and $K^{k}=(K^{k})_{2\times 2},(k=1,...,4)$. The zero-curvature equation yields $$\begin{aligned}
&&K^{1}=K^{4}-K^{2}_y-K^{2}B, \\
&&K^{3}=K^{4}_y-K^{2}_{yy}-(K^{2}B)_y+K^{2}A,\\
&&A_\tau=K^{3}_y-AK^{1}-BK^{3}+K^{4}A,\\
&&B_\tau=K^{4}_y-AK^{2}-BK^{4}+K^{3}+K^{4}B,\end{aligned}$$ which imply that $$\begin{aligned}
&&(i-\partial_y)(2K^4_{12}-K^2_{12y})+\lambda(K^2_{22}-K^2_{11})=0, \\
&&(i+\partial_y)(2K^4_{21}-K^2_{21y})+\lambda(K^2_{22}-K^2_{11})=0, \\
&&K^4_{11}+K^4_{22}-\frac{1}{2}[(\partial_y+i)K^2_{11}+(\partial_y-i)K^2_{22}]=0,\\
&&S_1-S_2-2\lambda(K^2_{11}+K^2_{22})_y=0, \\
&&S_1+S_2+\lambda[2K^4_{22}-2K^4_{11}+K^2_{11y}-K^2_{22y}+2i(K^2_{11}+K^2_{22})]=0, \\
&&[\frac{1}{2}(\partial_y^3+\partial_y^2 i-i\partial_y^2-i\partial_y i)-j\partial_y-\partial_y j](K^2_{11}-K^2_{22})
+\lambda(K^2_{12y}-K^2_{21y}+2K^4_{21}-2K^4_{12})=0,\end{aligned}$$ and $$\begin{aligned}
\label{fir1}
&&i_\tau=\lambda(K^2_{12}-K^2_{21})+\lambda^{-1}M_1, \\\label{fir2}
&&j_\tau=\lambda[\frac{1}{2}(3K^2_{12}+K^2_{21})_y+i(K^2_{21}-K^2_{12})]+\lambda^{-1}M_2,\end{aligned}$$ where $$\begin{aligned}
S_1 &=&(\frac{1}{2}\partial_y^3-i\partial_y^2-j\partial_y-\partial_y j-\frac{1}{2}\partial_y i\partial_y+i^2\partial_y+2ij)K^2_{12}, \\
S_2 &=&-(\frac{1}{2}\partial_y^3+\partial_y^2 i-j\partial_y-\partial_y j+\frac{1}{2}\partial_y i\partial_y+2i\partial_y i-i^2\partial_y-2ij)K^2_{21}, \\
M_1 &=& \frac{1}{4}[2(S_1+S_2)_y+\partial_y i\partial_y^{-1}(S_1-S_2)], \\
M_2 &=& \frac{1}{4}[(\partial_y^2-i\partial_y)(S_1+S_2)+(-\frac{1}{2}\partial_y^3+\frac{i}{2}\partial_y^2+j\partial_y+\partial_y j)\partial_y^{-1}(S_1-S_2)].\end{aligned}$$ Then the first negative flow of the modified DS III hierarchy can be obtained by taking $K^2_{12}=-2g\lambda^{-1}, K^2_{21}=2f\lambda^{-1}$ in (\[fir1\]-\[fir2\]) $$\label{flow}
\left(
\begin{array}{c}
i \\
j \\
\end{array}
\right)_\tau=
{\cal K}\left(
\begin{array}{c}
f \\
g \\
\end{array}
\right),\quad {\cal J}\left(
\begin{array}{c}
F_1 \\
F_2 \\
\end{array}
\right)=\left(
\begin{array}{c}
0 \\
0 \\
\end{array}
\right),$$ where $$\begin{aligned}
{\cal K} &=& \left(
\begin{array}{cc}
-2 & -2 \\
\partial_y+2i & -3\partial_y+2i \\
\end{array}
\right),\\
{\cal J} &=& \frac{1}{2}\left(
\begin{array}{cc}
\partial_y+\frac{i}{2}+\frac{i_y}{2}\partial_y^{-1} & \partial_y-\frac{i}{2}-\frac{i_y}{2}\partial_y^{-1} \\
\frac{1}{4}\partial_y^2-\frac{i}{4}\partial_y+j+\frac{j_y}{2}\partial_y^{-1} & \frac{3}{4}\partial_y^2-\frac{3i}{4}\partial_y-j-\frac{j_y}{2}\partial_y^{-1} \\
\end{array}
\right).\end{aligned}$$ and $$\left(
\begin{array}{c}
F_1 \\
F_2 \\
\end{array}
\right)=\Theta\left(
\begin{array}{c}
f \\
g \\
\end{array}
\right),\quad \quad \quad
\Theta=\left(
\begin{array}{cc}
0 & -\Theta_1 \\
\Theta_1^{*} & 0 \\
\end{array}
\right).$$ Straightforward calculation shows that ${\cal J}\Theta{\cal K}^{-1}=-\tilde{R}$, so the system (\[flow\]) is just the first negative flow of the modified DS III hierarchy.
RECIPROCAL TRANSFORMATION OF A TWO-COMPONENT CH TYPE EQUATION
=============================================================
In fact another two-component system $$\begin{aligned}
\label{coupleCH}
&&v_t=4vq_x+2v_xq+2vr,\nonumber\\
&&w_t=4wq_x+2w_xq-2wr,\\
&&v=q_{xx}-q+r_x,\quad w=q_{xx}-q-r_x,\nonumber\end{aligned}$$is also given by Geng and Wang [@geng1], this system and (\[cubicCH\]) share the same spectral problem, but auxiliary problem here is $$\begin{aligned}
\varphi_{t}=\left(
\begin{array}{cccc}
r-q_x & 0 & 2q & \frac{1}{\lambda} \\
0 & -r-q_x & \frac{1}{\lambda} & 2q \\
\frac{1}{2}(v+w+q)-q_{xx} & 2\lambda vq+\frac{1}{4\lambda} & r+q_x & 0 \\
2\lambda wq+\frac{1}{4\lambda} & \frac{1}{2}(v+w+q)-q_{xx} & 0 & q_x-r \\
\end{array}
\right)\varphi.\nonumber\end{aligned}$$ As points out in [@geng1], the two-component CH system (\[coupleCH\]) possesses a closed 1-form $$\omega=(vw)^{\frac{1}{4}}dx+2q(vw)^{\frac{1}{4}}dt,$$ which defines a reciprocal transformation as $$\label{trs2}
dy=udx+2qudt, \quad d\tau=dt.$$ Proceeding as before we obtain, the CH system (\[coupleCH\]) is reciprocally transformed to $$\begin{aligned}
\label{coutr1}
i_{\tau} &=& s-\frac{u^2}{s}, \quad \quad \quad \quad \quad \quad i=\frac{s_y}{s}-\frac{u_y}{u},\\\label{coutr2}
j_{\tau} &=& \frac{1}{2}s_y+\frac{u_y}{u}s+\frac{u^2s_y}{2s^2}, \quad j=\frac{1}{4u^2}+\frac{u_ys_y}{2us}-\frac{3s_y^2}{4s^2}+\frac{s_{yy}}{2s},\end{aligned}$$and the transformed Lax pair is $$\begin{aligned}
&&\Phi_y=\left(
\begin{array}{cccc}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
j & \lambda & i & 0 \\
\lambda & j-i_y & 0 & -i \\
\end{array}
\right)\Phi, \quad \Phi_\tau=\left(
\begin{array}{cccc}
0 & \frac{s}{\lambda}(\frac{s_y}{2s}-\frac{u_y}{u})& 0 & \frac{s}{\lambda} \\
-\frac{u^2s_y}{2\lambda s^2} & 0 & \frac{u^2}{\lambda s} & 0 \\
s & \frac{A_1}{\lambda}& 0 & \frac{s_y}{2\lambda} \\
\frac{u^2A_1}{s^2\lambda} & \frac{u^2}{s} & (\frac{u^2}{2\lambda s})_y& 0 \\
\end{array}
\right)\Phi,\end{aligned}$$where $A_1=s(\frac{1}{4u^2}+\frac{s_y^2}{4s^2}-\frac{u_ys_y}{2su})$.
It is not difficult to find that the transformed system (\[coutr1\]-\[coutr2\]) is a constraint of the first negative flow of modified DS III hierarchy, but the linear differential polynomials for $s,\frac{u^2}{s}$ like $f,g$ in $F_1,F_2$ are difficult to analyse, therefore the concrete relation between (\[coutr1\]-\[coutr2\]) and the first negative flow of the modified DS III hierarchy is open.
References {#references .unnumbered}
==========
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|
---
abstract: 'The importance of power-law distributions is attributed to the fact that most of the naturally occurring phenomenon exhibit this distribution. While exponential distributions can be derived by minimizing KL-divergence w.r.t some moment constraints, some power law distributions can be derived by minimizing some generalizations of KL-divergence (more specifically some special cases of Csiszár $f$-divergences). Divergence minimization is very well studied in information theoretical approaches to statistics. In this work we study properties of minimization of Tsallis divergence, which is a special case of Csiszár $f$-divergence. In line with the work by Shore and Johnson (IEEE Trans. IT, 1981), we examine the properties exhibited by these minimization methods including the Pythagorean property.'
author:
- |
Jithin Vachery and Ambedkar Dukkipati\
Dept of CSA, IISc\
{jithinvachery, ambedkar} @csa.iisc.ernet.in
bibliography:
- '1.bib'
title: 'On Shore and Johnson properties for a Special Case of Csiszár $f$-divergences'
---
INTRODUCTION
============
Shannon measure of information, also called entropy, is central to information theory which has wide range of applications spanning, communication theory, statistical mechanics, probability theory, statistical inference etc. [@CoverThomas:1991:ElementsOfInformationTheory]. It quantifies uncertainty or information that is associated with a discrete random variable by taking an average of uncertainty (Hartley information) associated with each state. The first generalization of this measure of information was suggested by R$\acute{e}$nyi [@Renyi:1965:OnTheFoundationsOfInformationTheory]. He replaced the linear averaging by K-N averages (Kolmogrov-Nagumo averages) and imposed additivity constraint. Havrda and Charvat [@HavrdaCharvat:1967:QuantificationMethodOfClassificationProcess] introduced one more generalization which is now known as nonextensive entropy or Tsallis entropy [@TatsuakiTakeshi:2001:WhenNonextensiveEntropyBecomesExtensive; @Suyari:2004:GeneralizationOfShannonKhinchinAxioms; @Furuichi:2005:OnUniquenessTheormForTsallisEntropyAndTsallisRelativeEntropy], which has been studied in statistical mechanics.\
Another important notion is that of finding the distance or divergence between two probability distributions. The information measure capturing this is KL-divergence, which is the directed distance between two probability distributions. KL-divergence is a special case of Tsallis divergence, which in turn is a special case of Csiszár $f$-divergence [@CsiszarShields:2004:InformationTheoryAndStatistics]. KL-divergence plays a central role in Kullback’s minimum divergence principle, Which is a means of estimating the probability distribution of a system. It suggests the minimization of KL-divergence using a given prior distribution, subject to moment constraints as the estimation technique. Kullback’s minimum divergence principle reduces to Jaynes maximum entropy principle when we use uniform distribution as the prior. Kullback’s minimum divergence principle can be extended to generalized divergences. When applied to classical KL-divergence, this yields a distribution from the exponential family. Whereas applying Kullback’s principle to Tsallis divergence gives a power-law distribution.\
Exponential distributions are very important class of distributions and many problems have been successfully modeled using this [@clark:2004:primer]. Though exponential distributions are used in many modeling problems [@bishop2006pattern] due to theoretical tractability, many naturally occurring phenomena exhibit power-law distributions. It is of great practical and theoretical interest to study both these family of distributions.\
In this work we have been able to establish many properties for Tsallis divergence. We have established the property of *transformation invariance* and *subset independence*. In addition we have found some properties for Tsallis divergence minimization in classical constraints viz. *uniqueness*, *reflexiveness*, *idempotence*, *invariance*, *weak subset independence* and *subset aggregation*. In this work we have also attempted to derive a *Pythagorean property*. In addition we have proposed a $q \leftrightarrow
2-q$ *additive transformation* for Tsallis divergence.\
The paper is organized as follows. In Section \[sec2\] we introduce the preliminaries and basics required for understanding the results. Sections \[sec3\] through \[snjmini\] are dedicated to the results and observations made. In these sections we perform Tsallis divergence minimization for classical constraints and we follow it up with the analysis of the properties exhibited by the same. In particular we are study about the Shore and Johnson properties. In the subsequent section we discuss about the a transformation relation which we established.
Preliminaries and Background {#sec2}
============================
Exponential family and KL Divergence
------------------------------------
In many of the problems we might have a prior estimate of the probability distribution and given such a prior we are interested in finding the probability distribution that is closest to this prior, which also satisfies the set of linear constraints. To define the notion of closeness we need a distance measure between two distributions. One such distance measure is KL divergence [@Kullback:1959:InformationTheoryAndStatistics] defined as $$I(p||r) = {\sum_{x \in \mathcal{X}}}p(x) \ln \left(\frac{p(x)}{r(x)}\right){\enspace,}$$ where $r$ is the prior. The minimization of KL-divergence results in a posterior which is from the exponential family.
Power-Law distribution and Generalized Divergence
-------------------------------------------------
$f$-divergence is a generalized measure of divergence, that was introduced by Csis$\acute{z}$ar [@CsiszarShields:2004:InformationTheoryAndStatistics] and independently by Ali & Silvey [@Ali_Silvey:1966:AGeneralClassOfCoefficientsOfDivergence]. Let $f (t)$ be a real valued convex function defined for $t > 0$, with $f (1) =
0$. The $f$-divergence of a distribution $p$ from $r$ is defined by $$D_f(p||r) = {\sum_{x \in \mathcal{X}}}r(x) f\left(\frac{p(x)}{r(x)}\right){\enspace.}$$ Here we take $0f\left(\frac{0}{0}\right) = 0, f(0) = \lim_{t \to 0} f(t).$ $f$-divergence has many important properties like non-negativity, monotonicity and convexity. This has been used in many applications like speech recognition [@qiao2010study], analysis of contingency tables [@CsiszarShields:2004:InformationTheoryAndStatistics], etc. By specializing $f$ to various functions we get different divergences like KL-divergence, $\chi^2$-divergence, Hellinger distance, variational distance, Tsallis-divergence, etc. On setting $f(t)= t\ln_qt$ we get Tsallis divergence [@TatsuakiTakeshi:2001:WhenNonextensiveEntropyBecomesExtensive], defined as $$I_q(p||r) = - {\sum_{x \in \mathcal{X}}}p(x) \ln_q\frac{r(x)}{p(x)}{\enspace,}$$ where $\ln_q$ is $q$-logarithm function [@Borges:2004:ApossibleDeformedAlgebra], defined as, $\ln_qx =
\frac{x^{(1-q)}-1}{1-q} \quad (x>0, q \in \mathbb{R})$. Tsallis divergence recovers KL-divergence for $q \rightarrow 1$ i.e., $
\lim_{q \to 1} I_q(p||r) = I(p||r)$. For values of $q>0$ we have $I_q(p||r) \ge 0 $ and Tsallis divergence becomes a convex function of both the parameters. Tsallis divergence also exhibits pseudo additivity property, i.e., $ I_q(X1\times X2||Y1\times Y2) = I_q(X1||X2)
\oplus_q
I_q(Y1||Y2)$, where $X1$ and $X2$ are independent, so are $Y1$ and $Y2$. Here $\oplus_q$ is addition in $q$-deformed algebra [@Borges:2004:ApossibleDeformedAlgebra] defined as, $x\oplus_qy = x+y+(1-q)xy$. In the minimization of Tsallis divergence the choice of constraints play an important role [@TsallisMendesPlastino:1998:TheRoleOfConstraints].\
Tsallis Divergence minimization with respect to $q$-expectation constraint has been studied by [@BorlandPlastinoTsallis:1998:InformationGainWithinNonextensiveThermostatistics]. In this case Pythagoras theorem is established by [@DukkipatiMurtyBhatnagar:2006:NonextensiveTriangleEquality; @Dukkipati:2007:NonextensivePythagorasTheorem; @DukkipatiMurtyBhatnagar:2005:PropertiesOfKullback-LeiblerCrossEntropyMinimization] and proved in differential geometric setup by Ohara [@ohara2007geometry].\
Tsallis divergence minimization with normalized constraints gives probability distribution which is self referential in nature, i.e., $p(x)$ depends of $p(x)$. Here too we have nonextensive Pythagoras property [@DukkipatiMurtyBhatnagar:2006:NonextensiveTriangleEquality; @Dukkipati:2007:NonextensivePythagorasTheorem] exhibited by Tsallis-divergence.\
In this paper we are going to study this minimization with respect to classical expectations, as it has the important property of convexity, ensuring a unique solution.
Basic Shore and Johnson Properties {#sec3}
==================================
Shore and Johnson [@Shore:1981:PropertiesOfCrossEntropyMinimization] in their work in 1981 had discussed many of the important properties of KL-divergence minimization. We have found that many of those properties hold in the case of Tsallis divergence. In this section we shall discuss about the properties that pertain to Tsallis divergence, i.e., regardless of minimization.\
In this section and section \[snjmini\] we shall be using the following notation.\
Let $p$ be a pmf. on random variable $X$ taking values from $\mathcal{X}$. We would like to impose the following linear equality and inequality constraints on it. $$\begin{aligned}
{\sum_{x \in \mathcal{X}}}p(x) &= 1 \label{const1}{\enspace,}\\
{\sum_{x \in \mathcal{X}}}p(x)u_m &= {\langleu_m\rangle}\quad m= 1\dots M \label{const2}{\enspace,}\\
{\sum_{x \in \mathcal{X}}}p(x)w_n &\ge {\langlew _n\rangle}\quad n= 1\dots N {\enspace.}\label{const3}
\end{aligned}$$ Equations , and constitute the constraint set. This can also be considered as the information available about the probability distribution. We shall denote a constraint set by ${\mathcal{C}}$, and a subscript to distinguish between different constraint sets.\
Hence the task of divergence minimization can be viewed as, given a prior probability distribution $q(x)$ and constraint set ${\mathcal{C}}$ finding the probability distribution $p_{min}$ such that $p_{min} = {\arg\min\limits_{p\in{\mathcal{C}}}}
I_q(p||r)$. It can be easily verified that the constraint set ${\mathcal{C}}$ constitutes a convex set. We would like to inform that some of these notation have been borrowed from [@Shore:1981:PropertiesOfCrossEntropyMinimization].\
Invariance of KL-divergence to coordinate transformations enables us to generalize KL-divergence to continious random variables. We have observed that the invariance property holds true in the case of Tsallis divergence too.
Let $\Gamma$ be a coordinate transformation from $x\in {\mathcal{X}}$ to $y\in
{\mathcal{X}}'$ with $(\Gamma p)(y) = J^{-1}p(x)$, where $J$ is the Jacobian $J =
\partial(y)/\partial(x)$. Let $\Gamma\mathcal{X}$ be the set of densities $\Gamma p$ corresponding to densities $p \in \mathcal{X}$. Let $(\Gamma{\mathcal{C}})
\subseteq (\Gamma{\mathcal{X}})$ correspond to ${\mathcal{C}}\subseteq {\mathcal{X}}$. Then, given a prior distribution $r$ $$\begin{aligned}
{\arg\min\limits_{p \in {\Gamma}{\mathcal{C}}}} I_q(p||{\Gamma}r) &= {\arg\min\limits_{s
\in{\mathcal{C}}}}I_q(s||r)\label{res1_1}{\enspace,}\\
\text{and }
I_q({\Gamma}p_{min}||{\Gamma}r) &= I_q(p_{min}||r)\label{res1_2}{\enspace,}\end{aligned}$$ hold. where ${\Gamma}p_{min} = {\arg\min\limits_{p \in {\Gamma}{\mathcal{C}}}} I_q(p||{\Gamma}r)$ and $p_{min} =
{\arg\min\limits_{s \in{\mathcal{C}}}}I_q(s||r)$.
We have $(\Gamma p)(y) = J^{-1}p(x)$, where $J$ is the Jacobian $J =
\partial(y)/\partial(x)$. $$\begin{aligned}
I_q({\Gamma}p|| {\Gamma}r) &= -\int_{{\Gamma}{\mathcal{X}}} {\Gamma}p(y) \;\ln_q \left(\frac{{\Gamma}r(y)}{{\Gamma}p(y)}\right) \mathrm{d}y\\
&= -{\int_{\mathcal{X}}J^{-1} p(x) \; \ln_q \left(\frac{J^{-1} r(x)}{J^{-1} p(x}\right)
\;J\mathrm{d}x}\\
&= -{\int_{\mathcal{X}}p(x)\;\ln_q\frac{r(x)}{p(x)}\mathrm{d}x}\\
&= I_q(p||r)\end{aligned}$$ This proves . From it also follows that the minimum in ${\Gamma}{\mathcal{C}}$ corresponds to the minimum in ${\mathcal{C}}$, which proves .
\[res2\] Let $S_1,S_2,\dots,S_n$ be a partition of ${\mathcal{X}}$. Let the new information ${\mathcal{C}}$ comprise about each of the conditional densities $p(x/x\in s_i),\; i= 1 \dots
n$. Thus, ${\mathcal{C}}= {\mathcal{C}}_1 \wedge {\mathcal{C}}_2 \wedge \dots \wedge {\mathcal{C}}_n$, where ${\mathcal{C}}_i$ is the constraint set on the conditional densities of $S_i$. Let $\mathcal{M}$ be the new information giving the probability of being in each of the $n$ subsets, which is the constraint $$\sum_{x\in S_i} p(x) = m_i,\quad i=1\dots n{\enspace,}$$ where $m_i$ are known values. Then given the prior distribution $r$, $$p_{\mathcal{CM}}^{min}(x/x\in S_i) = {\arg\min\limits_{p \in {\mathcal{C}}_i}}
I_q(p||r_i), \quad q\in(0,1)\label{res2_1}{\enspace,}$$ and $$\begin{aligned}
I_q(p_{\mathcal{CM}}^{min}||r) &= {\sum_{i=1}^n}m_i\; I_q(p_i||r_i) - {\sum_{i=1}^n}m_i
\;\ln_q \frac{s_i}{m_i} \nonumber\\
&+ (1-q) {\sum_{i=1}^n}\left( m_i\; \ln_q \frac{s_i}{m_i}\;
I_q(p_i||r_i)\right)\label{res2_2}\end{aligned}$$ hold, where $$\begin{aligned}
p_{\mathcal{CM}}^{min} &= {\arg\min\limits_{p\; \in {\mathcal{C}}\wedge \mathcal{M}}} I_q(p||r){\enspace,}\\
p_i(x) &= p_{\mathcal{CM}}^{min}(x/x \in S_i){\enspace,}\\
r_i(x) &= r(x/x \in S_i){\enspace,}\end{aligned}$$ and $s_i$ are the prior probability of being in each subset, given by $s_i = \sum_{x \in S_i} r(x)$.
$$I_q(p_{\mathcal{CM}}^{min}||r) = - {\sum_{i=1}^n}\sum_{x\in S_i} m_i p_i(x)
\;\ln_q \frac{s_ir_i(x)}{m_i
r_i(x)}{\enspace.}$$
Using the relation $\ln_q(xy) = \ln_qx + \ln_qy+(1-q)\ln_qx\;\ln_qy$, we get $$\begin{aligned}
I_q(p_{\mathcal{CM}}^{min}||r) &= - {\sum_{i=1}^n}\sum_{x\in S_i} m_i p_i(x)
\left(
{\ln_q \frac{s_i}{m_i}} + {\ln_q \frac{r_i(x)}{p_i(x)}} \right.\\
& \qquad \left.+ (1-q)\;{\ln_q \frac{s_i}{m_i}}\;{\ln_q \frac{r_i(x)}{p_i(x)}}
\right)\\
&= {\sum_{i=1}^n}m_i I_q(p_i||r_i) - {\sum_{i=1}^n}m_i\;{\ln_q \frac{s_i}{m_i}}\\
&\qquad + (1-q){\sum_{i=1}^n}\left( m_i\;{\ln_q \frac{s_i}{m_i}} I_q(p_i||r_i)\right){\enspace,}\end{aligned}$$ this proves . To prove it may be noted that each of the terms $m_i\;{\ln_q \frac{s_i}{m_i}}$ is a constant. Hence minimizing rhs of is independent of the values taken by it. i.e for $q\in (0,1)$ minimizing $I_q(p_{\mathcal{CM}}^{min}||r)$ is equivalent to minimizing each of the terms, $I_q(p_i||r_i)$.
Let us further analyze equation and try to interpret it. What this means is that, given a system which naturally partitions into subsets, we can find the posterior densities in two different ways
1. We can find the posterior $p_{\mathcal{CM}}^{min}$ and condition it on the different subsets $S_i$ or
2. We can condition the prior $r$ on the different subsets $S_i$ and use that as a prior to minimize in the constraint set ${\mathcal{C}}_i$
By both these approaches should give the same result.
Tsallis Divergence Minimization - Classical {#sec4}
===========================================
The task of minimization can be defined as follows: Minimize $I_q(p||r)$ subject to the constraints $$\begin{aligned}
{\sum_{x \in \mathcal{X}}}{p(x)} &= 1\label{pxsum}{\enspace,}\\
p(x) &\ge 0\nonumber{\enspace,}\\
{\sum_{x \in \mathcal{X}}}{u_m(x) p(x)} &= {\langleu_m\rangle}, \quad m = 1,\dots,M\nonumber{\enspace.}\end{aligned}$$ By choosing the Lagrangian for the minimization problem as $$\begin{aligned}
\lefteqn{\mathcal{L} =}\\
&{\sum_{x \in \mathcal{X}}}p(x)\frac{\left[\frac{p(x)}{r(x)}\right]^{q-1}-1}{q-1}
- {\left(\frac{q\lambda-1}{q-1}\right)} \bigl( {\sum_{x \in \mathcal{X}}}{p(x)} - 1\bigr) \\
&- {\sum_{m=1}^M}q\lambda\beta_m({\sum_{x \in \mathcal{X}}}{u_m(x) p(x)} - {\langleu_m\rangle}){\enspace.}\end{aligned}$$ The distribution that we get after minimization is $$\label{px1}
p(x) = r(x) \left[\lambda\Bigl(1+(q-1){\sum_{m=1}^M}\beta_m
u_m(x)\Bigr)\right]^{\frac{1}{q-1}}{\enspace.}$$ Substituting in we get $$\begin{aligned}
\lambda^{\frac{1}{q-1}} = \frac{1}{
{\sum_{x \in \mathcal{X}}}\left[
r(x)
\Bigl(
1-(1-q){\sum_{m=1}^M}\beta_m u_m(x)
\Bigr) ^ {\frac{1}{q-1}}
\right]
} {\enspace.}\end{aligned}$$ Substituting in we get $$\begin{aligned}
p(x) = \frac{r(x) \Bigl(1-(1-q)
{\sum_{m=1}^M}\beta_mu_m(x)\Bigr)^{\frac{1}{q-1}}}{{\widehat{Z}}}\label{px1_1}{\enspace,}\end{aligned}$$ where $$\begin{aligned}
{\widehat{Z}}= {\sum_{x \in \mathcal{X}}}\left[
r(x)
\Bigl(
1-(1-q){\sum_{m=1}^M}\beta_m u_m(x)
\Bigr) ^ {\frac{1}{q-1}}
\right]\nonumber{\enspace.}\end{aligned}$$ equation can be rewritten as $$\begin{aligned}
p(x) = \frac{r(x)}{{\widehat{Z}}\;\;\exp_q\Big(-{\sum_{m=1}^M}\beta_mu_m(x)\Big)}\label{px1_2}{\enspace.}\end{aligned}$$ Where Where $\exp_q$ is exponentiation in $q$-deformed algebra [@Borges:2004:ApossibleDeformedAlgebra], and is defined as, $$\exp_q(x) = \left\{
\begin{array}{l l}
[1+(1-q)x]^\frac{1}{1-q} & \quad \text{if } 1+(1-q)x \ge 0\\
0 & \quad \text{otherwise}{\enspace.}\end{array} \right.$$ using the relation $\frac{1}{\exp_q(x)} =
\exp_q\left(\frac{-x}{1+(1-q)x}\right)$, we get $$p(x) = \frac{r(x)}{{\widehat{Z}}}
\exp_q\left(\frac{{\sum_{m=1}^M}\beta_mu_m(x)}{1-(1-q){\sum_{m=1}^M}\beta_mu_m(x)}\right){\enspace.}$$ Note that we need an extra condition known as *Tsallis cut-off condition* to prevent negative values for $p(x)$. We have assumed this condition to be implicit.
Shore and Johnson Properties involving maximum entropy {#snjmini}
======================================================
In this section we shall discuss properties which depend on the formalism employed.
For $q>0$ given a prior, the posterior probability distribution is unique.
For $q>0$ Tsallis divergence is a convex function, for both its parameter. Since the constraint set ${\mathcal{C}}$ is a convex set, the minimization is always unique.
\[res4\] For $q > 0$, given a prior $r$ and constraint set ${\mathcal{C}}$, the posterior obtained by minimizing the Tsallis divergence is same as $r$ if and only if $r \in {\mathcal{C}}$
This property follows directly from the following facts $I_q(p||r) = 0
\quad\text{iff}\quad p = r$ and $I_q(p||r) > 0 \quad\text{for}\quad q > 0$.
Given a prior $r$ and constraint set ${\mathcal{C}}$, let $p$ be the posterior obtained, then ${\arg\min\limits_{u \in {\mathcal{C}}}} I_q(u||p) = p$, i.e., taking the same information into account twice has the same effect as taking it into account once.
This is a simple corollary of proposition \[res4\], since $p \in {\mathcal{C}}$ the posterior obtained by taking $p$ as prior and ${\mathcal{C}}$ as constraint, will also be $p$.
\[res7\] Given a prior $r$ consider the constraint sets ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$, let $p = {\arg\min\limits_{u \in {\mathcal{C}}_1}} I_q(u||r)$, then following relations hold $$\begin{aligned}
p &= {\arg\min\limits_{u\in {\mathcal{C}}_1 \wedge {\mathcal{C}}_2}}I_q(u||r)\label{res7_1}\\
&= {\arg\min\limits_{u \in {\mathcal{C}}_1 \wedge {\mathcal{C}}_2}} I_q(u||p)\label{res7_2}\\
&= {\arg\min\limits_{u \in {\mathcal{C}}_2}} I_q(u||p)\label{res7_3}{\enspace.}\end{aligned}$$
$p \in {\mathcal{C}}_1$ and $p \in {\mathcal{C}}_2$ hence $p \in {\mathcal{C}}_1 \wedge {\mathcal{C}}_2$ so from proposition \[res4\] both, and follow. We know that $p = {\arg\min\limits_{u \in {\mathcal{C}}_1}} I_q(u||r)$ and $p \in {\mathcal{C}}_1 \wedge
{\mathcal{C}}_2$ from the above two, follows.
The result shows that if the posterior obtained from ${\mathcal{C}}_1$ is an element of ${\mathcal{C}}_2$ then applying ${\mathcal{C}}_2$ on the posterior in different ways does not result in any change.
Let $S_1, S_2, \dots, S_n$ be a partition of ${\mathcal{X}}$. Let the new information ${\mathcal{C}}$ comprise about each of the conditional densities $p(x/x\in s_i),\; i= 1 \dots
n$. Thus, ${\mathcal{C}}= {\mathcal{C}}_1 \wedge {\mathcal{C}}_2 \wedge \dots \wedge {\mathcal{C}}_n$, where ${\mathcal{C}}_i$ is the constraint set on the conditional densities of $S_i$.Then given the prior distribution $r$ $$\begin{aligned}
p_{{\mathcal{C}}}^{min}(x/x\in S_i) = {\arg\min\limits_{p \in {\mathcal{C}}_i}}
I_q(p||r_i), \quad q\in(0,1)\label{res3_1}{\enspace,}\end{aligned}$$ and $$\begin{aligned}
I_q(p_{{\mathcal{C}}}^{min}||r) &= {\sum_{i=1}^n}u_i\; I_q(p_i||r_i) - {\sum_{i=1}^n}u_i
\;\ln_q \frac{s_i}{u_i} \nonumber\\
&\quad+ (1-q) {\sum_{i=1}^n}\left( u_i\; \ln_q \frac{s_i}{u_i}\;
I_q(p_i||r_i)\right)\label{res3_2}{\enspace,}\end{aligned}$$ hold where $$\begin{aligned}
p_{{\mathcal{C}}}^{min} &= {\arg\min\limits_{p\; \in {\mathcal{C}}}} I_q(p||r){\enspace,}\\
p_i(x) &= p_{{\mathcal{C}}}^{min}(x/x \in S_i){\enspace,}\\
r_i(x) &= r(x/x \in S_i){\enspace.}\end{aligned}$$ $s_i$ are the prior probability of being in each subset, given by $s_i = \sum_{x \in S_i} r(x)$, and $u_i$ are the posterior probability of being in each subset, given by $u_i = \sum_{x \in S_i} p_{\mathcal{C}}^{min}(x)$.
Let $\mathcal{R}$ be the information defined by the constraint $\sum_{x \in
S_i} p(x) = u_i$, then it follows from proposition \[res7\] that $${\arg\min\limits_{p\; \in {\mathcal{C}}}} I_q(p||r) = {\arg\min\limits_{p\; \in {\mathcal{C}}\wedge\mathcal{R}}}
I_q(p||r){\enspace.}$$ Now we can apply proposition \[res2\] to get and .
This result is same as proposition \[res2\] and has the same interpretation. This difference here lies in the fact that we do not have a prior information $\mathcal{M}$ regarding the total probability in each subset.
Let $S_1,S_2,\dots,S_n$ be a partition of ${\mathcal{X}}$. Let ${\Gamma}$ be a transformation which converts a given distribution $p$ to discrete distribution over $S_i$, the transformation is defined by $$p'(x_i) = {\Gamma}p = \int_{S_i} p(x) \mathrm{d}x{\enspace,}$$ where $x_i$ is a discrete state corresponding to $x \in S_i$. Let ${\mathcal{C}}$’ be the new information about the distribution ${\Gamma}p$. Then for a given prior $r$, then $$\begin{aligned}
r (x/x\in S_i) &= p_{min} (x/x\in S_i)\label{res8_1}{\enspace,}\\
{\Gamma}p_{min} &= {\Gamma}(p_{min})\label{res8_2}{\enspace,}\\
\text {and }
I_q({\Gamma}p_{min}||{\Gamma}r) &= I_q (p_{min}||r)\label{res8_3}{\enspace,}\end{aligned}$$ where $p_{min} = {\arg\min\limits_{p \in {\Gamma}^{-1}({\mathcal{C}}')}} I_q (p||r)$.
The constraint set ${\mathcal{C}}'$ is defined by a set of expectations $${\sum_{i=1}^n}p'(x_i)u_m(x_i) = {\langleu_m\rangle} \quad m=1\dots M{\enspace.}$$ In terms of $p = {\Gamma}p'$ the constraint set can be represented as $$\int_{{\mathcal{X}}} p(x)w_m(x) = {\langleu_m\rangle} \quad m=1\dots M{\enspace,}$$ where $w_m$ is defined as $$w_m(x) = u_m(x_i), \quad\text{for}\quad x\in S_i,\quad i=1\dots n{\enspace,}$$ i.e., $w_m$ is constant in each of the subsets $S_i$.\
From we get $$\label{xyz1}
p_{min}(x)=\frac{r(x)}{{\widehat{Z}}\;\;\exp_q\Big(-{\sum_{m=1}^M}\beta_m w_m(x)\Big)}{\enspace.}$$ Since $w_m$ is a constant within each subset $S_i$ and $iz$ is a constant in itself. So equation reduces to: $$\begin{aligned}
p_{min}(x) &= K_i\;r(x){\enspace,}\end{aligned}$$ where $K_i$ is a constant for each subset. Now we have $$\begin{aligned}
r (x/x\in S_i) &= r(x)\left/ \int_{y \in S_I} r(y)\right.\\
&= p_{min} (x/x\in S_i){\enspace.}\end{aligned}$$ This proves .\
Now consider the relation $$\begin{aligned}
I_q(p_{min}||r) &= {\sum_{i=1}^n}u_i\; I_q(p_i||r_i) - {\sum_{i=1}^n}u_i
\;\ln_q \frac{s_i}{u_i} \nonumber\\
&\quad+ (1-q) {\sum_{i=1}^n}\left( u_i\; \ln_q \frac{s_i}{u_i}\;
I_q(p_i||r_i)\right) {\enspace,}\label{xyz2}
\end{aligned}$$ which follows from . where $$\begin{aligned}
p_i(x) &= p_{min}(x/x \in S_i){\enspace,}\\
r_i(x) &= r(x/x \in S_i){\enspace,}\\
s_i &= \sum_{x \in S_i} r(x){\enspace,}\\
\text{and }
u_i &= \sum_{x \in S_i} p_{min}(x){\enspace.}\end{aligned}$$ From we have that $p_i(x)=r_i(x)$ and hence $I_q(p_i||r_i) = 0$. Now equation reduces to $$\begin{aligned}
I_q(p_{min}||r) &= - {\sum_{i=1}^n}u_i\;\ln_q \frac{s_i}{u_i}\nonumber\\
&= I_q({\Gamma}p_{min}||{\Gamma}r){\enspace.}\end{aligned}$$ This proves and .
Some Observations On Duality and Pythagoras
===========================================
Pythagorean Property
--------------------
Because of its extensive use in many problems, Pythagorean property is very important. It has been shown to exist for both second and third formalisms, involving $q$-expectation and normalized $q$-expectation respectively. In this section we have attempted to find the equivalent result for the classical expectation. The result we got is not promising but we present it here for future reference, and to introduce an alternative way to manipulate the Lagrange multipliers. Lets formally state our problem at hand:
#### Problem statement :
Let $r$ be the prior distribution and let $p$ be the posterior got by minimizing the Tsallis divergence subject to the constraint set ${\mathcal{C}}$ $$\begin{aligned}
{\sum_{x \in \mathcal{X}}}p(x) u_m(x) = {\langleu_m\rangle} \quad m = 1\dots M{\enspace.}\\
\intertext{Let $l$ be another distribution satisfying the constraint}
{\sum_{x \in \mathcal{X}}}l(x) u_m(x) = {\langlew_m\rangle} \quad m = 1\dots M{\enspace.}\end{aligned}$$ We are interested in finding the relation between ${\langleu_m\rangle}$ and ${\langlew_m\rangle}$ so as to minimize the divergence $I_q(l||p)$.
#### Solution
To find a solution to this problem we shall minimize the Tsallis divergence in a different manner. We start the minimization with the following Lagrangian $$\begin{aligned}
\lefteqn{\mathcal{L} =}\\
&{\sum_{x \in \mathcal{X}}}p(x)\frac{\left[\frac{p(x)}{r(x)}\right]^{q-1}-1}{q-1}
- (1-q\lambda) \bigl( {\sum_{x \in \mathcal{X}}}{p(x)} - 1\bigr) \\
&+ {\sum_{m=1}^M}q\beta_m({\sum_{x \in \mathcal{X}}}{u_m(x) p(x)} - {\langleu_m\rangle}){\enspace,}\end{aligned}$$ differentiating $\mathcal{L}$ with respect to $p(x)$ and equating to $0$, we get $$\begin{aligned}
\ln_q \left(\frac{r(x)}{p(x)}\right) & = \lambda - {\sum_{m=1}^M}\beta_mu_m(x)\label{xyz4}{\enspace,}\\
p_{min} &=p(x) = \frac{r(x)}{\exp_q(\lambda - {\sum_{m=1}^M}\beta_mu_m(x))}{\enspace.}\end{aligned}$$ Multiplying equation by $p(x)$ and summing it over ${\mathcal{X}}$ we get $$\begin{aligned}
{\sum_{x \in \mathcal{X}}}p(x) \;\ln_q \left(\frac{r(x)}{p(x)}\right) & ={\sum_{x \in \mathcal{X}}}p(x) \lambda\\
&\qquad-{\sum_{x \in \mathcal{X}}}{\sum_{m=1}^M}p(x) \;\beta_mu_m(x){\enspace,}\\
-I_q^{min}(p||r) &= \lambda - {\sum_{m=1}^M}\beta_m{\langleu_m\rangle}{\enspace.}\end{aligned}$$ Differentiating $I_q^{min}(p||r)$ with respect to ${\langleu_m\rangle}$ we get $${\frac{\partial\; I_q^{min}}{\partial {\langleu_m\rangle}}} = \beta_m\label{leg2_1}{\enspace.}$$ Substituting $$\beta_m = \beta_m' (1+(1-q)\lambda){\enspace,}$$ equation reduces to $$\begin{aligned}
\ln_q \left(\frac{r(x)}{p(x)}\right) &=\lambda \oplus_q
-{\sum_{m=1}^M}\beta_mu_m(x)\\
p(x) & = \frac{r(x)}{{\widehat{Z}}\;\exp_q(-{\sum_{m=1}^M}\beta_mu_m(x))}{\enspace,}\end{aligned}$$ where ${\widehat{Z}}= \exp_q(\lambda)$. Hence equation can be rewritten as $$\begin{aligned}
\ln_q \left(\frac{r(x)}{p(x)}\right) & = \ln_q{\widehat{Z}}- {\sum_{m=1}^M}\beta_mu_m(x){\enspace.}\end{aligned}$$ Multiplying this equation $p(x)$ and summing it over ${\mathcal{X}}$ we get $$\begin{aligned}
-I_q^{min} = \ln_q{\widehat{Z}}- {\sum_{m=1}^M}\beta_m {\langleu_m\rangle}{\enspace.}\end{aligned}$$ Differentiating $I_q^{min}$ with respect to $\beta_m$ and equating to $0$ we get $${\frac{\partial\; \ln_q{\widehat{Z}}}{\partial \beta_m}} = {\langleu_m\rangle}\label{leg2_2}{\enspace.}$$ Equations and are the Legendre transform relations. Given the relations and the divergence minimization let us look at the Pythagorean property.\
We want to minimize the divergence $I_q (l||p)$. For this we will proceed as follows $$\begin{aligned}
I_q(l||r) &- I_q(l||p) = -{\sum_{x \in \mathcal{X}}}l(x)\left[ \ln_q \frac{r(x)}{l(x)} - \ln_q
\frac{p(x)}{l(x)}\right]{\enspace,}\intertext{using the relation $\ln_q \left(\frac{x}{y}\right) = y^{q-1}(\ln_qx -
\ln_qy)$, we get}
I_q(l||r) &- I_q(l||p) \\
&= -{\sum_{x \in \mathcal{X}}}l(x)\left[\ln_q \frac{r(x)}{p(x)} \left[1+
(1-q) \ln_q\frac{p(x)}{l(x)}\right]\right]{\enspace,}\end{aligned}$$ using equation $$\begin{aligned}
& I_q(l||r) - I_q(l||p) \nonumber \\
&= -{\sum_{x \in \mathcal{X}}}l(x)\left[\left(\lambda - {\sum_{m=1}^M}\beta_mu_m(x)\right)\right.\nonumber
\\
&\qquad \qquad \qquad \qquad \left.\left(1+ (1-q)
\ln_q\frac{p(x)}{l(x)}\right)\right]\nonumber\\
&= \lambda - {\sum_{m=1}^M}\beta_m {\langlew_m\rangle} - (1-q) \lambda \; I_q(l||p)\nonumber\\
&\qquad-(1-q) {\sum_{x \in \mathcal{X}}}\left(l(x) \;\ln_q\frac{p(x)}{l(x)} \; {\sum_{m=1}^M}\beta_m
u_m(x)
\right) \label{xyz6}{\enspace.}\end{aligned}$$ The minimum of $I_q(l||p)$ is achieved for $$\begin{aligned}
{\frac{\partial\; I_q(l||p)}{\partial \beta_m}} = 0{\enspace.}\end{aligned}$$ Differentiating we get $$\begin{aligned}
&{\frac{\partial\; \lambda}{\partial \beta_m}} - {\langlew_m\rangle} - (1-q) I_q(l||p)
{\frac{\partial\; \lambda}{\partial \beta_m}} \\
&\;-(1-q) {\sum_{x \in \mathcal{X}}}l(x) {\frac{\partial\; }{\partial \beta_m}} \left[
\ln_q\frac{p(x)}{l(x)} \; {\sum_{m=1}^M}\beta_m u_m(x)\right] = 0{\enspace.}\end{aligned}$$ Using equation we get $$\begin{aligned}
{\langleu_m\rangle} &- {\langlew_m\rangle} - {\langleu_m\rangle}\;I_q(l||p) \nonumber\\
&=(1-q) {\sum_{x \in \mathcal{X}}}l(x) {\frac{\partial\; }{\partial \beta_m}} \left[
\ln_q\frac{p(x)}{l(x)} \; {\sum_{m=1}^M}\beta_m u_m(x)\right]\nonumber\\
&=(1-q) {\sum_{x \in \mathcal{X}}}l(x) \left[ \beta_m \ln_q\frac{p(x)}{l(x)} \right.\nonumber\\
&\qquad \qquad \qquad \left. + {\sum_{m=1}^M}\beta_m u_m(x) {\frac{\partial\; }{\partial \beta_m}} \left(
\ln_q\frac{p(x)}{l(x)}\right) \right]{\enspace.}\label{py1}\end{aligned}$$ Evaluating it further we by using the relations $\ln_q\left(\frac{x}{y}\right) =
\frac{\ln_qx - \ln_q y}{1 + (1-q)\ln_qy}$ and $
p(x) = \frac{r(x)}{\lambda - {\sum_{m=1}^M}\beta_mu_m(x)}$, We get $$\begin{aligned}
{\langlew_m\rangle} &=\nonumber\\
&{\langleu_m\rangle}(1 - I_q(l||p)) + (1-q)\beta_m I_q(l||p)\nonumber\\
&- {\sum_{x \in \mathcal{X}}}\frac{l^q(x)\;(u_m(x)- {\langleu_m\rangle}) \;\Psi}{\left[ 1 + (1-q) \ln_q
\frac{r(x)}{p(x)}\right]^2}{\enspace,}\end{aligned}$$ where $\Psi = (1+(1-q) \ln_qr(x)) {\sum_{m=1}^M}\beta_mu_m(x)$. Note that in this expression $r(x)$ can be replaced in terms of $p(x)$.\
Though this relation does not seem promising, we have mentioned it here for the sake of completion.\
Additive transformation - $\mathbf{q\leftrightarrow2-q}$
--------------------------------------------------------
In $q-$deformed algebra there exists a $q\leftrightarrow2-q$ duality. Which is the following: $$\begin{aligned}
\ln_q (1/x) &= \ln_{2-q} (x){\enspace,}\\
\exp_q(-x) & = \frac{1}{\exp_{2-q} (x)}\label{xyz3}{\enspace.}\end{aligned}$$ Using this duality Tsallis entropy has been well studied, i.e., various properties of $S_{2-q}$ has been studied. Initial observations regarding $S_{2-q}$ were made by Baldovin and Robledo [@baldovin:2004:nonextensive]. Naudts [@Naudts:2004:GeneralizedThermostatisticsAndMeanfieldTheory] has further analyzed both the dualities. More study has been carried forward by Wada and Scarfone [@WadaScarfone:2005:ConnectionsBetweenTsallisFormalismEtc]. they have found relations between the Lagrange multipliers of both the dualities. In this section we introduce a similar transformation for Tsallis divergence.\
Given a prior $r$ and the constraints set ${\mathcal{C}}$ defined by $$\begin{aligned}
{\sum_{x \in \mathcal{X}}}{p(x)} &= 1\nonumber{\enspace,}\\
p(x) &\ge 0\nonumber{\enspace,}\\
{\sum_{x \in \mathcal{X}}}{u_m(x) p(x)} &= {\langleu_m\rangle}, \quad m = 1,\dots,M\nonumber{\enspace.}\end{aligned}$$ from equation we have $$p(x) = \frac{r(x)}{{\widehat{Z}}\;\;\exp_q\Big(-{\sum_{m=1}^M}\beta_mu_m(x)\Big)}{\enspace,}$$ and using the relation it becomes $$p(x) = \frac{r(x)\;\;\exp_{2-q}\Big({\sum_{m=1}^M}\beta_mu_m(x)\Big)}{{\widehat{Z}}}{\enspace.}$$ This form for the posterior is very good and is the basis for the $q\leftrightarrow2-q$ transformation. Note that $$2 - (2-q) = q{\enspace,}$$ i.e if we minimize $I_{2-q} (p||r)$ instead of $I_q (p||r)$, we have. $$\begin{aligned}
p(x) &= {\arg\min\limits_{p \in {\mathcal{C}}}} I_{2-q} (p||r)\\
&=\frac{r(x)\;\;\exp_{q}\Big({\sum_{m=1}^M}\beta_mu_m(x)\Big)}{{\widehat{Z}}}{\enspace.}\end{aligned}$$
Conclusion
==========
In this work we explored Shore and Johnson properties for Tsallis formalism of the third kind involving normalized $q$-expectation, it was observed that none of these properties hold for the formalism. Whereas in the study of first formalism involving classical expectation, we have been able to establish substantial number of Shore and Johnson properties. We were also able to establish a crude form of Pythagorean relation. We have also been found a $q
\leftrightarrow 2-q$ additive transformation, which gives a very good form for the posterior distribution. We conclude from these observations that the first formalism is of stronger theoretical and practical significance; and these results along with the $q \leftrightarrow 2-q$ additive transformation also provides some ground work for definition of a power law family.
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abstract: 'We present new [[*XMM-Newton* ]{}]{}EPIC observations of the nuclei of the nearby radio galaxies 3C305, DA240, and 4C73.08, and investigate the origin of their nuclear X-ray emission. The nuclei of the three sources appear to have different relative contributions of accretion- and jet-related X-ray emission, as expected based on earlier work. The X-ray spectrum of the FRII narrow-line radio galaxy (NLRG) 4C73.08 is modeled with the sum of a heavily absorbed power law that we interpret to be associated with a luminous accretion disk and circumnuclear obscuring structure, and an unabsorbed power law that originates in an unresolved jet. This behavior is consistent with other narrow-line radio galaxies. The X-ray emission of the low-excitation FRII radio galaxy DA240 is best modeled as an unabsorbed power law that we associate with a parsec-scale jet, similar to other low-excitation sources that we have studied previously. However, the X-ray nucleus of the narrow-line radio galaxy 3C305 shows no evidence for the heavily absorbed X-ray emission that has been found in other NLRGs. It is possible that the nuclear optical spectrum in 3C305 is intrinsically weak-lined, with the strong emission arising from extended regions that indicate the presence of jet–environment interactions. Our observations of 3C305 suggest that this source is more closely related to other weak-lined radio galaxies. This ambiguity could extend to other sources currently classified as NLRGs. We also present [[*XMM-Newton* ]{}]{}and VLA observations of the hotspot of DA240, arguing that this is another detection of X-ray synchrotron emission from a low-luminosity hotspot.'
author:
- 'Daniel A. Evans, Martin J. Hardcastle, Julia C. Lee, Ralph P. Kraft, Diana M. Worrall, Mark Birkinshaw, Judith H.Croston'
title: 'XMM-Newton Observations of the Nuclei of the Radio Galaxies 3C305, DA240, and 4C73.08'
---
INTRODUCTION {#intro}
============
Radio galaxies consist of twin jets of particles that are ejected from a compact region in the vicinity of a supermassive black hole, feeding into large-scale ‘plumes’ or ‘lobes’. There are two principal morphological classes of radio galaxies, low-power (Fanaroff-Riley type I, hereafter FRI) sources and high-power (FRII) sources [@fr74]. FRI sources exhibit ‘edge-darkened’ large-scale radio structure, and modeling implies that initially supersonic jets in these sources decelerate to transonic speeds on $\sim$kpc scales before flaring into large plumes (e.g., @per07). FRII sources appear ‘edge-brightened’, and in these cases highly supersonic jets propagate out to large distances (often $>100$ kpc) from the core before terminating in bright hotspots and accompanying radio lobes. Observationally, the Fanaroff-Riley divide occurs at a 178-MHz radio power of $\sim$$10^{25}$ W Hz$^{-1}$ sr$^{-1}$.
It is important to understand whether the kpc-scale Fanaroff-Riley dichotomy is determined by the interaction between the jet and its external hot-gas environment (e.g., @bic95), or rather is nuclear in origin and governed by differences in the properties of the accretion flow (@rey96). The first observations of large samples of $z<0.1$ 3CRR radio-galaxy [*nuclei*]{} with [[*Chandra* ]{}]{}and [[*XMM-Newton* ]{}]{}(@don04 [@bal06; @evans06]) showed that FRI nuclei show no signs of heavily absorbed X-ray emission that would be expected from standard AGN unification models (@up95), are dominated by emission from an unresolved jet (e.g., @wb94 [@bal06; @evans06]), and have highly radiatively inefficient accretion flows. Narrow optical-line FRII sources show evidence for heavily obscured ($N_{\rm H}>10^{23}$ cm$^{-2}$) nuclear X-ray emission that is associated with a radiatively efficient accretion flow, together with an unabsorbed component of jet-related emission (@evans06 [@hec06]). FRII radio galaxies at higher redshift are consistent with such behavior (@bel06).
A significant breakthrough for understanding the physical origin of the FRI/FRII dichotomy came from [[*Chandra* ]{}]{}and [[*XMM-Newton* ]{}]{}observations of the population of [*low-excitation radio galaxies*]{} (LERGs), which have weak or no emission lines in their optical spectra (@hine79 [@jac97]). Almost all FRI radio galaxies are LERGs, but there is a significant population of FRII LERGs at $0.1 < z < 0.5$. The X-ray spectra of LERGs, irrespective of their FRI or FRII morphology, are dominated by unabsorbed emission that can be associated with a parsec-scale jet, with no obvious contribution from accretion-related emission. These sources are likely to accrete in a radiatively inefficient manner (@hec06). On the other hand, high-excitation radio galaxies (HERGs – i.e., NLRGs, BLRGs, and quasars), which display prominent narrow or broad optical emission lines, have X-ray spectra that are consistent with standard unification models: they show evidence for luminous, radiatively efficient accretion disks, together with circumnuclear tori when the source is oriented close to edge-on with respect to the observer. HERGs tend to show evidence for additional hot dust over and above that of LERGs in their mid-IR spectra (e.g., @ogle06; Birkinshaw et al., in preparation), which is again consistent with reprocessing of luminous accretion-related emission by torus-like structure. Most high radio-power (FRII) sources are high-excitation radio galaxies. The distinct X-ray nuclear properties of low- and high-excitation radio galaxies, regardless of their large-scale FRI or FRII morphology, could be interpreted as implying that the Fanaroff-Riley dichotomy remains principally influenced by jet power and environment. The excitation dichotomy, on the other hand, is interpreted to be attributed to the radiative efficiency of the accretion flow (e.g., @hec06) and possibly related to the nature of the accreting material (@hec07).
Here, we report new [[*XMM-Newton* ]{}]{}observations of the nuclei of three $z<0.1$ 3CRR radio galaxies — 3C305, DA240, and 4C73.08. The three sources have 178-MHz radio powers that lie close to the FRI/FRII dividing luminosity (Table \[sourcessummary\]), plus a range of radio morphologies and optical emission-line characteristics. They are therefore good candidates for examining possible connections between the central engine and large-scale radio characteristics. This paper is organized as follows. In Section 2, we describe the optical and radio properties of the three sources. Section 3 contains a description of the data and a summary of our analysis. In Section 4, we report the results of our spectroscopic analysis of the sources. In Section 5, we describe VLA and [[*XMM-Newton* ]{}]{}observations of the bright NE hotspot in DA240. In Section 6, we interpret the observations in the context of our previous [[*Chandra* ]{}]{}and [[*XMM-Newton* ]{}]{}observations of 3CRR radio galaxies and discuss the optical emission-line characteristics of the sources. We end with our conclusions in Section 7. All results presented in this paper use a cosmology in which $\Omega_{\rm m, 0}$ = 0.3, $\Omega_{\rm \Lambda, 0}$ = 0.7, and H$_0$ = 70 km s$^{-1}$ Mpc$^{-1}$. Errors quoted in this paper are 90 per cent confidence for one parameter of interest (i.e., $\chi^2_{\rm min}$ + 2.7), unless otherwise stated.
Overview Of The Sources
=======================
3C305
-----
3C305 is a $z=0.0416$ ($d_{\rm L}=183$ Mpc) narrow-line radio galaxy (@lrl83) with an unusually compact radio morphology that displays both FRI and FRII characteristics. MERLIN and VLA observations of the source (@hec82 [@jac03; @mor05]) show twin jets that each extend into radio lobes separated by $\sim$4$''$ (3.3 kpc). [@lrl83] classify the source as an FRI-type radio galaxy. Optical emission-line studies of the circumnuclear environment of 3C305 show an extended morphology (@hec82 [@jac95]), and a detailed comparison between the radio ejecta and \[O [ii]{}\] gas observed with [*HST*]{} (@jac95) suggests that the gas has been shock-excited by the jet.
DA240
-----
DA240 ($z=0.0356$, $d_{\rm L}=157$ Mpc) is a giant radio galaxy (GRG), the name given to the subclass of FRII sources with a projected radio extent in excess of 1 Mpc. Westerbork Synthesis Radio Telescope (WSRT) images of the source (@klein94 [@peng04]) show two hotspots, with the northeastern one 50 times brighter than the southwestern one. The northeastern hotspot has an unusual bifurcated structure, with what appears to be a radio jet entering a compact, primary hotspot, together with a fainter secondary hotspot feature. Optically, DA240 is classified as a low-excitation radio galaxy (@lrl83).
4C73.08
-------
4C73.08 ($z=0.0581$, $d_{\rm L}=258$ Mpc) is another example of an FRII GRG. WSRT images (@may79 [@klein94]) show a compact core accompanied by two hotspots. The brighter (western) hotspot is connected to the core by a bridge of radio emission. Both lobes show unusual protrusions toward the north and south. Optically, 4C73.08 is classified as a narrow-line radio galaxy (@lrl83).\
A summary of the properties of the three sources is given in Table \[sourcessummary\].
--------- ---------- ------------------- ------------ ------------------ ---------------------------- ------------------------
Optical Optical spectrum 178-MHz Luminosity Density Galactic
Source Redshift FR classification Excitation reference (W Hz$^{-1}$ sr$^{-1}$) absorption (cm$^{-2}$)
3C305 0.0416 I NLRG [@liu95] $5.50\times10^{24}$ $1.69\times10^{20}$
DA240 0.0356 II LERG [@sau89] $5.38\times10^{24}$ $4.36\times10^{20}$
4C73.08 0.0581 II NLRG [@sau89] $9.94\times10^{24}$ $2.33\times10^{20}$
--------- ---------- ------------------- ------------ ------------------ ---------------------------- ------------------------
\[sourcessummary\]
Observations And Data Reduction
===============================
Source Obs ID Observation date Filter Nominal exposure (ks) Screened exposure (ks)
--------- ------------ ------------------ -------- ----------------------- ------------------------
3C305 0404050301 2006 August 08 Medium 11.4, 11.4, 9.8 11.4, 11.4, 9.8
DA240 0404050101 2006 October 18 Medium 12.6, 12.6, 11.0 12.6, 12.6, 11.0
4C73.08 0404050601 2007 April 28 Medium 4.5, 4.5, 13.9 3.4, 3.4, 4.7
\[obslog\]
[*XMM-Newton*]{} observed 3C305, DA240, and 4C73.08 as part of AO-5. We reprocessed the data with version 7.1.0 of the Scientific Analysis Software (SAS) using the standard pipeline tasks [emchain]{} and [epchain]{}. The data were filtered for PATTERN values $\leq 12$ (MOS) and $\leq 4$ (pn) and the bit-mask flags 0x766a0600 (MOS) and 0xfa000c (pn). These flagsets are equivalent to the standard flagsets \#XMMEA\_EM/EP but include out of field-of-view events and exclude bad columns and rows.
To check for intervals of high particle background, we extracted light curves from the CCD on which the source is located. The events were filtered to include only those with PATTERN=0 attributes and an energy range of 10–15 keV. The background was relatively low during the observations of 3C305 and DA240, meaning that no further filtering was required, especially given that we are performing spectral analyses of point sources. However, the observation of 4C73.08 was heavily affected by flaring for almost the entire duration of the observation; indeed the MOS observation was truncated due to high background. We therefore chose filtering criteria of $<2.5$ s$^{-1}$ (MOS) and $<25$ s$^{-1}$ (pn) in the 10–15 keV band to remove the worst flaring but retain sufficient data to perform spectroscopy. Table \[obslog\] gives the details of the three [[*XMM-Newton* ]{}]{}observations. All spectral fits include Galactic absorption.
The unresolved nuclei of all three sources are detected with [*XMM-Newton*]{}; in addition, there is a weak but clear detection of an X-ray source coincident with the bright NE hotspot of DA240. We discuss the analysis of these X-ray features in the following two sections.
Spectroscopic Analysis of the nuclei
====================================
3C 305 {#3c305spec}
------
We extracted the nuclear X-ray spectrum of 3C305 from a source-centered circle of radius 35$''$, with background sampled from a large off-source region on the same CCD as the target. There were sufficient counts in the spectra from each of the MOS1, MOS2, and pn cameras to perform a joint analysis for all three datasets. The spectra were grouped to a minimum of 20 counts per bin.
We initially attempted to fit the spectrum with a single, unabsorbed power law, but this achieved a poor fit ($\chi^{2} = 68.0$ for 33 dof). We found an acceptable fit ($\chi^{2} = 29.3$ for 31 dof) with the combination of an unabsorbed power law and thermal emission, characterized by an [apec]{} model with $kT=0.69^{+0.10}_{-0.15}$ keV, abundance fixed at 0.3 times solar, and normalization $(3.78^{+1.06}_{-1.02})\times10^{-5}$ photons s$^{-1}$ cm$^{-2}$ keV$^{-1}$. The best fitting parameters of the power law are $\Gamma=1.61^{+0.37}_{-0.38}$ and 1 keV normalization $(1.45\pm0.44)\times10^{-5}$ photons s$^{-1}$ cm$^{-2}$ keV$^{-1}$. Adding additional components, such as allowing the power law to be modified by additional absorption, failed to improve the fit (the best-fitting intrinsic absorption tended to zero). The thermal interpretation is supported by a [[*Chandra* ]{}]{}observation of the source (PI: D. Harris), which shows resolved emission elongated 3$''$ either side of the nucleus along the direction of the jets. Indeed, the unresolved [[*Chandra* ]{}]{}nuclear flux is 3 times lower than that which we measured with [*XMM-Newton*]{}. The [[*XMM-Newton* ]{}]{}spectra and best-fitting model are shown in Figure \[3c305\_spectrum\].
![[[*XMM-Newton* ]{}]{}MOS1 [*(black)*]{}, MOS2 [*(red)*]{}, and pn [*(green)*]{} spectrum of 3C305. Also shown is the best-fitting model of an unabsorbed power law and thermal emission.[]{data-label="3c305_spectrum"}](f1.eps){height="8cm"}
DA 240
------
We extracted the spectrum of the nucleus of DA240 from a source-centered circle of radius 35$''$, and extracted a background spectrum from a large off-source circular region on the same CCD. Only the spectrum from the pn camera had sufficient counts for an adequate spectral analysis. With the data grouped to 10 counts per bin, we found an acceptable fit ($\chi^{2} = 10.5$ for 9 dof) with a single unabsorbed power law of photon index $1.91^{+0.54}_{-0.51}$ and 1 keV normalization $(6.48^{+1.48}_{-1.51})\times10^{-6}$ photons s$^{-1}$ cm$^{-2}$ keV$^{-1}$. Allowing the power law to be modified by intrinsic absorption failed to improve the fit (the best-fitting $N_{\rm H}$ is zero). Additional components to our model also led to no statistically significant improvement in the fit. The spectrum, and best-fitting model, of a single, unabsorbed power law, are shown in Figure \[da240\_spectrum\].
![[[*XMM-Newton* ]{}]{}pn spectrum of DA240, together with the best-fitting model of an unabsorbed power law.[]{data-label="da240_spectrum"}](f2.eps){height="8cm"}
4C 73.08
--------
We sampled the nuclear spectrum of 4C73.08 from a source-centered circle of radius 35$''$, and extracted a background spectrum from a large off-source circular region on the same chip. We initially attempted to model the spectrum with a single, unabsorbed power law, but this achieved a poor fit ($\chi^{2} = 63.2$ for 15 dof), and we noticed significant residuals above $\sim$4 keV that clearly indicated the presence of an additional, heavily absorbed component. We achieved a good fit ($\chi^{2} = 7.2$ for 11 dof) to the spectrum with the sum of a heavily absorbed \[$N_{\rm H}=(9.2^{+5.4}_{-2.9})\times10^{23}$ cm$^{-2}$\] power law of photon index frozen at 1.7, a Gaussian neutral, unresolved, Fe K$\alpha$ line of equivalent width $\sim$300 eV (the Gaussian line is significant at the 2$\sigma$ level), and a second, unabsorbed, power law of photon index frozen at 2. There are insufficient counts to fit the power-law slopes, so we adopted values consistent with canonical values found in radio galaxies (e.g., @evans04 [@evans06]). The 1 keV normalizations of the power laws are $(1.82^{+2.19}_{-1.00})\times10^{-3}$ photons s$^{-1}$ cm$^{-2}$ keV$^{-1}$ and $(1.90\pm0.47)\times10^{-5}$ photons s$^{-1}$ cm$^{-2}$ keV$^{-1}$, respectively. Replacing the unabsorbed power law with a thermal component resulted in a worse fit to the spectrum ($\chi^{2} = 16.9$ for 10 dof). The spectrum and our best-fitting model are shown in Figure \[4c73.08\_spectrum\]. The hotspots of 4C73.08 are not detected in this short observation.\
Table \[spectralfitting\] summarizes the best-fitting spectral models to the X-ray spectra of 3C305, DA240, and 4C73.08.
![[[*XMM-Newton* ]{}]{}pn spectrum of 4C73.08. Also shown is the best-fitting model of a heavily absorbed power law, a neutral Fe K$\alpha$ line, and a second, unabsorbed power law.[]{data-label="4c73.08_spectrum"}](f3.eps){height="8cm"}
---------- -------------------------- ------------------------------------ ------------------------ ----------------- ---------------- ------------------------ ------------------------------------ ----------------
$L_{\rm (2-10 keV)}$
(Power Law)
Source Spectrum $N_{\rm H}$ (cm$^{-2}$) $\Gamma$ E (keV) $\sigma$ (keV) $kT$ (keV) (ergs s$^{-1}$) $\chi^{2}$/dof
(1) (2) (3) (4) (5) (6) (7) (8) (9)
3C 305 PL+TH – $1.61^{+0.37}_{-0.38}$ – – $0.61^{+0.10}_{-0.15}$ $(2.6\pm0.8)\times10^{41}$ 29.3/31
DA 240 PL – $1.91^{+0.54}_{-0.51}$ – – – $(5.5\pm1.3)\times10^{40}$ 10.5/9
4C 73.08 $N_{\rm H}$(PL+Gauss)+PL $(9.2^{+5.4}_{-2.9})\times10^{23}$ $\Gamma_1 = 1.7$ (f); 6.32 $\pm$ 0.14 0.01 (f) – $(5.7^{+6.9}_{-3.2})\times10^{43}$ 7.2/11
– $\Gamma_2 = 2$ (f) – – – $(3.8\pm0.9)\times10^{41}$
---------- -------------------------- ------------------------------------ ------------------------ ----------------- ---------------- ------------------------ ------------------------------------ ----------------
\[spectralfitting\]
--------- ----------------------------------------------- --------------------------------- --------------------------------- ------------------------------------ -----------------
Radio core luminosity 1-keV soft luminosity 178-MHz luminosity 2–10 keV ‘accretion’ Core luminosity
Source density (W Hz$^{-1}$ sr$^{-1}$) and frequency density (W Hz$^{-1}$ sr$^{-1}$) density (W Hz$^{-1}$ sr$^{-1}$) luminosity (ergs s$^{-1}$) Reference
(1) (2) (3) (4) (5) (6)
3C 305 $<$$3.21\times10^{20}$ (1.4 GHz) $(3.1\pm0.9)\times10^{15}$ $5.50\times10^{24}$ $<1.0\times10^{42}$ [@sar97]
DA 240 $2.46\times10^{22}$ (5 GHz) $(1.4\pm0.3)\times10^{15}$ $5.38\times10^{24}$ $<5.3\times10^{41}$ [@tsi82]
4C73.08 $3.53\times10^{21}$ (5 GHz) $(4.1\pm1.0)\times10^{15}$ $9.94\times10^{24}$ $(5.7^{+6.9}_{-3.2})\times10^{43}$ [@sar97]
--------- ----------------------------------------------- --------------------------------- --------------------------------- ------------------------------------ -----------------
\[luminosities\]
The Hotspot Of DA240
====================
The bright NE hotspot of the giant radio galaxy DA240 lies in the [[*XMM-Newton* ]{}]{}field of view and is clearly detected in X-rays in our observation (Fig. \[da240-xmm\]). X-ray detections of the hotspots in relative low-luminosity radio sources like our targets are quite common in [*Chandra*]{} observations [e.g., @khwm05; @hc05; @kbhe07; @hck07; @efhk08] and recently some bright hotspots have also been detected with [[*XMM-Newton* ]{}]{}[@efbm07; @ghck08]. It has been argued that that X-ray detections of low-luminosity hotspots such as those of DA240 (whose NE hotspot has a 5-GHz radio luminosity of $9\times10^{22}$ W Hz$^{-1}$ sr$^{-1}$) are almost certainly due to synchrotron rather than inverse-Compton emission (see Fig. 5 of @hhwb04). If this is the case, X-ray detections of hotspots can give us important information about the relationship between the location of high-energy particle acceleration (traced by the X-ray) and the locations where low-energy particle and field energy densities are highest (traced by the radio synchrotron emission). The available evidence to date is that this relationship is complex; kpc-scale offsets are often found between the peaks of X-ray and radio emission [e.g., @hck07].
The only radio image for DA240 available to us at the start of our study was the WSRT 608-MHz image from the on-line atlas of low-$z$ 3CRR sources[^1]. This image has a resolution of $34''$, and so does not allow us to see details of the hotspot structure or its relationship to the X-ray emission. Accordingly we obtained a short observation of the hotspot with the VLA at 4.9 GHz under the exploratory time program. This observation (observation identifier AE163) was taken on 2007 May 18 when the VLA was in the process of moving between its D and A configurations. In addition, the EVLA antennas of the array were unavailable for most of the observation. As a result there were only 13 antennas available in the expected D-configuration arrangement, as opposed to the usual 27. Nevertheless we obtained an image with a resolution of $7.5''$ and were able to detect and resolve the compact hotspot (Fig. \[da240-zoom\]). The radio emission coincident with the X-ray emission is resolved into two compact components aligned roughly N-S, with the brighter southern component having a radio flux density of 270 mJy and the fainter northern component at a level of $\sim 90$ mJy, plus extended structure. Both the compact components are point-like at the resolution of our image; however, higher-resolution images (@tsi82) resolve the southern component, showing it contains at least two separate peaks. The peak of the X-ray emission is closest to the brighter component of the hotspot, but both components may be X-ray sources: the resolution of [[*XMM-Newton* ]{}]{}is not good enough (particularly at this off-axis distance) to separate them. We used the default astrometry for both the VLA and [[*XMM-Newton* ]{}]{}data to search for offsets between the radio and X-ray emission in the northern hotspot (the radio core is too far down the primary beam of the VLA for us to be able to use it to align the radio and X-ray frames). The X-ray emission appears to be offset by several kpc approximately in the direction of the nucleus. However the higher-resolution radio observation of [@tsi82], which shows both the nucleus and hotspot, indicates that the brightest radio hotspot subcomponent (marked A by Tsien et al.) is separated by approximately 8$''$ ($\sim$ 5.5 kpc) from the brightest X-ray hotspot emission. This is consistent with our VLA observation in both sense and approximate offset with respect to the X-ray emission. Offsets this large, or larger, have been observed in other powerful radio galaxies (e.g., @efbm07).
We extracted a spectrum for the hotspot from the pn data, using a 30$''$ radius circle as the source region and local background, and fitted it with a power-law model with Galactic absorption. It is well fitted ($\chi^2 = 0.29$ for 2 d.o.f.) with a model with photon index of $2.2 \pm 0.3$ and 1-keV unabsorbed flux density of $7 \pm
1$ nJy. This flux density puts it among the brighter known X-ray hotspots [@hhwb04], very similar in flux density (though not luminosity) to the bright hotspot detected by @efbm07 in the giant quasar 4C74.26.
The steep X-ray spectrum and the possible offset between the radio and X-ray peak favor a synchrotron rather than inverse-Compton origin for the X-rays in this source. In the absence of optical measurements it is easy to fit a curved or broken synchrotron spectrum through the radio and X-ray data. Moreover, if we model the hotspot (normalizing using the observed radio flux for the brighter component) as a uniform sphere at equipartition with a radius of 1 kpc (which is consistent with the size reported by @tsi82 for the most compact component of the hotspot only) then the predicted inverse-Compton flux density, using the code of @hbw98, is 3 orders of magnitude below that observed. All of the flux density from the hotspot would have to come from a region $<0.1$ pc in size, the magnetic field strength would have to be a factor $\sim 30$ below the equipartition value, or some combination of the two would have to apply in order for the observed X-ray flux density to be produced by the synchrotron self-Compton model. Since DA240 is a low-excitation radio galaxy, the nuclear emission-line classification gives us no information about the orientation with respect to the line of sight, and so it is possible in principle that inverse-Compton emission could be boosted by a process which requires beaming and small angles of the jet to the line of sight [e.g., @gk03]. However, an efficient role for beaming would imply a very large physical size ($\ga 4$ Mpc) for DA240, so we do not regard this model as probable, and we attribute the X-ray emission to synchrotron radiation. The relative brightness of the hotspot should make it a good target for follow-up radio and high-resolution X-ray observations aimed at understanding the details of high-energy particle acceleration in this source.
![[[*XMM-Newton* ]{}]{}observations of DA240 in the MOS and pn cameras, co-added taking account of exposure and smoothed with a Gaussian of FWHM $15.3''$. Overlaid are contours from the $34''$-resolution 608-MHz WSRT map described in the text, at $2 \times(1,4,16\dots)$ mJy beam$^{-1}$.[]{data-label="da240-xmm"}](f4.eps){width="8.5cm"}
![Co-added [[*XMM-Newton* ]{}]{}observations of DA240 as in Fig. \[da240-xmm\], but smoothed with a Gaussian with FWHM $6.1''$. Overlaid are contours of our VLA data with $7.5'' \times 6.7''$ resolution (major $\times$ minor axis of restoring elliptical Gaussian) at $4 \times (1, 2, 4\dots)$ mJy beam$^{-1}$. The 90% off-axis encircled energy radius of [[*XMM-Newton* ]{}]{}at this location is $\sim$1$'$.[]{data-label="da240-zoom"}](f5.eps){width="8.5cm"}
Interpretation of the nuclear spectra {#interp}
=====================================
![X-ray luminosity of the unabsorbed nuclear component for the three sources, together with combined $z<0.5$ sample (@evans06 [@hec06]) as a function of 5-GHz radio core luminosity (Table \[luminosities\]). Open circles are LERG, filled circles NLRG, open stars BLRG, and filled stars quasars. Large surrounding circles indicate that a source is an FRI. The sources studied in this paper are indicated by surrounding boxes. Note that the core luminosity of 3C305 is measured at 1.4 GHz. Where error bars are not visible they are smaller than symbols. Dotted lines show the regression line to all data and its 1$\sigma$ confidence range.[]{data-label="rx"}](f6.eps){width="8cm"}
![X-ray luminosity of the accretion-related component for the three sources, together with combined $z<0.5$ sample (@evans06 [@hec06]) as a function of 178-MHz total radio luminosity (Table \[luminosities\]). Symbols are as in Fig. \[rx\] .Dotted lines show the regression line to the NLRGs only and its 1$\sigma$ confidence range.[]{data-label="lum-lum"}](f7.eps){width="8cm"}
Overview of the spectra {#interp_spectra}
-----------------------
The unabsorbed X-ray spectrum of the low-excitation FRII radio galaxy DA240 is consistent with the other LERGs in the [@hec06] sample, which observationally encompass both FRI and FRII radio galaxies. The high-excitation (narrow-line) FRII radio galaxy 4C73.08 shows a heavily absorbed, luminous, component of X-ray emission, similar to the other narrow-line radio galaxies studied by [@hec06]. However, the spectrum of 3C305 shows no evidence for the heavily absorbed X-ray emission that is characteristic of narrow-line radio galaxies. We return to this in Section \[interp\_nlrgs\].
The Radio Core–X-Ray Core Correlation
-------------------------------------
Figure \[rx\] shows a plot of the 1-keV luminosity of the low-absorption power-law component against the core luminosity for our three sources (Table \[luminosities\]), together with the 3CRR sources presented in [@evans06] and [@hec06]. DA240 and 4C73.08 both lie close to the correlation between the radio and X-ray luminosities established by, e.g., [@fab84], [@wb94], and [@hec06]. On the other hand, 3C305 lies somewhat away from the other data, though this is almost certainly due to the extended X-ray emission detected with [*Chandra*]{}. Indeed, the unresolved [[*Chandra* ]{}]{}nuclear flux is 3 times lower than the [[*XMM-Newton* ]{}]{}value, which would bring 3C305 closer to established trendline in Figure \[rx\]. The radio–X-ray core correlation suggests a common origin of the two at the base of an unresolved jet, as has been extensively argued by, e.g., [@wb94], [@har99], [@evans06], [@bal06] and [@bel06].
Accretion-Related X-ray Emission
--------------------------------
We now wish to consider any accretion-related X-ray emission. For 4C73.08, as with the other narrow-line radio galaxies studied by [@evans06] and [@hec06], we can take the accretion-related luminosity to be the unobscured luminosity of the heavily absorbed power-law component: this is supported by the presence of Fe K$\alpha$ emission (e.g., @evans06). The 2–10 keV accretion luminosity of 4C73.08 ($\sim$6$\times10^{43}$ ergs s$^{-1}$) is substantially larger than its jet-related luminosity ($\sim$4$\times10^{41}$ ergs s$^{-1}$), as has been found with other NLRGs (e.g., @evans06 [@hec06]).
In the cases of the LERG DA240 and the (purported NLRG 3C305), we followed the method of [@evans06] and assumed that, in addition to the dominant jet component of X-ray emission, there exists an additional ‘hidden’ component of accretion-related emission of photon index 1.7 that is obscured by a torus of intrinsic absorption $10^{23}$ cm$^{-2}$. We added this component to the best-fitting model, refitted the spectra, and determined the 90%-confidence upper limit to the 2–10 keV accretion-related luminosity to be $5.3\times10^{41}$ ergs s$^{-1}$ for DA240 and $1.0\times10^{42}$ ergs s$^{-1}$ for 3C305.
Figure \[lum-lum\] shows a plot of the 178-MHz and 2–10 keV accretion-related luminosities of the three sources (Table \[luminosities\]), together with those of the other $z<0.5$ 3CRR sources studied by [@evans06] and [@hec06]. Figure \[lum-lum\] shows that the upper limit to the accretion-related components in the LERG DA240, given our assumed absorbing column of $10^{23}$ cm$^{-2}$, lies below the trendline established for high-excitation (narrow-line) radio galaxies such as 4C73.08. Of course, if no obscuring region is present in DA240, as seems to be the case in other LERGs, then the luminosity of any accretion-related emission will be substantially lower than that shown. Alternatively, the accretion-related X-ray luminosity of LERGs can be made to lie in the region occupied by the HERGs, but this requires extremely high values of intrinsic absorption (@evans04) that can be ruled out by infrared observations (e.g, @mul04). The upper limit to the accretion-related luminosity of 3C305 lies between the populations of low- and high-excitation sources in Figure \[lum-lum\], though as previously mentioned the [*XMM-Newton*]{}-measured unresolved core flux is overestimated by a factor $\sim$3 (see Section \[3c305spec\]).
Optical Emission Line Classifications and Relationships to the Central Engine {#interp_nlrgs}
-----------------------------------------------------------------------------
In our previous studies of the X-ray properties of 3CRR radio sources we have used the [@lrl83] optical emission-line classifications of low- and high-excitation radio galaxies. [@laing94] provided a quantitative definition of LERGs as having \[OIII\] equivalent widths of less than 3 Å and \[OIII\]/H$\alpha$ line ratios $>0.2$. A similar classification was given by [@jac97], with LERGs having \[OIII\] equivalent widths of less than 10 Å and/or \[OII\]/\[OIII\] line ratios $>1$. However, these definitions of low- and high-excitation sources do not necessarily take into account the potentially [*different*]{} sources of ionizing radiation, their size scale, or their relationship to the AGN itself. This may lead to occasional ambiguities where sources are classified based on their emission-line characteristics. We discuss some of the issues here.
[*HST*]{} observations of the nuclei of radio galaxies have revealed the origin of the optical continuum emission and its likely relationship to any unresolved emission lines. In the case of LERGs, [@chi99] and [@har00] showed that the correlations between the radio and optical continuum luminosities support the common origin of the two in the form of a jet.[*HST*]{} narrow-band imaging of LERGs (@cap05) showed that so-called Compact Emission Line Regions (CELRs) are commonplace, and that they are associated with the dominant source of ionizing photons, assumed to be the jet. Further, [@chi02] argued that the dominant contribution to the optical emission in obscured high-excitation radio galaxies is the accretion disk. There is likely to be a substantial ionizing field in these sources that is directly related to the accretion process.
On larger scales, high-resolution [*HST*]{} emission-line images of the extended environments in radio galaxies (@pri08) provide insights on the different components that constitute the kpc-scale narrow-line region (NLR) and $\sim$10 kpc-scale extended narrow-line region (ENLR). In addition to photoionization from the nucleus, jet–environment interactions may play a significant role in governing the energy budget of the NLR and ENLR, either in the form of collisional ionization or a radiative ’autoionizing’ shock (e.g., @ds95 [@ds96]).
The different physical origins for optical line-emission in radio galaxies illustrate the difficulties in disentangling genuine AGN emission from that which is not directly related to the accretion process. An excellent case in point is the purported NLRG 3C305, whose X-ray spectrum is consistent with that of a LERG, rather than a NLRG. [*HST*]{} \[OII\] observations of the extended emission-line environment in the source show that the majority of the \[OII\] emission lies just beyond the edge of the radio jet at a distance of 1.5$''$ from the core, and [@jac95] suggested that it has been shock-excited by the jet. Several other FRI radio sources studied by [@evans06] and [@hec06] also show optical spectra that may be attributed to their environments. Some of these lie at the centers of cooling-core clusters, in which significant amounts of optical line-emission might be expected that are not necessarily directly related to the central AGN. This may go some way to explaining the handful of other purported NLRGs in Figure \[lum-lum\] whose X-ray properties are more consistent with low-excitation sources.
The above arguments suggest that the emission-line classification of relatively weak-lined radio galaxies does not always reflect the nuclear accretion activity itself. We propose that only the combination of high-resolution optical spectroscopy, X-ray observations, and constraints from [*Spitzer*]{} mid-infrared observations of reprocessed emission in radio galaxies can reliably determine the structure of the central engine in radio-loud AGN. In the case of 3C305, [*Spitzer*]{} observations would enable us to distinguish between (1) a genuinely narrow-line radio galaxy that is obscured by a Compton-thick absorber (in which case the $<10$ keV X-ray continuum would show few, if any, signs of heavily absorbed emission), and (2) a low-excitation radio galaxy with a prominent extended emission-line environment. We will return to this point in subsequent publications (Birkinshaw et al., 2008, in prep.; Hardcastle et al. 2008, in prep.).
Conclusions
===========
We have presented results from [[*XMM-Newton* ]{}]{}observations of the nuclei of the radio galaxies 3C305, DA240, and 4C73.08. We have shown the following:
1. The X-ray spectrum of the narrow-line FRII radio galaxy 4C73.08 can be modeled as the sum of a heavily absorbed power law associated with a luminous accretion disk and circumnuclear obscuring structure, together with an unabsorbed component of X-ray emission that has a common origin with the radio emission at the base of an unresolved jet. This behavior is consistent with the other narrow-line FRII radio galaxies studied by [@evans06] and [@hec06].
2. The nuclear X-ray spectrum of the FRII giant radio galaxy DA240, optically classified as a low-excitation radio galaxy, can be modeled as a single, unabsorbed power law that is likely associated with emission from the parsec-scale jet. The upper limit to the X-ray luminosity of any additional, accretion-related emission suggests that the accretion process in DA240 is substantially sub-Eddington and likely radiatively inefficient in nature.
3. The X-ray emission in the nucleus of the narrow-line radio galaxy 3C305 can be modeled as an unabsorbed power law that originates at the base of the jet. However, it shows no evidence for heavily absorbed X-ray emission was found in the NLRGs studied by [@evans06].
4. We have discovered an X-ray counterpart to the NE hotspot of the giant radio galaxy DA240. We argue that the emission process is overwhelmingly likely to be synchrotron emission. Because of the high X-ray flux of the hotspot, it is a good candidate for followup high-resolution X-ray observations.
5. We have discussed the different origins of optical emission lines in the nuclear and circumnuclear gaseous environments of radio galaxies. These include photoionization from the AGN accretion flow or parsec-scale jet, shock-excitation by the radio jet, or cooling gas in the centers of clusters. This may lead to occasional misclassification of genuinely weak-lined sources such as 3C305 as high-excitation sources.
6. We therefore argue that there is not necessarily always a one-to-one correspondence between optical emission-line class (low- vs. high-excitation) and accretion-flow state (inefficient flow vs. standard thin disk), especially when low angular-resolution optical spectroscopy is used. We suggest that only the combination of high-resolution optical, X-ray, and infrared observations can reliably uncover the nature of the central engine in radio-loud AGN.
DAE gratefully acknowledges financial support for this work from NASA under grant number NNX06AG37G. MJH thanks the Royal Society for a Research Fellowship. We wish to thank the anonymous referee for valuable comments. We also thank Dan Harris and Francesco Massaro for useful discussions of the nuclear properties of 3C305. This work is based on observations obtained with [*XMM-Newton*]{}, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
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[^1]: http://www.jb.man.ac.uk/atlas/
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abstract: 'We consider how the swampland criteria might be applied to models in which scalar fields have nontrivial kinetic terms, particularly in the context of $P(\phi,X)$ theories, popularly used in approaches to inflation, to its alternatives, and to the problem of late-time cosmic acceleration. By embedding such theories in canonical multi-field models, from which the original theory emerges as a low-energy effective field theory, we derive swampland constraints, and study the circumstances under which these might be evaded while preserving cosmologically interesting phenomenology. We further demonstrate how these successes are tied to the phenomenon of [*turning*]{} in field space in the multi-field picture. We study both the general problem and specific examples of particular interest, such as DBI inflation.'
author:
- 'Adam R. Solomon'
- Mark Trodden
bibliography:
- 'refs.bib'
title: 'Non-canonical kinetic structures in the swampland'
---
Introduction
============
Cosmological model-building typically works, explicitly or otherwise, with “bottom-up" effective field theories (EFTs).[^1] According to the standard lore, one constructs a bottom-up EFT by deciding on some particle content and some set of symmetries, and then writing down the most general action consistent with those symmetries, organized in an appropriate expansion. This approach has a number of virtues: it allows one to work with low-energy theories in a good deal of generality, and it typically requires only a finite number of operators in order to compare to observations at a given level of precision. Nevertheless, the number of parameters an EFT has can be greater than can be constrained with a given set of observations, and it is interesting and useful to ask whether these parameters can be further restricted by theoretical considerations.
A variety of programs have been developed to constrain the space of low-energy effective theories using guidance from high energies. The general approach is to explore whether there are effective field theories that appear consistent, but nevertheless do not possess an ultraviolet (UV) completion.[^2] These programs include the weak gravity conjecture [@ArkaniHamed:2006dz], positivity bounds [@Adams:2006sv], and the swampland [@Vafa:2005ui].
In recent years, string theoretic arguments have been used to suggest that the string landscape may exclude a large “swampland" of low-energy effective field theories which, remarkably, would include most models with quasi-de Sitter vacua, which are typically used to obtain inflation, or to model dark energy [@Obied:2018sgi; @Agrawal:2018own]. Many such conjectures have been proposed, with two of the most prominent being [*the de Sitter conjecture*]{} and [*the distance conjecture*]{}, formulated as follows.
Consider a low-energy effective theory consisting of a set of scalar fields $\Phi^a$ with a field-space metric $\mathcal{G}_{ab}(\Phi)$ and a potential $V(\Phi)$, $$S = \int{\mathrm{d}}^4x{\sqrt{-g}}\left[\frac{{M_\mathrm{Pl}}^2}{2}R-\frac12\mathcal{G}_{ab}(\Phi)\partial_\mu\Phi^a\partial^\mu\Phi^b-V(\Phi)\right]. \label{eq:sigma-action}$$ The de Sitter conjecture proposes that the gradient of the potential $V$ is bounded as $$|\nabla V| \equiv \sqrt{\mathcal{G}^{ab}\partial_aV\partial_bV} \geq \frac c {M_\mathrm{Pl}}V \label{eq:dSconj}$$ for some $\mathcal{O}(1)$ constant $c$, and with $\partial_a\equiv\partial/\partial\Phi^a$. The distance conjecture states that the field excursion is sub-Planckian, $$|\Delta\Phi|\equiv\sqrt{\mathcal{G}_{ab}\Delta\Phi^a\Delta\Phi^b}<\mathcal{O}({M_\mathrm{Pl}}). \label{eq:distconj}$$ The combination of these conjectures has been argued to severely constrain single-field models of inflation and dark energy [@Agrawal:2018own; @Akrami:2018ylq; @Heisenberg:2018yae; @Hertzberg:2018suv], although multi-field models can evade these bounds through field-space effects [@Achucarro:2018vey]. Refinements of these conjectures exist; for example demanding that $$\operatorname{min}(\partial_a\partial_bV)\leq-\frac{c'}{{M_\mathrm{Pl}}^2}V$$ for some constant $c'\sim\mathcal{O}(1)$ [@Garg:2018reu; @Ooguri:2018wrx].
A large class of theories of interest in cosmology are not of the $\sigma$-model form , but instead involve non-canonical structures. Examples include $P(\phi,X)$ theories [@ArmendarizPicon:1999rj; @ArmendarizPicon:2000ah; @ArmendarizPicon:2000dh], galileons [@Nicolis:2008in; @Hinterbichler:2010xn], Horndeski models [@Horndeski:1974wa; @Deffayet:2011gz; @Kobayashi:2011nu], and DHOST theories [@Langlois:2015cwa; @BenAchour:2016fzp]. It is therefore natural to ask whether the swampland criteria can be applied to inflationary or dark energy models with such non-canonical kinetic terms. In this paper we demonstrate a method of applying these criteria to $P(\phi,X)$ theories, in which the action has a non-trivial, algebraic dependence on the scalar-field kinetic term $X\equiv-\frac12(\partial\phi)^2$, $$S = \int{\mathrm{d}}^4x{\sqrt{-g}}\left(\frac{{M_\mathrm{Pl}}^2}{2}R+ P(\phi,X)\right). \label{eq:action}$$ For generality, and to make contact with much of the literature, we allow arbitrary dependence on both $X$ and $\phi$. We can rewrite a $P(\phi,X)$ theory in a $\sigma$-model form by introducing an auxiliary field $\chi$ [@Tolley:2009fg; @Elder:2014fea], $$S = \int{\mathrm{d}}^4x{\sqrt{-g}}\left[\frac{{M_\mathrm{Pl}}^2}{2}R+P(\phi,\chi) + P_\chi\left(X-\chi\right)\right].$$ Since the $\chi$ equation of motion sets $\chi=X$,[^3] this action is dynamically equivalent to \[eq:action\]. We can therefore write a “UV extension" of the $P(\phi,X)$ theory by adding a small kinetic term for $\chi$, $$S = \int{\mathrm{d}}^4x{\sqrt{-g}}\left[\frac{{M_\mathrm{Pl}}^2}{2}R+P(\phi,\chi) - \frac{1}{2\Lambda^6}(\partial\chi)^2 + P_\chi\left(X-\chi\right)\right], \label{eq:action-2field}$$ where $\Lambda$ is a mass scale, introduced on dimensional grounds, which acts as a cutoff for the UV extension. At sufficiently low energies we can ignore the dynamics of $\chi$ and again obtain the $P(\phi,X)$ theory. The action is of the form , with $\Phi^a=(\phi,\chi)$ and $$\begin{aligned}
\mathcal{G}_{ab} &= \operatorname{diag}\left(P_\chi,\frac{1}{\Lambda^6}\right), \\
V(\Phi) &= -P+\chi P_\chi,\end{aligned}$$ and is therefore amenable to a swampland analysis.[^4]
The general idea here is to perform a swampland analysis for the theory , and then to consider the constraints on the field $\phi$ (which we might imagine as the inflaton or dark energy field) as we approach the decoupling limit of $\chi$. We will focus on applying the relevant constraints along solutions of cosmological interest, such as those describing inflation, although the general technique can be applied more broadly.
We start in \[sec:swamp\] by formulating the swampland criterion for general $P(\phi,X)$ theories and discussing the conditions on $P(\phi,X)$ under which it is satisfied. In \[sec:DBI\] we apply these general considerations to DBI inflation, a prominent class of inflationary models of the $P(\phi,X)$ type. Finally we provide a physical interpretation of these results from the two-field perspective in terms of field-space turning in \[sec:turning\], before concluding in \[sec:conc\].
Swampland estimates {#sec:swamp}
===================
Swampland criterion for $P(\phi,X)$
-----------------------------------
Our aim is to apply the de Sitter swampland criterion to $P(\phi,X)$ theories in the UV-extended form . We start by considering the as-yet-unfixed scale $\Lambda$, which must be chosen sufficiently large that the $P(\phi,\chi)$ theory is well-approximated by the original $P(\phi,X)$ theory on the low-energy solutions of interest. The original theory arises in the limit of infinite $\Lambda$. At first glance it appears as if the de Sitter conjecture, which depends on $\Lambda$ through the inverse field-space metric $\mathcal{G}^{ab}=\operatorname{diag}(1/P_\chi,\Lambda^6)$, is trivially satisfied in this limit. However, we are not allowed to increase $\Lambda$ arbitrarily, as we will run into strong coupling. Moreover, the distance conjecture has the opposite scaling with $\Lambda$, as it depends on $\mathcal{G}_{ab}$ rather than its inverse $\mathcal{G}^{ab}$.
To avoid such strong coupling problems and to be maximally conservative in applying the swampland conjecture, we choose to take $\Lambda$ as small as possible consistent with inflation remaining as a valid solution of the low-energy EFT. Consider the $\chi$ equation of motion, $$\frac{1}{\Lambda^6}\Box\chi + P_{\chi\chi}\left(X-\chi\right) = 0,$$ and expand $\chi$ in powers of the small parameter $1/\Lambda^6$,[^5] $$\chi = X + \frac{1}{\Lambda^6}\chi_1+\mathcal{O}\left(\frac{1}{\Lambda^{12}}\right).$$ The $\mathcal{O}(\Lambda^0)$ piece of the $\chi$ equation of motion is satisfied by our choice of the leading-order term $\chi = X+\cdots$, and the $\mathcal{O}(\Lambda^{-6})$ piece is $$\Box X -P_{XX}\,\chi_1=0\Longrightarrow\chi_1=\frac{\Box X}{P_{XX}}.$$ By insisting that this not dominate the solution for $\chi$, i.e., $\chi_1/\Lambda^6\ll X$, we obtain a lower bound on $\Lambda$, $$\Lambda^6\gtrsim\frac{\Box X}{X P_{XX}}. \label{eq:minLambda}$$ Note that the right-hand side of this expression can be evaluated entirely on the solution to the low-energy $P(\phi,X)$ theory, since by construction we have $\chi\approx X$ to this order. We will take $\Lambda$ to have the smallest value allowed, essentially saturating the inequality in \[eq:minLambda\].[^6] This allows us to obtain a conservative estimate of the swampland bound; $\Lambda$ could certainly be larger, in which case the de Sitter conjecture is more easily satisfied.
Let us phrase the de Sitter conjecture as $R\geq c$ for some $c\sim\mathcal{O}(1)$, where the ratio $R$ is $$R\equiv \frac{1}{{M_\mathrm{Pl}}^2}\frac{\mathcal{G}^{ab}\partial_a V\partial_bV}{V^2}.$$ Evaluating this for the two-field extension of $P(\phi,X)$ we have $$R = {M_\mathrm{Pl}}^2\frac{\frac{(P_\phi-\chi P_{\phi\chi})^2}{P_\chi}+\Lambda^6\chi^2 P_{\chi\chi}^2}{(P-\chi P_\chi)^2}.$$ We assume that on-shell $\chi\approx X$, so that we can evaluate the right-hand side in terms of $\phi$ and $X$ rather than $\phi$ and $\chi$, and write this along the solution of interest—such as an inflationary trajectory—as $$R \approx {M_\mathrm{Pl}}^2\frac{\frac{(P_\phi-X P_{\phi X})^2}{P_X}+X P_{XX}\Box X}{(P-X P_X)^2}, \label{eq:R}$$ where, as discussed above, we have taken $\Lambda$ to have its minimum allowed size.
Swampland criterion during inflation
------------------------------------
We now proceed to evaluate the expression on inflationary trajectories. The aim is to derive and collect useful expressions in terms of slow-roll parameters and model parameters that will reduce \[eq:R\] to the much simpler form . We assume a spatially homogeneous and isotropic solution, and a spatially flat metric, $$\phi=\phi(t),\qquad {\mathrm{d}}s^2 = -{\mathrm{d}}t^2+a^2(t){\mathrm{d}}\vec{x}^2.$$ The Friedmann equations are $$\begin{aligned}
3{M_\mathrm{Pl}}^2H^2 &= -P+2XP_X, \\
{M_\mathrm{Pl}}^2\dot H &= -XP_X,\end{aligned}$$ where overdots denote time derivatives, $H={\mathrm{d}}\ln a/{\mathrm{d}}t$, and $X=\dot\phi^2/2$, while the scalar equation of motion is $$\begin{aligned}
P_\phi &= a^{-3}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\left(a^3P_X\dot\phi\right) \nonumber\\
&= \ddot\phi\left(P_X+\dot\phi^2P_{XX}\right)+\dot\phi^2P_{\phi X}+3HP_X\dot\phi. \label{eq:scalareom}\end{aligned}$$ The Friedmann equations can be combined to obtain an exact expression for the Hubble slow-roll parameter, $$\varepsilon\equiv-\frac{\dot H}{H^2} = \frac{XP_X}{{M_\mathrm{Pl}}^2H^2}.\label{eq:SR-epsilon}$$ We demand a quasi-de Sitter phase, so this quantity should be small, $\varepsilon\ll1$.[^7] As usual we also insist that its first derivative be small enough to allow inflation to persist for a sufficient number of $e$-foldings, i.e., $\eta\ll1$ with $$\eta\equiv \frac{\dot\varepsilon}{H\varepsilon}.$$ Taking a derivative of \[eq:SR-epsilon\] and using the definition of $\eta$, we find an expression which will be useful later, $$\eta-2\varepsilon=\frac{\ddot\phi}{H}\left(\frac2{\dot\phi}+\frac{P_{XX}}{P_X}\dot\phi\right)+\frac{P_{\phi X}}{P_X}\frac{\dot\phi}H. \label{eq:SR-scalar}$$ We will also find it convenient to replace $P_X$ in the Friedmann equation using $\varepsilon$, obtaining $$(3-2\varepsilon)H^2=-\frac{P}{{M_\mathrm{Pl}}^2}. \label{eq:SR-fried}$$ Note that all of these equations are exact: we have yet to use the slow-roll approximation.
In addition to the slow-roll parameters, which measure the deviation from exact de Sitter, we will introduce a pair of parameters to quantify the deviation from the canonical scalar, for which $P(\phi,X)=X-V(\phi)$. We choose convenient dimensionless parametrizations of $P_{XX}$ and $P_{\phi X}$, $$\begin{aligned}
c_s^2 &\equiv \frac{P_X}{P_X+2XP_{XX}},\\
\beta &\equiv \frac{P_{\phi X}}{P_X}\frac{\dot\phi}{H}.\end{aligned}$$ The first of these, $c_s^2=P_X/\rho_X$, is the usual sound speed, while $\beta$ is chosen to simplify expressions involving $P_{\phi X}$. For a canonical kinetic term we have $\beta=0$ and $c_s=1$, and deviations from these values signal that the non-canonical kinetic structure is important. It is worth mentioning here that, as we pointed out earlier, technically, our UV-extension of the original theory is only valid as long as $P_{\chi\chi}\neq0$ along the trajectory of interest. To leading order in the EFT this condition reads $P_{XX}\neq0$, which we can see is satisfied as long as $P_X\neq0$ (which in turn implies $\varepsilon\neq0$), $X$ is finite, and $c_s<1$. All of these conditions are satisfied by the models under consideration.[^8]
These parameters allow us to write the scalar equation of motion in a simple form reminiscent of the usual canonical expression, $$P_X\left[c_s^{-2}\ddot\phi+(3+\beta)H\dot\phi\right]=P_\phi. \label{eq:scalareom2}$$ In canonical slow-roll inflation we usually drop the first term, as $\ddot\phi\ll H\dot\phi$. This is a consequence of \[eq:SR-scalar\], which for $P(\phi,X)=X-V(\phi)$ becomes $\ddot\phi=(\eta/2-\varepsilon)H\dot\phi$. It is important to understand the conditions under which this continues to hold in the more general $P(\phi,X)$ setting. Let us rewrite \[eq:SR-scalar\] in terms of $c_s^2$ and $\beta$, $$\frac{\ddot\phi}{H\dot\phi} = \frac{\eta-2\varepsilon-\beta}{1+c_s^{-2}}, \label{eq:ddotphi}$$ where, as already mentioned, we still have not used any slow-roll approximation. Note that, for $0<c_s^2\leq1$, the factor $1/(1+c_s^{-2})$ ranges monotonically from 0 to 1/2, and is approximately $c_s^2$ for $c_s^2\ll1$. This factor is therefore $\mathcal{O}(c_s^2)$ for all $c_s$. We see that $\ddot\phi\ll H\dot\phi$ still holds during slow roll as long as $\beta c_s^2$ is small.[^9]
Finally we address the factor $\Box X = \ddot X + 3H\dot X$ appearing in the expression for $R$. We will assume that $\ddot X\ll H\dot X$. This follows from \[eq:ddotphi\] during slow-roll if the factor $\beta(1+c_s^{-2})^{-1}\sim\mathcal{O}(\beta c_s^2)$ is small (so that $\ddot\phi\ll H\dot\phi$) and roughly constant over a Hubble time.[^10] Under these assumptions, during slow roll we find $$\Box X \approx -3H\dot X=-3H\dot\phi\ddot\phi.$$
Using the expressions presented in this section, we can simplify \[eq:R\] to find, at leading order in slow-roll, $$\boxed{R\approx \varepsilon\left(2-\frac{{\mathcal{B}}}3+\frac{{\mathcal{B}}^2}{18}\right),} \label{eq:swampland-PX}$$ where we have defined $${\mathcal{B}}\equiv\frac{1-c_s^2}{1+c_s^2}\beta.$$ Note that $R$ depends on $c_s^2$ and $\beta$ only through this particular combination. We emphasize that the only assumptions we have made in deriving \[eq:swampland-PX\] are Hubble slow-roll ($\varepsilon,\eta\ll1$) and slow scalar evolution ($\ddot\phi\ll H\dot\phi$).[^11]
The first term in \[eq:swampland-PX\] is $\mathcal{O}(\epsilon)$, so the only way to satisfy the swampland bound—that is, to have $R>\mathcal{O}(1)$—is to increase the value of ${\mathcal{B}}$, $${\mathcal{B}}>\mathcal{O}\left(\frac1{\sqrt{\varepsilon}}\right)\gg1,$$ so that the third (and possibly second) term is large. Our goal therefore becomes to understand the circumstances under which this condition is satisfied. Because we have assumed $c_s^2\beta\ll1$, which is required in order that $\dot\phi$ evolve slowly over a Hubble time, we can only have ${\mathcal{B}}\gg1$ if the sound speed is small, $c_s^2<\mathcal{O}(\sqrt\varepsilon)\ll1$. Then ${\mathcal{B}}\approx\beta$, so the swampland criterion requires $$\beta\gtrsim\frac{1}{\sqrt\varepsilon}.$$
In the next section we will consider DBI inflation. We will see that on inflationary solutions the sound speed is small, $c_s^2\ll1$, while $\beta$ is large, with the combination $c_s^2\beta\sim\mathcal{O}(\varepsilon)$, implying $$R\sim\mathcal{O}\left(\frac{\varepsilon^3}{c_s^4}\right).$$ We will find, therefore, that the swampland criterion is satisfied, $R\gtrsim\mathcal{O}(1)$, if $c_s$ is sufficiently small to overcome the slow-roll suppression in the numerator.
DBI {#sec:DBI}
===
In this section we apply the de Sitter conjecture to Dirac-Born-Infeld (DBI) inflation, a particularly prominent, physically-motivated example of inflation with a $P(\phi,X)$ action.[^12] One attraction of DBI inflation is that it can be realized as a brane inflation scenario in type IIB string theory, with a D3-brane traveling down a warped throat at relativistic speeds [@Silverstein:2003hf; @Alishahiha:2004eh]. Besides its motivation in string theory, this model is also appealing because it is a sensible EFT from the bottom-up perspective, since its structure is protected from radiative corrections by a nonlinear symmetry [@deRham:2010eu; @Goon:2011qf; @Babic:2019ify]. Note that the DBI model is constrained by observations, predicting for instance large non-Gaussianities when the sound speed is small [@Chen:2006nt]; our focus here is on a proof of concept rather than advocating for a specific inflationary model. As in the previous section we will focus on inflationary trajectories.
The action for DBI inflation is given by \[eq:action\] with $$P(\phi,X) = -\frac{1}{g_s}\left(\frac{1}{f(\phi)}\sqrt{1-2f(\phi)X}+V(\phi)\right).$$ Here $f(\phi)$ is the (squared) warp factor of the throat, and $V(\phi)$ is the potential. The form of $f(\phi)$ depends on the geometry of the throat; for example for a pure AdS$_5$ throat of radius $R$, we have $f(\phi)=\lambda/\phi^4$ with $\lambda\equiv R^4/\alpha'^2$ [@Silverstein:2003hf; @Alishahiha:2004eh]. We will leave both the warp factor and the potential general, and return to this specific example at the end.
The distinguishing feature of DBI inflation is “D-cceleration," an alternative mechanism to potential slow roll. The brane is taken to be moving near the bulk speed of light, imposing a speed limit that leads to slow roll, even if the potential is too steep to allow slow roll in the presence of a canonical kinetic term. From the perspective of the 4D theory this speed limit is imposed by the reality of the square-root term in the action, which requires $2f(\phi)X<1$.
It is convenient to introduce the 5D Lorentz factor $\gamma$, $$\gamma\equiv\frac{1}{\sqrt{1-2fX}},$$ in terms of which the condition for D-cceleration is $\gamma\gg1$. This implies that, to leading order in slow roll, $$X\approx\frac{1}{2f(\phi)}.$$ Calculating the sound speed we find $$c_s^2 = \frac{1}{\gamma^2}\ll1,$$ so that ${\mathcal{B}}\approx\beta$, which in turn is $$\begin{aligned}
\beta &= \gamma^2Xf'\frac{\dot\phi}{H} \nonumber\\
&\approx \frac12\frac{\gamma^2}{H}\frac{f'}{f^{3/2}},\end{aligned}$$ where in the second line we have used $X\approx1/(2f)$. The slow-roll parameter $\varepsilon$ is $$\begin{aligned}
\varepsilon &= \frac{XP_X}{{M_\mathrm{Pl}}^2H^2} \nonumber\\
&\approx \frac{1}{{M_\mathrm{Pl}}^2H^2}\frac\gamma{g_s}\frac{1}{2f},\end{aligned}$$ where we have used $P_X=\gamma/{g_s}$, which holds exactly.
These expressions can be written more clearly in terms of the free parameters and functions of the model. We can eliminate $H$ using the Friedmann equation: the potential energy dominates, so we have $$H^2 \approx \frac{1}{3{g_s}{M_\mathrm{Pl}}^2}V.$$ We can also write $\gamma$ in terms of the potential and warp factor. We write the acceleration equation as $$\begin{aligned}
\dot H &= -\frac{1}{2{g_s}{M_\mathrm{Pl}}^2f}\left(\gamma-\frac1\gamma\right)\nonumber \\
&\approx -\frac{\gamma}{2{g_s}{M_\mathrm{Pl}}^2f},\end{aligned}$$ where in the second line we have used $\gamma\gg1$. Taking a time derivative of the Friedmann equation we also find $$\dot H \approx \frac{V'}{2\sqrt{3{g_s}}{M_\mathrm{Pl}}\sqrt{Vf}}.$$ Comparing these, we then obtain an expression for $\gamma$, $$\gamma\approx\sqrt{\frac{g_s}3}{M_\mathrm{Pl}}\sqrt f\frac{|V'|}{\sqrt V}. \label{eq:gamma}$$
Note that we can now calculate $c_s^2\beta$ (which we have seen needs to be small in order to have $\ddot\phi\ll H\dot\phi$ and $\ddot X\ll H\dot X$), finding $$c_s^2\beta \sim \varepsilon \frac{f'}{f}\frac{V'}{V}.$$ In regions of field space where $f(\phi)$ and $V(\phi)$ are, or can be approximated by, power laws with order-unity exponents, then we find $c_s^2\beta\sim\mathcal{O}(\varepsilon)\ll1$.
Recall (cf. \[eq:swampland-PX\]) that the swampland criterion is given by $R>\mathcal{O}(1)$, with $$R\approx \varepsilon\left(2-\frac{\beta}3+\frac{\beta^2}{18}\right),$$ where we have used the fact that $c_s^2\ll1$ for D-ccelerating DBI theories implies that ${\mathcal{B}}\approx\beta$. The first term is clearly small during slow roll. The other two terms are, to leading order in slow roll, $$\begin{aligned}
\varepsilon\beta &\approx\frac14{g_s}^2{M_\mathrm{Pl}}^4\frac{f'}{f}\left|\frac{V'}{V}\right|^3,\\
\varepsilon\beta^2 &\approx\frac{1}{16}\frac{{g_s}^4{M_\mathrm{Pl}}^8}{\varepsilon}\left(\frac{f'}{f}\right)^2\left(\frac{V'}{V}\right)^6 \ .\end{aligned}$$ If the swampland criterion is satisfied, then the term proportional to $\varepsilon\beta^2$ must be dominant. If it were not, i.e., if $\varepsilon\beta^2<\varepsilon\beta$, then $\beta$ would be small, implying $R\approx2\varepsilon\ll1$. In this régime we therefore have $$R \approx \frac{1}{18}\varepsilon\beta^2 \approx \frac{1}{288}\frac{{g_s}^4{M_\mathrm{Pl}}^8}{\varepsilon}\left(\frac{f'}{f}\right)^2\left(\frac{V'}{V}\right)^6.$$
Of course, whether or not this is larger than unity is model-dependent. We can rephrase this result in two more physically-illuminating ways. One is to consider power-law forms for the warp factor and potential, $$f(\phi) = \lambda \phi^p,\qquad V(\phi) = \mu \phi^q,$$ in which case we have $$R \approx \frac{p^2q^6}{288}\frac{{g_s}^4}{\varepsilon}\left(\frac{M_\mathrm{Pl}}\phi\right)^8,$$ or, in terms of the canonically-normalized field $\phi_\mathrm{c}\equiv\phi/\sqrt{{g_s}}$, $$R \approx \frac{p^2q^6}{288}\frac1\varepsilon\left(\frac{{M_\mathrm{Pl}}}{\phi_\mathrm{c}}\right)^8,$$ The simplest example falls into this category: a pure AdS throat with $f(\phi)=\lambda/\phi^4$ and a standard mass term $V(\phi)=m^2\phi^2$, in which case the prefactor $p^2q^6/288$ becomes order unity, $$R \approx \frac{32}{9}\frac{1}{\varepsilon}\left(\frac{{M_\mathrm{Pl}}}{\phi_\mathrm{c}}\right)^8.$$ We see that when the warp factor and potential are power laws, the swampland criterion is satisfied when the canonically-normalized field takes on sub-Planckian values.[^13]
Another useful approach is to compare the swampland criterion to standard slow-roll inflation with the same potential, for which $$R_\mathrm{SR} = {M_\mathrm{Pl}}^2\left(\frac{V'}{V}\right)^2.$$ In the D-cceleration régime of DBI, $R$ is enhanced by a factor of $$\frac{R}{R_\mathrm{SR}} = \frac{{M_\mathrm{Pl}}^6{g_s}^4}{288}\frac1\varepsilon\left(\frac{f'}{f}\right)^2\left(\frac{V'}{V}\right)^4.$$ We see that the non-canonical kinetic structure, appearing through $f(\phi)$, is crucial for evading the swampland bound, since it is large gradients in $f(\phi)$ that enhance $R$. We emphasize that one should be careful when comparing $V'/V$ to its single-field slow-roll counterpart, as the D-cceleration mechanism allows for inflation with rather larger values of the inflaton mass (or, more generally, with steeper potentials) than does standard single-field slow roll [@Alishahiha:2004eh].
Turning estimates {#sec:turning}
=================
We have seen that $P(\phi,X)$ theories can have quasi-de Sitter phases which evade the swampland conjecture . The analysis requires these theories to be rephrased in a multi-field form via the introduction of an auxiliary field which is given (small) dynamics. In general, multi-field inflation theories are able to evade the swampland bound with what is known as large [*turning*]{} in field space [@Achucarro:2018vey]. In contrast to single-field inflation, in a multi-field context the fields can take a variety of trajectories through field space, rather than being required to follow gradients of the potential; trajectories develop angular momentum in field space, or turning, when the field trajectory is misaligned with the gradient flow of the potential. This severs the direct link between the Hubble and potential slow-roll parameters, allowing the field value to stay approximately constant even on a steep potential, driving acceleration. In this section we calculate the amount of turning on inflationary trajectories in the multi-field picture in $P(\phi,X)$ inflation, and show that, in agreement with , the evasion of the swampland bound is due to large turning in field space in the two-field UV extension.
Our analysis will follow the geometric approach summarized in . Let us package the fields into a field-space vector $\Phi^a=(\phi,\chi)$.[^14] We define the norm of the time derivative of $\Phi^a$ by $$\dot\Phi \equiv \sqrt{\mathcal{G}_{ab}\dot\Phi^a\dot\Phi^b} = \sqrt{P_X\dot\phi^2+\frac{1}{\Lambda^6}\dot X^2}. \label{eq:Phidot}$$ As in the previous sections, we assume we are in the low-energy régime of the EFT and freely use $\chi\approx X$. Next we define a covariant time derivative $D_t$ by $$D_t V^a \equiv \dot V^a + \Gamma^a_{bc}V^b\dot\Phi^c,$$ where $V^a$ is any vector in field space and $\Gamma^a_{bc}$ are the Christoffel symbols associated with the field-space metric, whose only nonzero components are $$\Gamma^\phi_{\phi\phi} = \frac12\frac{P_{\phi X}}{P_X},\qquad \Gamma^\phi_{\chi\phi} = \frac12\frac{P_{XX}}{P_X},\qquad \Gamma^\chi_{\phi\phi} = -\frac12\Lambda^6P_{XX}.$$ Defining the unit tangent vector, $$T^a \equiv \frac{\dot\Phi^a}{\dot\Phi},$$ we can finally define the angular velocity in field space, $$\begin{aligned}
\Omega &\equiv |D_tT| \nonumber\\
&= \sqrt{P_X (D_tT^\phi)^2 + \frac{1}{\Lambda^6}(D_tT^\chi)^2},\end{aligned}$$ where the components of $D_tT^a$ are $$\begin{aligned}
D_tT^\phi &= \left(\frac{\dot\phi}{\dot\Phi}\right)\dot{\bigg.} + \frac{1}{2P_X}\frac{\dot\phi}{\dot\Phi}\left(P_{\phi X}\dot\phi+2P_{XX}\dot X\right),\\
D_tT^\chi &= \left(\frac{\dot X}{\dot\Phi}\right)\dot{\bigg.} - \frac12\Lambda^6P_{XX}\frac{\dot\phi^2}{\dot\Phi}.\end{aligned}$$
We begin by estimating $\dot\Phi$. Per \[eq:Phidot\], this has two terms, one from the $\phi$ sector and one from the $\chi$ sector. Using the expressions and approximations presented in \[sec:swamp\] we find, to leading order in slow roll, $$\frac{\Lambda^{-6}\dot X^2}{P_X\dot\phi^2} = \frac{\mathcal{B}}6.$$ If the swampland criterion is satisfied then we have ${\mathcal{B}}>\mathcal{O}(\varepsilon^{-1/2})\gg1$, in which case we find $$\dot\Phi\approx\frac{\dot X}{\Lambda^3}.$$ We can perform a similar analysis for the turning rate, $\Omega=|D_tT|$. We start with the term $\dot\phi/\dot\Phi$ which appears in $D_tT^\phi$, and which in slow roll we can straightforwardly calculate as, $$\left(\frac{\dot\phi}{\dot\Phi}\right)^2 \approx\frac{6}{{\mathcal{B}}P_X}.$$ Assuming ${\mathcal{B}}$ is approximately constant, we find $$\begin{aligned}
\left(\frac{\dot\phi}{\dot\Phi}\right)\dot{\bigg.} &\approx -2\frac{\dot\phi}{\dot\Phi}\frac{\dot{P}_X}{P_X}\nonumber\\
&\approx-\frac{\beta}{1+c_s^{-2}}H\frac{\dot\phi}{\dot\Phi}.\end{aligned}$$ Recall from \[sec:swamp\] that the same combination of $\beta$ and $c_s^2$ appears in the ratio $\ddot\phi/(H\dot\phi)$, and so under the same assumptions we have been making, we see that $\dot\phi/\dot\Phi$ varies slowly over a Hubble time. Using this in our expression for $D_tT^\phi$ we see that the $(\dot\phi/\dot\Phi)\dot{}$ term is subdominant.[^15] The time-derivative term $D_tT^\chi$ is also negligible: $\dot X/\dot\Phi\approx \Lambda^3$, which is constant by definition. With these approximations we can then calculate the ratio of terms appearing in $\Omega$, $$\begin{aligned}
\frac{\Lambda^{-6}(D_tT^\chi)^2}{P_X (D_tT^\phi)^2} &\approx \frac{6}{\beta}\frac{1-c_s^4}{(1-3c_s^2)^2} \nonumber\\
&= \frac6{\mathcal{B}}\left(\frac{1-c_s^2}{1-3c_s^2}\right)^2.\end{aligned}$$ If the swampland criterion is satisfied then this is much smaller than unity, and so we find $$\Omega^2 \approx P_X(D_t T^\phi)^2.$$ Note that the $\chi$ contribution dominates $\dot\Phi$, while the $\phi$ contribution dominates $\Omega$.
To evaluate the magnitude of this we should compare it to the Hubble rate, which has the same dimensions, $$\begin{aligned}
\frac{\Omega^2}{H^2} &\approx \frac32\beta\frac{(3c_s^2-1)^2}{1-c_s^4}\nonumber\\
&= \frac32{\mathcal{B}}\left(\frac{1-3c_s^2}{1-c_s^2}\right)^2.\end{aligned}$$ If $R>\mathcal{O}(1)$ then $c_s^2\ll1$ and $\beta\gg1$, so the turning is large, $$\frac{\Omega^2}{H^2}\sim\mathcal{O}(\beta)\gg1.$$ We conclude that, from the perspective of the two-field picture, DBI inflation has large turning when the de Sitter swampland criterion is satisfied.
These considerations show that $P(\phi,X)$ theories fit into the analysis of , which studied, in greater generality, multi-field inflation and the swampland conjectures in the presence of large $\Omega$, taking into account both the de Sitter conjecture and the distance conjecture . They find that satisfying both of these swampland conjectures simultaneously requires $$\Omega\gtrsim180H.$$ In the context of $P(\phi,X)$ inflation this imposes $$\sqrt\beta\gtrsim150.$$
Conclusions {#sec:conc}
===========
We have applied the de Sitter swampland conjecture of to scalar field theories with non-canonical kinetic structures of the $P(\phi,X)$ type. This conjecture is phrased in terms of non-linear sigma models whose kinetic terms are quadratic in derivatives. We fit $P(\phi,X)$ theories into this framework by introducing an auxiliary field and giving it a small kinetic term, so that solutions to the $P(\phi,X)$ theory remain approximate solutions to the two-field theory. This can be seen as embedding the $P(\phi,X)$ theory in a multi-field UV extension.
The de Sitter conjecture translates into a bound on the function $P(\phi,X)$ and its derivatives. We have analyzed this condition on quasi-de Sitter inflationary solutions in conjunction with the Hubble slow-roll approximation,[^16] and have applied them to a prominent string-inspired inflationary model of the $P(\phi,X)$ form: DBI inflation. We find that DBI inflation can satisfy the swampland bound given potentials which would violate the bound in the presence of a canonical kinetic structure. We have shown that when these models satisfy the swampland conjecture, they do so because of *turning* in field space, in agreement with other examples of multi-field inflation which do not violate the swampland bound [@Achucarro:2018vey].
While the techniques we have used in this paper have proven extremely useful for extending swampland bounds to one set of theories with nontrivial kinetic terms, they are not applicable, at least in a simple way, to all such theories. In particular, it would be very interesting to study whether there are methods to extend the same reasoning to theories such as the galileon or Horndeski models, in which higher-derivative terms enter explicitly into the Lagrangians. Thus far we have not uncovered such techniques.
We thank Mark Hertzberg and particularly McCullen Sandora for useful discussions. A.R.S. is supported by DOE HEP grants DOE DE-FG02-04ER41338 and FG02-06ER41449 and by the McWilliams Center for Cosmology, Carnegie Mellon University. The work of M.T. is supported in part by US Department of Energy (HEP) Award DE-SC0013528, and by the Simons Foundation, grant number 658904.
[^1]: See, e.g., .
[^2]: Of course, this is a very general question, and caveats exist, e.g., the UV completion one has in mind may be required to satisfy certain properties, such as being Lorentz-invariant, or arising from a string theory.
[^3]: This requires $P_{\chi\chi}\neq0$, which is not a particularly restrictive condition, but one should check that it is not violated along the solution under consideration.
[^4]: See for further discussion of this idea and of other approaches to applying the swampland conjectures to $P(\phi,X)$ theories. Also see for previous discussions of auxiliary field techniques in the swampland context. For other approaches to applying the swampland bounds to $P(\phi,X)$ theories and their generalizations, see, e.g., . See for an alternative auxiliary-field forumulation of shift-symmetric $P(\phi,X)=P(X)$ theories.
[^5]: The requirement is that some characteristic energy scale be less than order $\Lambda$; strictly speaking this corresponds to some small dimensionless parameter.
[^6]: Technically we assume $\Lambda$ to take the minimum value of the function on the right hand side of throughout the inflationary trajectory, although this distinction will not be important.
[^7]: Its magnitude is model-dependent. For instance, $\varepsilon\sim n_s-1$ in standard slow-roll inflation, with $n_s$ the tilt of the scalar power spectrum, while in DBI inflation, a prominent example of inflation driven by non-trivial kinetic terms of a $P(\phi,X)$ form, we have instead $\varepsilon\sim\sqrt{n_s-1}$ [@Silverstein:2003hf; @Alishahiha:2004eh].
[^8]: Note also that, even if this were not the case, the fact that $P_{\chi\chi}$ differs from $P_{XX}$ by some higher-order terms in the EFT indicates that the analysis would probably be valid even if the trajectory did encounter a point with $P_{XX}=0$.
[^9]: In DBI inflation, which we discuss in the next section, this quantity is $\mathcal{O}(\varepsilon)$.
[^10]: To see this, define $\alpha\equiv\beta(1+c_s^{-2})^{-1}$. We then have $\dot X=\dot\phi\ddot\phi=2\alpha HX$, so $\ddot X\ll H\dot X$ as long as $\alpha\ll1$ and $\dot\alpha\ll H\alpha$.
[^11]: Since we are keeping $\beta$ and $c_s$ general, we may be keeping terms in \[eq:swampland-PX\] which are similar in size to slow-roll terms that we have dropped. The point is to remain agnostic about the size of these model parameters while making use of Hubble slow roll, which is required for a quasi-de Sitter phase.
[^12]: For earlier work applying the swampland conjectures to DBI inflation, see .
[^13]: The suppression by $1/\varepsilon$ does not significantly affect this conclusion: for $\varepsilon$ in the range of $10^{-1}$–$10^{-2}$, $\varepsilon^{1/8}$ is close to unity. Similarly, it is difficult for the prefactor $p^2q^6/288$ to affect this bottom line; note that $288^{1/8}\approx2.03$. On the other hand, if $\phi_c$ is even double the Planck mass, then $R$ is suppressed by a factor of $2^8$.
[^14]: Technically the units are mixed, since $\chi$ has units of mass$^4$, but our calculations will be self-consistent, since the field-space metric has correspondingly mixed units.
[^15]: In particular, it is suppressed by a factor of $c_s^2$, which we have seen is necessarily small if $R>1$ and $\ddot\phi\ll H\dot\phi$.
[^16]: Note that the bound should hold over all field space, not just on the inflationary trajectory.
|
---
abstract: 'This paper studies eigenvalues of the clamped plate problem on a bounded domain in an $n$-dimensional Euclidean space. We give an estimate for the gap between $\sqrt {\Gamma_{k+1}-\Gamma_{1}}$ and $\sqrt {\Gamma_{k}-\Gamma_{1}}$, for any positive integer $k$. According to the asymptotic formula of Agmon and Pleijel, we know, the gap between $\sqrt {\Gamma_{k+1}-\Gamma_{1}}$ and $\sqrt {\Gamma_{k}-\Gamma_{1}}$ is bounded by a term with a lower order $k^{\frac1n}$ in the sense of the asymptotic formula of Agmon and Peijel, where $\Gamma_j$ denotes the $j^{^{\text{th}}}$ eigenvalue of the clamped plate problem.'
address:
- |
Daguang Chen\
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China, dgchen@math.tsinghua.edu.cn
- |
Qing-Ming Cheng\
Department of Applied Mathematics, Faculty of Sciences, Fukuoka University, 814-0180, Fukuoka, Japan, cheng@fukuoka-u.ac.jp
- |
Guoxin Wei\
School of Mathematical Sciences, South China Normal University, 510631, Guangzhou, P. R. China, weiguoxin@tsinghua.org.cn
author:
- 'Daguang Chen, Qing-Ming Cheng and Guoxin Wei'
title: A gap for eigenvalues of a clamped plate problem
---
introduction
============
It is well-known that study on eigenvalues of the eigenvalue problem of elliptic operators is a very important subject in geometry and analysis.
Let $\Omega$ be a bounded domain with piecewise smooth boundary in an $n$-dimensional complete Riemannian manifold $M$. The following is called [*the Dirichlet eigenvalue problem of Laplacian*]{}: $$\begin{cases}
\Delta u=-\lambda u & \text{in $\Omega$}, \\
u=0 & \text{on $\partial \Omega$},
\end{cases}$$\
where $\Delta$ is the Laplacian on $M$. Many mathematicians study universal estimates of eigenvalues of the Dirichlet eigenvalue problem of Laplacian. As main developments for study on universal estimates of eigenvalues, Payne, Pólya and Weinberger [@PPW], Hile and Protter [@HP], Yang [@Y] makes very important contributions for bounded domains in Euclidean spaces (see Ashbaugh [@As1; @As2; @As3]). For domains in sphere, Cheng and Yang [@CY1] obtains optimal universal estimates on eigenvalues. For bounded domains in complete Riemannian manifolds, universal estimates on eigenvalues have been obtained by in Cheng and Yang [@CY2], Chen and Cheng [@CC], Chen, Zheng and Yang [@CZY] and El Soufi, Harrell and Ilias [@EHI], Cheng [@C1] and so on. By making use of the universal estimates on eigenvalues and the recursive inequality of Cheng and Yang [@CY4], Cheng and Yang [@CY5] obtain sharp lower bounds and upper bounds for the $k^{\text{th}}$ eigenvalues of the Dirichlet eigenvalue problem of Laplacian in the sense of order of $k$. For bounded domains in the Euclidean space, by making use of Fourier transform, Li and Yau [@LY] gives an optimal lower bound for the average of the first $k$ eigenvalues of the Dirichlet eigenvalue problem of Laplacian. Recently, in [@As3], Ashbaugh gives a very nice survey for estimates on eigenvalues of the Dirichlet eigenvalue problem of Laplacian for bounded domains in Euclidean space. For bounded domains in complete Riemannian manifolds, see the very nice book of Urakawa [@U].
In this paper, we consider an eigenvalue problem of the biharmonic operator $\Delta^2$ on a bounded domain with piecewise smooth boundary in an $n$-dimensional complete Riemannian manifold $M$, which is also called [*the clamped plate problem*]{}: $$\label{ccp}
\begin{cases}
\Delta^2 u=\Gamma u & \text{in $\Omega$} \\
u=\displaystyle{ \frac{\partial u}{\partial \nu}}=0 & \text{on $\partial \Omega$},
\end{cases}$$ where $\Delta^2$ denotes the biharmonic operator on $M$, and $\nu$ is the outward unit normal of $\partial \Omega$.
When $\Omega$ is a bounded domain in $\mathbf R^n$, Agmon and Pleijel give the following asymptotic formula of eigenvalues of the clamped plate problem (1.2): $$\Gamma_k\sim
\dfrac{16\pi^4}{\big(\omega_n\text{vol}(\Omega)\big)^{\frac{4}{n}}}k^{\frac{4}{n}},\
\ \ k\rightarrow\infty.$$ This implies that $$\frac{1}{k}\sum_{j=1}^k\Gamma_j
\sim\frac{n}{n+4}\dfrac{16\pi^4}{\big(\omega_n\text{vol}(\Omega)\big)^{\frac{4}{n}}}k^{\frac{4}{n}},
\ \ k\rightarrow\infty,$$ where $\Gamma_j$ denotes the $j^{\text{\rm th}}$ eigenvalue of the clamped plate problem \[ccp\], $\text{vol}(\Omega)$ and $\omega_n$ denote volumes of $\Omega$ and the unit ball in $\mathbf R^n$, respectively. Furthermore, by making use of the Fourier transform and a lemma due to Hörmander, Levine and Protter [@LP] proves that eigenvalues of the clamped plate problem \[ccp\] satisfy $$\dfrac{1}{k}\sum_{j=1}^k\Gamma_j
\geq\frac{n}{n+4}\dfrac{16\pi^4}{\big(\omega_n\text{vol}(\Omega)\big)^{\frac{4}{n}}}k^{\frac{4}{n}}.$$ The above formula shows that the coefficient of $k^{\frac{4}{n}}$ is the best possible constant. and the order of $k$ is optimal according to the asymptotic formula of Agmon and Peijel. Cheng and Wei [@CW1; @CW2] and Cheng, Qi and Wei [@CQW] generalize the result of Levine and Protter by adding the lower terms.
On the other hand, it is a very difficult problem to obtain a sharp estimate for the upper bound of eigenvalues with optimal order of $k$ of the clamped plate problem (\[ccp\]). For estimates for upper bounds of eigenvalues and estimates of two consecutive eigenvalues of the clamped plate problem, Payne, Pólya and Weinberger [@PPW] proves $$\Gamma_{k+1} - \Gamma_{k} \leq \frac{8(n+2)}{n^{2} k} \sum_{i=1}^{k} \Gamma_{i}.$$ Chen and Qian [@CQ] and Hook [@H], independently, extend the above inequality to $$\frac{n^{2} k^{2}}{8(n+2)} \leq
\sum_{i=1}^{k} \frac{\Gamma_{i}^{\frac{1}{2}}}{\Gamma_{k+1}
- \Gamma_{i}} \sum_{i=1}^{k} \Gamma_{i}^{\frac{1}{2}}.$$ Cheng and Yang [@CY2] and Wang and Xia [@WX] prove $$\sum_{i=1}^k (\Gamma_{k+1}-\Gamma_{i})^2 \leq \displaystyle{\frac{8(n+2)}{n^2}}\sum_{i=1}^k(\Gamma_{k+1}-\Gamma_{i})\Gamma_{i}$$ In the open problem section ( of the 6th International Chinese Congress of Mathematicians, July 9-14, 2013, Taiwan National University), the second author proposes the following problem:
[**Conjecture 1.1**]{}. Eigenvalues of the clamped plate problem (\[ccp\]) for a bounded domain in $\mathbf {R}^n$ satisfies $$\sum_{i=1}^k (\Gamma_{k+1}-\Gamma_{i})^2 \leq \displaystyle{\frac{8}{n}}\sum_{i=1}^k(\Gamma_{k+1}-\Gamma_{i})\Gamma_{i}.$$
In fact, if one may prove the conjecture 1.1, by making use of the recursive formula of Cheng and Yang [@CY4], one may obtain the sharp estimates on the upper bound of the $k^{\text{th}}$ eigenvalue, in the sense of the order of $k$, of the clamped plate problem.
In this paper, we study the gap of two consecutive eigenvalues of the clamped plate problem. We obtain the following:
Let $\Omega$ be a bounded domain in the Euclidean space $ \mathbf {R}^n $. Then, for any integer $k\geq 0$, we have $$(\sqrt {\Gamma_{k+1}-\Gamma_{1}} -\sqrt {\Gamma_{k}-\Gamma_{1}} )^2
\leq \dfrac{16\sqrt{\Gamma_1}}{n}\bigl\{(\Gamma_{k+1}-\Gamma_1)(\Gamma_{k}-\Gamma_1)\bigl\}^{\frac14}+C,$$ where $$C=\max\biggl\{\dfrac{8\int_{\Omega}|\nabla\Delta u_1|^2dv}{(n+2)\|\nabla u_1\|^2 }, \
\dfrac{4(n+12)\Gamma_1+16\int_{\Omega}\sum_{m=1}^n(\dfrac{\partial^2 u_1}{\partial x_m^2})^2dv}{n}\biggl\}.$$ is constant only depending on the dimension $n$, the first eigenvalue $\Gamma_1$ and the normalized first eigenfunction $u_1$.
According to the asymptotic formula of Agmon and Pleijel, we have $$\lim_{k\to \infty}\dfrac{\Gamma_k}{k^{\frac{4}{n}}}=
\dfrac{16\pi^4}{\big(\omega_n\text{\rm vol}(\Omega)\big)^{\frac{4}{n}}}.$$ From our theorem, we know, the gap between $\sqrt {\Gamma_{k+1}-\Gamma_{1}}$ and $\sqrt {\Gamma_{k}-\Gamma_{1}}$ is bounded by a term with a lower order $k^{\frac1n}$ in the sense of the asymptotic formula of Agmon and Peijel.
Since $$\Gamma_{k+1}-\Gamma_{k}=
(\sqrt {\Gamma_{k+1}-\Gamma_{1}}-\sqrt {\Gamma_{k}-\Gamma_{1}})
(\sqrt {\Gamma_{k+1}-\Gamma_{1}}+\sqrt {\Gamma_{k}-\Gamma_{1}}),$$ according to the asymptotic formula of Agmon and Pleijel, we know that the gap between $\Gamma_{k+1}$ and $\Gamma_{k}$ is bounded by a term with a lower order $k^{\frac3n}$.
A general result
================
Let $\Omega$ be a bounded domain with piecewise smooth boundary in an $n$-dimensional complete Riemannian manifold $M$. Let $u_i$ be an eigenfunction corresponding to the eigenvalue $\Gamma_{i}$ such that $$\begin{cases}
\Delta^2{u_i}=\Gamma_i u_i & \text{in $\Omega$} \\
u_i=\displaystyle{\frac{\partial u_i}{\partial\nu}}=0 & \text{on $\partial \Omega$} \\
\displaystyle{\int_\Omega u_i u_jdv}=\delta_{ij}, \ i, j=1, 2, \cdots,
\end{cases}$$ where eigenvalues are accounted according to their multiplicities. Thus, we know that $\{u_j\}_{j=1}^{\infty}$ forms an orthonormal base of $L^2(\Omega)$-function space. For any smooth function $g$, we can write $$gu_1=\sum_{j=1}^{\infty}r_{j}u_j, \quad \|gu_1\|^2=\int_{\Omega}(gu_1)^2dv=\sum_{j=1}^{\infty}r_{j}^2,$$ where $r_{j}=\displaystyle{\int_\Omega g u_1 u_jdv}$, for $j=1, 2, \cdots$. For any positive integer $k$, we define $$\varphi:=gu_1-\sum_{j=1}^k r_{j} u_j.$$ By a simple calculation, we obtain $$\displaystyle{\int_\Omega u_j\varphi dv=0}, \ \ j=1,\cdots,k.$$ Hence $$\|\varphi\|^2=\sum_{j=k+1}^{\infty}r_{j}^2.$$ Defining $$\label{p}
p=\Delta^2 g \cdot u_1
+2\nabla(\Delta g)\cdot \nabla u_1+2\Delta g\Delta u_1
+2\Delta(\nabla g \cdot \nabla u_1)
+2\nabla g \cdot \nabla(\Delta u_1),$$ we have $$p=\sum_{j=1}^{\infty}s_{j}u_j, \quad \|p\|^2=\sum_{j=1}^{\infty}s_{j}^2,$$ where $$s_{j}=\displaystyle{\int_{\Omega}}p u_jdv.$$ Since $$\begin{aligned}
& 2\displaystyle{\int_{\Omega}}(\Delta u_j \nabla g \cdot \nabla u_1 - \Delta u_1 \nabla g \cdot \nabla u_j)dv \\
& =(\Gamma_j - \Gamma_1)r_{j} - \displaystyle{\int_{\Omega}}u_1\Delta u_j \Delta gdv
+\displaystyle{\int_{\Omega}}u_j\Delta u_1\Delta gdv,\end{aligned}$$\
we can infer $$s_{j} =( \Gamma_j - \Gamma_1 ) r_{j}.$$\
Thus, we get $$\|p\|^2=\sum_{j=1}^{\infty}(\Gamma_j-\Gamma_1)^2r_{j}^2.$$ $$\int_{\Omega}gu_1pdv=\int_{\Omega}gu_1\sum_{j=1}^{\infty}s_{j}u_jdv=\sum_{j=1}^{\infty}s_{j}r_{j}
=\sum_{j=1}^{\infty}(\Gamma_{j}-\Gamma_1)r_{j}^2.$$ From the definition of $\varphi$, we have $$\int_{\Omega}\varphi p dv=\int_{\Omega}(gu_1-\sum_{j=1}^k r_{j}u_j)p dv
=\sum_{j=1}^{\infty}(\Gamma_{j}-\Gamma_1)r_{j}^2-\sum_{j=1}^{k}(\Gamma_{j}-\Gamma_1)r_{j}^2.$$ Hence, we obtain $$\int_{\Omega}\varphi pdv=
\sum_{j=k+1}^{\infty}(\Gamma_{j}-\Gamma_1)r_{j}^2.$$
The following algebraic lemma plays an important role in this paper, which may be found in Chen-Yang-Zheng [@CZY], essentially. For reader’s convenient, we give a detailed proof of it in the Appendix.
\[lemma1\] Let $\{\mu_j\}_{j=k+1}^{\infty}$ be a sequence satisfying $$0\leq \mu_{k+1}\leq \mu_{k+2}\leq \cdots \to \infty.$$ If a sequence $\{a_j\}_{j=k+1}^{\infty}$ satisfies $\sum_{j=k+1}^{\infty}\mu_j^2a_j^2=A<\infty $ and $\sum_{j=k+1}^{\infty}a_j^2=B<\infty $, then we have $$\sum_{j=k+1}^{\infty}\mu_ja_j^2\leq \dfrac{A+\mu_{k+1}\mu_{k+2}B}{\mu_{k+1}+\mu_{k+2}}.$$
By applying the lemma 2.1 with $\mu_j=\Gamma_j-\Gamma_1$ and $a_j=r_{j}$, we obtain $$\begin{aligned}
&\bigl\{(\Gamma_{k+1}-\Gamma_1)+(\Gamma_{k+2}-\Gamma_1)\bigl\}\int_{\Omega}\varphi pdv\\
&\leq \bigl(\|p\|^2-\sum_{j=1}^{k}(\Gamma_{j}-\Gamma_1)^2r_{j}^2\bigl)
+(\Gamma_{k+1}-\Gamma_1)(\Gamma_{k+2}-\Gamma_1)\|\varphi\|^2,
\end{aligned}$$ namely, $$\begin{aligned}
&\bigl\{(\Gamma_{k+1}-\Gamma_1)+(\Gamma_{k+2}-\Gamma_1)\bigl\}(\int_{\Omega}gu_1pdv-\sum_{j=1}^{k}(\Gamma_{j}-\Gamma_1)r_{j}^2)\\
&\leq \bigl(\|p\|^2-\sum_{j=1}^{k}(\Gamma_{j}-\Gamma_1)^2r_{j}^2\bigl)
+(\Gamma_{k+1}-\Gamma_1)(\Gamma_{k+2}-\Gamma_1)(\|gu_1\|^2-\sum_{j=1}^{k}r_{j}^2).
\end{aligned}$$ Since $$\begin{aligned}
&\bigl\{(\Gamma_{k+1}-\Gamma_1)+(\Gamma_{k+2}-\Gamma_1)\bigl\}\sum_{j=1}^{k}(\Gamma_{j}-\Gamma_1)r_{j}^2\\
&\leq \sum_{j=1}^{k}(\Gamma_{j}-\Gamma_1)^2r_{j}^2
+(\Gamma_{k+1}-\Gamma_1)(\Gamma_{k+2}-\Gamma_1)\sum_{j=1}^{k}r_{j}^2,
\end{aligned}$$ we have $$\begin{aligned}
&\bigl\{(\Gamma_{k+1}-\Gamma_1)+(\Gamma_{k+2}-\Gamma_1)\bigl\}\int_{\Omega}gu_1pdv
\leq \|p\|^2
+(\Gamma_{k+1}-\Gamma_1)(\Gamma_{k+2}-\Gamma_1)\|gu_1\|^2.
\end{aligned}$$ Thus, we have proved the following:
Let $\Omega$ be a bounded domain in an $n$-dimensional complete Riemannian manifold $M$. Assume that $\Gamma_{i}$ is the $i^{\text{th}}$ eigenvalue of the clamped plate problem [(\[ccp\])]{}. For any smooth function $g$, we have, for any integer $k$, $$\begin{aligned}
&\bigl\{(\Gamma_{k+2}-\Gamma_1)+(\Gamma_{k+1}-\Gamma_1)\bigl\}\int_{\Omega}gu_1pdv
\leq \|p\|^2
+(\Gamma_{k+2}-\Gamma_1)(\Gamma_{k+1}-\Gamma_1)\|gu_1\|^2,
\end{aligned}$$ where $p$ is defined by the formula [($\ref{p}$)]{} and $u_1$ is the normalized first eigenfunction corresponding to the first eigenvalue $\Gamma_1$.
$$\int_{\Omega}gu_1pdv=\displaystyle{\int_{\Omega}}\biggl\{(\Delta g)^2 u_1^2
+4(\nabla g \cdot \nabla u_1)^2 -2|\nabla g |^2 u_1 \Delta u_1
+4u_1\Delta g \nabla g\cdot \nabla u_1\biggl\}dv.$$
From Stokes’ theorem, we infer $$2\displaystyle{\int_{\Omega}}g u_1\nabla(\Delta g) \cdot\nabla u_1dv
=\displaystyle{\int_{\Omega}}\biggl\{2u_1\Delta g\nabla u_1\cdot \nabla g
+u_1^2(\Delta g)^2 -g u_1^2\Delta^2 g\biggl\}dv,$$ $$2\displaystyle{\int_{\Omega}} g u_1\Delta(\nabla g \cdot \nabla u_1)dv
=\displaystyle{\int_{\Omega}}\biggl\{2u_1\Delta g \nabla g \cdot \nabla u_1
+4(\nabla g \cdot \nabla u_1)^2
+2g \Delta u_1\nabla g \cdot \nabla u_1\biggl\}dv,$$ $$2\displaystyle{\int_{\Omega}}gu_1\nabla g \cdot \nabla (\Delta u_1)dv
=-2\displaystyle{\int_{\Omega}}\biggl(|\nabla g |^2 u_1 \Delta u_1
+g \Delta u_1\nabla g \cdot \nabla u_1
+ g \Delta g u_1 \Delta u_1\biggl)dv.$$\
From the definition of $p$, we obtain $$\int_{\Omega}gu_1pdv=\displaystyle{\int_{\Omega}}\biggl\{(\Delta g)^2 u_1^2
+4(\nabla g \cdot \nabla u_1)^2 -2|\nabla g |^2 u_1 \Delta u_1
+4u_1 \Delta g \nabla g\cdot \nabla u_1\biggl\}dv.$$
For any smooth function $f$ in $M$ and constant $a$, we consider $g_1=\cos (af)$. We have $$\nabla g_1=-a\sin (af)\nabla f, \quad \Delta g_1 =-a^2\cos (af)|\nabla f|^2-a\sin (af) \Delta f$$
$$\begin{aligned}
\nabla \Delta g_1 &=a^3\sin (af)|\nabla f|^2\nabla f-a^2\cos (af)\nabla(|\nabla f|^2)\\
&-a^2\cos (af) \Delta f\nabla f
-a\sin (af)\nabla(\Delta f)
\end{aligned}$$
$$\begin{aligned}
\Delta^2 g_1 &=a^4\cos (af)|\nabla f|^4+2a^3\sin (af) \nabla(|\nabla f|^2)\cdot\nabla f+2a^3\sin (af) |\nabla f|^2\Delta f\\
& - a^2\cos (af)\Delta (|\nabla f|^2) -2a^2\cos (af) \nabla(\Delta f)\cdot\nabla f\\
&-a^2\cos (af) (\Delta f)^2-a\sin (af) \Delta^2f.
\end{aligned}$$
In the same way, for $g_2=\sin (af)$, we have $$\nabla g_2=a\cos (af)\nabla f, \quad \Delta g_2 =-a^2\sin (af)|\nabla f|^2+a\cos (af) \Delta f$$
$$\begin{aligned}
\nabla \Delta g_2 &=-a^3\cos (af)|\nabla f|^2\nabla f-a^2\sin (af)\nabla(|\nabla f|^2)\\
&-a^2\sin (af) \Delta f\nabla f
+a\cos (af)\nabla(\Delta f)
\end{aligned}$$
$$\begin{aligned}
\Delta^2 g_2 &=a^4\sin (af)|\nabla f|^4-2a^3\cos (af) \nabla(|\nabla f|^2)\cdot\nabla f\\
&-2a^3\cos (af) |\nabla f|^2\Delta f- a^2\sin (af)\Delta (|\nabla f|^2) \\
&-2a^2\sin (af) \nabla(\Delta f)\cdot\nabla f-a^2\sin (af) (\Delta f)^2+a\cos (af) \Delta^2f.
\end{aligned}$$
Thus, we obtain the following:
If the function $f$ satisfies $|\nabla f|^2=1$ and $\Delta f =b=$constant, we have $$\nabla g_1=-a\sin (af)\nabla f, \quad \Delta g_1 =-a^2\cos (af)-ab\sin (af),$$ $$\begin{aligned}
\nabla \Delta g_1 &=a^3\sin (af)\nabla f-a^2b\cos (af) \nabla f,
\end{aligned}$$ $$\begin{aligned}
\Delta^2 g_1 &=a^4\cos (af)+2a^3b\sin(af)-a^2b^2\cos (af),
\end{aligned}$$ $$\nabla g_2=a\cos (af)\nabla f, \quad \Delta g_2 =-a^2\sin (af)+ab\cos (af),$$
$$\begin{aligned}
\nabla \Delta g_2 &=-a^3\cos (af)\nabla f-a^2b\sin (af) \nabla f,
\end{aligned}$$
$$\begin{aligned}
\Delta^2 g_2 &=a^4\sin (af) -2a^3b\cos (af)-a^2b^2\sin (af).
\end{aligned}$$
By defining $$p_1=\Delta^2 g_1 \cdot u_1
+2\nabla(\Delta g_1)\cdot \nabla u_1+2\Delta g_1\Delta u_1
+2\Delta(\nabla g_1 \cdot \nabla u_1)
+2\nabla g_1 \cdot \nabla(\Delta u_1),$$ and $$p_2=\Delta^2 g_2 \cdot u_1
+2\nabla(\Delta g_2)\cdot \nabla u_1+2\Delta g_2\Delta u_1
+2\Delta(\nabla g_2 \cdot \nabla u_1)
+2\nabla g_2 \cdot \nabla(\Delta u_1),$$ we have
If the function $f$ satisfies $|\nabla f|^2=1$ and $\Delta f =b=$constant, we have $$\begin{aligned}
&|p_1|^2+|p_2|^2\\
&=\biggl((a^4-a^2b^2\bigl)u_1-4a^2b\nabla f\cdot \nabla u_1-2a^2\Delta u_1
-4a^2\nabla f\cdot \nabla(\nabla f \cdot \nabla u_1)\biggl)^2\\
&+\biggl(2a^3bu_1+4a^3\nabla f\cdot \nabla u_1-2ab\Delta u_1
-2a\Delta(\nabla f \cdot \nabla u_1)
-2a\nabla f \cdot \nabla(\Delta u_1)\biggl )^2.
\end{aligned}$$
From the above lemma 2.3, we have $$\begin{aligned}
p_1&=\bigl(a^4\cos (af)+2a^3b\sin(af)-a^2b^2\cos (af)\bigl)u_1\\
&+2\bigl(a^3\sin (af)-a^2b\cos (af) \bigl)\nabla f\cdot \nabla u_1-2(a^2\cos (af)+ab\sin (af))\Delta u_1\\
&-2a\Delta(\sin (af)\nabla f \cdot \nabla u_1)
-2a\sin (af)\nabla f \cdot \nabla(\Delta u_1)
\end{aligned}$$ and $$\begin{aligned}
&\Delta(\sin (af)\nabla f \cdot \nabla u_1)=\sin (af)\Delta(\nabla f \cdot \nabla u_1)\\
&+2a\cos (af)\nabla f\cdot \nabla(\nabla f \cdot \nabla u_1))
-\biggl(a^2\sin (af)-ab\cos (af)\biggl)\nabla f \cdot \nabla u_1,
\end{aligned}$$ $$\begin{aligned}
p_2&=\bigl(a^4\sin (af) -2a^3b\cos (af)-a^2b^2\sin (af)\bigl)u_1\\
&-2\bigl(a^3\cos (af)+a^2b\sin (af) \bigl)\nabla f\cdot \nabla u_1-2(a^2\sin (af)-ab\cos (af))\Delta u_1\\
&+2a\Delta(\cos (af)\nabla f \cdot \nabla u_1)
+2a\cos (af)\nabla f \cdot \nabla(\Delta u_1)
\end{aligned}$$ and $$\begin{aligned}
&\Delta(\cos (af)\nabla f \cdot \nabla u_1)=\cos (af)\Delta(\nabla f \cdot \nabla u_1)\\
&-2a\sin (af)\nabla f\cdot \nabla(\nabla f \cdot \nabla u_1)
-\biggl(a^2\cos (af)+ab\sin(af)\biggl)\nabla f \cdot \nabla u_1.
\end{aligned}$$ Hence, we infer $$\begin{aligned}
&p_1=\biggl((a^4-a^2b^2)u_1-4a^2b \nabla f\cdot \nabla u_1-2a^2\Delta u_1
-4a^2\nabla f\cdot \nabla(\nabla f \cdot \nabla u_1)\biggl)\cos (af)\\
&+\biggl(2a^3bu_1+4a^3\nabla f\cdot \nabla u_1-2ab\Delta u_1
-2a\Delta(\nabla f \cdot \nabla u_1)
-2a\nabla f \cdot \nabla(\Delta u_1)\biggl)\sin (af),
\end{aligned}$$ $$\begin{aligned}
&p_2
=\biggl((a^4-a^2b^2\bigl)u_1-4a^2b\nabla f\cdot \nabla u_1-2a^2\Delta u_1
-4a^2\nabla f\cdot \nabla(\nabla f \cdot \nabla u_1)\biggl)\sin (af)\\
&-\biggl(2a^3bu_1+4a^3\nabla f\cdot \nabla u_1-2ab\Delta u_1
-2a\Delta(\nabla f \cdot \nabla u_1)
-2a\nabla f \cdot \nabla(\Delta u_1)\biggl )\cos (af).
\end{aligned}$$ From the above two equalities, we obtain $$\begin{aligned}
&|p_1|^2+|p_2|^2\\
&=\biggl((a^4-a^2b^2\bigl)u_1-4a^2b\nabla f\cdot \nabla u_1-2a^2\Delta u_1
-4a^2\nabla f\cdot \nabla(\nabla f \cdot \nabla u_1)\biggl)^2\\
&+\biggl(2a^3bu_1+4a^3\nabla f\cdot \nabla u_1-2ab\Delta u_1
-2a\Delta(\nabla f \cdot \nabla u_1)
-2a\nabla f \cdot \nabla(\Delta u_1)\biggl )^2.
\end{aligned}$$
If the function $f$ satisfies $|\nabla f|^2=1$ and $\Delta f =b=$constant, we have $$\begin{aligned}
\int_{\Omega}g_1u_1p_1dv+\int_{\Omega}g_2u_1p_2dv&=\displaystyle{\int_{\Omega}}\biggl\{(a^4-a^2b^2) u_1^2
+4a^2(\nabla f \cdot \nabla u_1)^2 -2a^2u_1 \Delta u_1\biggl\}dv.
\end{aligned}$$
Since $$\begin{aligned}
\int_{\Omega}g_1u_1p_1dv&=\displaystyle{\int_{\Omega}}\biggl\{(a^2\cos (af)+ab\sin (af))^2 u_1^2 \\
&+4a^2(\sin (af))^2(\nabla f \cdot \nabla u_1)^2 -2a^2(\sin (af))^2 u_1 \Delta u_1\\
&+4a\sin (af) (a^2\cos (af)+ab\sin (af))u_1 \nabla f\cdot \nabla u_1\biggl\}dv
\end{aligned}$$ and $$\begin{aligned}
\int_{\Omega}g_2u_1p_2dv&=\displaystyle{\int_{\Omega}}\biggl\{(a^2\sin (af)-ab\cos (af))^2 u_1^2 \\
&+4a^2(\cos (af))^2(\nabla f \cdot \nabla u_1)^2 -2a^2(\cos (af))^2 u_1 \Delta u_1\\
&-4a\cos (af)(a^2\sin (af)-ab\cos (af))u_1\nabla f\cdot \nabla u_1\biggl\}dv,
\end{aligned}$$ we infer $$\begin{aligned}
&\int_{\Omega}g_1u_1p_1dv+\int_{\Omega}g_2u_1p_2dv\\
&=\displaystyle{\int_{\Omega}}\biggl\{(a^4+a^2b^2) u_1^2
+4a^2(\nabla f \cdot \nabla u_1)^2
-2a^2u_1 \Delta u_1+4a^2bu_1 \nabla f\cdot \nabla u_1\biggl\}dv.
\end{aligned}$$ According to Stokes formula, we know $$\int_{\Omega}2u_1\nabla f\cdot \nabla u_1dv=-\int_{\Omega}bu^2_1dv.$$ Hence, we get $$\begin{aligned}
\int_{\Omega}g_1u_1p_1dv+\int_{\Omega}g_2u_1p_2dv&=\displaystyle{\int_{\Omega}}\biggl\{(a^4-a^2b^2) u_1^2
+4a^2(\nabla f \cdot \nabla u_1)^2 -2a^2u_1 \Delta u_1\biggl\}dv.
\end{aligned}$$
The proof of the theorem 1.1
============================
[*Proof of Theorem*]{} 1.1. Since $\Omega$ is a bounded domain in the Euclidean space $\mathbf {R}^n$. Let $(x_1, x_2, \cdots, x_n)$ be the standard coordinate. By taking $f=x_m$, for $m=1, 2, \cdots, n$, we know $$|\nabla f|^2=1, \quad \Delta f=0.$$ Thus, from the propositions 2.1, we obtain, for $m=1, 2, \cdots, n$,
$$\begin{aligned}
|p_1|^2+|p_2|^2
&=\biggl(a^4u_1-2a^2\Delta u_1
-4a^2\dfrac{\partial^2 u_1}{\partial x_m^2}
\biggl)^2\\
&+\biggl(4a^3\dfrac{\partial u_1}{\partial x_m}-2a\Delta(\dfrac{\partial u_1}{\partial x_m})
-2a\dfrac{\partial(\Delta u_1)}{\partial x_m}\biggl )^2\\
&=a^4\biggl(a^2u_1-2\Delta u_1
-4\dfrac{\partial^2 u_1}{\partial x_m^2}
\biggl)^2+16a^2\biggl(a^2\dfrac{\partial u_1}{\partial x_m}-\Delta(\dfrac{\partial u_1}{\partial x_m})\biggl )^2.
\end{aligned}$$
Hence, $$\begin{aligned}
\int_{\Omega}|p_1|^2dv+\int_{\Omega}|p_2|^2dv
&=\int_{\Omega}a^4\biggl(a^2u_1-2\Delta u_1
-4\dfrac{\partial^2 u_1}{\partial x_m^2}
\biggl)^2dv\\
&+\int_{\Omega}16a^2\biggl(a^2\dfrac{\partial u_1}{\partial x_m}-\Delta(\dfrac{\partial u_1}{\partial x_m})
\biggl )^2dv
\end{aligned}$$ holds. By a direct computation, we infer $$\begin{aligned}
&\int_{\Omega}\biggl(a^2u_1-2\Delta u_1
-4\dfrac{\partial^2 u_1}{\partial x_m^2}\biggl)^2dv
\\&=a^4+4\Gamma_1+16\int_{\Omega}(\dfrac{\partial^2 u_1}{\partial x_m^2})^2dv
\\
&+4a^2\int_{\Omega}|\nabla u_1|^2dv-8a^2\int_{\Omega}u_1\dfrac{\partial^2 u_1}{\partial x_m^2}dv
+16\int_{\Omega}\Delta u_1\dfrac{\partial^2 u_1}{\partial x_m^2}dv
\end{aligned}$$ and $$\begin{aligned}
&\int_{\Omega}\biggl(a^2\dfrac{\partial u_1}{\partial x_m}-\Delta(\dfrac{\partial u_1}{\partial x_m})
\biggl )^2dv\\
&=\int_{\Omega}\biggl(a^4(\dfrac{\partial u_1}{\partial x_m})^2+(\Delta(\dfrac{\partial u_1}{\partial x_m}))^2
-2a^2\dfrac{\partial u_1}{\partial x_m}\Delta(\dfrac{\partial u_1}{\partial x_m})\biggl )dv.\\
\end{aligned}$$ We derive $$\begin{aligned}
&\int_{\Omega}|p_1|^2dv+\int_{\Omega}|p_2|^2dv\\
&=a^4\biggl\{a^4+4\Gamma_1+16\int_{\Omega}(\dfrac{\partial^2 u_1}{\partial x_m^2})^2dv
+4a^2\int_{\Omega}|\nabla u_1|^2dv\\
&-8a^2\int_{\Omega}u_1\dfrac{\partial^2 u_1}{\partial x_m^2}dv
+16\int_{\Omega}\Delta u_1\dfrac{\partial^2 u_1}{\partial x_m^2}dv\biggl\}\\
&+16a^2\biggl\{\int_{\Omega}\biggl(a^4(\dfrac{\partial u_1}{\partial x_m})^2+(\Delta(\dfrac{\partial u_1}{\partial x_m}))^2
-2a^2\dfrac{\partial u_1}{\partial x_m}\Delta(\dfrac{\partial u_1}{\partial x_m})\biggl )dv\biggl\}.
\end{aligned}$$ From the proposition 2.2, we infer
$$\begin{aligned}
\int_{\Omega}g_1u_1p_1dv+\int_{\Omega}g_2u_1p_2dv&=\displaystyle{\int_{\Omega}}\biggl\{a^4 u_1^2
+4a^2(\dfrac{\partial u_1}{\partial x_m})^2 -2a^2u_1\Delta u_1\biggl\}dv\\
&=a^4
+2a^2\displaystyle{\int_{\Omega}}\biggl\{2(\dfrac{\partial u_1}{\partial x_m})^2 +|\nabla u_1|^2 \biggl\}dv.
\end{aligned}$$
We apply the theorem 2.1 to functions $g=g_1$ and $g=g_2$, respectively and take summation for them, we have
$$\begin{aligned}
&\bigl\{(\Gamma_{k+1}-\Gamma_1)+(\Gamma_{k+2}-\Gamma_1)\bigl\}\biggl(a^4
+2a^2\displaystyle{\int_{\Omega}}\biggl\{2(\dfrac{\partial u_1}{\partial x_m})^2 +|\nabla u_1|^2 \biggl\}dv\biggl)\\
&\leq a^4\biggl\{a^4+4\Gamma_1+16\int_{\Omega}(\dfrac{\partial^2 u_1}{\partial x_m^2})^2dv
+4a^2\int_{\Omega}|\nabla u_1|^2dv\\
&-8a^2\int_{\Omega}u_1\dfrac{\partial^2 u_1}{\partial x_m^2}dv
+16\int_{\Omega}\Delta u_1\dfrac{\partial^2 u_1}{\partial x_m^2}dv\biggl\}\\
&+16a^2\biggl\{\int_{\Omega}\biggl(a^4(\dfrac{\partial u_1}{\partial x_m})^2+(\Delta(\dfrac{\partial u_1}{\partial x_m}))^2
-2a^2\dfrac{\partial u_1}{\partial x_m}\Delta(\dfrac{\partial u_1}{\partial x_m})\biggl )dv\biggl\}\\
&+(\Gamma_{k+1}-\Gamma_1)(\Gamma_{k+2}-\Gamma_1),
\end{aligned}$$
Taking summation for $m$ from $1$ to $n$ and making use of Stokes formula, we have $$\begin{aligned}
&\bigl\{(\Gamma_{k+1}-\Gamma_1)+(\Gamma_{k+2}-\Gamma_1)\bigl\}\bigl(na^4
+2a^2(2+n)\|\nabla u_1\|^2\bigl )\\
&\leq a^4\biggl\{na^4+4(n+4)\Gamma_1
+4a^2(n+2)\|\nabla u_1\|^2+16\int_{\Omega}\sum_{m=1}^n(\dfrac{\partial^2 u_1}{\partial x_m^2})^2dv\biggl\}\\
&+16a^2\biggl\{a^4\|\nabla u_1\|^2+2a^2\Gamma_1+\int_{\Omega}\sum_{m=1}^n(\Delta(\dfrac{\partial u_1}{\partial x_m}))^2dv\biggl\}\\
&+n(\Gamma_{k+1}-\Gamma_1)(\Gamma_{k+2}-\Gamma_1),
\end{aligned}$$ that is, $$\begin{aligned}
&\bigl\{(\Gamma_{k+1}-\Gamma_1)+(\Gamma_{k+2}-\Gamma_1)\bigl\}(na^2
+2(n+2)\|\nabla u_1\|^2 )\\
&\leq a^2\biggl\{na^4
+4a^2(n+6)\|\nabla u_1\|^2+4(n+12)\Gamma_1+16\int_{\Omega}\sum_{m=1}^n(\dfrac{\partial^2 u_1}{\partial x_m^2})^2dv\biggl\}\\
&+16\int_{\Omega}\sum_{m=1}^n(\Delta(\dfrac{\partial u_1}{\partial x_m}))^2dv
+\dfrac{n}{a^2}(\Gamma_{k+1}-\Gamma_1)(\Gamma_{k+2}-\Gamma_1).\\
\end{aligned}$$
Thus, we obtain $$\begin{aligned}
&\bigl\{(\Gamma_{k+1}-\Gamma_1)+(\Gamma_{k+2}-\Gamma_1)\bigl\}\\
&\leq
a^2( a^2+2\frac{n+2}{n}\|\nabla u_1\|^2)
+\dfrac{(\Gamma_{k+1}-\Gamma_1)(\Gamma_{k+2}-\Gamma_1)}{a^2(a^2+2\frac{n+2}{n}\|\nabla u_1\|^2 )}\\
&+\dfrac{16a^4\|\nabla u_1\|^2}{n(a^2+2\frac{n+2}{n}\|\nabla u_1\|^2 )}\\
&+\dfrac{16\int_{\Omega}|\nabla\Delta u_1|^2
+\biggl(4(n+12)\Gamma_1
+16\int_{\Omega}\sum_{m=1}^n(\dfrac{\partial^2 u_1}{\partial x_m^2})^2dv\biggl)a^2}{na^2+2(n+2)\|\nabla u_1\|^2 }.\\
\end{aligned}$$ For $k_1\geq 0$, $k_2>0$ and $k_3>0$, the function $f(t)=\dfrac{k_1+tk_2}{nt+k_3}$, for $ t\geq 0$, satisfies $$f(t)\leq \max\{\dfrac{k_1}{k_3}, \ \dfrac{k_2}n\}.$$ Thus, we have $$\begin{aligned}
&\dfrac{16\int_{\Omega}|\nabla\Delta u_1|^2
+\biggl(4(n+12)\Gamma_1
+16\int_{\Omega}\sum_{m=1}^n(\dfrac{\partial^2 u_1}{\partial x_m^2})^2dv\biggl)a^2}{na^2+2(n+2)\|\nabla u_1\|^2 }
\leq C,\\
\end{aligned}$$ where $C$ is given by $$C=\max\biggl\{\dfrac{8\int_{\Omega}|\nabla\Delta u_1|^2dv}{(n+2)\|\nabla u_1\|^2 },
\dfrac{4(n+12)\Gamma_1+16\int_{\Omega}\sum_{m=1}^n(\dfrac{\partial^2 u_1}{\partial x_m^2})^2dv}{n}\biggl\}.$$ If we put $$a^2(a^2+2\dfrac{n+2}n\|\nabla u_1\|^2 )=\sqrt{(\Gamma_{k+1}-\Gamma_1)(\Gamma_{k+2}-\Gamma_1)},$$ we obtain $$a^4\leq \sqrt{(\Gamma_{k+1}-\Gamma_1)(\Gamma_{k+2}-\Gamma_1)},$$ $$\begin{aligned}
\bigl(\sqrt{\Gamma_{k+2}-\Gamma_1}-\sqrt{\Gamma_{k+1}-\Gamma_1}\bigl)^2\leq \dfrac{16\sqrt{\Gamma_1}}{n}\biggl\{(\Gamma_{k+1}-\Gamma_1)(\Gamma_{k+2}-\Gamma_1)\biggl\}^{\frac14}+C,
\end{aligned}$$ because $$\begin{aligned}
\|\nabla u_1\|^2\leq \sqrt{\Gamma_1}.
\end{aligned}$$ If we change $k+2$ and $k+1$ into $k+1$ and $k$, respectively, we know that the theorem 1.1 is proved.
Appendix
========
In this Appendix, we shall give a proof of the lemma 2.1.
[**Lemma 2.1.**]{} Let $\{\mu_j\}_{j=k+1}^{\infty}$ be a sequence satisfying $$0\leq \mu_{k+1}\leq \mu_{k+2}\leq \cdots \to \infty.$$ If a sequence $\{a_j\}_{j=k+1}^{\infty}$ satisfies $\sum_{j=k+1}^{\infty}\mu_j^2a_j^2=A<\infty $ and $\sum_{j=k+1}^{\infty}a_j^2=B<\infty $, then we have $$\sum_{j=k+1}^{\infty}\mu_ja_j^2\leq \dfrac{A+\mu_{k+1}\mu_{k+2}B}{\mu_{k+1}+\mu_{k+2}}.$$
From the Cauchy-Schwarz inequality, we know $$\mu_{k+1}\sum_{j=k+1}^{\infty}a_j^2\leq \sum_{j=k+1}^{\infty}\mu_ja_j^2\leq
\sqrt{\sum_{j=k+1}^{\infty}\mu_j^2a_j^2\sum_{j=k+1}^{\infty}a_j^2}=\sqrt{AB}.$$ Hence $$\mu_{k+1}\leq \sqrt{\dfrac{A}{B}}.$$ For any sequence $\{x_j\}_{j=k+1}^{\infty}$ with $\sum_{j=k+1}^{\infty}\mu_j^2x_j^2=A $ and $\sum_{j=k+1}^{\infty}x_j^2=B $, we consider the following function $$F(x_j)= \sum_{j=k+1}^{\infty}\mu_jx_j^2+\lambda(\sum_{j=k+1}^{\infty}\mu_j^2x_j^2-A) +\mu(\sum_{j=k+1}^{\infty}x_j^2-B),$$ where $\lambda$ and $\mu$ are Lagrange multipliers. Thus, the maximum $f_{max}$ of the function $f= \sum_{j=k+1}^{\infty}\mu_jx_j^2 $ is attained at critical points of $F$. If $\{c_j\}_{j=k+1}^{\infty}$ is a critical point of $F$, for any sequence $\{b_j\}_{j=k+1}^{\infty}$, we have $$\dfrac{dF(c_j+tb_j)}{dt}\biggl|_{t=0}=
2\sum_{j=k+1}^{\infty}\mu_jc_jb_j+2\lambda\sum_{j=k+1}^{\infty}\mu_j^2c_jb_j +2\mu\sum_{j=k+1}^{\infty}c_jb_j=0.$$ By taking $$b_j=\begin{cases} 1 &j=p,\\
0 & j\neq p,
\end{cases}$$ we have $$(\mu_p+\lambda\mu_p^2+\mu)c_p=0.$$ Since $\mu_p+\lambda\mu_p^2+\mu=0$ is a quadratic equation of $\mu_p$, if $\mu_p+\lambda\mu_p^2+\mu\neq 0$, we have $c_p=0$. Let $\mu_r$ and $\mu_s$, $r<s$, be solutions of $\mu_p+\lambda\mu_p^2+\mu=0$ with multiplicity $r_0+1$ and $s_0+1$, respectively, that is, $$\mu_r=\mu_{r+1}=\cdots=\mu_{r+r_0} \quad \mu_s=\mu_{s+1}=\cdots=\mu_{s+s_0}.$$ Therefore, we have $$\begin{aligned}
&A=\mu_r^2(c_r^2+c_{r+1}^2+\cdots+c_{r+r_0}^2)+\mu_s^2(c_s^2+c_{s+1}^2+\cdots+c_{s+s_0}^2),\\
&B=(c_r^2+c_{r+1}^2+\cdots+c_{r+r_0}^2)+(c_s^2+c_{s+1}^2+\cdots+c_{s+s_0}^2),\\
&f_{max}=\mu_r(c_r^2+c_{r+1}^2+\cdots+c_{r+r_0}^2)+\mu_s(c_s^2+c_{s+1}^2+\cdots+c_{s+s_0}^2)\\
\end{aligned}$$ Hence, we get $$f_{max}=\dfrac{A+\mu_r\mu_sB}{\mu_r+\mu_s}.$$ Since $f_{max}\leq \sqrt{AB}$ from the Cauchy-Schwarz inequality, we have $$f_{max}=\dfrac{A+\mu_r\mu_sB}{\mu_r+\mu_s}\leq \sqrt{AB}.$$ Thus, we obtain $$(\sqrt{\dfrac AB}-\mu_r)(\sqrt{\frac AB}-\mu_s)\leq 0,$$ that is, we have $$\sqrt{\dfrac AB}-\mu_r\geq 0, \quad \sqrt{\frac AB}-\mu_s\leq 0$$ because of $\mu_r\leq \mu_s$. Since $\sqrt{\dfrac AB}-\mu_r\geq 0$, we know that $G(t)=\dfrac{A+\mu_rtB}{\mu_r+t}$ is a decreasing function of $t$. Hence, we have $$f_{max}\leq \dfrac{A+\mu_r\mu_{k+2}B}{\mu_r+\mu_{k+2}}.$$ If $\mu_{k+2}\geq \sqrt{\frac AB}$, we have $\mu_r=\mu_{k+1}$ because of $r<s$ and $\mu_r\leq \sqrt{\dfrac AB}$, that is $$f_{max}\leq \dfrac{A+\mu_{k+1}\mu_{k+2}B}{\mu_{k+1}+\mu_{k+2}}.$$ If $\mu_{k+2}\leq \sqrt{\frac AB}$, we know that $G(t)=\dfrac{A+\mu_{k+2}tB}{\mu_{k+2}+t}$ is a decreasing function of $t$. Hence, we have $$f_{max}\leq \dfrac{A+\mu_r\mu_{k+2}B}{\mu_r+\mu_{k+2}}\leq \dfrac{A+\mu_{k+1}\mu_{k+2}B}{\mu_{k+1}+\mu_{k+2}}.$$ It completes the proof of the lemma.
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|
---
abstract: |
We describe a new algorithm to compute the geometric intersection number between two curves, given as edge vectors on an ideal triangulation. Most importantly, this algorithm runs in polynomial time in the bit-size of the two edge vectors.
In its simplest instances, this algorithm works by finding the minimal position of the two curves. We achieve this by phrasing the problem as a collection of linear programming problems. We describe how to reduce the more general case down to one of these simplest instances in polynomial time. This reduction relies on an algorithm by the first author to quickly switch to a new triangulation in which an edge vector is significantly smaller.
author:
- 'Mark C. Bell[^1]'
- 'Richard C. H. Webb[^2]'
bibliography:
- 'bibliography.bib'
title: Applications of fast triangulation simplification
---
Introduction
============
Let $S$ be an (orientable) punctured surface and let $\zeta = \zeta(S) \defeq -3 \chi(S)$. We will assume that $S$ is sufficiently complex that $\zeta \geq 3$ and so $S$ can be decomposed into an (ideal) triangulation. Any such triangulation of $S$ has exactly $\zeta$ edges.
We can use a triangulation to give a combinatorial description of a curve. The (essential, simple closed) curve $\gamma$ on $S$ is uniquely determined by its *edge vector*: $$\calT(\gamma) \defeq \left(\begin{array}{c} \intersection(\gamma, e_1) \\ \vdots \\ \intersection(\gamma, e_\zeta) \end{array} \right) \in \NN_0^\zeta$$ where $e_1, \ldots, e_\zeta$ are the (ordered) edges of $\calT$.
In this paper we describe a new algorithm for computing the geometric intersection number $\intersection(\alpha, \beta)$ from $\calT(\alpha)$ and $\calT(\beta)$. Most importantly, this algorithm runs in polynomial time in the bit-size of $\calT(\alpha)$ and $\calT(\beta)$.
To achieve this we focus on the simplest case when $\calT$ is *$\alpha$–minimal*, that is, when $\calT$ minimises $\intersection(\alpha, \calT)$. On such a triangulation the combinatorics of $\alpha$ are extremely restricted and so there are very few possibilities that we need to consider. This allows us to reduce finding the minimal position for $\alpha$ and $\beta$ down to a collection of linear programming problems. We show that these problems are sufficiently small that we can solve them, and so deduce $\intersection(\alpha, \beta)$, in polynomial time.
In the more general case, we apply a series of moves to convert the problem back to one on an $\alpha$–minimal triangulation. These moves consist of flipping edges of the triangulation and performing powers of a Dehn twist along a short curve. In [@BellSimplifying] the first author showed that there is always such a move which reduces $\intersection(\alpha, \calT)$ by a definite fraction. We use this to bound the number of moves needed to reach an $\alpha$–minimal triangulation.
There are several other simplification results in other models of curves on surfaces [@AHT Section 4] [@DynnikovBraids] [@EricksonNayyeri] [@SchaeferSedgwick]. However, in all of these other models it is very difficult to keep track of how another curve changes during the simplification process. This makes it extremely hard to reduce the generic problem down to the $\alpha$–minimal case as we are required to track $\beta$ through these moves too.
Minimal triangulations {#sec:minimal}
======================
In this section we consider the problem of putting $\alpha$ in minimal position with respect to $\beta$ when both of these are curves given on an $\alpha$–minimal triangulation $\calT$.
One case that is particularly straightforward is if $\alpha$ is *non-isolating*, that is, if every component of $S - \alpha$ contains a puncture. Here there is only one possibility for how $\alpha$ can appear on $\calT$.
If $\alpha$ is non-isolating then $\intersection(\alpha, \calT) = 2$ and so $\alpha$ must appear on $\calT$ as shown in Figure \[fig:non\_isolating\].
Hence in this configuration it is straightforward to put $\alpha$ in minimal position with respect to $\beta$ and so we can directly compute their intersection number.
If $\alpha$ is non-isolating then, following the notation of Figure \[fig:non\_isolating\], $$\intersection(\alpha, \beta) = \max(\mathbf{e} - \mathbf{b}, \mathbf{b} - \mathbf{e}, \mathbf{a} + \mathbf{c} - \mathbf{b} - \mathbf{e})$$ where $\mathbf{x} \defeq \intersection(\beta, x)$.
Thus we focus the remainder of this section on the case in which $\alpha$ is isolating.
Combinatorial restrictions
--------------------------
When $\alpha$ is isolating there are again many restrictions on its combinatorics. The argument of [@BellThesis Section 2.4.2] shows that $\intersection(\alpha, e) \in \{0, 2\}$ for every edge $e$ of $\calT$. This means that in each triangle $\alpha$ must appear either as a *tripod* or *corridor*, as shown in Figure \[fig:tripod\] and Figure \[fig:corridor\] respectively.
[0.45]{}
[0.45]{}
\[prop:tripod\_corridor\] Let $g'$ be the genus of the component of $S - \alpha$ which does not contain any punctures. Then $\alpha$ appears as a corridor in exactly one triangle of $\calT$ and as a tripod in exactly $4g' - 2$ triangles of $\calT$.
Clearly $\alpha$ must appear as a tripod in exactly $4g' - 2$ triangles due to the Euler characteristic of the unpunctured component. Furthermore, if $\alpha$ did not appear as a corridor in any triangle of $\calT$ then it would be peripheral. Hence it only remains to show that $\alpha$ appears as a corridor in at most one triangle.
Suppose instead that there are two triangles in which $\alpha$ appears as a corridor. Let $e$ be the arc shown in Figure \[fig:two\_corridors\], which follows around $\alpha$ from the outside of one corridor to the other before connecting to a puncture.
We follow along $e$, flipping each edge of $\calT$ that we meet along the way. Each flip reduces $\intersection(e, \calT)$ but does not increase $\intersection(\alpha, \calT)$ [@MosherFoliations Page 38]. Thus, after performing at most $2 \zeta$ flips, we finish with a triangulation $\calT'$ which contains $e$ as an edge. However, since $e$ is disjoint from $\alpha$ we must have that $\intersection(\alpha, \calT') < \intersection(\alpha, \calT)$. This contradicts the fact that $\calT$ was $\alpha$–minimal.
Linear programming
------------------
To finish the case in which $\alpha$ is isolating we formulate a collection of integer linear programming problems. An optimal solution over all of these problems will then correspond to a minimal position. The number of problems and, thanks to the fact that these curves are given on an $\alpha$–minimal triangulation, the number of variables involved will be bounded only in terms of $\zeta$.
To do this we first assign an orientation to each edge of $\calT$. Additionally, for ease of notation throughout this section let $\mathbf{e_i} \defeq \intersection(\beta, e_i)$.
Fix $\mathfrak{b} \in \beta$ to be a representative which meets $\calT$ minimally. Without loss of generality we may draw $\mathfrak{b}$ as a collection of straight line segments in each triangle.
Now for each edge $e_i$, choose $x_i, y_i \in \NN_0$ such that $x_i + y_i \leq \mathbf{e_i}$. We construct a representative $\mathfrak{a} \in \alpha$ from these variables as follows.
1. If $\alpha$ meets the edge $e_i$ then we place two marks on the edge. Walking along $e_i$ in the direction of its orientation, we place the first mark just after we encounter the $x_i\nth{}$ point of $\mathfrak{b}$. Similarly, walking along $e_i$ in the reverse direction, we place the second mark just after we encounter the $y_i\nth{}$ point of $\mathfrak{b}$.
2. Each triangle now has exactly $0$, $4$ or $6$ marks on its boundary. In the first case we do nothing in this triangle. In the second and third cases we connect these via straight line segments to form a corridor or tripod respectively.
3. The union of these segments is our representative $\mathfrak{a} \in \alpha$.
For example, see Figure \[fig:tripod\_constraints\] where $x_i = 5$, $y_i = 3$, $x_j = 4$, $y_j = 3$, $x_k = 3$ and $y_k = 3$. Alternatively, in the example shown in Figure \[fig:corridor\_constraints\] we have that $x_i = 0$, $y_i = 0$, $x_j = 4$, $y_j = 3$, $x_k = 2$ and $y_k = 2$ but $\intersection(\alpha, e_i) = 0$.
[0.45]{}
[0.45]{}
\[prop:intersection\_PL\] The intersection number $\intersection(\mathfrak{a}, \mathfrak{b})$ is a piecewise linear function of $x_1, y_1, \ldots, x_\zeta, y_\zeta$.
Consider a single triangle of $\calT$ with sides $e_i$, $e_j$ and $e_k$. Let $\mathbf{z_i}$, $\mathbf{z_j}$ and $\mathbf{z_k}$ denote the number of segments of $\mathfrak{b}$ running through this triangle parallel to the specified edge, as shown in Figure \[fig:corridor\_constraints\]. That is, $$\mathbf{z_i} \defeq \frac{1}{2}(\mathbf{e_j} + \mathbf{e_k} - \mathbf{e_i}), \;
\mathbf{z_j} \defeq \frac{1}{2}(\mathbf{e_i} + \mathbf{e_k} - \mathbf{e_j}) \; \textrm{and} \;
\mathbf{z_k} \defeq \frac{1}{2}(\mathbf{e_i} + \mathbf{e_j} - \mathbf{e_k}).$$ Now consider a single segment $I$ of $\mathfrak{a}$ in this triangle which, without loss of generality, connects from $e_j$ to $e_k$.
If $I$ is part of a tripod, as shown in Figure \[fig:tripod\_constraints\], or is the segment of a corridor which is furthest from $e_i$, as shown in Figure \[fig:corridor\_constraints\], then: $$\intersection(I, \mathfrak{b}) = \begin{cases}
|x_k - y_j| & \textrm{if} \; x_k \leq \mathbf{z_i} \; \textrm{or} \; y_j \leq \mathbf{z_i} \\
x_k + y_j - 2\mathbf{z_i} & \textrm{otherwise}.
\end{cases}$$ Similarly, if $I$ is the segment of a corridor which is closest to $e_i$ then: $$\intersection(I, \mathfrak{b}) = \begin{cases}
|(\mathbf{e_j} - y_k) - (\mathbf{e_j} - x_j)| & \textrm{if} \; \mathbf{e_j} - y_k \leq \mathbf{z_i} \; \textrm{or} \; \mathbf{e_j} - x_j \leq \mathbf{z_i} \\
(\mathbf{e_j} - y_k) + (\mathbf{e_j} - x_j) - 2\mathbf{z_i} & \textrm{otherwise}.
\end{cases}$$ Note that both formulae are dependent on the orientations of $e_i$, $e_j$ and $e_k$ matching those in Figure \[fig:tripod\_constraints\] and Figure \[fig:corridor\_constraints\]. In the event that the orientation on $e_i$ does not match, for example, the variables $x_i$ and $y_i$ must be interchanged.
In either case, this is a piecewise linear function of $x_1, y_1, \ldots, x_\zeta, y_\zeta$. Hence by summing these functions over all segments in all triangles we see that $\intersection(\mathfrak{a}, \mathfrak{b})$ is a piecewise linear function of $x_1, y_1, \ldots, x_\zeta, y_\zeta$ too.
We will denote this piecewise-linear function by $f$ and so $$\intersection(\mathfrak{a}, \mathfrak{b}) = f(x_1, y_1, \ldots, x_\zeta, y_\zeta).$$
Now any representative of $\alpha$ which is in minimal position with respect to $\mathfrak{b}$ is isotopic, relative to $\mathfrak{b}$, to some representative constructed by the above procedure. Therefore there is a choice of $x_1, y_1, \ldots, x_\zeta, y_\zeta$ such that the corresponding $\mathfrak{a}$ is in minimal position with respect to $\mathfrak{b}$. Thus the task of putting $\alpha$ in minimal position with respect to $\beta$ is equivalent to finding a minimum of $f$.
To find such a minimum we consider each piece of $f$ in turn. We can formulate the problem of finding a minimum of $f$ on a piece as an integer linear programming problem. There are many algorithms for solving such problems and, while they are $\mathbf{NP}$-complete in general [@GathenSieveking] [@GareyJohnson], these can be solved in polynomial time as we have a fixed number of variables:
\[thrm:integer\_LP\] Suppose that $m_0$ is fixed. There is an algorithm which, given a matrix $A \in \ZZ^{n \times m_0}$, vector $b \in \ZZ^{n \times 1}$ and vector $c \in \ZZ^{1 \times m_0}$ finds an optimal solution $x \in \ZZ^{m_0 \times 1}$ to the integer linear programming problem: $$\begin{aligned}
\textrm{Minimise} & c \cdot x \\
\textrm{Subject to} & A \cdot x \geq b.\end{aligned}$$ Moreover, this algorithm runs in polynomial time in the bit-size of $A$, $b$ and $c$.
The bit-size of each of the above integer linear programming problems is at most $\log(\intersection(\beta, \calT))$. Therefore by using the algorithm of Theorem \[thrm:integer\_LP\] we can solve each of these problems in at most $O(\poly(\log(\intersection(\beta, \calT))))$ operations.
Finally, note that $\intersection(I, \mathfrak{b})$ from the proof of Proposition \[prop:intersection\_PL\] is a piecewise linear function with $5$ pieces. Therefore the number of intersections occurring in a triangle is a piecewise linear function with at most $5^3$ pieces. Thus $f$ has at most $5^{2 \zeta}$ pieces and so there are at most $5^{2\zeta} \in O(1)$ such problems we must consider. Hence we can also find the minimal solution over all problems, and so a minimum of $f$, in polynomial time in the bit-size of $\calT(\beta)$.
Suppose we are given $\calT(\alpha)$ and $\calT(\beta)$ where $\calT$ is an $\alpha$–minimal triangulation. Then we can compute the minimal position for $\alpha$ relative to $\beta$, and so $\intersection(\alpha, \beta)$, in polynomial time in the bit-size of $\calT(\beta)$.
We may also view $\beta$ as a measured lamination. Then minimal position occurs when there are no bigons between $\alpha$ and the underlying lamination of $\beta$. Thus we can find the minimal position of $\alpha$ with respect to $\beta$ by solving a collection of linear programming problems instead of integer linear programming problems.
Flips and twists {#sec:flips_twists}
================
We now consider the more general case, in which $\alpha$ and $\beta$ are given on a triangulation $\calT$ which is not $\alpha$–minimal. To deal with this case, we introduce two basic moves for modifying triangulations; the *flip* and the *twist*. We use these moves to give a polynomial time reduction back to the case in Section \[sec:minimal\].
Firstly, we say that an edge of $\calT$ is *flippable* if it is contained in two distinct triangles. If $e$ is such an edge then we may flip it to obtain a new triangulation $\calT'$ as shown in Figure \[fig:flip\].
Secondly, if $\delta$ is a curve on $S$ then we may modify $\calT$ by performing the *Dehn twist* $T_\delta^k$ [@FM Chapter 3]. This move cuts the surface open along the curve $\delta$ and rotates one of the boundary components $k$ times to the right (or $|k|$ times to the left if $k$ is negative) before regluing the boundary components together.
In both cases it is straightforward to compute the edge vectors of $\alpha$ and $\beta$ on the new triangulation after performing such a move:
\[prop:flip\_intersection\] Suppose that $\gamma$ is a curve and $e$ is a flippable edge of a triangulation $\calT$ as shown in Figure \[fig:flip\] then $$\intersection(\gamma, f) = \max(\intersection(\gamma, a) + \intersection(\gamma, c), \intersection(\gamma, b) + \intersection(\gamma, d)) - \intersection(\gamma, e).$$ Hence we can compute $\calT'(\gamma)$ in at most $O(\log(\intersection(\gamma, \calT)))$ operations.
\[prop:twist\_intersection\] Suppose that $\delta$ and $\gamma$ are curves. Given $\calT(\delta)$, $\calT(\gamma)$ and $k \in \ZZ$ we can compute $\calT'(\gamma)$ where $\calT' \defeq T_\delta^k(\calT)$ in at most $$O(\poly(\log(\intersection(\gamma, \calT)) + \log(\intersection(\delta, \calT)) + \log(k)))$$ operations.
The usefulness of these moves comes from the fact that if a curve appears very complicated on $\calT$ then there is always such a move which reduces the number of intersections by a definite fraction:
\[thrm:fraction\_intersection\] Let $D \defeq 80 \zeta B (10B + 1)^C$ where $B \defeq 5^{2\zeta}$ and $C \defeq 2^{2\zeta}$. If $\intersection(\gamma, \calT) > D$ then there is a triangulation $\calT'$ such that either:
- $\calT$ and $\calT'$ differ by a flip, or
- $\calT' = T_\delta^k(\calT)$ where $|k| \leq \intersection(\gamma, \calT)$ and $\intersection(\delta, \calT) \leq 2 \zeta$
and $\intersection(\gamma, \calT') \leq (1 - 1/D) \intersection(\gamma, \calT)$.
Thus, by using Theorem \[thrm:fraction\_intersection\] at most $O(\log(\intersection(\alpha, \calT)))$ times we can obtain a triangulation $\calT'$ where $\intersection(\alpha, \calT') \leq D$. To continue simplifying further we use the following lemma:
\[lem:drop\_intersection\] If $\calT$ is not $\alpha$–minimal then by performing at most $2 \zeta$ flips we can reach a triangulation $\calT'$ such that $\intersection(\alpha, \calT') < \intersection(\alpha, \calT)$.
If $\intersection(\gamma, e) > 2$ for some edge $e$ of $\calT$ then there is an edge of $\calT$ which can be flipped in order to reduce the intersection number [@BellThesis Lemma 2.4.3]. Hence we may assume that $\intersection(\alpha, e) \leq 2$ for each edge $e$ of $\calT$.
Now if $\intersection(\alpha, e) = 1$ for some edge $e$ then by performing at most two flips we can reduce the intersection number [@BellThesis Lemma 2.4.4].
On the other hand, if $\intersection(\alpha, e) \in \{0, 2\}$ for every edge $e$ then $\alpha$ looks like a tripod or corridor in each triangle of $\calT$. Hence, by the same argument as in the proof of Proposition \[prop:tripod\_corridor\], by performing at most $2 \zeta$ flips we can reach a triangulation with fewer intersections with $\alpha$.
\[cor:general\_minimal\] Given $\calT(\alpha)$ and $\calT(\beta)$ we can compute an $\alpha$–minimal triangulation $\calT'$ together with $\calT'(\alpha)$ and $\calT'(\beta)$ in at most $$O(\poly(\log(\intersection(\alpha, \calT)) + \log(\intersection(\beta, \calT))))$$ operations. Furthermore, the bit-size of $\calT'(\beta)$ is at most $$O(\log(\intersection(\beta, \calT)) + \log^2(\intersection(\alpha, \calT))).$$ Thus we can compute minimal position representatives on $\calT'$, and so $\intersection(\alpha, \beta)$ in polynomial time in the bit-sizes of $\calT(\alpha)$ and $\calT(\beta)$ too.
By combining Theorem \[thrm:fraction\_intersection\] and Lemma \[lem:drop\_intersection\] we obtain a sequence of moves from $\calT$ to an $\alpha$–minimal triangulation $\calT'$. By Proposition \[prop:flip\_intersection\] and Proposition \[prop:twist\_intersection\], we can push $\calT(\alpha)$ and $\calT(\beta)$ through these moves and so obtain the edge vectors for these curves on $\calT'$ too. As there are only $O(\log(\intersection(\alpha, \calT)))$ such moves, we can perform this computation in at most $$O(\poly(\log(\intersection(\alpha, \calT)) + \log(\intersection(\beta, \calT))))$$ operations.
We now consider the effect of an individual move on $\intersection(\beta, \calT)$:
- Performing a flip increases $\intersection(\beta, \calT)$ by a factor of at most two by Proposition \[prop:flip\_intersection\].
- Performing $T_\delta^k$ increases $\intersection(\beta, \calT)$ by at most $|k| \intersection(\delta, \calT) \intersection(\delta, \beta)$ [@FM Proposition 3.4]. However for the twists that we will perform:
- $|k| \leq \intersection(\alpha, \calT)$,
- $\intersection(\delta, \calT) \leq 2 \zeta$, and
- $\intersection(\delta, \beta) \leq 2 \zeta \intersection(\beta, \calT)$.
Hence, performing this move increases $\intersection(\beta, \calT)$ by at most a factor of $4 \zeta^2 \intersection(\alpha, \calT)$.
Again, as only $O(\log(\intersection(\alpha, \calT)))$ such moves are performed, we have that $$\intersection(\beta, \calT') \in O\left(\intersection(\beta, \calT) \intersection(\alpha, \calT)^{\log(\intersection(\alpha, \calT))} \right)$$ and so the bound holds by taking logs.
We can now reapply the procedure of Section \[sec:minimal\] to compute minimal position for $\alpha$ and $\beta$ on $\calT'$ and so deduce $\intersection(\alpha, \beta)$. As $\intersection(\alpha, \calT')$ and $\intersection(\beta, \calT')$ are sufficiently small, this can also be done in polynomial time.
Further extensions and applications {#sec:extensions}
===================================
We finish with some further generalisations of the procedure of Corollary \[cor:general\_minimal\].
Multicurves {#sub:multicurves}
-----------
A slight variant of this procedure works even when $\alpha$ and $\beta$ have multiple components.
To handle this case we first use a polynomial time algorithm to extract the individual components and their multiplicities [@AHT Section 4] [@BellSimplifying Section 4] [@EricksonNayyeri Section 6.4]. That is, we find $a_i, b_j \in \NN$ and curves $\alpha_i$, $\beta_j$ such that $$\alpha = \bigcup_i a_i \cdot \alpha_i \inlineand \beta = \bigcup_j b_j \cdot \beta_j.$$ We proceed by computing $\intersection(\alpha_i, \beta_j)$ for each $i$ and $j$ by the above procedure and then $$\intersection(\alpha, \beta) = \sum_{i,j} a_i b_j \intersection(\alpha_i, \beta_j).$$
Multiarcs {#sub:multiarcs}
---------
The above procedure also works when $\alpha$ or $\beta$ is a *multiarc*, that is, the isotopy class of the image of a smooth proper embedding of a finite number of copies of $[0, 1]$ (whose endpoints connect into punctures) into $S$.
If $\alpha$ is an arc then an $\alpha$–minimal triangulation is one which contains $\alpha$ as an edge. Thus we perform the simplification routine to obtain $\alpha$ and $\beta$ on an $\alpha$–minimal triangulation $\calT'$. After this $\intersection(\alpha, \beta)$ is just the entry of $\calT'(\beta)$ associated to the edge $\alpha$.
If $\alpha$ is a multiarc then we extract its individual components and their multiplicities and proceed as in Section \[sub:multicurves\].
The only modification needed to enable this is a slight change to how these multiarcs are represented combinatorially. Since a non-trivial multiarc can have zero intersection with all edges, we make a slight modification to the standard definition of intersection number.
If $\alpha$ is a multiarc which contains $k$ copies of the edge $e$ of $\calT$ then their *intersection number* is defined to be $\intersection(\alpha, e) \defeq -k$.
This allows us to again represent a multiarc via its intersection numbers with the edges of a triangulation and for the procedure to work as before.
Boundaries of neighbourhoods {#sub:boundaries}
----------------------------
For curves $\alpha$ and $\beta$ let $\gamma \defeq \partial(N(\alpha \cup \beta))$. When $\alpha$ and $\beta$ are in minimal position on an $\alpha$–minimal triangulation, it is straightforward to compute $\calT(\gamma)$ from $\calT(\alpha)$ and $\calT(\beta)$. The same argument as in Section \[sec:flips\_twists\] allows us to easily reduce this calculation on any triangulation back to one on an $\alpha$–minimal triangulation. Thus, we can compute $\calT(\gamma)$ in polynomial time in the bit-sizes of $\calT(\alpha)$ and $\calT(\beta)$.
One key application for this is when $h$ is a reducible, aperiodic mapping class. In this case, suppose that we have found a multicurve $\alpha$ such that $h^n(\alpha) = \alpha$. We wish to upgrade from this $h^n$–invariant multicurve to an $h$–invariant one. One way to achieve this is to take the $h$–invariant multicurve $\gamma \defeq \partial N(\alpha \cup h(\alpha) \cup \ldots \cup h^{n-1}(\alpha))$. This multicurve is essential since $h$ is aperiodic and, by the above argument, we can compute $\calT(\gamma)$ in polynomial time in the size of $h$ and the bit-size of $\calT(\alpha)$. Here we think of $h$ as being given as a sequence of edge flips starting from $\calT$ and its size is just the number of flips.
The first author acknowledges support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network).
The second author is supported by EPSRC Fellowship Reference EP/N019644/1.
[^1]: Department of Mathematics, University of Illinois: `mcbell@illinois.edu`
[^2]: DPMMS, Centre for Mathematical Sciences, University of Cambridge: `rchw2@cam.ac.uk`
|
---
abstract: 'We demonstrate single-atom resolved imaging with a survival probability of $0.99932(8)$ and a fidelity of $0.99991(1)$, enabling us to perform repeated high-fidelity imaging of single atoms in tweezers for thousands of times. We further observe lifetimes under laser cooling of more than seven minutes, an order of magnitude longer than in previous tweezer studies. Experiments are performed with strontium atoms in $813.4~\text{nm}$ tweezer arrays, which is at a magic wavelength for the clock transition. Tuning to this wavelength is enabled by off-magic Sisyphus cooling on the intercombination line, which lets us choose the tweezer wavelength almost arbitrarily. We find that a single not retro-reflected cooling beam in the radial direction is sufficient for mitigating recoil heating during imaging. Moreover, this cooling technique yields temperatures below $5~\mu$K, as measured by release and recapture. Finally, we demonstrate clock-state resolved detection with average survival probability of $0.996(1)$ and average state detection fidelity of $0.981(1)$. Our work paves the way for atom-by-atom assembly of large defect-free arrays of alkaline-earth atoms, in which repeated interrogation of the clock transition is an imminent possibility.'
author:
- 'Jacob P. Covey'
- 'Ivaylo S. Madjarov'
- Alexandre Cooper
- Manuel Endres
bibliography:
- 'library.bib'
title: '2000-times repeated imaging of strontium atoms in clock-magic tweezer arrays'
---
Optical lattice clocks of alkaline-earth(-like) atoms (AEAs) have reached record precision [@Ludlow2015; @Campbell2017] for which the exploration of fundamental physics, such as geodesy [@Grotti2018], gravitational waves [@Kolkowitz2016], and even dark matter [@Wcislo2016] is now a possibility. Yet, despite the precise optical control of AEAs that has been demonstrated in a low-entropy array [@Campbell2017], the ability to address and detect single atoms is currently lacking. Such single-atom control techniques would provide new avenues for optical clock systems. Specifically, they are required for realizing a myriad of quantum computing protocols for AEAs using clock states [@Daley2008; @Gorshkov2009; @Daley2011; @Daley2011e; @Pagano2018] and could provide the foundation for generating and probing entanglement for quantum-enhanced metrology [@Gil2014; @Norcia2018a]. Optical tweezer (OT) techniques have matured into a powerful tool for single-atom control, e.g., they provide the versatility required for atom-by-atom assembly of defect-free arrays [@Barredo2016; @Endres2016; @Kim2018; @Kumar2018; @Robens2017] and they automatically position single atoms at distances such that interaction shifts on the clock-transition are expected to be strongly reduced [@Chang2004]. Further, OT experiments generally have fast experimental repetition rates and, as demonstrated below, enable repeated low-loss clock-state read-out without reloading atoms. Such techniques could provide a pathway for quasi-continuous interleaved clock operation in order to tame the Dick effect [@Schioppo2017]. Since state-insensitive ‘magic’ trapping conditions are required for clock operation [@Ye2008], tweezers operating at a clock-magic wavelength are highly suited for these directions.
![ **Low-loss imaging scheme for $^{88}$Sr.** [(a)]{} Our single atom imaging scheme requires only a single *not* retro-reflected cooling beam at $689$ nm for narrow-line Sisyphus cooling (red single arrow) to compensate recoil heating from fluorescence generated by exciting atoms with a single retro-reflected $461$ nm beam (blue double arrow). Microscope objectives with $\text{NA}=0.5$ are used to generate $813.4$ nm tweezers and to collect the fluorescence light. [(b)]{} Previous studies found an imaging loss channel via the decay of $\ce{^{1}P_{1}}$ to $\ce{^{1}D_{2}}$ with a branching ratio of $\sim 1:20000$, where atoms left the tweezers since $\ce{^{1}D_{2}}$ was strongly anti-trapped [@Cooper2018a; @Norcia2018b]. Crucially, the $\ce{^{1}D_{2}}$ state is now trapped in $813.4$ nm and our results show that atoms are recovered into the $\ce{^{3}P_{J}}$ manifold with very high probability. Two lasers ($679$ nm and $707$ nm) repump atoms to $\ce{^{3}P_{1}}$, which decays back into the ground state, thus closing the $\ce{^{1}D_{2}}$ loss channel. [(c)]{} We use narrow-line *attractive* Sisyphus cooling on the $m_J=\pm1$ states [@Chen2018], originally proposed in Refs. [@Taieb1994; @Ivanov2011]. This mechanism is based on the excited state being more strongly trapped than the ground state (in contrast to *repulsive* Sisyphus cooling demonstrated recently by us [@Cooper2018a]). Atoms at the bottom of the trap are excited and have to climb up a steeper potential than they would in the ground state, leading to an average reduction in energy after spontaneous emission. Cooling results from the trapping potential mismatch, and not from photon recoil, thus requiring only a single cooling beam. [(d)]{} Average image (top) and single-shot image (bottom) of atomic fluorescence from twenty-five uniformized tweezers with an imaging time of one second. []{data-label="FigOverview"}](figures/FigExperimentalSystem){width="\columnwidth"}
Recently, two-dimensional arrays of AEAs, specifically Sr [@Cooper2018a; @Norcia2018b] and Yb [@Saskin2018], in optical tweezers have been demonstrated, including single-atom resolved imaging. Cooling during imaging has been performed on the narrow $\ce{^{1}S_{0}}\leftrightarrow\ce{^{3}P_{1}}$ intercombination line (see Fig. \[FigOverview\]), similar to quantum gas microscopes for Yb [@Yamamoto2016]. To this end, trapping wavelengths have been chosen such that the differential polarizability on this transition is small, enabling motional sidebands to be spectrally resolved in the case of Sr [@Cooper2018a; @Norcia2018b], but precluding the possibility of achieving a magic trapping condition for the optical clock transition. Significantly, a more versatile Sisyphus cooling mechanism [@Wineland1992] has recently been observed for Sr atoms [@Cooper2018a; @Chen2018], providing a general pathway for cooling on narrow lines with strong polarizability mismatch. This observation combined with prior predictions [@Taieb1994; @Ivanov2011] should allow for tweezer trapping and cooling of AEAs – and more generally atoms with narrow transitions – in a very wide range of wavelengths.
Here, we demonstrate detection and cooling of single $^{88}$Sr atoms in clock-magic optical tweezer arrays of wavelength $813.4$ nm [@Katori2003; @Takamoto2003; @Takamoto2005; @Boyd2006; @Campbell2017; @Norcia2018a] where the loss during imaging is suppressed by two orders of magnitude compared to previous work for Sr [@Cooper2018a; @Norcia2018b]. Specifically, we find a survival probability of $0.99932(8)$ and a fidelity of $0.99991(1)$ for single atom detection, enabling us to perform repeated high-fidelity detection for thousands of times. We also observe lifetimes under laser cooling of more than seven minutes.
These values provide a benchmark for simultaneous low-loss and high-fidelity imaging as well as trapping lifetimes for single neutral atoms, including work with alkalis [@Kuhr2016; @Endres2016; @Barredo2016; @Kim2018; @Kumar2018; @Robens2017; @Wu2018; @Kwon2017]. We expect this development to be important for improved scalability of atom-by-atom assembly schemes [@Kuhr2016; @Endres2016; @Barredo2016; @Kim2018; @Kumar2018; @Robens2017] and for verifying high-fidelity quantum operations with neutral atoms [@Levine2018; @Wu2018]. For example, the success probability in atom-by-atom assembly is fundamentally limited by $p_s^M$, where $p_s$ is the combined survival probability for two images and hold time for rearrangement, and $M$ is the final array size [@Endres2016]. Our work improves this fundamental limitation of $p_s^M\sim 0.5$ to $M \gtrsim 1000$, enabling in principle assembly of arrays with thousands of atoms in terms of imaging- and vacuum-limited lifetimes. Finally, we demonstrate single-shot clock-state resolved detection with average fidelity of $0.981(1)$ and average atom survival probability of $0.996(1)$, which could be used for repeated clock interrogation and readout periods without reloading.\
*Experimental techniques -* Single atoms are loaded stochastically from a narrow-line magneto-optical trap into an array of tweezers as described in detail in Ref. [@Cooper2018a]. In contrast to our previous work, we use 813.4 nm light to generate tweezers (Fig. \[FigOverview\]a). While providing a magic-wavelength for the clock transition, this wavelength also closes a previously observed loss channel, providing the basis for the low-loss detection demonstrated here (Fig. \[FigOverview\]b) [@Cooper2018a; @Norcia2018b]. Further, the imaging scheme is simplified to a single *not* retro-reflected cooling beam at 689 nm and a retro-reflected imaging beam at 461 nm. Both beams propagate in the plane orthogonal to the tweezer propagation axis. The cooling (imaging) beam is polarized parallel (perpendicular) to the tweezer propagation axis. We modulate the retro-mirror of the imaging beam to wash out interference patterns [@Endres2016]. The tweezers are linearly polarized and have a depth of $\approx450$ $\mu$K and waist of $\approx700$ nm. The array of 25 tweezers has a spacing of $\approx7.4$ $\mu$m and is uniformized to within $\approx2\%$ [@Cooper2018a]. Tweezer arrays are generated with a bottom objective, while a second top objective is used to image both the tweezer light on a diagnostic camera and the fluorescence light on an electron multiplied charge-coupled device (EMCCD) camera.
![ **Low-loss high-fidelity imaging.** [(a)]{} Histogram of fluorescence counts from a single representative tweezer. We find a detection fidelity of 0.99991(1) and an average survival probability of 0.99932(8), demonstrating simultaneous high-fidelity and low-loss imaging. Results are for an imaging time of $t=50$ ms under simultaneous repumping and Sisyphus cooling. [(b)]{} The survival fraction as a function of hold time in minutes under these imaging conditions (blue squares and fitted line). Importantly, we find a lifetime of $\tau=126(3)$ s, while only needing $t \lesssim 50$ ms imaging time for reaching high detection fidelity, leading to small loss fractions consistent with $\exp(-t/\tau)$. Moreover, we find a lifetime of $434(13)$ s under Sisyphus cooling alone (without $461$ nm) demonstrating a vacuum-limited lifetime greater than seven minutes (red circles and fitted line). [(c)]{} Survival fraction versus image number for 2000 repeated images. The dark red line represents the mean over 40 realizations, with the lighter red lines showing the standard error of the mean. Atoms are imaged with high fidelity for 50 ms followed by a 29 ms cooling block. [(d)]{} A representative realization of atom detection over the course of the 2000 images. Detected atoms are plotted in red versus the image number, where the rows represent the 25 tweezers. Note that we find no occurrences of atoms returning after they are lost. A video of the 2000 images is available online. []{data-label="FigImagingLifetime"}](figures/FigFluorescenceImaging){width="\columnwidth"}
Cooling during imaging is based on a narrow-line *attractive* Sisyphus cooling scheme on the $7.4$ kHz transition at $689$ nm (Fig. \[FigOverview\]c), following the original proposals in Refs. [@Taieb1994; @Ivanov2011], which has been observed only very recently in continuous beam deceleration [@Chen2018]. In contrast to the *repulsive* Sisyphus cooling scheme demonstrated recently by our group [@Cooper2018a], the attractive scheme relies on the excited state experiencing a significantly stronger trapping potential than the ground state. For linearly polarized trapping light, this situation is realized for the the $m_J=\pm1$ sublevels of $\ce{^{3}P_{1}}$ in wavelengths ranging from $\approx700~\text{nm}$ to $\approx900~\text{nm}$. (For longer wavelengths, including $1064$ nm, repulsive Sisyphus cooling can be used.) This enables us to fine-tune the wavelength to $813.4$ nm, which is magic for the clock transition to $\ce{^{3}P_{0}}$, while providing cooling conditions for the transition to $\ce{^{3}P_{1}}$.\
*Imaging results -* Our results show simultaneous high-fidelity and low-loss detection of single atoms. First, we observe a histogram of photons collected in $50$ ms with clearly resolved count distributions corresponding to cases with no atom and a single atom (Fig. \[FigImagingLifetime\]a). Taking a second image, after $29$ ms hold time under Sisyphus cooling alone, we find a survival probability of 0.99932(8). At the same time, the fidelity of the scheme (defined by the accuracy of distinguishing no atom from a single atom [@Cooper2018a]) reaches a value of 0.99991(1), demonstrating simultaneous low-loss and high-fidelity imaging. These values are enabled by a lifetime under imaging conditions of more than two minutes (Fig. \[FigImagingLifetime\]b), while requiring only tens of milliseconds to acquire enough photons. We also find lifetimes under Sisyphus cooling (without $461$ nm light) that are longer than seven minutes.
These results enable us to repeatably image single atoms for thousands of times. Specifically, we alternate between $50$ ms long imaging blocks and $29$ ms pure cooling blocks for 2000 times, and collect photons on the camera in each imaging block. Recording the survival probability as a function of the number of images $N$, we find that even after 2000 high-fidelity images, the survival fraction stays above $\approx 0.5$. In more detail, the decay follows an approximate exponential trend with $p_1^N$, where the single image survival probability is $p_1 \approx 0.9997$, slightly higher than the above quoted value measured with only two images. We note that the success probability in atom-by-atom assembly schemes is also limited by $p_1^{2N}$ as a function of the final array size. (The factor of two appears because two consecutive images with interleaved hold times are needed for successful rearrangement.) Our results indicate that assembly of systems with thousands of atoms could be possible in terms of imaging fidelity, loss probability, and lifetime during the assembly step.\
![ **Sisyphus cooling during imaging.** [(a)]{} The survival probability of the atom versus the detuning (with respect to the free space resonance) of the $689$ nm Sisyphus cooling beam. A broad feature with high survival fraction is observed with a red-detuned edge approximately at the detuning for exciting atoms at the trap bottom, consistent with our interpretation of attractive Sisyphus cooling. Data is for a 50 ms imaging time with a $461$ nm scattering rate of $\approx 41$ kHz. For comparison, the results in Fig. \[FigImagingLifetime\] are for $-2.3$ MHz detuning. We perform cooling during imaging with an intensity of the $689$ nm beam of $I/I_s\approx1000$, where $I_s$ is the saturation intensity. [(b)]{} The survival fraction of single atoms versus the scattering rate from the $461$ nm imaging beam under simultaneous repumping and Sisyphus cooling for an imaging time of $50~\text{ms}$. For scattering rates $\gtrsim 80$ kHz, increased loss indicates that Sisyphus cooling is not able to compensate for $461$ nm recoil heating anymore. For comparison, the results in Fig. \[FigImagingLifetime\] are for $\approx 41$ kHz scattering rate.[]{data-label="FigCooling"}](figures/FigCooling){width="\columnwidth"}
*Cooling results -* These low-loss high-fidelity results are achieved by optimizing the Sisyphus cooling frequency and picking a conservative $461$ nm scattering rate as detailed in Fig. \[FigCooling\] and the corresponding caption.
In addition, attractive Sisyphus cooling without the 461 nm beam results in a radial temperature below $5~\mu$K (Fig. \[FigThermometry\]), which we quote as a conservative upper bound based on a release-and-recapture technique [@Tuchendler2008; @Thompson2013]. This technique is primarily sensitive to radial temperatures, and is compared to classical Monte-Carlo simulations to extract a temperature. However, comparison to classical simulation would overestimate actual temperatures that are close to or below the energy scale of the radial trapping frequency ($T \lesssim \frac{\hbar \omega}{2k_B}$), which for us is roughly $2.4~\mu$K. More precise measurement of lower temperatures could be done via resolved sideband spectroscopy, which we leave for further work. An open question in this context is whether cooling to the motional ground state can be achieved in the strongly off-magic cooling configuration used here.
As Sisyphus cooling occurs due to a trapping mismatch between ground and excited state, it is expected that cooling occurs in all directions even for a single radial cooling beam. The low loss we observe during imaging already provides evidence of this mechanism, as fluorescence recoil heating must be mitigated in all directions. Determination of the axial temperature after cooling can be achieved via techniques such as adiabatic rampdown [@Alt2003; @Tuchendler2008] or spectroscopy of thermally-broadened light shifts [@Cooper2018a]. Our preliminary results with such techniques are consistent with three-dimensional temperatures similar to our quoted radial temperatures; however, we leave a thorough investigation to future work. We note that we have made no explicit attempt to further cool the axial direction, and that doing so is likely possible by applying a beam in that direction. Finally, we note that the clock-magic condition of our tweezers opens the door to well-resolved sideband thermometry on the clock transition, which would be required to see resolved axial sidebands that are otherwise poorly resolved on the intercombination line at our trapping frequencies.\
![ **Sisyphus cooling to low temperatures** [(a)]{} The survival fraction in an array in a release-and-recapture measurement performed by diabatically turning off the traps for a variable time followed by a sudden switch-on [@Tuchendler2008]. Lower temperatures are indicated by higher survival rates at long off-times. We show data after imaging (blue squares) and after adding a dedicated cooling block with Sisyphus cooling alone (red circles). Results are compared with classical Monte-Carlo simulations for a three-dimensional thermal distribution at $5$ $\mu$K (dashed line). Note that the release-and-recapture method is mostly sensitive to the energy distribution in the radial direction. [(b)]{} Survival fraction in an array after release-and-recapture for 60 $\mu$s off-time versus the red frequency during Sisyphus cooling for $25$ ms with an intensity of $I/I_s\approx200$. The dashed line represents the case without a dedicated cooling block. We find that atoms are cooled for appropriately chosen red detunings, and heated for detunings further to the blue. This is consistent with an understanding of Sisyphus cooling as an *attractor* in energy space [@Taieb1994; @Ivanov2011]. Data in (a) is at $-2.6$ MHz detuning. []{data-label="FigThermometry"}](figures/FigThermometry){width="\columnwidth"}
*Clock-state resolved detection -* Finally, as an outlook we characterize our ability to perform low-loss state-resolved read-out of the optical clock states $\ce{^{1}S_{0}}\equiv {| g \rangle}$ and $\ce{^{3}P_{0}}\equiv {| e \rangle}$. As detailed below, our scheme relies on shelving techniques that are routinely used in ion trap experiments to realize low-loss, high-fidelity state-resolved detection [@Leibfried2003; @Myerson2008; @Harty2014]. They are also prevalent in optical lattice clocks with alkaline-earth atoms, but have not been extended to single-atom-resolved imaging [@Takamoto2005; @Boyd2006]. More generally, low-loss state-resolved detection of single *neutral* atoms has been realized only recently with alkali atoms [@Fuhrmanek2011; @Boll2016; @Kwon2017; @Martinez-Dorantes2017; @Wu2018]. Since in this case hyperfine states are used, simultaneous cooling during state-resolved detection in a tweezer is challenging and thus deep traps are required. This will limit scalability, and the approach has so far only been demonstrated in up to five traps [@Kwon2017]. Note that Stern-Gerlach detection of hyperfine spins via spatial separation in a lattice has been performed very recently [@Boll2016; @Wu2018], which provides an alternative route for high-fidelity lossless state-resolved detection.
Our scheme consists of two consecutive images (Fig. \[FigStateResolved\]). In the first image, we aim at detecting atoms in ${| g \rangle}$. To this end, we turn off the $679$ nm repump laser such that atoms in ${| e \rangle}$, in principle, do not scatter any photons. Hence, if we find a signal in the first image, we identify the state as ${| g \rangle}$. In the second image, we turn the $679$ nm repump laser back on to detect atoms in both ${| g \rangle}$ and ${| e \rangle}$. Thus, if an atom is not detected in the first image but appears in the second image, we can identify it as ${| e \rangle}$. If neither of the images shows a signal, we identify the state as “no-atom”.
![ **Low-loss state-resolved detection.** [(a)]{} A statistical mixture of optical clock qubit states is represented as a circle, where the green section at $\ce{^{1}S_{0}}\equiv|g\rangle$ represents the population in $|g\rangle$ and the purple section at $\ce{^{3}P_{0}}\equiv|e\rangle$ represents the population in $|e\rangle$. To measure the population in $|g\rangle$, we image without the 679 nm repumper during which $|e\rangle$ remains dark. The accuracy of measuring the population in $|g\rangle$ is limited both by off-resonant scattering of the tweezer light which pumps $|e\rangle$ back to $|g\rangle$, and by pathways that pump $|g\rangle$ to $|e\rangle$ such as the $\ce{^{1}D_{2}}$ decay channel and the off-resonant scattering from $\ce{^{3}P_{1}}$ during cooling. As a result, the average state detection fidelity is $0.981(1)$. [(b)]{} We perform a second image that includes the 679 nm repumper, which pumps $|e\rangle$ to $|g\rangle$ via the $\ce{^{3}S_{1}}$ state and the 707 nm repumper, such that both states are detected. The pumping process is illustrated by the purple arrow. This image measures the population in $|g\rangle$ and $|e\rangle$, and informs whether the atom has been lost as a result of the first image. We find that the average survival probability of this double-imaging sequence is $0.996(1)$. []{data-label="FigStateResolved"}](figures/FigStateSelectiveReadout){width="\columnwidth"}
We find that the inaccuracy of this scheme is dominated by off-resonant scattering of the tweezer light when atoms are shelved in $|e\rangle$ during the first image. Specifically, by pumping atoms into $|e\rangle$ before imaging, we observe that, at our trap depth of $\approx450$ $\mu$K, they decay back to $|g\rangle$ with a time constant of $\tau_p=470(30)$ ms. This leads to events in the first image, where $|e\rangle$ atoms are misidentified as $|g\rangle$ atoms. To minimize the probability of misidentification, the first imaging time should be as short as possible. To reduce the imaging time, we compromise slightly on the survival probability in order to work with higher $461$ nm scattering rates in the first image (see Fig. \[FigCooling\]). Specifically we use $t=15$ ms at a scattering rate of $\approx72$ kHz. The second image is performed with the same settings as in Fig. \[FigImagingLifetime\].
Additionally, atoms in $|g\rangle$ can be misidentified as $|e\rangle$ if they are pumped to $|e\rangle$ in the first image. This can occur either via the $\ce{^{1}D_{2}}$ leakage channel and subsequent scattering of $707$ nm photons, or via off-resonant scattering of the trap light from $\ce{^{3}P_{1}}$ during cooling. We identify this misidentification probability by initializing atoms in $|g\rangle$, and counting how often we identify them as $|e\rangle$ in the state-resolved imaging scheme.
In summary, we place a conservative upper-bound for the probability of misidentifying $|e\rangle$ as $|g\rangle$ by $e^{-t/\tau_p}=0.031(2)$ and we directly measure the probability of misidentifying $|g\rangle$ as $|e\rangle$ yielding $0.008(1)$. We define the average state detection infidelity for a generic initial state as the mean of these probabilities [@Wu2018], yielding an average state detection fidelity of $0.981(1)$. Further, we similarly define the average survival probability of the double imaging scheme in terms of the measured survival probabilities of $|g\rangle$ and $|e\rangle$, for which we obtain $0.996(1)$.
Our fidelity is comparable to recent measurements with alkali atoms in tweezers [@Fuhrmanek2011; @Kwon2017; @Martinez-Dorantes2017], yet our survival probability is substantially higher than any tweezer- or lattice-based schemes to our knowledge [@Fuhrmanek2011; @Kwon2017; @Martinez-Dorantes2017; @Boll2016; @Wu2018]. These results constitute an excellent setting for continuous measurement in an optical clock. However, we emphasize that this investigation was not exhaustive, and we expect that further optimization of these imaging parameters is possible. In general, these values could be further improved by either imaging in shallower tweezers or in tweezers at a wavelength further detuned from higher lying states. For instance, tweezers operating at 1064 nm are a promising possibility, and would be a convenient choice for operating a quantum gas microscope. Further, it is possible to switch between 813.4 nm tweezers/lattices for clock interrogation during which the trap depth can be orders of magnitude lower, and 1064 nm tweezers/lattices for imaging.
In conclusion, we have addressed two major limitations preventing the implementation of tweezer arrays for optical clock-based quantum information processing and metrology. By working at the magic wavelength for clock operation, we observe imaging with a fidelity of $0.99991(1)$ and a survival probability of $0.99932(8)$, and lifetimes under cooling of more than seven minutes. By employing a double imaging technique with specific combinations of repump lasers, we study low-loss state-resolved detection and observe an average fidelity of $0.981(1)$ with an average survival probability of $0.996(1)$. This work provides a setting for continuous measurement in an optical clock which can suppress laser fluctuations due to the Dick effect [@Lodewyck2009; @Schioppo2017]. Clock operation on bosonic isotopes of AEAs such as $^{88}$Sr used in this work has been performed with Sr [@Akatsuka2010] and Yb [@Taichenachev2006a; @Barber2006]. Moreover, the tools developed in this work enable excitation to highly-excited Rydberg states in the $\ce{^{3}S_{1}}$ series via the clock state $\ce{^{3}P_{0}}$. Engineering long-range Rydberg-mediated interactions will facilitate the generation of entanglement between optical clock qubits, which can be used for quantum information processing [@Saffman2010], quantum simulation [@Labuhn2016; @Bernien2017], and quantum-enhanced metrology via spin squeezing [@Gil2014].
We acknowledge funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF Grant PHY-1733907). This work was supported by the NSF CAREER award (number 1753386), the Sloan Foundation, by the NASA/JPL President’s and Director’s Fund, and by Fred Blum. JPC acknowledges support from the PMA Prize postdoctoral fellowship, and AC acknowledges support from the IQIM Postdoctoral Scholar fellowship.
|
---
abstract: |
We prove that for every nowhere dense class of graphs ${{\mathcal{C}}}$, positive integer $d$, and $\eps>0$, the following holds: in every $n$-vertex graph $G$ from ${{\mathcal{C}}}$ one can find two disjoint vertex subsets $A,B\subseteq V(G)$ such that
- $|A|{\geqslant}(1/2-\eps)\cdot n$ and $|B|=\Omega(n^{1-\eps})$; and
- either $\operatorname{dist}(a,b){\leqslant}d$ for all $a\in A$ and $b\in B$, or $\operatorname{dist}(a,b)>d$ for all $a\in A$ and $b\in B$.
We also show some stronger variants of this statement, including a generalization to the setting of First-Order interpretations of nowhere dense graph classes.
address:
- 'Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland'
- 'Institute of Informatics, University of Warsaw, Poland'
author:
- Marcin Briański
- Piotr Micek
- Michał Pilipczuk
- 'Michał T. Seweryn'
bibliography:
- 'sparsity.bib'
title: 'Erdős-Hajnal properties for powers of sparse graphs'
---
[[^1]]{}
[^1]: The work of Michał Pilipczuk is supported by the project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 677651).
{width="\textwidth"}
|
---
abstract: 'Giant radio haloes in galaxy clusters are the primary evidence for the existence of relativistic particles (cosmic rays) and magnetic fields over Mpc scales. Observational tests for the different theoretical models explaining their powering mechanism have so far been obtained through X-ray selection of clusters, e.g. by comparing cluster X-ray luminosities with radio halo power. Here we present the first global scaling relations between radio halo power and integrated Sunyaev-Zel’dovich (SZ) effect measurements, using the [*Planck*]{} all-sky cluster catalog and published radio data. The correlation agrees well with previous scaling measurements based on X-ray data, and offers a more direct probe into the mass dependence inside radio haloes. However, we find no strong indication for a bi-modal cluster population split between radio halo and radio quiet objects. We discuss the possible causes for this apparent lack of bi-modality, and compare the observed slope of the radio-SZ correlation with competing theoretical models of radio halo origin.'
author:
- |
Kaustuv Basu[^1]\
Argelander Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
date: 'Accepted 2012 January 4. Received 2012 January 3; in original form 2011 September 15'
title: 'A Sunyaev-Zel’dovich take on cluster radio haloes – I. Global scaling and bi-modality using [*Planck*]{} data'
---
\[firstpage\]
galaxies: clusters: intracluster medium – radiation mechanism: thermal – radiation mechanism: non-thermal – radio continuum: general
Introduction
============
The bulk of the baryonic mass in galaxy clusters exists in the form of a low density, diffused ionized gas filling up the space between cluster galaxies. This hot intra-cluster medium (ICM, $T \sim 2-10$ keV) emits in the X-rays through thermal Bremsstrahlung emission, and is also observable in the millimeter/sub-millimeter wavelengths through the Sunyaev-Zel’dovich (SZ) effect, which is a distortion in the intensity of the Cosmic Microwave Background (CMB) radiation caused by the same thermal component (Sunyaev & Zel’dovich 1980). Together, these two observables are central to the use of galaxy clusters as cosmological probes.
The ICM is also host to a large population of ultra-relativistic particles (cosmic rays) and magnetic fields, seen primarily through radio observations. The most spectacular evidence for this non-thermal population comes from observations of giant radio haloes, which are diffuse sources of radio synchrotron emission extending over $\sim 1$ Mpc scales. The haloes are not associated to any particular cluster galaxy, and are morphologically distinct from radio mini-haloes (residing in cluster cool cores), radio relics (formed at the edge of a merger shock) and radio lobes associated with active galactic nuclei. The similarity of their morphology with the ICM suggests a correspondence between their powering mechanism and the total cluster mass (e.g. Liang et al. 2000). They are relatively rare and are found mostly in clusters showing evidence of ongoing mergers. As such, they can prove to be essential in understanding cluster merger dynamics and associated heating processes in the ICM (see e.g. review by Ferrari et al. 2008).
Despite their importance, the powering mechanism of these giant radio haloes remains uncertain. There are two models for particle acceleration in a radio halo volume: the hadronic model which uses collisions between cosmic-ray protons and thermal protons for generating relativistic electrons (Dennison 1980), and the turbulence models where the electrons are re-accelerated through MHD turbulence in the ICM caused by cluster mergers (Brunetti et al. 2001, Petrosian 2001). The distinction between these two models is partly based upon the observed scaling between radio and X-ray power (the latter indicating the total cluster mass), and the fact that X-ray selection seems to indicate two distinct populations of clusters: the radio halo and “radio quiet" ones (e.g Brunetti et al. 2007). However, recent discoveries of radio haloes in clusters with very low X-ray luminosities (Giovannini et al. 2011), and the lack of radio haloes in some mergers (e.g. Russell et al. 2011) show that the X-ray selection may not be as clean as expected. These new observations and the underlying large scatter in the $L_{X}-P_{\mathrm{radio}}$ correlation suggest that a new observational window on the selection and mass estimation of clusters harboring radio haloes can bring some much needed clarity.
One further reason for expecting a robust correlation between radio power and SZ is the timescale argument: the boost in the X-ray luminosity during mergers happens in a relatively short timescale, compared to the gas thermalization in a modified potential well producing a more gradual and moderate increase in the SZ signal (Poole et al. 2007, Wik et al. 2008). This should correspond better with the radio halo time scale ($\sim 1$ Gyr), derived from the spatial extent of the haloes. The integrated SZ signal is also a more robust indicator of cluster mass than the X-ray luminosity, irrespective of cluster dynamical state (e.g. Motl et al. 2005, Nagai 2006). Thus SZ-selection might be able to find radio haloes in late mergers and other massive systems which are left out in X-ray selection.
This letter presents the first radio-SZ correlation for clusters with radio haloes. The radio data is a compilation of published results, and the SZ measurements are taken from the [[*Planck *]{}]{}all-sky cluster catalog (Planck collaboration 2011). All results are derived using the $\Lambda$CDM concordance cosmology with $\Omega_M = 0.26$, $\Omega_{\Lambda} = 0.74$ and $H_0 = 71$ km s$^{-1}$ Mpc$^{-1}$. The quantity ${Y_{\mathrm{SZ}}}$ is used throughout to denote the [*intrinsic*]{} Compton $Y$-parameter for a cluster, $Y d_A^2$, where $d_A$ is its angular diameter distance.
Radio & SZ data sets
====================
We do not attempt to define a new comprehensive sample for this work, rather use a set of available cluster catalogs with radio halo detections and non-detections to probe the robustness of the radio-SZ scaling. Published radio error estimates often ignore systematic effects like flux loss in interferometric imaging and contribution from unresolved point sources, which in turn create an over-estimation of the intrinsic scatter. Since the present work is mainly concerned with the mean slope of the radio-SZ scaling and not its dispersion, error underestimation in the literature will not affect the results as long as the measurements are unbiased.
The radio catalogs can be divided into two groups: those with and without a listing of non-detections. A comprehensive sample in the former category is by Giovannini et al. (2009, hereafter G09), presenting results at 1.4 GHz for $z<0.4$ clusters. Potentially problematic are its mixing of radio halos and mini halos, and not separating contributions from radio relics. More critically, the sizes of the radio haloes are approximated by the observed largest linear sizes (LLS), which is not a good approximation for radio halo diameter. To address the latter issue we use a smaller subsample by Cassano et al. (2007, hereafter C07), which provides a better measurement of radio halo sizes by averaging their minimum and maximum extensions.
The most systematic study of radio halo non-detections in an X-ray selected sample is by Venturi et al. (2008), using GMRT observation at 610 GHz. However, this sample contains too few clusters which have [[*Planck *]{}]{}SZ measurements to obtain any robust correlation. We therefore use the compilation by Brunetti et al. (2009, hereafter B09), which lists GMRT results scaled to 1.4 GHz with other unambiguous halo detections. The non-detection upper limits were obtained by simulating fake radio halos in the GMRT data and scaling to 1.4 GHz by using $\alpha=1.3$, a typical spectral index for radio haloes. Our final sample is from Rudnick & Lemmerman (2009, hereafter R09) who re-analyzed WENSS survey data at 327 MHz for an X-ray selected sample. The shallowness of WENSS data makes R09 ineffective in testing bi-modality, as the $3\sigma$ upper limits are not sufficiently below the detection level. Its use is mainly limited to testing possible changes in the scaling law at lower frequencies.
The Sunyaev-Zel’dovich effect measurements are taken from the Planck ESZ catalog (Planck collaboration 2011). This all-sky cluster catalog provides a list of 189 objects in the highest mass range out to a redshift $z\sim 0.6$, selected at $S/N > 6$ from the first year survey data. Out of these, 22 are either new cluster candidates or have no X-ray data. The remaining 167 clusters, spanning a redshift range $0.01 <z <0.55$, are cross-correlated against the radio catalogs. All radio halo clusters therefore have an $R_{500}$ estimate in the [[*Planck *]{}]{}catalog obtained from the $L_X-M_{500}$ relation. This is used to model the pressure profile for each cluster when scaling their integrated SZ signal, ${Y_{\mathrm{SZ}}}$, between different radii.
Proper regression analysis between the radio and SZ observables is fundamental to this work. We must allow for measurement errors and intrinsic scatter in both observables, which makes the regression analysis non-trivial (see Kelly 2007). We use the publicly available <span style="font-variant:small-caps;">idl</span> code by Kelly to perform the regression analysis using a Bayesian approach. An important advantages of this method is the provision for including non-detections.
Sample sub-category B (slope) A (norm.)
-------- --------------------- ---------------- ---------------
G09 global $1.84\pm 0.38$ $31.3\pm1.4$
inside LLS $0.95\pm 0.14$ $28.8\pm 0.5$
C07 global $1.88\pm 0.24$ $31.4\pm0.8$
inside $R_{H}$ $1.17\pm 0.18$ $29.7\pm0.8$
B09 global, haloes only $2.03\pm 0.28$ $32.1\pm 1.0$
$+$ non-detections $2.41\pm 0.44$ $33.4\pm 1.6$
R09 global, haloes only $0.81\pm 0.36$ $28.1\pm 1.4$
$+$ non-detections $1.38\pm 0.43$ $29.8\pm 1.8$
: Regression coefficients for the scaling relation $\log(P_{\nu}) = A + B ~\log({Y_{\mathrm{SZ}}})$. The term [*global*]{} implies correlation with the total SZ signal, as opposed to that scaled inside the halo radius. []{data-label="regtable"}
Samples: G09=Giovannini et al. 2009; C07=Cassano et al. 2007; B09=Brunetti et al. 2009; R09=Rudnick & Lemmerman 2009.
{width="0.92\columnwidth"} {width="0.92\columnwidth"}
Results
=======
Radio-SZ scaling
----------------
The radio-SZ scaling relation is obtained by performing linear regression in log-space: $\log(P_{\nu}) = A + B\log({Y_{\mathrm{SZ}}})$. The normalization $A$ and slope $B$ are obtained from the Markov Chains, as well as the intrinsic scatter $\sigma_{\log P|\log Y}$. A summary of the results are given in Table 1. The first correlation example is from the G09 sample, comparing the radio power with the [*global*]{} SZ signal (Fig. \[GCsamp\] [*left*]{}). Out of 32 objects in this sample 24 have [*Planck*]{} counterparts. The mean slope is $1.84\pm 0.38$. There is a lot of scatter in this correlation, with mean scatter 0.45 dex, i.e. roughly a factor $\sim 2.8$. Much of this scatter is driven by the low-redshift objects which are under-luminous (e.g. A401 and A754). This potentially indicates a systematic bias in their total flux and size measurements with interferometers. Only A1351 stands out as overtly radio luminous for its mass, although later revisions of its radio power (Giacintucci et al. 2009) moves it closer to the mean value.
The [[*Planck *]{}]{}catalog provides the integrated $Y$ parameter within radius $5 R_{500}$, obtained from a matched filtering algorithm assuming a universal gas pressure profile (see Planck collaboration 2011). At this radius, $Y_{5R_{500}}^{\mathrm{cyl}} \approx Y_{5R_{500}}^{\mathrm{sph}}$. This is nearly 3 times the cluster virial radius, and much larger than the extent of the radio emitting regions. Therefore, a tighter correlation can be expected if this [*global*]{} SZ signal is scaled down to that inside the radio halo volume. We do this conversion by assuming the universal pressure profile of Arnaud et al. (2010), as also used by the [*Planck*]{} team. In particular, the best fit profile for mergers/disturbed clusters from the appendix of Arnaud et al. (2010) is used, but the difference is negligible if the mean profile is used instead. This scaling of the SZ signal inside LLS changes the results significantly. Correlation between radio and SZ powers inside the halo volume becomes consistent with a linear relation, with mean slope $0.95\pm 0.14$ (Fig.\[GCsamp\] [*right*]{}) and a reduced mean intrinsic scatter in radio (0.35 dex).
The largest linear sizes are in general not a good approximation for radio halo diameters, so the above analysis is repeated with the C07 sample using their revised measurement of halo radius, $R_H$. The slope for the global correlation with this sample is $1.88\pm 0.24$, whereas after scaling the SZ signal inside $R_H$ it becomes $1.17\pm 0.18$, with mean intrinsic scatter 0.28 dex (Fig.\[GCsamp\] [*inset*]{}). Although this is statistically fully consistent with the scaled result of the G09 sample inside the LLS, we use this slightly super-linear correlation when making comparisons with theoretical models, due to the better definition of halo radius. We emphasize that from the current analysis using radio data from the literature, a linear correspondence between radio and SZ power is a fully valid result.
The R09 sample at 327 MHz indicates a flattening of the correlation slope at lower frequencies: the best fit value is $0.81\pm 0.36$ when considering the halo sample, with a scatter of only 0.21 dex. The large flux uncertainties (and correspondingly low intrinsic scatter) reflect on the shallowness of the WENSS data, which is more than an order of magnitude less sensitive compared to typical VLA measurements scaled to its frequency. However, the method used by R09 to detect haloes (and place upper limits) based on simulating sources in control regions safeguards against flux underestimation bias, which will otherwise occur in a visual inspection. There can be a residual bias from fluxes associated with small scale structures that are not recovered. If non-detection upper limits in R09 are included in the correlation, then the slope becomes $1.38\pm 0.43$, which is consistent at $1\sigma$ with the scaling result at 1.4 GHz.
{width="0.92\columnwidth"} {width="0.92\columnwidth"}
Lack of strong bi-modality
--------------------------
To test whether there are two distinct populations of clusters: those hosting powerful radio haloes and those without, we use the B09 sample. Actual flux measurements at 1.4 GHz are used for the clusters A209 and RXCJ1314 (Giovannini et al. 2009), instead of the extrapolated values from 610 MHz as given by B09. All non-detection upper limits are extrapolation of simulation results at 610 MHz using halo spectral index $\alpha=1.3$. Regression analysis for the two cases (with and without halo non-detections) yields slopes which are statistically consistent with each other, even though a bi-modal division appears to be emerging (Fig. \[Bsamp\] [*left*]{}). Significantly, we do not find high-${Y_{\mathrm{SZ}}}$ objects with radio non-detections, as in the case of highly X-ray luminous “radio quiet” cool core clusters. But the small number of non-detections makes it difficult to conclude whether the bi-modality is weaker or non-existent. All non-detection upper limits lie below the 95% confidence interval of the halo-only correlation, suggesting clusters with upper limits are generally radio under-luminous. This can be partly redshift-driven, as the samples are not SZ-complete.
An alternative, although not independent, way to visualize this result is to correlate the radio power directly against cluster gas mass. Rough estimates of $M_{\mathrm{gas}}$ inside $R_{500}$ were obtained by dividing the Planck ${Y_{\mathrm{SZ}}}$ values by the mean X-ray temperatures taken from the literature. The global ${Y_{\mathrm{SZ}}}$ measurements of [[*Planck *]{}]{}are scaled to that inside $R_{500}$ using the universal pressure profile, as this is the radius within which X-ray temperatures are typically obtained. The result is similar to the $P_{1.4}-{Y_{\mathrm{SZ}}}$ correlation, lacking a strong bi-modal division (Fig.\[Bsamp\], right panel). The scatter is increased by roughly 30% (from 0.5 dex to 0.6 dex), in line with the expectation that ${Y_{\mathrm{SZ}}}$ is a lower scatter mass proxy. The four non-detection clusters have generally lower $T_X$ values as compared to the halo detections (median $T_X$ is $7.3$ keV compared to $8.7$ keV). It should be made clear that by taking $T_X$ estimates from literature we ignore potential errors due to non-uniform radius for extracting $T_X$, systematic differences between XMM-Newton and Chandra measurements, etc. However, the mean slope for the $P_{1.4}-M_{\mathrm{gas}}$ scaling relation, $3.2\pm 0.7$ with the full sample, is consistent with the global mass scaling derived from the $Y-M$ relation (see §\[Msection\]), indicating that no additional biases are incurred while using this non-uniform selection of X-ray temperatures.
A tentative argument for a selection bias in X-ray complete samples and the ensuing bi-modality can be given by comparing the relative frequency with which radio haloes and non-detections occur in the [[*Planck *]{}]{}catalog. In Venturi et al. (2008), GMRT data were obtained for a complete X-ray selected sample, with 6 detection of radio haloes plus 20 non-detections. The [[*Planck *]{}]{}catalog contains 5 out of these 6 halo clusters, but only 4 out of 20 non-detection clusters. For the B09 sample this ratio is 16 out of 21 radio halo clusters and 4 out of 20 non-detections (the same non-detections as in the Venturi et al. sample). Since the [[*Planck *]{}]{}catalog should not have a significant bias towards mergers, this provides an indirect evidence for our hypothesis that being SZ-bright (hence massive) is a better indicator for clusters hosting radio haloes, as opposed to being X-ray luminous. Even though the R09 sample is too shallow to directly test bi-modality, it interestingly follows this same trend: [[*Planck *]{}]{}reports measurement of 12 out of 14 counterparts for haloes and other diffuse emissions, as opposed to only 15 out of 58 counterparts for non-detections.
Theoretical considerations
==========================
Mass scaling of the radio halo power {#Msection}
------------------------------------
The independent variable, ${Y_{\mathrm{SZ}}}$, is defined as the integral of the total pressure in a spherical volume, and hence is proportional to the total gas mass: $${Y_{\mathrm{SZ}}}\equiv Y d_A^2 \propto \int n_e T_e~ dV \propto M_{\mathrm{gas}} T_e = f_{\mathrm{gas}} M_{\mathrm{tot}} T_e.
\label{eq:szdef}$$ Here $T_e$ is the mean gas temperature within the integration radius, and $f_{\mathrm{gas}}$ is the gas-to-mass ratio. Assuming hydrostatic equilibrium and isothermality, the temperature scales to the total mass as $T_e \propto M_{\mathrm{tot}}^{2/3} E(z)^{2/3}$ (e.g. Bryan & Norman 1998), where $E(z)$ is the ratio of the Hubble parameter at redshift $z$ to its present value. Therefore, the scaling between the SZ observable and total mass is ${Y_{\mathrm{SZ}}}E(z)^{-2/3} \propto f_{\mathrm{gas}} M_{\mathrm{tot}}^{5/3}$. Numerical simulations, analytical models and SZ observations indicate that this mass scaling is extremely robust, with little scatter over a large range of cluster mass, dynamical state or other details of cluster physics (e.g. Motl et al. 2005, Reid & Spergel 2006, Andersson et al. 2011). We thus assume this scaling to be valid also [*inside*]{} a cluster at different radii, provided that the radius is sufficiently large to exclude complex physics at clusters cores. The large halo sizes measured by C07 ($\bar{R}_H \sim 600$ kpc, typically of the same order as $R_{2500}$), ensures that they encompass a representative cluster volume. The $E(z)^{-2/3}$ factor for self-similar evolution changes the scaling results only marginally, well within the statistical errors. The gas mass fraction, $f_{\mathrm{gas}}$, has a weak dependence on cluster mass: $f_{\mathrm{gas}} \propto M_{\mathrm{500}}^{~0.14}$ (Bonamente et al. 2008, Sun et al. 2009). Assuming the same mass dependence of $f_{\mathrm{gas}}$ for all radii, we therefore obtain $$P_{1.4} \propto M_H^{~~2.1\pm 0.3} \propto M{_{\mathrm{vir}}}^{~~3.4 \pm 0.4}.
\label{eq:mscale}$$ In the above, $M_H$ is the total mass inside radio haloes, and $M{_{\mathrm{vir}}}$ is the cluster virial mass which scales linearly with $M_{\mathrm{tot}}(<5R_{500})$. The scaling index inside haloes is in good agreement with previous X-ray hydrostatic mass estimates (e.g. Cassano et al. 2007). The global scaling with total cluster mass can be a useful parameter for estimating radio halo statistics, particularly in simulations.
The radio halo sizes are known to scale non-linearly with cluster radius, in a break from self-similarity (Kempner & Sarazin 2001, Cassano et al. 2007). Indeed, using the X-ray derived $R_{500}$ measurements from the [[*Planck *]{}]{}catalog, we obtain the empirical relation $R_H \propto R_{500}^{\ 3.1 \pm 0.2}$ with the C07 sample, consistent with the estimate by C07 using $L_X-M{_{\mathrm{vir}}}$ scaling relation ($R_H \propto R{_{\mathrm{vir}}}^{2.6\pm 0.5}$). A consequence of this rapid increase in radius is a drop of the mean gas density inside haloes with increasing halo mass. Our observed scaling between the halo radius and scaled SZ signal, $R_H \propto Y_H^{\ 0.31\pm 0.03}$, implies that the mean gas density ($\bar{n}_H$) scales down as roughly $\bar{n}_H \propto T_e^{-0.9}$, or assuming thermal equilibrium inside haloes, as $\bar{n}_H \propto M_H^{\ -0.6}$. This brings the observed non self-similar scaling between $R_H$ and $R{_{\mathrm{vir}}}$ in conformity with the mass scaling in Eq.(\[eq:mscale\]). It is worth mentioning at this point that radio halo size measurements with insufficient S/N will tend to show a steeper dependence on luminosity than the true scaling, since only the bright central regions will be picked up.
Comparison with radio/X-ray scaling
-----------------------------------
There is some confusion in the literature about the exact power in the X-ray/radio scaling: reported values using the luminosity in the soft X-ray band range between $P_{\nu} \propto L_{X[0.1-2.4]}^{1.6-2}$ (Brunetti et al. 2007, Kushnir et al. 2009). Using the regression method used in this work we find the scaling index in the middle of this range, e.g. from the B09 sample the mean slope for $\log P_{\nu} - \log L_{X[0.1-2.4]}$ correlation is $1.80\pm 0.21$, with mean intrinsic scatter 0.3 dex. The mass-luminosity relation for the X-ray soft band is well-established observationally. We use the result given by Zhang et al. (2011) for disturbed clusters in the HIFLUGCS sample: $L_{X[0.5-2]} \propto [M_{\mathrm{gas}, R_{500}} E(z)]^{1.16\pm 0.04}$, where the luminosities are core corrected. This combined with the weak mass dependence of $f_{\mathrm{gas}}$ produces a mass scaling of radio power as $P_{1.4} \propto M_{500}^{\ 2.4}$, which is much shallower than the virial mass scaling obtained in Eq.(\[eq:mscale\]) but roughly consistent with the halo mass power law. This indicates that the global X-ray emission acts as a relatively good proxy for radio halo masses due to its peaked profile, as most of the X-ray flux comes from within a radius that is $\lesssim R_H$.
Expectations from theoretical models
------------------------------------
The hadronic model for radio synchrotron emission postulates that electrons at ultra-relativistic energies are produced in the ICM by $p$-$p$ collisions between cosmic ray protons and thermal protons (see review by Ensslin et al. 2011 and references therein). For estimating the scaling relation between radio halo power and cluster mass, we follow the formulation by Kushnir et al. (2009). In this model, the total radio power is the volume integral of the cosmic ray energy density ($\epsilon_{\mathrm{CR}} = X_{\mathrm{CR}} ~n ~k T_e$) and the hadronic interaction rate ($\tau_{\mathrm{pp}}^{-1} \sim n ~\sigma_{\mathrm{pp}}$): $$P_{\nu} = \int \tau_{\mathrm{pp}}^{-1} ~\epsilon_{\mathrm{CR}} ~dV
\sim X_{\mathrm{CR}} ~n^2 ~k T_e ~\sigma_{\mathrm{pp}} ~f_B ~R_H^3.\\$$ Here $X_{\mathrm{CR}}$ is the ratio between cosmic ray pressure and thermal pressure, $n$ is the gas density, $\sigma_{\mathrm{pp}}$ is the $p$-$p$ collision cross-section, $f_B$ is the volume filling factor for magnetic fields, and $R_H$ is the halo radius. In the second step we have assumed the magnetic field energy density to be much larger than the CMB energy density, $B >> B_{\mathrm{CMB}} \approx 3.2(1+z)^2 \mu$G. Considering that ${Y_{\mathrm{SZ}}}(<R_H) \sim n ~k T_e ~R_H^3$, we thus obtain: $P_{\nu} / {Y_{\mathrm{SZ}}}\propto X_{\mathrm{CR}} ~f_B ~n ~\sigma_{\mathrm{pp}}$. Therefore, if the cosmic ray fraction and mean density do not depend on the halo mass, we recover the observed linear dependence between radio power and the SZ signal inside haloes. The latter assumption, however, is in conflict with our observed scaling of mean gas density (§\[Msection\]), which actually drops with increasing halo mass. Another potential problem is the assumption of strong magnetic fields, $B_H >> B_{\mathrm{CMB}}$, over the entire halo volume. A likely scenario with the hadronic model would therefore be to assume a more clumpy radio emission, where the regions contributing most of the radio power have constant densities and strong magnetic fields.
In the turbulent re-acceleration model a pre-existing population of electrons at lower energies gets re-accelerated by merger induced turbulence (see review by Ferrari et al. 2008 and references therein). The powering mechanism of radio haloes is complex, but for a simple estimate we can follow the formulation by Cassano & Brunetti (2005) and Cassano et al. (2007). In their model, the energy injection rate from turbulence depends on the mean density and velocity dispersion inside the radio haloes; $\dot{\varepsilon}_t \propto n ~\sigma_H^2$, where $\sigma_H^2 \equiv GM_H/R_H$. The total power of a radio halo is then $$P_{\nu} \sim \int \dot{\varepsilon}_t ~(\Gamma_{\mathrm{rel}}/\Gamma_{\mathrm{th}})
~dV \propto \dfrac{M_H~ \sigma_H^3}{{\cal F}(z, M_H, B_H)} ,
\label{eq:turb}$$ where $\Gamma$ is the turbulence damping rate transferring energy to the particles, and the function ${\cal F}(z, M_H, B_H)$ is constant in the asymptotic limit of strong magnetic fields. Thus to the first approximation, Eq.(\[eq:turb\]) implies $P_{\nu} \propto Y_H T_e^{1/2}$, in good agreement with the slightly super-linear slope inside haloes seen from the C07 sample ($P_{1.4} \propto Y_H^{\ 1.17 \pm 0.18}$). However, using the definition of $\sigma_H$ and the observed scaling between halo mass and radius, we find a mass dependence slightly shallower than observed: $P_{\nu} \propto M_H^{1.7-1.8}$, which is still consistent with Eq.(\[eq:mscale\]). This may indicate a preference for more realistic field strengths, e.g. figure 2 in C07 suggests approximately ${\cal F} \propto M_H^{-0.3}$ if the mean field strength is of the order $5-6 ~\mu$G inside a radio halo of mass $M_H \sim 10^{14.5}$ M$_{\odot}$, assuming $B_H \propto M_H^{0.5}$. A shallower $B_H-M_H$ relation will correspondingly imply an weaker field to explain the observed $P_{1.4}-M_H$ scaling.
Conclusions
===========
In this letter we presented the first radio-SZ correlation results for clusters hosting radio haloes, using published radio data and the [[*Planck *]{}]{}SZ catalog. There is a clear correspondence between these two thermal and non-thermal components as expected from the well-established radio/X-ray correlation. On the other hand, we found no strong bi-modal division in the cluster population split between radio halo and “radio quiet” objects. The halo non-detection clusters are generally radio under-luminous, but their occurrence in the [[*Planck *]{}]{}catalog is much less frequent as compared to all X-ray complete samples, and as such we can not conclude whether the bi-modality is weaker or non-existent when measured against SZ. A likely explanation for this difference can be that the bi-modality seen in the $L_X$ selection comes from a bias towards lower mass cool core systems (which are radio quiet), whereas SZ selection picks up the most massive systems irrespective of their dynamical states. A forthcoming work will purport to test this hypothesis using a complete SZ-selected sample.
The radio-SZ correlation results were compared with the simplified theoretical predictions from hadronic and turbulent re-acceleration models. Even though the observed correlation power can be explained from both these models under certain assumptions, the turbulent re-acceleration model can be considered a better fit to the data given the simple formulations used in this letter. The difference between the global radio-SZ scaling and the one within the halo volume is explainable from the non-linear scaling between radio halo mass and total cluster mass. An indicative flattening of the correlation slope was observed when considering a cluster sample at 327 MHz, but the result became consistent with 1.4 GHz observations when the haloes and non-detections were considered together as one population.
Acknowledgments {#acknowledgments .unnumbered}
===============
I am grateful to Klaus Dolag and Christoph Pfrommer for their help in clarifying the theory and observations of radio haloes. I thank Arif Babul, Luigi Iapichino, Silvano Molendi, Florian Pacaud and Martin Sommer for helpful discussions, Mariachiara Rossetti for providing a temperature estimate for the cluster AS780, and in particular the anonymous referee for a thorough reading of the manuscript and suggesting numerous improvements. I acknowledge the invitation to participate in the KITP Program “Galaxy clusters: the crossroads of astrophysics and cosmology”, supported by the NSF Grant No. PHY05-51164, where this research was initiated.
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\[lastpage\]
[^1]: E-mail: kbasu@astro.uni-bonn.de
|
---
abstract: |
The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar’s constraint qualification holds, which is called the “sum problem”.
In this paper, we establish the maximal monotonicity of $A+B$ provided that $A$ and $B$ are maximally monotone operators such that ${\ensuremath{\operatorname{dom}}}A\cap{\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B\neq\varnothing$, and $A+N_{\overline{{\ensuremath{\operatorname{dom}}}A}}$ is of type (FPV). This generalizes various current results and also gives an affirmative answer to a problem posed by Borwein and Yao. Moreover, we present an equivalent description of the sum problem.
author:
- 'Liangjin Yao[^1]'
date: 'June 29, 2014'
---
[**2010 Mathematics Subject Classification:**]{}\
[Primary 47H05; Secondary 49N15, 52A41, 90C25]{}
[**Keywords:**]{} Constraint qualification, convex function, convex set, Fitzpatrick function, maximally monotone operator, monotone operator, normal cone operator, operator of type (FPV), subdifferential operator, sum problem.
Introduction
============
Throughout this paper, we assume that $X$ is a real Banach space with norm $\|\cdot\|$, that $X^*$ is the continuous dual of $X$, and that $X$ and $X^*$ are paired by ${\langle{{\cdot},{\cdot}}\rangle}$. Let $A\colon X{\ensuremath{\rightrightarrows}}X^*$ be a *set-valued operator* (also known as point-to-set mapping or multifunction) from $X$ to $X^*$, i.e., for every $x\in X$, $Ax\subseteq X^*$, and let ${\ensuremath{\operatorname{gra}}}A := {\big\{{(x,x^*)\in X\times X^*} \mid {x^*\in Ax}\big\}}$ be the *graph* of $A$, and ${\ensuremath{\operatorname{dom}}}A:= {\big\{{x\in X} \mid {Ax\neq\varnothing}\big\}}$ be the *domain* of $A$. Recall that $A$ is *monotone* if $${\langle{{x-y},{x^*-y^*}}\rangle}\geq 0,\quad \forall (x,x^*)\in {\ensuremath{\operatorname{gra}}}A\;\forall (y,y^*)\in{\ensuremath{\operatorname{gra}}}A.$$ We say $A$ is *maximally monotone* if $A$ is monotone and $A$ has no proper monotone extension (in the sense of graph inclusion). Let $A:X\rightrightarrows X^*$ be monotone and $(x,x^*)\in X\times X^*$. We say $(x,x^*)$ is *monotonically related to* ${\ensuremath{\operatorname{gra}}}A$ if $$\begin{aligned}
\langle x-y,x^*-y^*\rangle\geq0,\quad \forall (y,y^*)\in{\ensuremath{\operatorname{gra}}}A.\end{aligned}$$ Let $A:X\rightrightarrows X^*$ be maximally monotone. We say $A$ is *of type (FPV)* [@FitzPh; @VV5] if for every open convex set $U\subseteq X$ such that $U\cap {\ensuremath{\operatorname{dom}}}A\neq\varnothing$, the implication $$x\in U\,\text{and}\,(x,x^*)\,\text{is monotonically related to ${\ensuremath{\operatorname{gra}}}A\cap (U\times X^*)$}
\Longrightarrow (x,x^*)\in{\ensuremath{\operatorname{gra}}}A$$ holds.
Monotone operators have proven important in modern Optimization and Analysis; see, e.g., the books [@BC2011; @BorVan; @BurIus; @ButIus; @ph; @Si; @Si2; @RockWets; @Zalinescu; @Zeidler2A; @Zeidler2B] and the references therein. We adopt standard notation used in these books. Given a subset $C$ of $X$, ${\ensuremath{\operatorname{int}}}C$ is the *interior* of $C$, $\overline{C}$ is the *norm closure* of $C$, and ${\ensuremath{\operatorname{conv}}}{C}$ is the *convex hull* of $C$. The *indicator function* of $C$, written as $\iota_C$, is defined at $x\in X$ by $$\begin{aligned}
\iota_C (x):=\begin{cases}0,\,&\text{if $x\in C$;}\\
+\infty,\,&\text{otherwise}.\end{cases}\end{aligned}$$
If $C,D\subseteq X$, we set $C-D:=\{x-y\mid x\in C, y\in D\}$. For every $x\in X$, the *normal cone* operator of $C$ at $x$ is defined by $N_C(x):= {\big\{{x^*\in
X^*} \mid {\sup_{c\in C}{\langle{{c-x},{x^*}}\rangle}\leq 0}\big\}}$, if $x\in C$; and $N_C(x):=\varnothing$, if $x\notin C$. For $x,y\in X$, we set $\left[x,y\right]:=\{tx+(1-t)y\mid 0\leq t\leq 1\}$.
Given $f\colon X\to {\ensuremath{\,\left]-\infty,+\infty\right]}}$, we set ${\ensuremath{\operatorname{dom}}}f:= f^{-1}({\ensuremath{\mathbb R}})$. We say $f$ is *proper* if ${\ensuremath{\operatorname{dom}}}f\neq\varnothing$. Let $f$ be proper. Then $\partial f\colon X{\ensuremath{\rightrightarrows}}X^*\colon
x\mapsto {\big\{{x^*\in X^*} \mid {(\forall y\in
X)\; {\langle{{y-x},{x^*}}\rangle} + f(x)\leq f(y)}\big\}}$ is the *subdifferential operator* of $f$. Thus $N_C=\partial\iota_C$. We also set $P_X: X\times X^*\rightarrow X\colon
(x,x^*)\mapsto x$. The *open unit ball* in $X$ is denoted by $U_X:= {\big\{{x\in X} \mid {\|x\|< 1}\big\}}$, the *closed unit ball* in $X$ is denoted by $B_X:= {\big\{{x\in X} \mid {\|x\|\leq 1}\big\}}$, and ${\ensuremath{\mathbb N}}:=\{1,2,3,\ldots\}$. We denote by $\longrightarrow$ and $\operatorname{\rightharpoondown_{\mathrm{w*}}}$ the norm convergence and weak$^*$ convergence of nets, respectively.
Let $A$ and $B$ be maximally monotone operators from $X$ to $X^*$. Clearly, the *sum operator* $A+B\colon X{\ensuremath{\rightrightarrows}}X^*\colon x\mapsto
Ax+Bx: = {\big\{{a^*+b^*} \mid {a^*\in Ax\;\text{and}\;b^*\in Bx}\big\}}$ is monotone. Rockafellar established the following significant result in 1970.
*(See [@Rock70 Theorem 1] or [@BorVan].)* Suppose that $X$ is reflexive. Let $A, B: X\rightrightarrows X^*$ be maximally monotone. Assume that $A$ and $B$ satisfy the classical *constraint qualification*:$${\ensuremath{\operatorname{dom}}}A \cap{\ensuremath{\operatorname{int}\operatorname{dom}}\,}B\neq
\varnothing.$$ Then $A+B$ is maximally monotone.
The generalization of Rockafellar’s sum theorem in the setting of a reflexive space can be found in [@AttRiaThe; @Si2; @SiZ; @BorVan; @FABVY].
The most famous open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators satisfying Rockafellar’s constraint qualification in general Banach spaces; this is called the “sum problem”. Some recent developments on the sum problem can be found in Simons’ monograph [@Si2] and [@Bor1; @Bor2; @Bor3; @BorVan; @BY4FV; @BY3; @BY2; @ZalVoi; @Voi1; @MarSva5; @VV2; @BWY4; @BWY9; @Yao3; @Yao2; @YaoPhD], and also see [@AtBrezis] for the subdifferential operators.
In this paper, we focus on the case when $A, B$ are maximally monotone with ${\ensuremath{\operatorname{dom}}}A\cap{\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B\neq\varnothing$, and $A+N_{\overline{{\ensuremath{\operatorname{dom}}}B}}$ is of type (FPV) (see Theorem \[TePGV:1\]).
Corollary \[CorPbA:1\] provides an affirmative answer to the following problem posed by Borwein and Yao in [@BY3 Open problem 4.5].
> Let $f:X\rightarrow {\ensuremath{\,\left]-\infty,+\infty\right]}}$ be a proper lower semicontinuous convex function, and let $B:X\rightrightarrows X^*$ be maximally monotone with ${\ensuremath{\operatorname{dom}}}\partial f\cap{\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B\neq\varnothing$. Is $\partial f +B$ necessarily maximally monotone?
The remainder of this paper is organized as follows. In Section \[s:aux\], we collect auxiliary results for future reference and for the reader’s convenience. In Section \[s:main\], our main result (Theorem \[TePGV:1\]) is presented. We also show that Problem \[OPRKM:1\] is equivalent to the sum problem.
Auxiliary Results {#s:aux}
=================
We first introduce the well known Banach-Alaoglu Theorem and the two of Rockafellar’s results.
\[BaAlo\] *(See [@Rudin Theorem 3.15] or [@Megg Theorem 2.6.18].)* The closed unit ball in $X^*$, $B_{X^*}$, is weakly$^*$ compact.
\[f:F4\] *(See [[@Rock66 Theorem 3]]{}, [[@Si2 Theorem 18.1]]{}, or [[@Zalinescu Theorem 2.8.7(iii)]]{}.)* Let $f,g: X\rightarrow{\ensuremath{\,\left]-\infty,+\infty\right]}}$ be proper convex functions. Assume that there exists a point $x_0\in{\ensuremath{\operatorname{dom}}}f \cap {\ensuremath{\operatorname{dom}}}g$ such that $g$ is continuous at $x_0$. Then $\partial (f+g)=\partial f+\partial g$.
*(See [@Rock69 Theorem 1] or [@Si2 Theorem 27.1 and Theorem 27.3].)* \[f:refer02c\] Let $A:X{\ensuremath{\rightrightarrows}}X^*$ be maximally monotone with ${\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}A\neq\varnothing$. Then ${\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}A={\ensuremath{\operatorname{int}}}\overline{{\ensuremath{\operatorname{dom}}}A}$ and ${\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}A$ and $\overline{{\ensuremath{\operatorname{dom}}}A}$ are both convex.
The Fitzpatrick function defined below is an important tool in Monotone Operator Theory.
*(See [[@Fitz88 Corollary 3.9]]{}.)* \[f:Fitz\] Let $A\colon X{\ensuremath{\rightrightarrows}}X^*$ be monotone, and set $$F_A\colon X\times X^*\to{\ensuremath{\,\left]-\infty,+\infty\right]}}\colon
(x,x^*)\mapsto \sup_{(a,a^*)\in{\ensuremath{\operatorname{gra}}}A}
\big({\langle{{x},{a^*}}\rangle}+{\langle{{a},{x^*}}\rangle}-{\langle{{a},{a^*}}\rangle}\big),$$ the *Fitzpatrick function* associated with $A$. Suppose also $A$ is maximally monotone. Then for every $(x,x^*)\in X\times X^*$, the inequality ${\langle{{x},{x^*}}\rangle}\leq F_A(x,x^*)$ is true, and the equality holds if and only if $(x,x^*)\in{\ensuremath{\operatorname{gra}}}A$.
The next result is the key to our arguments.
*(See [@Voi1 Theorem 3.4 and Corollary 5.6], or [@Si2 Theorem 24.1(b)].)* \[f:referee1\] Let $A, B:X{\ensuremath{\rightrightarrows}}X^*$ be maximally monotone operators. Assume $\bigcup_{\lambda>0} \lambda\left[P_X({\ensuremath{\operatorname{dom}}}F_A)-P_X({\ensuremath{\operatorname{dom}}}F_B)\right]$ is a closed subspace. If $$F_{A+B}\geq\langle \cdot,\,\cdot\rangle\;\text{on \; $X\times X^*$},$$ then $A+B$ is maximally monotone.
Applying Fact \[CoHull\], we can avoid computing the domain of the Fitzpatrick functions in Fact \[f:referee1\] (see Corollary \[VoiSimn:1\] below).
*(See [@BY2 Theorem 3.6] or [@BY3].)* \[CoHull\] Let $A:X{\ensuremath{\rightrightarrows}}X^*$ be a maximally monotone operator. Then $$\begin{aligned}
\overline{{\ensuremath{\operatorname{conv}}}\left[{\ensuremath{\operatorname{dom}}}A\right]}=\overline{P_X\left[{\ensuremath{\operatorname{dom}}}F_A\right]}.\end{aligned}$$
\[NonL:1\] Let $A, B\colon X{\ensuremath{\rightrightarrows}}X^*$ be maximally monotone, and suppose that $\bigcup_{\lambda>0}\lambda\left[{\ensuremath{\operatorname{dom}}}A-{\ensuremath{\operatorname{dom}}}B\right]$ is a closed convex subset of $X$. Then $$\begin{aligned}
\bigcup_{\lambda>0}\lambda\left[{\ensuremath{\operatorname{dom}}}A-{\ensuremath{\operatorname{dom}}}B\right]=
\bigcup_{\lambda>0}\lambda\left[P_{X}{\ensuremath{\operatorname{dom}}}F_A-P_{X}{\ensuremath{\operatorname{dom}}}F_B\right].\end{aligned}$$
By Fact \[f:Fitz\] and Fact \[CoHull\], we have $$\begin{aligned}
&\bigcup_{\lambda>0} \lambda\left[{\ensuremath{\operatorname{dom}}}A-{\ensuremath{\operatorname{dom}}}B\right]\subseteq
\bigcup_{\lambda>0} \lambda\left[P_{X}{\ensuremath{\operatorname{dom}}}F_A-P_{X}{\ensuremath{\operatorname{dom}}}F_B\right]
\subseteq\bigcup_{\lambda>0} \lambda\left[\overline{{\ensuremath{\operatorname{conv}}}{\ensuremath{\operatorname{dom}}}A}-\overline{{\ensuremath{\operatorname{conv}}}{\ensuremath{\operatorname{dom}}}B}\right]\\
&\subseteq\bigcup_{\lambda>0} \lambda\left[\overline{{\ensuremath{\operatorname{conv}}}{\ensuremath{\operatorname{dom}}}A-{\ensuremath{\operatorname{conv}}}{\ensuremath{\operatorname{dom}}}B}\right]=\bigcup_{\lambda>0} \lambda\left[\overline{{\ensuremath{\operatorname{conv}}}\left[{\ensuremath{\operatorname{dom}}}A-{\ensuremath{\operatorname{dom}}}B\right]}\right]\subseteq
\overline{\bigcup_{\lambda>0} \lambda{\ensuremath{\operatorname{conv}}}\left[{\ensuremath{\operatorname{dom}}}A-{\ensuremath{\operatorname{dom}}}B\right]}\\
&=\bigcup_{\lambda>0} \lambda\left[{\ensuremath{\operatorname{dom}}}A-{\ensuremath{\operatorname{dom}}}B\right]\quad \text{(by the assumption)}.\end{aligned}$$ Hence $\bigcup_{\lambda>0}\lambda\left[{\ensuremath{\operatorname{dom}}}A-{\ensuremath{\operatorname{dom}}}B\right]=
\bigcup_{\lambda>0}\lambda\left[P_{X}{\ensuremath{\operatorname{dom}}}F_A-P_{X}{\ensuremath{\operatorname{dom}}}F_B\right]$.
\[VoiSimn:1\] Let $A, B:X{\ensuremath{\rightrightarrows}}X^*$ be maximally monotone operators. Assume that $\bigcup_{\lambda>0} \lambda\left[{\ensuremath{\operatorname{dom}}}A-{\ensuremath{\operatorname{dom}}}B\right]$ is a closed subspace. If $$F_{A+B}\geq\langle \cdot,\,\cdot\rangle\;\text{on \; $X\times X^*$},$$ then $A+B$ is maximally monotone.
Apply Fact \[f:referee1\] and Lemma \[NonL:1\] directly.
Now we cite some results on operators of type (FPV).
*(See [@FitzPh Corollary 3.4], [@VV1 Theorem 3] or [@Si2 Theorem 48.4(d)].)* \[f:referee0d\] Let $f:X\rightarrow{\ensuremath{\,\left]-\infty,+\infty\right]}}$ be proper, lower semicontinuous and convex. Then $\partial f$ is of type (FPV).
*(See [@Si2 Theorem 44.2].)* \[SDMn:pv\] Let $A:X{\ensuremath{\rightrightarrows}}X^*$ be of type (FPV). Then $$\begin{aligned}
\overline{{\ensuremath{\operatorname{dom}}}A}=\overline{{\ensuremath{\operatorname{conv}}}\big({\ensuremath{\operatorname{dom}}}A\big)}=\overline{P_X\big({\ensuremath{\operatorname{dom}}}F_A\big)}.\end{aligned}$$
The following result presents a sufficient condition for a maximally monotone operator to be of type (FPV).
*(See [@Si2 Theorem 44.1], [@VV1] or [@Bor2].)* \[f:refer02a\] Let $A:X{\ensuremath{\rightrightarrows}}X^*$ be maximally monotone. Suppose that for every closed convex subset $C$ of $X$ with ${\ensuremath{\operatorname{dom}}}A \cap {\ensuremath{\operatorname{int}}}C\neq \varnothing$, the operator $A+N_C$ is maximally monotone. Then $A$ is of type (FPV).
*(See [@BY1 Fact 4.1].)*\[extlem\] Let $A:X\rightrightarrows X^*$ be monotone and $x\in{\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}A$. Then there exist $\delta>0$ and $M>0$ such that $x+\delta B_X\subseteq{\ensuremath{\operatorname{dom}}}A$ and $\sup_{a\in x+\delta B_X}\|Aa\|\leq M$. Assume that $(z,z^*)$ is monotonically related to ${\ensuremath{\operatorname{gra}}}A$. Then $$\begin{aligned}
\langle z-x, z^*\rangle
\geq \delta\|z^*\|-(\|z-x\|+\delta) M.\end{aligned}$$
We need the following bunch of useful tools from [@BY4FV].
*(See [@BY4FV Proposition 3.1].)*\[FProCVS\] Let $A:X{\ensuremath{\rightrightarrows}}X^*$ be of type (FPV), and let $B:X\rightrightarrows X^*$ be maximally monotone. Suppose that ${\ensuremath{\operatorname{dom}}}A \cap {\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B\neq\varnothing$. Let $(z,z^*)\in X\times X^*$ with $z\in\overline{{\ensuremath{\operatorname{dom}}}B}$. Then $$\begin{aligned}
F_{A+B}(z,z^*)\geq\langle z,z^*\rangle.\end{aligned}$$
*(See [@BY4FV Lemma 2.10].)*\[LeWExc:1\] Let $A:X{\ensuremath{\rightrightarrows}}X^*$ be monotone, and let $B:X\rightrightarrows X^*$ be maximally monotone. Let $(z, z^*)\in X\times X^*$. Suppose $x_0\in{\ensuremath{\operatorname{dom}}}A \cap {\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B$ and that there exists a sequence $(a_n, a^*_n)_{n\in{\ensuremath{\mathbb N}}}$ in ${\ensuremath{\operatorname{gra}}}A\cap\Big({\ensuremath{\operatorname{dom}}}B\times X^*\Big)$ such that $(a_n)_{n\in{\ensuremath{\mathbb N}}}$ converges to a point in $\left[x_0,z\right[$, and $$\begin{aligned}
\langle z-a_n, a^*_n\rangle\longrightarrow+\infty.\end{aligned}$$ Then $F_{A+B}(z,z^*)=+\infty$.
*(See [@BY4FV Lemma 2.12].)*\[LeWExc:3\] Let $A:X{\ensuremath{\rightrightarrows}}X^*$ be of type (FPV). Suppose $x_0\in{\ensuremath{\operatorname{dom}}}A$ but that $z\notin\overline{{\ensuremath{\operatorname{dom}}}A}$. Then there exists a sequence $(a_n, a^*_n)_{n\in{\ensuremath{\mathbb N}}}$ in ${\ensuremath{\operatorname{gra}}}A$ such that $(a_n)_{n\in{\ensuremath{\mathbb N}}}$ converges to a point in $\left[x_0,z\right[$ and $$\begin{aligned}
\langle z-a_n, a^*_n\rangle\longrightarrow+\infty.\end{aligned}$$
The proof of Fact \[FCTV:1\] and Fact \[FCTV:2\] is mainly extracted from the part of the proof of [@BY4FV Proposition 3.2].
\[FCTV:1\] Let $A:X{\ensuremath{\rightrightarrows}}X^*$ be maximally monotone and $z\in\overline{{\ensuremath{\operatorname{dom}}}A}\backslash{\ensuremath{\operatorname{dom}}}A$. Then for every sequence $(z_n)_{n\in{\ensuremath{\mathbb N}}}$ in ${\ensuremath{\operatorname{dom}}}A$ such that $z_n\longrightarrow
z$, we have $\lim_{n\rightarrow\infty}\inf\|A(z_n)\|=+\infty$.
Suppose to the contrary that there exists a sequence $z^*_{n_k}\in A(z_{n_k})$ and $L>0$ such that $\sup_{k\in{\ensuremath{\mathbb N}}}\|z^*_{n_k}\|\leq L$. By Fact \[BaAlo\], there exists a weak\* convergent subnet, $(z^*_{\beta})_{\beta\in J}$ of $(z^*_{n_k})_{k\in{\ensuremath{\mathbb N}}}$ such that $z^*_{\beta}\operatorname{\rightharpoondown_{\mathrm{w*}}}z^*_{\infty}\in X^*$. [@BY1 Fact 3.5] or [@BFG Section 2, page 539] shows that $(z, z^*_{\infty})\in{\ensuremath{\operatorname{gra}}}A$, which contradicts our assumption that $z\notin {\ensuremath{\operatorname{dom}}}A$. Hence we have our result holds.
\[FCTV:2\] Let $A, B:X{\ensuremath{\rightrightarrows}}X^*$ be monotone. Let $(z, z^*)\in X\times X^*$. Suppose that $x_0\in{\ensuremath{\operatorname{dom}}}A \cap {\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B$ and that there exist a sequence $(a_n, a^*_n)_{n\in{\ensuremath{\mathbb N}}}$ in ${\ensuremath{\operatorname{gra}}}A\cap\big({\ensuremath{\operatorname{dom}}}B\times X^*)$ and a sequence $(K_n)_{n\in{\ensuremath{\mathbb N}}}$ in ${\ensuremath{\mathbb R}}$ such that $(a_n)_{n\in{\ensuremath{\mathbb N}}}$ converges to a point in $\left[x_0,z\right[$, and that $$\begin{aligned}
\langle z- a_n, a^*_n\rangle\geq K_n.\label{FCTV:2:e1}\end{aligned}$$ Assume that there exists a sequence $b^*_n\in Ba_n$ such that $\frac{K_n}{\|b^*_n\|}\longrightarrow 0$ and $\|b^*_n\|\longrightarrow
+\infty$. Then $F_{A+B}(z,z^*)=+\infty$.
By the assumption, there exists $0\leq\delta<1$ such that $$\begin{aligned}
a_n\longrightarrow x_0+\delta(z-x_0).\label{FCTV:2:e2}\end{aligned}$$ Suppose to the contrary that $$\begin{aligned}
F_{A+B}(z,z^*)<+\infty.\label{FCTV:2:e3}\end{aligned}$$ By Fact \[BaAlo\], there exists a weak\* convergent subnet, $(\frac{b^*_i}{\|b^*_i\|})_{i\in I}$ of $\frac{b^*_n}{\|b^*_n\|}$ such that $$\begin{aligned}
\frac{b^*_i}{\|b^*_i\|}\operatorname{\rightharpoondown_{\mathrm{w*}}}b^*_{\infty}\in X^*.\label{PCSM:c4}\end{aligned}$$ By , we have $$\begin{aligned}
K_n+\Big\langle z-a_n, b^*_n\Big\rangle+\Big\langle z^*, a_n\Big\rangle&\leq
\Big\langle z-a_n, a_n^*\Big\rangle+
\Big\langle z-a_n, b^*_n\Big\rangle+\Big\langle z^*, a_n\Big\rangle\\
&\leq F_{A+B}(z,z^*)\end{aligned}$$ Thus $$\begin{aligned}
\frac{K_n}{\|b^*_n\|}
+\Big\langle z-a_n, \frac{b^*_n}{\|b^*_n\|}\Big\rangle+\frac{1}{\|b^*_n\|}\Big\langle z^*, a_n\Big\rangle
&\leq \frac{F_{A+B}(z,z^*)}{\|b^*_n\|}.\label{PCSM:c3}\end{aligned}$$
By the assumption that $\frac{K_n}{\|b^*_n\|}\longrightarrow 0$ and $\|b^*_n\|\longrightarrow
+\infty$, , and , we take the limit along the subnet in to obtain $$\begin{aligned}
\Big\langle z-x_0-\delta(z-x_0), b^*_{\infty}\Big\rangle\leq0.\end{aligned}$$ Since $\delta<1$, $$\begin{aligned}
\Big\langle z-x_0, b^*_{\infty}\Big\rangle\leq0.\label{PCSM:c6}\end{aligned}$$
On the other hand, since $x_0\in{\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B$ and $(a_n, b^*_n)\in{\ensuremath{\operatorname{gra}}}B$, Fact \[extlem\] implies that there exist $\eta>0$ and $M>0$ such that $$\begin{aligned}
\langle a_n-x_0, b_n^*\rangle
\geq\eta\|b^*_n\|-(\|a_n-x_0\|
+\eta)M.\end{aligned}$$ Thus $$\begin{aligned}
\langle a_n-x_0, \frac{b_n^*}{\|b^*_n\|}\rangle
\geq\eta-\frac{(\|a_n-x_0\|
+\eta)M}{\|b^*_n\|}.\end{aligned}$$ Since $\|b^*_n\|\longrightarrow +\infty$, by and , we take the limit along the subnet in the above inequality to obtain $$\begin{aligned}
\Big\langle x_0+\delta(z-x_0)-x_0, b^*_{\infty}\Big\rangle\geq\eta.\end{aligned}$$ Hence $$\begin{aligned}
\Big\langle z-x_0, b^*_{\infty}\Big\rangle\geq\frac{\eta}{\delta}>0,\end{aligned}$$ which contradicts . Hence $F_{A+B}(z,z^*)=
+\infty$.
Our main result {#s:main}
===============
The following result is the key technical tool for our main result (: Theorem \[TePGV:1\]). The proof of Proposition \[ProCVS:P1\] follows in part that of [@BY4FV Proposition 3.2].
\[ProCVS:P1\] Let $A:X{\ensuremath{\rightrightarrows}}X^*$ be of type (FPV), and let $B:X\rightrightarrows X^*$ be maximally monotone. Suppose $x_0\in{\ensuremath{\operatorname{dom}}}A\cap {\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B$ and $(z,z^*)\in X\times X^*$. Assume that there exist a sequence $(a_n)_{n\in{\ensuremath{\mathbb N}}}$ in ${\ensuremath{\operatorname{dom}}}A\cap\left[\overline{{\ensuremath{\operatorname{dom}}}B}\backslash{\ensuremath{\operatorname{dom}}}B\right]$ and $\delta\in\left[0,1\right]$ such that $a_n\longrightarrow\delta z+(1-\delta) x_0$. Then $F_{A+B}(z,z^*)\geq\langle z,z^*\rangle$.
Suppose to the contrary that $$\begin{aligned}
F_{A+B}(z,z^*)<\langle z,z^*\rangle.\label{EProF2:e1}\end{aligned}$$ By the assumption, we have $\delta z+(1-\delta) x_0\in\overline{{\ensuremath{\operatorname{dom}}}B}$. Since $a_n\notin{\ensuremath{\operatorname{dom}}}B$ and $x_0\in{\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B$, Fact \[FProCVS\] and imply that $$\begin{aligned}
0<\delta<1 \quad\text{and }\quad \delta z+(1-\delta) x_0\neq x_0.\label{PCSMaL:ec1}\end{aligned}$$ We set $$\begin{aligned}
y_0:=\delta z+(1-\delta) x_0.\label{PCSMaL:ea1}\end{aligned}$$ Since $a_n\in{\ensuremath{\operatorname{dom}}}A$, we let $$\begin{aligned}
(a_n, a^*_n)\in{\ensuremath{\operatorname{gra}}}A,\quad\forall n\in{\ensuremath{\mathbb N}}.\end{aligned}$$
Since $x_0\in{\ensuremath{\operatorname{dom}}}A\cap {\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B$, there exist $x^*_0, y^*_0\in X^*$ such that $(x_0, x^*_0)\in{\ensuremath{\operatorname{gra}}}A$ and $(x_0, y^*_0)\in{\ensuremath{\operatorname{gra}}}B$. By $x_0\in{\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B$, there exists $0<\rho_0\leq \|y_0-x_0\|$ by such that $$\begin{aligned}
x_0+\rho_0 U_X\subseteq{\ensuremath{\operatorname{dom}}}B.\label{PCSM:cc1}\end{aligned}$$ Now we show that there exists $\delta\leq t_n\in\left[1-\frac{1}{n},1\right[$ such that that $$\begin{aligned}
H_n\subseteq{\ensuremath{\operatorname{dom}}}B~\mbox{and~}
\inf\big\|B\big(H_n\big)\big\|\geq 4K^2_0 (\|a^*_n\|+1)n,
\label{PCSM:c1}\end{aligned}$$ where $$\begin{aligned}
H_n:&=t_na_n+(1-t_n) x_0+(1-t_n) \rho_0U_X\nonumber\\
K_0:&=\max\Big\{3\| z\|+2+3|x_0\|, \,
\frac{1}{\delta}\big(\frac{2\|y_0-x_0\|}{\rho_0}+1\big)\big(\|x^*_0\|+1\big)\Big\}.
\label{PCSMaL:ca1}\end{aligned}$$ For every $s\in \left]0,1\right[$, since $a_n\in\overline{{\ensuremath{\operatorname{dom}}}B}$, and Fact \[f:refer02c\] imply that $$\begin{aligned}
sa_n+(1-s) x_0+(1-s) \rho_0 U_X=sa_n+(1-s)\left[x_0+ \rho_0 U_X\right]\subseteq \overline{{\ensuremath{\operatorname{dom}}}B}.\end{aligned}$$ By Fact \[f:refer02c\] again, $sa_n+(1-s) x_0+(1-s) \rho_0 U_X\subseteq {\ensuremath{\operatorname{int}}}\overline{{\ensuremath{\operatorname{dom}}}B}={\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B$.
It directly follows from Fact \[FCTV:1\] and $a_n\in\overline{{\ensuremath{\operatorname{dom}}}B}\backslash{\ensuremath{\operatorname{dom}}}B$ that the second part of holds.
Set $$\begin{aligned}
r_n:= \frac{\frac{1}{2}(1-t_n)\rho_0}{t_n\|y_0-a_n\|+(1-t_n)\|y_0-x_0\|}
.\label{PCSMaL:ed1}\end{aligned}$$ Since $\rho_0\leq \|y_0-x_0\|$, we have $r_n\leq\frac{1}{2}$. Now we show that $$\begin{aligned}
v_n:&=r_n y_0+ (1-r_n)\left[t_na_n+(1-t_n) x_0\right]\nonumber\\
&=r_n\delta z+(1-r_n)t_n a_n+s_nx_0\in H_n,\label{PCSMaL:e5}\end{aligned}$$ where $s_n:=\left[1-t_n+r_n(t_n-\delta)\right]$.
Indeed, we have $$\begin{aligned}
&\Big\|v_n-t_na_n-(1-t_n) x_0\Big\|=
\Big\|r_n y_0+ (1-r_n)\left[t_na_n+(1-t_n) x_0\right]-t_na_n-(1-t_n) x_0\Big\|\\
&=\Big\|r_n y_0-r_n\left[t_na_n+(1-t_n) x_0\right]\Big\|
=r_n\Big\|t_ny_0+(1-t_n)y_0-\left[t_na_n+(1-t_n) x_0\right]\Big\|\\
&=r_n\Big\|t_n(y_0-a_n)+(1-t_n)(y_0-x_0)\Big\|\leq
r_n\Big(t_n\|y_0-a_n\|+(1-t_n)\|y_0-x_0\|\Big)\\
&=\frac{1}{2}(1-t_n)\rho_0\quad\text{(by \eqref{PCSMaL:ed1})}.\end{aligned}$$ Hence $v_n\in H_n$ and thus holds by .
Since $a_n\longrightarrow y_0$ and $v_n\in H_n$ by , $v_n\longrightarrow y_0$. Then we can and do suppose that $$\begin{aligned}
\|v_n\|\leq\|y_0\|+1\leq\|z\|+\|x_0\|+1,\quad\forall n\in{\ensuremath{\mathbb N}}\quad\text{(by \eqref{PCSMaL:ea1})}.\label{PCSMaL:cc2}\end{aligned}$$ Since $a_n\longrightarrow y_0$ and $\|y_0-x_0\|>0$ by , we can suppose that $$\begin{aligned}
\|y_0-a_n\|\leq\|y_0-x_0\|,\quad\forall n\in{\ensuremath{\mathbb N}}.\end{aligned}$$ Then by , $$\begin{aligned}
\frac{1-t_n}{r_n}\leq\frac{2\|y_0-x_0\|}{\rho_0},\quad\forall n\in{\ensuremath{\mathbb N}}. \label{PCSMaL:c1}\end{aligned}$$ Since $s_n=\left[1-t_n+r_n(t_n-\delta)\right]$, by and $\delta\leq t_n<1$, we have $$\begin{aligned}
\frac{s_n}{r_n}=\frac{1-t_n}{r_n}+t_n-\delta\leq \frac{2\|y_0-x_0\|}{\rho_0}+1,\quad\forall n\in{\ensuremath{\mathbb N}}. \label{PCSMaL:c2}\end{aligned}$$
Now we show there exists $(\widetilde{a_n}, \widetilde{a_n}^*)_{n\in{\ensuremath{\mathbb N}}}$ in ${\ensuremath{\operatorname{gra}}}A\cap (H_n\times X^*)$ such that $$\begin{aligned}
\big\langle z-\widetilde{a_n},\widetilde{a_n}^*\big\rangle\geq-4K^2_0 (\|a^*_n\|+1).\label{EProF2:e2}\end{aligned}$$ We consider two cases.
*Case 1*: $(v_n,(2-t_n)a^*_n)\in{\ensuremath{\operatorname{gra}}}A$.
Set $(\widetilde{a_n}, \widetilde{a_n}^*):=(v_n,(2-t_n)a^*_n)$. Then we have $$\begin{aligned}
&\langle z- \widetilde{a_n}, \widetilde{a_n}^*\rangle
=\langle z- v_n, (2-t_n){a_n}^*\rangle\geq-2\|z- v_n\|\cdot\|a^*_n\|\nonumber\\
&\geq-2\big(2\|z\|+\|x_0\|+1\big)\cdot\|a^*_n\|\geq-4K^2_0 (\|a^*_n\|+1)\quad\text{(by \eqref{PCSMaL:cc2} and
\eqref{PCSMaL:ca1})}.\end{aligned}$$ Hence holds since $v_n\in H_n$ by .
*Case 2*: $(v_n,(2-t_n)a^*_n)\notin{\ensuremath{\operatorname{gra}}}A$.
By Fact \[SDMn:pv\] and the assumption that $\{a_n, y_0, x_0\}\subseteq\overline{{\ensuremath{\operatorname{dom}}}A}$, shows that $v_n\in\overline{{\ensuremath{\operatorname{dom}}}A}$. Thus $H_n\cap{\ensuremath{\operatorname{dom}}}A\neq\varnothing$ by again. Since $\Big(v_n,(2-t_n)a^*_n\Big)\notin{\ensuremath{\operatorname{gra}}}A$, $v_n\in H_n$ by , and $A$ is of type (FPV), there exists $(\widetilde{a_n}, \widetilde{a_n}^*)\in{\ensuremath{\operatorname{gra}}}A\cap (H_n\times X^*)$ such that $$\begin{aligned}
\Big\langle v_n-\widetilde{a_n}, \widetilde{a_n}^*-(2-t_n)a^*_n\Big\rangle>0.
\end{aligned}$$ Thus by , we have $$\begin{aligned}
&\Big\langle v_n-\widetilde{a_n}, \widetilde{a_n}^*-(2-t_n)a^*_n\Big\rangle>0\nonumber\\
&\Longrightarrow \Big\langle r_n\delta z+(1-r_n)t_n a_n+s_nx_0 -\widetilde{a_n}, \widetilde{a_n}^*-a^*_n-(1-t_n)a^*_n\Big\rangle>0\nonumber\\
&\Longrightarrow \Big\langle r_n\delta z+(1-r_n)t_n a_n+s_nx_0 -\widetilde{a_n}, \widetilde{a_n}^*-a^*_n\Big\rangle\nonumber\\
&\quad>\Big\langle r_n\delta z+(1-r_n)t_n a_n+s_nx_0 -\widetilde{a_n}, (1-t_n)a^*_n\Big\rangle\nonumber\\
&\quad\geq- (1-t_n)\Big(\| z\|+\|a_n\|+\|x_0\| +\|\widetilde{a_n}\|\Big) \|a^*_n\|\label{PCSMaL:e12}
\end{aligned}$$ Note that $\widetilde{a_n}=r_n\delta\widetilde{a_n}+(1-r_n)t_n\widetilde{a_n}+s_n\widetilde{a_n}$. Thus implies that $$\begin{aligned}
&\Big\langle r_n\delta(z-\widetilde{a_n})+(1-r_n)t_n (a_n-\widetilde{a_n})+s_n(x_0-\widetilde{a_n}), \widetilde{a_n}^*-a^*_n\Big\rangle\nonumber\\
&\quad>- (1-t_n)\Big(\| z\|+\|a_n\|+\|x_0\| +\|\widetilde{a_n}\|\Big) \|a^*_n\|\nonumber\\
&\Longrightarrow
\Big\langle r_n\delta(z-\widetilde{a_n})+s_n(x_0-\widetilde{a_n}), \widetilde{a_n}^*-a^*_n\Big\rangle\nonumber\\
&\quad\geq(1-r_n)t_n\Big\langle a_n-\widetilde{a_n}, a^*_n-\widetilde{a_n}^*\Big\rangle- (1-t_n)\Big(\| z\|+\|a_n\|+\|x_0\| +\|\widetilde{a_n}\|\Big) \|a^*_n\|\nonumber\\
&\quad\geq- (1-t_n)\Big(\| z\|+\|a_n\|+\|x_0\| +\|\widetilde{a_n}\|\Big) \|a^*_n\|
\quad\text{(by the monotonicity of $A$)}\nonumber\\
&\Longrightarrow
\Big\langle r_n\delta(z-\widetilde{a_n})+s_n(x_0-\widetilde{a_n}), \widetilde{a_n}^*\Big\rangle\nonumber\\
&\quad> \Big\langle r_n\delta(z-\widetilde{a_n})+s_n(x_0-\widetilde{a_n}), a^*_n\Big\rangle- (1-t_n)\Big(\| z\|+\|a_n\|+\|x_0\| +\|\widetilde{a_n}\|\Big) \|a^*_n\|\nonumber\\
&\Longrightarrow
r_n\delta\Big\langle z-\widetilde{a_n}, \widetilde{a_n}^*\Big\rangle
> s_n\Big\langle \widetilde{a_n}-x_0, \widetilde{a_n}^*\Big\rangle+\Big\langle r_n\delta(z-\widetilde{a_n})+s_n(x_0-\widetilde{a_n}), a^*_n\Big\rangle\nonumber\\
&\quad- (1-t_n)\Big(\| z\|+\|a_n\|+\|x_0\| +\|\widetilde{a_n}\|\Big) \|a^*_n\|\label{PCSMaL:e13}
\end{aligned}$$ Since $\{(x_0, x^*_0), (\widetilde{a_n}, \widetilde{a_n}^*)\}\subseteq{\ensuremath{\operatorname{gra}}}A$, we have $\langle \widetilde{a_n}-x_0, \widetilde{a_n}^*\rangle\geq
\langle \widetilde{a_n}-x_0, x_0^*\rangle$ by the monotonicity of $A$. Thus, by , $$\begin{aligned}
&r_n\delta\Big\langle z-\widetilde{a_n}, \widetilde{a_n}^*\Big\rangle
> s_n\Big\langle \widetilde{a_n}-x_0, x_0^*\Big\rangle+\Big\langle r_n\delta(z-\widetilde{a_n})+s_n(x_0-\widetilde{a_n}), a^*_n\Big\rangle\nonumber\\
&\quad- (1-t_n)\Big(\| z\|+\|a_n\|+\|x_0\| +\|\widetilde{a_n}\|\Big) \|a^*_n\|\nonumber\\
&\geq -s_n\|\widetilde{a_n}-x_0\|\cdot\| x_0^*\|-\ r_n\|z-\widetilde{a_n}\|\cdot\|a^*_n\|-s_n\|x_0-\widetilde{a_n}\|\cdot \|a^*_n\|\nonumber\\
&\quad- (1-t_n)\Big(\| z\|+\|a_n\|+\|x_0\| +\|\widetilde{a_n}\|\Big) \|a^*_n\|\nonumber
\end{aligned}$$ Hence $$\begin{aligned}
&\Big\langle z-\widetilde{a_n}, \widetilde{a_n}^*\Big\rangle
> -\frac{s_n}{r_n\delta}\|\widetilde{a_n}-x_0\|\cdot\| x_0^*\|-\ \frac{1}{\delta}\|z-\widetilde{a_n}\|\cdot\|a^*_n\|-\frac{s_n}{r_n\delta}\|x_0-\widetilde{a_n}\|\cdot \|a^*_n\|\nonumber\\
&\quad- \frac{1-t_n}{r_n\delta}\Big(\| z\|+\|a_n\|+\|x_0\| +\|\widetilde{a_n}\|\Big) \|a^*_n\|\nonumber
\end{aligned}$$ Then combining and , we have $$\begin{aligned}
&\Big\langle z-\widetilde{a_n}, \widetilde{a_n}^*\Big\rangle
> -(\frac{2\|y_0-x_0\|}{\rho_0}+1)\frac{1}{\delta}\|\widetilde{a_n}-x_0\|\cdot\| x_0^*\|-\ \frac{1}{\delta}\|z-\widetilde{a_n}\|\cdot\|a^*_n\|
\nonumber\\
&\quad-(\frac{2\|y_0-x_0\|}{\rho_0}+1)\frac{1}{\delta}\|x_0-\widetilde{a_n}\|\cdot \|a^*_n\|- \frac{2\|y_0-x_0\|}{\rho_0\delta}\Big(\| z\|+\|a_n\|+\|x_0\| +\|\widetilde{a_n}\|\Big) \|a^*_n\|\nonumber\\
&\quad\geq
-K_0\|\widetilde{a_n}-x_0\|-K_0\|z-\widetilde{a_n}\|\cdot\|a^*_n\|
-K_0\|x_0-\widetilde{a_n}\|\cdot \|a^*_n\|\nonumber\\
&\quad- K_0\Big(\| z\|+\|a_n\|+\|x_0\| +\|\widetilde{a_n}\|\Big) \|a^*_n\|\quad\text{(by \eqref{PCSMaL:ca1})}\label{PCSMaL:e14}
\end{aligned}$$ Since $\widetilde{a_n}\in H_n$, $t_n\longrightarrow 1^{-}$ and $a_n\longrightarrow
y_0$, $$\widetilde{a_n}\longrightarrow y_0. \label{PCSMaL:e9}$$ Then we can and do suppose that $$\begin{aligned}
\max\big\{\|a_n\|,\|\widetilde{a_n}\|\big\}\leq \|y_0\|+1\leq\|x_0\|+\|z\|+1, \quad\forall n\in{\ensuremath{\mathbb N}}.\quad\text{(by \eqref{PCSMaL:ea1})}\label{PCSMaL:e15}
\end{aligned}$$ Then by , and , we have $$\begin{aligned}
&\Big\langle z-\widetilde{a_n}, \widetilde{a_n}^*\Big\rangle
>-K_0^2-K_0^2\|a^*_n\|
-K_0^2 \|a^*_n\|- K_0^2\|a^*_n\|\geq -4K^2_0 (\|a^*_n\|+1).\end{aligned}$$ Hence holds.
Combining the above two cases, we have holds.
Since $\widetilde{a_n}\in H_n$, implies that $\widetilde{a_n}\in{\ensuremath{\operatorname{dom}}}B$. Then combining , , and , Fact \[FCTV:2\] implies that $F_{A+B}(z,z^*)=+\infty$, which contradicts .
Hence $F_{A+B}(z,z^*)\geq\langle z,z^*\rangle$.
Now we come to our main result.
\[TePGV:1\] Let $A, B:X{\ensuremath{\rightrightarrows}}X^*$ be maximally monotone with ${\ensuremath{\operatorname{dom}}}A\cap{\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B\neq\varnothing$. Assume that $A+N_{\overline{{\ensuremath{\operatorname{dom}}}B}}$ is of type (FPV). Then $A+B$ is maximally monotone.
After translating the graphs if necessary, we can and do assume that $0\in{\ensuremath{\operatorname{dom}}}A\cap{\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B$ and that $(0,0)\in{\ensuremath{\operatorname{gra}}}A\cap{\ensuremath{\operatorname{gra}}}B$. By Corollary \[VoiSimn:1\], it suffices to show that $$\label{EOL:1}
F_{A+ B}(z,z^*)\geq \langle z,z^*\rangle,\quad \forall(z,z^*)\in X\times X^*.$$ Take $(z,z^*)\in X\times X^*$. Suppose to the contrary that $$\begin{aligned}
F_{A+B}(z,z^*)<\langle z,z^*\rangle.\label{FPCoS:e20}\end{aligned}$$ Since $B$ is maximally monotone, $B=B+N_{\overline{{\ensuremath{\operatorname{dom}}}B}}$. Thus $$\begin{aligned}
A+B= A+B+N_{\overline{{\ensuremath{\operatorname{dom}}}B}}= (A+N_{\overline{{\ensuremath{\operatorname{dom}}}B}})+B.\label{TePGVL:e1}\end{aligned}$$ Since $A+N_{\overline{{\ensuremath{\operatorname{dom}}}B}}$ is of type (FPV) and $0\in{\ensuremath{\operatorname{dom}}}\left[A+N_{\overline{{\ensuremath{\operatorname{dom}}}B}}\right]\cap{\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B$, Fact \[FProCVS\] and imply that $$\begin{aligned}
z\notin\overline{{\ensuremath{\operatorname{dom}}}B}\quad\text{and then}\quad
z\notin\overline{{\ensuremath{\operatorname{dom}}}\left[A+N_{\overline{{\ensuremath{\operatorname{dom}}}B}}\right]}
\label{TePGVL:e2}\end{aligned}$$ Then by Fact \[LeWExc:3\], there exist a sequence $(a_n, a^*_n)_{n\in{\ensuremath{\mathbb N}}}$ in ${\ensuremath{\operatorname{gra}}}(A+N_{\overline{{\ensuremath{\operatorname{dom}}}B}})$ and $\delta\in\left[0,1\right[$ such that $$\begin{aligned}
a_n\longrightarrow \delta z\quad\text{and}\quad
\langle z-a_n, a^*_n\rangle\longrightarrow+\infty.\label{TePGVL:e3}\end{aligned}$$ Thus $a_n\in{\ensuremath{\operatorname{dom}}}\left[A+N_{\overline{{\ensuremath{\operatorname{dom}}}B}}\right]\cap\overline{{\ensuremath{\operatorname{dom}}}B},\, \forall n\in{\ensuremath{\mathbb N}}$.
Now we consider two cases.
*Case 1*: There exists a subsequence of $(a_n)_{n\in{\ensuremath{\mathbb N}}}$ in ${\ensuremath{\operatorname{dom}}}B$.
We can and do suppose that $a_n\in{\ensuremath{\operatorname{dom}}}B$ for every $n\in{\ensuremath{\mathbb N}}$. Thus $(a_n)_{n\in{\ensuremath{\mathbb N}}}$ is in $ {\ensuremath{\operatorname{dom}}}\left[A+N_{\overline{{\ensuremath{\operatorname{dom}}}B}}\right]\cap{\ensuremath{\operatorname{dom}}}B$.
Combining Fact \[LeWExc:1\] and , $$\begin{aligned}
F_{A+B}(z,z^*)=F_{A+N_{\overline{{\ensuremath{\operatorname{dom}}}B}}+B}(z,z^*)
=+\infty,\end{aligned}$$ which contradicts .
*Case 2*: There exists $N_1\in{\ensuremath{\mathbb N}}$ such that $a_n\notin{\ensuremath{\operatorname{dom}}}B,\, \forall n\geq N_1$.
Then we can and do suppose that $a_n\notin{\ensuremath{\operatorname{dom}}}B$ for every $n\in{\ensuremath{\mathbb N}}$. Thus, $a_n\in{\ensuremath{\operatorname{dom}}}\left[A+N_{\overline{{\ensuremath{\operatorname{dom}}}B}}\right]\cap\left[\overline{{\ensuremath{\operatorname{dom}}}B}\backslash{\ensuremath{\operatorname{dom}}}B\right]$. By Proposition \[ProCVS:P1\] and , $$\begin{aligned}
F_{A+B}(z,z^*)=F_{A+N_{\overline{{\ensuremath{\operatorname{dom}}}B}}+B}(z,z^*)\geq\langle z,z^*\rangle,\end{aligned}$$ which contradicts .
Combing all the above cases, we have $F_{A+B}(z,z^*)\geq\langle z, z^*\rangle$ for all $(z,z^*)\in X\times X^*$. Hence $A+B$ is maximally monotone.
Theorem \[TePGV:1\] generalizes the main result in [@Yao3] (see [@Yao3 Theorem 3.4]).
\[CorPbA:1\] Let $f:X\rightarrow {\ensuremath{\,\left]-\infty,+\infty\right]}}$ be a proper lower semicontinuous convex function, and let $B:X\rightrightarrows X^*$ be maximally monotone with ${\ensuremath{\operatorname{dom}}}\partial f\cap{\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B\neq\varnothing$. Then $\partial f +B$ is maximally monotone.
By Fact \[f:refer02c\] and Fact \[f:F4\] (or [@AtBrezis Theorem 1.1]), $\partial f+N_{\overline{{\ensuremath{\operatorname{dom}}}B}}
=\partial (f+\iota_{\overline{{\ensuremath{\operatorname{dom}}}B}})$. Then Fact \[f:referee0d\] shows that $\partial f+N_{\overline{{\ensuremath{\operatorname{dom}}}B}}$ is of type (FPV). Applying Theorem \[TePGV:1\], we have $\partial f +B$ is maximally monotone.
Corollary \[CorPbA:1\] provides an affirmative answer to a problem posed by Borwein and Yao in [@BY3 Open problem 4.5].
Given a set-valued operator $A:X\rightrightarrows X^*$, we say $A$ is a *linear relation* if ${\ensuremath{\operatorname{gra}}}A$ is a linear subspace.
*(See [@BY3 Theorem 3.1] or [@BY4FV Corollary 4.5].)* Let $A:X{\ensuremath{\rightrightarrows}}X^*$ be a maximally monotone linear relation, and let $B: X\rightrightarrows X^*$ be maximally monotone. Suppose that ${\ensuremath{\operatorname{dom}}}A\cap{\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B\neq\varnothing$. Then $A+B$ is maximally monotone.
Apply Fact \[f:refer02c\], [@Yao2 Corollary 3.3] and Theorem \[TePGV:1\] directly.
*(See [@BY4FV Corollary 4.3].)* Let $A:X{\ensuremath{\rightrightarrows}}X^*$ be of type (FPV) with convex domain, and let $B: X\rightrightarrows X^*$ be maximally monotone. Suppose that ${\ensuremath{\operatorname{dom}}}A\cap{\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}B\neq\varnothing$. Then $A+B$ is maximally monotone.
Apply Fact \[f:refer02c\], [@Yao3 Corollary 2.10] and Theorem \[TePGV:1\] directly.
Applying Fact \[f:refer02a\] and Theorem \[TePGV:1\], we can obtain that the sum problem is equivalent to the following problem:
\[OPRKM:1\] Let $A:X{\ensuremath{\rightrightarrows}}X^*$ be maximally monotone, and $C$ be a nonempty closed and convex subset of $X$. Assume that ${\ensuremath{\operatorname{dom}}}A\cap{\ensuremath{\operatorname{int}}}C\neq\varnothing$. Is $A+N_C$ necessarily maximally monotone?
Clearly, Problem \[OPRKM:1\] is a special case of the sum problem. However, if we would have an affirmative answer to Problem \[OPRKM:1\] for every maximally monotone operator $A$ and every nonempty closed and convex set $C$ satisfying Rockafellar’s constraint qualification: ${\ensuremath{\operatorname{dom}}}A\cap{\ensuremath{\operatorname{int}}}C\neq\varnothing$. Then Fact \[f:refer02a\] implies that $A$ is of type (FPV), and then $\overline{{\ensuremath{\operatorname{dom}}}A}$ is convex by Fact \[SDMn:pv\]. Applying Fact \[f:refer02a\] again and using the technique similar to the proof of [@BY4FV Corollary 4.6] (or [@Yao3 Corollary 2.10]), we can obtain that $A+N_C$ is of type (FPV). Thus applying Theorem \[TePGV:1\], we have an affirmative answer to the sum problem.
Acknowledgment {#acknowledgment .unnumbered}
==============
The author thanks Dr. Anthony Lau for supporting his visit and providing excellent working conditions.
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[^1]: Department of Mathematical and Statistical Sciences, University of Alberta Edmonton, Alberta, T6G 2G1 Canada E-mail: `liangjin@ualberta.ca`.
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---
abstract: 'The universe should be dark at energies exceeding $\sim 5\times 10^{19}$ eV. This simple but solid prediction of our best known particle physics is not confirmed by observations, that seem to suggest a quite different picture. Numerous events have in fact been detected in this energy region, with spectra and anisotropy features that defy many conventional and unconventional explanations. Is there a problem with known physics or is this a result of astrophysical uncertainties? Here we try to answer these questions, in the light of present observations, while discussing which information future observations may provide on this puzzling issue.'
author:
- |
Pasquale Blasi\
[*Osservatorio Astrofisico di Arcetri, Largo E. Fermi, 5 - Firenze (ITALY)*]{}
title: ' THE HIGHEST ENERGY PARTICLES IN THE UNIVERSE: THE MYSTERY AND ITS POSSIBLE SOLUTIONS '
---
=11.6pt
Introduction
============
One of the major goals of cosmic ray physics has always been the discovery and understanding of the [*end of the cosmic ray spectrum*]{}. Until the end of the ’60s, this search was mainly aimed to understand the limits to the acceleration processes and nature of the sources responsible for the production of the particles with the highest energies. However, after the discovery of the cosmic microwave background (CMB), it became soon clear that the observed spectrum of the cosmic radiation had to be cut off at a “natural” energy, even if an ideal class of sources existed, able to accelerate particles to infinite energy. In fact, if the sources are distributed homogenously in the universe, the photopion production in the scattering of particles off the CMB photons imply a cutoff in the observed spectrum of the cosmic rays at an energy $E_{GZK}\sim (4-5)\times 10^{19}$ eV, close to the kinematic threshold for that process [@greisenzk66]. This cutoff has become known as the [*GZK cutoff*]{} and particles with $E>E_{GZK}$ are usually named ultra-high energy cosmic rays (UHECRs).
Several experiments have been operating to detect the flux of UHECRs, starting with Volcano Ranch [@linsley] and continuing with Haverah Park [@watson91] and Yakutsk [@efimov91] to the more recent experiments like AGASA [@agasa01; @tak99; @tak98; @ha94], Fly’s Eye [@bird93; @bird94; @bird95] and HiRes [@kieda99]. The search for the GZK cutoff, instead of confirming the simple picture illustrated above, has provided stronger and stronger evidence for the existence of events corresponding to energies well in excess of $E_{GZK}$: the GZK cutoff has not been found.
This problem hides many issues on plasma physics, particle physics and astrophysics, that in their whole represent the puzzle of UHECRs. In section \[sec:observations\] the present status of observations of UHECRs is summarized; in section \[sec:serious\] some caveats in the arguments that are often used to address the problem of UHECRs are considered, for the purpose of stating the problem in a clear way. In sections \[sec:acceleration\] and \[sec:td\] the bottom-up and top-down models of UHECR origin are summarized. In section \[sec:liv\] some speculations are discussed of new physics scenarios that might play a role not only for the explanation of UHECRs, but also for the understanding of other current puzzles in high energy astrophysics. Conclusions are reported in section \[sec:conclusion\].
Observations {#sec:observations}
============
The cosmic ray spectrum is measured from fractions of GeV to a (current) maximum energy of $3\times 10^{20}$ eV. The spectrum above a few GeV and up to $\sim 10^{15}$ eV (the knee) is measured to be a power law with slope $\sim 2.7$, while at higher energies and up to $\sim 10^{19}$ eV (the ankle) the spectrum has a steeper slope, of $\sim 3.1$. At energy larger than $10^{19}$ eV a flattening seems to be present.
The statistics of events is changing continuously: the latest analysis of the “all experiments” statistics was carried out in [@uchihori00] where 92 events were found above $4\times 10^{19}$ eV. 47 events were detected by the AGASA experiment. A more recent analysis [@agasa01] of the AGASA data, carried out expanding the acceptance angle to $\sim 60^o$, has increased the number of events in this energy region to 59.
In [@tak99] the directions of arrival of the AGASA events (with zenith angle smaller than $45^o$) above $4\times 10^{19}$ eV were studied in detail: no appreciable departure from isotropy was found, with the exception of a few small scale anisotropies in the form of doublets and triplets of events within an angular scale comparable with the angular resolution of the experiments ($\sim 2.5^o$ for AGASA). This analysis was repeated in [@uchihori00] for the whole sample of events above $4\times 10^{19}$ eV, and a total of 12 doublets and 3 triplets were found within $\sim 3^o$ angular scales. The attempt to associate these multiplets with different types of local astrophysical sources possibly clustered in the local supercluster did not provide evidence in that direction [@stanev].
Recently, the AGASA collaboration reported on the study of the small scale anisotropies in the extended sample of events with zenith angle $<60^o$: 5 doublets (chance probability $\sim 0.1\%$) and 1 triplet (chance probability $\sim 1\%$) were found.
The information available on the composition of cosmic rays at the highest energies is quite poor. A study of the shower development was possible only for the Fly’s Eye event [@bird95] and disfavors a photon primary [@halzen]. A reliable analysis of the composition is however possible only on statistical basis, because of the large fluctuations in the shower development at fixed type of primary particle.
The Fly’s Eye collaboration reports of a predominantly heavy composition at $3\times 10^{17}$ eV, with a smooth transition to light composition at $\sim 10^{19}$ eV. This trend was later not confirmed by AGASA [@ha94; @yoda98].
Recently in ref. [@zas2000] the data of the Haverah Park experiment on highly inclined events were re-analyzed: this new analysis results in no more than $30\%$ of the events with energy above $10^{19}$ eV being consistent with photons or iron (at $95\%$ confidence level) and no more than $55\%$ of events being photons above $4\times 10^{19}$ eV.
Recently a new mass of data has been presented by the HiRes experiment [@hiresICRC]. Only two events with energy above $10^{20}$ eV have been detected by this experiment insofar, compatible with the presence of a GZK cutoff. This discrepancy with the results of several years of AGASA operation needs further investigation. Several sistematics have been identified that might considerably affect the determination of the energies and fluxes of fluorescence experiments versus the ground array techniques [@agasa01]. These issues will not be discussed further in the present paper.
The GZK cutoff: how serious is its absence? {#sec:serious}
===========================================
The puzzle of UHECRs can be summarized in the following points:
- [*:*]{} the generation of particles of energy $\geq 10^{20}$ eV requires an excellent accelerator, or some new piece of physics that allows the production of these particles in a non-acceleration scenario.
- [*:*]{} observations show a remarkable large scale isotropy of the arrival directions of UHECRs, with no correlation with local structures (e.g. galactic disk, local supercluster, local group).
- [*:*]{} the small (degree) scale anisotropies, if confirmed by further upcoming experiments, would represent an extremely strong constraint on the type of sources of UHECRs and on magnetic fields in the propagation volume.
- [*:*]{} the GZK cutoff is mainly a geometrical effect: the number of sources within a distance that equals the pathlength for photopion production is far less than the sources that contribute lower energy particles, having much larger pathlength (comparable with the size of the universe). The crucial point is that the cutoff is present even if plausible nearby UHECR engines are identified.
- [*:*]{} it is crucial to determine the type of particles that generate the events at ultra-high energies. The composition can be really considered a smoking gun either in favor or against whole classes of models.
The five points listed above are most likely an oversimplification of the problem: some other issues could be added to the list, such as the spectrum, but at least at present this cannot be considered as a severe constraint. On the other hand, specific models make specific predictions on the spectral shape, so that when the results of future observations will be available, this information will allow a strong discrimination among different explanations for the origin of UHECRs. Any model that aims to the explanation of the problem of UHECRs must address all of the issues listed above (and possibly others).
In this section we consider in some more detail the issue of the GZK cutoff and the seriousness of its absence in the observed data.
It is often believed that the identification of one or a class of nearby UHECR sources would explain the observations and in particular the absence of the GZK cutoff. This is not necessarily true. The (inverse of the) lifetime of a proton with energy $E$ is plotted in fig. 1 (left panel) together with the derivative with respect to energy of the rate of energy losses $b(E)$ (right panel) \[the figure has been taken from ref. [@bereICRC]\]. The flux per unit solid angle at energy $E$ in some direction is proportional to $n_0 \lambda(E) \Phi(E)$, where $n_0$ is the density of sources (assumed constant), $\lambda(E)=c/((1/E)dE/dt)$ and $\Phi(E)$ is the source spectrum. This rough estimate suggests that the ratio of detected fluxes (multiplied as usual by $E^3$), at energies $E_1$ and $E_2$ is $${\cal R}=
\frac{E_1^3 F(E_1)}{E_2^3 F(E_2)}\sim \frac{\lambda(E_1) \Phi(E_1) E_1^3}
{\lambda(E_2)\Phi(E_2) E_2^3} =
\frac{\lambda(E_1)}
{\lambda(E_2)} \left(\frac{E_1}{E_2}\right)^{3-\gamma},
\label{eq:ratio}$$ where in the last term we assumed that the source spectrum is a power law $\Phi(E)\sim E^{-\gamma}$. If for instance one takes $E_1=10^{19}$ eV (below $E_{GZK}$) and $E_2=3\times 10^{20}$ eV (above $E_{GZK}$), from fig. 1 one obtains that ${\cal R}\sim 80$ for $\gamma=3$ and ${\cal R}\sim 10$ for $\gamma=2.4$. The ratio ${\cal R}$ gives a rough estimate of the suppression factor at the GZK cutoff and its dependence on the spectrum of the source. For flat spectra ($\gamma\leq 2$) the cutoff is less significant, but it is more difficult to fit the low energy data [@blanton] (at $E\sim 10^{19}$ eV). Steeper spectra make the GZK cutoff more evident, although they allow an easier fit of the low energy data. The simple argument illustrated above can also be interpreted in an alternative way: if there is a local overdensity of sources by a factor $\sim {\cal R}$, the GZK cutoff is attenuated with respect to the case of homogeneous distribution of the sources.
The question of whether we are located in such a large overdensity of sources was recently addressed, together with the propagation of UHECRs, in [@blanton]. Assuming that the density of the (unknown) sources follows the density of galaxies in large scale structure surveys like PSCz [@pscz] and Cfa2 [@cfa2], the authors estimate the local overdensity on scales of several Mpc to be of order $\sim 2$, too small to compensate for the energy losses of particles with energy above the threshold for photopion production.
There is however another issue that the calculations in [@blanton] address, which is related to the statistical fluctuations induced by the process of photopion production. The large inelasticity of this process can be taken into account properly only through the use of Montecarlo calculations. When the Montecarlo is applied to simulate the small statistics of events typical of current experiments, the fluctuations in the simulated fluxes above $\sim 10^{20}$ eV are very large, so that for flat spectra ($\sim E^{-2}$) the discrepancy between observations and simulations on the total number of events above $\sim 10^{20}$ eV is at the level of $\sim 2 \sigma$ (in agreement with the conclusions of Ref. [@bac_wax]). The situation is represented in fig. 2 [@blanton] where the hatched regions show the uncertainties in the simulated fluxes. The data points are from AGASA [@tak98; @hayashida99].
The bottom line of this section can be summarized in the following few points:
1\) the GZK cutoff is not avoided by finding sources of UHECRs that lie within the pathlength of photopion production, unless these sources are located only or predominantly nearby and are less abundant at large distances.
2\) The significance of the GZK feature depends on the fluctuations in the photopion production, and can be addressed properly only with a enhanced statistics of events with energy $\geq 10^{20}$ eV.
The UHECRs engines
==================
Models for the origin of UHECRs can be strongly constrained on the basis on the criteria illustrated in the previous section. The challenge to conventional acceleration models, that are supposed to work at lower energy scales, induced an increasing interest for more exotic generation mechanisms, eventually requiring new particle physics. In this section the main ideas on production scenarios and their signatures will be summarized.
{#sec:acceleration}
The accelerators able to reach maximum energies of order 1 ZeV have been named Zevatrons [@olinto]. The challenge for Zevatrons was recently discussed in detail by several authors [@blandford; @olinto]: the main concept in this class of models is that the energy flux embedded in a macroscopic motion or in magnetic fields is partly converted into energy of a few very high energy particles. This is what happens for instance in shock acceleration.
A discussion of all the models in the literature is not the purpose of the present paper, and in a sense we think it may not be very interesting. Nevertheless, it is instructive to understand at least which classes of models may have a chance to explain the acceleration to ZeV energies. In this respect, a pictorial way of proceeding is based on what is known as the Hillas plot [@hillas]. Our version of it is reported in fig. 4.
In the Hillas plot the maximum energy is taken to be in its simplest form, as determined by the local magnetic field $B$ and the size $L$ of the accelerator: $E_{max}=Ze B L$. Here $Ze$ is the electric charge of the accelerated particles. From fig. 4 it is evident that only 4 classes of sources have the potential to accelerate protons to ultra-high energies: 1) Neutron stars; 2) Radio Lobes; 3) Active Galactic Nuclei; 4) Clusters of Galaxies. In the case of Iron, the situation becomes more promising for other sources, like the galactic halo or extreme white dwarfs. These sources would however have other problems that make them unlikely sources of UHECRs.
The hillas plot does not include the effect of energy losses in the acceleration sites. Photopion production limits the maximum energy achievable in clusters of galaxies to $\leq \rm{a~few}~10^{19}$ eV. These sources will therefore not be considered any longer as sources of cosmic rays above the GZK energy. We also do not discuss here the so-called bursting sources. The prototypical example of these sources are gamma ray bursts, that have been proposed as sources of UHECRs [@GRB]. We refer the reader to recent literature treating this topic [@dar].
In the following we briefly summarize the situation with the other three classes of objects listed above.
The possibility that neutron stars may be accelerators of UHECRs was discussed in detail in Ref. [@bible] (and references therein). The main problem encountered in reaching the highest energies is related to the severe energy losses experienced by the particles in the acceleration site [@venkatesan; @bible]. Most of the mechanisms discussed in the literature refer to acceleration processes in the magnetosphere of the neutron star, where curvature radiation limits the maximum energy to a value much smaller that $10^{20}$ eV.
An alternative approach is to think of acceleration processes that occur outside the light cylinder of young neutron stars [@boe; @lazarian].
Rapidly rotating, newly formed neutron stars can induce the acceleration of iron nuclei through MHD winds outside the light cylinder [@boe]. Although the mechanism through which the rotation energy of the star is converted into kinetic energy of the wind is not yet completely understood, it seems from the observations of the Crab nebula that a relativistic wind does indeed exist, with a Lorentz factor of $\sim 10^7$ [@begelman]. Possible nuclei with charge $Z_{26}=Z/26$ can be accelerated in young neutron stars to a maximum energy $E_{max}=8\times 10^{20} Z_{26} B_{13}
\Omega_{3k}^2$, as estimated in Ref. [@boe]. Here $B_{13}$ is the surface magnetic field in units of $10^{13}$ G and $\Omega_{3k}=
\Omega/3000\rm{s}^{-1}$ is the rotation frequency of the star. Energies gradually smaller are produced while the star is spinning down, so that a spectrum $\sim E^{-1}$ is produced by a neutron star. The process of escape of the accelerated particles becomes efficient about a year after the neutron star birth. Particles that are generated earlier cannot escape, but can produce high energy neutrinos in collisions with the ambient particles and photons [@bedn]. The issue of the anisotropy due to the galactic disk is currently under investigation [@aniso].
Active galaxies are thought to be powered by the accretion of gas onto supermassive black holes. Acceleration of particles can occur in standing shocks in the infalling gas or by unipolar induction in the rotating magnetized accretion disk [@thorne]. In the former scenario energy losses and size of the acceleration region are likely to limit the maximum energy of the accelerated particles to $\ll 10^{20}$ eV. In the latter case, the main limiting factor in reaching the highest energies is represented by curvature energy losses, that are particularly severe [@bible] unless moderately high magnetic fields can be kept with a small accretion rate. This may be the case of dormant supermassive black holes, possibly related to the so-called dark massive objects (DMO). Some 32 of these objects have been identified in a recent survey [@magorrian] and 14 of them have been estimated to have the right features for acceleration of UHECRs [@boldt]. Had this model to be right, it would not be suprising that bright counterparts to the UHECR events were not found, since DMOs are in a quiescent stage of their evolution.
Very little is known of DMOs as cosmic ray accelerators: the spectrum is not known, and neither is known their spatial distribution. It is therefore hard to say at present whether DMOs can satisfy the criteria listed in section \[sec:serious\]. In [@levinson] an interesting prediction was proposed: if UHECRs are accelerated by unipolar induction, they have to radiate part of their energy by synchrotron emission, resulting in the sources to become observable at TeV energies.
One of the most powerful sites for the acceleration of UHECRs is the termination shock of gigantic lobes in radio galaxies. Of particular interest are a subclass of these objects known as Fanaroff-Riley class II objects (FR-II), that can in principle accelerate protons to $\sim 10^{20}-
10^{21}$ eV and explain the spectrum of UHECRs up to the GZK cutoff [@biermann]. These objects are on average on cosmological distances. The accidental presence of a nearby source of FR-II type might explain the spectral shape above the GZK energy, but it would not be compatible with the observed anisotropy [@slb; @blasiolinto99]. Nevertheless, it has been recently proposed that a nearby source in the Virgo cluster (for instance M87) and a suitable configuration of a magnetized wind around our own Galaxy might explain the spectrum and anisotropy at energies above $\sim 10^{20}$ eV [@ahn] as measured by AGASA. This conclusion depends quite sensibly on the choice of the geometry of the magnetic field in the wind. Several additional tests to confirm or disprove this model need to be carried out.
{#sec:td}
An alternative to acceleration scenarios is to generate UHECRs by the decay of very massive particles. In these [*particle physics inspired models*]{} the problem of reaching the maximum energies is solved [*by construction*]{}. The spectra of the particles generated in the decay are typically flatter than the astrophysical ones and their composition at the production point is dominated by gamma rays, although propagation effects can change the ratio of gamma rays to protons. The gamma rays generated at distances larger than the absorption length produce a cascade at low energies (MeV-GeV) which represents a powerful tool to contrain TD models [@bs00]. There are basically two ways of generating the very massive particles and make them decay at the present time: 1) trapping them inside topological defects; 2) making them quasi-stable (lifetime larger than the present age of the universe) in the early universe. We discuss these two possibilities separately in the next two sections.
### Topological Defects
Symmetry breakings at particle physics level are responsible for the formation of cosmic topological defects (for a review see Ref. [@vilshe94]). Topological defects as sources of UHECRs were first proposed in the pioneering work of Hill, Schramm and Walker [@hsw87]. The general idea is that the stability of the defect can be locally broken by different types of processes (see below): this results in the false vacuum, trapped within the defect, to fall into the true vacuum, so that the gauge bosons of the field trapped in the defect acquire a mass $m_X$ and decay.
Several topological defects have been studied in the literature: ordinary strings [@rana90], superconducting strings [@hsw87], bound states of magnetic monopoles [@hill83; @bs95], networks of monopoles and strings [@martin], necklaces [@berevile97] and vortons [@masperi]. Only strings and necklaces will be considered here, while a more extended discussion can be found in more detailed reviews [@bs00; @bbv98].
Strings can generate UHECRs with energy less than $m_X\sim \eta$ (the scale of symmetry breaking) if there are configurations in which microscopic or macroscopic portions of strings annihilate. It was shown [@shellard87; @gillkibble94] that self-intersection events provide a flux of UHECRs which is much smaller than required. The same conclusion holds for intercommutation between strings.
The efficiency of the process can be enhanced by multiple loop fragmentation: as a nonintersecting closed loop oscillates and radiates its energy away, the loop configuration gradually changes. After the loop has lost a substantial part of its energy, it becomes likely to self-intersect and fragment into smaller and smaller loops, until the typical size of a loop becomes comparable with the string width $\eta$ and the energy is radiated in the form of X-particles. Although the process of loop fragmentation is not well understood, some analytical approximations [@bbv98] show that appreciable UHECR fluxes imply utterly large gamma ray cascade fluxes (see however [@bs00]).
Another way of liberating X-particles is through cusp annihilation [@brand87], but the corresponding UHECR flux is far too low [@batta89; @gillkibble94] compared with observations.
The idea that long strings lose energy mainly through formation of closed loops was recently challenged in the simulations of Ref. [@vincent], which show that the string can produce X-particles directly and that this process dominates over the generation of closed loops. This new picture was recently questioned in Refs. [@ms98; @olum00].
Even if the results of Ref. [@vincent] are correct however, they cannot solve the problem of UHECRs [@bbv98]: in fact the typical separation between two segments of a long string is comparable with the Hubble scale, so that UHECRs would be completely absorbed. If by accident a string is close to us (within a few tens Mpc) then the UHECR events would appear to come from a filamentary region of space, implying a large anisotropy which is not observed. Even if the UHECR particles do not reach us, the gamma ray cascade due to absorption of UHE gamma rays produced at large distances imposes limits on the efficiency of direct production of X-particles by strings.
Necklaces are formed when the following symmetry breaking pattern is realized: $G\to H\times U(1) \to H\times Z_2$. In this case each monopole gets attached to two strings (necklace)[@berevile97].
The critical parameter that defines the dynamics of this network is the ratio $r=m/\mu d$ where $m$ is the monopole mass and $d$ is the typical separation between monopoles (e.g. the length of a string segment). If the system evolves toward a state where $r\gg 1$, the distance between the monopoles decreases and in the end the monopoles annihilate, with the production of X-particles and their decay to UHECRs. The rate of generation of X-particles is easily found to be ${\dot n}_X\sim r^2\mu/t^3 m_X$. The quantity $r^2\mu$ is upper limited by the cascade radiation, given by $\omega_{cas}=\frac{1}{2}
f_\pi r^2 \mu =\frac{3}{4} f_\pi r^2\mu/t_0^2$ ($f_\pi\sim 0.5-1$). The typical distance from the Earth at which the monopole-antimonopole annihilations occur is comparable with the typical separation between necklaces, $D\sim\left(\frac{3f_\pi\mu}{4t_0^2\omega_{cas}}\right)^{1/4} > 10
(\mu/10^6 GeV^2)^{1/4}$ kpc.
Clearly, necklaces provide an example in which the typical separation between defects is smaller than the pathlength of gamma rays and protons at ultra-high energies. Hence necklaces behave like a homogeneous distribution of sources, so that the proton component has the usual GZK cutoff. This component dominates the UHECR flux up to $\sim 10^{20}$ eV, while at higher energies gamma rays take over. The fluxes obtained in Ref. [@bbv98] are reported in Fig. 4, where the SUSY-QCD fragmentation functions [@berekac98] were used. The dashed lines are for $m_X=10^{14}$ GeV, the dotted lines for $m_X=
10^{15}$ GeV and the solid lines for $m_X=10^{16}$ GeV. The two curves for gamma rays refer to two different assumptions about the radio background at low frequencies [@protradio].
### Cosmological relic particles
Super heavy particles with very long lifetime can be produced in the early universe and generate UHECRs at present [@berekacvil97; @kr98; @ckr98; @kt98; @ktrev]. In order to keep the same symbolism used in previous sections, we will call these particles X-particles.
The simplest mechanism of production of X-particles in the early universe is the [*gravitational production*]{} [@zelsta72]: particles are produced naturally in a time variable gravitational field or indeed in a generic time variable classical field. In the gravitational case no additional coupling is required (all particles interact gravitationally). If the time variable field is the inflaton field $\phi$, a direct coupling of the X-particles to $\phi$ is needed. In the gravitational production inflation is not required a priori, and indeed it reduces the effect. It can be shown that at time t, gravitational production can only generate X-particles with mass $m_X\leq H(t)\leq m_\phi$, where $H(t)$ is the Hubble constant and $m_\phi$ is the inflaton mass. The authors in Ref. [@ckr98] and [@kt98] demostrated that the fraction of the critical mass contributed by X-particles with $m_X\sim 10^{13}$ GeV produced gravitationally may be $\Omega_X\sim 1$.
If the X-particles are directly coupled to the inflaton field, they can be effectively generated during preheating [@kls94; @felder98]. Alternative mechanisms for the production of X-particles are based on non-equilibrium thermal generation during the preheating stage [@berekacvil97].
As mentioned in the beginning of this section, in order for X-particles to be useful dark matter candidates and generate UHECRs they need to be long lived (for a possible annihilation scenario see Ref. [@dick]). The gravitational coupling by itself induces a lifetime much shorter than the age of the universe for the range of masses which we are interested in. Therefore, in order to have long lifetimes, additional symmetries must be postulated: for instance discrete gauge symmetries can protect X-particles from decay, while being very weakly broken, perhaps by instanton effects [@kr98]. These effects can allow decay times larger than the age of the universe, as shown in [@hama98]. The slow decay of X-particles produces UHECRs. The interesting feature of this model is that X-particles cluster in the galactic halo, as cold dark matter [@bbv98]. Hence UHECRs are expected to be produced locally, with no absorption, and as a consequence the observed spectra are nearly identical to the emission spectra, and therefore dominated by gamma rays. The very flat spectra and the gamma ray composition are two of the signatures. The calculations of the expected fluxes have been performed in [@bbv98; @bsarkar98; @blasi99; @sarkar]. The strongest signature of the model is a slight anisotropy due to the asymmetric position of the sun in the Galaxy [@dubo98; @bbv98; @sarkar1]. More recently a detailed evaluation of the amplitude and phase of the first harmonic has been carried out in [@beremik99] and [@medwat99]. The two papers agree that the present data is still consistent with the anisotropy expected in the model of X-particles in the halo. Small scale anisotropies do not find an easy explanation in TD models, with possibly the exception of the SH relics [@blsh00].
Hints of New Physics? {#sec:liv}
=====================
The possibility that at sufficiently high energies some deviations from known Physics may occur is of particular interest for UHECRs. Some attempts have been made to explain the events above $10^{20}$ eV as a manifestation of some kind of Physics beyond the Standard Model of particle interactions. These suggestions became even more interesting after the recent claims for correlations of the arrival directions of UHECRs with objects at large redshift. Some of these correlations are still subject of debate [@correla]. More recent studies result in a quite intriguing correlation with BL Lacs [@tkachev]. Two of the BL Lacs in the sample used in [@tkachev] are in the error box of the two triplets of events detected by AGASA, and correspond to a distance of $\sim 600$ Mpc, much larger than the pathlength for photopion production of protons (however more than half of the objects used in Ref. [@tkachev] have unknown redshift \[Tkachev, private communication\]).
The absence of bright nearby counterparts to the UHECR events has first inspired theoretical proposals of neutrinos as primary particles, since these particles are not affected by the presence of the CMB and can therefore propagate on cosmological distances without energy losses other than those due to the cosmic expansion. However, the small cross section of neutrinos makes them unlikely primaries: they simply fail to generate the observed showers in the atmosphere. There is one caveat in this argument: the neutrino nucleon cross section at center of mass energies above the electroweak (EW) scale has not been measured, so that the argument above is based on the extrapolation of the known cross sections, and simply limited by the weak unitarity bound [@weiler]. It has been proposed [@domokos] that an increase in the number of degrees of freedom above the EW scale would imply the increase of the neutrino nucleon cross section above the standard model prediction. In particular, in the theories that predict unification of forces at $\sim 1$ TeV scale with large extra dimensions, introduced to solve the hierarchy problem [@arkani], these additional degrees of freedom arise naturally, and imply a neutrino-nucleon cross section that increases linearly with energy, reaching hadronic levels for neutrino energies of $\sim 10^{20}$ eV (for string scale at 1 TeV). The calculations in Ref. [@plu] show that the cross section remains too small to explain vertical showers in the atmosphere (for possible upper limits on this cross section see Ref. [@tyler]).
An alternative, even more radical proposal to avoid the prediction of a GZK cutoff in the flux of UHECRs consists in postulating a tiny violation of Lorentz invariance (LI). The main effect of this violation is that some processes may become kinematically forbidden [@livio]. In particular, photon-photon pair production and photopion production may be affected by LI violation. The absence of the GZK cutoff would then result from the fact that the threshold for photopion production disappears and the process becomes kinematically not allowed. It is suggestive that the possibility of a small violation of LI has also been proposed to explain the apparent absence of an absorption cutoff in the TeV gamma ray emission from Markarian-like objects [@protheroe] (see [@bererecent] for a critical view of this possibility).
Conclusions {#sec:conclusion}
===========
The increasing evidence for a flux of UHECRs exceeding the theoretical expectations for extragalactic sources has fueled interest in several models. These models aim to satisfy all the requirements imposed by observations on fluxes, spatial anisotropies and composition. At present, however there is no obvious successful model. The situation might change with the availability, soon to come, of quite larger statistics of events, that will be achievable by experiments like the Pierre Auger Project [@cronin] and EUSO/Airwatch/OWL [@scarsi; @nasa].
These future experimental efforts will be crucial mainly in three respects: 1) the increase of statistics by a factor 100 for Auger and even more for the space based experiments will allow to strongly constrain theoretical models, and check whether the present excess is a $(2-3)\sigma$ fluctuation or a physical effect. Moreover the small scale anisotropies, if real, will become stronger and a correlation function approach will definitely become appropriate to the analysis of the events; 2) the full sky coverage will finally allow a test of models based on local extragalactic sources and on galactic sources of UHECRs, through the measurement of the large scale anisotropy, that is currently spoiled by the limited spatial exposure; 3) a better determination of the composition of the UHECR events will represent a smoking gun either in favour of or against models: TD models would be ruled out if no gamma rays are found or if heavy nuclei represent the main component. On the other hand, iron-dominated composition would point toward a possible galactic origin, possibly related to neutron stars.
The confirmation, on statistical grounds, of the association of UHECRs to distant cosmological objects like BL Lacs would represent a very strong indication of physics beyond the standard model. Either new interactions or some modification of fundamental physics would be needed to explain such a result.
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---
abstract: 'In this paper we study reusable magic states. These states are a special subset of the standard magic states. Once distilled, reusable magic states can be used repeatedly to apply some unitary U. Given this property, reusable magic states have the potential to greatly lower qubit and gate overhead in fault-tolerant quantum computation. While these states are promising, we provide a strong argument for their limited computational power. Specifically, we show that if reusable magic states can be used to apply non-Clifford unitaries, then we can exploit them to efficiently simulate poly-sized quantum circuits on a classical computer.'
author:
- 'Jonas T.'
date: ', '
title: On the Power of Reusable Magic States
---
Introduction
============
Magic states were introduced by Bravyi and Kitaev [@Bravyi:2005a] as a way to implement logical gates that were not available as transversal gates in an error-correcting code. Their idea was as follows: first prepare many initial magic states, then use these to create an encoded magic state. This encoding procedure will introduce noise, and the encoded magic state will only be close to the desired state. We repeat this process to obtain many noisy encoded magic states. We then put these through many rounds of a distillation protocol. This eventually produces an encoded magic state of the desired fidelity. Finally, we use gate teleportation to apply the gate corresponding to the magic state to our encoded state.
The procedure described above is the canonical way of completing a universal gate set. In fact, Eastin and Knill [@Eastin:2008a] proved that universal and transversal gate sets do not exist for any quantum code, thereby making magic state distillation not only a convenience, but a necessity. Note that there exist other possible ways of completing a universal gate set such as braiding of anyons [@Kitaev:1997a] or Dehn twists [@Koenig:2010a], but these will not be discussed here.
Once we have decided to use magic states, it is important to focus on lowering the immense overhead associated with them. This can be done in a variety of ways which will be discussed in turn below.
(1)
: We can choose a code with many transversal gates available. As mentioned above, it is not possible to have a universal and transversal gate set, but we can come close. For example, the 15-qubit Reed-Muller code [@MacWilliams:1977a; @Steane:1999a] needs only a Hadamard gate to achieve universality, while the toric codes [@Kitaev:1997a] need both the $T$ gate and the Hadamard gate. As a variant of this approach, we could also use codes for which the remaining non-transversal gates have a low overhead magic state implementation. For example, the Hadamard gate magic state protocol likely requires much less overhead than the $T$ gate magic state protocol.
(2)
: The encoding magic states typically introduces significant noise. If this is our primary concern, we can try to find procedures for distilling magic states that allow for very noisy magic states as input. There are known theoretical bounds for this approach. Magic state distillation protocols use only Clifford circuits, and therefore states within the stabilizer polyhedron can never be distilled to non-stabilizer states. This is simply because the stabilizer states are closed under Clifford operations. There are, however, other conditions which preclude non-stabilizer states from being distillable, such as positivity of the discrete Wigner function [@Veitch2012]. Also the existence of bound states [@Campbell:2010a; @Campbell:2009a; @Campbell:2009b; @Campbell:2010b] (states that cannot be distilled) have also been discovered. Nevertheless certain states that lie on the border of the stabilizer polyhedron have been shown to be distillable [@Reichardt:2005a]. Finding more states such as these should be our primary goal if we expect the initial magic states to be very noisy. The schemes discussed in this paragraph are illustrated in Fig. \[fig:protocolunencoded\]. We input some noisy states $\ket{\tilde{\rho}}$ into the distillation circuit (a). The approach outlined above seeks to find circuits (a) that allow very noisy inputs while still eventually distilling $\ket{M}$. If the system is noisy but qubits are abundant, this would be the appropriate paradigm to study.
(3)
: If it is not difficult to prepare noisy magic states that meet the criteria for distill-ability, then our primary concern is to reduce overhead. Two ideas for reducing overhead are discussed below. Note that (3a) and (3b) are not mutually exclusive.
(3a)
: For each magic state that we apply, we must first distill a high fidelity version of that magic state. This involves using many lower fidelity magic states as input to a distillation circuit with the goal of producing a higher fidelity magic state as output. Typically this process must be repeated for many rounds, feeding the output of one distillation circuit into the input of the next. The methods for reducing overhead in the distillation circuit involve either reducing the number of inputs needed at a given round or reducing the total number of rounds. Graphically, this approach focuses on improving the circuit in Fig. \[fig:protocolunencoded\](a) by reducing the number of inputs and/or rounds of distillation. Protocols to reduce the number of rounds in magic state distillation were discussed in [@Meier:2012a].
(3b)
: Another way of reducing the overhead is to reuse magic states. This is accomplished by modifying the gate teleportation procedure such that the magic state is available for reuse after the gate has been applied. We will refer to these magic states as reusable magic states. This approach would allow magic states $\ket{M}$ that are input into the box in Fig. \[fig:protocolunencoded\](b) to be reused without any additional distillation. If a code could be found such that its universal gate set was comprised of only transversal gates and reusable magic states, the savings in overhead would be immense. Once these magic states have been distilled there would be no need for any additional overhead ever! We will show that in most cases these reusable magic states are highly unlikely to exist.
Reusable Magic States
=====================
A quantum computing architecture that uses magic states consists of an encoded system $\mathcal{S}$ and a supply of encoded magic states in an auxiliary system $\mathcal{M}$. Here $\mathcal{M}$ refers to a fixed size, but otherwise arbitrary system. These systems should remain isolated until a magic state is needed in the computation. It is in this paradigm that we hope to implement gates with reusable magic states.
Below we will represent our system simply as $\ket{\psi}$. We will represent the auxiliary system containing the magic state as $\ket{M}$. The argument that follows applies to both single qubit and encoded qubit systems. Also, these states can be mixed or pure. To make notation simpler, we will represent the systems as pure states on single qubit systems.
Formally we define a [*reusable magic state*]{} as a state $\ket{M}$ such that after application of a Clifford circuit on the joint system $\mathcal{S}\otimes \mathcal{M}$ some gate $U_{M}$ has been applied to the system $\mathcal{S}$ and the state $\ket{M}$ of the system $\mathcal{M}$ is unchanged. The state $\ket{M}$ can therefore be used again. We will use this definition to prove that reusable magic states do not exist for non-Clifford gates. When defining reusable magic states for Clifford gates the above definition must be restricted to prevent Clifford gates of the type are attempting to implement re-usably.
A reusable magic state for the $S$ gate ($\sqrt{Z}$ gate) was shown in [@Aliferis:2007b; @Jones:2010a]. The $S$ gate is a Clifford gate; however the circuit uses only $CNOT$ and $H$ to implement the $S$ gate.
$$\Qcircuit @C=1.5em @R=1.5em {
\ket{\pi/2} & & \targ & \gate{H} & \targ & \gate{H} & \qw & \ket{\pi/2} \\
\ket{\psi} & & \ctrl{-1} & \qw & \ctrl{-1} & \qw & \qw & S\ket{\psi}
}$$
This gate can be modified to make a transversal $\sqrt{X}$ gate with the identity $\sqrt{X}=HSH$. Additionally, the combination of two reusable gates is itself a reusable gate. A reusable $\sqrt{Y}$ gate can be constructed by combining $\sqrt{X}$ and $S (\sqrt{Z})$ gates. It may seem that a reusable $H$ gate could be built through similar constructions; however all attempts by the author to date require that the $H$ gate be present in the circuit. This could only be the case when that gate is already available transversally, obviating the need for such a reusable magic state.
Non-Clifford Reusable Magic States
==================================
Most research on magic states focuses exclusively on non-Clifford magic states. In many technologies Clifford gates are considered to be easier to implement than general unitaries. They can be made universal with the addition of a single non-Clifford unitary. In other words, the gate set $<\{\mbox{Clifford}\},\; U>$ provides a dense set of unitaries in $SU(2^n)$. The canonical choice for $U$ is the $T$ gate ($\sqrt{S}$ gate).
$$\Qcircuit @C=1.5em @R=1.5em {
\ket{\psi} & & \ctrl{1} & \qw & \gate{S} & \qw & T\ket{\psi} \\
\ket{\pi/4} & & \targ & \measureD{Z} & \control \cw \cwx &
}$$
It is well-known that Clifford gates can be efficiently simulated on a classical computer in polynomial time [**P**]{}. In fact, the computational power of a Clifford gate computer is thought to be weaker than a polynomial-time classical computer.
The power of a polynomial-sized universal quantum gate set is, by definition, the class [**BQP**]{} and hence can solve any problem within this class. Using the Solvay-Kitaev algorithm [@Kitaev:2002a; @Nielsen:2000a], we can compile any gate from a universal gate set in time linear in $\log^{c}(1/\epsilon)$. Where $c$ is some constant (typically between 2 and 3), and $\epsilon$ is the desired precision of the compiled gate. While some gate sets may be more efficient (in terms of overhead) than others, any universal quantum gate set can be used to efficiently solve problems in the class [**BQP**]{}.
In the derivation below we will present a ‘proof by contradiction’. We will assume the existence of a non-Clifford reusable magic state circuit, and then show that if such a circuit could be constructed, it would imply that [**BQP**]{} $=$ [**P**]{}.
First, assume that the following circuit exists (see Fig. \[fig:reusableMS\]).
$$\Qcircuit @C=1.5em @R=1.5em {
\ket{M} & & \multigate{1}{\mathcal{C}} & \qw & \ket{M}\\
\ket{\psi} & & \ghost{\mathcal{C}} & \qw & M \ket{\psi}
}$$
Where $\mathcal{C}$ denotes some Clifford circuit and $\ket{M}$ is the reusable magic state. $\ket{M}$ may be comprised of many qubits and/or qudits as long as the size is fixed. $M$ is any non-Clifford unitary.
Now, since $<\{\mbox{Clifford}\},\; M>$ constitutes a universal gate set, we can write a general quantum circuit using only Clifford gates and $M$.
$$\Qcircuit @C=1.5em @R=1.5em {
\ket{0} & & \multigate{6}{\mathcal{C}_1} & \gate{M} & \qw & \multigate{6}{\mathcal{C}_2} & \qw & \qw \\
\ket{0} & & \ghost{\mathcal{C}_1} & \qw & \gate{M} & \ghost{\mathcal{C}_2} & \qw & \qw \\
\ket{0} & & \ghost{\mathcal{C}_1} & \qw & \qw & \ghost{\mathcal{C}_2} & \gate{M} & \qw \\
\ket{0} & & \ghost{\mathcal{C}_1} & \qw & \qw & \ghost{\mathcal{C}_2} & \gate {M} & \qw & ... \\
\ket{0} & & \ghost{\mathcal{C}_1} & \qw & \qw & \ghost{\mathcal{C}_2} & \qw & \qw & \\
\vdots \\
\ket{0} & & \ghost{\mathcal{C}_1} & \qw & \qw & \ghost{\mathcal{C}_2} & \qw & \qw \\
}$$
For example, Fig. \[fig:example\] depicts a general quantum circuit with $\mathcal{C}_n$ denoting a round of arbitrary poly-sized Clifford gates, and $M$ a non-Clifford unitary. The circuit is simulate-able by a [**BQP**]{} quantum computer as long as the circuit size (total number of circuit elements) is polynomial in the number of inputs.
However, since we assumed that circuits of the form shown in Fig. \[fig:reusableMS\] exist, we can execute the same computation shown in Fig. \[fig:example\] by replacing the $M$ gates with their magic state implementation. Note that this will only increase the number of inputs by a constant amount (the size of $\ket{M}$).
$$\Qcircuit @C=1.5em @R=1.5em {
\ket{M}& & \qw & \multigate{1}{\mathcal{C}} & \gate{\mathcal{C}} \qwx[1] & \qw & \gate{\mathcal{C}} \qwx[1] & \gate{\mathcal{C}} \qwx[1] & \qw & \ket{M}\\
\ket{0} & & \multigate{6}{\mathcal{C}_1} & \ghost{\mathcal{C}} & \qw \qwx[1] & \multigate{6}{\mathcal{C}_2} & \qw \qwx[1] & \qw \qwx[1] & \qw & \\
\ket{0} & & \ghost{\mathcal{C}_1} & \qw &\gate{\mathcal{C}} & \ghost{\mathcal{C}_2} & \qw \qwx[1] & \qw \qwx[1] & \qw & \\
\ket{0} & & \ghost{\mathcal{C}_1} & \qw & \qw & \ghost{\mathcal{C}_2} & \gate{\mathcal{C}} & \qw \qwx[1] & \qw & \\
\ket{0} & & \ghost{\mathcal{C}_1} & \qw & \qw & \ghost{\mathcal{C}_2} & \qw & \gate{\mathcal{C}} & \qw & ... \\
\ket{0} & & \ghost{\mathcal{C}_1} & \qw & \qw & \ghost{\mathcal{C}_2} & \qw & \qw & \qw & \\
\vdots \\
\ket{0} & & \ghost{\mathcal{C}_1} & \qw & \qw & \ghost{\mathcal{C}_2} & \qw & \qw & \qw & \\
}$$
We can continue this process and replace all gates $M$ by their magic state implementations. Now the entire body of the computation consists of only Clifford gates. We have only to prepare the state $\ket{M}$ which is unentangled with the rest of the system. This state could still be highly non-trivial; however we can always represent this state as a sum of stabilizer states. For example, the single qubit pure state $\ket{\psi} = \alpha\ket{0} + \beta\ket{1}$ can be written as $|\alpha|^{2}\ket{0}\bra{0} + \alpha\beta^{*}\ket{0}\bra{1} + \alpha^{*} \beta\ket{1}\bra{0} + |\beta|^{2}\ket{1}\bra{1} = a_{+}I + a_{-}Z + b_{+}X + (-i)b_{-}Y$. Where $a_{\pm} = \frac{|\alpha|^{2} \pm |\beta|^{2}}{2}$ and $b_{\pm} = \frac{\alpha\beta^{*} \pm \alpha^{*}\beta}{2}$. We have fixed the size of $\ket{M}$ to be independent of circuit size; therefore it can always assumed that the circuit size is large enough, such that the dimension of $\ket{M}$ is logarithmic in number of inputs to the circuit. We can then write $\ket{M}$ as a sum of stabilizers which will generally have a number of terms that grows as $\bigO(2^{d})$. Where $d$ is the dimension of $\ket{M}$. Again, this number is fixed and is independent of circuit size. This amounts to a constant overhead in our notation. Finally, since the entire body of the circuit consists of Clifford gates which map stabilizer states to stabilizer states, the number of terms in the initial sum of stabilizer states is fixed throughout the computation. We can simulate each of the terms in the initial sum of stabilizer states in time that grows polynomially with the number of input states. We can thus simulate the entire circuit in time $\bigO(2^{d}\times POLY(n)) = \bigO(POLY(n))$. Where $d=c$ (some constant) and $n$ is the number of input states.
In conclusion, we have shown that if a circuit such as that shown in Fig. \[fig:reusableMS\] exists for non-Clifford unitary $M$, then [**BQP**]{} $=$ [**P**]{}. In fact, since Clifford state computation is in the class [**ParityL**]{} [@Aaronson:2004a] (which is thought to be weaker than [**P**]{}), this would be of even greater consequence. In the highly unlikely event that such a circuit exists, it would not be useful since the entire endeavor of quantum computation would be obviated as a consequence.
Some open questions still linger such as: [*Does a reusable magic state exist for the $H$ gate?*]{} This circumvents the proof in this paper, since $H$ [*is*]{} a Clifford gate. Codes such as the 15-qubit Reed-Muller code can be made universal with the addition of such a gate; therefore finding such a state would drastically reduce the overhead for this and similar codes. As mentioned above, our definition of reusable magic states must be modified when the unitary we are trying to implement is a Clifford gate.
Qudit magic state distillation has recently been introduced in [@Campbell:2012a; @Anwar:2012a; @Veitch2012]. Our result applies to qudit codes as well. Namely, non-Clifford qudit gates cannot be implemented using reusable magic states, unless qudit quantum computation is efficiently simulate-able on a classical computer. The proof is briefly sketched here: The Clifford group for any prime number $p$ in $SU(p^n)$ is a maximal finite subgroup. The addition of any non-Clifford unitary generates an infinite group which is dense in $SU(p^n)$. As in the qubit case, we need only a single non-Clifford gate to complete a universal gate set. These properties of the Clifford group are not well known and were only recently mentioned in the physics literature (see appendix in [@Campbell:2012a] and references therein). Using this, the proof for the qudit case follows in exactly the same manner as the qubit case.
It is, however, possible that qudit analogues of the Reed-Muller or other similar codes can complete a universal gate set with the addition of some qudit Clifford gate. Therefore, it may be fruitful to search for these codes and for reusable qudit magic states.
The author would like to acknowledge many helpful conversations with Chris Cesare, Andrew Landahl, Rolando Somma, Adam Meier, Bryan Eastin, Jim Harrington, and Olivier Landon-Cardinal. Additionally, I would like to thank Earl Campbell for pointing out subtleties in the qudit Clifford group and providing ideas for extending this proof to the qudit case.
JTA was supported in part by the National Science Foundation through Grant 0829944. JTA was supported in part by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.
[20]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[`arXiv:#2`](http://arxiv.org/pdf/#2)]{} \[2\]\[\][[`doi:#2`](http://dx.doi.org/#2)]{}
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, , , **, p. (), , <http://arxiv.org/abs/1002.2816>.
, **, vol. of ** (, , ), ISBN .
, **, ****, (), ISSN , , [http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber%
=771249](http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber%
=771249).
, , , **, ().
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, **, p. ().
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, ** (), , , <http://arxiv.org/abs/1010.0104>.
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, , , **, p. (), , <http://arxiv.org/abs/1204.4221>.
, **, Ph.D. thesis, ().
, , , , , , , **, p. (), , <http://arxiv.org/abs/1010.5022>.
, , , **, vol. of ** (, , ), ISBN .
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, , , ** (), , <http://arxiv.org/abs/1205.3104>.
, , , **, p. (), , <http://arxiv.org/abs/1202.2326>.
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---
abstract: 'We establish the existence of weak solutions of a nonlinear radiation-type boundary value problem for elliptic equation on divergence form with discontinuous leading coefficient. Quantitative estimates play a crucial role on the real applications. Our objective is the derivation of explicit expressions of the involved constants in the quantitative estimates, the so-called absolute or universal bounds. The dependence on the leading coefficient and on the size of the spatial domain is precise. This work shows that the expressions of those constants are not so elegant as we might expect.'
address: 'Luisa Consiglieri, Independent Researcher Professor, European Union'
author:
- Luisa Consiglieri
title: 'Explicit estimates for solutions of nonlinear radiation-type problems'
---
Introduction
============
Thermal effects on steady-state physical and technological models, whatever they are from mechanical engineering, electrochemistry, biomedical engineering, to mention a few, appear as an additional elliptic equation with a nonlinear radiation-type boundary condition into the coupled PDE system under study [@zamm; @aduf; @lap; @epj]. These form a boundary value problem constituted by an elliptic quasilinear second order equation in divergence form with the leading coefficient depending on the spatial variable and on the solution itself. The problem of determining radiative effects provides an interesting special case of a conormal derivative boundary value problem for an elliptic divergence structure equation [@laitii]. Here, we deal with the radiation-type condition on a part of the boundary, and on the remaining part the Neumann condition is taken into account. Stationary heat conduction equation with the radiation boundary condition (fourth power law) has been studied in two-dimensional [@milka] and three-dimensional [@simon] Lipschitz domains.
In the existence theory, the quantitative estimates of solutions to a linear elliptic equation in divergence form, with bounded and measurable coefficient, play a crucial role. Indeed, they enjoy a large interest in the literature (see for instance [@ben; @caz; @gia93; @gt; @lu; @stamp63-64], and the references therein). Most mathematicians have bearing to keep abstract the universal bounds along one whole work. The values of the intervener constants are simply carried out. It is forgotten that their values are crucial on the real applications and/or the numerical analysis (see [@ko] and the references therein) of the problems under study. Our objective is to fill such gap.
The outline of the present paper is as follows. We begin by stating the problem under study and its functional framework in the next section. The Hilbert case is studied in Section \[diri\]. We derive $L^q$ (Section \[lpc\]), $L^\infty$ (Section \[linfty\]), and $W^{1,q}$ (under $L^1$-data in Section \[l1data\]) estimates for weak solutions. Finally, a $W^{1,p}$-estimate $(p<n/(n-1)$ for the Green kernel and a $W^{1,q}$-estimate for weak solutions of linear boundary value problem, the so-called mixed Robin-Neumann problem, are obtained in Sections \[secg\] and \[lqc\], respectively. Lipschitz domains, discontinuous leading coefficient, and $L^1$-data are the three mathematical shortcomings from the physical models on the real world. It is taken them into account that our results are stated.
Statement of the problem
========================
Set $\Omega$ a domain (that is, connected open set) in $\mathbb{R}^n$ ($n\geq 2$) of class $C^{0,1}$, and bounded. Its boundary $\partial\Omega$ is constituted by two disjoint open $(n-1)$-dimensional sets, $\Gamma_N$ and $\Gamma$, such that $\partial\Omega=\bar\Gamma_N\cup \bar\Gamma$. We consider $\Gamma_N$ over which the Neumann boundary condition is taken into account, and $\Gamma$ over which the radiative effects may occur.
We study the following boundary value problem, in the sense of distributions, $$\begin{aligned}
-\nabla\cdot( \mathsf{A}
\nabla u)=f-\nabla\cdot{\bf f}&\mbox{ in }&\Omega;\label{omega}\\
(\mathsf{A}\nabla u-{\bf f})\cdot{\bf n}+b(u)=h&\mbox{ on }&\Gamma;
\label{robin}\\
(\mathsf{A}\nabla u-{\bf f})\cdot{\bf n}=g&\mbox{ on }&\Gamma_N, \label{gama}\end{aligned}$$ where $\bf n$ is the unit outward normal to the boundary $\partial\Omega$. Whenever the $(n\times n)$-matrix of the leading coefficient is $\mathsf{A}=aI$, where $a$ is a real function and $I$ denotes the identity matrix, the elliptic equation stands for isotropic materials. Our problem includes the conormal derivative boundary value problem. For that, it is sufficient to consider the situation $\Gamma=\partial\Omega$ (or equivalently $\Gamma_N=\emptyset$). The problem (\[omega\])-(\[gama\]) is the so-called mixed Robin-Neumann problem if the boundary condition (\[robin\]) is linear, i.e. $$\label{bstar}
b(u)=b_*u,\qquad\mbox{for some }b_*>0.$$
Set for any $p,\ell\geq 1$ $$V_{p,\ell}:=\{ v\in
W^{1,p}(\Omega):\ v\in L^{\ell}(\Gamma)\}$$ the Banach space endowed with the norm $$\| v\|_{V_{p,\ell}}:=\| v\|_{p,\Omega}+
\|\nabla v\|_{p,\Omega}+\|v\|_{\ell,\Gamma}.$$ For the sake of simplicity, we denote by the same designation $v$ the trace of a function $v\in W^{1,1}(\Omega)$. For $p>1$, the space $V_{p,\ell}$ is reflexive by arguments given in [@dpz]. Observe that $V_{p,\ell}$ is a Hilbert space equipped with the inner product only if $p=\ell=2$. The above norm is equivalent to $$\label{norm}
\| v\|_{1,p,\ell}:=\|\nabla v\|_{p,\Omega}+\|v\|_{\ell,\Gamma},$$ due to a Poincaré inequality [@chakib Corollary 3]: $$\|v\|_{p,\Omega}\leq P_p\left(\sum_{i=1}^n
\|\partial_iv\|_{p,\Omega}+|\Gamma|^{1/p-1}
\left|\int_\Gamma v\mathrm{ds}\right|\right).$$ Here $|\cdot|$ stands for the $(n-1)$-Lebesgue measure. Throughout this work, the significance of $|\cdot|$ also stands for the Lebesgue measure of a set of $\mathbb{R}^n$.
By trace theorem, $$\begin{aligned}
V_{p,\ell}=W^{1,p}(\Omega),\quad \mbox{if } 1\leq\ell< p(n-1)/(n-p);\\
V_{p,\ell}\subset_{\not=}W^{1,p}(\Omega),\quad \mbox{if } \ell>p(n-1)/(n-p).\end{aligned}$$
For $1<q<n$, the best constants of the Sobolev and trace inequalities are, respectively, [@tale; @bond] $$\begin{aligned}
S_q&=&\pi^{-1/2}n^{-1/q}\left({q-1\over n-q}\right)^{1-1/q}\left[
{\Gamma(1+n/2)\Gamma(n)\over \Gamma (n/q)\Gamma(1+n-n/q)}\right]^{1/n};
\\
K_q&=&\pi^{(1-q)/2}\left({q-1\over n-q}\right)^{q-1}\left[
{\Gamma\left({q(n-1)\over 2(q-1)}\right)\Big/ \Gamma \left({
n-1\over 2(q-1)}\right)}\right]^{(q-1)/(n-1)},\end{aligned}$$ where $\Gamma$ stands for the Gamma function. For $1^*=n/(n-1)$, there exists the limit constant $S_1=\pi^{-1/2}n^{-1}[\Gamma(1+n/2)]^{1/n}$ [@tale]. Hence, we introduce $S_{q,\ell}=S_q\max\{1+ P_q2^{(n-1)(1-1/q)},
P_q |\Gamma |^{1/q-1/\ell}\}$ and $K_{q,\ell}=K_q \max\{1+ P_q
2^{(n-1)(1-1/q)},
P_q |\Gamma |^{1/q-1/\ell}\}$ that verify $$\begin{aligned}
\label{sob}
\|v\|_{nq/(n-q),\Omega}\leq S_{q,\ell}\|v\|_{1,q,\ell}
;\\ \label{sobt}
\|v\|_{(n-1)q/(n-q),\partial\Omega}\leq K_{q,\ell}\|v\|_{1,q,\ell}.\end{aligned}$$
\[def1\] We say that $u\in V_{p,\ell}$ is a weak solution to (\[omega\])-(\[gama\]), if it verifies $$\begin{aligned}
\label{pbu}
\int_{\Omega} ( \mathsf{A}\nabla u)\cdot
\nabla v \mathrm{dx}+\int_{\Gamma}b(u) v \mathrm{ds}
=\int_{\Omega}{\bf f}\cdot\nabla v \mathrm{dx}+\\
+\int_{\Omega}fv \mathrm{dx}
+\int_{\Gamma_N}gv \mathrm{ds}+\int_{\Gamma}hv \mathrm{ds},
\quad\forall v\in V_{p',\ell},\nonumber\end{aligned}$$ where ${\bf f}\in {\bf L}^{p}(\Omega)$, $f\in L^{t}(\Omega)$, with $t=pn/(n+p)$ if $p>n/(n-1)$ and any $t>1$ if $1<p\leq n/(n-1)$, $g\in L^{s}(\Gamma_N)$, with $s=p(n-1)/n$ if $p>n/(n-1)$ and any $s>1$ if $1<p\leq n/(n-1)$, and $h\in L^{\ell/(\ell-1)}(\Gamma)$.
All terms on the right hand side of (\[pbu\]) have sense, since the following embeddings hold: $$\begin{aligned}
W^{1,q}(\Omega)\hookrightarrow C(\bar\Omega)\quad\mbox{ for $q=p'>n,$
i.e. $p<n/(n-1)$};\\
\left.\begin{array}{l}
{W}^{1,q}(\Omega)
\hookrightarrow { L}^{q^*}(\Omega)\\
W^{1,q}(\Omega)\hookrightarrow L^{q_*}(\partial\Omega)
\end{array}\right\}\quad\mbox{ for $q=p'<n,$
i.e. $p>n/(n-1)$},\end{aligned}$$ with $q^*=qn/(n-q)$ and $q_*=q(n-1)/(n-q)$ being the critical Sobolev and trace exponents, respectively, and $p'$ accounts for the conjugate exponent $p'=p/(p-1)$. We observe that $q^*>1$ is arbitrary if $q=n$.
We emphasize that the existence of equivalence between the differential (\[omega\])-(\[gama\]) and variational (\[pbu\]) formulations is only available under sufficiently data. For instance, the Green formula may be applied if $\mathsf{A}\nabla u \in {\bf L}^p(\Omega)$ and $\nabla\cdot(\mathsf{A}
\nabla u)\in L^p(\Omega)$.
Assume
(A)
: $\mathsf{A}=[A_{ij}]_{i,j=1,\cdots,n}
\in [L^\infty(\Omega)]^{n\times n}$ is uniformly elliptic, and uniformly bounded: $$\begin{aligned}
\label{amin}
\exists a_\#>0,&&
A_{ij}(x)\xi_i\xi_j\geq a_\#|\xi|^2,
\quad\mbox{ a.e. }x\in\Omega,\ \forall \xi\in\mathbb{R}^n;\\
\exists a^\#>0,&&\|\mathsf{A}\|_{\infty,\Omega}\leq a^\#,\label{amax}\end{aligned}$$ under the summation convention over repeated indices.
(B)
: $b:\Omega\times \mathbb{R}\rightarrow \mathbb{R}$ is a Carathéodory function such that it is strictly monotone with respect to the last variable, and it has the following $(\ell-1)$-growthness properties: $$\begin{aligned}
\label{bmin}
\exists b_\#>0, &&b(x,T){\rm sign}(T)\geq b_\#|T|^{\ell-1};\\
\exists b^\# >0,&&|b(x,T)|\leq b^\#|T|^{\ell-1},\label{bmax}\end{aligned}$$ for a.e. $x\in\Omega$, and for all $T\in \mathbb{R}$.
If $b(T)=|T|^{\ell-2}T$, for all $T\in\mathbb{R}$, the property of strong monotonicity occurs with $b_\#=2^{(2-\ell)}$ [@dpz Lemma 3.3].
$V_{2,\ell}$-solvability ($\ell\geq 2$) {#diri}
=======================================
We establish the existence and uniqueness of weak solution as well as its quantitative estimate. Although their proof is quite standard, the explicit expression of the bound is unknown, as far as we known.
\[exist\] Let ${\bf f}\in {\bf L}^{2}(\Omega)$, $f\in L^{t}(\Omega)$, with $t=(2^*)'$, i.e. $t=2n/(n+2)$ if $n>2$ and any $t>1$ if $n=2$, $g\in L^{s}(\Gamma_N)$, with $s=2(n-1)/n$ if $n>2$ and any $s>1$ if $n=2$. Under the assumptions (A)-(B), there exists $u \in V_{2,\ell}$ being a weak solution to (\[omega\])-(\[gama\]), i.e. solving (\[pbu\]) for all $v\in V_{2,\ell}$. Moreover, the following estimate holds $$\begin{aligned}
\label{cotau}
{a_\#\over 2}\|\nabla u\|_{2,\Omega}^2+
{ b_\#(\ell -1)\over\ell}\| u\|_{\ell,\Gamma}^\ell\leq {1\over
2a_\#}\left(
\|{\bf f}\|_{2,\Omega}+\mathcal{F}_{n}(
\|f\|_{t,\Omega},\|g\|_{s,\Gamma_N})\right)^2+\\
+ {\ell-1\over\ell b_\#^{1/(\ell-1)}}
\left(\|h\|_{\ell /(\ell-1),\Gamma}
+\mathcal{H}_{n}(\|f\|_{t,\Omega}, \|g\|_{s,\Gamma_N})\right)^{\ell/(\ell-1)}
:=\mathcal{A},\nonumber\end{aligned}$$ where $\mathcal{F}_n(A,B)=\mathcal{H}_n(A,B)=S_{2,\ell}A+K_{2,\ell}B$ if $n>2$, $\mathcal{F}_2(A,B)=\mathcal{H}_2( |\Omega |^{1/t'}A,
|\Omega |^{1/(2s')}B)$, and $\mathcal{H}_2(A,B)= S_{2t/(3t-2),\ell} A+ K_{2s/(2s-1),\ell}B$ if $t<2$. In particular, if $t\geq 2=n$, the estimate (\[cotau\]) holds with $\mathcal{F}_2(A,B)=\mathcal{H}_2( |\Omega |^{1/2}A,
|\Omega |^{1/(2s')}B)$, and $\mathcal{H}_2(A,B)= S_{1,\ell}|\Omega |^{1/2-1/t}
A+K_{2s/(2s-1),\ell}B$.
The existence and uniqueness of a weak solution $u\in V_{2,\ell}$ is consequence of the Browder-Minty theorem, since the functional $T:V_{2,\ell}\rightarrow (V_{2,\ell})'$ defined by $$T(v)=
\int_{\Omega} ( \mathsf{A}\nabla u)\cdot
\nabla v \mathrm{dx}+\int_{\Gamma}b(u) v \mathrm{ds}$$ is strictly monotone, continuous, bounded and coercive.
Taking $v=u\in V_{2,\ell}$ as a test function in (\[pbu\]), using the Hölder inequality we obtain $$\begin{aligned}
\label{pbuab}
{a_\#}\|\nabla u\|_{2,\Omega}^2+
{ b_\#}\| u\|_{\ell,\Gamma}^\ell\leq
\|{\bf f}\|_{2,\Omega}\|\nabla u\|_{2,\Omega}+\\
+\|h\|_{\ell/(\ell-1),\Gamma}\| u\|_{\ell,\Gamma}
+\|f\|_{t,\Omega}\|u\|_{t',\Omega}
+\|g\|_{s,\Gamma_N}\|u\|_{s',\Gamma_N}.\nonumber\end{aligned}$$
For $n>2$, making use of (\[sob\]) and (\[sobt\]) with $q=2$, we get $$\begin{aligned}
{a_\#\over 2}\|\nabla u\|_{2,\Omega}^2+
{ b_\#\over\ell\,'}\| u\|_{\ell,\Gamma}^\ell\leq
{1\over 2a_\#}\left(\|{\bf f}\|_{2,\Omega}+
S_{2,\ell}\|f\|_{t,\Omega}+K_{2,\ell}
\|g\|_{s,\Gamma_N}\right)^2+\\
+{1\over \ell\,' b_\#^{1/(\ell-1)}}
\left(\|h\|_{\ell/(\ell-1),\Gamma}+S_{2,\ell}\|f\|_{t,\Omega}+K_{2,\ell}
\|g\|_{s,\Gamma_N}\right)^{\ell/(\ell-1)}.\end{aligned}$$ Therefore, (\[cotau\]) follows.
Consider the case of dimension $n=2$. For $t,s>1$, using the Hölder inequality in (\[sob\]) with $q=2t'/(t'+2)$ if $t'\geq 2$, and in (\[sobt\]) for any $s>1$, we have $$\begin{aligned}
\|u\|_{t',\Omega}\leq
S_{{2t\over 3t-2},\ell}\| u\|_{1,2t/(3t-2),\ell}\leq
S_{{2t\over 3t-2},\ell}\left(
|\Omega |^{1/t'}\|\nabla u\|_{2,\Omega}+\|u\|_{\ell,\Gamma}\right) ;\\
\|u\|_{s',\Gamma_N}\leq K_{{2s\over 2s-1},\ell}\|u\|_{1,2s/(2s-1),\ell} \leq
K_{{2s\over 2s-1},\ell}\left(|\Omega |^{1/(2s')}\|\nabla u\|_{2,\Omega}
+\|u\|_{\ell,\Gamma}\right) .\end{aligned}$$ Inserting the above inequalities in (\[pbuab\]), it results in (\[cotau\]).
Finally, if $t> 2$, we have $$\|u\|_{t',\Omega}\leq
|\Omega |^{1/2-1/t}\|u\|_{2,\Omega}\leq
|\Omega |^{1/2-1/t} S_{1,\ell}\left(
|\Omega |^{1/2}\|\nabla u\|_{2,\Omega}+\|u\|_{\ell,\Gamma}\right).$$ This concludes the proof of Proposition \[exist\].
\[sol\] Proposition \[exist\] remains valid if the assumption $h\in L^{\ell/(\ell-1)}(\Gamma)$ is replaced by $h\in L^{s}(\Gamma)$, with the estimate (\[cotau\]) being rewritten with $$\begin{aligned}
\label{cotausol}
{\mathcal A}={1\over 2a_\#}\left(
\|{\bf f}\|_{2,\Omega}+\mathcal{F}_{n}(
\|f\|_{t,\Omega},\|g\|_{s,\Gamma_N}+\|h\|_{s,\Gamma})\right)^2+\\
+ {\ell-1\over\ell b_\#^{1/(\ell-1)}}
\left[\mathcal{H}_{n}(\|f\|_{t,\Omega}, \|g\|_{s,\Gamma_N}+
\|h\|_{s,\Gamma})\right]^{\ell\,'}.\nonumber\end{aligned}$$
\[coruq\] Under the conditions of Proposition \[exist\], we have $$\begin{aligned}
\label{cotausup}\qquad
\|u\|_{2p/(p-2),\Omega}\leq
S_{{2pn\over 2p+n(p-2)},\ell}\left(
|\Omega |^{{1\over n}-{1\over p}}\left({2\mathcal{A}\over a_\#}\right)^{1/2}
+\left({\ell\;'\mathcal{A}\over b_\#}\right)^{1/\ell}\right);\\
\|u\|_{2s',\Omega}\leq
K_{{2sn\over 2s+(n-1)(s-1)},\ell}\left(
|\Omega |^{s-n+1\over 2ns}\left({2\mathcal{A}\over a_\#}\right)^{1/2}
+\left({\ell\;'\mathcal{A}\over b_\#}\right)^{1/\ell}\right),
\label{cotaus}\end{aligned}$$ for $p\geq n>2$, $p>n=2$, $s\geq n-1>1$, and $s>1$ ($n=2$).
Making use of (\[sob\]) with $q=2pn/[2p+n(p-2)]$ if $p>2$, and the Hölder inequality for $p\geq n$, we obtain $$\begin{aligned}
\|u\|_{2p/(p-2),\Omega}\leq
S_{{2pn\over 2p+n(p-2)},\ell}\| u\|_{1,{2pn\over 2p+n(p-2)},\ell}\leq\\
\leq S_{{2pn/[2p+n(p-2)]},\ell}\left(
|\Omega |^{1/n-1/p}\|\nabla u\|_{2,\Omega}+\|u\|_{\ell,\Gamma}\right).\end{aligned}$$ Applying (\[cotau\]) in the above inequality, we conclude (\[cotausup\]).
Making use of (\[sobt\]) with $q=2sn/[2s+(n-1)(s-1)]$ if $s>1$, and the Hölder inequality for $s\geq n-1$, we obtain $$\begin{aligned}
\|u\|_{2s/(s-1),\partial\Omega}\leq
K_{{2sn\over 2s+(n-1)(s-1)},\ell}\| u\|_{1,{2sn\over 2s+(n-1)(s-1)},\ell}\leq\\
\leq K_{{2sn/[2s+(n-1)(s-1)]},\ell}\left(
|\Omega |^{(s-n+1)/(2ns)}\|\nabla u\|_{2,\Omega}+\|u\|_{\ell,\Gamma}\right).\end{aligned}$$ Thus, (\[cotaus\]) holds as before.
$L^{q}$-estimates ($q<2(n-1)p/[2(n-1)-p]$, $2< p<2(n-1)$) {#lpc}
=========================================================
Section \[diri\] ensures the existence of a weak solution, $u\in L^{2p/(p-2)}(\Omega)$, to (\[omega\])-(\[gama\]) in accordance with Definition \[def1\], only if $p\geq n>2$. Let us improve that.
First, let us introduce the Marcinkiewicz space, $L^*_p(\Omega)$, which is Banach space of the measurable functions that have finite the following norm [@gt]: $$\|v\|_{*,p,\Omega}:=\sup _{t>0} t |
\Omega[|v|>t]|^{1/p},$$ for $p>1$ and $0<\varepsilon\leq p-1$, and $\Omega[|v|>t]:=\{x\in\Omega:
\ |v(x)|>t\}$. Moreover, we recall the following property $$\label{star}
\|v\|_{p-\varepsilon,\Omega}\leq
\left({p\over\varepsilon}\right)^{1/(p-\varepsilon)}|\Omega|^{
\varepsilon/[p(p-\varepsilon)]}
\|v\|_{*,p,\Omega},\quad\forall v\in L^*_p(\Omega).$$
We derive the explicit estimates via the analysis of the decay of the level sets of the solution [@stamp63-64], extending the global estimate established in [@da] of $(u,u|_{\partial\Omega})$ in $L^{np/(n-p)}(\Omega)\times
L^{(n-1)p/(n-p)}(\partial\Omega)$ if $f\in L^{p}(\Omega)$ with $2\leq p<n$.
\[qestimates\] Let $2<p,r<2(n-1)$, $u \in H^{1}(\Omega)$ be any weak solution to (\[omega\])-(\[gama\]) in accordance with Definition \[def1\], and (\[amin\]) and (\[bmin\]) be fulfilled. If ${\bf f}\in {\bf L}^{p}(\Omega)$, $f\in L^{np/(p+n)}(\Omega)$, $g\in L^{(n-1)p/n}(\Gamma_N)$, and $h\in L^{r}(\Gamma)$, then we have, for every $q<2(n-1)/[n-2-2(n-1)\delta]:=Q$, $$\label{cotaps}
\|u\|_{q,\Omega}+\|u\|_{q,\partial\Omega}\leq \mathcal{K}_{q,\delta}
\left(\mathcal{B}+2|\bar\Omega[|u|>1]|^{{n-2\over 2(n-1)}-\delta}
\right),$$ where $\delta=\min\{1/ 2-1/ p,1/2-1/r\}$, and the positive constants $\mathcal{K}$ and $\mathcal{B}$ are $$\begin{aligned}
&&
\mathcal{K}_{q,\delta}=
2^{n-2\over n-2-2(n-1)\delta}
\left({Q\over Q-q}\right)^{1/q}
\left( |\Omega|^{{1\over q}-{1\over Q}}+|\partial\Omega|^{
{1\over q}-{1\over Q}}\right);\\
\mathcal{B}&=&( |\Omega|^{n-2\over 2(n-1) n} S_{2,2}+K_{2,2})
\left[ \left({1\over a_\#}
+{1\over \sqrt{a_\#b_\#}}\right)
(\|{\bf f}\|_{p,\Omega}+ C_{n,p,r})|\Omega|^{{1\over 2}-{1\over p}-\delta}
\right.\\
&&\qquad\left. +\left({1\over b_\#}
+{1\over \sqrt{a_\#b_\#}}\right)
(\|h\|_{r,\Gamma}+ C_{n,p,r})
|\Gamma|^{1/2-1/r-\delta}
\right];\\&&
C_{n,p,r}=S_{p',r'}\|f\|_{np/(p+n),\Omega}+K_{p',r'}
\|g\|_{(n-1)p/n,\Gamma_N}
,\quad\forall n\geq 2.\end{aligned}$$
Let $k\geq k_0=1$. Hence forth we use the notation $A(k)=\{x\in A:\ |u(x)|>k\}$, with the set $A$ being either $\Omega$, $\Gamma_N$, $\Gamma$, $\partial\Omega$ or $\bar \Omega$. Choosing $v={\rm sign}(u)(|u|-k)^+={\rm sign}(u)\max\{|u|-k,0\}\in
H^1(\Omega)$ as a test function in (\[pbu\]), then $\nabla v=\nabla u\in {\bf L}^2(
\Omega (k))$. Since $|u|>1$ a.e. on $\Gamma(k)$, taking (\[amin\]) and (\[bmin\]) into account, we deduce $$\begin{aligned}
\label{varak}
a_\#\int_{ \Omega (k)}|\nabla u|^2\mathrm{dx}+
b_\#\int_{ \Gamma (k)}(|u|-k)^{2}\mathrm{ds}
\leq \\
\leq \|{\bf f}\|_{2,\Omega(k)}\|\nabla u\|_{2,\Omega(k)}
+\|f\|_{{np\over p+n},\Omega}\|(|u|-k)^+\|_{{np\over np-n-p},\Omega}
+\nonumber\\+
\|g\|_{{(n-1)p\over n}, \Gamma_N}\|(|u|-k)^+\|_{{(n-1)p\over np-n-p},\Gamma_N}
+\|h\|_{2, \Gamma(k)}\||u|-k\|_{2,\Gamma(k)}.\nonumber\end{aligned}$$ Using the Hölder inequality, it follows that ($ p,r>2$) $$\begin{aligned}
\|{\bf f}\|_{2,\Omega(k)}
\leq \|{\bf f}\|_{p,\Omega}|\Omega(k)|^{1/2-1/p};\\
\|h\|_{2,\Gamma(k)}
\leq \|h\|_{r,\Gamma}|\Gamma(k)|^{1/2-1/r}.\end{aligned}$$
Making use of (\[sob\])-(\[sobt\]) and $(|u|-k)^+\in V_{p',r'}$ with $p'<2\leq n$ and $r'<2$, and the Hölder inequality, we get $$\begin{aligned}
\|(|u|-k)^+\|_{{np\over n(p-1)-p},\Omega}
\leq S_{p',r'}\left(|\Omega(k)|^{{1/ p'}-{1/ 2}}
\|\nabla u\|_{2,\Omega(k)}+
\right. \\ \left.
+|\Gamma (k)|^{{1/ r'}-{1/2}}
\||u|-k\|_{2,\Gamma (k)}\right);\\
\|(|u|-k)^+\|_{{(n-1)p\over n(p-1)-p},\Gamma_N}
\leq K_{p',r'}\left(|\Omega(k)|^{{1/ p'}-{1/ 2}}
\|\nabla u\|_{2,\Omega(k)}+ \right. \\ \left.
+|\Gamma (k)|^{{1/r'}-{1/2}}
\||u|-k\|_{2,\Gamma (k)}\right). \end{aligned}$$ Inserting last four inequalities into (\[varak\]) we obtain $$\begin{aligned}
\label{energy}
{a_\#}\|\nabla u\|_{2,\Omega(k)}^2+{b_\#}\| |u|-k
\|_{2,\Gamma(k)}^2\leq {\left(
\|{\bf f}\|_{p,\Omega}+ C_{n,p,r}\right)^2
\over a_\#} |\Omega(k)|^{1-{2\over p}}\\
+{\left(
\|h\|_{r,\Gamma}+ C_{n,p,r}\right)^{2}
\over b_\#}|\Gamma(k)|^{1-2/r}
,\qquad \forall p,r>2.\nonumber\end{aligned}$$ It results in $$\begin{aligned}
\label{nuakn}
\|( |u|-k)^+\|_{1,2,2}\leq \left[ \left({1\over a_\#}
+{1\over \sqrt{a_\#b_\#}}\right)
{(
\|{\bf f}\|_{p,\Omega}+ C_{n,p,r})|\Omega(k)|^{1/2-1/p}
\over (|\Omega(k)|+|\Gamma(k)|)^\delta} +\right.\\
\left.+ \left({1\over b_\#}
+{1\over \sqrt{a_\#b_\#}}\right){(
\|h\|_{r,\Gamma}+ C_{n,p,r})|\Gamma(k)|^{1/2-1/r}
\over (|\Omega(k)|+|\Gamma(k)|)^\delta} \right]|\bar\Omega(k)|^\delta.\nonumber\end{aligned}$$
For $ h > k > k_0$, we have $$\label{aalfa1}
(h-k)|\bar \Omega (h)|^{1/\alpha}\leq
\||u|-k\|_{\alpha,\Omega(k)}+
\||u|-k\|_{\alpha,\partial\Omega(k)},\quad \forall\alpha\geq 1.$$
Choosing $\alpha= 2(n-1)/(n-2)$, we use (\[sob\]) and (\[sobt\]), $(|u|-k)^+\in W^{1,n\alpha/(\alpha+n-1)}(\Omega)$, with $n\alpha/(\alpha+n-1)<n$, and the Hölder inequality since $n\alpha/(\alpha+n-1)\leq 2$. Thus, we have $$\begin{aligned}
\||u|-k\|_{\alpha,\Omega(k)}
\leq |\Omega|^{1\over n\alpha}S_{2,2}
\left(\|\nabla u\|_{2,\Omega(k)}+\||u|-k\|_{2,\Gamma(k)}\right);\\
\||u|-k\|_{\alpha,\partial\Omega (k)}
\leq K_{2,2}
\left(\|\nabla u\|_{2,\Omega(k)}+\||u|-k\|_{2,\Gamma(k)}\right).\end{aligned}$$
Applying (\[nuakn\])-(\[aalfa1\]), we find $$(h-k)|\bar \Omega (h)|^{1/ \alpha}
\leq \mathcal{B}
|\bar\Omega(k)|^{\delta}.$$ Observing that $\beta=
\alpha\delta<1$ if and only if $p,r<2(n-1)$, we may appeal to [@stamp63-64 Lemma 4.1 (iii)], deducing $$h^{\alpha/(1-\beta)} |\bar\Omega(h)|\leq 2^{\alpha/(1-\beta)^2}\left(
\mathcal{B}^{\alpha/(1-\beta)}+ (2k_0)^{\alpha/(1-\beta)}
|\bar\Omega (k_0)|\right).$$ Considering $k_0=1$, using (\[star\]) the claimed estimate (\[cotaps\]) holds.
$L^\infty$-estimates {#linfty}
====================
In this section, we establish some maximum principles, by recourse to the De Giorgi technique [@stamp63-64], and the Moser iteration technique [@gt pp. 189-190]. New results are stated that provide $L^\infty$-estimates up to the boundary under any space dimension $n\geq 2$.
De Giorgi technique
-------------------
\[max\] Let $p>n\geq 2$, $r>2(n-1)$, $u \in H^{1}(\Omega)$ be any weak solution to (\[omega\])-(\[gama\]) in accordance with Definition \[def1\], and (\[amin\]) and (\[bmin\]) be fulfilled. Under the hypotheses ${\bf f}\in {\bf L}^{p}(\Omega)$, $f\in L^{np/(p+n)}(\Omega)$, $g\in L^{(n-1)p/n}(\Gamma_N)$, and $h\in L^{r}(\Gamma)$, we have $$\label{supess}
{\rm ess } \sup_{\Omega\cup \partial\Omega}|u|\leq 1
+\left\{\begin{array}{ll}
2^{\gamma/(\gamma-{1\over 2}+{1\over n})}
(|\Omega|+|\partial\Omega|)^{\gamma-{1\over 2}+{1\over n}}
\mathcal{Z}_n&\mbox{ if }n>2\\
2^{\alpha\gamma+1/2\over \alpha\gamma-1/2}
(|\Omega|+|\partial\Omega|)^{\gamma-1/(2\alpha)}\mathcal{Z}_2&\mbox{ if }n=2
\end{array}\right.$$ where $\alpha>1/(2\gamma)$, $\gamma=\min\{1/ 2-1/ p,(1/2-1/r)(n-1)/n\}$, and $\mathcal{Z}_n$ is $$\begin{aligned}
\mathcal{Z}_n=( S_{2,2}+K_{2,2})
\left[ \left({1\over a_\#}
+{1\over \sqrt{a_\#b_\#}}\right)
(\|{\bf f}\|_{p,\Omega}+ C_{n,p,r})|\Omega|^{{1\over 2}-{1\over p}-\gamma}
\right.\\
\left. +\left({1\over b_\#}
+{1\over \sqrt{a_\#b_\#}}\right)
(\|h\|_{r,\Gamma}+ C_{n,p,r})
|\Gamma|^{1/2-1/r-\gamma n/(n-1)}
\right]\quad \mbox{ if }n>2;\\
\mathcal{Z}_2=( S_{1,1}+K_{1,1})
\left[ \left({|\Omega|^{1/( 2\alpha)}\over a_\#}
+{1\over \sqrt{a_\#b_\#}}\right)
(\|{\bf f}\|_{p,\Omega}+ C_{2,p,r})|\Omega|^{{1\over 2}-{1\over p}-\gamma}
\right.\\
\left. +\left({1\over b_\#}
+{|\Omega|^{1/( 2\alpha)}\over \sqrt{a_\#b_\#}}\right)
(\|h\|_{r,\Gamma}+ C_{2,p,r})
|\Gamma|^{1/2-1/r-2\gamma }
\right],\end{aligned}$$ with $C_{n,p,r}$ being given in Proposition \[qestimates\].
Arguing as in the proof of Proposition \[qestimates\], (\[energy\]) holds. Defining $\Sigma(k)=|\Omega(k)|+|\partial\Omega(k)|^{n/(n-1)}$, we have $$\label{aalfa2}
(h-k)|\Sigma (h)|^{1/\alpha}\leq
\||u|-k\|_{\alpha,\Omega(k)}+
\||u|-k\|_{\alpha (n-1)/n,\partial\Omega(k)}:=I,$$ for every $ h > k > k_0=1$ and $\alpha\geq n/(n-1)$.
Next, taking $\alpha=2n/(n-2)$ if $n>2$ and any $\alpha>1/(2\gamma)$ if $n=2$, we get $(|u|-k)^+\in W^{1,n\alpha/(\alpha+n)}(\Omega)$. Thus, we use (\[sob\]) and (\[sobt\]) with $n\alpha/(\alpha+n)<n$, and the Hölder inequality, obtaining $$\begin{aligned}
\||u|-k\|_{\alpha,\Omega(k)}
\leq S_{{n\alpha\over \alpha+n},
{n\alpha\over \alpha+n}}
\left(|\Omega|^{\sf{z}}\|\nabla u\|_{2,\Omega(k)}+\||u|-k\|_{2,\Gamma(k)}\right)
\Sigma(k)^{\sf{z}};\\
\||u|-k\|_{{\alpha(n-1)\over n},\partial\Omega (k)}
\leq K_{{n\alpha \over\alpha+n},{n\alpha\over \alpha+n}}
\left(|\Omega|^{\sf{z}}\|\nabla u\|_{2,\Omega(k)}+\||u|-k\|_{2,\Gamma(k)}\right)
\Sigma(k)^{\sf{z}},\end{aligned}$$ where $\sf{z}=0$ if $n>2$, and $\sf{z}=1/(2\alpha)$ if $n=2$. Let us split these two situations.
Case $n>2$.
: Applying (\[energy\]), we obtain $$\begin{aligned}
I\leq (S_{2,2} +K_{2,2})
\left[ \left({1\over a_\#}
+{1\over \sqrt{a_\#b_\#}}\right)
{(
\|{\bf f}\|_{p,\Omega}+ C_{n,p,r})|\Omega(k)|^{1/2-1/p}
\over (|\Omega(k)|+|\Gamma(k)|^{n/(n-1)})^\gamma} +\right.\nonumber\\
\left.+ \left({1\over b_\#}
+{1\over \sqrt{a_\#b_\#}}\right){(
\|h\|_{r,\Gamma}+ C_{n,p,r})|\Gamma(k)|^{1/2-1/r}
\over (|\Omega(k)|+|\Gamma(k)|^{n/(n-1)})^\gamma} \right]\Sigma^\gamma.\end{aligned}$$
Case $n=2$.
: Applying (\[energy\]), we obtain $$\begin{aligned}
I\leq (S_{1,1} +K_{1,1})
\left[ \left({|\Omega|^{1\over 2\alpha} \over a_\#}
+{1\over \sqrt{a_\#b_\#}}\right)
{(
\|{\bf f}\|_{p,\Omega}+ C_{n,p,r})|\Omega(k)|^{{1\over 2}-{1\over p}}
\over (|\Omega(k)|+|\Gamma(k)|^{n/(n-1)})^\gamma} +\right.\nonumber\\
\left.+ \left({1\over b_\#}
+{|\Omega|^{1/( 2\alpha)} \over \sqrt{a_\#b_\#}}\right){(
\|h\|_{r,\Gamma}+ C_{n,p,r})|\Gamma(k)|^{1/2-1/r}
\over (|\Omega(k)|+|\Gamma(k)|^{n/(n-1)})^\gamma} \right]\Sigma^{\gamma+1/
(2\alpha)}.\end{aligned}$$
In both cases, we infer from (\[aalfa2\]) that $$|\Sigma(h)|\leq \left( \mathcal{Z}_n
\over h-k\right)^{\alpha}|\Sigma(k)|^\beta,
\qquad\beta=\left\{\begin{array}{ll}
\alpha\gamma&\mbox{ if }n>2\\
\alpha\gamma+ 1/2&\mbox{ if }n=2
\end{array}\right.$$ where $\beta>1$ if and only if $p>n$ and $r>2(n-1)$.
By appealing to [@stamp63-64 Lemma 4.1 (i)] we conclude $$|\Sigma(k_0+\mathcal{Z}_n|\Sigma(k_0)
|^{(\beta-1)/\alpha}2^{\beta/(\beta-1)})|=0.$$ This means that the essential supremmum does not exceed the well determined constant $k_0+
\mathcal{Z}_n|\Sigma(k_0)|^{(\beta-1)/\alpha}2^{\beta/(\beta-1)}$. This completes the proof of Proposition \[max\].
In particular, if $f=g=h=0$ on the corresponding domains and ${\bf f} \in {\bf L}^p(\Omega)$ for $p>n$, then $${\rm ess } \sup_{\Omega\cup\partial\Omega}|u|\leq
1+Z_n\|{\bf f}\|_{p,\Omega}\left\{\begin{array}{ll}
(|\Omega|+|\partial\Omega|)^{{1\over n}-{1\over p}}
\left({1\over a_\#}
+{1\over \sqrt{a_\#b_\#}}\right)&\mbox{ if }n>2\\
(|\Omega|+|\partial\Omega|)^{{\alpha-1\over 2\alpha}-{1\over p}}
\left({|\Omega|^{1\over 2\alpha}\over a_\#}
+{1\over \sqrt{a_\#b_\#}}\right)
&\mbox{ if }n=2
\end{array}\right.$$ for every $\alpha>p/(p-2)$, with $Z_n=( S_{2,2}+K_{2,2})2^{n(p-2)\over 2(p-n)}$ if $n>2$, and $Z_2=( S_{1,1}+K_{1,1})
2^{(\alpha+1)/2-1/p\over( \alpha-1)/2-1/p}$.
Moser iteration technique
-------------------------
\[max0\] Let $p>n\geq 2$, $\ell\geq 2$, $u \in V_{2,\ell}$ be any weak solution to (\[omega\])-(\[gama\]), in accordance with Definition \[def1\], under ${\bf f}\in {\bf L}^{p}(\Omega)$, $f\in L^{p/2}(\Omega)$, $g=0$ on $\Gamma_N$, and $h=0$ on $\Gamma$, and (\[amin\]) and (\[bmin\]) be fulfilled. Then, $u$ satisfies $${\rm ess } \sup_{\Omega\cup\partial\Omega}|u|\leq
E_n\chi^{\left(\sum_{m\geq 0}m\chi^{-m}\right)}
(\sqrt{2}\mathcal{E})^{\chi/(\chi-1)}
\|u\|_{2p/(p-2),\Omega},
\label{cotaubound}$$ with $E_n=S_{2,2}^{\chi/(\chi-1)}$ and $\chi=n(p-2)/[p(n-2)]$ if $n>2$, and $E_2= S_{p\chi/[p(\chi+1)-2],p\chi/[p(\chi+1)-2]}^{\chi/(\chi-1)}$ $\max\{|\Omega|^{(p-2)/[2p(\chi-1)]},|\Gamma|^{(p-2)/[2p(\chi-1)]}\}$ for any $\chi>1$, and $$\mathcal{E}=
\sqrt{(\|{\bf f}\|_{p,\Omega}^2/ a_\#
+2\|f\|_{p/2,\Omega})/ \min \{a_\#,b_\#\}}.$$
Let $\beta\geq 1$, and $k>1$. Defining the truncation operator $T_k(y)=\min\{y,k\}$, set $w=T_k(|u|)\in H^{1}(\Omega)\cap
L^{\infty}(\Omega)$ that satisfies $w\in
L^{\infty}(\partial\Omega)$. Choosing $v=\beta^2{\rm sign}(u)w^{2\beta-1}/(2\beta-1)\in
V_{2,\ell}$ as a test function in (\[pbu\]), then $\nabla v=
\beta^2u^{2(\beta-1)}\nabla u$ in $\Omega[|u|<k]$, and $\nabla v= {\bf 0}$ in $\Omega\setminus\overline{\Omega[|u|<k]}$. Thus, applying (\[amin\]) and (\[bmin\]) we deduce $$\begin{aligned}
\label{varab}
a_\#\int_{\Omega}|\nabla (w^\beta)|^2\mathrm{dx}+
{\beta^2\over 2\beta-1}
b_\#\int_{\Gamma}|u|^{\ell-1}|w|^{2\beta-1}\mathrm{ds}\leq\\
\leq\int_{\Omega} ( \mathsf{A}\nabla u)\cdot
\nabla v \mathrm{dx}+
\int_{\Gamma}b(u) v \mathrm{ds} \leq
{a_\#\over 2}\|\nabla (w^\beta)\|_{2,\Omega}^2+\nonumber\\
+{\beta^2\over 2a_\#}\|
{\bf f}w^{\beta-1}\|_{2,\Omega}^2+ {\beta^2\over 2\beta-1}
\|fw^{2\beta-1} \|_{1,\Omega},\nonumber\end{aligned}$$ using the Hölder inequality.
We may suppose that $w>1$. Otherwise, $w=|u|\leq 1<k$. Using the Hölder inequality, we separately compute, for $p>2$, $$\begin{aligned}
\int_{\Omega}|{\bf f}|^2w^{2(\beta-1)} \mathrm{dx}\leq
\|{\bf f}\|_{p,\Omega}^2
\|w^{\beta}\|_{q,\Omega}^2,&&q=2p/(p-2)>2;
\\
\int_{\Omega}|f|w^{2\beta-1} \mathrm{dx}\leq
\|f\|_{{p/2},\Omega}
\|w^{\beta}\|_{q,\Omega}^2.&&\end{aligned}$$ Inserting these two inequalities in (\[varab\]), and considering that the left hand side absorbs the corresponding term of the right hand side, we obtain $$\left(\|\nabla (w^\beta)\|_{2,\Omega}^2+\|w^\beta\|_{2,\Gamma}^2\right)^{1/2}
\leq {\beta} \mathcal{E}\|w ^\beta\|_{q,\Omega}.\label{varaw}$$
Let us split the proof of estimate (\[cotaubound\]) into two space dimension dependent cases.
[Case]{} $n>2$. Making use of (\[sob\]), $w^\beta\in
W^{1,2}(\Omega)\hookrightarrow L^{q\chi}(\Omega)$, with $q \chi =
2n/(n-2)$ i.e. $\chi=n(p-2)/[p(n-2)]>1$ considering that $p>n$, and after applying (\[varaw\]), we deduce $$\begin{aligned}
\|w\|_{q\beta\chi ,\Omega}^\beta\leq S_{2,2}
\|w^\beta\|_{1,2,2}\leq S_{2,2}\sqrt{2}\left(
\|\nabla (w^\beta)\|_{2,\Omega}^2+
\|w^\beta\|_{2,\Gamma}^2\right)^{1/2}\\
\leq S_{2,2}\sqrt{2} \mathcal{E}\beta
\|u\|_{q\beta,\Omega} ^\beta.\end{aligned}$$
Then, we may pass to the limit the resulting inequality as $k\rightarrow\infty$ by Fatou lemma, obtaining $$\|u\|_{q\beta\chi ,\Omega}\leq (\beta S_{2,2}\sqrt{2}
\mathcal{E})^{1/\beta}
\|u\|_{q\beta,\Omega} .$$
Taking $\beta=\chi^m>1$, by induction, we have $$\label{tesei}
\|u\|_{q\chi^{N},\Omega}\leq (S_{2,2}\sqrt{2} \mathcal{E})^{a_N}\chi^{b_N}
\|u\|_{q,\Omega},\quad\forall N\in\mathbb{N},$$ where $$a_N=\sum_{m=0}^{N-1}\chi^{-m}\quad
\mbox{and}\quad b_N=\sum_{m=0}^{N-1}m\chi^{-m}.$$ Therefore, by the definition $$\|u\|_{\infty,\Omega}=
\lim_{N\rightarrow\infty} \|u\|_{q\chi^{N},\Omega},$$ and observing that $\lim_{N\rightarrow\infty} a_N$ stands for the geometric series, we find $${\rm ess } \sup_{\Omega}|u|\leq
E_n\chi^{\left(\sum_{m\geq 0}m\chi^{-m}\right)}(\sqrt{2}
\mathcal{E})^{\chi/(\chi-1)}\|u\|_{2p/(p-2),\Omega}.$$
Next, making use of (\[sobt\]), $w^{\beta}\in
W^{1,2}(\Omega)\hookrightarrow L^{2(n-1)/(n-2)}(\partial\Omega)$, and (\[varaw\]), we deduce $$\begin{aligned}
\|w\|_{\beta 2(n-1)/(n-2) ,\partial\Omega}^\beta\leq K_{2,2}\sqrt{2}\left(
\|\nabla (w^\beta)\|_{2,\Omega}^2+
\|w^\beta\|_{2,\Gamma}^2\right)^{1/2} \leq \nonumber\\
\leq K_{2,2}\sqrt{2} \mathcal{E}\beta
\|u\|_{q\beta,\Omega} ^\beta .\end{aligned}$$ Taking $\beta=\chi^m>1$, and applying (\[tesei\]), we get $$\|w\|_{\chi^m 2(n-1)/(n-2) ,\partial\Omega}
\leq K_{2,2}^{\chi^{-m}}S_{2,2}^{a_m}(\sqrt{2} \mathcal{E}
)^{a_{m+1}}\chi^{b_{m+1}}
\|u\|_{q,\Omega}.$$ Thus, we may pass to the limit the above inequality first as $k\rightarrow\infty$ by Fatou lemma, and next as $m\rightarrow\infty$, concluding $${\rm ess } \sup_{\partial\Omega}|u|\leq
E_n\chi^{\left(\sum_{m\geq 0}m\chi^{-m}\right)}(\sqrt{2}
\mathcal{E})^{\chi/(\chi-1)}\|u\|_{2p/(p-2),\Omega},$$ which finishes (\[cotaubound\]).
[Case]{} $n=2$. Making use of $w^\beta\in
W^{1,2}(\Omega)\hookrightarrow
W^{1,2q\chi/(q\chi+2)}(\Omega)\hookrightarrow L^{q\chi}(\Omega)$, with $q\chi =2p\chi/(p-2)$ considering that $p>n=2$, and next applying (\[varaw\]), we deduce $$\begin{aligned}
\|w\|_{q\beta\chi ,\Omega}^\beta\leq S_{2q\chi/(q\chi+2),2q\chi/(q\chi+2)}
\|w^\beta\|_{1,2q\chi/(q\chi+2),2q\chi/(q\chi+2)}
\leq\\
\leq S_{2q\chi/(q\chi+2),2q\chi/(q\chi+2)}\left(|\Omega|^{1/(q\chi)}
\|\nabla (w^\beta)\|_{2,\Omega}+|\Gamma|^{1/(q\chi)}
\|w^\beta\|_{2,\Gamma}\right) \leq \nonumber\\
\leq S_{2q\chi/(q\chi+2),2q\chi/(q\chi+2)}
\max\{|\Omega|^{1/(q\chi)},|\Gamma|^{1/(q\chi)}\}
\sqrt{2} \mathcal{E}\beta \|u\|_{q\beta,\Omega} ^\beta.\end{aligned}$$ For the boundary bound, we use $w^\beta\in
W^{1,2}(\Omega)\hookrightarrow
W^{1,2q\chi/(q\chi+1)}(\Omega)\hookrightarrow L^{q\chi}(\partial\Omega)$, deducing $$\begin{aligned}
\|w\|_{q\beta\chi ,\partial\Omega}^\beta\leq
K_{{2q\chi\over q\chi+1},{2q\chi\over q\chi+1}}
\max\{|\Omega|^{1/(2q\chi)},|\Gamma|^{1/(2q\chi)}\}
\sqrt{2} \mathcal{E}\beta \|u\|_{q\beta,\Omega} ^\beta.\end{aligned}$$ Thus, we may proceed as in the above case, completing the proof of Proposition \[max0\].
In the following result stands for the particular case: $f=g=h=0$.
\[ccinf\] Under the conditions of Proposition \[max0\], there exists a $L^\infty$-constant, $C_\infty$, to the problem (\[omega\])-(\[gama\]), that is, for $p>n$, $${\rm ess } \sup_{\Omega\cup\partial\Omega}|u|\leq
C_\infty \|{\bf f}\|_{p,\Omega}^{1+\chi/(\chi-1)},$$ with $$\begin{aligned}
C_\infty=E_n\chi^{\left(\sum_{m\geq 0}m\chi^{-m}\right)}
\left({2\over a_\# \min \{a_\#,b_\#\}
}\right)^{\chi\over 2(\chi-1)}\times\\
\times S_{{2pn\over 2p+n(p-2)},\ell}\left(
{|\Omega |^{{1\over n}-{2\over p}+{1\over 2}}
\over a_\#}
+\left({\ell\,'|\Omega|^{1-1/p}\over 2a_\# b_\#}\right)^{1/\ell}\right).\end{aligned}$$
It suffices to insert the estimate (\[cotausup\]) into (\[cotaubound\]).
\[max2\] Let $n\geq 2$, $s>n-1$, $u \in H^{1}(\Omega)$ be any weak solution to (\[omega\])-(\[gama\]), in accordance with Definition \[def1\], and (\[amin\]) and (\[bmin\]) be fulfilled. If ${\bf f}={\bf 0}$ and $f=0$ in $\Omega$, $g\in L^{s}(\Gamma_N)$, and $h\in L^{s}(\Gamma)$, then $${\rm ess } \sup_{\bar\Omega}|u|\leq
G_n\chi^{\left(\sum_{m\geq 0}m\chi^{-m}\right)}
(\sqrt{2}\mathcal{G})^{\chi/(\chi-1)}
\|u\|_{2s/(s-1),\partial\Omega},\label{gh}$$ with $G_n=K_{2,2}^{\chi/(\chi-1)}$, $\chi=(s-1)(n-1)/[s(n-2)]$ if $n>2$, and $G_2= K_{4s\chi/(2s\chi+s-1),4s\chi/(2s\chi+s-1)}^{\chi/(\chi-1)}$ $\max\{|\Omega|^{(s-1)/[4s(\chi-1)]},|\Gamma|^{(s-1)/[4s(\chi-1)]}\}$ for any $\chi>1$, and $$\mathcal{G}=\sqrt{\left(\|g\|_{s,\Gamma_N}+
\|h\|_{s,\Gamma}
\right)/\min \{a_\#,b_\#\}}.$$
Let $\beta\geq 1$, and $k>1$. For $s>1$, and $q=2s/(s-1)$, proceeding as in the proof of Proposition \[max0\], we deduce $$\begin{aligned}
{a_\#}\int_{\Omega}|\nabla (w^\beta)|^2\mathrm{dx}+
{\beta^2\over 2\beta-1}
b_\#\int_{\Gamma}|u|^{\ell-1}|w|^{2\beta-1}\mathrm{ds}\leq\nonumber\\
\leq {\beta^2\over 2\beta-1}\left(
\|gw^{2\beta-1}\|_{1,\Gamma_N}
+ \|hw^{2\beta-1}\|_{1,\Gamma}\right)\leq\\
\leq\beta^2(\|g\|_{s, \Gamma_N} \|w^{\beta}\|_{q,\Gamma_N}^2+
\|h\|_{s, \Gamma} \|w^{\beta}\|_{q,\Gamma}^2).\end{aligned}$$ Thus, we obtain $$\left(\|\nabla (w^\beta)\|^2_{2,\Omega}+
\|w^\beta\|_{2,\Gamma}^{2} \right)^{1/2}
\leq {\beta}\mathcal{G} \| w^\beta\|_{q,\partial\Omega},\label{varak2}$$
Then, we obtain $$\begin{aligned}
\|u\|_{q_1\beta\chi ,\Omega}\leq (\beta\sqrt{2}\mathcal{G}M_1)^{1/\beta}
\|u\|_{q\beta,\partial\Omega} ;\\
\|u\|_{q\beta \chi ,\partial\Omega}\leq (\beta\sqrt{2}\mathcal{G}
M_2)^{1/\beta}\|u\|_{q\beta,\partial\Omega},\end{aligned}$$
Case $n>2$
: Setting $M_1= S_{2,2}$, and $M_2= K_{2,2}$, by using (\[sob\]), and $w^{\beta}\in
W^{1,2}(\Omega)\hookrightarrow L^{q_1\chi}(\Omega)$, (\[sobt\]), and $w^{\beta}\in
W^{1,2}(\Omega)\hookrightarrow L^{q\chi}(\partial\Omega)$ with $q\chi=2(n-1)/(n-2)$ i.e. $\chi=(s-1)(n-1)/[s(n-2)]>1$ if $s>n-1$.
Case $n=2$
: Setting $M_1=
S_{{2q_1\chi/( q_1\chi+2)},{2q_1\chi/(q_1\chi+2)}}
\max\{|\Omega|^{1/( q_1\chi)},|\Gamma|^{1/( q_1\chi)}\}$, and $M_2=
K_{{2q\chi/( q\chi+1)},{2q \chi/( q\chi+1)}}
\max\{|\Omega|^{1/(2q\chi)},|\Gamma|^{1/(2q\chi)}\}$, by using (\[sobt\]) with $w^{\beta}\in
W^{1,2q_1\chi/(q_1\chi+2)}(\Omega)\hookrightarrow L^{q_1\chi}(\Omega)$, and (\[sobt\]) with $w^{\beta}\in
W^{1,2q\chi/(q\chi+1)}(\Omega)\hookrightarrow L^{q\chi}(\partial\Omega)$.
In both cases, following the argument of the proof of Proposition \[max0\], we get $$\begin{aligned}
\|u\|_{q_1\chi ^{N+1},\Omega}\leq M_1^{\chi^{-N}}
(\sqrt{2}\mathcal{G})^{a_{N+1}}
M_2^{a_N}\chi^{b_{N+1}}
\|u\|_{q,\partial\Omega} ;&&\\
\|u\|_{q \chi ^N,\partial\Omega}\leq (\sqrt{2}\mathcal{G}
M_2)^{a_N}\chi^{b_N}\|u\|_{q,\partial\Omega},&&\quad\forall N\in\mathbb{N}.\end{aligned}$$ Therefore, we conclude (\[gh\]), finishing the proof of Proposition \[max2\].
Under the conditions of Propositions \[max0\] and \[max2\], if (\[bstar\]) is assumed then there exists a weak solution, $u\in H^1(\Omega)$, to (\[omega\])-(\[gama\]) in accordance with Definition \[def1\], such that $$\begin{aligned}
\label{cott}
{\rm ess } \sup_{\bar\Omega}|u|\leq\Xi_1
\left({\|{\bf f}\|_{p,\Omega}^2/ a_\#
+2\|f\|_{p/2,\Omega}\over \min \{a_\#,b_*\}
}\right)^{\chi_1\over 2(\chi_1-1)}{\|{\bf f}\|_{2,\Omega}
+{L}_n\|f\|_{t,\Omega}\over \min \{a_\#,b_*\}}+\\
+\Xi_2 \left({\|g\|_{s,\Gamma_N}+
\|h\|_{s,\Gamma}\over \min \{a_\#,b_*\}}
\right)^{\chi_2\over 2(\chi_2-1)}
{ {M}_n(\|g\|_{s,\Gamma_N}+
\|h\|_{s,\Gamma})\over \min \{a_\#,b_*\}
},\nonumber\end{aligned}$$ where $$\begin{aligned}
\Xi_1&=&
E_n\chi_1^{\left(\sum_{m\geq 0}m\chi_1^{-m}\right)}
\sqrt{2}^{\chi_1/(\chi_1-1)}
S_{{2pn\over 2p+n(p-2)},\ell}\left(|\Omega |^{p-n\over np}+1\right);\\
\Xi_2&=&
G_n\chi_2^{\left(\sum_{m\geq 0}m\chi_2^{-m}\right)}
\sqrt{2}^{\chi_2/(\chi_2-1)}
K_{{2sn\over 2s+(n-1)(s-1)},\ell}\left(|\Omega |^{s-n+1\over 2ns}+1\right),\end{aligned}$$ with $E_n$ and $\chi_1$ being the constants in accordance with Proposition \[max0\], $G_n$ and $\chi_2$ being the constants in accordance with Proposition \[max2\], $L_n=2S_{2,\ell}$ if $n>2$, $L_2=(|\Omega|^{1/t'}+1)S_{2t/(3t-2),\ell}$ if $t<2$, $L_2=(|\Omega|^{1/2}+1)|\Omega|^{1/2-1/t}S_{1,\ell}$ if $t\geq 2$, $M_n=2K_{2,\ell}$ if $n>2$, and $M_2=(|\Omega|^{1/(2s')}+1)K_{2s/(2s-1),\ell}$.
From Propositions \[exist\] and \[max0\], there exists $u_1\in H^1(\Omega)$ solving $$\int_{\Omega} ( \mathsf{A}\nabla u_1)\cdot
\nabla v \mathrm{dx}+\int_{\Gamma}u_1 v \mathrm{ds}
=\int_{\Omega}{\bf f}\cdot\nabla v \mathrm{dx}
+\int_{\Omega}fv \mathrm{dx},
\quad\forall v\in H^1(\Omega),$$ such that it verifies $${\rm ess } \sup_{\bar\Omega}|u_1|\leq
E_n\chi_1^{\left(\sum_{m\geq 0}m\chi_1^{-m}\right)}
(\sqrt{2}\mathcal{E})^{\chi_1/(\chi_1-1)}
\|u_1\|_{2p/(p-2),\Omega}.\label{pbuff}$$ From Propositions \[exist\] and \[max2\], and Remark \[sol\], there exists $u_2\in H^1(\Omega)$ solving $$\int_{\Omega} ( \mathsf{A}\nabla u_2)\cdot
\nabla v \mathrm{dx}+\int_{\Gamma}u_2 v \mathrm{ds}
=\int_{\Gamma_N} gv \mathrm{ds}+\int_{\Gamma} hv \mathrm{ds},
\quad\forall v\in H^1(\Omega),$$ such that it verifies $${\rm ess } \sup_{\bar\Omega}|u_2|\leq
G_n\chi_2^{\left(\sum_{m\geq 0}m\chi_2^{-m}\right)}
(\sqrt{2}\mathcal{G})^{\chi_2/(\chi_2-1)}
\|u_2\|_{2s/(s-1),\partial\Omega}.\label{pbugh}$$
Then, $u=u_1+u_2\in H^1(\Omega)$ is the required solution. Moreover, from (\[pbuff\])-(\[pbugh\]) gathered with Corollary \[coruq\] we find (\[cott\]), with ${L}_nA=\mathcal{F}_n(A,0)+\mathcal{H}_n(A,0)$, ${M}_n(B)=\mathcal{F}_n(0,B)+\mathcal{H}_n(0,B)$, and $\mathcal{F}_n$ and $\mathcal{H}_n$ being the functions in accordance with Proposition \[exist\].
$V_{q,\ell-1}$-solvability $(q< n/(n-1)$, $\ell\geq 2)$ {#l1data}
=======================================================
The $W^{1,q}$-solvability depends on the data regularity. In the presence of the boundary condition (\[robin\]), the duality theory is more straightforward than the $L^1$-theory when $L^1$ data are taken into account. In order to determine the explicit constant, the following result of the existence of a solution is based on the duality theory.
First let us recall that, for $q>1$, the $L^q$-norm may be defined as $$\label{dnorm}
\|{\bf u}\|_{q,\Omega}=\sup\left\{
|\int_{\Omega}{\bf u}\cdot {\bf g}\mathrm{dx}|
: \ {\bf g}\in {\bf L}^{q'}(\Omega), {\|{\bf g}\|
_{q',\Omega}}=1
\right\},$$ for all ${\bf u}\in{\bf L}^q(\Omega)$.
\[W1q\] Let ${\bf f}={\bf 0}$ a.e. in $\Omega$, $f\in L^1(\Omega)$, $g\in L^1(\Gamma_N)$, $h\in L^1(\Gamma)$, (A)-(B) be fulfilled, and $\mathsf{A}$ be symmetric. For any $\ell \geq 2$, there exists $u\in
V_{q,\ell -1}$ solving (\[pbu\]) for every $1< q < n/(n-1)$. Moreover, we have the following estimate $$\begin{aligned}
\label{cota1q}
\| \nabla u \|_{q,\Omega}&\leq &C_\infty\Big(|\Gamma|(1+b^\#)
+ \| f\|_{1,\Omega}+\|g\|_{1,\Gamma_N}+\|h\|_{1,\Gamma}+\\
&+&(1+b^\#)\left( \| f\|_{1,\Omega}+\|g\|_{1,\Gamma_N}+\|h\|_{1,\Gamma}
\right)/b_\#\Big); \nonumber\\
\| u\|_{\ell-1,\Gamma}^{\ell-1}&\leq &|\Gamma| + \left( \| f\|_{1,\Omega}+\|g\|_{1,\Gamma_N}+\|h\|_{1,\Gamma}
\right)/
b_\#,\label{cota1ql}\end{aligned}$$ with the constant $C_\infty$ being explicitly given in Corollary \[ccinf\].
For each $m\in \mathbb N$, take $
f_m=F_m(f)\in L^\infty(\Omega),$ $g_m=F_m (g)\in L^\infty(\Gamma_N),$ $h_m=F_m (h)\in L^\infty(\Gamma)$, with $$F_m(\tau)={m \tau\over m+|\tau |}.$$ Applying Proposition \[exist\], there exists a unique solution $u_m\in V_{2,\ell}
$ to the following variational problem $$\begin{aligned}
\label{pbum}
\int_\Omega (\mathsf{A}\nabla u_m)\cdot\nabla v\mathrm{dx}
+\int_\Gamma b(u_m) v\mathrm{ds}=
\int_\Omega f_mv\mathrm{dx}+\\
+\int_{\Gamma_N}g_mv\mathrm{ds}+\int_\Gamma
h_mv\mathrm{ds},\quad\forall v\in V_{2,\ell}.\nonumber\end{aligned}$$ In particular, (\[pbum\]) holds for all $v\in W^{1,q'}(\Omega)$ for $q'>n$. Defining the truncation operator $T_1(y)={\rm sign}(y)\min\{|y|,1\}$, let us choose $v=T_1(u_m)\in V_{2,\ell}$ as a test function in (\[pbum\]), obtaining $$\label{gum}
b_\#\left(\int_{ \Gamma [ |u_m|>1]}|u_m|^{\ell-1}\mathrm{ds}+
\int_{ \Gamma [ |u_m|\leq 1]}|u_m|^{\ell}\mathrm{ds}\right)\leq
\| f\|_{1,\Omega}+\| g\|_{1,\Gamma_N}+\|h\|_{1,\Gamma}.$$ Hence, we conclude that (\[cota1ql\]) is true for $u_m$.
In order to pass to the limit (\[pbum\]) on $m$ $(m\rightarrow\infty)$ let us establish the estimate (\[cota1q\]) for $ u_m$. Let $w\in V_{2,2}$ be the unique weak solution to the mixed Robin-Neumann problem (\[omega\])-(\[gama\]), under $f=g=h=0$, in accordance with Proposition \[exist\]. Since $\mathsf{A}$ is symmetric, we infer that $$\int_{\Omega} ( \mathsf{A}\nabla u_m)\cdot
\nabla w \mathrm{dx}=
\int_{\Omega} ( \mathsf{A}\nabla w)\cdot
\nabla u_m \mathrm{dx}
=\int_{\Omega}{\bf f}\cdot\nabla u_m \mathrm{dx}-
\int_{\Gamma}w u_m \mathrm{ds},$$ which gathered with (\[pbum\]) under $v=w$ reads $$\begin{aligned}
\label{fum}
\int_{\Omega}{\bf f}\cdot\nabla u_m \mathrm{dx}=
\int_{\Gamma}w u_m \mathrm{ds}-\int_{\Gamma}b(u_m) w \mathrm{ds}
+\int_{\Omega}f_mw \mathrm{dx}+\\
+\int_{\Gamma_N}g_mw \mathrm{ds}+\int_{\Gamma}h_mw \mathrm{ds}.\nonumber\end{aligned}$$
For ${\bf f}\in {\bf L}^{q'}(\Omega)$ with $q'>n$ such that $\|{\bf f}\|_{q',\Omega}=1$, Corollary \[ccinf\] guarantees that the existence of a $L^\infty$-constant $C_\infty$ such that $\|w\|_{\infty,\Omega}+
\|w\|_{\infty,\partial\Omega}\leq C_\infty $.
By (\[dnorm\]) with ${\bf u}=\nabla u_m $, and (\[fum\]), we obtain $$\begin{aligned}
\|\nabla u_m\|_{q,\Omega}\leq C_\infty\Big( |\Gamma|
(1+b^\#)+(1+b^\#)
\|u_m\|_{\ell-1,\Gamma[|u_m|>1]}^{\ell-1}+\\
+ \| f\|_{1,\Omega}+\|g\|_{1,\Gamma_N}+\|h\|_{1,\Gamma}
\Big).\end{aligned}$$ Applying (\[gum\]), then (\[cota1q\]) holds for $u_m$.
Therefore, the passage to the limit as $m$ tends to infinity is allowed, concluding the proof of Proposition \[W1q\].
Green kernel {#secg}
============
In this Section, we establish the existence of the Green kernel altogether some of its properties. Here, we follow the approach introduced in [@gw] in constructing Green’s function for the Dirichlet problem (see also [@lsw]).
\[def2\] For each $x\in\Omega$, we say that $G= G(x,\cdot)$ is a Green kernel associated to (\[omega\])-(\[gama\]), if it solves, in the distributional sense, $$\begin{aligned}
\label{pbud}
\nabla\cdot(\mathsf{A}\nabla G(x,\cdot))=\delta_x\quad\mbox{ in }\Omega;\\
\mathsf{A}\nabla G(x,\cdot)\cdot {\bf n}+b( G(x,\cdot))\chi_\Gamma=0
\quad\mbox{ on }\partial\Omega,
\end{aligned}$$ where $\delta_x$ is the Dirac delta function at the point $x$. That is, there is $q>1$ such that $G$ verifies the variational formulation $$\label{varf}
\int_{\Omega} \mathsf{A}\nabla G(x,\cdot)\cdot
\nabla v \mathrm{dy}+\int_\Gamma b(G(x,\cdot))v\mathrm{ds}
=v(x),\quad\forall v\in V_{q,\ell}.$$
\[green\] Let $n\geq 2$, $1\leq q<n/(n-1)$, (A)-(B) be fulfilled, and $\mathsf{A}$ be symmetric. Then, for each $x\in\Omega$ and any $r>0$ such that $r< {\rm dist}(x,\partial\Omega)$, there exists a unique Green function $G=
G(x,\cdot)\in V_{q,\ell-1}\cap H^1(\Omega \setminus B_r(x))$ according to Definition \[def2\], and enjoying the following estimates $$\begin{aligned}
\label{g1}
\|\nabla G\|_{q,\Omega} &\leq &
C_\infty \left(1+(1+b^\#)(|\Gamma|+1/b_\#)
\right);\\
\| G\|_{\ell-1,\Gamma} &\leq &( |\Gamma|+1/b_\#)^{1/(\ell-1)}
,\label{gg}\end{aligned}$$ with the constant $C_\infty$ being explicitly given in Corollary \[ccinf\]. Moreover, $G(x,y)\geq 0$ a.e. $x,y\in \Omega$.
Let $x\in\Omega$ and $\rho>0$ be such that $B_\rho(x)\subset\subset\Omega$. Thanks to Proposition \[exist\] with ${\bf f}={\bf 0}$ a.e. in $\Omega$, $g,h=0$ a.e. on, respectively, $\Gamma_N$ and $\Gamma$, and $f=\chi_{B_\rho(x)}/|B_\rho(x)| $ belonging to $ L^{2n/(n+2)}(\Omega)$ if $n>2$, and to $L^{2}(\Omega)$ if $n=2$, there exists $G^\rho=G^\rho(x,\cdot)\in V_{2,\ell}$ being the unique solution to $$\label{pbgreen}
\int_{\Omega} \mathsf{A}\nabla G^\rho\cdot
\nabla v \mathrm{dy}+\int_\Gamma b(G^\rho)v\mathrm{ds}
={1\over |B_\rho(x)|}\int_{B_\rho(x)}v \mathrm{dy},$$ for all $v\in V_{2,\ell} $. In particular, if $\ell=2$ (\[cotau\]) reads $$\label{grho2}
\|\nabla G^\rho\|_{2,\Omega}+
\| G^\rho\|_{2,\Gamma}\leq{{2}\over\min\{ a_\#,b_\#\}}\times\left\{
\begin{array}{ll}
S_{2,2} \omega_n^{{1\over n}-{1\over 2}}\rho^{1-n/2}&\mbox{ if } n>2\\
\sqrt{ |\Omega |+1\over \pi} S_{1,2}
\rho^{-1}&\mbox{ if } n=2
\end{array}\right. .$$ Therefore, for any $r>0$ such that $B_r(x)\subset\subset\Omega$, there exists $G\in H^1(\Omega\setminus B_r(x))$ such that $$G^\rho\rightharpoonup G\quad \mbox{in } H^1(\Omega\setminus B_r(x))\quad
\mbox{ as $\rho\rightarrow 0^+$.}$$
In order to $G$ verify $$\label{defgreen}
G(x,y)=\lim_{\rho\rightarrow 0}
G^\rho(x,y),\quad
\forall x\in\Omega, \ \mbox{a.e. }y\in\Omega, \ x\not=y,$$ we observe that the $V_{q,\ell-1}$-estimates (\[g1\])-(\[gg\]) are true for $G^\rho$ due to (\[cota1q\])-(\[cota1ql\]), by applying Proposition \[W1q\] with $g,h=0$, and $f=\chi_{B_\rho(x)}/|B_\rho(x)|\in L^{1}(\Omega)$. Then, we can extract a subsequence of $G^\rho,$ still denoted by $G^\rho,$ weakly converging to $G$ in $V_{q,\ell-1}$ as $\rho$ tends to $0$, with $G\in V_{q,\ell-1}$ solving (\[varf\]) for all $v\in W^{1,q'}(\Omega)$. A well-known property of passage to the weak limit implies (\[g1\])-(\[gg\]).
In order to prove the nonnegativeness assertion, first calculate $$a_\#
\int_{\Omega} |\nabla (G^\rho-|G^\rho|)|^2\mathrm{dy}
\leq {2\over |B_\rho(x)|}
\left(\int_{B_\rho(x)}G^\rho \mathrm{dy}-\int_{B_\rho(x)}|G^\rho|
\mathrm{dy}\right)\leq 0.$$ Then, $G^\rho=|G^\rho|$, and by passing to the limit as $\rho$ tends to $0$, the nonnegativeness claim holds, which completes the proof of Proposition \[green\].
Robin-Neumann problem ($\ell=2$) {#lqc}
================================
In the two-dimensional space, Proposition \[exist\] leads $H^1$ solution for the $L^p$-data, with an arbitrary $p>1$. Our concern is then the existence of weak solutions and the derivation of their estimates in the $n$-dimensional space: $n>2$.
\[W1qp\] Let ${\bf f}={\bf 0}$ a.e. in $\Omega$, $f\in L^t(\Omega)$ with $t\leq 2n/(n+2)$, $g\in L^s(\Gamma_N)$ and $h\in L^s(\Gamma)$ with $s\leq 2(n-1)/n$, and $\mathsf{A}$ be a symmetric matrix satisfying the assumption (A). Under the assumption (\[bstar\]) with $b_*=1$, there exists $u\in W^{1,q}(\Omega)$ solving (\[pbu\]) for every $1< q < 2(n-1)p/[2(n-1)-p]$ with $p=\min\{t,s\}$. Moreover, we have the following estimate $$\| \nabla u \|_{q,\Omega}
\leq\mathcal{M}_q\left({1\over a_\#}+{1\over\sqrt{a_\#}}
\right)\left(
\mathcal{K}_{t',q'}
\| f\|_{t,\Omega}+\mathcal{K}_{s',q'}(\|g\|_{s,\Gamma_N}+\|h\|_{s,\Gamma})
\right),
\label{li}$$ with $$\begin{aligned}
\mathcal{M}_q= |\Omega|^{(n-2)/[2(n-1) n]} S_{2,2}+K_{2,2}
+\\
+2|\Omega|^{{1\over q}-{1\over 2}}
\left( S_{{n\over n+1},{n\over n+1}}(|\Omega|^{{1\over 2}+{1\over n}}
+|\partial\Omega|^{{1\over 2}+{1\over n}})+K_{1,1}
(|\Omega|^{1\over 2}+
|\partial\Omega|^{{1\over 2}})\right)
.\end{aligned}$$
For each $m\in \mathbb N$, take the approximations $f_m$, $g_m$, and $h_m$ as in the proof of Proposition \[W1q\], and the corresponding unique solution $u_m\in V_{2,2}
$ to the variational problem (\[pbum\]). Moreover, (\[cota1ql\]) is true for $u_m$ ($\ell=2$).
Let $w\in V_{2,2}$ be the unique weak solution to the mixed Robin-Neumann problem (\[omega\])-(\[gama\]), under $f=g=h=0$, such that (\[fum\]) reads $$\int_{\Omega}{\bf f}\cdot\nabla u_m \mathrm{dx}=
\int_{\Omega}f_mw \mathrm{dx}
+\int_{\Gamma_N}g_mw \mathrm{ds}+\int_{\Gamma}h_mw \mathrm{ds}.\label{umfl}$$ Moreover, for $q'\geq 2$ (\[cotau\]) reads $$\label{cotau2}
\|w\|_{1,2,2}\leq \left({1\over a_\#}+{1\over\sqrt{a_\#}}
\right)|\Omega|^{1/q-1/2}
\|{\bf f}\|_{q',\Omega}.$$ Observe that $$\begin{aligned}
\label{ww}
\|w\|_{1,\Omega}+\|w\|_{1,\partial\Omega}\leq
\left(S_{{n\over n+1},{n\over n+1}}\left(|\Omega|^{1/2+1/n}
+|\Gamma|^{1/2+1/n}\right)+\right. \\
\left.+K_{1,1}\left(|\Omega|^{1/2}
+|\Gamma|^{1/2}\right)\right)\|w\|_{1,2,2}.\nonumber\end{aligned}$$
For any $t\leq 2n/(n+2)$, $s\leq 2(n-1)/n$, and $q<2(n-1)p/[2(n-1)-p]$ with $p=\min\{t,s\}$, which means $2n/(n-2)\leq t'<2(n-1)q'/[2(n-1)-q']$, and $2(n-1)/(n-2)\leq s'<2(n-1)q'/[2(n-1)-q']$ if $2<q'<2(n-1)$, Proposition \[qestimates\] (with $\delta=1/2-1/q'$ since $h\equiv 0$) yields $$\begin{aligned}
\|w\|_{t',\Omega}
\leq \mathcal{K}_{t',\delta}\left(
(|\Omega|^{n-2\over 2(n-1) n} S_{2,2}+K_{2,2})
\left({1\over a_\#}+{1\over\sqrt{a_\#}}\right)
\|{\bf f}\|_{q',\Omega}+\right.\\
\left.+2(
\|w\|_{1,\Omega}+\|w\|_{1,\partial\Omega})
\right)
\leq \mathcal{K}_{t',\delta}\mathcal{M}_q
\left({1\over a_\#}+{1\over\sqrt{a_\#}}\right)
\|{\bf f}\|_{q',\Omega},\end{aligned}$$ considering that $(2(n-1)-q')/[2(n-1)q']<1$ is taken into account, and applying (\[ww\]) accomplished with (\[cotau2\]). Analogously $$\| w\|_{s',\partial\Omega}
\leq \mathcal{K}_{s',\delta}\mathcal{M}_q
\left({1\over a_\#}+{1\over\sqrt{a_\#}}\right)
\|{\bf f}\|_{q',\Omega}.$$
By (\[dnorm\]) with ${\bf u}=\nabla u_m $, we infer from (\[umfl\]) that (\[li\]) holds for $u_m$. Therefore, by passage to the limit as $m$ tends to infinity, we conclude the claimed result.
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|
---
abstract: 'A dynamics between Newton and Langevin formalisms is elucidated within the framework of the generalized Langevin equation. For thermal noise yielding a vanishing zero-frequency friction the corresponding non-Markovian Brownian dynamics exhibits anomalous behavior which is characterized by ballistic diffusion and accelerated transport. We also investigate the role of a possible initial correlation between the system degrees of freedom and the heat-bath degrees of freedom for the asymptotic long-time behavior of the system dynamics. As two test beds we investigate (i) the anomalous energy relaxation of free non-Markovian Brownian motion that is driven by a harmonic velocity noise and (ii) the phenomenon of a net directed acceleration in noise-induced transport of an inertial rocking Brownian motor.'
author:
- 'Jing-Dong Bao,$^{1,}$[^1] Yi-Zhong Zhuo,$^2$ Fernando A. Oliveira,$^3$ and Peter Hänggi$^{4,5}$'
title: Intermediate dynamics between Newton and Langevin
---
introduction
============
The phenomenon of Brownian motion has assumed a fundamental and influential role in the development of thermodynamical and statistical theories and continues to do so as an inspiring source for active research in various fields of natural sciences [@HM2005]. The Brownian motion dynamics can conveniently be described by a generalized Langevin equation (GLE). The GLE was originally derived by Mori [@mori1965], Kawasaki [@kawasaki73], and Zwanzig [@zwanzig] by use of the Gram-Schmidt procedure. It was further investigated by Lee using the recurrence relations method [@lee2]. Starting out from the well-known system-plus-oscillator-reservoir model as e.g. detailed in Ref. [@zwanzig; @HALNP], one obtains the GLE derived from first principles. The validity of a thermal GLE is typically restricted to the case with a thermal equilibrium; e.g. see in Refs. [@HALNP; @kubo; @toda; @pot]. Specifically, such a GLE dynamics reads [@kawasaki73; @zwanzig; @HALNP]: $$m\dot{v}(t)+m\int_{0}^{t}\gamma (t-t^{\prime })v(t^{\prime
})dt^{\prime }+\partial _{x}U(x,t)=\varepsilon (t).$$ Notably, the thermal noise $ \varepsilon (t)$ is not correlated with the initial velocity, i.e., $\langle v(0)\varepsilon(t)\rangle=0$, see in Refs. [@zwanzig; @HALNP] and in section III below. In contrast, the initial position $x(0)$ typically is correlated with $ \varepsilon
(t)$. The memory friction $\gamma (t-t^{\prime })$ is in thermal equilibrium related to the correlation of stationary random forces [@kubo; @toda]. Kubo [@kubo] has addressed the common behavior of a classical equilibrium bath by setting $\langle \varepsilon
(t)\varepsilon (t^{\prime })\rangle =mk_{B}T\gamma (t-t^{\prime})$. Here $k_{B}$ is the Boltzmann constant and $T$ denotes the bath temperature. The one-sided Fourier transform of $\gamma(t)$ obeys Re$\tilde{\gamma}(\omega )\geq 0$ for real-valued $\omega$. This correlation result for the thermal noise $\varepsilon (t)$ is commonly termed the fluctuation-dissipation theorem (FDT) of the second kind [@toda]. The nonlinear GLE can also be extended to account for a nonlinear system - linear bath interaction [@zwanzig; @HALNP; @Illuminati], yielding a structure as in Eq. (1), but now with a nonlinear, coordinate-dependent friction function. It even can be generalized to arbitrary nonlinear system - nonlinear bath interactions containing then the potential of mean force [@GHT80].
With this work we aim at extending the theory of classical Brownian motion by focussing on the intricacies of a possible non-Markovian with an incomplete, non-Stokesian dissipative dynamics [@mor; @vai; @lee; @lutz; @mok; @rubi; @Dhar06]. We will demonstrate that the commonly stated conditions for the equilibrium bath are generally not complete within the framework of linear response theory. This is so, because the existence of anomalous diffusion has not been considered in the original treatment by Kubo and others. Moreover, we discuss also the influence of initial correlation preparation between the system and the heat bath upon the asymptotical behavior of the force-free system.
Biasing generalized Brownian motion
===================================
Let us first consider a free Brownian dynamics with $U(x,t)=0$, possessing via the FDT of the second kind a finite-valued zero-frequency friction. If this dynamics is next subjected to a constant external force, i.e. $-\partial U(x,t)/\partial x = F $, the acceleration vanishes in the case of a Stokesian friction, because the external force balances the friction force. A problem of broad interest is whether there exists an *intermediate situation* between the Newtonian mechanics and such an ordinary Langevin formalism. This in turn necessitates a non-Stokesian dissipation mechanism such that the asymptotic long-time statistical probability will typically approach a stationary state that explicitly depends on the initial preparation. It is thus of great practical interest to research what kind of heat bath can take on this role. Such non-ergodic non-equilibrium thermodynamics presents a timely subject that is presently hotly debated, both within theory [@mor; @vai; @lee; @lutz; @Muk2005; @lee2006; @Bai2005] and experiment [@Kutnjak99; @Fahri99; @Brok03].
For a GLE subjected to a constant force; i.e. $U(x,t)= - Fx$, the solution of (1) can be written as $$\begin{aligned}
x(t)&=&x(0)+v(0)H(t)+\frac{1}{m}\int^{t}_0dt'H(t-t')(\varepsilon
(t')+F),\nonumber\\
v(t)&=&v(0)h(t)+\frac{1}{m}\int^{t}_0dt'h(t-t')(\varepsilon (t')+F),\end{aligned}$$ where $H(0)=0$ and $h(0)=1$. The two response functions $H(t)$ and $h(t)$ are the inverse of the Laplace transforms $\hat{H}(s)=[s^2+s\hat{\gamma}(s)]^{-1}$ and $\hat{h}(s)=[s+\hat{\gamma}(s)]^{-1}$, respectively, where $\hat{\gamma}(s)$ is the Laplace transform of memory friction kernel, i.e. $\hat{\gamma}(s)=\int^{\infty}_0\gamma(t)\exp(-st)dt$. Under the assumption that the characteristic equation $s+\hat{\gamma}(s)=0$ possesses a zero root, i.e. $s=0$, the residue theorem implies that $h(t)=b+\sum_{i}\textmd{res}[\hat{h}(s_i)]\exp(s_it)$ and $H(t)=c+bt
+\sum_is^{-1}_i\textmd{res}[\hat{h}(s_i)]\exp(s_it)$. Here, $s_i$ (Re$s_i<0$) denote the non-zero roots of the above characteristic equation and res\[$\cdots$\] are the residues. Within this context, the two relevant, generally non-vanishing quantities $b$ and $c$ are determined to read: $$b=\frac{1}{1+\hat{\gamma}'(0)},\quad
c=-\frac{1}{2}b^2\hat{\gamma}''(0).$$ This result then requires that $\hat{\gamma}(0)=\int^{\infty}_0\gamma(t)dt=0$, i.e., implying a vanishing effective friction at zero frequency.
The average velocity and the average displacement under the external bias $F$ emerge at long times as $$\begin{aligned}
\{\langle v(t\to\infty)\rangle\}&=&b\left(\{
v(0)\}+\frac{F}{m}t\right)+\frac{F}{m}c,\\
\{\langle
x(t\to\infty)\rangle\}&=&\{x(0)\}+b\left(\{v(0)\}t+\frac{1}{2}\frac{F}{m}t^2\right)\nonumber\\
&+&c\left(\{v(0)\}+\frac{F}{m}t\right) +\frac{F}{m}d,\end{aligned}$$ where $d=-\sum_{i}\textmd{res}[\hat{h}(s_i)]/s^2_i$ is a noise-dependent quantity. Herein, we indicate by $\{\cdots \}$ the average with respect to the initial preparation of the state variables, i.e. an average over their initial values and $\langle
\cdots\rangle$ is the noise average. The dissipative [*acceleration*]{} of a Brownian particle of mass $m$ subjected to a constant force $F$ then reads: $$a=\frac{F}{m}b,$$ where generally $0\leq b\leq 1$. This quantity $b$ will be termed the *dissipation reducing* factor, the dissipation is reduced as $b$ is increased. This result is intermediate between a purely Newtonian mechanics (obeying $b=1$) and an ordinary Langevin dynamics (with $b=0$) including of GLEs.
The two limiting results for the asymptotic dynamics are found to read: (i) The Newton case with $b=1$ implying no dissipation, i.e. $\gamma(t)\equiv 0$ with $c=d=0$, yielding: $a=F/m$, $v(t)=v(0)+\frac{F}{m}t$, and $x(t)=x(0)+v(0)t+\frac{1}{2}\frac{F}{m}t^2$. (ii) The commonly known, ordinary Langevin situation is obtained with $b=0$; i.e. Eq. (3) then looses validity because of $\hat{\gamma}(0)\neq 0$. For this case $c=\hat{\gamma}^{-1}(0)$ where $\hat{\gamma}(0)$ denotes the Markovian friction strength, resulting in $a=0$, $\langle
v(t\to\infty)\rangle=F/(m\hat{\gamma}(0))$, and $\{\langle
x(t\to\infty)\rangle\}=\{x(0)\}+\{v(0)\}/\hat{\gamma}(0)+\frac{F}{m}t/\hat{\gamma}(0)+\frac{F}{m}d$.
In the unbiased case, we derive the two-time velocity correlation function (VCF) of free generalized Brownian motion in a generic form, i.e., $$\begin{aligned}
\{\langle
v(t_{1})v(t_{2})\rangle\}&=&\frac{k_BT}{m}h(|t_1-t_2|)+\left(\{v^2(0)\}-\frac{k_BT}{m}\right)\nonumber\\
&&h(t_1)h(t_2).\end{aligned}$$ Here we used only the condition: $\langle
v(0)\varepsilon(t)\rangle=0$. Note that depending on the specific choice for the initial preparation this velocity correlation generally is not time-homogeneous. The stationary velocity correlation function becomes again only a function of $|t_1-t_2|$ for the case that we use the equilibrium preparation with an initial velocity variance in accordance with the thermal equilibrium value, i.e. $\{v^2(0)\}=k_{B}T/m$ [@Bai2005]. The deduced asymptotic stationary VCF then reads: $C_{vv}(\infty)=b\frac{k_BT}{m}\neq 0$. This causes a breakdown of the ergodic equilibrium state because of the initial preparation-dependence, which is encoded in the $v(0)$-dependent asymptotic results: $\langle v\rangle_{st}=bv(0)\neq 0$ and $\{\langle v^2\rangle\}_{st}=k_{B}T/m+b^2(\{v^2(0)\}-k_BT/m)$.
Likewise, the mean square displacement (MSD) of the force-free particle is written as $$\begin{aligned}
\{\langle
x^2(t)\rangle\}&=&\{x^2(0)\}+\{v^2(0)\}H^2(t)+2\{x(0)v(0)\}H(t)\nonumber\\
&+&\frac{2}{m}\int^{t}_0dt'H(t-t')\{\langle
x(0)\varepsilon(t')\rangle\}\nonumber\\
&+&\frac{k_BT}{m}\left(2\int^t_0H(t')dt'-H^2(t)\right),\end{aligned}$$ where the fourth term denotes the effect of initial coupling between system and heat bath. We will discuss the correlation preparation in the following section. Note that here the largest power in the temporal variation of the MSD involves the square of time. The averaged displacement can be related to the MSD via the generalized Einstein relation, reading $$\kappa_2/ \mu_2(F\rightarrow 0)=k_BT_{\textmd{eff}} \;.$$ The [*ballistic*]{} diffusion coefficient is $
\kappa_2=\lim_{t\to\infty}\{\langle x^2(t)\rangle\}_{F=0}/(2t^2)$, being related to the increasing rate of linear mobility $\mu_2(F\rightarrow 0)=\lim_{t\to\infty}\{\langle x(t)\rangle\}/(F
t^2)_{F=0}$. Here, this effective temperature formally reads: $T_{\textmd{eff}}:=T+b(m\{v^2(0)\}/k_B - T)$, where $T$ is the temperature for the common case with $b=0$.
To assure the equilibrium behavior of this generalized Brownian motion the usual condition of Kubo’s FDT of the second kind for the thermal noise must be complemented as follows: Consider the Fourier transform $\tilde{\gamma}(\omega)$ of the memory damping kernel, i.e., $$\tilde{\gamma}(\omega)=\hat{\gamma}(s=-i\omega),$$ The real part of the former quantity is the spectral density of noise. A ballistic diffusion with $b\neq 0$ thus requires that the lowest power of $\hat{\gamma}(s)$ is of first-order in $s$, implying that the lowest power of Re$\tilde{\gamma}(\omega)$ is proportional to $\omega^2$ at low frequencies. Therefore, for a genuine thermal noise driven, force-free particle approaching at the equilibrium state the usual conditions must be completed by: $\lim_{s\to
0}(\hat{\gamma}(s)/s)\to\infty$, or $\lim_{\omega\to
0}(\textmd{Re}\tilde{\gamma}(\omega)/\omega^2)\to\infty$.
INITIAL CORRELATION BETWEEN SYSTEM AND BATH
===========================================
Starting from the system-plus-reservoir model, one knows that the coupling between the system and environmental degrees of freedom there exist four kinds of coupling forms which do not involve a renormalization of potential or mass of the system. In these cases the heat bath consists of a set of independent harmonic oscillators with masses $m_j$ and oscillation frequencies $\omega_j$. Pervious work [@Bai2005] has shown that the random force is [*independent*]{} of the system variables for the coordinate-velocity coupling. Nevertheless, however, the expression of thermal noise will depend on the initial preparation of system for the coupling between the system coordinate (velocity) and the environmental coordinates (velocities) considered here. To the best of our knowledge, only a small number of prior studies [@gra; @HT] have considered the general consequences of the detailed initial preparation procedure in view of the asymptotic statistical results of the system.
The coordinate-coordinate coupling
----------------------------------
For a bilinear coupling between the system coordinate $x$ and the heat bath’s coordinates $q_j$, the total Hamiltonian can be written as: $$H=\frac{P_x^2}{2m}+U(x,t)+\sum_j\left[\frac{p^2_j}{2m_j}+\frac{1}{2}m_j\omega^2_j\left(
q_j-\frac{c_j}{m_j\omega^2_j}x\right)^2\right]\;.$$ Here and below the momenta of the system and bath’s oscillators are related to $P_x=mv$ and $p_j=m_j\dot{q}_j$, respectively, and the set $c_j$ denote the coupling constants. The equation of motion of the system obeys the form of the GLE (1) and the thermal noise appearing in Eq. (1) emerges as [@zwanzig; @HALNP; @Bai2005] $$\varepsilon(t)=-m\gamma(t)x(0)+\xi_{\textmd{bath}}(t),$$ where $\gamma(t)=\frac{1}{m}\sum_j\frac{c^2_j}{m_j\omega^2_j}\cos\omega_jt$ and $\xi_{\textmd{bath}}(t)$ is determined by the initial coordinates and the velocities of the oscillators of the heat bath. The bath part $\xi_{\textmd{bath}}(t)$ of the noise explicitly reads: $$\xi_{\textmd{bath}}(t)=\sum_jc_j\left(q_j(0)\cos\omega_jt+\frac{\dot{q}_j(0)}{\omega_j}\sin\omega_jt\right).$$
Physically, this thermal noise $\varepsilon(t)$ obeys statistical properties that derive from the canonical, thermal equilibrium distribution of the total, combined system-plus-bath [@zwanzig; @HALNP; @ros]: this thermal noise then again yields a vanishing mean and its correlation obeys the thermal FDT [@HALNP]. The statistical quantities involving noise the system variables are strictly determined by the joint probability [@gra]. The correlation involving initial position of the system and the thermal noise $\varepsilon (t')$, i.e., the fourth term in Eq. (8) reads $$\begin{aligned}
&&\frac{2}{m}\int^{t}_0dt'H(t-t')\{\langle
x(0)\varepsilon(t')\rangle\}\nonumber\\&=&
-2\{x^2(0)\}\int^t_0dt'H(t-t')\gamma(t')\nonumber\\
&=&-2H(t)\ast\gamma(t)\{x^2(0)\}=-2(1-h(t))\{x^2(0)\},\nonumber\\\end{aligned}$$ wherein $``\ast"$ denotes the convolution integral. It reads, $H(t)\ast\gamma(t)=\frac{1}{2\pi i}\int
ds\hat{H}(s)\hat{\gamma}(s)\exp(st)=1-\dot{H}(t)=1-h(t)$. Here we have used the relation below Eq. (2), i.e., $\hat{H}(s)\hat{\gamma}(s)=s^{-1}-s\hat{H}(s)$. This contribution assumes a finite value, i.e., $-2(1-b)\{x^2(0)\}$ in the long-time limit.
Under the usual assumption that the thermal noise $ \varepsilon
(t)$ and the initial velocity $v(0)$ of the system are not correlated, we find from Eq. (12) that the initial coordinate of the system must be uncorrelated with its initial velocity for the coordinate-coordinate coupling case, namely, $\{x(0)v(0)\}=0$, being the case for a canonical thermal equilibrium, cf. the Hamiltonian in Eq. (11). Therefore, the third term in Eq. (8) also vanishes. In the following we shall not consider preparations with such initial correlations between the initial coordinate $x(0)$ and the initial velocity $v(0)$.
The velocity-velocity coupling
------------------------------
For a bi-linear coupling between the system velocity and the velocities of the bath oscillators the total Hamiltonian reads $$H=\frac{P_x^2}{2m}+U(x,t)+\sum_j\left[\frac{1}{2m_j}\left(p_j-\frac{d_j}{m}P_x\right)^2+\frac{1}{2}m_j\omega^2_j
q_j^2\right],$$ where $d_j$ denote the corresponding the coupling constants. We can derive again the GLE (1) describing the motion of the system with the thermal noise term now given by $$\begin{aligned}
\varepsilon(t)
&=&v(0)\sum_j\frac{d^2_j\omega_j}{m_j}\sin\omega_jt\nonumber\\
&+&\sum_j
d_j\omega^2_j\left[q_j(0)\cos\omega_jt+\frac{\dot{q}_j(0)}{\omega_j}\sin\omega_jt\right]\nonumber\\
&=& v(0)m\int^t_0\gamma(t')dt'+\xi_{\textmd{bath}}(t),\end{aligned}$$ where $\gamma(t)=\frac{1}{m}\sum_j\frac{d^2_j\omega^2_j}{m_j}\cos\omega_jt$ and in addition we have: $\{\langle
x(0)\xi_{\textmd{bath}}(t)\rangle\}=\{\langle
v(0)\xi_{\textmd{bath}}(t)\rangle\}=0$. We require that the FDT of the second kind is obeyed, namely that $\langle
\varepsilon(t)\rangle=0$ and $\langle
\varepsilon(t)\varepsilon(t')\rangle=k_BT\gamma(t-t')$. This is guaranteed when $\langle q_j(0)\rangle=\langle
\tilde{\dot{q}}_j(0)\rangle=\langle
q_i(0)\tilde{\dot{q}}_j(0)\rangle=0$, $\langle
q_i(0)q_j(0)\rangle=\frac{k_BT}{m\omega^2_j}\delta_{ij}$, and $\langle
\tilde{\dot{q}}_i(0)\tilde{\dot{q}}_j(0)\rangle=\frac{k_BT}{m_j}\delta_{ij}$, where $\tilde{\dot{q}}_i(0)=\dot{q}_i(0)-d_j/m_j v(0)$ [@Bai2005].
In the case of velocity-velocity coupling, the fourth term in Eq. (8) vanishes if again $\{x(0)v(0)\}=0$. An additional term emerges, however, for the mean squared displacement (MSD) of the force-free particle due to the thermal noise $\varepsilon(t)$ which now depends on the initial particle velocity $v(0)$. From Eq. (2), we obtain $$\begin{aligned}
\{\langle
x^2(t)\rangle\}_{\textmd{add}}&=&\frac{2}{m}H(t)\int^t_0dt'H(t-t')\{\langle
v(0)\varepsilon(t')\rangle\}\nonumber\\
&=&2H(t)\{v^2(0)\}\int^t_0dt'H(t-t')\int^{t'}_0du\gamma(u)\nonumber\\
&=&2(t-H(t))H(t)\{v^2(0)\}.\end{aligned}$$ Indeed, the ballistic diffusion arises also in this case. Notably, an additional contribution to the mean square velocity of the force-free particle \[c.f. Eq. (7) at $t_1=t_2$\] emerges in this case. It reads $$\begin{aligned}
\{\langle
v^2(t)\rangle\}_{\textmd{add}}&=&\frac{2}{m}h(t)\int^t_0dt'h(t-t')\{\langle
v(0)\varepsilon(t')\rangle\}\nonumber\\
&=&2\{v^2(0)\}h(t)\int^t_0dt'h(t-t')\int^{t'}_0\gamma(u)du\nonumber\\
&=&2h(t)(1-h(t))\{v^2(0)\},\end{aligned}$$ where we have used the inverse Laplace transform of the convolution integral in Eq. (18) by making use of the relation, $\hat{h}(s)\hat{\gamma}(s)=1-s\hat{h}(s)$.
In particular, the mean squared velocity of the force-free particle emerges in the long-time limit as $$\begin{aligned}
\{\langle
v^2(t\to\infty)\rangle\}&=&2b(1-b)\{v^2(0)\}+\frac{k_BT}{m}\nonumber\\
&+&\left(\{v^2(0)\}-\frac{k_BT}{m}\right)b^2.\end{aligned}$$ This result evidences that the system can not arrive at the equilibrium state for any initial preparation of the particle velocity if $b\neq 0$. Therefore, for the validity of FDT one has to use at initial time a preparation of thermal equilibrium for the system and the heat bath. Nevertheless, one needs not to worry that the noise is uncorrelated with the initial velocity of the system for a common non-Markovian dynamics with $b=0$. Then, the FDT is valid independent of the coupling form between system and bath whenever the effective Markovian damping of the system is finite at zero frequency. This is so because the first and the third term in Eq. (19) vanishes for $b=0$.
Test bed for non-Stokesian dissipative dynamics
===============================================
Given the FDT of the second kind by Kubo we investigate next unbiased, non-Markovian Brownian motion in (1) that is driven by colored noise known as the harmonic velocity noise (HVN) $\varepsilon(t)$ [@bao2005], which, however, does not obey the above additional requirements. The HVN itself is produced from a linear Langevin equation, namely, $$\dot{y}=\varepsilon, \quad \dot{\varepsilon}=-\Gamma
\varepsilon-\Omega ^{2}y+\xi(t),$$ where $\xi(t)$ denotes Gaussian white noise of vanishing mean with $\langle \xi (t)\xi (t^{\prime })\rangle =2\eta\Gamma
^{2}k_{B}T\delta (t-t^{\prime})$. The coefficient $\eta $ denotes the damping coefficient of the system corresponding to the thermal white noise; $\Gamma$ and $\Omega $ denote the damping and the frequency parameters. The second moments and the cross-variance of $y(0)$ and $\varepsilon(0)$ obey: $ \{y^{2}(0)\}=\eta \Gamma \Omega
^{-2}k_{B}T$, $\{\varepsilon^{2}(0)\}=\eta \Gamma k_{B}T$, and $\{y(0)\varepsilon(0)\}=0$. The Laplace transformation of the memory damping kernel reads $\hat{\gamma}(s)=\eta\Gamma s/(s^2+\Gamma
s+\Omega^2)$ with $$\textmd{Re}\tilde{\gamma}(\omega)=\frac{\eta\Gamma^2\omega
^{2}}{(\Omega ^{2}-\omega ^{2})^{2}+\Gamma ^{2}\omega ^{2}}\;,$$ respectively. The latter corresponds to the spectrum of HVN which indeed vanishes identically at zero-frequency. In this case the dissipation reducing factor emerges as $b=(1+\eta \Gamma \Omega
^{-2})^{-1}$, and likewise, $c=(1-b)^2/\eta$, $d=(1-b)^2[(\eta\Gamma)^{-1}-(1-b)\eta^{-2}]$.
Using models with a bi-linear coordinate system-bath coupling the dynamics can be characterized by the spectral density of bath modes, $J(\omega)$, being related to Re$\tilde{\gamma}(\omega)=J(\omega)/ m\omega$ [@HTB; @gra87; @RI; @chen]. Thus, for a weak coupling to a bath, as it can be realized either with optical-like bath modes [@mor; @RI], broadband colored noise [@bao2003], or also for the celebrated case of a black-body radiation field of the Rayleigh-Jeans type [@for] the static friction vanishes. Yet other physical situations that come to mind involve the vortex diffusion in magnetic fields [@ao], or open system dynamics with a velocity-dependent system-bath coupling [@for; @Bai2005; @bao06; @poll].
The non-Markovian Brownian motion can equivalently be recast as an embedded, higher-dimensional Markovian process. The Fokker-Planck equation (FPE) for the probability density $P(x,v,w,u,y,\varepsilon;t)$ corresponds then to the dynamics of a set of coupled Markovian LEs involving the auxiliary-variables $(w,u,y,\varepsilon)$: It obeys $\partial_{t}P = L_{FP}P$, where $L_{FP}$ is the associated FPE operator; when being formally supplemented here with a nonvanishing potential $U(x,t)$, it explicitly reads: $$\begin{aligned}
L_{FP}&=& -v\frac{\partial}{\partial
x}-m^{-1}(-U'(x,t)+w)\frac{\partial}{\partial v}\nonumber\\
&&+(\Gamma w+\eta\Gamma v+\Omega^2y+u)\frac{\partial}{\partial
w}\nonumber\\
&&-\Omega^2(w-\varepsilon)\frac{\partial}{\partial
u}+\varepsilon\frac{\partial}{\partial y}\nonumber\\
&&+(\Gamma \varepsilon+\Omega^2 y)\frac{\partial}{\partial
\varepsilon}+\eta\Gamma^2 k_BT\frac{\partial^2}{\partial
w^2}+\eta\Gamma^2 k_BT\frac{\partial^2}{\partial
\varepsilon^2}.\nonumber\\\end{aligned}$$
![(color online). Behavior of diverse mean energies (see text) for a nonergodic, force-free Brownian particle [*vs.*]{} time $t$. The parameters used are $m=1.0$, $k_BT=1.0$, $\eta=0.2$, $\Gamma=5.0$, $\Omega=1.0$, $x(0)=0$, and (i) $v(0)=\sqrt{2}$ and (ii) $1.0$, respectively, for the [*total*]{} energies given by the solid lines and the [*remnant*]{} energies by the dashed lines, from top to bottom. The open circles is the adsorbed energy from the bath for all cases and also the total energy for the $v(0)=0$ case.[]{data-label="1"}](Figbao1.eps){width="9cm" height="8cm"}
The energy relaxation
---------------------
We use this form for the analysis of the energy relaxation. The mean total energy of the thermal HVN-driven force-free particle reads $$\begin{aligned}
\{\langle E(t)\rangle\}&=&\frac{1}{2}m\{\langle
v^2(t)\rangle\}\nonumber\\
&=&\frac{1}{2}m\{\langle
v(t)\rangle^2\}+\frac{1}{2}m\{\langle(v(t)-\langle
v(t)\rangle)^2\rangle\}.\nonumber\\\end{aligned}$$ The first part describes the remnant initial kinetic energy of the particle, being dissipated partly by the heat bath environment. This part vanishes in the ordinary case with $b=0$. The second part denotes the energy provided from the heat bath. It is independent of the initial particle velocity, but does not relax, however, towards equilibrium. The absorbed power of the particle from the heat bath, namely, the rate of work being done by the fluctuation force [@li], is $$\begin{aligned}
P_{\textmd{abs}}&=&\lim_{t\to\infty}\int^t_0\gamma(t-t')\{\langle
v(t)v(t')\rangle\}dt'\nonumber\\
&=&\frac{\eta
k_{B}T}{1+\eta/(2\Gamma)+2\Omega^2/(\eta\Gamma)} \;.\end{aligned}$$ Note that it falls short of the equilibrium value $\eta k_{B}T$. All quantities plotted are dimensionless. Our numerical results are depicted with Fig. 1. These results are obtained via the simulation of a set of Markovian LE’s which are equivalent to the FPE in (22) with $U(x,t)=0$.
![(color online). Accelerated, (time and ensemble)- averaged Brownian motor velocity in a rocking ratchet that is driven by HVN. The used parameters are $x(0)=v(0)=0$, $m=1.0$, $k_BT=0.5$, $\protect\eta =5.0$, $\Gamma=22.0$, $\Omega^2=40.0$, $t_p=25.0$, $A_0=2.0$, $4.0$, $6.0$, $10.0$, and $15.0$, from top to bottom. The particles undergo a finite acceleration $a$ (times 100) [*vs.*]{} temperature $T$, being depicted with the inset for $A_0=10.0$.[]{data-label="1"}](Fignew2.eps){width="9cm" height="8cm"}
Brownian motor exhibiting accelerated transport
-----------------------------------------------
A most intriguing situation refers to Brownian motors [@BM] when driven by Brownian motion that exhibits ballistic diffusion. Take the case of thermal HVN driving a Brownian particle according to (1) with a periodic potential that breaks reflection symmetry, namely $$U(x,t)=U_0[\sin(2\pi x)+c_1\sin(4\pi x)+c_2\sin(6\pi x)]+A(t)x,$$ where $U_0=0.461$, $c_1=0.245$, and $c_2=0.04$ [@mac]. $A(t)$ is a square-wave periodic driving force that switches forth and back among $A(t)=A_{0}$ when $2nt_{p}\leq
t<(2n+1)t_{p}$ and $A(t)=-A_{0}$ when $(2n+1)t_{p}\leq
t<2(n+1)t_{p}$. The key challenge is whether a non-vanishing non-equilibrium current emerges that can be put to a constructive use in order to direct, separate or shuttle particles efficiently [@BM].
Figure 2 depicts the ensemble- and driving-phase-averaged velocity (with the latter being equivalent to an average over the temporal driving period, see in Ref. [@JH91]), i.e., $$\overline{\{\langle v(t)\rangle\}}=(2t_{p})^{-1}\int_{t}^{t+2t_{p}}
dt^{\prime}\{\langle v(t^{\prime })\rangle\}$$ for various strengths of the driving force $A_0$, as obtained via the simulation of the Markovian LE’s corresponding to the equivalent higher-dimensional FPE in (22). A startling finding is now that this averaged velocity is no longer a constant but rather increases [*linearly*]{} with time. This is in clear contrast to the behavior of ordinary Brownian motors [@BM]. This [*directed acceleration*]{} is presented by the slope (see dashed lines in Fig. 2) of the average velocity. For a weak rocking (i. e. small $A_{0}$) the phenomenon of directed motion involves the surmounting of barriers, thus hindering transport. In contrast, for a strong super-threshold rocking the averaged displacement is related to the mean square displacement via the modified Einstein relation in (9), yielding $\{\langle
x(t)\rangle\}\propto - F_{\textmd{eff}} \; t^2$ in the long-time limit. Here $F_{\textmd{eff}}$ is an effective tilting force stemming from the rocking of the ratchet potential. Amazingly, this Brownian motor can be accelerated because the driving and the noise-induced effective tilting force supersedes the acting friction force.
Conclusions
===========
We have researched within the GLE-formalism in (1) an intermediate dynamics proceeding between Newton and Langevin. The emerging non-equilibrium features are manifested by the initial preparation-dependent asymptotic stationary state, which is directly related to a non-Stokesian dissipative phenomenon which stems from a vanishing effective Markovian friction at zero-frequency. It has been found that the fluctuation-dissipation theory is valid and independent of the coupling form between system and bath when the effective Markovian damping is finite at zero frequency. In order to assure the equilibrium behavior of a system, the usual condition for the heat bath, i.e., the Kubo’s fluctuation-dissipation theorem of the second kind, must be completed by an additional requirement: $\lim_{s\to 0}(\hat{\gamma}(s)/s)\to\infty$ or $\lim_{\omega\to
0}(\textmd{Re}\tilde{\gamma}(\omega)/\omega^2)\to\infty$, where $\hat{\gamma}(s)$ and $\tilde{\gamma}(\omega)$ are the Laplace and Fourier transforms of the memory damping. However, the system can not arrive at the equilibrium state for any initial preparation of the system if the the condition is not obeyed for the bilinear coupling between the system velocity and the velocities of environmental oscillators.
Our findings exhibit anomalous super-diffusion in the form of a ballistic diffusion. Yet another riveting result is that the corresponding Brownian dynamics for a rocking Brownian motor exhibits a distinct, [*accelerated*]{} velocity, rather than the constant drift which typifies the situation with a Stokesian finite zero-frequency dissipation. We are also confident that our present results will serviceably impact other quantities of thermodynamic and quantum origin. Thus, this field is open for future studies that in turn may reveal further surprising findings.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
This work was supported by the NNSFC under 10674016 and 10475008 and the German Research Foundation (DFG), Sachbeihilfe HA1517/26-1 (P. H.) and the CNPq (Brazil) (F. A. O.).
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[^1]: Electronic address: jdbao@bnu.edu.cn
|
---
abstract: |
The paper consists of two parts. The first part introduces the representation ring for the family of compact unitary groups $U(1)$, $U(2)$, …. This novel object is a commutative graded algebra $R$ with infinite-dimensional homogeneous components. It plays the role of the algebra of symmetric functions, which serves as the representation ring for the family of finite symmetric groups. The purpose of the first part is to elaborate on the basic definitions and prepare the ground for the construction of the second part of the paper.
The second part deals with a family of Markov processes on the dual object to the infinite-dimensional unitary group $U(\infty)$. These processes were defined in a joint work with Alexei Borodin (J. Funct. Anal. 2012). The main result of the present paper consists in the derivation of an explicit expression for their infinitesimal generators. It is shown that the generators are implemented by certain second order partial differential operators with countably many variables, initially defined as operators on $R$.
address: 'Institute for Information Transmission Problems, Moscow, Russia; National Research University Higher School of Economics, Moscow, Russia'
author:
- Grigori Olshanski
title: The representation ring of the unitary groups and Markov processes of algebraic origin
---
Introduction {#sect1}
============
Preliminaries: the symmetric group case
---------------------------------------
The present paper deals with certain combinatorial and probabilistic aspects of the representation theory of the infinite-dimensional unitary group $U(\infty)$. A parallel theory also exists for the infinite symmetric group $S(\infty)$. That theory is simpler and better developed, and it served as a motivation for the present paper. So I start with a brief overview of some relevant results which hold in the symmetric group case.
In the modern interpretation, classical Frobenius’ construction [@Fr] of irreducible characters of the symmetric groups $S(N)$ relies on the isomorphism of graded algebras ${\operatorname{Rep}}(S(1),S(2),\dots)\simeq{\operatorname{Sym}}$, where ${\operatorname{Sym}}$ denotes the algebra of symmetric functions and ${\operatorname{Rep}}(S(1),S(2),\dots)$ is our notation for the *representation ring of the family $\{S(N): N=1,2,\dots\}$* of the finite symmetric groups.
The algebra ${\operatorname{Rep}}(S(1),S(2),\dots)$ can be described as follows: $$\label{eq1.A}
{\operatorname{Rep}}(S(1),S(2),\dots):=\bigoplus_{N=0}^\infty {\operatorname{Rep}}^S_N$$ where ${\operatorname{Rep}}^S_N$ is the space of class functions on $S(N)$, and the multiplication $${\operatorname{Rep}}^S_M\otimes {\operatorname{Rep}}^S_N\to {\operatorname{Rep}}^S_{M+N}$$ is given by the operation of induction from $S(M)\times S(N)$ to $S(M+N)$.
(This definition should not be confused with that of the representation ring of an individual group, see, e.g., Segal [@Segal]).
The algebra ${\operatorname{Rep}}(S(1),S(2),\dots)$ has a distinguished basis formed by the irreducible characters of the symmetric groups. Under the isomorphism ${\operatorname{Rep}}(S(1),S(2),\dots)\to{\operatorname{Sym}}$, called the *characteristic map*, this basis is transformed into the distinguished basis in ${\operatorname{Sym}}$ formed by the Schur symmetric functions.
These facts are well known, see e.g. Macdonald [@Ma95 Chapter I, Section 7].
The *infinite symmetric group* $S(\infty)$ is defined as the union of the infinite chain $$\label{eq1.L}
S(1)\subset S(2)\subset \dots \subset S(N-1)\subset S(N)\subset \cdots$$ of finite symmetric groups. For $S(\infty)$, the conventional notion of irreducible characters is not applicable. However, there exists a reasonable analog of *normalized* irreducible characters (that is, irreducible characters divided by dimension). These are the so-called *extreme* characters whose definition, first suggested by Thoma [@Tho64], was inspired by the Murray–von Neumann theory of factors. Thoma discovered that the extreme characters of $S(\infty)$ admit an explicit description: they are parameterized by the points of the *Thoma simplex* ${\Omega}^S$, a convex subset in the infinite-dimensional cube $[0,1]^\infty$. Note that ${\Omega}^S$ is compact in the product topology of $[0,1]^\infty$.
The *dual object* to the group $S(N)$ is defined as the set ${\widehat}{S(N)}$ of its irreducible characters, and it can be identified with the set ${\mathbb Y}_N$ of Young diagrams with $N$ boxes. Likewise, we regard the set of extreme characters of the group $S(\infty)$ as (one of the possible versions of) the dual object ${\widehat}{S(\infty)}$ and identify it with the Thoma simplex ${\Omega}^S$.
Vershik and Kerov [@VK81-Doklady], [@VK81] initiated the *asymptotic theory of characters* (see also Vershik’s foreword to [@Kerov-book]). They explained how the extreme characters of the group $S(\infty)$ arise from the normalized irreducible characters of the groups $S(N)$ in a limit transition as $N$ goes to infinity. In the asymptotic theory of characters, the algebra ${\operatorname{Sym}}$ still plays an important role. In particular, the so-called *ring theorem* of Vershik and Kerov says that the extreme characters of $S(\infty)$ are in a one-to-one correspondence with those linear functionals on ${\operatorname{Sym}}$ that are multiplicative, take nonnegative values on the basis of Schur functions, and vanish on the principal ideal $(e_1-1)\subset{\operatorname{Sym}}$, where $e_1$ is the first elementary symmetric function (see Vershik-Kerov [@VK-RingTheorem] and also Gnedin-Olshanski [@GO-zigzag]).
Now I proceed to probabilistic results. First, note that the embedding $S(N-1)\subset S(N)$ gives rise, by duality, to a canonical “link” ${\widehat}{S(N)}\dasharrow{\widehat}{S(N-1)}$. Here by a link $X\dasharrow Y$ between two spaces I mean a “generalized map” which assigns to every point of $X$ a probability distribution on $Y$; in other words, a link is given by a Markov kernel (which in our case is simply a stochastic matrix). As explained in Borodin-Olshanski [@BO-MMJ], the dual object ${\widehat}{S(\infty)}$ can be viewed as the projective limit of the chain $$\label{eq1.J}
{\widehat}{S(1)}\dashleftarrow{\widehat}{S(2)}\dashleftarrow\dots\dashleftarrow{\widehat}{S(N-1)}
\dashleftarrow{\widehat}{S(N)} \dashleftarrow\cdots$$ taken in an appropriate category with morphisms given by Markov kernels. Thus, $S(\infty)$ is an inductive limit group while its dual object ${\widehat}{S(\infty)}$ is obtained by taking a kind of projective limit.
In [@BO-PTRF09], Borodin and I constructed a two-parameter family of continuous time Markov processes on the Thoma simplex. Our work was inspired by our previous study of the problem of harmonic analysis on $S(\infty)$ and substantially used the canonical links from . We proved that the Markov processes in question have continuous sample trajectories and consequently are diffusion processes. The proof relied on the computation of the infinitesimal generators of the processes: we showed that the generators are given by certain second order differential operators initially acting on the the quotient algebra ${\operatorname{Sym}}/(e_1-1)$. To relate the?se operators to Markov processes we used the fact that there is a canonical embedding $$\label{eq1.K}
{\operatorname{Sym}}/(e_1-1) \hookrightarrow C({\Omega}^S),$$ where $C({\Omega}^S)$ denotes the Banach algebra of continuous functions on the compact space ${\Omega}^S$.
The results
-----------
Let us turn to the compact unitary groups. They are organized into a chain similar to , $$U(1)\subset U(2)\subset \dots \subset U(N-1)\subset U(N)\subset \cdots,$$ and we set $U(\infty):=\bigcup_{N=1}^\infty U(N)$. The extreme characters of the group $U(\infty)$ were first investigated by Voiculescu [@Vo76]. They are parameterized by the points of an infinite-dimensional space ${\Omega}$, which can be realized as a convex subset in the product of countably many copies of ${\mathbb R}_+$ (see Subsection \[sect3.A\] below). Note that ${\Omega}$ is locally compact. Like the dual object to $S(\infty)$, the space ${\Omega}={\widehat}{U(\infty)}$ can be identified with the projective limit of the dual chain $$\label{eq1.N}
{\widehat}{U(1)}\dashleftarrow{\widehat}{U(2)}\dashleftarrow\dots\dashleftarrow{\widehat}{U(N-1)}
\dashleftarrow{\widehat}{U(N)} \dashleftarrow\cdots$$
Although the groups $S(\infty)$ and $U(\infty)$ are structurally very different, there is a surprising similarity in the description of their characters. An explanation of this phenomenon is suggested in Borodin-Olshanski [@BO-MMJ].
Here is a brief description of what is done in the present paper.
1\. The attempt to extend the definition of the representation ring to the family of the unitary groups leads us to a novel object — a certain graded algebra $R$, which plays the role of the algebra ${\operatorname{Sym}}$.
2\. An analog of the embedding is found. As explained below, it may be viewed as a kind of Fourier transform on $U(\infty)$.
3\. The main result is the computation of the infinitesimal generators for the four-parameter family of Markov processes on ${\Omega}$, previously constructed in Borodin-Olshanski [@BO-GT-Dyn]. It is shown that the generators in question are implemented by certain second order partial differential operators, initially defined on $R$.
Now I will describe the results in more detail. As will be clear, for all the similarities between $S(\infty)$ and $U(\infty)$, the unitary group case turns out to be substantially more complicated.
The representation ring for the unitary groups: the algebra $R$
---------------------------------------------------------------
At first it was unclear to me if there is a good analog of the representation ring for the family $\{U(N)\}$. The difficulty here is that, in contrast to the case of finite symmetric groups, induced characters have infinitely many irreducible constituents. Therefore, directly following the definition of ${\operatorname{Rep}}(S(1),S(2),\dots)$ we see that products of basis elements are infinite sums; how to deal with them? The proposed solution is to enlarge the space and allow infinite sums. This leads to the following definition:
The algebra $R$, the suggested analog of the algebra ${\operatorname{Sym}}$, is the graded algebra of formal power series of bounded degree, in countably many variables each of which has degree 1. The variables are denoted by $\varphi_n$, where $n$ ranges over ${\mathbb Z}$.
Recall that ${\operatorname{Sym}}$ is the projective limit of polynomial algebras: $$\label{eq1.H}
{\operatorname{Sym}}=\varprojlim {\mathbb C}[e_1,\dots,e_k],$$ where $k\to\infty$ and $e_1,e_2,\dots$ are the elementary symmetric functions.
Likewise, $R$ also can be represented as the projective limit of polynomial algebras: $$\label{eq1.I}
R =\varprojlim {\mathbb C}[\varphi_{-l}, \dots,\varphi_{k}],$$ where $k, l\to+\infty$.
A substantial difference is that $\deg e_k=k$, while $\deg\varphi_n=1$ for all $n\in{\mathbb Z}$. Because of this, the homogeneous components of ${\operatorname{Sym}}$ have finite dimension, while those of $R$ are infinite-dimensional. Nevertheless, it turns out that the projective limit realization is a kind of finiteness property which can be efficiently exploited.
As in the case of the algebra ${\operatorname{Sym}}$, in $R$ there exist various interesting bases, but these are *topological bases*. Two bases are of particular importance for the purpose of this paper. They are denoted as $\{\varphi_{\lambda}\}$ and $\{{\sigma}_{\lambda}\}$, where the subscript ${\lambda}$ ranges over the set of highest weights of all unitary groups. The basis $\{\varphi_{\lambda}\}$ is formed by the monomials in letters $\varphi_n$ and is similar to the multiplicative basis in ${\operatorname{Sym}}$ generated by the elementary symmetric functions. The basis $\{{\sigma}_{\lambda}\}$ is an analog of the Schur functions. The interplay between these two bases plays an important role in the derivation of the main result.
By the Schur-Weyl duality, the representation ring for the *family* $\{S(N)\}$ is isomorphic to a certain representation ring of a *single* object — the Lie algebra $\mathfrak{gl}(\infty)$. Likewise, using the fermion version of the Howe duality one can identify the representation ring for the family $\{U(N)\}$ with a certain representation ring for the Lie algebra $\mathfrak{gl}(2\infty)$ (for more detail, see Subsection \[sect2.B\] below).
What is the Fourier transform on $U(\infty)$?
---------------------------------------------
Let us consider first a finite group $G$ and let ${M_{\operatorname{inv}}}(G)$ denote the space of complex measures on $G$, invariant with respect to inner automorphisms. Next, let ${\widehat}G$ stand for the set of normalized irreducible characters and ${{\operatorname{Fun}}}({\widehat}G)$ denote the space of functions on ${\widehat}G$. By integrating a character $\chi\in{\widehat}G$ against a measure $m\in{M_{\operatorname{inv}}}(G)$ we get a linear map $$F: {M_{\operatorname{inv}}}(G)\to {{\operatorname{Fun}}}({\widehat}G).$$ Using the functional equation for normalized irreducible characters one sees that $F$ turns the convolution product of measures into the pointwise product of functions. So $F$ is a reasonable version of Fourier transform.
More generally, the above definition of Fourier transform $F$ works perfectly when $G$ is a compact group. Then as ${M_{\operatorname{inv}}}(G)$ one can still take the space of invariant complex measures on $G$ or, if $G$ is a Lie group, the larger space of invariant distributions or else an appropriate subspace therein, depending on the situation.
But what happens for $G=S(\infty)$ or $G=U(\infty)$? The dual object ${\widehat}G$ has been defined, and one knows that it is large enough in the sense that the extreme characters of these groups separate the conjugacy classes. The problem is that the above definition of ${M_{\operatorname{inv}}}(G)$ no longer works. For instance, the only invariant *finite* measure on $S(\infty)$ is the delta measure at the unit element.
This difficulty can be resolved as follows. For a group $G$ which is an inductive limit of compact groups $G(N)$ we define $${M_{\operatorname{inv}}}(G):=\varinjlim {M_{\operatorname{inv}}}(G(N)),$$ where the map ${M_{\operatorname{inv}}}(G(N-1))\to{M_{\operatorname{inv}}}(G(N))$ is given by averaging over the action of the group of inner automorphisms of $G(N)$. In more detail, given a measure $M\in{M_{\operatorname{inv}}}(G(N-1))$, its image in ${M_{\operatorname{inv}}}(G(N))$ is defined as $$\int_{g\in G(N)}M^g dg,$$ where $M^g$ denotes the transformation of $M$ (which we transfer from $G(N-1)$ to $G(N)$) under the conjugation by an element $g\in G(N)$, and $dg$ denotes the normalized Haar measure on $G(N)$.
In the case of $G=S(\infty)$ it is readily verified that ${M_{\operatorname{inv}}}(S(\infty))$ can be identified, in a natural way, with the quotient algebra ${\operatorname{Sym}}/(e_1-1)$, and then the Fourier transform just defined coincides with the map .
In the case $G=U(\infty)$ the situation is more delicate. In the first approximation, the analog of ${\operatorname{Sym}}/(e_1-1)$ is the quotient algebra $R/J$, where $J$ is the following principal ideal $$\label{eq1.M}
J:=(\varphi-1), \qquad \varphi:=\sum_{n\in{\mathbb Z}}\varphi_n.$$ However, this algebra is too large and one has to narrow it in order for the Fourier transform to be well defined. We discuss two variants of doing this, both of which seem to be quite natural. Note that there are also many other possibilities: they depend on the concrete choice of the spaces ${M_{\operatorname{inv}}}(U(N))$. I did not go too far in this direction, because for the main result it was sufficient to dispose of the simplest way to relate the algebra $R$ to the space ${\widehat}{U(\infty)}={\Omega}$.
Note that in a number of cases involving those of $G=S(\infty)$ and $G=U(\infty)$, the set of conjugacy classes of $G$ can be endowed with a natural semigroup structure (see [@Ols-AA], [@Ols-Semigroups], [@Ols-GordonBreach]). Then one may endow ${M_{\operatorname{inv}}}(G)$ with a multiplication, which is an analog of convolution product and which turns into pointwise multiplication on ${\widehat}G$ under a suitable version of Fourier transform.
The Markov generators
---------------------
The Markov processes on ${\Omega}$ constructed in Borodin-Olshanski [@BO-GT-Dyn] depend on four complex parameters ${{z,z',w,w'}}$ subject to certain constraints (see Definition \[def6.A\]). Let us ignore for a moment the constraints, so that ${{z,z',w,w'}}$ are arbitrary complex numbers, and consider a formal second order partial differential operator $$\begin{gathered}
\label{eq1.C}
{\mathbb D}_{{{z,z',w,w'}}}=\sum_{n_1,n_2\in{\mathbb Z}} A_{n_1n_2}(\dots,\varphi_{-1},
\varphi_0,\varphi_1,\dots)\frac{{\partial}^2}{{\partial}\varphi_{n_1}{\partial}\varphi_{n_2}}\\
+\sum_{n\in{\mathbb Z}} B_n(\dots,\varphi_{-1}, \varphi_0,\varphi_1,\dots;
{{z,z',w,w'}})\frac{{\partial}}{{\partial}\varphi_n},\end{gathered}$$ where the variables $\varphi_n$ are indexed by integers $n\in{\mathbb Z}$, the second order coefficients $A_{n_1n_2}$ are certain (complicated) quadratic expressions in the variables, and the first order coefficients $B_n$ are certain linear expressions which involve the parameters, see the explicit formulas and below.
The main result of the paper can be informally stated as follows.
\[thm1.A\] Assume that the quadruple $({{z,z',w,w'}})$ satisfies the necessary constraints, so that the construction of [@BO-GT-Dyn] provides a Markov process $X_{{z,z',w,w'}}$ on ${\Omega}$. Then the generator of $X_{{z,z',w,w'}}$ is implemented by the differential operator ${\mathbb D}_{{z,z',w,w'}}$.
A rigorous version is given in Theorem \[thm7.B\].
Note that the Markov generator in question is defined on a dense subspace of $C_0({\Omega})$, the Banach space of continuous functions on ${\Omega}$ vanishing at infinity. To relate such an operator with an operator acting on $R$ we use the Fourier transform discussed in the preceding subsection. Here we use the fact that ${\mathbb D}_{{z,z',w,w'}}$ preserves the principal ideal $J\subset R$ (see above) and so also acts on $R/J$.
The operator ${\mathbb D}_{{z,z',w,w'}}$ is well adapted to the basis $\{\varphi_{\lambda}\}$ in $R$ while the Markov generators are initially defined by their action on another basis, $\{{\sigma}_{\lambda}\}$. This is the main source of difficulty in the proof of the main theorem: transition from one basis to another one is achieved by rather long computations.
The construction of the processes $X_{{z,z',w,w'}}$ in our work [@BO-GT-Dyn] is based on a limit transition along the chain : we find jump processes on the dual objects ${\widehat}{U(N)}$ which are consistent with the “links” ${\widehat}{U(N)}\dasharrow{\widehat}{U(N-1)}$. The key idea is very simple but the construction is formal and it drastically differs from the approaches used by probabilists. So the intriguing problem is to understand what is the nature of the processes $X_{{z,z',w,w'}}$ and what can be explicitly computed. The computation of the Markov generators in the present paper is the first step in this direction.
The fact that the Markov generators are implemented by differential operators makes plausible the conjecture that the sample trajectories of the processes are continuous (the diffusion property). In the symmetric group case (see Borodin-Olshanski [@BO-PTRF09]) we give a simple proof of the diffusion property for the processes on the Thoma simplex ${\Omega}^S$ using the realization of their generators as differential operators on ${\operatorname{Sym}}$. However, the structure of the differential operator ${\mathbb D}_{{z,z',w,w'}}$ is substantially more complicated, because, in contrast to the symmetric group case, the coefficients $A_{n_1n_2}$ are given by infinite series. This is an obstacle to extending the approach of [@BO-PTRF09].
It seems that the Markov generators cannot be written in terms of the natural coordinates on ${\Omega}$, and the same holds in the models related to $S(\infty)$, studied in [@BO-PTRF09] and [@BO-EJP] (a possible explanation is that the coordinate functions do not enter the domain of the generators, see in this connection the discussion in Petrov [@Petrov-FAA Remark 5.4] concerning a simpler model). This is why one needs to use a more involved construction using the algebra $R$ (or, in the symmetric group case, the algebra ${\operatorname{Sym}}$).
Lifting of multivariate Jacobi differential operators to algebra $R$
--------------------------------------------------------------------
Let $m=1,2,3,\dots$. The *Jacobi partial differential operator* in $m$ variables $t_1,\dots,t_m$ is given by $$\label{eq1.P}
D^{(a,b)}_m := \sum_{i=1}^m \left(t_i(1-t_i)\frac{\partial^2}{\partial
t_i^2}+\left[b+1-(a+b+2)t_i+\sum_{j:\, j\ne
i}\frac{2t_i(1-t_i)}{t_i-t_j}\right]\frac{\partial}{\partial t_i}\right).$$ Here $a$ and $b$ are parameters. In the simplest case $m=1$ this operator turns into the familiar hypergeometric ordinary differential operator $$D^{(a,b)}=t(1-t)\frac{d^2}{dt^2}+[b+1-(a+b+2)t]\frac{d}{dt}.$$ The operator $D^{(a,b)}$ is attached to the Jacobi orthogonal polynomials with the weight function $t^b(1-t)^a$ on the unit interval $0\le t\le 1$, that is, the Jacobi polynomials are just the polynomial eigenfunctions of $D^{(a,b)}$.
In the case of several variables, despite the singularities on the hyperplanes $t_i=t_j$, the operator $D^{(a,b)}_m$ is well defined on the space of symmetric polynomials in $t_1,\dots,t_m$ and is diagonalized in the basis of $m$-variate symmetric Jacobi polynomials. The latter polynomials are a particular case of the Heckman-Opdam orthogonal polynomials, which corresponds to the root system $BC_m$ and a special choice of the “Jack parameter” (see e.g. Heckman [@Heckman], Koornwinder [@Koornwinder]). The operator $D^{(a,b)}_m$ is well known; it appeared (in a more general form involving the Jack parameter) in many works, see, e.g., Baker-Forrester [@BF].
Given $m$, let us fix two nonnegative integers $k$ and $l$ such that $k+l=m$. We assume that $m+1$ variables $\varphi_{-l},\dots,\varphi_k$ are expressed through $m$ variables $t_1,\dots,t_m$ via $$\sum_{n=-l}^k\varphi_nu^n=\prod_{i=1}^k
(t_i+(1-t_i)u)\cdot\prod_{i=k+1}^m(1-t_i+t_iu^{-1}),$$ where the left-hand side should be viewed as a generating series for $\varphi_{-l},\dots,\varphi_k$ with an auxiliary indeterminate $u$ (then, by equating the coefficients of monomials $u^n$ in the both sides, we can write $\varphi_n$’s as polynomials in $t_i$’s). Setting $u=1$ one sees that the constraint $\sum_{n=-l}^k\varphi_n=1$ holds. Moreover, we may identify the algebra ${\operatorname{Sym}}_m$ of symmetric polynomials in variables $t_1,\dots,t_m$ with $${\widehat}R(k,-l):={\mathbb C}[\varphi_{-l},\dots,\varphi_k]\big/\left(\sum_{n=-l}^k\varphi_n-1\right),$$ the quotient by the principal ideal generated by the element $\sum_{n=-l}^k\varphi_n-1$.
In the next theorem we regard the same algebra ${\widehat}R(k,-l)$ as the quotient $R/J(k,-l)$, where $J(k,-l)$ denotes the ideal of $R$ generated by the elements $$\label{eq1.O}
\varphi_{k+1}, \varphi_{k+2}, \dots; \quad \varphi_{-l}, \varphi_{-l-1},
\dots;\quad \varphi_{-l}+\dots+\varphi_k-1.$$ Note that the ideal does not change if $\varphi_{-l}+\dots+\varphi_k-1$ is replaced by $\varphi-1$, where $\varphi$ is defined in above.
From the proof of Theorem \[thm1.A\] one can extract the following fact:
\[thm1.B\] Let us assume that parameters $z$ and $w$ are nonnegative integers, which are not both $0$. Let us denote them by $k$ and $l$, respectively.
In this special case the differential operator ${\mathbb D}_{{z,z',w,w'}}$ preserves the ideal $J(k,l)\subset R$ and so determines an operator on $R/J(k,-l)={\widehat}R(k,-l)$. The latter operator coincides with the $(k+l)$-variate Jacobi operator with parameters $a=z'-k$, $b=w'-l$.
This fact clarifies the nature of the differential operator ${\mathbb D}_{{z,z',w,w'}}$. Indeed, from Theorem \[thm1.B\] one can see that the sophisticated expression for ${\mathbb D}_{{z,z',w,w'}}$ appears as the result of formal analytic extrapolation, with respect to parameters $(k,l,a,b)$, of the Jacobi differential operators $D^{(a,b)}_{k+l}$ rewritten in a new set of variables. Note that as $k$ and $l$ increase, the ideals $J(k,-l)$ decrease and their intersection $\cap_{k,l=1}^\infty J(k,-l)$ coincides with the principal ideal $J\subset R$ generated by the sole element $\varphi-1$. Note also that the extrapolation procedure is purely formal, because the integers $k$ and $l$, whose sum $m=k+l$ initially represents the number of variables, finally turn into complex parameters.
It is interesting to compare this picture with what is done in the work of Sergeev and Veselov [@SV] which deals with the same Jacobi differential operators (involving the additional “Jack parameter”). However, in [@SV] the operators are lifted to the algebra ${\operatorname{Sym}}$, while our target space is the algebra $R$. The initial motivation of Sergeev and Veselov is also different: they used the lifting to ${\operatorname{Sym}}$ as a tool for constructing super versions of quantum integrable systems in finite dimensions, while our interest is in infinite-dimensional Markov dynamics. (See also the papers Desrosiers-Hallnäs [@DH], Olshanski [@Ols-LaguerreNote], [@Ols-Laguerre] — in all these works the target space is ${\operatorname{Sym}}$.)
Organization of the paper
-------------------------
Section \[sect2\] introduces the algebra $R$ and Section \[sect3\] relates it to the dual object ${\widehat}{U(\infty)}$. Section \[sect4\] introduces the differential operator ${\mathbb D}_{{z,z',w,w'}}$. In Sections \[sect5\] and \[sect6\] we recall some general facts about Feller Markov processes, next describe the “method of intertwiners” [@BO-GT-Dyn], and then explain how it produces a special family of Markov processes on ${\widehat}{U(\infty)}$ out of continuous time Markov chains on the discrete sets ${\widehat}{U(N)}$. In Section \[sect7\] we formulate the main theorem and outline the plan of its proof. The proof itself occupies Sections \[sect8\] and \[sect9\]. The last Section \[sect10\] is an appendix, where we prove the uniform boundedness of multiplicities in certain induced representations of compact groups; this fact was used in Section \[sect3\].
Acknowledgement
---------------
I am grateful to Igor Frenkel for an important comment which I used in Subsection \[sect2.B\], and to Vladimir L. Popov who confirmed that the statement of Proposition \[prop10.A\] is true and communicated its proof to me. I am also grateful to the anonymous referee for valuable suggestions. This research was partially supported by a grant from Simons Foundation (Simons-IUM Fellowship) and by the RFBR grant 13-01-12449.
The algebra $R$ {#sect2}
===============
Definition of algebra $R$ {#sect2.A}
-------------------------
Throughout the paper $\{\varphi_n\}$ stands for a doubly infinite collection of formal variables indexed by arbitrary integers $n\in{\mathbb Z}$.
We define $R$ as the commutative complex unital algebra formed by arbitrary formal power series of bounded degree, in variables $\varphi_n$, $n\in{\mathbb Z}$. Here we assume that $\deg\varphi_n=1$ for every $n$. The algebra $R$ is graded: we write $R=\bigoplus_{N=0}^\infty R_N$, where the elements of the $N$th homogeneous component $R_N$ have the form $$\label{eq2.A}
\psi=\sum_{n_1\ge \dots\ge
n_N}a_{n_1,\dots,n_N}\varphi_{n_1}\dots\varphi_{n_N}$$ with no restriction on the complex coefficients $a_{n_1,\dots,n_N}$.
Equivalently, $R$ can be defined as a projective limit of polynomial algebras. Namely, for a pair of integers $n_+\ge n_-$ we set $$R(n_+,n_-):={\mathbb C}[\varphi_{n_-},\varphi_{n_-+1},\dots,\varphi_{n_+-1},\varphi_{n_+}].$$ Then one can write $$R=\varprojlim R(n_+,n_-), \qquad n_+\to+\infty, \quad n_-\to-\infty,$$ where the limit is taken in the category of graded algebras.
We call the natural homomorphisms $R\to R(n_+,n_-)$ the *truncation maps*. Let $I(n_+,n_-)$ denote the kernel of the truncation $R\to R(n_+,n_-)$. As $n_\pm\to\pm\infty$, the ideals $I(n_+,n_-)$ decrease and their intersection equals $\{0\}$. We take these ideals as the base of a topology in $R$, which we call the *$I$-adic topology*.
Following Weyl [@Weyl] we define a *signature of length* $N$ as an arbitrary vector ${\lambda}=({\lambda}_1,\dots,{\lambda}_N)\in{\mathbb Z}^N$ with weakly decreasing coordinates: ${\lambda}\ge\dots\ge{\lambda}_N$. The set of all such vectors is denoted by ${{\mathbb{S}}}_N$. In particular, ${{\mathbb{S}}}_1={\mathbb Z}$. By agreement, ${{\mathbb{S}}}_0$ consists of a single element denoted by $\varnothing$.
With a signature ${\lambda}\in{{\mathbb{S}}}_N$ we associate a monomial of degree $N$, $$\varphi_{\lambda}:=\varphi_{{\lambda}_1}\dots\varphi_{{\lambda}_N},$$ and we agree that $\varphi_\varnothing=1$. With this notation, can be rewritten as $$\psi=\sum_{{\lambda}\in{{\mathbb{S}}}_N}a_{\lambda}\varphi_{\lambda}.$$ Initially, $\psi$ is a formal series, but, alternatively, the above sum can be interpreted as the limit, in the $I$-adic topology, of the truncated finite sums, $$\psi=\lim_{n_\pm\to\pm\infty}\;\sum_{{\lambda}\in{{\mathbb{S}}}_N:\; n_+\ge{\lambda}_1,\; {\lambda}_N\ge
n_-}a_{\lambda}\varphi_{\lambda}.$$
Therefore, one can say that the monomials $\varphi_{\lambda}$ form a homogeneous *topological* basis of $R$.
Bases in $R$
------------
We are going to describe a general recipe for constructing various topological bases in $R$ which are all consistent with the projective limit realization $R=\varprojlim R(n_+,n_-)$.
Let us introduce a partial order on signatures: two signatures ${\lambda}$, $\mu$ may be comparable only if they have the same length $N$, and then $${\lambda}\ge\mu \,
\Leftrightarrow\,{\lambda}-\mu\in{\mathbb Z}_+({\varepsilon}_1-{\varepsilon}_2)+\dots+{\mathbb Z}_+({\varepsilon}_{N-1}-{\varepsilon}_N),$$ where ${\varepsilon}_1,\dots,{\varepsilon}_N$ is the natural basis of the lattice ${\mathbb Z}^N$. In particular, ${\lambda}\ge\mu$ implies $\sum{\lambda}_i=\sum\mu_i$. We write ${\lambda}>\mu$ if ${\lambda}\ge\mu$ and ${\lambda}\ne\mu$. Note that the signatures of length $N$ are precisely the highest weights of the irreducible representations of $U(N)$, and the introduced order is nothing else than the standard *dominance partial order* on the set of weights of the reductive Lie algebra $\mathfrak{gl}(N,{\mathbb C})$, the complexified Lie algebra of $U(N)$.
We will be dealing with various symmetric *Laurent* polynomials in several variables $u_1,\dots,u_N$, $N=1,2,\dots$. The simplest example is the family of monomial sums $m_{\lambda}$. Here ${\lambda}\in{{\mathbb{S}}}_N$ and, by definition, $$m_{\lambda}=\sum_{(n_1,\dots,n_N)\in S(N)\cdot{\lambda}}u_1^{n_1}\dots u_N^{n_N},$$ where $S(N)\cdot{\lambda}$ denotes the orbit of ${\lambda}$ under the action of the symmetric group $S(N)$; in other words, the summation is over all *distinct* vectors $(n_1,\dots,n_N)\in{\mathbb Z}^N$ that can be obtained from $({\lambda}_1,\dots,{\lambda}_N)$ by permutations of the coordinates. By agreement, $m_\varnothing:=1$ (the same agreement is tacitly adopted for other families of polynomials that will appear below).
Assume we are given an arbitrary family $\{P_{\lambda}\}$ of homogeneous symmetric Laurent polynomials indexed by signatures and satisfying the following *triangularity condition*: $$\label{eq2.B}
P_{\lambda}=\sum_{\mu:\,\mu\le{\lambda}}{\alpha}({\lambda},\mu) m_\mu,\qquad {\alpha}({\lambda},\mu)\in{\mathbb C}, \quad
{\alpha}({\lambda},{\lambda})=1$$ (examples will be given shortly). In particular, the number of variables in $P_{\lambda}$ equals the length of ${\lambda}$.
With every such a family $\{P_{\lambda}\}$ we associate a family $\{\pi_{\lambda}\}$ of homogeneous elements of $R$ in the following way. We form a generating series for $\varphi_n$’s: $$\label{eq2.C}
\Phi(u):=\sum_{n\in{\mathbb Z}}\varphi_n u^n\in R[[u,u^{-1}]].$$ Then the elements $\pi_{\lambda}$ in question are obtained as the coefficients in the expansion $$\label{eq2.D}
\Phi(u_1)\dots\Phi(u_N)=\sum_{{\lambda}\in{{\mathbb{S}}}_N}\pi_{\lambda}P_{\lambda}(u_1,\dots,u_N), \qquad N=1,2,\dots,$$ and we agree that $$\pi_{\varnothing}=1.$$
If $P_{\lambda}=m_{\lambda}$ for all ${\lambda}$, then the meaning of is clear and we obtain $\pi_{\lambda}=\varphi_{\lambda}$. But in the general case one has to explain how to understand the sum in the right-hand side: the answer is that it converges coefficient-wise, in the $I$-adic topology of $R$.
Here is an equivalent definition. The relation is interpreted as an infinite system of linear equations, $$\label{eq2.E}
\sum_{{\lambda}:\, {\lambda}\ge\mu}{\alpha}({\lambda},\mu)\pi_{\lambda}=\varphi_\mu, \qquad \forall\mu.$$ The triangularity condition gives a sense to the infinite sum in the left-hand side of and guarantees that the infinite matrix $[{\alpha}({\lambda},\mu)]$ is invertible. Then we get $$\label{eq2.I}
\pi_{\lambda}=\sum_{\nu: \,\nu\ge{\lambda}}{\beta}(\nu,{\lambda})\varphi_\nu$$ with some new coefficients ${\beta}(\nu,{\lambda})$ such that ${\beta}({\lambda},{\lambda})=1$.
It is evident that $\{\pi_{\lambda}\}$ is a topological basis in $R$. Moreover, $\{\pi_{\lambda}\}$ is consistent with the ideals $I(n_+,n_-)$ meaning that $I(n_+,n_-)$ is (topologically) spanned by the basis elements that are contained in it, that is, by the elements $\pi_{\lambda}$, ${\lambda}\in{{\mathbb{S}}}_N$, such that ${\lambda}$ violates at least one of the inequalities $n_+\ge{\lambda}_1$, ${\lambda}_N\ge n_-$. The quotient algebra $R(n_+,n_-)$ is, on the contrary, spanned by the $\pi_{\lambda}$’s such that ${\lambda}$ satisfies the both inequalities.
Example: the basis $\{{\sigma}_{\lambda}\}$ related to the Schur rational functions
-----------------------------------------------------------------------------------
Let us turn now to concrete examples. The most important example is obtained when as $\{P_{\lambda}\}$ we take the *rational Schur functions* $s_{\lambda}$. These are symmetric Laurent polynomials given by the same ratio-of-determinants formula as the ordinary Schur polynomials, only the index ${\lambda}$ is an arbitrary signature, so that the integers ${\lambda}_i$ are not necessarily nonnegative: $$s_{\lambda}(u_1,\dots,u_N)=\frac{\det[u_i^{{\lambda}_j+N-j}]}{V(u_1,\dots,u_N)},$$ where the determinant in the numerator is of order $N$ and the denominator is the Vandermonde, $$V(u_1,\dots,u_N)=\prod_{1\le i<j\le N}(u_i-u_j).$$ The required triangularity condition holds because $s_{\lambda}$ is an irreducible character of $U(N)$. Another way to check is to use the combinatorial formula for the Schur polynomials.
Note that $$\label{eq2.J}
u_1\dots u_N s_{\lambda}(u_1,\dots,u_N)=s_{{\lambda}_1+1,\dots,{\lambda}_N+1}(u_1,\dots,u_N),$$ which makes it possible to reduce many claims concerning the rational Schur functions to the case of ordinary Schur polynomials.
For the basis $\{\pi_{\lambda}\}$ in $R$ corresponding to $P_{\lambda}=s_{\lambda}$ we use the special notation $\{{\sigma}_{\lambda}\}$. Thus, the elements ${\sigma}_{\lambda}\in R$ are defined as the coefficients of the expansion $$\label{eq2.H}
\Phi(u_1)\dots\Phi(u_N)=\sum_{{\lambda}\in{{\mathbb{S}}}_N}{\sigma}_{\lambda}s_{\lambda}(u_1,\dots,u_N).$$
Combining this with and the definition of $s_{\lambda}$, one gets a nice formula expressing ${\sigma}_{\lambda}$ through $\varphi_n$’s: $$\label{eq2.N}
{\sigma}_{\lambda}=\det[\varphi_{{\lambda}_i-i+j}]_{i,j=1}^N =\sum_{s\in
S(N)}{\operatorname{sgn}}(s)\varphi_{{\lambda}_1-1+s(1)}\dots\varphi_{{\lambda}_N-N+s(N)},$$ where $S(N)$ denotes the group of permutations of $\{1,\dots,N\}$ and ${\operatorname{sgn}}(s)=\pm1$ is the sign of a permutation $s$.
Thus, the expansion of the elements of the basis $\{{\sigma}_{\lambda}\}$ in the basis $\{\varphi_\nu\}$ has only finitely many nonzero terms. On the contrary, the expansion of the elements of the latter basis in the former basis has infinitely many terms (for $N\ge2$). For instance, $$\label{eq2.G}
{\sigma}_{{\lambda}_1,{\lambda}_2}=\varphi_{{\lambda}_1,{\lambda}_2}-\varphi_{{\lambda}_1+1,{\lambda}_2-1}$$ but $$\label{eq2.F}
\varphi_{{\lambda}_1,{\lambda}_2}=\sum_{n=0}^\infty{\sigma}_{{\lambda}_1+n,{\lambda}_2-n}.$$
Example: bases related to Macdonald polynomials
-----------------------------------------------
Observe that the Macdonald polynomials in finitely many variables (as well their degeneration, the Jack polynomials) have a natural Laurent version, because they satisfy the relation similar to , see Macdonald [@Ma95 chapter VI, (4.17)]. Moreover, they satisfy the condition , see [@Ma95 chapter VI, (4.7)]. Therefore, one may take $P_{\lambda}(u_1,\dots,u_N)=P_{\lambda}(u_1,\dots,u_N; q,t)$ (the Laurent version of Macdonald polynomials with two parameters $(q,t)$) or $P_{\lambda}(u_1,\dots,u_N)=P^{({\alpha})}(u_1,\dots,u_N)$ (the Laurent version of Jack polynomials with parameter ${\alpha}$), and then we get a certain topological basis in $R$. In particular, the case $q=t$ gives the Schur polynomials and the basis $\{{\sigma}_{\lambda}\}$, and the case $(q=0, t=1)$ give the monomial sums $m_{\lambda}$ and the basis $\{\varphi_{\lambda}\}$.
Structure constants of multiplication
-------------------------------------
Let, as above, $\{P_{\lambda}\}$ be a family of symmetric Laurent polynomials satisfying the triangularity condition and $\{\pi_{\lambda}\}$ be the corresponding topological basis in $R$. Then any homogeneous element $\psi\in
R_N$ can be uniquely represented in the form $\psi=\sum_{{\lambda}\in{{\mathbb{S}}}_N}a_{\lambda}\pi_{\lambda}$ with some complex coefficients $a_{\lambda}$. I am going to explain how to write the operation of multiplication in this notation.
Let $M$ and $N$ be two nonnegative integers and ${\lambda}\in{{\mathbb{S}}}_{M+N}$. Partitioning the variables in $P_{\lambda}$ into two groups, of cardinality $M$ and $N$, we get an expansion of the form $$\label{eq2.L}
P_{\lambda}(u_1,\dots,u_{M+N})=\sum_{\mu\in{{\mathbb{S}}}_M,
\,\nu\in{{\mathbb{S}}}_N}c({\lambda}\mid\mu,\nu)P_\mu(u_1,\dots,u_M)P_\nu(u_{M+1},\dots,u_{M+N}),$$ where $c({\lambda}\mid\mu,\nu)$ are certain coefficients. Indeed, the existence, finiteness, and uniqueness of this expansion is obvious in the case $P_{\lambda}=m_{\lambda}$, and the general case is reduced to that case using the triangularity property and the fact that for any signature ${\lambda}$, the set $\{\mu: \mu\le{\lambda}\}$ is finite.
Now it follows from that the same quantities $c({\lambda}\mid\mu,\nu)$ are the structure constants of multiplication in the basis $\{\pi_{\lambda}\}$. That is, $$\label{eq2.M}
\left(\sum a'_\mu\pi_\mu\right)\left(\sum a''_\nu\pi_\nu\right)=\sum
a_{\lambda}\pi_{\lambda}, \qquad a_{\lambda}:=\sum_{\mu,\nu}c({\lambda}\mid\mu,\nu)a'_\mu a''_\nu.$$ The latter sum makes sense because we know that the expansion is finite.
The isomorphism $R\to {\operatorname{Rep}}(\mathfrak{gl}(2\infty))$ {#sect2.B}
-------------------------------------------------------------------
The remark below is based on a comment by Igor Frenkel.
Let $\mathfrak{gl}(\infty)$ denote the Lie algebra of complex matrices of format $\infty\times\infty$ and finitely many nonzero entries. It has a natural basis formed by the matrix units $E_{ij}$ with indices $i,j$ ranging over $\{1,2,\dots\}$. The Schur-Weyl duality establishes a bijective correspondence $S_{\lambda}\leftrightarrow V_{\lambda}$ between the irreducible representations of various symmetric groups and a certain class of irreducible highest weight $\mathfrak{gl}(\infty)$-modules. Here ${\lambda}=({\lambda}_1,{\lambda}_2,\dots)$ is an arbitrary partition, $S_{\lambda}$ is the corresponding irreducible $S(N)$-module (where $N=|{\lambda}|:=\sum{\lambda}_i$), and $V_{\lambda}$ is the irreducible polynomial $\mathfrak{gl}(\infty)$-module whose highest weight is $({\lambda}_1,{\lambda}_2,\dots)$ with respect to the Borel subalgebra spanned by the $E_{ij}$ with $i\le j$. Under the Schur-Weyl correspondence, the multiplication in ${\operatorname{Rep}}(S(1),S(2),\dots)$ turns into the the tensor product of $\mathfrak{gl}(\infty)$-modules. In this sense the algebra ${\operatorname{Rep}}(S(1),S(2),\dots)={\operatorname{Sym}}$ can be identified with ${\operatorname{Rep}}(\mathfrak{gl}(\infty))$, the representation ring of polynomial $\mathfrak{gl}(\infty)$-modules.
A similar interpretation exists for the algebra $R$. Namely, we replace $\mathfrak{gl}(\infty)$ with its relative $\mathfrak{gl}(2\infty)$ — the latter Lie algebra has the basis $\{E_{ij}\}$ of matrix units with indices $i,j$ ranging over ${\mathbb Z}$. Instead of the Schur-Weyl duality we use a version of the “fermion” Howe duality [@Howe] between various unitary groups $U(N)$ and the Lie algebra $\mathfrak{gl}(2\infty)$. This duality establishes a different kind of correspondence of representations, $T_{\lambda}\leftrightarrow
V_{\lambda}$, where ${\lambda}$ ranges over the set of all signatures. Here, for ${\lambda}\in{{\mathbb{S}}}_N$, we denote by $T_{\lambda}$ the corresponding irreducible representation of $U(N)$, while $V_{\lambda}$ now stands for the irreducible $\mathfrak{gl}(2\infty)$-module with highest weight ${\widehat}{\lambda}=({\widehat}{\lambda}_i)_{i\in{\mathbb Z}}$ which is described as follows.
Recall that every signature ${\lambda}$ of length $N$ can be represented as a pair $({\lambda}^+,{\lambda}^-)$ of two partitions (=Young diagrams) such that $\ell({\lambda}^+)+\ell({\lambda}^-)\le N$, where $\ell(\,\cdot\,)$ is the conventional notation for the number of nonzero parts of a partition. Namely, $${\lambda}=({\lambda}^+_1,\dots,{\lambda}^+_{\ell({\lambda}^+)},
0,\dots,0,-{\lambda}^-_{\ell({\lambda}^-)},\dots,-{\lambda}^-_1).$$
In this notation, the weight correspondence ${\lambda}\to{\widehat}{\lambda}$ looks as follows $${\widehat}{\lambda}_i=({\lambda}^+)'_i, \quad i=1,2,\dots; \quad {\widehat}{\lambda}_{-(i-1)}=N-({\lambda}^-)'_i, \quad
i=1,2,\dots,$$ where $({\lambda}^\pm)'$ denotes the conjugate to ${\lambda}^\pm$ partition (=Young diagram).
Note that the coordinates ${\widehat}{\lambda}_i$, $i\in{\mathbb Z}$, weakly decrease; the fact that ${\widehat}{\lambda}_0\ge{\widehat}{\lambda}_1$ is equivalent to the inequality $\ell({\lambda}^+)+\ell({\lambda}^-)\le N$ mentioned above.
About this instance of Howe duality see also Olshanski [@Ols-HolExt Section 2] and [@Ols-GordonBreach Section 17].
As in the case of the Schur-Weyl duality, the multiplication in $R$ corresponds, on the Lie algebra side, to the tensor product of modules, so that we get an isomorphism $R\to{\operatorname{Rep}}(\mathfrak{gl}(2\infty))$, where ${\operatorname{Rep}}(\mathfrak{gl}(2\infty))$ is our notation for the representation ring for a special class of $\mathfrak{gl}(2\infty)$-modules. This class is generated by the weight modules that are locally nilpotent with respect to the upper triangular subalgebra and such that, for every weight ${\widehat}\mu=({\widehat}\mu_i)_{i\in{\mathbb Z}}$, the coordinates ${\widehat}\mu_i$ are nonnegative integers which stabilize to a nonnegative integer $N$ as $i\to-\infty$ and to $0$ as $i\to+\infty$. The irreducible modules $V_{\lambda}\in{\operatorname{Rep}}(\mathfrak{gl}(2\infty)$ correspond to the basis elements ${\sigma}_{\lambda}\in R$.
Comparison of $R$ with ${\operatorname{Sym}}$
---------------------------------------------
The two algebras have both similarities and differences. The homogeneous components in ${\operatorname{Sym}}$ have finite dimension while those in $R$ are not. The latter fact seems to be the most evident difference between $R$ and ${\operatorname{Sym}}$. On the other hand, both algebras are projective limits of polynomial algebras: $$\label{eq2.K}
{\operatorname{Sym}}=\varprojlim{\mathbb C}[e_1,\dots,e_n], \qquad
R=\varprojlim{\mathbb C}[\varphi_{n_-},\dots,\varphi_{n_+}].$$ These polynomial algebras can be viewed as *truncations* of the initial algebras.
All familiar homogeneous bases in ${\operatorname{Sym}}$ are parameterized by partitions, and those in $R$ are parameterized by signatures, which are relatives of partitions. However, these two kinds of labels, partitions and signatures, are related to the grading in a very different way: the degree of a basis element in ${\operatorname{Sym}}$ is given by the sum of parts of the corresponding partition, while the degree in $R$ corresponds to the length $N$ of a signature ${\lambda}$.
This is also seen from the comparison of the representation rings ${\operatorname{Rep}}(\mathfrak{gl}(\infty))$ and ${\operatorname{Rep}}(\mathfrak{gl}(2\infty))$. As abstract Lie algebras, $\mathfrak{gl}(\infty)$ and $\mathfrak{gl}(2\infty)$ are isomorphic, but the respective classes of modules are different, and the degrees of the irreducible modules are defined in a very different way.
Truncation in ${\operatorname{Sym}}$ and $R$ is also defined differently. Namely, a basis element in ${\operatorname{Sym}}$ is not contained in the kernel of the truncation map ${\operatorname{Sym}}\to{\mathbb C}[e_1,\dots,e_n]$ if and only if the length of the corresponding partition does not exceed $n$, while truncation in $R$ is controlled by the first and last coordinates of a signature ${\lambda}$. In the case when ${\lambda}_1>0$ and ${\lambda}_N<0$, one has ${\lambda}_1=\ell(({\lambda}^+)')$ and $|{\lambda}_N|=\ell(({\lambda}^-)')$.
To define a homomorphism of the algebra ${\operatorname{Sym}}$ in a commutative algebra $A$ (for instance, an algebra of functions on a space) it suffices to specialize, in an arbitrary way, the images of the generators $e_1, e_2,\dots$. In the case of $R$, the situation is more delicate. Although the elements $\varphi_n$ play the role similar to that of the $e_n$’s, to define a morphism $R\to A$ it does not suffice to specialize the image of the $\varphi_n$’s. The reason is that these elements are not generators of $R$ in the purely algebraic sense, but only *topological* generators. It may well happen that a given specialization of the $\varphi_n$’s can be extended only to a suitable subalgebra of $R$. Two examples of subalgebras are examined below.
The subalgebras ${\mathscr R}$ and ${\mathscr R}^0$
---------------------------------------------------
For ${\lambda}\in{{\mathbb{S}}}_N$, let $${\operatorname{Dim}}_N{\lambda}:=s_{\lambda}(1,\dots,1).$$ This is the dimension of the irreducible representation of $U(N)$ with highest weight ${\lambda}$. As is well known (Weyl [@Weyl], Zhelobenko [@Zhe]) $$\label{eq2.R}
{\operatorname{Dim}}_N{\lambda}=\prod_{1\le i<j\le N}\frac{{\lambda}_i-{\lambda}_j-i+j}{j-i}\,.$$
For a homogeneous element $\psi=\sum_{{\lambda}\in{{\mathbb{S}}}_N}a_{\lambda}{\sigma}_{\lambda}\in R_N$ we define its *norm* (which may be infinite) by $$\Vert\psi\Vert:=\sup_{{\lambda}\in{{\mathbb{S}}}_N}\frac{|a_{\lambda}|}{{\operatorname{Dim}}_N{\lambda}}\in{\mathbb R}_+\cup\{+\infty\}$$ and we extend this definition to non-homogeneous elements by setting $$\label{eq2.S}
\left\Vert\sum_{N=0}^M\psi_N\right\Vert:=\sum_{N=0}^M \Vert\psi_N\Vert, \qquad
\psi_N\in R_N, \quad N=0,1,\dots,M$$ with the understanding that $\Vert1\Vert=1$.
\[def2.B\] We define ${\mathscr R}\subset R$ as the subspace of elements with finite norm. Obviously, ${\mathscr R}$ is graded, so that we may write ${\mathscr R}=\sum_{N=0}^\infty{\mathscr R}_N$.
\[prop2.E\] ${\mathscr R}$ is a normed algebra.
We have to prove that for any elements $\psi', \psi''\in R$ one has $$\label{eq2.Q}
\Vert\psi'\psi''\Vert\le \Vert\psi'\Vert\Vert\psi''\Vert.$$
Indeed, assume first that $\psi'$ and $\psi''$ are homogeneous of degree $M$ and $N$, respectively, and write $\psi'=\sum a'_\mu{\sigma}_\mu$, $\psi''=\sum
a''_\nu{\sigma}_\nu$. By $$\Vert\psi'\psi''\Vert
=\sup_{{\lambda}\in{{\mathbb{S}}}_{M+N}}\frac{|\sum_{\mu,\nu}c({\lambda}\mid\mu,\nu)a'_\mu
a''_\nu|}{{\operatorname{Dim}}{\lambda}}.$$ Note that in our case, when $P_{\lambda}=s_{\lambda}$, the structure constants describe the expansion of irreducible characters restricted from $U(M+N)$ to $U(M)\times
U(N)$. It follows that these constants are nonnegative integers. Next, by counting dimensions one gets $$\sum_{\mu\in{{\mathbb{S}}}_M,\,\nu\in{{\mathbb{S}}}_N}c({\lambda}\mid\mu,\nu){\operatorname{Dim}}_M\mu{\operatorname{Dim}}_N\nu={\operatorname{Dim}}_{M+N}{\lambda}.$$ Therefore, for every ${\lambda}\in{{\mathbb{S}}}_{M+N}$, $$\frac{|\sum_{\mu,\nu}c({\lambda}\mid\mu,\nu)a'_\mu a''_\nu|}{{\operatorname{Dim}}_{M+N}{\lambda}}
\le\Vert\psi'\Vert \Vert\psi''\Vert
\frac{\sum_{\mu,\nu}c({\lambda}\mid\mu,\nu){\operatorname{Dim}}_M\mu{\operatorname{Dim}}_N\nu}{{\operatorname{Dim}}_{M+N}{\lambda}}
=\Vert\psi'\Vert \Vert\psi''\Vert.$$
This proves the desired inequality .
Now the general case, when $\psi'$ and $\psi''$ are not necessarily homogeneous, follows immediately, by taking into account the definition of the norm for non-homogeneous elements, .
\[def2.A\] For $N=1,2,\dots$ we define ${\mathscr R}^0_N\subset{\mathscr R}_N$ as the subspace of those elements $\psi=\sum_{{\lambda}\in{{\mathbb{S}}}_N}a_{\lambda}{\sigma}_{\lambda}\in R_N$ for which the ratio $|a_{\lambda}|/{\operatorname{Dim}}_N{\lambda}$ tends to 0 as ${\lambda}$ goes to infinity. In other words, for every ${\varepsilon}>0$ there should exist a finite subset of ${{\mathbb{S}}}_N$ outside of which $|a_{\lambda}|/{\operatorname{Dim}}_N{\lambda}\le{\varepsilon}$.
Next, we set $${\mathscr R}^0:=\bigoplus_{N=1}^\infty{\mathscr R}^0_N$$ and observe that ${\mathscr R}^0$ is a norm-closed subspace of ${\mathscr R}$.
Let $R^{\operatorname{fin}}$ denote the space of finite linear combinations of the basis elements ${\sigma}_{\lambda}$, where ${\lambda}\ne\varnothing$. By the very definition of ${\mathscr R}^0$, it coincides with the norm closure of $R^{\operatorname{fin}}$.
\[prop2.F\] ${\mathscr R}^0$ is closed under multiplication and so is a subalgebra in ${\mathscr R}$.
Note that, according to our definition, ${\mathscr R}^0$ does not contain the unity element $1={\sigma}_\varnothing$.
*Step* 1. For any fixed $\mu\in{{\mathbb{S}}}_M$ and $\nu\in{{\mathbb{S}}}_N$, where $M,N\ge1$, there exists a constant $C(\mu,\nu)$ such that $$\text{$c({\lambda}\mid\mu,\nu)\le C(\mu,\nu)$ for all ${\lambda}\in{{\mathbb{S}}}_{M+N}$}.$$ This is a nontrivial claim whose proof is postponed to Section \[sect10\].
*Step* 2. Let us fix $\mu$ and $\nu$ as above. We claim that $${\sigma}_\mu{\sigma}_\nu\in{\mathscr R}^0_{M+N}.$$
Indeed, by the definition of the multiplication in $R$, $${\sigma}_\mu{\sigma}_\nu=\sum_{{\lambda}\in{{\mathbb{S}}}_{M+N}}c({\lambda}\mid\mu,\nu){\sigma}_{\lambda}.$$ By the result of Step 1, the coefficients $c({\lambda}\mid\mu,\nu)$ are bounded from above. Therefore, to conclude that ${\sigma}_\mu{\sigma}_\nu\in{\mathscr R}^0_{M+N}$ it remains to show that ${\operatorname{Dim}}_N{\lambda}$ tends to infinity as ${\lambda}$ goes to infinity along the subset $$X:=\{{\lambda}\in{{\mathbb{S}}}_{M+N}: c({\lambda}\mid\mu,\nu)>0\}.$$
Observe that ${\lambda}\in X$ implies that the quantity ${\lambda}_1+\dots+{\lambda}_{M+N}$ remains fixed, because it is equal to $(\mu_1+\dots+\mu_M)+(\nu_1+\dots+\nu_N)$.
Therefore, as ${\lambda}$ goes to infinity along $X$, the difference ${\lambda}_1-{\lambda}_{M+N}$ tends to $+\infty$, so that ${\operatorname{Dim}}_N\to\infty$, as it is seen from Weyl’s dimension formula .
*Step* 3. Let us show that ${\mathscr R}^0$ is closed under multiplication. By the result of Step 2, $R^{\operatorname{fin}}R^{\operatorname{fin}}$ is contained in ${\mathscr R}^0$. Since $R^{\operatorname{fin}}\subset{\mathscr R}^0$ is dense with respect to the norm topology, we conclude that ${\mathscr R}^0{\mathscr R}^0\subset{\mathscr R}^0$.
Remarks on comultiplication
---------------------------
By Frobenius’ reciprocity, $$\operatorname{Ind}^{U(M+N)}_{U(M)\times U(N)} s_\mu\otimes
s_\nu=\sum_{{\lambda}\in{{\mathbb{S}}}_{M+N}} c({\lambda}\mid \mu,\nu)s_{\lambda},$$ where the left-hand side is the induced character. So, one could identify the formal symbols ${\sigma}_{\lambda}$ with the irreducible characters $s_{\lambda}$ and say that the multiplication $R_M\otimes R_N\to R_{M+N}$ mimics the operation of induction from $U(M)\times U(N)$ to $U(M+N)$. The reason to use the separate notation ${\sigma}_{\lambda}$ is that characters should be viewed as *functions* while elements of $R$ behave as *measures* (or, more generally, distributions), which are dual objects with respect to functions.
Of course, on a finite or compact group, one can use the normalized Haar measure $m_{\textrm{Haar}}$ to turn a function $f$ into a measure, $f
m_{\textrm{Haar}}$. However, one should not forget that functions and measures have different functorial properties, so that when we restrict a character $\chi$ to a subgroup, we regard $\chi$ as a function, while if we induct $\chi$ from a subgroup, we tacitly treat $\chi$ as a measure. In the case of finite groups, the assignment $f\mapsto f m_{\textrm{Haar}}$ is a linear isomorphism between the space of functions and the space of measures. Because of this, ${\operatorname{Rep}}(S(1),S(2),\dots)$ (the representation ring of the family of symmetric groups) possesses two dual operations, multiplication and comultiplication making it a selfdual Hopf algebra (Zelevinsky [@Zelevinsky]). For compact Lie groups $U(N)$, the situation is more delicate as the space of measures is much larger than the space of functions. This explains why the representation ring $R$, as we have defined it, is not a Hopf algebra.
Note that one can use the same structure constants $c({\lambda}\mid\mu,\nu)$ (in the basis $\{{\sigma}_{\lambda}\}$) to construct a coalgebra $R^\circ$ which is paired with $R$. Namely, a generic element of $R^\circ$ is a possibly infinite sum of homogeneous elements which in turn are finite linear combinations of symbols that we denote as $\chi_{\lambda}$; the comultiplication in $R^\circ$ is defined by setting, for ${\lambda}\in{{\mathbb{S}}}_N$, $$\triangle \chi_{\lambda}=\sum_{N_1,N_2:\, N_1+N_2=N}\,\sum_{\mu\in{{\mathbb{S}}}_{N_1},
\nu\in{{\mathbb{S}}}_{N_2}}c({\lambda}\mid \mu,\nu)\,\chi_\mu\otimes\chi_\nu.$$ Then the pairing $R\times R^\circ\to{\mathbb C}$ is defined in a natural way, by proclaming $\{{\sigma}_{\lambda}\}$ and $\{\chi_{\lambda}\}$ to be biorthogonal systems.
Likewise, one can also define a suitable coalgebra ${\mathscr R}\,^\circ$ which is paired with the algebra ${\mathscr R}$. However, in contrast to the case of the representation ring for the symmetric groups, I do not see any way to modify the definition of $R$ so that it becomes a selfdual Hopf algebra. Fortunately, for our purposes we do not need to have both operations, multiplication and comultiplication, to be defined on the same object.
Characters of $U(\infty)$ {#sect3}
=========================
Here we study a relationship between the representation ring $R$ and the dual object ${\Omega}={\widehat}{U(\infty)}$. In the symmetric group case, there is a homomorphism of the algebra ${\operatorname{Sym}}$ into the algebra of continuous functions on the dual object ${\widehat}{S(\infty)}$, and the kernel of that homomorphism is the principal ideal of ${\operatorname{Sym}}$ generated by $e_1-1$. The purpose of this section is to understand whether there exists something similar for the algebra $R$ and the dual object ${\Omega}$.
We exhibit three homomorphisms.
First, $R$ can be mapped into an algebra of functions defined on a certain subset ${\Omega}^0\subset{\Omega}$ (${\Omega}^0$ is composed from some finite-dimensional “faces” of ${\Omega}$). This map is far from being the desired analog but it is useful for some technical purposes.
Second, the subalgebra ${\mathscr R}$ can be mapped into $C({\Omega})$, the Banach algebra of bounded continuous functions on ${\Omega}$.
Third, the above map sends the subalgebra ${\mathscr R}^0\subset{\mathscr R}$ into the subalgebra $C_0({\Omega})\subset C({\Omega})$ formed by continuous functions vanishing at infinity. The space $C_0({\Omega})$ is of special interest for us because our main objects of study, the generators of Markov processes on ${\Omega}$, are operators on the Banach space $C_0({\Omega})$.
Description of extreme characters: the Edrei-Voiculescu theorem {#sect3.A}
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For every $N=1,2,\dots$, we identify $U(N)$ with the subgroup of the group $U(N+1)$ fixing the last basis vector in ${\mathbb C}^{N+1}$. This makes it possible to define the inductive limit group $U(\infty)=\varinjlim U(N)$. In other words, elements of $U(\infty)$ are infinite unitary matrices $[U_{ij}]_{i,j=1}^\infty$ such that $U_{ij}={\delta}_{ij}$ when $i$ or $j$ is large enough.
We endow $U(\infty)$ with the inductive limit topology, which plainly means that a function $f: U(\infty)\to{\mathbb C}$ is continuous if and only if for every $N$, the function $f_N:=f\big|_{U(N)}$ is continuous on $U(N)$.
Notice that $f$ is a class function (respectively, a positive definite function) if and only if so is $f_N$ for every $N$.
\[def3.A\] (i) By a *character* of $U(\infty)$ we mean a continuous class function $f:U(\infty)\to{\mathbb C}$ which is positive definite and normalized by $f(e)=1$.
\(ii) Note that the set of all characters in the sense of (i) is a convex set. Its extreme points are called *extreme* or *indecomposable* characters.
The extreme characters of $U(\infty)$ are analogs of the *normalized* irreducible characters $$\label{eq3.B}
\frac{s_{\lambda}(u_1,\dots,u_N)}{{\operatorname{Dim}}_N{\lambda}}, \qquad {\lambda}\in{{\mathbb{S}}}_N.$$
To describe the extreme characters we need to introduce some notation.
Let ${\mathbb R}_+\subset{\mathbb R}$ denote the set of nonnegative real numbers, ${\mathbb R}_+^\infty$ denote the product of countably many copies of ${\mathbb R}_+$, and set $${\mathbb R}_+^{4\infty+2}={\mathbb R}_+^\infty\times{\mathbb R}_+^\infty\times{\mathbb R}_+^\infty\times{\mathbb R}_+^\infty
\times{\mathbb R}_+\times{\mathbb R}_+.$$ Let ${\Omega}\subset{\mathbb R}_+^{4\infty+2}$ be the subset of sextuples $${\omega}=({\alpha}^+,{\beta}^+;{\alpha}^-,{\beta}^-;{\delta}^+,{\delta}^-)$$ such that $$\begin{gathered}
{\alpha}^\pm=({\alpha}_1^\pm\ge{\alpha}_2^\pm\ge\dots\ge 0)\in{\mathbb R}_+^\infty,\quad
{\beta}^\pm=({\beta}_1^\pm\ge{\beta}_2^\pm\ge\dots\ge 0)\in{\mathbb R}_+^\infty,\\
\sum_{i=1}^\infty({\alpha}_i^\pm+{\beta}_i^\pm)\le{\delta}^\pm, \quad {\beta}_1^++{\beta}_1^-\le 1.\end{gathered}$$ We observe that ${\Omega}$ is a locally compact space in the topology inherited from the product topology of ${\mathbb R}_+^{4\infty+2}$.
Instead of ${\delta}^\pm$ it is often convenient to use the quantities $${\gamma}^\pm:={\delta}^\pm-\sum_{i=1}^\infty({\alpha}_i^\pm+{\beta}_i^\pm).$$ Obviously, ${\gamma}^+$ and ${\gamma}^-$ are nonnegative. But, in contrast to ${\delta}^+$ and ${\delta}^-$, they are *not* continuous functions of ${\omega}\in{\Omega}$.
For $u\in{\mathbb C}^*$ and ${\omega}\in{\Omega}$ set $$\label{eqF.11}
\Phi(u;{\omega})= e^{{\gamma}^+(u-1)+{\gamma}^-(u^{-1}-1)}
\prod_{i=1}^\infty\frac{1+{\beta}_i^+(u-1)}{1-{\alpha}_i^+(u-1)}
\,\frac{1+{\beta}_i^-(u^{-1}-1)}{1-{\alpha}_i^-(u^{-1}-1)}.$$
For any fixed ${\omega}$, this is a meromorphic function in variable $u\in{\mathbb C}^*$ with possible poles on $(0,1)\cup(1,+\infty)$. The poles do not accumulate to $1$, so that the function is holomorphic in a neighborhood of the unit circle ${\mathbb T}:=\{u\in{\mathbb C}: |u|=1\}$.
Note that every conjugacy class of $U(\infty)$ contains a diagonal matrix with diagonal entries $u_1,u_2, \ldots\in{\mathbb T}$, where only finitely many of $u_n$’s are distinct from 1. These numbers are defined uniquely, within a permutation. Thus every class function on $U(\infty)$ can be interpreted as a symmetric function $\Psi(u_1,u_2,\dots)$.
\[thm3.A\] The extreme characters of the group $U(\infty)$ are precisely the functions of the form $$\label{eq3.A}
\Psi_{\omega}(u_1,u_2,\dots):=\prod_{k=1}^\infty\Phi(u_k;{\omega}),$$ where ${\omega}$ ranges over ${\Omega}$.
Note that the product actually terminates because $\Phi(1;{\omega})=1$ and $u_k=1$ for $k$ large enough. As compared with the normalized irreducible characters of the groups $U(N)$ given by , the extreme characters of $U(\infty)$ seem to be both more elementary and more sophisticated objects. They are more elementary because they are given by a product formula, but they are also more sophisticated as they depend on countably many continuous parameters.
About various proofs and different facets of this fundamental theorem see Edrei [@Edrei], Voiculescu [@Vo76], Boyer [@Boyer], Vershik-Kerov [@VK82], Okounkov-Olshanski [@OO-Jack], Borodin-Olshanski [@BO-GT-Appr], Petrov [@Petrov-MMJ].
\[prop3.C\] Given ${\omega}\in{\Omega}$, write the Laurent expansion of the function $u\mapsto
\Phi(u;{\omega})$ as $$\Phi(u;{\omega})=\sum_{n\in{\mathbb Z}}{\widehat}\varphi_n({\omega}) u^n.$$
For $n\in{\mathbb Z}$ fixed, the coefficient ${\widehat}\varphi_n({\omega})$ is a continuous function on ${\Omega}$ vanishing at infinity.
See Borodin-Olshanski [@BO-GT-Appr Proposition 2.10].
Recall that we denoted by $\Phi(u)$ the formal generating series assembling the variables $\varphi_n$, see above. The fact that we employ now a similar notation is not occasional. As explained below, the functions ${\widehat}\varphi_n({\omega})$ serve as the image of the generators $\varphi_n\in R$ under the maps mentioned in the preamble to the section.
The quotient algebra ${\widehat}R=R/J$
--------------------------------------
Observe that $\Phi(1;{\omega})\equiv1$, which implies $$\label{eq3.H}
\sum_{n\in{\mathbb Z}}{\widehat}\varphi_n({\omega})=1, \qquad {\omega}\in{\Omega}.$$ This relation motivates the following definitions.
Let us set $$\varphi:=\sum_{n\in{\mathbb Z}}\varphi_n.$$ and let $J:=(\varphi-1)\subset R$ be the principal ideal generated by the element $\varphi-1$. The ideal $J$ and the quotient algebra ${\widehat}R:=R/J$ play an important role in our theory, similar to that of the ideal $(e_1-1)\subset{\operatorname{Sym}}$ and the quotient ring ${\operatorname{Sym}}/(e_1-1)$ in Vershik-Kerov’s theory [@VK-RingTheorem], [@VK-GordonBreach].
The quotient ring ${\widehat}R$ is a filtered algebra: its filtration is inherited from the filtration in $R$, which in turn is determined from the grading; the latter is not inherited because the ideal $J$ is not homogeneous.
We will prove a few simple propositions concerning the algebra ${\widehat}R$.
\[prop3.D\] For every $N=0,1,2,\dots$, the intersection $J\cap R_N$ is trivial.
This is a formal consequence of the fact that $R$ has no zero divisors (which in turn follows from the isomorphism $R=\varprojlim R(n_+,n_-)$).
Indeed, assume $\psi\in J\cap R_N$ and show that $\psi=0$. There exists $\psi'\in R$ such that $\psi=(\varphi-1)\psi'$. Since $R$ has no zero divisors, the degree of $\psi'$ cannot be larger than $N-1$, so one can write $$\psi'=\psi_0+\dots+\psi_{N-1}, \qquad \psi_i\in R_i.$$ Then $$\psi=\sum_{i=0}^{N-1}(\varphi-1)\psi_i
=-\psi_0+(\varphi\psi_0-\psi_1)+\dots+(\varphi\psi_{N-2}-\psi_{N-1})+\varphi\psi_{N-1}.$$ Since $\psi$ is homogeneous of degree $N$, we have $\psi=\varphi\psi_{N-1}$ and $$-\psi_0=(\varphi\psi_0-\psi_1)=\dots=(\varphi\psi_{N-2}-\psi_{N-1})=0.$$ This implies $\psi_0=\dots=\psi_{N-1}=0$ and finally $\psi=0$.
Let, as above, $n_+\ge n_-$ be a couple of integers. We denote by $J(n_+,n_-)$ the ideal in $R$ generated by the ideals $J$ and $I(n_+,n_-)$. Under the homomorphism $R\to R(n_+,n_-)$, the image of $J$ is the principal ideal generated by the element $(\varphi_{n_-}+\dots+\varphi_{n_+})-1$. We set $$\label{eq3.C}
{\widehat}R(n_+,n_-):=R/J(n_+,n_-).$$ This algebra can be identified with the quotient $${\mathbb C}[\varphi_{n_-},\dots,\varphi_{n_+}]\big/(\varphi_{n_-}+\dots+\varphi_{n_+}-1)$$ and so is isomorphic to the algebra of polynomials with $n_+-n_-$ variables.
\[prop3.E\] As $n_\pm\to\pm\infty$, the intersection of the kernels of the composite homomorphisms $$R\to R(n_+,n_-)\to{\widehat}R(n_+,n_-)$$ coincides with $J$.
This is a trivial consequence of the absence of zero divisors. Indeed, the ideal $J$ lies in the intersection of the kernels in question. Conversely, assume $\psi\in R$ belongs to the intersections of the kernels and show that $\psi\in J$, that is, there exists $\psi'\in R$ such that $\psi=(\varphi-1)\psi'$.
By the assumption, for every couple $(n_+,n_-)$ there exists an element $\psi'_{n_+,n_-}\in R(n_+,n_-)$ such that the image of $\psi$ in $R(n_+,n_-)$ is equal to $$(\varphi_{n_-}+\dots+\varphi_{n_+}-1)\psi'_{n_+,n_-}.$$ Note that this element is unique and its degree is bounded from above by $\deg(\psi)-1$.
It follows that there exists an element $\psi'=\varprojlim \psi'_{n_+,n_-}$. The elements $\psi$ and $(\varphi-1)\psi'$ have the same image under the map $R\to R(n_+,n_-)$, for every $(n_+,n_-)$. Therefore, these elements are equal to each other.
\[cor3.A\] The algebra ${\widehat}R$ can be identified with the projective limit of filtered algebras ${\widehat}R(n_+,n_-)$ as $n_\pm\to\pm\infty$.
Since $R=\varprojlim R(n_+,n_-)$, there is a natural homomorphism ${\widehat}R\to
\varprojlim {\widehat}R(n_+,n_-)$. Proposition \[prop3.E\] shows that it is injective. Let us check that it is also surjective. Without loss of generality one can assume that $n_+>0>n_-$. Then we use the relation $\varphi_{n_-}+\dots+\varphi_{n_+}=1$ in ${\widehat}R(n_+,n_-)$ to eliminate $\varphi_0$ and to lift ${\widehat}R(n_+,n_-)$ into $R(n_+,n_-)$ as the subalgebra $R'(n_+,n_-)$ generated by $\varphi_{n_-}, \dots, \varphi_{-1},
\varphi_1,\dots,\varphi_{n_+}$. This makes it possible to identify $\varprojlim
{\widehat}R(n_+,n_-)$ with $\varprojlim R'(n_+,n_-)$, where both limits are taken in the category of filtered algebras. Then the surjectivity in question becomes obvious.
We say that two signatures $\mu\in{{\mathbb{S}}}_N$ and ${\lambda}\in{{\mathbb{S}}}_{N+1}$ *interlace* if $$\label{eq3.AA}
{\lambda}_i\ge\mu_i\ge{\lambda}_{i+1}, \qquad i=1,\dots,N,$$ and then we write $\mu\prec{\lambda}$ or, equivalently, ${\lambda}\succ\mu$. By agreement, any signature ${\lambda}\in{{\mathbb{S}}}_1$ is interlaced with the empty signature $\varnothing\in{{\mathbb{S}}}_0$.
\[prop3.AA\] For any $\mu\in{{\mathbb{S}}}_N$, where $N=0,1,2,\dots$, one has $$\label{eq3.AB}
\varphi{\sigma}_\mu=\sum_{{\lambda}\in{{\mathbb{S}}}_{N+1}:\,{\lambda}\succ\mu}{\sigma}_{\lambda}.$$
The classical Gelfand–Tsetlin branching rule says that for ${\lambda}\in{{\mathbb{S}}}_{N+1}$, $$s_{\lambda}(u_1,\dots,u_{N+1})=\sum_{\mu\in{{\mathbb{S}}}_N:\, \mu\prec{\lambda}}s_\mu(u_1,\dots,u_N) u_{N+1}^{|{\lambda}|-|\mu|},$$ where $|{\lambda}|:=\sum{\lambda}_i$, $|\mu|:=\sum\mu_j$. This gives us the structure constants $c({\lambda}\mid\mu,\nu)$ (see Section \[sect2\]) for the basis of rational Schur functions in the special case when ${\lambda}\in{{\mathbb{S}}}_{N+1}$, $\mu\in{{\mathbb{S}}}_N$, and $\nu=n\in{{\mathbb{S}}}_1={\mathbb Z}$. Namely, in this special case, $$c({\lambda}\mid\mu,\nu)=\begin{cases} 1, & \text{if $\mu\prec{\lambda}$ and $n=|{\lambda}|-|\mu|$}, \\
0, & \text{otherwise.}
\end{cases}$$ Combining this with the definition of the multiplication in $R$ we get .
The proposition shows that the ideal $J$ coincides with the closed linear span of the elements of the form $$-{\sigma}_\mu+\sum_{{\lambda}:\,{\lambda}\succ\mu}{\sigma}_{\lambda},$$ where $\mu$ ranges over the set ${{\mathbb{S}}}_0\cup{{\mathbb{S}}}_1\cup\dots$ of all signatures. This fact is used below in the proof of Proposition \[prop7.C\].
The simplices ${\Omega}(n_+,n_-)$ {#sect3.B}
---------------------------------
Let $(n_+, n_-)$ be a couple of integers such that $n_+\ge0\ge n_-$. We set $$\begin{gathered}
{\Omega}(n_+,n_-):=\big\{{\omega}=({\alpha}^\pm,{\beta}^\pm,{\delta}^\pm)\; : \;{\alpha}^\pm_i=0 \quad
\textrm{for all $i$}, \quad {\beta}^+_i=0 \quad \textrm{for $i>n_+$},\\ {\beta}^-_j=0
\quad \textrm{for $j>|n_-|$}, \quad {\delta}^+={\beta}^+_1+\dots+{\beta}^+_{n_+}, \quad
{\delta}^-={\beta}^-_1+\dots+{\beta}^-_{|n_-|} \big\} \subset {\Omega}\end{gathered}$$ This a compact subset of ${\Omega}$ whose elements depend only on $n_++|n_-|$ independent parameters ${\beta}^+_1,\dots,{\beta}^+_{n_+}, {\beta}^-_1\dots,{\beta}^-_{|n_-|}$. The conditions on these parameters can be written in the form $$1-{\beta}^+_{|n^+|}\ge\dots\ge1-{\beta}^+_1\ge{\beta}^-_1\ge\dots\ge{\beta}^-_{n^-}\ge0$$ (because ${\beta}^+_1+{\beta}^-_1\le1$). This means that ${\Omega}(n_+,n_-)$ can be viewed as a simplex of dimension $n_++|n_-|$.
If ${\omega}\in{\Omega}(n_+,n_-)$, then the function $\Phi(u;{\omega})$ drastically simplifies and takes the form $$\Phi(u;{\omega})=\prod_{i=1}^{n_+}(1-{\beta}^+_i+{\beta}^+_i u)\cdot
\prod_{j=1}^{|n_-|}(1-{\beta}^-_j+{\beta}^-_j u^{-1}).$$ The function $\varphi_n({\omega})$ vanishes identically on ${\Omega}(n_+,n_-)$ unless $n_+\ge n\ge n_-$.
Let $C({\Omega}(n_+,n_-))$ denote the algebra of continuous functions on the simplex ${\Omega}(n_+,n_-)$. By Proposition \[prop3.C\], every function ${\widehat}\varphi_n({\omega})$ is continuous on ${\Omega}(n_+,n_-)$.
Recall that $J(n_+,n_-)$ denotes the principal ideal in $R(n_+,n_-)$ generated by the element $(\varphi_{n_-}+\dots+\varphi_{n_+})-1$.
\[prop3.B\] The kernel of the homomorphism $$R(n_+,n_-)={\mathbb C}[\varphi_{n_-},\dots,\varphi_{n_+}]\to C({\Omega}(n_+,n_-))$$ assigning to $\varphi_n$ the function ${\widehat}\varphi_n({\omega})$ on ${\Omega}(n_+,n_-)$ coincides with the ideal $J(n_+,n_-)$.
Since ${\widehat}\varphi_n({\omega})$ vanishes on ${\Omega}(n_+,n_-)$ unless $n_+\ge n\ge n_-$, the equality shows that $$\sum_{n=n_-}^{n_+}{\widehat}\varphi_n\big|_{C({\Omega}(n_+,n_-))}=1.$$ It remains to prove that this is the only relation.
Let us examine the special case when $n_-=0$. To simplify the notation, set $n_+=m$ and $$(t_1,\dots,t_m):=(1-{\beta}^+_m,\dots,1-{\beta}^+_1)$$ Let us write ${\widehat}\varphi_n(t_1,\dots,t_m)$ instead of ${\widehat}\varphi_n({\omega})$, where $n=0,\dots,m$. These are symmetric polynomials in $t_1,\dots,t_m$ satisfying $$\prod_{i=1}^m(t_i+(1-t_i)u)=\sum_{n=0}^m{\widehat}\varphi_n(t_1,\dots,t_m)u^n.$$ For instance, for $m=2$, $${\widehat}\varphi_0(t_1,t_2)=t_1t_2, \quad {\widehat}\varphi_1(t_1,t_2)=(t_1+t_2)-2t_1t_2,
\quad {\widehat}\varphi_2(t_1,t_2)=(1-t_1)(1-t_2).$$
In the case under consideration, the claim of the proposition is equivalent to saying that the only algebraic relation between these $m+1$ polynomials is that their sum equals $1$. Let us prove the last assertion.
Evidently, our polynomials lie in the linear span of the elementary symmetric polynomials $e_n(t_1,\dots,t_m)$, where $n=0,\dots,m$ and $e_0:=1$. Therefore, it suffices to check that our polynomials are linearly independent.
To do this, we evaluate them in the following $m+1$ points of ${\mathbb R}^m$: $$x_k:=(\,\underbrace{1,\dots,1}_{m-k}, \underbrace{0,\dots,0}_{k}\,), \qquad
k=0,\dots,m.$$ At $x_k$, the product $\prod(t_i+(1-t_i)u)$ equals $u^k$. This implies that ${\widehat}\varphi_n(x_k)={\delta}_{nk}$, which concludes the proof in our special case.
Finally, the case $n_-<0$ is readily reduced to the special case $n_-=0$ by using the twisting transformation $\tau$ defined in the next subsection.
Proposition \[prop3.B\] shows that the quotient ring ${\widehat}R(n_+,n_-)=
R(n_+,n_-)/J(n_+,n_-)$ is embedded into the algebra $C({\Omega}(n_+,n_-))$ of continuous functions on the simplex ${\Omega}(n_+,n_-)$ as the subalgebra of polynomial functions.
Together with Proposition \[prop3.E\] this makes it possible to realize the quotient ring ${\widehat}R= R/J$ as an algebra of functions on the subset $$\label{eq3.I}
{\Omega}^0:=\bigcup_{n_+\ge n_-}{\Omega}(n_+,n_-)\subset{\Omega}.$$
Symmetries
----------
There exist natural transformations of characters of $U(\infty)$, which preserve the subset of extreme characters and thus induce transformations (or *symmetries*) ${\Omega}\to{\Omega}$ of the parameter space.
One such transformation is the operation of *conjugation* mapping a character $f(U)$ to the conjugate character $\overline{f(U)}$ (here $U$ ranges over $U(\infty)$). Conjugation induces the symmetry ${\omega}\mapsto{\omega}^*$ of ${\Omega}$ consisting in switching $({\alpha}^+,{\beta}^+,{\delta}^+)\leftrightarrow({\alpha}^-,{\beta}^-,{\delta}^-)$.
Another kind of transformation is the multiplication of $f(U)$ by $\det(U)$. In terms of the eigenvalues this amounts to multiplication by the product $u_1u_2\dots$. The corresponding symmetry of ${\Omega}$ leaves the parameters ${\alpha}^\pm$ intact and changes the remaining parameters in the following way: $$\begin{gathered}
({\beta}^+_1,{\beta}^+_2,\dots)\mapsto (1-{\beta}^-_1,{\beta}^+_1,{\beta}^+_2,\dots)\\
({\beta}^-_1,{\beta}^-_2,\dots)\mapsto ({\beta}^-_2,{\beta}^-_3,\dots)\\
{\delta}^+\mapsto{\delta}^++(1-{\beta}^-_1)\\
{\delta}^-\mapsto{\delta}^--{\beta}^-_1.\end{gathered}$$ Note that $1-{\beta}^-_1\ge {\beta}^+_1$ because of the condition ${\beta}^+_1+{\beta}^-_1\le1$.
We call this the *twisting* symmetry of ${\Omega}$ and denote it as ${\omega}\mapsto
\tau({\omega})$. Obviously, $\tau$ is invertible.
Under the symmetry ${\omega}\mapsto{\omega}^*$, the subset ${\Omega}(n_+,n_-)$ is mapped onto ${\Omega}(-n_-,-n_+)$. If $n_-\le-1$, then the twisting symmetry $\tau$ maps ${\Omega}(n_+,n_-)$ onto ${\Omega}(n_++1,n_-+1)$.
Recall that so far we assumed $n_+\ge0\ge n_-$. However, one can extend the definition of ${\Omega}(n_+,n_-)$ so that the equality $\tau({\Omega}(n_+,n_-))={\Omega}(n_++1,n_-+1)$ will be valid for every couple $n_+\ge
n_-$, dropping the assumption that $n_+\ge0$ and $n_-\le0$. For instance, if $n_-\ge1$, then the first $n_-$ coordinates in ${\beta}^+$ are equal to 1 and the actual parameters are ${\beta}^+_{n_-+1},\dots,{\beta}^+_{n_+}$.
The homomorphisms ${\mathscr R}\to C({\Omega})$ and ${\mathscr R}^0\to C_0({\Omega})$
-------------------------------------------------------------------------------------
Recall that the functions ${\widehat}\varphi_n({\omega})$ introduced in Proposition \[prop3.C\] belong to the Banach space $C_0({\Omega})$. At this moment we only exploit the fact that they belong to $C({\Omega})$. Let us assign to every generator $\varphi_n\in R$ the function ${\widehat}\varphi_n({\omega})$. We are going to extend this correspondence to a norm continuous homomorphism ${\mathscr R}\to C({\Omega})$.
Let us start by assigning to every basis element ${\sigma}_{\lambda}$ a suitable function ${\widehat}{\sigma}({\omega})$. This can be done in two equivalent ways.
*First way*. We use the determinantal formula and set for ${\lambda}\in{{\mathbb{S}}}_N$ and ${\omega}\in{\Omega}$ $$\label{eq3.E}
{\widehat}{\sigma}_{\lambda}({\omega}):=\det[{\widehat}\varphi_{{\lambda}_i-i+j}({\omega})].$$
*Second way*. Restricting the extreme character $\Psi_{\omega}$ defined in to the subgroup $U(N)\subset U(\infty)$ gives us a normalized positive definite class function on $U(N)$, which can be expanded into an absolutely and uniformly convergent series on the irreducible characters of $U(N)$. Then the desired quantities ${\widehat}{\sigma}_{\lambda}({\omega})$ arise as the coefficients of this expansion. Passing to matrix eigenvalues one can write this in the form $$\label{eq3.J}
\Phi(u_1;{\omega})\dots\Phi(u_N;{\omega})=\sum_{{\lambda}\in{{\mathbb{S}}}_N}{\widehat}{\sigma}_{\lambda}({\omega})
s_{\lambda}(u_1,\dots,u_N).$$
From it follows that the functions ${\widehat}{\sigma}_{\lambda}({\omega})$ belong to $C({\Omega})$ (even to $C_0({\Omega})$), and from we see that ${\widehat}{\sigma}_{\lambda}({\omega})\ge0$ (because the function in the left-hand side is positive definite). This is an important observation which will be exploited below.
Here is one more useful consequence of : setting $u_1=\dots=u_N=1$ we get the identity $$\label{eq2.O}
\sum_{{\lambda}\in{{\mathbb{S}}}_N}{\operatorname{Dim}}_N{\lambda}\,{\widehat}{\sigma}_{\lambda}({\omega})=1.$$
Next, given an element $\psi=\sum a_{\lambda}{\sigma}_{\lambda}\in{\mathscr R}$, we want to assign to it the function ${\widehat}\psi({\omega})=\sum a_{\lambda}{\widehat}{\sigma}_{\lambda}({\omega})$ on ${\Omega}$.
\[prop3.A\] [(i)]{} For every element $\psi=\sum a_{\lambda}{\sigma}_{\lambda}\in{\mathscr R}$, the series ${\widehat}\psi({\omega}):=\sum a_{\lambda}{\widehat}{\sigma}_{\lambda}({\omega})$ converges absolutely at every point ${\omega}\in{\Omega}$. Moreover, the resulting function on ${\Omega}$ is bounded and its supremum norm does not exceed $\Vert\psi\Vert$.
[(ii)]{} The map $\psi\mapsto{\widehat}\psi(\,\cdot\,)$ is an algebra homomorphism ${\mathscr R}\to C({\Omega})$.
[(iii)]{} The kernel of this homomorphism is the principal ideal ${\mathscr J}\subset
{\mathscr R}$ generated by the element $\varphi-1$. This ideal coincides with $J\cap{\mathscr R}$.
*Step* 1. Let us check (i). We will assume first that $\psi$ is homogeneous of degree $N$. Then we have (recall that ${\widehat}{\sigma}_{\lambda}({\omega})\ge0$) $$\label{eq3.D}
\sum_{\lambda}|a_{\lambda}|{\widehat}{\sigma}_{\lambda}({\omega})=\sum_{\lambda}\frac{|a_{\lambda}|}{{\operatorname{Dim}}_N{\lambda}}{\operatorname{Dim}}_N{\lambda}\,{\widehat}{\sigma}_{\lambda}({\omega})
\le\Vert\psi\Vert \sum_{\lambda}{\operatorname{Dim}}_N{\lambda}\,{\widehat}{\sigma}_{\lambda}({\omega})=\Vert\psi\Vert,$$ where the final equality follows from .
The same holds for arbitrary (not necessarily homogeneous) elements, by the very definition of the norm in ${\mathscr R}$.
*Step* 2. Let us check that the map $\psi\mapsto{\widehat}\psi(\,\cdot\,)$ is consistent with multiplication. That is, for any two elements $\psi',\psi''\in{\mathscr R}$ and any ${\omega}\in{\Omega}$ one has $${\widehat}{\psi'}({\omega}){\widehat}{\psi''}({\omega})={\widehat}\psi({\omega}), \qquad \psi:=\psi'\psi''.$$
Indeed, without loss of generality we may assume that $\psi'$ and $\psi''$ are homogeneous, of degree $M$ and $N$, respectively. Write $$\psi'=\sum_{\mu\in{{\mathbb{S}}}_M}a'_\mu{\sigma}_\mu, \qquad
\psi''=\sum_{\nu\in{{\mathbb{S}}}_N}a''_\nu{\sigma}_\nu, \qquad
\psi=\sum_{{\lambda}\in{{\mathbb{S}}}_{M+N}}a_{\lambda}{\sigma}_{\lambda}.$$ By virtue of , we have $$a_{\lambda}=\sum_{\mu,\nu}c({\lambda}\mid\mu,\nu)a'_\mu a''_\nu,$$ where the structure constants correspond to the choice $P_{\lambda}=s_{\lambda}$.
It readily follows that the desired statement is reduced to the following identity: for any fixed $\mu\in{{\mathbb{S}}}_M$ and $\nu\in{{\mathbb{S}}}_N$ one has $$\label{eq2.P}
{\widehat}{\sigma}_\mu({\omega}){\widehat}{\sigma}_\nu({\omega})=\sum_{{\lambda}\in{{\mathbb{S}}}_{M+N}}c({\lambda}\mid\mu,\nu){\widehat}{\sigma}_{\lambda}({\omega}),
\qquad {\omega}\in{\Omega}.$$
This identity, in turn, follows from the second definition of the quantities ${\widehat}{\sigma}_{\lambda}({\omega})$ (formula above) and the identity $$s_{\lambda}(u_1,\dots,u_{M+N})=
\sum_{\mu\in{{\mathbb{S}}}_M,\,\nu\in{{\mathbb{S}}}_N}c({\lambda}\mid\mu,\nu)s_\mu(u_1,\dots,u_M)
s_\nu(u_{M+1},\dots,u_{M+N}).$$ Necessary interchanges of the order of summation are justified because all the series are absolutely convergent.
*Step 3*. Let us show that the functions ${\widehat}\psi({\omega})$ are continuous on ${\Omega}$. We may assume that $\psi$ is homogeneous of degree $N$. Then the corresponding function ${\widehat}\psi({\omega})$ is given by the series $\sum_{{\lambda}\in{{\mathbb{S}}}_N}a_{\lambda}{\widehat}{\sigma}_{\lambda}({\omega})$. We know that the functions ${\widehat}{\sigma}_{\lambda}({\omega})$ are continuous, but one cannot immediately conclude that ${\widehat}\psi$ is also continuous because the series is not necessarily convergent in the norm topology of $C({\Omega})$. This difficulty is resolved in the following way. Since the space ${\Omega}$ is locally compact, it suffices to prove that the series for ${\widehat}\psi$ converges uniformly on compact subsets of ${\Omega}$. Looking at one sees that it suffices to do this for the series $\sum_{\lambda}{\operatorname{Dim}}_N{\lambda}\,{\widehat}{\sigma}_{\lambda}({\omega})$. By , it converges to the constant function $1$ at every point ${\omega}\in{\Omega}$. Since all the summands are nonnegative, the convergence is uniform on compact sets, as desired.
Thus, we completed the proof of (ii).
*Step* 4. Obviously, the element $\varphi$ belongs to ${\mathscr R}$, so that the principal ideal ${\mathscr J}\subset{\mathscr R}$ generated by $\varphi-1$ is well defined. Let us show that ${\mathscr J}=J\cap{\mathscr R}$. To do this we have to check that if $\psi\in R$ is such that $(\varphi-1)\psi\in{\mathscr R}$, then $\psi\in{\mathscr R}$. This is proved by the same argument as in the proof of Proposition \[prop3.D\].
*Step* 5. Finally, let us check that ${\mathscr J}$ coincides with the kernel of the homomorphism $\psi\mapsto{\widehat}\psi(\,\cdot\,)$. We know that the function ${\widehat}\varphi({\omega})$ is the constant function $1$, so ${\mathscr J}$ is contained in the kernel.
It remains to show that if, conversely, $\psi\in{\mathscr R}$ is such that ${\widehat}\psi({\omega})\equiv0$ on ${\Omega}$, then $\psi\in{\mathscr J}$. Here we apply the result stated at the very end of Subsection \[sect3.B\]. It suffices to use the fact that the function ${\widehat}\psi({\omega})$ vanishes on ${\Omega}^0$. Then that result says that $\psi\in J$. Because $J\cap{\mathscr R}={\mathscr J}$, we conclude that $\psi\in{\mathscr J}$.
\[cor3.B\] The homomorphism of Proposition \[prop3.A\] determines by restriction a homomorphism ${\mathscr R}^0\to C_0({\Omega})$.
By the definition of the subalgebra ${\mathscr R}^0\subset{\mathscr R}$, the linear span of the basis elements ${\sigma}_{\lambda}$ is dense in ${\mathscr R}^0$ with respect to the norm topology. On the other hand, as it was pointed above, the functions ${\widehat}{\sigma}_{\lambda}({\omega})$ belong to $C_0({\Omega})$. Since $C_0({\Omega})$ is closed in $C({\Omega})$ and the homomorphism ${\mathscr R}\to C({\Omega})$ is norm continuous, this shows that the image of the whole subalgebra ${\mathscr R}^0$ is contained in $C_0({\Omega})$.
Analog of the Vershik-Kerov ring theorem
----------------------------------------
Let ${\mathscr R}_+\subset{\mathscr R}$ denote the closed (in the norm topology) convex cone spanned by the elements ${\sigma}_{\lambda}$. For two elements $\psi_1, \psi_2\in{\mathscr R}_+$, write $\psi_1\le\psi_2$ if $\psi_2-\psi_1\in{\mathscr R}_+$.
The following result is similar to the so-called *ring theorem* due to Vershik and Kerov, see [@VK-RingTheorem Theorem 6] and [@Kerov-book Introduction, Theorem 4].
[(i)]{} The set of characters of $U(\infty)$ in the sense of Definition \[def3.A\] is in a natural one-to-one correspondence with linear functionals $F:{\mathscr R}\to{\mathbb C}$ satisfying the following properties[:]{}
- $F$ is norm-continuous and takes real nonnegative values on the cone ${\mathscr R}_+$.
- If $\psi\in{\mathscr R}_+$ is the least upper bound for a sequence $0\le\psi_1\le\psi_2\le\dots$, then $F(\psi)=\lim_{n\to\infty} F(\psi_n)$.
- $F(1)=1$ and $F(\varphi\psi)=F(\psi)$ for every $\psi\in{\mathbb R}$.
[(ii)]{} A character is extreme if and only if the corresponding functional $F$ is multiplicative, that is, $F(\psi_1\psi_2)=F(\psi_1)F(\psi_2)$ for any $\psi_1,\psi_2\in{\mathscr R}$.
The proof is similar to that given in [@VK-RingTheorem] (see also a more detailed version in Gnedin-Olshanski [@GO-zigzag Section 8.7]).
This result does not depend on the classification of the extreme characters and provides one more proof of their multiplicativity.
The operator $\mathbb D_{{z,z',w,w'}}$ {#sect4}
======================================
\[def4.A\] Fix an arbitrary quadruple $(z,z',w,w')$ of complex parameters and introduce the following formal differential operator in countably many variables $\{\varphi_n:n\in{\mathbb Z}\}$ $${\mathbb D}_{{z,z',w,w'}}=\sum_{n\in{\mathbb Z}}A_{nn}\frac{{\partial}^2}{{\partial}\varphi_n^2}+2\sum_{\substack{n_1,n_2\in{\mathbb Z}\\
n_1>n_2}} A_{n_1 n_2}\frac{{\partial}^2}{{\partial}\varphi_{n_1}{\partial}\varphi_{n_2}}
+\sum_{n\in{\mathbb Z}}B_n\frac{{\partial}}{{\partial}\varphi_n}, $$ where, for any indices $n_1\ge n_2$, $$\label{eq4.B}
\begin{gathered}
A_{n_1 n_2}=\sum_{p=0}^\infty(n_1-n_2+2p+1)(\varphi_{n_1+p+1}\varphi_{n_2-p}
+\varphi_{n_1+p}\varphi_{n_2-p-1})\\
-(n_1-n_2)\varphi_{n_1}\varphi_{n_2}
-2\sum_{p=1}^\infty(n_1-n_2+2p)\varphi_{n_1+p}\varphi_{n_2-p}
\end{gathered}$$ and, for any $n\in{\mathbb Z}$, $$\label{eq4.C}
\begin{gathered}
B_n=(n+w+1)(n+w'+1)\varphi_{n+1}+(n-z-1)(n-z'-1)\varphi_{n-1}\\
-\bigl((n-z)(n-z')+(n+w)(n+w')\bigr)\varphi_n.
\end{gathered}$$
Note that only coefficients $B_n$ depend on the parameters $(z,z',w,w')$.
The operator ${\mathbb D}_{{z,z',w,w'}}$ is correctly defined on $R$.
Note that not every formal differential operator in variables $\varphi_n$ can act on $R$. Here is a very simple example: application of $\sum_{n\in{\mathbb Z}}\frac{{\partial}}{{\partial}\varphi_n}$ to the element $\varphi=\sum_{n\in{\mathbb Z}}\varphi_n$ gives the meaningless expression $\sum_{n\in{\mathbb Z}}1$. As is seen from the argument below, the validity of the proposition relies on the concrete form of the coefficients of ${\mathbb D}_{{z,z',w,w'}}$.
\(i) Obviously, when ${\mathbb D}_{{z,z',w,w'}}$ is formally applied to a monomial in $R$, the result is a well-defined element of $R$. We have to prove that, more generally, the same holds when ${\mathbb D}_{{z,z',w,w'}}$ is applied to any homogeneous element $g\in R$. In other words, the infinite sum arising in ${\mathbb D}_{{z,z',w,w'}}g$ cannot contain infinitely many nonzero terms proportional to one and the same monomial.
\(ii) Given a monomial $\varphi_{\lambda}=\varphi_{{\lambda}_1}\dots\varphi_{{\lambda}_N}$ indexed by a signature ${\lambda}$, define its *support* ${\operatorname{supp}}\varphi_{\lambda}$ as the lattice interval $[a,b]:=\{a,\dots,b\}\subset{\mathbb Z}$, where $a={\lambda}_N=\min({\lambda}_1,\dots,{\lambda}_N)$ and $b={\lambda}_1=\max({\lambda}_1,\dots,{\lambda}_N)$.
From is is evident that for every monomial $\varphi_\mu$ entering $$A_{n_1 n_2}\frac{{\partial}^2 \varphi_{\lambda}}{{\partial}\varphi_{n_1}{\partial}\varphi_{n_2}},$$ one has ${\operatorname{supp}}\varphi_\mu\supseteq[a,b]$.
\(iii) Likewise, from it is clear that if a monomial $\varphi_\mu$ enters $$B_n\frac{{\partial}\varphi_{\lambda}}{{\partial}\varphi_n}$$ and $[a',b']:={\operatorname{supp}}\varphi_\mu$, then one has $|a'-a|\le1$, $|b'-b|\le1$.
\(iv) Let again, as in (i) above, $g$ be a homogeneous element of $R$, and examine the infinite sum ${\mathbb D}_{{z,z',w,w'}}g$ resulting from application of ${\mathbb D}_{{z,z',w,w'}}$ to $g$. Observe that there exist only finitely many monomials of a prescribed degree and with the support contained in a prescribed lattice interval. Therefore, (ii) and (iii) guarantee that the undesired accumulation of infinitely many proportional terms in ${\mathbb D}_{{z,z',w,w'}}g$ is excluded.
\[prop4.C\] If $z=n_+$ and $w=-n_-$, where $n_+\ge n_-$ are integers, then the operator ${\mathbb D}_{{z,z',w,w'}}$ preserves the ideal $I(n_+,n_-)$ and hence correctly determines an operator acting on the quotient ring $R(n_+,n_-)=R/I(n_+,n_-)$.
The ideal $I(n_+,n_-)$ consists of (possibly infinite) linear combinations of monomials whose support is not contained in the lattice interval $[n_-,n_+]$. Step (ii) of the argument above shows that the application of the second order terms in ${\mathbb D}_{{z,z',w,w'}}$ enlarges the supports and so preserves the ideal $I(n_+,n_-)$. Note that this holds for any values of the parameters.
Now let us examine the effect of the application of a first degree term $B_n\frac{{\partial}}{{\partial}\varphi_n}$. From it is seen that the only danger may come from the quantities $$(n+w+1)(n+w'+1)\varphi_{n+1}\big|_{n=n_--1}, \quad
(n-z-1)(n-z'-1)\varphi_{n-1}\big|_{n=n_++1}.$$ But these quantities vanish because, by our assumption, $w=-n_-$ and $z=n_+$.
\[prop4.A\] For any fixed integer $m$, the operator ${\mathbb D}_{{z,z',w,w'}}$ is invariant under the change of variables $\varphi_n\mapsto \varphi_{n+m}$ [($n\in{\mathbb Z}$)]{} combined with the shift of parameters $$z\to z+m, \quad z'\to z'+m, \quad w\to w-m, \quad w'\to w-m.$$
In connection with this proposition see also Remark 3.7 in [@BO-AnnMath].
Indeed, the indicated simultaneous shift of the variables and parameters does not change the coefficients $A_{n_1n_2}$ and $B_n$.
The next proposition is not so evident:
\[prop4.B\] The operator ${\mathbb D}_{{z,z',w,w'}}$ preserves the principal ideal $J\subset R$.
We will prove that ${\mathbb D}_{{z,z',w,w'}}$ commutes with the operator of multiplication by $\varphi$, which obviously implies that ${\mathbb D}_{{z,z',w,w'}}$ preserves $J$.
Take an arbitrary element $F\in R$ and observe that ${\mathbb D}_{{z,z',w,w'}}(\varphi F)-\varphi{\mathbb D}_{{z,z',w,w'}}F$ equals $$2\sum_{n\in{\mathbb Z}}\left(A_{nn}+\sum_{n_1:\,
n_1>n}A_{n_1n}+\sum_{n_2:\,n_2<n}A_{nn_2}\right)\frac{{\partial}F}{{\partial}\varphi_n}+\left(\sum_{n\in{\mathbb Z}}B_n\right)F.$$
We are going to check that this expression vanishes. More precisely, the $n$th summand in the first sum vanishes for every $n\in{\mathbb Z}$ and the sum $\sum B_n$ vanishes, too.
Indeed, by , one has $$\begin{gathered}
\sum_{n\in{\mathbb Z}}B_n=
\sum_{n\in{\mathbb Z}}(n+w+1)(n+w'+1)\varphi_{n+1}+\sum_{n\in{\mathbb Z}}(n-z-1)(n-z'-1)\varphi_{n-1}\\
-\sum_{n\in{\mathbb Z}}\bigl((n-z)(n-z')+(n+w)(n+w')\bigr)\varphi_n.\end{gathered}$$ By making the change $n\to n\pm1$ in the first two sums one sees that that the whole expression equals $0$.
Next, let us check that $$A_{nn}+\sum_{n_1:\, n_1>n}A_{n_1n}+\sum_{n_2:\,n_2<n}A_{nn_2}=0.$$ By virtue of Proposition \[prop4.A\], it suffices to do this for the particular value $n=0$, which slightly simplifies the notation. Then the identity in question can be written as $$\label{eq4.A}
A_{00}+\sum_{m>0}A_{m0}+\sum_{m>0}A_{0,-m}=0.$$
Let us write down explicitly all the summands: $$A_{00}=\sum_{p\ge0}(2p+1)[\varphi_{p+1}\varphi_{-p}+\varphi_p\varphi_{-p-1}]
-2\sum_{p\ge1}2p\varphi_p\varphi_{-p}.$$
$$A_{m0}=\sum_{p\ge0}(m+2p+1)[\varphi_{m+p+1}\varphi_{-p}+\varphi_{m+p}\varphi_{-p-1}]
-m\varphi_m\varphi_0-2\sum_{p\ge1}(m+2p)\varphi_{m+p}\varphi_{-p}.$$
$$A_{0,-m}=\sum_{p\ge0}(m+2p+1)[\varphi_{p+1}\varphi_{-m-p}+\varphi_{p}\varphi_{-m-p-1}]
-m\varphi_0\varphi_{-m}-2\sum_{p\ge1}(m+2p)\varphi_{p}\varphi_{-m-p}.$$ Then a slightly tedious but direct examination shows that in , all the terms are cancelled.
The method of intertwiners {#sect5}
==========================
This method was proposed in Borodin-Olshanski [@BO-GT-Dyn]. The method allows one to construct Markov processes on dual objects to inductive limit groups like $S(\infty)$ or $U(\infty)$ by essentially algebraic tools. Here we describe its idea. For more details, see [@BO-GT-Dyn], Borodin-Olshanski [@BO-EJP], and the expository paper Olshanski [@Ols-SPb].
Generalities on Markov kernels and Feller processes
---------------------------------------------------
Let $X$ and $Y$ be two measurable spaces. Recall that a *Markov kernel* with source space $X$ and target space $Y$ is a function $P(x,A)$, where the first argument $x$ ranges over $X$ and the second argument is a measurable subset of $Y$; next, one assumes that the following two conditions hold (see e.g. Meyer [@Mey66]):
$\bullet$ For $A$ fixed, $P(\,\cdot\,, A)$ is a measurable function on $X$.
$\bullet$ For $x$ fixed, $P(x,\,\cdot\,)$ is a probability measure on $Y$ (we will denote it by $P(x,dy)$).
When the second space $Y$ is a discrete space, it is convenient to interpret the kernel as a function on $X\times Y$ by setting $P(x,y):=P(x,\{y\})$. In the case when both spaces are discrete, $P(x,y)$ is a stochastic matrix of format $X\times Y$.
We regard a Markov kernel $P$ as a surrogate of map between $X$ and $Y$, denoted as $P: X\dasharrow Y$ and called a *link*. Here the dashed arrow symbolizes the fact that a link is not an ordinary map: it assigns to a given point $x\in X$ not a single point in $Y$ but a probability distribution on $Y$.
The superposition of two links $P':X\dasharrow Y$ and $P'': Y\dasharrow Z$ is the link $P=P'P''$ between $X$ and $Z$ defined by $$P(x,dz)=\int_{y\in Y}P'(x,dy)P''(y,dz).$$ If both $X$ and $Y$ are discrete, then the superposition becomes the matrix product.
Every link $P:X\dasharrow Y$ induces a contractive linear operator $f\mapsto Pf$ from the Banach space of bounded measurable functions on $Y$ to the similar function space on $X$: $$(P f)(x)=\int_{y\in Y}P(x,dy)f(y), \qquad x\in X.$$
Assuming $X$ and $Y$ are locally compact spaces, we say that $P: X\dasharrow Y$ is a *Feller link* if the above operator maps $C_0(Y)$ into $C_0(X)$. Note that the superposition of Feller links is a Feller link, too. (We recall that $C_0(X)$ consists of continuous functions on $X$ vanishing at infinity. If $X$ is a discrete space, then the continuity assumption is trivial and $C_0(X)$ consists of arbitrary functions vanishing at infinity.)
Now we recall a few basic notions from the theory of Markov processes (see Liggett [@Liggett], Ethier-Kurtz [@EK]).
A *Feller semigroup* on a locally compact space $X$ is a strongly continuous semigroup $P(t)$, $t\ge0$, of contractive operators on $C_0(X)$ given by Feller links $P(t;x, dy)$. A well-known abstract theorem says that a Feller semigroup gives rise to a Markov process on $X$ with transition function $P(t;x,dy)$. The processes derived from Feller semigroups are called *Feller processes*; they form a particularly nice subclass of general Markov processes.
A Feller semigroup $P(t)$ is uniquely determined by its *generator*. This is a closed dissipative operator $A$ on $C_0(X)$ given by $$Af=\lim_{t\to+0}\frac{P(t)f-f}{t}.$$ The *domain* of $A$, denoted by ${\operatorname{dom}}A$, is the (algebraic) subspace formed by those functions $f\in C_0(X)$ for which the above limit exists; ${\operatorname{dom}}A$ is always a dense subspace. Every subspace ${\mathscr F}\subset{\operatorname{dom}}A$ for which the closure of $A\big|_{{\mathscr F}}$ equals $A$ is called a *core* of $A$. One can say that a core is an “essential domain” for $A$. Very often, the full domain of a generator is difficult to describe explicitly, and then one is satisfied by exhibiting a core ${\mathscr F}$ with the explicit action of the generator on ${\mathscr F}$.
Stochastic links between dual objects
-------------------------------------
Here we introduce concrete examples of stochastic links we will dealing with.
For a compact group $G$, we denote by ${\widehat}G$ the set of irreducible characters of $G$ and call it the *dual object* to $G$. Given $\chi\in{\widehat}G$, we denote by ${\widetilde}\chi$ the corresponding normalized character: $${\widetilde}\chi(g)=\frac{\chi(g)}{\chi(e)}, \qquad g\in G.$$
In the special case when $G$ is commutative, ${\widetilde}\chi=\chi$ and ${\widehat}G$ is a discrete group, but in the general case (when $G$ is noncommutative), the dual object does not possess a group structure and we regard it simply as a discrete space.
To every morphism $\iota: G_1\to G_2$ of compact groups there corresponds a *canonical* “dual” link ${\Lambda}: {\widehat}G_2\dasharrow {\widehat}G_1$, defined as follows. For every irreducible character $\chi\in{\widehat}G_2$, its superposition with $\iota$ is a finite linear combination of irreducible characters $\chi'\in{\widehat}G_1$ with nonnegative integral coefficients. It follows that the superposition of ${\widetilde}\chi$ with $\iota$ is a convex linear combination of normalized irreducible characters of the group $G_1$; the coefficients of the latter expansion are just the entries of the stochastic matrix ${\Lambda}$. That is, $${\widetilde}\chi(\iota(g))=\sum_{\chi'\in {\widehat}G_1}{\Lambda}(\chi,\chi'){\widetilde}{\chi'}(g), \qquad
g\in G_1, \quad \chi\in{\widehat}G_2.$$
If $G_1\to G_2$ and $G_2\to G_3$ are two morphisms of compact groups, then it is evident that the superposition of the canonical dual links ${\widehat}G_3\dasharrow {\widehat}G_2$ and ${\widehat}G_2\dasharrow {\widehat}G_1$ coincides with the canonical link ${\widehat}G_3\dasharrow {\widehat}G_1$ corresponding to the composition morphism $G_1\to G_3$.
Consider now the infinite chain of groups $$U(1)\subset U(2)\subset U(3)\subset\dots$$ as defined in the beginning of Subsection \[sect3.A\]. For every $N<M$, this chain defines an embedding $U(N)\hookrightarrow U(M)$, and we denote by ${\Lambda}^M_N({\lambda},\mu):{{\mathbb{S}}}_M\dasharrow{{\mathbb{S}}}_N$ the corresponding dual link, which is a stochastic matrix of format ${{\mathbb{S}}}_M\times{{\mathbb{S}}}_N$. In particular, for $M=N+1$ this matrix takes the form $$\label{eq5.C}
{\Lambda}^{N+1}_N({\lambda},\mu)=\begin{cases}\dfrac{{\operatorname{Dim}}_N\mu}{{\operatorname{Dim}}_{N+1}{\lambda}}, & \text{if
$\mu\prec{\lambda}$}\\ 0, &\text{otherwise},\end{cases}$$ where $\mu\prec{\lambda}$ means that the two signatures *interlace* in the sense that $${\lambda}_i\ge\mu_i\ge{\lambda}_{i+1}, \qquad i=1,\dots,N,$$ see Borodin-Olshanski [@BO-GT-Dyn Section 1.1] for more details.
Next, consider the embedding $U(N)\hookrightarrow U(\infty)$ (the image of the former group in the latter group consists of the infinite unitary matrices $[U_{ij}]$ such that $U_{ij}={\delta}_{ij}$ unless both $i$ and $j$ are less or equal to $N$). We define the dual object ${\widehat}{U(\infty)}$ as the set of extreme characters and identify it with ${\Omega}$. Then the above definition of the dual link is still applicable with the extreme characters of $U(\infty)$ playing the role of the (nonexisting) normalized irreducible characters. The resulting Markov kernel ${\Omega}\dasharrow {{\mathbb{S}}}_N$ has the form $$\label{eq5.A}
{\Lambda}^\infty_N({\omega},{\lambda})={\operatorname{Dim}}_N{\lambda}\cdot {\widehat}{\sigma}_{\lambda}({\omega}), \qquad {\omega}\in{\Omega}, \quad
{\lambda}\in{{\mathbb{S}}}_N,$$ where ${\widehat}{\sigma}_{\lambda}({\omega})$ is defined in Section \[sect3\]. The derivation of this formula is simple: by , the restriction of the extreme character $\Psi_{\omega}$ to the subgroup $U(N)$ is given by the function $\Phi(u_1;{\omega})\dots\Phi(u_N;{\omega})$; the expansion of that function on the irreducible characters $\chi_{\lambda}=s_{\lambda}$ is given by , and we only need to introduce the factor ${\operatorname{Dim}}_N{\lambda}$ to get the required expansion on the normalized characters ${\widetilde}\chi_{\lambda}=s_{\lambda}/{\operatorname{Dim}}_N{\lambda}$.
\[prop5.A\] The canonical links ${\Lambda}^M_N:{{\mathbb{S}}}_M\dasharrow{{\mathbb{S}}}_N$ and ${\Lambda}^\infty_N:{\Omega}\dasharrow{{\mathbb{S}}}_N$ are Feller links.
For a proof, see Borodin-Olshanski [@BO-GT-Appr Corollary 2.11 and Proposition 2.12].
The method of intertwiners {#the-method-of-intertwiners}
--------------------------
Let $X$ and $Y$ be locally compact spaces, $P_X(t)$ and $P_Y(t)$ be Feller semigroups on $X$ and $Y$, respectively, and ${\Lambda}:X\dasharrow Y$ be a Feller link. We say that ${\Lambda}$ *intertwines* the semigroups $P_X(t)$ and $P_Y(t)$ if the following commutation relation holds $$P_X(t){\Lambda}={\Lambda}P_Y(t), \qquad t\ge0.$$ This relation can be understood as an equality of links or, equivalently, as an equality of operators acting from $C_0(Y)$ to $C_0(X)$.
\[prop5.B\] Assume we are given a family $\{P_N(t): N=1,2,3,\dots\}$ of Feller semigroups, where the $N$th semigroup acts on $C_0({{\mathbb{S}}}_N)$. Further, assume that these semigroups are intertwined by the canonical links ${\Lambda}^{N+1}_N$, so that $$P_{N+1}(t){\Lambda}^{N+1}_N={\Lambda}^{N+1}_N P_N(t), \qquad N=1,2,3,\dots, \quad t\ge0.$$
Then there exists a unique Feller semigroup $P_\infty(t)$ on $C_0({\Omega})$ characterized by the property $$P_\infty(t){\Lambda}^\infty_N={\Lambda}^\infty_N P_N(t), \qquad N=1,2,\dots, \quad t\ge0.$$
See Proposition 2.4 in Borodin-Olshanski [@BO-GT-Dyn]. The fact that the hypothesis of this proposition is satisfied in our concrete situation is established in Subsection 3.3 of that paper.
\[prop5.C\] We keep to the hypotheses of Proposition \[prop5.B\]. Let $A_N$ and $A_\infty$ denote the generators of the semigroups $P_N(t)$ and $P_\infty(t)$, respectively.
[(i)]{} For every $N=1,2,\dots$ and every $f\in{\operatorname{dom}}(A_N)$, the vector ${\Lambda}^\infty_Nf$ belongs to ${\operatorname{dom}}(A_\infty)$ and one has $$A_\infty{\Lambda}^\infty_N f={\Lambda}^\infty_N A_N f.$$
[(ii)]{} Assume additionally that for each $N=1,2,3,\dots$ we are given a core ${\mathscr F}_N\subseteq {\operatorname{dom}}(A_N)$ for the operator $A_N$. Then the linear span of the vectors of the form ${\Lambda}^\infty_N f$, where $N=1,2,\dots$ and $f\in{\mathscr F}_N$, is a core for $A_\infty$.
Claim (i) directly follows from the definition of the generator. Claim (ii) is established in Borodin-Olshanski [@BO-EJP Proposition 5.2].
The degenerate case {#sect5.A}
-------------------
Let us fix a couple of integers $n_+\ge n_-$ and set $$\label{eq5.B}
{{\mathbb{S}}}_N(n_+,n_-)=\{\nu\in{{\mathbb{S}}}_N: n_+\ge\nu_1\ge\dots\ge\nu_N\ge n_-\}.$$ Note that this is a finite set.
If $\mu\in{{\mathbb{S}}}_M(n_+,n_-)$ and $N<M$, then ${\Lambda}^M_N(\mu,\nu)$ vanishes unless $\nu\in{{\mathbb{S}}}_N(n_+,n_-)$. So, ${\Lambda}^M_N$ induces a link ${{\mathbb{S}}}_M(n_+,n_-)\dasharrow
{{\mathbb{S}}}_N(n_+,n_-)$. Likewise, if ${\omega}\in{\Omega}(n_+,n_-)$, then ${\Lambda}^\infty_N({\omega},\nu)$ vanishes unless $\nu\in{{\mathbb{S}}}_N(n_+,n_-)$. So, ${\Lambda}^\infty_N$ induces a link ${\Omega}(n_+,n_-)\dasharrow {{\mathbb{S}}}_N(n_+,n_-)$.
When ${{\mathbb{S}}}_N$ (with $N=1,2,3,\dots$) and ${\Omega}$ are replaced by ${{\mathbb{S}}}_N(n_+,n_-)$ and ${\Omega}(n_+,n_-)$, respectively, all the results of the present section remain valid. The proofs are extended automatically, and we only point out some simplifications:
In Proposition \[prop5.A\], the claim concerning the Feller property for the links ${\Lambda}^M_N$ becomes redundant as the links ${{\mathbb{S}}}_M(n_+,n_-)\dasharrow
{{\mathbb{S}}}_N(n_+,n_-)$ are finite matrices. Next, because ${\Omega}(n_+,n_-)$ is a compact space, the Feller property for the link ${\Lambda}^\infty_N:{\Omega}(n_+,n_-)\dasharrow
{{\mathbb{S}}}_N(n_+,n_-)$ simply means that the functions of the form ${\omega}\to
{\Lambda}^\infty_N({\omega},\nu)$ are continuous on ${\Omega}(n_+,n_-)$.
In Proposition \[prop5.B\], one should replace $C_0({\Omega})$ by $C({\Omega}(n_+,n_-))$, the Banach space of all continuous functions on the compact space ${\Omega}(n_+,n_-)$.
In Proposition \[prop5.C\], because the sets ${{\mathbb{S}}}_N(n_+,n_-)$ are finite, the generators $A_N$ are finite-dimensional, so that ${\operatorname{dom}}(A_N)$ is the whole space of functions on ${{\mathbb{S}}}_N(n_+,n_-)$.
Markov processes on ${\Omega}$ and their generators {#sect6}
===================================================
This section contains some necessary material from Borodin-Olshanski [@BO-GT-Dyn], together with a brief motivation. In that paper, we constructed a family $\{X_{{z,z',w,w'}}\}$ of continuous time Markov processes on the space ${\Omega}$, indexed by the quadruple of parameters $({{z,z',w,w'}})$ ranging over a certain subset of ${\mathbb C}^4$. The infinitesimal generator of $X_{{z,z',w,w'}}$, denoted by $A_{{z,z',w,w'}}$, is an unbounded operator on the Banach space $C_0({\Omega})$. The results of [@BO-GT-Dyn] tell us how $A_{{z,z',w,w'}}$ acts on a subspace ${\widehat}{\mathscr F}\subset
C_0({\Omega})$, the (algebraic) linear span of the functions ${\widehat}{\sigma}_{\lambda}({\omega})$, where ${\lambda}$ ranges over the set of all signatures except ${\lambda}=\varnothing$. The explicit formulas for this action are the starting point for the computations in the remaining part of the paper. Note that ${\widehat}{\mathscr F}$ serves as a core for the generator $A_{{z,z',w,w'}}$, so that it is uniquely determined by its restriction to ${\widehat}{\mathscr F}$.
Special bilateral birth-death processes
---------------------------------------
Birth-death processes form a well-studied class of continuous time Markov chains. The state space of every birth-death process is the set ${\mathbb Z}_+$ of nonnegative integers, and the process is determined by specifying the quantities $q(n,n\pm1)$, the *jump rates* from state $n\in{\mathbb Z}_+$ to the neighboring states $n\pm1$, with the understanding that $q(0,-1)=0$, which prevents from leaving the subset ${\mathbb Z}_+\subset{\mathbb Z}$. Under appropriate constraints on the jump rates the process is well defined (that is, does not explode, meaning that, with probability 1, one cannot escape to infinity in finite time).
The *bilateral* birth-death processes are defined in a similar way, only now the state space is the whole lattice ${\mathbb Z}$ and the jump rates $q(n,n\pm1)$ are assumed to be strictly positive for all $n\in{\mathbb Z}$. Again, one needs some restrictions to be imposed on these quantities in order that the process be non-exploding. Bilateral birth-death processes are not so widely known as the ordinary ones. However, they were also discussed in the literature.
We are interested in bilateral birth-death processes whose jump rates $q(n,n\pm1)$ are quadratic functions in variable $n$. We write them in the form $$\label{eq6.B}
q(n,n-1)=(w+n)(w'+n), \qquad q(n,n+1)=(z-n)(z'-n).$$ It is readily verified that these quantities are strictly positive for all $n\in{\mathbb Z}$ if and only if each of pairs $(z,z')$ and $(w,w')$ belongs to the subset $\mathscr Z\subset{\mathbb C}^2$ defined by $$\begin{gathered}
\label{eq6.A}
\mathscr Z:= \{(\zeta,\zeta')\in({\mathbb C}\setminus{\mathbb Z})^2\mid
\zeta'=\bar{\zeta}\}\\
\cup
\{(\zeta,\zeta')\in({\mathbb R}\setminus{\mathbb Z})^2\mid m<\zeta,\zeta'<m+1 \text{ for some }
m\in{\mathbb Z}\}.\end{gathered}$$
Note that if $(\zeta,\zeta')\in\mathscr Z$, then $\zeta+\zeta'$ is real.
\[def6.A\] We say that a quadruple $({{z,z',w,w'}})\in{\mathbb C}^4$ is *admissible* if $(z,z')\in\mathscr Z$, $(w,w')\in\mathscr Z$, and $z+z'+w+w'>-1$.
\[prop6.A\] Let $({{z,z',w,w'}})\in{\mathbb C}^4$ be admissible.
[(i)]{} There exists a non-exploding bilateral birth-process with the jump rates given by .
[(ii)]{} This process is a Feller process.
[(iii)]{} Its generator is implemented by the difference operator $D_{{z,z',w,w'}}$ on ${\mathbb Z}$ acting on functions $f(n)$, $n\in{\mathbb Z}$ by $$\begin{gathered}
\label{eq6.C}
(D_{{z,z',w,w'}}f)(n)=(z-n)(z'-n)(f(n+1)-f(n))\\
+(w+n)(w'+n)(f(n-1)-f(n)),\end{gathered}$$ and the domain of the generator consists of those functions $f\in C_0({\mathbb Z})$ for which $D_{{z,z',w,w'}}f\in C_0({\mathbb Z})$.
Statements (i) and (ii) are the subject of Theorem 5.1 in Borodin-Olshanski [@BO-GT-Dyn], and (iii) is their formal consequence, as explained in [@BO-GT-Dyn Proposition 4.6].
We refer to [@BO-GT-Dyn] for more details. Note that the property of non-explosion is the same as *regularity* of the so-called *Q-matrix* (or the *matrix of jump rates*), see [@BO-GT-Dyn Section 4] and references therein. In our case, this matrix is simply the matrix of the difference operator $D_{{z,z',w,w'}}$. This is a tridiagonal ${\mathbb Z}\times{\mathbb Z}$ matrix $Q=[q(n,n')]$ with the entries $q(n,n\pm1)$ given by , the diagonal entries $$q(n, n)=-q(n,n+1)-q(n,n-1),$$ and all remaining entries equal to 0.
Feller dynamics on ${{\mathbb{S}}}_N$ {#sect6.A}
-------------------------------------
As explained in Borodin-Olshanski [@BO-GT-Dyn Section 5.2], Proposition \[prop6.A\] admits an extension with ${\mathbb Z}$ replaced by ${{\mathbb{S}}}_N$, where $N=1,2,3,\dots$ (recall that ${{\mathbb{S}}}_1={\mathbb Z}$). To state it we need first to define a matrix $Q=[q(\nu,\mu)]$ of format ${{\mathbb{S}}}_N\times{{\mathbb{S}}}_N$. It depends on $({{z,z',w,w'}})$ and has the following form:
$\bullet$ the entries $q(\nu,\mu)$ equal 0 unless $\mu=\nu$ or $\mu=\nu\pm
{\varepsilon}_i$, where $i=1,\dots,N$ and ${\varepsilon}_1,\dots,{\varepsilon}_N$ stands for the canonical basis of ${\mathbb Z}^N$;
$\bullet$ the (nonzero) off-diagonal entries are given by $$\label{eq6.D}
q(\nu,\nu\pm {\varepsilon}_i)=\frac{{\operatorname{Dim}}_N(\nu\pm {\varepsilon}_i)}{{\operatorname{Dim}}_N\nu}\,r(\nu,\nu\pm
{\varepsilon}_i),$$ where $$\label{eq6.E}
r(\nu,\nu+{\varepsilon}_i)=(z-\nu_i+i-1)(z'-\nu_i+i-1), \qquad i=1,\dots,N,$$ and $$\label{eq6.F}
r(\nu,\nu-{\varepsilon}_i)=(w+\nu_i-i+N)(w'+\nu_i-i+N), \qquad i=1,\dots,N;$$
$\bullet$ the diagonal entries are given by $$\begin{gathered}
\label{eq6.G}
q(\nu,\nu)=-\sum_{\mu:\,
\mu\ne\nu}q(\nu,\mu)\\=(z+z'+w+w')\frac{N(N-1)}2+\frac{(2N-1)N(N-1)}3-\sum_{\mu:\,
\mu\ne\nu}r(\nu,\mu).\end{gathered}$$
When $N=1$, this agrees with the definition of the preceding subsection. (To compare the above formulas with those from [@BO-GT-Dyn Section 5.2], take into account a shift of parameters indicated in [@BO-GT-Dyn (6.1) and (6.2)].)
For $\nu=(\nu_1,\dots,\nu_N)\in{{\mathbb{S}}}_N$, we set $$\nu^*=(-\nu_N,\dots,-\nu_1).$$ The correspondence $\nu\mapsto\nu^*$ is an involutive bijection ${{\mathbb{S}}}_N\to{{\mathbb{S}}}_N$.
\[prop6.D\] One has $$q(\nu,\mu)=q^*(\nu^*,\mu^*),$$ where the matrix $[q^*(\,\cdot\,,\,\cdot\,)]$ is obtained from the matrix $[q^*(\,\cdot\,,\,\cdot\,)]$ by switching $(z,z')\leftrightarrow(w,w')$.
This is readily checked.
\[prop6.B\] Let $({{z,z',w,w'}})\in{\mathbb C}^4$ be admissible in the sense of Definition \[def6.A\].
[(i)]{} For every $N=1,2,3,\dots$, the ${{\mathbb{S}}}_N\times{{\mathbb{S}}}_N$ matrix $Q=[q(\nu,\mu)]$ defined above is regular, so that there exists a non-exploding continuous time Markov process on ${{\mathbb{S}}}_N$ with the jump rates given by the off-diagonal entries $q(\nu,\mu)$.
[(ii)]{} This process is a Feller process.
[(iii)]{} Its generator is implemented by the $N$-variate difference operator $D_{{{z,z',w,w'}}\mid N}$ on ${{\mathbb{S}}}_N\subset{\mathbb Z}^N$ acting on functions $f(\nu)$, $\nu\in{{\mathbb{S}}}_N$ by $$\label{eq6.H}
(D_{{{z,z',w,w'}}\mid N} f)(\nu)=\sum_{\mu\in{{\mathbb{S}}}_N}q(\nu,\mu)f(\mu)
=\sum_{\mu\in{{\mathbb{S}}}_N\setminus\{\nu\}}q(\nu,\mu)(f(\mu)-f(\nu)),$$ and the domain of the generator consists of those functions $f\in C_0({{\mathbb{S}}}_N)$ for which $D_{{{z,z',w,w'}}\mid N} f\in C_0({{\mathbb{S}}}_N)$.
Statements (i) and (ii) are proved in Borodin-Olshanski [@BO-GT-Dyn Theorem 5.4], and (iii) is their formal consequence, as explained in [@BO-GT-Dyn Proposition 4.6].
\[prop6.F\] For any $({{z,z',w,w'}})\in{\mathbb C}^4$ and any $N=0,1,2,\dots$ the following relation holds $$\label{eq6.O}
D_{{{z,z',w,w'}}\mid N+1}{\Lambda}^{N+1}_N={\Lambda}^{N+1}_N D_{{{z,z',w,w'}}\mid N}\qquad \forall
N=1,2,\dots\,.$$
(Recall that ${\Lambda}^{N+1}_N: {{\mathbb{S}}}_{N+1}\dasharrow{{\mathbb{S}}}_N$ are the canonical links defined in .)
In a slightly different notation, this is proved in [@BO-GT-Dyn Proposition 6.2].
This result serves as the basis for the construction described in the next subsection. It is also used in Section \[sect9\] below.
Feller dynamics on ${\Omega}$
-----------------------------
Throughout this subsection we assume, as before, that $({{z,z',w,w'}})$ is admissible (Definition \[def6.A\]).
\[prop6.C\] For $N=1,2,\dots$, we denote by $P_{{{z,z',w,w'}}\mid N}(t)$ the Feller semigroup on $C_0({{\mathbb{S}}}_N)$ afforded by Proposition \[prop6.B\].
[(i)]{} These semigroups $P_{{{z,z',w,w'}}\mid N}(t)$ satisfy the hypothesis of Proposition \[prop5.B\], that is, one has $$P_{{{z,z',w,w'}}\mid N+1}(t){\Lambda}^{N+1}_N={\Lambda}^{N+1}_N P_{{{z,z',w,w'}}\mid N}(t), \qquad t\ge0,$$ for every $N=1,2,3,\dots$.
[(ii)]{} There exists a unique Feller semigroup $P_{{{z,z',w,w'}}\mid\infty}(t)$ on $C_0({\Omega})$ characterized by the property $$P_{{{z,z',w,w'}}\mid\infty}(t){\Lambda}^\infty_N={\Lambda}^\infty_N P_N(t), \qquad N=1,2,\dots, \quad
t\ge0.$$
(Recall that ${\Lambda}^\infty_N:{\Omega}\dasharrow {{\mathbb{S}}}_N$ are the links defined in .)
Claim (i) is established in Borodin-Olshanski [@BO-GT-Dyn theorem 6.1]. Claim (ii) follows from Claim (i) by virtue of Proposition \[prop5.B\].
\[def6.B\] In what follows $A_{{{z,z',w,w'}}\mid N}$ denotes the generator of the semigroup $P_{{{z,z',w,w'}}\mid N}(t)$ on $C_0({{\mathbb{S}}}_N)$ and $A_{{z,z',w,w'}}$ denotes the generator of the semigroup $P_{{{z,z',w,w'}}\mid\infty}(t)$ on $C_0({\Omega})$.
In the next proposition and its proof we use the quantities $q(\nu,\mu)$ and $r(\nu,\mu)$ that were defined in the preceding subsection. Note that they depend on the parameters ${{z,z',w,w'}}$, and $N$.
\[prop6.E\] Let $N=1,2,\dots$. For every signature $\mu\in{{\mathbb{S}}}_N$, the function ${\widehat}{\sigma}_\mu\in C_0({\Omega})$ belongs to the domain of the generator $A_{{z,z',w,w'}}$ and $$\label{eq6.I}
A_{{z,z',w,w'}}{\widehat}{\sigma}_\mu=q(\mu,\mu){\widehat}{\sigma}_\mu+\sum_{\nu\in{{\mathbb{S}}}_N:\,
\nu\ne\mu}r(\nu,\mu){\widehat}{\sigma}_\nu.$$
For $\mu\in{{\mathbb{S}}}_N$, let ${\mathbf1}_\mu$ denote the function on ${{\mathbb{S}}}_N$ defined by ${\mathbf1}_\mu(\nu)={\delta}_{\mu\nu}$. By the definition of $D_{{{z,z',w,w'}}\mid N}$, see , $$\label{eq6.L}
D_{{{z,z',w,w'}}\mid N}{\mathbf1}_\mu=\sum_{\nu\in{{\mathbb{S}}}_N}q(\nu,\mu){\mathbf1}_\nu.$$
For any ${\lambda}\in{{\mathbb{S}}}_N$ we set $$\label{eq6.X}
{{\widetilde}{\mathbf1}}_{\lambda}=({\operatorname{Dim}}_N{\lambda})^{-1}\,{\mathbf1}_{\lambda}.$$ Then, by , formula can be rewritten as $$\label{eq6.M}
D_{{{z,z',w,w'}}\mid N}{{\widetilde}{\mathbf1}}_\mu=q(\mu,\mu){{\widetilde}{\mathbf1}}_\mu+\sum_{\nu\in{{\mathbb{S}}}_N:\,\nu\ne\mu}
r(\nu,\mu){{\widetilde}{\mathbf1}}_\nu.$$
Claim (iii) of Proposition \[prop6.B\] implies that all finitely supported functions on ${{\mathbb{S}}}_N$ belong to the domain of $A_{{{z,z',w,w'}}\mid N}$ and for every such function $f$ one has $A_{{{z,z',w,w'}}\mid N}f=D_{{{z,z',w,w'}}\mid N}f$. In particular, taking $f={{\widetilde}{\mathbf1}}_\mu$ we obtain from $$\label{eq6.K}
A_{{{z,z',w,w'}}\mid N}{{\widetilde}{\mathbf1}}_\mu=q(\mu,\mu){{\widetilde}{\mathbf1}}_\mu+\sum_{\nu\in{{\mathbb{S}}}_N:\,\nu\ne\mu}
r(\nu,\mu){{\widetilde}{\mathbf1}}_\nu.$$
Next, by virtue of Proposition \[prop6.C\] one can apply Proposition \[prop5.C\], claim (i). It implies that for every ${\lambda}\in{{\mathbb{S}}}_N$, the function ${\Lambda}^\infty_N{{\widetilde}{\mathbf1}}_{\lambda}$ on ${\Omega}$ belongs to the domain of the generator $A_{{z,z',w,w'}}$ and $$A_{{z,z',w,w'}}{\Lambda}^\infty_N{{\widetilde}{\mathbf1}}_{\lambda}={\Lambda}^\infty_N A_{{{z,z',w,w'}}\mid N}{{\widetilde}{\mathbf1}}_{\lambda}$$ (recall that the links ${\Lambda}^\infty_N:{\Omega}\dasharrow {{\mathbb{S}}}_N$ are defined in ). Together with this gives $$\label{eq6.N}
A_{{z,z',w,w'}}{\Lambda}^\infty_N{{\widetilde}{\mathbf1}}_\mu=q(\mu,\mu){\Lambda}^\infty_N
{{\widetilde}{\mathbf1}}_\mu+\sum_{\nu\in{{\mathbb{S}}}_N:\,\nu\ne\mu} r(\nu,\mu){\Lambda}^\infty_N{{\widetilde}{\mathbf1}}_\nu.$$
Finally, shows that for any ${\lambda}\in{{\mathbb{S}}}_N$ $${\Lambda}^\infty_N{{\widetilde}{\mathbf1}}_{\lambda}={\widehat}{\sigma}_{\lambda}.$$ Substituting this into gives the desired formula.
Let ${\widehat}{\mathscr F}\subset C_0({\Omega})$ denote the linear span of the functions ${\widehat}{\sigma}_{\lambda}$, where ${\lambda}$ range over ${{\mathbb{S}}}_1\sqcup
{{\mathbb{S}}}_2\sqcup{{\mathbb{S}}}_3\sqcup\dots$. As was shown in the proof of Proposition \[prop6.E\], ${\widehat}{\mathscr F}$ coincides with the linear span of the spaces ${\Lambda}^\infty_N C_c({{\mathbb{S}}}_N)$, where $N=1,2,3,\dots$ and $C_c({{\mathbb{S}}}_N)\subset
C_0({{\mathbb{S}}}_N)$ stands for the subspace of finitely supported functions. By Proposition \[prop6.E\], ${\widehat}{\mathscr F}$ is contained in the domain of the generator $A_{{z,z',w,w'}}$. Moreover, this proposition explains how the generator acts on ${\widehat}{\mathscr F}$. In particular, we see that ${\widehat}{\mathscr F}$ is invariant under the action of the generator.
\[thm6.A\] The subspace ${\widehat}{\mathscr F}\subset C_0({\Omega})$ is a core for the generator $A_{{z,z',w,w'}}$.
This fact is not used in the arguments below, but it is a substantial complement to our main result, Theorem \[thm7.B\], which describes explicitly the operator $A_{{z,z',w,w'}}\big|_{{\widehat}{\mathscr F}}$ (the restriction of the generator to ${\widehat}{\mathscr F}$). By virtue of Theorem \[thm6.A\], the latter operator uniquely determines the generator, so Theorem \[thm7.B\] contains, in principle, a complete information about the generator.
Theorem \[thm6.A\] is proved in [@Ols-FAA]. Here we only indicate the idea of the proof. By Proposition \[prop5.C\], it suffices to show that $C_c({{\mathbb{S}}}_N)$ is a core for $A_{{{z,z',w,w'}}\mid N}$ for every $N$. This, in turn, can be verified as in Borodin-Olshanski [@BO-EJP], by making use of a result due to Ethier and Kurtz (see its formulation in [@BO-EJP Theorem 2.3 (iv)]. (Note two misprints in [@BO-EJP]: the claims of Corollary 6.6 (ii) and Corollary 8.7 (ii) concern the subspace of finitely supported functions, so that instead of $C_0(\,\cdot\,)$ one should read $C_c(\,\cdot\,)$.)
The main theorem {#sect7}
================
Formulation of the main theorem
-------------------------------
In Section \[sect4\], we defined the differential operator ${\mathbb D}_{{z,z',w,w'}}$ which acts on $R$. It depends on an arbitrary quadruple $({{z,z',w,w'}})\in{\mathbb C}^4$. We also showed that it preserves the ideal $J\subset R$ and so determines an operator on the quotient ${\widehat}R=R/J$. Let us denote the latter operator by ${\widehat}{\mathbb D}_{{z,z',w,w'}}$.
Given $\psi\in R$, we will denote by ${\widehat}\psi\in{\widehat}R$ the image of $\psi$ under the canonical map $R\to{\widehat}R$. In particular, we may speak about the elements ${\widehat}{\sigma}_{\lambda}\in{\widehat}R$. Note that in Section \[sect3\], we already used the same notation: namely, given $\psi\in{\mathscr R}$, we denoted by ${\widehat}\psi({\omega})$ the corresponding function on ${\Omega}$ (its definition is given just before Proposition \[prop3.A\]). Formally, the two definitions of ${\widehat}\psi$ look differently, but the new definition is morally an extension of the old one, because, as shown in Proposition \[prop3.A\], the kernel of the homomorphism ${\mathscr R}\ni \psi\mapsto {\widehat}\psi(\,\cdot\,)$ coincides with $J\cap{\mathscr R}$.
\[thm7.B\] Let $({{z,z',w,w'}})\in{\mathbb C}^4$ be an admissible quadruple of parameters $({{z,z',w,w'}})\in{\mathbb C}^4$, see Definition \[def6.A\], and recall that $A_{{z,z',w,w'}}$ denotes the generator of the semigroup $P_{{{z,z',w,w'}}\mid\infty}(t)$, see Definition \[def6.B\]. We restrict $A_{{z,z',w,w'}}$ to the core ${\widehat}{\mathscr F}\subset C_0({\Omega})$ defined in the end of Section \[sect6\]. Finally, let ${\lambda}$ range over the set of all signatures, except ${\lambda}=\varnothing$.
Under the identification of the elements ${\widehat}{\sigma}_{\lambda}\in{\widehat}R$ with the functions ${\widehat}{\sigma}_{\lambda}({\omega})$ from the core ${\widehat}{\mathscr F}$, the action of the generator $A_{{z,z',w,w'}}$ on those functions coincides with the action of the operator ${\widehat}{\mathbb D}_{{z,z',w,w'}}$ on the corresponding elements ${\widehat}{\sigma}_{\lambda}\in{\widehat}R$.
\[remark7.A\]
As mentioned in the introduction, Theorem \[thm7.B\] gives a precise sense to the informal statement (Theorem \[thm1.A\]) that “the generator $A_{{z,z',w,w'}}$ is implemented by the differential operator ${\mathbb D}_{{z,z',w,w'}}$”. It is tempting to regard Theorem \[thm7.B\] as the indication that $X_{{z,z',w,w'}}$ are diffusion processes, and it would be very interesting to find out whether this is true. For instance, is it true that the operators $A_{{z,z',w,w'}}$ are diffusion generators as defined in Ledoux [@Ledoux Section 1.1].
Theorem \[thm7.B\] will be proved in a slightly stronger form (Theorem \[thm7.A\] below).
We are going to define a linear operator $R\to R$ that mimics the action of the generator $A_{{z,z',w,w'}}$ on ${\widehat}{\mathscr F}$. In the next proposition we use the $I$-adic topology in $R$, introduced in Subsection \[sect2.A\].
\[prop7.A\] For every quadruple $({{z,z',w,w'}})\in{\mathbb C}^4$ there exists a unique linear operator ${\mathbb A}_{{z,z',w,w'}}:R\to R$, continuous in the $I$-adic topology, annihilating the unity element $1\in R$, and such that for every $N=1,2,\dots$ and every $\mu\in{{\mathbb{S}}}_N$, $$\label{eq7.A}
{\mathbb A}_{{z,z',w,w'}}{\sigma}_\mu=q_{{{z,z',w,w'}}\mid N}(\mu,\mu){\sigma}_\mu+\sum_{\nu\in{{\mathbb{S}}}_N:\,
\nu\ne\mu}r_{{{z,z',w,w'}}\mid N}(\nu,\mu){\sigma}_\nu,$$ where $q_{{{z,z',w,w'}}\mid N}(\mu,\mu)$ and $r_{{{z,z',w,w'}}\mid N}(\nu,\mu)$ is a more detailed notation for the quantities $q(\mu,\mu)$ and $r(\nu,\mu)$ defined in the beginning of Subsection \[sect6.A\].
It is worth emphasizing that here we drop the admissibility condition on the parameters imposed in Section \[sect6\]: the operator ${\mathbb A}_{{z,z',w,w'}}$ is considered for any complex values of $({{z,z',w,w'}})$. This is possible because the formulas defining the quantities $q(\mu,\mu)$ and $r(\nu,\mu)$ make sense for arbitrary $({{z,z',w,w'}})\in{\mathbb C}^4$.
Together with the condition ${\mathbb A}_{{z,z',w,w'}}1=0$, formula determines ${\mathbb A}_{{z,z',w,w'}}$ on the linear span of the basis elements ${\sigma}_\mu$ including ${\sigma}_\varnothing=1$. The continuity of this operator immediately follows from the fact that ${\mathbb A}_{{z,z',w,w'}}{\sigma}_\mu$ is a linear combination of ${\sigma}_\mu$ and “neighboring” basis vectors of the form ${\sigma}_{\mu\pm {\varepsilon}_i}$. The explicit form of the coefficients is not important here.
The next claim will be used in Section \[sect9\].
\[prop7.C\] For any $({{z,z',w,w'}})\in{\mathbb C}^4$, the operator ${\mathbb A}_{{z,z',w,w'}}$ preserves the ideal $J\subset
R$.
It suffices to prove that ${\mathbb A}_{{z,z',w,w'}}$ commutes with the operator of multiplication by $\varphi$. We are going to show that the latter claim is merely a rephrasing of the commutation relation .
Indeed, for every $N=0,1,2,\dots$ we define a linear isomorphism $I_N$ between the space $R_N$ and the space ${{\operatorname{Fun}}}({{\mathbb{S}}}_N)$ of functions on the discrete set ${{\mathbb{S}}}_N$ by setting $$I_N: \sum_{\mu\in{{\mathbb{S}}}_N}a_\mu {\sigma}_\mu \mapsto \sum_{\mu\in{{\mathbb{S}}}_N}a_\mu{\widetilde}{\mathbf1}_\mu,$$ where $a_\mu$ are arbitrary complex coefficients. By the very definition of ${\mathbb A}_{{z,z',w,w'}}$, we have $${\mathbb A}_{{z,z',w,w'}}\big|_{R_N}=I_N^{-1}D_{{{z,z',w,w'}}\mid N}I_N.$$
On the other hand, Proposition \[prop3.AA\] says that for every $\mu\in{{\mathbb{S}}}_N$, $$\varphi{\sigma}_\mu=\sum_{{\lambda}:\, {\lambda}\succ\mu}{\sigma}_{\lambda}.$$ Comparing this with the definition of the canonical link ${\Lambda}^{N+1}_N$ (see ) and the definition of ${\widetilde}{\mathbf1}_\mu$ (see ) we conclude that the operator $R_N\to R_{N+1}$ given by multiplication by $\varphi$ coincides with the operator $I^{-1}_{N+1}{\Lambda}^{N+1}_N I_N$.
Therefore, the commutation relation just means that ${\mathbb A}_{{z,z',w,w'}}$ and multiplication by $\varphi$ commute.
\[thm7.A\] Let $({{z,z',w,w'}})$ be an arbitrary quadruple of complex parameters. The operator ${\mathbb A}_{{z,z',w,w'}}:R\to R$ from Proposition \[prop7.A\] coincides with the differential operator ${\mathbb D}_{{z,z',w,w'}}$ introduced in Definition \[def4.A\].
The theorem says that for every signature $\mu$, the element $\psi:={\mathbb D}_{{z,z',w,w'}}{\sigma}_\mu$ is a *finite* linear combination of basis elements ${\sigma}_\nu$ (which is not evident!) and the corresponding function ${\widehat}\psi$ coincides with $A_{{z,z',w,w'}}{\widehat}{\sigma}_\mu$. Obviously, this implies Theorem \[thm7.B\].
The rest of the paper is devoted to the proof of Theorem \[thm7.A\]. The main essence of difficulty is the fact that ${\mathbb A}_{{z,z',w,w'}}$ is defined by its action on the elements of the basis $\{{\sigma}_\mu\}$, whereas the action of ${\mathbb D}_{{z,z',w,w'}}$ is directly seen in another basis, $\{\varphi_\nu\}$. The transition coefficients between the two bases seem to be too complicated to allow a direct verification of the theorem.
In Subsection \[sect7.B\] we outline the plan of the proof, but first we need to recall a necessary formalism.
Abstract differential operators
-------------------------------
Let ${\mathscr A}$ be a commutative unital algebra and $\mathscr D:{\mathscr A}\to{\mathscr A}$ be a linear operator. For $x\in{\mathscr A}$, let $M_x:{\mathscr A}\to{\mathscr A}$ denote the operator of multiplication by $x$. Let us say that $\mathscr D$ has order $\le k$ (where $k=0,1,2,\dots$) if its $(k+1)$-fold commutator with operators of multiplication by arbitrary elements of the algebra vanishes: $$[M_{x_1},[M_{x_2,},\dots [M_{x_{k+1}},\mathscr D]\dots]]=0, \qquad
x_1,\dots,x_{k+1}\in{\mathscr A}.$$
Let $x_1,x_2,\dots$ be an arbitrary collection of elements of ${\mathscr A}$. If $\mathscr D:{\mathscr A}\to{\mathscr A}$ has order $\le k$, then its action on all monomials of any degree, formed from $\{x_i\}$, is uniquely determined provided one knows the action on the monomials of degree $\le k$, including the monomial of degree $0$, which is 1.
We give a proof for $k=2$ because we need this case only.
\[prop7.B\] Let, as above, ${\mathscr A}$ be a commutative unital algebra and ${\mathscr D}:{\mathscr A}\to{\mathscr A}$ be a linear operator of order $\le2$. For any elements $x_1,\dots,x_n\in{\mathscr A}$, where $n\ge3$, one has [(]{}below the indices range over $1,\dots,n$[)]{} $$\begin{gathered}
\label{eq7.B}
{\mathscr D}(x_1\dots x_n)\\=\sum_{i<j}\left(\prod_{k:\, k\ne i,j}x_k\right)
{\mathscr D}(x_ix_j)-\sum_{i}\left(\prod_{k:\, k\ne i}x_k\right) {\mathscr D}x_i
+\left(\prod_{k}x_k\right){\mathscr D}1.\end{gathered}$$
Assume first that ${\mathscr D}$ has order $\le0$. This means $[{\mathscr D},M_x]=0$ for any $x\in{\mathscr A}$. Then $$\label{eq7.C}
{\mathscr D}x={\mathscr D}M_x 1=M_x{\mathscr D}1=x{\mathscr D}1.$$
Next, assume ${\mathscr D}$ has order $\le1$. This means that $[{\mathscr D},M_x]$ has order $\le0$. Then, using , we have for any $x,y\in{\mathscr A}$ $$\label{eq7.D}
{\mathscr D}(xy)={\mathscr D}M_x y=x {\mathscr D}y+[{\mathscr D}, M_x]y=x {\mathscr D}y+y[{\mathscr D},M_x]1=x{\mathscr D}y+y{\mathscr D}x-xy\mathscr D1.$$
Finally, assume ${\mathscr D}$ has order $\le2$. We are going to show that for any $x,y,z\in{\mathscr A}$ $$\label{eq7.E}
{\mathscr D}(xyz)=x{\mathscr D}(yz)+y{\mathscr D}(xz)+z{\mathscr D}(xy)-xy{\mathscr D}z-xz{\mathscr D}y-yz{\mathscr D}x+xyz{\mathscr D}1.$$ Once this is established, the desired formula is verified by induction on $n$. Namely, is the base of the induction ($n=3$), and in order to pass from $n$ to $n+1$ one applies with $x=x_1\dots x_{n-1}$, $y=x_n$, $z=x_{n+1}$.
It remains to prove , which is achieved using the same trick. We have $${\mathscr D}(xyz)={\mathscr D}M_x(yz)=x{\mathscr D}(yz)+[{\mathscr D},M_x](yz).$$ As $[{\mathscr D},M_x]$ has order $\le1$, we may apply , which gives $$[{\mathscr D},M_x](yz)=y[{\mathscr D}, M_x]z+z[{\mathscr D}, M_x]y-yz[{\mathscr D},M_x]1.$$ Next, $$y[{\mathscr D}, M_x]z=y{\mathscr D}(xz)-xy{\mathscr D}z, \qquad z[{\mathscr D}, M_x]y=z{\mathscr D}(xy)-xz{\mathscr D}y$$ and $$-yz[{\mathscr D},M_x]1=-yz{\mathscr D}x+xyz{\mathscr D}1.$$ Putting all the pieces together we get .
Plan of proof {#sect7.B}
-------------
The proof of Theorem \[thm7.A\] is reduced to the following two claims.
\[claim1\] The operators ${\mathbb D}_{{z,z',w,w'}}$ and ${\mathbb A}_{{z,z',w,w'}}$ coincide on the monomials of degree $\le 2$.
\[claim2\] The operator ${\mathbb A}_{{z,z',w,w'}}: R\to R$ has order $\le2$ in the abstract sense.
Since both operators are continuous in the $I$-adic topology of $R$, it suffices to prove that they coincide on the monomials $\varphi_\nu=\varphi_{\nu_1}\dots\varphi_{\nu_N}$.
Since ${\mathbb D}_{{z,z',w,w'}}$ is a second order differential operator, it has order $\le2$ in the abstract sense. The same holds for the operator ${\mathbb A}_{{z,z',w,w'}}$, by virtue of Claim \[claim2\]. Thus, both operators have order $\le2$.
Therefore, by Proposition \[prop7.B\], it suffices to know that the two operators coincide on monomials of degree $N\le2$, and this holds by virtue of Claim \[claim1\].
Claims \[claim1\] and \[claim2\] are proved in Section \[sect8\] and \[sect9\], respectively.
The structure of the proof reflects the way of how the differential operator ${\mathbb D}_{{z,z',w,w'}}$ has been found. Namely, assuming that ${\mathbb A}_{{z,z',w,w'}}$ is a second order differential operator we may write down it explicitly by computing its action on the monomials of degree $\le2$, and this what we actually do in the proof of Claim \[claim1\].
The proof is indirect, but it seems to me that a direct verification of the equality ${\mathbb D}_{{z,z',w,w'}}={\mathbb A}_{{z,z',w,w'}}$, without recourse to Claim \[claim2\], is a difficult task.
Proof of Claim 7.7 {#sect8}
==================
Beginning of proof
------------------
The differential operator ${\mathbb D}_{{z,z',w,w'}}$ does not contain terms of order 0, so it annihilates the constants. The same holds for the operator ${\mathbb A}_{{z,z',w,w'}}$, by the very definition.
Let us verify that $${\mathbb D}_{{z,z',w,w'}}\varphi_n={\mathbb A}_{{z,z',w,w'}}\varphi_n, \qquad n\in{\mathbb Z}.$$
By the definition of ${\mathbb D}_{{z,z',w,w'}}$, the left-hand side equals $$\begin{gathered}
B_n=(n+w+1)(n+w'+1)\varphi_{n+1}+(n-z-1)(n-z'-1)\varphi_{n-1}\\
-\bigl((n-z)(n-z')+(n+w)(n+w')\bigr)\varphi_n.
\end{gathered}$$
To compute the right-hand side we observe that $\varphi_n={\sigma}_{(n)}$ and then use the definition of ${\mathbb A}_{{z,z',w,w'}}$ (see ). It says that $${\mathbb A}_{{z,z',w,w'}}\varphi_n=q(n,n)\varphi_n+r(n+1,n)\varphi_{n+1}+r(n-1,n)\varphi_{n-1}.$$ Here the quantities $r(n\pm1,n)$ and $q(n,n)$ are given by formulas , , and , where we take $N=1$, so that $n$ and $n\pm1$ denote signatures of length 1. We get first $$r(n,n+1)=(z-n)(z'-n), \quad r(n,n-1)=(w+n)(w'+n),$$ which implies $$r(n-1,n)=(z-n+1)(z'-n+1), \quad r(n+1,n)=(w+n+1)(w'+n+1).$$ Next, $$q(n,n)=-r(n,n+1)-r(n,n-1)=-(z+n)(z'+n)-(w+n)(w'+n).$$ This gives the same quantity $B_n$, as desired.
A more difficult task is to check that the two operators coincide on quadratic monomials. That is, $$\label{eq8.N}
{\mathbb D}_{{z,z',w,w'}}\varphi_{\varkappa}={\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa}, \qquad {\varkappa}=(k_1,k_2)\in{\mathbb Z}^2, \quad k_1\ge
k_2.$$ The rest of the section is devoted to the proof of this equality.
Below we use the notation: $${\delta}:={\varepsilon}_1-{\varepsilon}_2=(1,-1)\in{\mathbb Z}^2.$$
Step 1
------
By , $$\varphi_{\varkappa}=\sum_{p=0}^\infty {\sigma}_{{\varkappa}+p{\delta}}.$$
So far we used the notation $r(\nu,\mu)$ for $\nu=\mu\pm {\varepsilon}_i$ only, but now it will be convenient to write $r(\mu,\mu)$ instead of $q(\mu,\mu)$. With this agreement we have $${\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa}=\sum_{p=0}^\infty\sum_{\varepsilon}r({\varkappa}+p{\delta}+{\varepsilon},{\varkappa}+p{\delta}){\sigma}_{{\varkappa}+p{\delta}+{\varepsilon}},$$ where ${\varepsilon}$ ranges over $\{\pm {\varepsilon}_1, \pm {\varepsilon}_2, 0\}$.
Next, by , $${\sigma}_{{\varkappa}+p{\delta}+{\varepsilon}}=\varphi_{{\varkappa}+p{\delta}+{\varepsilon}}-\varphi_{{\varkappa}+(p+1){\delta}+{\varepsilon}}.$$ Consequently, $$\begin{gathered}
\label{eq8.A}
{\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa}=\sum_{{\varepsilon}}r({\varkappa}+{\varepsilon},{\varkappa})\varphi_{{\varkappa}+{\varepsilon}}
+\sum_{p=1}^\infty\sum_{\varepsilon}\big[r({\varkappa}+p{\delta}+{\varepsilon},{\varkappa}+p{\delta})\\-r({\varkappa}+(p-1){\delta}+{\varepsilon},
{\varkappa}+(p-1){\delta})\big]\varphi_{{\varkappa}+p{\delta}+{\varepsilon}}.\end{gathered}$$
The right-hand side is a linear combination of elements $\varphi_{\,l_1l_2}$ such that the difference $(l_1+l_2)-(k_1+k_2)$ takes only three possible values: $\pm1$ and $0$. According to this we write ${\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa}$ as the sum of three components, $${\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa}=({\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa})_1+({\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa})_{-1}+({\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa})_0.$$
On the other hand, it follows from and that ${\mathbb D}_{{z,z',w,w'}}\varphi_{\varkappa}$ has the same property, so we write $${\mathbb D}_{{z,z',w,w'}}\varphi_{\varkappa}=({\mathbb D}_{{z,z',w,w'}}\varphi_{\varkappa})_1+({\mathbb D}_{{z,z',w,w'}}\varphi_{\varkappa})_{-1}+({\mathbb D}_{{z,z',w,w'}}\varphi_{\varkappa})_0.$$
Thus we are led to check three equalities, $$\label{eq8.B}
\begin{gathered}({\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa})_1=({\mathbb D}_{{z,z',w,w'}}\varphi_{\varkappa})_1, \qquad
({\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa})_{-1}=({\mathbb D}_{{z,z',w,w'}}\varphi_{\varkappa})_{-1},\\
({\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa})_0=({\mathbb D}_{{z,z',w,w'}}\varphi_{\varkappa})_0.
\end{gathered}$$
The first two equalities are equivalent because of the symmetry consisting in switching $$(z,z') \leftrightarrow(w,w'), \quad (k_1,k_2)\leftrightarrow (-k_2,-k_1),
\qquad (l_1,l_2)\leftrightarrow(-l_2,-l_1).$$ Therefore, it suffices to check the first and third equalities in .
Step 2
------
On this step, we write down explicitly the component $({\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa})_1$ of . It collects the contribution from the terms with ${\varepsilon}={\varepsilon}_1$ and ${\varepsilon}={\varepsilon}_2$. Because ${\delta}={\varepsilon}_1-{\varepsilon}_2$, we have $$p{\delta}+{\varepsilon}_1=(p+1){\delta}+{\varepsilon}_2.$$ Using this relation one can write $({\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa})_1$ in the following form: $$\label{eq8.G}
({\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa})_1=X_1+X_2,$$ where $$\label{eq8.C}
\begin{gathered}
X_1:=r({\varkappa}+{\varepsilon}_1,{\varkappa})\varphi_{{\varkappa}+{\varepsilon}_1}+r({\varkappa}+{\varepsilon}_2,{\varkappa})\varphi_{{\varkappa}+{\varepsilon}_2}\\
+\big[r({\varkappa}+{\delta}+{\varepsilon}_2,{\varkappa}+{\delta})-r({\varkappa}+{\varepsilon}_2,{\varkappa})\big]\varphi_{{\varkappa}+{\delta}+{\varepsilon}_2}
\end{gathered}$$ and $$\label{eq8.D}
\begin{gathered}
X_2:=\sum_{p=1}^\infty\big[r({\varkappa}+p{\delta}+{\varepsilon}_1,{\varkappa}+p{\delta})-r({\varkappa}+(p-1){\delta}+{\varepsilon}_1,{\varkappa}+(p-1){\delta})\\
+r({\varkappa}+(p+1){\delta}+{\varepsilon}_2,{\varkappa}+(p+1){\delta})-r({\varkappa}+p{\delta}+{\varepsilon}_2,{\varkappa}+p{\delta})\big]\varphi_{{\varkappa}+p{\delta}+{\varepsilon}_1}.
\end{gathered}$$
To proceed further we need the explicit values of the jump rates: if ${\lambda}=(l_1,l_2)$ with $l_1\ge l_2$, then $$\begin{gathered}
r({\lambda}+{\varepsilon}_1,{\lambda})=(w+l_1+2)(w'+l_1+2), \label{eq8.E}\\
r({\lambda}+{\varepsilon}_2,{\lambda})=\begin{cases}(w+l_2+1)(w'+l_2+1), &\text{if $l_1>l_2$}\\ 0,
& \text{if $l_1=l_2$}\end{cases}.\label{eq8.F}\end{gathered}$$
Let us substitute this in . Then ${\lambda}={\varkappa}+p{\delta}$ or ${\lambda}={\varkappa}+(p+1){\delta}$ with $p\ge1$, and in both cases one has $l_1>l_2$. After a simple computation one finds $$X_2=2\sum_{p=1}^\infty(2p+1+k_1-k_2)\varphi_{{\varkappa}+p{\delta}+{\varepsilon}_1}.$$ It is convenient to extend the summation to $p=0$ and, to compensate this, subtract from $X_1$ the term $2(k_1-k_2+1)\varphi_{{\varkappa}+{\varepsilon}_1}$.
Then we rewrite the decomposition in a modified form: $$\label{eq8.H}
({\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa})_1=X'_1+X'_2,$$ where $$\label{eq8.I}
X'_2=2\sum_{p=0}^\infty(2p+1+k_1-k_2)\varphi_{{\varkappa}+p{\delta}+{\varepsilon}_1}
=2\sum_{p=0}^\infty(2p+1+k_1-k_2)\varphi_{k_1+p+1}\varphi_{k_2-m}$$ and $$X'_1=X_1-2(k_1-k_2+1)\varphi_{{\varkappa}+{\varepsilon}_1}.$$
Finally, using again and one can check that $$\label{eq8.J}
X'_1=(w+k_1+1)(w'+k_1+1)\varphi_{k_1+1}\varphi_{k_2}
+(w+k_2+1)(w'+k_2+1)\varphi_{k_1}\varphi_{k_2+1}$$
Step 3
------
Now let us turn to $({\mathbb D}_{{z,z',w,w'}}\varphi_{\varkappa})_1$. This quantity results from an appropriate truncation of the operator ${\mathbb D}$. Namely, we replace it by $${\mathbb D}^{(1)}_{{z,z',w,w'}}:=\sum_{n\in{\mathbb Z}}A^{(1)}_{nn}\frac{{\partial}^2}{{\partial}\varphi_n^2}+2\sum_{\substack{n_1,n_2\in{\mathbb Z}\\
n_1>n_2}} A^{(1)}_{n_1 n_2}\frac{{\partial}^2}{{\partial}\varphi_{n_1}{\partial}\varphi_{n_2}}
+\sum_{n\in{\mathbb Z}}B^{(1)}_n\frac{{\partial}}{{\partial}\varphi_n}, $$ where, for any indices $n_1\ge n_2$, $$A^{(1)}_{n_1
n_2}=\sum_{p=0}^\infty(n_1-n_2+2p+1)\varphi_{n_1+p+1}\varphi_{n_2-p}$$ and, for any $n\in{\mathbb Z}$, $$B^{(1)}_n=(n+w+1)(n+w'+1)\varphi_{n+1}.$$
We represent $({\mathbb D}_{{z,z',w,w'}}\varphi_{\varkappa})_1={\mathbb D}^{(1)}_{{z,z',w,w'}}\varphi_{\varkappa}$ as the sum of two components, the one coming from the action of the first order derivatives and the other coming from the action of the second order derivatives. From the explicit expressions above one can readily check that these two components coincide with $X'_1$ and $X'_2$, respectively.
This completes the proof of the identity $({\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa})_1=({\mathbb D}_{{z,z',w,w'}}\varphi_{\varkappa})_1$, which is the first equality in . Now we apply similar arguments to prove the third equality in .
Step 4 (cf. Step 2 above)
-------------------------
Here we compute $({\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa})_0$. From we obtain $$\label{eq8.K}
\begin{aligned}
({\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa})_0=r({\varkappa},{\varkappa})\varphi_{\varkappa}&+\sum_{p=1}^\infty\big[r({\varkappa}+p{\delta},{\varkappa}+p{\delta})\\
&-r({\varkappa}+(p-1){\delta}, {\varkappa}+(p-1){\delta})\big]\varphi_{{\varkappa}+p{\delta}}.
\end{aligned}$$
Recall that $r({\lambda},{\lambda}):=q({\lambda},{\lambda})$. By , for ${\lambda}=(l_1,l_2)$ with $l_1\ge l_2$, $$\begin{aligned}
r({\lambda},{\lambda})=&-(z-l_1)(z'-l_1)-(w+l_1+1)(w'+l_1+1)\\
&-(z-l_2+1)(z'-l_2+1)-(w+l_2)(w'+l_2)\\
&+z+z'+w+w'+2.
\end{aligned}$$
We substitute this into and obtain $$({\mathbb A}_{{z,z',w,w'}}\varphi_{\varkappa})_0=Y_1+Y_2,$$ where $$\label{eq8.L}
\begin{aligned}
Y_1=\big\{&-(z-l_1)(z'-l_1)-(w+l_1+1)(w'+l_1+1)\\
&-(z-l_2+1)(z'-l_2+1)-(w+l_2)(w'+l_2)\\
&+z+z'+w+w'+2\big\}\varphi_{\varkappa}\end{aligned}$$ and $$\label{eq8.M}
Y_2=-2\sum_{p=1}^\infty(k_1-k_2+p)\varphi_{{\varkappa}+p{\delta}}.$$
Step 5 (cf. Step 3 above)
-------------------------
Let us turn to $({\mathbb D}_{{z,z',w,w'}}\varphi)_0$. We write $$({\mathbb D}_{{z,z',w,w'}}\varphi)_0={\mathbb D}^{(0)}_{{z,z',w,w'}}\varphi_{\varkappa}$$ with ${\mathbb D}^{(0)}_{{z,z',w,w'}}$ being the following truncated operator: $${\mathbb D}^{(0)}_{{z,z',w,w'}}=\sum_{n\in{\mathbb Z}}A^{(0)}_{nn}\frac{{\partial}^2}{{\partial}\varphi_n^2}+2\sum_{\substack{n_1,n_2\in{\mathbb Z}\\
n_1>n_2}} A^{(0)}_{n_1 n_2}\frac{{\partial}^2}{{\partial}\varphi_{n_1}{\partial}\varphi_{n_2}}
+\sum_{n\in{\mathbb Z}}B^{(0)}_n\frac{{\partial}}{{\partial}\varphi_n}, $$ where, for any indices $n_1\ge n_2$, $$A^{(0)}_{n_1 n_2}= -(n_1-n_2)\varphi_{n_1}\varphi_{n_2}$$ and, for any $n\in{\mathbb Z}$, $$B^{(0)}_n=-\bigl((n-z)(n-z')+(n+w)(n+w')\bigr)\varphi_n.$$
It is readily seen that the result of the action on $\varphi_{\varkappa}$ of the first derivatives in ${\mathbb D}^{(0)}_{{z,z',w,w'}}$ coincides with $Y_1$ (see ), while the action of the second derivatives leads to $Y_2$ (see ).
This completes the proof of . Thus, Claim \[claim1\] is proved, too.
Proof of Claim 7.8 {#sect9}
==================
Reduction of the problem
------------------------
Let us fix two nonnegative integers $k$ and $l$, not equal both to $0$.
\[prop9.D\] If $z=k$ and $w=l$ as above, then the operator ${\mathbb A}_{{z,z',w,w'}}$ preserves the ideal $I(k,-l)$, the kernel of the canonical map $R\to R(k,-l)$.
Let us set ${{\mathbb{S}}}(k,-l)=\bigcup_{N=1}^\infty{{\mathbb{S}}}_N(k,-l)$ (recall that the definition of ${{\mathbb{S}}}_N(n_+,n_-)$ is given in ). The ideal $I(k,-l)$ is the closed linear span of the basis elements ${\sigma}_\nu$ such that $\nu\notin{{\mathbb{S}}}(k,-l)$, where the closure is taken in the $I$-adic topology. Therefore, it suffices to prove the following: if $\nu\notin{{\mathbb{S}}}_N(k,-l)$ and $\mu\in{{\mathbb{S}}}_N(k,-l)$, then the quantity $r_{k,z',l,w'\mid N}(\nu,\mu)$ vanishes.
Next, this claim is readily verified by using the definition of $r_{{{z,z',w,w'}}\mid
N}(\nu,\mu)$, see and .
By Proposition \[prop5.C\], ${\mathbb A}_{{z,z',w,w'}}$ preserves the ideal $J$ (for arbitrary $({{z,z',w,w'}})$). Therefore, if $z=k$ and $w=l$, then ${\mathbb A}_{{z,z',w,w'}}$ preserves the ideal $J(k,-l)$ generated by $J$ and $I(k,-l)$, and hence gives rise to an operator on the quotient algebra $${\widehat}R(k,-l)=R/J(k,-l)={\mathbb C}[\varphi_{-l},\dots,\varphi_k]\big/(\varphi_{-l}+\dots+\varphi_k-1),$$ (this quotient has already appeared in ). Let us denote the latter operator by $\bar{\mathbb A}_{k,z',l,w'}$.
\[prop9.E\] To prove Claim \[claim2\] it suffices to show that the operators $\bar{\mathbb A}_{k,z',l,w'}$ have order $\le2$.
Indeed, Proposition \[prop7.B\] says that Claim \[claim2\] is equivalent to the relation $$[M_{\psi_3},[M_{\psi_2},[M_{\psi_1},{\mathbb A}_{{z,z',w,w'}}]]]\psi_4=0,$$ which has to hold for arbitrary four elements $\psi_1,\psi_2,\psi_3, \psi_4\in
R$. Without loss of generality we may assume that all these elements are homogeneous. Then the left-hand side is homogeneous, too, as it follows from the definition of ${\mathbb A}_{{z,z',w,w'}}$. By virtue of Proposition \[prop3.D\], it suffices to prove that the left-hand side belongs to $J$. Because ${\mathbb A}_{{z,z',w,w'}}$ preserves $J$ (Proposition \[prop7.C\]), this allows us to pass from from $R$ to its quotient ${\widehat}R=R/J$.
Next, we want to specify $z=k$, $w=l$ and to reduce the desired relation modulo the ideal $I(k,-l)$. This is possible for the following reasons:
1\. ${\mathbb A}_{{z,z',w,w'}}$ depends quadratically on the parameters, which allows us to specialize $({{z,z',w,w'}})$ to any Zariski dense subset of ${\mathbb C}^4$ (or even any subset which is a set of uniqueness for quadratic polynomials);
2\. as $k,l\to +\infty$, the ideals $I(k,-l)$ decrease and their intersection is $\{0\}$;
3\. we know that the operator $A_{{z,z',w,w'}}$ can be reduced modulo $I(k,-l)$ provided that $z=k$ and $w=l$.
As the result of the factorization modulo both $J$ and $I(k,-l)$ the algebra $R$ is reduced to the algebra $${\widehat}R(k,-l):={\mathbb C}[\varphi_{-l},\dots,\varphi_k]\big/(\varphi_{-l}+\dots+\varphi_k-1),$$ which is isomorphic to the algebra of polynomials in $m:=k+l$ variables (we have $m+1$ variables subject to a linear relation). This substantially simplifies our task, because instead of the operators ${\mathbb A}_{{z,z',w,w'}}$ acting on the huge space $R$ we may deal with the operators $\bar{\mathbb A}_{k,z',l,w'}$ acting on algebras of polynomials.
We have a large freedom in the choice of parameters $(z',w')$, because the argument above allows us to restrict them to an arbitrary set which is a set of uniqueness for quadratic polynomials. For the reasons that will become clear below it is convenient to set $z'=k+a$, $w'=l+b$, where $a$ and $b$ are real numbers $>-1$.
Thus, we have to show that the operator $\bar{\mathbb A}_{k,k+a,l,l+b}$, which acts on the algebra ${\widehat}R(k,-l)$, is of order $\le2$.
As explained in Subsection \[sect3.B\], we may realize ${\widehat}R(k,-l)$ as the algebra of polynomial functions on the simplex ${\Omega}(k,-l)$. Our aim is to show that in this realization, $\bar{\mathbb A}_{k,k+a,l,l+b}$ is given by a second order partial differential operator (the Jacobi operator). This will evidently imply that it has order $\le2$ in the abstract sense.
Finally, it is readily seen that the operator ${\mathbb A}_{{z,z',w,w'}}$ behaves exactly as ${\mathbb D}_{{z,z',w,w'}}$ with respect to the shift of variables $\varphi_n\mapsto\varphi_{n+{\operatorname{const}}}$ (see Proposition \[prop4.C\]). This allows us to assume, without loss of generality, that $l=0$, which slightly simplifies the notation.
Thus, in what follows we assume that $$\label{eq9.M}
z=m, \quad z'=m+a, \quad w=0, \quad w'=b,$$ where $m=1,2,\dots$ and $a,b>-1$, and we are dealing with the operator $\bar{\mathbb A}_{m,m+a,0,b}$ acting on ${\widehat}R(m,0)$.
The Jacobi differential operators {#sect9.B}
---------------------------------
As in Subsection \[sect3.B\] we introduce new variables $t_1,\dots,t_m$ related to $\varphi_0,\dots,\varphi_m$ in the following way: $$\sum_{n=0}^m\varphi_nu^n=\prod_{i=1}^m (t_i+(1-t_i)u),$$ where $u$ is a formal variable. In other words, we substitute for $\varphi_0,\dots,\varphi_m$ certain symmetric polynomials in $t_1,\dots,t_m$. Then we may identify ${\widehat}R(m,0)$ with the algebra of symmetric polynomials in variables $t_1,\dots,t_m$ (see Proposition \[prop3.B\]). We also regard $(t_1,\dots,t_m)$ as coordinates on ${\Omega}(m,0)$ with the understanding that $$1\ge t_1\ge\dots\ge t_m\ge0.$$
Let us introduce the *Jacobi differential operator* on $[0,1]$: $$D^{(a,b)}=t(1-t)\frac{d^2}{dt^2}+[b+1-(a+b+2)t]\frac{d}{dt}.$$ Its connection with the classic Jacobi orthogonal polynomials is explained below (Subsection \[sect9.A\]). Let us observe that $$\label{eq9.I}
D^{{(a,b)}}t^n=-n(n+a+b+1) t^n+\,\text{lower degree terms}, \qquad n=0,1,2,\dots\,.$$
Let $$V_m=V_m(t_1,\dots,t_m):=\prod_{1\le i<j\le m}(t_i-t_j), \qquad m=1,2,\dots,$$ and let $$D^{{(a,b)}}_{\text{\rm variable $t_i$}}:=t_i(1-t_i)\frac{\partial^2}{\partial
t_i^2}+[b+1-(a+b+2)t_i]\frac{\partial}{\partial t_i}$$ be a copy of the Jacobi operator applied to the $i$th variable, $i=1,\dots,m$. From and the fact that $V_m$ is the Vandermonde determinant it follows that $$\label{eq9.E}
\left(\sum_{i=1}^m D^{{(a,b)}}_{\text{\rm variable $t_i$}}\right)V_m=-{\operatorname{const}}_{a,b,m}
V_m,$$ where $$\label{eq9.F}
{\operatorname{const}}_{a,b,m}:=\sum_{n=0}^{m-1}n(n+a+b+1).$$
Now we introduce the *$m$-variate Jacobi differential operator*, $m=2,3,\dots$, by $$\begin{gathered}
D^{(a,b)}_m :=\frac1{V_m}\circ\left(\sum_{i=1}^m D^{{(a,b)}}_{\text{\rm variable
$t_i$}}\right)\circ
V_m+{\operatorname{const}}_{a,b,m}\label{eq9.G}\\
=\sum_{i=1}^m \left(t_i(1-t_i)\frac{\partial^2}{\partial
t_i^2}+\left[b+1-(a+b+2)t_i+\sum_{j:\, j\ne
i}\frac{2t_i(1-t_i)}{t_i-t_j}\right]\frac{\partial}{\partial
t_i}\right).\label{eq9.H}\end{gathered}$$ The meaning of is that the partial differential operator $\sum_{i=1}^m D^{{(a,b)}}_{\text{\rm variable
$t_i$}}$ is conjugated by the operator of multiplication by the Vandermonde $V_m$, and adding ${\operatorname{const}}_{a,b,m}$ kills the constant term that arises after conjugation. The equality between and is verified directly (actually, in what follows, we use only ).
Note that, although the coefficients of the first order derivatives in have singularities along the diagonals $t_i=t_j$, the action of $D^{(a,b)}_m$ on the space of symmetric polynomials is well defined. Indeed, let us look at : the operator of multiplication by $V_m$ transforms symmetric polynomials into antisymmetric ones, then the application of the symmetric partial differential operator $\sum_{i=1}^m D^{{(a,b)}}_{\text{\rm variable
$t_i$}}$ leaves the space of antisymmetric polynomials invariant, and finally division by $V_m$ transforms it back into the space of symmetric polynomials.
(The construction of a partial differential (or difference) operator related to multivariate orthogonal polynomials that we used in (and also in below) is well known. The probabilistic meaning of this construction is related to Doob’s $h$-transform, see König [@Konig].)
The arguments of the preceding subsection reduce Claim \[claim2\] to the following theorem.
\[thm9.A\] As explained above, we identify ${\widehat}R(m,0)$ with the algebra of symmetric polynomials in $m$ variables $t_1,\dots,t_m$. Then the action of the operator $\bar{\mathbb A}_{m,m+a,0,b}$ on this algebra is implemented by the $m$-variate Jacobi differential operator $D^{(a,b)}_m$.
The proof occupies the rest of the section. Here is the scheme of proof.
As explained in Subsection \[sect5.A\], we dispose of finite stochastic matrices ${\Lambda}^{N+1}_N: {{\mathbb{S}}}_{N+1}(m,0)\dasharrow {{\mathbb{S}}}_N(m,0)$ and the links ${\Lambda}^\infty_N:{\Omega}(m,0)\dasharrow{{\mathbb{S}}}_N(m,0)$. Let, as above, $C({{\mathbb{S}}}_N(m,0))$ stand for the space of functions on the finite set ${{\mathbb{S}}}_N(m,0)$. The link ${\Lambda}^\infty_N$ maps $C({{\mathbb{S}}}_N(m,0))$ into $C({\Omega}(m,0))$, and the image is actually contained in ${\widehat}R(m,0)\subset C({\Omega}(m,0))$. As $N$ grows, this image enlarges (because of the relation ${\Lambda}^\infty_N={\Lambda}^\infty_{N+1}{\Lambda}^{N+1}_N$) and in the limit as $N\to\infty$ it exhausts the whole space ${\widehat}R(m,0)$. This point will be explained in more detail below.
Recall that the operator ${\mathbb A}_{{z,z',w,w'}}$ was defined through the difference operators $D_{{{z,z',w,w'}}\mid N}$. In the special case when $z=m$ and $w=0$, the $N$th difference operator is well defined on the subset ${{\mathbb{S}}}_N(m,0)$. From the definition of operator $\bar{\mathbb A}_{m,m+a,0,b}$ it follows that it is characterized by the commutation relations $$\bar{\mathbb A}_{m,m+a,0,b}{\Lambda}^\infty_N={\Lambda}^\infty_N D_{m, m+a,0,b\mid N},$$ where $N=1,2,\dots$ and the both sides are viewed as operators from the finite-dimensional space $C({{\mathbb{S}}}_N(m,0))$ to ${\widehat}R(m,0)$. We will prove that in these relations, $\bar{\mathbb A}_{m,m+a,0,b}$ can be replaced by the Jacobi operator $D^{(a,b)}_m$. That is, one has $$\label{eq9.L}
D^{(a,b)}_m{\Lambda}^\infty_N={\Lambda}^\infty_N D_{m, m+a,0,b\mid N},$$ This will imply the desired equality $\bar{\mathbb A}_{m,m+a,0,b}=D^{(a,b)}_m$.
The signatures ${\lambda}\in{{\mathbb{S}}}_N(m,0)$ can be viewed as Young diagrams contained in the rectangular diagram $$(m^N):=(\,\underbrace{m,\dots,m}_N\,).$$ Given such a diagram ${\lambda}$, we associate with it the complementary diagram ${\varkappa}\subseteq(N^m)$: it is obtained from the shape $(m^N)\setminus{\lambda}$ by rotation and conjugation.
The proof is divided into three steps:
*Step* 1. We express ${\Lambda}^\infty_N$ in terms of $(t_1,\dots,t_m)$ and ${\varkappa}$ (Proposition \[prop9.B\]).
*Step* 2. We show that under the correspondence ${\lambda}\leftrightarrow{\varkappa}$, the difference operator $D_{m, m+a,0,b\mid N}$ in the right-hand side of turns into the *$m$-variate Hahn difference operator* (Proposition \[prop9.A\]). As the result, takes the form $$\label{eq9.D}
D^{(a,b)}_m{\Lambda}^\infty_N={\Lambda}^\infty_N{\Delta}^{(a,b,N+m-1)}_m, \qquad N=1,2,\dots\,,$$ where ${\Delta}^{(a,b,N+m-1)}_m$ is the Hahn difference operator in question.
*Step* 3. We prove that ${\Lambda}^\infty_N$ transforms the $m$-variate symmetric Hahn polynomials into the respective $m$-variate symmetric Jacobi polynomials (Proposition \[prop9.C\]). Then the proof is readily completed.
We proceed to the detailed proof of the theorem.
Step 1: transformation of the link ${\Lambda}^\infty_N$
-------------------------------------------------------
Let ${\lambda}$ range over the set of Young diagrams contained in the rectangle $(m^N)$, and ${\varkappa}\subseteq(N^m)$ be the complementary diagram to ${\lambda}$. In more detail, $${\varkappa}=(N-{\lambda}'_m, \dots,N-{\lambda}'_1),$$ where the diagram ${\lambda}'$ is conjugate to the diagram ${\lambda}$. Next, we set $$l_i:={\lambda}_i+N-i, \quad i=1,\dots,N; \qquad k_j={\varkappa}_j+m-j, \qquad j=1,\dots,m.$$ Evidently, $l_1>\dots>l_N$ and $k_1>\dots>k_m$.
\[lemma9.C\] The set $\{0,\dots,N+m-1\}$ is the disjoint union of the sets ${\mathscr L}:=\{l_1,\dots,l_N\}$ and ${\mathscr K}:=\{k_1,\dots,k_m\}$.
This is a well-known fact, see e.g. Macdonald [@Ma95 ch. I, (1.7)].
Introduce a notation: $$M:=N+m-1, \quad {\mathbb I_M}=\{0,\dots,M\}.$$ Next, for a finite collection of numbers $X=\{x_1>\dots>x_n\}$ we set $$V(X)=V_n(x_1,\dots,x_n)=\prod_{1\le i<j\le n}(x_i-x_j).$$
\[lemma9.A\] One has $$\label{eq9.A}
V(l_1,\dots,l_N)=\frac{0!1!\dots M!\,V(k_1,\dots,k_m)}{\prod\limits_{j=1}^m
k_j!(M-k_j)!}$$
By the preceding lemma, ${\mathbb I_M}={\mathscr L}\sqcup{\mathscr K}$, whence $$\label{eq9.P}
V({\mathbb I_M})=V({\mathscr K}\sqcup {\mathscr L})=V({\mathscr K})\cdot V({\mathscr L})\cdot \prod_{x\in {\mathscr K}}\prod_{y\in {\mathscr L}}|x-y|.$$
For $x\in{\mathscr K}$, set $$f(x):=\prod_{z\in{\mathbb I_M}\setminus \{x\}}|x-z|$$ and observe that $$\prod_{x\in {\mathscr K}}\prod_{y\in
{\mathscr L}}|x-y|=\frac{\prod\limits_{x\in{\mathscr K}}f(x)}{(V({\mathscr K}))^2}.$$ Substituting this into gives $$V({\mathscr L})=\frac{V({\mathbb I_M})V({\mathscr K})}{\prod\limits_{x\in {\mathscr K}}f(x)}.$$
On the other hand, it is readily checked that $$f(x)=x!(M-x)!$$ and $V({\mathbb I_M})=0!1!\dots M!$. This completes the proof.
\[prop9.B\] Let ${\omega}={\omega}(t_1,\dots,t_m)$ be the point of the simplex ${\Omega}(m,0)$ with coordinates $(t_1,\dots,t_m)$. In the notation introduced above, $${\Lambda}^\infty_N({\omega};{\lambda})= {\operatorname{const}}_{m,M}\frac{V(k_1,\dots,k_m)}{V(t_1,\dots,t_m)}
\det\left[\binom{M}{k_j}t_i^{k_j}(1-t_i)^{M-k_j}\right]_{i,j=1}^m,$$ where $${\operatorname{const}}_{m,M}=\prod_{i=1}^m\frac{(M-i+1)!}{M!}.$$
In particular, in the simplest case $m=1$, there is a single coordinate $t=t_1\in[0,1]$, the diagram ${\lambda}$ has a single column, the complementary diagram has a single row whose length equals ${\varkappa}_1=k\in\{0,\dots,N\}$, and ${\Lambda}^\infty_N$ is represented as the link $[0,1]\dasharrow\{0,\dots,N\}$ that assigns to a point $t\in[0,1]$ the binomial distribution on $\{0,\dots,N\}$ with parameter $t$.
\(i) By the very definition of the link ${\Lambda}^\infty_N$ (see and the comment after it), $${\Lambda}^\infty_N({\omega},{\lambda})={\operatorname{Dim}}_N{\lambda}\cdot\left\{\text{\rm coefficient of
$s_{\lambda}(u_1,\dots,u_N)$ in $\Phi(u_1;{\omega})\dots\Phi(u_N;{\omega})$}\right\}.$$
We have $$\begin{gathered}
\Phi(u_1;{\omega})\dots\Phi(u_N;{\omega})=\prod_{i=1}^m\prod_{j=1}^N(1+{\beta}^+_i(u_j-1))
=\prod_{i=1}^m\prod_{j=1}^N(t_i+(1-t_i)u_j)\\
=\prod_{i=1}^m(1-t_i)^N \cdot
\prod_{i=1}^m\prod_{j=1}^N\left(\frac{t_i}{1-t_i}+u_j\right)\\
=\prod_{i=1}^m(1-t_i)^N \cdot \sum_{{\lambda}: {\lambda}\subseteq
(m^N)}s_{\varkappa}\left(\frac{t_1}{1-t_1},\dots,\frac{t_m}{1-t_m}\right)s_{\lambda}(u_1,\dots,u_N),\end{gathered}$$ where the last equality follows from the dual Cauchy identity, see [@Ma95 Chapter I, Section 4, Example 5]. Therefore, $${\Lambda}^\infty_N({\omega},{\lambda})={\operatorname{Dim}}_N{\lambda}\cdot \prod_{i=1}^m(1-t_i)^N \cdot
s_{\varkappa}\left(\frac{t_1}{1-t_1},\dots,\frac{t_m}{1-t_m}\right).$$
\(ii) By the definition of the Schur polynomials, $$\prod_{i=1}^m(1-t_i)^N\cdot
s_{\varkappa}\left(\frac{t_1}{1-t_1},\dots,\frac{t_m}{1-t_m}\right)
=\frac{\prod\limits_{i=1}^m(1-t_i)^N\cdot
\det\left[\left(\dfrac{t_i}{1-t_i}\right)^{k_j}\right]}
{V\left(\dfrac{t_1}{1-t_1},\dots,\dfrac{t_m}{1-t_m}\right)},$$ where the determinant in the numerator is of order $m$.
The denominator of this expression is equal to $$\prod_{i=1}^m(1-t_i)^{-m+1}\cdot V(t_1,\dots,t_m).$$ Therefore, the whole expression is $$\frac{\prod\limits_{i=1}^m(1-t_i)^M\cdot
\det\left[\left(\dfrac{t_i}{1-t_i}\right)^{k_j}\right]} {V(t_1,\dots,t_m)} =
\frac{\det\left[t_i^{k_j}(1-t_i)^{M-k_j}\right]} {V(t_1,\dots,t_m)},$$ so that $${\Lambda}^\infty_N({\omega},{\lambda})=\frac{{\operatorname{Dim}}_N{\lambda}}{V(t_1,\dots,t_m)}
\det\left[t_i^{k_j}(1-t_i)^{M-k_j}\right].$$
\(iii) It remains to handle ${\operatorname{Dim}}_N{\lambda}$. By Weyl’s dimension formula, $${\operatorname{Dim}}_N{\lambda}=\frac{V({\mathscr L})}{V(N-1,N-2,\dots,0)}=\frac{V(l_1,\dots,l_N)}{0!1!\dots(N-1)!}$$
The numerator has been computed in Lemma \[lemma9.A\]. Applying it we get $${\Lambda}^\infty_N({\omega},{\lambda})=\frac{0!1!\dots M!}{0!1!\dots
(N-1)!}\frac{V(k_1,\dots,k_m)}{V(t_1,\dots,t_m)}
\det\left[\frac1{k_j!(M-k_j)!}\,t_i^{k_j}(1-t_i)^{M-k_j}\right].$$
The constant factor in front equals $\prod_{j=1}^m(M-j+1)!$. Dividing it by $(M!)^m$ and introducing the same quantity inside the determinant we finally get the desired expression.
Step 2: transformation of the difference operator $D_{m, m+a,0,b}$
------------------------------------------------------------------
We continue to deal with two mutually complementary point configurations ${\mathscr L}=(l_1>\dots>l_N)$ and ${\mathscr K}=(k_1>\dots>k_m)$ on the lattice interval ${\mathbb I_M}=\{0,\dots,M\}$. Our next aim is to derive a convenient expression for the jump rates introduced in Subsection \[sect6.A\]. So far they were denoted as $q(\nu,\nu\pm {\varepsilon}_i)$. Now we rename $\nu$ to ${\lambda}$ and next we pass from ${\lambda}$ to the corresponding point configuration ${\mathscr L}$. In terms of ${\mathscr L}$, the transition ${\lambda}\to{\lambda}\pm {\varepsilon}_i$ can be written as $x\to x\pm1$, where $x=l_i$. According to this we change the former notation for the jump rates and will denote them by $q(x\to x\pm1)$, with the understanding that $x\in{\mathscr L}$.
Taking into account the values of the parameters (see ), the formulas of Subsection \[sect6.A\] can be rewritten as follows $$\begin{gathered}
q(x\to x+1)=\frac{V({\mathscr L}-\{x\}+\{x+1\})}{V({\mathscr L})}\,(M-x)(M+a-x), \label{eq9.B}\\
q(x\to x-1)=\frac{V({\mathscr L}-\{x\}+\{x-1\})}{V({\mathscr L})}\,x(b+x).\label{eq9.C}\end{gathered}$$ Here ${\mathscr L}-\{x\}+\{x\pm1\}$ denotes the configuration obtained from ${\mathscr L}$ by removing $x$ and inserting $x\pm1$ instead.
Note that the transition $x\to x+1$ is forbidden if the corresponding vector ${\lambda}+{\varepsilon}_i$ is not a signature, which happens when ${\lambda}_{i-1}={\lambda}_i$. In terms of ${\mathscr L}$, this means $x+1\in{\mathscr L}$, in which case the configuration ${\mathscr L}-\{x\}+\{x+1\}$ contains the point $x+1$ twice, and then $V({\mathscr L}-\{x\}+\{x+1\})$ should be understood as $0$. Likewise, if $x\to x-1$ is forbidden, then $V({\mathscr L}-\{x\}+\{x-1\})$ vanishes. Thus, and formally assign rate 0 to forbidden transitions, which is reasonable.
\[lemma9.D\] In terms of the complementary configuration ${\mathscr K}$, the jump rates take the form $$\begin{gathered}
{\widetilde}q(y\to y-1)=\frac{V({\mathscr K}-\{y\}+\{y-1\})}{V({\mathscr K})}y(M+1+a-y),\\
{\widetilde}q(y\to y+1)=\frac{V({\mathscr K}-\{y\}+\{y+1\})}{V({\mathscr K})}(M-y)(b+y+1).\end{gathered}$$
A jump $x\to x+1$ in ${\mathscr L}$ is possible if and only if $x\in{\mathscr L}$ and $x+1\notin{\mathscr L}$. This is equivalent to saying that $x+1\in{\mathscr K}$ and $x\notin {\mathscr K}$, which in turn means the possibility of the jump $y\to y-1$, where $y=x+1$. Therefore, ${\widetilde}q(y\to y-1)=q(x\to x+1)$.
Now we have to express the quantity $q(x\to x+1)$ given by in terms of ${\mathscr K}$. Lemma \[lemma9.A\] tell us that $$V({\mathscr L})={\operatorname{const}}\,\frac{V({\mathscr K})}{\prod\limits_{y\in{\mathscr K}}y!(M-y)!}.$$ It follows that $$\frac{V({\mathscr L}-\{x\}+\{x+1\})}{V({\mathscr L})}=\frac{V({\mathscr K}-\{y\}+\{y-1\})}{V({\mathscr K})}\frac{y}{M+1-y}$$ Next, $$(M-x)(M+a-x)=(M+1-y)(M+1+a-y).$$ Multiplying out these two quantities we get the desired expression for ${\widetilde}q(y\to y-1)$.
Likewise, the jump $x\to x-1$ is equivalent to $y\to y+1$, where $y=x-1$, so we rewrite the expression for $q(x\to x-1)$ given by . We have $$\frac{V({\mathscr L}-\{x\}+\{x-1\})}{V({\mathscr L})}=\frac{V({\mathscr K}-\{y\}+\{y+1\})}{V({\mathscr K})}\frac{M-y}{y+1}.$$ Next, $$x(b+x)=(y+1)(b+y+1).$$ Multiplying out these two quantities we get the desired expression for ${\widetilde}q(y\to y+1)$.
We introduce the *Hahn difference operator* ${\Delta}^{{(a,b,M)}}$ by $$\label{eq9.J}
\begin{aligned}
({\Delta}^{{(a,b,M)}}F)(y) =(y+b+1)(M-y)&[F(y+1)-F(y)]\\ +y(M+a-y+1)&[F(y-1)-F(y)],
\end{aligned}$$ where $F$ is a function in variable $y$. Note that ${\Delta}^{{(a,b,M)}}$ is well defined on ${\mathbb I_M}$. Indeed, the coefficient in front of $[F(y+1)-F(y)]$ vanishes at the point $y=M$, the right end of the interval; likewise, the coefficient in front of $[F(y-1)-F(y)]$ vanishes at the left end $y=0$.
The difference operator ${\Delta}^{{(a,b,M)}}$ is associated with the classic Hahn polynomials: see Koekoek-Swarttouw [@KS (1.5.5)] and the next subsection. Note that our parameters $(a,b,M)$ correspond to parameters $({\beta},{\alpha},N)$ from [@KS Section 1.5].
It is directly verified that $${\Delta}^{{(a,b,M)}}y^n=-n(n+a+b+1)y^n+\text{lower degree terms}, \qquad n=0,1,2,\dots\,.$$ Note that the factor in front of $y^n$ is exactly the same as in . In particular, it does not depend on the additional parameter $M$ that enters the definition of the difference operator.
Now we introduce the *$m$-variate Hahn difference operator* in the same way as we defined above the $m$-variate Jacobi operator: $$\label{eq9.Q}
{\Delta}^{{(a,b,M)}}_m=\frac1{V_m}\circ\left(\sum_{i=1}^m{\Delta}^{{(a,b,M)}}_{\text{\rm variable\,}
y_i}\right)\circ V_m+{\operatorname{const}}_{a,b,m}.$$ Here $y_1,\dots,y_m$ is an $m$-tuple of variables, $V_m=V_m(y_1,\dots,y_m)$ is the Vandermonde, ${\Delta}^{{(a,b,M)}}_{\text{\rm variable\,} y_i}$ denotes the one-variate Hahn operator acting on the $i$th variable, and the constant is given by . The same argument as above shows that the operator ${\Delta}^{{(a,b,M)}}_m$ is well defined on the space of symmetric polynomials and kills the constants.
Alternatively, ${\Delta}^{{(a,b,M)}}_m$ can be interpreted as an operator acting on the space of functions on $m$-point configurations ${\mathscr K}=(k_1>\dots>k_m)\subseteq(N^m)$ (here we write $(k_1,\dots,k_m)$ instead of $(y_1,\dots,y_m)$). This is just the interpretation that we need.
On the other hand, the difference operator $D_{m,m+1,0,b\mid N}$ acts on the functions defined on set of the diagrams ${\lambda}$ or, equivalently, on the set of configurations ${\mathscr L}$.
Now we use the correspondence ${\mathscr L}\leftrightarrow {\mathscr K}$ to compare the both operators.
\[prop9.A\] Under the correspondence ${\lambda}\leftrightarrow{\mathscr L}\leftrightarrow
{\mathscr K}\leftrightarrow{\varkappa}$, the operator $D_{m,m+a,0,b\mid N}$ turns into the operator ${\Delta}^{{(a,b,M)}}_m$.
Let us regard $D_{m,m+a,0,b\mid N}$ as an operator on the space of functions $F({\mathscr K})$. Then Lemma \[lemma9.D\] shows that $D_{m,m+a,0,b\mid N}$ acts as the following difference operator $$(D_{m,m+a,0,b\mid N} F)({\mathscr K})=\sum_{y\in{\mathscr K}}\sum_{{\varepsilon}=\pm1}{\widetilde}q(y\to
y+{\varepsilon})[F({\mathscr K}-\{y\}+\{y+{\varepsilon}\})-F({\mathscr K})].$$ Looking at the explicit expressions for the jump rates ${\widetilde}q(y\to y+{\varepsilon})$ given in Proposition \[prop9.A\] and comparing them with the definition of ${\Delta}^{{(a,b,M)}}_m$ (see ) we conclude that $D_{m,m+a,0,b\mid
N}={\Delta}^{{(a,b,M)}}_m$.
Step 3: The transformation Hahn $\to$ Jacobi {#sect9.A}
--------------------------------------------
Let us collect a few classic formulas about the Hahn and Jacobi orthogonal polynomials. They can be found, e.g., in Koekoek-Swarttouw [@KS].
The *Hahn polynomials* with parameters $(a,b,M)$, denoted here by $H^{{(a,b,M)}}_n(y)$, are the orthogonal polynomials on ${\mathbb I_M}=\{0,\dots,M\}$ with the weight $$W^{{(a,b,M)}}_{\text{\rm Hahn}}(y)=\binom{b+y}y \binom{a+M-y}{M-y}, \qquad y\in{\mathbb I_M}.$$ The subscript $n$ is the degree; it ranges also over ${\mathbb I_M}$. As was already pointed out, our notation slightly differs from that of [@KS]: our parameters $(a,b)$ correspond to parameters $({\beta},{\alpha})$ in [@KS Section 1.5].
The Hahn polynomials form an eigenbasis for the Hahn difference operator ${\Delta}^{{(a,b,M)}}$ defined in : $$\label{eq9.N}
{\Delta}^{{(a,b,M)}}H^{{(a,b,M)}}_n=-n(n+b+a+1)H^{{(a,b,M)}}_n.$$
Here is the explicit expression of the Hahn polynomials through a terminating hypergeometric series of type $(3,2)$ at point $1$: $$H^{{(a,b,M)}}_n(y)={}_3 F_2\left[\begin{matrix}-n,\, n+b+a+1,\, -y\\
b+1,\, -M\end{matrix}\,\Biggl|\,1\right], \qquad n=0,\dots,M.$$
Our notation for the Jacobi polynomials is $J^{{(a,b)}}_n(t)$; these are the orthogonal polynomials on the unit interval $[0,1]$ with the weight $$W^{{(a,b)}}_{\text{\rm Jacobi}}(t)=t^b(1-t)^a, \qquad 0\le t\le 1.$$ Note that many sources, including [@KS], take the weight function $(1-x)^a(1+x)^b$ with $x$ ranging over $[-1,1]$. The passage from $[0,1]$ to $[-1,1]$ is given by the change of variable $x=2t-1$.
The Jacobi polynomials form an eigenbasis for the Jacobi difference operator: $$\label{eq9.O}
D^{{(a,b)}}J^{{(a,b)}}_n=-n(n+b+a+1)J^{{(a,b)}}_n, \qquad n=0,1,2,\dots\,.$$
The Jacobi polynomials are expressed through the Gauss hypergeometric series: $$J^{{(a,b)}}_n(t)={}_2 F_1\left[\begin{matrix}-n,\, n+b+a+1\\
b+1\end{matrix}\,\Biggl|\,t\right], \qquad n=0,1,2,\dots\,.$$ Note that our normalization of the Jacobi polynomials differs from the conventional one, but this is convenient for the computation below.
\[lemma9.B\] The following relation holds $$\sum_{k=0}^M\binom{M}{k}t^k(1-t)^{M-k}H^{{(a,b,M)}}_n(k)=J^{{(a,b)}}_n(t), \qquad
n=0,\dots,M.$$
This is checked directly using the explicit expressions for the polynomials. Indeed, the sum in the left-hand side equals $$\sum_{k=0}^M\sum_{p=0}^n\frac{M!t^k(1-t)^{M-k}(-n)_p(n+b+a+1)_p(-k)_p}{k!(M-k)!(b+1)_p(-M)_pp!}.$$
Let us change the order of summation and observe that $(-k)_p$ vanishes unless $k\ge p$. Then the above expression can be rewritten as $$\sum_{p=0}^n\sum_{k=p}^M\frac{M!t^k(1-t)^{M-k}(-n)_p(n+b+a+1)_p(-k)_p}{k!(M-k)!(b+1)_p(-M)_pp!}.$$
Next, let us set $q=k-p$ and observe that $$\frac{M!(-k)_p}{k!(M-k)!(-M)_p}=\frac{M!k!(M-p)!}{k!(k-p)!M!(M-k)!}=\binom{M-p}q.$$
It follows that our double sum equals $$\sum_{p=0}^n\frac{(-n)_p(n+b+a+1)_p
}{(b+1)_pp!}\,t^p\,\sum_{q=0}^{M-p}\binom{M-p}qt^q(1-t)^{M-p-q}.$$
The interior sum equals 1, so that we finally get $$\sum_{p=0}^n\frac{(-n)_p(n+b+a+1)_p }{(b+1)_pp!}\,t^p=J^{{(a,b)}}_n(t),$$ as desired.
The *$m$-variate Hahn polynomials* are given by $$H^{{(a,b,M)}}_\nu(y_1,\dots,y_m)=\frac{\det\left[H^{{(a,b,M)}}_{n_j}(y_i)\right]}{V_m(y_1,\dots,y_m)}.$$ Here $\nu$ is an arbitrary Young diagram contained in $(N^m)$ and $$n_j:=\nu_j+m-j, \qquad j=1,\dots,m.$$ The definition is correct because the largest index $n_1$ does not exceed $M$ (recall that $M=N+m-1$; therefore, $\nu\subseteq(N^m)$ implies $n_1=\nu_1+m-1\le M$).
Likewise, the *$m$-variate Jacobi polynomials* are given by $$J^{{(a,b)}}_\nu(t_1,\dots,t_m)=\frac{\det\left[J^{{(a,b)}}_{n_j}(t_i)\right]}{V_m(t_1,\dots,t_m)}.$$ Here $\nu$ is an arbitrary Young diagram with at most $m$ nonzero rows.
\[prop9.C\] For every $N=1,2,\dots$ and every Young diagram $\nu\subseteq(N^m)$, the operator ${\Lambda}^\infty_N$ takes the Hahn polynomial $H^{{(a,b,M)}}_\nu$ to the respective Jacobi polynomial $J^{{(a,b)}}_\nu$, within a constant factor.
By virtue of Proposition \[prop9.B\], $$\begin{gathered}
\label{eq9.K}
{\Lambda}^\infty_N H^{{(a,b,M)}}_\nu)(t_1,\dots,t_m)\\
=\frac{{\operatorname{const}}_{m,M}}{V(t_1,\dots,t_m)}\sum_{M\ge
k_1>\dots>k_m\ge0}\det\left[\binom{M}{k_j}t_i^{k_j}(1-t_i)^{M-k_j}\right]
\det\left[H^{{(a,b,M)}}_{n_i}(k_j)\right].\end{gathered}$$
Now we apply a well-known identity, which is a consequence of the Cauchy-Binet identity: $$\sum_{M\ge k_1>\dots>k_m\ge0}\det[f_i(k_j)]_{i,j=1}^m\det[g_i(k_j)]_{i,j=1}^m=
\det[h_{ij}]_{i,j=1}^m,$$ where $$h_{ij}:=\sum_{k=0}^M f_i(k)g_j(k).$$ It tells us that the sum in equals the determinant of the $m\times m$ matrix whose $(i,j)$ entry is $$\sum_{k=0}^M\binom{M}{k}t_i^{k}(1-t_i)^{M-k}H_{n_j}(k).$$ By Lemma \[lemma9.B\], the last sum equals $J^{{(a,b)}}_{n_j}(t_i)$. This completes the proof of the proposition.
Completion of proof
-------------------
As pointed out above (see and ), the classic Hahn and Jacobi polynomials are eigenfunctions of the respective operators, and the $n$th eigenvalue in both cases is the same number $c(n):=-n(n+a+b+1)$.
By the very definition of the multivariate polynomials and operators, the similar assertion holds for arbitrary $m$ as well, and the eigenvalue corresponding to a given label $\nu$ is equal to $$\sum_{i=1}^m[c(\nu_i+m-i)-c(m-i)].$$
Combining this with the result of Step 3 (Proposition \[prop9.C\]) we obtain the desired commutation relation which says that the link ${\Lambda}^\infty_N$ intertwines the Jacobi differential operator $D^{(a,b)}_m$ with the Hahn difference operator ${\Delta}^{(a,b,N+m-1)}_m$.
Finally, as pointed out in the end of Subsection \[sect9.B\], the result of Step 2 (Proposition \[prop9.A\]) reduces Theorem \[thm9.A\] to that commutation relation.
This completes the proof of Theorem \[thm9.A\], which in turn implies Claim \[claim2\]. Thus, the proof of Theorem \[thm7.A\] is completed.
Appendix: uniform boundedness of multiplicities {#sect10}
===============================================
Here we prove the statement used in the proof of Proposition \[prop2.F\], step 1. We formulate the result in a greater generality, which seems to be more natural.
Let ${\widetilde}G$ be a connected reductive complex group and $G\subset{\widetilde}G$ be a reductive subgroup. We assume $G$ is spherical, meaning that for any simple ${\widetilde}G$-module $V$, the space $V^G$ of $G$-invariants has dimension at most 1. For a simple $G$-module $W$ we write $$[V:W]:=\dim \operatorname{Hom}_G(W,V).$$
\[prop10.A\] Let ${\widetilde}G$, $G$, $V$, and $W$ be as above. If $W$ is fixed, then for the multiplicity $[V:W]$ there exists a bound $[V:W]\le{\operatorname{const}}$, where the constant depends only on $W$ but not on $V$.
The fact that we needed in Proposition \[prop2.E\] is a particular case of Proposition \[prop10.A\] corresponding to ${\widetilde}G=GL(M+N,{\mathbb C})$ and $G=GL(M,{\mathbb C})\times GL(N,{\mathbb C})$.
Let us fix a Borel subgroup $B\subset{\widetilde}G$ and denote by $N$ the unipotent radical of $B$. Let $A={\mathbb C}[{\widetilde}G/N]$ be the algebra of regular functions on ${\widetilde}G/N$. In other words, $A$ consists of holomorphic functions on ${\widetilde}G/N$ which are ${\widetilde}G$-finite with respect to the action of ${\widetilde}G$ by left shifts. As a ${\widetilde}G$-module, $A$ is the multiplicity free direct sum of all simple ${\widetilde}G$-modules: $$\label{eq10.A}
A=\bigoplus_{{\lambda}\in{\Lambda}_+}A_{\lambda},$$ where ${\Lambda}_+$ denotes the additive semigroup of dominant weights with respect to $B$ and $A_{\lambda}$ denotes the subspace of $A$ carrying the simple ${\widetilde}G$-module with highest weight ${\lambda}$.
We fix a simple $G$-module $W$. Given a $G$-module $X$, we denote by $X^{(W)}$ the $W$-isotypic component in $X$. Using this notation, the desired claim can be reformulated as follows: as ${\lambda}$ ranges over ${\Lambda}_+$, the quantities $\dim
A_{\lambda}^{(W)}$ are uniformly bounded from above.
*Step* 1. Let $A^G\subset A$ be the subalgebra of $G$-invariants. Obviously, $A^{(W)}$ is a $A^G$-module. We claim that it is finitely generated.
Indeed, this is equivalent to saying that ${\operatorname{Hom}}_G(W,A)$ is finitely generated as a $A^G$-module.
Observe that the expansion is a grading of $A$. That is, $$A_{{\lambda}'}A_{{\lambda}''}\subseteq A_{{\lambda}'+{\lambda}''}, \qquad {\lambda}',{\lambda}''\in{\Lambda}_+.$$ Since the semigroup ${\Lambda}_+$ is finitely generated, the algebra $A$ is finitely generated.
This property together with the fact that $G$ is assumed to be reductive make it possible to apply the classic trick (used in Hilbert’s theorem on invariants) to the $A$-$G$-module ${\operatorname{Hom}}(W,A)$, see Popov-Vinberg [@PV Theorems 3.6 and 3.25]. Then we obtain that $({\operatorname{Hom}}(W,A))^G$ is a finitely generated $A^G$-module, as desired.
*Step* 2. By virtue of Step 1, there exists a finite collection of weights ${\lambda}(1),\dots, {\lambda}(n)\in{\Lambda}_+$ such that $A^{(W)}$ is generated over $A^G$ by the subspace $ A_{{\lambda}(1)}^{(W)}+\dots+A_{{\lambda}(n)}^{(W)}$. From this and we conclude that for every weight ${\lambda}\in{\Lambda}_+$, the subspace $
A_{\lambda}^{(W))}$ is contained in the sum of subspaces of the form $A_{{\lambda}-{\lambda}(i)}^G A_{{\lambda}(i)}^{(W)}$, where $i\in\{1,\dots,n\}$ should be such that ${\lambda}-{\lambda}(i)\in{\Lambda}_+$.
Because $G$ is a spherical subgroup of ${\widetilde}G$, every subspace $A_{{\lambda}-{\lambda}(i)}^G$ has dimension at most 1. This gives us the desired bound $$\dim A_{\lambda}^{(W)}\le\sum_{i=1}^n\dim A_{{\lambda}(i)}^{(W)},$$ uniform on ${\lambda}\in{\Lambda}_+$.
Given a finite-dimensional $G$-module $Y$, we can define the induced ${\widetilde}G$-module ${\operatorname{Ind}}(Y)$: its elements are holomorphic vector-functions $f:{\widetilde}G\to
Y$, which are ${\widetilde}G$-finite with respect to right shifts and such that $f(g{\widetilde}g)=gf({\widetilde}g)$ for any $g\in G$ and ${\widetilde}g\in{\widetilde}G$.
As above, we fix a simple $G$-module $W$. The desired claim is equivalent to the existence of a uniform bound for $[{\operatorname{Ind}}(W):V]$, the multiplicity of an arbitrary simple ${\widetilde}G$-module $V$ in the decomposition of ${\operatorname{Ind}}(W)$.
Given a finite-dimensional ${\widetilde}G$-module $X$, let us denote by $X_G$ the same space regarded as a $G$-module. One can choose $X$ in such a way that $X_G$ contained $W$. Then we obviously have $[{\operatorname{Ind}}(W):V]\le [{\operatorname{Ind}}(X_G):V]$.
The key observation is that ${\operatorname{Ind}}(X_G)$ is isomorphic to ${\operatorname{Ind}}({\mathbb C})\otimes X$, where ${\mathbb C}$ stands for the trivial one-dimensional $G$-module.
Now let $V$ be an arbitrary simple ${\widetilde}G$-module. We have $$[{\operatorname{Ind}}({\mathbb C})\otimes X:V]=\dim{\operatorname{Hom}}_{{\widetilde}G}(V\otimes X^*, {\operatorname{Ind}}({\mathbb C})),$$ where $X^*$ is the dual module to $X$. Observe that in the decomposition of $V\otimes X^*$ on simple components, every multiplicity does not exceed $\dim
X^*=\dim X$ (this follows from a well-known formula describing the decomposition of tensor products, see Zhelobenko [@Zhe end of §124] or Humphreys [@Humphreys §24, Exercise 9] or else can be easily proved directly). Since ${\operatorname{Ind}}({\mathbb C})$ is multiplicity free, we finally conclude that $[{\operatorname{Ind}}(W):V]\le\dim X$, which is the desired uniform bound.
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---
abstract: 'We present a novel algorithm for the completion of low-rank matrices whose entries are limited to a finite discrete alphabet. The proposed method is based on the recently-emerged framework of optimization theory, which is applied here to solve a regularized formulation of the completion problem that includes a term enforcing the discrete-alphabet membership of the matrix entries.'
author:
-
bibliography:
- 'listofpublications.bib'
title: |
Discrete-Aware Matrix Completion via\
Proximal Gradient
---
Introduction {#sect:intro}
============
With fair-winds of big data and , modern signal and information processing applications such as information filtering systems, networking, machine learning, and wireless communications often face a structured problem, which intends to infer a low-rank matrix $\mathbf{X}\in\mathbb{R}^{m\times n}$ given a partially observed incomplete matrix $\mathbf{O}\in\mathbb{R}^{m\times n}$ [@CandesMC09; @YuejieTSP19; @LuongArXiv19]. has therefore attracted much attention from both academic and industrial researchers, and has been applied to many different applications including recommender systems, localization, image compression and restoration, massive and millimeter wave channel estimation, and phase retrieval.
To address this challenge, effective strategies based on convex relaxation have been well-studied in the literature [@FazelPhD; @CandesMC09; @CandesSIAMOpt10] in terms of theoretical performance and complexity guarantees, of which crux is to replace the intractable non-convex rank function with its convex envelope ($i.e.,$ the ). To cite several milestones, one of the earliest works [@FazelPhD] proposed to convert such a nuclear-norm-based optimization problem into , which however is not suitable to large-scaled problems as seen in practical scenarios due to the fact that solvers require *at least* the cubic order complexity. To circumvent this issue, the as a proximal minimizer of the function was proposed in [@CandesSIAMOpt10], which has been later extended to its low-complexity alternative via the Lanczos algorithm. These methods in addition to other state-of-the-arts will be technically reviewed in Section \[sect:priorwork\].
In spite of intractability, structured non-convex optimization frameworks to address low-rankness have numerically shown successful performance improvements against its convex counterparts [@QuanmingIJCAI17], which have recently been guaranteed to possess lower complexities from a theoretical point of view [@YuejieTSP19]. Indeed, as recently shown in [@LiNIPS15; @QuanmingPAMI19; @BalcanJMLR19], non-convex approaches outperformed the state-of-the-art convex methods in terms of regardless of observation ratios.
Despite such intensive developments over the last decade, most of the algorithms have been designed for general problems at the cost of missing use of the most of the problem structure, leaving potential of further performance improvements. To elaborate, many existing algorithms including ones mentioned above or in Section \[sect:priorwork\] have assumed randomness or continuity of entries of the low-rank matrix $\mathbf{X}$, albeit in many practical situations those entries must belong to a certain finite discrete alphabet set.
In this article, we therefore introduce an additive discrete-aware regularizer that can be adopted for many different state-of-the-art algorithms, proposing a discrete-aware variate of Soft-Impute, one of the state-of-the-art methods for large-scaled problems, so as to illustrate the effectiveness of the proposed regularizer. Simulation results confirm the superior performance of the proposed method.
Prior Work {#sect:priorwork}
==========
In this section we briefly review major techniques studied over the last decade, which intend to recover unknown entries of a targeted low-rank matrix from partial observations, facilitating introduction to our proposed discrete-aware framework. To this end, we start with the original optimization problem, which can be written as the following intractable rank minimization problem:
\[eqn:MC\_original\] $$\begin{aligned}
{\mathop {\mathrm {argmin}} \limits_{\mathbf{X} \in \mathbb {R}^{m\times n}}} &&\!\!\!\! {\text{rank}\left(\mathbf{X}\right)}\\
\mathrm {s.t.} &&\!\!\!\! {P_\mathbb{\Omega}\left(\mathbf{X}\right)} = {P_\mathbb{\Omega}\left(\mathbf{O}\right)},
\end{aligned}$$
where ${\text{rank}\left(\mathbf{\cdot}\right)}$ denotes the rank of a given input matrix and ${P_\mathbb{\Omega}\left(\mathbf{\cdot}\right)}$ indicates the mask operator ($i.e.,$ projection) defined as $$\left[{P_\mathbb{\Omega}\left(\mathbf{A}\right)}\right]_{ij} =
\begin{cases}
\left[\mathbf{A}\right]_{ij} & \text{if}\:\: (i,j)\in\mathbb{\Omega}\\
0 & \text{otherwise}
\end{cases},$$ with $\left[\cdot\right]_{ij}$ being the $(i,j)$-th element of a given matrix and $\mathbb{\Omega}$ denoting the observed index set.
Although the global solution of equation corresponds to a matrix that has the lowest rank and matches observations corresponding to indexes belonging to the indicator set $\mathbb{\Omega}$, naively solving the above rank minimization problem is known to be -hard due to the non-convexity of the rank operator ${\text{rank}\left(\mathbf{\cdot}\right)}$. Taking advantage of the idea that the $\ell_0$-norm function can be replaced by its convex surrogate $\ell_1$-norm in -related problems, the above rank minimization problem can be relaxed by introducing the $\|\mathbf{A}\|_{*}$ ($i.e.,$ the sum of the singular values of $\mathbf{A}$) [@CandesMC09], namely,
\[eqn:MC\_nuclear\] $$\begin{aligned}
{\mathop {\mathrm {argmin}} \limits_{\mathbf{X} \in \mathbb {R}^{m\times n}}} &&\!\!\!\! \|\mathbf{X}\|_{*}\\
\label{eqn:MC_nuclear_constraint}
\mathrm {s.t.} &&\!\!\!\! {P_\mathbb{\Omega}\left(\mathbf{X}\right)} = {P_\mathbb{\Omega}\left(\mathbf{O}\right)},
\end{aligned}$$
where note that the is known to be the tightest convex lower bound of the rank operator [@RechtSIAM10].
Among various numerical optimization algorithms solving equation , one of the landmark attempts has been proposed in literature [@FazelPhD], which recasts equation as a [@VandenbergheSDP96]:
\[eqn:MC\_nuclear\_SDP\] $$\begin{aligned}
{\mathop {\mathrm {argmin}} \limits_{\mathbf{X}, \mathbf{W}_1, \mathbf{W}_2}} &&\!\!\!\! {{\rm Tr}\left({\mathbf{W}_1}\right)} + {{\rm Tr}\left({\mathbf{W}_2}\right)} \\
\mathrm {s.t.} &&\!\!\!\! {P_\mathbb{\Omega}\left(\mathbf{X}\right)} = {P_\mathbb{\Omega}\left(\mathbf{O}\right)}\\
&& \begin{bmatrix}
\mathbf{W}_1 & \mathbf{X}\\
\mathbf{X}^{\rm T} & \mathbf{W}_2
\end{bmatrix} \succeq 0
\end{aligned}$$
which can be solved by interior point methods available at various convex optimization solvers including SDPT3 [@TohOMS99], MOSEK [@AndersenMOSEK00], and SeDuMi [@SturmOMS99].
Since the aforementioned solvers suffer from prohibitive time and complexity due to the nature of second-order methods, however, the above approaches are only suitable for small-sized problems in spite of the fact that we are often interested in scenarios where the dimension of $\mathbf{X}$ is large. Aiming at reducing the computational burden while relaxing the equality constraint for cases where the observations contain noise or the targeted matrix to be recovered may only be regarded as approximately low-rank, various prior works including ones proposed in [@KeshavanTIT10; @CandesSIAMOpt10; @TohPJOpt10; @HastieJMLR15; @MazumderJMLR10; @QuanmingIJCAI15; @MaMP11; @TaoSIAMOpt11; @PrateekACM12] can be categorized as a solution to either the problem:
\[eqn:MC\_nuclear\_Fro\_bounded\] $$\begin{aligned}
{\mathop {\mathrm {argmin}} \limits_{\mathbf{X} \in \mathbb {R}^{m\times n}}} &&\!\!\!\! \|\mathbf{X}\|_{*}\\
\mathrm {s.t.} &&\!\!\!\! \underbrace{\frac{1}{2}\|{P_\mathbb{\Omega}\left(\mathbf{X-O}\right)}\|^2_{F}}_{\triangleq f(\mathbf{X})} \leq \varepsilon,
\end{aligned}$$
or its regularized form $$\label{eqn:MC_nuclear_regularized}
{\mathop {\mathrm {argmin}} \limits_{\mathbf{X} \in \mathbb {R}^{m\times n}}}\:\: f(\mathbf{X}) + \lambda \|\mathbf{X}\|_{*},$$ or with the rank information
\[eqn:MC\_nuclear\_rank\_info\] $$\begin{aligned}
{\mathop {\mathrm {argmin}} \limits_{\mathbf{X} \in \mathbb {R}^{m\times n}}} &&\!\!\!\! f(\mathbf{X})\\
\mathrm {s.t.} &&\!\!\!\! {\text{rank}\left(\mathbf{X}\right)} \leq s,
\end{aligned}$$
where $f(\cdot)$ is implicitly defined for notational convenience.
Although a great amount of efforts has been made to efficiently tackle the aforementioned convex problems[^1], there is growing progress on developing non-convex optimization algorithms for as first-order methods have shown via numerical studies remarkable success in practice, which is based on non-convex regularizers such as the capped $\ell_1$-norm, the , and the . To cite a few examples, [@LiNIPS15] proposed a algorithm for general non-convex and non-smooth optimization, named , which computes gradient steps in a forward-backward fashion and is further extended in [@QuanmingIJCAI17] to its accelerated variate, dubbed as . The authors in [@BalcanJMLR19] study the strong duality of non-convex matrix factorization problems, proving that under certain dual conditions, the global optimality of such non-convex problems can be achieved by solving its convex bi-dual problem, while [@SunTIT16] paves the way towards a theoretical guarantee for non-convex optimization frameworks to properly learn the targeted underlying low-rank matrix. For more information, please refer to a recent comprehensive survey [@YuejieTSP19] on non-convex problems and solutions.
Proposed method {#sect:proposed}
===============
As recently pointed out in [@LuongArXiv19], most of the techniques including ones mentioned above assume that entries of the targeted low-rank matrix are randomly generated ($i.e.,$ continuous random variables) in spite of the fact that many real data matrices including recommendation systems are composed of a finite set of discrete numbers, indicating potential to improve the recovery performance of the existing state-of-the-art algorithms. To this end, in this section we introduce a discreteness-aware additive regularizer recently studied in wireless communication and signal processing literature [@IimoriAsilomar2019; @HayakawaTWC2017; @AndreiAsilomar2019; @NagaharaSPL15; @IimoriTWC20], proposing a novel discrete-aware algorithm as a sequence of developments [@QuanmingIJCAI15; @QuanmingIJCAI17] stemming from Soft-Impute [@MazumderJMLR10]. Notice that the proposed regularizer can be employed in various other optimization frameworks, leaving such further extensions to future open problems due to the lack of space.
It is also worth noting that one may confuse the phrase “discrete-aware” with the existing similar research items [@ChenghaoAAAI19; @LianSIGKDD17], which exploit binary hashing codes for terminal user devices to reduce the storage volume and time complexity, and therefore are differentiated from the herein proposed method in the problem setup and optimization approach. Also, the proposed approach can be differentiated from [@ZhouyuanAAAI16; @HuangAAAI13; @NguyenSPL18] in terms of the applicability of the proposed regularizer and the optimization approach.
Brief Summary of Soft-Impute
----------------------------
Soft-Impute and its accelerated variates are state-of-the-art algorithms for large-scale problems, which aim at solving an optimization problem similar to equation and therefore to equation . To elaborate, Soft-Impute consists of the following recursion $$\begin{aligned}
\label{eqn:PG_iterates}
\mathbf{X}_{\rm t} = {\text{SVT}_{\lambda}{\left(\mathbf{X}_{\rm t-1} + {P_\mathbb{\Omega}\left(\mathbf{O-\mathbf{X}_{\rm t-1}}\right)}\right)}},\end{aligned}$$ where we utilized the fact that $f(\mathbf{X})$ is a convex function with $1$-Lipschitz constant, $t$ denotes the iteration index and the function is given by [@CandesSIAMOpt10 Theorem 2.1] as $$\label{eq:SVT}
{\text{SVT}_{\lambda}{\left(\mathbf{A}\right)}} \triangleq \mathbf{U}\left(\mathbf{\Sigma} - \lambda\mathbf{I}\right)_+\mathbf{V}^{\rm T},$$ with $\mathbf{A}\triangleq \mathbf{U}\mathbf{\Sigma}\mathbf{V}^{\rm T}$ and $(\cdot)_+$ being the positive part of the input.
It has recently been shown that Soft-Impute can be categorized as a algorithm [@QuanmingIJCAI15], and therefore, the well-known Nesterov-type momentum acceleration technique can be employed without loss of convergence guarantee [@PatrickPG2009; @AntonelloArxiv2018], leading to $$\begin{aligned}
\label{eqn:APG_iterates}
\mathbf{X}_{\rm t} = {\text{SVT}_{\lambda}{\left(\mathbf{Y}_{\rm t} + {P_\mathbb{\Omega}\left(\mathbf{O-\mathbf{Y}_{\rm t}}\right)}\right)}},\end{aligned}$$ with $\mathbf{Y}_{\rm t} \triangleq (1+ \beta_{\rm t})\mathbf{X}_{\rm t-1} + \beta_{\rm t} \mathbf{X}_{\rm t-2}$ where $\beta_{\rm t}$ is the momentum weight.
Discrete-Aware Matrix Completion
--------------------------------
Assuming that entries of the matrix to be recovered belong to a certain finite discrete alphabet set $\mathbb{A}\triangleq \{a_1,a_2,\cdots\}$ ($e.g.,$ integers in case of recommendation systems), we intend to tackle a variety of the following regularized minimization problem $$\label{eqn:MC_proposed}
{\mathop {\mathrm {argmin}} \limits_{\mathbf{X} \in \mathbb {R}^{m\times n}}}\:\: f(\mathbf{X}) + \lambda g(\mathbf{X}) + \xi r(\mathbf{X}|p),$$ where $g(\mathbf{X})$ denotes a non-smooth (possibly non-convex) low-rank regularizer [@QuanmingPAMI19], $\xi \geq 0$, and $$\label{eq:DAreg}
r(\mathbf{X}|p) \triangleq \sum^{|\mathbb{A}|}_{k=1}\|{\text{vec}_{\mathbb{\Omega}^c}{\left(\mathbf{X}\right)}} - a_k\mathbf{1}\|_p\vspace{-1ex}$$ where $r(\mathbf{X}|p)$ is the discrete-space regularizer[^2] with $0\leq p$, ${\text{vec}_{\mathbb{\Omega}^c}{\left(\mathbf{X}\right)}}$ denoting vectorization of entries of $\mathbf{X}$ corresponding to a given index set $\mathbb{\Omega}^c$, and $\mathbb{\Omega}^c$ being the complementary set of $\mathbb{\Omega}$.
Although non-convex scenarios where either $g(\mathbf{X})$, $r(\mathbf{X}|p)$ or both are non-convex regularizer(s) can be considered, we hereafter focus on the convex scenario ($i.e.,$ $g(\mathbf{X})= \|\mathbf{X}\|_{*}$ and $r(\mathbf{X}|1)= \sum^{|\mathbb{A}|}_{k=1}\|{\text{vec}_{\mathbb{\Omega}^c}{\left(\mathbf{X}\right)}} - a_k\mathbf{1}\|_1$) for the sake of simplicity and because of space constraints[^3]. The accelerated algorithm for a discrete-aware convex variate of Soft-Impute as described in equation , which hold the convergence rate $\mathcal{O}\big(\frac{1}{t^2}\big)$, can be summarized as the following recursion:
\[eq:proposed\_convexAPG\] $$\begin{aligned}
\mathbf{Y}_{\rm t} &=& (1+ \beta_{\rm t})\mathbf{X}_{\rm t-1} + \beta_{\rm t} \mathbf{X}_{\rm t-2}\\
\mathbf{Z}_{\rm t} &=& {\text{prox}_{\xi r}{\left(\mathbf{Y}_{\rm t}\right)}} \\
\mathbf{X}_{\rm t} &=& {\text{SVT}_{\lambda}{\left({P_{\mathbb{\Omega}^c} \left(\mathbf{\mathbf{Z}_{\rm t}}\right)}+{P_\mathbb{\Omega}\left(\mathbf{O}\right)}\right)}}\end{aligned}$$
where $ {\text{prox}_{\xi r}{\left(\mathbf{Y}_{\rm t}\right)}}$ is the proximal operator given by $$\label{eq:prox}
{\text{prox}_{\xi r}{\left(\mathbf{Y}_{\rm t}\right)}} \triangleq \underset{{\mathbf{U}}} {\mathrm{argmin}}\:\:\: r(\mathbf{U}|1) + \frac{1}{2\xi}{\left\lVert{\text{vec}_{\mathbb{\Omega}^c}{\left(\mathbf{U}-\mathbf{Y}_{\rm t}\right)}}\right\rVert}^2_{2}.\!\!$$
Taking into account the fact that the proximal operator of a sum of convex regularizers can be computed from a sequence of individual proximal operators [@PatrickPG2009], we readily obtain $$\label{eqn:prox_sum}
{\text{prox}_{\xi r}{\left(\mathbf{Y}_{\rm t}\right)}} = {\text{prox}_{\xi r_1}{\left({\text{prox}_{\xi r_2}{\left(\cdots{\text{prox}_{\xi r_{|\mathbb{A}|}}{\left(\mathbf{Y}_{\rm t}\right)}}\right)}}\right)}},$$ where $r_k(\mathbf{Y}_{\rm t})\triangleq \|{\text{vec}_{\mathbb{\Omega}^c}{\left(\mathbf{Y}_{\rm t}\right)}} - a_k\mathbf{1}\|_1$ for $k\in\{1,2,\ldots, |\mathbb{A}|\}$.
To this end, each proximal operator can be written as $$\label{eq:prox_each}
{\text{prox}_{\xi r_k}{\left(\mathbf{Y}_{\rm t}\right)}} \triangleq \underset{{\mathbf{U}}} {\mathrm{argmin}}\:\: \|\mathbf{u} - a_k\mathbf{1}\|_1 + \frac{1}{2\xi}{\left\lVert\mathbf{u}-\mathbf{y}_{\rm t}\right\rVert}^2_{2},$$ with $\mathbf{u}\triangleq {\text{vec}_{\mathbb{\Omega}^c}{\left(\mathbf{U}\right)}}$ and $\mathbf{y}_t\triangleq {\text{vec}_{\mathbb{\Omega}^c}{\left(\mathbf{Y}_t\right)}}$, which can be compactly written element-by-element as $$\label{eq:prox_each_element}
\underset{\bar{u}_\ell} {\mathrm{argmin}}\:\: |\bar{u}_\ell| + \frac{1}{2\xi}(\bar{u}_\ell- \bar{y}_{{\rm t},\ell})^2,$$ where $\bar{u}_\ell\triangleq \left[\mathbf{u}\right]_\ell - a_k$, $\bar{y}_{{\rm t},\ell}\triangleq \left[\mathbf{y}_t\right]_\ell - a_k$, $\bar{\mathbf{u}}\triangleq [\bar{u}_1, \bar{u}_2, \ldots, \bar{u}_{ |\mathbb{\Omega}^c|}]^{\rm T}$, $\bar{\mathbf{y}}_{\rm t} \triangleq [\bar{y}_{{\rm t},1}, \bar{y}_{{\rm t},2}, \ldots, \bar{y}_{{\rm t}, |\mathbb{\Omega}^c|}]^{\rm T}$ and $\ell\in\{\mathbb{Z}| 1\leq\ell\leq |\mathbb{\Omega}^c|\}$.
One readily notice that equation has a closed form solution ($i.e.,$ soft-thresholding function) given by $$\label{eqn:softthr}
\bar{\mathbf{u}} = \text{sign}\left(\bar{\mathbf{y}}_{\rm t} \right)\odot(|\bar{\mathbf{y}}_{\rm t}| - \xi\mathbf{1})_+$$ where $\odot$ is the Hadamard product and $\text{sign}\left(\cdot \right)$ denotes the (element-wise) sign function.
Notice that in equation , $|\bar{\mathbf{y}}_{\rm t}|$ performs the element-wise absolute operation. Finally we recover $\mathbf{u}$ by $$\mathbf{u} = \bar{\mathbf{u}} + a_k\mathbf{1},$$ and $\mathbf{U}$ by mapping $\mathbf{u}$ onto the unobserved indexes, namely, $$\label{eq:InvVec}
\mathbf{U} = {\text{vec}^{-1}_{\mathbb{\Omega}^c}{\left(\mathbf{u}\right)}},$$ where ${\text{vec}^{-1}_{\mathbb{\Omega}^c}{\left(\cdot\right)}}$ denotes the inverse function of ${\text{vec}_{\mathbb{\Omega}^c}{\left(\cdot\right)}}$.
Numerical Evaluation {#sect:results}
====================
![ performance behavior on the MovieLens-100k data set as a function of algorithmic iterations at $20$% observed ratio.[]{data-label="fig:Convergence"}](Fig/IimoriAsilomar2020_NMSE-crop.pdf){width="\columnwidth"}
![ performance behavior on the MovieLens-100k data set as a function of algorithmic iterations at $20$% observed ratio.[]{data-label="fig:Convergence"}](Fig/IimoriAsilomar2020_Convergence-crop.pdf){width="\columnwidth"}
In this section, we perform numerical experiments on discrete-valued real-world data sets to evaluate the proposed discrete-aware algorithm. To evaluate the robustness jointly with the recovery performance, we vary the observed ratio from $20$% to $60$%, while is utilized as the performance metric, which is given by $$\text{\ac{NMSE}} \triangleq \frac{\|{P_{\mathbb{\Omega}^c} \left(\mathbf{X-O}\right)}\|^2_F }{\|{P_{\mathbb{\Omega}^c} \left(\mathbf{O}\right)}\|^2_F}.$$
Besides the Soft-Impute algorithm [@MazumderJMLR10], we compare our proposed algorithm with other state-of-the-art methods such as AIS-Impute [@QuanmingIJCAI15], an accelerated variate of Soft-Impute, niAPG [@QuanmingIJCAI17], a non-convex variate of Soft-Impute with the non-convex regularizer.
The performance results comparing our proposed discrete-aware variates of Soft-Impute and the aforementioned state-of-the-art algorithms as a function of ratio of observed ratings for training are shown in Figure \[fig:NMSE\], where the red lines correspond to our proposed methods and black or gray lines are associated with the state-of-the-arts. For the sake of clarity, the performance gaps due to discreteness-awareness is highlighted by annotation arrows. It can be observed from the figure that most of the algorithms are able to successfully achieve less than $0.1$ in terms of for a wide range of observed ratios, albeit non-convex algorithms with can reduce the performance degradation in a severe scenario where only a few number of entries of the matrix can be observed. More interestingly, even in case of convex algorithms, the discreteness-awareness considerably decrease increment of the curve at the low observed ratio range, which indicates the robustness of the proposed discrete-aware regularizer.
In Figure \[fig:Convergence\], the convergence behavior of the algorithms with respect to the number of algorithmic iterations is presented, where we can perceive that most of the algorithms converge within $100$ iterations in case of with the convex regularizer and $180$ iterations in case of with the non-convex regularizer, respectively. Furthermore, the figure illustrates the accelerated convergence of the proposed algorithm with the convex regularizer. According to this observation, it may be concluded that the discreteness-awareness can not only improve the performance but also contribute to finding the optimality condition. However, the latter benefit is not necessary in case of non-convex scenarios due to multiple local minima, which rather results in slightly slower convergence.
Besides the above, we remark that the additional complexity due to the discreteness-aware regularizer in equation with $p=1$ is linear with respect to the cardinality of the unknown index set ($i.e.,$ $|\mathbb{\Omega}^c|$) as one may readily observe from the element-by-element operation in equations –. Therefore, one may conclude that the most expensive part of the algorithm in terms of complexity is the same as that of the state-of-the-art methods, $i.e.,$ , indicating that the proposed algorithm maintains the same complexity order.
In case of $p=0$, however, the regularizer may affect the convergence or the complexity of the proposed algorithm due to many different reasons such as expansiveness of $r_0(\mathbf{X})$ [@BurgerSMCISE17] or successive convex approximation to relax the $\ell_0$-norm function. Taking into account the aforementioned issues, we will provide a -based algorithm for $r_0(\mathbf{X})$ in the final version of the manuscript due to the space limitation at the submission.
In light of all the above, we conclude from the numerical performance evaluations that our proposed discreteness-aware algorithm may further accelerate the convergence and improve the completion performance in case of adopting convex functions for both regularizers ($i.e.,$ $g(\cdot)=\|\cdot\|_*$ and $r_1(\cdot)$), while enjoying the uniqueness of the solution due to the convexity of equation . In case of non-convex low-rank regularizer ($i.e.,$ ) while maintaining convex discreteness-aware regularizer ($i.e.,$ $r_1(\cdot)$), it has been shown that at the expense of slower convergence, the performance can be enhanced as shown in Figure \[fig:Convergence\].
Conclusion and Remarks {#sect:conclusion}
======================
In this article, we proposed a novel discrete-aware algorithm for structured practical problems where entries of the matrix to be recovered is subject to a certain finite discrete alphabet set such as recommender systems. To tackle this open problem indicated by a recent comprehensive survey [@LuongArXiv19], we introduce a discrete-aware additive regularizer that has been recently considered in signal processing and compressive sensing literature. Performance evaluations via software simulations demonstrate the superior performance of the proposed methods due to the awareness to such specific structure in the targeted matrix. We conclude this article by providing some possible applications of the proposed algorithm.
- The most important and obvious application is recommender systems for Netflix, Amazon, and so on with discrete scores, e.g., 1 of 5.
- Another application of the proposed MC algorithm is an estimation of connections among users in networks such as social networks, large wireless ad-hoc networks, etc where 0 means not-connected and 1 denotes connected. Although it is not impossible to obtain the whole adjacency matrix from the network, it would cost tons of resources for that. For example, if the proposed MC precisely estimates the whole matrix of the wireless ad-hoc network with the partial information, it enables the network to perform the optimal routing, network coding, distributed coding, and so on with even less overhead, which results in significant improvement of the throughput.
- An interesting field to which discreteness-aware algorithms can be applied is an index coding problem in a broadcast channel [@EsfahanizadehIITW14], where a single source communicates with multiple-users over a rate-limited channel.
[^1]: Although equation is not convex, it can be efficiently solved given an accurate rank information estimate by taking advantage of the fact that the set of $s$-dimensional subspaces belonging to $\mathbb{R}^{m\times n}$ with $r\leq m$ and $r\leq n$ is a differentiable Riemannian manifold as shown in OptSpace [@OptSpace09Arxiv; @KeshavanTIT10]
[^2]: Although it has been shown in the literature [@IimoriAsilomar2019; @HayakawaTWC2017; @IimoriICOIN20; @NagaharaSPL15; @IimoriTWC20; @HayakawaAccess2018; @AndreiAsilomar2019] that the base of the norm function is set to be $p=0$ or $p=1$ to enhance the discreteness of the inputs, the base $p$ can be any positive number in principle.
[^3]: A full description of algorithmic designs with non-convex regularizers will be given together with the associated pseudo-codes in the final version of the manuscript.
|
---
abstract: 'Digital mathematical libraries (DMLs) such as arXiv, Numdam, and [EuDML](https://eudml.org) contain mainly documents from STEM fields, where mathematical formulae are often more important than text for understanding. Conventional information retrieval (IR) systems are unable to represent formulae and they are therefore ill-suited for math information retrieval (MIR). To fill the gap, we have developed, and open-sourced the MIR system. MIaS is based on the full-text search engine Apache Lucene. On top of text retrieval, MIaS also incorporates a set of tools for preprocessing mathematical formulae. We describe the design of the system and present speed, and quality evaluation results. We show that MIaS is both efficient, and effective, as evidenced by our victory in the NTCIR-11 Math-2 task.=-1'
author:
- Petr Sojka
- Michal Ružička
- Vít Novotný
bibliography:
- 'main.bib'
- 'sojka.bib'
title: '*MIaS*: Math-Aware Retrieval in Digital Mathematical Libraries'
---
=1
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\# CIKM 2018 \#\# TODOs - For the review: - \*\*Convert to the \[ACM LaTeX template\](https://www.acm.org/publications/proceedings-template), use ‘sample-sigconf.tex‘ as example, aim for 4 pages.\*\* - \*\*Make sure the article is properly anonymized. (not necessary for Demo paper\*\* - \*\*Make sure the bibliography is properly formatted.\*\* - \*\*Extend the evaluation section with results from NTCIR\*\* - \*\*Describe query expansion, and striping\*\* - For the camera-ready:
\#\# Links - \[Topics of Interest\](http://www.cikmconference.org/\#topics): - Performance evaluation - Information storage and retrieval and interface technology - Digital libraries - \[Call for Demonstrations\](http://www.cikm2018.units.it/callfordemo.html)
\# MIaS Figures - WebMIaS web interface: - \[@dml:Liskaetal2011, Figure 2\] - \[@dl:liska2010eng, Figure 4.1\] - \[@MIR:MIRMU, Figure 1\] \*\[shows two extra webpage screenshots\]\* - \[@dml:cicm2014liskaetal, Figure 1\] - \[@Ruzicka17Math, Figure 4\] - Scheme of system workflow: - \[@dml:Liskaetal2011, Figure 1\] - \[@dl:liska2010eng, Figure 3.1\] \*\[in Slovak\]\* - \[@mir:LiskaMasters2013, Figures 4.1, and 4.2\] - \[@dml:sojkaliska2011, Figures 1, and 2\] - \[@dml:doceng2011SojkaLiska, Figure 1\] - \[@Ruzicka17Math, Figures 1, and 2\] \*\[shows extra detail\]\* - Formula preprocessing: - \[@dl:liska2010eng, Appendix B\] \*\[in Slovak\]\* - \[@dml:sojkaliska2011, Figure 3\] - \[@dml:doceng2011SojkaLiska, Figure 2\] - Unpublished CICM 2017 article \*„Towards Math-Aware Automated Classification and Similarity Search of Scientific Publications“\* (Figure 1) - \[@Ruzicka17Math, Figure 3\] - Relative number of results found using different subqueries in NTCIR-11 CMath run of MIR MU: - \[@LiskaSojkaRuzicka15Combining, Figure 1\] - \[@mir:MIaSNTCIR-11, Figure 1\] - \[@Ruzicka17Math, Figure 5\] - NTCIR-11 BPREF evaluation results: - \[@RuzickaSojkaLiska16Math, Figure 2\] - MathML structural unification: - \[@RuzickaSojkaLiska16Math, Figure 1\] - \[@Ruzicka17Math, Figure 7\] - Scalability diagrams: - \[@dml:sojkaliska2011, Figure 4\] \*\[MREC dataset\]\* - \[@mir:LiskaMasters2013, Figures 5.1–5.3\] \*\[MIR workshop dataset\]\* - MIaS UML diagrams: - \[@mir:LiskaMasters2013, Figures B.1–B.3\] - MSC topic modelling: - Unpublished CICM 2017 article \*“Towards Math-Aware Automated Classification and Similarity Search of Scientific Publications”\* (Figures 2, and 3) - \[@Ruzicka17Math, Figures 26, and 27\] - Relevance density estimates: - Unpublished COLING 2018 article \*“Weighted Averaging in Disjoint Passage Retrieval for Question Answering”\* (Figure 4).
\# Top-level outline - Abstract 1. Background 2. Aim 3. Methods 4. Results 5. Conclusion 1. Introduction (following the organizational pattern set forth by \*“Academic Writing for Graduate Students: Essential Tasks and Skills”\* by Swales, and Feak, 1994) - Move 1a: Establishing a research territory - Move 1b: Introducing previous research - Move 2: Establishing a niche - Move 3: Occupying a niche 2. System Description - MIaS - MathML Canonicalizer - MathML Unificator - MIaSMath - WebMIaS - Figure 1: WebMIaS web interface 3. Evaluation - MREC evaluation results \[@dml:Liskaetal2011, Section 5\] - MIR workshop evaluation results \[@mir:LiskaMasters2013, Chapter 5\] - NTCIR-10 evaluation results: - \[@MIR:MIRMU, Sections 4, and 5\] - \[@mir:NTCIR-10-Overview, Table 7\] - NTCIR-11 evaluation results: - \[@mir:MIaSNTCIR-11, Sections 4, and 5\] - \[@NTCIR11Math2overview, Table 8\] - NTCIR-12 evaluation results: - \[@RuzickaSojkaLiska16Math, Section 5\] - \[@ZanibbiEtAl16NTCIR, Tables 8, and 9\] 4. Conclusion and Future Work - Extending the techniques to general semi-structured text - Incorporating static relevance estimates based on the position of resulting documents - Figure 2: Relevance density estimates - Migrating MIaS to ElasticSearch - Extending ordering to a rewriting system in a computer algebra system (CAS) \[@cohl2017semantic\] - References
Introduction
============
In mathematical discourse, formulae are often more important than text for understanding. As a result, digital mathematical libraries (DMLs) require math information retrieval (MIR) systems that recognize both text and math in documents and queries. Conventional IR systems represent both text, and formulae using the bag-of-words vector-space model (VSM). However, the VSM captures neither the structural, nor the semantic similarity between mathematical formulae, which makes it ill-suited for MIR.
To fill the gap, new math-aware IR systems started to appear after the pioneering workshop on DMLs [@dml:dml2008proceedings]. Springer’s [[^1]]{} system takes formulae from papers with available LaTeX sources, and hashes the formulae to obtain a text representation. Zentralblatt Math uses the system[[^2]]{} [@kohlhase2008mathwebsearch], which represents formulae with substitution trees. We have developed and open-sourced the MIaS (Math Indexer and Searcher) system[[^3]]{} [@mir:MIaSNTCIR-11; @Ruzicka17Math] using the robust highly-scalable full-text search engine [Apache Lucene](https://lucene.apache.org/) [@bialecki12] and our own set of tools for the preprocessing of mathematical formulae. Since 2012, MIaS has been deployed in [the European Digital Mathematical Library (EuDML)[[^4]]{}]{}, making it historically the first system to be deployed in a DML.
{width="\textwidth"}
System Description
==================
MIaS processes text and math separately. The text is tokenized and stemmed to unify inflected word forms. Math is expected to be in [the MathML format[[^5]]{}]{}. Open tools such as [Tralics[[^6]]{}]{}, [LaTeXML[[^7]]{}]{} convert documents in the popular math authoring language of LaTeX to MathML. Other tools such as [InftyReader](http://www.inftyreader.org/) [@dml:suzukietal2003] , and [MaxTract](https://github.com/zorkow/MaxTract) [@DBLP:conf/aisc/BakerSS12] convert raster, and vector PDF documents, respectively, to MathML. The math is then canonicalized, ordered, tokenized, and unified (see Figure \[fig:system\]). We will describe each of these processing steps in detail in the following paragraphs.
#### Canonicalization
As explained above, MathML can originate from multiple sources and each can encode equivalent mathematical formulae a little differently. To obtain a single *canonical* representation, we initially used the third-party MathML canonicalizer from the UMCL library that converts math to a subset of MathML called the Canonical MathML [@acc:archambaultmoco06cmathml]. However, since the conversion speed and accuracy did not match our expectations, we have developed and open-sourced our own MathML canonicalizer[^8] [@FormanekEtAl:OpenMathUIWiP2012].
#### Ordering
MathML canonicalization only affects the encoding of mathematical formulae and does not result in any syntactic manipulation. We go a step further and reorder the operands of commutative operators alphabetically. For example, we convert the formulae $a+b$, and $b+a$ to a single canonical form $a+b$.=-1
#### Tokenization
A user of our system may not know the precise form of a formula they are searching for. To enable partial matches, we index not only the original formula, but also all its *subformulae*, which correspond to all the XML subtrees of the original formula XML tree. To penalize partial matches, the weight of subformulae is inversely proportional to their depth in the XML tree. [@dml:sojkaliska2011]
A user is likely interested in documents that contain either the query formula itself, or larger formulae with the query formula as a subformula. On the other hand, a user is unlikely to be interested in documents that contain only small parts of the query formula, such as isolated numbers, and symbols. For that reason, we only tokenize formulae in indexed documents, not in user queries.
#### Unification
In theory, the naming of variables does not affect the meaning of formulae. To match formulae in different notations, we replace each variable with a numbered identifier. For example, we convert the formulae $a+b^a,$ and $x+y^x$ to a single *unified* form $\text{id}_1+\text{id}_2^{\text{id}_1}$. In practice, many fields have an established notation and variable names are meaningful. To encourage precise matches, we keep the original formulae in addition to the unified formulae.
Two formulae that only differ in numeric constants are often related. For example, both $3x^2-2x+2,$ and $8x^2-3x+6$ are quadratic polynomials. We replace every numeric constant with a constant identifier. For example, we convert the above formulae to a single unified form $\text{const}x^2-\text{const}x+\text{const}$. To encourage precise matches, we keep the original formulae in addition to the unified formulae.
In predicate logic, a variable can represent an arbitrary formula. For example, the formulae $a^2+{}$, and $a^2+{}$ are equivalent if $x$ equals $\sqrt b$. Starting with the deepest subformulae, we replace all subformulae at a given depth with a unifying identifier. [@RuzickaSojkaLiska16Math] For example, we convert the formula $a^2+{}$ to a sequence of *structurally unified* formulae $a^2+{}$$,\Unif^{\unif}+{}$, and $\Unif+\Unif$ and the formula $a^2+{}$ to a sequence of structurally unified formulae $\Unif^{\unif}+{}$, and $\Unif+\Unif$. To penalize partial matches, the weight of the formulae is proportional to the depth of replacement. To encourage precise matches, we keep the original formulae in addition to the unified formulae. We have open-sourced [the MathML structural unificator[[^9]]{}]{}.
------------- ------- ------- ------- ------- -------
Subquery 1: $f_1$ $f_2$ $t_1$ $t_2$ $t_3$
Subquery 2: $f_1$ $f_2$ $t_1$ $t_2$
Subquery 3: $f_1$ $f_2$ $t_1$
Subquery 4: $f_1$ $f_2$
Subquery 5: $f_1$ $t_1$ $t_2$ $t_3$
Subquery 6: $t_1$ $t_2$ $t_3$
------------- ------- ------- ------- ------- -------
adddotafter
####
After preprocessing, a query consists of a weighted set of terms, and formulae. Since we are now going to search for documents that match at least one term, and at least one formula from the query, ill-posed terms, and formulae will negatively impact the recall of our system. To overcome this problem, we remove selected terms and formulae to produce a set of *subqueries*. Figure \[fig:query-expansion\] shows an example strategy for producing subqueries. @LiskaSojkaRuzicka15Combining describe other strategies that we use. We then submit the subqueries to Apache Lucene and receive ranked lists of resulting documents. Since the scores of the resulting documents are incomparable between subqueries, we cannot merge and rerank the individual result lists. Instead, we interleave them to obtain the final search results that we present to the user.
{width="\textwidth"}
--------- ------------- --------------- ---------- ----------
Docs Input Indexed Real CPU
10,000 3,406,068 64,008,762 35.75 35.05
50,000 18,037,842 333,716,261 189.71 181.19
100,000 36,328,126 670,335,243 384.44 366.54
200,000 72,030,095 1,326,514,082 769.06 733.44
300,000 108,786,856 2,005,488,153 1,197.75 1,116.64
350,000 125,974,221 2,318,482,748 1,386.66 1,298.10
439,423 158,106,118 2,910,314,146 1,747.16 1,623.22
--------- ------------- --------------- ---------- ----------
: Speed evaluation results on the NTCIR-11 Math-2 dataset using the same computer as above.[]{data-label="tab:speed-eval-ntcir11"}
----------- ------------ --------------- --------- ----------
Docs Input Indexed Real CPU
8,301,545 59,647,566 3,021,865,236 1940.07 3,413.55
----------- ------------ --------------- --------- ----------
: Speed evaluation results on the NTCIR-11 Math-2 dataset using the same computer as above.[]{data-label="tab:speed-eval-ntcir11"}
adddotafter
####
To provide a web user interface to MIaS, we have developed and open-sourced [WebMIaS[[^10]]{}]{}$^,$[[^11]]{} [@mir:MIaSNTCIR-11; @dml:cicm2014liskaetal]. Users can input their query in a combination of text, and math with a native support for LaTeX provided by Tralics, and [MathJax](https://www.mathjax.org/) [@cervone2012mathjax]. Matches are conveniently highlighted in the search results. The user interface of WebMIaS is shown in Figure \[fig:webmias\]. We have deployed a demo of [the latest development version of WebMIaS[[^12]]{}]{} using the [Apache Tomcat[[^13]]{}]{} implementation of the Java Servlet. The demo uses an index built from a subset of the arXMLiv dataset [@dml:arXMLiv2010] made available to the NTCIR-12 conference participants and will serve as the basis for our live demonstration at the conference.
Evaluation
==========
We performed a speed evaluation of MIaS on the MREC dataset of 439,423 documents [@dml:liska2011] (see Table \[tab:speed-eval-mrec\]), a quality and speed evaluation on the NTCIR-10 Math [@mir:NTCIR-10-Overview; @MIR:MIRMU] dataset of 100,000 documents, and a quality and speed evaluation on the NTCIR-11 [@NTCIR11Math2overview; @mir:MIaSNTCIR-11] (see Tables \[tab:speed-eval-ntcir11\], and \[tab:quality-eval-ntcir11\]), and NTCIR-12 MathIR [@ZanibbiEtAl16NTCIR; @RuzickaSojkaLiska16Math] dataset of 105,120 documents that were split into 8,301,578 paragraphs. Speed evaluation shows that the indexing time of our system is linear in the number of indexed documents and that the average query time is 469ms. With respect to quality evaluation, MIaS has notably won the NTCIR-11 Math-2 task.
Measure Level PMath CMath PCMath LaTeX
--------- ------- -------- ---------------- ---------------- --------
MAP 3 0.3073 **0.3630 /1/** 0.3594 0.3357
P@10 3 0.3040 **0.3520 /1/** 0.3480 0.3380
P@5 3 0.5120 **0.5680 /1/** 0.5560 0.5400
MAP 1 0.2557 **0.2807 /2/** 0.2799 0.2747
P@10 1 0.5020 0.5440 **0.5520 /1/** 0.5400
P@5 1 0.8440 **0.8720 /2/** 0.8640 0.8480
: Quality evaluation results on the NTCIR-11 Math-2 dataset. The mean average precision (MAP), and precisions at ten (P@10), and five (P@5) are reported for queries formulated using Presentation (PMath), and Content MathML (CMath), a combination of both (PCMath), and LaTeX. Two different relevance judgement levels of $\geq1$ (partially relevant), and $\geq3$ (relevant) were used to compute the measures. Number between slashes (/$\cdot$/) is our rank among all teams.[]{data-label="tab:quality-eval-ntcir11"}
Conclusion and Future Work
==========================
With the growing importance of DMLs, there is a growing demand for effective MIR systems. The evaluation shows that our open-source system is both efficient, and effective while building on industrial-strength full-text search engine Apache Lucene. The system allows low-latency responses even on the big math corpora as proved by its deployment in EuDML.
The speed of indexing and response latency of MIR will be further increased by the migration of from Apache Lucene to the distributed full-text search engine [ElasticSearch[[^14]]{}]{}. The idea of indexing structures rather than terms can be generalized from mathematical formulae to semi-structured text. Reordering the operands of associative operators is only a simple transformation. For example, to convert $\sqrt[n]{a}$, and $a^{1/n}$ to a single canonical representation, a general computer algebra system (CAS) can be used. We experiment [@rygletal16] with improving the vector space representations of document passages, aiming to add support for mathematics in the future. Embeddings can also be computed for equations [@mir:krstowski2018arXiv] now, which presents new possibilities of using language modeling for the semantic segmentation of STEM articles, and weighting the segments [@rygletal16]. Grasping the meaning of mathematical formulae is crucial: content is king. =-1
### Acknowledgements {#acknowledgements .unnumbered}
We gratefully acknowledge the support by the European Union under the FP7-CIP program, project 250,503 (EuDML), and by the ASCR under the Information Society R&D program, project 1ET200190513 (DML-CZ). We also sincerely thank three anonymous reviewers for their insightful comments.
[^1]: <https://www.ams.org/notices/201004/rnoti-apr10-cov4.pdf>
[^2]: <https://zbmath.org/formulae/>
[^3]: <https://github.com/MIR-MU/MIaS>
[^4]: <https://eudml.org/search>
[^5]: <https://www.w3.org/TR/MathML3/>
[^6]: <https://www-sop.inria.fr/marelle/tralics/>
[^7]: <https://dlmf.nist.gov/LaTeXML/>
[^8]: <https://github.com/MIR-MU/MathMLCan>
[^9]: <https://github.com/MIR-MU/MathMLUnificator>
[^10]: <https://mir.fi.muni.cz/webmias/>
[^11]: <https://github.com/MIR-MU/WebMIaS>
[^12]: <https://mir.fi.muni.cz/webmias-demo/>
[^13]: <https://tomcat.apache.org/>
[^14]: <https://elastic.co>
|
---
abstract: 'A new family of asymmetric matrices of Walsh-Hadamard type is introduced. We study their properties and, in particular, compute their determinants and discuss their eigenvalues. The invertibility of these matrices implies that certain character formulas are invertible, yielding expressions for the cardinalities of sets of combinatorial objects with prescribed descent sets in terms of character values of the symmetric group.'
address:
- |
Department of Mathematics\
Bar-Ilan University\
52900 Ramat-Gan\
Israel
- |
Department of Mathematics\
Bar-Ilan University\
52900 Ramat-Gan\
Israel
author:
- 'Ron M. Adin'
- Yuval Roichman
date: 'Sep. 11, ’13'
title: 'Matrices, Characters and Descents'
---
[^1]
Introduction {#section:intro}
============
Many character formulas involve the descent set of a permutation or of a standard Young tableau. We propose here a general setting for such formulas, involving a new family of asymmetric matrices of Walsh-Hadamard type. These matrices turn out to have fascinating properties, some of which are studied here using a transformation based on Möbius inversion. These include the evaluation of determinants, entries of transformed matrices and their inverses, and eigenvalues.
The inverse matrices lead to formulas expressing the cardinalities of sets of combinatorial objects with prescribed descent sets in terms of character values of the symmetric group. Examples of such objects include permutations of fixed length, involutions, standard Young tableaux, and more. It also follows that certain statements in permutation statistics have equivalent formulations in character theory. For example, the fundamental equi-distribution Theorem of Foata and Schützenberger, independently proved by Garsia and Gessel, is equivalent to a theorem of Lusztig and Stanley in invariant theory.
The organization of this paper is as follows: Section \[section:preliminaries\] contains the necessary definitions and background material, ending with a statement of the central motivating question. Section \[section:matrices\] introduces the main tool – a family (actually, two “coupled” families) of square matrices – and states some of their properties. Section \[section:AM\_BM\] contains a proof of the invertibility of these matrices, using a transformation corresponding to Möbius inversion. Properties of the transformed matrices are described in Section \[section:matrix\_entries\]. The application to character formulas, involving the concept of fine sets, is described in Section \[section:rep\_theory\_fine\].
Preliminaries and notation {#section:preliminaries}
==========================
Intervals, compositions, partitions and runs {#subs.prelim_compositions}
--------------------------------------------
\
For positive integers $m, n$ denote $$[m,n]:= \begin{cases}
\{m, m+1, \ldots, n\}, & \text{if $m \le n$;}\\
\emptyset, & \text{otherwise.}
\end{cases}$$ Denote also $[n] := [1,n] = \{1, \ldots, n\}$.
A [*composition*]{} of a positive integer $n$ is a vector $\mu=(\mu_1, \ldots, \mu_t)$ of positive integers such that $\mu_1+\cdots+\mu_t=n$. A [*partition*]{} of $n$ is a composition with weakly decreasing entries $\mu_1 \ge \ldots \ge \mu_t > 0$. The [*underlying partition*]{} of a composition is obtained by reordering the entries in weakly decreasing order.
For each composition $\mu = (\mu_1, \ldots, \mu_t)$ of $n$ define the set of its partial sums $$S(\mu) := \{ \mu_1, \mu_1+\mu_2, \ldots, \mu_1 + \ldots + \mu_t = n\} \subseteq [n],$$ as well as its complement $$I(\mu) := [n] \setminus S(\mu) \subseteq [n-1].$$ For example, for the composition $\mu=(3,4,2,5)$ of $14$: $S(\mu) = \{3,7,9,14\}$ and $I(\mu) = \{1,2,4,5,6,8,10,11,12,13\}$.
The correspondence $\mu \longleftrightarrow I(\mu)$ is a bijection between the set of all compositions of $n$ and the power set (set of all subsets) of $[n-1]$. The [*runs*]{} (maximal consecutive intervals) in $I(\mu)$ correspond to some of the components of $\mu$ – those satisfying $\mu_k > 1$. The length of the run corresponding to $\mu_k$ is $\mu_k-1$.
Permutations, Young tableaux and descent sets
---------------------------------------------
\
Let $S_n$ be the symmetric group on the letters $1,\dots,n$. For $1 \le i \le n-1$ denote $s_i := (i,i+1)$, a simple reflection (adjacent transposition) in $S_n$. For a composition $\mu = (\mu_1, \dots, \mu_t)$ of $n$ let $$s_\mu := (1, 2, \ldots, \mu_1)(\mu_1+1, \mu_1+2, \ldots, \mu_1+\mu_2) \cdots \in S_n,$$ a product of $t$ cycles of lengths $\mu_1, \mu_2, \ldots, \mu_t$ consisting of consecutive letters. The permutation $s_\mu$ may be obtained from the product $s_1 s_2 \cdots s_{n-1}$ of all simple reflections (in the usual order) by deleting the factors $s_{\mu_1+\ldots+\mu_k}$ for all $1 \le k < t$; equivalently, $$s_\mu = \prod_{i \in I(\mu)} s_i.$$
The [*descent set*]{} of a permutation $\pi\in S_n$ is ${\operatorname{Des}}(\pi):=\{i : \ \pi(i)>\pi(i+1)\}$.
The [*descent set*]{} of a standard Young tableaux $T$ is the set ${\operatorname{Des}}(T):=\{1 \le i \le n-1 : i+1 \hbox{ lies southwest of } i\}$.
$\mu$-unimodality {#subs.prelim_unimodality}
-----------------
\
A sequence $(a_1, \ldots, a_n)$ of distinct positive integers is [*unimodal*]{} if there exists $1 \le m\le n$ such that $a_1 > a_2 > \ldots > a_m < a_{m+1} < \ldots < a_n$. (This definiton differs slightly from the commonly used one, where all inequalities are reversed.)
Let $\mu=(\mu_1,\dots,\mu_t)$ be a composition of $n$. A sequence of $n$ positive integers is $\mu$-[*unimodal*]{} if the first $\mu_1$ integers form a unimodal sequence, the next $\mu_2$ integers form a unimodal sequence, and so on. A permutation $\pi \in S_n$ is $\mu$-[*unimodal*]{} if the sequence $(\pi(1), \ldots, \pi(n))$ is $\mu$-unimodal. For example, $\pi=936871254$ is $(4,3,2)$-unimodal, but not $(5,4)$-unimodal.
Let $U_\mu$ be the set of all $\mu$-unimodal permutations in $S_n$.
A family of character formulas {#section:character formulas}
------------------------------
\
Let $\lambda$ and $\mu$ be partitions of $n$. Let $\chi^\lambda$ be the $S_n$-character of the irreducible representation $S^\lambda$, and let $\chi^\lambda_\mu$ be its value on a conjugacy class of cycle type $\mu$. The following formula for the irreducible characters is a special case of [@Ro2 Theorem 4]. For a direct combinatorial proof see [@Ra2].
\[t.c1\][@Ro2 Theorem 4]$$\chi^\lambda_\mu =
\sum\limits_{\pi\in {\mathcal C}\cap U_\mu}
(-1)^{|{\operatorname{Des}}(\pi)\cap I(\mu)|},$$ where $\mathcal C$ is any Knuth class of RSK-shape $\lambda$.
Let $\chi^{(k)}$ be the $S_n$-character defined by the symmetric group action on the $k$-th homogeneous component of the coinvariant algebra. Then
\[t.c2\][@APR1 Theorem 5.1]$$\chi^{(k)}_\mu =
\sum\limits_{\pi\in L(k)\cap U_\mu}
(-1)^{|{\operatorname{Des}}(\pi)\cap I(\mu)|},$$ where $L(k)$ is the set of all permutations of length $k$ in $S_n$.
A complex representation of a group or an algebra $A$ is called a [*Gelfand model*]{} for $A$ if it is equivalent to the multiplicity free direct sum of all the irreducible $A$-representations. Let $\chi^G$ be the character of the Gelfand model of $S_n$ (or of its group algebra).
\[t.c3\][@APR2 Theorem 1.2.3] $$\chi^G_\mu =
\sum\limits_{\pi\in I_n\cap U_\mu}
(-1)^{|{\operatorname{Des}}(\pi)\cap I(\mu)|},$$ where $I_n:=\{\sigma\in S_n : \sigma^2=id\}$ is the set of all involutions in $S_n$.
More character formulas of this type are described in Subsection \[sec:fine\_sets\].
In this paper we propose a general setting for all of these results. In particular, we provide an answer to the following question.
\[q.invertible\] Are these character formulas invertible?
Two families of matrices {#section:matrices}
========================
It is well known that [*partitions*]{} of $n$ are the natural indices for the characters and conjugacy classes of $S_n$. It turns out that a major step towards an answer to Question \[q.invertible\] is to use, instead, [*compositions*]{} of $n$ (or, equivalently, subsets of $[n-1]$), in spite of the apparent redundancy. A surprising structure arises, in the form of a certain matrix $A_{n-1}$. In fact, it is convenient to define two “coupled” families of matrices, $(A_n)$ and $(B_n)$; for each nonnegative integer $n$, $A_n$ and $B_n$ are square matrices of order $2^n$, with entries $0$, $\pm 1$, which may be viewed as asymmetric variants of Walsh-Hadamard matrices. These matrices and some of their properties will be presented in this section.
We shall give two equivalent definitions for these matrices. The explicit definition is closer in spirit to the subsequent applications, but the recursive definition is very simple to describe and easy to use, and will therefore be presented first.
A recursive definition
----------------------
\
Recall the well known [*Walsh-Hadamard (Sylvester)*]{} matrices, defined by the recursion $$H_n = \left(\begin{array}{cc}
H_{n-1} & H_{n-1} \\
H_{n-1} & -H_{n-1}
\end{array}\right)
\qquad(n \ge 1)$$ with $H_0 = (1)$.
\[d.AB\_recursion\] Define, recursively, $$A_n = \left(\begin{array}{cc}
A_{n-1} & A_{n-1} \\
A_{n-1} & -B_{n-1}
\end{array}\right)
\qquad(n \ge 1)$$ with $A_0 = (1)$, and $$B_n = \left(\begin{array}{cc}
A_{n-1} & A_{n-1} \\
0 & -B_{n-1}
\end{array}\right)
\qquad(n \ge 1)$$ with $B_0 = (1)$.
Each of the matrices $A_n$ and $B_n$ may be obtained from the corresponding Walsh-Hadamard matrix $H_n$, all the entries of which are $\pm 1$, by replacing some of the entries by $0$.
$$A_1 = \left(\begin{array}{cc}
1 & 1 \\
1 & -1
\end{array}\right)
\qquad B_1 = \left(\begin{array}{cc}
1 & 1 \\
0 & -1
\end{array}\right)$$
$$A_2 = \left(\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & -1 & 1 & -1\\
1 & 1 & -1 & -1 \\
1 & -1 & 0 & 1
\end{array}\right)
\qquad B_2 = \left(\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & -1 & 1 & -1 \\
0 & 0 & -1 & -1 \\
0 & 0 & 0 & 1
\end{array}\right)$$
An explicit definition
----------------------
\
It will be convenient to index the rows and columns of $A_n$ and $B_n$ by subsets of the set $\{1,\ldots,n\}$.
\[def.Pn\] Let $P_n$ be the power set (set of all subsets) of $[n]:=\{1,\ldots,n\}$. Endow $P_n$ with the anti-lexicographic linear order: for $I, J \in P_n$, $I \ne J$, let $m$ be the largest element in the symmetric difference $I \triangle J := (I \cup J) \setminus (I \cap J)$, and define: $I < J \iff m \in J$.
The linear order on $P_3$ is $$\emptyset < \{1\} < \{2\} < \{1, 2\} < \{3\} < \{1, 3\} < \{2, 3\} < \{1, 2, 3\}.$$
\[def.intervals\] For $I \in P_n$ let $I_1, \ldots, I_t$ be the sequence of [*runs*]{} (maximal consecutive intervals) in $I$, namely: $I$ is the disjoint union of the $I_k$ ($1\le k \le t$), and each $I_k$ is a nonempty set of the form $\{ m_k+1, m_k+2, \ldots, m_k+\ell_k \}$ with $\ell_k\ge 1$ $(\forall k)$ and $0\le m_1 < m_1+\ell_1 < m_2 < m_2+\ell_2 < \ldots < m_t < m_t+\ell_t \le n$. In particular, $|I| = \ell_1 + \ldots + \ell_t$.
For $I = \{1,2,4,5,6, 8, 10\}\in P_{10}$: $I_1 = \{1,2\}$, $I_2 =
\{4,5,6\}$, $I_3 = \{8\}$, $I_4 = \{10\}$.
Order $P_n$ as in Definition \[def.Pn\]. The entries of the Walsh-Hadamard matrix $H_n = (h_{I,J})_{I, J \in P_n}$ are explicitly given by the formula $$h_{I,J} := (-1)^{|I \cap J|} \qquad(\forall I, J \in P_n).$$
A [*prefix*]{} of an interval $I =\{m+1, \ldots, m + \ell\}$ is an interval of the form $\{m +1, \ldots, m + p\}$, for $0 \le p \le \ell$.
\[t.AB\_explicit\][(Explicit definition)]{} Order $P_n$ as in Definition \[def.Pn\], and let $I_1, \ldots, I_t$ be the runs of $I\in P_n$. Then:
- $A_n = (a_{I,J})_{I, J \in P_n}$, where $$a_{I,J} =
\begin{cases}
(-1)^{|I \cap J|}, & \text{if $I_k \cap J$ is a prefix of $I_k$ for each $k$;}\\
0, & \text{otherwise.}
\end{cases}$$
- $B_n = (b_{I,J})_{I, J \in P_n}$, where: $$b_{I,J} =
\begin{cases}
(-1)^{|I \cap J|}, & \text{if $I_k \cap J$ is a prefix of $I_k$ for each $k$, and}\\
& n \not\in I \setminus J;\\
0, & \text{otherwise.}
\end{cases}$$
It will be convenient here to define $A_n$ and $B_n$ explicitly as in the lemma, and then show that they satisfy the recursions in Definition \[d.AB\_recursion\].
We shall start with $A_n$. Clearly $A_0 = (1)$.
For $I, J \in P_n$ ($n \ge 1$) denote $I' := I \setminus \{n\}$ and $J' := J \setminus \{n\}$.
The “upper left” quarter of $A_n$ corresponds to $I, J \in P_n$ such that $n \not\in I$ and $n \not\in J$. In this case, clearly $a_{I,J}$ in $A_n$ is the same as $a_{I',J'}$ in $A_{n-1}$.
Similarly when $n \not\in I$ and $n \in J$, and also when $n \in I$ and $n \not\in J$: $|I \cap J| = |I' \cap J'|$, and $I_k \cap J$ is a prefix of $I_k$ for all $k$ if and only if $I'_k \cap J$ is a prefix of $I'_k$ for all $k$.
The “lower right” quarter of $A_n$ corresponds to $I, J \in P_n$ such that $n \in I \cap J$. If $n-1 \not\in I$ then $I_k \cap J$ is a prefix of $I_k$ for all $k$ if and only if $I'_k \cap J$ is a prefix of $I'_k$ for all $k$. Also $|I \cap J| = |I' \cap J'| + 1$, so that $a_{I,J}$ in $A_n$ is equal to $-a_{I',J'}$ in $A_{n-1}$ and also to $-b_{I',J'}$ in $B_{n-1}$ (since $n-1 \not\in I'$ so $n-1 \not\in I' \setminus J'$). If $n-1 \in I \cap J$ then, again, $a_{I,J}$ in $A_n$ is equal to $-a_{I',J'}$ in $A_{n-1}$ and also to $-b_{I',J'}$ in $B_{n-1}$ (since $n-1 \in J'$ so $n-1 \not\in I' \setminus J'$). Finally, if $n-1 \in I$ but $n-1 \not\in J$ then, for the last run $I_t$ of $I$, $I_t \cap J$ is not a prefix of $I_t$, and thus $a_{I,J} = 0$ in $A_n$ as well as $-b_{I',J'} = 0$ in $B_{n-1}$ (since $n-1 \in I' \setminus J'$).
We have proved the recursion for $A_n$. The entries of $B_n$ are equal to the corresponding entries of $A_n$, except for those in the quarter corresponding to $(I,J)$ with $n \in I$ and $n \not\in J$, which are all zeros (since $n \in I \setminus J$). This proves the recursion for $B_n$ as well.
Determinants
------------
\
It turns out that the invertibility of $A_n$ is the key factor in an answer to Question \[q.invertible\].
\[t.An-determinant\] $A_n$ and $B_n$ are invertible for all $n \ge 0$. In fact, $$\det(A_n) = (n+1) \cdot \prod_{k=1}^{n} k^{2^{n-1-k} (n+4-k)} \qquad(n \ge 2)$$ while $\det(A_0) = 1$ and $\det(A_1) = -2$, and $$\det(B_n) = \prod_{k=1}^{n} k^{2^{n-1-k} (n+2-k)} \qquad(n \ge 2)$$ while $\det(B_0) = 1$ and $\det(B_1) = -1$.
A proof of Theorem \[t.An-determinant\] will be given in the next section. For comparison, $$\det(H_n) = 2^{2^{n - 1} n} \qquad(n \ge 2)$$ with $\det(H_0) = 1$ and $\det(H_1) = -2$.
Eigenvalues
-----------
\
Consider the matrix $$A_2 = \left(\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & -1 & 1 & -1\\
1 & 1 & -1 & -1 \\
1 & -1 & 0 & 1
\end{array}\right).$$ As an asymmetric matrix, it might conceivably have non-real eigenvalues. Surprisingly, computation shows that its characteristic polynomial is $$(x^2 - 3)(x^2 - 4),$$ and thus all its eigenvalues are (up to sign) square roots of positive integers!
This is not a coincidence. The following combinatorial description of the eigenvalues of $A_n$ and $B_n$, which was stated as a conjecture in an earlier version of this paper, has recently been proved by Gil Alon.
\[t.GilAlon\] [(G. Alon [@Alon])]{}
- The roots of the characteristic polynomial of $A_n$ are in $2:1$ correspondence with the compositions of $n$: each composition $\mu = (\mu_1, \ldots, \mu_t)$ of $n$ corresponds to a pair of eigenvalues $\pm \sqrt{\pi_\mu}$ of $A_n$, where $$\pi_\mu := \prod_{i=1}^{t} (\mu_i+1).$$
- The roots of the characteristic polynomial of $B_n$ are in $2:1$ correspondence with the compositions of $n$: each composition $\mu = (\mu_1, \ldots, \mu_t)$ of $n$ corresponds to a pair of eigenvalues $\pm \sqrt{\pi'_\mu}$ of $B_n$, where $$\pi'_\mu := \prod_{i=1}^{t-1} (\mu_i+1).$$
Surprising connections between the eigenvalues and the diagonal elements of $A_n^2$, as well as the column sums of some related matrices, appear in Theorem \[t.col\_sums\] below.
Möbius inversion {#section:AM_BM}
================
In this section we prove Theorem \[t.An-determinant\]. Our approach is to study certain matrices, with more transparent structure, obtained from $A_n$ and $B_n$ by a transformation corresponding to (poset theoretic) Möbius inversion.
Auxiliary definitions
---------------------
\
Let us define certain auxiliary families of matrices.
\[d.ZM\_recursion\] Define, recursively, $$Z_n = \left(\begin{array}{cc}
Z_{n-1} & Z_{n-1} \\
0 & Z_{n-1}
\end{array}\right)
\qquad(n \ge 1)$$ with $Z_0 = (1)$, as well as $$M_n = \left(\begin{array}{cc}
M_{n-1} & -M_{n-1} \\
0 & M_{n-1}
\end{array}\right)
\qquad(n \ge 1)$$ with $M_0 = (1)$.
$Z_n$ is the [*zeta matrix*]{} of the poset $P_n$ with respect to [*set inclusion*]{} (not with respect to its linear extension, described in Definition \[def.Pn\]). Thus $Z_n = (z_{I,J})_{I,J \in P_n}$ is a square matrix, with entries satisfying $$z_{I,J} =
\begin{cases}
1, & \text{if $I \subseteq J$;}\\
0, & \text{otherwise.}
\end{cases}$$
$M_n = Z_n^{-1}$ is the corresponding [*Möbius matrix*]{}, expressing the Möbius function (see [@Rota]) of the poset $P_n$. Thus $M_n = (m_{I,J})_{I,J \in P_n}$ has entries satisfying $$m_{I,J} =
\begin{cases}
(-1)^{|J \setminus I|}, & \text{if $I \subseteq J$;}\\
0, & \text{otherwise.}
\end{cases}$$
Denote $AM_n := A_n M_n$, $BM_n := B_n M_n$ and $HM_n := H_n M_n$.
It follows from Definitions \[d.AB\_recursion\] and \[d.ZM\_recursion\] that $$\label{eq.AMn}
AM_n = \left(\begin{array}{cc}
AM_{n-1} & 0 \\
AM_{n-1} & -(AM_{n-1} + BM_{n-1})
\end{array}\right)
\qquad(n \ge 1)$$ with $AM_0 = (1)$ and $$\label{eq.BMn}
BM_n = \left(\begin{array}{cc}
AM_{n-1} & 0 \\
0 & -BM_{n-1}
\end{array}\right)
\qquad(n \ge 1)$$ with $BM_0 = (1)$, as well as $$\label{eq.HMn}
HM_n = \left(\begin{array}{cc}
HM_{n-1} & 0 \\
HM_{n-1} & -2HM_{n-1}
\end{array}\right)
\qquad(n \ge 1)$$ with $HM_0 = (1)$.
The block triangular form of $AM_n$ and block diagonal form of $BM_n$ facilitate a recursive computation of the determinants of $A_n$ and $B_n$.
A proof of Theorem \[t.An-determinant\]
---------------------------------------
\
By recursion (\[eq.BMn\]), $$\det(BM_n) = \det(AM_{n-1}) \det(-BM_{n-1}) \qquad(n \ge 1).$$ Now $M_n$ is an upper triangular matrix with $1$-s on its diagonal, so that $$\det(M_n) = 1.$$ We conclude that $$\label{eq.Bn}
\det(B_n) = \delta_{n-1} \det(A_{n-1}) \det(B_{n-1}) \qquad(n\ge 1),$$ where $$\delta_n = (-1)^{2^{n}} =
\begin{cases}
-1, & \text{if $n = 0$;} \\
1, & \text{if $n \ge 1$.}
\end{cases}$$ Similarly, for any scalar $t$, $$AM_n + tBM_n = \left(\begin{array}{cc}
(t+1)AM_{n-1} & 0 \\ AM_{n-1} & -AM_{n-1} - (t+1)BM_{n-1}
\end{array}\right)
\qquad(n \ge 1)$$ and a similar argument yields $$\det(A_n + tB_n) = \delta_{n-1} \det((t+1)A_{n-1})
\det(A_{n-1} + (t+1)B_{n-1}) \qquad(n\ge 1).$$
It follows that $$\begin{aligned}
\det(A_n)
&= \delta_{n-1} \det(A_{n-1}) \det(A_{n-1} + B_{n-1}) \\
&= \delta_{n-1} \det(A_{n-1}) \delta_{n-2} \det(2 A_{n-2}) \det(A_{n-2} + 2B_{n-2}) \\
&= \ldots \\
& = \left( \prod_{k=1}^{n} \delta_{n-k} \det(k A_{n-k}) \right) \cdot \det(A_0 + n B_0) = \\
& = -(n+1) \cdot \prod_{k=1}^{n} k^{2^{n-k}} \cdot \prod_{k=1}^{n} \det(A_{n-k}) \qquad(n \ge 1).\end{aligned}$$ Since $A_0 = (1)$ it follows that $\det(A_n) \ne 0$ for any nonnegative integer $n$, and therefore $$\det(A_n) / \det(A_{n-1}) =
\frac{-(n+1)}{-n} \cdot n \cdot \prod_{k=1}^{n-1} k^{2^{n-1-k}} \cdot \det(A_{n-1}) \qquad(n\ge 2).$$ The solution to this recursion, with initial value $\det(A_1) = -2$, is $$\det(A_n) = (n+1) \cdot \prod_{k=1}^{n} k^{2^{n-1-k} (n+4-k)} \qquad(n \ge 2).$$ Recursion (\[eq.Bn\]) above, with initial value $\det(B_1) = -1$, now yields $$\det(B_n) = \prod_{k=1}^{n} k^{2^{n-1-k} (n+2-k)} \qquad(n \ge 2).$$ For comparison, $$\det(H_n) = 2^{2^{n-1}} \det(H_{n-1})^2 \qquad(n\ge 2)$$ with initial value $\det(H_1) = -2$, so that $$\det(H_n) = 2^{2^{n - 1} n} \qquad(n \ge 2).$$
We can also write $$\det(A_n) = \prod_{k=1}^{n+1} k^{a_{n+1-k}}\qquad(n \ge 2),$$ where the sequence $(a_0, a_1, \ldots) = (1, 2, 5, 12, 28, 64, \ldots)$ coincides with [@Sloane sequence A045623].
Properties of the transformed matrices {#section:matrix_entries}
======================================
In this section we describe some additional properties of the transformed matrices $AM_n$ and $BM_n$, namely: Explicit expressions for their entries, for the entries of their inverses, and for their row sums and column sums. The latter are surprisingly related to the eigenvalues of $A_n$ and $B_n$ described in Theorem \[t.GilAlon\] above. The proofs of most of the results in this section follow from recursion formulas (\[eq.AMn\]) and (\[eq.BMn\]), and are therefore indicated only when additional ingredients are present.
Matrix entries
--------------
\
We shall now compute explicitly the entries of $AM_n$ and $BM_n$, starting with $HM_n$ as a “baby case”.
$$HM_3 = \left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & -2 & 0 & 0 & 0 & 0 & 0 \\
1 & -2 & -2 & 4 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & -2 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 & -2 & 4 & 0 & 0 \\
1 & 0 & -2 & 0 & -2 & 0 & 4 & 0 \\
1 & -2 & -2 & 4 & -2 & 4 & 4 & -8
\end{array}\right)$$
This generalizes to an explicit description of the entries of $HM_n$, which follows easily from recursion (\[eq.HMn\]).
\[t.HMentries\]$$(HM_n)_{I,J} \ne 0 \iff J \subseteq I$$ and $$(HM_n)_{I,J} \ne 0 \,{\Longrightarrow}\, (HM_n)_{I,J} = (-2)^{|J|}.$$
The corresponding results for $AM_n$ and $BM_n$ are much more subtle (and interesting). Their proofs follow, in general, from recursions (\[eq.AMn\]) and (\[eq.BMn\]).
\[t.LT\] For every $n\ge 0$, the matrices $AM_n$ and $BM_n$ are lower triangular.
$(AM_n) \cdot Z_n$ is an $LU$ factorization of $A_n$; similarly for $B_n$.
$$AM_1 = \left(\begin{array}{cc}
1 & 0 \\
1 & -2
\end{array}\right)
\qquad BM_1 = \left(\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right)$$
$$AM_2 = \left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 \\
1 & 0 & -2 & 0 \\
1 & -2 & -1 & 3
\end{array}\right)
\qquad BM_2 = \left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right)$$
$$AM_3 = \left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & -2 & 0 & 0 & 0 & 0 & 0 \\
1 & -2 & -1 & 3 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & -2 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 & -2 & 4 & 0 & 0 \\
1 & 0 & -2 & 0 & -1 & 0 & 3 & 0 \\
1 & -2 & -1 & 3 & -1 & 2 & 1 & -4
\end{array}\right)$$ $$BM_3 = \left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & -2 & 0 & 0 & 0 & 0 & 0 \\
1 & -2 & -1 & 3 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 2 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -1
\end{array}\right)$$
Apparently, $AM_n$ has the same zero pattern and the same sign pattern as $HM_n$. $BM_n$ also has the same sign pattern, but has more zero entries. The absolute values of entries in both families are more intricate than in $HM_n$.
\[t.AB\_entries\] [(Entries of $AM_n$ and $BM_n$)]{}
- [**Zero pattern:**]{} $$(AM_n)_{I,J} \ne 0 \iff J \subseteq I$$ and $$(BM_n)_{I,J} \ne 0 \iff J \subseteq I \text{\ and\ } {\operatorname{maxout}}(J) =
{\operatorname{maxout}}(I),$$ where $${\operatorname{maxout}}(I) := \max\{0 \le i \le n \,|\, i \not\in I\} \qquad(\forall
I \in P_n).$$
- [**Signs:**]{} $$(AM_n)_{I,J} \ne 0 {\Longrightarrow}{\operatorname{sign}}((AM_n)_{I,J}) = (-1)^{|J|}$$ and $$(BM_n)_{I,J} \ne 0 {\Longrightarrow}{\operatorname{sign}}((BM_n)_{I,J}) = (-1)^{|J|}.$$
- [**Absolute values:**]{} For $I, J \in P_n$, let $J_1,
\ldots, J_t$ be the runs (maximal consecutive intervals) in $J$. For $J_k = \{m_k + 1, \ldots, m_k + \ell_k\}$ $(1 \le k \le t)$, let $$c_k(I) =
\begin{cases}
0, & \text{if $m_k \in I$;} \\
1, & \text{otherwise.}
\end{cases}$$ Then $$(AM_n)_{I,J} \ne 0 {\Longrightarrow}|(AM_n)_{I,J}| = \prod_{k=1}^{t} (|J_k| +
1)^{c_k(I)}$$ and $$(BM_n)_{I,J} \ne 0 {\Longrightarrow}|(BM_n)_{I,J}| = \prod_{k=1}^{t'} (|J_k|
+ 1)^{c_k(I)},$$ where $$t' =
\begin{cases}
t-1, & \text{if $n \in I$ (equivalently, $n \in J$);} \\
t, & \text{otherwise.}
\end{cases}$$
It is clear from recursion formulas (\[eq.AMn\]) and (\[eq.BMn\]) that all the entries in column $J$ of $AM_n$ (or $BM_n$) have sign $(-1)^{|J|}$ or are zero, exactly as in $HM_n$.
Comparison of the two recursions shows that wherever $AM_n$ has a zero entry so does $BM_n$, but not conversely. The zero pattern of $AM_n + BM_n$ is therefore the same as that of $AM_n$, and thus recursions (\[eq.AMn\]) and (\[eq.HMn\]) imply that $AM_n$ and $HM_n$ have the same zero pattern. The zero pattern of $BM_n$ now follows from recursion (\[eq.BMn\]).
Finally, the explicit formulas for the absolute values of entries are relevant, of course, only when $J \subseteq I$. They are a little difficult to come up with, but easy to confirm by recursion.
Let $I, J \in P_n$ satisfy $J \subseteq I$. Then:
- $$|(AM_n)_{I,J}| \le |(HM_n)_{I,J}| = 2^{|J|},$$ with equality if and only if $|J_k| = 1$ for each $k$ for which $m_k \not\in I$.
- $$|(BM_n)_{I,J}| \le |(AM_n)_{I,J}|,$$ with equality if and only if either $n \not\in I$ or $m_t \in I$.
$|J_k| + 1 \le 2^{|J_k|}$, with equality if and only if $|J_k| =
1$.
An alternative description of the entries may be given in terms of compositions, using the correspondence $\mu \longleftrightarrow I(\mu)$ described in Subsection \[subs.prelim\_compositions\] above.
\[t.AB\_entries\_comp\] [(Entries of $AM_n$ and $BM_n$, composition version)]{}\
Let ${\lambda}$ and $\mu$ be compositions of $n+1$. Write $(AM_n)_{{\lambda}, \mu}$ instead of $(AM_n)_{I({\lambda}), I(\mu)}$, and similarly for $BM_n$.
- [**Zero pattern:**]{} $$(AM_n)_{{\lambda},\mu} \ne 0 \iff \text{\rm $\mu$ is a refinement of ${\lambda}$}$$ and $$\begin{aligned}
(BM_n)_{{\lambda},\mu} \ne 0 &\iff& \text{\rm $\mu$ is a refinement of ${\lambda}$ and}\\
& & \text{\rm the last component of ${\lambda}$ is unrefined in $\mu$}.\end{aligned}$$
- [**Signs:**]{} $$(AM_n)_{{\lambda},\mu} \ne 0 \,{\Longrightarrow}\, {\operatorname{sign}}((AM_n)_{{\lambda},\mu}) = (-1)^{n+1-\ell(\mu)}$$ and $$(BM_n)_{{\lambda},\mu} \ne 0 \,{\Longrightarrow}\, {\operatorname{sign}}((BM_n)_{{\lambda},\mu}) = (-1)^{n+1-\ell(\mu)},$$ where $\ell(\mu)$ is the number of components of $\mu$.
- [**Absolute values:**]{} $$(AM_n)_{{\lambda},\mu} \ne 0 \,{\Longrightarrow}\, |(AM_n)_{{\lambda}, \mu}| = \prod_i \mu_{\rm init}({\lambda}_i)$$ and $$(BM_n)_{{\lambda},\mu} \ne 0 \,{\Longrightarrow}\, |(BM_n)_{{\lambda}, \mu}| = \prod_i ' \mu_{\rm init}({\lambda}_i),$$ where $\mu_{\rm init}({\lambda}_i)$ is the first component in the subdivision (in $\mu$) of the component ${\lambda}_i$ of ${\lambda}$, and $\prod_i '$ is a product over all values of $i$ except the last one.
Diagonal entries
----------------
\
The following corollary of Theorem \[t.AB\_entries\](3) and Theorem \[t.AB\_entries\_comp\](3) is stated, simultaneously, in terms of a composition $\mu$ of $n+1$ and the corresponding subset $J = I(\mu)$ of $[n]$. Different indices ($i$ and $k$) are used for the components of $\mu$ and the runs of $J$, since runs correspond only to the components statisfying $\mu_i > 1$.
\[t.AB\_diag\] [(Diagonal and last row of $AM_n$)]{}
- The diagonal entries of $AM_n$ are $$|(AM_n)_{J,J}| = \prod_i \mu_i = \prod_k (|J_k| + 1)$$ and the entries in its last row are $$|(AM_n)_{[n],J}| = \mu_1 =
\begin{cases}
|J_1| + 1, & \text{if $1 \in J$;}\\
1, & \text{otherwise.}
\end{cases}$$
- Each nonzero entry $(AM_n)_{I,J}$ divides the corresponding diagonal entry $(AM_n)_{J,J}$ and is divisible by the corresponding last row entry $(AM_n)_{[n],J}$.
For any finite set $J$ of positive integers, let ${\operatorname{ord}}(J) :=\sum_{j \in J} 2^{j-1}$. This is the ordinal number of $J$ in the anti-lexicographic order on $P_n$ (see Definition \[def.Pn\]), where the enumeration starts with $0$ for the empty set. The number ${\operatorname{ord}}(J)$ is independent of $n$, as long as $J \in P_n$. Define $$a_{{\operatorname{ord}}(J)} := |(AM_n)_{J,J}|, $$ and consider the resulting sequence $(a_m)_{m\ge 0}$ of absolute values of diagonal entries of the “limit matrix” $AM_{\infty} = \lim_{n \to \infty} AM_n$.
The sequence $(a_m)$ satisfies the recursion $$a_0 = 0,\ \ \ a_{2m} = a_m,\ \ \ a_{4m+1} = 2a_{2m},\ \ \ a_{4m+3}
= 2a_{2m+1} - a_m \quad(\forall m \ge 0).$$
Consider the formula in Corollary \[t.AB\_diag\](1) expressing a diagonal entry of $AM_n$ in terms of the corresponding run lengths. We shall not distinguish a set $J$ from its ordinal number ${\operatorname{ord}}(J)$.
(The set corresponding to) $2m$ has the same runs as $m$, shifted forward by $1$, so that $a_{2m} = a_m$. $4m+1$ has the same runs as $2m$, shifted forward by $1$, plus a singleton run $\{1\}$, so that $a_{4m+1} = 2a_{2m}$.
If $m$ is even then $a_{4m+3} = 3a_m$ and $a_{2m+1} = 2a_m$, so that $a_{4m+3} = 2a_{2m+1} - a_m$. If $m$ is odd, let $\ell$ be the length of the first run in $m$. The corresponding runs in $2m+1$ and in $4m+3$ have lengths $\ell+1$ and $\ell+2$, respectively, so that again $a_{4m+3} = 2a_{2m+1} - a_m$.
The sequence $(a_m)$ coincides with [@Sloane sequence A106737]. Thus, in particular, $$a_m = \sum_{k=0}^m \left[ {m+k \choose m-k} {m \choose k} \mod 2 \right],$$ where the expression in the square brackets is interpreted as either $0$ or $1$ and summed as an ordinary integer.
Row sums and column sums
------------------------
\
The following two results, regarding row and column sums of $AM_n$ and $BM_n$, are stated, for simplicity, almost entirely in the language of compositions.
\[t.row\_sums\] [(Row sums of $AM_n$, $BM_n$)]{}\
Let ${\lambda}$ be a composition of $n+1$, and let $I = I({\lambda})$ be the corresponding subset of $[n]$.
- The sum of all entries in row $I$ of $AM_n$ (or $BM_n$, or $HM_n$) is $(-1)^{|I|}$.
- The sum of absolute values of all entries in row $I$ of $AM_n$ is $$\prod_i (2^{{\lambda}_i} - 1).
$$ The sum of absolute values of all entries in row $I$ of $BM_n$ is $$\prod_i ' (2^{{\lambda}_i} - 1),
$$ where $\prod_i '$ is a product over all values of $i$ except the last. In $HM_n$ the corresponding sum is $3^{|I|}$.
\[t.col\_sums\] [(Column sums of $AM_n$, $BM_n$ and diagonal entries of $A_n^2$, $B_n^2$)]{}\
Let $\mu$ be a composition of $n+1$, and let $J = I(\mu)$ be the corresponding subset of $[n]$. Let $\mu^*$ be the composition of $n$ obtained from $\mu$ by reducing its first component by $1$, without changing the other components: $\mu_1^* = \mu_1 - 1$, $\mu_i^* = \mu_i$ $(\forall i > 1)$.
- The sum of absolute values (also: absolute value of the sum) of all the entries in column $J$ of $AM_n$ is equal to the diagonal entry $(A_n^2)_{J,J}$, which in turn is equal to $$\prod_i (\mu_i^* + 1).
$$
- The sum of absolute values (also: absolute value of the sum) of all the entries in column $J$ of $BM_n$ is equal to the diagonal entry $(B_n^2)_{J,J}$, which in turn is equal to $$\prod_i ' (\mu_i^* + 1),
$$ where $\prod_i '$ is a product over all values of $i$ except the last.
- For comparison, the sum of absolute values of all the entries in column $J$ of $HM_n$ is equal to the diagonal entry $(H_n^2)_{J,J}$, which in turn is equal to the constant $2^n$.
The recursions for $A_n^2$ and $B_n^2$ are $$A_n^2 = \left(\begin{array}{cc}
2 A_{n-1}^2 & A_{n-1}(A_{n-1} - B_{n-1}) \\
(A_{n-1} - B_{n-1})A_{n-1} & A_{n-1}^2 + B_{n-1}^2
\end{array}\right)
\qquad(n \ge 1)$$ and $$B_n^2 = \left(\begin{array}{cc}
A_{n-1}^2 & A_{n-1}(A_{n-1} - B_{n-1}) \\
0 & B_{n-1}^2
\end{array}\right)
\qquad(n \ge 1),$$ with $A_0^2 = B_0^2 = (1)$. Denoting $\alpha_n(J) := (A_n^2)_{J,J}$, $\beta_n(J) := (B_n^2)_{J,J}$ and $J' := J \setminus \{n\}$, it follows that $$\alpha_n(J) =
\begin{cases}
2 \alpha_{n-1}(J'), & \text{if $n \not\in J$;} \\
\alpha_{n-1}(J') + \beta_{n-1}(J'), & \text{otherwise}
\end{cases}$$ and $$\beta_n(J) =
\begin{cases}
\alpha_{n-1}(J'), & \text{if $n \not\in J$;} \\
\beta_{n-1}(J'), & \text{otherwise,}
\end{cases}$$ with $\alpha_0(\emptyset) = \beta_0(\emptyset) = 1$.
A short look at recursions (\[eq.AMn\]) and (\[eq.BMn\]) shows that the above recursions also hold if $\alpha_n(J)$ and $\beta_n(J)$ denote the sum of absolute values of all the entries in column $J$ of $AM_n$ and of $BM_n$, respectively. The same recursions also hold if $\alpha_n(J)$ and $\beta_n(J)$ stand for the explicit product formulas in the theorem, since if $J = I(\mu)$ and $J' = J \setminus \{n\} = I(\mu')$ then $n \not\in J$ means that $\mu$ is obtained from $\mu'$ by appending a new component of size $1$, while $n \in J$ means that $\mu$ is obtained from $\mu'$ by increasing the last component by $1$.
Comparing Theorem \[t.col\_sums\] with Theorem \[t.GilAlon\] gives a surprising conclusion.
The multiset of eigenvalues, counted by algebraic multiplicity, of $A_n^2$ (or $B_n^2$) is equal to the multiset of diagonal entries of this matrix.
This is remarkable since, apparently, for $n \ge 3$ the matrices $A_n^2$ are not even diagonalizable!
Inverse matrix entries
----------------------
\
We would like to have explicit expressions for the entries of $A_n^{-1}$, for use in Section \[section:rep\_theory\_fine\]. This turns out to be difficult to do directly, and we shall compute, as an intermediate step, the entries of $AM_n^{-1}$. Note that $A_n^{-1} = M_n \cdot AM_n^{-1}$.
$$A_3^{-1} = \left(\begin{array}{cccccccc}
1/24 & 1/24 & 1/12 & 1/12 & 1/8 & 1/8 & 1/4 & 1/4 \\
1/8 & -1/24 & 1/12 & -1/12 & 5/24 & -1/8 & 1/12 & -1/4 \\
5/24 & 5/24 & -1/12 & -1/12 & 1/8 & 1/8 & -1/4 & -1/4 \\
1/8 & -5/24 & -1/12 & 1/12 & 1/24 & -1/8 & -1/12 & 1/4 \\
1/8 & 1/8 & 1/4 & 1/4 & -1/8 & -1/8 & -1/4 & -1/4 \\
5/24 & -1/8 & 1/12 & -1/4 & -5/24 & 1/8 & -1/12 & 1/4 \\
1/8 & 1/8 & -1/4 & -1/4 & -1/8 & -1/8 & 1/4 & 1/4 \\
1/24 & -1/8 & -1/12 & 1/4 & -1/24 & 1/8 & 1/12 & -1/4
\end{array}\right)$$ $$AM_3^{-1} = \left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1/2 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\
1/2 & 0 & -1/2 & 0 & 0 & 0 & 0 & 0 \\
1/6 & -1/3 & -1/6 & 1/3 & 0 & 0 & 0 & 0 \\
1/2 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 \\
1/4 & -1/4 & 0 & 0 & -1/4 & 1/4 & 0 & 0 \\
1/6 & 0 & -1/3 & 0 & -1/6 & 0 & 1/3 & 0 \\
1/24 & -1/8 & -1/12 & 1/4 & -1/24 & 1/8 & 1/12 & -1/4
\end{array}\right)$$
We shall attempt an inductive computation of $AM_n^{-1}$. Recursion formulas (\[eq.AMn\]) and (\[eq.BMn\]) yield corresponding recursions for the inverse matrices: $$AM_n^{-1} = \left(\begin{array}{cc}
AM_{n-1}^{-1} & 0 \\
(AM_{n-1} + BM_{n-1})^{-1} & -(AM_{n-1} + BM_{n-1})^{-1}
\end{array}\right)
\qquad(n \ge 1)
$$ and $$BM_n^{-1} = \left(\begin{array}{cc}
AM_{n-1}^{-1} & 0 \\
0 & - BM_{n-1}^{-1}
\end{array}\right)
\qquad(n \ge 1),
$$ with $AM_0^{-1} = BM_0^{-1} = (1)$; however, the recursion for $AM_n^{-1}$ involves the inverse of a new matrix, $AM_{n-1} + BM_{n-1}$, which in turn involves the inverse of $AM_{n-2} + 2 BM_{n-2}$, and so forth. We are thus led to consider a more general situation.
\[def.Mnx\] For any real number $x$ let $$M_n(x) := x AM_n + (1-x) BM_n.$$
In particular, $M_n(0) = BM_n$ and $M_n(1) = AM_n$.
For each $n \ge 0$ and $x > 0$, $$M_n^{-1}(x)_{I,J} \ne 0 \iff J \subseteq I$$ and, for $J \subseteq I$, $$M_n^{-1}(x)_{I,J} = (-1)^{|J|} \prod_{i \in I} \frac{d_{I, J, x}(i)}{e_{I, J, x}(i)},$$ where $I_1, \ldots, I_t$ are the runs of $I$ and, for $i \in I_k$:
- If $n \not\in I_k$ then $$d_{I, J, x}(i) :=
\begin{cases}
\max(I_k) - i + 1, & \text{if $i \in J$;}\\
1, & \text{otherwise}
\end{cases}$$ and $$e_{I, J, x}(i) := \max(I_k) - i + 2.$$
- If $n \in I_k$ (and thus necessarily $k = t$) then $$d_{I, J, x}(i) :=
\begin{cases}
(\max(I_k) - i) \cdot x + 1, & \text{if $i \in J$;}\\
x, & \text{otherwise}
\end{cases}$$ and $$e_{I, J, x}(i) := (\max(I_k) - i + 1) \cdot x + 1.$$
Let $x > 0$. By Definition \[def.Mnx\] and recursion formulas (\[eq.AMn\]) and (\[eq.BMn\]), $$\begin{aligned}
M_n(x)
&= \left(\begin{array}{cc}
AM_{n-1} & 0 \\
x AM_{n-1} & -(x AM_{n-1} + BM_{n-1})
\end{array}\right) \\
&= \left(\begin{array}{cc}
M_{n-1}(1) & 0 \\
x M_{n-1}(1) & -(1+x) M_{n-1}\left(\frac{x}{1+x}\right)
\end{array}\right)
\qquad(n \ge 1)\end{aligned}$$ with $M_0(x) = (1)$. Invertibility of $M_{n-1}(x)$ for all $x > 0$ clearly implies the invertibility of $M_{n}(x)$ for all $x > 0$. The inverse satisfies $$\label{e.Mnx_inverse_rec}
M_n^{-1}(x) = \left(\begin{array}{cc}
M_{n-1}^{-1}(1) & 0 \\
\frac{x}{1+x} M_{n-1}^{-1}\left(\frac{x}{1+x}\right) & \frac{-1}{1+x} M_{n-1}^{-1}\left(\frac{x}{1+x}\right)
\end{array}\right)
\qquad(n \ge 1)$$ with $M_0^{-1}(x) = (1)$, for all $x > 0$.
Recursion (\[e.Mnx\_inverse\_rec\]) shows that, indeed, for $x > 0$: $M_n^{-1}(x)_{I,J} \ne 0 \iff J \subseteq I$, and that the sign of this entry is $(-1)^{|J|}$.
Regarding the absolute value of this entry, assume by induction that the prescribed formula holds for $M_{n-1}^{-1}(x)$, $\forall x > 0$.
If $n \not\in I$ then also $n \not\in J$, and clearly $M_{n}^{-1}(x)_{I,J} = M_{n-1}^{-1}(1)_{I,J}$ satisfies the required formula.
If $n \in I$, let $I' := I \setminus \{n\}$, $J' := J \setminus \{n\}$ and $x' := \frac{x}{1+x}$. The assumed formula for $M_{n-1}^{-1}(x')_{I',J'}$ and the claimed formula for $M_{n}^{-1}(x)_{I,J}$ have exactly the same factors for all $i \not\in I_t$, so we need only consider $i \in I_t$.
If $|I_t| = 1$ (i.e., $n-1 \not\in I$) then there is nothing else in $M_{n-1}^{-1}(x')_{I',J'}$, but according to (\[e.Mnx\_inverse\_rec\]) there is an extra factor $\frac{1}{1+x}$ or $\frac{x}{1+x}$ in $M_{n}^{-1}(x)_{I,J}$ (depending on whether or not $n \in J$), and this is exactly the missing $d_{I, J, x}(n)/e_{I, J, x}(n)$.
Finally, assume that $|I_t| > 1$. Again, the extra factor $\frac{1}{1+x}$ or $\frac{x}{1+x}$ is exactly $d_{I, J, x}(n)/e_{I, J, x}(n)$. The other factors in $M_{n-1}^{-1}(x')_{I',J'}$, corresponding to $i \in I'_t$, are (if $i \in J$) $$\frac{d_{I', J', x'}(i)}{e_{I', J', x'}(i)} =
\frac{(n-1-i)x'+1}{(n-i)x'+1} =
\frac{(n-1-i)x+1+x}{(n-i)x+1+x}=
\frac{d_{I, J, x}(i)}{e_{I, J, x}(i)}$$ or (if $i \not\in J$) $$\frac{d_{I', J', x'}(i)}{e_{I', J', x'}(i)} =
\frac{x'}{(n-i)x'+1} =
\frac{x}{(n-i)x+1+x}=
\frac{d_{I, J, x}(i)}{e_{I, J, x}(i)},$$ exactly as claimed for $M_{n}^{-1}(x)_{I,J}$.
We are especially interested, of course, in the special case $x = 1$.
[(Entries of $AM_n^{-1}$)]{}\[t.AM\_inverse\]\
For each $n \ge 0$ $$(AM_n^{-1})_{I,J} \ne 0 \iff J\subseteq I$$ and, for $J\subseteq I$, $$(AM_n^{-1})_{I,J} = (-1)^{|J|} \prod_{i \in I} \frac{d_{I, J}(i)}{e_{I, J}(i)},$$ where $I_1, \ldots, I_t$ are the runs of $I$ and, for $i \in I_k$: $$d_{I, J}(i) :=
\begin{cases}
\max(I_k) - i + 1, & \text{if $i \in J$;}\\
1, & \text{otherwise}
\end{cases}$$ and $$e_{I, J}(i) := \max(I_k) - i + 2.$$ Equivalently, for $J\subseteq I$, $$(AM_n^{-1})_{I,J} = (-1)^{|J|} \prod_{k=1}^{t} \frac{1}{(|I_k|+1)!} \prod_{i \in I_k \cap J} (\max(I_k) - i + 1).
$$
Note that the denominator $\prod_{k=1}^{t} (|I_k| + 1)!$ is the cardinality of the parabolic subgroup $\langle I \rangle$ of $S_{n+1}$ generated by the simple reflections $\{s_i\,:\, i \in I\}$.
\[t.rows\_of\_inverse\]
- Each nonzero entry of $AM_n^{-1}$ is the inverse of an integer.
- In each row of $AM_n^{-1}$, the sum of absolute values of all the entries is $1$.
- In each row $I$ of $AM_n^{-1}$, the first entry $$(AM_n^{-1})_{I, \emptyset} = \prod_{k=1}^{t} \frac{1}{(|I_k|+1)!}$$ divides all the other nonzero entries and the diagonal entry $$(AM_n^{-1})_{I, I} = (-1)^{|I|} \prod_{k=1}^{t} \frac{1}{|I_k|+1}$$ is divisible by all the other nonzero entries, where a rational number $r$ is said to [*divide*]{} a rational number $s$ if the quotient $s/r$ is an integer.
Fine sets {#section:rep_theory_fine}
=========
The concept {#sec:concept}
-----------
\
A general setting for character formulas is introduced in this section. It will serve as a framework for the answer to Question \[q.invertible\].
Recall from Subsection \[subs.prelim\_compositions\] the definition of $I(\mu)$ for a composition $\mu$.
\[def.unimodal\_set\] Let $\mu=(\mu_1, \ldots, \mu_t)$ be a composition of $n$. A subset $J \subseteq [n-1]$ is [*$\mu$-unimodal*]{} if each run of $J \cap I(\mu)$ is a prefix of the corresponding run of $I(\mu)$; in other words, if $J \cap I(\mu)$ is a disjoint union of intervals of the form $\left[ \sum_{i=1}^{k-1}\mu_i+1, \sum_{i=1}^{k-1}\mu_i + \ell_k \right]$, where $0 \le \ell_k \le \mu_k-1$ for every $1 \le k \le t$.
A permutation $\pi \in S_n$ is $\mu$-unimodal according to the definition in Subsection \[subs.prelim\_unimodality\] if and only if its descent set ${\operatorname{Des}}(\pi)$ is $\mu$-unimodal according to Definition \[def.unimodal\_set\].
\[defn-fine\] Let ${{\mathcal{B}}}$ be a set of combinatorial objects, and let ${\operatorname{Des}}: {{\mathcal{B}}}\to P_{n-1}$ be a map which associates with each element $b\in {{\mathcal{B}}}$ a subset ${\operatorname{Des}}(b) \subseteq [n-1]$. Denote by ${{\mathcal{B}}}^\mu$ the set of elements in ${{\mathcal{B}}}$ whose “descent set” ${\operatorname{Des}}(b)$ is $\mu$-unimodal. Let $\rho$ be a complex $S_n$-representation. Then ${{\mathcal{B}}}$ is called a [*fine set*]{} for $\rho$ if, for each composition $\mu$ of $n$, the character of $\rho$ at a conjugacy class of cycle type $\mu$ satisfies $$\label{defn-fine1}
\chi^\rho_\mu=\sum\limits_{b\in {{\mathcal{B}}}^\mu} (-1)^{|{\operatorname{Des}}(b)\cap I(\mu)|}.$$
It follows from Theorems \[t.c1\], \[t.c2\] and \[t.c3\] that
\[fine-examples\]
- Any Knuth class of RSK-shape $\lambda$ is a fine set for the Specht module $S^\lambda$.
- The set of permutations of a fixed Coxeter length $k$ in $S_n$ is a fine set for the $k$-th homogeneous component of the coinvariant algebra of $S_n$.
- The set of involutions in $S_n$ is a fine set for the Gelfand model of $S_n$.
More examples of fine sets are given in Subsection \[sec:fine\_sets\].
Distribution of descent sets
----------------------------
\
\[sec:descent\_sets\] We are now ready to state our main application.
\[t.main\] If ${{\mathcal{B}}}$ is a fine set for an $S_n$-representation $\rho$ then the character values of $\rho$ determine the distribution of descent sets over ${{\mathcal{B}}}$. In particular, for every $I\subseteq [n-1]$, the number of elements in ${{\mathcal{B}}}$ whose descent set contains $I$ satisfies $$|\{b\in B : {\operatorname{Des}}(b) \supseteq I\}| =
\frac{1}{|\langle I \rangle|}
\sum\limits_{J\subseteq I} (-1)^{|J|}\chi^\rho(c_J)
\prod_{k=1}^{t} \prod_{i \in I_k \cap J} (\max(I_k) - i + 1),$$ where $I_1, \ldots, I_t$ are the runs in $I$, $|\langle I \rangle|$ is the cardinality of the parabolic subgroup of $S_n$ generated by $\{s_i : i\in I\}$, and $c_I$ is any Coxeter element in this subgroup.
The mapping $\mu \mapsto I(\mu)$ (see Subsection \[subs.prelim\_compositions\]) is a bijection between the set of all compositions of $n$ and the set $P_{n-1}$ of all subsets of $[n-1]$. For a subset $J=\{j_1,\dots,j_k\}\subseteq [n-1]$ with $j_1<j_2<\cdots<j_k$ let $c_J$ be the product $s_{j_1}s_{j_2}\cdots s_{j_k} \in S_n$. This is a Coxeter element in the parabolic subgroup generated by $\{s_i : i\in J\}$, and its cycle type is (the partition corresponding to) the composition $\mu$, where $J = I(\mu)$. Let $x^\rho$ be the vector with entries $\chi^\rho(c_J)$, where the subsets $J \in P_{n-1}$ are ordered anti-lexicographically as in Definition \[def.Pn\].
Similarly, let $v^{{\mathcal{B}}}=(v_J^{{\mathcal{B}}})_{J \in P_{n-1}}$ be the vector with entries $$v^{{\mathcal{B}}}_J:=|\{b\in {{\mathcal{B}}}: {\operatorname{Des}}(b)=J\}| \qquad(\forall J \in P_{n-1}).$$
By Definition \[defn-fine\] and Lemma \[t.AB\_explicit\]($i$), ${{\mathcal{B}}}$ is a fine set for $\rho$ if and only if $$\label{e.xAv}
x^\rho = A_{n-1} v^{{\mathcal{B}}},$$ where $x^\rho$ and $v^{{\mathcal{B}}}$ are written as column vectors. By Theorem \[t.An-determinant\], $A_{n-1}$ is an invertible matrix, which proves that $x^\rho$ uniquely determines $v^{{\mathcal{B}}}$.
The explicit formula follows from Corollary \[t.AM\_inverse\], as soon as equation (\[e.xAv\]) is written in the form $$Z_{n-1} v^{{\mathcal{B}}}= AM_{n-1}^{-1} x^\rho.$$
The Inclusion-Exclusion Principle (namely, multiplication by $M_{n-1}$) gives an equivalent form of the explicit formula.
Let $B$ be a fine set for an $S_n$-representation $\rho$. For every $I\subseteq [n-1]$, the number of elements in $B$ with descent set exactly $D$ satisfies $$|\{b\in B : {\operatorname{Des}}(b) = D\}| =
\sum_J \chi^\rho(c_J) \sum_{I: D \cup J \subseteq I} (-1)^{|I \setminus D|} (AM_{n-1}^{-1})_{I,J}
$$ where $(AM_{n-1}^{-1})_{I,J}$ is as in Corollary \[t.AM\_inverse\] and the notation is as in Theorem \[t.main\].
Permutation statistics versus character theory {#sec:versus}
----------------------------------------------
\
By Theorem \[t.main\], certain statements in permutation statistics have equivalent statements in character theory. In particular,
\[ps-rt\] Given two symmetric group modules with fine sets, the isomorphism of these modules is equivalent to equi-distribution of the descent set on their fine sets.
Combining Theorem \[t.main\] with Definition \[defn-fine\].
Here is a distinguished example. Recall the major index of a permutation $\pi$, $${\operatorname{maj}}(\pi):=\sum\limits_{i\in {\operatorname{Des}}(\pi)} i.$$ For a subset $I\subseteq [n-1]$ denote ${{\mathbf x}}^I:=\prod_{i\in I} x_i$. The following is a fundamental theorem in permutation statistics.
\[FS-thm\] [(Foata-Schützenberger)]{} [@FS79] $$\sum_{\pi\in S_n} {{\mathbf x}}^{{\operatorname{Des}}(\pi)}q^{\ell(\pi)} =
\sum_{\pi\in S_n} {{\mathbf x}}^{{\operatorname{Des}}(\pi)}q^{{\operatorname{maj}}(\pi^{-1})}.$$
See also [@GG].
For $0\le k\le {n\choose 2}$ denote by $R_k$ the $k$-th homogeneous component of the coinvariant algebra of the symmetric group $S_n$. The following is a classical theorem in invariant theory.
\[St-thm\] [(Lusztig-Stanley)]{} [@St79 Prop. 4.11] For a partition $\lambda$ denote by $m_{k,\lambda}$ the number of standard Young tableaux of shape $\lambda$ with major index $k$. Then $$R_k \cong \bigoplus_{\lambda\vdash n} m_{k,\lambda} S^\lambda,$$ where the sum is over all partitions of $n$ and $S^\lambda$ denotes the irreducible $S_n$-module indexed by $\lambda$.
It follows from Corollary \[ps-rt\] that
The Foata-Schützenberger Theorem is equivalent to the Lusztig-Stanley Theorem.
First, notice that the set of permutations $B_k=\{\pi\in S_n:\
{\operatorname{maj}}(\pi^{-1})=k\}$ is a disjoint union of Knuth classes, where for each partition $\lambda\vdash n$, there are exactly $m_{k,\lambda}$ Knuth classes of RSK-shape $\lambda$ in this disjoint union. Combining this fact with Proposition \[fine-examples\](1) implies that $B_k$ is a fine set for the representation $\rho_k:=
\bigoplus_{\lambda\vdash n} m_{k,\lambda} S^\lambda$.
On the other hand, by Proposition \[fine-examples\](2), the set of permutations $L_k=\{\pi\in S_n:\ \ell(\pi)=k\}$ is a fine set for $R_k$.
Combining these facts with Corollary \[ps-rt\], $\rho_k\cong
R_k$ if and only if the distributions of the descent set over $B_k$ and $L_k$ are equal.
A combinatorial proof of the Lusztig-Stanley Theorem as an application of Foata-Schützenberger’s Theorem appears in [@Ro-Schubert]. The opposite implication is new.
More examples of fine sets {#sec:fine_sets}
--------------------------
\
The following criterion for fine sets is useful.
\[t.condition-fine\] Let $\rho$ be an $S_n$-representation, let $\{C_b : b\in {{\mathcal{B}}}\}$ be a basis for the representation space, and let ${\operatorname{Des}}: {{\mathcal{B}}}\to P_{n-1}$ be a map. If for every $1 \le i
\le n-1$ and $b, v \in {{\mathcal{B}}}$ there are suitable coefficients $a_i(b,v)$ such that $$s_i (C_b)=
\begin{cases}
-C_b, &\hbox{\rm if } i\in {\operatorname{Des}}(b);\\
C_b+\sum\limits_{v \in {{\mathcal{B}}}\text{\rm\ s.t.\ } i\in{\operatorname{Des}}(v)}
a_i(b,v) C_v, &\hbox{\rm otherwise}
\end{cases}$$ then ${{\mathcal{B}}}$ is a fine set for $\rho$.
The proof is a natural extension of the proof of [@Ro-Schubert Theorem 1] and is omitted.
Two well known bases which satisfy the assumptions of Proposition \[t.condition-fine\] are the Kazhdan-Lusztig basis for the group algebra [@KL (2.3.b), (2.3.d)] and the Schubert polynomial basis for the coinvariant algebra [@BGG Theorem 3.14(iii)][@APR11]. Since $S_n$ embeds naturally in classical Weyl groups of rank $n$, it follows that Kazhdan-Lusztig cells, as well as subsets of elements of fixed Coxeter length in these groups, are fine sets for the $S_n$-action.
Another useful criterion is the following. For a partition $\nu\vdash n$ let ${{\rm {SYT}}}(\nu)$ be the set of standard Young tableaux of shape $\nu$.
\[criterion1\] A subset ${{\mathcal{B}}}\subseteq S_n$ is a fine set if and only if for every partition $\nu\vdash n$ there exist a nonnegative integer $m(\nu,{{\mathcal{B}}})$ such that $$\sum\limits_{\pi\in {{\mathcal{B}}}} {\bf x}^{{\operatorname{Des}}(\pi)}=
\sum\limits_{\nu\vdash n} m(\nu,{{\mathcal{B}}}) \sum\limits_{T\in SYT(\nu)}
{\bf x}^{{\operatorname{Des}}(T)}.$$
Follows from Theorem \[t.main\].
It follows that a subset ${{\mathcal{B}}}\subseteq S_n$ is a fine set if and only if the sum of quasi-symmetric functions $\sum\limits_{\pi \in
{{\mathcal{B}}}} F_{{\operatorname{Des}}(\pi)}$ is Schur positive. By [@GR Thm. 2.1] (as reformulated in [@R13 Thm. 2.2]), conjugacy classes in the symmetric group satisfy this criterion. It follows that subsets of the symmetric group, which are closed under conjugation, are fine sets.
Another example which satisfies this criterion has been given recently. A permutation $\pi\in S_n$ is called an [*arc permutation*]{} if for every $1\le k\le n$ the set $\{\pi(1),\dots,\pi(k)\}$ forms an interval in the cyclic group ${\mathbb{Z}}_n$ (where $n$ is identified with 0). By [@ER Theorem 5], the subset of arc permutations in $S_n$ satisfies the criterion of Proposition \[criterion1\]; thus, the subset of arc permutations is a fine set in $S_n$.
We conclude with a list of the known fine subsets of the symmetric group.
The following subsets of $S_n$ are fine sets:
- Subsets closed under Knuth relations. In particular, Knuth classes, inverse descent classes and $321$-avoiding permutations.
- Subsets closed under conjugation.
- Permutations of fixed Coxeter length.
- The set of arc permutations.
The first two examples appear in [@GR Thm. 5.5]. It is a challenging problem to find a characterization of fine subsets in $S_n$.
[99]{}
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[^1]: Both authors were partially supported by Internal Research Grants from the Office of the Rector, Bar-Ilan University
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---
abstract: 'Zinc-blende CdSe semiconducting nanoplatelets (NPL) show outstanding quantum confinement properties thanks to their small, atomically-controlled, thickness. For example, they display extremely sharp absorption peaks and ultra-fast recombination rates that make them very interesting objects for optoelectronic applications. However, the presence of a ground-state electric dipole for these nanoparticles has not yet been investigated. We therefore used transient electric birefringence (TEB) to probe the electric dipole of 5-monolayer thick zinc-blende CdSe NPL with a parallelepipedic shape. We studied a dilute dispersion of isolated NPL coated with branched ligands and we measured, as a function of time, the birefringence induced by DC and AC field pulses. The electro-optic behavior proves the presence of a large dipolar moment (> 245 D) oriented along the length of the platelets. We then induced the slow face-to-face stacking of the NPL by adding oleic acid. In these stacks, the in-plane dipole components of consecutive NPL cancel whereas their normal components add. Moreover, interestingly, the excess polarizability tensor of the NPL stacks gives rise to an electro-optic contribution opposite to that of the electric dipole. By monitoring the TEB signal of the slowly-growing stacks over up to a year, we extracted the evolution of their average length with time and we showed that their electro-optic response can be explained by the presence of a 80 D dipolar component parallel to their normal. In spite of the $\bar{4}$3m space group of bulk zinc-blende CdSe, these NPL thus bear an important ground-state dipole whose magnitude per unit volume is twice that found for wurtzite CdSe nanorods. We discuss the possible origin of this electric dipole, its consequences for the optical properties of these nanoparticles, and how it could explain their strong stacking propensity that severely hampers their colloidal stability.'
author:
- Ivan Dozov
- Claire Goldmann
- Patrick Davidson
- Benjamin Abécassis
bibliography:
- 'paper.bib'
title: |
Probing Permanent Dipoles in CdSe Nanoplatelets\
with Transient Electric Birefringence
---
Introduction
============
Among colloidal nanocrystals, zinc-blende CdSe semiconducting nanoplatelets (NPL) have recently emerged as a new class of particles with ground-breaking optical properties. [@Ithurria2008; @Ithurria2011; @Nasilowski2016; @lhuillier2015] These nanoparticles are the two-dimensional equivalent of quantum dots but quantum confinement occurs only along the dimension parallel to their atomically controlled thickness. As a consequence, they display original optical features such as extremely sharp absorption peaks, large binding energies, and fast recombination rates. [@Ithurria2011] These properties could be useful in a variety of optoelectronic applications such as LED. [@chen2014; @Giovanella2018] Moreover, two key features of this class of materials have recently been discovered. Ultra-fast Forster Resonance Energy Transfer (FRET) occurs at the nanoscale between NPL [@rowland2015; @Guzelturk2015]. FRET is particularly fast in the case of nanoplatelets and even faster than Auger recombination [@kunneman2013] which plagues quantum dots performances in devices [@klimov2000; @Guzelturk2016]. Furthermore, NPL display strongly anisotropic, directed photo-luminescence while their absorption is isotropic. [@Scott2017; @Gao2017a] These properties can potentially overcome current limitations observed in devices through engineering exciton flow in assemblies of NPL. However, exploiting these new (directional) optical properties requires organizing NPL orderly in a three-dimensional fashion, both in solution and on a surface. Several different methods have been used to induce macroscopic alignment of NPL such as stretching unidirectionally a NPL-polymer elastic composite [@Beaudoin2015; @Beaudoin2017; @Cunningham2016] or the use of evaporation-mediated self-assembly on a liquid [@Gao2017a] or solid substrate [@Ma2018a]. Another way to orient particles is by applying an electric field. This strategy can be very efficient if the particles bear an electric dipole since the dipoles will orient parallel to the field. CdSe nanorods have been shown to orient over large distances in electric fields [@Ryan2006; @Carbone2007] but such a strategy has never been applied for NPL.
The presence of a ground state electric dipole in CdSe NPL is unknown. Such a dipole can arise due to the difference of electro-negativity between Cd and Se. Depending on the crystallographic structure and the shape of the particles, this modulated electronic distribution can yield a permanent dipole if the centers of mass of the negative and positive ionic lattices do not coincide. For wurtzite-type CdSe nanocrystals, theoretical calculations have shown that a permanent dipole should exist along the c-axis [@Rabani2001] and electro-optical experiments have measured important dipolar moments in CdSe wurtzite quantum dots [@Blanton1997] and nanorods. [@Li2003] This type of crystalline structure is intrinsically polar but important dipolar moments have also been reported for zinc-blende type nano-structures whose cubic symmetry theoretically forbids the emergence of a significant dipole. [@Shim1999] The values of dipoles for CdSe nanocrystals range from 41 to 98 D for particles with diameters between 2.7 and 5.6 nm and increase with size. [@Blanton1997; @Shim1999; @Kortschot2014] For wurtzite, the measured dipole ranged between 25 and 50 D for spheres and can be as large as 210 D for nanorods. [@Li2003] All these values have been measured using dielectric spectroscopy or transient electric birefringence in the case of nanorods. Larger values, up to 500 D, have been extracted from cluster-size distribution measurements [@Klokkenburg2007] but these values strongly depend on the type of model used for the inter-particle potential. Though some reports link the dipolar moment with the total volume of the nanoparticles, there is no universally accepted scaling relation between the two quantities and recent simulations have shown that other factors such as ligand structure could have a more important effect. [@Greenwood2018]
The existence of a permanent dipole can impact both the optical properties and the self-assembly of nanocrystals. Ground-state dipoles can induce important effects on optical properties since the electric field within the crystal breaks the inversion symmetry, which has important consequences on the electronic states and hence on the emission spectrum. [@Schmidt1997] Differences in absorption spectra between one and two-photon excitations can notably be difficult to explain without evoking ground-state dipoles due to the mixing of states. Dipolar coupling between adjacent colloidal nanocrystals can also induce giant enhancement of the absorption cross-section in close-packed films compared to isolated nanoparticles. [@Geiregat2013] Dipole-dipole interactions dramatically impact colloidal self-assembly at the nanoscale. [@Talapin2007; @Batista2015] For example, typical dipolar structures such as chains have already been observed in semi-conducting nanoparticle assemblies [@Dollefeld2001a; @Klokkenburg2007; @Shanbhag2007; @Cho2005; @Baskin2012].
It is thus important to determine whether or not CdSe NPL bear a permanent dipole. The crystallographic structure of CdSe NPL is zinc-blende. [@Ithurria2008; @Chen2015] The top and bottom planes of the NPL are \[001\] basal planes with cadmium atoms linked to carboxylates ligands. [@Singh2018] These ligands provide the NPL with an initially moderate colloidal stability [@Jana2015] which can be improved with an “entropic” ligand. [@Yang2016] We have previously shown that CdSe NPL can assemble into a variety of structures depending on the assembly conditions such as giant micro-needles [@Abecassis2014] or (twisted) nano-ribbons [@Jana2016; @Jana2017]. In these assemblies, NPL are stacked one on top of each other in a face to face fashion. [@Jana2015] Excitonic coupling in assemblies of NPL induces modifications to the fluorescence lifetime [@Guzelturk2014a] and causes the emergence of new peaks at low temperatures [@Tessier2013; @Diroll2018]
Transient electric birefringence (TEB) is a technique of choice to investigate dipoles in colloidal nanoparticles. This method consists in submitting a colloidal suspension of the particles of interest to time-dependent electric fields and in recording its field-induced birefringence. [@Benoit1951] The TEB signal carries a specific signature when the particles have a permanent electric dipole. Indeed, since the pioneering work of Peterlin and Stuart [@Peterlin1939], it was successfully used to answer similar questions in colloidal dispersions of various kinds of nanoparticles such as clays [@Shah1963; @Holzheu2002; @Dozov2011b; @Arenas-Guerrero2016; @Arenas-Guerrero2018], mineral nanorods [@delacotte2015], viruses [@Kramer1992], protein fibrils [@Rogers2006], macromolecules [@Frka-Petesic2014], and mixtures of rod-like and spherical colloids [@Mantegazza2005]. Recently, this technique has also been applied to investigate the field-induced alignment of CdSe nanorods [@Mohammadimasoudi2016] or the orientation of transition metal dichalcogenide nanodiscs. [@Rossi2015]
Here, we use TEB to measure ground-state dipoles in colloidal CdSe nanoplatelets. We first recall the theoretical background necessary to understand the experiments that were conducted. We then present our results on isolated CdSe NPL that are well dispersed in solution by the use of entropic ligands. The optical response of the dispersion to short square pulses of direct-current field and the relaxation dynamics when the field is switched off indicate that the most important contribution to the electric-field-induced orientation is due to a large permanent dipole in the plane of the NPL. This is confirmed by the low and high frequency alternating current experiments which are consistent with a permanent dipole larger than 245 D. However, these experiments alone cannot give an estimate of the out-of-plane component of the dipole. To do so, we induced the slow stacking of the NPL by the addition of oleic acid. When the NPL stack, the TEB signal not only increases in intensity with time but also changes sign. This is due to the fact that, in NPL stacks, the in-plane dipolar components cancel out since adjacent NPL assemble with in-plane dipoles in opposite directions whereas out-of-plane components add. Furthermore, the geometry and the polarizability of the objects responding to the field change when the NPL stack, which gives rise to an unusual situation where the major components of the permanent and induced dipoles are orthogonal.
Theoretical background
======================
Electric-field-induced birefringence of anisotropic objects
-----------------------------------------------------------
When a strong electric field $ \bm{E} $ is applied to a dispersion of anisotropic particles, they are aligned by the field, on average, along or perpendicular to its direction, depending on the anisotropy of the electric properties of the particles. There are two different mechanisms of orientation: one is due to the polarizability anisotropy of the particle and the other is present only if the particle bears a permanent dipole.
The total electric energy of a particle is then given by: $$U^{p}(\Omega)=-\bm{\mu \cdot E} - \frac{1}{2}\bm{E \cdot \alpha \cdot E},
\label{eq:energy}$$ where $ \bm{\alpha}$ is the (excess) polarizability tensor of the particles, $\bm{\mu}$ is their permanent dipole moment, and $ \Omega$ denotes the set of Euler angles defining the particle orientation in the laboratory frame.
When the dispersion is isotropic, as in our case, the particles reorient individually under the field action, with preferred average alignment defined by the sign of the particle electric anisotropy. Due to the revolution symmetry of the system around the field direction, $ \bm{e}$, the induced order is uniaxial. In the simplest case when the particles have cylindrical symmetry around some particle axis $ \bm{p}$, i.e. for rods and disks, the induced order is described by the scalar order parameter $ S(E)$ which depends on the field-induced and permanent dipoles of the particles (see supplementary information for the expression of $S(E)$).
At small particle volume fraction, $\Phi \ll 1$, the induced nematic-like orientational order can be probed by measuring the equilibrium value of the field-induced birefringence of the colloidal dispersion [@OKonski1959]: $$\Delta n(E)=\Phi \Delta n^pS(E).
\label{eq:biref}$$ Here, $\Delta n(E)=n_\parallel(E) - n_{\bot}(E) $ is the induced birefringence where $ \parallel$ and $\bot$ denote the direction of light polarization with respect to $\bm{e}$ and the refractive index $n_i$ is related to the polarizability of the dispersion, $\bm{\alpha^{opt}}$, in the optical frequency range by $n_i(E) \propto \sqrt{\alpha_i^{opt}(E)}$. Similarly, the specific birefringence of the particle, defined as the birefringence of the dispersion for perfectly oriented particles (S = 1) and extrapolated to $\Phi=1$, is $\Delta n^p=n_\parallel^p - n_{\bot}^p$ where $ \parallel$ and $\bot$ denote the direction of light polarization with respect to the symmetry axis $\bm{p}$ of the particle. The specific birefringence is defined by both the internal structure of the particle (the “intrinsic” birefringence, which is only related to the optical anisotropy of the particle material) and its shape (the “form” birefringence, which also depends on the solvent refractive index). For rods or disks in a weak field, the induced order is quadratic in the field [@OKonski1959]: $$S(E)\simeq \frac{1}{15} \Delta A E^2 \quad \textrm{with} \quad \Delta A = \frac{1}{(kT)^2}\left(\mu_{\parallel}^2-\frac{1}{2}\mu_{\bot}^2\right)+\frac{1}{kT}\Delta \alpha,
\label{eq:order}$$ where $\Delta \alpha=\alpha_\parallel - \alpha_{\bot} $ is the anisotropy of the excess polarizability and the subscripts denote the components of $\bm{\alpha}$ and $\bm{\mu}$ parallel and perpendicular to $\bm{p}$ (see figure 1.B.).
When the field-coupling coefficient $\Delta A $ is positive, the particles align with their symmetry axis parallel to the field. For $ \Delta A < 0$, however, the induced order is negative, $S<0$, and the symmetry axis of the particle tends to align perpendicular to the field. Thus, the sign of the induced birefringence gives important qualitative information about the strength and orientation of the permanent and induced dipole moments of the particle.
Time-dependent electric field
-----------------------------
More quantitative information about the dipole moments can be retrieved from the time-response of the birefringence when the field varies in time, e.g. under AC voltage with variable frequency [@Thurston1969] or under pulsed DC field [@Tinoco1959]. Indeed, the different contributions to $ \Delta A$ have different relaxation behaviors because the rotational diffusion coefficient $D^r$ depends on the orientation of the rotation axis, leading to parallel and perpendicular components, denoted $D^r_{\parallel}$ and $D^r_{\bot}$, respectively. Moreover, the transient behaviors of the dipole moment and the polarizability are qualitatively different because their couplings with the field are respectively linear and quadratic (see Eq.\[eq:energy\]). Indeed, upon fast inversion of the sign of the field, the contribution of the polarizability to the energy does not change; the particle would then keep the same orientation. However, the energy of the permanent dipole changes its sign, resulting in a “head-to-tail” reorientation of the particle and therefore a transient change of the birefringence.\
Here, we compare the rise and decay of the birefringence induced with square unipolar pulses [@Tinoco1959; @OKonski1957] and we study the frequency dependence of the DC and AC responses to bursts of sinusoidal voltage [@Peterlin1939; @Thurston1969]. These techniques have been widely used for the study of the permanent and induced dipoles of colloidal particles with effective rotational symmetry, i.e. for disks and rods. But, the CdSe platelets lack this symmetry and a more general approach should be used for the interpretation of the TEB data [@Kalmykov2009]. However, to simplify the interpretation, we will approximate in the following the rotational diffusion of the platelet by considering that it behaves like a cylindrical particle, e.g. with revolution symmetry around one of its axes. We assume that the rotational diffusion coefficients around the two other axes, $D^r_{\bot}$, are equal. Therefore, depending on the value of the diffusion coefficient for rotation around the effective symmetry axis, $D^r_{\parallel}$, the platelet or the stack of platelets can be considered either as a rod with $D^r_{\parallel} > D^r_{\bot}$ or as a disk with $D^r_{\parallel} \leq D^r_{\bot}$, which greatly simplifies the data analysis. The calculation of the diffusion coefficients (see SI for details) shows that the best uniaxial approximations, for both platelets and stacks, is that of a rod. For this geometry, the formulas describing the transient birefringence are well known. For square pulses and weak fields (Kerr regime), i.e. for small enough induced order parameter, $ S(E) \ll 1$, when the field is switched on, the birefringence increases and reaches an equilibrium value given by: $$\Delta n^{e}(E)= \Phi \Delta n^p S^e(E) = \frac{1}{15}\Phi \Delta n^p (p_{\parallel} - p_{\bot} + q) E^2=C_K^0 E^2
\label{eq:deltane}$$ with $$C_K^0 = \lim_{E \to 0} \frac{\Delta n^e(E)}{E^2}; \quad p_{\parallel}=\left( \frac{\mu_{\parallel}}{kT}\right) ^2; \quad p_{\bot}=\frac{1}{2}\left( \frac{\mu_{\bot}}{kT}\right) ^2; \quad q = \frac{\Delta \alpha}{kT}.
\label{eq:ck0}$$ $S^{e}(E)$ is the equilibrium order parameter reached after applying the field E for a very long time, $C_K^0$ is the DC-field Kerr constant of the colloidal dispersion, and the parameters $p_{\parallel},p_{\bot}$ and q describe the contributions of the permanent and induced dipole moments to the electric energy of the particle.
When the field is switched off, due to the rotational diffusion of the particles, the birefringence decays with time, following a simple exponential law: $$\Delta_n^{\textrm{off}}(t)=\Delta n^e(E) \exp \left(-6D_{\bot}^rt \right),$$ where the relaxation time $\tau_{\mathrm{off}}=1/(6D_{\bot}^r)$ corresponds to the rotational diffusion related to the second-rank tensor $\bm{\alpha^{\textrm{opt}}}$. The rise behavior of the induced birefringence when the field is switched on involves two more relaxation times: $ 1/(2D^r_{\bot}) $ and $1/(D^r_{\bot}+D^r_{\parallel})$, which are respectively related to the longitudinal and transverse components of the permanent dipole moment of the particle [@Tinoco1959] (see SI for more details). A simpler treatment of the data is possible when $D^r_{\parallel} \simeq D^r_{\bot}$ or when $D^r_{\parallel} \gg D^r_{\bot} $. In both cases, only two exponentials are needed to fit the rise curve: $$\Delta n^{on}(t)=\Delta n^e(E) \left[ 1-\frac{3\beta}{2(\beta+1)} \exp(-2D^r_{\bot}t) + \frac{\beta-2}{2(\beta+1)}
\exp(-6D^r_{\bot}t) \right],
\label{eq:2decays}$$ where $\beta$ is respectively $(p_{\parallel}-p_{\bot})/q$ and $p_{\parallel}/(q-p_{\bot})$ in each case.\
With bursts of sinusoidal field with frequency $f$ and amplitude $E_0$, $E(t)=E_0\cos (2\pi ft)$, the signal relaxes to a steady-state regime after a transient initial response. Then, the birefringence has two contributions: a stationary (DC) one, $\Delta n ^{\textrm{st}}(f)$, and an AC one, $\Delta n ^{\textrm{osc}}(f)$, oscillating at a frequency double of that of the field: $$\Delta n(t)=\Delta n ^{\textrm{st}}(f)+\Delta n ^{\textrm{osc}}(f)\cos (4\pi ft-\delta (f)).
\label{eq:nteb}$$
The information about the permanent and induced dipoles and the rotational diffusion of the particles is contained in the frequency dependences of the amplitudes of the two components. (The phase shift $\delta $ in eq. \[eq:nteb\] cannot be exploited with our external-electrode setup.) These dependences have been analyzed in detail by Thurston and Bowling (Th-B) [@Thurston1969] for a particle with revolution symmetry and with permanent dipole oriented only along the symmetry axis, i.e. with $p_{\bot}=0$ . The stationary response is given by: $$\Delta n ^{\textrm{st}}(f) = \Delta n ^{e}(E_{\textrm{rms}})\frac{1}{P+1}\left(P+\frac{1}{1+\left(\pi f /D_{\bot}^r \right)^2} \right)=C_K^{\textrm{st}}(f) E_\textrm{rms}^2.
\label{eq:stat_resp}$$ where $E_\textrm{rms}=E_0/\sqrt{2}$ is the root mean square (rms) value of the field and the parameter, $P=q/p_{\parallel}$ describes the relative weights of the induced and permanent dipoles.
The oscillating response is given by: $$\Delta n ^{\textrm{osc}}(f)=\Delta n^{e}(E_{\textrm{rms}})\left[1+\left(\frac{P}{P+1}\right)^2\left(\frac{\pi f}{D^r_{\bot}}\right)^2\right]^{\frac{1}{2}} \left[1+\left(\frac{\pi f}{D^r_{\bot}}\right)^2\right]^{-\frac{1}{2}}\left[1+\left(\frac{2\pi f}{3 D^r_{\bot}}\right)^2\right]^{-\frac{1}{2}}
\label{eq:oscill_resp}$$
At very low frequency, $\pi f /D_{\bot}^r \ll 1$, the particles always follow the field and $\Delta n ^{\textrm{st}}(f)$ remains constant, on a low-frequency plateau, $$\Delta n ^{\textrm{st}}(0) = \Delta n^{e}(E_{\textrm{rms}})= \frac{1}{15}\Phi \Delta n^p (p_{\parallel} - p_{\bot} + q) E_{\textrm{rms}}^2=C_K^0 E_{\textrm{rms}}^2.
\label{eq:lowf-plateau}$$
We note that the equilibrium value of the birefringence has this simple form even when $p_{\bot} \neq 0$. [@Thurston1969; @Kalmykov2009a]
Upon increasing frequency, the rotational relaxation of the particles takes place and they only partially follow the field. Through this process, the rms-value of the induced-dipole torque remains constant because it is quadratic in the field. In contrast, the permanent-dipole torque, which is linear in the field, decreases with increasing frequency and vanishes at $\pi f /D_{\bot}^r \gg 1$. In that limit, $\Delta n ^{\textrm{st}}(f) $ reaches a new, high-frequency, plateau: $$\Delta n ^{\textrm{st}}(\infty) = \Delta n^{e}(E_{\textrm{rms}})\frac{P}{P+1}= \frac{1}{15}\Phi \Delta n^p qE_{\textrm{rms}}^2=C_K^{\infty} E_{\textrm{rms}}^2,
\label{eq:highf-plateau}$$ which depends only on the polarizability of the particle. Therefore, the ratio of the two plateaus is given by: $$\frac{C_K^{0}}{C_K^{\infty}}=1+\frac{p_{\parallel}-p_{\bot}}{q},
\label{eq:ratiock}$$ and provides direct information about the relative importance of the permanent and induced dipoles and their orientation and anisotropy.
Results and discussion
======================
![A) Transmission electron microscopy image of CdSe nanoplatelets. B) Schematic representation of a CdSe platelet, shown as a parallelepiped, approximated as a cylinder with revolution symmetry along the **1**-axis. The components of the polarizability tensor are shown in blue and those of the electric dipole moment are shown in red. C) Transient electric birefringence of a colloidal dispersion of isolated CdSe platelets in hexane ($\Phi$ = 5.9$\times$10$^{-4}$). The black line shows the evolution with time of the applied field and the red circles are the data points of $\Delta n(t)$ induced by the short DC pulse (E = 950 V/mm). The blue lines are fits with the T-Y model (see equation \[eq:2decays\]). D) Equilibrium value of the birefringence $\Delta n^e$ as a function of the field squared, showing a linear behavior, with slope $C_K^0$, the Kerr constant.[]{data-label="fig:1"}](figure1.png){width="95.00000%"}
We synthesized CdSe nanoplatelets as described in the experimental section. Transmission electron microscopy images (Fig. 1.A) show that the NPL are parallelepipeds with “bare” (i.e. without the ligand brush) mean dimensions L$_1^b$=20 nm, L$_2^b$ = 9 nm, and L$_3^b$ = 1.5 nm. They are initially coated with the entropic ligand 2-hexyldecanoate which provides them with a longer colloidal stability [@Yang2016]. We assume that the short branched ligand measures 1.2 nm and is evenly located all over the particle. The addition of oleic acid slowly destabilizes the colloidal suspension as the CdSe platelets stack in wires whose average length increases with time over several months [@Jana2015; @Jana2016]. We therefore studied the TEB signal not only of the initial colloidal suspension of “isolated” platelets (i.e. of independent particles) but also of the slowly growing particle stacks. We first describe the results on isolated NPL and show what information can be extracted from these experiments. A second part of the paper deals with stacks of NPL that form over a few months upon addition of oleic acid.
Isolated Platelets
------------------
In the first few days after dispersion of the colloid, the TEB response is that expected for small isolated particles. The typical response to a short square pulse of DC field is shown on Fig. 1.C. When the field is switched on, there is a fast increase of the induced birefringence, $\Delta n(t)$, which then levels at a small positive equilibrium value, $\Delta n^{e}(E)$. This demonstrates that the NPL align in the field. When the field is switched off, the birefringence decreases back to zero with an even faster relaxation time. As we will show, in fact, this behavior agrees well with the model of Tinoco and Yamaoka (T-Y) [@Tinoco1959] for the TEB of particles with cylindrical symmetry of the polarizability tensor and of the rotational diffusion coefficient.\
In the rotational diffusion process, the particle and the ligand brush reorient together as a single rigid body that we call the “dressed” particle (i.e. including the ligand brush). Considering the brush thickness, the dressed particle dimensions along the three axes, $L_i$, are respectively 22.4, 11.4 and 3.9 nm. The calculation of the rotational coefficients $D^r_i$ around the axes i = **1**, **2**, **3**, of the dressed particle in hexane, using the Perrin formulae [@Perrin1934a; @Perrin1936], gives respectively $D_1 = 1.8 \times 10^6$ s$^{-1}$, $D_2 = 7 \times 10^5$ s$^{-1}$, $D_3 = 7.6 \times 10^5$ s$^{-1}$ (see SI for more details). Since the last two values are the closest, the best uniaxial approximation for the reorientation of an isolated CdSe platelet is a rod with length $L_{\parallel}=L_1$, diameter $L_{\bot}=\sqrt{4L_2L_3/\pi}$ and approximate rotational diffusion constants $D_{\parallel}^r=1.8 \times 10^6$ s$^{-1}$ and $D_{\bot}^r=7.3 \times 10^5$ s$^{-1}$. The components of the dipole moment and polarizability tensor of the equivalent rod are then obtained from those of the platelet (Fig. 1B): $\mu_{\parallel} = \mu_1; \mu_{\bot} = \sqrt{\mu_2^2+\mu_3^2}$ and $\alpha_{\parallel}=\alpha_{11}$; $\alpha_{\bot}=(\alpha_{22}+\alpha_{33})/2$. The Kerr constant of isolated platelets, deduced from the slope of the equilibrium value as a function of the field (Fig. 1.D.) is positive and very small, $C_K^0=6.0 \times 10^{-19}$ m$^2$/V$^{2}$. Because both $\Delta n^p$ and $\Delta \alpha$ are positive (see supplementary information for detailed calculation) for the isolated particle, approximated as an effective rod, we deduce from the sign of $C_K^0$ and Eq. \[eq:deltane\] that $p_{\parallel} - p_{\bot} + q > 0$, i.e. that the long axis of the NPL (the 1-axis of the equivalent rod) orients parallel to the field.\
The best fit of the signal decay (Fig. 1C) gives $\tau_{\textrm{off}}=$ 0.30 $\mu$s and $D_{\bot}^r = 5.5 \times 10^5$ s$^{-1}$ which is in fair agreement with the previously estimated value of $7.3 \times 10^5$ s$^{-1}$. We note that this extremely short relaxation time is close to the time-resolution of our experimental setup and is therefore overestimated. Indeed, the deconvolution of the data from the instrumental function, separately measured, gives $D_{\bot}^r = 6.6 \times 10^5$ s$^{-1}$, which agrees better with the theoretical prediction.\
For our CdSe particles, $D_{\parallel}^r \simeq 3D_{\bot}^r$, so that neither approximations required to describe the birefringence decay (Eq. \[eq:2decays\]) holds true. However, in our case, a simple approach consists in comparing the two areas, $I^{\textrm{on}}$ and $I^{\textrm{off}}$, limited by the on- and off- curves (see SI for more details). The experimental value, $\frac{I^{\textrm{on}}}{I^{\textrm{off}}} \simeq 3.57$, is much larger than 1, which shows the important contribution of the permanent dipoles to the birefringence. Indeed, in the opposite case where $q \gg p_{\parallel}, p_{\bot} $, the induced birefringence is mainly due to the polarizability of the particle and the ratio is close to 1. With the numerical estimations of the diffusion coefficients, we obtain $p_{\parallel}-p_{\bot}/2 \simeq 5(q-p_{\bot}/2)$. Taking into account that $p_{\parallel} - p_{\bot} + q>0$ (from the sign of the Kerr coefficient) and that the dipole moments are positive by definition, we obtain the inequalities $p_{\parallel} > q > p_{\bot}/2 \geq 0$. This means that the most important contribution to the TEB of isolated particles comes from a large permanent dipole along the 1-axis of the platelet. We also note that q > 0, in good agreement with our rod-like approximation for the isolated CdSe platelet.\
![ A, B: $\Delta n(t)$ induced by bursts of low- and high-frequency AC field (4 and 200 kHz). The blue lines are fits of the steady-state part of the curves with the Th-B model (equation \[eq:nteb\]). C: Double-logarithmic plot, versus frequency, of the Kerr constants corresponding to the steady (blue circles) and oscillating (red circles) TEB contributions. The lines are fits of the data with the Th-B model (Eq. \[eq:stat\_resp\], \[eq:oscill\_resp\])](figure2.png){width="\textwidth"}
. \[fig:2\]
Typical TEB responses to sinusoidal bursts, measured at low and high field frequency, are displayed in Fig. 2.A) and 2.B). The induced birefringence is positive at all frequencies, showing that the major components of the permanent and induced dipoles are parallel. [@Thurston1969]
Fig. 2.C) shows the logarithmic plot, versus frequency, of the Kerr constants corresponding to the steady and oscillating TEB contributions. The continuous lines on the figure show the best fits of the experimental data with the Th-B model (Eq. \[eq:stat\_resp\], \[eq:oscill\_resp\]). Qualitatively, both the steady and oscillating Kerr constants follow the expected trend, decreasing strongly at high frequencies. However, the fit is not quite satisfactory in the 20 – 80 kHz frequency range, suggesting the presence in this range of some additional relaxation process unrelated to rotational diffusion. $\Delta n ^{\textrm{st}}(f)$ remains positive in the whole frequency range accessible with our set-up. It decreases by almost one order of magnitude but does not yet reach the second plateau, showing that $C_K^{0}/C_K^{\infty} \geq 10 $. Since $q=\Delta \alpha / (kT) >0$ for the isolated particle, approximated as a rod, we conclude, using Eq. \[eq:ratiock\], that $p_{\parallel}>p_{\bot}$ and $(p_{\parallel}-p_{\bot})/q \geq 10$. This important results confirms the existence of a large permanent dipole moment parallel to the length of the particle, as already inferred from the slow birefringence rise induced by short square pulses. Therefore, to a very good approximation, the contribution of the induced dipoles to the TEB signal is negligible in front of that of the permanent dipoles, which greatly simplifies the following derivation of the absolute value of the dipoles of the CdSe platelets. For this purpose, one can use the low-field data acquired in the Kerr-regime, where the field-induced order is small and the birefringence is proportional to the square of the field. By neglecting the $q$ term, we obtain from Eq. \[eq:ck0\]: $$p_{\parallel}-p_{\bot}=(\mu_{\parallel}^2-\mu_{\bot}^2/2)/(kT)^2\simeq \frac{15}{\Phi \Delta n^p}C^0_K.$$ From the measurement of $C_K^0=6.0\times 10^{-19}$ m$^2$/V$^2$, the volume fraction $\Phi=5.9 \times 10^{-4}$ known through the absorption measurement [@Yeltik2015] and the calculated specific birefringence $\Delta n^p=0.39 $ (see SI), we obtain the effective value $\sqrt{\mu^2_{\parallel}-(1/2)\mu^2_{\bot}} \simeq 245 D$ for the dipole moment of the colloidal particles.
From these experiments, we demonstrate that the dipolar term is larger than the polarizability term and that the response to the field is mainly due to the dipolar component. Moreover, the dipolar component along the largest dimension of the NPL is much larger than the one perpendicular. Finally, we can extract from the Th-B model a lower bound of this component of the dipole: $\mu_{\parallel}$ is larger than 245 D.
Nanoplatelet stacks
-------------------
As mentioned above, we also studied the electro-optic behavior of assemblies of NPL. It is well known that upon the addition of oleic acid, NPL slowly assemble into stacks [@Jana2015] whose geometry and dynamics are expected to be very different from those of the NPL alone. For example, due to their large dimensions, they relax more slowly when the electric field is switched off.
![A) A stack of platelets is approximated as a cylinder (in grey) with revolution symmetry along the 3-axis. The subscripts $\parallel$ and $\bot$ refer to the orientation with respect to the revolution symmetry axis. The components of the polarizability tensor are shown in blue and those of the electric dipole moment are shown in red. On the left, the stack is made of platelets whose in-plane components of the dipole moment point alternatively in one direction (solid line) or the opposite one (dashed line). B) TEB of short stacks (25 days of aging) submitted to DC pulses (E = 950 V/mm). C) TEB of long stacks (105 days of aging) submitted to DC pulses, (E = 900 V/mm). In B) and C), the blue lines show fits of the data with the T-Y model (see equation 7) and, due to polydispersity effects, only the initial regions of the curves are fitted.[]{data-label="fig:3"}](figure3.png){width="50.00000%"}
At t=0, we added oleic acid to the dispersion and followed the optical response as a function of time. We expect the long-chain acid addition to trigger the slow destabilization of the NPL and the formation of stacks. In the first few days after dispersion of the colloid, the TEB response remains that expected for small isolated particles. A few days after the injection of oleic acid into the dispersion of NPL, the TEB signal started to evolve gradually from the fast response, with small amplitude, of isolated particles described previously to a slower response with larger amplitude, showing the occurrence of particle stacking. Moreover, the induced birefringence measured with short pulses or under bursts of low-frequency field changed its sign and became *negative*, which indicates a drastic change in the geometry of the reorienting objects.\
Fig. 3.B) shows the TEB signal of the short stacks (St1) that appear at an early stage of particle stacking (after 25 days). To interpret this TEB signal, the geometry and the physical properties of the stacks must first be discussed. Previous x-ray scattering and electron microscopy studies [@Jana2015] have shown that stacking takes place along the normal to the platelets, i.e. along their 3-axis (Fig. 3.A). Moreover, the condition of minimum electrostatic energy of the stack imposes that adjacent particles have parallel $\mu_3$ components of their dipole moment, but anti-parallel $\mu_1$ and $\mu_2$ components. Supposing that the particles are densely stacked, a stack of N particles will have the dimensions $L_3^N=N \times L_3$, $L_1^N=L_1$ and $L_2^N=L_2$. Because $L_3^N$ rapidly increases with $N$, for $N>10$, the equivalent shape of the stack transforms to a rod elongated along the 3-axis. Even the smallest stacks (St1, after 25 days) that we investigated electro-optically have $N \simeq 23$ (as shown in the following) and are moderately long rods, with rotational diffusion constants $(D^r_i)^N$ respectively of $1.6 \times 10^4$ s$^{-1}$, $1.8 \times 10^4 $ s$^{-1}$, $1.1 \times 10^5$ s$^{-1}$ (see supplementary information for details). Therefore, for the interpretation of the electro-optic data, we approximate the stack of N particles as a rod of length $L_{\parallel}^N=N\times L_3$ and diameter $L_{\bot}^N=\sqrt{4L_1L_2/\pi}$ with $(D^r_{\parallel})^N \gg (D^r_{\bot})^N$. For the dipole moment of the equivalent rod of the N-stack, we obtain a large longitudinal component, $\mu_{\parallel}^N = N \times \mu_3$ because the $\mu_3$ components add in the stack. On the contrary, the transverse component is very small: it is either $\mu_{\bot}^N=\sqrt{\mu_1^2+\mu_2^2}$ for odd $N$ or it vanishes for even $N$, due to the alternating 180$^{\circ}$ rotation around the **3**-axis of the particles in the stack. In contrast, the second-rank polarizability tensor, $\bm{\alpha}$, is invariant upon 180$^{\circ}$ rotation of the particle. Therefore, assuming that the CdSe particle cores in the stack are electrically insulated by their organic ligand brush (whose physical properties are similar to those of hexane), all three components of $\bm{\alpha}$ should be additive (see SI for more details).\
When the dressed particles are stacked, the polarizability density per unit volume and then the specific refractive indices remain unchanged after stacking: $(n^p_i)^N=n^p_i$ because the polarizability of a stack depends linearly on N and the total number of particles in the sample remains constant during stacking. However, contrary to the case of an isolated particle, the stack behaves hydrodynamically as a rod with revolution symmetry around the 3-axis. This leads to $(n^p_{\parallel})^N=n^p_3=1.85$, $(n^p_{\bot})^N=\sqrt{(n_1^p)^2+(n_2^p)^2/2}=2.44$, and $(\Delta n^p)^N=-0.58$, a *negative* value. Consequently, the equivalent optical polarizability tensor of the rod-like stack is oblate, i.e. as expected for a disk. This very unusual feature is due to the complex stack structure: the stack rotates as a rigid rod-like body but its optical response is that of a disk-like polarizable particle. In a similar way, based on the different known polarizability mechanisms, we expect that the electric polarizability at low frequency of the stacks is also additive, leading to: $\alpha_{\parallel}^N=N \times \alpha_{33}$ and $ \alpha_{\bot}^N \simeq N\times(\alpha_{11}+\alpha_{22})/2$ (see SI for details).\
Taking into account the effective rod geometry of the stacks, the negative sign of the induced birefringence (Fig. 3.B) indicates that, unlike the case of isolated particles, the major components of the permanent and induced dipoles of the stack are perpendicular. The TEB decay is not exponential, suggesting that the system is polydisperse. Moreover, the small and very fast overshoot at the beginning of the decay curve is due to isolated particles that still remain in the dispersion and that give a positive contribution to the birefringence. The best decay fit provides $\tau = 9.3 \, \mu$s and $D^r_{\bot}=1.8 \times 10^4$ s$^{-1}$, i.e. the rotational diffusion of the short stacks is about 30 times slower than for isolated particles. The T-Y fit of the TEB rise is reasonably good but it deviates from the data at both ends of the curve due to the polydispersity and the presence of isolated particles. The best fit parameters are $\beta=(p_{\parallel}-p_{\bot})/q \simeq -4$ and $C_K^0=-1.3 \times 10^{-18}$ m$^2$/V$^2$. Since the polarizability anisotropy $\Delta \alpha$ (and $q=\Delta \alpha/(kT)$) is negative for the rod-like stacks, the negative sign of $\beta$ and $C_K^0$ indicates that the permanent dipole of the stack is parallel to its long axis (i.e. the stacking axis).\
The TEB response of longer stacks under DC pulses (St4, after 105 days) is presented on Fig. 3.C). The experimental curves deviate strongly from the theoretical predictions and the fit with the T-Y model is good only in the initial regions. This behavior is most probably due to the stack polydispersity. The induced birefringence is again negative and is much stronger than for the shorter stacks (St1). The lack of overshoot of the decay curve suggests the absence of isolated particles. The best decay fit gives $\tau=57 \mu$s and $D^r_{\bot}=2.9 \times 10^3$ s$^{-1}$, i.e. the rotational diffusion of the long stacks is about 200 times slower than for isolated particles. The T-Y fit of the TEB rise is reasonable only in the first 400 $\mu$s of the signal, which is probably again due to the large stack polydispersity. The best fit parameters are $\beta=p_{\parallel}/q \simeq -9 $ and $C_K^0=-1.6 \times 10^{-17}$m$^2$/V$^2$, showing again a large and dominant contribution of the permanent dipole along the stack long axis. We note also that when $p_{\parallel} \gg \lvert q \lvert $, which is actually the case for the stacks, the T-Y fit is rather indiscriminative for the precise value of $\beta$. Therefore, the previous results, $\beta \simeq -9$, is only qualitative and just means that the TEB response is dominated by the large permanent dipole of the stack.\
Qualitatively, these conclusions are confirmed by the TEB signal (Fig. 4.A-C) of the same long stacks under bursts of low-, medium-, and high-frequency AC field. The steady component of the induced birefringence is negative at low frequency, vanishes at around 14 kHz and is positive at higher frequency. This behavior shows clearly that both the electrical, $\bm{\alpha}$, and the optical, $\bm{\alpha}^{\mathrm{opt}}$, polarisabilities of the stack are oblate tensors ($\Delta \alpha = \alpha_{\parallel}-\alpha_{\bot} < 0$) and that the permanent dipole of the stack is parallel to its long axis.\
The same salient features of the TEB behavior were observed throughout the growth of the CdSe platelet stacks: (i) the rise and decay times increased because of the decrease of the rotational diffusion coefficient $ D^r_{\bot}$; (ii) the Kerr constant at low frequency, $C_K^0$, remained negative and its absolute value increased with aging time (and hence with the stack length); (iii) the Kerr constant at high frequency, $C_K^{\infty}$, remained positive and much smaller than $\mid C_K^0 \mid$ and (iv) the frequency $f_0$ defined by $C_K^{\mathrm{st}}(f_0)=0, $ decreased with increasing stack length because of the decrease in $ D^r_{\bot}$.
![A, B, and C: TEB of the long stacks (105 days) submitted to bursts of low-, medium-, and high-frequency AC field: 70 Hz, 14 kHz and 70 kHz respectively. The blue lines are the fits of the relaxed part of the curves with the Th-B model (Eq. \[eq:nteb\]). D: Double-logarithmic plot, versus frequency, of the Kerr constants corresponding to the steady (red circles) and oscillating (blue circles) TEB contributions for the largest stacks (415 days of aging). The lines are fits of the data with the Th-B model (Eq. \[eq:stat\_resp\], \[eq:oscill\_resp\]).[]{data-label="fig:4"}](figure4.png){width="\textwidth"}
Fig. 4.D. displays the frequency dependence of the steady ($C_K^{\textrm{st}}$) and oscillating ($C_K^{\textrm{osc}}$) Kerr constants of the largest stacks (St5, after 415 days). Qualitatively, the behavior of the two curves follows the trend expected for a rod with $\Delta \alpha < 0$, $\Delta \alpha^{\mathrm{opt}} < 0$ and a large longitudinal permanent dipole moment. However, the theoretical model describes the experimental curve better or worse in the different frequency domains. Below $f=10$ Hz, both $C_K^{\mathrm{st}}$ and $C_K^{\textrm{osc}}$ decrease instead of remaining constant. This artifact is simply due to our external-electrodes technique for applying the field. Indeed, the field penetrating in the sample at these low frequencies is partially screened by the conductive charges in the solvent. However, despite this difficulty, the low-frequency plateau is well-enough pronounced, corresponding to a Kerr coefficient $C_K^0 = -5.8 \times 10^{-16}$ m$^2$/V$^2$, 400 times larger than for the shortest stacks (St1). The Th-B model does not describe the experimental curves well in the region between 30 and 800 Hz. This other discrepancy may be due to the large polydispersity of the stacks or to the fact that the Th-B model assumes that there is no other relaxation process than rotational diffusion in the frequency range under study. However, at higher frequencies, both around the sign-inversion frequency of $C_K^{\textrm{st}}$, $f_0=5.9$ kHz, and above, on the high-frequency plateau, the theory is in good agreement with the experimental curve. The Th-B fit of $C_K^{\textrm{st}}$ in this region provides $D^r_{\bot}=1020$ s$^{-1}$, $\beta=$ -330 and $C_K^0=-5.8 \times 10^{-16}$ m$^2$/V$^2$, indicating that the strong induced birefringence is due to the huge permanent dipole moment along the long axis of the stack. The main experimental results for stacks of different ages are presented in Table I. The values of $C_K^0$ obtained from the sinus bursts are very close to the pulse values, but are slightly more dispersed for the small stacks due to their weak TEB signal. The results for $\beta=p_{\parallel}/q$ and $D^r_{\bot}$ are those obtained from the value of the frequency $f_0$ at which $C_K^{\textrm{st}}$ changes sign and from the Th-B fit of $C_K^{\textrm{st}}(f)$ in the vicinity of $f_0$. Actually, these values are less influenced by the stack polydispersity and non-rotational relaxation processes.
------------ -------- ------------------------- -------------- ------------------------------ ---------------------------- -------- -------- ----
Experiment Age $C_K^0$ $D^r_{\bot}$ $(p_{\parallel}-p_{\bot})/q$ $(p_{\parallel}-p_{\bot})$ dipole S N
(Days) (10$^{-18}$m$^2$/V$^2$) (10$^3$ Hz) (10$^{-14}$m$^2$/V$^2$) (D)
a) b) c) c) d) e) f)
Stacks 1 25 - 1.3 16.1 - 11.5 8.1 350 0.0059 23
Stacks 2 35 - 6.0 13.9 - 17.7 38 750 0.027 27
Stacks 3 62 - 11.2 10.6 - 38.4 70 1030 0.051 32
Stacks 4 105 - 15.9 4.13 - 119 99 1230 0.073 46
Stacks 5 415 - 500 1.02 - 330 3120 6900 0.49 92
Stacks 5 415 -500 -16.1 3470 7260 0.49
Saturation g) g) g) g) g)
------------ -------- ------------------------- -------------- ------------------------------ ---------------------------- -------- -------- ----
: Electro-optic properties of CdSe platelet stacks directly measured (experimental data) or deduced from data interpretation (results). **a)** Kerr constant calculated from the DC pulses data; **b)** Rotational diffusion coefficient; the values are calculated from the sign-inversion frequency $f_0$; **c)** Ratio of the contributions to the TEB from the dipole moment and polarizability: $p_{\parallel}-p_{\bot} = \left(\mu_{\parallel}^2-\mu_{\bot}^2/2\right)/(kT)^2, q=\Delta \alpha/(kT)$; the values are calculated from the sign-inversion frequency $f_0$; **d)** dipole: $\sqrt{\mu_{\parallel}^2-\mu_{\bot}^2/2}$; **e)** Orientational order parameter measured at $E=1 V/\mu m, S(E)=\Delta n(E)/\Delta n^{\mathrm{sat}}$; **f)** Number of particles in the stack calculated by comparing the experimental and theoretical D$^r$ values; **g)** Values obtained from the fit of the saturation curve.
Saturation of the birefringence for long stacks
-----------------------------------------------
![Saturation of the induced birefringence at high field (10 ms long DC pulses) for the largest stacks (415 days of aging). The blue and red circles correspond to two independent experiments (see text). The red line is the best fit with the theoretically predicted behavior (see text). The inset shows the same information in log-lin representation. []{data-label="fig:saturation"}](figure5.png){width="50.00000%"}
At large field, away from the Kerr regime, the induced order should be strong enough to lead to the saturation of the TEB signal. This is indeed observed with the largest stacks, for which $ \Delta n (E)$ significantly deviates from the $E^2$ law, even though it does not reach complete saturation for the fields ($ E \leq 1 $ V/$\mu$m) accessible with our set-up in usual conditions (Fig. 5). However, using our “double-field” trick (see experimental section), we managed to apply inside the sample, in a transient way, fields up to 2 V/$\mu$m and, therefore, reach the complete saturation of $ \Delta n (E)$ (figure \[fig:saturation\]). The usual treatment of this kind of TEB data [@Shah1963; @OKonski1959] based on the series expansion in $E^2$ of $S(E)$ up to the $E^4$ term, works well for the case $(p_{\parallel}-p_{\bot})/q \geq 0$, when the $\Delta n(E)$ curve is monotonous. However, for large negative $(p_{\parallel}-p_{\bot})/q $ ratios, as in our case, the series converges too slowly and a large number of terms should be included, making this approach impractical. Therefore, we fitted our experimental data with the function $\Delta n(E) = \Delta n^{\textrm{sat}}S(E)$, where the order parameter $S(E)$ is calculated numerically, assuming $\mu_{\bot}=0$. The fit of the data is excellent and provides values presented in the last row of Table 1.
We note that these values of $ \mu_{\parallel} $, $\Delta \alpha$ and $C_K^0$ are self-consistent and independent of the Kerr-regime measurements under DC pulses and AC bursts. The maximum value of the order parameter is $S^{\mathrm{max}} = S(E=1.8)$ V/$\mu$m) = 0.57 which corresponds to a “saturated” birefringence value of $\Delta n^{\textrm{sat}}=\Delta n(S=1)=-2.19 \times 10^{-4}$. This value is of the same order of magnitude but smaller than the one estimated from the measured volume fraction and the calculated specific birefringence of the stack ($\Delta n^{\textrm{sat}} = \Phi \Delta n^p=-3.14 \times 10^{-4} $). However, the former value is directly derived from a self-consistent experiment and is not based on any approximation. Therefore, we used it to calculate $ \mu_{\parallel}$ and $\Delta \alpha$ from the experiments on the stacks at low field.\
Altogether, our TEB experiments provide a measurement of the dipole component $\mu_{3}^N$ for different nanoplatelet stacks with increasing size $N$, ranging from 350 D for the first and smallest stacks to 7260 D for the largest ones after more than a year (Table 1). These huge values are the physical origin of the very important TEB signal that we measured after addition of oleic acid. We stress that these results are very complementary with those obtained with isolated NPL which provided a value of $\mu_{\parallel}$. With the two sets of measurements, we can estimate the two orthogonal dipolar components if we manage to extract the value of $\mu_{\bot}$ for individual platelets from our measurements of $\mu_{\bot}^N$ of stacks. We describe a method to do so in the following paragraph.
Growth of nanoplatelet stacks with aging time
---------------------------------------------
![Evolution with time of the component of the electric dipole moment (red symbols) parallel to the main axis of the stacks and of the number of platelets per stack determined by modelling the TEB of stacks submitted to DC pulses (blue symbols).[]{data-label="fig:6"}](figure6.png){width="50.00000%"}
The approximately linear dependence of $\mu_{\parallel}^N$ on aging time (Figure 6) reflects the increase in average number of particles $N_{\textrm{av}}(t)$ and in length of the stack with time. $N_{\textrm{av}}(t)$ can be estimated by comparing the values of $D^r_{\bot}$ with the values calculated numerically [@Perrin1934a; @Perrin1936] for different values of $N_{\textrm{av}}$. We calculate the rotational diffusion constants $D_i^r$ (i=1,2,3), using the known dimensions $L_i$ of the dressed platelet and approximating the stack as a rigid biaxial ellipsoid with the same volume and axial ratios as those of a rectangular prism of dimensions $L_1, L_2, N_{\textrm{av}} \times L_3 $. The calculated value of $D^r_{\bot}=(D^r_1+D^r_2)/2$ is plotted versus $N_{\textrm{av}}$ in figure S2. The $D^r_{\bot} $ data was derived from the decay time of the TEB signals of the stacks, which were assumed to be monodisperse. The $N_{\textrm{av}}(t)$ values, called $N^{\textrm{DC}}$ were obtained by comparison of the experimental and calculated $D^r_{\bot}$ and are plotted in Fig. 6 as a function of aging time. N is about 23 in the first experiment where the stacks were detected (25 days of aging) and is 3 – 4 times larger for the longest stacks. Despite the linear increase in time of both $N^{\textrm{DC}}$ and $\mu_{\parallel}^N$, these quantities are not really proportional (Fig. S2), as would be expected from the relation $\mu_{\parallel}^N/N=\mu_3^1$. We explain in Supplementary Information how this discrepancy can arise from the presence of a fraction of isolated platelets coexisting with the stacks but slowly disappearing with aging time. This analysis leads us to the best estimate for the permanent dipole along the platelet normal: $\mu_3 \simeq 80 D$.
Discussion
==========
We thus find that CdSe NPL bear an important dipole whose magnitude is larger than 300 D. The in-plane component is larger than 245 D while the component along the thickness is around 80 D. These values are very large in comparison with previous direct measurements on dots and rods which yield typical dipoles ranging from 20 to 250 D. [@Greenwood2018] When scaled with the volume of the particles, this corresponds to 1.1 D/nm$^{-3}$, twice the value measured for wurtzite CdSe nanorods. [@Li2003] If we reason in the bulk, this is surprising in the first place since wurzite in known to be pyroelectric. The space group 6mm to which this structure belongs has a unique polar axis parallel to the 6-fold symmetry axis. In this direction, there is an alternation of short and long Cd-Se bonds. Furthermore, this axis is unique in the sense that it is not repeated by any symmetry element so that elementary dipolar moments add up. The zinc blende structure ($\bar{4}$3m) also displays 4 polar axes (in the <111> directions) but they are related by the $\bar{4}$ roto-inversion axis in such a way that the dipoles cancel. If the nanoparticles bear a permanent electric dipole moment, it is likely to be parallel to one of these directions. The basal planes of the CdSe NPL are of the [001]{} type and are neither parallel nor perpendicular to the <111> directions. Therefore, it is not a priori surprising that the permanent electric dipole has both parallel and perpendicular components with respect to the normal to the platelet. However, there is no symmetry reason for a particular <111> axis to be privileged compared with the other similar directions, so there must be another source of asymmetry in the system.\
There are several phenomena that can induce a spontaneous symmetry breaking and make a permanent dipole emerge. First, a NPL has limited dimensions and cutting a crystal into a given shape can reduce its symmetry. If we consider that the NPL adopt a perfect parallelepipedic shape, this argument does not hold since there are still multiple polar axis along the diagonals of the parallelepiped. Though the zinc-blende CdSe structure presents an asymmetric alternation of long and short Cd-Se bonds along the <111> directions [@Khurgin1998], any permanent electrical dipole along these particular axes will still be compensated within the perfect parallelepipedic shape. Hence, the symmetry reduction caused by cutting the crystal into a NPL can not explain alone the emergence of a permanent dipole.\
However, the zinc-blende structure is piezoelectric. In the presence of stress, the bonds will deform and, depending on the orientation of the stress with respect to the crystalline structure, this will yield a dipole. For example, if the zinc-blende lattice is strained along the <111> axis a net polarization will appear in this direction since the Cd-Se bonds will be deformed in such a way that dipoles will not compensate anymore [@vonHippel1952]. Not all stresses will yield a polarization though. This is apparent from the shape of the piezoelectricity tensor which links stress and polarization. In the zinc-blende case, only three terms are non-zero [@Nye1984] and a deviatoric component to the stress is needed for polarization to emerge. For example, a simple deformation of the NPL along the direction perpendicular to the basal [001]{} plane is not enough to make a polar axis unique. The piezoelectric constant $e_{14}$ relates the polarization to the strain. [@Huong1998] For zinc-blende CdSe, it is estimated [@Berlincourt1963; @Xin2007a] to be 0.2 C/m$^2$ . The dipole scaled to the volume that we measured (1.1 D/nm$^{-3}$) corresponds to a polarization of 3.63$\times$10$^{-3}$ C/m$^2$. Thus, a strain of only 1.8% can explain our result with the literature value of $e_{14}$. It is well known that surface ligands induce stress at the surface of semi-conducting colloidal nanocrystals due to incompatibility between their preferred conformation and the lattice of the inorganic core. [@Meulenberg2004; @Huxter2009] X-ray diffraction studies of CdSe spherical nanocrystals have shown that strain increases when the size of the nanocrystals decrease, reaching 0.5 % for 2.2 nm CdSe nanoparticles. In the case of NPL, their even smaller thickness and their high ligand density [@Singh2018] are likely to generate larger strains. Ligand exchange from the native oleic acid to phosphonic acid or thiols has been shown to distort the crystal lattice significantly with relative variations of lattice parameters which could reach 4%. [@Antanovich2017] These important strains are consistent with previous studies which have shown that CdSe NPL could adopt various curved conformations depending on the surface ligand and their crystallographic structure. [@Bouet2013a; @Hutter2014; @Jana2017] Due to the very thin nature of the NPL, even the small stress exerted by the ligands at their surface can result in large deformations. Atomic arrangements are modified by the surface stress and depart from their highly symmetric configurations. Consequently, the physical origin of the dipole could be the stress imposed by the organic ligands at the surface of the NPL.\
We now discuss the consequences of the presence of a large permanent dipole in CdSe nanoplatelets. Such an important permanent dipole moment will affect the colloidal interactions between NPL and strongly impact their colloidal stability in suspension. [@Israelachvili2010] With the particle dimensions and the values of the components of the dipole moment that we derived above, an order of magnitude of this dipolar interaction energy can be estimated for different relative orientations of two platelets (keeping in mind that the finite size of the dipoles may not be neglected in front of their separation). The largest attraction energy, of about -3 kT, is found for two stacked platelets at contact (i.e. at 4 nm separation) when the $\mu_3$ components are in line and the $\mu_1$ components are anti-parallel. However, at room temperature, thermal averaging of the relative orientations of the dipoles should also be considered, a process leading to the Keesom interactions for freely-rotating point-like dipoles. Thermal fluctuations will also induce deviations from the ideal stacked configuration, resulting in an increase of the average separation between platelets. This thermal averaging will sharply decrease the magnitude of the interaction energy since the potential strongly depends on the platelet separation. This reasoning may qualitatively explain the marginal colloidal stability of CdSe nanoplatelets in hexane but a more rigorous statistical physics treatment of this question is required to reach a more quantitative description.\
A large ground state dipole should also impact the optical properties of CdSe NPL. By breaking the inversion symmetry of the NPL, the internal electric field will mix odd and even quantum states. [@Schmidt1997] This should be visible in the difference between one photon and two-photon absorption spectra at low temperature and parity-forbidden transitions should be allowed. Such effects have already been shown to occur for CdSe spherical nanocrystals [@Schmidt1997] but their relevance for nanoplatelets is still to be assessed.
Conclusion
==========
Using transient electrical birefringence on dispersion of CdSe NPL and their self-assembled stacks, we demonstrated that these nanoparticle bear an important permanent dipole larger than 300 D with components perpendicular (> 245 D) and parallel ($\simeq 80 D$) to the NPL normal. This corresponds to a very large polarization, almost twice larger than what has been previously observed in wurtzite nanoparticles though the zinc-blende structure is not polar. The dipole could arise from deformation of the crystalline lattice from its cubic structure due to ligand induce surface stress. Variation of the particle thickness might help rationalizing these results further. These results have important implications on the self-assembly of NPL into larger scale structures. It also highlights that an electric field could be used to orient very efficiently NPL in space to harness their outstanding anisotropic optical properties and their directed emission [@Scott2017]. Finally, this ground state dipole should be taken into account in order to understand 2-photon and Stark spectra at low temperatures.
Experimental
============
Synthesis and purification of CdSe nanoplatelets
------------------------------------------------
All chemical were purchased at Sigma-Aldrich.\
**Synthesis of cadmium oleate**\
40 mmol of sodium oleate is dissolved in 200 ml of ethanol and 50 ml H$_2$O mixture and stirred for 30 min at 60-70$^{\circ}$C until a clear transparent solution is obtained. The solution is then cooled to around 40$^{\circ}$C . In another beaker 20 mmol of cadmium nitrate is dissolved in 50-60 ml of ethanol. This solution is slowly added to the Na-Oleate solution with constant stirring. After complete addition the mixture is kept stirring for another 30 minutes. A white precipitate is formed and the supernatant is discarded. Fresh ethanol is added and the precipitate is retrieved after a centrifugation at 3000 rpm for 5 min. The white product is washed 3/4 times by hot ethanol and finally washed with hot methanol. The final product is kept under vacuum overnight to dry. It should have the aspect of a white slightly sticky powder.\
**Synthesis and purification of the NPLS**\
404 mg of cadmium oleate, 27 mg of selenium powder (100 mesh), and 25 mL of octadecene (ODE, 90%) were introduced into a 50 ml three-neck round bottom flask, equipped with a septum, a temperature controller and a condenser, and were kept under vacuum for 30 minutes. Afterwards, the flask was purged with argon and the temperature was set to 240$^{\circ}$C. At 180-190 $^{\circ}$C, the selenium started to dissolve and the solution turned clear yellow. When the temperature reached 205 ${\circ}$C, the septum was withdrawn and 140 mg of cadmium acetate (Cd(OAc)$_2$, 2H$_2$O, Aldrich) was swiftly added into the flask. After the temperature reached 240$^{\circ}$C, the reaction continued for 12 minutes and 1 mL of oleic acid was injected at the end. The flask was immediately cooled down to room temperature. At this stage, the reaction product was a mixture of 5 monolayers (ML) NPL, a few 3ML NPL and quantum dots in solution. The 5 ML NPL were collected using size-selective precipitation by addition of ethanol and re-dispersion in 3mL of hexane. 50 uL of 2-hexyldecanoic acid were added. After 45 minutes, they were precipitated with 15 mL of acetone and centrifuged at 4000 rpm. The clear supernatant was discarded and the entire operation was repeated again. To finish, the platelets were re-dispersed in hexane.
Transient electric birefringence
--------------------------------
The experimental set-up for the TEB measurements was previously described in detail [@Dozov2011b; @Paineau2012; @Paineau2012a]. It is mostly inspired by classic TEB experiments [@OKonski1959], except for one important modification: instead of the classic Kerr cell, with electrodes immersed in the liquid and long light-path (several centimeters) of the probe beam, the sample in our case was contained in a flame-sealed cylindrical glass capillary of diameter D = 1 mm. The electric field was applied parallel to the capillary axis by a pair of external electrodes (2 mm apart) placed directly on the outer surface of the capillary wall. The voltage applied to the electrodes was either as bursts of sinusoidal alternating current (AC) voltage (from 1 to 10$^4$ periods in one burst) with variable frequency f, ranging from 1 Hz to 400 kHz, or as short direct current (DC) pulses (duration $\tau_{imp}$ from 10 $\mu$s to 10 ms). The numerical simulation of the field penetration into the capillary [@Dozov2011b; @Antonova2012] shows that the field inside the colloidal dispersion is uniform and that the screening losses due to accumulation of charges on the inner side of the capillary wall are negligible at high enough frequency (here, f > 10 Hz).\
Low-voltage (< 10 V) AC bursts and DC pulses with the required repetition rate were generated by an Arbitrary Waveform Generator (TGA 1241, TTi) and sent to an amplifying block. This block consisted of several different instruments, depending on the required voltage amplitude, U, and response time of the amplifier, $\tau_r$: (i) a Wide-Band Amplifier (WBA, Krohn-Hite 7602M) for U < 400 V and $\tau_r$ > 0.2 $\mu$s; (ii) a high-voltage (HV) amplifier (Trek 2220) for 0.4 kV < U < 2 kV and $\tau_r$ > 50 $\mu$s; (iii) a double-output HV switch (PVM-4210, Directed Energy) for unipolar DC pulses with 0.4 kV < U < 1.9 kV and $\tau_r$ > 0.02 $\mu$s; (iv) finally, a set of home-made transformers adapted to different frequency ranges were used to amplify the WBA output voltage up to about 2 kV for AC bursts with frequency f > 1 kHz. In this way, we could apply fields up to 1 kV/mm to the sample in the whole frequency range of interest. This field limit is imposed not only by the available amplifiers but also by the dielectric breakdown of air in usual laboratory humidity conditions. However, one feature of our external-electrodes setup allowed us to apply inside the sample, in a transient way, a field twice as large as this limit: Indeed, when a DC voltage U is applied to the external electrodes for a time much longer than the charge relaxation time of the solvent ($\tau_{ch} \simeq 20$ms for our sample), the conductivity charges of the solvent move and accumulate on the inner side of the capillary glass wall facing the electrodes. This process proceeds up to the complete screening of the field within the suspension because of the opposite field created by the accumulated charges. If now the voltage applied to the electrodes is rapidly reverted, from U to –U, the external field and the field due to the accumulated charges have the same sign, resulting in a twice stronger transient field in the suspension, 2U/L$_e$. This field relaxes back to zero with the same characteristic time $\tau_{ch}$ because of the migration of the charges to the opposite wall. Since the rise-time of the TEB signal ($\tau_{on} \simeq 2 ms$ for the longest stacks) is much smaller than $\tau_{ch}$, we can measure, during the transient regime, the induced birefringence under the internal field $E_\textrm{int}=2U/L_e$ i.e. up to 2 kV/mm. We call this field-inversion procedure the “double-field” trick in the main text.\
The field-induced birefringence was measured in real time, under polarizing microscope (Leitz Ortholux II), with the apparatus described in detail in references [@Dozov2011b; @Buluy2018]. It consists of a stabilized light source, an optical compensator introducing an additional constant phase shift, a photo-multiplier tube (PMT), a load resistor R$_L$ transforming the PMT anode current in a voltage difference, a differential amplifier with band-pass filters (AM 502, Tektronix), and a digital oscilloscope (DSO-X 2004A, Agilent Technologies) that accumulates the signal up to 64000 counts. Nevertheless, this setup was modified in several ways to achieve the high sensitivity and fast response time required for some of the measurements. For large particle stacks, the signal was strong enough, with good Signal-to-Noise (S/N) ratio, and we used, as previously, a Berek compensator introducing a $\lambda/4$ phase-shift, resulting in a transmitted intensity linearly proportional to the induced birefringence. However, this simple optical configuration is not convenient when the induced birefringence is very small because the PMT current is a sum of the small time-dependent induced-birefringence signal and a large constant term coming from the $\lambda/4$ phase-shift introduced by the compensator. Therefore, the residual noise after removal of the constant term is too high. To remedy this issue, for weak signals, we replaced the Berek compensator with a Senarmont compensator and we uncrossed the analyzer by just a few degrees. In this way, we obtained a more sensitive (quadratic) optical response and a smaller constant term, resulting in significantly better S/N ratio. Moreover, the Senarmont compensator introduces the same phase-shift over the whole field of view of the microscope, allowing us to use a much larger measurement window, which also improves the S/N ratio significantly. The response time of the set-up is mainly defined by the R$_L$C$_A$ constant of the PMT anode. For measurements with suspensions of isolated platelets, due to their large rotational diffusion constant, D$^r$ = 6.6$\times$10$^5$s$^{-1}$, the response time must be kept as short as possible. Since the anode capacitance, C$_A\simeq$300 pF, is fixed, we used a load resistor $R_L\leq 1 k\Omega$. For measurements of stacks (and for the static measurements of isolated particles), we used $R_L = 1 k\Omega$, which affords both a good S/N ratio and an acceptable response time, R$_L$C$_A\simeq$300 ns. For the dynamic experiments with the isolated particles, we improved the time-resolution of the set-up to less than 50 ns by deconvolution of the measured response with the instrumental function measured in a separate experiment.\
Our external-electrodes technique allowed for long-term, in-situ, time-resolved studies of suspensions of CdSe platelets and their stacks without sample degradation. Indeed, solvent evaporation is impossible in sealed capillaries and the glass wall separating the electrodes from the colloidal dispersion prevents any electrochemical degradation of the sample despite the repeated application of strong fields over several hours. To study the stacking kinetics, we monitored the same capillary over 15 months by measuring its TEB in exactly the same experimental conditions. The first measurement, made at t$_0$ = 0 days after sample preparation and addition of oleic acid, revealed only the presence of isolated platelets (no change was observed in the next experiments for about one week). Later measurements, made at times t$_1$ = 25, t$_2$ = 35, t$_3$ = 62, t$_4$ = 105, and t$_5$ = 415 days, revealed the presence of stacks (labelled respectively St$_i$ for i = 1, 2, …5) of increasing size. Between the measurements, the capillary was kept horizontal and, due to its small diameter D = 1 mm, no sedimentation was observed, even at time t$_5$. The path length of the probe light in the sample is also 1 mm, which drastically reduces light absorption and scattering from the dichroic platelets. We note that, for a classic Kerr cell, this length is at least ten times larger, so that much smaller concentrations are required, which would drastically lower the stacking rate and therefore make the experiment impossible.\
We thank Dr Santanu Jana for the synthesis of samples at early stages of the projects and ANR NASTAROD for funding.
|
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abstract: 'Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of Special Relativity (SR) is shown to provide a unified origin for: i) the complex vector space formulation of Quantum Mechanics (QM); ii) the momentum and space commutation relations and the corresponding representations; iii) the Dirac Hamiltonian in the formulation of Relativistic Quantum Mechanics (RQM); iv) the existence of a self adjoint Time Operator that circumvents Pauli’s objection.'
author:
- |
C.A. Aguillón\*, M. Bauer\*\* and G.E. García\*\
\*Instituto de Ciencias Nucleares, \*\*Instituto de Física\
Universidad Nacional Autónoma de México\
e-mail: bauer@fisica.unam.mx
title: Time and energy operators in the canonical quantization of special relativity
---
Introduction
============
Quantum mechanics (QM) fails to treat time and space coordinates on the almost equal footing accorded by Special Relativity (SR), as it does with momentum and energy. In QM time appears as a parameter, not as a dynamical variable. It is a c-number, following Dirac’s designation[@Dirac]. This is the Problem of Time (PoT) in QM, that results in the extensive discussion of the existance and meaning of a time operator[@Muga; @Muga2], and of a time energy uncertainty relation[@Busch; @Bauer] in view of Pauli’s objection[@Pauli].
The procedure usually termed “canonical quantization”. arises from applying to the Hamiltonian formulation of classical physics the rule of substituing dynamical variables by self adjoint operators acting on normalized vectors representing the physical system. In addition to considering the existance of Lorentz invariants, the Born reciprocity principle is brought into play[@Born]. This proposed principle arises from noting that the Hamiltonian formulation of classical mechanics is invariant under the transformations $x_{i}\rightarrow p_{i}$ , $p_{i}\rightarrow -x_{i}$ and from the equivalence of the configuration and momentum representations in QM. Although Born acknowledges to be unsuccessful in his intended applications[^1], the reciprocity principle is currently receiving a renewed interest[@Morgan; @Govaerts; @Freidel].In the present paper it is shown that it complements the required Lorentz invariance in the canonical quantization of Special Relativity (SR) to provide a unified origin for: i) the complex vector space formulation of QM; ii) the momentum and position operators’ commutation relations and their corresponding representations; iii) the Hamiltonian in Dirac’s formulation of Relativistic Quantum Mechanics (RQM)[@Dirac; @Thaller; @Greiner; @Messiah]; iv) the existence of a self adjoint Time Operator that circumvents Pauli’s objection[@Bauer1; @Bauer2; @Bauer3].
Lorentz and reciprocity invariants in the canonical quantization of special relativty
=====================================================================================
In SR the invariants under Lorentz transformations for a free particle are the scalar products of the fourvectors in a Minkowski space with metric $\eta _{_{^{\mu \nu }}}=diag(1,-1,-1,-1)$ , namely:$$p_{\mu }p^{\mu }=\eta ^{\mu \nu }p_{\mu }p_{\nu }=p_{0}^{2}-\mathbf{p}^{2}=(m_{0}c)^{2}\text{ \ \ \ }x_{\mu }x^{\mu }=\eta ^{\mu \nu }x_{\mu
}x_{\nu }=x_{0}^{2}-\mathbf{r}^{2}=s_{0}^{2}$$where $c$ is the constant light velocity and the constants $m_{0}$ (the rest mass) and $s_{0}$ (to be interpreted) are internal properties of the physical system (Einstein’s summation convention is assumed). To be included in addition are the constant products:$$O^{\pm }=x_{\mu }p^{\mu }\pm p^{\mu }x_{\mu }$$where the symetrization is introduced as these dynamical variables will be transformed to operators, where order matters.
i\) *The Dirac free particle Hamiltonian*
From the momentum invariant, first relation in Eq.1, one obtains upon quantization the QM constraint$$\lbrack \hat{p}_{\mu }\hat{p}^{\mu }-(m_{0}c)^{2}]\left\vert \Psi
\right\rangle =0$$This can be factorized as$$\lbrack \rho ^{\mu }\hat{p}_{\mu }+m_{0}c][\rho ^{\nu }\hat{p}_{\nu
}-m_{0}c]\left\vert \Psi \right\rangle =0$$provided that, to cancel the cross terms, the momentum operators satisfy the commutation relation $[\hat{p}_{\mu },\hat{p}_{\nu }]=0$ and the coefficients $\rho ^{\mu }\ \ $the anticommutation relation$\ \{\rho ^{\mu
}\rho ^{\nu }+\rho ^{\nu }\rho ^{\mu }\}=2\eta ^{\mu \nu }\mathbf{I}_{4}$.where $\mathbf{I}_{4}$ is the $4\times 4$. identity matrix. Thus the coefficients $\ \rho ^{\mu }$ obey a Clifford algebra and are represented by matrices.
Then the constraint is satisfied with the linear equation:$$\lbrack \rho ^{\nu }\hat{p}_{\nu }-m_{0}c]\left\vert \Psi \right\rangle =0$$Multiplying by $c\rho ^{0}$ and defining $\ \rho ^{0}:=\beta ,$ $\ \rho
^{0}\rho ^{i}:=\alpha ^{i}$ one obtains :$$c\hat{p}_{0}\left\vert \Psi \right\rangle =\{c\mathbf{\alpha .\hat{p}}+\beta
m_{0}c^{2}\}\left\vert \Psi \right\rangle$$that exhibits the Dirac Hamiltonian $H_{D}=c\mathbf{\alpha .\hat{p}}+\beta
m_{0}c^{2}$. One recognizes here the procedure followed by Dirac to obtain a first order linear equation in energy and momentum that agrees with the second order one resulting from the energy momentum relation in Eq.1[Dirac,Thaller,Greiner,Messiah]{}.
ii\) *The free particle time operator*
In exactly the same way, from the second relation in Eq.1, the displacement invariant yields upon quantization the QM constraint$$\lbrack \hat{x}_{\mu }\hat{x}^{\mu }-s_{0}^{2}]\left\vert \Psi \right\rangle
=0$$This can be factorized as$$\lbrack \rho ^{\mu }\hat{x}_{\mu }+s_{0}][\rho ^{\nu }\hat{x}_{\nu
}-s_{0}]\left\vert \Psi \right\rangle =0$$provided now $[\hat{x}_{\mu },\hat{x}_{\nu }]=0$ and again$\ \ \{\rho ^{\mu
}\rho ^{\nu }+\rho ^{\nu }\rho ^{\mu }\}=2\eta ^{\mu \nu }\mathbf{I}_{4}$ The constraint is then satisfied with the linear equation:$$\lbrack \rho ^{\nu }\hat{x}_{\nu }-s_{0}]\left\vert \Psi \right\rangle =0$$or, denoting $s_{0}=c\tau _{0}$ where $\tau _{0}$ would be an internal time property of the sysytem:$$(\hat{x}_{0}/c)\left\vert \Psi \right\rangle =\{\mathbf{\alpha .\hat{r}}/c+\beta \tau _{0}\}\left\vert \Psi \right\rangle$$Here $T=\mathbf{\alpha .\hat{r}}/c+\beta \tau _{0}$ is the time operator introduced earlier by analogy to the Dirac Hamiltonian[@Bauer2].
iii\) *The Born reciprocity invarian*t
Upon quantization, only $\hat{O}^{-}=\eta _{\mu \nu }[\hat{x}_{\mu },\hat{p}^{\nu }]=[\hat{x}_{0},\hat{p}_{0}]-[\mathbf{\hat{r},\hat{p}}]$ of the two operators in Eq.2 is also a Born reciprocity invariant. Noting that $(\hat{O}^{-})^{\dagger }=-(\hat{O}^{-})$, it follows that $\hat{O}^{-}$ is a purely imaginary constant. It will be satisfied by the familiar commutation relations:$$\lbrack \hat{x}_{\mu },\hat{p}_{\nu }]=i\hbar \eta _{\mu \nu }\mathbf{I}$$where $\hbar $ is the reduced Planck constant.
Thus reciprocity invariance complements Lorentz invariance to yield the commutation relations of the operators $\hat{x}_{\mu }$ and $\hat{p}_{\nu }$ , namely:$$\lbrack \hat{x}_{\mu },\hat{x}_{\nu }]=0,\ \ \ [\hat{p}_{\mu },\hat{p}_{\nu
}]=0\ \ \ \ \ [\hat{x}_{\mu },\hat{p}_{\nu }]=i\hbar \eta _{\mu \nu }\mathbf{I}$$ as an alternative to the postulate of transforming Poisson brackets to quantum commutators. Additionnally, in Appendix A it is shown that these commutation relations are sufficient to derive: a) the continuity from $-\infty $ to $+\infty $ of the spectra of $\ \hat{x}_{\mu }$ and $\hat{p}_{\mu }$ ; b) the representations of $\hat{x}_{\mu }$ and $\hat{p}_{\mu }$ in the corresponding orthogonal eigenvector basis: c) the Fourier transformation between the configuration and momentum representation of the system vector and d) the Heisenberg uncertainty relations, including Bohr’s interpretation of the time-energy uncertainty relation (Appendix B). Such unified relationship is unfortunately not present in QM textbooks, where some of these elements are usually introduced as independent* antzats*.
Configuration and momentum representations
==========================================
Considering Eq.6.in the configuration representation where $\hat{p}_{\nu
}\rightarrow -i\hbar \frac{\partial }{\partial x_{\nu }}$ this equation reads:$$i\hbar c\frac{\partial }{\partial x_{0}}\Psi (\mathbf{r,}x_{0})=\{-i\hbar
c\alpha ^{i}\frac{\partial }{\partial x_{i}}+\beta m_{0}c^{2}\}\Psi (\mathbf{r,}x_{0})$$Substituting, from SR, $x_{0}=ct$ , the result is Dirac’s relativistic equation as usually formulated, namely:$$i\hbar \frac{\partial }{\partial t}\Psi (\mathbf{r,}t)=\{-i\hbar c\alpha ^{i}\frac{\partial }{\partial x_{i}}+\beta m_{0}c^{2}\}\Psi (\mathbf{r,}t)$$
On the other hand, in the momentum representation where $\hat{x}_{\mu
}\rightarrow i\hbar \frac{\partial }{\partial p_{\mu }}$ , Eq.10 yields:$$i\hbar \frac{\partial }{\partial cp_{0}}\Phi (\mathbf{p},p_{0})=\{(i\hbar
/c)\alpha ^{i}\frac{\partial }{\partial p_{i}}+\beta \tau _{0}\}\Phi (\mathbf{p},p_{0})$$Substituting $cp_{0}=e$, on obtains for the time operator the equation:$$i\hbar \frac{\partial }{\partial e}\Phi (\mathbf{p},e)=\{(i\hbar /c)\alpha
^{i}\frac{\partial }{\partial p_{i}}+\beta \tau _{0}\}\Phi (\mathbf{p},e)$$
The energy and time spectra, and Pauli’s objection
==================================================
As is well known, the energy spectrum of the Dirac Hamiltonian has both positive and negative real values, namely $e(p)=\pm \sqrt{(cp)^{2}+m_{0}c^{2}}$ ,.separated by a $2m_{0}c^{2}$ gap. In the same way, the spectrum of the time operator contains positive and negative real values separated by a gap $2\tau _{0}$, as $\tau (r)=\pm \sqrt{(r/c)^{2}+\tau _{0}}$.
Now, the actual interpretation of the effect of the Dirac Hamiltonian $\
H_{D}$ and the time operator $\ T$ is seen from the fact that they are self adjoint. By Stone-vonNewmann’s theorem[@Jordan] they are generators of unitary transformations of the state vectors. Then it can be shown that for infinitesimal changes[@Bauer3]:
a\) $U_{T}=\exp \{i(\delta e)T/\hbar \}\thickapprox \exp \{i(\delta e)\mathbf{\alpha .r}/c\hbar \}\exp \{i(\delta e)\beta \tau _{0}\}$ generates displacements in momentum $\delta \mathbf{p}=(\mathbf{\alpha /}c\mathbf{)}\delta e=$ $c\mathbf{\alpha (\delta e}/c^{2})$ and changes in phase $\delta
\phi =\beta \tau _{0}\delta e/\hbar $. A $2\pi $ finite phase change for positive energy waves ($\left\langle \beta \right\rangle =1$) is obtained from setting:$$\tau _{0}=h/m_{0}c^{2}=T_{B}\text{ \ \ \ \ \ }\Delta E=m_{0}c^{2}=h/T_{B}\text{\ }$$Then $\tau _{0}$ is seen to be the deBroglie period $T_{B}$, in agreement with deBroglie’s daring assumptions[@Broglie; @Baylis]. It is an intrinsic time property associated with the rest mass[@Lan].
b\) $U_{H_{D}}=\exp \{i(\delta t)H_{D}/\hbar \}$ $\thickapprox \exp
\{i(\delta t)c\mathbf{\alpha .p}/\hbar \}\exp \{i(\delta t)\beta
m_{0}c^{2}\} $ generates displacements in space $\delta \mathbf{r}=c\mathbf{\alpha \delta }t$ and changes in phase $\delta \phi =\beta m_{0}c^{2}\delta
t/\hbar $. For $\left\langle \beta \right\rangle =1$, a $2\pi $ finite change of phase requires a time lapse:$$\Delta t=2\pi \hbar /m_{0}c^{2}=h/m_{0}c^{2}=T_{B}$$
For wave packets the expectation value $\left\langle c\mathbf{\alpha }\right\rangle $ is the group velocity $\mathbf{v}_{gp}$ and the space displacement in a time lapse $\Delta t$ generated by $H_{D}$ corresponds to the classical $\mathbf{v}_{gp}\Delta t$. On the other hand $T$ acts on the continous momentum space, generating a change of momentum $\Delta
\mathbf{p}=m\mathbf{v}_{gp}=m_{0}\gamma \mathbf{v}_{gp}$, where $\gamma
=[1-(v/c)^{2}]^{-1}$ is the Lorentz factor, and consequently an energy change from $E(\mathbf{p})$ to $E(\mathbf{p}+m\mathbf{v}_{gp})$ in both branches of the relativistic energy spectrum. This circumvents Pauli’s correct objection that a commutation relation $[T,H]=i\hbar $ where $T$ acts on the energy spectrum, necessarily implies a continuum energy spectrum from $-\infty $ to $+\infty $ , contradicting the fact that the energy expectation value is expected to be positive and that there also may be discrete eigenvalues[@Pauli].
To be remarked finally is that, from Eq.18, the energy gap $2m_{0}c^{2}$and the time gap $2h/m_{0}c^{2}$ are complementary of each other. The mass dependence of the Zitterbewegung period (twice the deBroglie period) has been correctly exhibited in the experimental simulation of the Dirac equation with trapped ions[@Gerritsma; @Bauer4].
Conclusion
==========
It has been shown that the canonical quantization of SR that preserves the Lorentz and reciprocity invariants, is at the origin of the (usually postulated or inferred separately) commutation relations of the configuration and momentum dynamical operators, as well as of the Dirac relativistic Hamiltonian together with a self adjoint relativistic “time operator”. Furthermore, it brings about the derived properties - infinite continuous space and momentum spectra, ensuing representations, uncertainty relation as shown in Appendix A and B, that unfortunately in most QM textbooks are introduced as independent* antzats*.
To be stressed also, it is the reciprocity invariance which introduces an imaginary constant, opening the formulation to complex functions which are necessary to allow for “a non-negative probability function that is constant in time when integrated over the whole space ”[@Pauli], the basis for a probabilistic interpretation of QM.
The problem of time is very much present in the canonical quantization of GR, with many facets: indefinition of the spacetime foliation (“many fingered time”), disappearence of time (“frozen formalism”), and so on[Isham,Kuchar,Butterfield,Anderson2]{}. However one condition to be satisfied is local concordance wit SR, i.e., any acceptable theory of QG must allow to recover the classical spacetime in the appropiate limit[@Bonder]. It follows that a venue to be explored is wether this bottom up completion of Dirac’s RQM with a time operator as derived above helps to resolve some of the issues noted[@Bauer4].
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[Appendix A. The lore of ]{}$[\hat{x},\hat{p}]=i\hbar $
=========================================================
To represent observables the operators $\hat{x}_{\mu }$ and $\hat{p}_{\mu }$ are selfadjoint $(\hat{x}_{\mu }=\hat{x}_{\mu }^{\dagger }$ , $\hat{p}_{\mu
}=\hat{p}_{\mu }^{\dagger })$ , which insures real eigenvalues. Then, for each component $\ \hat{x}_{\mu }$ and $\ \hat{p}_{\mu }$ it follows:
**1) Spectrum[@Messiah]**
Consider the eigenvalue equation :$$\hat{x}\ \left\vert x\right\rangle =x\ \left\vert x\right\rangle \tag{A.1}$$By Stone-von Newmann’s theorem the operator $U(\alpha )=\exp (-i\alpha \hat{p}/\hslash )$ with $a$ real is unitary[@Jordan]. Then it can be shown that:$$\hat{x}\ \{U(\alpha )\left\vert x\right\rangle \}=(x+\alpha )\{U(\alpha
)\left\vert x\right\rangle \} \tag{A.2}$$Therefore $\{U(\alpha )\left\vert x\right\rangle \}=C\left\vert x+\alpha
\right\rangle $. As $\alpha $ is arbitrary, it follows that the eigenvalues of $\hat{x}\ $are continous from $-\infty $ to $+\infty $ , and that the eigenvectors satisfy:$$\left\langle x^{\prime }\mid x\right\rangle =\delta (x^{\prime }-x)\ \ \ \ \
\ \ \ \ \ \ \int dx\ \left\vert x\right\rangle \left\langle x\right\vert =I
\tag{A.3}$$where $\delta (x^{\prime }-x)$ is the Dirac delta function and $I$ is the identity operator.
In the same way one can prove that the eigenvalues in $\hat{p}\ \left\vert
p\right\rangle =p\ \left\vert p\right\rangle $ span a continuum from $-\infty $ to $+\infty $ and that the eigenvectors satisfy:$$\left\langle p^{\prime }\mid p\right\rangle =\delta (p^{\prime }-p)\ \ \ \ \
\ \ \ \ \ \ \int dp\ \left\vert p\right\rangle \left\langle p\right\vert =I
\tag{A.4}$$
**2) Representations**
From Eq.A.3:$$\left\langle x^{\prime }\left\vert \hat{x}\right\vert x\right\rangle
=x\delta (x^{\prime }-x)$$and$$\begin{aligned}
\left\langle x^{\prime }\right\vert \left[ \hat{x},\hat{p}\right] \left\vert
x\right\rangle &=&i\hslash \delta (x^{\prime }-x)=\left\langle x^{\prime
}\right\vert \hat{x}\hat{p}-\hat{p}\hat{x}\left\vert x\right\rangle \\
&=&x^{\prime }\left\langle x^{\prime }\right\vert \hat{p}\left\vert
x\right\rangle -x\left\langle x^{\prime }\right\vert \hat{p}\left\vert
x\right\rangle =(x^{\prime }-x)\left\langle x^{\prime }\right\vert \hat{p}\left\vert x\right\rangle\end{aligned}$$It follows:$$\left\langle x^{\prime }\right\vert \hat{p}\left\vert x\right\rangle =\frac{i\hslash \delta (x^{\prime }-x)}{(x^{\prime }-x)}\Longrightarrow _{x^{\prime
}\rightarrow x}i\hslash \frac{d}{dx^{\prime }}\delta (x^{\prime }-x)
\tag{A,5}$$
Introducing the vectors:$$\left\vert \Theta \right\rangle =\hat{x}\left\vert \Psi \right\rangle \text{
\ \ \ \ }\left\vert \Phi \right\rangle =\hat{p}\left\vert \Psi \right\rangle
\text{\ }$$their representations in configuration space are:$$\Theta (x)=\left\langle x\right\vert \hat{x}\left\vert \Psi \right\rangle
=x\left\langle x\mid \Psi \right\rangle =x\Psi (x) \tag{A.6}$$and using Eq.A.5:$$\begin{aligned}
\Phi (x) &=&\left\langle x\mid \Phi \right\rangle =\left\langle x\mid \hat{p}\mid \Psi \right\rangle = \nonumber \\
&=&\int dx^{\prime }\left\langle x\right\vert \hat{p}\left\vert x^{\prime
}\right\rangle \left\langle x^{\prime }\mid \Psi \right\rangle =i\hslash
\int dx^{\prime }[\frac{d}{dx^{\prime }}\delta (x^{\prime }-x)]\left\langle
x^{\prime }\mid \Psi \right\rangle \TCItag{A.7} \\
&=&\left[ \delta (x^{\prime }-x)\Psi (x^{\prime })\right] _{-\infty
}^{+\infty }-i\hslash \int dx^{\prime }\delta (x^{\prime }-x)\frac{d}{dx^{\prime }}\Psi (x^{\prime })=-i\hslash \frac{d}{dx}\Psi (x) \nonumber\end{aligned}$$i.e., the representation in configuration space of the vector $\left\vert
\Phi \right\rangle =\hat{p}\left\vert \Psi \right\rangle $ is obtained by taking the derivative of the representation of the vector $\left\vert \Psi
\right\rangle $, while the representation in configuration space of the vector $\left\vert \Theta \right\rangle =\hat{x}\left\vert \Psi
\right\rangle $ is obtained multiplying by $x$ the representation of $\left\vert \Psi \right\rangle $.
To conclude, in configuration space one has:$$\hat{x}\Longrightarrow x\ \ \ \ \ \ \ \ \ \ \ \ \ \hat{p}\Longrightarrow
-i\hslash \frac{d}{dx} \tag{A.8}$$In the same way in momentum space:$$\hat{x}\Longrightarrow i\hslash \frac{d}{dp}\ \ \ \ \ \ \ \ \ \ \ \ \ \hat{p}\Longrightarrow \ p \tag{A.9}$$
3\) **Transformation between representations**
Consider:$$\left\langle x\mid \left[ \hat{x},\hat{p}\right] \mid p\right\rangle
=i\hslash \left\langle x\mid p\right\rangle$$Developing:$$\begin{aligned}
\left\langle x\mid \left[ \hat{x},\hat{p}\right] \mid p\right\rangle
&=&\left\langle x\mid \hat{x}\hat{p}\mid p\right\rangle -\left\langle x\mid
\hat{p}\hat{x}\mid p\right\rangle \\
&=&xp\left\langle x\mid p\right\rangle -\int dx^{\prime }\left\langle x\mid
\hat{p}\mid x^{\prime }\right\rangle \left\langle x^{\prime }\mid \hat{x}\mid p\right\rangle \\
&=&xp\left\langle x\mid p\right\rangle -i\hslash \int dx^{\prime }[\frac{d}{dx^{\prime }}\delta (x^{\prime }-x)]x^{\prime }\left\langle x^{\prime }\mid
p\right\rangle \\
&=&xp\left\langle x\mid p\right\rangle +i\hslash \lbrack \left\langle x\mid
p\right\rangle +i\hslash x\frac{d}{dx}\left\langle x\mid p\right\rangle ]\end{aligned}$$one obtains:$$xp\left\langle x\mid p\right\rangle +i\hslash \lbrack \left\langle x\mid
p\right\rangle +i\hslash x\frac{d}{dx}\left\langle x\mid p\right\rangle
]=i\hslash \left\langle x\mid p\right\rangle$$Thus:$$i\hslash \frac{d}{dx}\left\langle x\mid p\right\rangle =-p\left\langle x\mid
p\right\rangle \tag{A.10}$$which is satisfied if:$$\left\langle x\mid p\right\rangle =Ce^{ipx/\hslash }\ \ \ \ \ \ \ \ \ \
\left\langle p\mid x\right\rangle =C^{\ast }e^{-ipx/\hslash } \tag{A.11}$$
Finally:$$\Phi (p)=\left\langle p\mid \Psi \right\rangle =\int dx\left\langle p\mid
x\right\rangle \left\langle x\mid \Psi \right\rangle =C^{\ast }\int dx\
e^{-ipx/\hslash }\ \Psi (x) \tag{A.12}$$and:$$\Psi (x)=\left\langle x\mid \Psi \right\rangle =\int dp\left\langle x\mid
p\right\rangle \left\langle p\mid \Psi \right\rangle =C\int dx\
e^{ipx/\hslash }\ \Phi (p) \tag{A.13}$$ i.e., the representations of the state vector in the configuration and momentum spaces are* Fourier transforms* of each* *other*.* To preserve normalization one requires $C=C^{\ast }=1/\sqrt{2\pi
\hslash }$.
4\) **Uncertainty relation**
Consider the state vectors$$\left\vert \Phi \right\rangle =(\hat{x}-\left\langle x\right\rangle
)\left\vert \Psi \right\rangle \ \ \ \ and\ \ \left\vert \Xi \right\rangle =(\hat{p}-\left\langle p\right\rangle )\left\vert \Psi \right\rangle
\tag{A.14}$$
Then$$\left\langle \Phi \mid \Phi \right\rangle =\left\langle \Psi \right\vert
\hat{x}^{2}\left\vert \Psi \right\rangle -\left\langle \Psi \right\vert \hat{x}\left\vert \Psi \right\rangle ^{2}=(\Delta x)_{\Psi }^{2}.....\left\langle
\Xi \mid \Xi \right\rangle =\left\langle \Psi \right\vert \hat{p}^{2}\left\vert \Psi \right\rangle -\left\langle \Psi \right\vert \hat{p}\left\vert \Psi \right\rangle ^{2}=(\Delta p)_{\Psi }^{2} \tag{A.15}$$
By Schawrz inequality one has$$\begin{aligned}
\left\langle \Phi \mid \Phi \right\rangle \left\langle \Xi \mid \Xi
\right\rangle &\geq &\left\vert \left\langle \Phi \mid \Xi \right\rangle
\right\vert ^{2}= \\
&=&\left\vert \left\langle \Psi \right\vert
{\frac12}[\hat{x},\hat{p}]+{\frac12}\{\hat{x},\hat{p}\}-\left\langle x\right\rangle \left\langle p\right\rangle
\left\vert \Psi \right\rangle \right\vert ^{2}\geq \\
&\geq &\left\vert \left\langle \Psi \right\vert
{\frac12}[\hat{x},\hat{p}]\left\vert \Psi \right\rangle \right\vert ^{2}=(\hslash
/2)^{2}\end{aligned}$$
Finally$$(\Delta x)_{\Psi }(\Delta p)_{\Psi }\geq \hslash /2 \tag{A.16}$$
Appendix B. The time-energy uncertainty relation
================================================
The Dirac Hamiltonian and the time operator satisfy the commutation relation$$\lbrack T,H_{D}]=i\hbar \{I+2\beta K\}+2\beta \{\tau _{0}H_{D}-m_{0}c^{2}T\}
\tag{B.1}$$where $K=\beta (2\mathbf{s.l}/\hbar ^{2}+1)$ is a constant of motion[Thaller]{}. In the usual manner an uncertainty relation follows, namely:$$(\Delta T)(\Delta H_{D})\geq (\hbar /2)\left\vert \{1+2<\beta
K>\}\right\vert =(\hbar /2)\left\vert \{3+4\left\langle \mathbf{s.l}/\hbar
^{2}\right\rangle \}\right\vert \tag{B.2}$$
Consider:$$\begin{aligned}
(\Delta T)^{2} &=&\left\langle T^{2}\right\rangle -\left\langle
T\right\rangle ^{2}=\left\langle r^{2}/c^{2}+\tau _{0}^{2}\right\rangle
-\left\langle \mathbf{\alpha .r}/c+\beta \tau _{0}\right\rangle ^{2}=
\TCItag{B.3} \\
&=&(1/c^{2})(\Delta r)^{2}+[\left\langle r/c\right\rangle ^{2}+\tau
_{0}^{2}]-[\left\langle \mathbf{\alpha .r}/c+\beta \tau _{0}\right\rangle
^{2}]\gtrapprox (1/c^{2})(\Delta r)^{2} \nonumber\end{aligned}$$In the same way:$$(1/c^{2})(\Delta H_{D})^{2}\gtrapprox (\Delta p)^{2} \tag{B.4}$$It follows finally:$$(\Delta T)^{2}(\Delta H_{D})^{2}\gtrapprox (\Delta r)^{2}(\Delta p)^{2}
\tag{B.5}$$This correspponds to Bohr’s interpretation that the uncertainty in the time of passage at a certain point is given by the width of the wave packet, which is complementary to the momentum uncertainty, and thus to the energy uncertainty[@Bohr].
[^1]: Born considered the phase space reciprocity invariant $x_{\mu }x^{\mu
}+p_{\mu }p^{\mu }$ as the base to deduce the elemetary particle masses. It was too early.
|
---
abstract: |
****
Graphene samples can have a very high carrier mobility if influences from the substrate and the environment are minimized. Embedding a graphene sheet into a heterostructure with hexagonal boron nitride (hBN) on both sides was shown to be a particularly efficient way of achieving a high bulk mobility [@Wang1DContacts]. Nanopatterning graphene can add extra damage and drastically reduce sample mobility by edge disorder [@PhysRevLett.102.056403; @PhysRevB.82.041413; @PhysRevB.85.195432]. Preparing etched graphene nanostructures on top of an hBN substrate instead of SiO$_2$ is no remedy, as transport characteristics are still dominated by edge roughness [@BischoffBNRibbons]. Here we show that etching fully encapsulated graphene on the nanoscale is more gentle and the high mobility can be preserved. To this end, we prepared graphene antidot lattices [@PhysRevLett.66.2790] where we observe magnetotransport features stemming from ballistic transport. Due to the short lattice period in our samples we can also explore the boundary between the classical and the quantum transport regime.
author:
- Andreas Sandner
- Tobias Preis
- Christian Schell
- Paula Giudici
- Kenji Watanabe
- Takashi Taniguchi
- Dieter Weiss
- Jonathan Eroms
title: Ballistic transport in graphene antidot lattices
---
In single layer graphene the charge carriers are completely exposed to the environment, which limits their mobility. Placing graphene on hexagonal boron nitride (hBN) was shown to improve the carrier mobility [@Dean2010], allowing the observation of ballistic transport or the fractional quantum Hall effect in bulk graphene [@Dean2011]. Recently, a dry stacking technique was introduced, which allows complete encapsulation of graphene into layers of hBN and excludes any contamination from process chemicals such as electron beam resist [@Wang1DContacts]. To obtain graphene nanodevices, chemically prepared graphene nanostructures [@Jiao2009; @Kosynkin2009; @Cai2010] are a potential route for certain applications, however, the high flexibility of a top down patterning approach is extremely desirable. Graphene antidot lattices can help circumventing the problem of the missing band gap in transistor applications [@Bai2010], and were even predicted to serve as the technological basis for spin qubits [@PhysRevLett.100.136804]. Clearly, for advanced graphene nanodevices, not only the bulk mobility has to be improved, but the nanopatterning has to be optimised.
Here we present experiments on graphene antidot lattices [@PhysRevLett.66.2790; @Eroms2009; @Shen2008] etched into hBN/graphene/hBN heterostructures with lattice periods going down to $a=50$ nm. Magnetotransport on those samples shows commensurability features stemming from ballistic orbits around one or several antidots. This allows us to prove that the high carrier mobility is preserved in the nanopatterning step even though the zero field resistance is dominated by scattering on the artificial nanopattern, giving an apparent reduction of the mobility. The small feature size of our samples also allows us to approach the region where the classical picture of cyclotron orbits no longer applies. This classical to quantum crossover is governed by the ratio between the Fermi wavelength $\lambda_F$ of the carriers and the dimensions of the nanopattern.
To obtain embedded graphene samples, hBN/graphene/hBN stacks were prepared using the dry stacking technique, patterned into Hall bar shape, and contacted using Cr/Au [@Wang1DContacts]. In hBN/graphene/hBN samples prepared by this method, we routinely obtained carrier mobilities in excess of $\mu = 100\,000$ cm$^2$/Vs, showing all integer quantum Hall states starting from a few Tesla. In one sample without antidots and a mobility of $\mu = 300\,000$ cm$^2$/Vs we also observed the fractional quantum Hall effect at $T=1.4$ K. This shows that our fabrication procedure is mature and consistently yields high sample qualities. The samples presented in this study did not show any signs of a moiré superlattice [@Dean2013; @GeimMoire]. Afterwards, an antidot lattice was patterned. (For fabrication details, see Methods section). Fig. 1 shows an optical micrograph of a finished sample and a scanning electron micrograph of a sample after measuring. A sketch of the antidot lattice, etched into the stack is also shown. The antidot lattice period $a$ was varied between 50 nm and 250 nm. The antidot diameter $d$ was lithographically defined to be about 40 nm, but due to the conical etching profile, the actual diameter in the graphene plane is smaller. Using SEM inspection, we estimate it to be about $25\dots 30$ nm.
In Fig. 2, we show data for a sample with a lattice period of $a=200$ nm. From the gate response of the conductivity at a magnetic field $B=0$, shown in Fig. 2a, we calculate an apparent field effect mobility of $\mu = 35\,000$ cm$^2$/Vs. At a carrier density $n_S = 2.3\times 10^{12}$ cm$^{-2}$ this corresponds to an apparent mean free path of about $l_{mfp}=\frac{\hbar}{e}\sqrt{\pi n_S}\mu=620$ nm. We estimate the intrinsic mean free path to be about 1400 nm [@Ishizaka1999](see Supplementary Information). Magnetotransport traces of this device (see Fig. 2b) show pronounced peaks at field values where the cyclotron diameter $2R_C = \frac{\hbar}{eB}\sqrt{\pi n_S}$ is commensurate to the square antidot lattice. The peak belonging to $2R_C = a$, the fundamental antidot peak, is most pronounced. Additional peaks appearing at lower fields correspond to orbits encircling 2, and 4 antidots [@PhysRevLett.66.2790] (see Fig. 1d), confirming a mean free path which spans several lattice periods. While in a simple picture only the unperturbed orbits encircling the antidots are responsible for the magnetotransport features, a more detailed analysis based on the Kubo formula shows that velocity correlations in the chaotic trajectories, which occupy the largest part of the phase space, result in the magnetoresistance peaks [@PhysRevLett.68.1367; @PhysRevB.55.16331]. Most of the orbits therefore hit the antidot edges several times within a mean free path. Hence, the visibility of the antidot peaks not only proves a high bulk mobility, but also shows that scattering at the edges does not cut off the trajectories and we can conclude that the high carrier mobility also survives after nanopatterning.
At higher fields, the cyclotron diameter $2R_C$ is reduced below the neck width $a-d$ in between the antidots. We can observe Shubnikov-de Haas oscillations, eventually resulting in a well-defined quantum Hall effect. At $B=14$ T we clearly observe the $\nu=1$ plateau, which again shows the high sample quality (Fig. 2c). We evaluated the carrier density dependence of the magnetoresistance peaks corresponding to orbits around 1, 2 and 4 (Fig. 2d) and found that the peaks were always well described by a square root dependence of the cyclotron diameter on the carrier density down to $n_S = 3.2\times 10^{11}$ cm$^{-2}$. Quantitatively, we confirmed the formula for the cyclotron diameter for graphene given above, which contains spin and valley degeneracy.
Fig. 3a shows the magnetoresistance of a sample with $a=100$ nm at $n_S = 2.8\times 10^{12}$ cm$^{-2}$. The apparent Hall mobility at this density is about $\mu = 8\,000$ cm$^2$/Vs. Again, scattering at the antidot potential limits the apparent mobility [@Ishizaka1999], but the intrinsic mobility is higher as we clearly observe magnetoresistance peaks for $n=1, 2, 4$ antidots, and a fourth peak at lower fields is weakly visible. Ishizaka and Ando studied how the visibility of the higher order antidot peaks depends on the mobility [@PhysRevB.55.16331]. From their data, we estimate that the intrinsic mean free path must be at least 400 nm, well in excess of the apparent mean free path of 160 nm (see Supplementary Information).
The good visibility of the $n=2$ peak confirms the small aspect ratio $d/a \le 0.3$ [@PhysRevB.55.16331], in agreement with our SEM analysis and also with the onset of the well-defined Shubnikov-de Haas oscillations in our magnetotransport data. All these approaches give an antidot diameter of $d=25\dots 30$ nm.
In experiments in GaAs based antidot lattices it was found that due to depletion at the antidot boundaries, the potential can be very soft and small lattice periods are hard to realize. In our case the data compares well to hard-wall potential lattices in GaAs, which could be realized in GaAs only at much larger lattice periods [@PhysRevLett.66.2790]. We also compared data for similar carrier densities in the electron and hole regime in Fig. 3b and found the graphs to be virtually identical. This proves that there is no edge doping at the antidot boundaries, which would have led to different potential shapes in the electron and hole regime due to Fermi level pinning at the edges.
Now let us discuss the transition between the quantum and the classical transport regime. In GaAs-based heterostructures, the smallest lattice period realised so far was $a=80$ nm, and required critical tuning of the etch depth [@Antidots80nm]. In contrast, due to the lack of a depletion region in graphene the fabrication of samples with a very small lattice period is less critical, and the carrier density is widely tunable. Also, due to valley degeneracy, the Fermi wavelength in graphene, $\lambda_F = 2\sqrt{\frac{\pi}{ n_S}}$ is a factor of $\sqrt{2}$ larger than in GaAs based 2DEGs at the same carrier density. Thus we can explore the transition from the semi-classical to the quantum regime [@Brack1997], where a description in terms of classical orbits is no longer justified. In the samples with $a\le 100$ nm we are able to study this transition. Fig. 4a shows the disappearance of the main antidot peak in a sample with $a=75$ nm as the carrier density is lowered, making $\lambda_F$ longer. We find that this peak is only visible at densities above $n_S = 4.3\times 10^{11}$ cm$^{-2}$, corresponding to $\lambda_F= 54$ nm. Also, in two samples with $a=100$ nm, we observe that the main antidot peak becomes visible for densities larger than $n_S = 2.2\times 10^{11}$ cm$^{-2}$, which corresponds to $\lambda_F = 75$ nm. In a sample with $a = 50$ nm, we observed a weak antidot peak only at $n_S = 2.5\times 10^{12}$ cm$^{-2}$($\lambda_F= 22$ nm). To be in the classical limit of a quantum system, the Fermi wavelength must satisfy a condition $\frac{\lambda_F}{2\pi}
\ll l$ [@SakuraiQM], where $l$ is a typical dimension of the system. In our case, the neck width $a-d$ of the constriction between the antidots is the shortest length scale in the problem, and we find that when $\lambda_F \approx a-d$ the classical regime sets in and the antidot peak becomes visible.
The fact that the antidot peaks disappear at low densities can be either due to a limited mean free path or the breakdown of the classical picture. In Fig. 2d (lattice period $a=200$ nm) all the antidot peaks disappear at roughly the same magnetic field, $B\approx 0.5$ T (where $\mu B$ exceeds some constant), but different carrier density. This behaviour is clearly governed by a limited mean free path. In contrast, in the sample of Fig. 4a ($a=100$ nm), we find that the classical features at both $B\approx 1$ T and $B\approx 2.5$ T disappear at the same carrier densities, making a $\lambda_F$-driven scenario more realistic.
Finally, at low densities, we can observe a weak localization (WL) feature at low temperatures: a peak in the magnetoresistance at $B=0$ (see Fig. 4b). Using a standard analysis for WL in graphene [@PhysRevLett.97.146805] that we employed in earlier work on graphene antidot lattices on SiO$_2$ [@Eroms2009], we extracted the phase coherence length $L_\phi$. For the sample with $a=100$ nm (same as in Fig. 3a) we found it to be between 120 nm and 300 nm (see Fig. 4c). It clearly exceeds the lattice period, unlike in graphene antidot samples on SiO$_2$ where $L_\phi$ was significantly below $a$ [@Eroms2009]. We therefore again conclude that nanopatterning of embedded graphene leads to greatly reduced scattering at the sample edges.
In summary, we prepared antidot lattices in stacks of hBN/graphene/hBN and observed well-developed commensurability features in samples with lattice periods from $a=50$ nm to $a=250$ nm. This shows that the etching procedure preserves the high sample quality. In the short-period graphene samples, we could observe the disappearance of classical features when the Fermi wavelength $\lambda_F$ exceeds $a-d$, marking a classical to quantum transition. Our experiments therefore pave the way for well-controlled graphene based nanodevices.
Methods
=======
Single crystalline hexagonal boron nitride (hBN)[@Kubota2007; @Taniguchi2007] was exfoliated onto a stack of PMGI and PMMA polymers spin coated on an oxidised Si-Wafer [@Dean2010]. Suitable hBN flakes, serving later as the top hBN layer, were located in an optical microscope by using different bandpass filters. The PMGI sacrificial layer was dissolved in photoresist developer and DI water, leaving the PMMA with the hBN floating on DI water. Then the PMMA film was transferred to a microscope slide into which a hole had been cut [@Dean2010]. Single layer graphene was exfoliated from HOPG (Momentive Performance, ZYA grade) onto oxidised Si, and picked up by van-der-Waals interaction using the first hBN flake [@Wang1DContacts]. Using a home-made setup in an optical microscope, this stack was transferred to a second hBN flake, residing on an oxidised Si substrate with prepatterned markers. Subsequently, the finished stacks were annealed in forming gas flow at 320$^\circ\text{C}$ for several hours. The hBN/graphene/hBN heterostructure was then patterned into Hall bar shape using electron beam lithography (EBL) and CHF$_3$/O$_2$ (40 sccm/6 sccm, 60 Watt power) based reactive ion etching (RIE) [@Wang1DContacts]. Cr/Au side contacts were defined with EBL and deposited by thermal evaporation and lift-off after brief oxygen plasma cleaning of the contact areas [@Wang1DContacts]. Finally, the antidot lattice was defined in a separate EBL and RIE step. Samples were glued into chip carriers with silver filled epoxy to contact the back gate, wire-bonded and measured in a helium cryostat with variable temperature insert, using low frequency lock in techniques with a bias current of 10 nA.
Acknowledgments
===============
The authors thank A. Geim and R. Jalil for sharing details of the graphene transfer procedure, R. Fleischmann, T. Geisel and K. Richter for helpful discussions, and the Deutsche Forschungsgemeinschaft (DFG) for funding through projects GRK 1570 and GI 539/4-1.
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Figures
=======
![**a**, Optical micrograph of a finished graphene Hall bar etched out of an hBN/graphene/hBN heterostructure, and contacted with Cr/Au leads. Scale bar length 5 $\mu$m. **b**, False-color scanning electron micrograph of a sample with lattice period $a=100$ nm. Here, the heterostructure is shaded in green, the Cr/Au contacts yellow, and the Si/SiO$_2$ substrate violet. Scale bar length 500 nm. **c**, Sketch of the antidot lattice in an hBN/graphene/hBN heterostructure. The antidot lattice period $a$ ranges from 50 to 250 nm, and the antidot diameter $d$ is about $25\dots 30$ nm. **d**, The most prominent cyclotron orbits fitting into the lattice, giving rise to magnetoresistance peaks.[]{data-label="Fig:SEM"}](Fig1_neu2.png){width="8.3cm"}
![**a**, Gate dependence of the sheet conductivity of a sample with $a=200$ nm. The linear fit (red line) gives an apparent mobility of $\mu = 35\,000$ cm$^2$/Vs. **b**, Magnetoresistance (black) and Hall resistance (red). The arrows correspond to the expected magnetic field positions of the orbits sketched in Fig 1d. The fine structure in $R_{xx}$ is not noise, but phase-coherent oscillations [@Nihey1993; @PhysRevLett.70.4118; @PhysRevB.49.8510] that disappear at higher temperatures. **c**, Gate dependence of $R_{xx}$ and $R_{xy}$ at $B=14$ T, showing a clear $\nu=1$ quantum Hall plateau. This feature is only observed in high-mobility graphene devices. **d**, The magnetic field positions of the three antidot peaks scale with the square root of the carrier density, confirming the classical nature of those peaks.[]{data-label="Fig:200nm"}](Fig2_neu2.png){width="100.00000%"}
![**a**, Magnetoresistance and Hall resistance data taken on a sample with $a=100$ nm. The three well-defined antidot peaks correspond to orbits around 1, 2 and 4 antidots. **b**, $R_{xx}$ taken at similar electron and hole density. Here, $T=80$ K to show the classical features more clearly. There is virtually no difference between those graphs, proving that the potential profile is the same for electrons and holes.[]{data-label="Fig:100nm"}](Fig3_neu.png){width="100.00000%"}
![**a**, $R_{xx}$ data of a sample with $a=75$ nm taken at very low densities, at the transition into the regime of classical transport. The densities $n_S$ are given in units of $10^{11}$ cm$^{-2}$, shown next to the corresponding graphs. The expected position of the main antidot peak is marked with a triangle for each density. A higher order antidot peak is also visible at lower magnetic fields. As the carrier density is lowered, the antidot peaks disappear. Inset: Sketch of the Fermi wavelength corresponding to the red graph. **b**, Weak localization (WL) peak in the sample with $a=100$ nm, taken at $n_S=1.3\times 10^{11}$ cm$^{-2}$. The antidot peak is not visible at this low density, the big peaks at $B=\pm 1.4$ T are a Shubnikov-de Haas oscillation. **c**, Phase coherence length taken from the weak localization fits of a sample with $a=100$ nm at various low densities and $T=1.4$ K. The phase coherence length exceeds the lattice period, showing that the etched boundaries do not lead to severe phase-breaking.[]{data-label="Fig:lowdens"}](Fig4_neu2.png){width="100.00000%"}
|
---
abstract: 'We employ isolated N-body simulations to study the response of self-interacting dark matter (SIDM) halos in the presence of the baryonic potentials. Dark matter self-interactions lead to kinematic thermalization in the inner halo, resulting in a tight correlation between the dark matter and baryon distributions. A deep baryonic potential shortens the phase of SIDM core expansion and triggers core contraction. This effect can be further enhanced by a large self-scattering cross section. We find the final SIDM density profile is sensitive to the baryonic concentration and the strength of dark matter self-interactions. Assuming a spherical initial halo, we also study evolution of the SIDM halo shape together with the density profile. The halo shape at later epochs deviates from spherical symmetry due to the influence of the non-spherical disc potential, and its significance depends on the baryonic contribution to the total gravitational potential, relative to the dark matter one. In addition, we construct a multi-component model for the Milky Way, including an SIDM halo, a stellar disc and a bulge, and show it is consistent with observations from stellar kinematics and streams.'
author:
- |
Omid Sameie$^{1}$[^1][^2], Peter Creasey$^1$, Hai-Bo Yu$^1$[^3], Laura V. Sales$^1$[^4], Mark Vogelsberger$^2$[^5] and Jesús Zavala$^3$\
$^1$ Department of Physics and Astronomy, University of California, Riverside, California 92507, USA\
$^2$ Department of Physics, Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA\
$^3$ Center for Astrophysics and Cosmology, Science Institute, University of Iceland, Dunhagi 5, 107 Reykjavik, Iceland
bibliography:
- 'paper.bib'
title: 'The impact of baryonic discs on the shapes and profiles of self-interacting dark matter halos'
---
methods: numerical-galaxies: evolution-galaxies: formation-galaxies: structure-cosmology: theory.
Introduction
============
The $\Lambda$ cold dark matter ([[$\Lambda$CDM]{}]{}) model, where DM is assumed to be collisionless, is the leading theory for the growth of structure and formation of galaxies in our Universe. It fits the spectrum of matter fluctuations in the early universe with extraordinary precision [@planck2014] and explains many important aspects of galaxy formation and evolution [@Springel:2006vs; @TrujilloGomez:2010yh; @frenk2012; @2014MNRAS.444.1518V; @Vogelsberger:2014kha]. However, the success of [[$\Lambda$CDM]{}]{} does not preclude the possibility that DM may have strong self-interactions. When DM particles have a scattering cross section per unit mass, $\sigma_{\rm x}/m_{\rm x}\sim1~{\rm cm^2/g}$, DM collisions occur multiple times in the inner halo over the cosmological timescale and make distinct departures from the CDM predictions, while the outer halo remains collisionless, retaining the large-scale predictions of CDM [@spergel1999; @yoshida2000; @dave2001; @colin2002; @Volgesberger2012; @rocha2013]. In fact, SIDM has been motivated to address outstanding discrepancies between observations on galactic scales and CDM predictions, see [@bullock2017] for a recent review on the CDM challenges and [@2017arXiv170502358T] on their solutions within SIDM. In particular, it has been shown that the diverse shapes of galactic rotation curves [@Oman; @deNaray:2009xj] can be explained naturally in the SIDM model [@2017PhRvL.119k1102K; @creasey2017], while being consistent with observations in galaxy clusters [@kaplinghat2015].
DM self-interactions kinematically thermalize the inner halo and lead to distinct features in the halo properties. For dwarf galaxies, where DM dominates over all radii, SIDM thermalization leads to a large density core, and the stellar distribution is more extended in SIDM than CDM and there is tight correlation between the DM core size and the stellar one [@Vogelsberger_2014]. While for baryon-dominated systems, the thermalization can significantly increase the central SIDM density and the inner halo shape follows the baryon distribution due to the influence of the baryonic potential [@kaplinghat2014a], a radical deviation from SIDM-only simulations. Moreover, @kaplinghat2014a argued that once the inner halo reaches equilibrium, the inner SIDM profile can be modeled as an isothermal distribution that is sensitive to the final baryonic potential, but not the formation history. This has motivated a number of isolated simulations to test the response of the SIDM halo to the baryonic potential [@elbert2016; @creasey2017], where they assumed a CDM halo and a stellar potential as the initial condition. Recently, [@2017arXiv171109096R] performed cosmological hydrodynamical SIDM simulations of galaxy clusters and explicitly confirmed this expectation. In addition, although in the presence of strong baryonic feedback both CDM and SIDM could lead to similar density profiles [@Fry:2015rta], the internal structure of the SIDM halo is more robust to the inclusion of baryonic feedback, compared to its CDM counterpart, due to the rapid energy redistribution caused by the DM collisions [@2017MNRAS.472.2945R].
In this paper, we utilize isolated simulations to study the response of the SIDM halo in the presence of the baryonic potential for Milky Way (MW)-sized galaxies, where the baryonic contribution to the potential is important. Our goal is to understand the interplay between the DM self-scattering strength and the baryonic concentration in shaping the SIDM distribution, and the significance of the potential in altering evolution history of the SIDM halo. In the first two sets of simulations, we vary both the baryonic concentration and $\sigma_{\rm x}/m_{\rm x}$ in the range of $0.5\textup{--}5~{\rm cm^2/g}$, and study the variation of the SIDM predictions in the density profile and shape as a function of the cross section. In the presence of baryons, the central density of an SIDM halo no longer decreases monotonically with increasing $\sigma_{\rm x}/m_{\rm x}$, as expected in the SIDM-only case for $\sigma_{\rm x}/m_{\rm x}$ we take. Accordingly, the SIDM halo shape varies with $\sigma_{\rm x}/m_{\rm x}$ even for the same baryonic potential. Our results indicate that inferring $\sigma_{\rm x}/m_{\rm x}$ from stellar kinematics of luminous galaxies, where the baryons dominate the potential, could be challenging.
In our third set of simulations, we construct a realistic MW mass model, including an SIDM halo, a stellar bulge and disc. We fix $\sigma_{\rm x}/m_{\rm x}=1~{\rm cm^2/g}$ and carefully adjust the model parameters to reproduce the mass model inferred from the stellar kinematics. We then make a detailed comparison between the halo shape predicted in our model and those inferred from observations.
The structure of this paper is as follows. In Sec. \[sec:section2\], we discuss the numerical details of our simulations and the methodology used to quantify the halo shapes. In Sec. \[sec:halo-disc\], we use our code to explore the evolution of a MW-sized halo with a stellar disc and measure the effect of the radial length scale of the disc. In Sec. \[sec:MW\], we compare our predictions for an SIDM MW halo against those from CDM simulations and those inferred from observations of stellar streams. We conclude and summarize our results in Sec. \[sec:conclusion\].
Simulations and halo shape algorithms {#sec:section2}
=====================================
Numerical Simulations
---------------------
We carry out N-body simulations using the code [Arepo]{} [@Springel2010]. Gravity modules in Arepo are a modified version of [GADGET-2]{} and [Gadget-3]{} [@springel2005]. We use the algorithm developed in @Volgesberger2012 [@2016MNRAS.460.1399V] to model DM self-interactions. This is a Monte Carlo-based method, where at each time step a particle may pairwise scatter with any of its nearest neighbors. We assume a velocity-independent constant cross section in our simulations. This is a good approximation, since the observationally self-scattering cross section varies mildly across galactic scales [@kaplinghat2015] and we mainly focus on isolated simulations for a given halo mass. We evolve our simulations for $10\;{\rm Gyr}$, slightly shorter than Hubble time scale ($H_{0}^{-1}\approx 13.96~{\rm Gyr}$) in order to account for the assembly of the primordial galactic halo.
Following @creasey2017, we model the baryonic component in our simulations as a static potential. This approach ignores the back-reaction of the halo evolution on the baryons, an effect expected to be sub-dominant, since we are interested in the systems that the final baryonic distribution is known. We consider two models for the baryonic potential. One is the Miyamoto-Nagai (MN) disc [@Miyamoto-Nagai],
$$\Phi_{\text{MN}}(R,z)\ =\ \frac{-G\ M_{\text{d}}}{\sqrt{R^2\ +\left( R_{\text{d}}
+ \sqrt{z_{\text{d}}^2+z^2}\right)^2}},
\label{eq:Miyamoto_nagai}$$
where $M_{\rm d}$ is the disc mass, $R_{\rm d}$ the disc scale length and $z_{\rm d}$ the disc scale height. The implementation of the MN disc in [Arepo]{} is as described in @creasey2017. We also consider a Hernquist bulge potential [@hernquist-1990] $$\Phi_{\rm Hernquist} = -\frac{G M_{\rm H}}{r+r_{\rm H}} \, ,$$ where $M_{\rm H}$ is the bulge mass and $r_{\rm H}$ is the scale length.
We run three sets of simulations, varying the baryonic component and the strength of the cross section. In the first two sets, we only include the MN disc ($M_{\rm d}=6.4\times10^{10}\;{M}_{\odot}$) with two disc scale lengths, $R_{\rm d}=3\;{\rm kpc}$ (compact disc) and $R_{\rm d}=6\;{\rm kpc}$ (extended disc). In both cases, we fix $z_{\rm d}=0.3 R_{\rm d}$. The DM self-scattering cross section is chosen to be $\sigma_{\rm x}/m_{\rm x}\ =0,\ 0.5,\ 1,\ 3$, and $5\; {\rm cm^2/g}$, i.e., $10$ simulations in total. For the [*initial*]{} halo component, we assume a spherical NFW profile [@NFW], and take the halo parameters as $r_{\rm s}=37.03~{\rm kpc}$ and $\rho_{\rm s}=2.95\times10^6M_\odot/{\rm kpc^3}$. The mass ratio of the disc to the halo is motivated by the baryonic Tully-Fisher relation [@Lelli]. We use the publicly available code [SpherIC]{}, introduced in [@garrison-kimmel2013], to generate the initial conditions. It truncates the outer halo profile exponentially at $r_{\rm cut}$ to avoid mass divergence. We take $r_{\rm cut}\approx 250~{\rm kpc}$, close to the virial radius of our initial CDM halo. We fix the gravitational softening length to be $\epsilon= 125~\rm{pc}$ and the mass resolution $m_{\rm p}=1.32\times 10^{6} M_{\odot}$. We include $2$ million mass particles in our simulations, necessary for resolving the innermost regions, resulting in a halo mass of $2.64\times10^{12}M_{\odot}$.
For the third set, we include both an MN disc and a Hernquist bulge to model the baryon distribution in the MW. The disc parameters are $M_{\rm d}=6.98\times10^{10}\ M_{\odot}$, $R_{\rm d}=3.38~{\rm kpc}$ and $z_{\rm d}=0.2 R_{\rm d}$ for the disc. The bulge ones are $M_{\rm H}=1.05\times10^{10}\ M_{\odot}$ and $r_{\rm H}=0.46~{\rm kpc}$. The [*initial*]{} halo parameters are $r_{\rm s}=42.18~{\rm kpc}$ and $\rho_{\rm s}=1.39\times10^6M_\odot/{\rm kpc^3}$. We have chosen these parameters to reproduce the MW mass model presented in [@mcmillan] (hereafter McM11), see Sec. \[sec:MW\] for details. The baryon-model parameters used in our simulations are summarized in Table \[table:bar\_param\]. We choose $r_{\rm cut}=100$ kpc, $\epsilon=125~\rm{pc}$ and $m_{\rm p}=5.76\times 10^{5} M_{\odot}$. We simulate $2$ million mass particles and the total halo mass is $1.15\times10^{12}M_{\odot}$.
Additionally, we have run cosmological zoom-in SIDM simulations for $5$ MW-mass Aquarius halos [@springel2008] with the initial conditions taken from [@Volgesberger2012; @zavala2013]. We will present these simulation results in Sec. \[sec:MW\] for comparison.
$\text{Component}$ Mass ($10^{10}\ M_{\odot}$) Length scale (kpc)
-------------------- ----------------------------- --------------------
Extended Disc 6.4 6.0
Compact Disc 6.4 3.0
MW-like Disc 6.98 3.38
MW-like Bulge 1.05 0.46
: Parameters of static potentials used in the three sets of simulations.[]{data-label="table:bar_param"}
Halo shape algorithm {#sub:algo}
--------------------
![Comparison between ellipsoids (dashed) and isodensity contours (solid) for the simulation with $\sigma_{\rm x}/m_{\rm x}=1~{\rm cm^2/g}$ and $R_{\rm d}=6~{\rm kpc}$. The horizontal axis is the major axis, while the vertical one is the minor axis aligned with the symmetry axis of the baryonic disc. The color bar shows the density scaled to $\rho_{\rm max}$, the maximal central DM density.[]{data-label="fig:iso_dens_vs_ellipsoids"}](surface_density_vs_ellipsoid.pdf){width="\columnwidth"}
We use the method introduced in @Dubinsky_carlberg (see also [@allgood2005]) to calculate the ellipticity of the simulated halos. It constructs the axial ratio for the best-fitting ellipsoid as a function of the major axis length. This method is an iterative one where at each iteration the reduced inertia tensor is determined for the set of particles within the previous ellipsoid, and then a new ellipsoid is determined from this tensor. Specifically, if we denote the major axis length $a$, then at each iteration the reduced inertia tensor is given by $$I_{\rm ij} = \sum_{k:d_{\rm k} < a} \frac{r_{\rm k,i}\times r_{\rm k,j}}{d_{\rm k}^2}$$ where $r_{\rm k,i}$ denotes the coordinate ${\rm i}$ of particle ${\rm k}$, and $d_{\rm k}$ is the elliptical radius found from the previous inertia tensor. We have $d_{\rm k} = \sqrt{x_{k}^2 + (y_k/q)^2 + (z_k/s)^2}$, where $x$, $y$ and $z$ are the coordinates along the major-, intermediate- and minor-axes of the ellipsoid, and $q=b/a$, $s=c/a$ are the axial ratios of the intermediate- and minor-to-major axes, respectively. After diagonalizing the inertia tensor with eigenvalues (ascending) $\{\lambda_1, \lambda_2, \lambda_3\}$, we have $q=\sqrt{\lambda_2/\lambda_3}$ and $s=\sqrt{\lambda_1/\lambda_3}$. In the initial iteration, the ellipsoid is set to a sphere, i.e. $q=s=1$. This process is continued until some convergence criteria, which we take it to be $10^{-6}$ on the difference between successive iterations, is satisfied. We note that if the number of DM particles in an ellipsoid is too small, typically less than $1000$, the result from this method is not accurate (see Appendix \[sec:A1\]).
Fig. \[fig:iso\_dens\_vs\_ellipsoids\] shows a comparison between isodensity contours and ellipsoids for an example, where $R_{\rm d} = 6$ kpc and $\sigma_{\text{x}}/m_{\text{x}}= 1\;{\rm cm^2/g}$. We see the overall agreement between the two methods is excellent.
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SIDM halo properties with a stellar disc {#sec:halo-disc}
========================================
Density profiles {#sub:density}
----------------
Fig. \[fig:density\] shows the DM density (top) and velocity dispersion (bottom) profiles for $R_{\rm d}=3~{\rm kpc}$ (left) and $6~{\rm kpc}$ (right). The solid curves are from our simulations for different values of $\sigma_{\rm x}/m_{\rm x}$ in the presence of the stellar disc. For comparison, we also plot the SIDM density profiles (dashed) without the disc potential, calculated with the analytical method in [@kaplinghat2014a; @kaplinghat2015].
In both cases, the presence of a baryonic potential increases the SIDM density profile and reduces the core size, and the effect is more significant if the baryonic concentration is higher. For $R_{\rm d}=3~{\rm kpc}$, a larger cross section leads to a higher DM density ($0.5\textup{--}5\;{\rm cm^2/g}$), opposite to the case without baryons. For the extended disc with $R_{\rm d}=6\;{\rm kpc}$, the SIDM density profiles are almost identical, even though the $\sigma_{\rm x}/m_{\rm x}$ value changes by a factor of $10$. A deep baryonic potential also increases the DM velocity dispersion in the inner halo, as shown in the bottom panels. In the case of compact disc, it is evident that all SIDM halos are close to the threshold of mild core collapse, as the velocity dispersion profiles start to develop a negative gradient from $1$ to $10~{\rm kpc}$ at the $3\textup{--}5\%$ level. The significance is continuously enhanced when $\sigma_{\rm x}/m_{\rm x}$ changes from $0.5$ to $5~{\rm cm^2/g}$. We have checked simulation results using the analytical method, where we assume a thin disc model and use the numerical templates developed in [@2017PhRvL.119k1102K]. Overall, they agree well. As an example, we show the density profiles derived from the analytical method for $\sigma_{\rm x}/m_{\rm x}=0.5~{\rm cm^2/g}$ ($1~{\rm cm^2/g}$) in Fig. \[fig:density\] (top), and the corresponding central DM velocity dispersions are $170~{\rm km/s}$ ($170~{\rm km/s}$) and $145~{\rm km/s}$ ($154~{\rm km/s}$) for the compact and extended cases, respectively. In the analysis, we match the inner isothermal distribution to the initial NFW profile such that the density and mass are continuous within $\sim5\%$ at the radius, where scattering occurs once over $10~{\rm Gyr}$.
The SIDM halo has a distinct evolution history. It first undergoes a core expansion phase, during which the DM collisions transport heat towards the inner region and a central density core forms. Since a self-gravitating system has a negative heat capacity, the core will eventually contract and collapse to a singular state [@2002ApJ...568..475B]. In cosmological SIDM-only simulations, mild core collapse is observed within $10~{\rm Gyr}$ when $\sigma_{\rm x}/m_{\rm x}\gtrsim10~{\rm cm^2/g}$ [@Volgesberger2012; @elbert2015]. We have also checked that for isolated SIDM-only ones with an NFW profile as the initial condition, the core contraction does not occur within $10~{\rm Gyr}$ for $\sigma_{\rm x}/m_{\rm x}= 0.5\textup{--}5~{\rm cm^2/g}$, consistent with the results in [@koda2011]. However, the SIDM thermalization with a deep baryonic potential can speed up this process, as shown in our simulations [see also @elbert2016].
We see that the presence of the stellar potential breaks the monotonic relation between the value of $\sigma_{\rm x}/m_{\rm x}$ and the central SIDM density. The effect depends on the baryonic concentration and the size of the self-scattering cross section. Our results indicate that it could be challenging to extract the $\sigma_{\rm x}/m_{\rm x}$ information from stellar kinematics of galaxies dominated by baryons.
Halo shapes {#sub:shape}
-----------
In Fig. \[fig:shape1\], we show the SIDM halo surface densities for $R_{\rm d}=3\;{\rm kpc}$ (left) and $R_{\rm d}=6\;{\rm kpc}$ (right). The density contrast for the compact case is higher for different cross sections, compared to the extended one, as expected from the density profiles shown in Fig. \[fig:density\]. It is also evident that the simulated halos are not spherically symmetric, although their initial conditions are exactly spherical.
Fig. \[fig:shape2\] shows the ratio of minor-to-major axes vs. elliptical radius $\sqrt{R^2 + (z/s)^2}$ for $R_{\rm d}=3\;{\rm kpc}$ and $R_{\rm d}=6\;{\rm kpc}$ with different cross sections (solid). In all cases, the $c/a$ value deviates from $1$ and decreases towards the center ($b/a$ remains close to $1$). However, the SIDM halos are more responsive to the presence of the baryonic disc than their collisionless counterpart, and their shapes are more aligned with the axisymmetric disc potential (dashed). Interestingly, for the compact case, $c/a$ increases when the cross section increases from $0.5$ to $5\;{\rm cm^2/g}$ and the inner halo becomes [*rounder*]{} mildly. We can see a similar trend in the case of $R_{\rm d}=6~{\rm kpc}$, although the errors in measuring $c/a$, calculated using bootstrap method, for $r\lesssim R_{\rm d}$ are large due to the lack of enough DM particles in the central region of the halos.
The behavior in Fig. \[fig:shape2\] can be understood as follows. Since the DM self-interactions thermalize the inner halo, the DM density can modeled by the isothermal distribution [@kaplinghat2014a], $\rho_{\rm DM}\propto\exp{\left[-(\Phi_{\rm DM}+\Phi_{\rm MN})/\sigma^2_{\rm v}\right]}$, where $\Phi_{\rm DM}$ and $\Phi_{\rm MN}$ are the DM and disc potentials, respectively. $\Phi_{\rm MN}$ induces the deviation from spherical symmetry of the initial NFW halo, as indicated in Fig. \[fig:shape2\] (dashed), and the significance depends on its magnitude relative to $\Phi_{\rm DM}$ and $\sigma^2_{\rm v}$. In the compact-disc case, the central DM density increases when $\sigma_{\rm x}/m_{\rm x}$ increases from $0.5$ to $5~{\rm cm^2/g}$, as well as the DM dispersion (very mildly), as shown in Fig. \[fig:density\] (left). Accordingly, the baryonic potential becomes less dominant and the inner halo becomes more spherical. Note in the compact-disc case the simulated $c/a$ profile for $\sigma_{\rm x}/m_{\rm x}=0.5~{\rm cm^2/g}$ agrees well with the isothermal profile due to the baryonic potential, because of the strong dominance of the disc in the inner regions. In addition, for $\sigma_{\rm x}/m_{\rm x}=5~{\rm cm^2/g}$ (compact), both the inner DM density and velocity dispersion are higher, compared to the CDM case, but the SIDM halo is more aspherical and aligned with the disc than the CDM one. In the extended-disc case, the halo $c/a$ profiles also follow the disc one, but not as close as the compact case, since the disc does not dominate the potential at all radii, as shown in Fig. \[fig:density\] (right).
Evolution history {#sub:evolution}
-----------------
In this section, we take a closer look at the evolution of the SIDM halo and explicitly show that the presence of the baryonic potential does speed up core contraction and shorten the expansion phase.
Fig. \[fig:time\_evol\] shows the density and $c/a$ profiles at different epochs for three examples: the lowest (top) and highest (middle) cross sections in the simulations with the extended disc, and the highest cross section for the compact disc case (bottom). For $R_{\rm d}=6~{\rm kpc}$ and $\sigma_{\rm x}/m_{\rm x}= 0.5\;~{\rm cm^2/g}$, the simulated halo is on the core expansion phase over the $10\;{\rm Gyr}$ span of the simulation. In this case both central DM density and the $c/a$ ratio decrease continuously. When we increase the cross section to $5\;{\rm cm^2/g}$ (middle), the duration of the core expansion phase becomes much shorter. After about $1~{\rm Gyr}$, the halo enters the core contraction phase and the central DM density increases, as well as the ratio of minor-to-major axes. A more compact stellar disc can change the halo evolution even more dramatically, as shown in the bottom panel. The simulated halo almost never gets into the expansion phase and the central density and $c/a$ in the regions increases over time monotonically. In this case, the inner SIDM halo contains even more DM mass than its CDM counterpart.
We conclude that the evolution history of the SIDM halo is sensitive to the presence of the baryonic potential. The final halo properties, such as the density profile and the ellipticity, depend on the baryonic concentration and the strength of DM self-interactions.
Implications for the shape of the Milky Way Halo {#sec:MW}
================================================
The effect of baryons on the SIDM halo can potentially be tested with observations of the MW. Here, we construct a model for the MW potential consisting of an SIDM halo, a baryonic bulge and disc. The DM halo is chosen initially as a spherical NFW profile with $r_{\rm s}=42.18~{\rm kpc}$ and $\rho_{\rm s}=1.39\times10^6M_\odot/{\rm kpc^3}$. We model the disc following an MN potential as in Eq. \[eq:Miyamoto\_nagai\], with disc length scale and mass specified in Table \[table:bar\_param\] and bulge following a spherical Hernquist profile. We take the cross section as $\sigma_{\rm x}/m_{\rm x}=1\;{\rm cm^2/g}$.
![Top: DM density profiles of simulated SIDM halo with baryons (blue squares), best-fitting halo model in (black circles), and the NFW initial condition (gray dashed). The data point with error bars indicates the local DM density near the solar position from @Bovy2012. Bottom: Total circular velocity profiles of our MW mass model (blue solid) and best-fitting model in (black solid) with $10\%$ uncertainties (shaded band). Plotted also DM halo (dashed) and baryon (thin dashed) contributions.[]{data-label="fig:lowconcen"}](den_prof_rot_curve_low_concen.pdf){width="\columnwidth"}
Top panel of Fig. \[fig:lowconcen\] shows the DM density profiles for our SIDM halo (blue squares), the initial NFW model (gray dashed), and the halo model in (black circles). Our MW model reproduces well the estimates for the local DM density near the solar neighborhood from @Bovy2012 (red). Note the initial halo concentration is about $1.5~\sigma$ lower than the average for the MW mass object according to @Dutton2014 and also lower than that in . This is a necessary choice to be consistent with observations, since SIDM thermalization significantly increases the DM density in the inner regions due to the presence of the baryonic potential. Although the inner density profile of the SIDM halo deviates from the one, our MW mass model produces a circular velocity profile, consistent with the one in within $10~\%$ uncertainties, as shown in Fig. \[fig:lowconcen\] (bottom).
To quantify the baryonic influence, we estimate the logarithmic slope of the SIDM density profile as $\alpha\sim-1.8$ for the range of $r=1\textup{--}2\% r_{\rm vir}$. Interestingly, this is consistent with the prediction in NIHAO hydrodynamical CDM simulations within the $1\sigma$ range [@tollet2016]. Thus, for galaxies like the MW, where the baryons dominate the inner regions, both SIDM and CDM can lead to similar predictions. The result shown in Fig. \[fig:lowconcen\] (top) is based on $\sigma_{\rm x}/m_{\rm x}=1~{\rm cm^2/g}$. Increasing the cross section will speed up the transition from the core expansion to contraction phases, resulting a denser inner halo, as illustrated in Fig. \[fig:time\_evol\].
![Halo shape measurement in different numerical simulations: our MW SIDM halo model (blue solid), cosmological CDM-only (black dashed) and hydrodynamical simulations (orange solid) from the NIHAO project [@butsky2016] (the shaded area represents the $1\sigma$ scatter for the halos with mass $\sim10^{12}~M_{\odot}$), and zoom-in cosmological SIDM-only simulations of $5$ Aquarius halos (long dashed). Data points with error bars are the measurements of the MW halo shape using the stellar streams, GD-1 ($c/a=1.3^{+0.5}_{-0.3}$ at $r\approx14~{\rm kpc}$, square), Pal 5 ($c/a=0.93\pm0.16$ at $r\approx19~{\rm kpc}$, pentagon), and the combined analysis of the two ($c/a=1.05\pm0.14$ for $r\lesssim20~{\rm kpc}$, diamond), taken from @bovy2016. []{data-label="fig:MW_axial_ratio"}](shape_MW_vs_obs_7.pdf){width="\columnwidth"}
Fig. \[fig:MW\_axial\_ratio\] shows the ratio of minor-to-major axes as a function of the elliptical radius, predicted in our SIDM MW halo model with $\sigma_{\rm x}/m_{\rm x}=1~{\rm cm}^2/{\rm g}$ (blue solid). Since we assume a spherical NFW initial halo, the deviation from $c/a=1$ is caused by the disc potential. We see the disc induces a mild asphericity in the inner regions of the halo, $c/a\sim0.7\textup{--}0.8$, in good agreement with the result based on the analytical model in [@kaplinghat2014a], where the spherical boundary condition is imposed at $10~{\rm kpc}$. Our simulations also show the effect is quite extended, with $c/a$ converging back to unity only at the distance $\sim 50~{\rm kpc}$. We also present cosmological SIDM-only simulations of $5$ Aquarius halos with $\sigma_{\rm x}/m_{\rm x}=1~{\rm cm^2/g}$ (blue long dashed). Our results are in agreement with the previous ones [@A.Peter; @2018MNRAS.474..746B]. For the roundest halos (Aq-A and Aq-C), $c/a\gtrsim0.85$ for $r\lesssim30~{\rm kpc}$, lending support to our assumption of an initially spherical NFW halo. Since the DM self-scattering rate increases by a factor of $100$ from $30~{\rm kpc}$ to inner few ${\rm kpc}$, as indicated by the DM density profile shown in Fig. \[fig:lowconcen\] (top), we expect the spherical assumption is well-justified in the inner halo.
The MW halo shape has been inferred from observations of stellar streams such as GD-1 and Pal 5 [e.g., @koposov2010; @bowden2015; @pearson2015; @kupper2015; @bovy2016]. In Fig. \[fig:MW\_axial\_ratio\], we show the results presented in @bovy2016 for GD-1, Pal 5 and the combined one, where an axisymmetric NFW density profile with $b/a=1$ was used to model the DM distribution in the MW. We see that the phase-space tracks of the streams are consistent with a spherical DM halo in the MW at intermediate radii, $r\sim20~{\rm kpc}$, which indicates any asphericity either intrinsic to the DM distribution or induced by the disc should be at most weak on that scale in order to accommodate the measurements. Our model predicts $c/a\sim0.85$ at $r\approx20~{\rm kpc}$, consistent with the combined constraint on $c/a$ within $\sim1.5\sigma$. Note that our scale height in the MN disc is $z_{\rm d}\approx0.68~{\rm kpc}$, higher than the best-fit value $\sim0.3~{\rm kpc}$ in @bovy2016. We have estimated that taking their $z_{\rm d}$ value would reduce $c/a$ by $5\%$ at most in the inner regions and the difference becomes negligible around $10~{\rm kpc}$. In addition, in our MW model, the total halo mass within $20~{\rm kpc}$ is $1.28\times10^{11}M_\odot$, consistent with $M_{\rm halo}(r<20~{\rm kpc})=1.1\pm0.1\times10^{11}M_{\odot}$ measured in @bovy2016, although our initial NFW halo has lower concentration, compared to theirs ($r_s=18.0\pm7.5~{\rm kpc}$). It would be interesting to analyze the stream data with the SIDM halo model.
For comparison, we also plot cosmological CDM-only (gray) and hydrodynamical (red) simulations from the NIHAO project [@butsky2016]. Overall, the shape of individual NIHAO halos follows the median trend shown in Fig. \[fig:MW\_axial\_ratio\]. As is well-known, CDM halos from cosmological simulations are strongly triaxial [@frenk1988; @suto2002; @hayashi2007; @kuhlen2007; @Vera-Ciro1; @diemand2011]. Taken at face value, CDM predictions (gray) could look at odds with the observations. However, baryons can make the CDM halo shapes more spherical (red) [see also, @Dubinski1994; @debattista2008; @kazantzidis2010; @abadi2010; @tissera2010], an effect partially attributed to the change of orbits from boxy to tube or rounder loop as a result of the central concentration of baryons [@debattista2008]. In the NIHAO simulations, the mean value of $c/a$ is $0.7$ for the CDM halos after including baryons, and it can reaches $0.8$ at the $1\sigma$ level of the scatter, consistent with the observations reasonably well.
Although the sphericity created by the baryons helps CDM to accommodate more easily the observational constraints, the trend with radius could be different in the two models. It seems that CDM halos plus baryons become more spherical at all radii, whereas the effects explored here in SIDM plus baryons would anticipate a flattening of the shapes towards the inner regions that follows that of the disc. Such premise, of course, ignores any effect of feedback or cosmological assembly, which may cause deviations of the system from equilibrium. Therefore, the exciting premise of using halo shape profiles to differentiate DM candidates awaits confirmation from cosmological hydrodynamical SIDM simulations. We hope such experiments will become available in the near future.
Conclusions {#sec:conclusion}
===========
We use isolated N-body simulations of DM halos with static disc potentials to explore the gravitational effect of baryons on SIDM halos. We model the disc as a Miyamoto-Nagai potential embedded within an initially NFW halo with mass $\sim 10^{12}~M_\odot$. We consider different self-scattering cross sections, $\sigma_{\rm x}/m_{\rm x}=0.5,~1,~3$, and $5~{\rm cm^2/g}$ besides the special case, $\sigma_{\rm x}/m_{\rm x}=0~{\rm cm^2/g}$. In addition, we vary the radial length scale of the disc, $R_{\rm d}$, to study in detail how the DM halo responds to the baryons as a function of how relevant their contribution is to the total potential at a given radius.
In the absence of baryons, SIDM halos develop a central flat core with its density and size that depend mostly on the self-scattering cross section. We confirm that the inclusion of a disc potential can change this behavior due to SIDM thermalization with the potential, resulting in a higher core density and a smaller core than expected without the disc, a crucial effect in solving the diversity problem in SIDM [@2017PhRvL.119k1102K; @creasey2017],
We highlight two phases of evolution during our numerical experiments: a first stage of [*core expansion*]{}, during which the density core gets established due to the turn-on of the self-interactions, and a second stage of [*core contraction*]{} due to the gravitational effects of the baryons. The timescale for these two phases of evolution is a function of both, the cross section and the relative importance of the baryons inside the core. Higher cross sections and more compact baryonic discs (encoded in a smaller length scale) speed up the transition between the two phases and make the timescale of core expansion shorter.
We have also studied the role of the disc potential in shaping the SIDM halo. To explore this subtle effect, we assumed an exact spherical initial NFW profile such that any departure from sphericity is due to the influence of the baryonic potential. Compared to the case of $\sigma_{\rm x}/m_{\rm x}=0~{\rm cm^2/g}$, the SIDM halos are more responsive to the potential due to the thermalization, and their final flattening is more aligned with the orientation of the disc, consistent with the expectation from the analytical method. Our simulations clearly demonstrate that the induced asphericity is mainly sensitive to the contribution of the disc to the total potential, relative to the DM one. We further confirmed this by checking the evolution history of the SIDM halos. The flattening effect is maximized during the epoch when the core has the lowest density, which coincides with the time when the disc contribution to the total potential is also maximized.
We have constructed a mass model for the MW and explored the shape prediction with observations. The model consists of a stellar disc and a bulge, embedded within a spherical SIDM halo. It reproduces observed stellar kinematics of the Galaxy within the uncertainties and the local DM density reported in the solar neighborhood. We find that the baryons are able to induce a mild flattening ($c/a \sim 0.7\textup{--}0.8$) in the inner regions but the effect weakens at larger radii. At $r \sim 20$ kpc where observational constraints seem to suggest an almost spherical halo, the effects of the disc are not strong, in agreement with the observations. We propose that the quasi sphericity of the halo at large distance is easier to accommodate in SIDM models than within the strongly triaxial structures predicted by CDM, although considering the effects of baryons might help to reconcile CDM models with observed spherical halos. Furthermore, we argue that a study of halo shapes as a function of radius might be able to help distinguish the nature of DM, although a more stringent comparison to cosmological simulations are needed to confirm this last point. On the observational side, future surveys capable of inferring the shape of the Galactic halo within the inner $20~{\rm kpc}$ regions are promising avenues to make progress on establishing the non-canonical nature of DM.
Acknowledgments {#acknowledgments .unnumbered}
===============
OS acknowledges support by NASA MUREP Institutional Research Opportunity (MIRO) grant number NNX15AP99A and HST grant HST-AR-14582. HBY acknowledges support from U. S. Department of Energy under Grant No. de-sc0008541 and the Hellman Fellows Fund. LVS is grateful for support from the Hellman Fellows Foundation and HST grant HST-AR-14582. MV acknowledges support through an MIT RSC award, the support of the Alfred P. Sloan Foundation, and support by NASA ATP grant NNX17AG29G. JZ acknowledges support by a Grant of Excellence from the Iceland Research Fund (grant number 173929-051).
Convergence test for the halo shape algorithm {#sec:A1}
=============================================
![Ratio of minor- (solid) and intermediate-to-major (dashed) axes for the convergence test runs. []{data-label="fig:conv_test"}](resolution_test_cdm_Rd6_Md_eqaul_Mh.pdf){width="\columnwidth"}
To evaluate accuracy of the shape measurement, we follow @Vera-Ciro1 to determine the convergence radius where shape measurements are robust in our simulations. We consider the convergence radius $r_{\rm conv}$ [@power2003; @Navarro2008], $$\nonumber\kappa (r_{\rm conv}) \equiv \frac{t_{\rm relax}}{t_{\rm cir}(r_{200})} = \frac{\sqrt{200}}{8}
\frac{N(r_{\rm conv})}{\ln{N(r_{\rm conv})}}
\left(\frac{\overline{\rho}(r_{\rm conv})}{\rho_{\rm crit}}\right)^{-1/2}$$ where $t_{\rm relax}$ is the two-body relaxation time scale due to gravitational encounters, $t_{\rm cir}$ is the circular orbit timescale at $r_{200}$, $N$ is number of DM particles and $\overline{\rho}$ is the average density inside the convergence radius. We take $\kappa (r_{\rm conv})=7$ as in @Vera-Ciro1. In addition, we require $\sim 70\%$ of the particles to be inside the virial sphere and at least $2000$ DM particles inside of convergence radius.
The first requirement is satisfied if we choose a large cutoff radius in the [SpherIC]{} code, at which the density profile transits from an NFW one to an exponentially decaying one to avoid the divergence of the mass. Then we run three simulations with different values of the DM particle number and the gravitational softening length as a convergence test with details summarized in Table \[table:IC\_param\]. The convergence radius decreases with increasing the number of total DM particles. Thus, to probe the shape of the inner halo, down to few ${\rm kpc}$, we need at least $2$ million particles in simulations. Fig. \[fig:conv\_test\] shows the $b/a$ (top) and $c/a$ (bottom) profiles for different resolutions, we take a static MN potential as stellar disc similar to Sec. \[sec:halo-disc\], with a MW-sized halo with $M_{200} \simeq\ 2.6\times 10^{12}\ M_{\odot}$. We see that the convergence improves when $N_{\rm tot}$ increases. We take high-resolution run in the results presented in Sec. \[sec:halo-disc\] and \[sec:MW\].
[^1]: E-mail: <osame001@ucr.edu>
[^2]: NASA MIRO FIELDS Fellow
[^3]: Hellman Fellow
[^4]: Hellman Fellow
[^5]: Alfred P. Sloan Fellow
|
---
abstract: 'The evolution problem for a quantum particle confined in a 1D box and interacting with one fixed point through a time dependent point interaction is considered. Under suitable assumptions of regularity for the time profile of the Hamiltonian, we prove the existence of strict solutions to the corresponding Schrödinger equation. The result is used to discuss the stability and the steady-state local controllability of the wavefunction when the strenght of the interaction is used as a control parameter.'
author:
- 'Andrea Mantile[^1]'
title: 'Stability and control of a 1D quantum system with confining time dependent delta potentials.'
---
Introduction
============
We consider the evolution problem for a quantum particle confined in a 1D bounded domain and interacting with one fixed point through a delta shaped potential whose strength varies in time. The Hamiltonian associated to this class of potentials, denoted in the following with $H_{\alpha(t)}$, is defined in terms of a time dependent parameter, $\alpha(t)$, which describes the time profile of the interaction. Under suitable regularity assumptions, we study the dynamical system defined by $H_{\alpha(t)}$, its stability properties and the local controllability of the dynamics when $\alpha$ is used as a control function.
General conditions for the solution to the quantum evolution equation related to non autonomous Hamiltonians, $H(t)$, have been long time investigated (e.g. in [@Simon], [@Reed] and [@Fattorini]). When the operator’s domain $D(H(t))$ depends on time (as in the case of time dependent point interactions), the Cauchy problem$$\left\{
\begin{array}
[c]{l}\frac{d}{dt}\psi=-iH(t)\psi\\
\psi_{t=0}=\psi_{0}\end{array}
\right. , \label{Cauchy}$$ was explicitely considered in [@Kys] by Kisyński using coercivity and $C_{loc}^{2}$-regularity of $t\rightarrow H(t)$. A similar assumption, $\alpha\in C_{loc}^{2}(t_{0},+\infty)$ in the above notation, is used in the works of Yafaev [@Yaf1], [@Yaf2] and [@Yaf3] (with M. Sayapova) to prove the existence of a strongly differentiable time propagator for the scattering problem with time dependent delta interactions in $\mathbb{R}^{3}$. Such a condition, however, can be relaxed by exploiting the explicit character of point interactions. The dynamics associated to this class of operators, indeed, is essentially described by the evolution of a finite dimensional variable related to the values taken by the regular part of the state in the interaction points (e.g. in [@Albeverio], [@Dell'Anto], [@AdTe]). This leads to simplified evolution equations allowing rather explicit estimates under Fourier transform. Using this approach, the quantum evolution problem for a 1D time dependent delta interaction has been considered in [@Taoufik]. In particular, it is shown that $\alpha\in H^{\frac{1}{4}}(0,T)$ allows to define a strongly continuous dynamical system in $L^{2}(\mathbb{R})$, while $\alpha\in H^{\frac{3}{4}}(0,T)$ leads to the existence of strong solutions to$$\left\{
\begin{array}
[c]{l}\frac{d}{dt}\psi=-iH_{\alpha(t)}\psi\\
\psi_{t=0}=\psi_{0}\in D(H_{\alpha(0)})
\end{array}
\right. . \label{Cauchy 1}$$ The same strategy has also been adopted to study the diffusion problem with nonautonomous delta potentials in $\mathbb{R}^{3}$ [@Dell'Anto], and the quantum evolution problem for a 1D nonlinear model where the parameter $\alpha$ is assigned as a function of the particle’s state [@AdTe].
The techniques used in these works can be adapted to the 1D confining case. However, the lack of a simple explicit expression for the free propagator kernel and the use of eigenfunction expansions, which replaces the Fourier transform analysis in this setting, necessarily requires some additional efforts to obtain a result. In the perspective of the stability and controllability analysis, we restrict ourselfs to the case $\alpha\in
H^{1}(0,T)$ and: $H_{\alpha}=-\frac{d^{2}}{dx^{2}}+\alpha\delta$ with Dirichlet conditions on the boundary of $I=\left[ -\pi,\pi\right] $. Under these assumptions, we prove that the evolution problem (\[Cauchy 1\]) admits a strongly differentiable time propagator preserving the regularity of the initial state. This turns out to be a key point to study the controllability of the dynamics.
The controllability of a quantym dynamics through an external field has attracted an increasing attention due to possible applications in nuclear magnetic resonance, laser spectroscopy, photochemestry and quantum information. This problem has been considered for confining Schrödinger operators with regular potentials of the form: $H=-\Delta_{\Omega}^{D}+V(x)+u(t)W(x)$, where $\Delta_{\Omega}^{D}$ is the Dirichlet Laplacian in the bounded domain $\Omega\subset\mathbb{R}^{m}$, while $u$ is used as a control function. The particular setting: $m=1$, $V=0$ and $W(x)=x$, corresponding to a quantum particle confined in a 1D box and moving under the action of a time dependent uniform eletric field, has been considered in [@Beau]. The exact controllability of the quantum state was proved, in this case, in $H^{7}$-neighbourhoods of the steady states by using $u\in
L^{2}(0,T)$ controls with $T>T_{m}>0$ (actually a simpler version of this proof holding for all $T>0$ is given in [@BeLa]). For the same system, the exact controllability between neighbourhoods of any couple of eigenstates is discussed in [@BeCo]. In the more general setting, with $V,W\in C^{\infty
}\left( \bar{\Omega},\mathbb{R}\right) $ and $m$ being any space dimension, differents approximate controllability results in $L^{2}$ have been presented in [@ChMaSiBo] and [@Ner]. An example concerned with singular pointwise potentials is presented in [@AdBo], where point interaction Hamiltonians are used to construct a general scheme allowing to steer the system between the eigenstates of a drift operator $H_{0}$ (the 1D Dirichlet Laplacian). The idea is to introduce adiabatic perturbations of the spectrum of $H_{0}$ producing eigenvalues intersection, while controlling the state’s evolution through the adiabatic theorem. The result is the approximate state to state controllability in infinite time. The particular case of 1D point interactions is also related to control problems on quantum graphs where these Hamiltonians naturally arise.
In Section \[Sec\_control\], the nonlinear map $\alpha\rightarrow\psi_{T}$, associating to the function $\alpha$ the solution at time $T$ of (\[Cauchy 1\]), is considered for a fixed initial state $\psi_{0}\in
D(H_{\alpha(0)})$. This can be regarded as a control system where the control parameter is $\alpha$ and the target state is $\psi_{T}$. Using the regularity properties of the time propagator, we show that: for $\psi_{0}\in H^{2}\cap
H_{0}^{1}(I)$, the map $\alpha\rightarrow\psi_{T}$ is of class $C^{1}(H_{0}^{1}(0,T),\,H^{2}\cap H_{0}^{1}(I))$. Then, the local controllability relies on the surjectivity of the corresponding linearized map according to an inverse function argument. This point is considered in Section \[Sec\_control\_lin\] where a general condition for the solution of the linearized control problem is given in (\[Lin1\])-(\[Lin2\]) and proposition \[Prop\_moment\]. When $\psi_{0}$ coincides with an even eigenstate of the Diriclet Laplacian, this scheme provides with a result of local steady state controllability in finite time.
The Model
=========
Point interactions form a particular class of singular perturbations of the Laplacian supported by finite set of points. In $\mathbb{R}^{d}$, $d\leq3$, these Hamiltonians have been rigorously defined using the theory of selfadjoint extensions of symmetric operators (e.g. in [@Albeverio]). The definition easily extends to the case of bounded regions by taking into account the Dirichlet conditions on the boundary for the functions in the operator’s domain (e.g. in [@A.M.]). In what follows we consider a delta perturbation of the Dirichlet Laplacian, centered in the origin of the interval $I=\left[ -\pi,\pi\right] $. In terms of quadratic forms, this family of Hamiltonians acts on $H_{0}^{1}(I)$ as$$H_{\alpha}=-\Delta_{I}^{D}+\alpha\,\delta\label{operatore 0}$$ where $\Delta_{I}^{D}$ is the Dirichlet Laplacian on the interval, $\delta$ is the Dirac distribution centered in the origin, while $\alpha$ is a real parameter. The operator’s domain $D(H_{\alpha})$ extends to all those vectors $\psi\in L^{2}(I)$, such that: $H_{\alpha}\psi\in L^{2}(I)$. The description of $D(H_{\alpha})$ is strictly related to the properties of the Green’s kernel, $\mathcal{G}_{0}^{z}$, for the resolvent of the unperturbed operator $\left( -\Delta_{I}^{D}+z\right) ^{-1}$. Fix $x^{\prime}\in I$; whenever $z$ does not belong to the spectrum of $\Delta_{I}^{D}$, $\mathcal{G}_{0}^{z}$ satisfies the equation$$\left( -\Delta_{I}^{D}+z\right) \mathcal{G}_{0}^{z}(x,x^{\prime})=\delta(x-x^{\prime}),\quad z\in\mathbb{C}\backslash\sigma_{\Delta_{I}^{D}}
\label{Green 0}$$ The solution to (\[Green 0\]) can be represented as the sum of the ’free’ Green’s function plus an additional term taking into account the conditions at the boundary$$\mathcal{G}_{0}^{z}(x,x^{\prime})=\frac{e^{-\sqrt{z}\left\vert x-x^{\prime
}\right\vert }}{2\sqrt{z}}-h(x,x^{\prime},z) \label{Green 1}$$$$\left\{
\begin{array}
[c]{l}\left( -\Delta_{I}^{D}+z\right) h(\cdot,x^{\prime},z)=0\\
\left. h(\cdot,x^{\prime},z)\right\vert _{x=\pm\pi}=\left. \frac
{e^{-\sqrt{z}\left\vert \cdot-x^{\prime}\right\vert }}{2\sqrt{z}}\right\vert
_{x=\pm\pi}\end{array}
\right. \label{h 1}$$ The solution of (\[h 1\]) is$$\mathcal{G}_{0}^{z}(x,x^{\prime})=\frac{e^{-\sqrt{z}\left\vert x-x^{\prime
}\right\vert }}{2\sqrt{z}}-\frac{1}{\sqrt{z}}\frac{e^{-\pi\sqrt{z}}}{e^{2\pi\sqrt{z}}-e^{-2\pi\sqrt{z}}}\left[ e^{\sqrt{z}x}\sinh\left( \sqrt
{z}(\pi+x^{\prime})\right) +e^{-\sqrt{z}x}\sinh\left( \sqrt{z}(\pi
-x^{\prime})\right) \right] \label{Green 1.1}$$ Another useful representation of $\mathcal{G}_{0}^{z}$ is in terms of Fourier expansion w.r.t. the Laplacian’s eigenfunctions. Denoting with $\psi_{k}$ and $\lambda_{k}$ respectively the eigenfunctions and the eigenvalues of $-\Delta_{I}^{D}$, we have$$\psi_{k}(x)=\left\{
\begin{array}
[c]{l}\frac{1}{\sqrt{\pi}}\,\sin\frac{k}{2}x\quad k\ even\\
\frac{1}{\sqrt{\pi}}\,\cos\frac{k}{2}x\quad k\ odd
\end{array}
\right. ;\quad\lambda_{k}=\frac{k^{2}}{4};\quad k\in\mathbb{N}
\label{stati stazionari}$$ and$$\mathcal{G}_{0}^{z}(x,x^{\prime})=\sum_{k\in\mathbb{N}}\frac{1}{\lambda_{k}+z}\,\psi_{k}^{\ast}(x^{\prime})\,\psi_{k}(x),\quad z\notin\sigma_{\Delta
_{I}^{D}} \label{Green 2.0}$$
Using (\[operatore 0\]) and (\[Green 0\]), a straightforward calculation shows that $H_{\alpha}\psi\in L^{2}(I)$ whenever $\psi$ has the form$$\psi=\phi+q\mathcal{G}_{0}^{z}(\cdot,0),\ \phi\in D(\Delta_{I}^{D}),\ -q=\alpha\,\psi(0) \label{b.c. 1}$$ This is a general characterization of $D(H_{\alpha})$. It can be derived by using the von Neumann theory of selfadjoint extensions to identify $H_{\alpha
}$ as the extension of the symmetric operator $H_{0}$$$\left\{
\begin{array}
[c]{l}D(H_{0})=\left\{ \psi\in H^{2}\cap H_{0}^{1}(I)\,\left\vert \,\psi
(0)=0\right. \right\} \\
H_{\alpha}\psi=-\frac{d^{2}}{dx^{2}}\psi
\end{array}
\right. \label{H_0}$$ to those vectors $\psi\in D(H_{0}^{\ast})$ fulfilling the selfadjoint boundary conditions (e.g. in [@Albeverio; @2] and [@Reed])$$\left\{
\begin{array}
[c]{l}\psi(0^{+})-\psi(0^{-})=0\\
\psi^{\prime}(0^{+})-\psi^{\prime}(0^{-})=\alpha\,\psi(0)
\end{array}
\right. \label{b.c. 2}$$ The following proposition is a rephrasing of the result in [@Albeverio] in the bounded domain case.
\[Proposition 0\]Let $\alpha\in\mathbb{R}$, $\lambda\in\mathbb{C}\backslash\sigma_{\Delta_{I}^{D}}$ and denote with $H_{\alpha}$ the family of selfadjoint extensions of $H_{0}$ associated to the boundary condition (\[b.c. 2\]). The following representation holds:$$D(H_{\alpha})=\left\{ \psi\in H^{2}(I\backslash\left\{ 0\right\} )\cap
H_{0}^{1}(I)\ \left\vert \ \psi=\phi^{\lambda}+q\mathcal{G}_{0}^{\lambda
}\left( \cdot,0\right) ;\ \phi^{\lambda}\in H^{2}\cap H_{0}^{1}(I);\ -q=\alpha\psi(0)\right. \right\} \label{dominio1}$$$$H_{\alpha}\psi=-\frac{d^{2}}{dx^{2}}\phi^{\lambda}-\lambda q\mathcal{G}_{0}^{\lambda}(\cdot,0) \label{operatore}$$ Moreover, for $z\in\mathbb{C}\backslash\mathbb{R}$, $\varphi\in L^{2}(I)$, the resolvent $\left( H_{\alpha}+z\right) ^{-1}$ writes as:$$\left( H_{\alpha}+z\right) ^{-1}\varphi=\left( -\Delta_{I}^{D}+z\right)
^{-1}\varphi-\frac{\alpha}{1+\alpha\,\mathcal{G}_{0}^{z}(0)}\left(
-\Delta_{I}^{D}+z\right) ^{-1}\varphi(0)\,\mathcal{G}_{0}^{z}(\cdot,0)
\label{Resolvent}$$
Although the representation: $\psi=\phi^{\lambda}+q\mathcal{G}_{0}^{\lambda
}\left( \cdot,0\right) $ may change with different choices of $\lambda$, the operator $H_{\alpha}$ depends only on the value of the parameter $\alpha$ related to the interaction’s strength.
Next we assign $\alpha$ as a function of time and consider the non autonomous system defined by $H_{\alpha(t)}$$$\left\{
\begin{array}
[c]{l}\medskip i\frac{d}{dt}\psi(x,t)=H_{\alpha(t)}\psi(x,t)\,,\\
\psi(x,0)=\psi_{0}(x)\in D(H_{\alpha(0)})\,.
\end{array}
\right. \label{Schroedinger}$$ The mild solutions are$$\psi(\cdot,t)=e^{it\Delta_{I}^{D}}\psi_{0}+i\int_{0}^{t}q(s)\,e^{i\left(
t-s\right) \Delta_{I}^{D}}\delta\,ds\,, \label{Schroedinger 2-1}$$ where $e^{it\Delta_{I}^{D}}$ is the time propagator associated to $-\Delta
_{I}^{D}$ and $-q(t)=\alpha(t)\psi(0,t)$ fixes the boundary condition of the operator’s domain. The action of the operator $e^{it\Delta_{I}^{D}}$ on $L^{2}(I)$ is$$e^{it\Delta_{I}^{D}}f=\sum\limits_{k\in\mathbb{N}}\left( \psi_{k},f\right)
_{L^{2}(I)}\,e^{-i\lambda_{k}t}\psi_{k}\quad\forall f\in L^{2}(I)
\label{propagatore}$$ where the scalar product in $L^{2}$ is defined according to: $\left(
u,v\right) _{L^{2}(I)}={\displaystyle\int_{-\pi}^{\pi}}
\bar{u}\,v$. This relation suggests to replace the distributional part at the r.h.s. of (\[Schroedinger 2-1\]) with$$\frac{i}{\sqrt{\pi}}\sum\limits_{k\in\mathbb{N}}\int_{0}^{t}q(s)\,e^{-i\lambda
_{k}\left( t-s\right) }\,ds\,\psi_{k}$$ It follows$$\psi(\cdot,t)=e^{it\Delta_{I}^{D}}\psi_{0}+\frac{i}{\sqrt{\pi}}\sum
\limits_{\substack{k\in\mathbb{N}\\k\ odd}}\int_{0}^{t}q(s)\,e^{-i\lambda
_{k}\left( t-s\right) }\,ds\,\psi_{k} \label{Schroedinger 3}$$$$q(t)=-\alpha(t)\left[ e^{it\Delta_{I}^{D}}\psi_{0}(0)+\frac{i}{\pi}\sum\limits_{\substack{k\in\mathbb{N}\\k\ odd}}\int_{0}^{t}q(s)\,e^{-i\lambda
_{k}\left( t-s\right) }\,ds\right] \label{carica 1}$$ The essential informations about the dynamics described in (\[Schroedinger 3\])-(\[carica 1\]) are contained in the auxiliary variable $q$, usually referred to as the *charge* of the particle (e.g. in [@Albeverio]). In what follows, the solution of the above problem is considered under suitable regularity assumptions for $\alpha$. The equivalence with the original Cauchy problem (\[Schroedinger\]) is further addressed. Finally, the dependence of this solution from $\alpha$, considered as a control parameter, is investigated. Our result is exposed in the following theorem.
\[Th\]Assume: $\alpha\in H^{1}(0,T)$, $\psi_{0}\in D(H_{\alpha(0)})$ and let $\psi_{T}(\psi_{0},\alpha)$, $q(t,\psi_{0},\alpha)$ respectively denote the solution at time $T$ and the charge as functions of the intial state and of the parameter $\alpha$. The following properties hold.$1)$ The system (\[Schroedinger 3\]) - (\[carica 1\]) admits an unique solution $\psi_{t}\in C(0,T;\,H^{1}(I))\cap C^{1}(0,T;\,L^{2}(I))$ with: $\psi_{t}\in
D(H_{\alpha(t)})$ and $i\partial_{t}\psi(\cdot,t)=H_{\alpha(t)}\psi(\cdot,t)$ at each $t$.$2)$ The map $\alpha\rightarrow\psi_{T}(\psi
_{0},\alpha)$ is $C^{1}(H^{1}(0,T),\,H_{0}^{1}(I))$ in the sense of Fréchet. Moreover, the ’regular part’ of the solution at time $T$, defined by: $\psi_{T}(\psi_{0},\alpha)-q(T,\psi_{0},\alpha)\mathcal{G}_{0}^{\lambda}$ and considered as a function of $\alpha$, is of class $C^{1}(H^{1}(0,T),\,H^{2}\cap H_{0}^{1}(I))$. If $\alpha\in H_{0}^{1}(0,T)$, then $\psi_{T}(\psi_{0},\alpha)$ coincides with its regular part and $\alpha
\rightarrow\psi_{T}(\psi_{0},\alpha)\in C^{1}(H_{0}^{1}(0,T),\,H^{2}\cap
H_{0}^{1}(I))$.$3)$ Let $\alpha\in H_{0}^{1}(0,T)$ and $\mathcal{W}$ be the subspace of $H^{2}\cap H_{0}^{1}(I)$ generated by the system: $\left\{ \psi_{k},\ k\text{ odd}\right\} $. If $\psi_{0}\in\mathcal{W}$, then: $\alpha\rightarrow\psi_{T}(\psi_{0},\alpha)\in
C^{1}(H_{0}^{1}(0,T),\,\mathcal{W})$. If $\psi_{0}=\psi_{\bar{k}}$ for a fixed $\bar{k}$ odd and $T\geq8\pi$, then: it exist an open neighbourhood $V\times
P\subseteq H_{0}^{1}(0,T)\times\mathcal{W}$ of the point $\left( 0,\psi
_{\bar{k}}\right) $ such that $\left. \alpha\rightarrow\psi_{T}(\psi
_{0},\alpha)\right\vert _{V}:V\rightarrow P$ is surjective.
The proof is developed in the Propositions \[Proposition 3\], \[Proposition 4\], \[Proposition 4.1\] and in the Theorem \[local controllability\] of sections 3 and 4.
*Notation:* The Sobolev spaces are denoted with $H^{\nu}$ or $H_{0}^{\nu
}$ in the case of Dirichlet boundary conditions. The notation ’$\lesssim$’ is to be intended as ’$\leq C$’ where $C$ is a suitable positive constant.
The existence of the dynamics
=============================
In what follows the existence of solutions to (\[Schroedinger\]) is discussed under the assumption: $\alpha\in H^{1}(0,T)$. Our strategy consists in using the mild approach described by (\[Schroedinger 3\])-(\[carica 1\]). It articulates in two steps: First we prove that (\[carica 1\]) admits an unique solution in $H^{1}(0,T)$, for any finite time $T$ and any $\alpha\in
H^{1}(0,T)$. This result is then used to obtain estimates for the solution of (\[Schroedinger\]).
The charge equation
-------------------
Let consider the linear map $U$$$Uq=\sum_{k\in D}\int_{0}^{t}q(s)e^{-i\lambda_{k}\left( t-s\right) }ds,\quad
D=\left\{ k\in\mathbb{N},\ k\ odd\right\} ,\ \lambda_{k}=\frac{k^{2}}{4}
\label{U}$$
\[Rem\_base\]Depending on the value of $T$, some of the functions $e^{-i\lambda_{k}t}$ may belongs to the standard basis $\left\{ e^{i\omega
nt};\,\omega=\frac{2\pi}{T}\right\} _{n\in\mathbb{Z}}$ of the space $L^{2}(0,T)$. In particular, for: $T=8\pi N$, $N\in\mathbb{N}$, it follows$$\left\{ e^{-i\lambda_{k}t}\right\} _{k\in\mathbb{N}}\subset\left\{
e^{i\omega nt};\,\omega=\frac{2\pi}{T}\right\} _{n\in\mathbb{Z}}
\label{aux_condition}$$ The (\[aux\_condition\]) will be often used as an auxiliary condition in the forthcoming analysis.
\[Lemma 1.1\]Let $\mathcal{H}_{T}$ denotes the Hilbert subspace$$\mathcal{H}_{T}\mathcal{=}\left\{ q\in H^{1}(0,T),\ q(0)=0\right\} \label{H}$$ equipped with the $H^{1}$-norm. The map $U$ (\[U\]) is bounded in $\mathcal{H}_{T}$ with the estimate$$\left\Vert Uq\right\Vert _{\mathcal{H}_{T}}\lesssim\left( T+1\right)
\,\left\Vert q\right\Vert _{\mathcal{H}_{T}} \label{St_0}$$
We use the standard basis $\left\{ e^{i\omega nt};\,\omega=\frac{2\pi}{T}\right\} _{n\in\mathbb{Z}}$ of $L^{2}(0,T)$. The Fourier coefficients of $\int_{0}^{t}q(s)e^{-i\lambda_{k}\left( t-s\right) }ds$ w.r.t. the vectors $\,e^{i\omega nt}$ write as$$\int_{0}^{T}dt\int_{0}^{t}ds\,q(s)e^{-i\lambda_{k}\left( t-s\right)
}e^{-i\omega nt}=\int_{0}^{T}ds\,q(s)e^{i\lambda_{k}s}\int_{s}^{T}dt\,e^{-i\left( \lambda_{k}+\omega n\right) t}\,.$$ Assume that $T$ fulfills the condition (\[aux\_condition\]) and, consequently, $\left\{ e^{-i\lambda_{k}t}\right\} $ forms a subset of $\left\{ e^{i\omega nt};\,\omega=\frac{2\pi}{T}\right\} _{n\in\mathbb{Z}}$. This choice allows to avoid small divisors, while it does not imply any loss of generality. With this assumption, the above expression writes as$$\int_{0}^{T}dt\int_{0}^{t}ds\,q(s)e^{-i\lambda_{k}\left( t-s\right)
}e^{-i\omega nt}=\left\{
\begin{array}
[c]{l}\bigskip\int_{0}^{T}ds\,q(s)e^{i\lambda_{k}s}\left( T-s\right)
,\quad\text{for }\lambda_{k}+\omega n=0\,,\\
\frac{i}{\lambda_{k}+\omega n}\left[ \int_{0}^{T}ds\,q(s)e^{i\lambda_{k}s}-\int_{0}^{T}ds\,q(s)e^{-i\omega ns}\right] ,\quad\text{for }\lambda
_{k}+\omega n\neq0\,.
\end{array}
\right.$$ and the expansion$$\begin{gathered}
\int_{0}^{T}dt\int_{0}^{t}ds\,q(s)e^{-i\lambda_{k}\left( t-s\right)
}e^{-i\omega nt}=\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\nonumber\\
=\left( Tq_{-Nk^{2}}-\left( tq\right) _{-Nk^{2}}\right) e^{-i\lambda_{k}t}+\sum_{\substack{n\in\mathbb{Z}\\\lambda_{k}+\omega n\neq0}}\frac{i}{\lambda_{k}+\omega n}\left[ q_{-Nk^{2}}-q_{n}\right] e^{i\omega nt}
\label{Fourier}$$ follows, $q_{n}$ denoting the $n$-th Fourier coefficient of $q$ (in particular: $q_{-Nk^{2}}=\left( q,e^{-i\lambda_{k}t}\right) _{L^{2}(0,T)}$). Let $S=\left\{ n\in\mathbb{Z}:\omega n\notin\left\{ -\lambda_{k}\right\}
_{k\in D}\text{ }\right\} $; we have$$\int_{0}^{T}\left( te^{-i\lambda_{k}t}\right) e^{-i\omega nt}=\frac
{iT}{\lambda_{k}+\omega n},\quad\forall\,n\in S\,.$$ This allows to write$$\sum_{\substack{n\in\mathbb{Z}\\\lambda_{k}+\omega n\neq0}}\frac{i}{\lambda_{k}+\omega n}q_{-Nk^{2}}e^{i\omega nt}=q_{-Nk^{2}}\Pi_{S}\left(
\frac{t}{T}e^{-i\lambda_{k}t}\right) \,,$$ where $\Pi_{S}$ is the projector over the subspace spanned by $\left\{
e^{i\omega nt}\right\} _{n\in S}$. Using (\[Fourier\]) and the above relation, the Fourier expansion of $Uq$ writes as$$Uq=\sum_{k\in D}\left( Tq_{-Nk^{2}}-\left( tq\right) _{-Nk^{2}}\right)
e^{-i\lambda_{k}t}+\sum_{k\in D}q_{-Nk^{2}}\Pi_{S}\left( \frac{t}{T}e^{-i\lambda_{k}t}\right) -\sum_{k\in D}\sum_{n\in S}\frac{iq_{n}}{\lambda_{k}+\omega n}\,e^{i\omega nt}\,. \label{L1 1.2}$$ Consider the contributions to $Uq$:$i)$ The norm of the first term at the r.h.s. of (\[L1 1.2\]) is bounded by$$\left\Vert \sum_{k\in D}\left( Tq_{-Nk^{2}}-\left( tq\right) _{-Nk^{2}}\right) e^{-i\lambda_{k}t}\right\Vert _{L^{2}(0,T)}\leq2T\left\Vert
q\right\Vert _{L^{2}(0,T)}\,. \label{L1 1.3}$$ $ii)$ For the second term, one has$$\begin{gathered}
\left\Vert \sum_{k\in D}q_{-Nk^{2}}\Pi_{S}\left( \frac{t}{T}e^{-i\lambda
_{k}t}\right) \right\Vert _{L^{2}(0,T)}=\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad\qquad\nonumber\\
=\left\Vert \Pi_{S}\sum_{k\in D}q_{-Nk^{2}}\left( \frac{t}{T}e^{-i\lambda
_{k}t}\right) \right\Vert _{L^{2}(0,T)}\leq\sum_{k\in D}\left\vert
q_{-Nk^{2}}\right\vert ^{2}\leq\left\Vert q\right\Vert _{L^{2}(0,T)}\,.
\label{L1 1.4}$$ $iii)$ The remaining term is a superposition of $L^{2}$-functions: $\sum_{n\in
S}\frac{iq_{n}}{\lambda_{k}+\omega n}\,e^{i\omega nt}$ parametrized by the index $k\in D$. Its $n$-th Fourier coefficient is formally expressed by: $\sum_{k\in D}\frac{iq_{n}}{\lambda_{k}+\omega n}$ and the $L^{2}$-norm is$$\left\Vert \sum_{k\in D}\sum_{n\in S}\frac{iq_{n}}{\lambda_{k}+\omega
n}\,e^{i\omega nt}\right\Vert _{L^{2}(0,T)}^{2}=\sum_{n\in S}\left\vert
\sum_{k\in D}\frac{q_{n}}{\lambda_{k}+\omega n}\right\vert ^{2}\,.
\label{L1 2.0}$$ To study the sum at the r.h.s, the relation$$\frac{1}{\pi}\sum_{k\in D}\frac{1}{\lambda_{k}+z}=\mathcal{G}_{0}^{z}(0,0)\,,
\label{Green_formula}$$ arising from (\[stati stazionari\]) and (\[Green 2.0\]), is used. According to (\[Green 1.1\]) and (\[Green\_formula\]), it follows$$\sum_{k\in D}\frac{1}{\lambda_{k}+\omega n}=\frac{1}{2\sqrt{\omega n}}-\frac{1}{\sqrt{\omega n}}\frac{e^{-\pi\sqrt{\omega n}}}{e^{2\pi\sqrt{\omega
n}}-e^{-2\pi\sqrt{\omega n}}}2\sinh\left( \pi\sqrt{\omega n}\right)
\,,\qquad n\in S\,. \label{Green 2.1}$$ This leads to$$\sum_{k\in D}\frac{1}{\lambda_{k}+\omega n}\underset{\left\vert n\right\vert
\rightarrow\infty}{\sim}\left\{
\begin{array}
[c]{ll}\medskip\mathcal{O}\left( \frac{1}{\sqrt{\omega n}}e^{-\sqrt{\omega n}\pi
}\right) & \quad\text{for }n>0\text{, }n\in S\\
\mathcal{O}\left( \frac{1}{\sqrt{\left\vert \omega n\right\vert }}\right) &
\quad\text{for }n<0\text{, }n\in S
\end{array}
\right. \,. \label{Green 2.2}$$ Consider the map $T:f_{n}\rightarrow f_{n}\left( \sum_{k\in D}\frac
{i}{\lambda_{k}+\omega n}\right) $ in $\ell_{2}(S)$. Using (\[Green 2.2\]), $T$ can be identified with the limit of the sequence of finite rank maps: $\left. T_{N}f_{n}=\left\{ f_{n}\left( \sum_{k\in D}\frac{i}{\lambda
_{k}+\omega n}\right) \right\} _{n\leq N}\right. $, in the $\ell_{2}(S)$-operator norm. Thus, it defines a compact operator in $\ell_{2}(S)$ (e.g. in Theorem VI.12, [@Simon1]), and we get$$\sum_{n\in S}\left\vert \sum_{k\in D}\frac{q_{n}}{\lambda_{k}+\omega
n}\right\vert ^{2}=\left\Vert \left( \sum_{k\in D}\frac{q_{n}}{\lambda
_{k}+\omega n}\right) \right\Vert _{\ell_{2}\left( S\right) }^{2}=\left\Vert T\left( q_{n}\right) \right\Vert _{\ell_{2}\left( S\right)
}^{2}\lesssim\left\Vert q_{n}\right\Vert _{\ell_{2}\left( S\right) }^{2}\,,
\label{L1 2.1}$$$$\left\Vert \sum_{k\in D}\sum_{n\in S}\frac{iq_{n}}{\lambda_{k}+\omega
n}\,e^{i\omega nt}\right\Vert _{L^{2}(0,T)}\lesssim\left\Vert q\right\Vert
_{L^{2}(0,T)}\,. \label{L2 2.1}$$
The estimates (\[L1 1.3\]), (\[L1 1.4\]) and (\[L2 2.1\]) yield$$\left\Vert Uq\right\Vert _{L^{2}(0,T)}\lesssim\left( T+1\right) \,\left\Vert
q\right\Vert _{L^{2}(0,T)} \label{L1 3}$$ This can be obviously extended to any finite time $\bar{T}\in\left(
0,+\infty\right) $ by fixing $N_{\bar{T}}\in\mathbb{N}$ such that: $8\pi
N_{\bar{T}}\geq\bar{T}$ and setting $T=8\pi N_{\bar{T}}$ in (\[aux\_condition\]).
Next, consider $q\in\mathcal{H}_{T}$. An integration by part of (\[U\]) gives$$Uq=\sum_{k\in D}\frac{1}{i\lambda_{k}}q(t)-\sum_{k\in D}\frac{1}{i\lambda_{k}}\int_{0}^{t}q^{\prime}(s)e^{-i\lambda_{k}\left( t-s\right) }ds\,.
\label{C2-1}$$ Due to the asymptotic behaviour of the coefficients $\frac{1}{i\lambda_{k}}\sim\frac{1}{k^{2}}$, this sum uniformly converges to a continuous function of $t\in\left[ 0,T\right] $, with $$\frac{d}{dt}Uq=\sum_{k\in D}\int_{0}^{t}\dot{q}(s)e^{-i\lambda_{k}\left(
t-s\right) }ds\,. \label{C1 1}$$ From (\[L1 3\]), we get$$\left\Vert \frac{d}{dt}Uq\right\Vert _{L^{2}(0,T)}\lesssim\left( T+1\right)
\left\Vert \dot{q}\right\Vert _{L^{2}(0,T)} \label{St_1_dUf}$$ and$$\left\Vert Uq\right\Vert _{H^{1}(0,T)}\lesssim\left( T+1\right) \,\left\Vert
q\right\Vert _{H^{1}(0,T)}\,. \label{St_2_dUf}$$
In the next Lemma we give an iteration scheme for the solution of charge-like equations.
\[Lemma 1.3\]Let $f,\varphi\in H^{1}(0,T)$, $\varphi$ real valued, and $\mathcal{G}_{0}^{\lambda}(\cdot,0)$ be the Green’ function defined in (\[Green 1\])-(\[h 1\]) with $\lambda\in\mathbb{C}\backslash\mathbb{R}$. The equation$$v=f-\varphi\left( v(0)\left. e^{it\Delta_{I}^{D}}\mathcal{G}_{0}^{\lambda
}(\cdot,0)\right\vert _{x=0}+\frac{i}{\pi}Uv\right) \label{charge_scheme}$$ has a unique solution in $H^{1}(0,T)$ allowing the estimate$$\left\Vert v\right\Vert _{H^{1}(0,T)}\lesssim\left\Vert f\right\Vert
_{H^{1}(0,T)}\left( 1+\left\Vert \varphi\right\Vert _{H^{1}(0,T)}+\mathcal{P}\left( \left\Vert \varphi\right\Vert _{H^{1}(0,T)}\right)
\right) \label{charge_scheme_est}$$ where $\mathcal{P}(\cdot)$ is a suitable positive polynomial.
Since Lemma \[Lemma 1.1\] provides with an estimate of $Uf$ in the $\mathcal{H}_{T}$-norm, we introduce the variable $w(t)=v(t)-v(0)$. In this setting, our problem writes as$$w=R-\varphi\frac{i}{\pi}Uw\,, \label{charge_resc}$$$$R(t)=f(t)-v(0)-\varphi(t)\left( v(0)\left. e^{it\Delta_{I}^{D}}\mathcal{G}_{0}^{\lambda}(\cdot,0)\right\vert _{x=0}+\frac{i}{\pi}U\left[
v(0)\right] (t)\right) \,. \label{R_0}$$ Next, the regularity of $R(t)$ is considered. Using the definition of $U$, and the explicit expansion$$e^{it\Delta_{I}^{D}}\mathcal{G}_{0}^{\lambda}(0,0)=\frac{1}{\pi}\sum_{k\in
D}\frac{e^{-i\lambda_{k}t}}{\lambda_{k}+\lambda}\,, \label{source_green}$$ $R(t)$ writes as$$\begin{aligned}
R(t) & =f(t)-v(0)-v(0)\varphi(t)\left( \left. e^{it\Delta_{I}^{D}}\mathcal{G}_{0}^{\lambda}(\cdot,0)\right\vert _{x=0}+\frac{1}{\pi}\sum_{k\in
D}\frac{1}{\lambda_{k}}\left( 1-e^{-i\lambda_{k}t}\right) \right)
\nonumber\\
& =f(t)-v(0)-v(0)\varphi(t)\frac{1}{\pi}\sum_{k\in D}\left( \frac{1}{\lambda_{k}}-e^{-i\lambda_{k}t}\frac{\lambda}{\lambda_{k}\left( \lambda
_{k}+\lambda\right) }\right) \,,\end{aligned}$$ where the last term at the r.h.s. is $H^{1}(\mathbb{R})$ for any $-\lambda\notin\sigma_{-\Delta_{I}^{D}}$. Recalling that $H^{1}(0,T)$ is an algebra w.r.t. the product, the above relation and $f,\varphi\in H^{1}(0,T)$ imply: $R\in H^{1}(0,T)$. Taking into account the initial condition $R(0)=0$ (following from the definition of the rescaled variable and the operator $U$ in (\[charge\_resc\])), we get: $R\in\mathcal{H}_{T}$ (see (\[H\])).
Next consider the term $\varphi\frac{i}{\pi}Uw$ in (\[charge\_resc\]). For $w\in\mathcal{H}_{T}$ and $\varphi\in H^{1}(0,T)$, the Lemma \[Lemma 1.1\] yields$$\left\Vert \varphi\frac{i}{\pi}Uw\right\Vert _{\mathcal{H}_{t}}\leq
C\frac{\left( 1+t\right) \,}{\pi}\left\Vert \varphi\right\Vert _{H^{1}(0,t)}\left\Vert w\right\Vert _{\mathcal{H}_{t}} \label{contraction_est}$$ for any $t\in\left[ 0,T\right] $ and for a suitable $C>0$. To construct a global solution, we notice that, for fixed $T,C>0$, $\varphi\in H^{1}(0,T)$, there exist $\delta>0$ and a finite set $\left\{ t_{j}\right\} _{j=1}^{N}$, $t_{0}=0$ and $t_{j}>t_{j-1}$, fulfilling the conditions$$\left[ 0,T\right] =\cup_{j=1}^{N}\left[ t_{j-1},t_{j}\right] \,,\qquad
t_{j}-t_{j-1}<\delta\,, \label{partition_con}$$$$C\left( 1+\delta\right) \,\left\Vert \varphi\right\Vert _{H^{1}(t_{j-1},t_{j})}<\pi\,c_{\delta}\qquad\text{for all }j\,,
\label{contraction_con}$$ with $c_{\delta}<1$, depending on $\delta$. According to (\[contraction\_est\]) and (\[contraction\_con\]) with $j=1$, $\varphi
\frac{i}{\pi}U$ is contraction map in $\mathcal{H}_{t_{1}}$ and the equation (\[charge\_resc\])-(\[R\_0\]) admits an unique solution, $w_{1}=1_{\left[
0,t_{1}\right] }w$, bounded by$$\left\Vert w_{1}\right\Vert _{\mathcal{H}_{t_{1}}}\lesssim\left\Vert
R\right\Vert _{\mathcal{H}_{t_{1}}}\lesssim\left\Vert f\right\Vert
_{H^{1}(0,t_{1})}+\left\vert v(0)\right\vert \left( 1+\left\Vert
\varphi\right\Vert _{H^{1}(0,t_{1})}\right)$$ Using (\[charge\_scheme\]), the initial value of $v$ if formally related to $f(0)$ by: $v(0)\left( 1+\varphi(0)\mathcal{G}_{0}^{\lambda}(0,0)\right)
=f(0)$; this allows to define $v(0)$ privided that $\left( 1+\varphi
(0)\mathcal{G}_{0}^{\lambda}(0,0)\right) $ is not null. According to the resolvent’s formula (\[Resolvent\]), the condition: $\left( 1+\varphi
(0)\mathcal{G}_{0}^{z}(0,0)\right) =0$, with $z\notin\sigma_{-\Delta_{I}^{D}}$, corresponds to the eigenvalue equation for the Hamiltonian $H_{\varphi}$. Since this is a selfadjoint model, the assumption $\lambda\in\mathbb{C}\backslash\mathbb{R}$ implies: $\left( 1+\varphi(0)\mathcal{G}_{0}^{\lambda
}(0,0)\right) \neq0$. It follows that: $v(0)=\left( 1+\varphi(0)\mathcal{G}_{0}^{\lambda}(0,0)\right) ^{-1}f(0)$ and $\left\vert v(0)\right\vert
\lesssim\left\vert f(0)\right\vert \,.$This leads to$$\left\Vert w_{1}\right\Vert _{\mathcal{H}_{t_{1}}}\lesssim\left\Vert
f\right\Vert _{H^{1}(0,t_{1})}\left( 1+\left\Vert \varphi\right\Vert
_{H^{1}(0,t_{1})}\right) \,.$$ The definition: $w_{1}=1_{\left[ 0,t_{1}\right] }\left( v-v(0)\right) $, provides with a similar estimate for the function $v$$$\left\Vert v\right\Vert _{H^{1}(0,t_{1})}\lesssim\left\Vert f\right\Vert
_{H^{1}(0,t_{1})}\left( 1+\left\Vert \varphi\right\Vert _{H^{1}(0,t_{1})}\right) \,. \label{charge_bound3}$$
This can be extended to larger times by the following iteration scheme:$$w_{j+1}(t)=v(t)-v(t_{j})\,,\qquad t\in\left[ t_{j},t_{j+1}\right] \,,\qquad
j=1,...,N-1 \label{charge_iteration}$$$$w_{j+1}(t^{\prime}+t_{j})=R_{j+1}(t^{\prime}+t_{j})-\varphi(t^{\prime}+t_{j})\frac{i}{\pi}\left[ Uw_{j+1}(\cdot+t_{j})\right] (t^{\prime
})\,,\qquad t^{\prime}\in\left[ 0,t_{j+1}-t_{j}\right]
\label{charge_iteration_eq}$$$$\begin{aligned}
R_{j+1}(t) & =f(t)-v(t_{j})-\varphi(t)\left( v(0)\left. e^{it\Delta
_{I}^{D}}\mathcal{G}_{0}^{\lambda}(\cdot,0)\right\vert _{x=0}\right)
\nonumber\\
& -\frac{i}{\pi}\varphi(t)\left( U\left[ v(t_{j})\right] +\sum_{k\in
D}\int_{0}^{t_{j}}\left( v(s)-v(t_{j})\right) e^{-i\lambda_{k}\left(
t-s\right) }ds\right) \label{source_j+1}$$ with the source term $R_{j+1}$ depending at each step on the past solution $1_{\left[ 0,t_{j}\right] }q$. From this definition, it follows: $R_{j+1}(t_{j})=0$, while an integration by part gives$$R_{j+1}(t)=f(t)-v(t_{j})-\frac{\varphi(t)}{\pi}\sum_{k\in D}\left(
\frac{v(t_{j})}{\lambda_{k}}+v(0)\left( \frac{-\lambda e^{-i\lambda_{k}t}}{\lambda_{k}\left( \lambda_{k}+\lambda\right) }\right) -\frac{1}{\lambda_{k}}\int_{0}^{t_{j}}\dot{v}(s)e^{-i\lambda_{k}\left( t-s\right)
}ds\right) \label{source_j+1_exp}$$ where (\[source\_green\]) have been taken into account. If $v\in
H^{1}(0,t_{j})$, all the contributions at the r.h.s. of (\[source\_j+1\_exp\]) are $H^{1}(0,T)$ and $R_{j+1}\in\mathcal{H}_{t_{j+1}-t_{j}}$. Then, the contractivity property of the operator, expressed by (\[charge\_iteration\_eq\]), implies the existence of an unique solution $w_{j+1}\in\mathcal{H}_{t_{j+1}-t_{j}}$ with the bound$$\left\Vert w_{j+1}\right\Vert _{\mathcal{H}_{t_{j+1}-t_{j}}}\lesssim\left\Vert
R_{j+1}\right\Vert _{\mathcal{H}_{t_{j+1}-t_{j}}}\lesssim\left\Vert
f\right\Vert _{H^{1}(t_{j},t_{j+1})}+\left\Vert v\right\Vert _{H^{1}(0,t_{j})}+\left\Vert \varphi\right\Vert _{H^{1}(0,T)}\left\Vert v\right\Vert
_{H^{1}(0,t_{j})}\,. \label{charge_bound1}$$ Starting from (\[charge\_bound3\]), an induction argument based on the itarated use of (\[charge\_bound1\]) leads to$$\left\Vert v\right\Vert _{H^{1}(0,T)}\lesssim\left\Vert f\right\Vert
_{H^{1}(0,T)}\left( 1+\left\Vert \varphi\right\Vert _{H^{1}(0,T)}+\mathcal{P}\left( \left\Vert \varphi\right\Vert _{H^{1}(0,T)}\right)
\right)$$ where $\mathcal{P}(\cdot)$ is a positive polynomial.
A first application of this Lemma gives an $H^{1}$-bound for the solutions to the charge equation.
\[Coroll\_1.3\_1\]Let $\alpha\in H^{1}(0,T;\mathbb{R})$ and $\psi_{0}\in
D(H_{\alpha(0)})$ and denote with $\phi_{0}^{\lambda}$ the regular part of $\psi_{0}$ defined according to the representation (\[dominio1\]). Then:$i)$ The equation (\[carica 1\]) admits an unique solution $q\in H^{1}(0,T)$ such that$$\left\Vert q\right\Vert _{H^{1}(0,T)}\leq C_{\lambda,\phi_{0}^{\lambda}}\left\Vert \alpha\right\Vert _{H^{1}(0,T)}\left( 1+\mathcal{P}\left(
\left\Vert \alpha\right\Vert _{H^{1}(0,T)}\right) \right)
\label{charge_bound}$$ where $\mathcal{P}\left( \cdot\right) $ is a positive polynomial, $C_{\lambda,\phi_{0}^{\lambda}}$ is a positive constant depending on $\lambda\in\mathbb{C}\backslash\mathbb{R}$ and $\phi_{0}^{\lambda}$.$ii)$ For a fixed $\psi_{0}\in D(H_{\alpha(0)})$, the map $\alpha\rightarrow q$ is locally Lipschitzian in $H^{1}(0,T)$.
$i)$ With the notation introduced in (\[U\]), the charge equation is$$q=-\alpha e^{it\Delta_{I}^{D}}\psi_{0}(0)-\alpha\frac{i}{\pi}Uq\,.
\label{charge_eq}$$ Using the decomposition: $\psi_{0}=\phi_{0}^{\lambda}+q(0)\mathcal{G}_{0}^{\lambda}(\cdot,0)$, $\phi_{0}^{\lambda}\in H^{2}\cap H_{0}^{1}(I)$, holding for $\lambda\in\mathbb{C}\backslash\mathbb{R}$ (we refer to Proposition \[Proposition 0\]), this equation rephrases as follows$$q=-\alpha e^{it\Delta_{I}^{D}}\phi_{0}^{\lambda}-\alpha\left( q(0)e^{it\Delta
_{I}^{D}}\left. \mathcal{G}_{0}^{\lambda}(\cdot,0)\right\vert _{x=0}+\frac
{i}{\pi}Uq\right) \,.$$ Consider $e^{it\Delta_{I}^{D}}\phi_{0}^{\lambda}(0)$. As a consequence of the Stone’s Theorem (e.g. in [@Simon])), the operator $e^{it\Delta_{I}^{D}}$ defines a continuous flow on $H^{2}\cap H_{0}^{1}(I)$ strongly differentiable in $\,L^{2}(I)$. For $\left. \phi_{0}^{\lambda}\in H^{2}\cap H_{0}^{1}(I)\right. $, we use the Fourier expansion $\phi_{0}^{\lambda}=\sum\limits_{k\in\mathbb{N}}a_{k}\psi_{k}$, where the coefficients $a_{k}=\left( \phi_{0}^{\lambda},\psi_{k}\right) _{L^{2}(I)}$ are characterized by: $a_{k}\in\ell_{2}(\mathbb{N})$, $\lambda_{k}a_{k}\in\ell
_{2}(\mathbb{N})$ (as it follows integrating by parts and exploiting the boundary conditions of $\psi_{k}$). The time propagator $e^{it\Delta_{I}^{D}}$ acts like: $e^{it\Delta_{I}^{D}}\phi_{0}^{\lambda}=\sum\limits_{k\in
\mathbb{N}}a_{k}e^{-i\lambda_{k}t}\psi_{k}$. In particular, we have$$e^{it\Delta_{I}^{D}}\phi_{0}^{\lambda}(0)=\sum\limits_{k\in D}a_{k}e^{-i\lambda_{k}t}\,. \label{termine noto}$$ According to: $\lambda_{k}a_{k}\in\ell_{2}$, it follows: $\partial
_{t}e^{it\Delta_{I}^{D}}\phi_{0}^{\lambda}(0)=-i\sum_{k\in D}a_{k}\lambda
_{k}e^{-i\lambda_{k}t}$, in the weak sense and: $e^{it\Delta_{I}^{D}}\phi
_{0}^{\lambda}(0)\in H^{1}(0,T)$. Then the first statement follows as an application of Lemma \[Lemma 1.3\] with $f=-\alpha e^{it\Delta_{I}^{D}}\phi_{0}^{\lambda}(0)$ and $\varphi=\alpha$.
$ii)$ Let $\alpha$, $\tilde{\alpha}\in H^{1}(0,T)$, $\psi_{0}$ and $\tilde{\psi}_{0}$ defined with the same regular part$$\psi_{0}=\phi_{0}^{\lambda}+q(0)\mathcal{G}_{0}^{\lambda}(\cdot,0)\,,\qquad
\tilde{\psi}_{0}=\phi_{0}^{\lambda}+\tilde{q}(0)\mathcal{G}_{0}^{\lambda
}(\cdot,0)\,, \label{charge_cont_psi}$$ and consider the corresponding solutions to (\[charge\_eq\]) $q$ and $\tilde{q}$. The initial values depends on $\alpha$ and $\tilde{\alpha}$ through the conditions$$q(0)=\frac{\phi_{0}^{\lambda}(0)}{1+\alpha(0)\mathcal{G}_{0}^{\lambda}(0,0)}\,,\qquad\tilde{q}(0)=\frac{\phi_{0}^{\lambda}(0)}{1+\tilde{\alpha
}(0)\mathcal{G}_{0}^{\lambda}(0,0)}\,, \label{charge_cont_init}$$ while the difference $u=q-\tilde{q}$ solves the equation$$u=S-\alpha\left( u(0)\,e^{it\Delta_{I}^{D}}\left. \mathcal{G}_{0}^{\lambda
}(\cdot,0)\right\vert _{x=0}+\frac{i}{\pi}Uu\right) \,,
\label{charge_diff_eq}$$$$S(t)=-\left( \alpha-\tilde{\alpha}\right) \left[ e^{it\Delta_{I}^{D}}\phi_{0}^{\lambda}(0)-\frac{\tilde{q}}{\tilde{\alpha}}\right] \,.
\label{charge_diff_source}$$ From the previous point, we have: $e^{it\Delta_{I}^{D}}\phi_{0}^{\lambda
}(0),q,\tilde{q}\in H^{1}(0,T)$. The same inclusion holds for the ratios $\frac{q}{\alpha},\frac{\tilde{q}}{\tilde{\alpha}}$, according to the equation’s structure. An application of Lemma \[Lemma 1.3\] with: $f=S$ and $\varphi=\alpha$, gives$$\left\Vert u\right\Vert _{H^{1}(0,T)}\lesssim\left\Vert \alpha-\tilde{\alpha
}\right\Vert _{H^{1}(0,T)} \label{charge_diff_est}$$
Solution of the evolution problem
---------------------------------
Next we consider the system (\[Schroedinger 3\])-(\[carica 1\]) with$$\alpha\in H^{1}(0,T),\quad\psi_{0}\in D(H_{\alpha(0)})\,.
\label{stato iniziale}$$ Due to the result of Lemma \[Lemma 1.3\], the equation (\[carica 1\]) has a unique solution $q\in H^{1}(0,T)$. This can be used to show that the corresponding evolution, defined by (\[Schroedinger 3\]), is $C\left(
\left[ 0,T\right] ,H_{0}^{1}(I)\right) \cap C^{1}(\left[ 0,T\right]
,L^{2}(I))$ and solves the Cauchy problem (\[Schroedinger\]). Fix $\lambda\in\mathbb{C}\backslash\sigma_{\Delta_{I}^{D}}$; from the definition of the operator’s domain (see (\[dominio1\])), any $\psi\in D(H_{\alpha
(t)})$ can be represented as the sum of a ’regular’ part plus a Green’s function$$\psi(\cdot,t)=\phi^{\lambda}(\cdot,t)+q(t)\mathcal{G}_{0}^{\lambda}
\label{state}$$ with $q$ fulfilling (\[carica 1\]). In the mild sense, the evolution of the regular part is$$\phi^{\lambda}(x,t)=e^{it\Delta_{I}^{D}}\psi_{0}(x)+\frac{i}{\sqrt{\pi}}\sum\limits_{k\in D}\int_{0}^{t}q(s)e^{-i\lambda_{k}\left( t-s\right)
}\,ds\,\psi_{k}(x)-q(t)\mathcal{G}_{0}^{\lambda}(x)\,. \label{phi 1}$$ Next we consider the operator$$F(q,t)=\frac{i}{\sqrt{\pi}}\sum\limits_{k\in D}\left( \int_{0}^{t}q(s)e^{-i\lambda_{k}\left( t-s\right) }\,ds\right) \,\psi_{k}\,. \label{F}$$
\[Lemma 1.2\]For $q\in H^{1}(0,T)$, the map $F(q,t)$ is bounded in $C\left( \left[ 0,T\right] ,\,H_{0}^{1}(I)\right) $.
Let $q_{t}=1_{\left( 0,t\right) }q$ and $N_{T}$ be the smallest integer such that: $8\pi N_{T}\geq T$. By definition (see (\[aux\_condition\])), $\left\{
e^{-i\lambda_{k}s}\right\} _{k\in\mathbb{N}}$ forms a subset of the standard basis in $L^{2}(0,8\pi N_{T})$. The Fourier coefficients of $q_{t}$ along these frequencies are$$\int_{0}^{8\pi N_{T}}q_{t}(s)e^{i\lambda_{k}s}\,ds=\int_{0}^{t}q(s)e^{i\lambda
_{k}s}\,ds=c_{k}(t) \label{L1.2 0}$$ and the inequality$$\sum_{k\in\mathbb{N}}\left\vert c_{k}(t)\right\vert ^{2}\leq\left\Vert
q_{t}\right\Vert _{L^{2}(0,8\pi N_{T})}^{2}=\int_{0}^{t}\left\vert
q(s)\right\vert ^{2}\,ds \label{L1.2 1}$$ holds. It follows$$\left\Vert F(q,t)\right\Vert _{L^{2}(I)}^{2}=\frac{1}{\pi}\sum_{k\in
D}\left\vert c_{k}(t)\right\vert ^{2}\leq\frac{1}{\pi}\int_{0}^{t}\left\vert
q(s)\right\vert ^{2}\,ds \label{L1.2 2}$$ and$$\sup_{t\in\left[ 0,T\right] }\left\Vert F(q,t)\right\Vert _{L^{2}(I)}^{2}\leq\frac{1}{\pi}\int_{0}^{T}\left\vert q(s)\right\vert ^{2}\,ds=\frac
{1}{\pi}\left\Vert q\right\Vert _{L^{2}(0,T)}^{2}\,. \label{L1.2 3}$$ For $q\in H^{1}(0,T)$, an integration by part gives$$F(q,t)=\frac{1}{\sqrt{\pi}}\sum\limits_{k\in D}\frac{1}{\lambda_{k}}\left(
q(t)-q(0)e^{-i\lambda_{k}t}-\int_{0}^{t}q^{\prime}(s)e^{-i\lambda_{k}\left(
t-s\right) }\,ds\,\right) \psi_{k}\,, \label{C1.2 0}$$ while, using the relation: $\frac{d}{dx}\psi_{k}=\lambda_{k}^{\frac{1}{2}}\psi_{k}$, the weak derivative $\frac{d}{dx}F(q,t)$ can be written as$$\frac{d}{dx}F(q,t)=\frac{1}{\sqrt{\pi}}\sum\limits_{k\in D}\frac{1}{\lambda_{k}^{\frac{1}{2}}}\left( q(t)-q(0)e^{-i\lambda_{k}t}-\int_{0}^{t}q^{\prime}(s)e^{-i\lambda_{k}\left( t-s\right) }\,ds\,\right) \psi
_{k}\,. \label{C1.2 1}$$ For the first term at the r.h.s. of (\[C1.2 1\]), the asymptotic behaviour of the coefficients $\lambda_{k}^{\frac{1}{2}}=\frac{k}{2}$ and the assumption: $q\in H^{1}(0,T)$ implies: $\left. \sum_{k\in D}\lambda
_{k}^{-\frac{1}{2}}\left( q(t)-q(0)e^{-i\lambda_{k}t}\right) \psi_{k}\in
C(\left( \left[ 0,T\right] ,L^{2}(I)\right) )\right. $. For the remaining term, we proceed as before: let $q_{t}^{\prime}=1_{\left( 0,t\right)
}q^{\prime}$ and denote with $c_{k}^{\prime}$ the Fourier coefficients of $q_{t}^{\prime}$ along the freqencies $\left\{ e^{-i\lambda_{k}s}\right\}
_{k\in\mathbb{N}}$$$c_{k}^{\prime}=\int_{0}^{t}q^{\prime}(s)e^{-i\lambda_{k}\left( t-s\right)
}\,ds\,.$$ It follows$$\left\Vert \sum\limits_{k\in D}\frac{1}{\lambda_{k}^{\frac{1}{2}}}\int_{0}^{t}q^{\prime}(s)e^{-i\lambda_{k}\left( t-s\right) }\,ds\,\psi
_{k}\right\Vert _{L^{2}(I)}^{2}=\sum_{k\in D}\frac{\left\vert c_{k}^{\prime
}\right\vert ^{2}}{\lambda_{k}}\leq\int_{0}^{t}\left\vert q^{\prime
}(s)\right\vert ^{2}\,ds \label{C1.2 2}$$ and$$\sup_{t\in\left[ 0,T\right] }\left\Vert \sum\limits_{k\in D}\frac{1}{\lambda_{k}^{\frac{1}{2}}}\int_{0}^{t}q^{\prime}(s)e^{-i\lambda_{k}\left(
t-s\right) }\,ds\,\psi_{k}\right\Vert _{L^{2}(I)}^{2}\leq\left\Vert
q^{\prime}\right\Vert _{L^{2}(0,T)}^{2}\,. \label{C1.2 3}$$ The above estimates lead us to$$\sup_{t\in\left[ 0,T\right] }\left\Vert F(q,t)\right\Vert _{H_{0}^{1}(I)}\lesssim\left\Vert q\right\Vert _{H^{1}(0,T)}\,. \label{bound1.0}$$
\[Proposition 3\]Let $\alpha\in H^{1}(0,T)$ and $\psi_{0}\in
D(H_{\alpha(0)})$. The system (\[Schroedinger 3\])-(\[carica 1\]), has an unique solution $\psi_{t}\in C\left( \left[ 0,T\right] ,\,H_{0}^{1}(I)\right) \cap C^{1}(\left[ 0,T\right] ,\,L^{2}(I))$ such that: $\psi_{t}\in D(H_{\alpha(t)})$ and $i\partial_{t}\psi_{t}=H_{\alpha(t)}\psi_{t}$ at each $t$.
The proof articulates in two steps. We first consider the conditions $\psi
_{t}\in C\left( \left[ 0,T\right] ,\,H_{0}^{1}(I)\right) $ and $\psi
_{t}\in C^{1}(\left[ 0,T\right] ,\,L^{2}(I))$. Then we discuss the equivalence of the system (\[Schroedinger 3\])-(\[carica 1\]) with the initial problem.
$1)$ Using the notation introduced in (\[F\]), the solution $\psi_{t}$ of (\[Schroedinger 3\])-(\[carica 1\]) writes as$$\psi_{t}=e^{it\Delta_{I}^{D}}\psi_{0}+F(q,t)$$ with $q\in H^{1}(0,T)$ solving the charge equation (\[carica 1\]), and $\psi_{0}\in D(H_{\alpha(0)})$. Due to the domain’s structure, $\psi_{0}\in
H_{0}^{1}(I)$: in this case, the term $e^{it\Delta_{I}^{D}}\psi_{0}$ defines a $C(\left[ 0,T\right] ,\,H_{0}^{1}(I))$ map (as a consequence of the Stone’s theorem, e.g. in [@Simon]). Moreover, following the result of Lemma \[Lemma 1.2\], one has: $F(q,t)\in C\left( \left[ 0,T\right] ,\,H_{0}^{1}(I)\right) $. In order to study the $C^{1}(\left[ 0,T\right]
,\,L^{2}(I))$ regularity of $\psi_{t}$, one can use the decomposition stated in Proposition \[Proposition 0\]: $\psi_{0}=\phi_{0}^{\lambda}+q(0)\mathcal{G}_{0}^{\lambda}(\cdot,0)$, with $\phi_{0}^{\lambda}\in
H^{2}\cap H_{0}^{1}(I)$ and $\lambda\in\mathbb{C}\backslash\sigma_{\Delta
_{I}^{D}}$. For $w(t)=q(t)-q(0)$, $\psi_{t}$ writes as$$\psi_{t}=e^{it\Delta_{I}^{D}}\phi_{0}^{\lambda}+F(w,t)+Z(t)\,, \label{ref1}$$$$Z(t)=q(0)\left[ e^{it\Delta_{I}^{D}}\mathcal{G}_{0}^{\lambda}(\cdot
,0)+F(1,t)\right] \,.$$ Due to the regularity of $\phi_{0}^{\lambda}$, one has: $e^{it\Delta_{I}^{D}}\phi_{0}^{\lambda}\in C^{1}(\left[ 0,T\right] ,\,L^{2}(I))$. Next consider $F(w,t)$: an integration by part gives$$F(w,t)=\frac{1}{\sqrt{\pi}}\sum\limits_{k\in D}\frac{1}{\lambda_{k}}\left(
w(t)-\int_{0}^{t}w^{\prime}(s)e^{-i\lambda_{k}\left( t-s\right)
}\,ds\,\right) \psi_{k}$$ and$$\frac{d}{dt}F(w,t)=\frac{i}{\sqrt{\pi}}\sum_{k\in D}\int_{0}^{t}w^{\prime
}(s)e^{-i\lambda_{k}\left( t-s\right) }ds\,\psi_{k}=F(w^{\prime},t)\,.
\label{ref2}$$ Using the estimate (\[L1.2 3\]) in Lemma \[Lemma 1.2\], this relation gives: $F(w,t)\in C^{1}(\left[ 0,T\right] ,\,L^{2}(I))$. Concerning the term $Z(t)$, the formula$$e^{it\Delta_{I}^{D}}\mathcal{G}_{0}^{\lambda}(\cdot,0)=\frac{1}{\sqrt{\pi}}\sum_{k\in D}\frac{e^{-i\lambda_{k}t}}{\lambda_{k}+\lambda}\psi_{k}$$ and an explicit computation lead to$$Z(t)=\frac{q(0)}{\sqrt{\pi}}\sum_{k\in D}\frac{1}{\lambda_{k}}\psi_{k}-\frac{q(0)}{\sqrt{\pi}}\sum_{k\in D}\frac{\lambda e^{-i\lambda_{k}t}}{\lambda_{k}\left( \lambda_{k}+\lambda\right) }\psi_{k}\,. \label{Zeta}$$ It follows: $Z(t)\in C^{1}(\left[ 0,T\right] ,\,H_{0}^{1}(I))$ and: $\psi_{t}\in C\left( \left[ 0,T\right] ,\,H_{0}^{1}(I)\right) \cap
C^{1}(\left[ 0,T\right] ,\,L^{2}(I))$.
$2)$ The regular part of $\psi_{t}$ writes as$$\phi_{t}^{\lambda}=e^{it\Delta_{I}^{D}}\psi_{0}+F(q,t)-q(t)\mathcal{G}_{0}^{\lambda}(\cdot,0) \label{phi 0}$$ Using once more the decomposition: $\psi_{0}=\phi_{0}^{\lambda}+q(0)\mathcal{G}_{0}^{\lambda}(\cdot,0)$, $\phi_{0}^{\lambda}\in H^{2}\cap
H_{0}^{1}(I)$, $\lambda\in\mathbb{C}\backslash\sigma_{\Delta_{I}^{D}}$ and $w(t)=q(t)-q(0)$, we get$$\phi_{t}^{\lambda}=e^{it\Delta_{I}^{D}}\phi_{0}^{\lambda}+F(w,t)-w(t)\mathcal{G}_{0}^{\lambda}(\cdot,0)+q(0)Q(t) \label{phi 0.1}$$$$Q(t)=\left[ \left( e^{it\Delta_{I}^{D}}-1\right) \mathcal{G}_{0}^{\lambda
}(\cdot,0)+F(1,t)\right] \label{Q(t)}$$ The regularity of $\phi_{0}^{\lambda}$, yields: $e^{it\Delta_{I}^{D}}\phi
_{0}^{\lambda}\in C(\left[ 0,T\right] ,\,H^{2}\cap H_{0}^{1}(I))$, while, according to the explicit form of $e^{it\Delta_{I}^{D}}\mathcal{G}_{0}^{\lambda}(\cdot,0)$ and $F(1,t)$, $Q(t)$ is$$Q(t)=\frac{1}{\sqrt{\pi}}\sum_{k\in D}\left( 1-e^{-i\lambda_{k}t}\right)
\frac{\lambda}{\lambda_{k}\left( \lambda_{k}-\lambda\right) }\psi_{k}
\label{Qu}$$ This yields $Q(t)\in C(\left[ 0,T\right] ,\,H^{2}\cap H_{0}^{1}(I))$. For the remaining term, $F(w,t)-w(t)\mathcal{G}_{0}^{\lambda}(\cdot,0)$, it has already been noticed that $F(w,t)$ is continuously embedded into $H_{0}^{1}(I)$; the same holds for $w(t)\mathcal{G}_{0}^{\lambda}(\cdot,0)$, since $\mathcal{G}_{0}^{\lambda}\left( \cdot,0\right) \in H_{0}^{1}(I)$ and $w$ is continuous. Moreover, a direct computation of the Fourier coefficients of $\frac{d^{2}}{dx^{2}}\left[ F(q,t)-q(t)\mathcal{G}_{0}^{\lambda}\left(
\cdot,0\right) \right] $ w.r.t. the system $\left\{ \psi_{k}\right\}
_{k\in\mathbb{N}}$ gives$$b_{k}=\left\{
\begin{array}
[c]{c}\frac{1}{\sqrt{\pi}}\int_{0}^{t}w^{\prime}(s)e^{-i\lambda_{k}\left(
t-s\right) }\,ds\,-\lambda w(t)\left( \mathcal{G}_{0}^{\lambda}\left(
\cdot,0\right) ,\,\psi_{k}\right) _{L^{2}(I)}\qquad k\ odd\\
0\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad otherwise
\end{array}
\right. \label{Fourier coefficients 4}$$ from which it follows$$-\frac{d^{2}}{dx^{2}}\left[ F(w,t)-w(t)\mathcal{G}_{0}^{\lambda}\left(
\cdot,0\right) \right] =iF(w^{\prime},t)+\lambda w(t)\mathcal{G}_{0}^{\lambda}\left( \cdot,0\right) \,. \label{phi 4}$$ Then, the estimate (\[L1.2 3\]) in Lemma \[Lemma 1.2\] yield: $\frac
{d^{2}}{dx^{2}}\left[ F(q,t)-q(t)\mathcal{G}_{0}^{\lambda}\left(
\cdot,0\right) \right] \in C\left( \left[ 0,T\right] ,\,L_{2}(I)\right)
$ and $\left[ F(w,t)-w(t)\mathcal{G}_{0}^{\lambda}\left( \cdot,0\right)
\right] \in C\left( \left[ 0,T\right] ,\,H^{2}\cap H_{0}^{1}(I)\right) $, which implies$$\phi_{t}^{\lambda}\in C\left( \left[ 0,T\right] ,\,H^{2}\cap H_{0}^{1}(I)\right) \,;$$ this, together with the boundary condition $-q(t)=\alpha(t)\psi_{t}(0)$ (arising from the equation (\[carica 1\])), assures that $\psi_{t}\in
D(H_{\alpha(t)})$ at each $t$.
Next we discuss the last point of the Proposition. The continuity in time of the map $\psi_{t}$ and (\[Schroedinger 3\]) allow to write: $\psi
(\cdot,t=0)=\psi_{0}$. According to (\[ref1\])-(\[Zeta\]), the derivative $i\partial_{t}\psi_{t}$ is$$i\frac{d}{dt}\psi_{t}=-\frac{d^{2}}{dx^{2}}e^{it\Delta_{I}^{D}}\phi
_{0}^{\lambda}+iF(w^{\prime},t)+iZ^{\prime}(t) \label{P2 1}$$$$iZ^{\prime}(t)=-\frac{q(0)}{\sqrt{\pi}}\sum_{k\in D}\frac{\lambda
e^{-i\lambda_{k}t}}{\lambda_{k}+\lambda}\psi_{k}$$ On the other hand, from definition (\[operatore\]) one has$$H_{\alpha(t)}\psi_{t}=-\frac{d^{2}}{dx^{2}}\phi_{t}^{\lambda}-\lambda
q(t)\mathcal{G}_{0}^{\lambda}\left( \cdot,0\right) \,. \label{P2 2}$$ Making use of (\[phi 0.1\])-(\[Qu\]) and (\[phi 4\]), this writes as$$H_{\alpha(t)}\psi_{t}=-\frac{d^{2}}{dx^{2}}e^{it\Delta_{I}^{D}}\phi
_{0}^{\lambda}+iF(w^{\prime},t)+\lambda q(0)\mathcal{G}_{0}^{\lambda}\left(
\cdot,0\right) -q(0)\frac{d^{2}}{dx^{2}}Q(t)$$ with$$-q(0)\frac{d^{2}}{dx^{2}}Q(t)=iF(w^{\prime},t)+\lambda q(0)\mathcal{G}_{0}^{\lambda}\left( \cdot,0\right) \,.$$ Combining the above relations, it follows$$i\frac{d}{dt}\psi_{t}-H_{\alpha(t)}\psi(x,t)=0\quad\text{in }L^{2}(I).$$
\[Sec\_control\]Stability and local controllability
===================================================
The evolution problem (\[Schroedinger\]) defines a map $\Gamma(\alpha
,\psi_{0})$ associating to the coupling parameter, $\alpha$, and the initial state, $\psi_{0}$, the solution at time $T$. According to the notation introduced in (\[U\]) and (\[F\]), it writes as$$\Gamma(\alpha)=e^{iT\Delta}\psi_{0}+F\left( V(\alpha),T\right) \,,
\label{G-def1}$$$$V(\alpha)=q:q(t)=-\alpha(t)\left( e^{it\Delta_{I}^{D}}\psi_{0}(0)+\frac
{i}{\pi}Uq(t)\,\right) \,. \label{G-def2}$$ The local controllability in time $T$ of the dynamics described by (\[Schroedinger\]), when $\alpha$ is considered as a control parameter, is connected with the following question: for $\psi_{0}\in D(H_{\alpha(0)})$, does $\alpha$ exists such that $\Gamma(\alpha,\psi_{0})=\psi_{T}$ for arbitrary $\psi_{T}$ in a neighbourhood of $\psi_{0}$? In what follows, we fix the initial state $\psi_{0}$ and discuss the regularity of the map $\alpha\rightarrow\Gamma(\alpha,\psi_{0})$. Hence, this control problem will be considered in the local setting where $\psi_{0}$ concides with a steady state of the Dirichlet Laplacian and $\alpha$ is small in $H_{0}^{1}(0,T)$. To simplify the notation, we use $\Gamma(\alpha)$ instead of $\Gamma(\alpha
,\psi_{0})$.
Regularity of $\Gamma$
----------------------
Recall a standard result in the theory of differential calculus in Banach spaces. Let $X$ and $Y$ denote two Banach spaces, $W$ an open subset of $X$, $d_{\alpha}F$ and $d_{\alpha}^{G}F$ respectively the Fréchet and the Gâteaux derivatives of $F:W\rightarrow Y$ evaluated in the point $\alpha$. A functional $F:W\rightarrow Y$ is of class $C^{1}$ if the map$$F^{\prime}:W\rightarrow\mathcal{L}(X,Y),\ F^{\prime}(\alpha)=d_{\alpha}F
\label{C1_def}$$ is continuous.
\[(Theorem 1.9 in [@Ambrosetti])\]\[prodi\]Suppose $F:W\rightarrow Y$ is Gâteaux-differentiable in $U$. If the map:$$F_{G}^{\prime}:W\rightarrow\mathcal{L}(X,Y),\ F_{G}^{\prime}(\alpha
)=d_{\alpha}^{G}F$$ is continuous at $\alpha^{\ast}\in W$, then $F$ is Fréchet-differentiable at $\alpha^{\ast}$ and its Fréchet derivative evaluated in $\alpha^{\ast}$ results$$d_{\alpha^{\ast}}F=d_{\alpha^{\ast}}^{G}F$$
\[Lemma 4\]The map $V$ defined by (\[G-def2\]) is $C^{1}(H^{1}(0,T);\,H^{1}(0,T))$.
The continuity of $V$ has been discussed in Lemma \[Lemma 1.3\]. Next we consider the differential map $V^{\prime}:H^{1}(0,T)\rightarrow\mathcal{L}(H^{1}(0,T),\,H^{1}(0,T))$. Due to Theorem \[prodi\], it is enough to prove that $\alpha\rightarrow d_{\alpha}^{G}V$ is continuous w.r.t. the operator norm in $\mathcal{L}(H^{1}(0,T),\,H^{1}(0,T))$. The action of $d_{\alpha}^{G}V$ on $u\in H^{1}(0,T)$ is$$d_{\alpha}^{G}V(u)=q\,, \label{dV_def1}$$$$q=-u\,\frac{V(\alpha)}{\alpha}-\alpha\left( u(0)e^{it\Delta_{I}^{D}}\left.
\mathcal{G}_{0}^{\lambda}(\cdot,0)\right\vert _{x=0}+\frac{i}{\pi}Uq\right)
\,. \label{dV_def2}$$ Then, the $H^{1}$-bound$$\left\Vert q\right\Vert _{H^{1}(0,T)}\lesssim\left\Vert u\right\Vert
_{H^{1}(0,T)} \label{bound1.1}$$ and the continuous dependence from $\alpha$ follows from Lemma \[Lemma 1.3\] and proceeding as in Corollary \[Coroll\_1.3\_1\].
\[Proposition 4\]For $\psi_{0}\in D(H_{\alpha(0)})$, the map $\alpha\rightarrow\Gamma(\alpha)$ is of class $C^{1}(H^{1}(0,T),\,H_{0}^{1}(I))$. Moreover, the regular part of the solution at time $T$, $\Gamma(\alpha)-V(\alpha)(T)\mathcal{G}_{0}^{\lambda}$, considered as a function of $\alpha$, is of class $C^{1}(H^{1}(0,T),\,H^{2}\cap H_{0}^{1}(I))$.
The continuity of $\Gamma(\alpha)$ follows directly from the continuity of the map $\alpha\rightarrow V(\alpha)$ and the estimate (\[bound1.0\]) in Lemma \[Lemma 1.2\]. Next consider the differential map $\Gamma^{\prime}:H^{1}(0,T)\rightarrow\mathcal{L}(H^{1}(0,T),\,H_{0}^{1}(I))$. Our aim is to prove that $\Gamma^{\prime}$ is continuous in the operator norm. Due to the Theorem \[prodi\], it is enough to prove that $\alpha\rightarrow d_{\alpha
}^{G}\Gamma$ is continuous. The action of $d_{\alpha}^{G}\Gamma$ on $u\in
H^{1}(0,T)$ is: $d_{\alpha}^{G}\Gamma(u)=F(d_{\alpha}^{G}V(u),T)$; for $\alpha,\bar{\alpha}\in H^{1}(0,T)$ the difference: $d_{\alpha}^{G}\Gamma-d_{\bar{\alpha}}^{G}\Gamma$ is expressed by$$d_{\alpha}^{G}\Gamma-d_{\bar{\alpha}}^{G}\Gamma=F(d_{\alpha}^{G}V(u)-d_{\bar{\alpha}}^{G}V(u),T)$$ Using (\[bound1.0\]), one has$$\begin{gathered}
\left\vert \left\vert \left\vert d_{\alpha}^{G}\Gamma-d_{\bar{\alpha}}^{G}\Gamma\right\vert \right\vert \right\vert =\sup_{\substack{u\in
H^{1}(0,T)\\\left\Vert u\right\Vert _{H^{1}(0,T)}=1}}\left\Vert F(d_{\alpha
}^{G}V(u)-d_{\bar{\alpha}}^{G}V(u),T)\right\Vert _{H_{0}^{1}(I)}\lesssim\\
\lesssim\sup_{\substack{u\in H^{1}(0,T)\\\left\Vert u\right\Vert _{H^{1}(0,T)}=1}}\left\Vert d_{\alpha}^{G}V(u)-d_{\bar{\alpha}}^{G}V(u)\right\Vert
_{H^{1}(0,T)}=\left\vert \left\vert \left\vert d_{\alpha}^{G}V-d_{\bar{\alpha
}}^{G}V\right\vert \right\vert \right\vert\end{gathered}$$ then, the continuity of $\alpha\rightarrow d_{\alpha}^{G}\Gamma$ follows from the continuity of $\alpha\rightarrow d_{\alpha}^{G}V$ proved in Lemma \[Lemma 4\].
Let introduce the map$$\mu(\alpha)=\Gamma(\alpha)-V(\alpha)(T)\mathcal{G}_{0}^{\lambda}(\cdot,0)
\label{mu_alpha}$$ representing the regular part of the state at time $T$. Combining the result of Lemma \[Lemma 4\] and the first part of this proof, one has: $\mu
(\alpha)\in C^{1}(H^{1}(0,T),\,H_{0}^{1}(I))$. To achive our result it remains to prove that: $-\partial_{x}^{2}\mu(\alpha)$ belongs to $C^{1}(H^{1}(0,T),\,L^{2}(I))$. According to (\[phi 0.1\])-(\[Q(t)\]), this can be written as$$-\frac{d^{2}}{dx^{2}}\left[ e^{iT\Delta_{I}^{D}}\phi_{0}^{\lambda
}+F(w,T)-w(T)\mathcal{G}_{0}^{\lambda}(\cdot,0)+\left[ V(\alpha)\right]
(0)Q(T)\right]$$ where $w(t)=\left[ V(\alpha)\right] (t)-\left[ V(\alpha)\right] (0)$, while the regularity of the terms: $-\frac{d^{2}}{dx^{2}}e^{iT\Delta_{I}^{D}}\phi_{0}^{\lambda}$ and $Q(T)$ is discussed in Proposition \[Proposition 3\]. For $\alpha,\tilde{\alpha}\in H^{1}(0,T)$, the difference $\partial_{x}^{2}\left( \mu(\alpha)-\mu(\tilde{\alpha})\right) $ is$$\begin{aligned}
\mu(\alpha)-\mu(\tilde{\alpha}) & =-\frac{d^{2}}{dx^{2}}\left[ \left(
F\left( w,T\right) -F\left( \tilde{w},T\right) \right) -\left(
w(T)-\tilde{w}(T)\right) \mathcal{G}_{0}^{\lambda}\right] +\\
& -\left( \left[ V(\alpha)\right] (0)-\left[ V(\tilde{\alpha})\right]
(0)\right) \frac{d^{2}}{dx^{2}}Q(T)\,.\end{aligned}$$ Using (\[phi 4\]), one has$$\begin{aligned}
\mu(\alpha)-\mu(\tilde{\alpha}) & =i\left( F\left( w^{\prime},T\right)
-F\left( \tilde{w}^{\prime},T\right) \right) +\lambda\left( w(T)-\tilde
{w}(T)\right) \mathcal{G}_{0}^{\lambda}+\\
& -\left( \left[ V(\alpha)\right] (0)-\left[ V(\tilde{\alpha})\right]
(0)\right) \frac{d^{2}}{dx^{2}}Q(T)\end{aligned}$$ $w^{\prime}$ and $\tilde{w}^{\prime}$ respectively denoting the time derivatives of the functions $V(\alpha)$ and $V(\tilde{\alpha})$. The continuity of $\mu(\alpha)$, then, follows from the continuity of $F(\cdot,T)$ (expressed by (\[bound1.0\])) and of the map $V(\cdot)$. The Gâteaux derivative $d_{\alpha}^{G}\mu$ acts on $u\in H^{1}(0,T)$ as$$d_{\alpha}^{G}\mu(u)=-\frac{d^{2}}{dx^{2}}\left[ F(d_{\alpha}^{G}w(u),T)-\left[ d_{\alpha}^{G}w(u)\right] (T)\mathcal{G}_{0}^{\lambda}(\cdot,0)+\left[ d_{\alpha}^{G}V(\alpha)\right] (0)Q(T)\right] \,.$$ Proceeding as before, the difference $d_{\alpha}^{G}\mu(u)-d_{\tilde{\alpha}}^{G}\mu(u)$ is$$\begin{aligned}
d_{\alpha}^{G}\mu(u)-d_{\tilde{\alpha}}^{G}\mu(u) & =i\left( F\left(
\partial_{t}d_{\alpha}^{G}w(u)-\partial_{t}d_{\tilde{\alpha}}^{G}w(u),T\right) \right) +\lambda\left( \left[ d_{\alpha}^{G}w(u)\right]
(T)-\left[ d_{\tilde{\alpha}}^{G}w(u)\right] (T)\right) \mathcal{G}_{0}^{\lambda}+\\
& -\left( \left[ d_{\alpha}^{G}w(u)\right] (0)-\left[ d_{\tilde{\alpha}}^{G}w(u)\right] (0)\right) \frac{d^{2}}{dx^{2}}Q(T)\,.\end{aligned}$$ Using (\[bound1.0\]) and recalling, from Lemma \[Lemma 4\], that $\alpha\rightarrow d_{\alpha}^{G}V$ is continuous in the $\mathcal{L}\left(
H^{1}(0,T),\,H^{1}(0,T)\right) $ operator norm, this relation leads to continuity of $\alpha\rightarrow d_{\alpha}^{G}\mu$ in the $\mathcal{L}\left(
H^{1}(0,T),\,L^{2}(I)\right) $-norm.
\[Sec\_control\_lin\]The linearized map
---------------------------------------
Next we discuss the surjectivity of the linearized map $d_{\alpha}^{G}\Gamma$ for $\alpha\in H_{0}^{1}(0,T)$. Since the transformations $V$ and $d_{\alpha
}^{G}V$ preserves Dirichlet boundary conditions, in this framework the result of Lemma \[Lemma 4\] raphrases as:
\[Lemma 4.1\]The map $V$ defined by (\[G-def2\]) is $C^{1}(H_{0}^{1}(0,T);\,H_{0}^{1}(0,T))$.
This also implies that the state at the initial and finial times possesses only regular part. Thus, Proposition \[Proposition 4\] become:
\[Proposition 4.1\]For $\psi_{0}\in H^{2}\cap H_{0}^{1}(I)$, the map $\alpha\rightarrow\Gamma(\alpha)$ is of class $C^{1}(H_{0}^{1}(0,T),\,H^{2}\cap H_{0}^{1}(I))$.
It is whorthwhile to notice that, when $\psi_{0}$ is given by a linear superposition of eigenstates of odd kind (i.e. $\psi_{k}=\sin\frac{k}{2}x,\ k\ $even), the source term in (\[G-def2\]), $e^{it\Delta}\psi_{0}(0)$, is null at each $t$, and the charge, $V(\alpha)$ in the above notation, is identically zero. In these conditions, the particle does not ’feel’ the interaction and the evolution is simply determined by the free propagator $e^{it\Delta_{I}^{D}}$. This implies: $\Gamma(\alpha)=e^{iT\Delta_{I}^{D}}\psi_{0}$. In order to avoid this situation, we assume the initial state $\psi_{0}$ to be fixed in the subspace of even functions$$\mathcal{W}=\left\{ \varphi\in H^{2}\cap H_{0}^{1}(I)\,\left\vert
\,\varphi=\sum_{k\in D}c_{k}\psi_{k}\right. \right\}
\label{stato iniziale 1}$$ This choice also implies that $\Gamma$ takes values in $\mathcal{W}$, as can be cheked by its explicit formula. Due to Proposition \[Proposition 4.1\], one has: $\Gamma\in C^{1}(H_{0}^{1}(0,T),\,\mathcal{W})$.
The linearized map $d_{\alpha}^{G}\Gamma$ is defined according to$$d_{\alpha}^{G}\Gamma(u)=\frac{i}{\sqrt{\pi}}\sum\limits_{k\in D}\left(
\int_{0}^{T}q(s)e^{-i\lambda_{k}\left( T-s\right) }\,ds\right) \,\psi_{k}
\label{Lin1}$$$$-q(t)=\frac{i}{\pi}\alpha(t)Uq(t)+u(t)\left( e^{it\Delta}\psi_{0}(0)+\frac
{i}{\pi}U\left[ V(\alpha)\right] (t)\right) \label{Lin2}$$ Let $\psi_{T}\in\mathcal{W}$ be a target state for the linear transformation (\[Lin1\])-(\[Lin2\]). We address the following question: does $u\in
H_{0}^{1}(0,T)$ exists such that $d_{\alpha}^{G}\Gamma(u)=\psi_{T}$ ?Consider at first the equation$$\psi_{T}=\frac{i}{\sqrt{\pi}}\sum\limits_{k\in D}\left( \int_{0}^{T}\rho(s)e^{-i\lambda_{k}\left( T-s\right) }\,ds\right) \,\psi_{k}
\label{Lin1.1}$$ w.r.t. the variable $\rho$. Denoting with $c_{k}$ the Fourier coefficients of $\psi_{T}$ in the basis $\left\{ \psi_{k}\right\} _{k\in D}$, the above equation is equivalent to$$c_{k}=\frac{i}{\sqrt{\pi}}\int_{0}^{T}\rho(s)e^{-i\lambda_{k}\left(
T-s\right) }\,ds,\qquad k\in D \label{Lin1.2}$$ This is a trigonometric moment problem; the existence of solutions is established by K. Beauchard and C. Laurent in [@BeLa] for arbitrary values of $T>0$ according to the following result.
\[Th\_K\]Let $T>0$ and $\left\{ \omega_{k}\right\} _{k\in\mathbb{N}_{0}}$, with $\mathbb{N}_{0}=\mathbb{N\cup}\left\{ 0\right\} $, be an increasing sequence of $\left[ 0,+\infty\right) $ such that $\omega_{0}=0$ and$$\lim_{k\rightarrow+\infty}\left( \omega_{n+1}-\omega_{n}\right) =+\infty$$ For every $c\in\ell^{2}(\mathbb{N}_{0},\mathbb{C})$, there exist infinitely many $v\in L^{2}\left( \left( 0,T\right) ,\mathbb{C}\right) $ solving$$c_{k}=\int_{0}^{T}v(s)e^{i\omega_{n}s}\,ds,\qquad n\in\mathbb{N}_{0}\,.
\label{moment_eq}$$
The proof rephrases the one given in Corollary 1 of [@BeLa], where the authors discuss the existence in $L^{2}\left( \left( 0,T\right)
,\mathbb{R}\right) $ under the particular condition: $c\in\ell^{2}(\mathbb{N}_{0},\mathbb{C})$, $c_{0}\in\mathbb{R}$. For sake of completeness we give a sketch of it.
Let define: $\omega_{-n}=-\omega_{n}$ for $n\in\mathbb{N}_{0}$. According to the result of [@Haraux] (referred as Theorem 1 in [@BeLa]), the family $\left\{ e^{i\omega_{n}t}\right\} _{n\in\mathbb{Z}}$ is a Riesz basis of the subspace $\mathcal{F}$ defined by the clousure in $L^{2}(0,T)$ of the $Span\left\{ e^{i\omega_{n}t}\right\} _{n\in\mathbb{Z}}$. This condition is equivalent to the existence of an isomorphism $J$ (we refer to Proposition 20 in [@BeLa]):$$\begin{array}
[c]{llll}\medskip J: & \mathcal{F} & \mathcal{\longrightarrow} & \ell^{2}(\mathbb{Z},\mathbb{C})\\
& v & \mathcal{\longrightarrow} & \int_{0}^{T}v(s)e^{i\omega_{n}s}\,ds
\end{array}$$ Given $c\in\ell^{2}(\mathbb{N}_{0},\mathbb{C})$, we consider $\tilde{c}\in
\ell^{2}(\mathbb{Z},\mathbb{C})$ such that: $\tilde{c}_{k}=c_{k}$ for $k\in\mathbb{N}_{0}$. To any choice of $\tilde{c}$ there corresponds an unique solution to (\[moment\_eq\]) defined by $J^{-1}(\tilde{c})$.
When the $\lambda_{k}$ coincide with the frequencies of the standard basis $\left\{ e^{i\omega nt},\ \omega=\frac{2\pi}{T}\right\} _{n\in\mathbb{Z}}$ in $L^{2}(0,T)$, equation (\[Lin1.2\]) can be also interpreted as a constraint over the Fourier coefficients of the function $\rho$. In this case, it is possible to give a solution of the problem respecting Dirichlet boundary conditions. These remarks are summarized in the following proposition.
\[Prop\_moment\]The following properties hold:$1)$ For $\left\{ c_{k}\right\} \in\ell_{2}(D)$ and $T>0$, (\[Lin1.2\]) admits infinitely many solutions $\rho\in L^{2}(0,T)$.$2)$ Let $T\geq8\pi$. For any $\psi_{T}\in\mathcal{W}$, it is possible to find at least one $\rho\in H_{0}^{1}(0,T)$ solving (\[Lin1.1\]).
$1)$ Let $\psi_{T}=\sum_{k\in D}c_{k}\psi_{k}$, and consider the corresponding moment equation (\[Lin1.2\]). For $k\in D$, we set: $k=2n-1$, $\omega
_{n}=\lambda_{2n-1}$ with $n\in\mathbb{N}$, and $\omega_{0}=0$; with this notation, our problem writes as$$\tilde{c}_{n}=\int_{0}^{T}\tilde{\rho}(s)e^{i\omega_{n}s}\,ds,\qquad
n\in\mathbb{N}_{0} \label{control_1}$$$$\tilde{\rho}=\frac{i}{\sqrt{\pi}}\rho\,,\qquad\left. \tilde{c}_{n,T}\medskip\right\vert _{n\in\mathbb{N}}=e^{i\lambda_{k}T}c_{2n-1}\quad
\text{and\quad}\tilde{c}_{0}=0 \label{control_1.1}$$ According to theorem \[Th\_K\], (\[control\_1\]) admits infinitely many solutions $\tilde{\rho}\in L^{2}(0,T)$, which are determined by fixing the extensions of the families $e^{i\omega_{n}s}$ and $\tilde{c}_{n,T}$ to $n\in\mathbb{Z}$.
$2)$ For $T=8\pi$, the functions $\left\{ e^{-i\lambda_{k}t}\right\} _{k\in
D}$ form a subsystem of the standard basis $\left\{ e^{i\omega nt},\ \omega=\frac{2\pi}{T}\right\} _{n\in\mathbb{Z}}$ in $L^{2}(0,T)$. Thus, the equation (\[Lin1.2\]) partially determines the Fourier coefficients of the function $\rho$. A particular solution $\rho_{0}$ is obtained by a superposition respecting Dirichlet conditions on the boundary$$\frac{i}{\sqrt{\pi}}\rho_{0}(t)=\sum_{k\in D}e^{i\lambda_{k}T}c_{k}\left(
e^{-i\lambda_{k}t}-e^{i\lambda_{k}t}\right) \,. \label{solution}$$ Since the regularity of $\psi_{T}\in\mathcal{W}$ implies: $\left\{
\,k^{2}c_{k}\right\} \in\ell_{2}(D)$, it follows $\rho\in H_{0}^{1}(0,T)$. For $T>8\pi$, the target $\psi_{T}$ is attained by $\rho$ such that: $\left.
\rho\right\vert _{t\in\left[ 0,8\pi\right] }=\rho_{0}$ is a solution of (\[Lin1.1\]) in $H_{0}^{1}(0,8\pi)$ and $\rho(t\geq8\pi)=0$.
In order to solve the inverse problem related to (\[Lin1\])-(\[Lin2\]), we proceed as follows: for a target state $\psi_{T}\in\mathcal{W}$, Proposition \[Prop\_moment\] allows to determine (at least one) $q\in H_{0}^{1}(0,T)$ solving (\[Lin1\]); then, this function is replaced in (\[Lin2\]) in the attempt of finding a suitable $u\in H_{0}^{1}(0,T)$ such that$$-q(t)=\frac{i}{\pi}\alpha(t)Uq(t)+u(t)\left( e^{it\Delta}\psi_{0}(0)+\frac
{i}{\pi}U\left[ V(\alpha)\right] (t)\right)$$ The solvability of this equation w.r.t. $u$ is strictly connected with the specific choice of the initial state, $\psi_{0}$, and of the linearization point $\alpha$. In particular, if the function $\left( e^{it\Delta}\psi
_{0}(0)+\frac{i}{\pi}U\left[ V(\alpha)\right] (t)\right) $ is different from zero at each $t$, one can easly determine $u$ as$$u(t)=\left( -q(t)-\frac{i}{\pi}\alpha(t)Uq(t)\right) \left( e^{it\Delta
}\psi_{0}(0)+\frac{i}{\pi}U\left[ V(\alpha)\right] (t)\right) ^{-1}$$ Consider a simplified framework where $\psi_{0}$ is a Laplacian’s eigenstate in $\mathcal{W}$$$\psi_{0}=\psi_{\bar{k}}\quad\bar{k}\ odd \label{stato iniziale 3}$$ and $\alpha=0$. In this setting, the equation (\[Lin2\]) writes as$$-q(t)=\frac{1}{\sqrt{\pi}}u(t)\,e^{-i\lambda_{\bar{k}}t} \label{Lin4}$$ and the inverse problem$$\psi_{T}=-\frac{i}{\pi}\sum\limits_{k\in D}\left( \int_{0}^{T}u(s)\,e^{-i\lambda_{\bar{k}}s}e^{-i\lambda_{k}\left( T-s\right)
}\,ds\right) \,\psi_{k} \label{Lin3}$$ is solved in $H_{0}^{1}(0,T)$ by the last point of Proposition \[Prop\_moment\]. This result, and the regularity of the map $\Gamma$ leads to the local steady state controllability of the system (\[G-def1\])-(\[G-def2\]) around the point $\alpha=0$.
\[Local controllability\]\[local controllability\]Let $T\geq8\pi$ and assume (\[stato iniziale 3\]). It exist an open neighbourhood $V\times P\subseteq
H_{0}^{1}(0,T)\times\mathcal{W}$ of the point $\left( 0,\psi_{\bar{k}}\right) $ such that: $\left. \Gamma\right\vert _{V}:V\rightarrow P$ is surjective
As remarked at the beginning of this section, for $\psi_{0}\in\mathcal{W}$, the map $\Gamma$ is $C^{1}(H_{0}^{1}(0,T),\,\mathcal{W})$. Moreover, under our assumptions, $d_{\alpha}\Gamma$ is surjective for $\alpha=0$. Then, by the Local Surjectivity Theorem (e.g. [@Ratiu]) we know that there exists an open neighbourhood $V\times P\subseteq H_{0}^{1}(0,T)\times\mathcal{W}$ of $\left( 0,\Gamma(0)\right) $ such that the restriction $\left.
\Gamma\right\vert _{V}:V\rightarrow P$ is surjective. By definition, $\Gamma(0)=e^{-i\lambda_{\bar{k}}T}\psi_{\bar{k}}$ and $P$ is a neighbourhood of $\psi_{\bar{k}}$ in $\mathcal{W}$.
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[^1]: IRMAR, UMR - CNRS 6625, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France.
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---
abstract: 'We study the positivity properties of Hermitian (or even Finsler) holomorphic vector bundles in terms of $L^p$-estimates of $\bar\partial$ and $L^p$-extensions of holomorphic objects. To this end, we introduce four conditions, called the optimal $L^p$-estimate condition, the multiple coarse $L^p$-estimate condition, the optimal $L^p$-extension condition, and the multiple coarse $L^p$-extension condition, for a Hermitian (or Finsler) vector bundle $(E,h)$. The main result of the present paper is to give a characterization of the Nakano positivity of $(E,h)$ via the optimal $L^2$-estimate condition. We also show that $(E,h)$ is Griffiths positive if it satisfies the multiple coarse $L^p$-estimate condition for some $p>1$, the optimal $L^p$-extension condition, or the multiple coarse $L^p$-extension condition for some $p>0$. These results can be roughly viewed as converses of Hörmander’s $L^2$-estimate of $\bar\partial$ and Ohsawa-Takegoshi type extension theorems. As an application of the main result, we get a totally different method to Nakano positivity of direct image sheaves of twisted relative canonical bundles associated to holomorphic families of complex manifolds.'
address:
- |
Fusheng Deng: School of Mathematical Sciences, University of Chinese Academy of Sciences\
Beijing 100049, P. R. China
- 'Jiafu Ning: Department of Mathematics, Central South University, Changsha, Hunan 410083, P. R. China.'
- |
Zhiwei Wang: School of Mathematical Sciences\
Beijing Normal University\
Beijing\
100875\
P. R. China
- |
Xiangyu Zhou: Institute of Mathematics\
Academy of Mathematics and Systems Sciences\
and Hua Loo-Keng Key Laboratory of Mathematics\
Chinese Academy of Sciences\
Beijing\
100190\
P. R. China
- 'School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China'
author:
- Fusheng Deng
- 'Jiafu Ning^\*^'
- 'Zhiwei Wang^\*^'
- Xiangyu Zhou
title: 'Positivity of holomorphic vector bundles in terms of $L^p$-conditions of $\bar\partial$'
---
[^1]
Introduction
============
The present paper is to study positivity properties of Hermitian (or even Finsler) holomorphic vector bundles via $L^p$-estimates of $\bar\partial$ and $L^p$-extensions of holomorphic objects, which can be roughly viewed as converses of Hörmander’s $L^2$-estimate of $\bar\partial$ and Ohsawa-Takegoshi type extension theorems. This is a continuation of the previous work [@DNW1] on characterizations of plurisubharmonic functions.
To state the main results, we first introduce some notions.
\[def:Lp estimate\] Let $(X,\omega)$ be a Kähler manifold of dimension $n$, which admits a positive Hermitian holomorphic line bundle, $(E,h)$ be a (singular) Hermitian vector bundle (maybe of infinite rank) over $X$, and $p>0$.
- $(E,h)$ satisfies *the optimal $L^p$-estimate condition* if for any positive Hermitian holomorphic line bundle $(A,h_A)$ on $X$, for any $f\in\mathcal{C}^\infty_c(X,\wedge^{n,1}T^*_X\otimes E\otimes A)$ with $\bar\partial f=0$, there is $u\in L^p(X,\wedge^{n,0}T_X^*\otimes E\otimes A)$, satisfying $\bar\partial u=f$ and $$\int_X|u|^p_{h\otimes h_A}dV_\omega\leq \int_X\langle B_{A,h_A}^{-1}f,f\rangle^{\frac{p}{2}} dV_\omega,$$ provided that the right hand side is finite, where $B_{A,h_A}=[i\Theta_{A,h_A}\otimes Id_E,\Lambda_\omega]$.
- $(E,h)$ satisfies *the multiple coarse $L^p$-estimate condition* if for any $m\geq 1$, for any positive Hermitian holomorphic line bundle $(A,h_A)$ on $X$, and for any $f\in\mathcal{C}^\infty_c(X,\wedge^{n,1}T^*_X\otimes E^{\otimes m}\otimes A)$ with $\bar\partial f=0$, there is $u\in L^p(X,\wedge^{n,0}T_X^*\otimes E^{\otimes m}\otimes A)$, satisfying $\bar\partial u=f$ and $$\int_X|u|^p_{h^{\otimes m}\otimes h_A}dV_\omega\leq C_m\int_X\langle B_{A,h_A}^{-1}f,f\rangle^{\frac{p}{2}} dV_\omega,$$ provided that the right hand side is finite, where $C_m$ are constants satisfying the growth condition $\frac{1}{m}\log C_m{\rightarrow}0$ as $m{\rightarrow}\infty$.
\[def:Lp extension\] Let $(E,h)$ be a Hermitian holomorphic vector bundle (maybe of infinite rank) over a domain $D\subset{\mathbb{C}}^n$ with a singular Finsler metric $h$, and $p>0$.
- $(E,h)$ satisfies *the optimal $L^p$-extension condition* if for any $z\in D$, and $a\in E_{z}$ with $|a|=1$, and any holomorphic cylinder $P$ with $z+P\subset D$, there is $f\in H^0(z+P, E)$ such that $f(z)=a$ and $$\frac{1}{\mu(P)}\int_{z+P}|f|^p\leq 1,$$ where $\mu(P)$ is the volume of $P$ with respect to the Lebesgue measure. (Here by a holomorphic cylinder we mean a domain of the form $A(P_{r,s})$ for some $A\in U(n)$ and $r,s>0$, with $P_{r,s}=\{(z_1,z_2,\cdots,z_n):|z_1|^2<r^2,|z_2|^2+\cdots+|z_n|^2<s^2\}$).
- $(E,h)$ satisfies *the multiple coarse $L^p$-extension condition* if for any $z\in D$, and $a\in E_{z}$ with $|a|=1$, and any $m\geq 1$, there is $f_m\in H^0(D, E^{\otimes m})$ such that $f_m(z)=a^{\otimes m}$ and satisfies the following estimate: $$\int_D|f_m|^{p}\leq C_m,$$ where $C_m$ are constants independent of $z$ and satisfying the growth condition $\frac{1}{m}\log C_m{\rightarrow}0$ as $m{\rightarrow}\infty$.
(See §\[subsec:finsler metric\] for the definition of singular Finsler metrics.)
Similarly, one can define the optimal (resp. multiple coarse) $L^p$-extension condition for a Hermitian vector bundle $(E,h)$ over a Kähler manifold $X$. But it is clear that if $(E,h)$ satisfies the optimal (resp. multiple coarse) $L^p$-extension condition on $X$, then it admits the same condition when restricted on any open set $D$ of $X$. So we just focus on bounded domains in Definition \[def:Lp extension\]. However, it is not the case for the optimal (resp. multiple coarse) $L^p$-estimate condition since a positive Hemitian line bundle over an open domain in $X$ may not extend to $X$.
The conditions defined in Definition \[def:Lp estimate\], \[def:Lp extension\] for trivial line bundles were studied in [@DNW1]. The multiple coarse $L^p$-extension condition for vector bundles with singular Finsler metrics was introduced in [@DWZZ1], and the multiple coarse $L^2$-estimate condition for Hermitian vector bundles was introduced in [@HI], which was named as the twisted H" ormander condition there. A property (called “minimal extension property”) that is related to the optimal $L^2$-extension condition was introduced in [@HPS16].
The first and the main result of this paper is the following characterization of Nakano positivity in terms of optimal $L^2$-estimate condition.
\[thm:theta-nakano text\_intr\] Let $(X,\omega)$ be a Kähler manifold of dimension $n$ with a K" ahler metric $\omega$, which admits a positive Hermitian holomorphic line bundle, $(E,h)$ be a smooth Hermitian vector bundle over $X$, and $\theta\in C^0(X,\Lambda^{1,1}T^*_X\otimes End(E))$ such that $\theta^*=\theta$. If for any $f\in\mathcal{C}^\infty_c(X,\wedge^{n,1}T^*_X\otimes E\otimes A)$ with $\bar\partial f=0$, and any positive Hermitian line bundle $(A,h_A)$ on $X$ with $i\Theta_{A,h_A}\otimes Id_E+\theta>0$ on $\text{supp}f$, there is $u\in L^2(X,\wedge^{n,0}T_X^*\otimes E\otimes A)$, satisfying $\bar\partial u=f$ and $$\int_X|u|^2_{h\otimes h_A}dV_\omega\leq \int_X\langle B_{h_A,\theta}^{-1}f,f\rangle_{h\otimes h_A} dV_\omega,$$ provided that the right hand side is finite, where $B_{h_A,\theta}=[i\Theta_{A,h_A}\otimes Id_E+\theta,\Lambda_\omega]$, then $i\Theta_{E,h}\geq\theta$ in the sense of Nakano. On the other hand, if in addition $X$ is assumed to have a complete Kähler metric, the above condition is also necessary for that $i\Theta_{E,h}\geq\theta$ in the sense of Nakano. In particular, if $(E,h)$ satisfies the optimal $L^2$-estimate condition, then $(E,h)$ is Nakano semi-positive.
We prove Theorem \[thm:theta-nakano text\_intr\] by connecting $\Theta_{E,h}$ with the optimal $L^2$-estimate condition through the Bochner-Kodaira-Nakano identity, and then using a localization technique to produce a contradiction if $i\Theta_{E,h}\geq\theta$ is assumed to be not true.
Does Theorem \[thm:theta-nakano text\_intr\] still hold if the optimal $L^2$-estimate condition is replaced by the optimal $L^p$-estimate condition for some $p\neq 2$?
Establish results analogous to Theorem \[thm:theta-nakano text\_intr\] for holomorphic vector bundles with singular Hermitian metrics.
\[thm:coarse estimate text-intr\] Let $(X,\omega)$ be a Kähler manifold, which admits a positive Hermitian holomorphic line bundle, and $(E,h)$ be a holomorphic vector bundle over $X$ with a continuous Hermitian metric $h$. If $(E,h)$ satisfies the multiple coarse $L^p$-estimate condition for some $p>1$, then $(E,h)$ is Griffiths semi-positive.
\[rem:reduce to trivial bundle\] If $X$ admits a strictly plurisubharmonic function, it is obviously from the proof that, in Theorem \[thm:theta-nakano text\_intr\] and Theorem \[thm:coarse estimate text-intr\], we can take $A$ to be the trivial bundle (with nontrivial metrics).
The case that $p=2$ and $h$ is Hölder continuous for Theorem \[thm:coarse estimate text-intr\] was proved in [@HI], by showing that the multiple coarse $L^2$-estimate condition implies the multiple coarse $L^2$-extension condition and then applying [@DWZZ1 Theorem 1.2]. The case that $E$ is a trivial line bundle was proved in [@DNW1]. Theorem \[thm:coarse estimate text-intr\] is proved by modifying the technique in [@HI; @DNW1].
\[thm: optimal Lp extension : Griffiths positive-intr\] Let $E$ be a holomorphic vector bundle over a domain $D\subset{\mathbb{C}}^n$, and $h$ be a singular Finsler metric on $E$, such that $|s|_{h^*}$ is upper semi-continuous for any local holomorphic section $s$ of $E^*$. If $(E,h)$ satisfies the optimal $L^p$-extension condition for some $p>0$, then $(E,h)$ is Griffiths semi-positive.
A first related result in this direction was given by Guan-Zhou in [@GZh15d], where they showed that Berndtsson’s plurisubharmonic variation of the relative Bergman kernels [@Bob06] can be deduced from the optimal $L^2$-extension condition. By developing Guan-Zhou’s method, Hacon-Popa-Schnell in [@HPS16] proved that a Hermitian vector bundle $(E,h)$ with singular Hermitian metric is Griffiths semi-positive if it satisfies the so called minimal extension condition, a notion defined there as mentioned above. Theorem \[thm: optimal Lp extension : Griffiths positive-intr\] is proved by combining the ideas in [@GZh15d; @HPS16] and a lemma in [@DNW1].
\[thm: multiple coarse Lp: Griffiths positivity-intr.\] Let $E$ be a holomorphic vector bundle over a domain $D\subset{\mathbb{C}}^n$, and $h$ be a singular Finsler metric on $E$, such that $|s|_{h^*}$ is upper semi-continuous for any local holomorphic section $s$ of $E^*$. If $(E,h)$ satisfies the multiple coarse $L^p$-extension condition for some $p>0$, then $(E,h)$ is Griffiths semi-positive.
Theorem \[thm: multiple coarse Lp: Griffiths positivity-intr.\] was originally proved in [@DWZZ1]. In this paper, we give a new proof based on the idea in the proof of [@DNW1 Theorem 1.5].
In the case for trivial line bundles over bounded domains, Theorem \[thm:theta-nakano text\_intr\]-\[thm: optimal Lp extension : Griffiths positive-intr\] were proved in [@DNW1]. We should emphasize that Theorem \[thm:theta-nakano text\_intr\]-\[thm: multiple coarse Lp: Griffiths positivity-intr.\] are true for vector bundles of infinite rank, as well as for those of finite rank.
In applications, it is possible to prove that $(E, e^{\phi}h)$ satisfies the optimal $L^p$-extension condition for some $\phi\in C^0(D)$, by Theorem \[thm:coarse estimate text-intr\] /Theorem \[thm: optimal Lp extension : Griffiths positive-intr\]/Theorem \[thm: multiple coarse Lp: Griffiths positivity-intr.\], which implies that $$i\Theta_E\geq i\partial\bar\partial\phi\otimes Id_E$$ in the sense that $i\partial\bar\partial \log|s|^2_{h^*}\geq i\partial\bar\partial\phi$ in the sense of currents, for any nonvanishing local holomorphic section $s$ of $E^*$.
We now explain why Theorem \[thm:theta-nakano text\_intr\]-\[thm: multiple coarse Lp: Griffiths positivity-intr.\] can be roughly viewed as converses of $L^2$-estimate of $\bar\partial$ and Ohsawa-Takegoshi type $L^2$-extensions.
Let $(X,\omega)$ be a weakly pseudoconvex Kähler manifold, and $(E,h)$ be a Hermitian holomorphic vector bundle over $X$. If the curvature of $(E,h)$ is Nakano semi-positive, then $(E,h)$ satisfies the optimal $L^2$-estimate condition and the multiple coarse $L^2$-estimate condition by works of H" ormander [@Hor65] and Demailly [@Dem82]. Combining Theorem \[thm:theta-nakano text\_intr\], we see in this setting that *Nakano positivity for $(E,h)$ is equivalent to the optimal $L^2$-estimate condition*. Considering Theorem \[thm:coarse estimate text-intr\] which shows that the multiple coarse $L^2$-estimate condition for $(E,h)$ implies Griffiths positivity of $(E,h)$, it is natural to propose the following
Does the multiple coarse $L^2$-estimate condition of $(E,h)$ imply the Nakano positivity of $(E,h)$?
Let $(E,h)$ be a Hermitian holomorphic vector bundle over a bounded pseudoconvex domain $D$. If the curvature of $(E,h)$ is Nakano semi-positive, then $(E,h)$ satisfies the multiple coarse $L^2$-extension condition by Ohsawa-Takegoshi [@OT1], and satisfies the optimal $L^2$-extension condition by Błocki [@Bl] and Guan-Zhou [@GZh12][@GZh15d]. Theorem \[thm: optimal Lp extension : Griffiths positive-intr\] and Theorem \[thm: multiple coarse Lp: Griffiths positivity-intr.\] show that the optimal $L^2$-extension condition and the multiple coarse $L^2$-extension condition imply Griffiths positivity of $(E,h)$. It is observed in [@HI] that the optimal $L^2$-extension condition and the multiple coarse $L^2$-extension condition are strictly weaker than the Nakano positivity in some cases. Motivated by this and the above theorems, we believe that the optimal $L^2$-estimate condition and the multiple coarse $L^2$-estimate condition are equivalent to the Nakano positivity, while, the optimal $L^2$-extension condition and the multiple coarse $L^2$-extension condition are equivalent to the Griffiths positivity. So we propose the following problem.
Does the Griffiths positivity imply the optimal $L^2$-extension condition and the multiple coarse $L^2$-extension condition?
By the proof of [@Ber-Pau10 Proposition 0.2], one can see that the optimal $L^2$-extension condition and the multiple coarse $L^2$-extension condition imply the optimal $L^p$-extension condition and the multiple coarse $L^p$-extension condition for $0<p<2$.
The second part of this paper is to apply the above theorems to study the curvature positivity of direct image sheaves of twisted relative canonical bundles associated to holomorphic fibrations, which is a topic of extensive study in recent years (see [@Bob06; @Bob09a; @BP08; @Ber-Pau10; @LS14; @GZh15d; @PT18; @Cao141; @HPS16; @ZZ17; @ZhouZhu; @Bob18; @DWZZ1]). The main novelty here is that Theorem \[thm:theta-nakano text\_intr\] provides a natural explanation and a very simple unified proof of the Nakano positivity of direct image bundles associated to families of both Stein manifolds and compact Kähler manifolds, with an effective estimate of the lower bound of the curvatures.
We first consider a family of bounded domains. Let $\Omega=U\times D\subset {\mathbb{C}}^n_t\times{\mathbb{C}}^m_z$ be a bounded pseudoconvex domain and $p:\Omega{\rightarrow}U$ be the natural projection. Let $h$ be a Hermitian metric on the trivial bundle $E=\Omega\times{\mathbb{C}}^r$ that is $C^2$-smooth to $\overline\Omega$. For $t\in U$, let $$F_t:=\{f\in H^0(D, E|_{\{t\}\times D}):\|f\|^2_{t}:=\int_D|f|^2_{h_t}<\infty\}$$ and $F:=\coprod_{t\in U}F_t$. Since $h$ is continuous to $\overline\Omega$, $F_t$ are equal for all $t\in U$ as vector spaces. We may view $(F, \|\cdot\|)$ as a trivial holomorphic Hermitian vector bundle of infinite rank over $U$.
\[thm: direc im stein-intr\] Let $\theta$ be a continuous real $(1,1)$-form on $U$ such that $i\Theta_{E}\geq p^*\theta\otimes Id_E$, then $i\Theta_{F}\geq \theta\otimes Id_F$ in the sense of Nakano. In particular, if $i\Theta_{E}>0$ in the sense of Nakano, then $i\Theta_{F}>0$ in the sense of Nakano.
Let $\pi:X\rightarrow U$ be a proper holomorphic submersion from a K" ahler manifolds $X$ of complex dimension $m+n$, to a bounded pseudoconvex domain $U$, and $(E,h)$ be a Hermitian holomorphic vector bundle over $X$, with Nakano semi-positive curvature. From the Ohsawa-Takegoshi extension theorem, the direct image $F:=\pi_*(K_{X/U}\otimes E)$ is a vector bundle, whose fiber over $t\in U$ is $F_t=H^0(X_t, K_{X_t}\otimes E|_{X_t})$. There is a hermtian metric $\|\cdot\|$ on $F$ induced by $h$: for any $u\in F_t$, $$\|u(t)\|^2_t:=\int_{X_t}c_mu\wedge \bar u,$$ where $m=\dim X_t$, $c_m=i^{m^2}$, and $u\wedge \bar u$ is the composition of the wedge product and the inner product on $E$. So we get a Hermitian holomorphic vector bundle $(F, \|\cdot\|)$ over $U$.
\[thm: direct image-optimal L2 estimate-intr\] The Hermitian holomorphic vector bundle $(F, \|\cdot\|)$ over $U$ defined above satisfies the optimal $L^2$-estimate condition. Moreover, if $i\Theta_{E}\geq p^*\theta\otimes Id_E$ for a continuous real $(1,1)$-form $\theta$ on $U$, then $i\Theta_{F}\geq \theta\otimes Id_F$ in the sense of Nakano.
Theorem \[thm: direc im stein-intr\] in the case that $E$ is a line bundle is a result of Berndtsson [@Bob09a Theorem 1.1], the case for vector bundles $E$ without lower bound estimate was proved by Raufi [@Raufi; @13 Theorem 1.5] with the same method of Berndtsson. Theorem \[thm: direct image-optimal L2 estimate-intr\] in the case that $L$ is a line bundle is due to Berndtsson [@Bob09a], and the case for vector bundles was proved in [@MoTa08] and [@LiYa14] by developing the method of Berndtsson.
Our method to Theorem \[thm: direc im stein-intr\] and Theorem \[thm: direct image-optimal L2 estimate-intr\] is very different. In fact, taking Theorem \[thm:theta-nakano text\_intr\] for granted, one can clearly see why Theorem \[thm: direc im stein-intr\] and Theorem \[thm: direct image-optimal L2 estimate-intr\] should be true, since it is obvious that the bundles $F$ in Theorem \[thm: direc im stein-intr\] and Theorem \[thm: direc im stein-intr\] satisfy the optimal $L^2$-estimate condition by Hörmander’s $L^2$-estimate of $\bar\partial$.
In this paper, we also provide some new methods to show that $(F, \|\cdot\|)$ is Griffiths semi-positive, via Theorem \[thm:coarse estimate text-intr\], \[thm: optimal Lp extension : Griffiths positive-intr\], and \[thm: multiple coarse Lp: Griffiths positivity-intr.\]. By applying the tensor-power technique introduced in [@DWZZ1], we show that $(F, \|\cdot\|)$ satisfies the multiple coarse $L^2$-estimate condition; by applying the Ohsawa-Takegoshi extension theorem with optimal estimate for vector bundles ([@GZh15d], [@ZhouZhu192]), we show that $(F, \|\cdot\|)$ satisfies the optimal $L^2$-extension condition; and by applying the tensor-power technique mentioned above and the Ohsawa-Takegoshi extension theorem, we show that $(F, \|\cdot\|)$ satisfies the multiple coarse $L^2$-extension condition.\
$\mathbf{Acknowlegements.}$ The authors are partially supported respectively by NSFC grants (11871451, 11801572, 11701031, 11688101). The first author was partially supported by the University of Chinese Academy of Sciences. The third author is partially supported by Beijing Natural Science Foundation (Z190003, 1202012).
Preliminaries
=============
An extension property of Hermitian metrics on a line bundle
-----------------------------------------------------------
In this section, we present a basic property of K" ahler manifolds, which admit positive Hermitian holomorphic line bundles.
\[prop: construct function 1\]Let $X$ be a K" ahler manifold, which admits a positive Hermitian holomorphic line bundle, and $(A,h_A)$ be a positive Hermitian holomorphic line bundle over $X$. Let $(U\subset X, z=(z_1,\cdots, z_n))$ be a coordinate chart on $X$, such that $A|_U$ is trivial, and $B\Subset U$ be a coordinate ball. Then for any smooth strictly plurisubharmonic function $\psi$ on $U$, there is a positive integer $m$, and a Hermitian metric $h_m$ on the line bundle $A^{\otimes m}$, such that $h_m=e^{-\psi_m}$ on $U$ with $\psi_m|_B=\psi$.
Assume that $h_A|_U=e^{-\phi}$ for some smooth strictly plurisubharmonic function $\phi$ on $U$. We may assume that $\phi>0$. We may assume that $B=B_1$ is the unit ball, and the ball $B_{1+3\delta}$ with radius $1+3\delta$ is also contained in $U$, for $0<\delta\ll 1$. Let $\chi$ be a cut-off function on $U$, such that $\chi$ is identically equal to $1$ near $\overline B_{1+\delta}$ and vanishes outside $B_{1+2\delta}$. Let $\phi_{m}:=m\phi+\chi\log(\|z\|^2-1)$ on $U\backslash B_{1}$, where $m\gg 1$ is an integer such that $\phi_m$ is strictly p.s.h on $U$ and $\phi_{m} >\psi$ on $\partial B_{1+\delta}$.
Now we define a function $\psi_m$ on $U$ as follows:
$$\begin{aligned}
\psi_m=\left\{
\begin{array}{ll}
\phi_m, & \hbox{outside $B_{1+\delta}$;} \\
\max_\epsilon\{\phi_{m}, \psi\} , & \hbox{on $B_{1+\delta}\setminus B_{1}$;}\\
\psi, & \hbox{on $B_{1}$.}
\end{array}
\right.
\end{aligned}$$
Then for $0<\epsilon\ll 1$, $\psi_m$ is strictly p.s.h on $U$, $\psi_m|_B=\psi$, and equals to $m\phi$ on $U\backslash B_{1+2\delta}$. So $\psi_m$ gives a Hermitian metric on $A^{\otimes m}|_U$ which coincides with $h^{\otimes m}$ on $U\backslash B_{1+2\delta}$.
Basics of Hermitian holomorphic vector bundles
----------------------------------------------
Let $(X,\omega)$ be a complex manifold of complex dimension $n$, equipped with a Hermitian metric $\omega$, and $(E,h)$ be a Hermitiann holomorphic vector bundle of rank $r$ over $X$. In this subsection, we assume $r<\infty$.
Let $D=D'+\bar{\partial}$ be the Chern connection of $(E,h)$, and $\Theta_{E,h}=[D',\bar\partial]=D'\bar{\partial}+\bar{\partial}D'$ be the Chern curvature tensor. Denote by $(e_1,\cdots, e_r)$ an orthonormal frame of $E$ over a coordinate patch $\Omega\subset X$ with complex coordinates $(z_1,\cdots, z_n)$, and $$i\Theta_{E,h}=i\sum_{1\leq j,k\leq n,1\leq \lambda,\mu\leq r}c_{jk\lambda\mu}dz_j\wedge d\bar z_k\otimes e^*_\lambda\otimes e_{\mu}, ~~\bar c_{jk\lambda\mu}=c_{kj\mu\lambda}.$$ To $i\Theta_{E,h}$ corresponds a natural Hermitian form $\theta_{E,h}$ on $TX\otimes E$ defined by $$\begin{aligned}
\label{eqn: chern curvature form}
\theta_{E,h}(u,u)=\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}(x)u_{j\lambda}\bar u_{k\mu}, ~~~~~u\in T_xX\otimes E_x
.\end{aligned}$$
\[defn:positivities\]
- $E$ is said to be Nakano positive (resp. Nakano semi-positive) if $\theta_{E,h}$ is positive (resp. semi-positive) definite as a Hermitian form on $TX\otimes E$, i.e. for every $u\in TX\otimes E$, $u\neq 0$, we have $$\theta(u,u)>0 ~~~(\mbox{resp.} \geq 0).$$
- $E$ is said to be Griffiths positive (resp. Griffiths semi-positive) if for any $x\in X$, all $\xi\in T_xX$ with $\xi\neq 0$, and $s\in E_x$ with $s\neq 0$, we have $$\theta(\xi\otimes s,\xi\otimes s)>0~~~(\mbox{resp.} \geq 0).$$
- Nakano negative (resp. Nakano semi-negative) and Griffiths negative (resp. Griffiths semi-negative) are similarly defined by replacing $>0$ (resp. $\geq 0$) by $<0$ (resp. $\leq 0$) in the above definitions respectively.
The following are basic facts about Griffiths positivity and Nakano positivity.
- It is a well-known fact that, a Hermitian holomorphic vector bundle $(E,h)$ is Griffiths positive (resp. semi-positive) if and only if $(E^*,h^*)$ is Griffiths negative (resp. semi-negative). However, Nakano positivity does not share this duality condition, see [@Dem Chapter VII, Page 339, Example 6.8] for an example.
- It is a fact that, Griffiths positivity can be explained as a several complex variables property, see Definition \[def:positivity of finsler\] in §\[subsec:finsler metric\]. However, Nakano positivity does not have such a characterization.
\[rem: Nakano tensor product\] Let $(E_1,h_1) $ and $(E_2,h_2)$ be two Hermitian holomoprhic vector bundles over a complex $n$-dimensional manifold $X$. It is a basic fact that the Chern connection $D_{E_1\otimes E_2}$ of $(E_1\otimes E_2, h_1\otimes h_2)$ is just $D_{E_1}\otimes \text{Id}_{E_2}+\text{Id}_{E_1}\otimes D_{E_2}$, and we have the following Chern curvature formula
$$\Theta_{E_1\otimes E_2, h_1\otimes h_2}=\Theta_{E_1,h_1}\otimes \text{Id}_{E_2}+\text{Id}_{E_1}\otimes \Theta_{E_2,h_2}.$$
From and Remark \[rem: Nakano tensor product\], we get the following
\[lem: Nakano tensor product\] Let $(E_1,h_1)$ and $(E_2,h_2)$ be two Hermitian holomorphic vector bundles over a complex manifold $X$. Let $(E,h):=(E_1\otimes E_2, h_1\otimes h_2)$. Then if $(E_1, h_1)$ and $(E_2,h_2)$ are Nakano positive (resp. Nakano semi-positive), then $(E,h)$ is Nakano positive (resp. Nakano semi-positive).
\[lem: Nakano fiber tensor product\] Let $\pi_i:X_i\rightarrow Y$ be two holomorphic submersions for $j=1,2$. Let $X\subset X_1\times X_2$ be the fiberwise product of $X_1$ and $X_2$ with respect to $\pi_1$ and $\pi_2$, and $pr_j:X\rightarrow X_j$ be the natural projections from $X$ to $X_j$ for $j=1,2$. Let $(E_1,h_1)$ and $(E_2,h_2)$ be two Hermitian holomorphic vector bundles over $X_1$ and $X_2$, respectively. Denote by $(E, h)$ the Hermitian holomorphic vector bundle $(pr_1^*E_1\otimes pr_2^*E_2, pr_1^*h_1\otimes pr_2^* h_2)$ on $X$. If $(E_1,h_1)$ and $(E_2,h_2)$ are Nakano positive (resp. Nakano semi-positive), then $E$ is also Nakano positive (resp. Nakano semi-positive).
For any $u\in \Lambda^{p,q}T^*_X\otimes E$, we consider the global $L^2$-norm $$\begin{aligned}
\|u\|^2=\int_X|u|_{\omega,h}^2dV_\omega,
\end{aligned}$$ where $|u|_{\omega,h}$ is the pointwise Hermitian norm and $dV_\omega=\omega^n/n!$ is the volume form on $X$. This $L^2$-norm induces an $L^2$-inner product on $\Lambda^{p,q}T^*_X\otimes E$, and thus we can define $D'^*$ and $\bar{\partial}^*$ operators as the (formal) adjoint of $D'$ and $\bar{\partial}$, respectively. Let $$\Delta'=D'D'^*+D'^*D', \ \ \Delta''=\bar{\partial}\bar{\partial}^*+\bar{\partial}^*\bar{\partial}$$ be the corresponding $D'$ and $\bar{\partial}$-Laplace operators.
\[lem: BKN identity\] Let $(X,\omega)$ be a Kähler manifold, $(E,h)$ be a Hermitian vector bundle over $X$. The complex Laplacian operators $\Delta'=D'D'^*+D'^*D'$ and $\Delta''=\bar{\partial}\bar{\partial}^*+\bar{\partial}^*\bar{\partial}$ acting on $E$-valued forms satisfy the identity $$\Delta''=\Delta'+[i\Theta_{E,h},\Lambda_\omega].$$
Let us say more on the Hermitian operator $[i\Theta_{E,h},\Lambda_\omega]$. Let $x_0\in X$ and $(z_1,\cdots, z_n)$ be local coordinates centered at $x_0$, such that $(\partial/\partial z_1,\cdots, \partial/\partial z_n)$ is an orthonormal basis of $TX$ at $x_0$. One can write $$\omega=i\sum dz_j\wedge d\bar z_j+O(\|z\|),$$ and $$i\Theta_{E,h}(x_0)=i\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}dz_j\wedge d\bar z_k \otimes e^*_\lambda\otimes e_\mu,$$ where $(e_1,\cdots, e_r)$ is an orthonormal basis of $E_{x_0}$. Let $u=\sum{u_{K,\lambda}}dz\wedge d\bar z_K\otimes e_\lambda\in \Lambda^{n,q}T^*_X\otimes E$ , where $dz=dz_1\wedge \cdots\wedge dz_n$. In [@Dem Chapter VII, Page 341, (7.1)], it is computed that $$\begin{aligned}
\label{eqn: computation of B 1}
\langle [i\Theta_{E,h},\Lambda_\omega]u,u\rangle=\sum_{|S|=q-1}\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}u_{jS,\lambda}\bar u_{kS,\mu}.
\end{aligned}$$ In particular, if $q=1$, becomes $$\begin{aligned}
\label{eqn: computation of B 2}
\langle [i\Theta_{E,h},\Lambda_\omega]u,u\rangle=\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}u_{j,\lambda}\bar u_{k,\mu}
\end{aligned}$$ Comparing and , we obtain the following
\[lem: Hermitian operator formula\] Let $(X,\omega)$ be a Kähler manifold, $(E,h)$ be a Hermitian vector bundle over $X$. Then $(E,h)$ is Nakano positive (resp. semo-positive) if and only if the Hermitian operator $[i\Theta_{E,h},\Lambda_\omega]$ is positive definite (resp. semi-positive definite) on $\Lambda^{n,1}T^*_X\otimes E$.
Basic concepts and conditions of Hermitian vector bundles of infinite rank {#subsec:infinite rank v.b.}
--------------------------------------------------------------------------
In this subsection, we will briefly discuss some concepts and basic conditions of Hermitian holomorphic vector bundles of infinite rank, and explain why the above mentioned Bochner-Kodaira-Nakano identity also holds in this framework.
Let $H$ be a Hilbert space (separable, say over ${\mathbb{C}}$) with inner product $(,)$. Let $U\subset \mathbb R^n$ be open. Let $f:U\rightarrow H$ be a map. If $$\begin{aligned}
\frac{\partial f}{\partial x_j}=\lim_{\Delta x_j\rightarrow 0}\frac{f(x_1,\cdots,x_j+\Delta x_j,\cdots, x_{n})-f(x_1,\cdots,x_{n})}{\Delta x_j}\in H\end{aligned}$$ exists and is continuous on $U$ for any $j=1,2,\cdots,n$, $f$ is called of $C^1$. We say $f$ is of $C^r$ if all partial derivatives $\frac{\partial^rf}{\partial x_1^{r_1}\cdots \partial x_{n}^{r_{n}}}: U\rightarrow H$ of order $r$ exist and continuous, and $f$ is smooth if $f$ is of $C^r$ for any $r$.
Let $\{e_\lambda\}^{\infty}_{\lambda=1}$ be an orthonormal basis of $H$. Then a map $f:U{\rightarrow}H$ can be written as $(f_1, f_2,\cdots)$, where $f_\lambda$ are functions on $U$ such that $$\|f(z)\|^2=\sum_\lambda|f_\lambda(z)|^2.$$ If $f$ is continuous, by Dini’s theorem, one can see that the series $\sum_\lambda|f_\lambda(z)|^2$ converges uniformly locally on $U$ to $\|f(z)\|^2$; and if $f$ is smooth, then $\sum_\lambda |\frac{\partial^r f_\lambda}{\partial x_1^{r_1}\cdots \partial x_n^{r_n}}|^2$ locally uniformly converges to $\|\frac{\partial^r f}{\partial x_1^{r_1}\cdots \partial x_n^{r_n}}\|^2$.
Now assume $U\subset{\mathbb{C}}^n$ be an open set. A map $f:U{\rightarrow}H$ is called holomorphic if $f$ is smooth and satisfies the Cauchy-Riemann equation $$\frac{\partial}{\partial\bar z_j}f:=\frac{1}{2}(\frac{\partial}{\partial x_j}+i\frac{\partial}{\partial y_j})f=0,\ j=1, \cdots, n.$$
We now consider holomorphic vector bundles of infinite rank with $H$ as the model of the fibers. In the present paper, we will focus on local conditions of holomorphic vector bundles, So we just consider the trivial bundle $E:=U\times H\rightarrow U$, here we view $H$ as a locally convex topological vector space.
A Hermitian metric on $E$ is a map $$\begin{aligned}
h:U\rightarrow Herm(H)\end{aligned}$$ which satisfies the following conditions:
- $h$ is smooth, and
- $h(z)\geq \delta(z) Id$ for some positive continuous function $\delta$ on $U$,
where $Herm(H)$ is the space of self-adjoint bounded operators on $H$.
Given $h$ as above, we get a smooth family of inner products on $H$ as $$(u,v)_z=(h(z)u,v),\ z\in U.$$ So our definition of the Hermitian metric on $E$ matches to the definition of Hermitian metrics for holomorphic vector bundles of finite rank.
Given a Hermitian metric $h$ on $E$, we can define a unique connection $D=D'+\bar\partial$ on $E$ which is compatible with $h$ and whose $(0,1)$-part is $\bar\partial$, as in the finite rank case. We view a section of $E$ as a map from $U$ to $H$. Assume $u$ is a smooth section of $E$, and $v\in H$ viewed as a constant section of $E$, then from the condition that $$\partial (hu,v)=(hD'u,v),$$ we get $$D'u=h^{-1}\partial (hu).$$ This formula shows that $D'u$ is a smooth section of $\Lambda^{1,0}T^*U\otimes E$.
The curvature operator of $(E,h)$ is give by $$\Theta_{E,h}=[D',\bar\partial],$$ which is an operator that maps smooth sections of $E$ to smooth sections of $\Lambda^{1,1}T^*U\otimes E$. In the same way as in the case of finite rank vector bundles, Nakano positivity and Griffiths positivity can be defined for $(E,h)$.
We now show that, at any point $z_0\in U$, the metric $h$ coincides with a flat metric up to order 1. To show this, we may assume $z_0=0$ and $h_0=Id$. Let $h_j=\frac{\partial h}{\partial z_j}(0),\ j=1,\cdots, n$, and define $$\tilde e_\lambda=e_\lambda-\sum_j z_jh_j(e_\lambda),$$ then $$(\tilde{e}_{\lambda},\tilde{e}_{\mu})_z=\delta_{\lambda\mu}+O(\|z\|^2),$$ where the bound for $O(\|z\|^2)$ is uniform for $\lambda, \mu$.
With the above preparation, following the line of the proof of [@Dem Theorem 1.1, Theorem 1.2, Chapter VII, §1], we see that the Bochner-Kodaira-Nakano identity also holds for $(E,h)$.
Singular Finsler metrics on holomorphic vector bundles {#subsec:finsler metric}
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In this subsection, we recall the notions of singular Finsler metrics on holomorphic vector bundles and positively curved singular Finsler metrics on coherent analytic sheaves, introduced in [@DWZZ1], see also [@DWZZ2].
\[def:finsler on v.b.\] Let $E\rightarrow X$ be a holomorphic vector bundle over a complex manifolds $X$. A (singular) Finsler metric $h$ on $E$ is a function $h:E\rightarrow [0, +\infty]$, such that $|cv|^2_h:=h(cv)=|c|^2h(v)$ for any $v\in E$ and $c\in \mathbb C$.
For a singular Finsler metric $h$ on $E$, its dual Finsler metric $h^*$ on the dual bundle $E^*$ of $E$ is defined as follows. For $f\in E^*_x$, the fiber of $E^*$ at $x\in X$, $|f|_{h^*}$ is defined to be $0$ if $|v|_h=+\infty$ for all nonzero $v\in E_x$; otherwise, $$|f|_{h^*}:=\sup\{|f(v)|: v\in E_x, |v|_h\leq 1\}\leq +\infty.$$
\[def:positivity of finsler\] Let $(E,h)$ be a holomorphic vector bundle over a complex manifold $X$, equipped with a singular Finsler metric $h$. We call $h$ is negatively curved (in the sense of Griffiths) if for any local holomorphic section $s$ of $E$, the function $\log|s|^2_h$ is plurisubharmonic, and we call $h$ is positively curved (in the sense of Griffiths) if its dual metric $h^*$ is negatively curved.
\[def:finsler on sheaf\] Let $\mathcal F$ be a coherent analytic sheaf on a complex manifold $X$. Let $Z\subset X$ be an analytic subset of $X$ such that $\mathcal F|_{X\setminus Z}$ is locally free. A positively curved singular Finsler metric $h$ on $\mathcal F$ is a singular Finsler metric on the holomorphic vector bundle $\mathcal F|_{X\setminus Z}$, such that for any local holomorphic section $s$ of the dual sheaf $\mathcal F^*$ on an open set $U\subset X$, the functionn $\log |s|_{h^*}$ is plurisubharmonic on $U\setminus Z$, and can be extended to a plurisubharmonic function on $U$.
Suppose that $\log|s|_{h^*}$ is p.s.h. on $U\setminus Z$. It is well-known that if codim$_{\mathbb{C}}(Z)\geq 2$ or $\log|s|_{H^*}$ is locally bounded above near $Z$, then $\log|s|_{h^*}$ extends across $Z$ to $U$ uniquely as a p.s.h function. Definition \[def:finsler on sheaf\] matches Definition \[def:finsler on v.b.\] and Definition \[def:positivity of finsler\] if $\mathcal F$ is a vector bundle.
$L^2$ theory of $\bar\partial$
------------------------------
In this section, we recall H" ormander’s $L^2$-estimate of $\bar\partial$ and Ohsawa-Takegoshi type $L^2$-extension of holomorphic sections, for holomorphic vector bundles.
We first clarify some notions and notations. Let $H$ be a Hilbert space with an inner product $(\cdot, \cdot)$, and $A:H{\rightarrow}H$ be a bounded semi-positive self-adjoint operator with closed range $Im A$. The we have an orthogonal decomposition $$H=Im A\oplus \ker A$$ and $A|_{Im A}:Im A{\rightarrow}Im A$ is a linear isomorphism. In the remaining of the paper, we always denote $A|^{-1}_{Im A}$ by $A^{-1}$, as in general references about complex geometry, and define $(A^{-1}v, v)=+\infty$ if $v\not\in ImA$.
\[thm: L2 estimate Nakano\]Let $(X,\omega)$ be a complete K" ahler manifold, with a K" ahler metric which is not necessarily complete. Let $(E,h)$ be a Hermitian vector bundle of rank $r$ over $X$, and assume that the curvature operator $B:=[i\Theta_{E,h},\Lambda_\omega]$ is semi-positive definite everywhere on $\Lambda^{p,q}T_X^*\otimes E$, for some $q\geq 1$. Then for any form $g\in L^2(X,\Lambda^{p,q}T^*_{X}\otimes E)$ satisfying $\bar{\partial}g=0$ and $\int_X\langle B^{-1}g,g\rangle dV_\omega<+\infty$, there exists $f\in L^2(X,\Lambda^{p,q-1}T^*_X\otimes E)$ such that $\bar{\partial}f=g$ and $$\int_X|f|^2dV_\omega\leq \int_X\langle B^{-1}g,g\rangle dV_\omega.$$
The following $L^2$-extension theorem for Kähler families is due to Zhou-Zhu [[@ZhouZhu192 Theorem 1.1]]{}. The same result for projective families is due to Guan-Zhou [@GZh12; @GZh15d].
\[thm: optimal L2 extension\] Let $\pi:X\rightarrow B$ be a proper holomorphic submersion from a complex $n$-dimensional K" ahler manifold $(X,\omega)$ onto a unit ball in $\mathbb C^m$. Let $(E,h=h_E)$ be a Hermitian holomorphic vector bundle over $X$, such that the curvature $i\Theta_{E,h_E}\geq 0$ in the sense of Nakano. Let $t_0\in B$ be an arbitrarily fixed point. Then for every section $u\in H^0(X_{t_0}, K_{X_{t_0}}\otimes E|_{X_{t_0}})$, such that $$\begin{aligned}
\int_{X_{t_0}}|u|_{\omega, h}^2dV_{\omega_{X_{t_0}}}<+\infty,\end{aligned}$$ there is a section $\widetilde u\in H^0(X,K_X\otimes E)$, such that $\widetilde u|_{X_{t_0}}=\widetilde u\wedge dt$, with the following $L^2$-estimate $$\begin{aligned}
\int_X|\widetilde u|^2_{\omega,h}dV_{X,\omega}\leq \mu({B})\int_{X_{t_0}}|u|_{\omega, h}^2dV_{\omega_{X_{t_0}}},\end{aligned}$$ where $dt=dt_1\wedge\cdots\wedge dt_m$, and $t=(t_1,\cdots, t_m)$ are the holomorphic coordintes on $\mathbb C^m$, and $\mu(B)$ is the volume of the unit ball in $\mathbb C^m$ with respect to the Lebesgue measure on $\mathbb C^m$.
We take $R(t)=e^{-t}$, $\alpha_0=\alpha_1=0$, and $\psi=m\log\|t-t_0\|^2$ in [@ZhouZhu192 Theorem 1.1], and from [@GZh15d Lemma 4.14], [@ZhouZhu192 Remark 1.2], we can get the precise form of Theorem \[thm: optimal L2 extension\].
Positivities of holomorphic vector bundles via $L^p$-conditions of $\bar\partial$
=================================================================================
The aim of this section is to prove Theorem \[thm:theta-nakano text\_intr\]– Theorem \[thm: multiple coarse Lp: Griffiths positivity-intr.\].
Characterizations of Nakano positivity in term of optimal $L^2$-estimate condition
----------------------------------------------------------------------------------
\[thm:theta-nakano text\] Let $(X,\omega)$ be a Kähler manifold of dimension $n$ with a K" ahler metric $\omega$, which admits a positive Hermitian holomorphic line bundle, $(E,h)$ be a smooth Hermitian vector bundle over $X$, and $\theta\in C^0(X,\Lambda^{1,1}T^*_X\otimes End(E))$ such that $\theta^*=\theta$. If for any $f\in\mathcal{C}^\infty_c(X,\wedge^{n,1}T^*_X\otimes E\otimes A)$ with $\bar\partial f=0$, and any positive Hermitian line bundle $(A,h_A)$ on $X$ with $i\Theta_{A,h_A}\otimes Id_E+\theta>0$ on $\text{supp}f$, there is $u\in L^2(X,\wedge^{n,0}T_X^*\otimes E\otimes A)$, satisfying $\bar\partial u=f$ and $$\int_X|u|^2_{h\otimes h_A}dV_\omega\leq \int_X\langle B_{h_A,\theta}^{-1}f,f\rangle_{h\otimes h_A} dV_\omega,$$ provided that the right hand side is finite, where $B_{h_A,\theta}=[i\Theta_{A,h_A}\otimes Id_E+\theta,\Lambda_\omega]$, then $i\Theta_{E,h}\geq\theta$ in the sense of Nakano. On the other hand, if in addition $X$ is assumed to have a complete Kähler metric, the above condition is also necessary for that $i\Theta_{E,h}\geq\theta$ in the sense of Nakano. In particular, if $(E,h)$ satisfies the optimal $L^2$-estimate condition, then $(E,h)$ is Nakano semi-positive.
The second statement is a corollary of Theorem \[thm: L2 estimate Nakano\]. We now give the proof of the first statement. We give the proof in the case that $\theta$ is $\mathcal C^1$, and the general case follows the proof by an approximation argument.
To illustrate the main idea more clearly, we may assume that there is a smooth strictly plurisubharmonic function on $X$, which corresponds to the existence of a positive Hermitian trivial holomophic line bundle on $X$. For general case, the same proof goes through by replacing data related to $e^{-\psi}$ by $h_A$, and using Proposition \[prop: construct function 1\].
Let $\psi$ be any smooth strictly plurisubharmonic function on $X$. By assumption, we can solve the equation $\bar{\partial}u=f$ for any $\bar\partial$-closed $f\in\mathcal{C}^\infty_c(X,\wedge^{n,1}T^*_X\otimes E)$, with the estimate $$\int_X|u|^2e^{-\psi}dV_\omega\leq \int_X\langle B_{\psi,\theta}^{-1} f,f\rangle e^{-\psi}dV_\omega,$$ where $B_{\psi,\theta}:=[i\partial\bar\partial \psi\otimes Id_E+\theta, \Lambda_\omega]$. For any $\alpha\in\mathcal{C}^\infty_c(X,\wedge^{n,1}T^*_X\otimes E)$, we have $$\begin{aligned}
|\langle\langle\alpha,f\rangle\rangle_{\psi}|&=|\langle\langle\alpha,\bar{\partial}u\rangle\rangle_{\psi}|\\
&=|\langle\langle {\bar{\partial}}^{*} \alpha,u\rangle\rangle_{\psi}|\\
&\leq||u||_{\psi}||{\bar{\partial}}^{*} \alpha||_{\psi},\end{aligned}$$ where $\bar\partial^*$ is the adjoint of $\bar\partial$ with respect to $\omega$, $e^{-\psi}h$.
From Lemma \[lem: BKN identity\], we obtain $$\label{eq1}
\begin{split}
&|\langle\langle\alpha,f\rangle\rangle_{\psi}|^2\\
\leq& \int_ X\langle B_{\psi,\theta}^{-1}f,f\rangle e^{-\psi}dV_\omega\\
&\times\left(||D'\alpha||_\psi^2+||{D'}^*\alpha||_\psi^2+\langle\langle[i\Theta_{E,h}+
i\partial\bar\partial\psi\otimes Id_E,\Lambda_\omega]\alpha,\alpha\rangle\rangle_\psi-||{\bar{\partial}}\alpha||_\psi^2\right)\\
\leq& \int_ X\langle B_{\psi,\theta}^{-1}f,f\rangle e^{-\psi}dV_\omega\times\left(\langle\langle[i\Theta_{E,h}+
i\partial\bar\partial\psi\otimes Id_E,\Lambda_\omega]\alpha,\alpha\rangle\rangle_\psi+||{D'}^*\alpha||_\psi^2\right),
\end{split}$$ where $D'$ is the $(1,0)$ part of the Chern connection on $E$ with respect to the metric $e^{-\psi}h$.
Let $\alpha=B_{\psi,\theta}^{-1}f$, i.e., $f=B_{\psi,\theta} \alpha.$ Then inequality (\[eq1\]) becomes $$\begin{split}
&\left(\langle\langle B_{\psi,\theta} \alpha,\alpha\rangle\rangle_{\psi}\right)^2\\
\leq&\langle\langle B_{\psi,\theta} \alpha,\alpha\rangle\rangle_{\psi}
\left(\langle\langle[i\Theta_{E,h},\Lambda_\omega]\alpha,\alpha\rangle\rangle_\psi+
\langle\langle B_{\psi,0} \alpha,\alpha\rangle\rangle_{\psi}+||{D'}^*\alpha||_\psi^2\right).
\end{split}$$ Therefore, we can get $$\label{eq2}
\langle\langle[i\Theta_{E,h}-\theta,\Lambda_\omega]\alpha,\alpha\rangle\rangle_\psi+||{D'}^*\alpha||_\psi^2\geq0.$$ We argue by contradiction. Suppose that $i\Theta_{E,h}-\theta$ is not Nakano semi-positive on $X$. By Lemma \[lem: Hermitian operator formula\], there is $x_0\in X$ and $\xi_0\in \Lambda^{n,1}T^*_{X,x_0}\otimes E_{x_0}$ such that $|\xi_0|=1$ and $\langle[i\Theta_{E,h}-\theta,\Lambda_{\omega}]\xi_0,\xi_0\rangle=-2c$ for some $c>0$.
Let $(U;z_1,z_2,\cdots,z_n)$ be a holomorphic coordinate on $X$ centered at $x_0$ such that $\omega=i\sum dz_j\wedge d\bar z_j+O(|z|^2)$, and assume $\{e_1,e_2,\cdots,e_r\}$ is a holomorphic frame of $E$ on $U$. Let $\xi=\sum \xi_{j\lambda}dz_1\wedge\cdots\wedge dz_n\wedge d\bar{z}_j\otimes e_\lambda$, with constant coefficients such that $\xi(x_0)=\xi_0$. We may assume $$\langle[i\Theta_{E,h}-\theta,\Lambda_\omega]\xi,\xi\rangle<-c$$ on $U$. Choose $R>0$ such that $B(0,R):=\{z:|z|<R\}\subset U$, and write $B(0,R)$ as $B_R$.
Choose $\chi\in \mathcal{C}^\infty_c(B_R)$, satisfying $\chi(z)=1$ for $z\in B_{R/2}$. Let $f=\bar{\partial}\nu$ with $$\nu(z)=(-1)^n \sum_{j,\lambda}\xi_{j\lambda}\bar{z}_j\chi(z)dz_1\wedge\cdots\wedge dz_n\otimes e_\lambda.$$ Then $$f(z)=\sum \xi_{j\lambda}dz_1\wedge\cdots\wedge dz_n\wedge d\bar{z}_j\otimes e_\lambda$$ for $z\in B_{R/2}$. From Proposition \[prop: construct function 1\], we can construct a smooth strictly plurisubharmonic function $\psi$ on $X$, such that $\psi|_{B_R}(z)=|z|^2-\frac{R^2}{4}$. For any integer $m>0$, set $$\psi_m(z)=m\psi(z).$$ As before, set $\alpha_m=B_{\psi_m,\theta}^{-1}f=\frac{1}{m}B_{\psi,\theta/m}^{-1}f$. By [@Dem Chapter VII, Theorem 1.1], we have $${D'}^*B_{\psi,0}^{-1}f(0)=0.$$ So after shrinking $R$, we can get $|{D'}^*\alpha_m(z)|\leq\frac{\sqrt c}{2m}$ for $z\in B_{R/2}$ and any $m$. Since $f$ has compact support in $B_R$, there is a constant $C>0$, such that $|\langle[i\Theta_{E,h}-\theta,\Lambda_\omega]\alpha_m,\alpha_m\rangle|\leq \frac{C^2}{m^2}$ and $|{D'}^*\alpha_m|\leq \frac{C}{m}$ hold for any $m>0$.
We now estimate both terms in with $\alpha$ and $\psi$ replaced by $\alpha_m$ and $\psi_m$ defined as above.
$$\label{eq4}
\begin{split}
&m^2\left(\langle\langle[i\Theta_{E,h}-\theta,\Lambda_\omega]\alpha_m,\alpha_m\rangle\rangle_{\psi_m}+||{D'}^*\alpha_m||_{\psi_m}^2\right)\\
=&m^2\left(\int_{B_{R/2}}\langle[i\Theta_{E,h}-\theta,\Lambda_\omega]\alpha_m,\alpha_m\rangle e^{-\psi_m}dV_\omega+
\int_{B_{R/2}}| {D'}^*\alpha_m|^2e^{-\psi_m}dV_\omega\right)\\
&+m^2\left(\int_{ B_R\setminus B_{R/2}} \langle[i\Theta_{E,h}-\theta,\Lambda_\omega]\alpha_m,\alpha_m\rangle e^{-\psi_m}dV_\omega
+\int_{ B_R\setminus B_{R/2}} | {D'}^*\alpha_m|^2e^{-\psi_m}dV_\omega\right)\\
\leq& -\frac{3c}{4}\int_{B_{R/2}}e^{-\psi_m}dV_\omega+2C^2\int_{ B_R\setminus B_{R/2}} e^{-\psi_m}dV_\omega.
\end{split}$$
Since $\lim_{m\rightarrow+\infty}\psi_m(z)=+\infty$ for $z\in B_R\setminus \overline{B}_{R/2}$, and $\psi_m(z)\leq 0$ for $z\in B_{R/2}$ and all $m$. Therefore, we obtain from that $$\langle\langle[i\Theta_{E,h}-\theta,\Lambda_\omega]\alpha_m,\alpha_m\rangle\rangle_{\psi_m}+||{D'}^*\alpha_m||_{\psi_m}^2<0$$ for $m>>1$, which contradicts to the inequality (\[eq2\]).
With the discussion in §\[subsec:infinite rank v.b.\], the above proof holds for vector bundles of infinite rank.
Griffiths positivity in terms of multiple coarse $L^p$-estimate condition
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\[thm:coarse estimate text\] Let $(X,\omega)$ be a Kähler manifold, which admits a positive Hermitian holomorphic line bundle, and $(E,h)$ be a holomorphic vector bundle over $X$ with a continuous Hermitian metric $h$. If $(E,h)$ satisfies the multiple coarse $L^p$-estimate condition for some $p>1$, then $(E,h)$ is Griffiths semi-positive.
We prove the theorem by modifying the idea in [@HI; @DNW1]. For the same reason as in the proof of Theorem \[thm:theta-nakano text\], we may assume that there is a strictly smooth plurisubharmonic function on $X$.
We will show that $(E,h)$ satisfies the multiple coarse $L^p$-extension condition. We assume that $D', z=(z_1, \cdots, z_n)$ is an arbitrary coordinate chart on $X$, and let $D$ be an arbitrary relatively compact subset of $D'$. We assume that $E|_{D'}=D'\times{\mathbb{C}}^r$ is trivial and $\omega|_D\leq C\ i/2\sum_{j=1}^{n}dz_j\wedge d\bar{z}_j$ with some $C>0$.
Fix an integer $m>0$, $w\in D$ (we identify $w$ with its coordinate $z(w)$) and $a\in E_w$ with $|a|_h=1$. We will construct $f \in H^0(X,E^{\otimes m})$ such that $f(w) = a^{\otimes m}$ and $$\int_{X} |f|^p _{h^{\otimes m}}dV_\omega\leq C'_m ,$$ where $C'_m$ are uniform constants independent of $w$ that satisfy $$\lim_{m\rightarrow\infty}\frac{\log C'_m}{m}=0.$$
Let $\chi = \chi(t)$ be a smooth function on $\mathbb{R}$, such that
- $\chi(t) = 1$ for $t \leq 1/4$,
- $\chi(t) = 0$ for $t \geq 1$, and
- $|\chi'(t)|\leq 2$ on $\mathbb{R}$.
Viewing $a$ as a constant section of $E|_D$, we define an $E^{\otimes m}$-valued $(n,1)$-form $\alpha_\epsilon$ by $$\begin{aligned}
\alpha_\epsilon &:= {\bar{\partial}}\chi\left({|z-w|^2 \over \epsilon^2 }\right)dz \otimes a^{\otimes m}\\
&= \chi'\left(\frac{|z-w|^2}{\epsilon^2} \right) \sum_j \frac{z_j-w_j}{\epsilon^2} d\bar{z}_j\wedge dz\otimes a^{\otimes m},\end{aligned}$$ where $dz=dz_1\wedge\cdots\wedge dz_n$, and from Proposition \[prop: construct function 1\], we can choose a smooth strictly plurisubharmonic function $\psi_\delta$ on $X$ such that $$\psi_{\delta}|_D = |z|^2 + n \log(|z-w|^2 + \delta^2),$$ where $0<\epsilon, \delta\ll 1$ are parameters. From the multiple coarse $L^p$-estimate condition, we obtain a smooth section $u_{\epsilon, \delta}$ of $E^{\otimes m}$-valued $(n,0)$-form on $ X$ such that ${\bar{\partial}}u_{\epsilon,\delta} = \alpha_\epsilon$ and $$\int_{X} |u_{\epsilon,\delta}|^p _{h^{\otimes m}}e^{-\psi_\delta}dV_\omega\leq C_m\int_{X}\langle B^{-1}_{\psi_{\delta} } \alpha_\epsilon,\alpha_\epsilon\rangle^{\frac{p}{2}} e^{-\psi_\delta}dV_\omega.\label{eqn:dbar-est}$$
On $D$, we have the following estimate: $$\begin{split}\langle B^{-1}_{\psi_{\delta} } \alpha_\epsilon,\alpha_\epsilon\rangle
& = | \chi' (\frac{|z-w|^2}{\epsilon^2} ) |^2 \cdot \frac{1}{\epsilon^4}
\langle B^{-1}_{\psi_{\delta} }\sum_j (z_j-w_j) d\overline{z}_j\wedge dz\otimes a^{\otimes m},\sum_j (z_j-w_j) d\overline{z}_j\wedge dz\otimes a^{\otimes m}\rangle\\
& \leq C_1| \chi' (\frac{|z-w|^2}{\epsilon^2} ) |^2 \cdot \frac{1}{\epsilon^4}|z-w|^2|a|_{h(z)}^{2m},
\end{split}$$ where $C_1$ depends only on $\omega$. Note that $$\text{supp}\ \chi'\left(\frac{|z-w|^2}{\epsilon^2} \right)\subset\{1/4 \leq |z-w|^2 /\epsilon^2 \leq 1 \}$$ and $\psi_{\delta}\geq 2n\log|z-w|$, we have $$\label{ineq t3}
\begin{split}
(\text{RHS of (\ref{eqn:dbar-est})}) &\leq C_m C_1^{\frac{p}{2}}\int_{\{\epsilon^2 / 4 \leq |z-w|^2 \leq \epsilon^2\}}| \chi' (\frac{|z-w|^2}{\epsilon^2} )|^p \frac{1}{\epsilon^{2p}}|z-w|^p e^{- \psi_{\delta}}|a|_{h(z)}^{mp}dV_\omega\\
& \leq C_m C_1^{\frac{p}{2}}\frac{2^p}{\epsilon^{2p}} \int_{\{\epsilon^2 / 4 \leq |z-w|^2 \leq \epsilon^2\}} |z-w|^pe^{- \psi_{\delta}}|a|_{h(z)}^{mp}dV_\omega\\
&\leq C_m C_1^{\frac{p}{2}}\frac{2^p}{\epsilon^{2p}}\int_{\{\epsilon^2 / 4 \leq |z-w|^2 \leq \epsilon^2\}} \epsilon^p
\sup_{B(w,\epsilon)}|a|_{h(z)}^{mp} e^{-2n\log|z-w|}dV_\omega\\
&\leq C_2C_m \frac{\sup_{B(w,\epsilon)}|a|_{h(z)}^{mp}}{\epsilon^p},
\end{split}$$ where $C_2=2^{p+2n}C_1^{\frac{p}{2}}C^n\mu(B_1)$ and $\mu(B_1)$ is the volume of the unit ball $B_1$ with respect to the Lebesgue measure.
To summarize, we have obtained a smooth section $u_{\epsilon,\delta}$ of $E^{\otimes m}$-valued $(n,0)$-form on $ X$ such that
- ${\bar{\partial}}u_{\epsilon,\delta} = \alpha_\epsilon$, and
- the following estimate holds: $$\int_{D} |u_{\epsilon,\delta}|_{h^{\otimes m}}^p e^{-\psi_{\delta}}dV_\omega \leq C_2C_m\frac{\sup_{B(w,\epsilon)}|a|_{h(z)}^{mp}} {\epsilon^p}.\label{eqn:matome}$$
Note that the weight function $\psi_{\delta}$ is decreasing when $\delta \searrow 0$, $e^{-\psi_{\delta}}$ is increasing when $\delta \searrow 0$. Fix $\delta_0>0$. Then, for $\delta < \delta_0$, we have that $$\int_{D} |u_{\epsilon,\delta}|_{h^{\otimes m}}^p e^{-\psi_{\delta_0}}dV_\omega \leq \int_{D} |u_{\epsilon,\delta}|_{h^{\otimes m}}^p e^{- \psi_{\delta}}dV_\omega \leq C_2C_m\frac{\sup_{B(w,\epsilon)}|a|_{h(z)}^{mp}} {\epsilon^p}.$$ Thus $\{u_{\epsilon,\delta}\}_{\delta < \delta_0}$ forms a bounded sequence in $L^p(X, K_X\otimes E^{\otimes m}, e^{-\psi_{\delta_0}})$. Note that $p>1$, we can choose a sequence $\{u_{\epsilon,\delta^{(k)}}\}_k$ in $L^p(X, e^{-\delta_0})$ which weakly converges to some $u_\epsilon\in L^p(X, K_X\otimes E^{\otimes m}, e^{-\psi_{\delta_0}})$, satisfying $$\int_{D} |u_\epsilon|_{h^{\otimes m}}^p e^{-\psi_{\delta_0}} dV_\omega\leq C_2C_m\frac{\sup_{B(w,\epsilon)}|a|_{h(z)}^{mp}} {\epsilon^p}.$$ Repeating this argument for a sequence $\{\delta_j\}$ decreasing to $0$, by diagonal argument, we can select a sequence $\{u_{\epsilon,\delta^k}\}_k$ which weakly converges to $u_\epsilon$ in $L^p(X, K_X\otimes E^{\otimes m}, e^{-\psi_{\delta_j}})$ with $u_\epsilon$ satisfying $$\int_{D} |u_\epsilon|_{h^{\otimes m}}^p e^{-\psi_{\delta_j}}dV_\omega \leq C_2C_m\frac{\sup_{B(w,\epsilon)}|a|_{h(z)}^{mp}} {\epsilon^p}$$ for all $j$. By the monotone convergence theorem, $$\int_{ D} |u_\epsilon|_{h^{\otimes m}}^p e^{- \psi_{0}}dV_\omega \leq C_2C_m\frac{\sup_{B(w,\epsilon)}|a|_{h(z)}^{mp}} {\epsilon^p}.$$ Since $\bar\partial$ is weakly continuous, we also have ${\bar{\partial}}u_\epsilon = \alpha_\epsilon.$
Since $\frac{1}{|z-w|^{2n}}$ is not integrable near $w$, $u_\epsilon(w)$ must be $0$. Let $$f_\epsilon:= \chi(|z-w|^2/\epsilon^2)dz\otimes a^{\otimes m} - u_\epsilon.$$ Then $f_\epsilon\in H^0(X,\wedge^{(n,0)}T^*_X\otimes E^{\otimes m})$, $f_\epsilon(0) = dz \otimes a^{\otimes m}$ and $$\label{eqn:4}
\begin{split}
\int_{D} |f_\epsilon|_{h^{\otimes m}}^pdV_\omega
&\leq \left(\left(\int_{D} \left|\chi(|z-w|^2/\epsilon^2)dz\otimes a^{\otimes m}\right|_{h^{\otimes m}}^pdV_\omega\right)^{1/p}
+ \left(\int_{D} |u_\epsilon|_{h^{\otimes m}}^pdV_\omega\right)^{1/p}\right)^p\\
&\leq 2^p\left(\int_{D} \left|\chi(|z-w|^2/\epsilon^2)dz\otimes a^{\otimes m}\right|_{h^{\otimes m}}^pdV_\omega +\int_{D} |u_\epsilon|_{h^{\otimes m}}^pdV_\omega \right)
\end{split}$$
Since $\chi \leq 1$ and the support of $\chi(|z-w|^2 / \epsilon^2) $ is contained in $\{|z-w|^2 \leq \epsilon^2 \}$ and $0<\epsilon\leq1$, we have $$\int_{D} \left|\chi(|z-w|^2/\epsilon^2)dz\otimes a^{\otimes m}\right|_{h^{\otimes m}}^pdV_\omega\leq C^n\mu( B_1)\sup_{B(w,\epsilon)}|a|_{h(z)}^{mp}.$$ We also have $$\begin{aligned}
\int_{D}|u_\epsilon|_{h^{\otimes m}}^pdV_\omega &\leq \sup_{z \in D} e^{\psi_0(z)} \cdot \int_{D}
|u_\epsilon|_{h^{\otimes m}}^p e^{-\psi_0 }dV_\omega \\
&\leq\sup_{z \in D}e^{\psi_0(z)} \cdot C_2C_m\frac{\sup_{B(w,\epsilon)}|a|_{h(z)}^{mp}}{\epsilon^p} \\
&\leq C_3C_m\frac{\sup_{B(w,\epsilon)}|a|_{h(z)}^{mp} }{\epsilon^p},\end{aligned}$$ where $C_3$ is a constant depends only on $D$. We may assume $C_m\geq1$. Combining these estimates with , we obtain that $$\int_{D} |f_\epsilon|_{h^{\otimes m}}^p dV_\omega \leq C_4 C_m\frac{\sup_{B(w,\epsilon)}|a|_{h(z)}^{mp}}{\epsilon^p},$$ where $C_4$ is a constant independent of $m$ and $w$.
Let $$O_\epsilon=\sup\limits_{z,w\in D, |z-w|\leq\epsilon} \left|\log |a|_{h(z)}-\log |a|_{h(w)}\right|.$$ By the uniform continuity of $\log |a|_{h(z)}$ on $D$, $O_\epsilon$ is finite and goes to 0 as $\epsilon{\rightarrow}0$. Let $\epsilon := 1/m$. We have $\left|mp\log |a|_{h(z)} - mp\log |a|_{h(w)}\right| \leq mp O_{1/m}$ for $|z-w| \leq 1/m$. Then $$\begin{aligned}
\int_{D} |f_{1/m}|^p e^{-m\phi}dV_\omega
&\leq C_4C_m m^{p} e^{mp\log |a|_{h(w)}+mp O_{1/m}}\nonumber\\
&= C_4C_mm^{p} e^{mp O_{1/m}}.\label{eqn:last}\end{aligned}$$ Let $C'_m=C_4C_mm^{p} e^{mp O_{1/m}}$, we have $$\frac{\log C'_m}{m}=\frac{\log (C''C_mm^{p})}{m}+ pO_{1/m}\to 0.$$ Considering $f_{1/m}/dz$, we see that $(E,h)$ satisfies the multiple coarse $L^p$-extension condition on $D$, and hence $(E,h)$ is Griffiths semi-positive on $D$ by [@DWZZ1 Theorem 1.2]. Since $D$ is arbitrary, $(E,h)$ is Griffiths semi-positive on $X$.
Griffiths positivity in terms of optimal $L^p$-extension condition
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\[thm: optimal Lp estimate : Griffiths positive\] Let $E$ be a holomorphic vector bundle over a domain $D\subset{\mathbb{C}}^n$, and $h$ be a singular Finsler metric on $E$, such that $|s|_{h^*}$ is upper semi-continuous for any local holomorphic section $s$ of $E^*$. If $(E,h)$ satisfies the optimal $L^p$-extension condition for some $p>0$, then $(E,h)$ is Griffiths semi-positive.
Let $u$ be a holomorphic section of $E^*$ over $D$. Let $z\in D$ and $P$ be any holomorphic cylinder such that $z+P\subset D$. Take $a\in E_z$ such that $|a|_h=1$ and $|u|_{h^*}(z)=|\langle u(z),a\rangle|$. Since $(E,h)$ satisfies the optimal $L^p$-extension condition, there is a holomorphic section $f$ of $E$ on $z+P$, such that $f(z)=a$ and satisfies the estimate $$\begin{aligned}
\label{eqn: optimal Lp extension a}
\frac{1}{\mu(P)}\int_{z+P}|f|^p_h\leq 1.
\end{aligned}$$ Note that $|u|_{h^*}\geq |\langle u,f\rangle|/|f|_h$ on $z+P$, it follows that $$\log|u|_{h^*}\geq \log|\langle u,f\rangle|-\log|f|_h.$$ Taking integration, we get that $$\begin{aligned}
p\left(\frac{1}{\mu(P)}\int_{z+P}\log |u|_{h^*}\right)
&\geq p\left(\frac{1}{\mu(P)}\int_{z+P}\log|\langle u,f\rangle|\right)- \frac{1}{\mu(P)}\int_{z+P}\log|f|^p_h\\
&\geq p\left(\frac{1}{\mu(P)}\int_{z+P}\log|\langle u,f\rangle|\right)-\log\left(\frac{1}{\mu(P)}\int_{z+P}|f|^p_h\right)\\
&\geq p\log|\langle u(z),f(z)\rangle|\\
&=p\log|\langle u(z), a\rangle|=p\log|u(z)|_{h^*},
\end{aligned}$$ where the second inequality follows from Jensen’s inequality and , and the third inequality follows from the fact that $\log|\langle u,f\rangle|$ is a plurisubharmonic function, and from [@DNW1 Lemma 3.1]. Dividing by $p$, we obtain that $$\begin{aligned}
\log|u(z)|_{h^*}\leq \frac{1}{\mu(P)}\int_{z+P}\log |u|_{h^*}.
\end{aligned}$$ Again from [@DNW1 Lemma 3.1], we see that $\log |u|_{h^*}$ is plurisubharmonic on $D$.
Griffiths positivity in terms of multiple coarse $L^p$-extension condition
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The following theorem was originally given in [@DWZZ1 Theorem 6.4]. In the present paper, we give a new proof of it based on Guan-Zhou’s idea [@GZh15d] about connecting optimal $L^2$-extension condition to Berndtsson’s plurisubharmonic variation of relative Bergman kernels [@Bob06].
\[thm: multiple coarse Lp: Griffiths positivity.\] Let $E$ be a holomorphic vector bundle over a domain $D\subset{\mathbb{C}}^n$, and $h$ be a singular Finsler metric on $E$, such that $|s|_{h^*}$ is upper semi-continuous for any local holomorphic section $s$ of $E^*$. If $(E,h)$ satisfies the multiple coarse $L^p$-extension condition, then $(E,h)$ is Griffiths semi-positive.
Let $u$ be a holomorphic section of $E^*$ over $D$. Then $u^{\otimes m}\in H^0(D,({E^*})^{\otimes m})$.
Let $z\in D$ and $P$ be any holomorphic cylinder such that $z+P\subset D$. Take $a\in E_z$ such that $|a|_h=1$ and $|u|_{h^*}(z)=|\langle u(z),a\rangle|$. Since $(E,h)$ satisfies the multiple coarse $L^p$-extension condition, there is $f_m\in H^0(D, E^{\otimes m})$, such that $f_m(z)=a^{\otimes m}$ and satisfies the following estimate $$\int_D|f_m|^p\leq C_m,$$ where $C_m$ are constants independent of $z$ and satisfy the growth condition $\frac{1}{m}\log C_m\rightarrow 0$ as $m\rightarrow \infty$. Since $|u^{\otimes m}|_{(h^*)^{\otimes m}}=|u|^m_{h^*}\geq \frac{|\langle u^{\otimes m},f_m\rangle|}{|f_m|_{h^{\otimes m}}}$, we have that $$m\log |u|_{h^*}\geq \log|\langle u^{\otimes m},f_m\rangle|-\log |f_m|.$$ Taking integration, we get that $$\begin{aligned}
m\left(\frac{1}{\mu(P)}\int_{z+P}\log |u|_{h^*}\right)&\geq \frac{1}{\mu(P)}\int_{z+P}\log |\langle u^{\otimes m},f_m\rangle|-\frac{1}{p}\left(\frac{1}{\mu(P)}\int_{z+P}\log |f_m|^p\right)\\
&\geq m\log|u(z)|_{h^*}-\frac{1}{p}\log\left(\frac{1}{\mu(P)}\int_{z+P}|f_m|^p\right)\\
&\geq m\log|u(z)|_{h^*}-\frac{1}{p}\log\left(\frac{1}{\mu(P)}\int_{D}|f_m|^p\right)\\
&\geq m\log|u(z)|_{h^*}-\frac{1}{p}\log(C_m/\mu(P)),
\end{aligned}$$ where the first inequality follows from the fact that $\log |\langle u^{\otimes m},f_m\rangle|$ is a plurisubharmonic function, and [@DNW1 Lemma 3.1], and Jensen’s inequality, and the second inequality follows from the fact that $z+P\subset D$. Dividing by $m$ in both sides, we obtain that $$\begin{aligned}
\frac{1}{\mu(P)}\int_{z+P}\log |u|_{h^*}\geq \log|u(z)|_{h^*}-\frac{1}{mp}\log(C_m/\mu(P)).
\end{aligned}$$ Letting $m\rightarrow \infty$, we see that $\log|u|_{h^*}$ satisfies the following inequality $$\log|u(z)|_{h^*}\leq \frac{1}{\mu(P)}\int_{z+P}\log |u|_{h^*},$$ since $\frac{1}{m}\log C_m\rightarrow 0$ as $m\rightarrow \infty$. Then from [@DNW1 Lemma 3.1], we get that $\log|u|_{h^*}$ is plurisubharmonic on $D$.
Positivities of direct images of twisted relative canonical bundles
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Optimal $L^2$-estimate condition and Nakano positivity
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The aim of this subsection is to prove Theorem \[thm: direc im stein-intr\] and Theorem \[thm: direct image-optimal L2 estimate-intr\].
To avoid some complicated geometric quantities and highlight the main idea, we first consider a simple case of Theorem \[thm: direc im stein-intr\] as a warm-up.
\[thm direc im stein\] Let $U$ and $D$ be bounded domains in $\mathbb{C}_t^{n}$ and $\mathbb{C}_z^{m}$ respectively, and $\phi\in\mathcal{C}^2(\overline{U}\times \overline{D})\cap PSH(U\times D)$. Assume that $D$ is pseudoconvex. For $t\in U$, let $A^2_t:=\{f\in\mathcal{O}(D):||f||^2_t:=\int_{D}|f|^2e^{-\phi(t,\cdot)}<\infty\}$ and $F:=\coprod_{t\in U}A^2_t$. We may view $F$ as a Hermitian holomorphic vector bundle on $U$. Then $(F,||\cdot||_t)$ is Nakano semi-positive.
We will first prove that $(F,||\cdot||_t)$ satisfies the $\bar\partial$ optimal $L^2$-estimate for pseudoconvex domains contained in $U$. We may assume $U$ is pseudoconvex.
For any smooth strictly plurisubharmonic function $\psi$ on $U$, for any $\bar\partial$ closed $f\in\mathcal{C}^\infty_c(T^*_{U}\Lambda^{(0,1)}\otimes F)$ (We identify $\mathcal{C}^\infty_c(T^*_{U}\Lambda^{(0,1)}\otimes F)$ with $\mathcal{C}^\infty_c(T^*_{U}\Lambda^{(n,1)}\otimes F)$). We may write $f=\sum_{j=1}^{n}f_j(t,z)d\bar t_j$ with $f_j(t,\cdot)\in F_t$ for $t\in U$ and $j=1,2,\cdots,n$. Therefore, we may view $f$ as a $\bar\partial$-closed $(0,1)$-form on $U\times D$. By Lemma \[thm: L2 estimate Nakano\], there exists a function $u$ on $U\times D$, satisfying $\bar\partial u=f$ and $$\begin{split}
&\int_{U\times D}|u|^2e^{-(\phi+\psi)}\\
\leq&\int_{U\times D}|f|^2_{i\partial\bar\partial(\phi+\psi)}e^{-(\phi+\psi)}\\
\leq&\int_{U\times D}|f|^2_{i\partial\bar\partial\psi}e^{-(\phi+\psi)}\\
=&\int_{U}\sum_{j,k=1}^{n}\psi^{j\bar k}\langle f_j(t,\cdot),f_k(t,\cdot)\rangle_te^{-\psi},
\end{split}$$ where $(\psi^{j\bar k})_{n\times n}:=(\frac{\partial^2\psi}{\partial t_j\partial_{\bar t_k}})^{-1}_{n\times n}$. Note that $\int_{U\times D}|u|^2e^{-(\phi+\psi)}=\int_{U}||u||_t^2e^{-\psi}<\infty$ and $\frac{\partial u}{\partial\bar z_j}=0$ for $j=1,2,\cdots,m$, we may view $u$ as a $L^2$-section of $F$ on $U$. By Theorem \[thm:theta-nakano text\_intr\] and Remark \[rem:reduce to trivial bundle\], $(F,||\cdot||_t)$ is Nakano semi-positive.
Let $\Omega=U\times D\subset {\mathbb{C}}^n_t\times{\mathbb{C}}^m_z$ be a bounded pseudoconvex domains and $p:\Omega{\rightarrow}U$ be the natural projection. Let $h$ be a Hermitian metric on the trivial bundle $E=\Omega\times{\mathbb{C}}^r$ that is $C^2$-smooth to $\overline\Omega$. For $t\in U$, let $$F_t:=\{f\in H^0(D, E|_{\{t\}\times D}):\|f\|^2_{t}:=\int_D|f|^2_{h_t}<\infty\}$$ and $F:=\coprod_{t\in U}F_t$. Since $h$ is continuous to $\overline\Omega$, $F_t$ are equal for all $t\in U$ as vector spaces. We may view $(F, \|\cdot\|)$ as a trivial holomorphic Hermitian vector bundle of infinite rank over $U$.
(=Theorem \[thm: direc im stein-intr\])\[thm: direc im stein\] Let $\theta$ be a continuous real $(1,1)$-form on $U$ such that $i\Theta_{E}\geq p^*\theta\otimes Id_E$, then $i\Theta_{F}\geq \theta\otimes Id_F$ in the sense of Nakano. In particular, if $i\Theta_{E}>0$ in the sense of Nakano, then $i\Theta_{F}>0$ in the sense of Nakano.
By Theorem \[thm:theta-nakano text\], it suffices to prove that $(F,\|\cdot\|)$ satisfies: for any $f\in\mathcal{C}^\infty_c(U,\wedge^{n,1}T^*_U\otimes F\otimes A)$ with $\bar\partial f=0$, and any positive Hermitian line bundle $(A,h_A)$ on $U$ with $i\Theta_{A,h_A}+\theta>0$ on $\text{supp}f$, there is $u\in L^2(U,\wedge^{n,0}T_U^*\otimes F\otimes A)$, satisfying $\bar\partial u=f$ and $$\int_U|u|^2_{h\otimes h_A}dV_\omega\leq \int_U\langle B_{h_A,\theta}^{-1}f,f\rangle_{h\otimes h_A} dV_\omega,$$ provided that the right hand side is finite, where $B_{h_A,\theta}=[(i\Theta_{A,h_A}+\theta)\otimes Id_F,\Lambda_\omega]$.
We may write $f=\sum_{j=1}^{n}f_j(t,z)dt\wedge d\bar t_j$ with $f_j(t,\cdot)\in F_t\otimes A$ for $t\in U$ and $j=1,2,\cdots,n$. Therefore, we may view $f$ as a $\bar\partial$-closed $E\otimes p^*A$-valued $(n,1)$-form on $\Omega$. Let $\tilde{f}=f\wedge dz$, then $\tilde{f}$ is a $\bar\partial$-closed $E\otimes p^*A$-valued $(m+n,1)$-form on $\Omega$. By assumption, $i\Theta_{E}\geq p^*\theta\otimes Id_E$. We get $$i\Theta_{E}+ip^*(\Theta_{A,h_A})\otimes Id_E\geq p^*(\theta+i\Theta_{A,h_A})\otimes Id_E.$$ Therefore, $$\begin{split}
&\langle[i\Theta_{E}+ip^*(\Theta_{A,h_A})\otimes Id_E,\Lambda_\omega]^{-1}\tilde{f},\tilde{f}\rangle_{h\otimes h_A}\\
\leq&\langle[p^*(\theta+i\Theta_{A,h_A})\otimes Id_E,\Lambda_\omega]^{-1}\tilde{f},\tilde{f}\rangle_{h\otimes h_A}
\end{split}$$
By Lemma \[thm: L2 estimate Nakano\], we can find an $E\otimes p^*A$-valued $(n+m,0)$-form $\tilde u$ on $\Omega$, satisfying $\bar\partial\tilde u= \tilde f$ and $$\begin{split}
&\int_{\Omega}|\tilde u|_{h\otimes h_A}^2\\
\leq&\int_{\Omega}\langle[p^*(\theta+i\Theta_{A,h_A})\otimes Id_E,\Lambda_\omega]^{-1}\tilde{f},\tilde{f}\rangle_{h\otimes h_A}\\
=&\int_U\langle B_{h_A,\theta}^{-1}f,f\rangle_{h\otimes h_A},
\end{split}$$ where the last equality holds by the Fubini theorem. Since $\frac{\partial\tilde u}{\partial z_j}=0$, $\tilde u$ is holomorphic along fibers and we may view $u=\tilde u/dz$ as a section of $K_U\otimes F\otimes A$. Also by the Fubini theorem, we have $$\int_{\Omega}|\tilde u|^2_{h\otimes h_A}=\int_{U}||u||_{h\otimes h_A}^2<\infty.$$ We also have $\bar\partial u=f$. Hence $(F,\|\cdot\|)$ satisfies the optimal $L^2$-estimate condition and is Nakano semi-positive by \[thm:theta-nakano text\].
Let $\pi:X\rightarrow U$ be a proper holomorphic submersion from K" ahler manifold $X$ of complex dimension $m+n$, to a bounded pseudoconvex domain $U\subset \mathbb C^n$, and $(E,h)$ be a Hermitian holomorphic vector bundle over $X$, with the Chern curvature Nakano semi-positive. From Lemma \[thm: optimal L2 extension\], the direct image $F:=\pi_*(K_{X/U}\otimes E)$ is a vector bundle, whose fiber over $t\in U$ is $F_t=H^0(X_t, K_{X_t}\otimes E|_{X_t})$. There is a hermtian metric $\|\cdot\|$ on $F$ induced by $h$: for any $u\in F_t$, $$\|u(t)\|^2_t:=\int_{X_t}c_mu\wedge \bar u,$$ where $m=\dim X_t$, $c_m=i^{m^2}$, and $u\wedge \bar u$ is the composition of the wedge product and the inner product on $E$. So we get a Hermitian holomorphic vector bundle $(F, \|\cdot\|)$ over $U$.
(=Theorem \[thm: direct image-optimal L2 estimate-intr\])\[thm: direct image-optimal L2 estimate\] The Hermitian holomorphic vector bundle $(F, \|\cdot\|)$ over $U$ defined above satisfies the optimal $L^2$-estimate condition. Moreover, if $i\Theta_{E}\geq p^*\theta\otimes Id_E$ for a continuous real $(1,1)$-form $\theta$ on $U$, then $i\Theta_{F}\geq \theta\otimes Id_F$ in the sense of Nakano.
Similar to the proof of Theorem \[thm: direc im stein\], we may assume $\theta=0$. From Theorem \[thm:theta-nakano text\], it suffices to prove that $(\pi_*(K_{X/Y}\otimes E), \|\cdot\|)$ satisfies the optimal $L^2$-estimate condition with the standard Kähler metric $\omega_0$ on $U\subset{\mathbb{C}}^n$. Let $\omega$ be an arbitrary Kähler metric on $X$.
Let $f$ be a $\bar\partial$-closed compact supported smooth $(n,1)$-form with values in $F$, and let $\psi$ be any smooth strictly plurisubharmonic function on $U$.
We can write $f(t)=dt\wedge(f_1(t)d\bar t_1+\cdots +f_n(t)d\bar t_n)$, with $f_i(t)\in F_t=H^0(X_t, K_{X_t}\otimes E)$. One can identify $f$ as a smooth compact supported $(n+m,1)$-form $\tilde f(t,z):=dt\wedge (f_1(t,z)d\bar t_1+\cdots+f_n(t,z)d\bar t_n)$ on $X$, with $f_i(t,z)$ being holomorphic section of $K_{X_t}\otimes E|_{X_t}$. We have the following observations:
- $\bar\partial_zf_i(t,z)=0$ for any fixed $t\in B$, since $f_i(t,z)$ are holomorphic sections $K_{X_t}\otimes E|_{X_t}$.
- $\bar\partial_tf=0$, since $f$ is a $\bar\partial$-closed form on $B$.
It follows that $\tilde f$ is a $\bar\partial$-closed compact supported $(n+m,1)$-form on $X$ with values in $E$. We want to solve the equation $\bar\partial u=\tilde f$ on $X$ by using Lemma \[thm: L2 estimate Nakano\]. Now we equipped $E$ with the metric $\tilde h:=he^{-\pi^*\psi}$, then $i\Theta_{E,\tilde h}=i\Theta_{E,h}+i\partial\bar\partial\pi^*\psi\otimes Id_{E}$, which is also semi-positive in the sense of Nakano.
We consider the integration $$\int_X\langle [i\Theta_{E,h}+i\partial\bar\partial \pi^*\psi\otimes Id_E, \Lambda_{\omega}]^{-1}\tilde f,\tilde f\rangle e^{-\pi^*\psi}dV_\omega.$$ Note that, acting on $\Lambda^{n+m,1}T^*_X\otimes E$, by Lemma \[lem: Hermitian operator formula\], we have $$\begin{aligned}
[i\Theta_{E,h}+i\partial\bar\partial \pi^*\psi\otimes Id_E, \Lambda_{\omega}]
\geq [i\partial\bar\partial \pi^*\psi\otimes Id_E, \Lambda_{\omega}].\end{aligned}$$ Thus we obtain that $$\begin{aligned}
[i\Theta_{E,h}+i\partial\bar\partial \pi^*\psi\otimes Id_E, \Lambda_{\omega}]^{-1}\leq [i\partial\bar\partial \pi^*\psi\otimes Id_E, \Lambda_{\omega}]^{-1}.\end{aligned}$$
For any $p\in X$, we use Lemma \[lem:inverse cur. indep of metric\] to modify $\omega$ at $p$. We take a local coordinate $(t_1, \cdots, t_n, z_1, \cdots, c_m)$ on $X$ near $p$, where $t_1,\cdots, t_n$ is the standard coordinate on $U\subset{\mathbb{C}}^n$. Let $\omega'=i\sum_{j=1}^n dt_j\wedge d\bar{t}_j+i\sum_{l=1}^{m}dz_l\wedge d\bar{z}_l$.
Note that $$i\partial\bar\partial \pi^*\psi=\sum_{j=1}^{n}\frac{\partial^2\psi}{\partial t_j\partial\bar{t}_k}dt_j\wedge d\bar{t}_k,$$ we have $$[i\partial\bar\partial \pi^*\psi\otimes Id_E, \Lambda_{\omega'}]\tilde f
=\sum_{j,k}\frac{\partial^2\psi}{\partial t_j\partial\bar{t}_k}f_j(t,z)dt\wedge d\bar{t}_k,$$ and $$[i\partial\bar\partial \pi^*\psi\otimes Id_E, \Lambda_{\omega'}]^{-1}\tilde f
=\sum_{j,k}\psi^{jk}f_j(t,z)dt\wedge d\bar{t}_k$$ at $p$, where $(\psi^{jk})=(\frac{\partial^2\psi}{\partial t_j\partial\bar{t}_k})^{-1}.$ By Lemma \[lem:inverse cur. indep of metric\], we have $$\begin{split}
&\langle[i\partial\bar\partial \pi^*\psi\otimes Id_E, \Lambda_{\omega}]^{-1}\tilde f,\tilde f\rangle_{\omega}dV_{\omega}\\
=&\langle[i\partial\bar\partial \pi^*\psi\otimes Id_E, \Lambda_{\omega'}]^{-1}\tilde f,\tilde f\rangle_{\omega'}dV_{\omega'}\\
=&\sum_{j,k}\psi^{jk}c_m f_j\wedge \bar{f}_kc_ndt\wedge d\bar{t}.
\end{split}$$
By Fubini’s theorem, we get that $$\begin{aligned}
&\int_X\langle [i\partial\bar\partial \pi^*\psi\otimes Id_E, \Lambda_{\omega}]^{-1}\tilde f,\tilde f\rangle_\omega e^{-\pi^*\psi} dV_\omega\\
=&\int_X\sum_{j,k}\psi^{jk}c_m f_j\wedge \bar{f}_k e^{-\pi^*\psi}c_ndt\wedge d\bar{t}\\
=&\int_U<f_j,f_k>_t\psi^{jk}e^{-\psi}c_ndt\wedge d\bar t\\
=&\int_U\langle [i\partial\bar\partial \psi, \Lambda_{\omega_0}]^{-1}f,f\rangle_te^{-\psi}dV_{\omega_0},\end{aligned}$$ where by $\langle\cdot,\cdot\rangle_t$, we mean that pointwise inner product with respect to the Hermitian metric $\|\cdot\|$ of $F$.
From Lemma \[thm: L2 estimate Nakano\], there is $\tilde u\in \Lambda^{m+n,0}(X,E)$, such that $\bar\partial\tilde u=\tilde f$, and satisfies the following estimate $$\begin{aligned}
\label{eqn: optimal L2 estimate 1}
&\int_Xc_{m+n}\tilde u\wedge \bar{\tilde u}e^{-\pi^*\psi}\notag\\
\leq &\int_X\langle [i\Theta_{E,h}+i\partial\bar\partial \pi^*\psi\otimes Id_E, \Lambda_{\omega}]^{-1}\tilde f,\tilde f\rangle e^{-\pi^*\psi}dV_\omega\\
\leq &\int_U\langle [i\partial\bar\partial \psi, \Lambda_{\omega_0}]^{-1}f,f\rangle_te^{-\psi}dV_{\omega_0}.\notag\end{aligned}$$
We observe that $\bar\partial\tilde u|_{X_t}=0$ for any fixed $t\in U$, since $\bar\partial\tilde u=\tilde f$. This means that $\tilde u_t:=\tilde u(t,\cdot)\in F_t$. Therefore we may view $\tilde u$ as a section $u$ of $F$. It is obviously that $\bar\partial u=f$.
From Fubini’s theorem, we have that $$\begin{aligned}
\label{eqn:optimal L2 estimate 2}
\int_Xc_{m+n}\tilde u\wedge \bar{\tilde u}e^{-\pi^*\psi}=\int_U\|u\|_t^2e^{-\psi}dV_{\omega_0}.\end{aligned}$$
Combining , we have $$\begin{aligned}
\int_U\|u\|_t^2e^{-\psi}dV_{\omega_0}\leq \int_U\langle [i\partial\bar\partial \psi, \Lambda_{\omega_0}]^{-1}f,f\rangle_te^{-\psi}dV_{\omega_0}.\end{aligned}$$ We have proved that $F$ satisfies the optimal $L^2$-estimate condition, thus from Theorem \[thm:theta-nakano text\] (and Remark \[rem:reduce to trivial bundle\]), $F$ is Nakano semi-positive.
Multiple coarse $L^2$-estimate condition and Griffiths positivity
-----------------------------------------------------------------
We apply Theorem \[thm:coarse estimate text-intr\] and the fiber product technique introduced in [@DWZZ1] to provide a new method to study the Griffiths positivity of direct images.
\[thm: direct image-coarse L2 estimate\] The Hermitian holomorphic vector bundle $(F, \|\cdot\|)$ over $U$ as in Theorem \[thm: direct image-optimal L2 estimate\] satisfies the multiple coarse $L^2$-estimate condition. In particular, $F$ is Griffiths semipositive.
Let $\omega_0$ be the standard K" ahler metric on $U$ and $\omega$ be an arbitrary K" ahler metric on $X$. We have the following constructions:
- Let $X_k:=X\times_\pi\cdots\times_\pi X$ be the $k$ times fiber product of $X$ with respect to the map $\pi: X\rightarrow U$.
- The induced map $X_k\rightarrow U$ by $\pi$ is denoted by $\pi_k: X_k\rightarrow U$, and $X_{k,t}:=\pi^{-1}_k(t)=X_t^k$ for every $t\in U$.
- There are natural holomorphic projections $pr_j$ from $X_k$ to its $j$-th factor $X$.
- The induced K" ahler metric $\omega_k:=pr_1^*\omega+\cdots+pr_k^*\omega$ on $X_k$.
- Set $E_j:=pr_j^*E$, and $E^k:=E_1\otimes \cdots \otimes E_k$. Then $E^k$ can be equipped with the induced metric $h^k:=pr_1^*h\otimes \cdots\otimes pr_k^*h$.
We have the following observations:
- From Lemma \[lem: Nakano fiber tensor product\], $E^k$ equipped with the Hermitian metric $h^k$ is Nakano semi-positive.
- From [@DWZZ1 Lemma 9.2], the direct image bundle $F^k:=(\pi_k)_*(K_{X_k/U}\otimes E^k)=(\pi_*(K_{X/U}\otimes E))^{\otimes k}=F^{\otimes k}$, as Hermitian holomorphic vector bundles. (In fact, [@DWZZ1 Lemma 9.2] was proved for line bundles, but it is clear that the proof also works for vector bundles.)
Let $f$ be an arbitrarily fixed smooth compactly supported $(m,1)$-form on $U$ with valued in $F^k$, such that $\bar\partial f=0$. Let $\psi$ be an arbitrary smooth strictly plurisubharmonic function on $U$. To prove that $F$ satisfies the multiple coarse $L^2$-estimate, we need to show that one can solve the equation $\bar\partial u=f$ on $U$, with the estimate $\int_U |u|^2_{h^k}e^{-\psi}\leq \int_U \langle B_{\psi}^{-1}f,f \rangle e^{-\psi}$, where $B_\psi=[i\partial\bar\partial \psi, \Lambda_{\omega_0}]$.
As in the proof of Theorem \[thm: direct image-optimal L2 estimate\], we may consider $f$ as a smooth compactly supported $K_{X_k}\otimes E^k$ valued $(0,1)$-form $\tilde f$ on $X_k$. Then it is clear that $\bar\partial \tilde f=0$. We consider the following integration $$\int_{X_k}\langle [i\Theta_{E^k,h^k}+i\partial\bar\partial \pi_k^*\psi\otimes Id_{E^k}, \Lambda_{\omega_k}]^{-1}\tilde f,\tilde f\rangle_{\omega_k} e^{-\pi_k^*\psi}dV_{\omega_k}.$$ By the same analysis as in the proof of Theorem \[thm: direct image-optimal L2 estimate\], we can get that $$\begin{aligned}
&\int_{X_k}\langle [i\Theta_{E^k,h^k}+i\partial\bar\partial \pi_k^*\psi\otimes Id_{E^k}, \Lambda_{\omega_k}]^{-1}\tilde f,\tilde f\rangle_{\omega_k} e^{-\pi_k^*\psi}dV_{\omega_k}\\
&\leq \int_{X_k}\langle [i\partial\bar\partial \pi_k^*\psi\otimes Id_{E^k}, \Lambda_{\pi_k^*\omega_0}]^{-1}\tilde f, \tilde f \rangle_{\omega_k} e^{-\pi_k^*\psi}dV_{\omega_k}\\
&=\int_{X_k} \sum_{j,k}\psi^{jk}c_{km}f_j\wedge\bar f_k e^{-\pi_k^*\psi} c_ndt\wedge d\bar t \\
&= \int_U\langle B_\psi^{-1}f,f\rangle_te^{-\psi}dV_{\omega_0}.\end{aligned}$$
Now from Lemma \[thm: L2 estimate Nakano\], we can solve the equation $\bar\partial\tilde u=\tilde f$ with the estimate $$\begin{aligned}
\int_{X_k}|\tilde u|^2_{h^k}e^{-\pi_k^*\psi}dV_{\omega_k}&\leq \int_{X_k}\langle [i\Theta_{E^k,h^k}+i\partial\bar\partial \pi_k^*\psi\otimes Id_{E^k}, \Lambda_{\omega_k}]^{-1}\tilde f,\tilde f\rangle_{\omega_k} e^{-\pi_k^*\psi}dV_{\omega_k}\\
&\leq \int_U\langle B_\psi^{-1}f,f\rangle_te^{-\psi}dV_{\omega_0}.\end{aligned}$$
Similarly, $\bar\partial\tilde u|_{X_t}=0$ for any fixed $t\in U$, since $\bar\partial\tilde u=\tilde f$. This means that $\tilde u_t:=\tilde u(t,\cdot)\in F^k_t$. Therefore we may view $\tilde u$ as a section $u$ of $F^k$. It is obviously that $\bar\partial u=f$.
Applying Fubini’s theorem to the L.H.S of above inequality, we get that $$\begin{aligned}
\int_U|u_t|_t^2e^{-\psi}dV_{\omega_0}\leq \int_U\langle B_\psi^{-1}f,f\rangle_te^{-\psi}dV_{\omega_0},\end{aligned}$$ which implies that $(F, \|\cdot\|)$ satisfies the multiple coarse $L^2$-estimate on $U$.
Optimal $L^2$-extension condition and Griffiths positivity
----------------------------------------------------------
\[thm: direct image-optimal L2 extension\]The Hermitian holomorphic vector bundle $(F, \|\cdot\|)$ over $U$ as in Theorem \[thm: direct image-optimal L2 estimate\] satisfies the optimal $L^2$-extension condition. In particular, $F$ is Griffiths semipositive.
For any $t_0\in U$, any holomorphic cylinder $P$ such that $t_0+P\subset U$, and any $a_{t_0}\in F_{t_0}$, which is a holomorphic section of $K_{X_{t_0}}\otimes E|_{X_{t_0}}$ on $X_{t_0}$. Since $E$ is Nakano semi-positive, from Lemma \[thm: L2 estimate Nakano\], we get a homolomorphic extension $ a\in H^0(X,K_X\otimes E)$ such that $ a|_{X_{t_0}}=a_{t_0}\wedge dt$, and with the estimate $$\begin{aligned}
\int_{\pi^{-1}(t_0+P)}c_{m+n} a\wedge \bar a\leq \mu(P)\int_{X_{t_0}}c_m a_{t_0}\wedge \bar a_{t_0}=\mu(P)|a_{t_0}|_{t_0}^2,\end{aligned}$$ where $\mu(P)$ is the volume of $P$ with respect to the Lebesgue measure $d\mu$ on $\mathbb C^m$. Since $ a_t:=(a/dt)|_{X_t}\in H^0(X_t,K_{X_t}\otimes E|_{X_t})$, $a/dt$ can be seen as a holomorphic section of the direct image bundle $F $ over $t_0+P$, and from Fubini’s theorem, we can obtain that $$\begin{aligned}
\int_{t_0+P}|a_t|^2_tdV_{\omega_0}\leq \mu({P})|a_{t_0}|^2_{t_0},\end{aligned}$$ which is the desired optimal $L^2$-extension.
Multiple coarse $L^2$-extension condition and Griffiths positivity
------------------------------------------------------------------
In this subsection, we will prove the following
\[thm: direct image-coarse L2 extension\] The Hermitian holomorphic vector bundle $(F, \|\cdot\|)$ over $U$ as in Theorem \[thm: direct image-optimal L2 estimate\] satisfies the multiple coarse $L^2$-extension condition. In particular, $F$ is Griffiths semipositive.
Let $(X_k, \pi_k, \omega_k, F^k)$ be as in the proof of Theorem \[thm: direct image-coarse L2 estimate\]. For any $t_0\in U$, $a_{t_0}\in F_{t_0}$, $a_{t_0}^{\otimes k}$ is a holomorphic section of $K_{X_{k,t_0}}\otimes E^k$. Since $E^k$ with the induced metric $h^k$ is semi-positive in the sense of Nakano on $X_k$, by Lemma \[thm: optimal L2 extension\], there exists $ a\in H^0(X_k,K_{X_k}\otimes E^k)$, such that $ a|_{X_{k,t_0}}=a_{t_0}^{\otimes k}\wedge dt$ and satisfies the following estimate $$\begin{aligned}
\int_{X_k}| a|^2_{h^k}dV_{\omega_k}\leq C |a_{t_0}^{\otimes k}|^2_{t_0},\end{aligned}$$ where $C$ is a universal constant which only depends on the diameter and dimension of $U$. We can view $ a_t:= (a/dt)|_{X_t}, t\in U$ as a holomorphic section of $F^k$. From Fubini’s theorem, we have that $$\begin{aligned}
\int_{X_k}| a_t|^2_{h^k}dV_{\omega_k}=\int_{U}| a_t|_t^2dV_{\omega_0}.\end{aligned}$$ In conclusion, we get a holomorphic extension $ a/dt$ of $a_{t_0}^{\otimes k}$ , with the estimate $$\begin{aligned}
\int_{U}| a_t|_t^2dV_{\omega_0}\leq C|a_{t_0}^{\otimes k}|^2_{t_0}.\end{aligned}$$ This completes the proof of Theorem \[thm: direct image-coarse L2 extension\].
\[rem: Griffiths vector bundle map\] Let $\pi:X\rightarrow Y$ be a proper holomorphic map between Kähler manifolds which may be not regular. Let $(E,h)$ be a Hermitian holomorphic vector bundle on $X$ whose Chern curvature is Nakano semi-positive. Then the direct image sheaf $\mathcal F:=\pi_*(K_{X/Y}\otimes E)$ can be equipped with a natural singular metric which is positively curved in the sense of Definition \[def:finsler on sheaf\]. In fact, let $Z\subset Y$ be the singular locus of $\pi$, then on $X\setminus \pi^{-1}(Z)$, $\pi$ is a submersion, and $\mathcal F$ is locally free and can be viewed as a vector bundle $F$ on $Y':=Y\backslash Z$, with $F_t=H^0(K_{X_t}\otimes E|_{X_t})$. The induced Hermitian metric $\|\cdot\|$ on $F$ is as follows: for any holomorphic section $u\in H^0(Y', F)$, $$\|u\|_t^2:=\int_{X_t}c_{m}u\wedge\bar u.$$ From one of Theorem \[thm: direct image-coarse L2 estimate\], and Theorem \[thm: direct image-optimal L2 extension\], Theorem \[thm: direct image-coarse L2 extension\], we see that $\|\cdot\|_t$ is a Hermitian metric on $F$ with Griffiths semi-positive curvature. Moreover, by similar argument as in [@HPS16 Propositionn 23.3] (see also [@DWZZ1 Step 3 in the proof of Theorem 9.3]), one can show that the metric on $F$ extends to a positively curved metric on $\mathcal F$. In the special case that $E$ is a line bundle, the same conclusion is true if $h$ is singular and pseudoeffective (see [@BP08; @PT18; @HPS16; @DWZZ1; @ZhouZhu]).
Appendix {#appendix .unnumbered}
========
We prove a result used in the proof of Theorem \[thm: direct image-optimal L2 estimate-intr\], which seems to be already known.
\[lem:inverse cur. indep of metric\] Let $U\subset\mathbb{C}^n$ be a domain, $\omega_1$, $\omega_2$ be any two Hermitian forms on $U$, and $E=U\times \mathbb{C}^r$ be trivial vector bundle on $U$ with a Hermitian metric. Let $\Theta\in C^0(X,\Lambda^{1,1}T^*_X\otimes End(E))$ such that $\Theta^*=-\Theta$. Then $$Im[i\Theta,\Lambda_{\omega_1}]=Im [i\Theta,\Lambda_{\omega_2}],$$ and for any $E$-valued $(n,1)$ form $u\in Im[i\Theta,\Lambda_{\omega_1}]$, $$\langle[i\Theta,\Lambda_{\omega_1}]^{-1}u,u\rangle_{\omega_1}dV_{\omega_1}
=\langle[i\Theta,\Lambda_{\omega_2}]^{-1}u,u\rangle_{\omega_2}dV_{\omega_2}.$$
For any $z_0\in U$, after a linearly transformation, we may assume $\omega_1=i\sum_{j=1}^{n}dz_j\wedge d\bar{z}_j$ and $\omega_2=i\sum_{j=1}^{n}\lambda_j^2 dz_j\wedge d\bar{z}_j$ at $z_0$ with $\lambda_j>0$. Let $w_j=\lambda_j z_j$ for $j=1,2,\cdots,n$, then $\omega_2=i\sum_{j=1}^{n}dw_j\wedge d\bar{w}_j$. We may write $$\label{eq p1}
i\Theta=i\sum_{jk\alpha\beta}c_{jk\alpha\beta}dz_j\wedge d\bar{z}_k\otimes e^*_\alpha\otimes e_\beta
=i\sum_{jk\alpha\beta}c'_{jk\alpha\beta}dw_j\wedge d\bar{w}_k\otimes e^*_\alpha\otimes e_\beta$$ with $c'_{jk\alpha\beta}=\frac{c_{jk\alpha\beta}}{\lambda_j\lambda_k}$.
Denote $\lambda=\prod_{j=1}^{n}\lambda_j$. Let $u=\sum_{j,\alpha}u_{j\alpha}dz\wedge d\bar{z}_j\otimes e_\alpha$, then $u=\sum_{j,\alpha}u'_{j\alpha}dw\wedge d\bar{w}_j\otimes e_\alpha$ with $u'_{j\alpha}=\frac{u_{j\alpha}}{\lambda\lambda_j}$. Note that $$\label{eq p2}
[i\Theta,\Lambda_{\omega_1}]u=\sum_{jk\alpha\beta}u_{j\alpha}c_{jk\alpha\beta}dz\wedge d\bar{z}_k\otimes e_{\beta},$$ and $$\label{eq p3}
[i\Theta,\Lambda_{\omega_2}]u=\sum_{jk\alpha\beta}u'_{j\alpha}c'_{jk\alpha\beta}dw\wedge d\bar{w}_k\otimes e_{\beta}.$$ So it is easy to see $Im[i\Theta,\Lambda_{\omega_1}]=Im [i\Theta,\Lambda_{\omega_2}]$. We write $$[i\Theta,\Lambda_{\omega_1}]^{-1}u=\sum_{jk\alpha\beta}u_{j\alpha}d_{jk\alpha\beta}dz\wedge d\bar{z}_k\otimes e_{\beta},$$ $$[i\Theta,\Lambda_{\omega_2}]^{-1}u=\sum_{jk\alpha\beta}u'_{j\alpha}d'_{jk\alpha\beta}dw\wedge d\bar{w}_k\otimes e_{\beta},$$ Then from equations ,,, we can get $$d'_{jk\alpha\beta}=\lambda_j\lambda_k d_{jk\alpha\beta}.$$ We now assume that $\{e_\alpha\}$ are orthonormal at $z_0$. Then $$\langle[i\Theta,\Lambda_{\omega_1}]^{-1}u,u\rangle_{\omega_1}dV_{\omega_1}=
\sum_{jk\alpha\beta}d_{jk\alpha\beta}u_{j\alpha}\bar{u}_{k\beta}c_ndz\wedge d\bar{z},$$ $$\langle[i\Theta,\Lambda_{\omega_2}]^{-1}u,u\rangle_{\omega_2}dV_{\omega_2}=
\sum_{jk\alpha\beta}d'_{jk\alpha\beta}u'_{j\alpha}\bar{u'}_{k\beta}c_ndw\wedge d\bar{w}.$$ Note also that $$c_ndw\wedge d\bar{w}=\lambda^2c_ndz\wedge d\bar{z},$$ We get $$\langle[i\Theta,\Lambda_{\omega_1}]^{-1}u,u\rangle_{\omega_1}dV_{\omega_1}=
\langle[i\Theta,\Lambda_{\omega_2}]^{-1}u,u\rangle_{\omega_2}dV_{\omega_2}.$$
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[^1]: (\*) The second author and the third author are both corresponding authors.
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---
abstract: 'Recently, the stochastic series expansion (SSE) has been proposed as a powerful MC-method, which allows simulations at low $T$ for quantum-spin systems. We show that the SSE allows to compute the magnetic conductance for various one-dimensional spin systems without further approximations. We consider various modifications of the anisotropic Heisenberg chain. We recover the Kane-Fisher scaling for one impurity in a Luttinger-liquid and study the influence of non-interacting leads for the conductance of an interacting system.'
author:
- Kim Louis and Claudius Gros
title: 'Quantum Monte Carlo simulation for the conductance of one-dimensional quantum spin systems'
---
Introduction
============
In general, integrable one-dimensional models show an ideally conducting behavior in contrast to most real three-dimensional materials. [@Cas95] There has been intensive investigation of the influence of the integrability on the conductivity for various model systems.[@Cas95; @Zot96; @Zot97; @Alv02] The conductance of conducting, nearly one-dimensional devices is, on the other hand, of substantial experimental interest. Over the last years it has become possible to fabricate mesoscopic devices,[@Imry] such as carbon nanotubes which can be viewed as a realization of systems with ballistic transport properties. Therefore, the computation of dynamical transport quantities has received considerable interest.
A basic approach for the study of the conductance has been in the past via the bosonization of appropriate model systems,[@ApelRice; @KaneFish; @GogNerTsve] valid in the low-temperature limit. Numerical studies have so far involved the density-matrix-renormalization-group (DMRG) technique [@Marston; @Molina] and Monte Carlo (MC).[@Moon; @Leung] In the case of Ref. a reduced set of states was used in order to evaluate the dynamics, in Ref. , a phenomenological formula by Sushkov[@Sushkov] was used to compute the conductance (see also Ref. ). The simulations by Refs. use an effective bosonized Hamiltonian as a starting point.
Here we will discuss how to obtain the conductance with quantum-Monte-Carlo (QMC) on the original lattice Hamiltonian. For this purpose the conductance will be calculated on the imaginary frequency axis. We will show that a reliable extrapolation to zero frequency can be performed at finite but low temperatures. We will thus obtain approximation-free results for the dynamics of inhomogeneous quantum-spin systems at low but finite temperatures, within a well defined numerical accuracy defined by the statistics of the MC-sampling and the accuracy of the zero-frequency extrapolation.
By the Jordan-Wigner transform a one-dimensional spinless fermionic system can be mapped to a hard-core boson model. Hence, it is possible to calculate the conductance for a fermionic system in a bosonic one. This is vitally important since boson models can be easily analyzed by Monte Carlo simulations where the sign problem is absent. However, for an evaluation of the conductance one requires a highly efficient simulation method which performs well at low temperatures. Recently,[@Sandvik] such a powerful method has been proposed: the Stochastic Series Expansion (SSE). In this paper we will compute the conductance in a hard-core boson lattice model by the aid of this new method.
explanation of the method
==========================
Definition of the Conductance {#subSec_def_g}
-----------------------------
We consider the anisotropic $xxz$-Hamiltonian $$H_{xxz}=\sum_{n=1}^{N-1}
J_x\left(S_n^+S_{n+1}^-+S_n^-S_{n+1}^+\right)/2+J_zS_n^zS_{n+1}^z,$$ where the $S_{n}^\pm=S_n^x\pm i S_n^y$ are the raising/lowering operators for spin-1/2 Heisenberg spins. The spin-current operator $j_n$ at a given site $n$ follows from the continuity equation and is given by (see e.g. Ref. ) $$j_n=iJ_xe\left(S_n^+S_{n+1}^--S_n^-S_{n+1}^+\right)/(2\hbar).$$ As a perturbation we will use a local “voltage drop”. In the hard-core boson notation this corresponds to a step in chemical potential—at site $m$— which is equivalent to $$P_m=e\sum_{n>m}S^z_n,$$ in terms of the Heisenberg-spin operators.
The conductance is then defined as the dynamical response of the current operator at site $x$ to the voltage drop at site $y$: $$\label{defg}
g:=\lim_{z\to 0}{\rm Re}\frac{i}{\hbar}\int_0^\infty e^{izt}\langle
[j_x(t),P_y]\rangle \,dt.$$ For open boundary conditions (OBC) the relation $i[H,P_x]=\hbar\, j_x$ holds and a partial integration of (\[defg\]) yields: $$\begin{aligned}
g%&=&\!{\rm Re}\Bigl[\frac{i}{\hbar}\langle [j_x,P_y(-t)]\rangle
%\frac{e^{izt}}{iz}\Bigr|_0^\infty\!\!\!+{\rm Re}\frac{i}{\hbar}
%\int_0^\infty\! \frac{e^{izt}}{iz}\langle[j_x(t),j_y]\rangle \\
&=&\!{\rm Re}\left[(-iz)^{-1}\frac{i}{\hbar}\Bigl\{\langle [j_x,P_y]\rangle -
\int_0^\infty\!\! e^{izt}\langle[j_x(t),j_y]\rangle dt\Bigr\}\right]. \\\end{aligned}$$ Using ${\rm Re}(ab)={\rm Re}a{\rm Re}b-{\rm Im}a{\rm Im}b$ in the above equation gives two contributions to the conductance. With the definition of the generalized Drude weight for two operators $A$ and $B$: $$\langle\langle AB\rangle\rangle \ \equiv\
\lim_{z\to 0}(-iz)\int_0^\infty
e^{itz}\langle \Delta A(t)\Delta B\rangle\, dt~.
%\label{<<_>>}\end{equation}$$ where $\Delta A=A-\langle A\rangle$ one can show using the Lehmann representation[@louis] that the first contribution (${\rm Re}a{\rm Re}b$) reads $$\langle\langle j_xj_y\rangle\rangle\,{\rm Re}(-iz)^{-1} =
\langle\langle j_xj_y\rangle\rangle\,\pi\,\delta({\rm Re}z).
\label{the_first_term}$$ The second factor in the expression (\[the\_first\_term\]) \[namely, ${\rm Re}(-iz)^{-1}$\] gives rise to a delta-function, such that (\[the\_first\_term\]) should be the dominating contribution to $g$. The first factor of expression (\[the\_first\_term\]) (namely, $\langle\langle j_xj_y\rangle\rangle$) is closely related to the Drude Peak $D=\langle\langle JJ\rangle\rangle/N$ where $J=\sum_n j_n$ is the total current operator.
From the discussion of the Drude Peak[@Kohn] we know that under OBC’s the Drude Peak is zero \[because $D$ can be written as the response to a static twist which can be removed by a gauge transformation of the form $\exp(i\sum_n nS_n^z)$ (see Ref. ) \] whereas it is non-zero for periodic boundary conditions (PBC’s). In our case $\langle\langle j_xj_y\rangle\rangle$ is zero under OBC’s \[use a gauge transform $\exp(i\sum_{n>y} S_n^z)$\]. Under PBC’s we find, because of translational invariance and because of the continuity equation, that $\langle\langle
j_xj_y\rangle\rangle$ does depend neither on $x$ nor $y$. This implies $D=N\langle\langle j_xj_y\rangle\rangle$. Since the Drude peak is finite (at least for the models we are interested in) we conclude that expression (\[the\_first\_term\]) vanishes even under PBC’s in the Thermodynamic limit. So we obtain $$g={\rm Re}[(z\hbar)^{-1}]{\rm Re}\int_0^\infty e^{izt}
\langle[j_x(t),j_y]\rangle dt.
\label{intbypart1}$$
Restarting from Eq. (\[defg\]) we may—again by partial integration—arrive at another formula. $$\begin{aligned}
g%&=&\!{\rm Re}\Bigl[\frac{i}{\hbar}\langle [j_x,P_y(-t)]\rangle
%\frac{e^{izt}}{iz}\Bigr|_0^\infty\!\!\!+{\rm Re}\frac{i}{\hbar}
%\int_0^\infty\! \frac{e^{izt}}{iz}\langle[j_x(t),j_y]\rangle \\
&=&\!{\rm Re}\left[\frac{i}{\hbar}\Bigl\{-\langle [P_x,P_y]\rangle -(iz)
\int_0^\infty\!\! e^{izt}\langle[P_x(t),P_y]\rangle dt\Bigr\}\right]. \\\end{aligned}$$ The first term in the square brackets does not contribute—as the potentials $P_x$ and $P_y$ commute—and if we restrict ourselves to ${\rm Re}z=0$ we obtain: $$g=-{\rm Im}z{\rm Im}\left(
\frac{1}{\hbar}\int_0^\infty e^{izt}\langle[P_x(t),P_y]\rangle dt\right).
\label{intbypart2}$$ The latter formula is especially useful for MC-simulations as it allows the computation of the conductance in terms of the diagonal $S^z$-$S^z$-correlators (under OBC’s).
According to its definition as it is given by Eq. (\[defg\]) the conductance might in principle depend on the actual choice of the positions of the voltage drop $y$ and the current measurement $x$. Here, we point out that in the limit $z\to 0$ this is not the case. In a rather general situation one can show (using the continuity equation) that the right hand side of Eq. (\[defg\]) gives the same result for any choice of $x$ and $y$. (see Appendix \[proofxy\].)
It is instructive to consider the free fermion case [*en détail*]{}. We denote the—formal—dependence on $x$ and $y$ by corresponding subscripts. Of course, in a translational invariant system $g_{xy}$ depends only on the difference $x-y$. The conductance $g$ as a function of $\omega={\rm Im}z$ (here and in the sequel ${\rm Re}z=0$) is plotted for the free fermion case in Fig. \[gxy\]. One sees that the conductance in the limit $\omega\to 0$ approaches the universal value $e^2/h$. Here we emphasize that a spatial separation of voltage drop and current measurement leads to an exponential decrease in $g(\omega)$ at small $\omega$. Therefore, we will restrict our attention to the cases $|x-y|\leq 1$ for the rest of the paper.
Technical details of the MC-method
----------------------------------
We now turn to some technical details of our simulations. The second formula for the conductance, Eq. (\[intbypart2\]), lends itself to a study with Monte Carlo simulations (it requires OBC’s). At the Matsubara frequencies $\omega_M=2\pi
M(\beta\hbar)^{-1},\; M\in{\mathbb N}$ we may use the equivalent expression $$g(\omega_M)=-\omega_M/\hbar {\rm Re}\,\int_0 ^{\hbar\beta} \langle P_x
P_y(i\tau)\rangle e^{i\omega_M \tau}d\tau.$$ We employ a standard QMC method (SSE) to compute the conductance. Since $P_x$ is diagonal in the $S^z$-Basis, the simulation of $\langle H^kP_x H^{L-k}P_y\rangle$ can be easily performed with the help of the SSE.[@Sandvik; @Dorneich; @Sandvik2] Here, $L$ is the approximation order. One may simply obtain $\langle P_xP_y(i\tau)\rangle$ as a linear combination of the terms $\langle H^kP_x H^{L-k}P_y\rangle$ with binomial weight factors $B(\tau,k)$. We found it convenient to assume a Gaussian distribution for the $B(\tau,k)$ instead of a binomial one, because the former is easier to evaluate. The error that we introduce by this replacement is smaller than the statistical error if $L>100$. (Note that in our simulations $L$ is typically of the order of $10^4-10^5$.)
What remains to be done in order to get $$g(\omega_M)=-\omega_M/\hbar\int_0^{\hbar\beta} \cos(\omega_M
\tau)\langle P_xP_y(i\tau)\rangle d\tau$$ is an integration in the final step. We performed it with the Simpson rule and a grid of 800 $\tau$-values.
We are now left with the standard problem of extrapolating $g(\omega)$ from the Matsubara frequencies $\omega_M$ to $\omega=0$. Unfortunately, the spacing of the Matsubara frequencies is linear in $T$, so our method becomes unstable when we increase $T$.
To see that an application of our MC-method makes only sense at low temperatures we compare it to a simpler method: exact diagonalization. Fig. \[cond\] shows $g(\omega)$ for various system sizes $N$ at $T=J_x/k_B$. One sees that convergence with $N$ is rather fast (at high temperatures). Hence, one can determine $g(\omega)$ for $\omega > J_x/\hbar$ with exact diagonalization. To compute $g(\omega)$ at some $\omega\leq J_x/\hbar$ with MC-methods one needs to work at a temperature $T<J_x/(2\pi k_B)$, and even then exact diagonalization is preferable as it yields $g(\omega)$ in a continuous interval rather than on a discrete set of points. Hence, at high temperatures the MC-method is inferior to a simple exact diagonalization.
In our simulation we make one “MC-sweep” between two measurements which consists of one diagonal update and several loop-updates[@Sandvik] between two measurements. We are able to run approximately $10^5$ sweeps.
Test with Jordan-Wigner
------------------------
If $J_z=0$ then $g(\omega)$ can be exactly evaluated—for arbitrary system size—not only at $\omega=0$ (see below).
We can exploit this fact in two regards: Firstly, we test our MC-method by comparing it with the exact curve obtained by Jordan-Wigner. The result can be seen in Fig. \[condmc\].
Secondly, we can test which frequencies (and hence which temperatures) we need such that a linear extrapolation can be carried out without introducing a larger error than the statistical one. One sees also which system sizes are needed to determine $g(\omega)$ without detectable finite size error. Our conclusion is that our method works for $T<0.02$. At $T\approx 0.01$ a system size of $N\approx 200-300$ is appropriate.
The conductance in various systems
===================================
Results for a dimerized Jordan-Wigner-chain
-------------------------------------------
One may derive a simple analytical result for the conductance in the free fermion case. Here, we consider the slightly more difficult case—but also more interesting as the system has a gap[@Orignac]—of a dimerized chain with magnetic field $B$, i.e., $J_z=0$ and the hopping parameter alternates: $(J_x)_{2n,2n+1}=J_1$ and $(J_x)_{2n+1,2n+2}=J_2$. The energy dispersion is given by $$E_k^\pm=B\pm\sqrt{J_1^2+J_2^2+2J_1J_2\cos(2k)}\,/\,2$$ and we assume a positive dispersion $E_k=E_k^+$ if $k\in [0,\pi/2]\cup[3\pi/2,2\pi]$ and $E_k=E_k^-$ else. The gap $E_g$ is $E_g=E_{\pi/2}^+-E_{\pi/2}^-$ at $k=\pm\pi/2$. The evaluation of the formula for the conductance Eq. (\[defg\]) is in principle straightforward. (see Appendix \[Apniceres\].) One obtains the compact result: $$\begin{aligned}
g=&&
\frac{e^2}{2h}\big[\tanh(E_0\beta/2)-\tanh(E_\pi\beta/2)\nonumber\\
&&-\tanh(E_{\pi/2}^+
\beta/2)+\tanh(E_{\pi/2}^-\beta/2)\big].
\label{niceresult}\end{aligned}$$ In the case of zero magnetic field this reduces to: $$g={ \frac{e^2}{h}\big[\tanh(E_0\beta/2)-\tanh(E_{\pi/2}^+
\beta/2)\big]}.$$ The $T=0$-value of the conductance as a function of magnetic field is quantized: It is zero if $|B|$ is smaller than the zero field gap $E^+_{\pi/2}$ or larger than the zero field band width $E_0$, and it is $1$ between these values, and precisely at these values it is $1/2 $ (all values in units of $e^2/h$).
Comparison with the Apel-Rice-formula
-------------------------------------
At low temperatures the $xxz$-chain can be described by a Luttinger liquid. For this model the conductance was first obtained by Apel and Rice in the eighties:[@ApelRice; @Giamarchi] $$\label{ApelRice}
g_{\rm ApelRice}=\frac{e^2}{h}\frac{\pi}{2(\pi-\theta)},$$ where $\cos\theta=J_z/J_x$. This formula may be derived from Eq. (\[intbypart2\]) if we use concrete expressions for $\langle S^z_nS_m^z(i\tau)\rangle$ which are available from conformal field theory[@Affleck]. Fig. \[QmcgJz\] shows our QMC-results for $g(\omega)$ on the imaginary axis for the $xxz$-model, in Fig. \[apelrice\] we display a comparison, as a function of $J_z$, between the $g(\omega=0)$ extrapolated from Fig. \[QmcgJz\], and the exact Bosonization result, Eq. (\[ApelRice\]).
We note, that the statistical error of the QMC-results presented in Fig. \[apelrice\] does not increase much with the parameter $J_z$. This is due to our using the “directed loops” as described in Ref.. The choice for transition probabilities which was proposed there makes the SSE-algorithm more effective. Here the improvement is remarkable.
System with one impurity
-------------------------
Now, we consider the Hamiltonian $$H=H_{xxz}+B_{\rm Imp}S^z_{N/2},$$ i.e., we add an impurity in the middle of the system. This model was first studied in a paper by Kane and Fisher.[@KaneFish] Later, this kind of model received considerable attention by other authors.[@Leung; @Weiss; @Fendley; @Tsvelik; @Qin] By an RG approach Kane and Fisher found that the perturbation $B_{\rm Imp}$ is relevant for repulsive interactions (i.e., $J_z>0$) . This means that at zero $T$ the chemical potential anomaly “cuts” the system into two halves, such that the conductance is zero. This result cannot be directly confirmed by Monte Carlo methods since these are necessarily finite-temperature methods.
Fortunately, the scaling behavior (with temperature) of the conductance is also known. For $K=1/2$ one may derive an exact formula for the conductance by a refermionization technique:[@Weiss; @GogNerTsve] $$\label{WES}
g=e^2/h\left[1-\frac{B_{\rm Imp}^2}{2\pi^2 T}\psi^\prime
\left(1/2+\frac{B_{\rm Imp}^2}{2\pi^2T}\right)\right]$$ where $\psi$ is the Digamma function. So we can compare our MC-data once again with an exact result. In Fig. \[KFG2\] we present two different QMC-results for the conductance on the imaginary axis of the Heisenberg-chain with one impurity, for different impurity strengths.
For the upper set of curves in Fig. \[KFG2\] the position of voltage drop and the position of the current measurement are $x=y=N/2$; and for the lower set of curves they are $x=N/2,\;y=N/2-1$. The curves are not as smooth as the ones in Fig. \[QmcgJz\], so we use a quadratic fit from the first three Matsubara frequencies instead of a linear extrapolation to estimate $g(\omega=0)$. We also note that the curves with $x-y=1$ are better suited for extrapolation than those with $x=y$ because the slope at $\omega=0$ is smaller. The statistical error is less than one percent. The result from the extrapolation is given in Fig. \[sassetti\] along with the exact curves from Eq. (\[WES\]). We used system sizes of $N=400$ for $T\geq 0.01J_x/k_B$ and $N=800$ for $T=0.005J_x/k_B$. We performed $2\cdot 10^5$ MC-sweeps. The error bars are smaller than the symbol size, so the error that we see in the figure is mainly due to our extrapolation method \[and to possible logarithmic finite-temperature-corrections to Eq. (\[WES\])\]. We see that the quadratic fit tends to underestimate the correct value.
Inhomogeneous systems
---------------------
Several years after the publication of the Eq. (\[ApelRice\]) by Apel and Rice it was generally agreed upon[@Safi; @MaslovStone] that it does not reflect the (correct) physical behavior one would encounter in an experimental realization. Experiments are never performed on a closed system but on one coupled to reservoirs which make it possible for the particles to leave and enter the system. These reservoirs can be modeled by attaching two leads consisting of infinite non-interacting spin half-chains to our model.
The complete Hamiltonian reads then: $$H=H_{xxz}(J_z=0) +\!\!\!\! \sum_{n=(N-N_I+1)/2}^{(N+N_I-3)/2}\!\!\!\!
S_n^zS_{n+1}^z,
\label{H_leads}$$ i.e., the interaction is confined to a small region in the middle consisting of $N_I$ sites. This approach has been followed by many authors, e.g. Refs.. Generally, the presence of leads yields a conductance which is independent of $J_z$,[@Safi; @MaslovStone] namely, $$g=e^2/h$$ in sharp contrast to Eq. (\[ApelRice\]). The non-interacting semi-chains—which we call leads—play the rôle of reservoirs.
We note that the behavior of $g(\omega)$ depends on the parity of $N_I$ a fact which was already reported in Ref. . (A similar effect was also found in the Hubbard model.[@Oguri]) In the following we will only consider the case of $N_I$ odd. A detailed discussion of $N_I$ even/odd and a comparison with Ref. will be presented elsewhere.
The “natural” choice for current measurement and voltage drop would be at the two ends of the interacting region. But here is caution advised. From Fig. \[gxy\]—which is again for the free Fermion case—one learns two things: Firstly, $g$ does not depend on the choice of $x$ and $y$. Secondly, if $x$ and $y$ are some distance apart, convergence with $\omega$ becomes slow, hence one needs to go to lower $T$ and larger system sizes if one wants to extract $g(\omega=0)$ reliably. A simple phenomenological explanation for this is the following: If the place of the measurement is far from the voltage drop the particles have to travel a long distance and hence one has to wait a long time, before one can determine $g$. We can now place both the voltage drop and our current measurement at the middle of the system, but this should not help much. The problem is that the particles still have to travel a long distance until they see the leads. Hence $g(\omega)$ will be unaffected by the introduction of leads if $\omega$ is sufficiently large. This expectation is confirmed by the QMC-data presented in Fig. \[leads\]. In this figure the exponential decay of $g(\omega)$ at small $\omega$ is apparent, and the curves illustrate clearly that this decay is induced by the length scale $N_I$—because it becomes stronger with increasing $N_I$. It is this decay that prevents us from discussing larger $N_I$. If we increase $N_I$ the decay becomes stronger, hence we need to evaluate $g(\omega)$ for more (and smaller) frequencies in order to extract $g(\omega=0)$ reliably. But smaller frequencies are only available at smaller temperatures. As we cannot decrease $T$ much below $0.01J_x/k_B$ we restrict ourselves to $N_I<20$. If we considered a system with $N_I=200$ (at $T=0.01J_x/k_B$) we would not see any difference from a system without leads, because the difference occurs at small $\omega$.
The data presented in Fig. \[leads\] clearly indicates an upturn of $g(\omega)$ for $\omega\to0$, indicating that $g(\omega=0)$ is unaffected by the interaction in the low-temperature limit. Our result may, however, not be totally convincing, since we can only analyze relatively small interacting regions. One might argue that the enhanced conductance is not due to the leads but simply to the reduced “mean” interaction—which is close to zero as only few sites interact. To invalidate this argument we considered another model. We have performed QMC-simulation of a system where we attach a lead only at [*one*]{} side such that we obtain a chain which is non-interacting in one half and interacting in the other. For this system we found no deviation at all imaginary frequencies from the situation where the interacting region extends over the whole chain, even though there are as many interacting as interaction-free bonds.
Spin-Hamiltonian with third-nearest-neighbor interaction
--------------------------------------------------------
Monte Carlo simulations allow the inclusion of long-range hopping, and thus breaking the integrability of the pure $xxz$-Hamiltonian, as long as the resulting system is not frustrated. We thus consider a Hamiltonian with a third-nearest-neighbor interaction: $$\begin{aligned}
\nonumber
H\ =\ H_{xxz}
&+&J_{x3}\sum_n \big[\,(S_n^+S_{n+3}^-+S_n^-S_{n+3}^+)/2\\
&&\ \ +\,J_{z}/J_xS_n^zS_{n+3}^z\,\big].
\label{Hamilhop3}\end{aligned}$$ For simplicity we assumed that the anisotropy is independent of the hopping range (i.e., $J_{z3}=J_{x3}J_{z}/J_{x}$). In this context we emphasize, that the long-range hopping in the spin system does not transform under Jordan-Wigner to a long-range hopping in a fermionic system but to a more complicated four-sites operator.
In general, adding a new term to the Hamiltonian changes the current operator which is defined via the continuity-equation, $\nabla j:=(j_{n+1}-j_n)=i[S_{n+1}^z,H]/\hbar$. In a one-dimensional system with OBC’s the continuity-equation is solved however by the relation $ j_x=i[H,P_x]/\hbar$ (see Sec. \[subSec\_def\_g\]) such that Eq. (\[intbypart2\]) still applies. Nonetheless, it is useful to look at the current operator for this case. It reads: $$j_n=j_{n,1}+j_{n,3}+j_{n-1,3}+j_{n-2,3},$$ where $j_{n,k}=iJ_xe\left(S_n^+S_{n+k}^--S_n^-S_{n+k}^+\right)/(2\hbar).$ If we compare it with the current operator of the $xxz$ chain we see that the long range hopping $J_{x3}$ gives rise to three additional terms which are analogous to the first one.
We compute the conductance as a function of the hopping amplitudes $J_{x3}/J_{x}$ and present the result in Fig. \[hop3\]. If $J_{x3}=0$ the conductance is of course given by the Apel-Rice-result Eq. (\[apelrice\]). However, if $J_{x3}>>J_x$ we may eventually neglect the nearest-neighbor-hopping-term such that we end up with three uncoupled chains. Thus, we conclude that the conductance will grow towards three times the Apel-Rice-result when we increase $J_{x3}$. >From the figure we see that the crossover between these two values is shifted to smaller values of $J_{x3}$ when the anisotropy is increased. (In fact, at the isotropic point the increasing of $g$ is barely visible.)
In conclusion we have developed a QMC-technique which allows the evaluation of the DC-conductance for a wide range of non-frustrating quantum-spin chains at low but finite temperatures. We have presented several stringent tests for this technique, like the Kane-Fisher scaling for the conductance through an impurity in a Luttinger-liquid.
Proof that the right hand side of Eq. (\[defg\]) is independent of $x$ and $y$ {#proofxy}
==============================================================================
Here we provide a general argument which relies on the (physical) assumption that some linear response functions are finite in the Thermodynamic limit.
[[**Theorem:**]{}]{}
[**Proof:**]{}
First we consider $\lim_{z\to 0}z\int_0^\infty e^{izt}\langle
[S^z_n(t),P_m]\rangle dt.$ This expression corresponds to a plateau value of the response function $\varphi(t):= i\langle [P_m,S_n^z(t)]\rangle$, i.e., $$\lim_{z\to 0}z\int_0^\infty e^{izt}\left\langle
[S^z_n(t),P_m]\right\rangle dt=\lim_{t\to \infty}\varphi(t).$$ The response function may be written in the following way (by Kubo’s identity)[@ZMR] $$\varphi(t)=\beta(P_m,iLe^{iLt}S_n^z) =:\dot\Phi(t).$$ Here $(\cdot,\cdot )$ is the Mori scalar product \[for operators $A$ and $B$: $\beta(B,A)=
\int_0^\beta{\rm Tr} B^\dag\exp(-\tau H)A\exp((\tau-\beta) H)d\tau/{\rm Tr} \exp(-\beta H)$\] and $L$ is the Liouville operator. To prove that $\varphi(t)\to 0$ as $t\to\infty$, it is sufficient to show that $\Phi$ as a function of $t$ is bounded.
But this is just the assumption that we made in the statement of the theorem because $(S_n^z(t),P_m)$ represents the linear response of the operator $S^z_n$ to the perturbation $P_m$. Finally, we can prove our main assertion. We want to show $g(x,y)=g(x^\prime,y^\prime)\;\forall x,y,x^\prime,y^\prime$. This follows easily from $g(x,y)=g(x+1,y)\;\forall x,y$ and $g(x,y)=g(x,y+1)\;\forall x,y$. We will only consider only the second equality \[the proof of the first equality is analogous by the symmetric structure of Eq. (\[intbypart2\])\].
Using Eq. (\[intbypart2\]) and our previous result we see: $$g(x,y)-g(x,y+1)=%\sum_{m=y+1}^{y_1}
\lim_{z\to 0}{\rm Im }z\int_0^\infty\!\! e^{izt}\langle
[S_{y+1}^z(t), P_x]\rangle dt=0.$$
Q.E.D.
Derivation of Eq. (\[niceresult\]) {#Apniceres}
==================================
For eigenvalues $E_n$ and $E_k$ the respective one-particle-eigenstates will be denoted by $|n\rangle$ and $|k\rangle$, the annihilation operators by $c_n$ and $c_k$, and the occupation numbers by $n_n$ and $n_k$.
For the current operator we find: $N\langle k,j_xk\rangle =ev_k:=e/\hbar\frac{dE}{dk}$ and $\langle -k,j_xk\rangle=0$.
In a free Fermion system one can derive a simple expression for $\langle A(i\omega_M)B\rangle=\int_0^\beta d\tau e^{i\omega_M\tau}\langle
AB(i\tau)\rangle$ \[with $i\omega_M=\omega+i\delta=2\pi iM/(\hbar\beta)$\] when $A$ and $B$ are one-particle operators (i.e., $A=\sum_{n,k}
A_{nk}c_n^\dag c_k$):
$$\langle AB\rangle(i\omega_M)= \sum_{n\neq k}
\frac{A_{nk}B_{kn}}{i\hbar\omega_M+E_n-E_k}f_{kn}$$ where $f_{kn}=
\frac
{\sinh(\beta(E_n-E_k)/2)}{2\cosh(\beta E_n/2)\cosh(\beta E_k/2)}=
\left(
\langle n_k\rangle -\langle n_n\rangle\right).$ $\langle A(z=\omega+i\delta)B\rangle$ may then be obtained by analytic continuation. In our case $A$ and $B$ are local current operators; for the conductance we use Eq. (\[intbypart1\]):
$$\begin{aligned}
g&=& \frac{1}{\omega}{\rm Im}\sum_{n\neq k}
\frac{\langle n,j_x k\rangle\langle k,j_y n\rangle}
{i\hbar\omega_M+E_n-E_k}f_{kn} \\
&=& -\frac{1}{\omega}\sum_{n\neq k}\langle n,j_x k\rangle\langle k,j_y n\rangle
\frac{\delta}{(\hbar\omega+E_n-E_k)^2+\delta^2}f_{kn}. \end{aligned}$$
For $\delta\to 0$ $$\frac{\delta}{(\hbar\omega+E_n-E_k)^2+\delta^2}\to
\pi\delta(\hbar\omega+E_n-E_k).$$ In the continuum limit we replace $\sum_kN^{-1}\to\int \frac{dk}{2\pi}=\int \frac{dE_k}{2\pi|\hbar v_k|} $
Performing the integration over the variable $n$ and then taking $\omega\to 0$ yields $$\begin{aligned}
g%&=& \frac{e^2}{2}\int \frac{dk}{2\pi}\frac{v(k)^2}{|v(k)|}\frac{\sinh(\beta(\hbar\omega)/2)}{\hbar\omega}\frac{1}{2\cosh(\beta E_n/2)\cosh(\betaE_k/2)} \\
&=& \frac{e^2}{2\hbar}\int \frac{dk}{2\pi}
|\hbar v(k)|\frac{\beta/2}{2\cosh^2(\beta
E_k/2)}\\
&=&\frac{e^2}{\hbar}\int_{-\pi}^\pi \frac{dk}{2\pi}
\frac{\hbar v_k{\rm sgn}(k)\beta/2}{4\cosh^2(\beta
E_k/2)} =\frac{e^2}{h}\int_{k=0}^{k=\pi}\frac{dE\beta/2}{2\cosh^2(\beta
E_k/2)} \\
&=&\frac{e^2}{2h}\int_{E_{\pi/2}^+\beta/2}^{E_0\beta/2}
dx\frac{1}{\cosh^2(x)}+\frac{e^2}{2h}\int_{E_{\pi}\beta/2}^{E_{\pi/2}^-\beta/2}
dx\frac{1}{\cosh^2(x)} \\
&=& \frac{e^2}{2h}[\tanh(E_0\beta/2)-\tanh(E_\pi\beta/2) \\
&&-\tanh(E_{\pi/2}^+
\beta/2)+\tanh(E_{\pi/2}^-\beta/2)].\end{aligned}$$
Note that the conductance does not depend on the energy dispersion but only on the band width $E_0$ and the gap $E_{\pi/2}^+$.
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|
---
abstract: |
In this paper we study the computational complexity of the game of Scrabble. We prove the PSPACE-completeness of a certain natural model of the game, showing NP-completeness of Scrabble solitaire game as an intermediate result.\
**Keywords:** *Scrabble, PSPACE-completeness, combinatorial games, computational complexity*
author:
- |
Karolina Sołtys\
Faculty of Mathematics, Informatics and Mechanics\
University of Warsaw\
`ksoltys@students.mimuw.edu.pl`\
title: 'Scrabble is PSPACE-complete'
---
Introduction {#intro}
============
The computational complexity of games and puzzles has been studied since the very begining of complexity theory. Several important complexity classes admit a very natural characterization in terms of games, with PSPACE being a typical example. Many real-life two-player games, such as Amazons, Go, Hex or Othello, have been proven to be PSPACE-complete when generalized to arbitrary board sizes [@epp]. In this paper we will add to this list the popular game of Scrabble, by proving the following theorem.
The problem of determining the winner of a Scrabble game, given an arbitrary Scrabble board position and an arbitrary sequence of letters left in the bag, is PSPACE-complete.
As an intermediate result we will obtain the result.
The problem of determining whether a given instance of a Scrabble solitaire game can be completed with using up the whole sequence of lettered tiles in the bag is NP-complete.
Our model of Scrabble {#model}
=====================
To study the complexity of Scrabble in terms of classical complexity classes we have to derandomize the only element of Scrabble based on chance, i.e. the drawing of lettered tiles. In our model we will assume that each player knows in advance the sequence of tiles she will draw[^1], which will make the game totally deterministic. Due to the same reason we will also require that the tiles exchanged during the play, which in a normal game would be shuffled back into the bag, return to the end of the sequence in the order in which they appeared on the rack of the player making the exchange. This is the only change to the original rules of the game of Scrabble.
In our model we will also ingore the notion of points and premiums, and we will assume that the player with the greatest sum of the lengths of words formed during play will be the winner (which is equivalent to the assumpition that all the letters in the words formed during play are worth one point and that there are no premium tiles on the board). The proof can be adapted to alphabets in which different letters have different point values.
Obviously, to perform the reductions in the proof we will need arbitrarily large boards: we can either assume that the game is played on an $n \times n$ board for a given $n$, or that it is played on an infinite board, on which only a finite number of squares are initially non-empty, as edges of the board play no role in our proof.
We also assume that the initial position may contain words, which are not in the dictionary. It is a reasonable assumption, because in a standard Scrabble game a malformed word may stay on the board if the opponent does not challenge it. It is nevertheless quite easy to get rid of this assumption, but it is technical and we will not show it in this paper.
We define a *Scrabble game* $\mathcal{S}$ to be an ordered nonuple $(\Sigma, \Delta, k, \mathbf{B}_0, \\
\sigma_0, u_0, v_0, q_0, r_0)$ where:
- $\Sigma$ is a finite *alfabet*
- $\Delta \subset \Sigma^*$ is a finite *dictionary*
- $k \in \mathbb{N}$ is the size of the rack
- $\mathbf{B}_0: \mathbf{M}_{n \times n}(\Sigma)$ is the initial *board*
- $\sigma_0 \in \Sigma^*$ is the initial sequence of lettered tiles called the *bag*
- $u_0, v_0 \in \Sigma^{[0-k]}$ are the initial contents of the rack of the first and the second player, respectively.
- $q_0, r_0 \in \mathbb{N}$ are the initial points of the first and the second player, respectively.
A $(\Sigma, \Delta, k)$-Scrabble game will be a Scrabble game with first three parameters fixed.
A *position* $\pi$ in the game is an ordered septuple $(p, \mathbf{B}, \sigma, u, v, q, r)$, where $p \in {1, 2}$ is the number of the active player, and the other parameters describe the current state analogously as above. A *play* $\Pi = \pi_1 \dots \pi_l$ is a sequence of positions such that (for all $i$) $\pi_{i+1}$ is attainable from $\pi_i$ by the active player by:
- forming a *proper play* on the board. Proper play uses any number of the player’s tiles from the rack to form a single continuous word (*main word*) on the board, reading either left-to-right or top-to-bottom. The main word must either use the letters of one or more previously played words, or else have at least one of its tiles horizontally or vertically adjacent to an already played word. If words other than the main word are newly formed by the play, they are scored as well, and are subject to the same criteria for acceptability [@wiki]. All the words thus formed must belong to the dictionary $\Delta$. After forming a proper play, the sum of the lengths of all words formed is added to the active player’s points, letters used are removed from the player’s rack and the rack is refilled up to $k$ letters (or less, if $|\sigma_i|<k$) with the appropriate number of letters forming the prefix of $\sigma_i$.
- passing. In this case, $\pi_{i+1} = (\bar p_i, \mathbf{B}, \sigma, u, v, q, r)$.
- exchanging letters. The active player chooses a subsequence $w$ of letters from his rack. Like in the case of forming a proper play, these letters are removed from the rack and the rack is refilled. The letters exchanged are added to the bag: $\sigma_{i+1} = \sigma_i \cdot w$.
A play $\Pi = \pi_1 \dots \pi_l$ is *finished* if $\pi_l$ was attained as a result of 6-th pass in a row, or if $\sigma_l = \epsilon$ (i.e. the bag is empty). A *winner* of a finished play is the player with the greater number of points (draws are possible).\
For the sake of simplifying the proof, we define additionaly two variants of Scrabble game:
- *No-Exchange-Scrabble* is a variant of Scrabble, where exchanges are not allowed.
- *Separate-Bags-Scrabble* is a variant of Scrabble, where players have two separate bags and each player can use only her own bag for drawing letters. Exchanges are not allowed.
A *Scrabble solitaire* game is defined analogously, but with only a single player. Exchanging of tiles is not allowed, and the player *solves* the solitaire if she manages to get rid of all the letters from the bag. We define *$(\Sigma, \Delta, k)$-Scrabble solitaire* as above.
In the proof we will see that it is sufficient to take $\Sigma$ such that $|\Sigma| \leq 10 $ and $k \leq 7 $, which means that the reduction can be implemented with standard English-language scrabble set by using only 1-point letters (there are 10 such letters in the English language set).
Outline of the proof {#outline}
====================
Let <span style="font-variant:small-caps;">Scrabble-Variant</span> be the problem of determining the winner of a given game of a given variant of Scrabble, and let <span style="font-variant:small-caps;">Scrabble-Solitaire</span> be the problem of determining if a given instance of Scrabble solitaire is solvable.
Our goal is the following theorem.
There exist $\, \Sigma, \, \Delta, \, k$ such that <span style="font-variant:small-caps;">$(\Sigma, \Delta, k)$-Scrabble</span> is PSPACE-complete.
The proof of this theorem, although based only on a few basic ideas, will be quite involved technically. To make it more accessible to the reader, we will reach the proof in several steps, in each proving some complexity results about simplified variants of Scrabble. These lemmas will not be necessary for the final proof, but will help us to expose the main idea of the proof, which otherwise would be hidden behind technical difficulties.
1. [**Step** . ]{}<span style="font-variant:small-caps;">Scrabble-Solitaire</span> is NP-complete. We will show it in section \[np\] using a reduction from <span style="font-variant:small-caps;">3-CNF-SAT</span>.
2. [**Step** . ]{}<span style="font-variant:small-caps;">Separate-Bags-Scrabble</span> is PSPACE-complete. We will prove it in section \[pspace\] using a reduction from <span style="font-variant:small-caps;">3-CNF-QBF</span>.
3. [**Step** . ]{}There exist $\Sigma, \, \Delta, \, k$ such that <span style="font-variant:small-caps;">$(\Sigma, \, \Delta, \, k)$-Separate-Bags-Scrab-ble</span> is PSPACE-complete. We will show it in section \[alphabet\] by finding some technical means of imposing the constraint that the value-assigning word may be used only in the gadget corresponding to a given variable, without the aid of variable-specific symbols in the alphabet.
4. [**Step** . ]{}There exist $\Sigma, \, \Delta, \, k$ such that <span style="font-variant:small-caps;">$(\Sigma, \, \Delta, \, k)$-Scrabble</span> is PSPACE-complete. This is our goal, which we obtain in section \[bag\]. Here, we will have to address the technical difficulties posed by exchanging letters and using a common bag for both players.
NP-completeness of Scrabble solitaire {#np}
=====================================
In this section we will prove the following lemma.
<span style="font-variant:small-caps;">Scrabble-Solitaire</span> is NP-complete.
Proving that the problem is in NP is straightforward – to check whether a game is solvable we simply guess the sequence of moves leading to solving the game. The length of this sequence will be polynomial (in the terms of the size of the description of the game), and each move can be described by some polynomial information.
To estabilish the NP-hardness of <span style="font-variant:small-caps;">Scrabble-Solitaire</span>, we will construct a reduction to this problem from a typical NP-complete problem, <span style="font-variant:small-caps;">3-CNF-SAT</span>. Given an instance $\phi$ of <span style="font-variant:small-caps;">3-CNF-SAT</span> we will construct in polynomial time a polynomial-sized game Scrabble-Solitaire game $\mathcal{S}$ such that $\phi$ is satisfiable iff $\mathcal{S}$ is solvable.
The general idea of the proof is as follows. We will create gadgets associated to variables, where the player will assign values to this variables. We will ensure that the state of the game after the value-assigning phase completes will correspond to a consistent valuation. Then the player will proceed to the testing phase, when for each clause she will have to choose one literal from this clause, which should be true according to the gadget of the respective variable. If she cannot find such a literal, she will be unable to complete a move. Thus we will obtain an immediate correspondence between the satisfiability of the formula and the outcome of the game.
\[gadgetfig\]
{width="70.00000%"}
The gadget for variable $x_i$ is shown in Figure \[gadgetfig\]. The construction of the dictionary and the sequence in the bag will ensure that at some point during the value-assigning the only way for the player to move on is to form a word like in Figure \[assgn:false\] or to form a horizontally symmetrical arrangement (Fig. \[assgn:true\]).
During the test phase, for each clause $c_i = (l_1 \vee l_2 \vee l_3)$ in every play there will be a position, when the player will be obliged to choose one of the literals from the clause, in whose gadget she will try to play a word. She will be able to form a word there iff the value of the corresponding variable, which has been set in the earlier phase, agrees with the literal.
\[testfig\]
{width="70.00000%"}
Let us describe the game more formally. The alphabet $\Sigma$ of $\mathcal{S}$ will contain:
- a symbol $\mathbf{x_i}$ for every variable $x_i$,
- a symbol $\mathbf{c_i^{l_j}}$ for every clause $c_i$ and a literal $l_j$ appearing in this clause,
- auxilliary symbols: **\$**, **\#** and **$\ast$**.
Let $r$ be such that no literal appears in more than $r$ clauses.\
The rack size will be $2r$.\
The dictionary will contain the following words:
- a word $\,\mathbf{\#x_i x_i\$}^{2r-1}$ for every variable $x_i$,
- a word $\,\#\mathbf{c_i^{l_e} c_i^{l_e} c_i^{l_f} c_i^{l_g}\{\ast\}}^{2r-3}$ for every permutation $(e, j, f)$ of the indexes of the literals appearing in the clause $c_i$.
The sequence in the bag will be a concatenation of the following, in the given order:
1. a word $\,\mathbf{x_i\$}^{2r-1}$ for every variable $x_i$, $i \in {1, \dots, n}$[^2],
2. a word $\,\mathbf{c_i^{l_e} c_i^{l_f} c_i^{l_g}\{\ast\}}^{2r-3}$ for every clause $c_i=(l_e \vee l_f \vee l_g), i \in {1, \dots, m}$.
We can now prove several facts.
\[eatsall\] During each turn, to form a word a player must use all the letters from her rack.
Let us notice first that at every round for each word on the board, its length is either 2 or it is greater or equal than $2r+2$. At the start of the game this is obviously true, and at every round we cannot form a word shorter than $2r+2$ (because all the words in the dictionary are of length $2r + 2$). Therefore the only way to form a legal word is to extend an existing 2-letter word on the board, which uses up all $2r$ letters from the rack.
During the value-assigning phase, at each turn the player performs an action that is in our setting equivalent to a correct valuation of a variable, as shown in Figure \[fig:assignments\].
From the previous fact we gather that during each round in the value-assigning phase, the contents of the player’s rack are $\,\mathbf{x_i\$}^{2r-1}$ for some $i$. A simple case by case analysis shows that the player can form a word from this letters only in one of the two ways shown in Figure \[fig:assignments\].
During the test phase, at each turn the player’s actions are equivalent to checking whether a clause, that had not been checked before, is satisfied by a literal of the player’s choice, as shown in Figure \[testfig\].
Basing on the previous two fact we know that during each round in the value-assigning phase, the contents of the player’s rack are $\,\mathbf{c_i^{l_e} c_i^{l_f} c_i^{l_g}\{\ast\}}^{2r-3}$ for subsequent $i$’s. One can easily see that the player can form a legal word from these letters only by extending one of the words $\#\mathbf{c_i^{l_{e'}}}$ on the board by the word $\,\mathbf{c_i^{l_{e'}} c_i^{l_{f'}} c_i^{l_{g'}}\{\ast\}}^{2r-3}$ where $(e', j', f')$ is some permutation of $(e, j, f)$. The player can choose any of such permutations, which means she can choose the literal, in whose gadget she will play the word. A simple analysis shows that the player can play this word in that position iff the valuation of the variable agrees with the chosen literal (i. e. if the chosen literal reads $\neg x_j$, then $x_j$ must have been set to false etc.). \[fig:assignments\].
The above facts imply that the game correctly simulates assigning some valuation to a 3-CNF formula and checking whether it is satisfied. It is easy to check that the size instance of the Scrabble solitaire game obtained by the reduction is polynomial in terms of the size of the input formula and that the instance can be computed in polynomial time. We have thus shown that <span style="font-variant:small-caps;">Scrabble-Solitaire</span> is NP-complete.
PSPACE-completeness of Scrabble {#pspace}
===============================
To prove the PSPACE-completeness of <span style="font-variant:small-caps;">Scrabble</span> it suffices to notice that the above reduction from <span style="font-variant:small-caps;">3CNF-SAT</span> to <span style="font-variant:small-caps;">Scrabble-Solitaire</span> easily translates to the analogous reduction from <span style="font-variant:small-caps;">3CNF-QBF</span> to <span style="font-variant:small-caps;">Separate-Bags-Scrabble</span>.
*Details coming soon.*
Reduction of alphabet, dictionary and rack size {#alphabet}
===============================================
*Details coming soon.*
Shared bag and exchanges {#bag}
========================
*Details coming soon.*
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C. H. Papadimitriou, *Computational complexity*, Addison Wesley, 1994.
D. Eppstein, *Computational Complexity of Games and Puzzles*, [http://www.ics.uci.edu/ eppstein/cgt/hard.html](http://www.ics.uci.edu/~eppstein/cgt/hard.html)
<http://en.wikipedia.org/wiki/Scrabble>
[^1]: At first we will consider the easier case when each player draws from a separate sequence of tiles, then, in section \[bag\], we will adapt the proof to cover the case when both player draw from the same sequence.
[^2]: The time period, when at least one of the letters from this prefix is still on the rack, will be called the *value-assigning phase*. The following time period will be called the *test phase*.
|
---
abstract: 'We studied the diluted magnetic semiconductor by the self-consistent Green’s function approach, which treats the spin-wave kinematics appropriately at finite temperatures. Our approach leads to a simple formula for the critical temperature in a wide range of parameter space. In addition, the magnetization curve versus temperature in some regimes is concave, which is dramatically different from the usual convex shape. Finally, we discuss the possibility of generalizing the current theory to include the realistic band structure, electronic correlations and disorders in a systematic way.'
author:
- 'Shih-Jye Sun'
- 'Hsiu-Hau Lin'
title: Diluted Magnetic Semiconductor at Finite Temperature
---
Diluted magnetic semiconductor (DMS) has attracted intense attentions[@Ohno98a; @Prinz98] for its potential applications in spintronics devices. Robust ferromagnetic order has been observed in (Ga$_{1-x}$Mn$_x$)As up to 110 K[@Ohno96; @Ohno99]. Magnetically doped wide bandgap semiconductors and oxides such as GaN, ZnO, TiO$_2$ even exhibit ferromagnetism at room temperature[@Matsumoto01; @Theodoropoulou01; @Lee02], although the magnetization is less robust than the doped III-V semiconductors. While lots of efforts are focused on the search of optimal materials with enhanced critical temperatures[@Datta90], it remains a challenging task to describe the coupled localized moments and the itinerant carriers in an appropriate way.
Part of the difficulty lies in the fact that the spatial fluctuations of the ferromagnetic order is large at finite temperature and the usual Weiss mean-field theory does not work. To account for the spatial fluctuations, numerical approaches[@Dietl00; @Schliemann01; @Sham01; @Sun02], such as local density functional approximation and Monte Carlo simulations, are quite helpful in estimation of various thermodynamical properties. However, it is rather difficult to study the electronic transport by these numerical approaches. Analytical approaches[@Akai98; @Konig00; @Litvinov01; @Berciu01] provide partial descriptions in several particular limits by treating the impurity spin semiclassically or ignoring the interactions and the kinematic constraints among spin waves. At finite temperatures, the average spin-wave density is large so that these approximations are no longer appropriate. It is therefore desirable to develop a spin-wave theory which works at finite temperature.
In this Letter, we adapt the self-consistent Green’s function approach to describe the fluctuating spin correlations at finite temperature. Since the kinematic constraint of spin waves are treated exactly in the equation of motion, this method can be applied to a wide range of parameter space, even very close to the critical temperature[@Vogt01; @Yang01]. Indeed, the critical temperature is determined by the simple formula, $$k_{B} T_c = \frac{S+1}{3} \left[\frac{1}{cV} \sum_{p} \frac{\alpha_{c}}{\Omega_{c}(p)}\right]^{-1},
\label{CriticalT}$$ where $c$ is the density of impurity spins. The spin-wave dispersion $\Omega_{c}(p)$ and the impurity spin polarization $\alpha_c$ are both determined self-consistently near the critical temperature $T \to T_c$. The trends of the critical temperature upon the change of parameters are studied in detail later.
We model the DMS by the Hamiltonian, containing only the kinetic energy of itinerant carriers and the exchange interaction between the itinerant and the localized impurity spins, $$H =H_0 + J \int d^{3} r\: {\mbox{\boldmath$S$}}(r) \cdot {\mbox{\boldmath$s$}}(r),$$ where $J > 0$ is the strength of exchange interaction. The impurity spin density is ${\mbox{\boldmath$S$}}(r)=\sum_{I} \delta^{(3)}(r-R_I) {\mbox{\boldmath$S$}}_{I}$ while the itinerant spin density is ${\mbox{\boldmath$s$}}(r) = \psi^{\dag}(r) ({\mbox{\boldmath$\sigma$}}/2) \psi(r)$. The band structure of the itinerant carriers is described by $H_0$, which depends on the host semiconductors. Since our emphasis here is how to cope with spatially fluctuations appropriately, the dispersion is taken as the simplest parabolic band, $H_0 = p^2/2m^{*}$. Generalization to more realistic but complex band structures, such as the Luttinger model, can be achieved straightforwardly.
Since the impurity spins are randomly doped into the host semiconductor, their positions are random. If the disorder is strong, the itinerant electrons are localized and the percolation approach[@Litvinov01; @Berciu01] would be more appropriate. However, we are interested in the metallic regime where itinerant carriers are delocalized. The disorder also plays a crucial role in smoothing out the impurity spin density ${\mbox{\boldmath$S$}}(r)$. For instance, the magnitude of the spin density is smeared after coarse-graining, $$\langle {\mbox{\boldmath$S$}}^2(r) \rangle_{R_I} \approx c^2 S(S+1),$$ where $c$ is the impurity spin density and the average is taken over random locations of the impurity spins. Therefore, the presence of weak disorder allows a field-theory description in the continuous limit.
The dynamics of the impurity spins is described by the thermal Green’s function, $$\begin{aligned}
D(r,\tau) &\equiv& \langle\langle S^{+}(r,\tau); S^{-}(0,0) \rangle\rangle
\nonumber\\
&\equiv& -\Theta(\tau) \langle [S^{+}(r,\tau), S^{-}(0,0)] \rangle. \end{aligned}$$ The equation of motion for $D(r, \tau)$ would involve more Green’s functions of higher orders. The exact solution then involves the Green’s functions of all orders and is not feasible in general. However, within mean-field approximation, the higher-order Green’s functions can be decomposed into simpler ones and eventually a self-consistent solution is possible. For the spin-wave propagator $D(q,i\nu_n)$ in momentum space, the mean-field decomposition simplifies the equation of motion, $$i\nu_n D(q, i\nu_n) = 1+ J \langle s_z \rangle +\frac{J\langle S_z \rangle}{c^* V} \sum_{k} F(k,k+q,i\nu_n),
\label{EOM1}$$ where $c^*$ is the density of itinerant carriers. Notice that it only involves one additional Green’s function $F(k,k+q,i\nu_n) \equiv \langle\langle \psi^{\dag}_{\uparrow}(k) \psi_{\downarrow}(k+q); S^{-}(0,0)\rangle\rangle$. Applying the same trick again, the equation of motion for $F(k, k+q, i\nu_n)$ is $$F(k, k+q, i\nu_n) = \frac{Jc^*}{2} \frac{f_{\uparrow}(k) -f_{\downarrow}(k+p)}{i\nu_n +\Delta + \epsilon_{k} - \epsilon_{k+p}} D(q, i\nu_n),
\label{EOM2}$$ where $f_{\uparrow, \downarrow}(k) = [e^{\beta(\epsilon_{k} \pm \Delta/2 -\mu)}+1]^{-1}$ is the Fermi distribution for itinerant carriers with different spins. The Zeeman gap $\Delta \equiv J \langle S_z \rangle$ in the electronic band structure is due to the ferromagnetic order of the impurity spin and has to be determined self-consistently.
![\[f1\] The magnon excitation spectrum at zero temperature. The impurity density $c=1.0/{nm^3}$ and $c^*=0.01/{nm^3}$. The shaded area represents the regime where the imaginary part of the self energy is not zero but negligibly small.](Dispersion){width="8cm"}
From Eqs. (\[EOM1\]) and (\[EOM2\]), we can solve for the spin-wave propagator. Upon the Wick rotation, $i \nu_n \to \Omega + i\eta$, the spin-wave dispersion is identified as the simple pole in the Greens’ function, $$\Omega - J \langle s_z \rangle -\frac{J\Delta}{2V} \sum_{k} \frac{f_{\uparrow}(\epsilon_{k}) -f_{\downarrow}(\epsilon_{k+p})}{\Omega +\Delta + \epsilon_{k} - \epsilon_{k+p} + i\eta} =0.
\label{Dispersion}$$ The average spin densities $\langle s_z \rangle$ and $\langle S_z \rangle$ in the above equation remain unknown and need to be determined self-consistently.
The average itinerant spin density $\langle s_z \rangle$ is just the difference of the spin densities in majority and minority bands, split by the Zeeman gap $\Delta$. The relation between the average localized spin density $\langle S_z \rangle$ and the spin-wave dispersion $\Omega(k)$ is more subtle due to the non-trivial spin kinematic constraint. From Callen’s formula,[@Callen63] $$\frac{1}{c}\langle S_z \rangle = S - \langle n_{sw} \rangle + \frac{(2S+1) \langle n_{sw} \rangle^{2S+1}}{(1+ \langle n_{sw} \rangle)^{2S+1} - \langle n_{sw}\rangle^{2S+1}}.
\label{Magnetization}$$ Here $\langle n_{sw} \rangle = (1/cV) \sum_{k}[e^{\beta \Omega(k)}-1]^{-1}$ is the average number of spin waves over all momenta. At low temperatures, the average spin-wave number is small and the last term in Eq. (\[Magnetization\]) can be safely ignored. The magnetization show the $T^{3/2}$ behavior as in the independent spin-wave approximation. As the temperature approaches the critical regime, the spin-wave density becomes large and the kinematic constraint becomes important. Solving Eqs. (\[Dispersion\]) and (\[Magnetization\]), both the spin-wave dispersion $\Omega(k)$ and the averaged impurity spin density $\langle S_z \rangle$ are obtained self-consistently.
The dispersion from Eq. (\[Dispersion\]) has two branches due to the presence of both itinerant and localized spins. Because the gapless spin-wave fluctuations dominates, the optical branch can be safely ignored. Besides, the spectral weight of the optical mode is also small due to the dilute density of itinerant carriers. In Fig. \[f1\], the lower branch of spin-wave dispersion at zero temperature is shown, corresponding to the Goldstone excitation of the ferromagnetic order. Our numerical results show that the imaginary part of the spin-wave self energy, due to the presence of Stöner continuum (the optical branch), is negligible in all regimes and the dispersion is sharp. Thus, it is a reasonable approximation to assume the spectral weight is carried by the gapless spin-wave excitations only.
![\[f3\] Magnetization curves at different itinerant spin densities $c^*$, while the impurity spin density is fixed at $c=1$ nm$^{-3}$. The critical temperature $T_c$, as shown in the inset, reaches the maximum at an optimal concentration of the itinerant spin density.](Doping){width="8cm"}
![\[f3\] Magnetization curves at different itinerant spin densities $c^*$, while the impurity spin density is fixed at $c=1$ nm$^{-3}$. The critical temperature $T_c$, as shown in the inset, reaches the maximum at an optimal concentration of the itinerant spin density.](MCurve){width="8cm"}
The magnetization curves at different spin densities $c^{*}, c$ are plotted in Figs. \[f2\] and \[f3\]. Throughout this Letter, the exchange coupling and the effective mass are fixed at typical values $J = 0.15$ eV nm$^3$ and $m^* = 0.5 m_e$[@Ohno99]. First of all, we study the temperature dependence of the magnetization at different impurity spin densities $c$, while the ratio of itinerant and localized spin densities fixed at $c^*/c =0.1$. It is quite interesting to notice that the magnetization drops more dramatically near the critical regime when the impurity spin density is large. As shown in Fig. \[f2\], the critical temperature, computed by Eq. \[CriticalT\], increases monotonically as the densities increase[@Sun03]. This monotonic increase in the critical temperature is qualitatively (not quantitatively) the same as the Weiss mean-field theory and also agree with the experiments.
On the other hand, if the impurity spin density is hold constant, say $c=1$ nm$^{-3}$, both the magnetization curve and the trend of the Curie temperatures show interesting behaviors, which deviate from the Weiss mean-field theory. Starting from extremely dilute density of the itinerant carriers, the magnetization curve is concave as shown in Fig. \[f3\], in contrast to the usual convex curvature in Weiss mean-field theory. This concave shape resembles the magnetization curve in percolation theory near the boundary of the metal-insulator transition and is often used to be an indication of localization in the presence of disorder[@Litvinov01; @Berciu01]. However, our results show that a concave magnetization curve is not necessarily tied up to the localization tendency. We emphasize that the disorder of impurity spin is included in our approach only through the coarse-graining procedure and the transport of the itinerant carriers is assumed ballistic here. So it is rather surprising, by including the spin-wave fluctuations appropriately at finite temperatures, the magnetization curve is concave at dilute densities.
In the regime where the magnetization curve is concave, the critical temperature increases with the carrier concentration $c^*$. Beyond an optimal density, $c^* \sim 0.01$ nm$^{-3}$ (the impurity spin density is fixed at $c = 1$ nm$^{-3}$), the critical temperature reaches the maximum and starts to fall back as in the inset of Fig. \[f3\]. The suppression of the critical temperature is mainly due to the oscillatory RKKY interaction at higher concentrations. The effective coupling between impurity spins is frustrated and not longer purely ferromagnetic. The existence of an optimal density is totally missed in the Weiss mean-field theory, where both the spatially varying fluctuations and the quantum frustrations are ignored.
Beyond the optimal density, not only the critical temperature falls back, the shape of the magnetization also undergoes an interesting change. There are two significant features. One is the magnetization curve becomes convex again. Another feature is that the magnetization has a very sharp decrease near the critical temperature, making it look almost like the first-order phase transition. To make sure that the sharp drop is not an artifact of numerical errors, the data points near this regime are chosen very closely to ensure we still have enough points in the steep regime. In particular, for $c^* =0.05$, it is spectacular that the magnetization drops 50% within 1K. It is not clear at this point what is the underlying mechanism behind this dramatic suppression of magnetization. A more sophisticated theory starting from the critical point might be able to address this interesting dive of magnetization near the critical temperature.
Since the global trend of the critical temperature is important, it is plotted in Fig. \[f4\]. Comparing with Fig. 2 in Ref.[@Dietl00], the self-consistent Green’s function approach produces a more complicated landscape. In the Weiss mean-filed theory, the critical temperature is $$T^{\rm MF}_c = \frac{\chi_{P}}{(g^{*} \mu_B/2)^2} \frac{S(S+1) J^2 N}{12},$$ where $g^*$ is the $g$-factor of the carriers and $\chi_{P}$ is their Pauli susceptibility, which is proportional to the effective band mass. The above formula would produce a profile with monotonically increasing $T_c$ as the densities become larger. It is clear from Fig. \[f4\] that the profile of the Curie temperature has a ridge, roughly along the curve $c \sim 100 c^{*}$ with the given parameters in this Letter, and is [*qualitatively*]{} different from previous studies. A closer check would find that our approach also produces quantitative differences. Comparing with the estimates of Curie temperatures, previously studied by one of the authors in Refs. [@Schliemann01] and [@Konig00], the present approach gives a lower $T_c$ in all densities regimes. So it seems that the inclusion of spin kinematics at finite temperatures is not only important to get the right trend of the critical temperatures, but also crucial in estimating their values.
![\[f4\] The trend of the critical temperatures at different densities.](Trend){width="8cm"}
Finally, we address the important aspects of physics which are left out so far. To make quantitative comparison with the experiments, it is crucial to adapt the realistic band structure of the host semiconductors, for instance, the six-band Luttinger model for GaAs. The inclusion of the more complex band structure would not cause formidable messes in calculating the Green’s function self-consistently. While most of the conclusions drawn from the simple parabolic band should remain valid, a more realistic band structure is desirable for making quantitative predictions. The electronic correlation can also be included at the mean-field level in a systematic way. Since Coulomb repulsive interaction stabilizes the ferromagnetic phase, the Curie temperature is expected to be higher. To include the disorder is more subtle. In the diffusive regime, one can replace the ballistic propagator of the itinerant carriers by the diffusive one to account for the disorder effects.
In conclusion, we employ the self-consistent Green’s function approach to study the DMS and derive a general formula for the Curie temperature. In addition, we demonstrate the interesting crossover of the magnetization curve from concave to convex, which is not driven by the localization effects.
We thanks Chung-Yu Mou and Ming-Fong Yang for fruitful discussions and the support of National Science Council in Taiwan. The hospitality of the National Center for Theoretical Science, where the work was initiated, is greatly acknowledged.
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---
abstract: 'To enable efficient exploration of Web-scale scientific knowledge, it is necessary to organize scientific publications into a hierarchical concept structure. In this work, we present a large-scale system to (1) identify hundreds of thousands of scientific concepts, (2) tag these identified concepts to hundreds of millions of scientific publications by leveraging both text and graph structure, and (3) build a six-level concept hierarchy with a subsumption-based model. The system builds the most comprehensive cross-domain scientific concept ontology published to date, with more than 200 thousand concepts and over one million relationships.'
author:
- |
Zhihong Shen\
Microsoft Research\
Redmond, WA, USA\
Hao Ma\
Microsoft Research\
Redmond, WA, USA\
[{zhihosh,haoma,kuansanw}@microsoft.com]{}\
Kuansan Wang\
Microsoft Research\
Redmond, WA, USA\
bibliography:
- 'acl2018.bib'
title: 'A Web-scale system for scientific knowledge exploration'
---
Introduction
============
Scientific literature has grown exponentially over the past centuries, with a two-fold increase every 12 years [@dong2017century], and millions of new publications are added every month. Efficiently identifying relevant research has become an ever increasing challenge due to the unprecedented growth of scientific knowledge. In order to assist researchers to navigate the entirety of scientific information, we present a deployed system that organizes scientific knowledge in a hierarchical manner.
To enable a streamlined and satisfactory semantic exploration experience of scientific knowledge, three criteria must be met:
- a comprehensive coverage on the broad spectrum of academic disciplines and concepts (we call them *concepts* or *fields-of-study*, abbreviated as *FoS*, in this paper);
- a well-organized hierarchical structure of scientific *concepts*;
- an accurate mapping between these *concepts* and all forms of academic *publications*, including books, journal articles, conference papers, pre-prints, etc.
![Three modules of the system: *concept discovery*, *concept-document tagging*, and *concept hierarchy generation*.[]{data-label="fig:FoSTagging_artchitecture"}](FoSTagging.pdf){width="8cm"}
--------------------- --------------------------- ------------------------ --------------------
[**Concept** ]{} [**Concept** ]{} [**Hierarchy** ]{}
[**discovery**]{} [**tagging**]{} [**building**]{}
[**Main** ]{} scalability / trustworthy scalability / stability /
[**challenges**]{} representation coverage accuracy
[**Problem** ]{} knowledge base multi-label topic hierarchy
[**formulation**]{} type prediction text classification construction
[**Solution /** ]{} Wikipedia / KB / word embedding / extended
[**model(s)**]{} graph link analysis text + graph structure subsumption
[**Data scale**]{} $10^5$ – $10^6$ $10^9$ – $10^{10}$ $10^6$ – $10^7$
[**Data update**]{}
[**frequency**]{} monthly weekly monthly
--------------------- --------------------------- ------------------------ --------------------
\[tab:System\_summary\]
To build such a system on Web-scale, the following challenges need to be tackled:
- **Scalability:** Traditionally, academic discipline and concept taxonomies have been curated manually on a scale of hundreds or thousands, which is insufficient in modeling the richness of academic concepts across all domains. Consequently, the low concept coverage also limits the exploration experience of hundreds of millions of scientific publications.
- **Trustworthy representation:** Traditional concept hierarchy construction approaches extract concepts from unstructured documents, select representative terms to denote a concept, and build the hierarchy on top of them [@sanderson1999deriving; @liu2012automatic]. The concepts extracted this way not only lack authoritative definition, but also contain erroneous topics with subpar quality which is not suitable for a production system.
- **Temporal dynamics:** Academic publications are growing at an unprecedented pace (about 70K more papers per day according to our system) and new concepts are emerging faster than ever. This requires frequent inclusion on latest publications and re-evaluation in tagging and hierarchy-building results.
In this work, we present a Web-scale system with three modules—concept discovery, concept-document tagging, and concept-hierarchy generation—to facilitate scientific knowledge exploration (see Figure \[fig:FoSTagging\_artchitecture\]). This is one of the core components in constructing the Microsoft Academic Graph (MAG), which enables a semantic search experience in the academic domain[^1]. MAG is a scientific knowledge base and a heterogeneous graph with six types of academic entities: publication, author, institution, journal, conference, and field-of-study (i.e., *concept* or *FoS*). As of March 2018, it contains more than 170 million publications with over one billion paper citation relationships, and is the largest publicly available academic dataset to date[^2].
To generate high-quality *concepts* with comprehensive coverage, we leverage Wikipedia articles as the source of concept discovery. Each Wikipedia article is an *entity* in a general knowledge base (*KB*). A *KB* entity associated with a Wikipedia article is referred to as a Wikipedia entity. We formulate concept discovery as a knowledge base type prediction problem [@neelakantan2015inferring] and use graph link analysis to guide the process. In total, 228K academic concepts are identified from over five million English Wikipedia entities.
During the tagging stage, both textual information and graph structure are considered. The text from Wikipedia articles and papers’ meta information (e.g., titles, keywords, and abstracts) are used as the *concept*’s and *publication*’s textual representations respectively. Graph structural information is leveraged by using text from a *publication*’s neighboring nodes in MAG (its citations, references, and publishing venue) as part of the *publication*’s representation with a discounting factor. We limit the search space for each publication to a constant range, reduce the complexity to $O(N)$ for scalability, where $N$ is the number of publications. Close to one billion concept-publication pairs are established with associated confidence scores.
Together with the notion of subsumption [@sanderson1999deriving], this confidence score is then used to construct a six-level directed acyclic graph (DAG) hierarchy with over 200K nodes and more than one million edges.
Our system is a deployed product with regular data refreshment and algorithm improvement. Key features of the system are summarized in Table \[tab:System\_summary\]. The system is updated weekly or monthly to include fresh content on the Web. Various document and language understanding techniques are experimented with and incorporated to incrementally improve the performance over time.
System Description
==================
Concept Discovery
-----------------
As top level disciplines are extremely important and highly visible to system end users, we manually define 19 top level (“L0") disciplines (such as *physics*, *medicine*) and 294 second level (“L1") sub-domains (examples are *machine learning*, *algebra*) by referencing existing classification[^3] and get their correspondent Wikipedia entities in a general in-house knowledge base (*KB*).
It is well understood that entity types in a general *KB* are limited and far from complete. Entities labeled with *FoS* type in *KB* are in the lower thousands and noisy for both in-house *KB* and latest Freebase dump[^4]. The goal is to identify more *FoS* type entities from over 5 million English Wikipedia entities in an in-house KB. We formulate this task as a knowledge base type prediction problem, and focus on predicting only one specific type—*FoS*.
In addition to the above-mentioned “L0" and “L1" FoS, we manually review and identify over 2000 high-quality ones as initial seed FoS. We iterate a few rounds between a *graph link analysis* step for candidate exploration and an *entity type based filtering and enrichment* step for candidate fine-tuning based on *KB* types.
***Graph link analysis***: To drive the process of exploring new FoS candidates, we apply the intuition that if the majority of an entity’s nearest neighbours are FoS, then it is highly likely an FoS as well. To calculate nearest neighbours, a distance measure between two Wikipedia entities is required. We use an effective and low-cost approach based on Wikipedia link analysis to compute the semantic closeness [@milne2008effective]. We label a Wikipedia entity as an FoS candidate if there are more than $K$ neighbours in its top $N$ nearest ones are in a current FoS set. Empirically, $N$ is set to 100 and $K$ is in \[35, 45\] range for best results.
***Entity type based filtering and enrichment***: The candidate set generated in the above step contains various types of entities, such as *person*, *event*, *protein*, *book topic*, etc.[^5] Entities with obvious invalid types are eliminated (e.g. *person*) and entities with good types are further included (e.g. *protein*, such that all Wikipedia entities which have labeled type as *protein* are added). The results of this step are used as the input for *graph link analysis* in the next iteration.
More than 228K FoS have been identified with this iterative approach, based on over 2000 initial seed FoS.
Tagging Concepts to Publications
--------------------------------
We formulate the concept tagging as a multi-label classification problem; i.e. each publication could be tagged with multiple FoS as appropriate. In a naive approach, the complexity could reach $M\cdot N$ to exhaust all possible pairs, where $M$ is 200K+ for FoS and $N$ is close to 200M for publications. Such a naive solution is computationally expensive and wasteful, since most scientific publications cover no more than 20 FoS based on empirical observation.
We apply heuristics to cut candidate pairs aggressively to address the scalability challenge, to a level of 300–400 FoS per publication[^6]. Graph structural information is incorporated in addition to textual information to improve the accuracy and coverage when limited or inadequate text of a *concept* or *publication* is accessible.
We first define ***simple** representing text* (or *SRT*) and ***extended** representing text* (or *ERT*). *SRT* is the text used to describe the academic entity itself. *ERT* is the extension of *SRT* and leverages the graph structural information to include textual information from its neighbouring nodes in MAG.
A publishing venue’s full name (i.e. the journal name or the conference name) is its *SRT*. The first paragraph of a concept’s Wikipedia article is used as its *SRT*. Textual meta data, such as title, keywords, and abstract is a publication’s *SRT*.
We sample a subset of publications from a given venue and concatenate their *SRT*. This is used as this venue’s *ERT*. For broad disciplines or domains (e.g. L0 and L1 FoS), Wikipedia text becomes too vague and general to best represent its academic meanings. We manually curate such concept-venue pairs and aggregate *ERT* of venues associated with a given concept to obtain the *ERT* for the concept. For example, *SRT* of a subset of papers from *ACL* are used to construct *ERT* for *ACL*, and subsequently be part of the *ERT* for *natural language processing* concept. A *publication*’s *ERT* includes *SRT* from its citations, references and *ERT* of its linked publishing *venue*.
We use $h_{s}^{p}$ and $h_{e}^{p}$ to denote the representation of a *publication* ($p$)’s *SRT* and *ERT*, $h_{s}^{v}$ and $h_{e}^{v}$ for a *venue* ($v$)’s *SRT* and *ERT*. Weight $w$ is used to discount different neighbours’ impact as appropriate. Equation \[eq:Pub\_aggre\] and \[eq:Venue\_aggre\] formally define publication *ERT* and venue *ERT* calculation.
$$\label{eq:Pub_aggre}
h_{e}^{p} = h_{s}^{p} + \sum_{i \in Cit}w_{i}h_{s}^{p}(i) +
\sum_{j \in Ref}w_{j}h_{s}^{p}(j) + w_{v}h_{e}^{v}$$
$$\label{eq:Venue_aggre}
h_{e}^{v} =\sum_{i \in V}h_{s}^{p}(i) + h_{s}^{v}$$
Four types of features are extracted from the text: bag-of-words (BoW), bag-of-entities (BoE), embedding-of-words (EoW), and embedding-of-entities (EoE). These features are concatenated for the vector representation $h$ used in Equation \[eq:Pub\_aggre\] and \[eq:Venue\_aggre\]. The *confidence score* of a concept-publication pair is the cosine similarity between these vector representations.
We pre-train the word embeddings by using the skip-gram [@mikolov2013distributed] on the academic corpus, with 13B words based on 130M titles and 80M abstracts from English scientific publications. The resulting model contains 250-dimensional vectors for 2 million words and phrases. We compare our model with pre-trained embeddings based on general text (such as Google News[^7] and Wikipedia[^8]) and observe that the model trained from academic corpus performs better with higher accuracy on the concept-tagging task with more than 10% margin.
Conceptually, the calculation of publication and venue’s *ERT* is to leverage neighbours’ information to represent itself. The MAG contains hundreds of millions of nodes with billions of edges, hence it is computationally prohibitive by optimizing the node latent vector and weights simultaneously. Therefore, in Equation \[eq:Pub\_aggre\] and \[eq:Venue\_aggre\], we initialize $h_{s}^{p}$ and $h_{s}^{v}$ based on textual feature vectors defined above and adopt empirical weight values to directly compute $h_{e}^{p}$ and $h_{e}^{v}$ to make it scalable.
After calculating the similarity for about 50 billion pairs, close to 1 billion are finally picked based on the threshold set by the confidence score.
{width="7cm"}
\[fig:Hierarchy\_explain\]
Concept Hierarchy Building
--------------------------
In this subsection, we describe how to build a concept hierarchy based on concept-document tagging results. We extend Sanderson and Croft’s early work which uses the notion of subsumption—a form of co-occurrence—to associate related terms. We say term $x$ subsumes $y$ if $y$ occurs only in a subset of the documents that $x$ occurs in. In the hierarchy, $x$ is the parent of $y$. In reality, it is hard for $y$ to be a strict subset of $x$. Sanderson and Croft’s work relaxed the subsumption to 80% (e.g. $P(x|y) \geq 0.8, P(y|x) < 1$).
In our work, we extend the concept co-occurrence calculation weighted with the concept-document pair’s confidence score from previous step. More formally, we define a *weighted relative coverage* score between two concepts $i$ and $j$ as below and illustrate in Figure \[fig:Hierarchy\_explain\].
$$RC(i,j)=
\frac{\sum_{k \in (I \cap J)} w_{i,k}}{\sum_{k \in I} w_{i,k}}
-
\frac{\sum_{k \in (I \cap J)} w_{j,k}}{\sum_{k \in J} w_{j,k}}$$
![Deployed system homepage at March 2018, with all six types of entities statistics: over 228K *fields-of-study*.[]{data-label="fig:System_Deploy"}](System_Deploy){width="8cm"}
Set $I$ and $J$ are documents tagged with concepts $i$ and $j$ respectively. $I \cap J$ is the overlapping set of documents that are tagged with both $i$ and $j$. $w_{i,k}$ denotes the confidence score (or weights) between concept $i$ and document $k$, which is the final *confidence score* in the previous concept-publication tagging stage. When $RC(i,j)$ is greater than a given positive threshold[^9], $i$ is the child of $j$. Since this approach does not enforce single parent for any FoS, it results in a directed acyclic graph (DAG) hierarchy.
With the proposed model, we construct a six level FoS hierarchy (from L0 and L5) on over 200K concepts with more than 1M parent-child pairs. Due to the high visibility, high impact and small size, the hierarchical relationships between L0 and L1 are manually inspected and adjusted if necessary. The remaining L2 to L5 hierarchical structures are produced completely automatically by the extended subsumption model.
One limitation of subsumption-based models is the intransitiveness of parent-child relationships. This model also lacks a type-consistency check between parents and children. More discussions on such limitations with examples will be in evaluation section \[sec:Evaluation\].
Deployment and Evaluation
=========================
{width="11cm"}
Deployment
----------
The work described in this paper has been deployed in the production system of Microsoft Academic Service[^10]. Figure \[fig:System\_Deploy\] shows the website homepage with entity statistics. The contents of MAG, including the full list of FoS, FoS hierarchy structure, and FoS tagging to papers, are accessible via API, website, and full graph dump from *Open Academic Society*[^11].
Figure \[fig:Word2vec\_example\] shows the example for *word2vec* concept. Concept definition with linked Wikipedia page, its immediate parents (*machine learning*, *artificial intelligence*, *natural language processing*) in the hierarchical structure and its related concepts[^12] (*word embedding*, *artificial neural network*, *deep learning*, etc.) are shown on the right rail pane. Top tagged publications (without *word2vec* explicitly stated in their text) are recognized via graph structure information based on citation relationship.
[**Step**]{} [**Accuracy**]{}
------------------------ ------------------
`1. Concept discovery` [94.75%]{}
`2. Concept tagging` [81.20%]{}
`3. Build hierarchy` [78.00%]{}
: Accuracy results for each step.[]{data-label="tab:eval_accuracy_results"}
\[!t\]
[**L5**]{} [**L4**]{} [**L3**]{} [**L2**]{} [**L1**]{} [**L0**]{}
----------------------- --------------- --------------------- ---------------- ------------------- -------------
Convolutional Deep Deep belief Deep Artificial Machine Computer
Belief Networks network learning neural network learning Science
(Methionine synthase) Methionine Amino Biochemistry / Chemistry /
reductase synthase Methionine acid Molecular biology Biology
(glycogen-synthase-D) Phosphorylase Glycogen
phosphatase kinase synthase Glycogen Biochemistry Chemistry
Fréchet Generalized extreme Extreme
distribution value distribution value theory Statistics Mathematics
Hermite’s Hermite Spline Mathematical
problem spline interpolation Interpolation analysis Mathematics
\[tab:example\_results\]
Evaluation {#sec:Evaluation}
----------
For this deployed system, we evaluate the accuracy on three steps (*concept discovery*, *concept tagging*, and *hierarchy building*) separately.
For each step, 500 data points are randomly sampled and divided into five groups with 100 data points each. On *concept discovery*, a data point is an FoS; on *concept tagging*, a data point is a concept-publication pair; and on *hierarchy building*, a data point is a parent-child pair between two concepts. For the first two steps, each 100-data-points group is assigned to one human judge. The concept hierarchy results are by nature more controversial and prone to individual subjective bias, hence we assign each group of data to three judges and use majority voting to decide final results.
The accuracy is calculated by counting positive labels in each 100-data-points group and averaging over 5 groups for each step. The overall accuracy is shown in Table \[tab:eval\_accuracy\_results\] and some sampled hierarchical results are listed in Table \[tab:example\_results\]. Most hierarchy dissatisfaction is due to the intransitiveness and type-inconsistent limitations of the subsumption model. For example, most publications that discuss the *polycystic kidney disease* also mention *kidney*; however, for all publications that mentioned *kidney*, only a small subset would mention *polycystic kidney disease*. According to the subsumption model, *polycystic kidney disease* is the child of *kidney*. It is not legitimate for a *disease* as the child of an *organ*. Leveraging the entity type information to fine-tune hierarchy results is in our plan to improve the quality.
Conclusion
==========
In this work, we demonstrated a Web-scale production system that enables an easy exploration of scientific knowledge. We designed a system with three modules: concept discovery, concept tagging to publications, and concept hierarchy construction. The system is able to cover latest scientific knowledge from the Web and allows fast iterations on new algorithms for document and language understanding.
The system shown in this paper builds the largest cross-domain scientific concept ontology published to date, and it is one of the core components in the construction of the Microsoft Academic Graph, which is a publicly available academic knowledge graph—a data asset with tremendous value that can be used for many tasks in domains like data mining, natural language understanding, science of science, and network science.
[^1]: The details about where and how we obtain, aggregate, and ingest academic publication information into the system is out-of-scope for this paper and for more information please refer to [@sinha2015overview].
[^2]: <https://www.openacademic.ai/oag/>
[^3]: <http://science-metrix.com/en/classification>
[^4]: <https://developers.google.com/freebase/>
[^5]: Entity types are obtained from the in-house KB, which has higher type coverage compared with Freebase, details on how the in-house KB produces entity types is out-of-scope and not discussed in this paper.
[^6]: We include all L0s and L1s and FoS entities spotted in a publication’s ***extended** representing text*, which is defined later in this section
[^7]: <https://code.google.com/archive/p/word2vec/>
[^8]: <https://fasttext.cc/docs/en/pretrained-vectors.html>
[^9]: It is usually in \[$0.2$, $0.5$\] based on empirical observation.
[^10]: <https://academic.microsoft.com/>
[^11]: <https://www.openacademic.ai/oag/>
[^12]: Details about how to generate related entities are out-of-scope and not included in this paper.
|
---
abstract: 'The angular power spectrum of the cosmic microwave background (CMB) temperature anisotropies is a good probe to look into the primordial density fluctuations at large scales in the universe. Here we re-examine the angular power spectrum of the Wilkinson Microwave Anisotropy Probe data, paying particular attention to the fine structures (oscillations) at $\ell=100 \sim 150$ reported by several authors. Using Monte-Carlo simulations, we confirm that the gap from the simple power law spectrum is a rare event, about 2.5–3$\sigma$, if these fine structures are generated by experimental noise and the cosmic variance. Next, in order to investigate the origin of the structures, we examine frequency and direction dependencies of the fine structures by dividing the observed QUV frequency maps into four sky regions. We find that the structures around $\ell \sim 120$ do not have significant dependences either on frequencies or directions. For the structure around $\ell \sim 140$, however, we find that the characteristic signature found in the all sky power spectrum is attributed to the anomaly only in the South East region.'
author:
- Kohei Kumazaki
- Kiyotomo Ichiki
- Naoshi Sugiyama
- Joseph Silk
bibliography:
- 'reference.bib'
title: Exploring the origin of the fine structures in the CMB temperature angular power spectrum
---
Introduction
============
The inflationary cosmology is a successful paradigm in explaining the generation of primordial density fluctuations and solving essential problems of the classical Big Bang cosmology [@1981PhRvD..23..347G; @1981MNRAS.195..467S; @1980PhLB...91...99S; @1982PhLB..115..295H; @1982PhLB..117..175S; @1982PhRvL..49.1110G]. The primordial density fluctuations are transformed into the anisotropy of the Cosmic Microwave Background (CMB) and the Large Scale Structure of the universe (LSS). Thanks to the high angular resolution and longer-term observations of the anisotropy of the temperature fluctuations, such as by the Wilkinson Microwave Anisotropy Probe (WMAP) [@Jarosik:WMAP7yr], the South Pole Telescope [@2011ApJ...743...28K], the Atacama Cosmology Telescope [@2011ApJ...739...52D], the Arcminute Cosmology Bolometer Array Receiver [@2009ApJ...694.1200R], the Cosmic Background Imager [@2009arXiv0901.4540S] and so on, it has been found that the angular power spectrum of temperature fluctuations conforms to the prediction from the $\Lambda$CDM model with slow roll inflation.
While the observed angular power spectrum is globally consistent with a smooth power-law primordial spectrum of density fluctuations [@Wang:1998gb; @Tegmark:2002cy; @Spergel:2006hy; @Vazquez:2011xa; @Hlozek:2011pc], some gaps between the prediction from the simplest power-law model and the observed data have been reported. They are discussed recently by grace of the detailed observations, including a small bump and dip at $\ell = 20$ – $40$ [@Mortonson:2009qv; @Dvorkin:2009ne; @Hazra:2010ve] or oscillation around $\ell = 100$ – $150$ [@Ichiki:2009zz; @Ichiki:2009xs; @Nakashima:2010sa; @Kumazaki:2011eb].
Possible causes of these anomalous structures are discussed in the literature. The small bump and dip at $\ell=20$ – $40$, for example, may be explained by the mass variation of the inflaton during the inflation phase [@Adams:2001vc; @Mortonson:2009qv; @Dvorkin:2009ne; @Hazra:2010ve]. When inflaton obeys single slow-roll inflation dynamics, the power spectrum of primordial density fluctuations shows a power-law feature, and the angular power spectrum of the CMB is expected to be a smooth curve. If the inflaton mass has changed during inflation, however, some oscillating structures emerge in the power-law primordial power spectrum because the inflaton field is forced to accelerate and/or decelerate rapidly during the slow roll inflation phase. In such cases the bump and dip structure rises up in the angular power spectrum of the CMB.
On the other hand, it seems difficult to explain the oscillating structures around $\ell = 100$ – $150$ on the firm theoretical background, though some works have tried to explain the origin [@Ichiki:2009zz; @Ichiki:2009xs; @Nakashima:2010sa; @Kumazaki:2011eb]. In the previous paper, some of us have tried to explain the oscillating structures with the inflaton mass variation [@Kumazaki:2011eb]. The condition considered in that paper is that inflaton mass changes with oscillations. We found that oscillating structures can be generated at the arbitrary scale by adjusting oscillation number and time scale of the mass variation. However, the width of this structure tends to become so wide, and to match the observed data we need some fine-tunings. If we force to explain the oscillation matching with the observed data, the parameters become unrealistic values. Therefore, we have concluded that it is difficult to explain the oscillating structure on the angular power spectrum found at multipole range $\ell=100$–150 with the inflaton mass variation.
Nakashima et al. [@Nakashima:2010sa] have proposed a sudden change of the sound velocity of the inflaton field during inflation. Based on their model, the authors found oscillating structures in the primordial power spectrum. However, the oscillating structures tend to extend in a wide range of wavenumber up to $\ell \leq 300$ which includes the first acoustic peak. Therefore, this model may not match with observed data which shows the oscillation only at the confined region of $100\leq \ell\leq 150$.
In light of the difficulty in explaining the structure on the theoretical background, in this paper we closely explore the origin of the structure with the data taken by WMAP, particularly paying attention to the frequency and direction dependences. If the observed structure is really from the cosmological origin, we expect the structures should be independent of them.
This paper is organized as follows. In section II, we review the method to estimate the angular power spectrum from the two point correlation function. Following the method, we examine the probability of the fine structures being generated from noise and/or cosmic variance in section III, and estimate the angular power spectrum with each frequency band in section IV. In section V, we give the angular power spectrum with the partial sky, with an explanation of some difficulties in estimating the angular power spectrum with the partial sky map [@Ansari:2009ys; @Ko:2011ut]. Section VI is devoted to the summary of this work.
The angular power spectrum
==========================
The angular power spectrum $C_\ell$ is a good indicator for quantifying the temperature fluctuations of the CMB, and defined as $$\langle\,a^\ast_{\ell m}a_{\ell^\prime m^\prime}\,\rangle
= \delta_{\ell\ell^\prime}\delta_{m m^\prime}C_\ell~,$$ with $$a_{\ell m} = \int \sin{\theta} d\theta \int d\phi
\Delta T(\theta, \phi)Y_{\ell m}^\ast(\theta, \phi)\,,$$ where $Y_{\ell m}^\ast(\theta, \phi)$ is the spherical harmonics function evaluated at the position $(\theta, \phi)$ of spherical coordinate, $\Delta T(\theta, \phi)$ is the value of temperature fluctuations and $\langle$...$ \rangle$ means ensemble average. In practice, however, we can observe only one unique sky and need to estimate the angular power spectrum from one realization. We can obtain the estimator $\tilde C_\ell$ of the $C_\ell$ by $$\tilde C_\ell = \frac{1}{2\ell +1}\sum^\ell_{\ell=-m}|a_{\ell m}|^2.$$ In this work, we use a fast CMB analysis named Spice introduced by Szapudi et al. for temperature [@Szapudi:2000xj] and Chon et al. for polarization [@2004MNRAS.350..914C]. This software calculates the angular power spectrum from the CMB temperature/polarization data and can convert the angular power spectrum to two point correlation function or vice versa, if we need. We review the relation between the angular power spectrum and the angular correlation function for later discussion.
The CMB temperature angular correlation function $\xi(\theta)$ is defined by $$\xi(\theta) = \langle\,\Delta T(\vec n) \Delta T(\vec n^\prime)\,\rangle~,$$ where $\vec n$ and $\vec n^\prime$ denote sky directions, and $\cos
\theta = \vec n \cdot \vec n^\prime$. The relation between the angular correlation function and the angular power spectrum can be written as, $$\begin{aligned}
\xi(\theta) = \frac{1}{4\pi}\sum_{\ell=0}^{\infty}\,(2\ell+1)\,C_\ell\,P_\ell(\cos\theta)~, \\
C_\ell = 2\pi \int_{0}^\pi \sin\theta\,d\theta\,\xi(\theta)\,P_\ell(\cos\theta)~, \end{aligned}$$ where $P_\ell(\cos\theta)$ is the Legendre polynomial.
Validity of the fine structures {#sec:vldty}
===============================
Monte-Carlo method
------------------
As a possible origin of the fine structures, instrumental noise and cosmic variance effects should be called into question first. In this section, we examine the possibility of the fine structures being generated by these noise effects. For this purpose, we generate a number of sky maps of CMB temperature fluctuations using the HEALPix function, [**isynfast**]{} [@Gorski:1998vw]. The routine generates a temperature fluctuation map from an input angular power spectrum $C_\ell$. For the $C_\ell$, we adopt the best-fitting $\Lambda$CDM model of the WMAP seven year parameter table [@Komatsu:WMAP7yr]. The resolution of this map is taken as $N_{\rm
side}=512$. This value is same as that of the released sky map given by the WMAP team.
Next, we add the Gaussian noise expected for the WMAP observation on these simulated sky maps. The variance of the instrumental noise can be written by $$N^i=\frac{\sigma_0}{\sqrt{N_{\rm obs}^i}}~,$$ where superscript $i$ represents the pixel number on the sky, $\sigma_0$ the rms noise per an observation, and $N_{\rm obs}^i$ the number of observation of $i$th pixel. The values of $\sigma_0$ and $N^i_{\rm
obs}$ are also given by the WMAP team [@Jarosik:WMAP7yr]. In this way, we can prepare the sky maps of CMB temperature fluctuations of different universes with the same cosmological parameters and noise properties.
In the present analysis, we prepare 3,000 sky maps and calculate the angular power spectrum for each sky map. These $C_\ell$’s are denoted by $C_\ell^j$’s using a superscript of the map number $j$. We estimate the average and variance of these $C_\ell^j$’s at each multipole moment $\ell$, and compare the angular power spectra with the WMAP angular power spectrum, $C_\ell^{\rm WMAP}$. In order to emphasize the differences from the $\Lambda$CDM model, we show the distributions of the simulated $C_{\ell}$’s as residuals from the $\Lambda$CDM model, $C_{\ell}^{\rm \Lambda CDM}$ in Fig. \[fig:val\]. The variances of $C_\ell$’s at $1\sigma$, $2\sigma$, and $3\sigma$ are shown as the boxes. We can see that the average of the simulated $C_\ell$’s (the black line in the figure) is on the zero horizontal axis. The red line represents the residuals between $C_\ell^{\rm \Lambda CDM}$ and $C_\ell^{\rm
WMAP}$.
![Differences between the WMAP angular power spectrum $C_\ell^{\rm WMAP}$ and the power spectrum of the $\Lambda$CDM model $C_\ell^{\rm \Lambda CDM}$ (red line). The boxes represent the variance of $1,2,3\sigma$ at each $\ell$ estimated from the 3,0000 simulations. The black line represents the average value of residuals between $C_\ell^j$’s and $C_\ell^{\rm \Lambda CDM}$.[]{data-label="fig:val"}](graph/lcdm_alldiff_sum.eps){width="45.00000%"}
significance of the structure
-----------------------------
In the multipole region of $\ell = 100$ – $150$, the distributions of the $C_\ell$’s are well approximated by the Gaussian one. In that case the probability that the $C_\ell$ is within $1\sigma$ and $2\sigma$ should correspond to about 68.3% and 95.5%, respectively. In our analysis, there are 51 data points because we focus only between $\ell=100$ – $150$, and the expected number of data over $1\sigma$ or $2\sigma$ is estimated as 16.2 or 2.3 points, respectively. Thus, it is natural that some data points deviate from the theoretical curve to this extent. However, from Fig. \[fig:val\], we find that 19 and 7 points of data are over $1\sigma$ and $2\sigma$ levels, respectively. That indicates the $C_\ell^{\rm WMAP}$ somewhat deviates from the standard Gaussian distribution more than expected.
We roughly estimate the probability $P$ as a function of $N_{2\sigma}$ and $N_{1\sigma\mbox{-}2\sigma}$ which are the numbers of realization that exceeds $2\sigma$ and lies between 1$\sigma$ and 2$\sigma$, respectively, assuming the realization follows the Gaussian distribution. We show the result as contours in Figure. \[fig:possibility\]. In the figure the red, green and blue lines represent $1\sigma$, $2\sigma$ and $3\sigma$, respectively. From the figure, we can see that the peak is at the expected value ($N_{1\sigma\mbox{-}2\sigma}$, $N_{2\sigma}$)=(13.9, 2.3), and the probability decreases with distance away from the expected value. The yellow point represents the value from the WMAP data; ($N_{1\sigma\mbox{-}2\sigma}$, $N_{2\sigma}$)=(12, 7). It lies at the location between 2.5–$3\sigma$, which is consistent with the result of [@Ichiki:2009zz]. Therefore, we can understand that the fine structures at the range of $\ell=100$–150 of the angular power spectrum which are observed by WMAP team seem a rare event, if these structures are generated only by the noise and the cosmic variance effects.
![Contours of equal probabilities as a function of $N_{1\sigma\mbox{-}2\sigma}$ and $N_{2\sigma}$. The red, green and blue lines correspond to $1\sigma$, $2\sigma$ and $3\sigma$, respectively. The (yellow) point represents the WMAP data.[]{data-label="fig:possibility"}](graph/contour2.eps){width="50.00000%"}
If the structures were not related to the noise nor cosmic variance, the other possibilities would be foreground effects and/or real cosmological signal. From the next section, we examine frequency and direction dependences of the fine structures in order to investigate whether the structures come from cosmological or astronomical phenomena.
Frequency dependence
====================
If any astronomical phenomena such as syncrotron emission, dust emission and radio galaxies infiltrate the CMB temperature fluctuation data and create the fine structures, we expect that they have a characteristic frequency dependence. Therefore, we estimate the angular power spectra for three different frequency bands, namely Q, V and W bands. We find that the shapes of the fine structures at each frequency band are very similar to each other and to the all sky one (Fig. \[fig:val\]). Thus it is improbable that the fine structures originate from astrophysical phenomena nor objects, because they will have different frequency dependences from the CMB blackbody spectrum.
Analysis with the partial skies
===============================
In this section, we look into the differences in the angular power spectra for different sky directions. For this purpose we prepare four masks, namely the North West mask ($0\leq \phi\leq \pi$, $0\leq
\theta\leq \pi/2$), the North East mask ($\pi\leq \phi\leq 2\pi,\,0\leq
\theta\leq \pi/2$), the South West mask ($0\leq \phi\leq \pi,\,\pi/2\leq
\theta\leq \pi$) and the South East mask ($\pi\leq \phi\leq 2\pi,\,\pi/2\leq
\theta\leq \pi$). We show these masks in Fig. \[fig:mask\_maps\].
[cc]{}
![Masks of the partial skies used in our analysis in the galactic coordinate. The left top panel refers to North West mask, the right top panel to North East mask, the left bottom panel to South West mask and, finally, the right bottom to South East mask.[]{data-label="fig:mask_maps"}](graph/mask_nw-bw.eps){width="\textwidth"}
![Masks of the partial skies used in our analysis in the galactic coordinate. The left top panel refers to North West mask, the right top panel to North East mask, the left bottom panel to South West mask and, finally, the right bottom to South East mask.[]{data-label="fig:mask_maps"}](graph/mask_ne-bw.eps){width="\textwidth"}
[cc]{}
![Masks of the partial skies used in our analysis in the galactic coordinate. The left top panel refers to North West mask, the right top panel to North East mask, the left bottom panel to South West mask and, finally, the right bottom to South East mask.[]{data-label="fig:mask_maps"}](graph/mask_sw-bw.eps){width="\textwidth"}
![Masks of the partial skies used in our analysis in the galactic coordinate. The left top panel refers to North West mask, the right top panel to North East mask, the left bottom panel to South West mask and, finally, the right bottom to South East mask.[]{data-label="fig:mask_maps"}](graph/mask_se-bw.eps){width="\textwidth"}
Using the masks, first let us calculate the gap between the angular power spectra of the simulated maps and that of the observed map by WMAP for the partial skies. Further, we compare the partial sky maps with each other, in order to see the dependence of the angular power spectrum on the sky directions. However, some difficulties arise when analyzing the CMB map with a partial sky mask. Because a mask narrows down the effective area of the spherical surface, effective information is reduced. Furthermore, if a mask has a sharp cutoff at the boundary, it will lead a truncated correlation function. Computing the angular power spectrum from the truncated correlation function causes spurious oscillations inherent to the Fourier transformation, which is known as the “Gibbs phenomenon”. However, these oscillations have nothing to do with the anisotropy of the CMB nor the fine structures, and they can be removed by apodization and binning techniques as we describe below.
In the next two subsections, we review these techniques that we used in order to lighten and avoid these nuisance effects, and in the final subsection, we show the direction dependence of the CMB angular power spectrum.
Apodization
-----------
In fact, we find that spurious oscillatory structure in the angular power spectrum that arises when the masks are used is caused by the large amplitude oscillations in the two point correlation function at large angular scales. The oscillations are caused by the statistical errors. Indeed, if we use a map with a maximal angular size $\theta_{\rm max}$, nothing can be known about the correlation function for $\theta \gtrsim \theta_{\rm max}$. However, simply truncating the correlation function at $\theta=\theta_{\rm max}$ does not help the situation as we mentioned above. Instead, the technique uses an appropriate function, called the “apodization function” $F(\theta)$, and multiplies it with the correlation function $\xi(\theta)$ for the product to go to zero smoothly [@Szapudi:2000xj; @2004MNRAS.350..914C]. Angular power spectrum is then given by $$\begin{aligned}
C^{\rm apd}_\ell =
2\pi\int_{\theta=0}^{\theta_{\rm max}}\,\sin\theta d\theta \,\,
\xi(\theta)F(\theta)P_\ell(\cos\theta)~,
\label{eq:7}\end{aligned}$$ where $\theta_{\rm max}$ is the maximum angle set by the mask. We can adopt any function as an apodization function which takes $F(\theta)=1$ at $\theta=0$ and decreases as $\theta$ increases. The Gaussian type $F_{\rm G}(\theta)$ and Cosine type $F_{\rm C}(\theta)$ are often adopted, and they are defined as $$\begin{aligned}
F_{\rm G}(\theta) &=& \exp\left[-\frac{\theta^2}{2(\sqrt{8\ln 2}\,\sigma_{\rm apd})^2}\right],\\[0.2cm]
F_{\rm C}(\theta) &=& \frac12\left[1+\cos\left(\pi\frac{\theta}{\sigma_{\rm apd}}\right)\right]~,\end{aligned}$$ where $\sigma_{\rm apd} = \pi\theta_{\rm apd}/180^\circ$ and $\theta_{\rm apd}$ represents the angle in degree where the apodization function becomes close to zero. Typical value of $\theta_{\rm apd}$ should be close to the cut off angle $\theta_{\rm max}$.
We calculate the angular power spectrum with apodization and show the results in Fig. \[fig:apornot\]. In the upper panel of the figure, in the case of $\theta_{\rm max} =100^\circ$ when the North West mask is used, the oscillatory feature with a large amplitude can be seen on the angular power spectrum, because we truncate the integral in Eq.(7) at $\theta=100^\circ$. In the case of $\theta=180^\circ$ when KQ75 mask is used, on the other hand, the oscillation is suppressed because $\xi$ decreases as $\theta \to 180^\circ$. The lower panel shows the results with apodization using the Gaussian and Cosine type functions, setting $\theta_{\rm apd}$ to $100^\circ$. As is shown in the lower panel, the resultant power spectra become smooth compared to the case without apodization (see the red line in the upper panel). Also it is noted that the shape of the apodized spectra is similar to the power spectrum of the full sky ($\theta_{\rm
max}=180^\circ$; the black line), though small scale oscillations have smoothed out. This is a bad news because we are interested in the fine structures.
![Angular power spectra estimated from two-point correlation functions. In the upper panel, the red line represents the power with the North West mask and with $\theta_{\rm max}=100^\circ$ in Eq.(\[eq:7\]), (which corresponds to a truncated correlation function) and the black line the case with KQ75 mask. The theoretical curve of the $\Lambda$CDM model is also shown (the magenta line). When we naively analyze the partial sky, the large spurious oscillations arise on the angular power spectrum. In the lower panel, we show the cases with apodization, with $\theta_{\rm
apd} = 100^\circ$. The black and magenta lines are same as in the upper panel. The red and blue lines represent the apodized $C_\ell$ with Gaussian and Cosine types, respectively. For both cases, the North West mask is used. The suppression of the spurious oscillations is clearly seen due to the apodization function. []{data-label="fig:apornot"}](graph/Noapodization.eps){width="90.00000%"}
![Angular power spectra estimated from two-point correlation functions. In the upper panel, the red line represents the power with the North West mask and with $\theta_{\rm max}=100^\circ$ in Eq.(\[eq:7\]), (which corresponds to a truncated correlation function) and the black line the case with KQ75 mask. The theoretical curve of the $\Lambda$CDM model is also shown (the magenta line). When we naively analyze the partial sky, the large spurious oscillations arise on the angular power spectrum. In the lower panel, we show the cases with apodization, with $\theta_{\rm
apd} = 100^\circ$. The black and magenta lines are same as in the upper panel. The red and blue lines represent the apodized $C_\ell$ with Gaussian and Cosine types, respectively. For both cases, the North West mask is used. The suppression of the spurious oscillations is clearly seen due to the apodization function. []{data-label="fig:apornot"}](graph/Apodization100.eps){width="90.00000%"}
As we take $\theta_{\rm apd}$ smaller, more significant the smoothing effect for the fine structures in the angular power spectrum becomes. Therefore we should take the angle $\theta_{\rm apd}$ as large as possible to keep the fine structures. The maximum apodization angles $\theta_{\rm apd}$ for our masks with a sufficient suppression of the spurious oscillations are found to be $140^\circ$ and $100^\circ$ for the Cosine and Gaussian types, respectively. In Fig. \[fig:apd\_fun\] we depict the apodization functions with those $\theta_{\rm apd}$. From the figure, we can notice that the Gaussian type reduces the information in the two-point correlation function at large angles more than the Cosine one. In this work, we should keep the information as much as possible, because we focus on the fine structures. Therefore, we use the Cosine type apodization function in the following analysis. In this case, the structure at $\ell = 100$–150 can survive the apodization because the amplitude of the apodization function at this scale is over 0.99.
As the loss of information about high frequency structure in the angular power spectrum is significant, components of $C_\ell$ become correlated with each other. This effect and coping technique are discussed in the next subsection.
Binned angular power spectrum
-----------------------------
In the previous subsection, we see the apodization technique suppresses the high frequency oscillations. However, this technique is unavoidably accompanied with the loss of information. The loss appears as correlations between two different multipole moments. In order to see the degree of the correlations, let us calculate the covariance matrix ${\cal
C}_{\ell\ell^{\prime}}$, $$\label{eq:cov_mat}
{\cal C}_{\ell\ell^{\prime}}\equiv
\frac{1}{\bar C_{\ell}^2}\frac{1}{N}
\sum_{i=1}^N \left(C_\ell^i - \bar C_\ell\right)
\left(C_{\ell^\prime}^i - \bar C_{\ell^\prime}\right)~,$$ where $$\label{eq:clave}
\bar C_\ell = \frac{1}{N}\sum_{i=1}^NC_\ell^i~.$$ Here, $C_\ell^i$ is the estimated angular power spectrum from the $i$th Monte Carlo simulation sky, N is the total number of the samples. From this matrix, we can estimate the strength of correlations between each multipole moment. If there are no correlations, which means the $C_\ell$ can be estimated independently, the covariance matrix should be diagonal.
On the contrary, the covariance matrix has off-diagonal elements if the $C_\ell$ depends on another multipole moment component $C_{\ell^\prime}$. The correlations between the $C_\ell$’s are caused by the mask. If a mask is applied on the CMB sky, the estimated angular power spectrum is given by a convolution between the power spectra of the mask and the true temperature anisotropies that we want to estimate. The convolution generates correlations between the multipoles of $\Delta \ell \simeq$ 2–3, which makes it complicated to estimate the statistical significance of the angular power spectrum. A simple way to obtain independent observable is to bin the angular power spectrum with the comparable bin width.
The covariance matrices of the angular power spectrum with KQ75 mask are depicted in Fig. \[fig:cov\]. The upper panel in the figure shows the matrix for the case without apodization. Each component of this matrix is practically vanishing except for the diagonal ones. This result indicates that the correlations caused by the KQ75 mask are not significant at the multipole region of $100\leq \ell \leq 150$. This is because the condition that $\Delta\ell \geq
\frac{\pi}{\theta_{\rm max}}\sim 1$ can still be satisfied with the mask, where $\theta_{\rm max}$ is the maximum separation angle in the pixel domain. On the other hand, in the middle panel, we use the KQ75 mask with apodization. In this case, the components around diagonal elements do not vanish. This manifests the information loss due to apodization, though the variances (diagonal elements) become smaller values.
![Covariance matrices of angular power spectra ${\cal C}_{\ell\ell^{\prime}}$. The upper and middle panels show the covariance matrices without apodization and with Cosine type apodization, respectively. Non-zero off-diagonal components indicate the presence of correlation between $\ell$ and $\ell^\prime$. In the bottom panel we show the covariance matrix of the binned angular power spectrum. We can see that the correlation between components becomes weak. []{data-label="fig:cov"}](graph/lcdm-all_map.eps "fig:"){width="50.00000%"} ![Covariance matrices of angular power spectra ${\cal C}_{\ell\ell^{\prime}}$. The upper and middle panels show the covariance matrices without apodization and with Cosine type apodization, respectively. Non-zero off-diagonal components indicate the presence of correlation between $\ell$ and $\ell^\prime$. In the bottom panel we show the covariance matrix of the binned angular power spectrum. We can see that the correlation between components becomes weak. []{data-label="fig:cov"}](graph/lcdm-nw_map.eps "fig:"){width="50.00000%"} ![Covariance matrices of angular power spectra ${\cal C}_{\ell\ell^{\prime}}$. The upper and middle panels show the covariance matrices without apodization and with Cosine type apodization, respectively. Non-zero off-diagonal components indicate the presence of correlation between $\ell$ and $\ell^\prime$. In the bottom panel we show the covariance matrix of the binned angular power spectrum. We can see that the correlation between components becomes weak. []{data-label="fig:cov"}](graph/lcdm-nw_3binmap.eps "fig:"){width="50.00000%"}
The correlations appear only between the neighboring multipoles. To reduce the correlations we should gather the neighboring components in one component, namely, bin the data. The contributions from the neighboring modes $\ell^\prime$ to the pivot scale $\ell$ are about 50% ($\ell^\prime = \ell \pm 1$) and -10% ($\ell^\prime = \ell \pm 2, \ell
\pm 3$), which is shown in the middle panel of Fig. \[fig:cov\]. We find that a binning with $\Delta \ell=3$ is sufficient because the positive and negative correlations nicely compensate with each other to reduce the correlations between the binned data. In this case the number of data points reduces to 17 from 51, and this reduction could lead some information loss. The covariance matrix of this case is depicted in the lower panel of Fig. \[fig:cov\]. Looking at this figure, we can see that the correlation becomes weak enough thanks to the binning. Then, each components can be considered approximately independent.
Although binning is useful to simplify the statistics, some information should be lost in the process, especially if there exist fine structures in the data. In the next subsection we show how the binning procedure affects the fine structures in multipole $\ell =100$–150 we are interested in.
We calculate the binned $C_\ell$’s and verify the fine structures using the same steps we followed in Sec. \[sec:vldty\]. There are three ways to select the pivot scale for binning. We calculate the binned spectra with the three patterns $C_\ell^{\rm B} =
\sum_{i=-1}^1C_{\ell+i}/3$ for case I ($\ell = 99+3n$), case II ($\ell = 100+3n$), and case III ($\ell = 101+3n$) where $n=1,2, \cdots,
17$. The results are shown in Fig. \[fig:apdbincl\]. The right panels represent the probability that the fine structures come from the noise and the cosmic variance effects, as we have shown in Fig. \[fig:possibility\]. The result slightly depends on the binning pattern, but in any cases the significance is above 2.5 $\sigma$. In the left panels of Fig. \[fig:apdbincl\], we show the residuals between binned $C_\ell^{\rm WMAP}$ and $C_\ell^{\rm \Lambda CDM}$ as red lines. The black line is the average of $C_\ell$ with the 3,000 simulations. The dispersion of the simulation data is also shown as the grey boxes. We can still see the oscillatory feature that is found in the case without binning for all sky, and some points are over 1$\sigma$ box or 2$\sigma$ box. The numbers of data points above 1 or 2 $\sigma$ are larger than expected from Gaussian distribution for all cases. These results indicate that the fine structures can survive even if we apodize and bin the angular power spectrum.
We consider these three power spectra as the fiducial binned power spectra of WMAP for the full sky. To figure out the origin of the fine structures we compare them with the spectra from the partial skies obtained with the same apodization function and binning, as we shall show in the next section.
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The angular power spectrum with partial sky
-------------------------------------------
In the previous sections we have seen that the apodization and binning can successfully remove the spurious oscillations and the correlations between multipoles due to the partial sky mask. Here we apply the same technique to the four different partial skies and see whether there exists the directional dependence of the fine structures in the angular power spectrum. In the following discussion, we separate the structures into two characteristic structures. First one is the oscillatory structure around $\ell = 100$ – $120$. The two peaks have a significant deviation from the smooth spectrum at around 3$\sigma$. The other is the bump around $\ell = 140$.
We show the results for the binning case I in the upper panel of Fig. \[fig:lcdm-areaz\]. The primary effect of the partial sky masks can be seen in the error bars. The estimated error bars depend on the number of available modes $N$ as $1/\sqrt{N}$. In our analysis the effective sky coverage for the partial sky becomes about one-quarter compared to the all sky map. Therefore the error bar should be twice of the case of the all sky (Fig. \[fig:apdbincl\]) and we can confirm this fact from the figure.
We find a distinct anisotropic structure which is above 3$\sigma$ error bar at $\ell = 138$ in the South East area (bottom right panel of Fig. \[fig:lcdm-areaz\]), which is highlighted by a double circle in figure. Also, the histogram of the difference between $\Lambda$CDM and the simulated data set at $\ell = 138$ is shown in the lower panel of Fig. \[fig:lcdm-areaz\] for each direction. The vertical red lines show the differences of the power between the $\Lambda$CDM and the observed one by WMAP. The anisotropic structure in the South East can be seen in the other binning cases, as in Figs. \[fig:lcdm-areao\], and \[fig:lcdm-areat\].
The peculiarity of South East area as a whole at the multipole range of $100 \leq \ell \leq 150$ can be quantified with the value of $\chi^2$, defined by $$\chi^2 = \sum_{i=1}^{17}\left(
C_\ell^{\rm B}-\bar{C}_\ell^{\rm B}
\right)
{\cal C}_{\ell\ell^\prime}^{{\rm B}^{-1}}
\left(
C_{\ell^\prime}^{\rm B}-\bar{C}_{\ell^\prime}^{\rm B}
\right)/\bar{C}^{\rm B}_{\ell} \bar{C}^{\rm B}_{\ell^\prime}~,$$ where ${\cal C}_{\ell\ell^\prime}^{{\rm B}^{-1}}$ is the inverse of the covariance matrix of the binned angular power spectrum. We show the probability distribution function of $\chi^2$ , written by $$F(\chi^2; n=17) = \frac{(\chi^2)^{n/2-1}}{2^{n/2}\Gamma(\frac{n}{2})}e^{-\chi^2/2}~,$$ and the values of $\chi^2$ for the four partial skies in Fig. \[fig:chi2\]. The value of $\chi^2$ away from the peak position indicates the overall deviation of the observed angular power spectrum from the average value of the mock power spectrum. From this figure, the South East area has especially peculiar $\chi^2$ value. The probability that the $\chi^2$ takes larger value than the South East area by chance can be estimated as 1.50 (case I), 1.56 (case II), 8.70 (case III) %.
These results may suggest a possibility of existence of characteristic features only at South East area. The structure which comes from cosmological origin can be assumed to be isotropic, and therefore this anisotropic structure around $\ell \approx 138$ might be attributed to some astronomical origin, which have the scale about $0.6^\circ$. It may be interesting to note that there has been a report about a power spectrum anomaly around the third acoustic peak at the same sky direction [@Ko:2011ut], although the authors argued that the origin could be the WMAP instrumental noise because the third acoustic peak is located at the limit of the WMAP angular resolution.
The oscillatory feature (peak and dip) that is found in the all sky analysis around $\ell = 100$ – $120$, on the other hand, can be found clearly in the North West area, and also found at all the other directions regardless of the binning cases. Thus, the oscillation seems to be caused by some cosmological origin, not the astrophysical one.
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![The $\chi^2$ distribution derived from the 3,000 simulations and the $\chi^2$ values of WMAP at each direction for each binning case.[]{data-label="fig:chi2"}](graph/xi_square_z.eps "fig:"){width="50.00000%"} ![The $\chi^2$ distribution derived from the 3,000 simulations and the $\chi^2$ values of WMAP at each direction for each binning case.[]{data-label="fig:chi2"}](graph/xi_square_o.eps "fig:"){width="50.00000%"} ![The $\chi^2$ distribution derived from the 3,000 simulations and the $\chi^2$ values of WMAP at each direction for each binning case.[]{data-label="fig:chi2"}](graph/xi_square_t.eps "fig:"){width="50.00000%"}
Summary
=======
In this work we have re-examined the temperature angular power spectrum of the cosmic microwave background (CMB) at three frequency bands and for four restricted sky directions, in order to explore the origin of the fine structures in the power at the multipole range of $100\leq \ell
\leq 150$.
We prepared 3,000 mock sky maps of CMB temperature fluctuations with anisotropic instrumental WMAP noise, and verified the noise and cosmic variance effect on the fine structures of the WMAP data. By comparing the angular power spectra from each mock data with the real data by the WMAP, we found that the probability of the fine structures to be realized by chance is about 2 - 3$\sigma$. We checked whether the fine structures of the power spectrum depend on the Q, U, and V band maps, and found no evidence for the frequency dependence.
In contrast, we obtained some interesting suggestions from the angular power spectra derived from the partial skies. We found that the characteristic bump around $\ell=130$ – $140$ seen in the all sky angular power spectrum is solely attributed to the anomalous power at the South East area. We have already known the existence of the cold spot as one of the anisotropy structures in South East area [@Cruz:2006fy; @Inoue:2006rd]. We have doubt if the cold spot will affect at $\ell =140$ as a substrucure, thought the cold spot is few degree scale. However, we confirm that this effect is too tiny to generate characteristic structures at $\ell = 140$. This result may indicate the existence of unknown peculiar structure in that area as the origin of the fine structure. The fine oscillating structures found around $\ell = 100$ – $120$, on the other hand, have no significant directional dependences, suggesting that the oscillations come from some cosmological origin.
If the observed feature is not a statistical fluctuation but has a primordial origin, a straight but important test is to look into the polarization of the CMB and/or the distribution of the large scale structure, because they should show the same characteristic structure in their power spectra. The coming data by PLANCK and future galaxy survey such as the Large Synoptic Survey Telescope [@2009arXiv0912.0201L] can be used for the cross check and to accept (or reject) the feature with high significance.
The authors would like to thank T. Matsumura, O. Tajima and M. Sato for helpful suggestions. This work is supported in part by the Grant-in-Aid for the Scientific Research Fund Nos. 24005235 (KK), 24340048 (KI) and 22340056 (NS) of the Ministry of Education, Sports, Science and Technology (MEXT) of Japan and also supported by Grant-in-Aid for the Global Center of Excellence program at Nagoya University “Quest for Fundamental Principles in the Universe: from Particles to the Solar System and the Cosmos” from the MEXT of Japan. This research has also been supported in part by World Premier International Research Center Initiative, MEXT, Japan.
|
---
abstract: 'The locking effect is a phenomenon which is unique to quantum information theory and represents one of the strongest separations between the classical and quantum theories of information. The Fawzi-Hayden-Sen (FHS) locking protocol harnesses this effect in a cryptographic context, whereby one party can encode $n$ bits into $n$ qubits while using only a constant-size secret key. The encoded message is then secure against any measurement that an eavesdropper could perform in an attempt to recover the message, but the protocol does not necessarily meet the composability requirements needed in quantum key distribution applications. In any case, the locking effect represents an extreme violation of Shannon’s classical theorem, which states that information-theoretic security holds in the classical case if and only if the secret key is the same size as the message. Given this intriguing phenomenon, it is of practical interest to study the effect in the presence of noise, which can occur in the systems of both the legitimate receiver and the eavesdropper. This paper formally defines the *locking capacity* of a quantum channel as the maximum amount of locked information that can be reliably transmitted to a legitimate receiver by exploiting many independent uses of a quantum channel and an amount of secret key sublinear in the number of channel uses. We provide general operational bounds on the locking capacity in terms of other well-known capacities from quantum Shannon theory. We also study the important case of bosonic channels, finding limitations on these channels’ locking capacity when coherent-state encodings are employed and particular locking protocols for these channels that might be physically implementable.'
author:
- 'Saikat Guha[^1]'
- 'Patrick Hayden[^2]'
- Hari Krovi
- 'Seth Lloyd[^3]'
- 'Cosmo Lupo[^4]'
- 'Jeffrey H. Shapiro'
- 'Masahiro Takeoka[^5] $^{\ ,}$'
- 'Mark M. Wilde[^6]'
bibliography:
- 'Ref.bib'
title: |
**Quantum enigma machines and the locking capacity**\
**of a quantum channel**
---
Introduction
============
The security of a cryptographic primitive can be assessed according to different security criteria. Most modern cryptosystems are *computationally* secure—that is, their security relies on the difficulty of breaking them in a reasonable amount of time given available technologies. This is also the case for the *enigma machines*, a family of historical polyalphabetic ciphers in use during the earlier half of the previous century—their security relied on the difficulty of uncovering patterns hidden in pseudorandom sequences [@Bruen].
A stronger security criterion requires that an encrypted message is close to being statistically independent of the corresponding unencrypted message, in which case one speaks of *information-theoretic* security. For the case of classical systems, a good measure of correlation is the mutual information between the unencrypted and encrypted message. If the mutual information vanishes, the chance of successfully decrypting the message is exponentially small in the length of the message. Any encryption scheme with such a property cannot perform any better than one-time pad encryption, where a truly random key is used to encrypt (and decrypt) the message [@S49]. The one-time pad guarantees information-theoretic security as long as the key is kept secret, it has the same length as the message, and can be used only once. However, the fact that the secret key should be the same length as the message imposes severe practical limitations on the use of the one-time pad protocol.
On the other hand, it is now known that quantum mechanics gives a way around these limitations. The *locking effect* is a phenomenon which is unique to quantum information theory [@DHLST04] and represents one of the most striking separations between the classical and quantum theories of information. It is responsible for important revisions to security definitions for quantum key distribution [@KRBM07] and might even help to explain how both unitarity could be preserved and most of the information leaking from an evaporating black hole could be inaccessible until the final stages of evaporation [@SO06; @DFHL10]. Quantum data locking occurs when the accessible information about a classical message encoded into a quantum state decreases by an amount that is much larger than the number of qubits of a small subsystem that is discarded [@DHLST04]. A device that realizes a quantum data locking protocol is called a quantum enigma machine [@QEM].
Impressive locking schemes exist [@HLSW04; @DFHL10; @FHS11]. Suppose that a sender and receiver share a constant number of secret key bits. Using these secret key bits, they can then encode an $n$-bit classical message into $n$ qubits such that an adversary who gains access to these $n$ qubits, but who does not know the secret key, cannot do much better than to randomly guess the message after performing an arbitrary measurement on these $n$ qubits.
However, the cryptographic applications of quantum data locking have to be taken with a grain of salt, as they are only applicable if the distribution of the message is completely random from the perspective of the adversary. Otherwise, the key size should increase by an amount necessary to ensure that the distribution of the message becomes uniform. Moreover, one might say that the strength of quantum data locking also exposes a weakness. Indeed, as a small key is sufficient for encrypting a long message, the leakage of a small part of the secret key may allow an adversary to uncover a disproportionate amount of information. For this reason, any cryptographic primitive based on the locking effect (called a locking scheme), does not necessarily guarantee composable security [@KRBM07]. This also implies that quantum data locking cannot necessarily be used for secure key distribution. The only exception is if the adversary has no option other than to perform a collective measurement on the qubits in her possession just after she receives them.
As stated above, Shannon proved that such a locking effect is impossible classically [@S49]. That is, when using only classical resources, a sender and receiver require a secret key whose size is proportional to the size of the message in order for the eavesdropper to have a negligible amount of information about the encrypted message. Thus, after Shannon’s result, information scientists looked in a different direction in order to determine ways for communication systems to provide secrecy in addition to reliable transmission. In reality, all communication systems suffer from physical-layer noise, and one might be able to determine the characteristics of the noise to a legitimate receiver and to an untrusted eavesdropper. Such a model is known as the wiretap channel [@W75], and it is well known now that if the noise to the eavesdropper is stronger than the noise to the legitimate receiver, then it is possible to communicate error-free at a positive rate such that the eavesdropper obtains a negligible amount of information about the messages being transmitted.
Summary of results
==================
In this paper, we consider the performance of locking protocols in the presence of noise, and as an important application, we consider locking protocols for bosonic channels. There are two types of noise to consider in any realistic locking protocol: that which affects the transmission to a legitimate receiver and that which affects the eavesdropper’s system. Both are important to consider in any realization of a quantum enigma machine.
We begin in Section \[QDL\] by reviewing the locking effect and a recently introduced quantum enigma machine (QEM) from Ref. [@QEM]. This QEM encodes a classical message into a single-photon state spread over a collection of discrete modes and then decodes it by direct photodetection. The encryption and decryption are realized by applying and inverting, respectively, a single multi-mode passive linear-optical unitary transformation, selected uniformly at random from a set of such transformations. Similar to historical enigma machines, QEMs can encrypt a long message using an exponentially shorter secret key. However, unlike historical enigma machines which were only computationally secure, quantum data locking implies security in the sense that the outcomes of any eavesdropper measurement will be essentially independent of the message.
After the review, Section \[sec:locking-capacity\] provides a formal definition of the locking capacity of a quantum channel. In short, a locking protocol uses a quantum channel $n$ times (for some arbitrarily large integer $n$) and has three requirements:
1. The receiver should be able to decode the transmitted message with an arbitrarily small error probability.
2. The eavesdropper can recover only an arbitrarily small number of the message bits after performing a quantum measurement on her systems.
3. The secret key rate is no more than sublinear in the number $n$ of channel uses (for example, logarithmic in $n$).
We define the locking capacity of a quantum channel to be the maximum rate at which it is possible to lock classical information according to the above requirements. Changing the systems to which the adversary has access leads to different notions of locking capacity, and we distinguish the notions by naming them the [*weak locking capacity*]{} and the [*strong locking capacity*]{}. The difference between the two is that, in the weak notion, the adversary is assumed to have access to only the channel environment, while, in the strong case, we allow her access to the channel input. We emphasize that when we use the term (weak or strong) “locking capacity” without any other modifiers, we refer to the locking capacity of a quantum channel without additional resources such as classical feedback. Most of the results reported here correspond to such a forward locking capacity. The locking capacity of channels with additional resources such as classical feedback remains largely open. However, at the very least, we can already say that quantum key distribution protocols provide lower bounds on the locking capacity in this setting.
We then find operational bounds on the locking capacity in terms of other well known capacities studied in quantum Shannon theory, and we find other information-theoretic upper bounds on the locking capacity. We prove that the locking capacity of an entanglement-breaking channel is equal to zero, which demonstrates that a quantum channel should have some ability to preserve entanglement in order for it to be able to lock information according to the above requirements. We also show that any achievable locking rate is equal to zero whenever a given locking protocol has a classical simulation. Furthermore, we find a class of channels for which the weak locking capacity is equal to both the private capacity and quantum capacity. Finally, we discuss locking protocols for some simple exemplary channels.
Section \[CSQEM\] establishes several important upper bounds on the locking capacity of channels when restricting to coherent-state encodings. If it were possible to exploit coherent-state encodings to perform locking at high rates, then this would certainly turn the locking effect from an interesting theoretical phenomenon into one with practical utility. However, we are able to show that there are fundamental limitations on the locking capacity when restricting to coherent-state encodings. In particular, we prove that the strong locking capacity of any channel is no larger than $\log_{2}(e)$ locked bits per channel use whenever the encoding consists of coherent states (where $e$ is the base for the natural logarithm). We also prove that the weak locking capacity of a pure-loss bosonic channel is no larger than the sum of its private capacity and $\log_{2}(e)$. In Section \[weakQEM\], we discuss an explicit protocol that uses a pulse position modulation encoding of coherent states. We derive bounds on the security and key efficiency of this coherent-state locking protocol and find that it has qualitative features analogous to the single-mode quantum enigma machine in the presence of linear loss.
Finally, Section \[end\] presents our conclusions, a discussion of the scaling of the required physical resources, and open questions for future research.
Notation
========
We briefly review some notation that we use in the rest of the paper. Let $\mathcal{B}\left(
\mathcal{H}\right) $ denote the algebra of bounded linear operators acting on a Hilbert space $\mathcal{H}$. The $1$-norm of an operator $X$ is defined as$$\left\Vert X\right\Vert _{1}\equiv\text{Tr}\{ \sqrt{X^{\dag}X}
\} .$$ Let $\mathcal{B}\left( \mathcal{H}\right) _{+}$ denote the subset of positive semidefinite operators (we often simply say that an operator is positive if it is positive semidefinite). We also write $X\geq0$ if $X\in\mathcal{B}\left(
\mathcal{H}\right) _{+}$. An operator $\rho$ is in the set $\mathcal{D}\left( \mathcal{H}\right) $ of density operators if $\rho\in\mathcal{B}\left( \mathcal{H}\right) _{+}$ and Tr$\left\{ \rho\right\} =1$. The tensor product of two Hilbert spaces $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ is denoted by $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. Given a multipartite density operator $\rho_{AB}\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, we unambiguously write $\rho_{A}=\ $Tr$_{B}\left\{ \rho_{AB}\right\} $ for the reduced density operator on system $A$.
A linear map $\mathcal{N}_{A\rightarrow B}:\mathcal{B}\left( \mathcal{H}_{A}\right) \rightarrow
\mathcal{B}\left( \mathcal{H}_{B}\right) $ is positive if $\mathcal{N}_{A\rightarrow B}\left( \sigma_{A}\right) \in\mathcal{B}\left(
\mathcal{H}_{B}\right) _{+}$ whenever $\sigma_{A}\in\mathcal{B}\left(
\mathcal{H}_{A}\right) _{+}$. Let id$_{A}$ denote the identity map acting on $\mathcal{B}\left( \mathcal{H}_{A}\right)$. A linear map $\mathcal{N}_{A\rightarrow B}$ is completely positive if the map id$_{R}\otimes\mathcal{N}_{A\rightarrow B}$ is positive for a reference system $R$ of arbitrary size. A linear map $\mathcal{N}_{A\rightarrow B}$ is trace-preserving if Tr$\left\{ \mathcal{N}_{A\rightarrow B}\left( \tau_{A}\right) \right\} =\ $Tr$\left\{ \tau
_{A}\right\} $ for all input operators $\tau_{A}\in\mathcal{B}\left(
\mathcal{H}_{A}\right) $. If a linear map is completely positive and trace-preserving, we say that it is a quantum channel or quantum operation. For simplicity, we denote a quantum channel $\mathcal{N}: \mathcal{B}({\mathcal{H}}_A) \mapsto \mathcal{B}({\mathcal{H}}_B)$ simply as $\mathcal{N}_{A\to B}$. Similarly, we denote an isometry $U: {\mathcal{H}}_A \mapsto {\mathcal{H}}_B \otimes {\mathcal{H}}_C$ simply as $U_{A\to BC}$.
The variational distance between two probability distributions $p(x)$ and $q(x)$ is defined as $$\sum_x | p(x) - q(x)| .$$ The trace distance between two quantum states $\rho$ and $\sigma$ is defined as follows: $$\Vert \rho - \sigma \Vert_1 ,$$ and it is a conventional measure used in quantum information theory to quantify the distinguishability of two quantum states. Clearly, when the two states are commuting, the trace distance is equal to the variational distance between the two probability distributions corresponding to the eigenvalues of $\rho$ and $\sigma$.
The von Neumann entropy of a state $\rho\in\mathcal{D}(\mathcal{H}_A)$ is given by $H(A)_{\rho} := - \text{Tr} \{\rho\log \rho\}$. Throughout this paper we take the logarithm base $2$. For a tripartite state $\rho_{ABC} \in \mathcal{D}(\mathcal{H}_{ABC})$, the quantum mutual information and the conditional quantum mutual information are respectively given by: $$\begin{aligned}
I(A;B)_\rho & \equiv H(A)_\rho + H(B)_\rho - H(AB)_\rho, \\
I(A;B|C)_\rho & \equiv I(A;BC)_\rho - I(A;C)_\rho,\end{aligned}$$ where $ H(A)_\rho$ denotes the von Neumann entropy of the reduced state $\rho_A$, for example.
Review of quantum data locking {#QDL}
==============================
A quantum data locking scheme can be implemented by a set of $|\mathcal{K}|$ unitary transformations $\{U_{k}\}_{k\in\mathcal{K}}$ acting on a Hilbert space $\mathcal{H}_{M}$ of finite dimension $|\mathcal{M}|$ [@DHLST04; @HLSW04; @DFHL10; @FHS11]. (For the moment, we restrict ourselves to finite-dimensional Hilbert spaces, but Definitions \[def:weak-lock-protocol\] and \[def:strong-lock-protocol\] appearing later on allow for encoding information into infinite-dimensional Hilbert spaces.) Alice encodes $|\mathcal{M}|$ equiprobable messages by means of a set of orthonormal states $\{|m\rangle\}_{m\in\mathcal{M}}$ defining a standard basis in $\mathcal{H}_{M}$. The encryption is then made by applying a particular unitary $U_{k}$ with $k$ chosen uniformly at random from $\mathcal{K}$, and this unitary maps a standard basis state $|m\rangle$ into a state $U_{k}|m\rangle$. The label $k$ identifies the choice of the basis and plays the role of a secret key.
It is helpful to consider a particular classical-quantum state when reasoning about a quantum data locking protocol. For such a state, we have two classical systems, the first associated with Alice’s message and the second associated with the secret key, and a quantum system $Q$ of dimension $|\mathcal{M}|$ corresponding to the quantum-encoded message of Alice. This classical-quantum state is given by the following density matrix:$$\rho_{MKQ}=\frac{1}{|\mathcal{M}||\mathcal{K}|}\sum_{m,k}|m,k\rangle\langle
m,k|_{MK}\otimes\left( U_{k}|m\rangle\langle m|U_{k}^{\dag}\right)
_{Q}\,,\label{rhoAB}$$ where the sets $\{|m\rangle\}$ and $\{|k\rangle\}$ are comprised of orthonormal states representing the message and the secret key, respectively. The receiver Bob has access to the quantum system $Q$ and the key system $K$. We assume that an eavesdropper Eve only has access to the quantum system $Q$ (for example, before it gets passed along to the receiver Bob). The classical correlations between Alice’s message $M$ and Bob’s systems $K$ and $Q$ can be quantified by the *accessible information* [@S90]. This is defined as the maximum classical mutual information that can be extracted by performing local measurements on the bipartite state:$$I_{\mathrm{acc}}(M;KQ)_{\rho}=\max_{\mathcal{M}_{KQ\rightarrow Y}}I(M;Y)\,,\label{eq:bob-acc-info}$$ where the maximization is taken over local measurement maps $\mathcal{M}_{KQ\rightarrow Y}$ and $I(X;Y)=H(X)+H(Y)-H(XY)$ is the mutual information, with $H(Z)$ denoting the Shannon entropy of the random variable $Z$ [@Cover].
The accessible information in (\[eq:bob-acc-info\]) can never be larger than $\log_{2}|\mathcal{M}|$, due to the bound $I(M;Y)\leq\log_{2}|\mathcal{M}|$ which holds for any random variable $Y$. A particular strategy for achieving this upper bound is for Bob first to perform the controlled unitary $\sum
_{k}|k\rangle\langle k|_{K}\otimes(U_{k}^{\dag})_{Q}$, leaving the state$$\frac{1}{|\mathcal{M}||\mathcal{K}|}\sum_{m,k}|m,k\rangle\langle
m,k|_{MK}\otimes|m\rangle\langle m|_{Q}.$$ He then simply measures in the basis $\{|m\rangle\}$ to recover the message $m$ perfectly, so that his accessible information is maximal, equal to $\log_{2}|\mathcal{M}|$.
To assess the security of the communication, let us consider the accessible information for a party Eve who does not have access to the secret key. We consider the following reduced state:$$\rho_{MQ}=\frac{1}{|\mathcal{M}|}\sum_{m}|m\rangle\langle m|_{M}\otimes
\frac{1}{|\mathcal{K}|} \sum_{k} \left( U_{k}|m\rangle\langle m|U_{k}^{\dag
}\right) _{Q}\,, \label{rhoAE}$$ obtained by taking the partial trace over the key system in (\[rhoAB\]). The aim of Eve is to find an optimal positive operator-valued measure (POVM) to maximize the classical mutual information. It is sufficient to consider a POVM $\mathcal{M}_{Q\to Y}$ with rank-one measurement operators, i.e.,$$\{\mu_{y}|\phi_{y}\rangle\langle\phi_{y}|\}, \label{eq:rank-1-meas}$$ where each $|\phi_{y}\rangle$ is a normalized vector and $\mu_{y}>0$ (the sufficiency of rank-one POVMs follows by a data processing argument). We then find the following expression for Eve’s accessible information about Alice’s message [@DHLST04]:$$I_{\mathrm{acc}}(M;Q)_{\rho}=\log_{2}{|\mathcal{M}|}-\min_{\mathcal{M}_{E\to
Y}}\sum_{y}\frac{\mu_{y}}{|\mathcal{M}||\mathcal{K}|}\sum_{k}H(q_{yk})\,,
\label{acc}$$ where the probability distributions $q_{yk}$ have components $q_{yk}^{m}=|\langle\phi_{y}|U_{k}|m\rangle|^{2}$. Notice that Eve’s accessible information is written in terms of the minimum of the Shannon entropies $H(q_{yk})=-\sum_{m} q_{yk}^{m}\log_{2}{q_{yk}^{m}}$ averaged over $y$ and $k$.
While finding Eve’s optimal POVM is generally a difficult problem, one can obtain a good upper bound by a convexity argument [@DHLST04]. Furthermore, one can choose the encoding unitaries uniformly at random according to the Haar measure [@HLSW04; @FHS11; @Buhrman; @DFHL10], and if one also adjoins to the message a small ancilla system in a maximally mixed state [@FHS11], then it is possible to reduce the adversary’s accessible information to become arbitrarily small. These latter results show that for large enough $|\mathcal{M}|$ there exist data locking schemes with $\log_{2}|\mathcal{K}|$ negligibly small in comparison to $\log_{2}|\mathcal{M}|$ and for which $$I_{\mathrm{acc}}(M;Q)_{\rho}\ll I_{\mathrm{acc}}(M;KQ)_{\rho}.$$ That means that a relatively short secret key can be used to encrypt an exponentially longer message. To be more precise, consider the results of [@FHS11], according to which for $\left\vert \mathcal{M}\right\vert$ large enough there exist choices of $|\mathcal{K}|$ unitaries, with $$\log_{2}{|\mathcal{K}|} = 4\log_{2}{(\varepsilon^{-1})}
+ O(\log_2\log_{2}{(\varepsilon^{-1})})\,,$$ such that $$I_{\mathrm{acc}}(M;Q)_{\rho} \leq \varepsilon \log_{2}{|\mathcal{M}|} \,,$$ for any $\varepsilon>0$. Moreover, if one randomly chooses the $|\mathcal{K}|$ unitaries according to the Haar distribution on the unitary group, the probability of picking up a set with this property approaches one exponentially fast in the limit as $\left\vert \mathcal{M}\right\vert
\rightarrow\infty$.
In quantum data locking, the removal of a subsystem reduces the accessible information by an amount larger than the number of qubits removed. This is a purely quantum feature which has no classical analog. For comparison, consider a classical counterpart of the quantum data locking setting, in which Alice has access to a message variable $M$, Bob to an output random variable $Y$ and key variable $K$, while Eve has access only to $Y$. In the classical framework, the following inequality holds$$I(M;YK)-I(M;Y)=I\left( M;K|Y\right) \leq H(K)\leq\log_{2}{|\mathcal{K}|}\,.
\label{cl-ineq}$$ This inequality shows that in the classical framework, removal of the key variable $K$ reduces the mutual information by no more than $\log_{2}{|\mathcal{K}|}$.
In the quantum case as discussed above, this inequality can be violated by an arbitrarily large amount by replacing the classical mutual information with the accessible information. A violation of the classical inequality in (\[cl-ineq\]) can be quantified in terms of the following ratios [@DHLST04; @HLSW04]:$$\begin{aligned}
r_{1} & =\frac{I_{\mathrm{acc}}(M;Q)_{\rho}}{I_{\mathrm{acc}}(M;KQ)_{\rho}}\,,\label{r1}\\
r_{2} & =\frac{\log_{2}{|\mathcal{K}|}}{I_{\mathrm{acc}}(M;KQ)_{\rho
}-I_{\mathrm{acc}}(M;Q)_{\rho}}\,. \label{r2}$$ The first is the ratio of the accessible information without the secret key to that with the secret key. The second is the ratio of the key length to the amount of information that Bob can unlock by having access to the key. For a good locking scheme, both of these quantities should be small, and the quantum data locking schemes discussed above are such that both $r_{1}$ and $r_{2}$ can be made arbitrarily small. On the other hand, the inequality in (\[cl-ineq\]) implies that $r_{2}\geq1$ for any locking scheme that uses classical resources only. Notice that the one-time pad protocol has $r_{2}=1$ because the number of bits in the key is equal to the amount of unlocked information for Bob.
Quantum enigma machine {#unaryQEM}
----------------------
A particular example of a QEM was proposed in Ref. [@QEM]. This QEM implements an optical realization of quantum data locking, in which Alice exploits a pulse position modulation (PPM) encoding using single-photon states over $n$ optical modes [@QEM]. The message states $|m\rangle=a_{m}^{\dag}|0\rangle$ represent the states of a single photon occupying one out of a set of $n$ bosonic modes with canonical operators $\{a_{m},a_{m}^{\dag}\}_{m\in\{1,\dots,n\}}$. Thus, for this case, we have $n=|\mathcal{M}|$. The unitaries $\{U_{k}\}_{k\in\mathcal{K}}$ are realized as passive linear-optical transformations acting on $n$ modes. The encryption through a passive linear-optical unitary $U_{k}$ transforms the message states into $$|m\rangle_{k}:=U_{k}|m\rangle=\sum_{m^{\prime}=1}^{n} \widetilde{U}_{k}^{\left( m,m^{\prime}\right) } |m^{\prime}\rangle\, ,$$ where $\widetilde{U}_{k}$ is the corresponding $n\times n$ unitary matrix acting on the mode labels. The effect of the encryption is to spread a single photon coherently over $n$ modes.
Let us first assume that Alice and Bob communicate via a noiseless quantum channel. Then Bob receives the state prepared by Alice unperturbed. He decrypts the message by first applying the inverse transformation $U_{k}^{\dag}$ and then by performing photodetection on the modes $\{a_{m}\}$. We assume that Eve may intercept the signal but she does not know which unitary has been used for encryption. Then, a direct application of the results of [@FHS11] shows that Eve’s accessible information can be made arbitrary small using a pre-shared secret key of length logarithmic in the length of the message.
One natural application of a QEM is in synergy with standard quantum key distribution (QKD) [@BB84; @RevModPhys.81.1301]—that is, a relatively short secret key can be first established by QKD and then used to encrypt a much (exponentially) longer message through the QEM. This combination of QKD and QEM in an all-quantum-optical cryptosystem could possibly overcome the bit-rate limitations of standard QKD, but more work is necessary to determine if this is the case.
Let us now suppose that Alice and Bob communicate through a pure-loss bosonic channel with transmissivity $\eta\in(0,1)$ and Eve makes a passive wiretap attack on the communication line, hence getting the photon lost in the channel with probability no larger than $1-\eta$. A simple feedback-assisted strategy allows for Alice and Bob to use the same scheme even for transmissivity values below $50\%$. Notice that the only effect of the pure-loss channel is to induce a probabilistic leakage of the photon. Hence, each time Bob detects a photon (which happens with probability $\eta$) he can be sure that he has correctly decrypted Alice’s message. On the other hand, if Bob’s photodetectors do not produce a click (which happens with probability $1-\eta
$) he can request for Alice to resend. This shows that with the help of a classical feedback channel Alice and Bob can attain the accessible information $$I_{\mathrm{acc}}(M;KQ)_{\rho}=\eta\log_{2}{n}\,.$$ Although reduced, this value of the accessible information equals the maximum value achievable through a pure-loss bosonic channel with a mean value of $n^{-1}$ photons per mode [@QEM; @GiovPRL]. Lloyd argues that such a scheme should be secure in principle [@QEM]. However, a critical assumption for this security to hold is that Eve should attack each block that she receives independently, in which case her accessible information is reduced by a factor $1-\eta$ when compared to the lossless case. Indeed, an important assumption for the security of any locking protocol is that the distribution of the message is uniform from the perspective of the adversary. If this is not the case (as for repeated transmission of the same message when it does not show up at the receiver’s end), then the secret key needs to be large enough so that the distribution of the message becomes uniform (see Proposition 4.16 in Ref. [@F12]).
Concerning the key efficiency of the protocol, we can estimate the key efficiency ratio as $$r_{2} \simeq \frac{4\log_{2}{(\varepsilon^{-1}})}{\eta\log_{2}{n}}\,.$$ This expression implies that, although $r_{2}$ can be made arbitrarily small by increasing $n$, the number of bosonic modes needed to fulfill the key efficiency condition $r_{2}<1$ grows exponentially with decreasing $r_{2}$ and $\eta$. This feature is first of all a consequence of the fact that the quantum data locking scheme in [@FHS11] (similar conclusions are also obtained using the results of [@HLSW04; @DFHL10]) requires a high-dimensional Hilbert space. On top of that, there is the fact that the PPM encoding, as remarked above, is highly inefficient as it encodes $\log_2{n}$ qubits into $n$ optical modes.
According to Definition \[def:weak-lock-protocol\] below, this QEM is an instance of an $(n,R,\varepsilon)$ weak locking protocol (assisted by classical feedback) for the pure-loss bosonic channel with transmissivity $\eta$, with a locking rate $R=\left[ \eta\log_{2}{n}\right]/n$. It is worthwhile to notice that, due to the inefficiency of PPM encoding, the rate of this QEM approachs zero as $n$ increases.
The locking capacity of a quantum channel {#sec:locking-capacity}
=========================================
In this section, we take a more general approach to quantum data locking than that pursued in prior work by defining the *locking capacity* of a quantum channel. Our goal is to understand the locking effect in the setting of quantum Shannon theory, where a sender and receiver are given access to $n$ independent uses of a noisy quantum channel (where $n$ is an arbitrarily large integer). Their aim is to exploit some sublinear (in $n$) amount of secret key in order to lock classical messages from an adversary, in the sense that this adversary will not be able to do much better than random guessing when performing a quantum measurement to learn about the transmitted message. Also, we demand that the legitimate receiver (who knows the value of the secret key) be able to recover the classical message with an arbitrarily small probability of error. This leads us naturally to the following formal definition of a locking protocol for a noisy channel:
\[Weak locking protocol\]\[def:weak-lock-protocol\]An $\left(
n,R,\varepsilon\right) $ weak locking protocol for a channel $\mathcal{N}_{A\rightarrow B}$ consists of encoding and decoding maps $\mathcal{E}_{MK\rightarrow A^{n}}$ and $\mathcal{D}_{B^{n}K\rightarrow\hat{M}}$, respectively. The encoding $\mathcal{E}_{MK\rightarrow A^{n}}$ acts on a message system $M$ and a key system $K$ and outputs the system $A^{n}$ for input to $n$ uses of the channel. The decoding map $\mathcal{D}_{B^{n}K\rightarrow\hat{M}}$ acts on the output systems $B^{n}$ and the key system $K$ to produce a classical system $\hat{M}$ containing the receiver’s estimate of the message. Without loss of generality, the encoding consists of $\left\vert \mathcal{M}\right\vert \left\vert \mathcal{K}\right\vert $ quantum states $\rho_{m,k}$, where $\left\vert \mathcal{M}\right\vert $ is the number of messages and $\left\vert \mathcal{K}\right\vert $ is the number of key values. Furthermore, the decoding consists of $\left\vert \mathcal{K}\right\vert $ POVMs $\{\Lambda_{m}^{\left( k\right) }\}_{m\in\mathcal{M}}$. The rate $R=\log_{2}\left\vert
\mathcal{M}\right\vert /n$ and the parameter $\varepsilon >0$. The protocol should satisfy the following requirements:
1. Given the key, the receiver can decode the transmitted message well on average:$$\frac{1}{\left\vert \mathcal{M}\right\vert \left\vert \mathcal{K}\right\vert
}\sum_{m,k}\operatorname{Tr}\left\{ \Lambda_{m}^{\left( k\right) }\left(
\mathcal{N}_{A\rightarrow B}\right) ^{\otimes n}\left( \rho_{m,k}\right)
\right\} \geq1-\varepsilon.$$
2. Let $\{\Gamma_y\}$ be a POVM that Eve can perform in an attempt to learn about the message $M$. After she performs this measurement, the joint classical-classical state of the message and her measurement outcome is as follows: $$\frac{1}{\left\vert \mathcal{M}\right\vert }\sum_{m}\left\vert m\right\rangle
\left\langle m\right\vert _{M}\otimes \sum_y \operatorname{Tr}
\left\{ \Gamma_y \left(\frac{1}{\left\vert \mathcal{K}\right\vert }\sum_{k}\left( \mathcal{N}_{A\rightarrow E}\right) ^{\otimes
n}\left( \rho_{m,k}\right)\right) \right\} |y\rangle\langle y|_Y,$$ where $\mathcal{N}_{A\rightarrow E}$ is the channel complementary to $\mathcal{N}_{A\rightarrow B}$. Equivalently, the joint probability distribution $p_{M,Y}(m,y)$ is equal to $$p_{M,Y}(m,y) = \frac{1}{\left\vert \mathcal{M}\right\vert }
\operatorname{Tr}
\left\{ \Gamma_y \left(\frac{1}{\left\vert \mathcal{K}\right\vert }\sum_{k}\left( \mathcal{N}_{A\rightarrow E}\right) ^{\otimes
n}\left( \rho_{m,k}\right)\right) \right\} ,$$ Our security criterion (see also Ref. [@FHS11]) is that, for any measurement outcome $y$ of Eve, the variational distance between the message distribution $p_M(m)$ and the distribution $p_{M|Y}(m|y)$ for the message conditioned on any particular measurement outcome should be no larger than $\varepsilon$: $$\sum_m |p_M(m) - p_{M|Y}(m|y) | \leq \varepsilon .
\label{eq:sec-crit-var-dist}$$ The interpretation here is that Eve cannot do much better than to randomly guess the message if all of the conditional distributions $p_{M|Y}(m|y)$ are indistinguishable from the message distribution.
3. The secret key consumption grows sublinearly in the number $n$ of channel uses.
In a weak locking protocol, it is assumed that the eavesdropper has access to the channel environment only. A stronger locking protocol is obtained if we allow for the eavesdropper to have access to the channel input (or, equivalently, to both the channel output and environment):
\[Strong locking protocol\]\[def:strong-lock-protocol\]An $\left(
n,R,\varepsilon\right) $ strong locking protocol is similar to a weak locking protocol, except that we allow for Eve to have access to the $A^n$ systems, so that she can perform a measurement on the $A^n$ systems of the following state: $$\frac{1}{\left\vert \mathcal{M}\right\vert }\sum_{m}\left\vert m\right\rangle
\left\langle m\right\vert _{M}\otimes\frac{1}{\left\vert \mathcal{K}\right\vert }\sum_{k}\left( \rho_{m,k}\right) _{A^{n}}.$$ We then demand that the variational distance as in (\[eq:sec-crit-var-dist\]) can be made less than an arbitrarily small positive constant $\varepsilon$.
One could alternatively allow for the adversary to have access to the output of the channel, but we do not explore such a possibility in this paper.
\[rem:FA-acc-bound\] The Fannes-Audenaert inequality [@Fannes73; @A07] for continuity of entropy implies that if (\[eq:sec-crit-var-dist\]) holds, then we get the following bound on Eve’s accessible information: $$I_{\operatorname{acc}}\left( M;E^{n}\right) \leq
h_2(\varepsilon/2) + \varepsilon nR/2,
\label{eq:acc-info-bound}$$ where $h_2$ is the binary entropy and $n$ and $R$ are as in Definition \[def:weak-lock-protocol\]. In more detail, recall the Fannes-Audenaert inequality for continuity of entropy: $$T \equiv \tfrac{1}{2}\Vert \rho - \sigma \Vert_1 \implies
|H(\rho) - H(\sigma)| \leq h_2(T) + T \log (d-1) ,$$ where $h_2$ is the binary entropy and $d$ is the dimension of the states. Applying this to the condition in (\[eq:sec-crit-var-dist\]) gives $$\begin{aligned}
H(M) - H(M|Y=y) & \leq h_2(\varepsilon/2) + \tfrac{\varepsilon}{2}
\log(|\mathcal{M}|-1) \\
& \leq h_2(\varepsilon/2) + \varepsilon
nR/2.\end{aligned}$$ Since the above inequality holds for any measurement of Eve, averaging it with respect to the distribution $p_Y(y)$ gives the inequality in (\[eq:acc-info-bound\]).
If desired, one can demand further for the secret key rate of an $\left(
n,R,\varepsilon\right) $ weak or strong locking protocol to be consumed at a particular sublinear rate (for example, a logarithmic number of secret key bits or perhaps $\sqrt{n}$ secret key bits for $n$ channel uses). However, the present paper establishes several upper bounds on locking capacity in an IID setting, and these bounds converge to the same quantity in the large $n$ limit regardless of which sublinear rate is chosen. Also, the FHS protocol [@FHS11] is very strong, in the sense that it uses such a small amount of secret key. Thus, in light of these two observations it seems reasonable to define locking capacity in such a coarse-grained manner. However, other characterizations of locking capacity in a finite blocklength setting or in a one-shot setting might change depending on the amount of secret key allowed (so it would be necessary to specify in more detail the amount of secret key allowed).
\[rem:strong-implies-weak-lock\]Observe that an $\left( n,R,\varepsilon
\right) $ strong locking protocol is also an $\left( n,R,\varepsilon\right)
$ weak locking protocol, but the other implication is not necessarily true.
The security and key efficiency ratios become arbitrarily small for a strong locking protocol. Indeed, from the fact that $I_{\operatorname{acc}}\left(
M;A^{n}\right) \leq h_2(\varepsilon/2) + \varepsilon
nR/2$ and the fact that the receiver can decode with the key, so that $I_{\operatorname{acc}}\left( M;B^{n}K\right)
\approx\log_{2}\left\vert \mathcal{M}\right\vert $, it follows that the security ratio $r_{1}\leq h_2(\varepsilon/2)/(nR) + \varepsilon
/2$. Also, since we require the key to be sublinear in the message length, it follows that the key efficiency ratio $r_{2}=o\left( n\right) /O\left( n\right) $, which vanishes in the limit as $n\rightarrow\infty$.
In what follows, we use the modifier weak or strong only when we need to distinguish between them.
\[Achievable rate for locking\]\[def:achievable-rate-locking\]A rate $R$ is achievable if $\forall\,\delta,\varepsilon>0$ and sufficiently large $n$, there exists an $\left( n,R-\delta,\varepsilon\right) $ locking protocol.
\[Locking capacity\]\[def:locking-cap\]The locking capacity $L\left(
\mathcal{N}\right) $ of a quantum channel is the supremum of all achievable rates:$$L\left( \mathcal{N}\right) \equiv\sup\left\{ R\ |\ R\text{ is
achievable}\right\} .$$ Let $L_{W}\left( \mathcal{N}\right) $ and $L_{S}\left( \mathcal{N}\right)
$ denote the weak and strong locking capacity, respectively.
Relation of the locking capacity to other capacities
----------------------------------------------------
Let $Q\left( \mathcal{N}\right) $, $P\left( \mathcal{N}\right) $, and $C\left( \mathcal{N}\right) $ denote the quantum [@PhysRevA.54.2614; @PhysRevA.54.2629; @BNS98; @BKN98; @PhysRevA.55.1613; @capacity2002shor; @ieee2005dev], private [@ieee2005dev; @1050633], and classical [@Hol98; @PhysRevA.56.131] capacities of a quantum channel $\mathcal{N}$, respectively. By employing operational arguments, we can determine that the following bounds hold$$Q\left( \mathcal{N}\right) \leq P\left( \mathcal{N}\right) \leq
L_{W}\left( \mathcal{N}\right) \leq C\left( \mathcal{N}\right) .
\label{eq:weak-lock-bounds}$$ Indeed, for any channel, its quantum capacity is less than the private classical capacity because any scheme for quantum communication can be used for private classical communication such that the classical information is protected from the environment of the channel. Furthermore, the inequality $P\left( \mathcal{N}\right) \leq L_{W}\left( \mathcal{N}\right) $ holds because any $\left( n,R,\varepsilon\right) $ private classical communication protocol satisfies the three requirements of a weak locking protocol [@ieee2005dev; @1050633]. Finally, the requirements of a weak locking protocol are more restrictive than those for classical communication, so that $L_{W}\left( \mathcal{N}\right) \leq C\left( \mathcal{N}\right) $.
Operational arguments and the existence of the Fawzi-Hayden-Sen (FHS) locking protocol [@FHS11] also lead to the following bounds on the strong locking capacity:$$Q\left( \mathcal{N}\right) \leq L_{S}\left( \mathcal{N}\right) \leq
L_{W}\left( \mathcal{N}\right) . \label{eq:strong-lock-bounds}$$ We first justify the bound $Q\left( \mathcal{N}\right) \leq L_{S}\left(
\mathcal{N}\right) $, already observed in some sense in Ref. [@F12]. The strong locking capacity of the noiseless qubit channel is equal to one, due to the existence of the FHS locking protocol (see Example \[ex:noiseless-qudit\] below). By concatenating the FHS locking protocol with a family of capacity-achieving quantum error correcting codes, we obtain a family of strong locking protocols that achieve a strong locking rate equal to the quantum capacity of the channel. The bound $L_{S}\left(
\mathcal{N}\right) \leq L_{W}\left( \mathcal{N}\right) $ follows because a strong locking protocol always meets the demands of a weak locking protocol (recall Remark \[rem:strong-implies-weak-lock\]). The relationship between the private capacity and the strong locking capacity is less clear. Indeed, a private communication protocol for a quantum channel protects information only from the environment of the channel (which we think of as the eavesdropper’s system). For this reason, it does not meet the demands of a strong locking protocol. However, we could consider a strong privacy protocol in which the goal is to protect a message from both the environment and output of the channel, under the assumption that the party controlling these systems does not have access to the shared key. In this case, the strong private capacity would always be equal to zero because a sublinear amount of secret key is insufficient to get any strong private capacity out of the channel. For this reason, the bounds in (\[eq:weak-lock-bounds\])-(\[eq:strong-lock-bounds\]) are the best simple ones that we can derive from operational considerations.
We can also consider the case in which a classical feedback channel is available for free from the receiver to the sender. In this case, we denote the resulting capacities with a superscript $^{\left( \leftarrow\right) }$. By employing the same operational arguments as above, we find that the following inequalities hold$$\begin{aligned}
Q^{\left( \leftarrow\right) }\left( \mathcal{N}\right) & \leq P^{\left(
\leftarrow\right) }\left( \mathcal{N}\right) \leq L_{W}^{\left(
\leftarrow\right) }\left( \mathcal{N}\right) \leq C^{\left( \leftarrow
\right) }\left( \mathcal{N}\right) ,\\
Q^{\left( \leftarrow\right) }\left( \mathcal{N}\right) & \leq
L_{S}^{\left( \leftarrow\right) }\left( \mathcal{N}\right) \leq
L_{W}^{\left( \leftarrow\right) }\left( \mathcal{N}\right) .
\label{eq:feedback-op-inequalities}$$ Capacities assisted by classical feedback need not be equal to the unassisted capacities. For example, it is known that the quantum and private capacities assisted by classical feedback can be strictly larger than the corresponding unassisted capacities [@LLS09], and this is true even for the classical capacity [@SS09]. The locking capacity of quantum channels with classical feedback remains largely an open question.
Upper bounds on the locking capacity
------------------------------------
Let us define the information quantity $L_{W}^{\left( u\right) }\left(
\mathcal{N}\right) $ as follows:$$L_{W}^{\left( u\right) }\left( \mathcal{N}\right) \equiv\max_{\left\{
p\left( x\right) ,\rho_{x}\right\} } \left[I\left( X;B\right) -I_{\text{acc}}\left( X;E\right) \right] ,\label{eq:weak-lock-upper-quantity}$$ where the above information quantities are evaluated with respect to a state of the following form:$$\sum_{x}p_{X}\left( x\right) \left\vert x\right\rangle \left\langle
x\right\vert _{X}\otimes U_{A\rightarrow BE}^{\mathcal{N}}\left( \rho
_{x}\right) ,\label{eq:locking-cap-up-bnd-state}$$ $U_{A\rightarrow BE}^{\mathcal{N}}$ is an isometric extension of the channel $\mathcal{N}$, and the superscript $\left( u\right) $ indicates that this quantity will function as an upper bound on the locking capacity. The following theorem establishes that the regularization of $L_{W}^{\left( u\right) }\left( \mathcal{N}\right) $ provides an upper bound on the weak locking capacity of a quantum channel. This bound is nontrivial given that the regularization of $L_{W}^{\left(
u\right) }\left( \mathcal{N}\right) $ does not depend on the secret key used in a given locking protocol.
\[thm:WLC\]The weak locking capacity $L_{W}\left( \mathcal{N}\right) $ of a quantum channel $\mathcal{N}$ is upper bounded by the regularization of $L_{W}^{\left( u\right) }\left( \mathcal{N}\right) $:$$L_{W}\left( \mathcal{N}\right) \leq\lim_{n\rightarrow\infty}\frac{1}{n}L_{W}^{\left( u\right) }\left( \mathcal{N}^{\otimes n}\right) .$$
The proof below places an upper bound on the weak locking capacity of a quantum channel by considering the most general protocol for this task. Suppose that the task is to generate shared, locked randomness rather than to send a locked message (placing an upper bound on achievable rates for this task gives an upper bound on achievable rates for the latter task, since a protocol for the latter task can be used to accomplish the former task). The most general protocol has Alice input her share of the key $K$ and her variable $M$ into an encoder that outputs some systems $A^{n}$ to be fed into the inputs of the channels. She then transmits these systems $A^{n}$ over the channel, so that Bob receives the output systems $B^{n}$. Let the following state describe all systems at this point in the protocol:$$\omega_{MKB^{n}}\equiv\frac{1}{\left\vert \mathcal{M}\right\vert \left\vert
\mathcal{K}\right\vert }\sum_{m,k}\left\vert m\right\rangle \left\langle
m\right\vert _{M}\otimes\left\vert k\right\rangle \left\langle k\right\vert
_{K}\otimes\mathcal{N}_{A\rightarrow B}^{\otimes n}\left( \rho_{k,m}\right)
.$$ Bob inputs his share of the key $K$ and the systems $B^{n}$ into a decoder $\mathcal{D}_{KB^{n}\rightarrow\hat{M}}$ to recover $\hat{M}$, which is his estimate of Alice’s variable $M$. The final state of the protocol is given by$$\omega_{M\hat{M}}^{\prime}\equiv\frac{1}{\left\vert \mathcal{M}\right\vert
\left\vert \mathcal{K}\right\vert }\sum_{m,k}\left\vert m\right\rangle
\left\langle m\right\vert _{M}\otimes\mathcal{D}_{KB^{n}\rightarrow\hat{M}}\left[ \left\vert k\right\rangle \left\langle k\right\vert _{K}\otimes\mathcal{N}_{A\rightarrow B}^{\otimes n}\left( \rho_{k,m}\right)
\right] .$$ If the protocol is any good for locking the message $M$, then the ideal distribution of $M$ and $\hat{M}$ deviates from the actual distribution of these variables by no more than $\varepsilon$, in the sense that$$\Big\Vert \overline{\Phi}_{M\hat{M}}-\omega_{M\hat{M}}^{\prime}\Big\Vert_{1}\leq\varepsilon,$$ where$$\overline{\Phi}_{M\hat{M}}\equiv\frac{1}{\left\vert \mathcal{M}\right\vert
}\sum_{m}\left\vert m\right\rangle \left\langle m\right\vert _{M}\otimes\left\vert m\right\rangle \left\langle m\right\vert _{\hat{M}}.$$ The above condition is equivalent to the condition that $\Pr\{\hat{M}\neq
M\}\leq\varepsilon/2$ because $$\frac{1}{2}\Big\Vert \overline{\Phi}_{M\hat
{M}}-\omega_{M\hat{M}}^{\prime}\Big\Vert _{1}=\Pr\{\hat{M}\neq M\} .$$ Also, from Remark \[rem:FA-acc-bound\], Eve’s accessible information $I_{\text{acc}}\left(
M;E^{n}\right) $ about the variable $M$ is bounded from above by $\varepsilon''n$, where $\varepsilon''\equiv h_2(\varepsilon/2)/n + \varepsilon R/2$, whenever (\[eq:sec-crit-var-dist\]) is satisfied. We can now proceed with bounding achievable rates for any locking protocol:$$\begin{aligned}
nR & =H\left( M\right) _{\overline{\Phi}}\\
& =I(M;\hat{M})_{\overline{\Phi}}\\
& \leq I(M;\hat{M})_{\omega^{\prime}}+n\varepsilon^{\prime}\\
& \leq I\left( M;B^{n}K\right) _{\omega}+n\varepsilon^{\prime}\\
& =I\left( M;B^{n}\right) _{\omega}+I\left( M;K|B^{n}\right) _{\omega
}+n\varepsilon^{\prime}\\
& \leq I\left( M;B^{n}\right) _{\omega}-I_{\text{acc}}\left(
M;E^{n}\right) _{\omega}+o\left( n\right) +n\varepsilon^{\prime
}+n\varepsilon''\\
& \leq L_{W}^{(u)}\left( \mathcal{N}^{\otimes n}\right) +o\left( n\right)
+n\varepsilon^{\prime}+n\varepsilon''.\end{aligned}$$ The first equality follows from the assumption that the random variable $M$ is a uniform random variable. The second equality is an identity because $H(M|\hat{M})=0$ for the ideal distribution on $M$ and $\hat{M}$. The first inequality follows from an application of the Alicki-Fannes-Audenart inequality (continuity of entropy) [@AF04; @A07], where $\varepsilon
^{\prime}$ is a function of $\varepsilon$ that approaches zero as $\varepsilon\rightarrow0$. The second inequality follows from an application of quantum data processing (both $B^{n}$ and $K$ are fed into the decoder to produce $\hat{M}$). The third equality follows from an application of the chain rule for mutual information. The third inequality follows from the upper bound $$I\left( M;K|B^{n}\right) \leq H\left( K|B^{n}\right) \leq H\left(
K\right) \leq o\left( n\right) ,$$ (the assumption that the secret key rate is sublinear) and from the accessible information bound $I_{\text{acc}}\left( M;E^{n}\right) \leq\varepsilon$. The final inequality follows from optimizing over all distributions, so that we have$$R\leq\lim_{n\rightarrow\infty}\frac{1}{n}L_{W}^{\left( u\right) }\left(
\mathcal{N}^{\otimes n}\right) .$$ in the limit as $n$ becomes large and as $\varepsilon\rightarrow0$.
\[thm:up-bnd-strong-locking-cap\]The strong locking capacity $L_{S}\left(
\mathcal{N}\right) $ of a quantum channel $\mathcal{N}$ is upper bounded as$$L_{S}\left( \mathcal{N}\right) \leq\lim_{n\rightarrow\infty}\frac{1}{n}L_{S}^{\left( u\right) }\left( \mathcal{N}^{\otimes n}\right) ,$$ where$$L_{S}^{\left( u\right) }\left( \mathcal{N}\right) \equiv\max_{\left\{
p\left( x\right) ,\rho_{x}\right\} } \left[I\left( X;B\right)
-I_{\operatorname{acc}}\left( X;BE\right) \right] ,$$ and the information quantities are with respect to the state in (\[eq:locking-cap-up-bnd-state\]).
The proof of this theorem is nearly identical to the proof of the one above. However, we employ the bound on the accessible information $I_{\text{acc}}\left( M;A^{n}\right)
=I_{\text{acc}}\left( M;B^{n}E^{n}\right)$ from Definition \[def:strong-lock-protocol\] and Remark \[rem:FA-acc-bound\] instead.
Observe that the bounds in the above theorem hold even if the key is allowed to be a sublinear size quantum system, as in the locking schemes discussed in [@DFHL10].
It is an interesting and important open question to determine if the upper bounds given in the above theorems are achievable.
### Entanglement-breaking channels have zero locking capacity
The above theorems and a further analysis allow us to determine that both the strong and weak locking capacities of an entanglement-breaking channel are equal to zero.
\[Entanglement-breaking channel [@HSR03]\]A channel $\mathcal{N}_{\operatorname{EB}}$ is entanglement-breaking if the output state is separable whenever it acts on one share of an entangled state:$$\left( \operatorname{id}_{R}\otimes\mathcal{N}_{\operatorname{EB}}\right)
\left( \rho_{RA}\right) =\sum_{x}p_{X}\left( x\right) \sigma_{R}^{x}\otimes\omega_{B}^{x},$$ where $p_{X}\left( x\right) $ is a probability distribution, each $\sigma_{R}^{x}$ is a state on the reference system $R$, and each $\omega
_{B}^{x}$ is a state on the channel output system $B$.
Both the strong and weak locking capacities of an entanglement-breaking channel $\mathcal{N}_{\operatorname{EB}}$ are equal to zero:$$L_{W}(\mathcal{N}_{\operatorname{EB}})=L_{S}(\mathcal{N}_{\operatorname{EB}})=0.$$ \[thm:ent-break\]
The proof of this theorem exploits the upper bound derived in Theorem \[thm:WLC\] and the fact that $L_{W}(\mathcal{N}_{\text{EB}})\geq
L_{S}(\mathcal{N}_{\text{EB}})$. We know from Ref. [@HSR03] that any entanglement-breaking channel has a representation with rank-one Kraus operators, so that its action on an input density operator is given by$$\mathcal{N}_{\text{EB}}\left( \rho\right) =\sum_{y}\left\vert \phi
_{y}\right\rangle _{B}\left\langle \psi_{y}\right\vert _{A}\rho\left\vert
\psi_{y}\right\rangle _{A}\left\langle \phi_{y}\right\vert _{B},$$ for some set of vectors $\left\{ \left\vert \psi_{y}\right\rangle
_{A}\right\} $ such that $\sum_{y}\left\vert \psi_{y}\right\rangle
\left\langle \psi_{y}\right\vert _{A}=I_A$ and a set of states $\left\{
\left\vert \phi_{y}\right\rangle _{B}\right\} $. An isometric extension of the channel is then given by$$U_{A\rightarrow BE}^{\mathcal{N}_{\text{EB}}}\equiv\sum_{y}\left\vert \phi
_{y}\right\rangle _{B}\left\langle \psi_{y}\right\vert _{A}\otimes\left\vert
y\right\rangle _{E},$$ with $\left\{ \left\vert y\right\rangle _{E}\right\} $ an orthonormal basis for the environment. From this representation, it is clear that the channel to the environment is of the form:$$\mathcal{N}_{\text{EB}}^{c}\left( \rho\right) =\sum_{y,z}\left\langle
\psi_{y}\right\vert \rho\left\vert \psi_{z}\right\rangle _{A} \ \left\langle
\phi_{z}|\phi_{y}\right\rangle _{B}\ \left\vert y\right\rangle \left\langle
z\right\vert _{E},$$ and the environment can simulate the channel to the receiver by first performing a von Neumann measurement in the basis $\left\{ \left\vert
y\right\rangle \right\} $ followed by a preparation of the state $\left\vert
\phi_{y}\right\rangle _{B}$ conditioned on the measurement outcome being $y$.
Now consider the information quantity $L_{W}^{\left( u\right) }(\mathcal{N}_{\text{EB}})$ defined in (\[eq:weak-lock-upper-quantity\]). Theorem \[thm:WLC\] states that the regularization of this quantity is an upper bound on the weak locking capacity. For any finite $n$, we can always pick the measurement to be a tensor-product von Neumann measurement of the form mentioned above, giving that$$I_{\text{acc}}\left( X;E^{n}\right) \geq I\left( X;Y^{n}\right) ,$$ where $Y^{n}$ is the random variable corresponding to the measurement outcomes. Due to the structural relationship given above (the fact that the environment can simulate the channel to the receiver by preparing $n$ quantum states $\left\vert \phi_{y_{1}}\right\rangle \otimes\cdots\otimes\left\vert
\phi_{y_{n}}\right\rangle $ from the measurement outcomes $y^{n}$), we find that$$I\left( X;Y^{n}\right) \geq I\left( X;B^{n}\right) ,$$ by an application of the quantum data processing inequality. This is equivalent to $I\left( X;B^{n}\right) -I\left( X;Y^{n}\right) \leq0$, which implies that $\lim_{n\rightarrow\infty}\frac{1}{n}L_{W}^{\left(
u\right) }\left( \mathcal{N}_{\text{EB}}^{\otimes n}\right) = 0$ and thus that the weak locking capacity vanishes for any entanglement-breaking channel.
The importance of the above theorem is the conclusion that a channel should be able to preserve entanglement between a purification of the channel input and its output in order for it to be able to lock information. If it is not able to (i.e., if it is entanglement-breaking), then the locking capacity is equal to zero. Ref. [@Boixo] suggested that entanglement does not play a role in quantum data locking, but this theorem shows that it does in any realistic implementation of a locking protocol.
It should be possible to provide a rigorous generalization of this result to entanglement-breaking channels defined over general infinite-dimensional spaces using the techniques from Ref. [@H08]. For example, it is known that a lossy bosonic channel becomes entanglement-breaking when the environment injects a thermal state with sufficiently high photon number [@H08]. However, we leave this question open for future work.
### Protocols with classical simulations have zero strong locking rate
It is important to determine the conditions for when the locking rate of a given protocol is zero, so that we can distinguish between the classical and quantum regimes for locking. In this regard, we can exclude all protocols that have a classical simulation in the following sense:
\[Classical simulation\]We say that a locking protocol has a classical simulation if the receiver’s decoding consists of performing a measurement on the output of the channel that is independent of the key $K$, followed by a classical post-processing of the measurement output and the key to produce an estimate of the transmitted message.
\[thm:classical-sim\]The strong locking rate of any locking protocol with a classical simulation is equal to zero.
The fact that this theorem should hold might be obvious, but nevertheless we provide a proof. The setup for this proof is similar to that in the proof of Theorems \[thm:WLC\] and \[thm:up-bnd-strong-locking-cap\], with the exception that the decoder first performs a key-independent measurement of the channel output to produce a random variable $Y$. The decoder then processes the random variable $Y$ and the key $K$ to produce an estimate $\hat{M}$ of the sender’s message. We can bound the rate $R$ of this protocol as follows:$$\begin{aligned}
nR & =H\left( M\right) _{\overline{\Phi}}\\
& =I(M;\hat{M})_{\overline{\Phi}}\\
& \leq I(M;\hat{M})_{\omega^{\prime}}+n\varepsilon^{\prime}\\
& \leq I\left( M;YK\right) _{\omega}+n\varepsilon^{\prime}\\
& =I\left( M;Y\right) _{\omega}+I\left( M;K|Y\right) _{\omega
}+n\varepsilon^{\prime}\\
& \leq I\left( M;Y\right) _{\omega}-I_{\text{acc}}\left( M;B^{n}E^{n}\right) _{\omega}+o\left( n\right) +n\varepsilon^{\prime}+n\varepsilon''\\
& \leq I\left( M;Y\right) _{\omega}-I\left( M;Y\right) _{\omega}+o\left(
n\right) +n\varepsilon^{\prime}+n\varepsilon''\\
& =o\left( n\right) +n\varepsilon^{\prime}+n\varepsilon''.\end{aligned}$$ The first three lines above are exactly the same as those in the proof of Theorem \[thm:WLC\]. The second inequality follows from quantum data processing. The third equality is the chain rule. The third inequality follows from the condition $I_{\text{acc}}\left( M;B^{n}E^{n}\right) _{\omega}\leq\varepsilon''n$, with $\varepsilon''\equiv h_2(\varepsilon/2)/n + \varepsilon R/2$, whenever (\[eq:sec-crit-var-dist\]) is satisfied, which should hold for any strong locking protocol. Also, it follows because $I\left( M;K|Y\right) _{\omega}\leq H\left( K\right) \leq
o\left( n\right) $. Finally, the adversary can choose her processing of the $B^{n}E^{n}$ systems to be a discarding of $E^{n}$ followed by whatever key-independent measurement of $B^{n}$ that the receiver is performing to produce $Y$. Thus, it follows that $I_{\text{acc}}\left( M;B^{n}E^{n}\right)
_{\omega}\geq I(M;Y)$. The statement that the strong locking rate is equal to zero follows by taking the limit as $n\rightarrow\infty$ and $\varepsilon
\rightarrow0$.
As a corollary of the above theorem, we find the following:
If a protocol does not consume any secret key at all, then the strong locking rate is equal to zero.
This follows simply because the receiver’s measurement on the channel outputs does not depend on a key for a scheme that does not use any key at all.
### The private and quantum capacity are equal to the weak locking capacity for particular Hadamard channels
In this section, we prove that if the channel is such that the map from the input to the environment is a quantum-to-classical channel, i.e., of the following form: $$\rho \to \sum_x \text{Tr}\{A_x \rho A_x^\dag\} \, \vert x \rangle \langle x \vert,
\label{eq:q-c-env}$$ for some orthonormal basis $\{ \vert x \rangle \}$ and where $\sum_x A_x^\dag A_x = I$, then the weak locking capacity of such a channel is equal to its private and quantum capacity. This result follows simply because the systems received by the environment are already classical, so that the best measurement for the adversary to perform is given by $\{\vert x \rangle \langle x \vert \}$ on each channel use. Any measurement other than this one will have a mutual information with the message lower than this measurement’s mutual information by a simple data processing argument. Furthermore, since the systems given to the environment are classical, the Holevo information of the environment with the input is equal to the accessible information of the environment with the input for such channels.
For such channels, the map from the input to the output is of the following form: $$\rho \to \sum_x A_x \rho A_x^\dag \otimes \vert x \rangle \langle x \vert
\label{eq:Hadamard-qc-env}$$ because the operator $\sum_x A_x (\cdot) \otimes \vert x \rangle \otimes \vert x \rangle$ is an isometric extension of the channel in (\[eq:q-c-env\]). A notable example of such a channel is the “photon detected-jump” channel, described in Ref. [@GJWZ10]. Channels of the form in (\[eq:Hadamard-qc-env\]) are examples of Hadamard channels, which are generally defined as channels complementary to entanglement-breaking ones [@K03had; @KMNR07].
We state the above result as the following theorem:
\[thm:Hadamard-qc-env\]The weak locking capacity of a channel of the form in (\[eq:Hadamard-qc-env\]) is equal to its private and quantum capacity and is given by the following expression: $$\max_{\left\{ p_{X}\left(
x\right) ,\rho_{x}\right\} }\left[ I\left( X;B\right) -I\left(
X;E\right) \right],$$ where the information quantities are evaluated with respect to the following state: $$\sum_x p_X(x) \vert x \rangle \langle x \vert_X \otimes U^{\mathcal{N}}_{A\to BE}(\rho_x)$$ with $U^{\mathcal{N}}_{A\to BE}$ an isometric extension of the channel $\mathcal{N}$.
A proof of this theorem follows the intuition mentioned above. In particular, we know from Refs. [@ieee2005dev; @1050633] that the following formula is equal to the private capacity of any channel: $$P\left( \mathcal{N}\right)
=\lim_{n\rightarrow\infty}\frac{1}{n}\left[ \max_{\left\{ p_{X}\left(
x\right) ,\rho^{(n)}_{x}\right\} }\left[ I\left( X;B^{n}\right) -I\left(
X;E^{n}\right) \right] \right]$$ Now, since we are assuming the channel to the environment to have the form given in (\[eq:q-c-env\]), the systems given to the environment are classical so that the accessible information $I_\text{acc}(X;E^n)$ is equal to the Holevo information $I\left(
X;E^{n}\right)$ for any finite $n$. Thus, our upper bound from Theorem \[thm:WLC\] on the weak locking capacity of such a channel is equal to the expression given above for its private capacity. Furthermore, all Hadamard channels are degradable [@BHTW10], meaning that the receiver can simulate the map from the input to the environment by acting with a degrading map on his system. Finally, it is known that the expression for the private capacity “single-letterizes” to the form in the statement of the theorem for degradable channels and that the quantum capacity is equal to the private capacity for such channels [@S08].
Theorem \[thm:Hadamard-qc-env\] demonstrates that it suffices to use a private capacity achieving code for channels of the form in (\[eq:Hadamard-qc-env\]), with the benefit that these private communication codes do not require the consumption of any secret key. That is, there is no need to devise an exotic information locking protocol for such channels in order to achieve their weak locking capacity.
### Quantum discord-based upper bound on the gap between weak locking capacity and private capacity
The quantum discord is an asymmetric measure that quantifies the quantum correlation in a bipartite quantum state [@HZ01]. For a given bipartite quantum state $\rho_{AB}$, the quantum mutual information $I\left(
A;B\right) _{\rho}$ quantifies all of the bipartite correlations in $\rho_{AB}$, while $\max_{\Lambda_{A\rightarrow X}}I\left( X;B\right) $ is meant to capture the classical correlations in the state that are recoverable by performing a local measurement on the $A$ system [@HV01]. Thus, the idea behind the quantum discord $D\left( A,B\right) _{\rho}$ is to quantify the quantum correlations in a state by subtracting out the classical correlation from the total correlation:$$D\left( A,B\right) _{\rho}\equiv I\left( A;B\right) _{\rho}-\max
_{\Lambda_{A\rightarrow X}}I\left( X;B\right) .$$ Ollivier and Zurek originally described the quantum discord as the correlations lost during a measurement process [@HZ01].
Our upper bound on the weak locking capacity from Theorem \[thm:WLC\] appears similar to the above formula for quantum discord. Indeed, we can place an upper bound on the gap between the weak locking capacity and the private capacity of a quantum channel in terms of the discord between the environment of the channel and the classical variable sent into the channel. We can also interpret this merely as the gap between the Holevo information of the environment and its accessible information. It is clear why this gap is related to quantum discord. In a private communication protocol, the security guarantee is with respect to the Holevo information, while in a locking protocol, the guarantee is with respect to the accessible information. Thus, the gap between the two capacities should be related to the correlations lost during Eve’s measurement.
The gap between the weak locking capacity and the private capacity of a quantum channel is no larger than$$\begin{aligned}
L_{W}\left( \mathcal{N}\right) -P\left( \mathcal{N}\right)
& \leq
\lim_{n\rightarrow\infty}\frac{1}{n}\left[ \max_{\left\{ p_{X}\left(
x\right) ,\rho_{x}\right\} }I\left( X;E^{n}\right) -I_{\operatorname{acc}}\left( X;E^{n}\right) \right] \\
& = \lim_{n\rightarrow\infty}\frac{1}{n}\left[ \max_{\left\{ p_{X}\left(
x\right) ,\rho_{x}\right\} } D(E^n,X) \right] ,\end{aligned}$$ where the entropies for any finite $n$ are with respect to a state of the following form:$$\sum_{x}p_{X}\left( x\right) \left\vert x\right\rangle \left\langle
x\right\vert _{X}\otimes\mathcal{N}_{A\rightarrow E}^{\otimes n}\left(
\rho_{x}\right) ,$$ and $\mathcal{N}_{A\rightarrow E}$ is the channel complementary to $\mathcal{N}_{A\rightarrow B}=\mathcal{N}$.
Consider that for any finite $n$, we have the bound$$\begin{aligned}
I\left( X;B^{n}\right) -I_{\text{acc}}\left( X;E^{n}\right) & =I\left(
X;B^{n}\right) -I\left( X;E^{n}\right) +I\left( X;E^{n}\right)
-I_{\text{acc}}\left( X;E^{n}\right) \\
& \leq\max_{\left\{ p_{X}\left( x\right) ,\rho_{x}\right\} }\left[
I\left( X;B^{n}\right) -I\left( X;E^{n}\right) \right] +\max_{\left\{
p_{X}\left( x\right) ,\rho_{x}\right\} }\left[ I\left( X;E^{n}\right)
-I_{\text{acc}}\left( X;E^{n}\right) \right] .\end{aligned}$$ Then by using the bound from Theorem \[thm:WLC\], the inequality above, and the characterization of the private capacity as $P\left( \mathcal{N}\right)
=\lim_{n\rightarrow\infty}\frac{1}{n}\left[ \max_{\left\{ p_{X}\left(
x\right) ,\rho_{x}\right\} }\left[ I\left( X;B^{n}\right) -I\left(
X;E^{n}\right) \right] \right] $, the bound in the statement of the theorem follows.
Examples
--------
\[Noiseless qudit channel\]\[ex:noiseless-qudit\]The noiseless qudit channel trivially has weak locking capacity equal to $\log_{2}d$, where $d$ is the dimension of the input and output for the channel. The reason for this is that an isometric extension of this channel has the following form:$$\sum_{i}\left\vert i\right\rangle _{B}\left\langle i\right\vert _{A}\otimes\left\vert \phi\right\rangle _{E}.$$ In this case, Eve’s state is independent of the input, so that her accessible information is always equal to zero (even without coding in any way).However, the noiseless qudit channel nontrivially has strong locking capacity also equal to $\log_{2}d$. This follows from the results of Fawzi *et al*. [@FHS11], in which they demonstrated the existence of a locking protocol that locks $n$ dits using $4 \log_{2}\left(
1/\varepsilon\right) +O\left( \log_{2}\log_{2}\left(
1/\varepsilon\right) \right) $ bits of key while having the variational distance in (\[eq:sec-crit-var-dist\]) for any eavesdropper measurement no larger than $\varepsilon$, for an eavesdropper who obtains the full output of the noiseless channel. Thus, this scheme is an $\left( n,\log_{2}d,\varepsilon\right) $ locking protocol that consumes secret key at a rate equal to$$\frac{1}{n}\left[ 4 \log_{2}\left( 1/\varepsilon\right)
+O\left( \log_{2}\log_{2}\left( 1/\varepsilon\right) \right) \right] .$$ So, for any fixed $\varepsilon>0$, we can take $n$ large so that the secret key rate vanishes in this limit, while the eavesdropper will not be able to do much better than to randomly guess the message. Thus, this construction gives a scheme to achieve the rate $\log_{2}d$. Since the strong locking capacity of the noiseless qudit channel cannot be any larger than $\log_{2}d$, this proves that it is equal to $\log_{2}d$ for this channel.
In reality, one does not ever have access to perfectly independent uses of a quantum channel, as this is just an idealization. As such, it can be helpful to define the “one-shot” locking capacity for a single use of a quantum channel. We provide such a definition below:
\[One-shot locking capacity\]\[def:one-shot-locking-cap\]The $\varepsilon
$-one-shot locking capacity of a quantum channel is the maximum number of locked bits that a sender can transmit to a receiver such that the receiver can recover the message with average error probability less than $\varepsilon>0$ and such that the total variational distance of the message distribution conditioned on the eavesdropper’s measurement outcome$~x$ with the unconditioned message distribution $p_{M}$ is no larger than $\varepsilon$: $$\sum_m \vert p_{M|X}(m|x) - p_{M}(m)\vert \leq\varepsilon.$$ We also demand that the number of secret key bits used is $O\left( \log
_{2}\log_{2}\left\vert \mathcal{M}\right\vert \right) $. Similar to the IID case, we can distinguish between weak and strong locking capacities.
\[Depolarizing channel\]Recall that the quantum depolarizing channel is defined as$$\rho\rightarrow\left( 1-p\right) \rho+p\frac{I}{d},$$ where $p\in\left[ 0,1\right] $ characterizes the noisiness of the channel and $d$ is its dimension. For sufficiently large $d$, the $\varepsilon
$-one-shot strong locking capacity of the depolarizing channel is equal to its $\varepsilon$-one-shot classical capacity (defined similarly as above—see Ref. [@WR12], for example). This result follows simply because any unitary encoding commutes with the action of the depolarizing channel on the input state, and we can employ the FHS protocol combined with an $\varepsilon
$-one-shot classical capacity achieving code, in order to achieve the same $\varepsilon$-one-shot strong locking capacity of the depolarizing channel. While easy to prove, this example illustrates the subtle interplay between locking, entanglement and classical communication. The fact that the depolarizing channel’s one-shot strong locking and classical capacities match regardless of the strength of the noise would seem to leave little room for quantum correlations to play any role. Indeed, it seems hard to square this result with Theorem \[thm:ent-break\]’s statement that entanglement-breaking channels have zero strong locking capacity, which is easily adapted to the one-shot setting. The resolution is that for any fixed but arbitrarily large amount of noise $p$, the depolarizing channel eventually ceases to be entanglement breaking for some sufficiently large $d = \text{poly}(1/p)$ [@gurvits2002largest]. Our best known characterization of the locking capacity of the IID memoryless depolarizing channel is in terms of the operational inequalities given in (\[eq:weak-lock-bounds\])-(\[eq:strong-lock-bounds\]).
\[Erasure channel\]Consider a $d$-dimensional quantum erasure channel defined as$$\rho\rightarrow\left( 1-p\right) \rho+p\left\vert e\right\rangle
\left\langle e\right\vert ,$$ where $\left\vert e\right\rangle $ is an erasure flag state that is orthogonal to the $d$-dimensional input state. For this channel, a unitary acting on the input commutes with the action of the channel, so that the same argument as above demonstrates that the $\varepsilon$-one-shot strong locking capacity of this channel is equal to its $\varepsilon$-one-shot classical capacity for sufficiently large $d$.The feedback-assisted weak and strong locking capacities of the memoryless erasure channel are at least $\left(
1-p\right) ^{2}$ for $p\leq1/2$ and $\left( 1-p\right) /\left(
1+2p\right) $ for $p\geq1/2$. Furthemore, they are no larger than $1-p$. These results follow from the best known lower bounds on the quantum capacity of the erasure channel assisted by classical feedback [@LLS09], the fact that the feedback-assisted classical capacity of the erasure channel cannot exceed $1-p$, and the operational inequalities in (\[eq:feedback-op-inequalities\]).
\[Parallelized locking protocols\]A simple parallelized protocol (as mentioned in Ref. [@QEM]) is to employ the FHS protocol for each use of a memoryless depolarizing or erasure channel. However, the best known statement regarding the parallel composition of locking protocols is given by Proposition 2.4 of Ref. [@FHS11]. That is, if one locking protocol guarantees that the total variational distance of a message distribution conditioned on the eavesdropper’s measurement outcome$~x_{1}$ with the unconditioned message distribution $p_{M}$ is no larger than $\varepsilon
_{1}$:$$\sum_{m_1} \vert p_{M_{1}|X_{1}}(m_1|x_1) - p_{M_{1}}(m_1) \vert \leq\varepsilon_{1},$$ and another guarantees it is no larger than $\varepsilon_{2}$:$$\sum_{m_2} \vert p_{M_{2}|X_{2}}(m_2|x_2) - p_{M_{2}}(m_2) \vert \leq\varepsilon_{2},$$ then the parallel composition of these protocols guarantees a total variational distance no larger than $\varepsilon_{1}+\varepsilon_{2}$:$$\sum_{m_1,m_2} \vert p_{ M_{1},M_{2} |X}(m_1,m_2|x) - p_{M_{1},M_{2} }(m_1,m_2) \vert \leq\varepsilon_{1}+\varepsilon_{2}.$$ Then consider a simple parallelized protocol consisting of $n$ uses of a $d$-dimensional channel, where we suppose that each channel use has a guarantee that the variational distance (as above) is no larger than $\gamma>0$. Parallel composition of the locking protocols guarantees that the variational distance for the $n$ channel uses is no larger than $\gamma n$. By applying the Fannes-Audenaert inequality [@Fannes73; @A07] as in Proposition 3.2 of Ref. [@FHS11], one finds the following bound on the accessible information of the adversary:$$\left( \gamma n\right) \log d_{E}^{n}+h_{2}\left( \gamma n\right) ,$$ where $d_{E}$ is the dimension of the environment for a single channel use. Thus, the number of secret key bits needed to guarantee that Eve’s accessible information is no larger than $n \varepsilon$ is equal to $O\left( n\log
_{2}\left( 1/\varepsilon\right) \right) $, so that the rate of key used in this scheme grows linearly with the number of channel uses. Clearly, this approach is less desirable than simply using a one-time pad combined with a classical capacity achieving code. For this latter protocol, the rate of key is a fixed constant independent of the number of channel uses and the protocol guarantees perfect secrecy from an adversary with access to a quantum memory. In information theory, results for memoryless channels usually follow straightforwardly from their one-shot counterparts. The linear key growth incurred when parallelizing locking protocols prevents us from quickly concluding that the non-one-shot strong locking capacities of the depolarizing and erasure channels match their classical capacities. Moreover, the covariance argument used to draw that conclusion does not translate directly to the setting of many channel uses. We therefore leave it as an open question to determine whether the equivalence persists beyond the one-shot setting.
Upper bounds on the locking capacity when restricting to coherent-state encodings {#CSQEM}
=================================================================================
In this section, we prove that there are fundamental limitations on the locking capacity of channels when we restrict ourselves to coherent-state encodings. In particular, we prove that the strong locking capacity of any quantum channel cannot be any larger than$$g\left( N_{S}\right) -\log_{2}\left( 1+N_{S}\right) ,$$ where $g(x) \equiv (x+1) \log_2(x+1) - x \log_2 x$, when restricting to coherent-state encodings with mean input photon number $N_{S}$. Observe that $g\left( N_{S}\right) -\log_{2}\left( 1+N_{S}\right)
\leq\log_{2}\left( e\right) $, and this latter bound is independent of the photon number used for the coherent-state codewords. An intuitive (yet not fully rigorous) reason for why we obtain this bound is that $\log_{2}\left(
1+N_{S}\right) $ is the rate of information that an adversary can recover about the message simply by performing heterodyne detection on each input to the channel, while $g\left(
N_{S}\right) $ is an upper bound on the classical capacity of any channel with mean input photon number $N_{S}$. Thus, the difference of these two quantities should be a bound on the strong locking capacity.
We also prove that the weak locking capacity of a pure-loss bosonic channel cannot be any larger than the sum of its private capacity and$$g\left( \left( 1-\eta\right) N_{S}\right) -\log_{2}\left(
1+\left(
1-\eta\right)N_{S} \right) ,$$ when restricting to coherent-state encodings with mean photon number $N_{S}$, where $\eta\in\left[ 0,1\right] $ is the transmissivity of the channel. As before, $g\left( \left( 1-\eta\right) N_{S}\right) -\log_{2}\left(
1+\left(
1-\eta\right)N_{S} \right) \leq
\log_{2}\left( e\right) $, which is independent of the photon number.
We consider a coherent-state locking protocol in which the encrypted states $\{U_{k}|m\rangle\}$ are generalized to a set of $n$-mode coherent states $\{|\alpha^{n}\left( m,k\right) \rangle\}_{m\in\mathcal{M},k\in\mathcal{K}}$, where $|\alpha^{n}\left( m,k\right) \rangle$ is an $n$-fold tensor product of coherent states:$$\left\vert \alpha^{n}\left( m,k\right) \right\rangle \equiv\left\vert
\alpha_{1}\left( m,k\right) \right\rangle \otimes\cdots\otimes\left\vert
\alpha_{n}\left( m,k\right) \right\rangle .$$
\[Coherent-state locking protocol\]\[def:coh-state-QEM\]A coherent-state locking protocol consists of coherent-state codewords $\left\{ |\alpha
^{n}\left( m,k\right) \rangle\right\} _{m\in\mathcal{M},k\in\mathcal{K}}$ depending upon the message $m$ and the key value$~k$. These codewords are then transmitted over a quantum channel to be decoded by a receiver.
The strong locking capacity of any channel when restricting to coherent-state encodings with mean photon number $N_{S}$ is upper bounded by $g\left(
N_{S}\right) -\log_{2}\left( 1+N_{S}\right) $.
As described in Definition \[def:coh-state-QEM\], the encoder for such a scheme prepares a coherent-state codeword $|\alpha^{n}\left( m,k\right)
\rangle$ at the input of $n$ uses of a quantum channel $\mathcal{N}$, depending upon the message $m$ and the key value $k$. It is useful for us to consider the following classical-quantum state, which describes the state of the message, key, and input to many uses of the channel:$$\rho_{MKA^{n}}=\frac{1}{|\mathcal{M}||\mathcal{K}|}\sum_{m,k}|m,k\rangle
\langle m,k|_{MK}\otimes|\alpha^{n}\left( m,k\right) \rangle\langle
\alpha^{n}\left( m,k\right) |_{A^{n}}\,.$$ The state after the isometric extension of the channel (unique up to unitaries acting on the environment) acts is then as follows:$$\rho_{MKB^{n}E^{n}}=\frac{1}{|\mathcal{M}||\mathcal{K}|}\sum_{m,k}|m,k\rangle\langle m,k|_{MK}\otimes U_{A^{n}\rightarrow B^{n}E^{n}}^{\mathcal{N}}\left( |\alpha^{n}\left( m,k\right) \rangle\langle\alpha
^{n}\left( m,k\right) |_{A^{n}}\right) ,$$ where $U_{A^{n}\rightarrow B^{n}E^{n}}^{\mathcal{N}}$ is the isometry corresponding to $n$ uses of the given channel. Recall from the proof of Theorem \[thm:up-bnd-strong-locking-cap\] that we obtain the following upper bound on the strong locking capacity of $\mathcal{N}$:$$I\left( M;B^{n}\right) -I_{\text{acc}}\left( M;B^{n}E^{n}\right) +o\left(
n\right) +n2\varepsilon^{\prime}. \label{eq:strong-locking-upper-bound}$$ (Recall that this bound holds for any $\left( n,R,\varepsilon\right)
$ strong locking protocol, with $\varepsilon^{\prime}$ a function of $\varepsilon$ that vanishes as $\varepsilon\rightarrow0$.) Consider that the information quantity $I\left( M;B^{n}\right) $ is upper bounded as follows:$$\begin{aligned}
I\left( M;B^{n}\right) _{\rho} & \leq I\left( M;A^{n}\right) _{\rho}\\
& =I\left( MK;A^{n}\right) _{\rho}-I\left( K;A^{n}|M\right) _{\rho},\end{aligned}$$ where the first inequality follows from quantum data processing, and the equality follows from the chain rule for quantum mutual information. We then find that$$\begin{aligned}
I\left( MK;A^{n}\right) _{\rho} & =H\left( A^{n}\right) _{\rho}-H\left(
A^{n}|MK\right) _{\rho}\nonumber\\
& =H\left( A^{n}\right) _{\rho}, \label{eq:mut-eq-vN-ent}$$ where the second equality follows because the state on $A^{n}$ is a pure coherent state when conditioned on systems $M$ and $K$.
On the other hand, we obtain a lower bound on the accessible information $I_{\text{acc}}\left( M;B^{n}E^{n}\right) =I_{\text{acc}}\left(
M;A^{n}\right) $ by having the adversary perform heterodyne detection (a particular measurement that is not necessarily the optimal one) on each of the systems $A^{n}$, giving$$\begin{aligned}
I_{\mathrm{acc}}\left( M;A^{n}\right) _{\rho} & \geq I_{\mathrm{het}}(M;A^{n})_{\rho}\\
& =I_{\mathrm{het}}\left( MK;A^{n}\right) _{\rho}-I_{\mathrm{het}}\left(
K;A^{n}|M\right) _{\rho},\end{aligned}$$ where in the second line we again apply the chain rule for mutual information. An ideal $n$-mode heterodyne measurement is described by a POVM $\{\frac
{d^{2n}\beta^{n}}{\pi^{n}}|\beta^{n}\rangle\langle\beta^{n}|\}$, where $\beta^{n}$ is the amplitude of the $n$-mode coherent state $|\beta^{n}\rangle\equiv\left\vert \beta_{1}\right\rangle \cdots\left\vert \beta
_{n}\right\rangle $ and $d^{2n}\beta^{n}$ denotes the Lebesgue measure on $\mathbb{C}^{n}$. We can then compute the heterodyne mutual information $I_{\mathrm{het}}\left( MK;A^{n}\right) _{\rho}$ as$$I_{\mathrm{het}}\left( MK;A^{n}\right) _{\rho}=W\left( A^{n}\right)
_{\rho}-W\left( A^{n}|MK\right) _{\rho}\ ,$$ where$$W(Q)_{\sigma}=-\int\frac{d^{2n}\beta^{n}}{\pi^{n}}\langle\beta^{n}|\sigma
|\beta^{n}\rangle\log_{2}{\langle\beta}^{n}{|\sigma|\beta}^{n}{\rangle}$$ denotes the Wehrl entropy for a state $\sigma$ defined on system $Q$ [@Wehrl] and its conditional version follows in the natural way. It is easy to see that the Wehrl entropy of an $n$-mode coherent state is equal to $n\log_{2}\left( e\right) $, so we find that$$I_{\mathrm{het}}\left( MK;A^{n}\right) _{\rho}=W\left( A^{n}\right)
_{\rho}-n\log_{2}\left( e\right) .\label{eq:heterodyne-info}$$ We are now in a position to derive an upper bound on (\[eq:strong-locking-upper-bound\]). Observe that our development above implies that$$\begin{aligned}
I\left( M;B^{n}\right) -I_{\text{acc}}\left( M;B^{n}E^{n}\right) & \leq
I\left( MK;A^{n}\right) _{\rho}-I\left( K;A^{n}|M\right) _{\rho
}\nonumber\\
& \ \ \ \ -\left[ I_{\mathrm{het}}\left( MK;A^{n}\right) _{\rho
}-I_{\mathrm{het}}\left( K;A^{n}|M\right) _{\rho}\right] \nonumber\\
& \leq I\left( MK;A^{n}\right) _{\rho}-I_{\mathrm{het}}\left(
MK;A^{n}\right) _{\rho}\nonumber\\
& \leq\max_{p_{X}\left( x\right) }\left[ I\left( X;A^{n}\right)
_{\omega}-I_{\mathrm{het}}\left( X;A^{n}\right) _{\omega}\right]
\nonumber\\
& \leq n\ \max_{p_{X}\left( x\right) }\left[ I\left( X;A\right)
_{\sigma}-I_{\mathrm{het}}\left( X;A\right) _{\sigma}\right] \nonumber\\
& =n\ \left( \log_{2}\left( e\right) +\max_{p_{X}\left( x\right)
}\left[ H\left( A\right) _{\sigma}-W\left( A\right) _{\sigma}\right]
\right) \nonumber\\
& \leq n\ \left[ g\left( N_{S}\right) -\log_{2}\left( 1+N_{S}\right)
\right] .\label{eq:heterodyne-bound-1}$$ The second inequality follows from data processing: $I\left( K;A^{n}|M\right) _{\rho}\geq I_{\mathrm{het}}\left( K;A^{n}|M\right) _{\rho}$ (the system $M$ is classical, and performing heterodyne detection on $A^{n}$ can only reduce the mutual information). The third inequality follows by taking a maximization over all distributions $p_{X}\left( x\right) $ where $\omega_{XA^{n}}$ is a state of the following form:$$\omega_{XA^{n}}\equiv\sum_{x}p_{X}\left( x\right) \left\vert x\right\rangle
\left\langle x\right\vert _{X}\otimes\left\vert \alpha_{x}^{n}\right\rangle
\left\langle \alpha_{x}^{n}\right\vert _{A^{n}},$$ such that the mean input photon number to the channel for each $x$ is $N_{S}$. The fourth inequality follows by realizing that the difference between the mutual information and the heterodyne information is equal to the private information of a quantum wiretap channel in which the state $\left\vert \alpha_{x}^{n}\right\rangle $ is prepared for the receiver while the heterodyned version of this state (a classical variable) is prepared for the eavesdropper. Such a quantum wiretap channel has pure product input states (they are coherent states) and it is degraded. Thus, we can apply Theorem \[thm:private-info-additive\] from the appendix to show that this private information is subadditive, in the sense that$$\max_{p_{X}\left( x\right) }\left[ I\left( X;A^{n}\right) _{\omega
}-I_{\mathrm{het}}\left( X;A^{n}\right) _{\omega}\right] \leq
n\ \max_{p_{X}\left( x\right) }\left[ I\left( X;A\right) _{\sigma
}-I_{\mathrm{het}}\left( X;A\right) _{\sigma}\right] ,$$ where we define the state $\sigma_{XA}$ as follows:$$\sigma_{XA}\equiv\sum_{x}p_{X}\left( x\right) \left\vert x\right\rangle
\left\langle x\right\vert _{X}\otimes\left\vert \alpha_{x}\right\rangle
\left\langle \alpha_{x}\right\vert _{A}.$$ The last equality follows from the observation in (\[eq:heterodyne-info\]) and because $I\left( X;A\right) _{\sigma}=H\left( A\right) _{\sigma
}-H\left( A|X\right) _{\sigma}=H\left( A\right) _{\sigma}$ (since the states are pure when conditioned on $X$).
We now show that the maximizing distribution for $\max_{p_{X}\left( x\right)
}\left[ H\left( A\right) _{\sigma}-W\left( A\right) _{\sigma}\right] $ is given by a circularly-symmetric Gaussian distribution with variance $N_{S}$, so that the optimal ensemble is a Gaussian ensemble of coherent states. Indeed, let $\varrho$ be a single-mode quantum state with $\mathrm{Tr}[a\varrho]=0$ and $\mathrm{Tr}[a^{\dagger}a\varrho]=N_{S}$ where $a^{\dagger}$ and $a$ are creation and annihilation operators, respectively. The von Neumann entropy is given by $H(\varrho)=-\mathrm{Tr}[\varrho\log_{2}\varrho]$. We show that $$H(\varrho)-W(\varrho)\label{eq:h-w}$$ is maximized when $\varrho$ is a thermal state. Our approach is based on a technique used in the appendix of Ref. [@HSO99], which in turn is based on classical approaches to this problem [@Cover]. Let$$\widetilde{\varrho}=\frac{1}{N_{S}+1}\sum_{m=0}^{\infty}\Bigg( \frac{N_{S}}{N_{S}+1}\Bigg)^m |m\rangle\langle m|,\label{eq:gaussification}$$ be a thermal state with mean photon number $N_{S}$. We will show that$$H(\widetilde{\varrho}) - W(\widetilde{\varrho}) -
\left( H(\varrho)-W(\varrho )\right)
\geq 0 \, , \label{eq:h-w2}$$ holds for any $\varrho$ with $\mathrm{Tr}[a\varrho]=0$ and $\mathrm{Tr}[a^{\dagger}a\varrho]=N_{S}$. Putting $$\mathcal{Q}_{\varrho}(\beta) = \langle\beta|\varrho|\beta\rangle \, ,\label{eq:Q_function}$$ the left hand side of (\[eq:h-w2\]) is equal to $$\begin{aligned}
& -\mathrm{Tr}[\widetilde{\varrho}\log_{2}\widetilde{\varrho}] + \mathrm{Tr}[\varrho\log_{2}\varrho]
+ \int \frac{d^{2}\beta}{\pi} \mathcal{Q}_{\widetilde{\varrho}}(\beta)\log_{2}\mathcal{Q}_{\widetilde{\varrho}}(\beta)
- \int \frac{d^{2}\beta}{\pi} \mathcal{Q}_{\varrho}(\beta)\log_{2}\mathcal{Q}_{\varrho}(\beta) \nonumber \label{eq:h-w3}\\
& = \mathrm{Tr}[\varrho(\log_{2}\varrho-\log_{2}\widetilde{\varrho})]+\mathrm{Tr}[(\varrho-\widetilde{\varrho})\log_{2}\widetilde{\varrho
}]\nonumber\\
& \ \ \ \ \ -\left\{ \int \frac{d^{2}\beta}{\pi} \mathcal{Q}_{\varrho}(\beta)(\log_{2}\mathcal{Q}_{\varrho}(\beta)-\log_{2}\mathcal{Q}_{\tilde{\rho}}(\beta) )
+ \int \frac{d^{2}\beta}{\pi} (\mathcal{Q}_{\varrho}(\beta)-\mathcal{Q}_{\widetilde{\varrho}}(\beta))\log_{2}\mathcal{Q}_{\widetilde{\varrho}}(\beta)\right\} \nonumber\\
& = D(\varrho||\widetilde{\varrho})-D(\mathcal{Q}_{\varrho}||\mathcal{Q}_{\widetilde{\varrho}})
+ \mathrm{Tr}[(\varrho-\widetilde{\varrho})\log_{2}\widetilde{\varrho}]
- \int \frac{d^{2}\beta}{\pi} (\mathcal{Q}_{\varrho}(\beta)-\mathcal{Q}_{\widetilde{\varrho}}(\beta))\log
_{2}\mathcal{Q}_{\widetilde{\varrho}}(\beta),\end{aligned}$$ where $D(\varrho||\widetilde{\varrho})$ and $D(\mathcal{Q}_{\varrho}||\mathcal{Q}_{\widetilde
{\varrho}})$ are quantum and classical relative entropies, respectively. We can easily show that their difference is positive by the monotonicity property of the relative entropy. The third term is$$\begin{aligned}
\mathrm{Tr}[(\varrho-\widetilde{\varrho})\log_{2}\widetilde{\varrho}] &
=\mathrm{Tr}\left[ (\varrho-\widetilde{\varrho})\sum_{m=0}^{\infty}\log
_{2}\left\{ \frac{1}{N_{S}+1}\Bigg( \frac{N_{S}}{N_{S}+1}\Bigg)
^{a^{\dagger}a}\right\} |m\rangle\langle m|\right]
\nonumber\label{eq:third_term}\\
& =-\log_{2}(N_{S}+1)\mathrm{Tr}[\varrho-\widetilde{\varrho}]+\log_{2}\left(
\frac{N_{S}}{N_{S}+1}\right) \mathrm{Tr}[(\varrho-\widetilde{\varrho
})a^{\dagger}a]\nonumber\\
& =0.\end{aligned}$$ Similarly, the fourth term is$$\begin{aligned}
& \int \frac{d^{2}\beta}{\pi} \left( \mathcal{Q}_{\varrho}(\beta)-\mathcal{Q}_{\widetilde{\varrho}}(\beta)\right) \log_{2}\mathcal{Q}_{\widetilde{\varrho}}(\beta
)\nonumber\label{eq:fourth_term}\\
& = \int \frac{d^{2}\beta}{\pi} \left( \mathcal{Q}_{\varrho}(\beta)-\mathcal{Q}_{\widetilde{\varrho}}(\beta)\right) \left( -\log_{2}(N_{S}+1)-\frac{|\beta|^{2}}{\ln(2) (N_{S}+1)}\right) \\
& =0.\end{aligned}$$ Note that $\mathcal{Q}_{\widetilde{\varrho}}(\beta)=\frac{1}{(N_{S}+1)}\exp\left[
-\frac{|\beta|^{2}}{N_{S}+1}\right] $ and we used the fact that if $\mathrm{Tr}[a^{\dagger}a\varrho]=\mathrm{Tr}[a^{\dagger}a\tau]$ then$$\int d^{2}\beta\ \mathcal{Q}_{\varrho}(\beta)\ |\beta|^{2}=\int d^{2}\beta\ \mathcal{Q}_{\tau}(\beta)\ |\beta|^{2}.$$ As a consequence, we have$$H(\widetilde{\varrho})-W(\widetilde{\varrho})-\left( H(\varrho)-W(\varrho)\right) =D(\varrho||\widetilde{\varrho})-D(\mathcal{Q}_{\varrho}||\mathcal{Q}_{\widetilde{\varrho}})\geq0,$$ which completes the proof that $\max_{p_{X}\left( x\right) }\left[ H\left(
A\right) _{\sigma}-W\left( A\right) _{\sigma}\right] $ is optimized by a circularly symmetric complex Gaussian distribution with variance $N_{S}$.
Finally, we can rewrite $\log_{2}\left( e\right) +\max_{p_{X}\left(
x\right) }\left[ H\left( A\right) _{\sigma}-W\left( A\right) _{\sigma
}\right] $ as $I\left( X;A\right) _{\sigma}-I_{\mathrm{het}}\left(
X;A\right) _{\sigma}$ for $X$ complex Gaussian, and these information quantities evaluate to $g\left( N_{S}\right) -\log_{2}\left( 1+N_{S}\right) $ in such a case. By combining the bounds in (\[eq:strong-locking-upper-bound\]) and (\[eq:heterodyne-bound-1\]), we deduce the following upper bound on the rate$~R$ of any strong locking protocol that employs coherent-state codewords with mean photon number$~N_{S}$:$$R\leq g\left( N_{S}\right) -\log_{2}\left( 1+N_{S}\right) +\frac{o\left(
n\right) }{n}+2\varepsilon^{\prime},$$ which converges to $g\left( N_{S}\right) -\log_{2}\left( 1+N_{S}\right) $ in the limit as $n\rightarrow\infty$ and $\varepsilon\rightarrow0$.
The weak locking capacity of a pure-loss bosonic channel with transmissivity $\eta\in\left[ 0,1\right] $ when restricting to coherent-state encodings with mean input photon number $N_{S}$ is upper bounded by$$\max\left\{ 0,g\left( \eta N_{S}\right) -g\left( \left( 1-\eta\right)
N_{S}\right) \right\} +\left[ g\left( \left( 1-\eta\right) N_{S}\right)
-\log_{2}\left(
1+\left(
1-\eta\right)N_{S} \right)
\right] .$$ The term $\max\left\{ 0,g\left( \eta N_{S}\right) -g\left( \left(
1-\eta\right) N_{S}\right) \right\} $ is equal to the private capacity of the pure-loss bosonic channel, while the second term is limited by the bound$$\left[ g\left( \left( 1-\eta\right) N_{S}\right) -\log_{2}\left(
1+\left(
1-\eta\right)N_{S} \right) \right] \leq\log_{2}\left(
e\right) .$$
The proof of this theorem is somewhat similar to the proof of the previous theorem. Nevertheless, there are some important differences, and so we give the full proof for completeness.
In the proof of Theorem \[thm:WLC\], we obtained the following upper bound on the weak locking capacity:$$I\left( M;B^{n}\right) -I_{\text{acc}}\left( M;E^{n}\right) +o\left(
n\right) +n2\varepsilon^{\prime}.\label{eq:WLC-bosonic-bound}$$ (Recall that this bound holds for any $\left( n,R,\varepsilon\right)
$ strong locking protocol, with $\varepsilon^{\prime}$ a function of $\varepsilon$ that vanishes when $\varepsilon\rightarrow0$.) We begin by bounding the quantity $I\left( M;B^{n}\right) -I_{\text{acc}}\left(
M;E^{n}\right) $:$$\begin{aligned}
& I\left( M;B^{n}\right) -I_{\text{acc}}\left( M;E^{n}\right) \\
& \leq I\left( MK;B^{n}\right) -\left[ I_{\text{het}}\left(
MK;E^{n}\right) -I_{\text{het}}\left( K;E^{n}|M\right) \right] \\
& \leq I\left( MK;B^{n}\right) -I_{\text{het}}\left( MK;E^{n}\right)
+o\left( n\right) \\
& =H\left( B^{n}\right) -W\left( E^{n}\right) +n\log_{2}\left( e\right)
+o\left( n\right) \\
& =H\left( B^{n}\right) -H\left( E^{n}\right) +H\left( E^{n}\right)
-W\left( E^{n}\right) +n\log_{2}\left( e\right) +o\left( n\right) \\
& \leq n\left[ \max\left\{ 0,g\left( \eta N_{S}\right) -g\left( \left(
1-\eta\right) N_{S}\right) \right\} \right] \\
& \ \ \ \ \ +n\left[ g\left( \left( 1-\eta\right) N_{S}\right) -
\log_{2}\left(
1+\left(
1-\eta\right)N_{S} \right) \right]
+o\left( n\right) .\end{aligned}$$ The first inequality follows from data processing $I\left( M;B^{n}\right)
\leq I\left( MK;B^{n}\right) $, the fact that $I_{\text{acc}}\left(
M;E^{n}\right) \geq I_{\text{het}}\left( MK;E^{n}\right) $, and the identity $I_{\text{het}}\left( M;E^{n}\right) =I_{\text{het}}\left(
MK;E^{n}\right) -I_{\text{het}}\left( K;E^{n}|M\right) $. The second inequality follows because $I_{\text{het}}\left( K;E^{n}|M\right) \leq
H\left( K\right) \leq o\left( n\right) $. The first equality follows from the fact that $I\left( MK;B^{n}\right) =H\left( B^{n}\right) $ for the pure-loss bosonic channel and from the fact that $I_{\text{het}}\left(
MK;E^{n}\right) =W\left( E^{n}\right) -n\log_{2}\left( e\right) $. The second equality is a simple identity. The final inequality follows because the entropy difference $H\left( B^{n}\right) -H\left( E^{n}\right) $ is equal to a coherent information of the $n$-use pure-loss bosonic channel. The only relevant property of the input state for which the coherent information is evaluated is that it has a mean photon number $N_{S}$, and so the coherent information is always lower than $n\max\left\{ 0,g\left( \eta N_{S}\right)
-g\left( \left( 1-\eta\right) N_{S}\right) \right\} $, which is equal to $n$ times the quantum and private capacity of this channel [@WPG07; @WHG12]. We also employ an argument similar to that in the previous theorem to bound $H\left( E^{n}\right) -W\left( E^{n}\right) +n\log_{2}\left( e\right) $ from above by $n\left[ g\left( \left( 1-\eta\right) N_{S}\right)
-\log_{2}\left(
1+\left(
1-\eta\right)N_{S} \right)
\right] $. Finally, by combining the above bound with the bound in (\[eq:WLC-bosonic-bound\]), we deduce the following upper bound on the rate $R$ of any weak locking protocol that employs coherent-state codewords for transmission over a pure-loss bosonic channel:$$R\leq\max\left\{ 0,g\left( \eta N_{S}\right) -g\left( \left(
1-\eta\right) N_{S}\right) \right\} +\left[ g\left( \left(
1-\eta\right) N_{S}\right) -\log_{2}\left(
1+\left(
1-\eta\right)N_{S} \right) \right] +\frac{o\left( n\right) }{n}+2\varepsilon^{\prime},$$ which converges to $\max\left\{ 0,g\left( \eta N_{S}\right) -g\left(
\left( 1-\eta\right) N_{S}\right) \right\} +\left[ g\left( \left(
1-\eta\right) N_{S}\right) -\log_{2}\left(
1+\left(
1-\eta\right)N_{S} \right) \right] $ in the limit as $n\rightarrow\infty$ and $\varepsilon\rightarrow0$.
Given that the private capacity of a pure-loss bosonic channel with mean input photon number $N_{S}$ is equal to $\max\left\{ 0,g\left( \eta N_{S}\right)
-g\left( \left( 1-\eta\right) N_{S}\right) \right\} $, the above theorem implies a strong limitation on the weak locking capacity of a pure-loss bosonic channel when restricting to coherent-state encodings with mean input photon number $N_{S}$. That is, the weak locking capacity when restricting to coherent-state encodings cannot be more than 1.45 bits larger than the channel’s private capacity.
These bounds apply in particular to channels that use a coherent-state locking protocol in which there is a fixed codebook $\left\{ \left\vert \alpha^{n}\left( m\right)
\right\rangle \right\} $ and the coherent states are encrypted according to passive mode transformations $U_{k}$ that transform $n$-mode coherent states as $\left\vert \alpha^{n}\right\rangle \rightarrow|\widetilde{U}_{k}\alpha
^{n}\rangle$, where $\widetilde{U}_{k}\alpha^{n}$ is understood to be a label for a coherent state vector with the following complex amplitudes:$$\begin{bmatrix}
\widetilde{U}_{k}^{\left( 1,1\right) } & \cdots & \widetilde{U}_{k}^{\left(
1,n\right) }\\
\vdots & \ddots & \vdots\\
\widetilde{U}_{k}^{\left( n,1\right) } & \cdots & \widetilde{U}_{k}^{\left(
n,n\right) }\end{bmatrix}\begin{bmatrix}
\alpha_{1}\\
\vdots\\
\alpha_{n}\end{bmatrix}
.\label{eq:passive-mode-trans}$$
If the coherent-state locking protocol consists of passive mode transformations (as defined above) for the encryption and the receiver performs heterodyne detection to recover the message after decrypting with a passive mode transformation, then the strong locking capacity of a channel using such a scheme is equal to zero. This result follows because such a scheme has a classical simulation—passive mode transformations commute with heterodyne detection in such a way that heterodyne detection can be performed first followed by a classical postprocessing of the measurement data with a matrix multiplication as in (\[eq:passive-mode-trans\]). That is, the decoding in such a scheme is equivalent to first performing key-independent heterodyne detection measurements followed by classical post-processing of the key and the measurement results. Thus, Theorem \[thm:classical-sim\] applies so that the strong locking capacity of a channel using such a scheme is equal to zero. However, this theorem does not apply if the receiver performs photodetection because passive mode transformations do not commute with such a measurement.
From our upper bounds on the locking capacity of channels restricted to coherent-state encodings, it is clear that there are strong limitations on the rates that are achievable when employing bright coherent states. That is, it clearly would not be worthwhile to invest a large mean input photon number per transmission given the above limitations on locking capacity that are independent of the photon number. In spite of this result, it might be possible to achieve interesting locking rates with weak coherent states, but we should keep in mind that the above bounds were derived by considering the information that an adversary can gain by performing heterodyne detection—the information of the adversary can only increase if she performs a better measurement. Nevertheless, we can determine values of the mean input photon number $N_{S}$ such that the difference $g\left( N_{S}\right) -\log_{2}\left(
1+N_{S}\right) $ becomes relatively large. By considering $N_{S}\ll1$, we find the following expansions of $g(N_{S})$ and $\log_{2}{(1+N_{S})}$, respectively:$$\begin{aligned}
g(N_{S}) & \approx\left( -N_{S}\ln{N_{S}}+N_{S}+\frac{N_{S}^{2}}{2}\right)
\log_{2}\left( {e}\right) \,,\\
\log_{2}{(1+N_{S})} & \approx\left( N_{S}-\frac{N_{S}^{2}}{2}\right)
\log_{2}\left( {e}\right) \,,\end{aligned}$$ so that the difference $g\left( N_{S}\right) -\log_{2}\left( 1+N_{S}\right) \approx\left[ -N_{S}\ln{N_{S}+}N_{S}^{2}\right] \log_{2}\left(
e\right) $ for $N_{S}\ll1$. Indeed, in the limit as $N_{S}\rightarrow0$, we find that the relative ratio of our upper bound on the strong locking capacity of a coherent-state protocol to the classical capacity $g\left( N_{S}\right)
$ of the noiseless bosonic channel approaches one:$$\lim_{N_{S}\rightarrow0}\frac{g(N_{S})-\log_{2}{(1+N_{S})}}{g(N_{S})} = 1,$$ so that there is some sense in which the rate at which we can lock information becomes similar to the rate at which we can insecurely communicate information if the bound $g(N_{S})-\log_{2}{(1+N_{S})}$ is in fact achievable. However, this remains an important open question.
One-shot PPM coherent-state locking protocol {#weakQEM}
============================================
In spite of the previous section’s limitations on the locking capacity of coherent-state protocols, we still think it is interesting to explore what kind of locking protocols are possible using coherent-state encodings. To this end, we now discuss a one-shot strong locking protocol (in the sense of Definition \[def:one-shot-locking-cap\]) that employs a coherent-state encoding. For simplicity, we consider the case of a noiseless channel, with the generalization to the pure-loss bosonic channel and weak locking being straightforward.
An explicit scheme for locking using weak coherent states can be obtained by analogy with the PPM encryption presented in Ref. [@QEM] and reviewed in Section \[unaryQEM\]. Similar to the single-photon scheme, to encode a message $m$ Alice prepares an $n$-mode coherent state $|\alpha_{m}\rangle$ which is a tensor product of a single-mode coherent state of amplitude $\alpha
$ on the $m$th mode and the vacuum on the remaining $n-1$ modes: $$|\alpha_{m}\rangle\equiv|0\rangle_{1}\ldots|0\rangle_{m-1}|\alpha\rangle
_{m}|0\rangle_{m+1}\ldots|0\rangle_{n}.$$ (Notice that, as in the single-photon case, the PPM encoding is highly inefficient in terms of number of modes, as it encodes only $\log_{2}{n}$ bits into $n$ bosonic modes.) Let us fix $N_{\mathrm{tot}}=|\alpha|^{2}$ to be the total mean number of photons involved in the protocol, and $$N_{S}\equiv\frac{N_{\mathrm{tot}}}{n}$$ to be the mean photon number per mode. Before sending anything to Bob, Alice encrypts a message by applying a unitary selected uniformly at random (according to the shared secret key) from a set of $\left\vert \mathcal{K}\right\vert $ $n$-mode linear-optical passive transformations. If the unitary $U_{k}$ is used, then the final state is the $n$-mode coherent state $$U_{k}|\alpha_{m}\rangle=\bigotimes_{m^{\prime}=1}^{n} | \widetilde{U}_{k}^{\left( m^{\prime},m\right) } \alpha\rangle\,.$$ Bob, who knows which unitary has been chosen by Alice, applies the inverse transformation and performs photodetection on the received modes. He will detect (one or more) photons only in the $m$th mode, hence successfully decrypting the message in case of a detection.
Different from the single-photon architecture of Ref. [@QEM], there is a non-zero probability that Bob’s detector does not click. Analogous to the case of the single-photon locking protocol in the presence of loss, if no photon is detected, Bob may use a public classical communication channel to ask Alice to resend, yielding $$I_{\mathrm{acc}}(M;KQ)_{\rho} = N_\mathrm{tot} \log_{2}{n}\,.$$ However, the same observations from Section \[unaryQEM\] apply here. That is, locking is only known to be secure when the message distribution is uniform, and this is certainly not the case for a feedback-assisted scheme unless Eve attacks each PPM block independently. If she attacks collectively, then it is necessary for Alice and Bob to exploit an amount of key necessary to ensure that the message distribution is uniform.
Assuming that Eve independently attacks each block that she receives, we have to evaluate her accessible information with respect to the following state: $$\rho_{MKQ} = \frac{1}{n \left\vert
\mathcal{K}\right\vert }\sum_{m,k}|m,k\rangle\langle m,k|_{MK}\otimes\left(
U_{k}|\alpha_{m}\rangle\langle\alpha_{m}|U_{k}^{\dag}\right) _{Q}\,.$$ If the set of $n$-mode unitaries are selected uniformly at random according to the Haar measure, one might expect that such a set of unitaries scrambles phase information of $\alpha$ so that it is not accessible to Eve. Thus, a presumably clever strategy for Eve is to perform a measurement that commutes with the total number of photons. Such a POVM has elements$$\{|0\rangle\langle0|,\{\mu_{y}^{(1)}|\phi_{y}^{(1)}\rangle\langle\phi
_{y}^{(1)}|\}_{y},\{\mu_{y}^{(2)}|\phi_{y}^{(2)}\rangle\langle\phi_{y}^{(2)}|\}_{y},\dots\}, \label{eq:photon-number-POVM-elements}$$ where $|0\rangle$ is the $n$-mode vacuum, and for any $k\geq1$ each vector $|\phi_{y}^{(k)}\rangle$ belongs to the $k$-photon subspace. This suboptimal measurement allows Eve to achieve a mutual information $I_{\mathrm{num}}(M;Q)_{\rho}$ such that$$I_{\mathrm{num}}(M;Q)_{\rho}\leq I_{\mathrm{acc}}(M;Q)_{\rho}.$$ For small values of $N_{\mathrm{tot}}\ll1$, the probability of having more than one photon is of order $N_{\mathrm{tot}}^{2}$. We can hence argue that for $N_{\mathrm{tot}}^{2}\ll1$, the main contribution to $I_{\mathrm{num}}(M;Q)_{\rho}$ comes from POVM elements in (\[eq:photon-number-POVM-elements\]) with $k=0,1$ and that the contribution of those with $k\geq2$ is, in the worst case, of order $N_{\mathrm{tot}}^{2}\log_{2}{n}$. Noticing that the projection of the coherent state $U_{k}|\alpha_{m}\rangle$ in the subspace spanned by the vacuum and the single-photon subspace is $$e^{-\frac{|\alpha|^{2}}{2}}(|0\rangle+\alpha U_{k}|m\rangle)=e^{-\frac
{|\alpha|^{2}}{2}}(|0\rangle+\alpha\sum_{m^{\prime}} \widetilde{U}_{k}^{\left( m^{\prime},m\right) } |m^{\prime}\rangle)\,,$$ where $|m^{\prime}\rangle$ is the state of a single photon on the $m^{\prime}$th mode, a straightforward calculation leads to the following expression for the lower bound $I_{\mathrm{num}}(M;Q)_{\rho}$: $$I_{\mathrm{num}}(M;Q)_{\rho}=N_{\mathrm{tot}}\left[ \log_{2}{n}-\min_{\mathcal{M}_{E}^{(1)}}\sum_{y}\frac{\mu_{y}^{(1)}}{n
\left\vert \mathcal{K}\right\vert }\sum_{k}H(q_{yk})\right] +O(N_{\mathrm{tot}}^{2}\log_{2}{n})\,, \label{Inum}$$ where the optimization is over the POVM $\mathcal{M}_{E}^{(1)}$ defined on the single-photon subspace, with elements $\{\mu_{y}^{(1)}|\phi_{y}^{(1)}\rangle\langle\phi_{y}^{(1)}|\}_{i}$, and $q_{yk}^{m}=|\langle\phi_{y}^{(1)}|U_{k}|m\rangle|^{2}$.
The expression in square brackets in (\[Inum\]) is formally the same as that in (\[acc\]). We can hence bound $I_{\mathrm{num}}(M;Q)_{\rho}$ using the results of Ref. [@FHS11]. It follows that there exist choices of $\left\vert \mathcal{K}\right\vert $ $n$-mode passive linear-optical unitaries with $$\log_{2}{|\mathcal{K}|} = 4\log_{2}{(\varepsilon^{-1})} + O(\log_2\log_{2}{(\varepsilon^{-1})})\,,$$ such that$$I_{\mathrm{num}}(M;Q)_{\rho} \leq \varepsilon N_{\mathrm{tot}} \log_{2}{n} + O(N_{\mathrm{tot}}^{2}\log_{2}{n})\,,$$ with $\varepsilon$ arbitrarily small if $n$ is large enough.
Clearly, the security condition $r_{1}\ll1$ can be satisfied only if $I_{\mathrm{num}}(M;Q)_{\rho}\ll I_{\mathrm{acc}}(M;QK)_{\rho}$. A necessary condition for $r_{1}\ll1$ to hold is $$\varepsilon + O(N_{\mathrm{tot}}) \ll 1 \,,
\label{sec-cond}$$ which can be fulfilled in the case of weak coherent states, where $N_{\mathrm{tot}} \ll 1$. In this regime, the key-efficiency condition $r_{2}<1$ can be satisfied only if $$4\log_{2}{(\varepsilon^{-1} )} < N_{\mathrm{tot}} \log_{2}{n} \,.$$ This implies that the value of $N_{\mathrm{tot}}$ has to be in the range $$\label{effweak}
1\gg N_{\mathrm{tot}} > \frac{4\log_{2}{(\varepsilon^{-1})}}{\log_{2} {n}} \, .$$
In conclusion, this weak coherent state PPM protocol is analogous to the single-photon one in the presence of linear loss. In principle the condition in (\[effweak\]) can always be fulfilled for $n$ large enough, yet the minimum value of $n$ increases exponentially with decreasing key-efficiency ratio $r_2$ and with decreasing $N_\mathrm{tot}$.
Conclusion {#end}
==========
In this paper, we formally defined the locking capacity of a quantum channel in order to establish a framework for understanding the locking effect in the presence of noise. We can distinguish between a weak locking capacity and a strong one, the difference being whether the adversary has access to the environment of the channel or to its input. We related these locking capacities to other well known capacities from quantum Shannon theory such as the quantum, private, and classical capacity. The existence of the FHS locking protocol [@FHS11] establishes that both the weak and strong capacity locking capacities are not smaller than the quantum capacity, while the weak locking capacity is not smaller than the private capacity because a private communication protocol always satisfies the demands of a weak locking protocol. Furthermore, the classical capacity is a trivial upper bound on both locking capacities. We also proved that the strong locking capacity is equal to zero whenever a locking protocol has a classical simulation and that both locking capacities are equal to zero for an entanglement-breaking channel. This latter result demonstrates that a channel should have some ability to preserve entanglement in order for non-zero locking rates to be achievable. Moreover, we found a class of channels for which the weak locking capacity is equal to both the private capacity and the quantum capacity.
As an important application, we considered the case of the pure-loss bosonic channel and the locking capacities for channels restricted to coherent-state encodings. We note that a particular example of such a protocol is the $\alpha\eta$ protocol (also known as $Y00$) [@Y00]. We found limitations of the locking capacity for these coherent-state schemes: the strong locking capacity of any channel is not larger than $\log_{2}\left( e\right) $ locked bits per channel use while the weak locking capacity of the pure-loss bosonic channel is not larger than the sum of its private capacity and $\log_{2}\left(
e\right) $ locked bits per channel use. If the scheme exploits passive mode transformations and the receiver uses heterodyne detection, the restrictions are as severe as they can be: the strong locking capacity is equal to zero because there is a classical simulation of such a protocol.
As a final contribution, we discussed locking schemes that exploit weak coherent states and that might be physically implementable. They are similar to the single-photon quantum enigma machine (QEM) of Ref. [@QEM], with the exception that information is encoded by pulse position modulation (PPM) of a coherent state of a given amplitude $\alpha$, with $|\alpha|^{2}\ll1$, over $n$ modes. The necessary conditions for security and key efficiency of this scheme are qualitatively equivalent to that of the single-photon QEM.
The realization of a proof of principle demonstration of a quantum enigma machine is a tremendous experimental challenge. The main difficulty to overcome concerns the scaling of the physical resources required for key efficient encryption. Notice that for single-photon PPM encoding, $n$ optical modes are needed to encode $\log_{2}{n}$ bits, while the required secret key has length of the order of $\log_{2}{\log_{2}{n}}$. As a consequence, the number of modes increases very quickly if one requires small values of the key efficiency ratio $r_{2}$, as defined in (\[r2\]). Although keeping the same scaling law, the resources required for a key efficient single-photon QEM become even more demanding when one introduces loss in the single-photon scheme and when one moves from the single-photon PPM to the weak coherent-state PPM. On the other hand, in the case of a weak coherent-state QEM, one can trade off the increase in the number of modes (and/or the reduction in the key efficiency level) with the fact that coherent states can be prepared deterministically and are much easier to handle than single-photon states. However, it seems that one has to go beyond PPM encoding to overcome the key efficiency limitations.
There are many open questions to consider going forward from here. Perhaps the most pressing question is to determine a formula that serves as a good lower bound on the locking capacities (that is, one would need to demonstrate locking protocols with nontrivial achievable rates according to the requirements stated in Definitions \[def:weak-lock-protocol\], \[def:strong-lock-protocol\], and \[def:achievable-rate-locking\]). One might suspect that the formulas given in Theorems \[thm:WLC\] and \[thm:up-bnd-strong-locking-cap\] are in fact achievable, but it is not clear to us if this is true. Furthermore, it is important to determine if there is an example of channel (perhaps many?) for which its weak locking capacity is strictly larger than its private classical capacity, and similarly, if there is a channel for which its strong locking capacity is strictly larger than its quantum capacity. We also suspect that the locking capacity is non-additive, as is the case for other capacities in quantum Shannon theory [@science2008smith; @LWZG09; @SS09]. If it is the case that the 50% quantum erasure channel has a weak locking capacity equal to zero, then it immediately follows from the results of Ref. [@SS09] and our operational bounds in (\[eq:weak-lock-bounds\]) and (\[eq:strong-lock-bounds\]) that both the weak and strong locking capacities are non-additive.
Another intriguing question is the relationship between the strong locking and quantum identification capacities [@Win13] of a quantum channel. Both seem to involve a weak form of coherent data transmission from a sender to receiver. FHS even established an explicit connection between locking using unitary encodings and quantum identification over a channel built out of the inverses of those unitaries [@FHS11]. It is tempting to speculate that the single-letter formula for the amortized quantum indentification capacity found in Ref. [@HW12] could thereby be recruited as a tool to study the locking capacity.
**Acknowledgements**. We are grateful to Omar Fawzi, Graeme Smith, and Andreas Winter for helpful discussions. This research was supported by the DARPA Quiness Program through US Army Research Office award W31P4Q-12-1-0019. PH was supported by the Canada Research Chairs program, CIFAR, NSERC and ONR through grant N000140811249.
Private capacity of degraded quantum wiretap channels
=====================================================
This appendix contains a proof that the private capacity of a degraded quantum wiretap channel when restricted to product-state encodings is single-letter. Also, we show by an appeal to Hastings’ counterexample to the additivity conjecture [@H09] that there exists two quantum wiretap channels that are degraded but nevertheless have non-additive private information. This latter result provides a simple answer to a question that has been open since the introduction of weakly degradable channels [@CG06].
A quantum wiretap channel is defined as a completely positive trace-preserving map $\mathcal{N}_{A\rightarrow BE}$ from an input system $A$ to a legitimate receiver’s system $B$ and an eavesdropper’s system $E$. Such a map has an isometric extension $U_{A\rightarrow BEF}$ with the property that$$\mathcal{N}_{A\rightarrow BE}\left( \rho\right) =\text{Tr}_{F}\left\{
U_{A\rightarrow BEF}\rho U_{A\rightarrow BEF}^{\dag}\right\} .$$ The private capacity of a quantum wiretap channel is given by [@ieee2005dev; @1050633]$$\lim_{n\rightarrow\infty}\frac{1}{n}P\left( \mathcal{N}_{A\rightarrow
BE}^{\otimes n}\right) ,$$ where $P\left( \mathcal{N}_{A\rightarrow BE}\right) $ is the private information, defined as$$P\left( \mathcal{N}_{A\rightarrow BE}\right) \equiv\max_{\left\{
p_{X}\left( x\right) ,\rho_{x}\right\} }I\left( X;B\right) _{\rho
}-I\left( X;E\right) _{\rho}, \label{eq:private-info-formula}$$ with the entropies taken with respect to the following classical-quantum state:$$\rho_{XBE}\equiv\sum_{x}p_{X}\left( x\right) \left\vert x\right\rangle
\left\langle x\right\vert _{X}\otimes\mathcal{N}_{A\rightarrow BE}\left(
\rho_{x}\right) . \label{eq:private-code-state}$$ Such a wiretap channel is degraded if there exists a degrading map $\mathcal{D}_{B\rightarrow E}$ such that$$\mathcal{D}_{B\rightarrow E}\circ\mathcal{N}_{A\rightarrow B}=\mathcal{N}_{A\rightarrow E}.$$
The private information formula in (\[eq:private-info-formula\]) for two degraded quantum wiretap channels is generally non-additive. That is, there exists degraded quantum wiretap channels $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ such that$$P\left( \mathcal{N}_{1}\otimes\mathcal{N}_{2}\right) >P\left(
\mathcal{N}_{1}\right) +P\left( \mathcal{N}_{2}\right) .$$
This result follows by exploiting the counterexample of Hastings [@H09] for the Holevo information formula. Let $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ be the channels from Hastings’ counterexample, i.e., they satisfy$$\chi\left( \mathcal{M}_{1}\otimes\mathcal{M}_{2}\right) >\chi\left(
\mathcal{M}_{1}\right) +\chi\left( \mathcal{M}_{2}\right) ,
\label{eq:hastings-inequality}$$ where $\chi\left( \mathcal{N}\right) $ is the Holevo information of a channel $\mathcal{N}$. We then construct our quantum wiretap channels $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ as $\mathcal{N}_{1}\left(
\rho\right) =\mathcal{M}_{1}\left( \rho\right) \otimes\sigma_{E}$ and $\mathcal{N}_{2}\left( \rho\right) =\mathcal{M}_{2}\left( \rho\right)
\otimes\sigma_{E}$. Both channels are obviously degraded wiretap channels because the channel to the environment simply prepares a constant state $\sigma_{E}$. Also, there is no dependence of the environment’s output on the input state, so that the private informations of these channels reduce to Holevo informations:$$\begin{aligned}
P\left( \mathcal{N}_{1}\otimes\mathcal{N}_{2}\right) & =\chi\left(
\mathcal{M}_{1}\otimes\mathcal{M}_{2}\right) ,\\
P\left( \mathcal{N}_{1}\right) & =\chi\left( \mathcal{M}_{1}\right) ,\\
P\left( \mathcal{N}_{2}\right) & =\chi\left( \mathcal{M}_{2}\right) .\end{aligned}$$ Thus, the inequality in the statement of the theorem follows from (\[eq:hastings-inequality\]).
\[thm:private-info-additive\]The private information of a degraded quantum wiretap channel is additive when restricted to product state encodings.
First, consider that we can always restrict the optimization in the private information formula to be taken over pure input states whenever the quantum wiretap channel $\mathcal{N}_{A\rightarrow BE}$ is degraded. Indeed, consider the extension state$$\rho_{XYBE}\equiv\sum_{x,y}p_{X}\left( x\right) p_{Y|X}\left( y|x\right)
\left\vert x\right\rangle \left\langle x\right\vert _{X}\otimes\left\vert
y\right\rangle \left\langle y\right\vert _{Y}\otimes\mathcal{N}_{A\rightarrow
BE}\left( \psi_{x,y}\right) , \label{eq:ext-state}$$ where we are using a spectral decomposition for each state $\rho_{x}$:$$\rho_{x}=\sum_{y}p_{Y|X}\left( y|x\right) \psi_{x,y}.$$ Thus, the state in (\[eq:private-code-state\]) is a reduction of the state in (\[eq:ext-state\]). Now consider that$$\begin{aligned}
I\left( X;B\right) _{\rho}-I\left( X;E\right) _{\rho} & =I\left(
XY;B\right) -I\left( XY;E\right) -\left[ I\left( Y;B|X\right) -I\left(
Y;E|X\right) \right] \\
& \leq I\left( XY;B\right) -I\left( XY;E\right) \\
& \leq P\left( \mathcal{N}_{A\rightarrow BE}\right) .\end{aligned}$$ The first equality is from the chain rule for mutual information. The first inequality follows by exploiting the degrading condition and from the fact that $X$ is classical. The final inequality follows by considering $XY$ as a joint classical system, so that the private information of the channel can only be larger than $I\left( XY;B\right) -I\left( XY;E\right) $.
Now consider an isometric extension $U_{A\rightarrow BEF}$ of a quantum wiretap channel $\mathcal{N}_{A\rightarrow BE}$. By using the fact that the private information is optimized for pure state ensembles, we can always rewrite it as$$\begin{aligned}
I\left( X;B\right) -I\left( X;E\right) & =H\left( B\right) -H\left(
E\right) -H\left( B|X\right) +H\left( E|X\right) \nonumber\\
& =H\left( B\right) -H\left( E\right) -H\left( B|X\right) +H\left(
BF|X\right) \nonumber\\
& =H\left( B\right) -H\left( E\right) +H\left( F|BX\right) ,
\label{eq:useful-identity-private-info-wiretap}$$ where in the second line we used the fact that $H\left( E|X\right) =H\left(
BF|X\right) $ for pure-state ensembles.
Now we show the additivity property for product-state ensembles. Consider the following state on which we evaluate information quantities:$$\sigma_{XB_{1}E_{1}F_{1}B_{2}E_{2}F_{2}}\equiv\sum_{x}p_{X}\left( x\right)
\left\vert x\right\rangle \left\langle x\right\vert _{X}\otimes U_{A_{1}\rightarrow B_{1}E_{1}F_{1}}\left( \phi_{x}\right) \otimes U_{A_{2}\rightarrow B_{2}E_{2}F_{2}}\left( \psi_{x}\right) ,$$ where we are restricting the signaling states to be product and without loss of generality we can take them to be pure as shown above. Consider that$$\begin{aligned}
& I\left( X;B_{1}B_{2}\right) _{\sigma}-I\left( X;E_{1}E_{2}\right)
_{\sigma}\\
& =H\left( B_{1}B_{2}\right) _{\sigma}-H\left( E_{1}E_{2}\right)
_{\sigma}+H\left( F_{1}F_{2}|B_{1}B_{2}X\right) \\
& =H\left( B_{1}\right) _{\sigma}+H\left( B_{2}\right) _{\sigma}-H\left(
E_{1}\right) _{\sigma}-H\left( E_{2}\right) _{\sigma}-\left[ I\left(
B_{1};B_{2}\right) _{\sigma}-I\left( E_{1};E_{2}\right) _{\sigma}\right]
+H\left( F_{1}F_{2}|B_{1}B_{2}X\right) \\
& \leq H\left( B_{1}\right) _{\sigma}+H\left( B_{2}\right) _{\sigma
}-H\left( E_{1}\right) _{\sigma}-H\left( E_{2}\right) _{\sigma}+H\left(
F_{1}|B_{1}X\right) +H\left( F_{2}|B_{2}X\right) \\
& =\left[ I\left( X;B_{1}\right) -I\left( X;E_{1}\right) \right]
+\left[ I\left( X;B_{2}\right) -I\left( X;E_{2}\right) \right] .\end{aligned}$$ The first equality follows from the identity in (\[eq:useful-identity-private-info-wiretap\]). The second equality follows from entropy identities. The first inequality follows from the degraded wiretap channel assumption, so that $I\left( B_{1};B_{2}\right) _{\sigma
}-I\left( E_{1};E_{2}\right) _{\sigma}\geq0$ and by applying strong subadditivity of entropy [@LR73] three times to get that $H\left(
F_{1}F_{2}|B_{1}B_{2}X\right) \leq H\left( F_{1}|B_{1}X\right) +H\left(
F_{2}|B_{2}X\right) $. The last equality follows from the identity in (\[eq:useful-identity-private-info-wiretap\]) and the fact that we are restricting to product-state signaling ensembles.
[^1]: Quantum Information Processing Group, Raytheon BBN Technologies, Cambridge, Massachusetts 02138, USA
[^2]: Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, California 94305-4060, USA
[^3]: Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
[^4]: Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
[^5]: National Institute of Information and Communications Technology, 4-2-1 Nukuikita, Koganei, Tokyo 184-8795, Japan
[^6]: Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA
|
---
abstract: 'We have numerically studied the statics and dynamics of a model three-dimensional vortex lattice at low magnetic fields. For the statics we use a frustrated $3D$ $XY$ model on a stacked triangular lattice. We model the dynamics as a coupled network of overdamped resistively-shunted Josephson junctions with Langevin noise. At low fields, there is a weakly first-order phase transition, at which the vortex lattice melts into a line liquid. Phase coherence parallel to the field persists until a sharp crossover, conceivably a phase transition, near $T_{\ell} > T_m$ which develops at the same temperature as an [*infinite*]{} vortex tangle. The calculated flux flow resistivity in various geometries near $T=T_{\ell}$ closely resembles experiment. The local density of field induced vortices increases sharply near $T_\ell$, corresponding to the experimentally observed magnetization jump. We discuss the nature of a possible transition or crossover at $T_\ell$(B) which is distinct from flux lattice melting.'
address: 'Department of Physics, Ohio State University, Columbus, OH 43210'
author:
- 'Seungoh Ryu[@presentaddress] and David Stroud'
title: Nature of the Low Field Transition in the Mixed State of High Temperature Superconductors
---
Introduction
============
Ever since their discovery, the behavior of high-T$_c$ materials in a magnetic field has seemed mysterious[@malozemoff89]. Unlike the conventional low-T$_c$ type-II materials, high-T$_c$ superconductors (HTSC’s) show a broad region in the magnetic-field/temperature (H-T) plane where the Abrikosov lattice has apparently melted into a [*liquid*]{} state[@nelson88].
Considerable recent evidence now suggests that flux lattice (FL) melting is a [*first-order*]{} phase transition. On the experimental side, a local magnetization jump has been measured by network of Hall microprobes[@zeldov95] on and has been associated with the melting transition. The transition thus observed seems to lie quite near the melting curve as determined from low angle neutron diffraction[@cubitt93] and $\mu$SR experiments[@lee93]. More recently, Schilling [*et al*]{}[@schilling96; @schilling97] have directly observed the latent heat of the transition in along a line $T_m(H)$ in the $H-T$ phase diagram which agrees well with mechanical and transport measurements[@safar92; @farrell91; @farrell95]. Numerical evidence for a first order melting transition has been obtained from simulations based on a frustrated $XY$ model[@hetzel92; @dominguez95], and from a lowest Landau level model which is expected to be most accurate at high magnetic field[@sasik95]. First order melting has also been observed numerically in a system of unbreakable flux lines described by a Lawrence-Doniach model[@ryu96]. All these simulations are based on a large density of flux lines ($\sim {\cal O}(1-10) {\rm tesla}$).
An anomalous feature of the local Hall probe measurements is that the apparent first order transition line seems to terminate at a critical point above which the latent heat vanishes[@zeldov95]. Since on symmetry grounds a first order “melting” line cannot terminate in a critical point[@landau], this critical point may suggest that the first order melting line is instead intersecting another phase transition line related to the disorder. A related issue is the entropy released per vortex per layer across the transition line. This entropy increases very rapidly as the field decreases. Such behavior is difficult to accounted for within a model based only on the field induced vortices.
In the presence of disorder, the lattice becomes unstable at high fields against proliferation of quenched-in topological defects[@ryu96_2; @gingras96], possibly through a first order phase transition across a horizontal (constant H) line in the H-T plane. This line then may meet the temperature-driven melting line, causing it to terminate. Somewhere along this line, the melting transition may be converted into the universality class of the continuous vortex glass transition[@fisher89], characterized by divergent correlation lengths and times.
Another unresolved issue regarding the phase diagram is the possibility of reentrant melting at low fields. Reentrant flux lattice melting is expected because of screening of the widely-separated vortex lines at low fields[@nelson88; @ryu92]. It has been recently reported in single-crystal ${\rm NbSe_2}$ sample[@sabu], based on tracking of the so-called “peak effect”[@kwok94; @ling95]. Such reentrance behavior has been observed only at the limited field range by Ling [*et al*]{}[@ling97]. On the other hand, the melting curve tracked by the micro-Hall probe[@zeldov95] seems to monotonically approach the zero-field superconducting transition at $T_c(H=0)$ even for fields as low as a few Gauss.
FL melting can also be probed by transport measurement. But since such measurements are non-equilibrium, they offer only an indirect means of studying [*equilibrium*]{} FL melting. In real materials with disorder, the interpretation of transport measurement is further complicated by the many competing energy scales. In single-crystal , the in-plane resistivity exhibits a discontinuous jump and hysteresis which have been identified with a first-order melting transition[@safar92]. Nonetheless, the peak effect in the critical current occurs at slightly lower temperatures than the resistivity jump, leading some workers to postulate that there is a “premelting” phenomenon[@kwok94] in this material, in addition to melting. In , simultaneous transport and local magnetization measurements[@fuchs96] show that the jump in local magnetization $M$ coincides (at high fields) with a jump in the resistivity $\rho_{ab}$ from zero to a finite value, or (in low fields) the continuous development of a finite $\rho_{ab}$. In addition, at high fields, the jumps in $\rho$ and $M$ are accompanied by hysteresis. Together, these phenomena strongly suggest first order flux lattice melting at high fields. At low fields, the experiments are more ambiguous.
FL melting has been widely studied numerically. The possibility of [*two stage*]{} melting was first suggested by Li and Teitel[@li93_1; @li93_2] for a model with infinite penetration depth $\lambda$, and later for a system with finite $\lambda$[@chen95; @chen97]. The calculations of Li and Teitel are based on the so-called frustrated XY model with fairly low flux per plaquette of $f = 1/25$ (in units of the flux quantum $\Phi_0=hc/2e$) on a simple cubic lattice. They find that the three-dimensional flux line lattice (3D FLL) melts first into a “line liquid” characterized by disentangled flux lines, which become entangled at a second, higher-$T$ phase transition. Current-voltage (IV) measurements in the so-called “flux transformer” geometry[@safar94; @lopez94; @keener96; @keener97] provide some support for this picture. Specifically, they suggest that FL melting is signaled by the onset of finite in-plane resistance, while in an applied current, phase coherence is lost in the $c$ direction only at a distinctly higher temperature. On the other hand, simulations of dense (f=1/6) flux lines on a stacked triangular grid favor a single transition[@hetzel92; @dominguez95]. Dynamical calculations[@jagla96] on a triangular lattice at $f = 1/6$ suggest that if there are two separate transitions, they arise from pins, either intrinsic to the discrete cubic grid, or put in by hand. Yet more recent studies based on a London vortex loop model on a simple cubic lattice show that superconducting order disappears apparently in two steps, the sequence of which depends on the lattice anisotropy[@nguyen96], although it is argued to be a finite size effect by the same authors[@nguyen97].
In this paper, we attempt to resolve some of these issues by considering the frustrated $XY$ model over a [*wide range of flux densities*]{}, using both static and dynamic simulations but with no quenched disorder. By examining this model on a stacked triangular lattice, we minimize the unphysical periodic pinning due to the lattice. By working at relatively low densities, we focus on the regime, now being probed experimentally, where the XY phase fluctuations (vortex loops) are as important as those of [*field induced*]{} vortex lines[@tesanovic95]. Our main conclusion is that there are, in fact signatures of two separate transitions at low fields, which are not artifacts of pinning by the discrete grid. The transition at lower temperature is unambiguously associated with vortex lattice melting. The second transition may be a sharp crossover rather than a true phase transition. Nevertheless, it is responsible for several experimental features (such as sharp increases in local magnetization and in resistance) which are often identified as evidence for a first order melting transition.
The remainder of this paper is organized as follows. In Section \[modelsec\], we describe our model and its numerical solution. The following sections present our numerical results, which are followed by a discussion and then summarized in a concluding section.
Model {#modelsec}
=====
Hamiltonian and Thermodynamics
------------------------------
We study the standard frustrated $XY$ model described by the Hamiltonian $$\label{modeleq}
{\cal H} = -J\sum_{\langle ij \rangle}\cos(\theta_i-\theta_j -
A_{ij}),$$ where $A_{ij} = \frac{2\pi}{\Phi_0}\int_{i}^j{\bf A}\cdot{\bf dl}$, ${\bf A}$ is the vector potential associated with a uniform magnetic field ${\bf B} = B\hat{z}$ applied parallel to $\hat{z}$, $\Phi_0 = hc/2e$ is the flux quantum, $\theta_i$ is the phase of the order parameter on site $i$, and the sum runs over nearest neighbor pairs. We use a stacked triangular grid with ${\bf B}\| z$, the direction perpendicular to the triangular network, with periodic boundary conditions (PBC) in all directions except where stated otherwise.
To allow a wider range of frustrations compatible with the boundary conditions, we use a variant of the Landau gauge[@haldane]. Note that there are four bonds per grain: three in the xy-plane and one along $\hat{z}$. We label these by their unit vectors ${\hat{\alpha} = \hat{x},
\hat{y_1},
\hat{y_2},
\hat{z}}$ where $\hat{y_1} =(1/2)\hat{x} + (\sqrt{3}/2) \hat{y}$ and $\hat{y_2} = -(\sqrt{3}/2)\hat{x} + (1/2) \hat{y}$. The phase factors $A_{ij}$ connecting a grain located at (x,y,z) to its four nearest neighbors are given by $0$ along $\hat{x}$ or $\hat{z}$, $2 \pi f (2 x + 1/2)$ along $\hat{y_1}$, and $2\pi f (2 x - 1/2)$ along $\hat{y_2}$. There are exceptions to this form for grains lying on the boundaries: All grains lying on the $x = {\rm L}_x$ boundary plane have $A_{ij} = - 2\pi \cdot 2 {\rm L}_x y$ for bonds in the $x$ direction. Bonds at $x={\rm L}_x$ boundary such that ${\rm mod}(j,2) = 1$ have $A_{ij} = 2 \pi f [2 x + 1/2 - 2 {\rm L}_x ( y +1 ) ] $ in the $y_1$ direction. For bonds on the $x=0$ boundary with ${\rm mod}(j,2) = 0$, $A_{ij} = 2 \pi f [ 2 x - 1/2 + 2 {\rm L}_x (y+1) ] $ for bonds in the $y_2$ direction. In contrast to the usual Landau gauge (which is compatible with frustrations only in integer multiples of $1/(2N_x)$), this generalized gauge is compatible with any $f$ which is an integer multiple of $1 / (2N_xN_y) $ under periodic boundary conditions.
We have considered networks of sizes ${\cal N} = N_x \times N_y \times N_z$. For $f=1/24$, we have studied $N_z = 12, 24, 48$ and $N_x = N_y = 24$, and for other values of $f$ (1/2592, 1/1648, 1/81, and 1/6) we have considered $N_x=N_y=N_z=18$. In two dimensions, the vortices lie on the vertices of a honeycomb grid of unit length of $(1/\sqrt{3})a_B$ which is dual to the triangular grid of unit length $a_B$. Assuming that vortices form perfect triangular lattice on this grid, and equating the area per vortex to $(\sqrt{3}/4) a_B^2 /f,$ we obtain the following necessary condition for a triangular vortex lattice to form without [*geometric frustration*]{} of the FL:$
2/f = (n_1^2 / 3 + n_2^2 / 4)$ with integers $n_1, n_2$. The values of $1/f$ satisfying this condition are then $2,6,8,14,18,24,32,38,42 \ldots, 648,
\ldots$. For $f=1/162,$ two distinct pairs \[$(n_1,n_2)$ = (0,36), and (27,18)\] satisfy the condition. $f=1/648$ has the lowest possible nominal vortex density of one per $18 \times
18$ system compatible with our chosen gauge, and allows either of the pairs $(n_1,n_2)$ = (0,72) and (54,36). $f=1/2592 ( = 1/4 \times {1 \over 2\times18\times18}),$ represents less than a single vortex line, and the system is [*gauge-frustrated*]{}.
In practice, for $f \leq 1/81$, there are too few field-induced flux lines to study FL melting. Nonetheless, the dilute regime is still of interest, since in these cases, the flux lines behave independently and the thermodynamics is dominated by the zero-field phase degrees of freedom[@tesanovic95]. Except for f = 1/2592, we study only gauge-unfrustrated values allowing only an integer number of vortices in the simulation box.
We calculate the thermodynamics using a standard Monte Carlo (MC) algorithm, with up to $10^6$ MC steps at each temperature $T$. To ensure equilibration in the ground state for all values of $f$, we performed simulated annealing runs for the two-dimensional (2D) version at each $f$ with the same lateral dimensions. We then form the ground state of the 3D system by stacking the 2D ground states thus found uniformly along the $z-$direction. This enables us to find the ground state configuration of a perfect triangular lattice for low values of $1/24 \le f \le 1/18$ within a reasonable time. Starting from these 3D ground states, we warm up the system in steps of $\triangle T / J = 0.05$ or $0.1$, allowing at least $4-5 \times 10^4$ Monte Carlo sweeps for each $T$. The final configuration for each $T$ is then saved to be used as a starting configuration in some of the longer calculations as well in the dynamic simulations.
From these calculations, we extract a range of thermodynamic quantities. One of these is the specific heat $C_V = (\langle H^2 \rangle - \langle H\rangle^2)/(k_BT)$ at temperature $T$. We also calculate the local vorticity vector field $n_\alpha(p)$ defined for each Cartesian direction $\alpha$ and each point $p$ of the stacked honeycomb dual lattice. At each instant during the simulation, $n_\alpha(p)$ is determined from $$\sum^\alpha_p \rm{mod} [ \phi_i - \phi_j - A_{ij}, 2\pi ] =
2 \pi [ n_\alpha(p) - f_p ].
\label{vortdefeq}$$ Here the summation runs along the bonds belonging to the plaquette labeled $p, \alpha$ (a triangle in the $xy$ plane, a square in planes parallel to the $z$ axis); and $f_p
\equiv \sum^\alpha_p A_{ij}/(2\pi).$ From $n_{\alpha}$, one can also compute the [*structure factor*]{}, $S_{\alpha\beta}({\bf k}) = \langle n_{\alpha}({\bf k})n_{\beta}(-{\bf
k})\rangle$, where $n_{\alpha}({\bf k})$ is the Fourier transform of the local vorticity vector n$_{\alpha}({\bf r})$. We also calculate the principal components $\gamma_{xx}$ and $\gamma_{zz}$ of the helicity modulus tensor[@fisher73], in the directions perpendicular and parallel to the applied field. To within a constant factor, ${\bf \gamma}$ represents the phase rigidity or the superfluid density tensor of the system; its derivation in terms of equilibrium thermodynamic averages has been given elsewhere[@shih84].
Dynamics
--------
To treat the dynamics, we model each link between grains as an overdamped resistively shunted Josephson junction (RSJ) with critical current $I_c = 2eJ/\hbar$, shunt resistance $R$, and Langevin white noise to simulate temperature effects. The effective $IV$ characteristics are then obtained by numerically integrating the coupled $RSJ$ equations, as described elsewhere[@chung89], using a time constant typically of $0.1 t_0$ and obtaining voltages by averaging over an interval of $\sim 600 t_0 - 2000 t_0$. Since direct solution of these equations would involve inverting and storing an ${\cal N}\times {\cal N}$ matrix, where ${\cal N}$ is the total number of grains$ \sim {\cal O}(5000),$ we instead solve them iteratively[@recipe], incurring a speed penalty of a factor of $\ln {\cal N}.$ We verified that our solutions converge by comparing them with those from direct inversion for time steps of $0.01, 0.04$ and $0.1$ on an $8 \times 8 \times 8$ system.
The most obvious approach to the dynamics of this model would be to use free boundary conditions, injecting current into one face of the lattice and extracting it from the opposite, with periodic transverse boundary conditions[@lee93a]. But this has the following disadvantage. Once the flux lattice is depinned from its underlying periodic pinning potential, it will drift along [*as a whole*]{} under the influence of the Lorentz force provided by the driving current. Since this occurs equally in the solid and the liquid state, such a geometry may not distinguish clearly between flux lattice and flux liquid (in the absence of spatially inhomogeneous pinning centers). This problem may be even more conspicuous in our stacked triangular geometry, since the critical current $I_{dp}$ for depinning a single vortex pancake from the underlying triangular grid at zero temperature (T = 0) is smaller ($I_{dp} \approx 0.037I_c$) than in a square grid ($I_{dp} \approx 0.1I_c$)[@lobb].
We therefore adopt a different geometry for injecting and extracting current, as shown in Fig. \[geomfig\]. Fig. \[geomfig\] (a) corresponds to injecting current $I/I_c$ into each grain in the $yz$ plane at x = 0, and extracting it from each grain at $x = N_x/2$ (with periodic boundary conditions in all three directions). In this geometry, the Lorentz forces acting on the vortices in the two halves of the volume are oppositely directed, as indicated by the arrows. Thus, in this geometry, we are effectively probing the [*shear modulus*]{} $\mu$ of the vortex lattice, on a length scale ${\rm L}_x/2$. Similar geometries have been previously discussed in the context of possible experiments[@pastoriza95; @nelson91]. In Fig. \[geomfig\] (b), we show a geometry which is designed to probe the [*c-axis resistivity*]{}, $\rho_c$. In this case, we inject a current $I$ into each grain on the $xy$ plane at $z = 0$ and extract it from each grain at $z = N_z/2$. There is on average no Lorentz force on the vortex lines.
Zero-Field XY Model: f = 0
==========================
Thermodynamics
--------------
The 3D unfrustrated XY model on a [*cubic grid*]{} has been extensively studied[@gottlob93]. Near the phase transition, the specific heat $C_V \sim |T - T_{XY}|^{-\alpha}$ with $\alpha \sim 0$ and the correlation length $\xi \sim |T - T_{XY}|^{-\nu}$ with $\nu \sim 0.66-0.67$. For a cubic lattice, $T_{XY} \sim 2.203 J$. In a stacked triangular grid, where each grain has more nearest neighbors, the transition is shifted to a higher temperature. Numerically, we find that $T_{XY} \sim 3.04 J$.
The $XY$ phase transition is best characterized by the [*helicity modulus tensor*]{} $\gamma_{\alpha\beta}$, which measures the phase rigidity of the system[@fisher73]. In stacked triangular lattice, this tensor is diagonal with elements $$\begin{aligned}
\gamma_{\alpha\alpha} &=& {1\over V} {\delta^2 {\cal F} \over
\delta A_{\alpha}^*\delta A_{\alpha}^* }
\nonumber \\ &=& {J \over V} \Big< \sum_{ij} \cos(\Theta_{ij})
\hat{n}_{ij}\cdot
\hat{n}_\alpha \Big> - {J \over k_B T} \times \nonumber \\
& & \Big( \Big< [ \sum_{ij}\sin(\Theta_{ij})\hat{n}_{ij}\cdot
\hat{n}_\alpha ]^2 \Big> - \Big<
\sum_{ij}\sin(\Theta_{ij})\hat{n}_{ij}\cdot
\hat{n}_\alpha \Big>^2 \Big).
\label{heldefeq}\end{aligned}$$ Here $A^*_{\alpha}$ is a fictitious uniform vector potential in the $\alpha$ direction, $V$ is the volume, $\Theta_{ij}=\theta_i- \theta_j - A_{ij}$ is the gauge-invariant phase difference, $\hat{n}_{ij}$ and $\hat{n}_{\alpha}$ are unit vectors along the $ij^{th}$ bond and in the $\alpha$ direction.
It is useful to distinguish two contributions to $\Theta_{ij}$: one due to spin waves, and one due to vortices[@nelson81]. The former is dominant when the $\sin\Theta_{ij} \approx \Theta_{ij}$, while the other is nonzero when the vorticities $n_{\alpha}(p) \neq 0$. Thus we write $\gamma_{\alpha\alpha}= \gamma_{\alpha\alpha}^{SW}+\gamma_{\alpha\alpha}^V$, where the two terms on the right hand side are respectively the spin wave and vortex contributions to $\gamma_{\alpha\alpha}$. The spin-wave degrees should predominate at low temperatures, while the vortex degrees of freedom are the dominant excitations near the phase transition[@onsager; @feynman55; @khoring86; @shenoy90; @williams]. The spin-wave contribution $\gamma^{SW}$ can be estimated within a self-consistent harmonic approximation with the result[@kleinert89] $$\label{sqeq}
\gamma^{SW} \sim J \exp \left[ {- k_BT \over 2 D \gamma^{SW}} \right]$$ where $D=4$ for a stacked triangular lattice.
To characterize the vortex contribution, we introduce the [*net global vorticity vector*]{} by[@chaikin] $$\begin{aligned}
{\cal M}_\alpha &=& \hat{n}_\alpha \cdot \int_\Sigma d\hat{\sigma} \theta_v
\nonumber \\ &=& \int_{\Sigma^+} \theta_v d\sigma - \int_{\Sigma^-}
\theta_v d\sigma\end{aligned}$$ where $\Sigma^+$ and $\Sigma^-$ are the two bounding planes normal to $\hat{n}_\alpha$, with normal vectors parallel or anti-parallel to $\hat{n}_\alpha$. We assume that the singular portion of the phase variable $\theta_v$ has been selected out. ${\cal M}_\alpha$ is sensitive to existence of unbound vortex lines [*perpendicular*]{} to $\hat{n}_\alpha$. This is illustrated in the left panel of Fig. \[mdeffig\] for the case of a single infinite vortex line piercing the sample normal to the $\alpha$ direction. In this case, the phase integrals on the planes $\Sigma^+$ and $\Sigma^-$ give nearly equal but opposite values. Thus ${\cal M}_{\alpha}$ has large fluctuations, leading to a reduction in the value of $\gamma_{\alpha\alpha}$ (see below). Closed vortex loops, such as shown in the right panel, give a zero contribution to ${\cal M}_\alpha$. In general, for $f = 0$, the thermal average $\Big< {\cal M_{\alpha}} \Big> \sim 0$. For an applied field $\parallel \hat{z}$, $\Big< {\cal M}_\alpha \Big> \sim 0$ for $\alpha = x$ or $y$. It can be shown that $\gamma_{\alpha\alpha}$ and ${\cal M}_{\alpha}$ are related by $$\label{gameq}
\gamma_{\alpha\alpha} \sim \gamma^{SW}_{\alpha\alpha} -
{1\over V} {J^2 \over k_BT} \Big\{ \Big< {\cal M}^2_\alpha
\Big> - \Big< {\cal M}_\alpha \Big>^2 \Big\}.$$ Thus, the vortices make a negative contribution to $\gamma_{\alpha\alpha}$ arising from fluctuations in ${\cal M}$. They may be said to predominate over the spin waves when their fractional contribution to the helicity modulus is of order unity, that is $$\label{sqcriteq}
{J \over k_BT} \Big\{ \Big< {\cal M}^2_\alpha
\Big> - \Big< {\cal M}_\alpha \Big>^2 \Big\} / {\cal N} \sim {\cal O}(1)$$ where ${\cal N}$ is total number of grains.
In Fig. \[f0gammafig\], we show the calculated $\gamma_{zz}$, as well as the value $\gamma_{zz}^{SW}$ determined from the self-consistent harmonic approximation (SCHA), eq. (\[sqeq\]). The other principal components of $\gamma$ behave similarly. $\gamma_{zz}$ and $\gamma_{zz}^{SW}$ begin to differ for temperatures as low as $T \sim 0.3 T_{XY}$, where vortex loops start to be excited. The SCHA predicts a discontinuous jump in $\gamma_{SW}$ from a value of about $0.37$ at $T_{XY}$ to zero. This jump is an artifact of the approximation, which neglects the periodicity of the Hamiltonian in the angle variables and the vortex fluctuations. The inset shows a finite size scaling analysis to locate $T_{XY}$. The helicity modulus $\gamma \sim |T - T_{XY}|^v$ with $v = (d-2)\nu.$ Therefore, the scaled quantity $\gamma L$ for an $L\times L \times L$ system should cross a single point at $T_{XY}$. Based on this criterion, our numerical results give $T_{XY} = 3.04 J \pm 0.02$. We also observe that $\gamma(T)$ approaches zero with $v \sim 2/3$ for $0.02 < |T_{XY} - T| / T_{XY} < 0.1$ and deviates from this outside the range.
Eq. (\[gameq\]) is equivalent to that derived in Fourier space by Chen and Teitel[@chen97] for $\lambda \rightarrow \infty$: $$\label{chengammaz2}
\gamma_{zz} (q\hat{x}) = J \Big[ 1 - {4\pi^2 J \over V T}
{< n_y(q\hat{x}) n_y(-q\hat{x}) >\over q^2} \Big].$$ if we take $q \rightarrow \pi / {\rm L}_x$.
An alternative form based on the vortex loop scaling picture has been obtained by Williams[@williams95] in the limit $\lambda \rightarrow \infty$ where $\gamma_{\alpha\alpha}$ is given in terms of vortex loop diameter distribution. To check the importance of large loops, we show in Fig. \[f0plfig\], our calculated size distribution of [*connected vortex loops*]{} near $T_{XY}$. Two vortex segments are considered [*connected*]{} if they cross within a single unit cell. Such crossings become very extensive at $T_{XY}$, suggesting that the energy barrier for vortex line intersections vanishes near $T_{XY}$ For $T < T_{XY}$, the maximum vortex loop size is finite, while for $T \ge T_{XY},$ there start to occur vortex lines spanning the entire simulation cell. Thus $T_{XY}$ somewhat resembles a bond-percolation transition, although $T_{XY}$ does not correspond to the percolation threshold, but to a point where connected cluster first forms a $D-1$ dimensional manifold if the system is in dimension $D$. Because such infinite clusters occur, it appears that the behavior seen in Fig. \[f0plfig\] is not a consequence of the finite simulation cell, but persists in the thermodynamic limit. A similar picture, but with no long-range interactions among the vortex segments (the so-called polymer limit), has been discussed by Akao[@akao96] and by Kultanov [*et al*]{}[@kultanov95].
Dissipation near $T_{XY}$
-------------------------
We have also calculated the dissipation near $T_{XY}(f=0)$, using the periodic current injection geometry of Fig. \[geomfig\], In this case, since there are no field-induced vortex lines, the calculated dissipation can be unambiguously related to the resistivity. Our results are shown in Fig. \[f0rvstfig\] for several values of the bias current density. \[More details of the method are described in section \[f24rvstsection\].\] To calculate the differential resistance, $dV/dI,$ we carried out two separate runs at $I/I_c = 0.083$ and $0.043$, to obtain $dV/dI = V(0.083) - V(0.043)/0.04$. Fig. \[f0rvstfig\] shows this result as well as $R \equiv V(0.083) / I$ in the inset. These bias currents are low enough to show sharp features at $T_{XY}(f=0)$ while not significantly disrupting that transition. At higher current densities (not shown), there are numerous current-induced vortex loops. These increasingly round out the sharp jump at T$_{XY}(f=0)$ shown in the Figure, which eventually washes away entirely.
Fig. \[f0nvfig\] shows the average number of vortex segments per plaquette as calculated both by Monte Carlo simulations (with no driving current) and by coupled RSJ dynamics (with a finite bias current). Evidently, just at the XY transition, the system becomes filled with thermally generated vortex segments, [*one per grain, or elemental cell*]{}. Below $T_{XY}(f=0)$, while there may exist vortex loops of arbitrary size, the number of these, ${\cal P}(l)$, falls off exponentially for large clusters (cf. Fig. \[f0plfig\]). By contrast, for $T\ge T_{XY},$ ${\cal P}(l)$ diminishes algebraically with $l$. This subtle change in ${\cal P}(l)$ implies that the average size of the connected vortex tangle remains finite for $T < T_{XY}$, but diverges above it[@onsager]. We find that there are numerous [*finite*]{} vortex loops at temperatures as low as 0.5$T_{XY}$. Moreover, at any given temperature, we find that a large bias current further enhances both their number and their size.
A simple argument suggests that the dissipation below $T_{XY}(f=0)$ may be exponentially activated. The effective energy for a vortex loop of radius $r$ oriented normal to a uniform driving current density $j$ is $U(r) \sim 2\pi r \cdot {\rm Min}[ \ln (r) , \ln (\lambda) ]
- c \cdot \pi r^2 j$, where $\lambda$ is the penetration depth. Thus, there is a barrier to loop expansion with a critical radius $r_c \sim \ln \lambda / j$ and height $U_{max} \sim (\ln \lambda )^2 /
j.$ For sufficiently small j, only those vortex loops of size $r > r_c$ will expand and contribute to dissipation. From our simulations, for $T < T_{XY}$, ${\cal P}(l)$ decays exponentially. Therefore, the small dissipation involving the expansion of thermally nucleated vortex rings should have an activated temperature-dependence for $T < T_{XY}$, resulting in highly nonlinear IV characteristics.
Dense limit: f = 1/6
====================
This is the largest value we studied which allows a triangular vortex lattice commensurate with the underlying triangular grid. This value yields a strong first order transition[@hetzel92; @dominguez95], with an entropy of melting $\Delta S$ of about 0.3 k$_B$ per vortex pancake.
In the present MC simulation, the lattice was gradually warmed up from a perfect triangular lattice in temperature steps of $dT = 0.1 J$, with $50,000$ MC sweeps for each $T$. The insets show the in-plane density-density correlation $< n_z({\bf r}_\perp, z) n_z(0,0) >$ for $T/J = 1.175$ and $1.3$, slightly below and slightly above the melting temperature. At the melting temperature $T_m$, our MC histogram for the internal energy distribution agrees with that of [@hetzel92]. We have also calculated both $\gamma_{zz}(T)$ and the Bragg intensity $S_{zz}({\bf G}_{1})$ at the smallest reciprocal lattice vector ${\bf G}_{1}$ for the triangular vortex lattice. The results are shown in Fig. \[f6sqfig\]. To within $\delta T/J \sim 0.025$, both quantities vanish close to $T = T_m(1/6) \sim 1.175 J$, the melting temperature as determined from the double peaks in the energy histogram[@hetzel92]. The apparent melting $T \approx 1.2 J$, slightly higher than inferred from the energy histogram, seems to be due to a superheating effect.
The occurrence of only a single phase transition at $f = 1/6$[@hetzel92; @jagla96] is not surprising: at this field, there is one vortex pancake for every three grains, and hence, only three phase degrees of freedom per pancake. Thus, most potential vortex excitations are already exhausted by the lateral fluctuations of field-induced vortex lines in the liquid phase. This can be seen in Fig.\[f6linesfig\], where we show two typical vortex configurations, one slightly below and the other above the melting transition. Clearly, the transverse line fluctuations quickly dominate the thermodynamics above $T_m$. The strong first order transition at $f=1/6$ can be then understood by the close connection between the lateral line fluctuations and incipient vortex loops. We conclude, with an accuracy of $dT/J = 0.025$, that in the dense limit of $f = 1/6$, superconducting coherence is destroyed in all directions as soon as the lattice melts.
Dilute limit
============
By [*dilute limit*]{}, we mean the regime where the number density of thermally excited vortex line segments $n_c$ equals or exceed the density of field-induced vortex line segments. To make this more quantitative, we first consider a perfect line lattice at $T=0$ with a given $f$. The number of unit vortex line segments [*per grain*]{} will be $1/(3f)$, or $1/(15f)$ per plaquette (since there are five plaquettes per grain). Thus, for $f=1/6$, there are 0.4 vortex segments per plaquette. In Fig. \[f0nvfig\], we show the density n$_c$ of [*thermally excited*]{} vortex segments at $f = 0$, including all three directions. Note that at the XY transition, $n_c \equiv 0.15$. Thus, $f=1/6$ is clearly in the dense regime, while $f < 1/18$ is roughly in the dilute regime.
Fig. \[shfig\] shows the specific heat $C_V$ per grain, as calculated from energy fluctuations for several values of f \[1/162 (4 flux lines), 1/81, 1/24, 1/18 and 1/6 (108 lines)\] in both the dilute and dense regime. At low $T$, all the $C_V$’s approach $k_B/2$ per grain, as expected from the Dulong-Petit law. The overall behavior of the peaks in $C_V$ up to $f = 1/24$ is remarkably similar to that seen in [@overend94]. For $f = 0$, it is known that $C_V$ has a weak divergence ($\propto |t|^{-\alpha}$ with $\alpha \approx 0.0$[@gottlob93]), as expected for the $3D$ $XY$ model. At finite $f$, this peak is rounded as seen experimentally[@overend94; @roulin96]. As discussed below, these broad peaks in $C_V$ generally occur [*well above*]{} the melting transition in this field range. Note that our results for the dense ($f=1/6$) case differ qualitatively from all those at lower $f$. The sharp peak for $f=1/6$ is actually a delta-function singularity, consistent with the finite heat of fusion of a first-order transition known to occur at this density[@hetzel92].
Fig. \[cpeakfig\] shows that the height of the peak for $f \le 1/18$ roughly follows a logarithmic dependence on the magnetic length $L_B$ defined as $L_B = 1 / \sqrt{f},$ the average vortex spacing. Furthermore, as we show in Fig. \[deltatcfig\], the position of the peak at a finite $f$ shifts from $T_{XY}(f=0)$ by an amount $\delta T_c (L_B)$ which closely follows the law $\sim L_B^{-3/2}.$ As we will show in more detail for $f=1/24$ below the phase rigidity along the applied field, as measured by $\gamma_{zz}$, vanishes for all $f$ near the broad maximum in specific heat. At the temperatures \[which we denote $T_\ell(f)$\] where $\gamma_{zz}(f)$ vanishes, we observe that the average number of thermally generated vortex segments [*per grain*]{} normal to the $\hat{z}$ direction closely follows the law $n^c_{xy}(T_\ell) \sim 0.1 \cdot L_B^{0.6 \pm 0.1}$. All this behavior is discussed in more detail below.
f=1/24
======
Statics: Melting and $\gamma_{zz}$
----------------------------------
In the dilute regime, such as $f \leq 1/24$, there are numerous phase degrees of freedom per field-induced vortex pancake. Thus, a double transition, if there is one, might be more plausible here than at $f = 1/6$, one transition being the melting of the field induced flux lattice, the other connected to the XY-degrees of freedom[@tesanovic95].
To check this possibility, we have studied $f = 1/24$ (a field which allows for a commensurate triangular flux lattice of 48 lines) on a stacked triangular grid of $24\times 24 \times 24$ grains. We first did an extensive simulated annealing run on a single layer, verifying that the vortices freeze into a perfect triangular lattice. We then stacked 24 such layers to form a three dimensional ground state. Next, the lattice was gradually warmed up in intervals of $dT/J = 0.1$ or $0.05$, typically with 50,000 MC steps for each temperature. For several $T$ close to a transition, we ran up to $10^6$ MC steps to ensure equilibration. The resulting Bragg intensity $S({\bf G}_{1})$ and helicity modulus component $\gamma_{zz}(T)$ are plotted in Fig. \[sq\_g\_24fig\]. \[The transverse components $\gamma_{xx}(T)$ and $\gamma_{yy}(T)$ fluctuate around zero for most $T > 0$, as expected for a very weakly pinned vortex lattice which is free to slide in the $ab$ plane.\]
The results do indeed suggest the possibility of [*two*]{} phase transitions. The first - the melting of the vortex lattice - occurs near $T = 1.5 J \equiv T_m$, where the Bragg intensity drops sharply. At higher temperatures, there is a broad dip in the normalized Bragg intensity which reaches a plateau at around $T/J \sim 2.1.$ The possible upper transition, near $T = 2.0J \equiv T_{\ell}$, is the point where $\gamma_{zz}(T)$ vanishes. Essentially the same behavior, but with an even wider temperature separation, has previously been observed on a cubic grid by Li and Teitel[@li93_1].
We have carried out several checks to see if the separation of these two transitions is an artifact due to a finite-size effect. First, as shown in the inset, we monitored the dependence of $\gamma_{zz}$ on accumulation time $\tau$ up to $10^6$ MC steps. More precisely, we define $\Big< A \Big>_\tau \equiv
{1\over \tau} \int_0^\tau A(t) dt$ and use this in calculating the averages which define $\gamma_{zz}$ in eq. (\[heldefeq\]). The $<\gamma_{zz}(T)>_\tau$ thus defined generally evolves approximately logarithmically in $\tau$[@logt] until it reaches its apparent equilibrium value. For temperatures $T_m < T < T_\ell$, the system tends very slowly towards an apparently [*finite*]{} limiting value. We have also checked the size dependence up to $24\times 24 \times 48$, verifying that the $24 \times 24\times 24$ behavior represents the asymptotic limit. Li and Teitel have carried out similar checks up to $200$ layers in the cubic model[@li93_2]. Nonetheless, $\gamma_{zz}$ has some size- dependence to a degree dependent strongly on anisotropy of the system.
If the ratio $J_z/J_{xy}$ is increased to 4.0 (where $J_z$ and $J_{xy}$ are the couplings perpendicular and parallel to the triangular plane), the separation $(T_{\ell} - T_m)/J_{xy}$ between the melting transition and the upper possible transition actually grows for a given size. For these values, the smallest-${\bf Q}$ Bragg peak vanishes at $T_{m} \sim 2.9 J_{xy}$, while $\gamma_{zz}$ vanishes at $T_{\ell} \sim 4.0 J_{xy}$. On the other hand, for weakly coupled layers with $J_z/J_{xy}=0.1$, the two transitions merge to within less than $0.1 J_{xy}$, as in the the [*isotropic*]{} dense $f=1/6$ case. In this weakly coupled case, $T_{m} \sim 0.6 J_{xy}$.
These observations suggest that phase coherence at finite $f$ in a disorder-free system may possibly be destroyed in two steps. First, coherence transverse to the average field direction is lost through melting of the lattice. But longitudinal coherence persists until it is destroyed, along with line-like correlations of the individual vortex segments, at a slightly higher temperature $T_\ell$. This is most apparent for the [*isotropic*]{} system only when $f \le 1/18.$
Vortex Analysis of Possible Transition at $T_\ell$ {#vortexsection}
--------------------------------------------------
There are several ways to look at general phase correlation function $<\Theta({\bf \rho}, z) \Theta({\bf 0}, 0) >$ where $\Theta$ is the gauge-invariant local phase[@glazman91] of the superconducting order parameter. To probe the longitudinal phase coherence, we only consider $c(0;z) \equiv {1 \over A}
\int d^2\rho
<\exp [ i (\Theta({\bf \rho}, z) -
\Theta({\bf \rho},0) ) ] >
$, where $A$ is the sample area - that is, the correlation function in the $\hat{z}$ direction. Glazman and Koshelev have pointed out that phonon-like fluctuations in the vortex lattice lead to a power law decay of $c(0;z)$[@glazman91] (not explicitly shown in the Figure \[czfig\]). We observe that this holds true for $T < T_m.$ For higher temperatures, this dependence changes to an exponential decay $c(0, z) \propto
\exp (- z / \xi_{0z})$. The correlation length $\xi_{0z} \sim [ \log ({T^2\over T_{0}^2} {\log
(T/T_0) \over
2\pi^2}) ]^{-1}$ where $T_0$ is the temperature scale such that $<[\Theta(0,z) -
\Theta(0,z+1)]^2> \sim 1$. This behavior is shown for $T/T_m = 1.1 $ in Fig.\[czfig\] for systems of several thicknesses. To ensure equilibration, we ran 86000 MC sweeps before accumulating data over the following 30000 MC sweeps. To check the effect of boundary conditions, we used both open boundary (OBC) and periodic boundary conditions (PBC) along $\hat{z}$; periodic boundary condition was used in the xy plane for both cases. In all cases, we observe a robust exponential dependence over a limited range $1
< dz < \xi_x$ where deviation sets in at $dz \sim \xi_x \sim 12 $ for $T/T_m = 1.1$, for example. In this temperature range, we do not find significant dependence of c(0, z) on either system size or boundary conditions. For a given temperature above $T_m$, we can use this robust temperature regime to extract the phase correlation length $\xi_{0z}$ from our numerical simulation.
The result is shown in Fig. \[xizfig\]. For all $T > T_m$, and for separations less than $\xi_x(T)$, we observe that $c(0, z) \sim \exp[ - z / \xi_{z0} ]$. $\xi_{z0}$ gradually decreases with increasing temperature approximately as $(T-T_m)^{-1}$, becoming equal to the unit layer spacing near $T/J = 3.0 \sim T_{XY}(0)$ as shown in the inset. We also observe(not shown in the Figure) that $\int d{\bf \rho} c({\bf \rho}, z)$ has far milder dependence on $z$. The exponential decay in $c(0, z)$ is accounted for by random walk-like excursions of the vortex lines and the presence of dislocation or disclination loops in this temperature regime. Note that spinwave excitations in the vortex lattice usually lead to an algebraic decay of $c(0, z)$. Note that given a [*static*]{} deformed vortex line configuration, we may still find a coordinate transformation $\{x, y, z\}\rightarrow \{x',y',z\}$ into a curved space in which the vortex line is straight. In that coordinate system, we will have a long range phase coherence along the straight line in the $\hat{z}$ direction. Therefore, the apparent exponential decay of $c(0, z)$ is not an equivocal indicator for the destruction of phase coherence along $\hat{z}$, but gives information about the deformation of vortex lines from the straight configuration.
In the large-distance tail of $c(0;z)$, where $z > {\rm L}_z/2$, $c(0, z)$ does depend on the boundary conditions and system size, having an upturn for the periodic boundary condition as expected. Moreover, $\xi_{z0}(T)$ rapidly falls as $T$ decreases toward $T_{XY}$ from above, leaving a large interval $\xi_x(T) < dz \ll {\rm L}_z/2$ in which the behavior deviates from simple exponential decay and is independent of the boundary condition used. We also observe that the deviations have relatively poorer statistics due to slow kinetics. It is likely to originate from vortex entanglement and cutting/reconnection, which develop on length scales larger than $\xi_{z0}$. Each of these rare and slow vortex crossings produces a drastic and long-lasting impact on the local phase correlations. The rarity of these events is due to the sizable barrier for vortex cutting and to the subdiffusive nature of vortex lines motion. Because of this rarity, a rapid thermal cycling across this sluggish region may lead to hysteresis.
We now look more closely into the vortex configurations near $T_\ell$ for $f=1/24.$ Fig. \[cor24fig\] shows the density-density correlation function $n_{2,z}({\bf r}_{\perp}, {\rm L}_z/2) = < n_z({\bf r}_{\perp},{\rm L}_z/2)n_z(0,0)>$ describing the $\hat{z}$-component of local vorticity at separations equal to half the total thickness ${\rm L}_z$ (=24). The most prominent feature in $n_{2,z}$ is the disappearance of triangular correlations in the $xy$ plane at melting ($T_m
\sim 1.5 J).$ This behavior is consistent with the disappearance of the Bragg spots in Fig. \[sq\_g\_24fig\]. Note, however, that the central spot, corresponding to the self-correlation between the two ends of the same line, persists well above melting until it vanishes near $T/J = 2.0$, close to $T_\ell.$ This is consistent with the fact that line-like correlations are maintained over at least 12 layers up to $T/J = 2.0$ as we already noted in Fig. \[xizfig\].
In a sample of truly macroscopic thickness, the lines in the liquid phase should carry out random-walk-like excursions, leading to loss of top-to-bottom vortex density correlations over a finite correlation length denoted $\xi_{vz}(T)$, which may be of the same order of magnitude as $\xi_{0z}$ defined above. A finite $\xi_{vz}(T)$ means that the underlying lines are [*flexible*]{}, not that they break up into 2D vortices. This breakup becomes relevant only for $T>T_\ell$. The objects which break apart into 2D objects above the melting transition are not the lines themselves, but the [*topological defects of the lattice*]{}, such as disclinations, which tend to appear as [*well-aligned*]{} line defects near $T_m$[@ryu96; @kim96]. Topological defects look well aligned only when $|{\bf r}_{i}(z) - {\bf r}_{i}(z+1)| / a_B
\ll 1$ where ${\bf r}_{i}(z)$ is the position of the segment of vortex line $i$ in the $z$-th layer. Note that the relevant minimum length scale for alignment of defect lines is the mean vortex spacing, $a_B$. The destruction of the [*lattice order*]{} along the field, which may be detected by vanishing Bragg peak in neutron diffraction, is related to proliferation and unbinding of these defects. On the other hand, the destruction of phase coherence along ${\bf B}$, as we discuss in more detail later, is related to the presence of fluctuations in the transverse vorticity. As the vortex density decreases, it is not [*a priori*]{} obvious if the energy scales for these different types of defects should remain the same.
Thus the line-liquid regime, if it is really a distinct thermodynamic phase, may possibly be described as a neutral gas of topological defects (disclinations of both signs) within the triangular lattice in each plane, which are correlated over a finite length in the $\hat{z}$ direction. Above melting, one expects unbound disclinations to proliferate. Hence, the long-range structural correlations of the vortex lattice are lost in all directions upon melting. However, the [*phase rigidity*]{}, as measured by $\gamma_{zz}$, may persist even above melting, but scaled down by the factor of $\xi_{z0}/{\rm L}_z$, the fraction of the volume of the sample into which the applied twist penetrates. Presumably, this continuous suppression, unless pre-empted by a first order transition (for high densities), persists until the condition ${\rm L}_z/\xi_{z0} \rightarrow \infty$ is met via proliferation of “unbound vortex loops” (vortex lines extending an infinite distance in the transverse direction). At this point, the phase coherence even between neighbouring planes normal to ${\bf B}$ will be lost.
In Fig. \[f24clusterfig\], we show snapshots of vortex configurations at T$_m$, T$_{\ell}$ and a temperature between $T_m$ and $T_{\ell}$. In this regime, by using a bond-searching algorithm, we have identified three distinct classes of vortex lines. The first consists of small vortex loops which close on themselves without crossing either of the two opposite bounding surfaces. The second class contains all isolated lines beginning at the bottom $xy$ plane and ending at the top one. Most of the disentangled field-induced vortex lines fall into this group. Finally, there occur “vortex tangles”. These are lines connected at a given time to one another by the crossing of two vortex segments in the same unit cell. Such tangles are formed either by collision of two flux lines or by interactions of such lines with the vortex loop excitations. This tangle is not static: the collisions which produce it are more and more frequent with increasing temperature and its overall shape will evolve with more rapidity as $T$ increases.
The three columns of the Figure represent the fraction of the vortices belonging to each class at a given instant. On melting ($T/J = 1.5$), the fluctuating lines in our finite sample still remain largely disentangled and separated from each other. As $T \rightarrow T_\ell,$ the density of loop excitations increases (left column), while the field-induced lines (central column) have stronger lateral fluctuations . Both of these effects cause more and more “connected" clusters (i. e., vortex tangles) to appear. Finally, at $T_\ell,$ an [*infinite*]{} tangle, connected by crossing vortex lines, forms. At this temperature, the connected tangle of vortices form a $(D-1)$ dimensional manifold of a tortuous shape, transverse to ${\bf B}$, and cut the original $D(=3)$- dimensional coherent XY system into halves.
Fig. \[f24plfig\] shows an instantaneous vortex cluster size distribution for various temperatures at $f = 1/24$. To generate this distribution, we define the [*projected transverse length*]{} of each vortex loop (or tangle) by $\ell_{xy} \equiv \oint | \hat{z}\times {\bf n}_v | $ for each isolated cluster composed of unit vortex segments ${\bf n}_v$, and accumulate a histogram, ${\cal P}(\ell_{xy})$. \[We consider only the size distribution projected onto the xy plane because the field induced lines (which are infinite along $\hat{z}$) could mask the loops with large extent in the z direction. We also believe that these fluctuations are more relevant to the vanishing of $\gamma_{zz}$.\]
In the first panel of Fig. \[f24plfig\], we plot ${\cal P}(\ell_{xy})$ for several $T \le T_m$. Each plot has a sharp maximum cutoff and a pronounced peak, which is due to the finite average lateral fluctuations of the field-induced vortex lines. For $T_m < T < T_\ell$ (second panel), the weight of distribution is shifted towards larger $\ell_{xy}$, because lines in the liquid phase undergo larger transverse fluctuations. Closed vortex loops also begin to appear in this region. As $T$ increases, the distribution is cut off at progressively larger values, as more and more lines join the connected clusters. Finally, for $T> T_\ell$, all curves are characterized by the appearance of “infinite" clusters, with no obvious length cutoff. The distribution appears to fall off algebraically in this regime - i. e., ${\cal P}(\ell_{xy}) \sim
\ell_{xy}^{-\mu}$ with $\mu \sim 1.0 < 2$ - suggesting $<\ell_{xy}>\rightarrow
\infty$ for $T > T_{\ell}$.
In Fig. \[f24plmaxfig\], we show the maximum value $\ell_{xy}$ occurring over $10,000$ MC sweeps for each temperature. The size is normalized by the linear system dimension in the $xy$ plane (48), and also by the number of xy-planes (24). Although $\ell_{xy}$ grows monotonically for $T>T_m,$ it seems to jump discontinuously from $\sim 1$ to $\sim 2$ between $T/J = 2.0$ and 2.1 (near $T_\ell$ for samples of this size). Qualitatively similar behavior occurs for the isotropic $f=0$ XY transition near $T_{XY}(f=0)$ \[cf. Fig. \[f0plfig\]\].
Entanglement, Winding Number and Other Exotica
----------------------------------------------
We now discuss a possible extension of the vortex loop picture of the zero field XY transition to the hypothetical $T_\ell$ transition or crossover for $f \le 1/24$. Such loop excitations have received far less attention in 3D systems[@feynman55] than their 2D counterparts, possibly because they require more energy to excite and therefore matter only very close to the mean field transition. But in high-T$_c$ materials, the short correlation lengths, high anisotropy, and high T$_c$ broadens the vortex-loop-dominated regime[@carneiro92; @chatto94], before amplitude fluctuations set in.
In a cubic sample with periodic boundary conditions, all vortex lines naturally close on themselves to form loops. These loops are of two topological types: those which can continuously shrink to a point (“trivial class”) and those which cannot (“nontrivial”). The latter are said to have a nonzero “winding number,” i. e., number of infinite lines in a given direction. In the 2D periodic case, the loops lies on the surface of a torus. Here, there are two distinct subclasses of non-trivial loops: one which winds around its circumference, and another which runs transverse to it. In the infinite 2D geometry, these correspond to lines infinite in either the $\hat{x}$ or the $\hat{y}$ direction. On the 3D hypertorus, there are infinite lines in any of [*three*]{} directions.
These notions play critical role in the description of dissipation via vortex motion, i. e. phase slips[@langer67_2]. For current flowing in a given direction, the dissipation may occur either through the expansion of loops, or through motion of an infinite line. In either case, the dissipation arises from fluctuations in the winding number of vortex lines perpendicular to the current(c. f. Equations (\[gameq\]) and (\[sqcriteq\])). Note that when a finite field is applied along $\hat{z}$ direction, the hypertorus already contains many “windings" along that direction even without an applied current.
In the absence of pinning, dissipation in the plane normal to $\hat{z}$ is governed by fluctuations of the winding number in the $\hat{z}$ direction. This dissipation should not depend directly on whether or not the “windings" - that is, the vortex lines - form a lattice, but may depend on the [*density*]{} and mobility of windings.
Dissipation parallel to the field direction (c-axis resistance) depends mainly on winding number fluctuations transverse to $\hat{z}$. Clearly, the average winding in this direction vanishes, unless the field-induced lines themselves, while winding along $\hat{z}$ as required, also wind along another direction like wires around a solenoid. For this to occur, the lines would have to break a chiral symmetry, spontaneously generating a global surface current with a net magnetization normal to $\hat{z}$ - an effect which should be prohibited energetically in the ground state.
It may occur, however, if there exist entangled field induced vortex lines which collide with each other to switch connections(a process we may call “cutting and reconnection”). In Fig. \[donutfig\], we show two field induced lines residing on the surface of a torus (left panel) going through such a cutting and reconnection \[(panels (a)-(b)\]. The right column of the figure shows an alternative view of the same process in an infinite space with open boundary conditions. Initially, both vortex lines wind only along the $\hat{z}$ axis. After the cutting and a special reconnection process in which one strand circles around the torus before meeting its other end, a net transverse winding number has been created. This “global” process is, however, energetically expensive because it involves a spatially extended excursion \[panel (b)\] and should occur very rarely, even in the melt.
If we introduce an “entanglement length" $\ell_c$, defined as the average distance along $\hat{z}$ required for any two vortex lines to wind around each other, we expect $\ell_c$ to be infinite for $T < T_m$, but to become finite in the liquid phase. Because of the finite line tension and repulsive interactions between vortex line segments, such entanglement events along the flux lines are costly in energy and hence rare, in the liquid near melting. Deeper into the liquid phase, as the repulsive interaction between vortex lines is overcome by entropic forces of attraction, $\ell_c$ should become much shorter, leading to a much denser entanglement pattern. The now numerous local transverse fluctuations, and local cutting and reconnection events (i. e., collisions) generate fluctuations in the “global" transverse winding number and cause $\gamma_{zz} = 0.$ On symmetry grounds, the average transverse vorticities $< n_x > $ and $< n_y >$ have to be zero at all temperatures. But possibly $< |n_x|^2 +|n_y|^2 > $ acquires a finite value for $T>T_{\ell}$, suggesting that this quantity could be used as another “order parameter" for the hypothetical phase transition at $T = T_{\ell}$ (with a nonzero value at [*higher*]{} temperatures).
Alternatively, we may view the upper transition in the context of a bond percolation transition. The field induced lines provide a kind of backbone network. With increasing $T$, vortex lines undergo more and more transverse collisions. At $T = T_{\ell}$, these collisions induce the entire ensemble of field-induced vortex lines to form an infinite connected $D-1$ dimensional structure transverse to the applied field, causing large fluctuations in the transverse winding number \[thick gray line in panel (c)\], thereby wiping out any superconducting path connecting the top and the bottom layers normal to the field.
Let the mean-square transverse displacement of field-induced vortices per layer be denoted $\ell_T^2\equiv < |{\bf r}_i(z) - {\bf r}_i(z-1)|^2 >.$ Then $\ell_c / d$ is defined as the number of layers along $\hat{z}$ over which a line wanders transversely by the average intervortex distance. We write this condition as $\ell_T^2 \cdot [\ell_c/d]^{2\zeta} = a_B^2$, where we introduce an unspecified “wandering" exponent $\zeta$. In the limit of dilute (independent) lines, we expect $\zeta \approx 1/2$, corresponding to a random walk of each vortex line segment. Long-range intervortex repulsion is known to renormalize the unit step size $\ell_T$ from $c(T) \cdot (T/J_z)^{0.5}$ down to a smaller value with a similar form with an unspecified T dependence encoded in $c(T) < 1
$[@monica94; @ryu_thesis]. It is not clear how $\zeta$ is affected by intervortex repulsion, but possibly the interactions with other fluctuating lines are equivalent to the line of interest being in a random environment. For a flexible line in a 3D random environment, $\zeta \sim 0.6$ [@kardar87; @balents93]. For $D \ge 2$, in the actual system of interacting fluctuating lines, no exact result is available for $\zeta$, although several numerical results and conjectures give $\zeta \sim 0.2-0.6$[@krug].
We can use these crude estimates to make a guess at the field dependence of $T_{\ell}$, interpreted as a bond percolation transition. Along a given field-induced vortex line, the probability per unit length that a transverse connection is made to a neighbouring vortex line at any position along the $\hat{z}$-axis is $p = d/\ell_c = (\ell_T / a_B )^{1/\zeta}.$ Since $\ell_T \propto (T/J_z)^0.5$ and $a_B \propto B^{-0.5},$ the percolation threshold is reached roughly when $ c(T)^2 TB/J_z > [p_c]^{2\zeta}$, where $p_c$ is an appropriate percolation threshold. This condition defines a [*lower bounds*]{} for a possible transition at $B_{\ell}(T)$ which approximately follows $$\label{dceq}
B_{\ell} \propto { [p_c]^{2\zeta} \over c(T)^2}{J_z(T) \over T}.$$ For a dense lattice or large anisotropy (i. e., small $J_z$), this condition is probably satisfied immediately upon melting, as at $f=1/6.$ For dilute systems, however, the second transition is not automatically triggered by melting and may occur only deep into the liquid phase, at a temperature where the entanglement barrier is sufficiently weak to allow an infinite vortex tangle to form. Whether this percolation transition is a true phase transition or only a sharp crossover remains to be determined.
This same picture hints at how correlated pins such as columnar damage tracks[@civale91] may increase $T_{\ell}$. Such columnar disorder will encourage the vortex lines to stay straight along the defect track, reducing the effective unit step $\ell_T$ by a factor $c_p \ll c.$ As a result, the wandering exponent $\zeta_p$ may also change from its thermal value $\zeta$. Consequently, $T_\ell$ will be enhanced by an overall factor of $({c \over c_p})^2({1 \over p_c})^{2(\zeta-\zeta_p)}$.
In summary, the upper transition is characterized by the following set of equivalent criteria:[@li93_2; @jagla96] Disappearance of finite transverse diamagnetism; disappearance of phase rigidity along the field direction; appearance of an infinite [*transverse*]{} vortex cluster; large fluctuations in the global transverse winding number or the net vorticity ${\cal M}$; onset of finite c-axis phase-slip resistance in the limit of vanishing bias current in the $c$ direction, which is equivalent to saying no superconducting path exists over macroscopic distances.
Dissipation for $f=1/24$ {#f24rvstsection}
------------------------
While Bitter decoration serves as a detailed probe of spatial vortex configurations[@kim96; @murray90; @yao94], it yields ambiguous information about freezing, and is restricted to very low flux densities. Cubitt [*et al*]{} obtained evidence of a melting transition in from low angle neutron diffraction[@cubitt93]. Similar results were obtained by a $\mu$SR technique, which probes the local magnetic field distribution[@lee93]. NMR[@recchia95] and atomic beam[@harald] techniques have also been used to study both the static properties and the melting of the vortex lattice. More recently, Schilling [*et al*]{} employed a differential thermometry to search for the latent heat of melting in and to obtain a melting curve[@schilling96; @schilling97].
Far more information has been accumulated from transport measurements, but this is much less easily interpreted in terms of vortex lattice melting. The interpretation is complicated by disorder, as well as by the fact that the measurements are nonequilibrium and usually involve nonuniform current distributions. Safar [*et al*]{}[@safar92] measured a sharp jump in resistivity in the mixed state of . The resistivity also showed a hysteretic behavior upon thermal cycling, indicating a first order transition. The transition line thus obtained seems to coincide with “melting curves" obtained by torque measurements[@farrell91], and more recently, by differential specific heat measurements[@schilling96]. Kwok [*et al*]{} have carefully demonstrated the effect of twin boundary pinning on the melting transition in a series of transport measurements which track the so-called “peak effect" associated with vortex lattice softening [@kwok94]. They find that the peak effect sets in at a few degrees below the melting curve determined from a sharp kink in resistivity. This sharp resistivity kink, as observed by both Safar [*et al*]{}[@safar92] and Kwok [*et al*]{}[@kwok92], tends to become less pronounced both at very high$(B > 10 \, {\rm tesla})$ or low flux densities$(B < 1 \, {\rm tesla})$[@sharpjump].
An ideal, but impractical, transport measurement to determine the melting curve would consist of applying an infinitesimal current to induce a net Lorentz force on the lattice, which is held in place by a balancing pinning force. As soon as the lattice melts, individual lines would begin to drift, inducing “flux flow" resistance. Most real materials, however, are complicated by disorder, and even the static properties of the lattice with disorder are incompletely understood[@bouchaud92_3; @giam95]. In the presence of disorder, varying the field density produces changes in both the effective pinning strength and the effective flux lattice anisotropy. Depending on relative strengths of all these competing effects, many complications may arise in probing thermodynamic properties using transport experiments[@danna95; @koshelev94; @hellerqvist96; @jensen96; @ryu97].
A number of recent transport measurements have, nonetheless, produced rich information about flux lattice melting. Pastoriza [*et al*]{} have used a non-uniform distribution of pinning strength to probe the shear modulus directly[@pastoriza95], giving direct information about the lattice stability. Zeldov [*et al*]{}[@zeldov95] have used local Hall probes in to monitor the local field density. They found a very sharp jump, again interpreted as a signature of a first order melting transition. More recently[@fuchs96], Fuchs [*et al*]{} performed simultaneous measurements of the resistance and local magnetization, confirming that that the jump in local magnetic density coincides with a sharp increase in resistance (but not necessarily a discontinuous [*jump*]{}). The heat of melting [*per vortex per layer*]{} inferred from this local magnetization jump, however, shows rather peculiar features: it vanishes continuously as the field is increased, while steeply increasing as the field is lowered toward zero.
These results raise several outstanding questions: How can the seemingly first order transition line terminate apparently at a point in the H-T plane? Does the melting line monotonically approach $T_c(H=0)$ or follow a reentrant melting curve? Another important issue is the longitudinal phase coherence probed by c-axis resistivity vs. T measurements[@briceno91; @gray93; @monica94], which show a striking series of broad peaks in single crystals. The nonlocal conductivity associated with this phase coherence can be probed in the so-called flux transformer geometry. The experimental data of Keener [*et al*]{}[@keener96] showed that phase coherence over a finite correlation length along ${\bf B}$ persists above the melting transition of the vortex lattice in some region of the H-T phase diagram.
A simple and natural model for probing the dynamics of the mixed state is a network of resistively shunted Josephson junctions with Langevin noise. In this section, we present some results of simulations using this model, and to connect these to the analogous static XY results. Our calculations are carried out as follows. At any given temperature, the final snapshots from the MC simulations are used as the initial dynamical phase configurations. We use an integration time step $\triangle t = 0.1t_0$. After the current is switched on, $1000-5000 \triangle t$ is allowed for the system to reach a steady state, following which the voltage is averaged over the next $6000- 12000$ steps of $\triangle t$.
Fig. \[rvstfig\] shows the “bulk in-plane resistance” at $f = 1/24$. The measurement geometry is as shown in Fig. \[geomfig\] (a); thus, these calculations probe the shear rigidity of the lattice in contrast to the usual transport experiment in which random pins play an essential and complicating role. Through the rest of this paper, we will call these calculated quantities $R_{ab}$ and $R_c$, even though they differ from the resistivity measured in most transport experiments. At the highest bias current of 2.83$I_c$ per grain(equivalent to $1.4 I_c$ per bond), we have a smooth curve without any noticeable changes either at $T_m$ or at $T_\ell.$ For lower values of driving current, sharper features emerge. There is a slope discontinuity near $T_m = 1.5 J$ for both $I/I_c = 0.83$ and $0.083$, but the most dramatic change occurs near $T_\ell = 2.0 J$. The entangled line liquid for $T_m < T < T_\ell$ seems to have a sizable viscosity. This viscosity impedes the motion of the flux lines in the liquid, the two halves of which are driven past each other by opposing Lorentz forces. As a result, the lines move slowly, and dissipation (defined as a “resistance” R$_{ab}$) is small. The steady increase of $R_{ab}$ with temperature in this region is due to screening by vortex loops which gradually lowers the viscosity. For $T>T_{\ell},$ this viscosity vanishes, leading to a steep increase in $R_{ab}$ This increase at $T_\ell$ is enhanced by additional (and probably dominant) dissipation produced as the system goes through an XY-like transition or crossover. The large viscosity for $T_m < T < T_\ell$ is also consistent with the slow ($\ln t$) equilibration seen in the Monte Carlo measurement of $\gamma_{zz}$ for $T_{m} < T < T_{\ell}$[@logt]. We believe that the change in $R_{ab}$ near $T_\ell$ shares the same mechanism as that seen near $T_{XY}(f=0)$ shown in Fig. \[f0rvstfig\].
Fig. \[caxisfig\] shows the “$c$ axis resistivity" $R_c$ at f = 1/24, as calculated using the geometry of Fig. \[geomfig\] (b). For comparison, we also show the calculated $\gamma_{zz}$. As the driving current is reduced, $R_c$ seems to approach a curve which vanishes asymptotically as $T\rightarrow T_\ell^+$, coinciding with the vanishing $\gamma_{zz}.$ Our numerical results thus suggest that the dramatic increase in $R_c$ results from the $T_\ell$ transition rather than melting. The increase in $R_c$ is thus correlated with massive vortex line cutting, as put forward earlier[@ryu_thesis; @monica94], and with an increase in the density of transverse vortex segments $<|n_{xy}|>$, the frequency of vortex line crossings, and fluctuations in the transverse net vorticity, $\delta {\cal M}_{z}^2$. This distinction between $T_{e\\}$ and $T_m$ may be most important for at low fields, and in disordered dense systems[@jagla96], where the temperatures may be most separated.
We can draw some conclusions relevant to experiment from the current dependence of both $R_{ab}$ and $R_c$ observed in our simulations. First, the “melting line,” [*as detected via a voltage criterion at constant current*]{} in a bulk resistance measurement, should be sensitive to the driving current, even at a very low bias. Existence of pinning force is essential in getting distinct transport behaviors for the lattice and the liquid. On the other hand, the transition at $T_\ell$, whether monitored by the vanishing of $R_c$ in the limit of small current or by a jump in $R_{ab}$ between two [*finite*]{} values, should be relatively insensitive to applied current density, since the main mechanism of dissipation(presence of transverse vorticity) in this case is switched off below $T_\ell$ and sets in above $T_\ell$ irrespective of whether we have pins or not. Indeed, just such an observation has been made by Keener [*et al*]{}[@keener97] in describing their curves for $T_m(H)$ (melting) and $T_D(H)$ (“decoupling transition”), as obtained by flux-transformer measurements on single crystals. It is plausible that their $T_D(H)$ at very low fields ($B <
100$ Gauss) corresponds to $T_\ell$ in our model. True melting line is presumably the limiting value of the current-dependent $T_m(H, J)$ as $J\rightarrow 0.$ It is not experimentally verified whether such a limiting value coincides with $T_D$ or not.
Let us briefly comment on the experimental possibility of distinguishing between $T_m$ and $T_{\ell}$ If we imagine a hypothetical [*isotropic*]{} high temperature superconductor, the coupling constant $J$ in our model calculation is related to the parameters of the superconductor via $J \sim {d\phi_0^2 \over 16 \pi^3 \lambda^2(0)}
\times ( 1 - [T / T_{c0}]^4)$ (assuming the two-fluid model). Taking $T_{c0} = 92 K$, $d = 10 \AA$, and $\lambda(0) = 1000 \AA$, we find $T_m \sim 89.7 K$ and $T_\ell \sim 90.3 K$ \[eq. (11)\]. Thus the two transitions are remarkably close even for the isotropic case. In real materials such as and , the separation between the two will be further reduced by an anisotropy factor, although pinning disorder may tend to separate them. Therefore, in many cases, it will be nearly impossible to separate the two transitions experimentally.
Local magnetization jump and heat of melting {#zeldovsec}
============================================
A striking result of the micro-Hall probe measurements is the sharp jump in local magnetization (vortex density) across the phase transition[@zeldov95]. At “high” fields ($\sim 200 G$), the jump occurs at nearly constant $T$, and even for lower fields, still within $\delta T \sim 3 mK.$ The heat of melting per vortex per layer, $T_m \triangle S = - {T_m \triangle B \over 4 \pi} {dH_m \over dT}$, as obtained from the Clausius-Clapeyron relation, increases monotonically from $0$ at $B \sim 400 G$ to about $0.6 k_B$ at $B \sim 55 G$, beyond which the slope ${dH_m \over dT}$increases very sharply (cf. Fig. 6 of [@zeldov95]).
Similar jumps also seem to occur in , as reported in recent calorimetric[@schilling96] and magnetization measurements[@liang96; @welp96]. The estimated latent heat of melting yields $\triangle S$ (per vortex per layer) $\sim 0.4 k_B$ for $1 < B < 8$(tesla). The data (cf. Fig. 1 of Ref. [@schilling96]) shows that the jump $\Delta M$ at $T=85 K$ ($B\sim 3.7$ tesla) for is spread over a field range $\delta B \sim 0.1T$, or, for a given field, over a temperature range $\delta T \sim 0.1 K$. This jump decreases rather abruptly for flux densities $B \leq 1 T$. The estimated entropy of melting ($\sim 0.4 k_B$ per vortex pancake) is quite close to that numerically obtained by Hetzel[*et al*]{}[@hetzel92] ($\sim 0.3k_B$ per pancake), and also to the values obtained in model calculations based on the lowest Landau level and London approximations[@errata].
As the field decreases, the jump in $M$ occurs over a broader temperature range (cf. Fig. 3 of [@zeldov95]). The resistance jumps measured by Kwok [*et al*]{}[@kwok92], attributed to the melting transition, also become broader with decreasing field. By contrast, the height of the resistivity kink seems quite uniform over a wide range of fields. These last two observations are consistent, however, if we interpret the jump in resistance as a signal that $\gamma_{zz} \rightarrow 0$. In view of all these facts, it is plausible that, at least at relatively high fields which corresponds to $f J_{xy} / J_z > 1/18$, the experimental jumps observed in local magnetization and resistance[@zeldov95; @schilling96; @kwok92] shows the combined effects of two distinct processes, occurring within $\triangle T < 10 mK.$ As a corollary, the very low field measurements($f J_{xy} / J_z < 1/18$) actually may not track the melting transition itself, but various manifestations of the predominant XY fluctuations (vortex loops) which are most conspicuous near $T_\ell.$.
Observation of the [*reentrant*]{} melting curve has not been reported in any high-T$_c$ materials. In NbSe$_2$, the melting line detected by the peak effect was reported to be non-monotonic in field[@sabu], consistent with the reentrant melting curve proposed for the more anisotropic high-T$_c$ superconductors[@nelson88]. But if one interprets the magnetization jump in as evidence for melting, then the melting curve for apparently approaches $T_{c0}$ [*monotonically*]{} at field as low as $\sim 1 G$. This behavior is surprising since, at these fields, the vortex separation far exceeds the magnetic screening length. Furthermore, in , the peak effect at $0.35 -1.5$ T is observed to lie below the resistivity kinks[@kwok94] (about $0.8 K$ below at $0.5$ T). These measurements suggest that, at least for low flux densities, transport measurements may actually not be probing flux lattice melting.
We propose that low-field melting is indeed reentrant. Most low-field experiments which probe magnetization[@zeldov95; @fuchs96], thermal properties[@schilling96], and transport coefficients [@safar92; @fuchs96; @keener96; @kwok92] actually track $T_{\ell}$, which is progressively more separated from $T_m$ and approaches $T_{XY}(0)$ as the field is reduced. We have already shown that dramatic changes in $R_{ab}$ and $R_c$ occur at $T_{\ell}.$ Moreover, the broad peaks in $C_V$ are centered at $T_\ell$ and they, too, approach $T_{XY}(0)$ as $B$ decreases. Our estimated upper bound for the total entropy release in the temperature range $T_m < T < T_\ell$, as estimated from the $T-$dependence of the internal energy, qualitatively resembles that of Zeldov [*et al*]{} (Fig. 6 of [@zeldov95]) in that it steeply increases as $f$ decreases. At higher fields, the observed (and also calculated) $\triangle S \sim 0.3-0.5 k_B$ is consistent with the destruction of phase coherence at a [*single*]{} transition. By contrast, at low fields, phase rigidity is lost in a two-step process. Most of the entropy release ($\triangle S \geq 0.5 k_B$ per vortex per layer) occurs near $T \sim
T_\ell > T_m$, whether or not this is a true phase transition. Note that a hysteresis in the resistivity may be observed near $T_{\ell}$ due to finite vortex-cutting barriers below $T_\ell$. This is not necessarily an evidence for a first-order melting transition at very low fields.
The melting line in the dilute limit may be quite difficult to detect experimentally. Conceivably it may be tracked by the peak effect, by high-resolution IV measurements, or by direct measurement of the shear modulus [@pastoriza95]. Of course, direct observation of a vanishing neutron diffraction pattern as in [@cubitt93] would be ideal, but this technique is of limited applicability in this density range.
To shed further light on this problem, we have carried out calculations [*with mixed boundary conditions*]{}. That is, we allow local density fluctuations in the net z-component of vorticity by using free boundary conditions in $x-$ and $y-$ directions, while retaining periodic boundary conditions along the $z-$axis. Of course, surface effect are now stronger, possibly reducing the melting temperature. Another point is that our uniform-frustration model assumes that $\lambda = \infty$. Therefore, we should proceed with some caution in relating our numerical results to experimental data.
To study the system with these mixed boundary conditions, we again did a simulated annealing run for a single layer of the triangular grid to find the lowest energy configuration. By stacking the resulting state layer by layer, we form the ground state lattice, which, because of incommensurability and the free boundaries, now consists of an imperfect triangular lattice with some defects. This lattice melts at $T/J \le 1.4$ for the nominal density of $f=1/24$ on a $26\times 26\times 12$ grid. This is slightly below the value $T_m/J \sim 1.5$ found for the fixed density system of 24 layers with periodic boundary conditions. For a nominal density of $f=1/6$, melting occurs near $T/J \le 1.15$ with these mixed boundary conditions.
One might think of defining the “magnetization” $M_z$ as the average net vortex density $n\equiv \int n_z({\bf r}) d{\bf r} / A$, where $n_z({\bf r})$ is the local vortex density and $A$ is the total area. However, $M_z$ defined in this way suffers from spurious boundary effects, arising from the depletion of vortices near the boundaries in the lattice phase[@ebner]. Upon freezing, the lattice develops a [*rigid*]{} free surface of irregular shape, expelling some of the vortices from the rectangular bounding box. The resulting change in density, $\delta n / n$, is an artifact of the open boundary conditions, and we find that it vanishes for large samples as $1 / \sqrt{A}$, confirming that it originates from a surface effect.
Instead, we define $M_z$ by a criterion involving the [*local Voronoi cell area*]{} ${\cal A}_i$, i. e., the area of the generalized Wigner-Seitz cell for vortex $i$ (the shaded area shown in Fig. \[vorfig\]). Before applying the procedure, we first eliminate the thermally induced vortex loops, which are present in addition to the field-induced vortices for $T \ge 0.5 \times T_\ell$. To do this, we pair each antivortex with the nearest vortex in each plane, identifying the resulting pairs as bound dipoles to be excluded from the count (cf. left panel of Fig. \[dipolefig\]). Since most such dipole pairs have linear dimensions much smaller than $1/\sqrt{<n>}$, this criterion is justified. We then perform a Delaunay triangulation on the field induced vortices to determine topological neighbors for each vortex. From the bond configuration thus determined, we obtain its dual, which is the desired Voronoi diagram. A local vortex density at a point ${\bf R}$ may then be defined as $$\label{localdensityeq}
n({\bf R}) = \sum_i \delta( {\bf R} \in {\cal A}_i )
/ {\cal A}_i$$ where $\delta ({\bf R} \in {\cal A}_i) = 1$ if the point ${\bf R}$ lies in the Voronoi cell associated with vortex $i$, and zero otherwise. Next, the local magnetization $M_z$, which we interpret as the [*bulk average density*]{} $<n>$ is calculated from $$<n> = \sum_{{\bf r}_i \in {\cal C}} { 1 \over {\cal A}_i },$$ i. e., as the average of the inverse Voronoi area for vortices lying within a measurement area ${\cal C}$ suitably distant from the sample boundary.
In Fig. \[jumpsfig\], we show the relative average vortex density (filled circles) along $\hat{z}$, $<n_z> / n_0$, normalized to the nominal density per layer at $f=1/6$ and $f=1/24.$ For $f = 1/6$, $<n_z>$ shows a sharp jump at $T_m$ to a value about 7 % larger than $n_0$. For $f=1/24,$ the there is a similar change of about 15 % which is less sharp than at $f = 1/6$ and is centered at $T_\ell.$ From these two data points, we observe that $[<n>(T_\ell) - n_0] / n_0 \sim
f^{-1/2}.$ Comparing the result for two different sample areas ${\rm L}_x{\rm L}_y$ for $f=1/24$, we have verified that the observed change is indeed a [*bulk*]{} phenomenon, independent of any surface influence. Note that we have about the same number of flux lines($\sim {\cal
O}(200)$) for both $f=1/6$ and $f = 1/24$ for our chosen sample sizes of $24 \times 24 \times 12$ and $48 \times 48 \times 12$. While the jump occurs at $T_m$ for $f=1/6$, we do not observe a similar feature near melting ($T_m = 1.35 J$) at $f=1/24.$ Therefore, the cause of the jump in the local vortex density should be sought in the nature of transition at $T_\ell$, rather than in the mechanism for flux lattice melting. Note that these jumps resemble those in Fig. 5 of [@zeldov95] in the “anomalous low-field regime” $(1 < B < 55G)$ in the following sense: the fractional change in vortex density decreases, and the jump becomes sharper, as the field increases. Probably, the line density $\langle n_z \rangle$ increases with increasing T for $T_m < T \le T_\ell$ because the repulsive intervortex interaction is screened by polarizable vortex loops. The 2D analog of this effect is the screening of the repulsion between field-induced vortices by thermally excited vortex-antivortex pairs[@doniach79].
Does the jump in flux density occur exactly at the melting transition, or is it more closely connected to the other “transition” at $T_\ell$, i. e., to a transition between two liquids with different compressibilities? This question may actually be rather academic, since $T_m$ and $T_{\ell}$ may practically merge in real, anisotropic materials at high fields. Eq. (\[dceq\]) gives a rough criterion for $T_{\ell}$ in isotropic systems: $1/\ell_c = (c^2 BT/J_z)^{1/\zeta} > p_c$. For anisotropic systems such as and , a given value of $f$ corresponds to a field which is reduced, relative to the isotropic system, by a factor of $J_z / J_{xy}.$ Therefore, the merging of $T_{\ell}$ and $T_m$, which in isotropic systems occurs around $f \sim 1/18$, should in anisotropic materials occur around f = (1/18)$J_z/J_{xy}$, This anisotropy factor $J_z / J_{xy}$ could be as small as ${\cal
O}(0.0001)$ in .
Discussion
==========
Analogy to XY Transitions of Slabs of Finite Thickness
------------------------------------------------------
In the previous sections, we made following observations from numerical simulations: (i) $\gamma_{zz}$ vanishes at $T_\ell\ne T_m$ for isotropic system with $f >
1/18$ and $T_\ell(B)$ appears to terminate at $T_{XY}$ for $B \rightarrow 0$; (ii) As $B$ changes, it tracks the broad peak in $C_V$ which behaves $$\label{cpeakeq}
C_V^{max}(B) \sim - \ln (L_B),$$ and $$\label{deltatceq}
|T_{\ell} - T_{XY}| \sim L_B^{-2/3},$$ where $L_B \equiv f^{-1/2} \sim B^{-1/2}$; (iii), $T_{\ell}$, and not $T_m$, appears to coincide with the principal change in local vortex density which follows $$\label{deltameq}
\triangle M / B \sim B^{-1/2}.$$ There is a corresponding change in bulk resistances $R_{ab}$ and $R_c$ over the same temperature range; (iv) $T_\ell$ and $T_m$ seem to merge at a sufficiently high field.
We now summarize some recent experimental observations which appear to be consistent with these numerical results. Schilling et al[@schilling97] have reported high resolution calorimetric evidence for a first order transition in , which they interpret as a melting transition. They observe a very sharp delta-function like peak lying on the left shoulder of the broad peak in specific heat in the range of $0.75 - 9 $ tesla, which roughly corresponds to $1/81 <
f < 1/6$ in our isotropic sample. The delta-function appears to vanish for densities lower than about 0.5 tesla. This remarkable experiment thus establishes the existence of a first order melting transition line, which empirically follows $|T_m - T_c(0)| \sim L_B^{-1.61}$ over $0.75 - 9 $ tesla. These data are consistent with our numerical results on the following points: (i) a first order melting transition exists, and becomes weaker as the flux density is lowered; (ii) the melting transition is located on the left shoulder of a broader peak in $C_V$; and (iii) the height of this broad peak and its position generally follow the behavior described in equations \[cpeakeq\], \[deltatceq\] and \[deltameq\]. As further evidence of the correspondence, we show in Fig. \[schillingfig\], the data extracted from Fig. 1 of [@schilling97] for fields as high as 6 T. At higher fields ($>7$ T), the points deviate from the observed power law behavior shown in the Figure.
Another experimental data which agrees well with our numerical results is that of Roulin et al[@roulin96]. These workers have reported that both the melting curve $T_m(H)$ and a point they label the “superconducting-normal” (SN) transition as monitored by tracking the maximum in $C_V$ as a function of $H$ both follow the equation $|T - T_c(0)| \sim L_B^{-3/2}$, consistent with our numerical results and the scaling analysis discussed above (to within logarithmic corrections). Welp [*et al*]{}[@welp96] have reported a detailed study of $\Delta M$ as a function of $T$ and $H$. The data presented in Fig. 2 of their paper shows that $\triangle M / B \sim B^{-1/2}$ for $1.8
\le B \le 5.6$ tesla, once again in agreement with both our numerical results and the scaling data From this data, we conclude that our numerical observations, based on a frustrated 3D XY model, are generally consistent with recent experimental observations.
Most interpretations of these experimental results focused on only one true phase transition in the low field regime, namely a first-order liquid-solid transition. This viewpoint is consistent with our numerical results only if we assume that $T_{\ell}-T_m$ (where $T_{\ell}$ is defined as the temperature where $\gamma_{zz}$ vanishes) will go to zero in the thermodynamic limit. In the following, we will briefly review a multicritical scaling approach which assumes a single melting transition which happens to be in the vicinity of the zero field XY critical point. Our data are not sufficient to determine without ambiguity whether or not this assumption is correct. Therefore, we follow it by giving an alternative discussion based on the hypothesis that there are actually two separate phase transition lines: $T_m(H)$ for flux lattice melting and $T_\ell(H)$ for complete destruction of any superconducting path (phase coherence) in all directions.
Friesen and Muzikar[@moloni96] describe the SN transition at a finite ${\bf B}$ in the vicinity of the $f=0$ XY critical point. Their scaling hypothesis takes the form $$\label{scaleq}
f_s(B,T) \sim |t|^{2-\alpha} \phi_\pm (B |t|^{-2\nu})$$ for the singular portion of the free energy density in the XY critical region. Here $\alpha$ and $\nu$ are the standard critical exponents describing the specific heat and correlation length of the $f=0$ critical point, $t = T-T_{XY}(0)$, and $\phi_{\pm}$ are appropriate scaling functions. $B$ is put in by hand based on the assumption that it is the only relevant length scale. It is plausible, but does not have rigorous justification. Since $\alpha \sim 0$ and $\nu \sim 2/3$ for the $d = 3$ $XY$ model, this expression can be rewritten as $$\label{scaleq1}
f_s(B,T) \sim |t|^2\ln |t|\phi_{\pm}(B|t|^{-4/3}).$$
The singular part of $C_V \sim -\partial^2f_s/\partial t^2$ can now be shown to satisfy the relation (for $T < T_{XY})$ $$\label{cveq}
C_V(B, T) \sim {\cal C}(x)\ln t$$ where $x = B|t|^{-4/3}$ is the appropriate scaling variable and ${\cal C}(x)$ is another scaling function. From this we find (i) that the quantity $C_V/\ln |t|$ has a maximum at some fixed value of $x$; and (ii) at that fixed value of x, the maximum value $C_v^{max} \sim \ln |t|$. Both (i) and (ii) are in agreement with our numerical data. Similarly, the magnetization is given by $M(B,T) \sim (\partial f/\partial B)_T$. It is readily shown to satisfy $$\label{mageq}
M \sim {\cal M}(x)(B^{1/2}\ln |x| - \ln B),$$ where ${\cal M}$ is another scaling function. A reasonable interpretation of the “jump” $\Delta M$ in magnetization is the difference in M between two fixed values $x_1$ and $x_2$ of the scaling variable. Then, if the term involving $\ln B$ can be neglected, we have $\Delta M/B \approx B^{-1/2}$ in agreement with our numerical results.
This same scaling picture can be used to interpret the heat of fusion at the first-order melting transition at $T_m(B)$. Assuming that $T_m(B)$ happens to be in the XY critical region, we write the free energy densities below and above $T_m(B)$ as $-t^2\ln |t|f_s(B|t|^{-4/3})$ and $-t^2\ln |t|f_{\ell}(B|t|^{-4/3})$, where $f_s$ and $f_{\ell}$ are two different scaling forms for the free energy density above and below the melting transition. At the melting point, these free energy densities must be equal. Then a little algebra shows that the jump $\Delta s$ in the entropy density $S = -(\partial f/\partial T)_B$ takes the form $\Delta s = -t^2\ln |t|(f_{s}^{\prime}(x_m)-f_{\ell}^{\prime}(x_m))$, where $x_m = B|t_m|^{-4/3}$ is the value of the scaling parameter at the melting point. Using $B \sim |t|^{4/3}$ along the melting curve, we find that along the melting curve $$\label{dseq}
\Delta s \sim B^{3/2}\ln B.$$ Thus the melting transition should have an entropy jump which gets smaller as the field is reduced.
If $T_\ell(B)$ represents a true phase transition as we desribed using the idea of vortex tangles, how can it be understood in terms of the phase coherence? One possibility is a [*line of critical points*]{} for a continuous phase transition similar to the XY transition in a semi-infinite slab. This view provides a natural explanation why the scaling theory with the scaling variable $B\xi^2$ should be successful. Mathematically, one can attach a “phantom” cut-line to the core of each vortex segment across which phase slips by $2 \pi.$ These are benign since continuity and single valuedness of the phase $\Theta$ at every point is ensured. However, their shape and motion can be monitored most conveniently to keep track of the spatial and temporal disturbance of the phase coherence which have important consequences such as phase slip dissipation in superconductors. For an isolated vortex segment placed at origin with positive vorticity along $\hat{z}$, the cut-line may lie straight along the positive $\hat{x}$-axis. A negative vortex will then have the cut-line on the negative $\hat{x}$-axis. Once we choose a cut-line for a particular vortex by fixing the reference phase angle, it provides the reference for all other vortices. Note that these cut-lines can only terminate either at the sample boundary or at the core of vortices of opposite charges. When there are several interacting vortices, these cut-lines are no longer straight, and their tortuosity reflects the phase disturbances induced by deformation of the vortex configuration away from perfect lattice. If the vortex lines were straight, we will observe that the cut-lines associated with each segment all line up as we move along a vortex line. Therefore, a cut-line associated with each vortex line will form a semi-infinite cut-sheet, separated from other sheets by roughly $L_B\sim B^{-1/2}$. As the vortex lines become tortuous in the liquid phase above $T_m$, the cut-sheets will become wrinkled and our system will look like a three dimensional maze walled by these sheets. Both in the vortex solid and line-liquid phases, this maze will allow a arbitrarily curved path connecting both sides(either along $\hat{z}$ or $\hat{x}$-axes) of the sample, and the average width of the path free of the walls will be ${\cal O}(L_B).$ We conjecture that it is possible that the phase variables may maintain long range coherence. At low fields(large $L_B$), this tortuous slab contains many XY phase degrees of freedom which, being confined within the walls, [*do not feel*]{} the presence of free vortices, and therefore could conceivably undergo a phase transition in the universality class of a zero-field $XY$ model in a “film" of thickness $\sim L_B$, i. e., a quasi 3D-XY transition. This crosses over to a bulk XY transition as $B \rightarrow 0$.
There are two possible objections to this picture. First, our numerical results only hint at, and certainly do not prove, two separate phase transitions. Secondly, the “film” mentioned above is a dynamical rather than an equilibrium film, in the sense that its boundaries are not fixed. It is not clear that such a dynamical object could have an XY phase transition. The boundaries (i. e., cut-sheets of the deformed vortex lines) are, of course, moving subdiffusively[@ryu96; @dorsey92] as long as $\xi_{vz}/d > {\cal O}(10) $. This condition, as we confirmed numerically in section \[vortexsection\], holds true in the range $T_m < T < T_\ell$ and makes the above picture more plausible. It also greatly enhances the chance if we consider pins in real material, since even a single vortex line then becomes collectively pinned into a glassy state.
We now discuss a 3D XY-like transition for the infinite slab of thickness $L_B$. Such a slab belongs to the $G_2$-class in Barber’s classification of finite size systems[@barber83]). From this identification, we can derive many characteristics of the phase transition. First, consider a thermodynamic quantity for an infinite system in 3D, which varies as $P_\infty(T) \sim C_\infty t^{-\rho}$, where $t = (T-T_c)/T_c$ with $T_c$ the transition temperature for an infinite system and $\rho$ an appropriate critical exponent. For a slab of thickness $L_B$, a general finite size scaling ansatz dictates that $$\label{ambeq}
P_{L_B}(T) \sim L_B^\omega Q(L_B^\theta \dot{t})$$ as $L_B \rightarrow \infty, \dot{t} \rightarrow 0$ with $\theta = 1 / \nu.$ The exponent $\omega$ is determined by requiring bulk behavior in the limit $L_B \rightarrow
\infty$; this condition gives $\omega = \rho / \nu.$ The transition temperature for a finite ${\bf B}$ is shifted $$(T_c - T_c(L_B)) / T_c \sim L_B^{-\lambda}$$ and the shift exponent $\lambda $ is generally equal to $1 / \nu$, as has been discussed for the superfluid transition in bulk $^4$He of finite thickness by Ambegaokar [*et al*]{}[@amb80]. For our purposes, it is sufficiently accurate to take $\nu \sim 2/3$, which then agrees very well with our numerical result(Eq. \[deltatceq\]). Eq. \[ambeq\] needs to be modified for a quantity with a logarithmic divergence; it becomes $P_\infty (T) \sim C_\infty \ln t$ as $t\rightarrow 0,$ one modifies the ansatz[@fisher71] so that we have $P_{L_B}(T) - P_{L_B}(T_0) \sim Q( L_B^\theta \dot{t} ) - Q( L_B^\theta
\dot{t_0} )$, where $T_0$ is some non-critical temperature. For $\dot{t}
\rightarrow 0$ at a fixed $L_B$, we obtain for such a variable $$P_{L_B} ( T_c(L_B) ) \sim - C_\infty \theta \ln L_B$$ where we have assumed that $Q(z) = {\cal O}(1)$ for $z \rightarrow 0$. This prediction is in good agreement with the calculated maximum height of specific heat peak(Eq. \[cpeakeq\]), which for the 3D XY model, has a weak divergence with $\alpha \sim 0.$ Similar results have been discussed for the superfluid transition in He II of finite thickness by Ambegaokar et al[@amb80].
As the field increases, one may eventually reach the limit $L_B / \xi_{XY}(T) < 1$, at which the transition at $T_c(L_B)$ will crossover to a 2D KT universality class and we expect the merging of the two transitions, $T_\ell = T_m.$ In our model, we believe this happens for a value of $f$ between $1/6$ and $1/18.$
Other Recent Simulations
========================
Towards the completion of this work, we became aware of some of the more recent studies based on similar models. We briefly discuss them in comparison with our main results and interpretations. To avoid confusion, we use our own conventions for the flux density given in terms of the frustration $f$ defined earlier and introduce the anisotropy factor $\Gamma^2 \equiv J_{xy} / J_z$. For an isotropic system, $\Gamma = 1$ while for ${\rm YBa_2Cu_3O_{7-\delta}}$ it is $\sim {\cal O}(10)$ and for ${\rm Bi_2Sr_2CaCu_2O_8},$ it is $>> {\cal O}(100).$ We also use the same notation $T_\ell$ for the temperature where $\gamma_{zz}$ drops to zero. Some researchers opted to use $T_z.$
Nguyen and Sudb[ø]{}[@nguyen97] have extended their earlier work on the anisotropic London loop model[@nguyen96]. Their numerical results in both the vortex structure factor and $\gamma_{zz}$ for $\Gamma = 1$ with $f=1/32$ follow a pattern qualitatively similar to our main results for $\Gamma = 1$, $f=1/24$ \[See Figure 6 of [@nguyen97]\] as well as those of Li and Teitel[@li93_1; @li93_2]. By looking at the dependence of $T_\ell(N_z)$ on the thickness of the system $16 \le N_z \le 96$, and linearly extrapolating the finite size effect, they conclude that $T_\ell^\infty = T_m$ in the thermodynamic limit($N_z
\rightarrow \infty)$. Their argument is based on the following observations:1) $T_\ell$ decreases with an approximately linear dependence on increasing $N_z$, $T_\ell(N_z+\delta N_z)
\sim T_\ell(N_z) - c \cdot \delta N_z $ with a positive number $c$. 2) In the thermodynamic limit, below the melting transition($T \sim T_m$), the energy scale for the interlayer phase fluctuation $T^* \sim \xi^2 J_z$ diverges due to long range lattice order. Therefore, the linear progression can not continue below $T_m$. From these, they conclude that $T_\ell \rightarrow T_m$ in the thermodynamic limit.
We agree with the validity of the second assumption on general grounds. However, this does not exclude the possible existence of an intermediate phase in which the interlayer fluctuation may be suppressed due to the quasi-long range phase correlations in a line liquid “phase”. With this possibility open, $T_m$ is only a lower bound for the $T_\ell.$ It should also be noted that $\gamma_{zz}$ does not show a significant dependence on size in the region where $0.8 < \gamma_{zz} < 1,$ in temperatures $T_m < T < 1.8T_m.$ It is only at higher temperatures, near where $\gamma_{zz} \rightarrow 0$, that the linear dependence of the shift in $T_\ell$ on $N_z$ is observable. In other words, the size dependence of $\gamma_{zz}$ is not trivial as temperature varies and one should not expect the same size dependence be uniformly applied over the whole temperature range $T_m < T < T_{XY}.$
Furthermore, the linear dependence of shift in $T_\ell$ on $N_z$ in the region where $\gamma_{zz} \sim 0$ is anticipated on more general grounds. In the London limit, $\gamma_{zz}$ measured in the simulation under periodic boundary conditions is $$\label{lineareq}
\gamma_{zz}(q_x=\pi/\sqrt{N_xN_y}) \sim {J \over V\Gamma^2}
\Big[ 1 - {4J\over N_z \Gamma^2 T \pi} < n_y (q_x=\pi/N_x)
n_y(q+x = -\pi/N_x) > \Big]$$ where $n_y(q_x=\pi/N_x)$ is the Fourier component of the vorticity vector field lying along $\hat{y}$ direction. Position of $T_\ell$ is governed by the condition that the vortex fluctuations make the factor in the bracket vanish. Let us assume that there is a characteristic number of layers $N_z^*$ for which the true thermodynamic transition is realized at $T_\ell^*.$ For a size $N_z$ smaller than $N_z^*$, $N_z = N_z^* + \delta N_z$($\delta
N_z < 0)$, linearization of the condition gives $$T_\ell(N_z) \sim T_\ell^* + g(T_\ell^*) \delta N_z$$ with $g(T) = \Gamma^2 \partial [<n_y(\pi/N_x)n_y(-\pi/N_x)>/T ] / \partial T.$ What is the temperature dependence of $<n_y n_y>$? Near $T_m,$ where thermally activated vortex loops [*begin*]{} to appear, it is dominated by the vortex loop fugacity factor and is steeply increasing function of $T$ following an $S$-shaped curve. However, near the foot of $\gamma_{zz}$, where our $T_\ell$ is located, it is numerically observed that it has reached the plateau and the temperature dependence is dominated by the $1/T$ factor. It is also consistent with the interpretation of $T_\ell$ in terms of XY-type unbinding transition. Since $g(T_\ell) < 0$ and $\delta N_z < 0$ in this region, we then reach the conclusion that $T_\ell(N_z)$ linearly increases away from $T_\ell^*$ with $c\dot |N_z^* - N_z|$ as $N_z$ decreases from the asymptotic(thermodynamic) limit. Note that we do not assume $T_\ell^* = T_m$ in reaching this conclusion. If one should follow Nguyen and Sudb[ø]{} and extrapolate the observation of the linear dependence in the limited range of the system size and conclude that $T_\ell^*(B)=T_m(B)$, it also follows that we have a paradoxical consequence of predicting $T_{XY} \rightarrow 0$ for the zero field, based on the similar bahavior numerically observed for $\gamma_{\alpha\alpha}$ for $f=0.$
Following the convention of Koshelev, we employ the [*scaled field variable*]{} $h = \Gamma^2 f$ which characterizes the thermodynamics of the mixed state as long as $\lambda \rightarrow \infty$. As we pointed out earlier[@ryu97], $f=1/6$ with $\Gamma^2 = 1$ ($h = 1/6$) represents a situation where the interlayer decoupling sets in right at the melting transition. From our numerical results with $h$ varying from $1/6$ to $0$, we believe that there is a universal crossover value of scaled density $h$ between $1/6$ and $1/18$ which separates a [*low field*]{} regime from the [*high field*]{} regime for which $T_m = T_\ell = T_c(B).$ Koshelev[@koshelev97] made a numerical observation that $T_\ell \rightarrow
T_m.$ However, it is again made for $\Gamma^2 = J_{xy} / J_z =36, f =
1/36$, i. e. $h = 1$, equivalent to an extremely dense limit. Recent calculations by Hu and Tachiki[@hu97] based on $f=1/25, \Gamma^2
= 10$ in a larger system size of $50\times 50\times 40$ falls into the same category with $h=10/25 \gg 1/6.$ For this dense limit, they observe that $\gamma_{zz}$ drops sharply to zero at the melting transition, as we had observed earlier for the $h=1/6$ case in the stacked triangular XY model[@ryu97].
The technical difficulty of simulating the very low field limit $(h < 1/18)$ with large number of field induced vortex lines remains largely unsurmounted. To minimize the artificial grid pinning effect, one is required to choose a fairly small value of $f$( $< 1/32$ for the square grid, $< 1/16$ for the triangular grid ). However, choice of a large anisotropy factor to mimic HTSC then tends to push the model into the high field regime as pointed out earlier. To access the truly low field regime($B \ll 1 $ tesla for , $\ll 200 $ gauss for ), we had to employ an [isotropic]{} model for a practical value of $f < 1/18$. Summarizing this section, we make educated guess on the thermodynamics of extremely low field limit, where the field-induced vortex degrees of freedom are no longer viable[@tmzero]. Here, the transition at $T_\ell$ takes over the first order melting transition as the [*S-N*]{} transition which possibly belongs to the universality class of [*zero field transition of XY film*]{} with thickness $a_B$. The phase for $T_m < T < T_\ell$ may still be considered superconducting in the sense that a superconducting path, however narrow, may exist across a macroscopic distance along ${\bf B}$, which defines a tiny, but finite critical current. This may be enhanced by collective pinning of the single lines, but resistance measured with a large driving will show a non-linear IV characteristics and hysteresis as current has to distribute itself among the superconducting paths and normal channels.
The field induced lines in the low field limit, if we take the world-line analogy, are equivalent to extremely massive bosons and eventually drop out of the thermodynamics. They become localized charges which tend to polarize the underlying vacuum and induce a [*dielectric breakdown*]{} as the XY medium become more and more polarizable as T increases toward $T_\ell^*$. It is somewhat similar to a [*metal-insulator*]{} transition in a narrow band-gap semiconductor in which local field gradient(due to the field induced vortices) and shrinking band gap (as $T\rightarrow T_\ell$) conspire to a massive generation of screening dipole pairs (vortex loops) leading to a metallic state.
Conclusions
===========
To summarize, we confirmed the existence of a single first order melting transition at high vortex densities. We also showed that upon melting the local vortex density increases due to the screening effect of thermally generated vortex loops. More significantly, however, we observed that, in the absence of disorder, destruction of phase coherence in a superconductor may proceed by two separate transitions at low magnetic fields: a quasi-long range phase coherence parallel to the field disappears at a temperature $T_\ell$ higher than $T_m$ at which the lattice periodicity disappears and true long range phase coherence is lost. In this low-field regime, the lattice first melts into a liquid of lines with a finite entanglement length along the applied field. These lines eventually disappear through increasing entanglement, and through their interaction with thermally induced vortex and antivortex loops. While the melting transition is best characterized by the disappearance of Bragg peaks for the vortex lines and a delta function peak in the specific heat, there is a narrow region above $T_m$ where we observe dramatic changes in dissipation tensor which coincide with jump in the local vortex density and disappearance of the longitudinal phase rigidity, $\gamma_{zz}=0$. Instead of being a gradual crossover, we propose that a possible transition at $T_\ell \ne T_m$ sets in at low densities. It tracks the broad peak in specific heat as B increases, obeying the behavior of 3D zero field XY system confined to a semi-infinite slab of finite thickness $L_B \sim B^{-1/2}.$ It can alternatively described in terms of appearence of connected vortex tangle which effectively leads to decoupling of neighbouring layers. Within this picture, origins of several puzzling and conflicting anomalies recently obtained on and may be understood.
Acknowledgments
===============
This work has been supported by the Midwest Superconductivity Consortium at Purdue University, through Grant DE-FG02-90ER45427. Calculations were carried out, in part, on the network of SP-2 computers of the Ohio Supercomputer Center.
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a_B$.
|
---
abstract: |
In this work we apply model averaging to parallel training of deep neural network (DNN). Parallelization is done in a model averaging manner. Data is partitioned and distributed to different nodes for local model updates, and model averaging across nodes is done every few minibatches.
We use multiple GPUs for data parallelization, and Message Passing Interface (MPI) for communication between nodes, which allows us to perform model averaging frequently without losing much time on communication. We investigate the effectiveness of Natrual Gradient Stochasitc Gradient Descent (NG-SGD) and Restricted Boltzmann Machine (RBM) pretraining for parallel training in model-averaging framework, and explore the best setups in term of different learning rate schedules, averaging frequencies and minibatch sizes. It is shown that NG-SGD and RBM pretraining benefits parameter-averaging based model training. On the 300h Swithboard dataset, a 9.3 times speedup is achieved using 16 GPUs and 17 times speedup using 32 GPUs with limited decoding accuracy loss. [^1]
address: |
$^1$ International Computer Science Institute, Berkeley, California, US\
$^2$ Dept. of Electrical Engineering & Computer Science, University of California, Berkeley, CA, USA\
$^3$ Nanyang Technological University, Singapore\
[{suhang3240@gmail.com}]{}
bibliography:
- 'paper.bib'
title: Experiments on Parallel Training of Deep Neural Network using Model Averaging
---
Parallel training, model averaging, deep neural network, natural gradient
Introduction {#sec:intro}
============
Deep Neural Networks (DNN) has shown its effeciveness in several machine learning tasks, espencially in speech recognition. The large model size and massive training examples make DNN a powerful model for classification. However, these two factors also slow down the training procedure.
Parallelization of DNN training has been a popular topic since the revival of neural networks. Several different strategies have been proposed to tackle this problem. Multiple thread CPU parallelization and single GPU implementation are compared in [@scanzio2010parallel; @vesely2010parallel], and it is shown that single GPU could beat multi-threaded CPU implementation by a factor of 2.
Optimality for parallelization of DNN training was analyzed in [@seide2014parallelizability], and based on the analysis, a gradient quantization approach (1-bit SGD) was proposed to minimize communication cost [@seide20141]. It shows that 1 bit quantization can effectively reduce data exchange in an MPI framework, and a 10 times speed-up is achieved using 40 GPUs.
DistBelief proposed in [@dean2012large] reports that 8 CPU machines train 2.2 times faster than a single GPU machine on a moderately sized speech model. Asynchronous SGD using multiple GPUs achieved a 3.2x speed-up on 4 GPUs [@zhang2013asynchronous].
A pipeline training approach was propoased in [@chen2012pipelined] and a 3.3x speedup was achieved using 4 GPUs, but this method does not scale beyond number of layers in the neural network.
A speedup of 6x to 14x was achieved using 16 GPUs on training convolutional neural networks [@coates2013deep]. In this approach, each GPU is responsible for a partition of the neural network. This approach is more useful for image classification where local structure of the neural network could be exploited. For a fully connected speech model, a model partition approach may not be able to contribute as much.
Distributed model averaging is used in [@zhang2014improving; @miao2014distributed], and a further improvement is done using NG-SGD [@povey2014parallel]. In this approach, separate models are trained on multiple nodes using different partitions of data, and model parameters are averaged after each epoch. It is shown that NG-SGD can effectively improve convergence and ensure a better model trained using the model averaging framework.
Our approach is mainly based on the NG-SGD with model averaging. We utilize multiple GPUs in neural networks training via MPI, which allows us to perform model averaging more frequently and efficiently. Unlike the other approach [@seide20141], we do not use a warm-up phase where only single thread is used for model update. (Admittedly, this might lead to further improvement). In this work, we conduct a lot of experiments and compare different setups in model averaging framework.
In Section 2, we introduce related works on NG-SGD. Section 3 describe the model averaging approach and some intuition on the analysis. Section 4 records experimental results on different setups and Section 5 concludes.
Relationship to Prior Works
===========================
To avoid confusion, we should mention that Kaldi[@kaldi11] contains two neural network recipes. The first implementation [^2] is described in [@vesely2013sequence] which supports Restricted Boltzmann Machine pretraining [@hinton2006fast] and sequence-discriminative training [@povey2008boosted]. It uses single GPU for SGD training. The second implementation [^3] [@zhang2014improving] was originally designed to support parallel training on multiple CPUs. Now it also supports multiple GPUs for training using model averaging. By default, it uses layer-wise discriminative pretraining.
Our work extends the first implementation so that it can utilize multiple GPUs using model averaging. We use MPI in implementation, so file I/O is avoided during model averaging. This allows us to perform model averaging much more frequently.
Data parallelization and Model Averging
=======================================
SGD is a popular method for DNN training. Even though neural network training objectives are usually non-convex, mini-batch SGD has been shown to be effective for optimizing the objective[@seide2011conversational]. Roughly speaking, a bigger minibatch size gives a better estimate of the gradient, resulting in a better the converge rate. Thus, a straight forward idea for parallellization would be distributing the gradient computation to different computing nodes. In each step, gradients of minibatches on different nodes are reduced to a single node, averaged and then used to update models in each node. This method, i.e. gradient averaging, can compute the gradient accurately, but it requires heavy communication between nodes. Also, it is shown that increasing minibatch size does not always benefit model training[@seide2011conversational], especially in early stage of model training.
On the other hand, if we choose to average the parameters rather than gradients, it is not necessary to exchange data that often. Currently, there is no straight forward theory that guarantees convergence, but we would like to explore a bit why this strategy should work, just as we observe in the experiments.
First, in the extreme case where model parameters are averaged after each weight update, model averaging is equivalent to gradient averaging. Furthermore, if model averaging is done every $n$ minibatch based weight update, model update formula could be written as $$\begin{split}
\theta_{t+n} &= \theta_{t} +\sum_{i=0}^{n-1} \alpha g_{t+i}\\
&= \theta_{t} +\sum_{i=0}^{n-1} \alpha \frac{\partial}{\partial\theta} F(x;\theta_{t+i})
\end{split}$$ $${\mkern 1.5mu\overline{\mkern-1.5mu\theta_{t+n}\mkern-1.5mu}\mkern 1.5mu} = {\mkern 1.5mu\overline{\mkern-1.5mu\theta_{t}\mkern-1.5mu}\mkern 1.5mu} +\sum_{i=0}^{n-1} \alpha \frac{\partial}{\partial\theta} {\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu}(x;\theta_{t+i})$$ where $\theta$ is the model parameter and $\alpha$ is learning rate. If changes in model parameter $\theta$ is limited within $n$ updates, this approach could be seen as an approximation to gradient averaging.
Second, it is shown that model averging for convex models is guaranteed to converge [@mcdonald2010distributed; @mcdonald2009efficient]. It is suggested that unsupervised pretraining guides the learning towards basins of attraction of minima that support better generalization from the training data set; [@erhan2010does].
Fig \[fig:allreduce\] is an example of all-reduce with 4 nodes. This operation could be easily implemented by MPI\_Allreduce.
![All-reduce network[]{data-label="fig:allreduce"}](allreduce.png){width="50.00000%"}
Natural Gradient for Model Update
=================================
This section introduces the idea proposed in [@povey2014parallel].
In stochastic gradient descent (SGD), the learning rate is often assumed to be a scalar $\alpha_t$ that may change over time, the update formula for model parameters $\theta_{t}$ is $$\theta_{t+1} = \theta_{t} + \alpha_t g_t$$ where $g_t$ is the gradient.
However, according to Natural Gradient idea [@murata1999statistical; @roux2008topmoumoute], it is possible to replace the scalar with a symmetric positive definite matrix $E_t$, which is the inverse of the Fisher information matrix. $$\theta_{t+1} = \theta_{t} + \alpha_t E_t g_t$$
Suppose $x$ is the variable we are modeling, and $f(x;\theta)$ is the probability or likelihood of $x$ given parameters $\theta$, then the Fisher information matrix $I(\theta)$ is defined as $$E\bigg[\bigg(\frac{\partial}{\partial\theta}\log f(x;\theta)\bigg) \bigg(\frac{\partial}{\partial\theta}\log f(x;\theta)\bigg)^\top\bigg]$$
For large scale speech recognition, it is impossible to estimate Fisher information matrix and perform inversion, so it is necessary to approximate the inverse Fisher information matrix directly. Details about the theory and implementation of NG-SGD could be found in [@povey2014parallel].
Experimental Results
====================
Setup
-----
In this work, we report speech recognition results on the 300 hour Switchboard conversational telephone speech task (Switchboard-1 Release 2). We use MSU-ISIP release of the Switchboard segmentations and transcriptions (date 11/26/02), together with the Mississippi State transcripts2 and the 30Kword lexicon released with those transcripts. The lexicon contains pronunciations for all words and word fragments in the training data. We use the Hub5 ’00 data for evaluation. Specifically, we use the the development set and Hub5 ’01 (LDC2002S13) data as a separate test set.
The Kaldi toolkit[@kaldi11] is used for speech recognition framework. Standard 13-dim PLP feature, together with 3-dim Kaldi pitch feature, is extracted and used for maximum likelihood GMM model training. Features are then transformed using LDA+MLLT before SAT training. After GMM training is done, a tanh-neuron DNN-HMM hybrid system is trained using the the 40-dimension transformed fMLLR (also known as CMLLR [@gales1996generation]) feature as input and GMM-aligned senones as targets. fMLLR is estimated in an EM fashion for both training data and test data. A trigram language model (LM) is trained on 3M words of the training transcripts only.
Work in this paper is built on top of the Kaldi nnet1 setup and the NG-SGD method introduced in nnet2 setup. Details of DNN training follows Section 2.2 in [@vesely2013sequence]. In this work, we use 6 hidden layers, where each hidden layer has 2048 neurons with sigmoids. Input layer is 440 dimension (i.e. the context of 11 fMLLR frames), and output layer is 8806 dimension. Mini-batch SGD is used for backpropagation and the minibatch is set to 1024 for all the experiments. By defult, DNNs are initialized with stacked restricted Boltzmann machines (RBMs) that are pretrained in a greedy layerwise fashion [@hinton2006fast]. Comparison between random initialization and RBM-initialization in model averaging framework is reported in Section \[sec:init\].
The server hardware used in this work is Stampede (TACC) (URL: https://portal.xsede.org/tacc-stampede). It is a Dell Linux cluster provided as an Extreme Science Engineering Discovery Environment (XSEDE) digital service by the Texas Advanced Computing Center (TACC). Stampede is configured with 6,400 Dell DCS Zeus compute nodes, the majority of which are configured with two 2.7 GHz E5-2680 Intel Xeon (Sandy Bridge) processors and one Intel Xeon Phi SE10P coprocessor. 128 of the nodes are augmented with an NVIDIA K20 GPU and 8 GB of GDDR5 memory each, which we use for neural network training in this work. Stampede nodes run Linux 2.6.32 with batch services managed by the Simple Linux Utility for Resource Management (SLURM).
Switchboard Results
-------------------
Fig. \[fig:scaling\] shows the scaling factor and speedup plot for model averaging experiments. As is shown in the graph, a speedup of 17 could be achieved when 32 GPUs are used.
![Scaling factor and speedup factor v.s. number of gpus[]{data-label="fig:scaling"}](scaling.png){width="50.00000%"}
Table \[tab:wer\] shows the main decoding results for DNNs trained using different number of GPUs. In general, decoding results of DNNs trained model averaging degrades 0.30.4 WER, depending on the number of GPUs used.
1 2 4 8 16 32
---------- ------ --- --- ------ ------ ------
SWB 14.7 – – 15.1 15.1 15.2
CallHM 26.8 – – 27.4 27.0 27.1
SWB 16.1 – – 16.4 16.2 16.4
SWB2P3 21.0 – – 21.8 21.7 21.7
SWB-Cell 27.4 – – 27.3 27.4 27.8
: Comparison of WERs using different number of GPUs[]{data-label="tab:wer"}
Initialization Matters {#sec:init}
----------------------
Table \[tab:init\] compares random initialization with Restricted Boltzmann Machine (RBM) based initialization.
----------------- ------ ------ ------ ------
Nodes 1 32 1 32
random init 15.6 16.4 27.4 28.8
RBM pretraining 14.7 15.2 26.8 27.1
----------------- ------ ------ ------ ------
: Comparing RBM pretraining with random initialization[]{data-label="tab:init"}
As we can see in the table, random initialization is worse than DNN with RBM pretraining by 0.9/0.6 in single GPU case. While in model averaging setup, random initialization becomes even worse – 0.3/0.9 point more degradation on WER.
Averaging frequency
-------------------
Averaging frequency here is defined as the number of minibatch-SGD performed per model averaging. Due to the limitation of computing resource, we only did preliminary experiments on this. Minibatch size of 1024 is set as default, and we compare averaging frequency of 10 and 20. It is shown in Table \[tab:freq\] that an averaging frequency of 10 gives slight worse speedup but a better decoding WER. The tradeoff between lower averaging frequency (i.e. better speedup) and better training accuracy is within expectation in that frequent model averaging means steady gradient estimation.
frequency Speedup SWB CallHome
----------- --------- ------ ----------
baseline – 14.7 26.8
10 9.32 15.1 27.0
20 10.07 15.8 28.0
: Comparing different averaging frequencies[]{data-label="tab:freq"}
Minibatch Size
--------------
Table \[tab:mbsize\] compares two different minibatch size in model averaging setup.
Speedup
------- ------ ------ --------- ------ ------
nodes 1 16 1 16
256 15.3 15.6 26.8 27.3 –
1024 14.7 15.1 26.8 27.0 9.32
: Comparing different minibatch size[]{data-label="tab:mbsize"}
Learing Rate Schedule
---------------------
Initial learning rate is increased in porportion to number of threads in model averaging setup. The reason for this is straight forward: Assume we have $n$ minibatches of data for model training. When the model is trained using single thread, it gets updated $n$ times. When data is distributed to $m$ machines, then each model gets updated $n/m$ times. Since the effect of model averaging is mostly aggregating knowledge learnt from different data partition, the absolute change of model shall be compensated by $m$ times.
We compare two learning rate schedules in this section. The first one is the default setup used in Kaldi nnet1 (Newbob). It starts with a initial learning rate of 0.32 and halves the rate when the improvement in frame accuracy on a cross-validation set between two successive epochs falls below 0.5%. The optimization terminates when the frame accuracy increases by less than 0.1%. Cross-validation is done on 10% of the utterances that are held out from the training data.
The second learing rate schedule is exponentially decaying. This method is used in [@senior2013empirical; @povey2014parallel] and is shown to be superior to performance scheduling and power scheduling. In this work, it starts with the same initial learning rate as the first method (Newbob), and decrease to the final learning rate (which is set to be 0.01 \* initial learning rate). The number of epochs is set to $15$ in this task, which is set to be the same as Newbob scheduling.
As is shown in Table \[tab:lr\], these two learning rate scheduling methods give similar decoding results. However, exponential learning rate might need more tuning since it requires a initial learning rate, a final learning rate and predefined number of epochs to train.
------------- ------ ------ ------ ------
Nodes 1 16 1 16
Newbob 14.9 15.4 26.6 27.2
exponential 14.7 15.1 26.8 27.0
------------- ------ ------ ------ ------
: Comparing learning rate schedule[]{data-label="tab:lr"}
Online NG-SGD Matters
---------------------
Table \[tab:ngsgd\] compares plain SGD with NG-SGD in model averaging mode, and it shows NG-SGD is crucial to model training with parameter-averaging.
-------- ------ ------ ------ ------
Nodes 1 16 1 16
SGD 14.9 16.3 26.9 28.3
NG-SGD 14.7 15.1 26.8 27.0
-------- ------ ------ ------ ------
: Comparing NG-SGD and naive SGD[]{data-label="tab:ngsgd"}
Conclusion and Future Work
==========================
In this work, we show that neural network training can be efficiently speeded up using model averaging. on a 300h Switchboard dataset, a 9.3x / 17x speedup could be achieved using 16 / 32 GPUs respectively, with limited decoding accuracy loss. We also show that model averaging benefits a lot from NG-SGD and RBM based pretraining. Preliminary experiments on minibatch size, averaging frequency and learning rate schedules are also presented.
Further accuracy improvement might be achieved if parallel training runs on top of serial training initialization. It would be interesting to see if sequence-discriminative training combines well with model averaging. Speedup factor could be further improved if CUDA aware MPI is used. Theory on convergence using model averaging is to be explored, which might be useful for guiding future development.
Acknowledgements
================
We would like to thank Karel Vesely and Daniel Povey who wrote the original “nnet1” neural network training code and natural gradient stochastic gradient descent upon which the work here is based. We would also like to thank Nelson Morgan, Forrest Iandola and Yuansi Chen for their helpful suggestions.
We acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this work (URL: http://www.tacc.utexas.edu).
[^1]: This work is not submitted to peer-review conferences because the authors think it needs more investigation. The authors are in lack of resources to perform further exploration. However, we welcome any comments and suggestions.
[^2]: Location in code: src/{nnet,nnetbin}
[^3]: Location in code: src/[nnet2,nnet2bin]{}
|
---
abstract: 'Local reasoning about programs exploits the natural local behaviour common in programs by focussing on the footprint - that part of the resource accessed by the program. We address the problem of formally characterising and analysing the notion of footprint for abstract local functions introduced by Calcagno, O’Hearn and Yang. With our definition, we prove that the footprints are the only essential elements required for a complete specification of a local function. We formalise the notion of small specifications in local reasoning and show that, for well-founded resource models, a smallest specification always exists that only includes the footprints. We also present results for the non-well-founded case. Finally, we use this theory of footprints to investigate the conditions under which the footprints correspond to the smallest safe states. We present a new model of RAM in which, unlike the standard model, the footprints of every program correspond to the smallest safe states. We also identify a general condition on the primitive commands of a programming language which guarantees this property for arbitrary models.'
address: |
Department of Computing,\
Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK\
author:
- Mohammad Razaa
- Philippa Gardnerb
title: Footprints in Local Reasoning
---
Introduction
============
Local reasoning about programs focusses on the collection of resources directly acted upon by the program. It has recently been introduced and used to substantial effect in [*local*]{} Hoare reasoning about memory update. Researchers previously used Hoare reasoning based on First-order Logic to specify how programs interacted with the [ *whole*]{} memory. O’Hearn, Reynolds and Yang instead introduced [ local]{} Hoare reasoning based on Separation Logic [@ORY01; @IO01]. The idea is to reason only about the local parts of the memory—the [*footprints*]{}—that are accessed by a program. Intuitively, the footprints form the pre-conditions of the [*small*]{} axioms, which provide the smallest complete specification of the program. All the true Hoare triples are derivable from the small axioms and the general Hoare rules. In particular, the [ *frame rule*]{} extends the reasoning to properties about the rest of the heap which has not been changed by the command.
O’Hearn, Reynolds and Yang originally introduced Separation Logic to solve the problem of how to reason about the mutation of data structures in memory. They have applied their reasoning to several memory models, including heaps based on pointer arithmetic [@ORY01], heaps with permissions [@BCOP05], and the combination of heaps with variable stacks which views variables as resource [@BCY05; @PBC06]. In each case, the basic soundness and completeness results for local Hoare reasoning are essentially the same. For this reason, Calcagno, O’Hearn and Yang [@COY07] recently introduced abstract local functions over abstract resource models which they call separation algebras. They generalised their specific examples of local imperative commands and memory models in this abstract framework. They introduced Abstract Separation Logic to provide local Hoare reasoning about such functions, and give general soundness and completeness results.
We believe that the general concept of a local function is a fundamental step towards establishing the theoretical foundations of local reasoning, and Abstract Separation Logic is an important generalisation of the local Hoare reasoning systems now widely studied in the literature. However, Calcagno, O’Hearn and Yang do not characterise the footprints and small axioms in this general theory, which is a significant omission. O’Hearn, Reynolds and Yang, in one of their first papers on the subject [@ORY01], state the local reasoning viewpoint as:
> ‘to understand how a program works, it should be possible for reasoning and specification to be confined to the cells that the program actually accesses. The value of any other cell will automatically remain unchanged.’
A complete understanding of the foundations of local Hoare reasoning therefore requires a formal characterisation of the footprint notion. O’Hearn tried to formalise footprints in his work on Separation Logic (personal communication with O’Hearn). His intuition was that the footprints should be the smallest states on which the program is safe - the [*safety footprint*]{}, and that the [*small axioms*]{} arising from these footprints should give rise to a complete specification using the general rules for local Hoare reasoning. However, Yang discovered that this notion of footprint does not work, since it does not always yield a [*complete*]{} specification for the program. Consider the program[^1] $$\mathit{AD} ::= \quad x := new();dispose(x)$$ This *allocate-deallocate* program allocates a new cell, stores its address value in the stack variable $x$, and then deallocates the cell. It is local because all its atomic constituents are local. This tiny example captures the essence of a common type of program; there are many programs which, for example, create a list, work on the list, and then destroy the list.
The smallest heap on which the *AD* program is safe is the empty heap $emp$. The specification using this pre-condition is: $$\begin{aligned}
\{emp\} \quad \mathit{AD} \quad \{emp\}\end{aligned}$$ We can extend our reasoning to larger heaps by applying the frame rule: for example, extending to a one-cell heap with arbitrary address $l$ and value $v$ gives $$\begin{aligned}
\{l \mapsto v \} \quad \mathit{AD} \quad \{l \mapsto v\}\end{aligned}$$ However, axiom (1) does not give the complete specification of the *AD* program. In fact, it captures very little of the spirit of allocation followed by de-allocation. For example, the following triple is also true: $$\begin{aligned}
\{l \mapsto v\} \quad \mathit{AD} \quad \{l \rightarrow v \wedge x \neq l\} \end{aligned}$$ This triple (3) is true because, if $l$ is already allocated, then the new address cannot be $l$ and hence $x$ cannot be $l$. It cannot be derived from (1). However, the combination of axiom (1) and axiom (3) for arbitrary one-cell heaps does provide the smallest complete specification. This example illustrates that O’Hearn’s intuitive view of the footprints as the minimal safe states just does not work for common imperative programs.
In this paper, we introduce the formal definition of the footprint of a local function that does yield a complete specification for the function. For our *AD* example, our definition identifies $emp$ and the arbitrary one-cell heaps $l \mapsto v$ as footprints, as expected. We prove the general result that, for any local function, the footprints are the only elements which are [*essential*]{} to specify completely the behaviour of this function.
We then investigate the question of [*sufficiency*]{}. For well-founded resource, we show that the footprints are also always sufficient: that is, a complete specification always exists that only uses the footprints. We also explore results for the non-well-founded case, which depend on the presence of [*negativity*]{}. A resource has negativity if it is possible to combine two non-unit elements to get the unit, which is like taking two non-empty pieces of resource and joining them to get nothing. For non-well-founded models without negativity, such as heaps with infinitely divisible fractional permissions, either the footprints are sufficient (such as for the *write* command in the permissions model) or there is no smallest complete specification (such as for the [*read*]{} command in the permissions model). For models with negativity, such as the integers under addition, we show that there do exist smallest complete specifications based on elements that are not essential and hence not footprints.
In the final section, we apply our theory of footprints to the issue of regaining the safety footprints. We address a question that arose from discussions with O’Hearn and Yang, which is whether there is an alternative model of RAM in which the safety footprint does correspond to the actual footprint, yielding complete specifications. We present such a model based on an examination of the cause of the *AD* problem in the original model. We prove that in this new model the footprint of [*every*]{} program, including *AD*, does correspond to the safety footprint. Moreover, we identify a general condition on the primitive commands of a programming language which ensures that this property holds in arbitrary models.
A preliminary version of this paper was presented at the FOSSACS 2008 conference. The final section reports on work that is new to this journal version. This paper also contains the proofs which were excluded from the conference paper.
Background {#sec:separationalgebras}
==========
The discussion in this paper is based on the framework introduced in [@COY07], where the approach of local reasoning about programs with separation logic was generalised to local reasoning about *local* functions that act on an abstract model of resource. Our objective in this work is to investigate the notion of footprint in this abstract setting, and this section gives a description of the underlying framework.
Separation Algebras and Local Functions
---------------------------------------
We begin by describing separation algebras, which provide a model of resource which generalises over the specific heap models used in separation logic works. Informally, a separation algebra models resource as a set of elements that can be ‘glued’ together to create larger elements. The ‘glueing’ operator satisfies properties in accordance with this resource intuition, such as commutativity and associativity, as well as the cancellation property which requires that, if we are given an element and a subelement, then ‘ungluing’ that subelement gives us a unique element.
A [**separation algebra**]{} is a cancellative, partial commutative monoid $(\Sigma,\bullet,u)$, where $\Sigma$ is a set and $\bullet$ is a partial binary operator with unit $u$. The operator satisfies the familiar axioms of associativity, commutativity and unit, using a partial equality on $\Sigma$ where either both sides are defined and equal, or both are undefined. It also satisfies the cancellative property stating that, for each $\sigma \in \Sigma$, the partial function $\sigma\bullet (\cdot ): \Sigma {\!\mapsto\!}\Sigma $ is injective.
We shall sometimes overload notation, using $\Sigma$ to denote the separation algebra $(\Sigma,\bullet,u)$. Examples of separation algebras include multisets with union and unit $\emptyset$, the natural numbers with addition and unit $0$, heaps as finite partial functions from locations to values ( [@COY07] and example \[locsepexamples\]), heaps with permissions [@COY07; @BCOP05], and the combination of heaps and variable stacks enabling us to model programs with variables as local functions ( [@COY07], [@PBC06] and example \[locsepexamples\]). These examples all have an intuition of resource, with $\sigma_1 \bullet \sigma_2$ intuitively giving more resource than just $\sigma_1$ and $ \sigma_2$ for $\sigma_1 , \sigma_2 \neq u$. However, notice that the general notion of a separation algebra also permits examples which may not have this resource intuition, such as $\{a,u\}$ with $a \bullet a = u$. Since our aim is to investigate general properties of local reasoning, our inclination is to impose minimal restrictions on what counts as resource and to work with a simple definition of a separation algebra.
Given a separation algebra $(\Sigma,\bullet,u)$, the [**separateness**]{} ($\#$) relation between two states $\sigma_0, \sigma_1 \in \Sigma$ is given by $\sigma_0\# \sigma_1 \;\mbox{iff}\; \sigma_0 \bullet \sigma_1\;\mbox{is defined}$. The [**substate**]{} ($\preceq$) relation is given by $\sigma_0\preceq \sigma_1 \;\mbox{iff}\; \exists \sigma_2.\, \sigma_1=\sigma_0\bullet \sigma_2$. We write $\sigma_0\prec \sigma_1$ when $\sigma_0\preceq \sigma_1$ and $\sigma_0 \neq \sigma_1$.
\[subtraction\] For $\sigma_1, \sigma_2 \in \Sigma$, if $\sigma_1 \preceq \sigma_2$ then there exists a unique element denoted $\sigma_2 - \sigma_1 \in \Sigma$, such that $(\sigma_2 - \sigma_1) \bullet \sigma_1 = \sigma_2$.
Existence follows by definition of $\preceq$. For uniqueness, assume there exist $\sigma', \sigma'' \in \Sigma$ such that $\sigma' \bullet \sigma_1 = \sigma_2$ and $\sigma'' \bullet \sigma_1 = \sigma_2$. Then we have $\sigma' \bullet \sigma_1 = \sigma'' \bullet \sigma_1$, and thus by the cancellation property we have $\sigma' = \sigma''$. We consider functions on separation algebras that generalise imperative programs operating on heaps. Such programs can behave non-deterministically, and can also *fault*. To model non-determinism, we consider functions from a separation algebra $\Sigma$ to its powerset $\mathcal{P}(\Sigma)$. To model faulting, we add a special top element $\top$ to the powerset. We therefore consider total functions of the form $f:\Sigma\rightarrow \mathcal{P}(\Sigma)^\top$. On any element of $\Sigma$, the function can either map to a set of elements, which models *safe* execution with non-deterministic outcomes, or to $\top$, which models a faulting execution. Mapping to the empty set represents divergence (non-termination).
The standard subset relation on the powerset is extended to $\mathcal{P}(\Sigma)^\top$ by defining $p \sqsubseteq \top$ for all $p \in \mathcal{P}(\Sigma)^\top$. The binary operator $\ast$ on $\mathcal{P}(\Sigma)^\top$ is given by $$\begin{aligned}
p*q &=& \{\sigma_0\bullet \sigma_1 \mid \sigma_0 \# \sigma_1 \wedge
\sigma_0 \in p \wedge \sigma_1 \in q\} \quad \mathit{if}\; p, q \in \mathcal{P}(\Sigma)\\
&=& \top \quad otherwise\end{aligned}$$ $\mathcal{P}(\Sigma)^\top$ is a total commutative monoid under $\ast$ with unit $\{u\}$.
For functions $f, g:\Sigma\rightarrow \mathcal{P}(\Sigma)^\top$, $f \sqsubseteq g$ iff $f(\sigma) \sqsubseteq g(\sigma)$ for all $\sigma \in \Sigma$.
We shall only consider functions that are *well-behaved* in the sense that they act *locally* with respect to resource. For imperative commands on the heap model, the locality conditions were first characterised in [@YO02], where a soundness proof for local reasoning with separation logic was demonstrated for the specific heap model. The conditions identified were
1. *Safety monotonicity*: if the command is safe on some heap, then it is safe on any larger heap.
2. *Frame property*: if the command is safe on some heap, then in any outcome of applying the command on a larger heap, the additional heap portion will remain unchanged by the command.
In [@COY07], these two properties were amalgamated and formulated for abstract functions on arbitrary separation algebras.
\[def:localaction\] A [**local function on $\Sigma$**]{} is a total function $f:\Sigma\rightarrow \mathcal{P}(\Sigma)^\top$ which satisfies the [**locality condition**]{}: $$\sigma \# \sigma'\;\;\mbox{implies}\;\; f(\sigma' \bullet \sigma) \sqsubseteq \{\sigma'\} * f(\sigma)$$ We let $LocFunc$ be the set of local functions on $\Sigma$.
Intuitively, we think of a command to be local if, whenever the command executes safely on any resource element, then the command will not ‘touch’ any additional resource that may be added. Safety monotonicity follows from the above definition because, if $f$ is safe on $\sigma$ ($f(\sigma) \sqsubset \top$), then it is safe on any larger state, since $f(\sigma'\bullet\sigma) \sqsubseteq \{\sigma'\} *
f(\sigma) \sqsubset \top$.
The frame property follows by the fact that the additional state $\sigma'$ is preserved in the output of $f(\sigma'\bullet\sigma)$. Note, however, that the $\sqsubseteq$ ordering allows for reduced non-determinism on larger states. This, for example, is the case for the $AD$ command from the introduction which allocates a cell, assigns its address to stack variable $x$, and then deallocates the cell. On the empty heap, its result would allow all possible values for variable $x$. However, on the larger heap where cell 1 is already allocated, its result would allow all values for $x$ except 1, and we therefore have a more deterministic outcome on this larger state.
Locality is preserved under sequential composition, non-deterministic choice and Kleene-star, which are defined as $$\eqalign{
(f;g)(\sigma)
&= \left\{ \begin{array}{ll}
\begin{array}{l} \top \end{array} & \begin{array}{l}\mbox{ if } f(\sigma) = \top \end{array} \\
\begin{array}{l} \bigsqcup \{ g(\sigma') \mid \sigma' \in f(\sigma)\} \end{array} & \begin{array}{l}\mbox{ otherwise} \end{array}
\end{array}\right.\cr
(f+g)(\sigma) &= f(\sigma) \sqcup g(\sigma)\cr
f^\ast(\sigma)&= \displaystyle\bigsqcup_{n} f^n(\sigma)\cr}$$
\[locsepexamples\]
(1) [**Plain heap model**]{}. A simple example is the separation algebra of heaps $(H, \bullet, u_H)$, where $H = L \rightharpoonup_{{\mathit{fin}}} Val$ are finite partial functions from a set of locations $L$ to a set of values $Val$ with $L \subseteq Val$, the partial operator $\bullet$ is the union of partial functions with disjoint domains, and the unit $u_H$ is the function with the empty domain. For $h \in H$, let $dom(h)$ be the domain of $h$. We write $l \mapsto v$ for the partial function with domain $\{l\}$ that maps $l$ to $v$. For $h_1, h_2 \in H$, if $h_2 \preceq h_1$ then $h_1 - h_2 = h_1\!\mid_{dom(h_1) - dom(h_2)}$. An example of a local function is the $dispose[l]$ command that deletes the cell at location $l$: $$dispose[l](h)
= \left\{ \begin{array}{ll}
\begin{array}{l}\{h - (l {\!\mapsto\!}v)\} \end{array} & \; h \succeq (l {\!\mapsto\!}v) \\
\begin{array}{l} \top \end{array} & \; \mbox{otherwise}
\end{array}\right.$$ The function is local: if $h \not \succeq (l {\!\mapsto\!}v)$ then $dispose[l](h) = \top$, and $dispose[l](h' \bullet h) \sqsubseteq \top$. Otherwise, $dispose[l](h' \bullet h)
= \{(h'\bullet h) - (l {\!\mapsto\!}v)\}
\sqsubseteq \{h'\} * \{h - (l {\!\mapsto\!}v)\}
= \{h'\} * dispose[l](h)$.
(2) [**Heap and stack**]{}. There are two approaches to modelling the stack in the literature. One is to treat the stack as a total function from variables to values, and only combine two heap and stack pairs if the stacks are the same. The other approach, which we use here, is to allow splitting of the variable stack and treat it as part of the resource. We can incorporate the variable stack into the heap model by using the set $H = L \cup Var
\rightharpoonup_{{\mathit{fin}}} Val$, where $L$ and $Val$ are as before and $Var$ is the set of stack variables $\{x, y, z, ...\}$. The $\bullet$ operator combines heap and stack portions with disjoint domains, and is undefined otherwise. The unit $u_H$ is the function with the empty domain which represents the empty heap and empty stack. Although this approach is limited to disjoint reference to stack variables, this constraint can be lifted by enriching the separation algebra with *permissions* [@BCOP05]. However, this added complexity using permissions can be avoided for the discussion in this paper. For a state $h \in H$, we let $loc(h)$ and $var(h)$ denote the set of heap locations and stack variables in the domain of $h$ respectively. In this model we can define the allocation and deallocation commands as $$\quad\enspace\enspace\eqalign{
new[x](h)
&= \left\{ \begin{array}{ll}
\begin{array}{l}\{ h' \bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}w \mid w \in Val, l \in L \backslash loc(h')\} \end{array} & \begin{array}{l}h = h'\bullet x {\!\mapsto\!}v \end{array} \\
\begin{array}{l}\top \end{array} & \begin{array}{l}\mbox{otherwise} \end{array}
\end{array}\right.\cr
dispose[x](h)
&= \left\{ \begin{array}{ll}
\begin{array}{l}\{ h' \bullet x {\!\mapsto\!}l \} \end{array} & \begin{array}{l}h = h'\bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}v \end{array} \\
\begin{array}{l}\top \end{array} & \begin{array}{l}\mbox{otherwise} \end{array}
\end{array}\right.\cr}$$
Commands for heap mutation and lookup can be defined as $$\eqalign{
mutate[x,v](h)
&= \left\{ \begin{array}{ll}
\begin{array}{l}\{ h' \bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}v \} \end{array} & \begin{array}{l}h = h'\bullet x {\!\mapsto\!}l\bullet l {\!\mapsto\!}w \end{array} \\
\begin{array}{l}\top \end{array} & \begin{array}{l}\mbox{otherwise} \end{array}
\end{array}\right.\cr
lookup[x,y](h)
&= \left\{ \begin{array}{ll}
\begin{array}{l}\{ h' \bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}v \bullet y {\!\mapsto\!}v \} \end{array} & \begin{array}{l}h = h'\bullet x {\!\mapsto\!}l\bullet l {\!\mapsto\!}v \bullet y {\!\mapsto\!}w \end{array} \\
\begin{array}{l}\top \end{array} & \begin{array}{l}\mbox{otherwise} \end{array}
\end{array}\right.\cr}$$ The *AD* command described in the introduction, which is the composition $new[x];dispose[x]$, corresponds to the following local function $$AD(h)
= \left\{ \begin{array}{ll}
\begin{array}{l}\{h' \bullet x {\!\mapsto\!}l \mid l \in L\backslash loc(h')\} \end{array} & \begin{array}{l}h = h'\bullet x {\!\mapsto\!}v \end{array} \\
\begin{array}{l}\top \end{array} & \begin{array}{l}\mbox{otherwise} \end{array}
\end{array}\right.\qquad\enspace$$ Note that in all cases, any stack variables that the command refers to should be in the stack in order for the command to execute safely, otherwise the command will be acting non-locally.
(3) [**Integers**]{}. The integers form a separation algebra under addition with identity 0. In this case we have that any ‘adding’ function $f(x) = \{x + c\}$ that adds a constant $c$ is local, while a function that multiplies by a constant $c$, $f(x) = \{cx\}$, is non-local in general. However, the integers under multiplication also form a separation algebra with identity 1, and in this case every multiplying function is local but not every adding function. This illustrates the point that the notion of locality of commands depends on the notion of separation of resource that is being used.
Predicates, Specifications and Local Hoare Reasoning
----------------------------------------------------
We now present the local reasoning framework for local functions on separation algebras. This is an adaptation of Abstract Separation Logic [@COY07], with some minor changes in formulation for the purposes of this paper. Predicates over separation algebras are treated simply as subsets of the separation algebra.
\[definition:predicates\] A [**predicate**]{} $p$ over $\Sigma$ is an element of the powerset $\mathcal{P}(\Sigma)$.
Note that the top element $\top$ is not a predicate and that the $*$ operator, although defined on $\mathcal{P}(\Sigma)^\top \times \mathcal{P}(\Sigma)^\top \rightarrow \mathcal{P}(\Sigma)^\top$, acts as a binary connective on predicates. We have the distributive law for union that, for any $X \subseteq \mathcal{P}(\Sigma)$, $$(\bigsqcup X) * p = \bigsqcup \{ x*p\mid x\in X\}$$ The same is not true for intersection in general, but does hold for $precise$ predicates. A predicate is precise if, for any state, there is at most a single substate that satisfies the predicate.
\[precise\] A predicate $p \in \mathcal{P}(\Sigma)$ is [**precise**]{} iff, for every $\sigma \in \Sigma$, there exists at most one $\sigma_p\in p$ such that $\sigma_p\preceq\sigma$.
Thus, with precise predicates, there is at most a unique way to break a state to get a substate that satisfies the predicate. Any singleton predicate $\{\sigma\}$ is precise. Another example of a precise predicate is $\{l {\!\mapsto\!}v \mid v \in Val\}$ for some $l$, while $\{l {\!\mapsto\!}v \mid l \in L\}$ for some $v$ is not precise.
\[precisecharacterization\] A predicate $p$ is precise iff, for all $X \subseteq \mathcal{P}(\Sigma)$, $(\bigsqcap X) * p = \bigsqcap \{ x * p \mid x\in X\}$
We first show the left to right direction. Assume $p$ is precise. We have to show that for all $X \subseteq \mathcal{P}(\Sigma)$, $(\bigsqcap X) * p = \bigsqcap \{ x * p \mid x\in X\}$. Assume $\sigma \in (\bigsqcap X) * p$. Then there exist $\sigma_1, \sigma_2$ such that $\sigma = \sigma_1\bullet\sigma_2$ and $\sigma_1 \in \bigsqcap X$ and $\sigma_2 \in p$. Thus for all $x \in X$, $\sigma \in x * p$, and hence $\sigma \in \bigsqcap \{ x * p \mid x\in X\}$. Now assume $\sigma \in \bigsqcap \{ x * p \mid x\in X\}$. Then $\sigma \in x * p$ for all $x \in X$. Hence there exists $\sigma_1 \preceq \sigma$ such that $\sigma_1 \in p$. Since $p$ is precise, $\sigma_1$ is unique. Let $\sigma_2 = \sigma - \sigma_1$. Thus we have $\sigma_2 \in x$ for all $x \in X$, and so $\sigma_2 \in \bigsqcap X$. Hence we have $\sigma \in (\bigsqcap X) * p$.
For the other direction, we assume that $p$ is not precise and show that there exists an $X$ such that $(\bigsqcap X) * p \neq \bigsqcap \{ x * p \mid x\in X\}$. Since $p$ is not precise, there exists $\sigma \in \Sigma$ such that, for two distinct $\sigma_1, \sigma_2 \in p$, we have $\sigma_1 \preceq \sigma$ and $\sigma_2 \preceq \sigma$. Let $\sigma'_1 = \sigma - \sigma_1$ and $\sigma'_2 = \sigma - \sigma_2$. Now let $X = \{\{\sigma'_1\},\{\sigma'_2\}\}$. Since $\sigma \in \{\sigma'_1\} * p$ and $\sigma \in \{\sigma'_2\} * p$, we have $\sigma \in \bigsqcap \{ x * p \mid x\in X\}$. However, because of the cancellation property, we also have that $\sigma'_1 \neq \sigma'_2$, and so $(\bigsqcap X) * p = \emptyset * p = \emptyset$. Hence, $\sigma \not \in (\bigsqcap X) * p$, and we therefore have $(\bigsqcap X) * p \neq \bigsqcap \{ x * p \mid x\in X\}$.
Our Hoare reasoning framework is formulated with tuples of pre- and post- conditions, rather than the usual Hoare triples that include the function as in [@COY07]. In our case the standard triple shall be expressed as a function $f$ [*satisfying*]{} a tuple $(p, q)$, written $f \models (p, q)$. The reason for this is that we shall be examining the properties that a pre- and post- condition tuple may have with respect to a given function, such as whether a given tuple is complete for a given function. This approach is very similar to the notion of the *specification statement* (a Hoare triple with a ‘hole’) introduced in [@M88], which is used in refinement calculi, and was also used to prove completeness of a local reasoning system in [@YO02].
Let $\Sigma$ be a separation algebra. A [**statement**]{} on $\Sigma$ is a tuple $(p, q)$, where $p, q \in \mathcal{P}(\Sigma)$ are predicates. A [**specification**]{} $\phi$ on $\Sigma$ is a set of statements. We let $\Phi_{\Sigma} = \mathcal{P}(\mathcal{P}(\Sigma) \times \mathcal{P}(\Sigma))$ denote the set of all specifications on $\Sigma$. We shall exclude the subscript when it is clear from the context. The [**domain**]{} of a specification is defined as $D(\phi) = \bigsqcup \{p \mid (p, q) \in \phi\}$. [**Domain equivalence**]{} is defined as $\phi \cong_{D} \psi \mbox{ iff } D(\phi) = D(\psi)$.
Thus the domain is the union of the preconditions of all the statements in the specification. It is one possible measure of *size*: how much of $\Sigma$ the specification is referring to. We also adapt the notion of precise predicates to specifications.
\[precisesaturatedspecification\] A specification is precise iff its domain is precise.
A local function $f$ satisfies a statement $(p,q)$, written $f \models (p,q)$, iff, for all $\sigma \in p$, $f(\sigma) \sqsubseteq q$. It satisfies a specification $\phi \in \Phi$, written $f \models \phi$, iff $f \models (p, q)$ for all $(p, q) \in \phi$.
\[semanticconsequence\] Let $p, q, r, s \in \mathcal{P}(\Sigma)$ and $\phi, \psi \in \Phi$. Each judgement $(p, q) \models (r, s), \phi \models (p, q)$, $(p, q) \models \phi$, and $\phi \models \psi$ holds iff all local functions that satisfy the left hand side also satisfy the right hand side.
\[ordercharacterization\] $f \sqsubseteq g$ iff, for all $p,q \in \mathcal{P}(\Sigma)$, $g \models (p, q)$ implies $f \models (p, q)$.
For every specification $\phi$, there is a ‘best’ local function satisfying $\phi$ (lemma \[blalemma\]), in the sense that all statements that the best local function satisfies are satisfied by any local function that satisfies $\phi$. For example, in the heap and stack separation algebra of example \[locsepexamples\].2, consider the specification $$\phi_{new} = \{(\{x {\!\mapsto\!}v\},\{x {\!\mapsto\!}l \bullet l {\!\mapsto\!}w \mid l \in L, w \in Val\}) \mid v \in Val\}$$ There are many local functions that satisfy this specification. Trivially, the local function that always diverges satisfies it. Another example is the local function that assigns the value $w$ of the newly allocated cell to be 0, rather than any non-deterministically chosen value. However, the best local function for this specification is the $new[x]$ function described in example \[locsepexamples\].2, as it can be checked that for any local function $f$ satisfying $\phi_{new}$, we have $f \sqsubseteq new[x]$. The notion of the best local function shall be used when addressing questions about completeness of specifications. It is adapted from [@COY07], except that we generalise to the best local function of a specification rather than a single pre- and post-condition pair.
For a specification $\phi \in \Phi$, the best local function of $\phi$, written ${\mathit{bla}}[\phi]$, is the function of type $\Sigma \rightarrow \mathcal{P}(\Sigma)^\top$ defined by $${\mathit{bla}}[\phi](\sigma) = \bigsqcap \{\{\sigma'\} * q \mid \sigma = \sigma'\bullet\sigma'', \sigma''\in p, (p, q) \in \phi\}$$
As an example, it can be checked that the best local function ${\mathit{bla}}[\phi_{new}]$ of the specification $\phi_{new}$ given above is indeed the function $new[x]$ described in example \[locsepexamples\].2. The following lemma presents the important properties which characterise the best local function.
\[blalemma\] Let $\phi \in \Phi$. The following hold:
1. ${\mathit{bla}}[\phi]$ is local
2. ${\mathit{bla}}[\phi] \models \phi$
3. if $f$ is local and $f \models \phi$ then $f \sqsubseteq {\mathit{bla}}[\phi]$
To show that ${\mathit{bla}}[\phi]$ is local, consider $\sigma_1,\sigma_2$ such that $\sigma_1\#\sigma_2$. We then calculate $$\begin{array}{rll}
{\mathit{bla}}[\phi](\sigma_1\bullet \sigma_2)\enspace
= & \bigsqcap \{\{\sigma'\} * q \mid \sigma_1\bullet \sigma_2 =\sigma'\bullet \sigma'', \sigma''\in p, (p, q) \in \phi\} \\
\sqsubseteq & \bigsqcap \{\{\sigma_1\bullet \sigma'''\} * q \mid \sigma_2=\sigma'''\bullet \sigma'', \sigma''\in p, (p, q) \in \phi\} \\
= & \bigsqcap \{\{\sigma_1\} * \{\sigma'''\} * q \mid \sigma_2=\sigma'''\bullet \sigma'', \sigma''\in p, (p, q) \in \phi\} \\
= & \{\sigma_1\} * \bigsqcap \{\{\sigma'''\} * q \mid \sigma_2=\sigma'''\bullet \sigma'', \sigma''\in p, (p, q) \in \phi\} \\
= & \{\sigma_1\} * {\mathit{bla}}[\phi](\sigma_2)
\end{array}$$ In the second-last step we used the property that $\{\sigma_1\}$ is precise (lemma \[precisecharacterization\]).\
To show that ${\mathit{bla}}[\phi]$ satisfies $\phi$, consider $(p, q) \in
\phi$ and $\sigma\in p$. Then ${\mathit{bla}}[\phi](\sigma) \sqsubseteq \{u\} *
q = q$.
For the last point, suppose $f$ is local and $f \models \phi$. Then, for any $\sigma$ such that $\sigma=\sigma_1\bullet \sigma_2$ and $\sigma_2\in p$ and $(p, q) \in \phi$, $$\begin{array}{lcll}
f(\sigma) &= & f(\sigma_1\bullet \sigma_2) \\
&\sqsubseteq & \{\sigma_1\} * f(\sigma_2) \\
&\sqsubseteq & \{\sigma_1\} * q \\
\end{array}$$ Thus $f(\sigma) \sqsubseteq {\mathit{bla}}[\phi](\sigma)$.\
In the case that there do not exist $\sigma_1,\sigma_2$ such that $\sigma=\sigma_1\bullet \sigma_2$ and $\sigma_2 \in D(\phi)$, then $$\begin{array}{lcll}
{\mathit{bla}}[\phi](\sigma) &= & \bigsqcap \emptyset \\
&= & \top \\
\end{array}$$ So in this case also $f(\sigma) \sqsubseteq {\mathit{bla}}[\phi](\sigma)$.
\[blaconsequence\] For $\phi \in \Phi$ and $p, q \in \mathcal{P}(\Sigma)$, ${\mathit{bla}}[\phi] \models (p, q) \Leftrightarrow \phi \models (p, q)$.
$$\begin{array}{cll}
& {\mathit{bla}}[\phi] \models (p, q) \\
\hbox to50 pt{\hfill}
\Leftrightarrow
& \mbox{for all local functions } f,\;
f \models \phi \Rightarrow f \models (p, q)
& \mbox{ (by lemma \ref{blalemma})} \\
\hbox to50 pt{\hfill}
\Leftrightarrow
& \phi \models (p, q)
& \mbox{ (by definition \ref{semanticconsequence})}.
\hbox to51 pt{\hfill\qEd}
\end{array}$$
------------------------------------------------------------------------
$\,$ $$\begin{array}{cccc}
\infer
{
(p*r, q*r)
}
{ (p, q) }
&\quad
\infer
{
(p', q')
}
{ p' \sqsubseteq p & (p,q) & q \sqsubseteq q'}
\quad&\quad
\infer
{
\left(\bigsqcup_{i\in I} p_i, \bigsqcup_{i\in I} q_i\right)
}
{ (p_i,q_i),\;\mbox{all}\;i\in I}
\quad&\quad
\infer
{
\left(\bigsqcap_{i\in I} p_i, \bigsqcap_{i\in I} q_i\right)
}
{ (p_i, q_i),\;\mbox{all}\;i\in I, I \neq \emptyset}
\\[2ex]
Frame & Consequence & Union & Intersection
\end{array}$$
------------------------------------------------------------------------
The inference rules of the proof system are given in figure \[figure:rules\]. Consequence, union and intersection are adaptations of standard rules of Hoare logic. The frame rule is what permits local reasoning, as it codifies the fact that, since all functions are local, any assertion about a separate part of resource will continue to hold for that part after the application of the function. We omit the standard rules for basic constructs such as sequential composition, non-deterministic choice, and Kleene-star which can be found in [@COY07].
\[proofconsequence\] For predicates $p, q, r, s$ and specifications $\phi, \psi$, each of the judgements $(p, q) \vdash (r, s), \phi \vdash (p, q)$, $(p, q) \vdash \phi$, and $\phi \vdash \psi$ holds iff the right-hand side is derivable from the left-hand side by the rules in figure \[figure:rules\].
The proof system of figure \[figure:rules\] is sound and complete with respect to the satisfaction relation.
\[completeness\] $\phi \vdash (p, q) \Leftrightarrow \phi \models (p, q)$
Soundness can be checked by checking each of the proof rules in figure \[figure:rules\]. The frame rule is sound by the locality condition, and the others are easy to check.
For completeness, assume we are given $\phi \models (p, q)$. By lemma \[blaconsequence\], we have ${\mathit{bla}}[\phi] \models (p, q)$. So for all $\sigma \in p$, ${\mathit{bla}}[\phi](\sigma) \sqsubseteq q$, which implies $$\displaystyle\bigsqcup_{\sigma \in p} {\mathit{bla}}[\phi](\sigma) \sqsubseteq q \quad (*)$$ Now we have the following derivation:
[$$\infer
{
(p, q)
}
{
\infer
{
(\displaystyle\bigsqcup_{\sigma \in p} \{\sigma\}, \displaystyle\bigsqcup_{\sigma \in p} {\mathit{bla}}[\phi](\sigma))
}
{
\infer
{
(\{\sigma\}, {\mathit{bla}}[\phi](\sigma)) \quad \mbox{\tiny{for all $\sigma \in p$}}
}
{
\infer
{
\big( \displaystyle\bigsqcap_{\substack{\sigma' \preceq \sigma \\ \sigma' \in r \\ (r, s) \in \phi}} \{\sigma - \sigma'\} * \{\sigma'\}, \displaystyle\bigsqcap_{\substack{\sigma' \preceq \sigma \\ \sigma' \in r \\ (r, s) \in \phi}} \{\sigma - \sigma'\} * s \big) \quad \mbox{\tiny{for all $\sigma \in p$}}
}
{
\infer
{
( \{\sigma - \sigma'\} * \{\sigma'\}, \{\sigma - \sigma'\} * s) \quad \mbox{\tiny{for all $\sigma' \in r, (r, s) \in \phi, \sigma' \preceq \sigma, \sigma \in p$}}
}
{
\infer
{
(\{\sigma'\}, s) \quad \mbox{\tiny{for all $\sigma' \in r, (r, s) \in \phi$}}
}
{
\infer
{
(r, s) \quad \mbox{\tiny{for all $(r, s) \in \phi$}}
}
{
\phi
}
}
}
}
}
}
}$$]{} The last step in the proof is by $(*)$ and the rule of consequence. Note that the intersection rule can be safely applied because the argument of the intersection is necessarily non-empty (if it were empty then ${\mathit{bla}}[\phi](\sigma) = \top$, which contradicts ${\mathit{bla}}[\phi](\sigma) \sqsubseteq q$).
Properties of Specifications {#completespecifications}
============================
We discuss certain properties of specifications as a prerequisite for our main discussion on footprints in Section 4. We introduce the notion of a [*complete*]{} specification for a local function, which is a specification from which follows every property that holds for the function. However, a function may have many complete specifications, so we introduce a canonical form for specifications. We show that of all the complete specifications of a local function, there exists a unique canonical complete specification for every domain. As discussed in the introduction, an important notion of local reasoning is the *small specification* which completely describes the behaviour of a local function by mentioning only the footprint. Thus, as a prerequisite to investigating their existence, we formalise small specifications as complete specifications with the smallest possible domain. Similarly, we define [*big*]{} specifications as complete specifications with the biggest domain.
A specification $\phi \in \Phi$ is a [**complete specification**]{} for $f$, written $complete(\phi, f)$, iff, for all $p, q \in \mathcal{P}(\Sigma)$,$ f \models (p, q) \Leftrightarrow \phi \models (p, q) $. Let [**$\Phi_{comp(f)}$**]{} be the set of all complete specifications of f.
$\phi$ is complete for $f$ whenever the tuples that hold for $f$ are [*exactly*]{} the tuples that follow from $\phi$. This also means that any two complete specfications $\phi$ and $\psi$ for a local function are semantically equivalent, that is, $\phi {\mathrel{\reflectbox{$\vDash$}}}\vDash \psi$. The following proposition illustrates how the notions of best local action and complete specification are closely related.
\[blaiffcomplete\] For all $\phi \in \Phi$ and local functions $f$, $complete(\phi, f) \Leftrightarrow f = {\mathit{bla}}[\phi]$.
Assume $f$ = $bla[\phi]$. Then, by lemma \[blaconsequence\], we have that $\phi$ is a complete specification for $f$.
For the converse, assume $complete(\phi, f)$. We shall show that for any $\sigma \in \Sigma$, $f(\sigma) = {\mathit{bla}}[\phi](\sigma)$.
[**case 1: $f(\sigma) = \top$**]{}. If ${\mathit{bla}}[\phi](\sigma) \neq \top$, then ${\mathit{bla}}[\phi] \models (\{\sigma\}, {\mathit{bla}}[\phi](\sigma))$. This means that $\phi \models (\{\sigma\}, {\mathit{bla}}[\phi](\sigma))$ (by lemma \[blaconsequence\]), and so $f \models (\{\sigma\}, {\mathit{bla}}[\phi](\sigma))$, but this is a contradiction. Therefore, ${\mathit{bla}}[\phi](\sigma) = \top$
[**case 2: ${\mathit{bla}}[\phi](\sigma) = \top$**]{}. If $f(\sigma) \neq \top$, then $f \models (\{\sigma\}, f(\sigma))$. This means that $\phi \models (\{\sigma\}, f(\sigma))$, and so ${\mathit{bla}}[\phi] \models (\{\sigma\}, f(\sigma))$, but this is a contradiction. Therefore, $f(\sigma) = \top$
[**case 3: ${\mathit{bla}}[\phi](\sigma) \neq \top$ and $f(\sigma) \neq \top$**]{}. We have $$\begin{array}{cl}
&f \models (\{\sigma\}, f(\sigma)) \\
\Rightarrow & {\mathit{bla}}[\phi] \models (\{\sigma\}, f(\sigma)) \\
\Rightarrow & {\mathit{bla}}[\phi](\sigma) \sqsubseteq f(\sigma)
\\\\
&{\mathit{bla}}[\phi] \models (\{\sigma\}, {\mathit{bla}}[\phi](\sigma)) \\
\Rightarrow & f \models (\{\sigma\}, {\mathit{bla}}[\phi](\sigma)) \\
\Rightarrow & f(\sigma) \sqsubseteq {\mathit{bla}}[\phi](\sigma)
\end{array}$$
Therefore $f(\sigma) = {\mathit{bla}}[\phi](\sigma)$
Any specification is therefore only complete for a unique local function, which is its best local action. However, a local function may have lots of complete specifications. For example, if $\phi$ is a complete specification for $f$ and $(p, q) \in \phi$, then $\phi \cup \{(p, q')\}$ is also complete for $f$ if $q \subseteq q'$. For this reason it will be useful to have a canonical form for specifications.
The [**canonicalisation**]{} of a specification $\phi$ is defined as $\phi_{can} = \{ (\{\sigma\}, {\mathit{bla}}[\phi](\sigma)) \mid \sigma \in D(\phi) \} $. A specification is in [**canonical**]{} form if it is equal to its canonicalisation. Let $\Phi_{can(f)}$ denote the set of all canonical complete specifications of $f$.
Notice that a given local function does not necessarily have a *unique* canonical complete specification. For example, both $\{(\{u\}, \{u\})\}$ and $\{(\{u\}, \{u\}), (\{\sigma\}, \{\sigma\})\}$, for some $\sigma \in \Sigma$, are canonical complete specifications for the identity function.
\[canproposition\] For any specification $\phi$, we have $\phi {\mathrel{\reflectbox{$\vDash$}}}\vDash \phi_{can}$.
We first show $\phi \vDash \phi_{can}$. For any $(p, q) \in \phi_{can}$, $(p, q)$ is of the form $(\{\sigma\}, {\mathit{bla}}[\phi](\sigma))$ for some $\sigma \in D(\phi)$. So we have ${\mathit{bla}}[\phi] \models (p, q)$, and so $\phi \models (p, q)$ by lemma \[blaconsequence\].
We now show $\phi_{can} \vDash \phi$. For any $(p, q) \in \phi$, we have ${\mathit{bla}}[\phi] \models (p, q)$. So for all $\sigma \in p$, ${\mathit{bla}}[\phi](\sigma) \sqsubseteq q$, which implies $$\displaystyle\bigsqcup_{\sigma \in p} {\mathit{bla}}[\phi](\sigma) \sqsubseteq q \quad (*)$$ Now we have the following derivation: $$\infer
{
(p, q)
}
{
\infer
{
(\displaystyle\bigsqcup_{\sigma \in p} \{\sigma\}, \displaystyle\bigsqcup_{\sigma \in p} {\mathit{bla}}[\phi](\sigma))
}
{
\infer
{
(\{\sigma\}, {\mathit{bla}}[\phi](\sigma)) \quad \mbox{\scriptsize{for all $\sigma \in p$}}
}
{
\phi_{can}
}
}
}$$ The last step is by $(*)$ and consequence. So we have $\phi_{can} \vdash \phi$, and by soundness $\phi_{can} \models \phi$.
Thus, the canonicalisation of a specification is logically equivalent to the specification. The following corollary shows that all complete specifications that have the same domain have a unique canonical form, and specifications of different domains have different canonical forms.
\[candomainisomorphism\] $\Phi_{can(f)}$ is isomorphic to the quotient set $\Phi_{comp(f)}/\cong_{D}$, under the isomorphism that maps $[\phi]_{\cong_{D}}$ to $\phi_{can}$, for every $\phi \in \Phi_{comp(f)}$. By proposition \[blaiffcomplete\], all complete specifications for $f$ have the same best local action, which is $f$ itself. So by the definition of canonicalisation, it can be seen that complete specifications with different domains have different canonicalisations, and complete specifications with the same domain have the same canonicalisation. This shows that the mapping is well-defined and injective. Every canonical complete specification $\phi$ is also complete, and $[\phi]_{\cong_{D}}$ maps to $\phi_{can} = \phi$, so the mapping is surjective.
\[bigsmallspec\] $\phi$ is a [**small specification**]{} for $f$ iff $\phi \in \Phi_{comp(f)}$ and there is no $\psi \in \Phi_{comp(f)}$ such that $D(\psi) \sqsubset
D(\phi)$. A [**big specification**]{} is defined similarly.
[*Small*]{} and [*big*]{} specifications are thus the specifications with the smallest and biggest domains respectively. The question is if/when small and big specifications exist. The following result shows that a canonical big specification exists for every local function.
\[bigspec\] For any local function $f$, the canonical big specification for $f$ is given by $\phi_{big(f)} = \{ (\{\sigma\}, f(\sigma)) \mid f(\sigma) \sqsubset \top \}$.
$f \models \phi_{big(f)}$ is trivial to check. To show $complete(\phi_{big(f)}, f)$, assume $f \models (p, q)$ for some $p, q \in \mathcal{P}(\Sigma)$. Note that, for any $\sigma \in p$, $f(\sigma) \sqsubseteq q$ and so $\displaystyle\bigsqcup_{\sigma \in p} f(\sigma) \sqsubseteq q$. We then have the derivation $$\infer
{
(p, q)
}
{
\infer
{
(\displaystyle\bigsqcup_{\sigma \in p} \{\sigma\}, \displaystyle\bigsqcup_{\sigma \in p} f(\sigma))
}
{
\infer
{
(\{\sigma\}, f(\sigma)) \quad \mbox{\scriptsize{for all $f(\sigma) \sqsubset \top$}}
}
{
\phi_{big(f)}
}
}
}$$ By soundness we get $\phi_{big(f)} \models (p, q)$. $\phi_{big(f)}$ has the biggest domain because $f$ would fault on any element not included in $\phi_{big(f)}$.
The notion of a small specification has until now been used in an informal sense in local reasoning papers [@ORY01; @BCOP05; @CGZ05] as specifications that completely specify the behaviour of an update command by only describing the command’s behaviour on the part of the resource that it affects. Although these papers present examples of such specifications for specific commands, the notion has so far not received a formal treatment in the general case. The question of the existence of small specifications is strongly related to the concept of footprints, since finding a small specification is about finding a complete specification with the smallest possible domain, and therefore enquiring about which elements of $\Sigma$ are essential and sufficient for a complete specification. This requires a formal characterisation of the footprint notion, which we shall now present.
Footprints {#sec:footprints}
==========
In the introduction we discussed how the *AD* program demonstrates that the footprints of a local function do not correspond simply to the smallest safe states, as these states alone do not always yield complete specifications. In this section we introduce the definition of footprint that does yield complete specifications. In order to understand what the footprint of a local function should be, we begin by analysing the definition of locality. Recall that the definition of locality (definition \[def:localaction\]) says that the action on a certain state $\sigma_1$ imposes a *limit* on the action on a bigger state $\sigma_2\bullet\sigma_1$. This limit is $\{\sigma_2\} * f(\sigma_1)$, as we have $f(\sigma_2\bullet \sigma_1) \sqsubseteq \{\sigma_2\} * f(\sigma_1)$.
Another way of viewing this definition is that for any state $\sigma$, the action of the function on that state has to be within the limit imposed by *every* substate $\sigma'$ of $\sigma$, that is, $f(\sigma) \sqsubseteq \{\sigma - \sigma'\} * f(\sigma')$. In the case where $\sigma' = \sigma$, this condition is trivially satisfied for any function (local or non-local). The distinguishing characteristic of local functions is that this condition is also satisfied by every strict substate of $\sigma$, and thus we have $$f(\sigma) \sqsubseteq \displaystyle\bigsqcap_{\sigma' \prec \sigma} \{\sigma - \sigma'\} * f(\sigma')$$ We define this overall constraint imposed on $\sigma$ by all of its strict substates as the *local limit* of $f$ on $\sigma$, and show that the locality definition is equivalent to satisfying the local limit constraint.
For a local function $f$ on $\Sigma$ and $\sigma \in \Sigma$, the [**local limit**]{} of $f$ on $\sigma$ is defined as $$L_f(\sigma) = \displaystyle\bigsqcap_{\sigma' \prec \sigma} \{\sigma - \sigma'\} * f(\sigma')$$
\[locallimitproposition\]$f \mbox{ is local} \quad \Leftrightarrow \quad f(\sigma) \sqsubseteq L_f(\sigma) \quad \mbox{for all $\sigma \in \Sigma$}$
Assume $f$ is local. So for any $\sigma$, for every $\sigma' \prec \sigma$, $f(\sigma) \sqsubseteq \{\sigma - \sigma'\} * f(\sigma')$. $f(\sigma)$ is therefore smaller than the intersection of all these sets, which is $L_f(\sigma)$.
For the converse, assume the rhs and that $\sigma_1 \bullet \sigma_2$ is defined. If $\sigma_1 = u$ then $f(\sigma_1 \bullet \sigma_2) \sqsubseteq \{\sigma_1\} * f(\sigma_2)$ and we are done. Otherwise, $\sigma_2 \prec \sigma_1 \bullet \sigma_2$ and we have $f(\sigma_1 \bullet \sigma_2) \sqsubseteq L_f(\sigma_1 \bullet \sigma_2) \sqsubseteq \{\sigma_1\} * f(\sigma_2)$.
Thus for any local function $f$ acting on a certain state $\sigma$, the local limit determines a *smallest upper bound* on the possible outcomes on $\sigma$, based on the outcomes on all smaller states. If this smallest upper bound does correspond exactly to the set of all possible outcomes on $\sigma$, then $\sigma$ is ‘large enough’ that just the action of $f$ on smaller states and the locality of $f$ determines the complete behaviour of $f$ on $\sigma$. In this case we will not think of $\sigma$ as a footprint of $f$, as smaller states are sufficient to determine the action of $f$ on $\sigma$. With this observation, we define footprints as those states on which the outcomes cannot be determined only by the smaller states, that is, the set of outcomes is a *strict* subset of the local limit.
\[def:footprint\] For a local function $f$ and $\sigma \in \Sigma$, $\sigma$ is a footprint of $f$, written $F_f(\sigma)$, iff $f(\sigma) \sqsubset L_f(\sigma)$. We denote the set of footprints of $f$ by $F(f)$.
Note that an element $\sigma$ is therefore not a footprint if and only if the action of $f$ on $\sigma$ is at the local limit, that is $f(\sigma) = L_f(\sigma)$.
\[minstates\] For any local function $f$, the smallest safe states of $f$ are footprints of $f$. Let $\sigma$ be a smallest safe state for $f$. Then for any $\sigma' \prec \sigma$, $f(\sigma') = \top$. Therefore $L_f(\sigma) = \top$ and so $f(\sigma) \sqsubset L_f(\sigma)$.
However, the smallest safe states are not always the *only* footprints. An example is the *AD* command discussed in the introduction. The empty heap is a footprint as it is the smallest safe heap, but the heap cell $l {\!\mapsto\!}v$ is also a footprint.
\[dispose\] The footprints of the $dispose[l]$ command in the plain heap model (example \[locsepexamples\].1) are the cells at location $l$. We check this by considering the following cases
(1) The empty heap, $u_H$, is not a footprint since $L_{dispose[l]}(u_H) = \top = dispose[l](u_H)$
(2) [Every cell $l {\!\mapsto\!}v$ for some $v$ is a footprint]{} $$\begin{array}{l}
L_{dispose[l]}(l {\!\mapsto\!}v) = \{l {\!\mapsto\!}v\} * dispose[l](u_H) = \{l {\!\mapsto\!}v\} * \top = \top\\
dispose[l](l {\!\mapsto\!}v) = \{u_H\} \sqsubset L_{dispose[l]}(l {\!\mapsto\!}v)
\end{array}$$
(3) Every state $\sigma$ such that $\sigma \succ (l {\!\mapsto\!}v)$ for some $v$ is not a footprint $$L_{dispose[l]}(\sigma) \sqsubseteq \{\sigma - (l {\!\mapsto\!}v)\} * dispose[l](l {\!\mapsto\!}v) = \{\sigma - (l {\!\mapsto\!}v)\} = dispose[l](\sigma)$$ By proposition \[locallimitproposition\], we have $L_{dispose[l]}(\sigma) = dispose[l](\sigma)$. The intuition is that $\sigma$ does not characterise any ‘new’ behaviour of the function: its action on $\sigma$ is just a consequence of its action on the cells at location $l$ and the locality property of the function.
(4) Every state $\sigma$ such that $\sigma \not \succ (l {\!\mapsto\!}v)$ for some $v$ is not a footprint $$L_{dispose[l]}(\sigma) \sqsubseteq \{\sigma\} * dispose[l](u_H) = \{\sigma\} * \top = \top = dispose[l](\sigma)$$ Again by proposition \[locallimitproposition\], $L_{dispose[l]}(\sigma) = dispose[l](\sigma)$.
\[ad\] The *AD* (Allocate-Deallocate) command was defined on the heap and stack model in example \[locsepexamples\].2. We have the following cases for $\sigma$.
(1) [$\sigma \not \succeq x {\!\mapsto\!}v_1$ for some $v_1$ is not a footprint, since $L_{AD}(\sigma) = \top = AD(\sigma)$]{}.
(2) [$\sigma = x {\!\mapsto\!}v_1$ for some $v_1$ is a footprint since $L_{AD}(\sigma) = \top$ (by case (1)) and $AD(\sigma) = \{x {\!\mapsto\!}w \mid w \in L\} \sqsubset L_{AD}(\sigma)$]{}.
(3) [$\sigma = l {\!\mapsto\!}v_1 \bullet x {\!\mapsto\!}v_2$ for some $l, v_1, v_2$ is a footprint.]{} $$\begin{array}{ll}
L_{AD}(\sigma) &= \{l {\!\mapsto\!}v_1\} * AD(x {\!\mapsto\!}v_2)\\
&\quad \mbox{(AD faults on all other elements strictly smaller than $\sigma$)}\\
&= \{l {\!\mapsto\!}v_1\} * \{ x {\!\mapsto\!}w \mid w \in L\}\\
&= \{l {\!\mapsto\!}v_1 \bullet x {\!\mapsto\!}w \mid w \in L\}\\\\
AD(\sigma) &= \{l {\!\mapsto\!}v_1\bullet x {\!\mapsto\!}w \mid w \in L, w \neq l\} \sqsubset L_{AD}(\sigma)
\end{array}$$
(4) [$\sigma = h\bullet x {\!\mapsto\!}v_1$ for some $v_1$, and where $|loc(h)| > 1$, is not a footprint]{}. $$\eqalign{
L_{AD}(\sigma) &\sqsubseteq \displaystyle\bigsqcap_{h \succ l \;{\!\mapsto\!}\; v} \{(h - l {\!\mapsto\!}v\} * AD(l {\!\mapsto\!}v\bullet x {\!\mapsto\!}v_1) \cr
&= \{h\bullet x {\!\mapsto\!}w \mid w \not \in loc(h)\} = AD(\sigma)}$$ By proposition \[locallimitproposition\], we get $L_{AD}(\sigma) = AD(\sigma)$.
Our footprint definition therefore works properly for these specific examples. Now we give the formal general result which captures the underlying intuition of local reasoning, that the footprints of a local function are the only essential elements for a complete specification of the function.
\[essentialitytheorem\] The footprints of a local function are the essential domain elements for any complete specification of that function, that is, $$F_f(\sigma) \quad \Leftrightarrow \quad \forall \phi \in \Phi_{comp(f)}.\: \sigma \in D(\phi)$$
Assume some fixed $f$ and $\sigma$. We establish the following equivalent statement : $$\neg F_f(\sigma) \quad \Leftrightarrow \quad \exists \phi\in \Phi_{comp(f)}.\: \sigma \not \in D(\phi)$$ We first show the right to left implication. So assume $\phi$ is a complete specification of $f$ such that $\sigma\not\in D(\phi)$. Since $complete(\phi, f)$, by proposition \[blaiffcomplete\], we have $f = {\mathit{bla}}[\phi]$. So $$f(\sigma) = \bigsqcap_{\sigma_1 \preceq \sigma, \sigma_1\in p, (p, q) \in \phi} \{\sigma - \sigma_1\} * q$$ Now for any set $\{\sigma - \sigma_1\} * q$ in the above intersection, we have that $\sigma_1 \in p$, and $(p, q) \in \phi$ for some $p$. Since $\sigma_1 \in p$, we have $f(\sigma_1) \sqsubseteq q$, and therefore $\{\sigma - \sigma_1\} * f(\sigma_1) \sqsubseteq \{\sigma - \sigma_1\} * q$. Also, $\sigma_1 \neq \sigma$, because otherwise we would have $\sigma \in p$, which would contradict the assumption that $\sigma \notin D(\phi)$. So $\sigma_1 \prec \sigma$ and we have $$L_f(\sigma) \sqsubseteq \{\sigma - \sigma_1\} * f(\sigma_1) \sqsubseteq \{\sigma - \sigma_1\} * q$$ So the local limit is smaller than each set $\{\sigma - \sigma_1\} * q$ in the intersection, and therefore it is smaller than the intersection itself: $ L_f(\sigma) \sqsubseteq f(\sigma)$. We know from proposition \[locallimitproposition\] that $f(\sigma) \sqsubseteq L_f(\sigma)$, so we get $f(\sigma) = L_f(\sigma)$ and therefore $\neg F_f(\sigma)$.
We now show the left to right implication. Assume that $\sigma$ is not a footprint of $f$. We shall use the big specification, $\phi_{big(f)}$, to construct a complete specification of $f$ which does not contain $\sigma$ in its domain. If $f(\sigma) = \top$ then the big specification itself is such a specification, and we are done. Otherwise assume $f(\sigma) \sqsubset \top$. Let $\phi = \phi_{big(f)} / \{(\{\sigma\}, f(\sigma))\}$. It can be seen that $\sigma \notin D(\phi)$. Now we need to show that $\phi$ is complete for $f$. For this it is sufficient to show $\phi \dashv \vdash \phi_{big(f)}$ because we know that $\phi_{big(f)}$ is complete for $f$. The right to left direction, $\phi \dashv \phi_{big(f)}$, is trivial.
For $\phi \vdash \phi_{big(f)}$, we just need to show $\phi \vdash (\{\sigma\}, f(\sigma))$. We have the following derivation: $$\infer
{
(\{\sigma\}, L_f(\sigma))
}
{
\infer
{
(\{\sigma\}, \displaystyle\bigsqcap_{\sigma' \prec \sigma, f(\sigma') \sqsubset \top} \{\sigma - \sigma'\} * f(\sigma'))
}
{
\infer
{
(\{\sigma - \sigma'\} * \{\sigma'\}, \{\sigma - \sigma'\} * f(\sigma')) \quad \mbox{\scriptsize{for all $\sigma' \prec \sigma$, $f(\sigma') \sqsubset \top$}}
}
{
\infer
{
(\{\sigma'\}, f(\sigma')) \quad \mbox{\scriptsize{for all $\sigma' \prec \sigma$, $f(\sigma') \sqsubset \top$}}
}
{
\phi
}
}
}
}$$ The intersection rule can be safely applied as there is at least one $\sigma' \prec \sigma$ such that $f(\sigma') \sqsubset \top$. This is because $f(\sigma) \sqsubset \top$, so if there were no such $\sigma'$ then $\sigma$ would be a footprint, which is a contradiction. Note that the last step uses the fact that $$\displaystyle\bigsqcap_{\sigma' \prec \sigma, f(\sigma') \sqsubset \top} \{\sigma - \sigma'\} * f(\sigma') =
\displaystyle\bigsqcap_{\sigma' \prec \sigma} \{\sigma - \sigma'\} * f(\sigma') = L_f(\sigma)$$ because adding the top element to an intersection does not change its value. Since $\sigma$ is not a footprint, $f(\sigma) = L_f(\sigma)$, and so $\phi \vdash (\{\sigma\}, f(\sigma))$.
Sufficiency and Small Specifications
====================================
We know that the footprints are the only elements that are [*essential*]{} for a complete specification of a local function in the sense that every complete specification must include them. Now we ask when a set of elements is [*sufficient*]{} for a complete specification of a local function, in the sense that there exists a complete specification of the function that only includes these elements. In particular, we wish to know if the footprints alone are sufficient. To study this, we begin by identifying the notion of the *basis* of a local function.
Bases
-----
In the last section we defined the local limit of a function $f$ on a state $\sigma$ as the constraint imposed on $f$ by all the strict substates of $\sigma$. This was used to identify the footprints as those states on which the action of $f$ cannot be determined by just its action on the smaller states. We are now addressing the question of when a set of states is [*sufficient*]{} to determine the behaviour of $f$ on any state. We shall do this by identifying a fixed set of states, which we call a [*basis*]{} for $f$, such that the action of $f$ on any state $\sigma$ can be determined by just the substates of $\sigma$ taken from this set (rather than all the strict substates of $\sigma$). Thus we first generalise the local limit definition to consider the constraint imposed by only the substates taken from a given set.
For a subset $A$ of a separation algebra $\Sigma$, the [**local limit**]{} imposed by $A$ on the action of $f$ on $\sigma$ is defined by $$L_{A, f}(\sigma) = \displaystyle\bigsqcap_{\sigma' \preceq \sigma, \sigma' \in A} \{\sigma - \sigma'\} * f(\sigma')$$
Sometimes, the local limit imposed by $A$ is enough to completely determine $f$. In this case, we call $A$ a *basis* for $f$.
$A \sqsubseteq \Sigma$ is a [**basis**]{} for $f$, written $basis(A, f)$, iff $L_{A, f} = f$.
This means that, when given the action of $f$ on elements in A alone, we can determine the action of $f$ on any element in $\Sigma$ by just using the locality property of $f$. Every local function has at least one basis, namely the trivial basis $\Sigma$ itself. We next show the correspondence between the bases and complete specifications of a local function.
\[basisspeclemma\] Let $\phi_{A, f} = \{ (\{\sigma\}, f(\sigma)) \mid \sigma \in A, f(\sigma) \sqsubset \top\}$. Then we have $basis(A, f) \Leftrightarrow complete(\phi_{A, f}, f)$. We have $L_{A, f} = {\mathit{bla}}[\phi_{A, f}]$ by definition. The result follows by proposition \[blaiffcomplete\] and the definition of basis.
For every canonical complete specification $\phi \in \Phi_{can(f)}$, we have $\phi = \phi_{D(\phi), f}$. By the previous lemma it follows that $D(\phi)$ forms a basis for $f$. The lemma therefore shows that every basis determines a complete canonical specification, and vice versa. This correspondence also carries over to all complete specifications for $f$ by the fact that every domain-equivalent class of complete specifications for $f$ is represented by the canonical complete specification with that domain (corollary \[candomainisomorphism\]). By the essentiality of footprints (theorem \[essentialitytheorem\]), it follows that the footprints are present in every basis of a local function.
\[footprintbasislemma\] The footprints of $f$ are included in every basis of f. Every basis $A$ of $f$ determines a complete specification for $f$ the domain of which is a subset of $A$. By the essentiality theorem (\[essentialitytheorem\]), the domain includes the footprints.
The question of sufficiency is about how small the basis can get. Given a local function, we wish to know if it has a smallest basis.
Well-founded Resource
---------------------
We know that every basis must contain the footprints. Thus if the footprints alone form a basis, then the function will have a *smallest* complete specification whose domain are just the footprints. We find that, for well-founded resource models, this is indeed the case.
\[sufficiencytheorem\] If a separation algebra $\Sigma$ is well-founded under the $\preceq$ relation, then the footprints of any local function form a basis for it, that is, $f = L_{F(f), f}$.
Assume that $\Sigma$ is well-founded under $\preceq$. We shall show by induction that $ f(\sigma) = L_{F(f), f}(\sigma)$ for all $\sigma \in
\Sigma$. The induction hypothesis is that, for all $\sigma' \prec
\sigma$, $ f(\sigma') = L_{F(f), f}(\sigma')$
[**case 1:**]{} Assume $\sigma$ is a footprint of $f$. We have $f(\sigma) = \{u\} * f(\sigma)$ is in the intersection in the definition of $L_{F(f), f}(\sigma)$, and so $L_{F(f), f}(\sigma)
\sqsubseteq f(\sigma)$. We have by locality that $f(\sigma)
\sqsubseteq L_{F(f), f}(\sigma)$, and so $f(\sigma) = L_{F(f),
f}(\sigma)$.
[**case 2:**]{} Assume $\sigma$ is not a footprint of $f$. We have $$\eqalign{
f(\sigma)
&=L_f(\sigma) \quad \mbox{\emph{(because $\sigma$ is not a footprint of f)}}\cr
&=\bigsqcap_{\sigma' \prec \sigma} \{\sigma - \sigma'\} * f(\sigma')\cr
&=\bigsqcap_{\sigma' \prec \sigma} \big(\{\sigma - \sigma'\}*\bigsqcap_{\sigma'' \preceq \sigma', F_f(\sigma'')} \{\sigma' - \sigma''\} * f(\sigma'')\big) \quad \mbox{\emph {(by the induction hypothesis)}} \cr
&=\bigsqcap_{\sigma' \prec \sigma, \sigma'' \preceq \sigma', F_f(\sigma'')} \{\sigma - \sigma'\} * \{\sigma' - \sigma''\} * f(\sigma'') \quad \mbox{\emph{(by the precision of $\{\sigma - \sigma'\}$)}}\cr
&=\bigsqcap_{\sigma'' \prec \sigma, F_f(\sigma'')} \{\sigma - \sigma''\} * f(\sigma'')\cr
&=\bigsqcap_{\sigma'' \preceq \sigma, F_f(\sigma'')} \{\sigma - \sigma''\} * f(\sigma'') \quad \mbox{\emph{(because $\sigma$ is not a footprint of f)}}\cr
&= L_{F(f), f}(\sigma)\rlap{\hbox to319 pt{\hfill}\qEd}}$$In section \[completespecifications\], the notions of big and small specifications were introduced (definition \[bigsmallspec\]), and the existence of a big specification was shown (proposition \[bigspec\]). We are now in a position to show the existence of the small specification for well-founded resource. If $\Sigma$ is well-founded, then every local function has a small specification whose domain is the footprints of the function.
\[smallspeccorollary\] For well-founded separation algebras, every local function has a small specification given by $\phi_{F(f), f}$. $\phi_{F(f), f}$ is complete by theorem \[sufficiencytheorem\] and lemma \[basisspeclemma\]. It has the smallest domain by the essentiality theorem.
Thus, for well-founded resource, the footprints are always essential and sufficient, and specifications need not consider any other elements. In practice, small specifications may not always be in canonical form even though they always have the same domain as the canonical form. For example, the heap dispose command can have the specification $\{(\{l {\!\mapsto\!}v \mid v \in Val\}, \{u_H\})\}$ rather than the canonical one given by $\{(\{l {\!\mapsto\!}v\}, \{u_H\}) \mid v \in Val\}$.
In practical examples it is usually the case that resource is well-founded. A notable exception is the fractional permissions model [@BCOP05] in which the resource includes ‘permissions to access’, which can be indefinitely divided. We next investigate the non-well-founded case.
Non-well-founded Resource
-------------------------
If a separation algebra is non-well-founded under the $\preceq$ relation, then there is some infinite descending chain of elements $\sigma_1 \succ \sigma_2 \succ \sigma_3 ...$. From a resource-oriented point of view, there are two distinct ways in which this could happen. One way is when it is possible to remove non-empty pieces of resource from a state indefinitely, as in the separation algebra of non-negative real numbers under addition. In this case any infinite descending chain does not have more than one occurrence of any element. Another way is when an infinite chain may exist because of repeated occurrences of some elements. This happens when there is *negativity* present in the resource: some elements have inverses in the sense that adding two non-unit elements together may give the unit. An example is the separation algebra of integers under addition, where $1 + (-1) = 0$, so adding -1 to 1 is like adding negative resource. Also, since $1 = 0 + 1$, we have that $1 \succ 0 \succ 1 ...$ forms an infinite chain.
A separation algebra $\Sigma$ has [**negativity**]{} iff there exists a non-unit element $\sigma \in \Sigma$ that has an inverse; that is, $\sigma \neq u$ and $\sigma \bullet \sigma' = u$ for some $\sigma' \in \Sigma$. We say that $\Sigma$ is [**non-negative**]{} if no such element exists.
All separation algebras with negativity are non-well-founded because, for elements $\sigma$ and $\sigma'$ such that $\sigma\bullet\sigma' = u$, the set $\{\sigma, u\}$ forms an infinite descending chain (there is no least element). All well-founded models are therefore non-negative. For the general non-negative case, we find that either the footprints form a basis, or there is no smallest basis.
\[nonnegativetheorem\] If $\Sigma$ is non-negative then, for any local $f$, either the footprints form a smallest basis or there is no smallest basis for f.
Let $A$ be a basis for $f$ (we know there is at least one, which is the trivial basis $\Sigma$ itself). If $A$ is the set of footprints then we are done. So assume $A$ contains some non-footprint $\mu$. We shall show that there exists a smaller basis for $f$, which is $A/\{\mu\}$. So it suffices to show $f(\sigma) = L_{A/\{\mu\},
f}(\sigma)$ for all $\sigma \in \Sigma$. [**case 1:**]{} $\mu \not \preceq \sigma$. We have $$f(\sigma)
=L_{A, f}(\sigma)
=\bigsqcap_{\sigma'\preceq\sigma,\sigma'\in A}\{\sigma-\sigma'\}*f(\sigma')
=\bigsqcap_{\sigma'\preceq\sigma,\sigma'\in A/\{\mu\}}
\{\sigma-\sigma'\}*f(\sigma')= L_{A/\{\mu\}, f}(\sigma)$$ as desired
[**case 2:**]{} $\mu \preceq \sigma$. This implies $$f(\sigma)=\Bigl(\bigsqcap_{\sigma'\preceq\sigma,\sigma'\in
A/\{\mu\}}\{\sigma-\sigma'\}*f(\sigma')\Bigr)\enspace\sqcap\enspace
(\{\sigma-\mu\}*f(\mu))$$ It remains to show that the right hand side of this intersection contains the left hand side: $$\eqalign{
\{\sigma - \mu\} * f(\mu)
&=\{\sigma - \mu\} * L_f(\mu)
\quad\mbox{{(because $\mu$ is not a footprint of f)}} \cr
&=\{\sigma-\mu\}*\bigsqcap_{\sigma' \prec \mu}\{\mu-\sigma'\}*f(\sigma')\cr
&=\{\sigma-\mu\}*\displaystyle\bigsqcap_{\sigma'\prec\mu}\big(\{\mu-\sigma'\}*
\bigsqcap_{\sigma''\preceq\sigma',\sigma''\in A/\{\mu\}}\{\sigma'-\sigma''\}*
f(\sigma'') \big)\cr
&\enspace\quad\mbox{{(case 1 applies because $\Sigma$ is non-negative,
so $\sigma' \prec \mu \Rightarrow \mu \not \preceq \sigma'$)}} \cr
&=\bigsqcap_{\sigma' \prec \mu}\bigsqcap_{\sigma'' \preceq \sigma', \sigma'' \in A/\{\mu\}} \{\sigma - \mu\} * \{\mu - \sigma'\} * \{\sigma' - \sigma''\} * f(\sigma'') \quad\mbox{{(by precision)}} \cr
&=\bigsqcap_{\sigma' \prec \mu}\bigsqcap_{\sigma'' \preceq \sigma', \sigma'' \in A/\{\mu\}} \{\sigma - \sigma''\} * f(\sigma'') \cr
&=\bigsqcap_{\sigma'' \prec \mu, \sigma'' \in A/\{\mu\}} \{\sigma - \sigma''\} * f(\sigma'') \cr
&\sqsupseteq\bigsqcap_{\sigma'' \preceq \sigma, \sigma'' \in
A/\{\mu\}} \{\sigma - \sigma''\} * f(\sigma'')
\rlap{\hbox to194 pt{\hfill}\qEd}\cr}$$
\[nonnegativecorollary\] If $\Sigma$ is non-negative, then every local function either has a small specification given by $\phi_{F(f), f}$ or there is no smallest complete specification for that function.
\[permissions\] The fractional permissions model [@BCOP05] is non-well-founded and non-negative. It can be represented by the separation algebra $\mathit{HPerm} = L \rightharpoonup_{{\mathit{fin}}} Val\times P$ where $L$ and $Val$ are as in example \[locsepexamples\], and $P$ is the interval (0, 1\] of rational numbers. Elements of $P$ represent ‘permissions’ to access a heap cell. A permission of 1 for a cell means both read and write access, while any permission less than 1 is read-only access. The operator $\bullet$ joins disjoint heaps and adds the permissions together for any cells that are present in both heaps only if the resulting permission for each heap cell does not exceed 1; the operation is undefined otherwise. In this case, the write function that updates the value at a location requires a permission of at least 1 and faults on any smaller permission. It therefore has a small specification with precondition being the cell with permission 1. The read function, however, can execute safely on any positive permission, no matter how small. Thus, this function can be completely specified with a specification that has a precondition given by the cell with permission $z$, for all $0 < z \leq 1$. However, this is not a *smallest* specification, as a smaller one can be given by further restricting $0 < z \leq 0.5$. We can therefore always find a smaller specification by reducing the value of $z$ but keeping it positive.
For resource with negativity, we find that it is possible to have small specifications that include non-essential elements (which by theorem \[essentialitytheorem\] are not footprints). These elements are non-essential in the sense that complete specifications exist that do not include them, but there is no complete specification that includes only essential elements.
\[negativeresource\] An example of a model with negativity is the separation algebra of integers $({\mathbb{Z}}, +, 0)$. In this case there can be local functions which can have small specifications that contain non-footprints. Let $f:{\mathbb{Z}} \rightarrow \mathcal{P}({\mathbb{Z}})^\top$ be defined as $f(n) = \{n + c\}$ for some constant $c$, as in example \[locsepexamples\]. $f$ is local, but it has no footprints. This is because for any $n$, $f(n) = 1 + f(n - 1)$, and so $n$ is not a footprint of $f$. However, $f$ does have small specifications, for example, $\{(\{0\}, \{c\})\}$, $\{(\{5\}, \{5 + c\})\}$, or indeed $\{(\{n\}, \{n + c\})\}$ for any $n \in {\mathbb{Z}}$. So although every element is non-essential, some element is required to give a complete specification.
Regaining Safety Footprints {#safety}
===========================
In the introduction we discussed how the notion of footprints as the smallest safe states - the *safety footprint*- is inadequate for giving complete specifications, as illustrated by the *AD* example. For this reason, so far in this paper we have investigated the general notion of footprint for arbitrary local functions on arbitrary separation algebras. Equipped with this general theory, we now investigate how the regaining of safety footprints may be achieved with different resource modelling choices. We start by presenting an alternative model of RAM, based on an investigation of why the $AD$ phenomenon occurs in the standard model. We then demonstrate that the footprints of the $AD$ command in this new model do correspond to the safety footprints. In the final section we identify, for arbitrary separation algebras, a condition on local functions which guarantees the equivalence of the safety footprint and the actual footprint. We then show that if this condition is met by all the primitive commands of a programming language then the safety footprints are regained for every program in the language, and finally show that this is indeed the case in our new RAM model.
An alternative model {#altmodel}
--------------------
In this section we explore an alternative heap model in which the safety footprints do correspond to the actual footprints. We begin by taking a closer look at why the *AD* anomaly occurs in the standard heap and stack model described in example \[locsepexamples\].2. Consider an application of the allocation command in this model: $$\mathit{new[x]}(42 \mapsto v \bullet x \mapsto w) = \{42 \mapsto v \bullet x \mapsto l \bullet l \mapsto r \mid l \in L\backslash \{42\}, r \in Val\}$$
The intuition of locality is that the initial state $42 \mapsto v \bullet x \mapsto w$ is only describing a local region of the heap and the stack, rather than the whole global state. In this case it says that the address 42 is initially allocated, and the definition of the allocation command is that the resulting state will have a new cell, the address of which can be anything other than 42. However, we notice that the initial state is in fact not just describing only its local region of the heap. It does state that 42 is allocated, but it also implicitly states a very global property: that *all other addresses are not allocated*. This is why the allocation command can choose to allocate any location that is not 42. Thus in this model, every local state implicitly contains some global allocation information which is used by the allocation command. In contrast, a command such as mutate does not require this global ‘knowledge’ of the allocation status of any other cell that it is not affecting. Now the global information of which cells are free [*changes*]{} as more resource is added to the initial state, so this can lead to program behaviour being sensitive to the addition of more resource to the initial state, and this sensitivity is apparant in the case of the *AD* program.
Based on this observation, we consider an alternative model. As before, a state $l \mapsto v$ will represent a local allocated region of the heap at address $l$ with value $v$. However, unlike before, this state will say nothing about the allocation status any locations other than $l$. This information about the allocation status of other locations will be represented explicitly in a *free* set, which will contain every location that is not allocated in the *global heap*. The model can be interpreted from an ownership point of view, where the free set is to be thought of as a unique, atomic piece of resource, ownership of which needs to be obtained by a command if it wants to do allocation or deallocation. An analogy is with the permissions model: a command that wants to read or write to a cell needs ownership of the appropriate permission on that cell. In the same way, in our new model, a command that wants to do allocation or deallocation needs to have ownership of the free set: the ‘permission’ to see which cells are free in the global heap so that it can choose one of them to allocate, or update the free set with the address that it deallocates. On the other hand, commands that only read or write to cells shall not require ownership of the free set.
\[freesetmodel\] Formally, we work with a separation algebra $(H,\bullet,u_H)$. Let $L$, $Var$ and $Val$ be locations, variables and values, as before. States $h \in H$ are given by the grammar: $$h ::= u_H \mid l {\!\mapsto\!}v \mid x {\!\mapsto\!}v \mid F \mid h \bullet h$$ where $l \in L$, $v \in Val$, $x \in Var$ and $F \in \mathcal{P}(L)$. The operator $\bullet$ is undefined for states with overlapping locations or variables. Let $loc(h)$ and $var(h)$ be the set of locations and variables in state $h$ respectively. The set $F$ carries the information of which locations are free. Thus we allow at most one free set in a state, and the free set must be disjoint from all locations in the state. So $h \bullet F$ is only defined when $loc(h) \cap F = \emptyset$ and $h \neq h'\bullet F'$ for any $h'$ and $F'$. We assume $\bullet$ is associative and commutative with unit $u_H$.
In this model, the allocation command requires ownership of the free set for safe execution, since it chooses the location to allocate from this set. It removes the chosen address from the free set as it allocates the cell. It is defined as $$new[x](h)
= \left\{ \begin{array}{ll}
\begin{array}{l}\{ h' \bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}w \bullet F\backslash \{l\}\mid w \in Val, l \in F\} \end{array} & \begin{array}{l}h = h'\bullet x {\!\mapsto\!}v \bullet F \end{array} \\
\begin{array}{l}\top \end{array} & \begin{array}{l}\mbox{otherwise} \end{array}
\end{array}\right.$$ Note that the output states $h' \bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}w \bullet F\backslash \{l\}$ are defined, since we have $l \not \in F\backslash\{l\}$ and the input state $h'\bullet x {\!\mapsto\!}v \bullet F$ implies that $loc(h')$ is disjoint from $F\backslash\{l\}$. The deallocation command also requires the free set, as it updates the set with the address of the cell that it deletes: $$dispose[x](h)
= \left\{ \begin{array}{ll}
\begin{array}{l}\{ h' \bullet x {\!\mapsto\!}l \bullet F \cup \{l\}\} \end{array} & \begin{array}{l}h = h'\bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}v \bullet F \end{array} \\
\begin{array}{l}\top \end{array} & \begin{array}{l}\mbox{otherwise} \end{array}
\end{array}\right.$$ Again, the output states are defined, since the input state implies that $loc(h') \cup \{l\}$ is disjoint from $F$, and so $loc(h')$ is disjoint from $F \cup \{l\}$. Notice that in this model, only the allocation and deallocation commands require ownership of the free set, since commands such as mutation and lookup are completely independent of the allocation status of other cells, and they are defined exactly as in example \[locsepexamples\].2: $$\eqalign{
mutate[x,v](h)
&= \left\{ \begin{array}{ll}
\begin{array}{l}\{ h' \bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}v \} \end{array} & \begin{array}{l}h = h'\bullet x {\!\mapsto\!}l\bullet l {\!\mapsto\!}w \end{array} \\
\begin{array}{l}\top \end{array} & \begin{array}{l}\mbox{otherwise} \end{array}
\end{array}\right.\cr
lookup[x,y](h)
&= \left\{ \begin{array}{ll}
\begin{array}{l}\{ h' \bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}v \bullet y {\!\mapsto\!}v \} \end{array} & \begin{array}{l}h = h'\bullet x {\!\mapsto\!}l\bullet l {\!\mapsto\!}v \bullet y {\!\mapsto\!}w \end{array} \\
\begin{array}{l}\top \end{array} & \begin{array}{l}\mbox{otherwise} \end{array}
\end{array}\right.\cr}$$
\[newmodellocal\]The functions $new[x]$, $dispose[x]$, $mutate[x,v]$ and $lookup[x,y]$ are all local in the separation algebra $(H,\bullet,u_H)$ from example \[freesetmodel\]. Let $f = new[x]$ and assume $h'\#h$. We want to show $f(h'\bullet h) \sqsubseteq \{h'\} * f(h)$. Assume $h = h'' \bullet x {\!\mapsto\!}v \bullet F$ for some $h''$, $x$, $l$, $v$ and $F$, because otherwise $f(h) = \top$ and we are done. So we have $$\begin{array}{lll}
f(h'\bullet h) &=& \{h' \bullet h'' \bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}w \bullet F\backslash \{l\}\mid w \in Val, l \in F\} \\
&=& \{h'\} * \{h'' \bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}w \bullet F\backslash \{l\}\mid w \in Val, l \in F\}\\
&=& \{h'\} * f(h)
\end{array}$$ The other functions can be checked in a similar way.
Safety footprints for *AD*
--------------------------
We consider the footprint of the *AD* command in the new model. In this model the sequential composition $new[x];dispose[x]$ gives the function $$AD(h)
= \left\{ \begin{array}{ll}
\begin{array}{l}\{h' \bullet x {\!\mapsto\!}l \bullet F \mid l \in F\} \end{array} & \begin{array}{l}h = h'\bullet x {\!\mapsto\!}v \bullet F \end{array} \\
\begin{array}{l}\top \end{array} & \begin{array}{l}\mbox{otherwise} \end{array}
\end{array}\right.$$ The smallest safe states are given by the set $\{x {\!\mapsto\!}v \bullet F \mid v \in Val, F \in \mathcal{P}(L)\}$. By lemma \[minstates\], these smallest safe states are footprints. However, unlike before, in this model these are the *only* footprints of the $AD$ command. To see this, consider a larger state $h \bullet x {\!\mapsto\!}v \bullet F$ for non-empty $h$. We have $$\begin{array}{lll}
AD(h \bullet x {\!\mapsto\!}v \bullet F) &=& \{h \bullet x {\!\mapsto\!}l \bullet F \mid l \in F\}\\
&=& \{h\} * \{x {\!\mapsto\!}l \bullet F \mid l \in F\}\\
&=& \{h\} * AD(x {\!\mapsto\!}v \bullet F)
\end{array}$$ Since the local limit $L_{AD}(h \bullet x {\!\mapsto\!}v \bullet F) \sqsubseteq \{h\} * AD(x {\!\mapsto\!}v \bullet F)$ by definition, we have by proposition \[locallimitproposition\] that $L_{AD}(h\bullet x {\!\mapsto\!}v \bullet F) = AD(h\bullet x {\!\mapsto\!}v \bullet F)$, and so $h\bullet x {\!\mapsto\!}v \bullet F$ is not a footprint of $AD$.
Thus the footprints of $AD$ in this model do not include any non-empty heaps. By corollary \[smallspeccorollary\], in this model the $AD$ command has a smallest complete specification in which the pre-condition only describes the empty heap. This specification is $$\{(\{x {\!\mapsto\!}v \bullet F\}, \{x {\!\mapsto\!}l \bullet F\}) \mid v \in Val, F \in \mathcal{P}(L), l \in F\}$$
Intuitively, it says that if initially the heap is empty, the variable $x$ is present in the stack, and we know which cells are free in the global heap, then after the execution, the heap will still be empty, exactly the same cells will still be free, and $x$ will point to one of those free cells. This completely describes the behaviour of the command for all larger states using the frame rule. For example, we get the complete specification on the larger state in which 42 is allocated: $$\{(\{42 {\!\mapsto\!}w\} * \{x {\!\mapsto\!}v \bullet F\}, \{42 {\!\mapsto\!}w\}*\{x {\!\mapsto\!}l \bullet F\}) \mid v,w \in Val, F \in \mathcal{P}(L), l \in F\}$$
In the pre-condition, the presence of location 42 in the heap means that 42 is not in the free set $F$ (by definition of $*$). Therefore, in the post-condition, $x$ cannot point to 42.
Notice that in order to check that we have ‘regained’ safety footprints, we only needed to check that the footprint definition (definition \[def:footprint\]) corresponds to the smallest safe states. The desired properties such as essentiality, sufficiency, and small specifications then follow by the results established in previous sections.
Safety footprints for arbitrary programs
----------------------------------------
Now that we have regained the safety footprints for *AD* in the new model, we want to know if this is generally the case for *any program*. We consider the abstract imperative programming language given in [@COY07]: $$\begin{array}{rcl}
C & ::= & c \mid \mathtt{skip} \mid C;C \mid C+C \mid C^\star
\end{array}$$ where $c$ ranges over an arbitrary collection of primitive commands, $+$ is nondeterministic choice, $;$ is sequential composition, and $(\cdot)^\star$ is Kleene-star (iterated $;$). As discussed in [@COY07], conditionals and while loops can be encoded using $+$ and $(\cdot)^\star$ and assume statements. The denotational semantics of commands is given in Figure \[fig:densemantics\].
------------------------------------------------------------------------
$$\begin{array}{c}
{\llbracket c \rrbracket} \in LocFunc \quad \quad
{\llbracket \mathtt{skip} \rrbracket}(\sigma)= \{\sigma\} \\[1.5ex]
{\llbracket C_1;C_2 \rrbracket} = {\llbracket C_1 \rrbracket};{\llbracket C_2 \rrbracket} \quad \quad
{\llbracket C_1+C_2 \rrbracket} = {\llbracket C_1 \rrbracket} \sqcup {\llbracket C_2 \rrbracket} \quad \quad
{\llbracket C^{\star} \rrbracket} = \bigsqcup_n {\llbracket C^{\,n} \rrbracket}
\end{array}$$
------------------------------------------------------------------------
Taking the primitive commands to be $new[x]$, $dispose[x]$, $mutate[x,v]$, and $lookup[x,y]$, our original aim was to show that, for every command $C$, the footprints of ${\llbracket C \rrbracket}$ in the new model are the smallest safe states. However, in attempting to do this, we identified a general condition on primitive commands under which the result holds for arbitrary separation algebras.
Let $f$ be a local function on a separation algebra $\Sigma$. If, for $A \in \mathcal{P}(\Sigma)$, we define $f(A) = \displaystyle\bigsqcup_{\sigma \in A} f(\sigma)$, then the locality condition (definition \[def:localaction\]) can be restated as $$\forall \sigma', \sigma \in \Sigma.\; f(\{\sigma'\}*\{\sigma\}) \sqsubseteq \{\sigma'\} * f(\{\sigma\})$$
The $\sqsubseteq$ ordering in this definition allows local functions to be more deterministic on larger states. This sensitivity of determinism to larger states is apparant in the *AD* command in the standard model from example \[locsepexamples\].2. On the empty heap, the command produces an empty heap, and reassigns variable $x$ to *any* value, while on the singleton cell 1, it disallows the possibility that $x = 1$ afterwards. In the new model, the $AD$ command does not have this sensitivity of determinism in the output states. In this case, the presence or absence of the cell 1 does not affect the outcomes of the $AD$ command, since the command can only assign $x$ to a value chosen from the free set, which does not change no matter what additional cells may be framed in. With this observation, we consider the general class of local functions in which this sensitivity of determinism is not present.
\[def:detconst\] Let $f$ be a local function and $\mathit{safe}(f)$ the set of states on which $f$ does not fault. $f$ has the determinism constancy property iff, for every $\sigma \in \mathit{safe}(f)$, $$\forall \sigma' \in \Sigma.\; f(\{\sigma'\}*\{\sigma\}) = \{\sigma'\} * f(\{\sigma\})$$
Notice that the determinism constancy property by itself implies that the function is local, and it can therefore be thought of as a form of ‘strong locality’. Firstly, we find that local functions that have determinism constancy always have footprints given by the smallest safe states.
\[detconstfootprint\] If a local function $f$ has determinism constancy then its footprints are the smallest safe states. Let $min(f)$ be the smallest safe states of $f$. These are footprints by lemma \[minstates\]. For any larger state $\sigma'\bullet\sigma$ where $\sigma \in min(f)$, $\sigma' \in \Sigma$ and $\sigma$ is non-empty, we have $$f(\sigma'\bullet \sigma) = f(\{\sigma'\}*\{\sigma\}) = \{\sigma'\} * f(\sigma)$$ Since $L_f(\sigma'\bullet\sigma) \sqsubseteq \{\sigma'\} * f(\sigma)$, by proposition \[locallimitproposition\] we have that $L_f(\sigma'\bullet\sigma) = f(\sigma'\bullet\sigma)$, and so $\sigma'\bullet\sigma$ is not a footprint of $f$.
We now demonstrate that the determinism constancy property is preserved by all the constructs of our programming language. This implies that if all the primitive commands of the programming language have determinism constancy, then the footprints of every program are the smallest safe states.
\[detconstprograms\] If all the primitive commands of the programming language have determinism constancy, then the footprint of every program is given by the smallest safe states.
Assuming all primitive commands have determinism constancy, we shall show by induction that every composite command has determinism constancy and the result follows by lemma \[detconstfootprint\]. So for commands $C_1$ and $C_2$, let $f = {\llbracket C_1 \rrbracket}$ and $g = {\llbracket C_2 \rrbracket}$ and assume $f$ and $g$ have determinism constancy. For sequential composition we have, for $\sigma \in \mathit{safe}(f;g)$ and $\sigma' \in \Sigma$, $$\quad\eqalign{
&(f;g)(\{\sigma'\}*\{\sigma\})\cr
=\,\enspace&g(f(\{\sigma'\}*\{\sigma\}))\cr
=\,\enspace&g(\{\sigma'\}*f(\{\sigma\}))
\qquad\!\vbox to 7 pt{\baselineskip=12 pt\noindent ($f$ has determinism constancy and
\phantom{(}$\sigma\in\mathit{safe}(f)$ since
$\sigma\in\mathit{safe}(f;g)$)\vss}\cr
=\,\enspace&g(\bigsqcup_{\sigma_1 \in f(\sigma)} \{\sigma'\}*\{\sigma_1\})\cr
=\,\enspace&\bigsqcup_{\sigma_1 \in f(\sigma)} g(\{\sigma'\}*\{\sigma_1\})\cr
=\,\enspace&\bigsqcup_{\sigma_1 \in f(\sigma)} \{\sigma'\}* g(\sigma_1)
\qquad\vbox to 7 pt{\baselineskip=10 pt\noindent ($g$ has determinism constancy and\\
\phantom{(}$\sigma_1\in\mathit{safe}(g)$ since $\sigma\in\mathit{safe}(f;g)$
and $\sigma_1\inf(\sigma)$)}\cr
=\,\enspace&\{\sigma'\}*\bigsqcup_{\sigma_1\in f(\sigma)}g(\sigma_1) \qquad\mbox{(distributivity)}\cr
=\,\enspace&\{\sigma'\}* (f;g)(\sigma)\cr}$$
For non-deterministic choice, we have for $\sigma \in \mathit{safe}(f + g)$ and $\sigma' \in \Sigma$, $$\quad\eqalign{
&(f + g)(\{\sigma'\}*\{\sigma\})\cr
=\,\enspace&f(\{\sigma'\}*\{\sigma\}) \sqcup g(\{\sigma'\}*\{\sigma\})\cr
=\,\enspace&\{\sigma'\}*f(\{\sigma\}) \sqcup \{\sigma'\}*g(\{\sigma\})
\qquad\vbox to7 pt{\baselineskip=10 pt\noindent ($f$ and $g$ have determinism constancy and \\
\phantom{(}$\sigma \in \mathit{safe}(f)$ and $\sigma \in \mathit{safe}(g)$
since $\sigma \in \mathit{safe}(f+g)$)}\cr\cr
=\,\enspace& \{\sigma'\}* (f(\{\sigma\}) \sqcup g(\{\sigma\}))\quad \mbox{(distributivity)}\cr
=\,\enspace& \{\sigma'\}* (f+g)(\{\sigma\})\cr}$$
For Kleene-star, we have for $\sigma \in \mathit{safe}(f^{\star})$ and $\sigma' \in \Sigma$, $$\quad\eqalign{
&(f^{\star})(\{\sigma'\}*\{\sigma\})\cr
=\,\enspace& \displaystyle \bigsqcup_{n} f^{n}(\{\sigma'\}*\{\sigma\})\cr
=\,\enspace& \displaystyle \bigsqcup_{n} \{\sigma'\}*f^{n}(\{\sigma\})
\qquad\vbox to7 pt{\baselineskip=10 pt\noindent (determinism constancy preserved under sequential composition and \\
\phantom{(}$\sigma \in \mathit{safe}(f^n)$)}\cr
=\,\enspace& \{\sigma'\}*\bigsqcup_{n} f^{n}(\{\sigma\})
\qquad \mbox{\mathstrut(distributivity)}\cr
=\,\enspace& \{\sigma'\}* (f^{\star})(\{\sigma\})\rlap{\hbox to320 pt{\hfill}\qEd}}$$
Now that we have shown the general result, it remains to check that all the primitive commands in the new model of section \[altmodel\] do have determinism constancy.
\[rammodelsprop\] Let $H_1$ be the stack and heap model of example \[locsepexamples\].2 and $H_2$ be the alternative model of section \[altmodel\]. The commands $new[x]$, $mutate[x,v]$ and $lookup[x,y]$ all have determinism constancy in both models. The $dispose[x]$ command has determinism constancy in $H_2$ but not in $H_1$.
We give the proofs for the new and dispose commands in the two models, and the cases for mutate and lookup can be checked in a similar way. For $dispose[x]$ in $H_1$, the following counterexample shows that it does not have determinism constancy. $$\begin{array}{lll}
&& dispose[x](\{l{\!\mapsto\!}v\}*\{x{\!\mapsto\!}l\bullet l {\!\mapsto\!}w\})\\
&=& dispose[x](\emptyset)\\
& = &\emptyset \\
&\sqsubset& \{l{\!\mapsto\!}v \bullet x{\!\mapsto\!}l\}\\
& =& \{l{\!\mapsto\!}v\}* dispose[x](x{\!\mapsto\!}l\bullet l {\!\mapsto\!}w)
\end{array}$$ For $new[x]$ in $H_1$, any safe state is of the form $h \bullet x {\!\mapsto\!}v$. For any $h' \in H_1$, we have $$\{h'\} * new[x](h \bullet x{\!\mapsto\!}v) = \{h'\} * \{ h \bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}w \mid w \in Val, l \in L \backslash loc(h)\} \quad (\dagger)$$
If $h'\bullet h \bullet x {\!\mapsto\!}v$ is undefined then $h'$ shares locations with $loc(h)$ or variables with $var(h) \cup \{x\}$. This means that the RHS in $\dagger$ is the empty set. We have $new[x](\{h'\}*\{h \bullet x{\!\mapsto\!}v\}) = new[x](\emptyset) = \emptyset = \{h'\}* new[x](h \bullet x{\!\mapsto\!}v)$. If $h'\bullet h \bullet x {\!\mapsto\!}v$ is defined, then $$\begin{array}{lll}
&& new[x](\{h'\}* \{h \bullet x{\!\mapsto\!}v\})\\
&=& new[x](h' \bullet h \bullet x{\!\mapsto\!}v)\\
&=& \{ h' \bullet h \bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}w \mid w \in Val, l \in L \backslash loc(h'\bullet h)\}\\
&=& \{ h'\} * \{h \bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}w \mid w \in Val, l \in L \backslash loc(h'\bullet h)\}\\
&=& \{ h'\} * \{h \bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}w \mid w \in Val, l \in L \backslash loc(h)\}\\
&=& \{h'\} * new[x](h \bullet x{\!\mapsto\!}v)
\end{array}$$ For $dispose[x]$ in $H_2$, any safe state is of the form $h \bullet x {\!\mapsto\!}l\bullet l {\!\mapsto\!}v \bullet F$. Let $h' \in H_2$. We have $$\{h'\} * dispose[x](h \bullet x {\!\mapsto\!}l\bullet l {\!\mapsto\!}v \bullet F) = \{h'\} * \{ h \bullet x {\!\mapsto\!}l \bullet F\cup\{l\}\} \quad (\dagger\mbox{\!}\dagger)$$
If $h'\bullet h \bullet x {\!\mapsto\!}l\bullet l {\!\mapsto\!}v \bullet F$ is undefined then either $h'$ contains a free set or it contains locations in $loc(h) \cup \{l\}$ or variables in $var(h) \cup \{x\}$. If $h'$ contains a free set or it contains locations in $loc(h)$ or variables in $var(h) \cup \{x\}$, then the RHS in $\dagger\mbox{\!}\dagger$ is the empty set. If $h'$ contains the location $l$ then also the RHS in $\dagger\mbox{\!}\dagger$ is the empty set since the free set $F \cup \{l\}$ also contains $l$. Thus in both cases the RHS in $\dagger\mbox{\!}\dagger$ is the empty set, and we have $dispose[x](\{h'\}*\{h \bullet x {\!\mapsto\!}l\bullet l {\!\mapsto\!}v \bullet F\}) = \emptyset = \{h'\} * dispose[x](h \bullet x {\!\mapsto\!}l\bullet l {\!\mapsto\!}v \bullet F)$.
If $h'\bullet h \bullet x {\!\mapsto\!}l\bullet l {\!\mapsto\!}v \bullet F$ is defined then we have $$\begin{array}{lll}
&& dispose[x](\{h'\}*\{h \bullet x {\!\mapsto\!}l\bullet l {\!\mapsto\!}v \bullet F\})\\
&=& dispose[x](h' \bullet h \bullet x {\!\mapsto\!}l\bullet l {\!\mapsto\!}v \bullet F)\\
&=& \{h' \bullet h \bullet x {\!\mapsto\!}l \bullet F\cup\{l\}\}\\
&=& \{h'\} * \{ h \bullet x {\!\mapsto\!}l \bullet F\cup\{l\}\}\\
&=& \{h'\} * dispose[x](h \bullet x {\!\mapsto\!}l\bullet l {\!\mapsto\!}v \bullet F)
\end{array}$$
For $new[x]$ in $H_2$, any safe state is of the form $h \bullet x {\!\mapsto\!}v \bullet F$. Let $h' \in H_2$. We have $$\{h'\} * new[x](h \bullet x {\!\mapsto\!}v \bullet F) = \{h'\} * \{ h \bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}w \bullet F\backslash\{l\} \mid w \in Val, l \in F\} \quad (\dagger\mbox{\!}\dagger\mbox{\!}\dagger)$$
If $h'\bullet h \bullet x {\!\mapsto\!}v \bullet F$ is undefined then either $h'$ contains a free set or it contains locations in $loc(h)$ or variables in $var(h) \cup \{x\}$. In all these cases the RHS in $\dagger\mbox{\!}\dagger\mbox{\!}\dagger$ is the empty set, and so we have $new[x](\{h'\}*\{h \bullet x {\!\mapsto\!}v \bullet F\}) = \emptyset = \{h'\} * new[x](h \bullet x {\!\mapsto\!}v \bullet F)$.
If $h'\bullet h \bullet x {\!\mapsto\!}v \bullet F$ is defined then we have $$\begin{array}{lll}
&&new[x](\{h'\}*\{h \bullet x {\!\mapsto\!}v \bullet F\})\\
&=& new[x](h' \bullet h \bullet x {\!\mapsto\!}v \bullet F)\\
&=& \{h' \bullet h \bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}w \bullet F\backslash\{l\} \mid w \in Val, l \in F\} \\
&=& \{h'\} * \{ h \bullet x {\!\mapsto\!}l \bullet l {\!\mapsto\!}w \bullet F\backslash\{l\} \mid w \in Val, l \in F\}\\
&=& \{h'\} * new[x](h \bullet x {\!\mapsto\!}v \bullet F)
\end{array}$$
Thus theorem \[detconstprograms\] and proposition \[rammodelsprop\] tell us that using the alternative model of example \[freesetmodel\], the footprint of every program is given by the smallest safe states, and hence we have regained safety footprints for all programs. In fact, the same is true for the original model of example \[locsepexamples\].2 if we do not include the dispose command as a primitive command, since all the other primitive commands have determinism constancy. This, for example, would be the case when modelling a garbage collected language [@Park06].
Conclusions
===========
We have developed a general theory of footprints in the abstract setting of local functions that act on separation algebras. Although central and intuitive concepts in local reasoning, the notion of footprints and small specifications had evaded a formal general treatment until now. The main obstacle was presented by the *AD* problem, which demonstrated the inadequacy of the safety footprint notion in yielding complete specifications. In addressing this issue, we first investigated the notion of footprint which does not suffer from this inadequacy. Based on an analysis of the definition of locality, we introduced the definition of the footprint of a local function, and demonstrated that, according to this definition, the footprints are the only essential elements necessary to obtain a complete specification of the function. For well-founded resource models, we showed that the footprints are also sufficient, and we also presented results for non-well-founded models.
Having established the footprint definition, we then explored the conditions under which the safety footprint does correspond to the actual footprint. We introduced an alternative heap model in which safety footprints are regained for *every* program, including *AD*. We also presented a general condition on local functions in arbitrary models under which safety footprints are regained, and showed that if this condition is met by all the primitive commands of the programming language, then safety footprints are regained for every program. The theory of footprints has proven very useful in exploring the situations in which safety footprints could be regained, as one only needs to check that the smallest safe states correspond to the footprint definition \[def:footprint\]. This automatically gives the required properties such as essentiality and sufficiency, which, without the footprint definition and theorems, would need to be explicitly checked in the different cases.
Finally, we comment on some related work. The discussion in this paper has been based on the static notion of footprints as [*states*]{} of the resource on which a program acts. A different notion of footprint has recently been described in [@HO08], where footprints are viewed as [*traces*]{} of execution of a computation. O’Hearn has described how the [*AD*]{} problem is avoided in this more elaborate semantics, as the allocation of cells in an execution prevents the framing of those cells. Interestingly, however, the heap model from example \[freesetmodel\] illustrates that it is not essential to move to this more elaborate setting and incorporate dynamic, execution-specific information into the footprint in order to resolve the [*AD*]{} problem. Instead, with the explicit representation of free cells in states, one can remain in an extensional semantics and have a purely static, resource-based (rather than execution-based) view of footprints.
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Acknowledgement {#acknowledgement .unnumbered}
===============
The authors wish to thank Cristiano Calcagno, Peter O’Hearn and Hongseok Yang for detailed discussions on footprints. Raza acknowledges support of an ORS award. Gardner acknowledges support of a Microsoft Research Cambridge/Royal Academy of Engineering Senior Research Fellowship.
[^1]: Yang’s example was the ‘allocate-deallocate-test’ program *ADT* ::= ‘x := new();dispose(x); if (x=1) then z:=0 else z:=1;x=0’. Our *AD* program provides a more standard example of program behaviour.
|
6.0in 8.6in
-0.25truein 0.30truein
=1.5pc
epsf =1 \#1 \#1\#2\#3
\#1[fig. ]{}
by1
RU–98–13\
[**G. R. Farrar, G. Gabadadze and M. Schwetz**]{}
[ Department of Physics and Astronomy, Rutgers University\
Piscataway, New Jersey 08855, USA]{}\
[emails: farrar, gabad, myckola@physics.rutgers.edu]{}
[**Abstract**]{}
We study the spectrum of the softly broken [*generalized*]{} Veneziano-Yankielowicz effective action for $N=1$ SUSY Yang-Mills theory. Two dual formulations of the effective action are given. The spurion method is used for the soft SUSY breaking. Masses of the bound states are calculated and mixing patterns are discussed. Mass splittings of pure gluonic states are consistent with predictions of conventional Yang-Mills theory. The results can be tested in lattice simulations of the SUSY Yang-Mills model.
PACS numbers: 11.30.Pb; 12.60.Jv; 11.15.Tk.
Keywords: SUSY Yang-Mills theory; effective action; bound states.
[**Introduction**]{}
Some time ago great progress was made in understanding the ground state structure of many supersymmetric gauge theories [@S], [@SW]. It is highly desirable to have direct non-perturbative tests of those results. There is a possibility that these models can be simulated on the lattice. Some preliminary work toward this complicated task has already been performed (see refs. [@Lattice], [@lattice:cernrome]).
The lattice regularization violates supersymmetry [@CV]. Thus, some special fine-tuning is required to recover the SUSY limit on the lattice. Away from the SUSY point, the continuum limit of the lattice theory is described by a model with explicit SUSY breaking terms. In some cases those terms may trigger only soft SUSY breaking [@giradello], although this is not guaranteed in general.
Softly broken SUSY models can be studied using the spurion technique [@spurion]. Some “exact” results were obtained within that approach [@soft1; @soft2; @soft3]. In this paper we consider softly broken supersymmetric Yang-Mills theory, the model which is relevant for lattice simulations. At the classical level supersymmetric Yang-Mills (SYM) is a theory with only one parameter, the gauge coupling constant. The lowest-dimensional renormalizable SUSY breaking term allowed by gauge invariance is the gaugino mass term. Therefore, we consider SYM with a gaugino mass term as a theory describing the continuum limit of the lattice regularized action.
In analogy with QCD, one expects that the spectrum of this model consists of colorless bound states of gluinos and gluons. Among those are: pure gluonic bound states (glueballs), gluino-gluino mesons and gluon-gluino composites. These states fall into the lowest-spin representations of the $N=1$ SUSY algebra written in the basis of parity eigenstates [@us]. The masses and interactions of these bound states can be given within the effective Lagrangian approach. The effective action for $N=1$ SYM was proposed by Veneziano and Yankielowicz (VY) [@VY]. The VY action [@VY] involves fields for gluino-gluino and gluino-gluon bound states. However, it does not include dynamical degrees of freedom which would correspond to pure gluonic composites (glueballs).
We argued in ref. [@us] that there are no physical reasons to expect glueballs to be heavier and decoupled from gluino-gluino and gluino-gluon bound states in $N=1$ SUSY YM theory. Moreover, there are SQCD sum rule based arguments indicating that the low-energy spectrum of SYM theory is not exhausted by the gluino containing bound states only [@Shap]; glueball degrees of freedom should also be taken into account.
The generalization of the VY effective action that includes pure gluonic degrees of freedom was given in ref. [@us]. The generalized VY effective Lagrangian of ref. [@us] describes mixed states of glueballs, gluino-gluino and gluino-gluon bound states. The fundamental superfield upon which that construction of the generalized VY action is based [@us] is a constrained tensor superfield [@Gates]. The set of components of that superfield includes as a subset the VY chiral supermultiplet.
The aim of the present paper is twofold. First we propose a new representation of the generalized VY effective action of ref. [@us]. This action is equivalent to the previously proposed one [@us], but it uses two different chiral supermultiplets instead of the tensor supermultiplet approach adopted in [@us].
Then we introduce soft SUSY breaking terms in the generalized VY Lagrangian and study mass splittings and mixing patterns in the softly broken theory. These results can be directly tested in lattice calculations. Predictions for the masses of the gluino-gluino and gluon-gluino bound states and their splittings in the broken theory were made in ref. [@NSM] using the original VY effective action [@VY] and the spurion technique. We will see that the presence of the glueball degrees of freedom changes the vacuum state of the broken theory. As a result, the mass splittings are also modified.
The paper is organized as follows. In section 1 we briefly review the generalized VY effective Lagrangian and recall some results obtained in ref. [@us]. In section 2 we explain how one can reformulate the generalized VY Lagrangian in terms of two independent chiral superfields using the chiral-tensor superfield duality [@Gates], [@GGRZ]. In section 3 we show how the effective action is modified when the gaugino mass term is introduced in SYM through the spurion method. Section 4 reports the masses and mixings for physical eigenstates of the broken theory.\
[**1. The Generalized VY Effective Action** ]{}
The on-shell Lagrangian of SYM for an $SU(N_c)$ gauge group is $$\begin{aligned}
{\cal L} ~=~ \frac{1}{g^2}\left[ \,-\frac{1}{4}
G_{\mu\nu}^a G_{\mu\nu}^a
~+~ i\lambda_{\dot\alpha}^\dagger D^{\dot\alpha\beta}\lambda_\beta
\right] ~.
\nonumber\end{aligned}$$ In terms of superfields the expression above can be written $$\begin{aligned}
{\cal L} ~=~ \int d^2 \theta~
\frac{1}{8\pi} Im\, \tau W^{\alpha}W_{\alpha}~+h.c.,
\label{lagrangian}\end{aligned}$$ where the gauge coupling is defined to be $\tau = \frac{4\pi i}{g^2} + \frac{\theta_0}{2\pi}$. For the purposes of this paper we set the theta term to be equal to zero, $\theta_0 =0$.
The classical action of $N=1$ SYM theory is invariant under $U(1)_R$, scale and superconformal transformations. In the quantum theory these symmetries are broken by the chiral, scale and superconformal anomalies respectively. Composite operators that appear in the expressions for the anomalies can be thought of as component fields of a chiral supermultiplet $S$ [@WessZumino] $$\begin{aligned}
S\equiv { \beta (g) \over 2 g} W^{\alpha}W_{\alpha}\equiv
A(y)+\sqrt{2} \theta \Psi(y) + \theta^2 F(y), \nonumber\end{aligned}$$ where $\beta(g)$ is the SYM beta function for which the exact expression is known [@beta]. The lowest component of the $S$ supefield is bilinear in gluino fields and has the quantum numbers of the scalar and pseudoscalar gluino bound states. The fermionic component in $S$ describes the gluino-gluon composite and the $F$ component of the chiral superfield includes operators corresponding to both the scalar and pseudoscalar glueballs ($G_{\mu\nu}^2$ and $G_{\mu\nu}{\tilde G^{\mu\nu}}$ respectively) [@VY].
Assuming that the effective action (more precisely, the generating functional for one-particle-irreducible (1PI) Green’s functions [@GoldstoneSalamWeinberg]) of the model can be written in terms of the single superfield $S$, and requiring also that the effective action respects all the global continuous symmetries and reproduces the anomalies of the SYM theory, one derives the Veneziano-Yankielowicz effective action [@VY]. Let us mention that the actual variables, in terms of which the generating functional for the 1PI Green’s functions (or effective action in our conventions) is written, are the VEV’s of composite operators calculated at nonzero values of external sources [@Shore]. In this paper, as well as in ref. [@us], we use a simpified notation where the VEV’s are denoted by the corresponding composite operators.
It was noticed in ref. [@ShifmanKovner] that the VY action does not respect the discrete $Z_{2N_c}$ symmetry – the nonanomalous remnant of anomalous $U(1)_R$ transformations. The VY action was amended by an appropriate term which makes the action invariant under the discrete $Z_{2N_c}$ group [@ShifmanKovner].
However, as we mentioned above, the VY action does not include all possible lowest-spin bound states of SYM theory. Glueballs are missing in that description because they are only present in the auxilliary component of the $S$ superfield and can be integrated out. In ref. [@us], in order to account for glueball degrees of freedom, we proposed to formulate the effective action in terms of a more general superfield, the real tensor superfield $U$ [@Gates]. The superfield $U$ can be written in component form as follows: $$\begin{aligned}
U=B+i\theta \chi -i {\bar \theta} {\bar \chi}+{1\over 16}\theta^2 {A^*}+
{1\over 16} {\bar \theta}^2 A+{1\over 48 }\theta \sigma^\mu {\bar
\theta} \varepsilon_{\mu\nu\alpha\beta}C^{\nu\alpha\beta}+
\nonumber \\
{1\over 2} \theta^2 {\bar \theta} \left ( {\sqrt{2} \over 8}{\bar
\Psi} +{\bar \sigma}^\mu \partial_\mu \chi \right )+
{1\over 2}{\bar \theta}^2 \theta \left ( {\sqrt{2} \over 8}
\Psi - \sigma ^\mu \partial_\mu {\bar \chi }\right )+{1\over 4}
\theta^2 {\bar \theta^2} \left ( {1\over 4} \Sigma -\partial^2 B\right ).
\label{U}\end{aligned}$$ It is straightforward to show that the real superfield $U$ satisfies the relation[^1] $$\begin{aligned}
S=-4 {\bar D}^2 U,
\nonumber\end{aligned}$$ where the $F$ term of the chiral supermultiplet $S$ is related to the fields $\Sigma$ and $C_{\mu\nu\alpha}$ in the following way $$\begin{aligned}
F=\Sigma+i{1\over 6}\varepsilon_{\mu\nu\alpha\beta}\partial^{\mu}
C^{\nu\alpha\beta}, \nonumber\end{aligned}$$ and $A$ and $\Psi$ are respectively the scalar and fermion components of the superfield $S$.
We argued [@us] that the effective Lagrangian for the lowest-spin multiplets of the $N=1$ SYM theory can be written in terms of the $U$ field only. That Lagrangian takes the following form [@us] $$\begin{aligned}
{\cal L}={1\over \alpha} (S^{+}S)^{1/3}\Big|_D+
\gamma \Big [( S \log {S\over \mu^3}-S)\Big|_F+{\rm h.c.}\Big ]+
{1\over \delta} \left ( -{U^2\over (S^+S)^{1/3}} \right )\Big|_D,
\label{NewA}\end{aligned}$$ where $\alpha~{\rm and}~\delta$ are arbitrary positive constants and $\gamma =- (N_c g/16 \pi^2 \beta(g))>0$. Notice that the superfield $S$ is not an inpendent variable in this Lagrangian. It is rather related to the $U$ superfield through the formula $$\begin{aligned}
S-\langle S \rangle= -4 {\bar D}^2 U.
\nonumber\end{aligned}$$ In the above equation we took into account that the $S$ superfield has a nonzero VEV in the phase where chiral symmetry is broken, $\langle S \rangle \equiv \mu^3$. Thus, the only independent superfield in the Lagrangian (\[NewA\]) is the $U$ field.
In this approach the following fields become dynamical [@us]:
- The $B$ field propagates and it represents one massive real scalar degree of freedom (identified with the scalar glueball).
- The three-form potential $C_{\mu\nu\alpha}$, which becomes massive, also propagates. It represents one physical degree of freedom (identified with the pseudoscalar glueball).
- The complex field $A$, being decomposed into parity eigenstates, describes the massive gluino-gluino scalar and pseudoscalar mesons.
- $\chi$ and $\Psi$ describe the massive gluino-gluon fermionic bound states.
Studying the potential of the model, we found that the physical eigenstates fall into two different mass “multiplets” (see ref. [@us] for details). Neither of them contain pure gluino-gluino, gluino-gluon or gluon-gluon bound states. Instead, the physical excitations are mixed states of these composites. The heavier set of states contains:
- A pseudoscalar meson, which without mixing reduces to the $0^{-+}$ gluino-gluino bound state (the analog of the QCD $\eta'$ meson).
- A scalar meson that without mixing is a $0^{++}$ $(l=1)$ gluino-gluino excitation.
- A fermionic gluino-gluon bound state.
These heavier states form the chiral supermultiplet described by the VY action. That action is recovered in the $\delta \rightarrow \infty$ limit. The new states which appear as a result of our generalization form the lighter multiplet:
- A scalar meson, which without mixing is a $0^{++}$ ($l=0$) glueball.
- A pseudoscalar state, which for zero mixing is identified as a $0^{-+}$ ($l=1$) glueball.
- A fermionic gluino-gluon bound state.
Notice, that although the physical states fall into multiplets whose $J^P$ quantum numbers correspond to two chiral supermultiplets, the action was written in terms of the one real tensor supermultiplet $U$. In particular, the pseudoscalar glueball in this approach is described by the only physical component of the massive three-form potential $C_{\mu\nu\alpha}$. The field strength of that potential couples to the pseudoscalar gluino-gluino bound state as it would couple to the $\eta'$ meson in QCD [@veneziano].
Since the physical spectrum of the mixed states fall into multiplets whose spin-parity quantum numbers correspond to two chiral supermultiplets, one might be wondering about the possibility to rewrite the whole action it terms of two different chiral superfields. If that is possible it would be crucial to study what peculiarities of the two-chiral-multiplet action allow it to be written in terms of only a real supermultiplet $U$, as was done in ref. [@us]. In the next section we address these questions.\
[**2. The Two Chiral Supermultiplet Action**]{}
The relation between a real tensor and chiral supermultiplets (the so called chiral-linear duality) was established in ref. [@Gates]. For SYM theory the chiral-linear duality was used in ref. (see also discussions in refs. [@Binetruy]). Applied to our problem the results of refs. [@Gates], [@Derendinger] and [@Binetruy] can be stated as follows. One introduces into the effective Lagrangian a new chiral superfield, let us denote it by $\chi$ $$\begin{aligned}
\chi (y, \theta) \equiv \phi_{\chi}(y)+ \sqrt {2}\theta
\Psi_{\chi}(y)+\theta^2 F_{\chi}(y).
\label{chi} \end{aligned}$$ One can find an effective Lagrangian written in terms of two chiral superfields, $S$ and $\chi$ which is equivalent to the expression given in (\[NewA\]). In our case $$\begin{aligned}
{\cal L}~=~{1\over \alpha}\, (S^{+}\,S)^{1/3} \Big|_D ~+~
{\delta \over 4 }\, (S^{+}\,S)^{1/3}\, (\chi~+~ \chi^{+})^2 \Big|_D
~+~ \nonumber \\
\Big[\gamma \,( S \,\log {S\over \mu^3}~-~S)\Big|_F ~+~
{1 \over 16} \,\chi\, (S ~-~ \mu^3)\Big|_F~+~{\rm h.c.}\Big]~.
\label{2s}\end{aligned}$$ Comparing this expression to the VY Lagrangian one notices that both the K[ä]{}hler potential and the superpotential are modified by new terms. The multiplets $S$ and $\chi$ are independent.
We would like to relate this expression to the Lagrangian of the theory written in terms of the $U$ field (\[NewA\]). If the $U$ field is postulated as a fundamental degree of freedom, then the $S$ field is a derivative superfield $$\begin{aligned}
S= \mu^3-4 {\bar D}^2 U.
\label{USrelation}\end{aligned}$$ Using this relation the Lagrangian (\[2s\]) can be rewritten as $$\begin{aligned}
{\cal L}~=~{1\over \alpha}\, (S^{+}\,S)^{1/3} \Big|_D ~+~
{\delta \over 4 }\, (S^{+}\,S)^{1/3}\, (\chi~+~ \chi^{+})^2 \Big|_D
~+~ \nonumber \\
\Big[\gamma \,( S \,\log {S\over \mu^3}~-~S)\Big|_F ~+~{\rm
h.c.}\Big]~+~ U (\chi~+~ \chi^{+})\Big|_D~.
\label{2cs}\end{aligned}$$ This expression depends on two superfields $U$ and $\chi$ ($S$ is expressed through $U$ in accordance with (\[USrelation\])). However, the dependence on the chiral superfield $\chi$ is trivial, the combination $\chi +\chi^{+}$ can be integrated out from the Lagrangian (\[2cs\]). As a result one derives $$\begin{aligned}
\chi + \chi^{+}= -{ 2U\over \delta (S^{+}S)^{1/3}}.
\label{Uchi}\end{aligned}$$ Substituting this expression back into the Lagrangian (\[2cs\]) one arrives at the original expression (\[NewA\]) where the $S$ field is a derivative field satisfying the relation (\[USrelation\]).
Let us stress again that the descriptions in terms of the Lagrangian (\[NewA\]) and (\[2s\]) are equivalent on the mass-shell. In the Lagrangian (\[NewA\]) the dynamical degrees of freedom are assigned to the only superfield $U$, while in the Lagrangian (\[2s\]) the physical degrees of freedom are found as components of two chiral supermultiplets $S$ and $\chi$. The peculiarity of the expression (\[2cs\]) is that the chiral superfield $\chi$ enters only through the real combination $\chi+\chi^+$. That is why it was possible to formulate the action in terms only of the real superfield $U$. It is essential from a physical point of view since the component glueball field must be real.
Using the Lagrangian (\[2s\]) one calculates the potential of the supersymmetric model. Integrating out the auxiliary fields of both chiral multiplets one finds $$\begin{aligned}
V_0= {2\over \delta (16)^2} {|\phi^3-\mu^3|^2
\over |\phi |^2}~+~
{9\alpha |\phi |^4 \over 1- {\alpha\over \delta}
{B^2\over |\phi |^4}}\cdot \left |{\phi_{\chi}\over 16}
~+~3\gamma \log {\phi \over
\mu}~+~{B(\phi^3-\mu^3)\over 24 \delta |\phi|^2 \phi^3}\right |^2,
\label{Potential}\end{aligned}$$ where the following notations are adopted $$\begin{aligned}
\phi_{\chi}={1\over \sqrt {2}} (\sigma +i\pi),~~~~~~~~~
\sigma \equiv -{\sqrt {2} B\over \delta |\phi|^2}. \nonumber
\nonumber\end{aligned}$$ The minimum of this potential is located at the point in field space where $\langle \phi \rangle =\mu$, $\langle B \rangle =
\langle \phi_{\chi}\rangle=0 $. The potential (\[Potential\]) is positive definite for field configurations satisfying $\alpha B^2 < \delta |\phi|^4$. Since the VEV of the $\phi$ field is nonzero and the VEV of the $B$ field is zero the positivity condition is satisfied for small oscillations about the SUSY minimum specified above. Notice that all SUSY field configurations are confined within a valley with infinite potential walls encountered at $\alpha B^2=\delta |\phi|^4$. Thus, the potential (\[Potential\]) and the Lagrangian (\[2s\]) itself describe only small oscillations about the SUSY minimum. In general, some higher order polynomials in the $\chi$ (or $U$) field could be present in the effective Lagrangian. In this work we are interested only in the mass spectrum of the model, so the approximation we used above is good enough for our goals.
In the next section we introduce soft SUSY breaking terms in the effective Lagrangian and study minima and the spectrum of the corresponding potential.\
[**3. Soft SUSY Breaking** ]{}
The gaugino mass term can be introduced in the Lagrangian (\[lagrangian\]) by means of the parameter $\tau$. One regards $\tau$ as a chiral superfield [@spurion]. A nonzero VEV of the $F$ component of $\tau$ yields a SUSY breaking gaugino mass term in (\[lagrangian\]). Thus, one performs the following substitution in expression (\[lagrangian\]) $$\tau \rightarrow \tau + F_{\tau} \theta \theta.$$ As a result, the following new term appears in the Lagrangian of SYM $$\begin{aligned}
- {1 \over 8 \pi}\, Im\, [\, F_{\tau}\, \lambda \lambda \,]+{\rm h.c.}.
\nonumber\end{aligned}$$ To make the gaugino mass canonically normalized one sets $F_{\tau} = i\, 8 \pi\, m_\lambda /g^2$. In the low-energy theory the $\tau$ parameter enters through the dynamically generated scale of the theory $\mu=\mu_0 {\rm exp}(-{8\pi^2\over \beta_0 g^2(\mu_0)})=
\mu_0 {\rm exp}(i{2\pi \tau \over \beta_0 })$. After the $\tau$ parameter is claimed to be a chiral superfield one should regard the $\mu$ parameter as a chiral superfield too. Thus, one also makes the following substitution in the low-energy effective Lagrangian of the model $$\begin{aligned}
\mu \rightarrow \mu~{\rm exp}\left (-{16\pi^2 m_\lambda \over g^2 \beta_0}
\theta \theta \right ),
\nonumber\end{aligned}$$ where $\beta_0$ stands for the first coefficient of the beta function. Performing this redefinition of the $\mu$ parameter in the Lagrangian (\[2s\]) one finds the following additional term in the scalar potential of the model $$\begin{aligned}
\Delta V ~=~- \widetilde{m}_\lambda \,Re \Big({\mu^3 \over 16}\, \phi_\chi
~+~ \gamma \,\phi^3 \Big),
\label {deltaV}\end{aligned}$$ where $\widetilde{m}_\lambda \equiv {32 \pi^2 \over g^2 N_c}\, m_\lambda$.
The expression (\[deltaV\]) is the only correction to the effective potential to leading order in $m_{\lambda}$. All higher order corrections are suppressed by powers of $m_\lambda / \mu$. Those corrections are neglected in this work.\
[**4. The Mass Spectrum** ]{}
Having derived the potential of the broken theory one turns to the calculation of the mass spectrum. The potential consists of two parts, $V_0$ defined in (\[Potential\]) and the SUSY breaking term (\[deltaV\]) $$V=V_0+\Delta V. \nonumber$$ One calculates minima of the full scalar potential $V$. Explicit though tedious calculations yield the following results. The VEV of the $\phi$ field does not get shifted when the soft SUSY breaking terms are introduced. Thus, even in the broken theory $\langle \phi \rangle =\mu $. However, the $\phi_\chi$ (and $B$ ) fields acquire nonzero VEVs in the broken case $$\begin{aligned}
\langle \phi_\chi \rangle =
{8 \over 9 \alpha \mu} \widetilde{m}_\lambda ~~~~~{\rm and }~~~~~~
\langle B \rangle=-{8 \delta \over 9 \alpha }\widetilde{m}_\lambda \mu .
\label{VEV}\end{aligned}$$ The shift of the vacuum energy causes the spectrum of the model to be also rearranged. Explicit calculations of the masses of all lowest-spin states yield the following results $$\label{mscalar}
M^2_{scalar\,\pm}~ ~=~~ M^2_{\pm} ~-~
{3 \over 4}\, \alpha \gamma\, \mu \,\widetilde{m}_\lambda \Bigg (
1~ \pm ~ \sqrt {1+x}~ \Bigg ) \Bigg (2~\pm~ {1\over \sqrt {1+x}}~
\Bigg),$$
$$\label{mfermion}
M^2_{fermion \,\pm} ~=~M^2_{\pm} ~-~
{3 \over 4}\, \alpha \gamma\, \mu \,\widetilde{m}_\lambda \Bigg(
1 \pm \sqrt {1+x}~ \Bigg ) \Bigg (3 ~\pm~ {1\over \sqrt {1+x}}~
\Bigg),$$
$$\label{mpseudo}
M^2_{p-scalar \,\pm} ~=~M^2_{\pm} ~-~
{3 \over 4}\, \alpha \gamma\, \mu \,\widetilde{m}_\lambda \Bigg (
1 \pm \sqrt {1+x} ~ \Bigg ) \Bigg (4~\pm~ {1\over \sqrt {1+x}}~
\Bigg),$$
where $M^2_{\pm}$ denote the masses in the theory with unbroken SUSY [@us] $$\label{msusy}
M^2_{\pm} ~=~ {18 \over (16)^2}{\alpha \over \delta}\mu^2 ~+~ {81 \over 2}
(\alpha \gamma)^2 \mu^2 \Bigg[1 ~\pm~ \sqrt{1 ~+~ x}~\Bigg],
~~~{\rm and}~~~ x\equiv {1 \over 288}
{\alpha \over \delta} {1\over(\alpha \gamma)^2}.
\nonumber$$ In these expressions the plus sign refers to the heavier supermultiplet and the minus sign to the lighter set of states . One can verify that these values satisfy the mass sum rule to leading order in $O(m_{\lambda})$: $$\begin{aligned}
\label{sum}
\sum_{j} \,(-1)^{2j+1}\, (2j+1)\, M_j^2~=~ 0~,
\nonumber\end{aligned}$$ where the summation goes over the spin $j$ of particles in the supermultiplet.
Let us discuss the mass shifts given in eqs. (\[mscalar\]-\[mpseudo\]). Consider the light supermultiplet. In accordance with eqs. (\[mscalar\]-\[mpseudo\]), the masses in the light multiplet are increased in the broken theory. The biggest mass shift is found in the pseudoscalar channel. The smallest shift is observed in the scalar channel. The fermion mass falls in between these two meson states. Thus, the lightest state in the spectrum of the model is the particle which without mixing would have been the scalar glueball. There is a fermion state above that scalar. Finally, the pseudoscalar glueball is heavier than those two states.
Let us now turn to the heavy supermultiplet. In the broken theory the masses in that multiplet get pulled down. However, all states of the heavy multiplet are still heavier than any state of the light multiplet in the domain of validity of our approximations. The ordering of the states in the heavy supermultiplet is just the opposite as in the light supermultiplet: the lightest state is the pseudoscalar meson, the heaviest is the scalar, and the fermion, as required, falls between them. The qualitative features of the spectrum are shown in fig. 1. =24.0 cm
It is not surprising that the lowest mass state obtained in (\[mscalar\]-\[mpseudo\]) is a scalar particle. This is in agreement with the result of ref. [@West] where it was shown that the mass of the lightest state which couples to the operator $G_{\mu\nu}^2$ is less than the mass of the lightest state that couples to $G \tilde{G}$, in pure Yang-Mills theory. As a result, the lightest glueball turns out to be the scalar glueball [@West]. One can apply the method of ref. [@West] to the SYM theory as well. Due to the positivity of the gluino determinant (see ref. [@steve]) one also deduces that the lightest state in softly broken SYM spectrum should be a scalar particle. The pseudoscalar of that multiplet is therefore heavier.
Our result that the multiplet containing glueballs is split in such a way that the scalar is lighter than the pseudoscalar, and vice versa for the multiplet containing gluino-gluino bound states, is consistent with expectations from quark-model lore. In ordinary mesons the $l=1$ states are heavier than their $l=0$ counterparts and the $l=0$ gluino-gluino bound state is a pseudoscalar, while an $l=0$ gluon-gluon bound state is a scalar. It is interesting that in SYM with massless gluinos the $l=0$ and $l=1$ bound states are degenerate, but when the gluino masses are turned on one recovers the expected ordering seen in $q \bar q $ states.
In the $\delta \rightarrow \infty$ limit one recovers the VY effective action. The spectrum of the softly broken VY Lagrangian was studied in [@NSM]. In that limit only the heavy multiplet of the spectrum survives. It is interesting that in the limit $\delta
\rightarrow \infty$, the ratio of the mass-shifts of the surviving states in (\[mscalar\]-\[mpseudo\]) is $5:4:3$, which differs from the prediction of ref. [@NSM]. The seeming discrepancy is resolved because in the limit $\delta \rightarrow \infty$ the vacuum expectation value of the glueball field $B$ tends to infinity. Thus, perturbing states about that vacuum is not a well defined procedure. The right way to obtain the $\delta \rightarrow \infty$ limit would be to decouple the “glueball” modes first, and then minimize the potential. This leads to a shift of the VEV of the $\phi$ field in the broken theory (as in ref. [@NSM]). As a result, the mass shifts calculated within this new vacuum state are in agreement with the values reported in the second work of ref. [@NSM]. We stress, however, that on physical grounds we do not expect SYM to realize the $\delta \rightarrow \infty $ limit of the general effective Lagrangian (5).\
[**5. Summary and Discussion** ]{}
We have shown that the generalized VY effective action can be written in two different ways. In one case the fundamental superfield upon which the action is constructed is the real tensor superfield $U$. In another approach all degrees of freedom of the model are described by two chiral superfields $\chi$ and $S$. In both cases the spectrum consists of two multiplets which are not degenerate in masses even when SUSY is unbroken. The spin-parity quantum numbers of these multiplets are identical to those of certain chiral supermultiplets.
The physical mass eigenstates are not pure gluon-gluon, gluon-gluino or gluino-gluino composites; rather, the physical particles are mixtures of them. The multiplet which without mixing would have been the glueball multiplet is lighter. As a result, those states cannot be decoupled from the effective Lagrangian. This means that comparisons of lattice results to analytic predictions based on the original VY action are not justified.
We introduced a soft SUSY breaking term in the Lagrangian of the $N=1$ SUSY Yang-Mills model. The spurion method was used to calculate the corresponding soft SUSY breaking terms in the generalized VY Lagrangian. These soft breaking terms cause a shift of the vacuum energy of the model. The physical eigenstates which are degenerate in the SUSY limit, are split when SUSY breaking is introduced. We studied these mass splittings in detail. We have confirmed that the spectrum of the broken theory is in agreement with some low energy theorems [@West], namely the scalar glueball turns out to be lighter than the pseudoscalar one. The results of the present paper can be directly tested in lattice studies of $N=1$ supersymmetric Yang-Mills theory ([@Lattice]).
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[^1]: Despite a seeming similarity, the tensor multiplet $U$ should not be interpreted as a usual vector multiplet. The vector field which might be introduced in this approach as a Hodge dual of the three-form potential $C_{\mu\nu\alpha}$ would give mass terms with the wrong sign in our approach (see ref. [@us]), thus, the actual physical variable is the three-form potential $C_{\mu\nu\alpha}$ rather than its dual vector field (the Chern-Simons current).
|
---
abstract: 'Using new and archival radio data, we have measured the proper motion of the black hole X-ray binary V404 Cyg to be $9.2\pm0.3$masyr$^{-1}$. Combined with the systemic radial velocity from the literature, we derive the full three-dimensional heliocentric space velocity of the system, which we use to calculate a peculiar velocity in the range 47–102kms$^{-1}$, with a best fitting value of 64kms$^{-1}$. We consider possible explanations for the observed peculiar velocity, and find that the black hole cannot have formed via direct collapse. A natal supernova is required, in which either significant mass ($\sim 11M_{\odot}$) was lost, giving rise to a symmetric Blaauw kick of up to $\sim65$kms$^{-1}$, or, more probably, asymmetries in the supernova led to an additional kick out of the orbital plane of the binary system. In the case of a purely symmetric kick, the black hole must have been formed with a mass $\sim 9M_{\odot}$, since when it has accreted 0.5–1.5$M_{\odot}$ from its companion.'
date: 'Accepted 2008 December 15. Received 2008 December 11; in original form 2008 November 3'
title: 'The formation of the black hole in the X-ray binary system V404 Cyg'
---
\[firstpage\]
X-rays: binaries – astrometry – radio continuum: stars – stars: individual (V404 Cyg) – stars: supernovae: general – stars: kinematics
Introduction {#sec:intro}
============
The proper motions of X-ray binary systems can be used to derive important information on the birthplaces and formation mechanisms of their compact objects. Since typical proper motions are of order a few milliarcseconds per year, high-resolution observations and long time baselines are required to measure the transverse motions of such systems across the sky. Proper motions have only been measured for a handful of X-ray binary systems to date [@Mir01; @Mir02; @Rib02; @Mir03; @Mir03b; @Dha06; @Dha07], and only one X-ray binary, Sco X-1, has a measured proper motion, parallactic distance, and radial velocity [@Bra99; @Cow75]. With the position, proper motion, radial velocity, and distance to the system, the full three-dimensional space velocity of the system can be derived. Along with system parameters such as the component masses, orbital period, donor temperature and luminosity, these parameters may be used to reconstruct the full evolutionary history of the binary system back to the time of compact object formation, as done for the systems GRO J1655-40 [@Wil05] and XTE J1118+480 [@Fra08].
By studying the distribution of black hole X-ray binary velocities with compact object masses, we can derive constraints on theoretical models of black hole formation [e.g. @Fry01]. The two most common theoretical scenarios for creating a black hole involve either a massive star collapsing directly into a black hole with very little or no mass ejection, or delayed formation in a supernova, as fallback onto the neutron star of material ejected during the explosion creates the black hole. In the latter case, constraints on the magnitude and symmetry of any natal kick can also be derived. When the primary star reaches the end of its life and explodes as a supernova, the centre of mass of the ejected material continues to move with the velocity the progenitor had immediately before the explosion. The centre of mass of the binary system then recoils in the opposite direction. Such a kick [@Bla61] is thus constrained to lie in the orbital plane. In addition, in the presence of asymmetries in the supernova explosion, a further, asymmetric, kick, which need not be in the orbital plane, may be imparted to the binary [see, e.g., @Bra95a; @Por98; @Lai01 for more detailed overviews]. While pulsar proper motions [@Lyn94] and the high eccentricities of Be/X-ray binaries provide strong evidence for asymmetric kicks in neutron star formation [@van97; @van00], less attention has been paid to black hole binary systems. In two of the black hole X-ray binaries with some of the best observational constraints on the system parameters, XTE J1118+480 and GRO J1655-40, there is evidence for an asymmetric supernova kick [@Gua05; @Wil05]. There is also some evidence suggesting an asymmetric kick in GRS1915+105 [@Dha07], although in that case the uncertainty in the distance to the system precludes the derivation of strong constraints on the kick velocity. However, the black hole in Cygnus X-1 is inferred to have formed via direct collapse [@Mir03]. More black hole sources need to be studied to measure black hole kicks and investigate consequent formation mechanisms in order to constrain theoretical models of black hole formation.
V404 Cyg
--------
V404 Cyg is a dynamically-confirmed black hole X-ray binary system, with a mass function of $6.08\pm0.06M_{\odot}$ [@Cas94]. The system comprises a black hole accretor of mass $12^{+3}_{-2}M_{\odot}$ [@Sha94] in a 6.5-d orbit with a $0.7^{+0.3}_{-0.2}M_{\odot}$ K0 IV stripped giant star [@Cas93; @Kin93]. The orbit is highly circular, with an eccentricity of $e<3\times10^{-4}$, in agreement with the fact that tidal forces just before the onset of mass transfer should have recircularised the orbit following the supernova. Its low radial velocity [$-0.4\pm2.2$kms$^{-1}$; @Cas94] has been taken as evidence for a small natal kick [@Bra95; @Nel99].
In this paper we present measurements of the proper motion of V404 Cyg, and go on to derive its full three-dimensional space velocity, infer its Galactocentric orbit, and discuss the implications for the formation of the black hole in the light of the inferred natal kick from the supernova.
Observations and data reduction {#sec:obs}
===============================
In order to investigate the proper motion of the source, we interrogated the Very Large Array (VLA) archives for high-resolution observations of V404 Cyg. We selected only A-configuration observations at frequencies of 8.4GHz and higher, to obtain the highest possible astrometric accuracy. We further restricted the dataset to observations where the phase calibrator was J2025+3343, the phase reference source used in the High Sensitivity Array (HSA) observations of @Mil08. Since all positions are measured relative to the phase reference source, observations using a different secondary calibrator would potentially have been subject to a systematic positional offset.
The VLA data were reduced using standard procedures within the 31Dec08 version of [aips]{} [@Gre03]. A script was written in [ParselTongue]{}, the Python interface to [aips]{}, to automate the bulk of the calibration. We corrected the derived positions for shifts in the assumed calibrator position, using a reference position of 20$^{\rm h}$25$^{\rm m}$108421050 334300214430 (J2000) for the calibrator source. All co-ordinates were precessed to J2000 values using the [aips]{} task [uvfix]{}. Using data from the USNO ([`h`ttp://maia.usno.navy.mil/ser7/finals2000A.all]{}), we corrected for offsets in (UT1-UTC), the difference between Universal Time (UT1) as set by the rotation of the Earth and measured by very long baseline interferometry (VLBI), and co-ordinated Universal Time (UTC), the atomic time (TAI) adjusted with leap seconds. Where measured offsets were available, we also corrected for shifts in antenna positions using the [aips]{} task [vlant]{}. Source positions were measured by fitting an elliptical Gaussian to the source in the image plane, using the deconvolved, phase-referenced image prior to any self-calibration. The source was not resolved in any of the images. We added an extra positional uncertainty of 10mas to the measured VLA positions, to account for systematic uncertainties in the astrometry. The list of observations and derived source positions is given in Table \[tab:vla\_obs\].
The dataset was enhanced by the use of two high-resolution VLBI measurements of the source position. We used the 8.4-GHz position of @Mil08, and also obtained a second measurement using global VLBI at 22GHz, under proposal code GM064. Eight European stations (Cambridge, Effelsberg, Jodrell Bank Mk[ii]{}, Medicina, Metsahovi, Noto, Onsala, and Robledo), all ten Very Long Baseline Array (VLBA) stations, the phased VLA and the Green Bank Telescope (GBT) participated in the experiment. Data were taken from 21:30:00 [ut]{} on 2008 May 31 until 16:00:00 [ut]{} on 2008 June 1, with the European stations being on source for the first 12h of the run (Robledo from 02:20:00 until 08:45:00 on 2008 June 1) and the North American stations for the second 12h (Mauna Kea from 07:30:00, when the source rose in Hawaii). The overlap time, when both sets of stations were on source, was 5.5h. The phase reference and fringe finder source was J2025+3343. We observed with a total bit rate of 512Mbs$^{-1}$, with a bandwidth of 64MHz per polarization. We observed in 150-s cycles, spending 1.5min on the target and 1min on the calibrator in each cycle. The VLA was phased up at the start of each calibrator scan, and we made referenced pointing observations with the larger dishes (the VLA, GBT, Effelsberg and Robledo) every 1–2h. Data were reduced using standard procedures within [aips]{}. We detected the source at a significance level of $4.7\sigma$.
------- ------------- ---------- ------- --------- ----------- ---------------------------------- ---------- ------------ ---------- -- --
Code Date MJD Error Config. Frequency RA Error Dec Error
(d) (GHz) (sec) (arcsec)
AH348 1990 Feb 16 47938.71 0.18 A 14.94 20$^{\rm h}$24$^{\rm m}$0382854 0.00085 3352020348 0.0103
AH385 1990 Mar 08 47958.70 0.12 A 14.94 20$^{\rm h}$24$^{\rm m}$0382896 0.00083 3352020379 0.0102
AH390 1990 Mar 25 47975.55 0.09 A 8.44 20$^{\rm h}$24$^{\rm m}$0382865 0.00081 3352020439 0.0101
AH424 1991 Sep 25 48525.03 0.05 AB 14.94 20$^{\rm h}$24$^{\rm m}$0382733 0.00276 3352019563 0.0235
AH424 1991 Sep 25 48525.04 0.05 AB 8.44 20$^{\rm h}$24$^{\rm m}$0382321 0.00148 3352020143 0.0136
AH424 1992 Oct 20 48916.03 0.07 A 14.94 20$^{\rm h}$24$^{\rm m}$0382752 0.00096 3352020208 0.0122
AH424 1992 Oct 20 48916.04 0.07 A 8.44 20$^{\rm h}$24$^{\rm m}$0382673 0.00084 3352020196 0.0105
AH390 1993 Jan 28 49015.68 0.04 AB 14.94 20$^{\rm h}$24$^{\rm m}$0382779 0.00222 3352019964 0.0173
AH390 1993 Jan 28 49015.68 0.02 AB 8.44 20$^{\rm h}$24$^{\rm m}$0382427 0.00219 3352020317 0.0158
AH641 1998 May 04 50937.60 0.04 A 8.46 20$^{\rm h}$24$^{\rm m}$0382493 0.00086 3352019820 0.0109
AH669 1999 Jul 04 51363.41 0.01 A 8.46 20$^{\rm h}$24$^{\rm m}$0382479 0.00144 3352019721 0.0156
AH669 1999 Jul 13 51372.37 0.01 A 8.46 20$^{\rm h}$24$^{\rm m}$0382431 0.00115 3352019984 0.0143
AH669 1999 Jul 26 51385.30 0.01 A 8.46 20$^{\rm h}$24$^{\rm m}$0382508 0.00260 3352019470 0.0343
AH669 1999 Aug 22 51412.18 0.01 A 8.46 20$^{\rm h}$24$^{\rm m}$0382443 0.00163 3352019429 0.0176
AH669 1999 Sep 01 51422.21 0.01 A 8.46 20$^{\rm h}$24$^{\rm m}$0382417 0.00152 3352020063 0.0207
AH669 1999 Sep 04 51426.02 0.01 A 8.46 20$^{\rm h}$24$^{\rm m}$0382397 0.00194 3352019464 0.0188
AH669 2000 Oct 20 51837.33 0.01 A 8.46 20$^{\rm h}$24$^{\rm m}$0382718 0.00595 3352019676 0.0405
AR476 2002 Feb 03 52308.69 0.01 A 8.46 20$^{\rm h}$24$^{\rm m}$0382325 0.00088 3352019481 0.0108
AR476 2002 Mar 01 52334.54 0.01 A 8.46 20$^{\rm h}$24$^{\rm m}$0382586 0.00257 3352019457 0.0224
AH823 2003 Jul 29 52849.30 0.29 A 8.46 20$^{\rm h}$24$^{\rm m}$0382440 0.00101 3352019317 0.0120
BG168 2007 Dec 02 54436.84 0.09 HSA 8.42 20$^{\rm h}$24$^{\rm m}$0382129 0.000010 3352018993 0.0003
GM064 2008 May 31 54618.42 0.25 Global 22.22 20$^{\rm h}$24$^{\rm m}$03821082 0.000009 3352018957 0.0001
------- ------------- ---------- ------- --------- ----------- ---------------------------------- ---------- ------------ ---------- -- --
Results
=======
Fig. \[fig:ra\_dec\] shows the measured Right Ascension and Declination as a function of time, over the $\sim20$ years since the 1989 outburst of V404 Cyg. The best fitting proper motions, in Right Ascension and Declination respectively, are $$\begin{aligned}
\mu_{\alpha}\cos\delta &= -4.99\pm0.19\quad {\rm mas}\quad {\rm y}^{-1}\\
\mu_{\delta} &= -7.76\pm0.21\quad {\rm mas}\quad {\rm y}^{-1}.\end{aligned}$$ Thus the total proper motion is $\mu = 9.2\pm0.3$masy$^{-1}$. All uncertainties are 68 per cent confidence limits, and unless otherwise noted we will henceforth quote $1\sigma$ error bars on all measurements.
The fit gives a reference position (prior to correcting for the effects of the unknown parallax) of $20^{\rm h}24^{\rm m}03\fs82177(2)$ $33\degr52^{\prime}01\farcs9088(3)$ (J2000) on MJD54000.0, from which we can use the proper motion to determine the predicted position at any future time.
![Top panel: Measured Right Ascension as a function of time, from 1990 to 2007. Bottom panel: Declination as a function of time. The dotted lines are the best fitting proper motions, $\mu_{\alpha}
= (-4.99\pm0.19)\times10^{-4}$secy$^{-1}$ in R.A. and $\mu_{\delta} = (-7.76\pm0.21)$masy$^{-1}$ in Dec.. The black dots are the VLA 8.4GHz points and the light grey triangles are the 15-GHz VLA measurements. The dark grey squares labelled ‘H’ and ‘G’ are from the 8.4-GHz HSA observations of @Mil08 and the 22-GHz global VLBI observations reported in this paper, respectively. \[fig:ra\_dec\]](millerjones_v404_f1.eps){width="\columnwidth"}
Converting to Galactic Space Velocity co-ordinates
--------------------------------------------------
With the systemic radial velocity of $-0.4\pm2.2$kms$^{-1}$ measured from optical H$\alpha$ studies [@Cas94], we can calculate, for a given source distance, the full three-dimensional space velocity of the system. The best constraint on the source distance is $4.0^{+2.0}_{-1.2}$kpc [@Jon04]. Using the transformations of @Joh87, and the standard solar motion of ($U_{\odot}=10.0\pm0.36$, $V_{\odot}=5.25\pm0.62$, $W_{\odot}=7.17\pm0.38$) kms$^{-1}$ [@Deh98], we can compute the heliocentric Galactic space velocity components $U$, $V$ and $W$ (defined as $U$ positive towards the Galactic Centre, $V$ positive towards $l=90$, and $W$ positive towards the North Galactic Pole). The derived system parameters are given in Table \[tab:uvw\].
Parameter Value
------------------------------------------------------ -------------------------------
Galactic longitude $l$ 73.12
Galactic latitude $b$ -2.09
Distance $d$ (kpc) $4.0^{+2.0}_{-1.2}$
Systemic velocity $\gamma$ (kms$^{-1}$) $-0.4\pm2.2$
Proper motion $\mu_{\alpha}\cos\delta$ (masy$^{-1}$) $-4.99\pm0.19$
Proper motion $\mu_{\delta}$ (masy$^{-1}$) $-7.76\pm0.21$
$U$ (kms$^{-1}$) $177.1\pm3.8^{+83.7}_{-50.2}$
$V$ (kms$^{-1}$) $-46.1\pm2.4_{-25.5}^{+15.3}$
$W$ (kms$^{-1}$) $0.2\pm3.7^{+2.1}_{-3.5}$
$U-<U>$ (kms$^{-1}$) $62.0^{+39.4}_{-17.4}$
$V-<V>$ (kms$^{-1}$) $-16.1_{-0.0}^{+6.5}$
$W-<W>$ (kms$^{-1}$) $0.2^{+2.1}_{-3.5}$
$v_{\rm pec}$ (kms$^{-1}$) $64.1\pm3.7^{+37.8}_{-16.6}$
: \[tab:uvw\]Measured and derived parameters. $U$, $V$ and $W$ are the Galactic space velocity components in the direction of the Galactic Centre, $l=90^{\circ}$ and $b=90^{\circ}$ respectively. The first set of error bars accounts for uncertainties in the measured space velocities only, and the second takes into account the distance uncertainty. $U-<U>$, $V-<V>$ and $W-<W>$ are the discrepancies from the velocities that would be expected for Galactic rotation. Summing these discrepancies in quadrature gives the peculiar velocity, $v_{\rm
pec}$. The major source of error in these values is the distance uncertainty.
For a given distance, the expected values of $U$ and $V$ can be calculated, assuming the source participates in the Galactic rotation. @Rei04 determined the angular rotation rate of the LSR at the Sun, $\Theta_0/R_0 = 29.45\pm0.15$kms$^{-1}$kpc$^{-1}$. Assuming a Galactocentric distance of 8.0kpc [@Rei93], this implies a circular velocity of 236kms$^{-1}$. For circular rotation about the Galactic Centre, the $W$ component of the velocity is expected to be zero. As done by @Dha07 for GRS1915+105, we can transform the measured values of $U$ and $V$ into radial and circular velocities about the Galactic Centre, expected to be $v_{\rm
rad} = 0$kms$^{-1}$ and $v_{\rm circ} = 236$kms$^{-1}$ respectively. Fig. \[fig:v\_pec\] shows the derived radial, circular and $W$ velocities and the peculiar velocity of V404 Cyg. The peculiar velocity is defined to be the difference between the measured 3-dimensional space velocity and that expected for a source participating in the Galactic rotation, $$v_{\rm pec} = \left(v_{\rm rad}^2+(v_{\rm circ}-236)^2+W^2\right)^{1/2}
\label{eq:vpec}$$
![Derived Galactocentric velocities of V404 Cyg, as a function of source distance. (a) shows the radial velocity, $v_{\rm rad}$, (b) shows the circular velocity, $v_{\rm circ}$, (c) shows the velocity out of the Galactic plane, $W$, and (d) shows the peculiar velocity, $v_{\rm pec}$, i.e. the difference from the values expected for a source participating in the Galactic rotation. Dotted lines show the expected values of 0kms$^{-1}$ for the radial, $W$ and peculiar velocities, and 236kms$^{-1}$ for the circular velocity. Error bars only account for uncertainties in the space velocity components, assuming zero error on the distance at each plotted point.\[fig:v\_pec\]](millerjones_v404_f2.eps){width="\columnwidth"}
For the range of distances found by @Jon04, 2.8–6.0kpc, it is clear that the peculiar velocity is non-zero. We derive a value of $v_{\rm pec} = 64.1\pm3.7^{+37.8}_{-16.6}$kms$^{-1}$, where the first error bar accounts for statistical error in the space velocities, and the second for the distance uncertainty. The predominant component of the peculiar velocity is radial, with a circular velocity slightly faster than expected, and a velocity out of the Galactic plane consistent with zero.
The orbital trajectory in the Galactic potential {#sec:orbit}
================================================
From the known source position and the measured spatial velocity components, we can integrate backwards in time to compute the orbital trajectory of the system in the potential of the Galaxy. Using a fifth-order Runge-Kutta algorithm [@Pre92] to perform the integration, we compared the predictions of several different models for the Galactic potential [Carlberg & Innanen 1987, using the revised parameters of @Kui89; @Pac90; @Joh95; @Wol95; @Fly96; @deO02], all using some combination of one or more disc, spherical bulge and halo components. A representative orbit reconstruction using the model of @Joh95 is shown in Fig. \[fig:gal\_orbit\].
{width="\textwidth"}
While the errors in the space velocity components make little difference to the computed orbital trajectory for a given model, the uncertainty in the distance has much more of an effect. Fig. \[fig:distances\] compares the trajectories computed for 2.8, 4.0 and 6.0kpc, the lower, mean and upper bounds to the possible range of distances. A larger source distance implies a more elliptical orbit, which reaches further from the Galactic Centre at apogalacticon.
![The effect of the distance uncertainty on the computed Galactocentric orbit of V404 Cyg. Trajectories have been computed for the minimum, best fitting and maximum distances derived by @Jon04, using the measured space velocities and the Galactic potential of @Joh95. For each distance, the spread in the trajectories due to uncertainties in the space velocity components has been plotted. In all panels, the cross marks the Galactic Centre, and the open circle and triangles mark the current positions of the Sun and V404 Cyg (at its different distances) respectively. \[fig:distances\]](millerjones_v404_f4.ps){width="\columnwidth"}
Comparing the different models for the Galactic potential, we find that the predicted orbital trajectories in the Galactic plane begin to diverge significantly after only 25–30Myr. In the perpendicular direction, the predictions diverge even faster, within 2–3Myr. Given the uncertainty in the source distance and Galactic potential, it is clearly impossible to integrate back in time over 0.4–0.8Gyr (Section \[sec:kicks\]) to locate the birthplace of the system. However, for a given source distance, we can average over the ensemble of model predictions to derive some generic properties of the orbit. The eccentricity $e$, semi-major axis $a$, and the distances of the apsides, $r_{\rm max}$ and $r_{\rm min}$, for the motion in the Galactic plane, the maximum height reached above or below the plane, $z_{\rm max}$, and the minimum and maximum values of the peculiar velocity, $v_{\rm pec,min}$ and $v_{\rm pec,max}$, are given in Table \[tab:orbit\_parms\].
We can compare these values with those of the thick and thin disk populations. @Bin98 give the velocity dispersions in the radial, azimuthal and vertical directions (relative to the Galactic plane) for both stellar populations, based on the data of @Edv93, for stars within 80pc of the Sun. A set of Monte Carlo simulations established that the typical planar eccentricities of the thin and thick disk populations were $0.12\pm0.06$ and $0.29\pm0.15$ respectively, while the typical maximum height reached above the plane was $0.17\pm0.14$ and $0.56\pm0.61$kpc respectively. Unless V404 Cyg is at the maximum possible distance, it is likely to have originated as a member of the thin disk population (as also suggested by the age, metallicity and component masses of the system), and received some sort of kick which increased the planar eccentricity and the component of velocity out of the Galactic plane.
Distance 2.8kpc 4.0kpc 6.0kpc
-------------------------------- --------------- --------------- ----------------
$r_{\rm max}$ (kpc) $9.8\pm0.4$ $11.7\pm0.7$ $20.2\pm4.4$
$r_{\rm min}$ (kpc) $7.0\pm0.1$ $7.2\pm0.1$ $7.9\pm0.1$
$z_{\rm max}$ (kpc) $0.13\pm0.02$ $0.20\pm0.03$ $0.47\pm0.14$
$a$ (kpc) $8.4\pm0.2$ $9.4\pm0.4$ $14.1\pm2.2$
$e$ $0.16\pm0.02$ $0.24\pm0.03$ $0.42\pm0.07$
$v_{\rm pec,min}$ (kms$^{-1}$) $21.9\pm5.1$ $38.9\pm6.4$ $82.4\pm7.4$
$v_{\rm pec,max}$ (kms$^{-1}$) $56.5\pm4.9$ $78.8\pm5.7$ $134.3\pm10.3$
: \[tab:orbit\_parms\]Derived parameters of the Galactocentric orbit, averaged over the ensemble of models for the Galactic potential. The parameters are the maximum and minimum distances from the Galactic Centre measured in the plane ($r_{\rm
max}$ and $r_{\rm min}$), the maximum distance reached above or below the plane ($z_{\rm max}$), the semi-major axis $a$, the orbital eccentricity $e$ (both calculated in the plane), and the minimum and maximum peculiar velocity ($v_{\rm pec,min}$ and $v_{\rm pec,max}$ respectively). Uncertainties are the scatter due to both the errors on the measured velocities and the differing models for the Galactic Potential.
The peculiar velocity {#sec:kicks}
=====================
The current peculiar velocity of the system is $64.1\pm3.7^{+37.8}_{-16.6}$kms$^{-1}$. However, owing to its orbit in the Galactic potential, this is not a conserved quantity. For a distance of 4kpc, we find that the peculiar velocity varies between 39 and 79kms$^{-1}$ (see Table \[tab:orbit\_parms\] and Fig. \[fig:velocities\]). We go on to examine potential explanations for this peculiar velocity.
![Variation in Galactocentric radial velocity $v_{\rm rad}$, circular velocity $v_{\rm circ}$, perpendicular velocity $v_z$, and peculiar velocity $v_{\rm pec}$, as a function of time while integrating the orbit backwards over the last Gyr, using the potential of @Joh95. Solid, dotted and dashed lines are for source distances of 4.0, 2.8 and 6.0kpc respectively. The grey lines show the uncertainty arising from the error bars on the measured space velocities, for a distance of 4.0kpc. The peculiar velocity varies as a function of time. \[fig:velocities\]](millerjones_v404_f5.ps){width="\columnwidth"}
Symmetric supernova kick
------------------------
If it formed with a natal supernova, the system can receive a Blaauw kick [@Bla61], whereby the binary recoils to conserve momentum after mass is instantaneously ejected from the primary. For the binary to remain bound after the supernova, the ejected mass $\Delta
M$ must be less than half the total mass of the system. A maximum ejected mass translates to a maximum recoil velocity of the binary, for which an expression was derived by @Nel99, $$\begin{gathered}
v_{\rm max} = 213 \left(\frac{\Delta M}{M_{\odot}}\right)
\left(\frac{m}{M_{\odot}}\right) \left({\frac{P_{\rm re-circ}}{{\rm
d}}}\right)^{-1/3}\\
\times\left(\frac{M_{\rm BH}+m}{M_{\odot}}\right)^{-5/3} {\rm km\,s}^{-1},
\label{eq:vmax}\end{gathered}$$ where $M_{\rm BH}$ and $m$ are the masses of the black hole and the secondary immediately after the supernova, before mass transfer has begun, and $P_{\rm re-circ}$ is the period of the orbit once it has recircularized following the supernova, related to the orbital period immediately after the supernova has occurred by $P_{\rm
re-circ}=P_{\rm post-SN}(1-e_{\rm post-SN}^2)^{3/2}$. Re-circularization occurs before the start of mass transfer from the secondary to the black hole, due to the strong tidal forces present once the secondary has evolved and expanded sufficiently to come close to filling its Roche Lobe.
During its X-ray binary phase, the system undergoes mass transfer from the donor star to the black hole, increasing the black hole mass and orbital period and reducing the donor mass. In the case of conservative mass transfer, angular momentum conservation and Kepler’s Third Law imply that these parameters evolve as $$P\propto(M_{\rm BH}m)^{-3}.
\label{eq:P-evolution}$$ While we have no exact constraint on how long ago mass transfer began, we can use the current orbital period and component masses [@Sha94] together with Equation \[eq:P-evolution\] to find the system parameters at any point in the past, and the implied maximum ejected mass and recoil velocity of a supernova which would have created a system with those parameters (from Equation \[eq:vmax\]). This is shown in Fig. \[fig:vmax\] for three possible current configurations of donor and accretor mass.
As mass is transferred from donor to accretor, the orbital period increases (Equation \[eq:P-evolution\]). Thus for a larger transferred mass, $M_{\rm trans}$, (a smaller initial black hole mass and a less recent onset of mass transfer) the orbital period at the onset of mass transfer is smaller. The pre-supernova orbital period, which is smaller due to the mass loss in the explosion (middle line in the top left hand panel of Fig. \[fig:vmax\]), cannot be so short that either the black hole progenitor or its companion fills its Roche lobe, setting a minimum possible pre-supernova orbital period (lower line in the top left hand panel of Fig. \[fig:vmax\]). For small values of $M_{\rm trans}$, the maximum mass lost in the supernova is equal to half the total system mass, and the maximum kick velocity $v_{\rm max}$ increases via Equation \[eq:vmax\] because both the post-supernova donor mass increases and the post-supernova period decreases with increasing $M_{\rm trans}$. Where the period constraint becomes important, the maximum mass lost in the supernova decreases, being set by the shortest allowed pre-supernova orbital period. The decreasing mass loss in the explosion then offsets the increasing post-supernova donor mass, and the maximum possible kick velocity decreases as $M_{\rm trans}$ increases further.
![Evolution of the post-supernova binary orbital period (top panels), maximum possible mass ejected in the supernova (middle panels), and maximum possible velocity kick (lower panels), as a function of total mass transferred since the supernova from the secondary to the black hole. Left-hand panels are all for the case of conservative mass transfer, for current component masses of $(M_{\rm BH}, m) = $ (12, 0.7) $M_{\odot}$. The top left plot shows the post-supernova orbital period (thick solid line), the pre-supernova orbital period (middle line) and the minimum permitted pre-supernova orbital period in which neither the helium star progenitor nor the main sequence companion fills its Roche lobe (bottom line). The right-hand panels show the situation for conservative mass transfer using different current component masses, and one case of non-conservative mass transfer. Thick solid lines are for current component masses of $(M_{\rm BH}, m) = $ (12, 0.7) $M_{\odot}$, thin solid lines are for (10, 1) $M_{\odot}$ and grey lines are for (14, 0.5) $M_{\odot}$, assuming conservative mass transfer in all cases. Dashed lines indicate non-conservative mass transfer for the (10, 1) $M_{\odot}$ case, with ten per cent of the mass transferred being lost from the system. \[fig:vmax\]](millerjones_v404_f6.ps){width="\columnwidth"}
Equation \[eq:P-evolution\] is valid only for the case of conservative mass transfer. However, there are strong indications that this assumption is not valid for V404 Cyg. At the very least, the radio outbursts [e.g. @Han92] suggest that material is being lost to jet outflows. @Pod03 calculated binary evolution tracks for a $10M_{\odot}$ black hole in orbit with donors of mass 2–$17M_{\odot}$ (for there to have been any symmetric kick, Fig. \[fig:vmax\] shows that the transferred mass must be $<2.5M_{\odot}$, implying that the initial secondary must have been less massive than $\sim3.5M_{\odot}$), some of which they found could reproduce the system parameters of V404 Cyg extremely well. They took into account non-conservative mass transfer, whereby mass transferred in excess of the Eddington rate is lost from the system. A comparison with the equations for conservative mass transfer suggested that the orbital period increases more slowly in the non-conservative case, by a factor of at most 2. As an example of how this could affect the maximum Blaauw kick velocity, the dashed lines in Fig. \[fig:vmax\] are for the current case of a $10M_{\odot}$ accretor with a $1M_{\odot}$ donor, with 10 per cent of the transferred mass being lost from the system and the orbital period increasing a factor 2 more slowly than predicted by Equation \[eq:P-evolution\].
Our calculations show that for V404 Cyg, it is just possible to achieve a kick of 64kms$^{-1}$ using only symmetric mass loss in the supernova (a Blaauw kick). If the explosion occurred at a point in the Galactocentric orbit where the peculiar velocity was minimized (Fig. \[fig:velocities\]), this could be sufficient to explain the observed peculiar velocity, particularly if the system is towards the lower end of the possible range of distances. However, to achieve such a large kick requires fine-tuning the parameters, to give a relatively low initial black hole mass (closer to $9.2M_{\odot}$ than the best fitting value of $12M_{\odot}$) and an ejection of $\sim11M_{\odot}$ during the supernova. Such a large mass loss, in combination with the system parameters (a $\sim9M_{\odot}$ black hole with a low-mass donor), do not make this scenario very plausible [see, e.g., @Fry01 for theoretical estimates of the amount of mass ejected in a supernova as a function of progenitor mass; to form a $9M_{\odot}$ black hole requires a progenitor of 25–30$M_{\odot}$, in which case the mass ejected in the supernova is expected to be $<3M_{\odot}$]. We find it unlikely that a symmetric kick alone is sufficient to explain the observed peculiar velocity.
Velocity dispersion in the disc {#sec:diffusion}
-------------------------------
The donor star in V404 Cyg is a K0 subgiant [@Cas94] of mass $0.7^{+0.3}_{-0.2}M_{\odot}$ [@Sha94]. @Kin93 demonstrated that the system is a stripped giant, which evolves on the nuclear timescale of the donor star, which is in the range 0.4–0.8Gyr. Thus the secondary was initially significantly more massive, and has transferred mass to the black hole. From Fig. \[fig:vmax\], the donor star is unlikely to have transferred more than $2.5M_{\odot}$ to the black hole during the mass transfer phase, implying an initial donor mass of $<3.5M_{\odot}$. The evolutionary tracks of @Pod03 show that such a system would take of order 0.7–0.8Gyr to evolve to an orbital period of 6.5d, in agreement with the range given by @Kin93. In 0.8Gyr, the system has made 3.5–5 orbits in the potential of the Galaxy (depending on the model used for the Galactic potential; see Section \[sec:orbit\]), so could well have received some component of peculiar velocity in the Galactic plane owing to non-axisymmetric forces such as scattering from the potentials of spiral arms or interstellar cloud complexes [@Wie77]. However, estimates of the velocity dispersion of the thin disk population [@Deh98; @Mig00] show that for F0–F5 stars (with initial masses comparable to that estimated for the donor star in V404 Cyg prior to the onset of mass transfer), it is of order 28kms$^{-1}$.
@Mig00 fitted the Sun’s peculiar motion and the differential velocity field caused by the Galactic rotation to the parallaxes and proper motions measured by Hipparcos. Examining the fit residuals showed the peculiar velocities to follow a three-dimensional Gaussian distribution, with the predicted scatter in the velocity in Galactic longitude being given by $(0.7<u^2>)^{1/2}$, where $<u^2>$ is the velocity dispersion in the radial direction, determined as $<u^2>^{1/2}=22.5\pm0.3$kms$^{-1}$ for F0-F5 stars. The measured peculiar velocity of V404 Cyg in the longitudinal direction is $-64.0^{+18.5}_{-37.7}$kms$^{-1}$, implying a probability of $7\times10^{-4}$ of being caused by Galactic velocity diffusion if the source is at 4kpc, and only $1.6\times10^{-2}$ even if the source is at 2.8kpc. Thus this mechanism is unlikely to account for the observed and inferred range of peculiar velocities.
An asymmetric supernova kick?
-----------------------------
If neither the effects of stellar diffusion nor a symmetric kick can explain the observed peculiar velocity, it could instead be the effect of an asymmetric kick during the supernova. The smaller dispersion in $z$-distance (distance above or below the Galactic plane) of black hole X-ray binaries when compared to neutron star systems had been interpreted as evidence for smaller kicks when forming black holes [@Whi96]. However, @Jon04, using a larger sample and revised distance estimates for the sources, found no evidence for such a discrepancy, suggesting that black holes can receive natal kicks comparable with those seen in neutron star systems. Recent analysis of the space velocities of the black hole X-ray binaries XTE J1118+480 [@Gua05] and GRO J1655-40 [@Wil05] found evidence for asymmetric kicks in these two systems, with such a kick being mandated in the case of XTE J1118+480 [@Fra08].
An asymmetric kick is not constrained to lie in the orbital plane, as in the case of a Blaauw kick. While the inclination angle of the system to the line of sight is well-constrained to be $56^{\circ}\pm4^{\circ}$ [@Sha94], the longitude of the ascending node, $\Omega$, with respect to the sky plane is not known, so we cannot determine the absolute orientation of the binary orbital plane. However, for a given value of $\Omega$, we can determine the orientation of the orbital plane with respect to the Galactic axes $x$, $y$ and $z$ (corresponding to the velocity components $U$, $V$ and $W$). We assume that the orientation of the orbital plane does not change with time. For any point during its Galactocentric orbit, we can use the positions and velocities computed in Section \[sec:orbit\] to calculate the component of the peculiar motion perpendicular to the orbital plane, $v_{\perp}$, which provides a lower limit to the asymmetric kick velocity should the supernova have occurred at that point in time. Assuming all values $0<\Omega<2\pi$ are equally likely, we can run Monte Carlo simulations to find the probability that the component of the peculiar velocity perpendicular to the orbital plane is equal to or less than would be expected for the progenitor from the typical velocity dispersion of massive stars [taken as 10kms$^{-1}$ in one component; see @Mig00]. Fig. \[fig:asymmetric\_kick\] shows the probability that $v_{\perp}$ is less than 10kms$^{-1}$ for the mean and extreme values of the distance, as a function of time. The probability is low, of order 10–20 per cent, with little variation in the mean probability as a function of source distance. It is therefore unlikely (although possible, except for certain short periods of time) that the measured peculiar velocity perpendicular to the binary orbital plane can be attributed to the velocity dispersion of the system prior to the supernova. Since a symmetric kick cannot give rise to velocity out of the orbital plane, it is probable that there was an asymmetric kick during the formation of the black hole.
![Probability that the component of the peculiar velocity perpendicular to the orbital plane is less than 10kms$^{-1}$. We assumed a uniform probability for the longitude of the ascending node, for angles in the range $0<\Omega<2\pi$, and derived the probabilities for the best-fitting distance (4.0kpc; solid line), and the upper (6.0kpc; dotted line) and lower (2.8kpc; dashed line) limits to the distance. A low probability implies that it is likely that the velocity perpendicular to the orbital plane is high, so the system has received an asymmetric kick at formation. \[fig:asymmetric\_kick\]](millerjones_v404_f7.eps){width="\columnwidth"}
Discussion
==========
It appears that a supernova is required to explain the peculiar velocity of V404 Cyg. Thus the black hole in this system did not form via direct collapse. From the range of peculiar velocities inferred during the Galactocentric orbit of the system, a symmetric supernova kick could just be sufficient to explain the observations if the source is at the lower end of the allowed range of distances. However, the mass loss required, as well as the component of peculiar velocity inferred to lie perpendicular to the orbital plane, make it likely that the system was subject to an asymmetric kick during black hole formation.
The full three-dimensional space velocities have been measured for a handful of other black hole systems. XTE J1118+480 [@Mir01] and GRO J1655-40 [@Mir02] are both in highly eccentric orbits around the Galactic Centre. An asymmetric natal kick is required for XTE J1118+480 [@Fra08], and believed to have been likely in GRO J1655-40 [@Wil05]. The high-mass X-ray binary Cygnus X-1 [@Mir03] is moving at only $9\pm2$kms$^{-1}$ with respect to its parent association Cyg OB3, implying that $<1M_{\odot}$ was ejected in the natal supernova, and that the black hole formed by direct collapse. GRS1915+105 on the other hand, has been inferred to have an orbit and peculiar velocity [@Dha07] fairly similar to that of V404 Cyg. @Dha07, using a Galactocentric distance of $R_0=8.5$kpc and a circular velocity of $\Theta_0=220$kms$^{-1}$ for the LSR, found that a symmetric supernova kick and stellar diffusion were insufficient to explain the peculiar velocity unless the source was located at 9–10kpc, where the peculiar velocity is minimized. However, using the values of $R_0=8.0$kpc and $\Theta_0=236$kms$^{-1}$ assumed in this paper gives a minimum peculiar velocity of 23kms$^{-1}$ for a source distance of 10kpc, which could then in principle be explained by a symmetric supernova explosion or stellar diffusion. While a distance at the lower end of the allowed range would give a higher value for the current peculiar velocity, the peculiar velocity would still be sufficiently low at certain points in the Galactocentric orbit for an asymmetric kick not to be required. However, if the source is at the upper end of the possible distance range ($\gtrsim 12$kpc), the peculiar velocity is high enough throughout the orbit that an asymmetric kick becomes necessary. Without knowing the source distance, we cannot definitively determine its formation mechanism. We also note that the momentum imparted by a kick of 23kms$^{-1}$ to such a $14M_{\odot}$ black hole [@Gre01] is similar to that gained by a neutron star receiving a kick of a few hundred kms$^{-1}$ [e.g. @Lyn94; @Han97], so owing to its large mass, even such a low peculiar velocity for GRS1915+105 would not necessarily rule out a natal kick.
For V404 Cyg, the derived components of the system velocity in the Galactic plane, $U$ and $V$, are much larger than $W$, the velocity out of the plane (Table \[tab:uvw\]). While this is to be expected since the Galactic rotation of 236kms$^{-1}$ forms a component of both $U$ and $V$, but not $W$, accounting for the Galactic rotation (giving the $U-<U>$ and $V-<V>$ terms in Table \[tab:uvw\]) does not remove this discrepancy between the components of the peculiar velocity in and out of the plane. A similar discrepancy is observed for the other four black hole systems with measured three-dimensional space velocities [@Mir01; @Mir02; @Mir03; @Dha07], even accounting for the ranges of values allowed by the uncertainties in the system parameters. However, as shown in Fig. \[fig:velocities\], the velocity components change with time, so we reconstructed the Galactocentric orbits of all five sources. This showed that the $W$ component of velocity can be significantly greater than the peculiar velocity in the plane for XTE J1118+480, and can be of a similar magnitude to the component in the plane at certain points in the orbits of Cygnus X-1 and GRS1915+105. With only five systems, we are dealing with small number statistics, but these results are consistent with there being no preferred orientation of the peculiar velocity relative to the Galactic plane. Indeed, considering the known population of black hole X-ray binaries, including those with no measured space velocity, the distribution in $z$ [@Jon04] demonstrates that a number of systems must have a significant $W$ velocity at certain points in their orbits, in order to reach the distances of several hundred parsecs above or below the plane at which they are currently observed. Thus there is no observational evidence to suggest that the natal kick distribution should not be isotropic.
Thus the two systems with the lowest black hole masses, XTE J1118+480 and GRO J1655-40 [both of order 6–7$M_{\odot}$ @Oro03], are found to have required asymmetric kicks during the formation of the black hole. For the higher-mass systems, the situation is less clear-cut. Cygnus X-1, V404 Cyg and GRS1915+105 all have masses of $>10M_{\odot}$. While Cygnus X-1 appears to have formed by direct collapse, the peculiar velocity of V404 Cyg implies that a supernova must have occurred, and an asymmetric kick seems likely to have been required in this system, contrary to assertions that black holes of $10M_{\odot}$ form by direct collapse [e.g. @Mir08]. For GRS1915+105, we cannot definitively determine its formation mechanism without an accurate distance to the source. Regardless, with only five systems, no clear trends with black hole mass can be identified. In order to better constrain the formation mechanisms of stellar-mass black holes, the space velocities of more such systems with a range of black hole masses need to be measured, to constrain the frequency of occurrence of natal kicks, and hence supernova explosions.
Conclusions
===========
We have measured the proper motion of V404 Cyg using 20 years’ worth of VLA and VLBI radio observations. Together with the radial velocity and constraints on the distance of the system, this translates to a peculiar motion of $64.1\pm3.7^{+37.8}_{-16.6}$kms$^{-1}$. Given the measured proper motion, the black hole cannot have been formed via direct collapse. A supernova is required to achieve the observed peculiar velocity, with either a large amount of mass ($\sim11M_{\odot}$) being lost in the explosion, or, more probably, the system being subject to an asymmetric kick. In the case of a pure Blaauw kick, $\sim 1M_{\odot}$ must have been transferred from the donor to the black hole since the onset of mass transfer, implying an initial black hole mass of $\sim 9M_{\odot}$ with a donor mass of $\sim 2M_{\odot}$ prior to the onset of mass transfer.
Acknowledgments {#acknowledgments .unnumbered}
===============
J.C.A.M.-J. is a Jansky Fellow of the National Radio Astronomy Observatory. E.G. is supported through Chandra Postdoctoral Fellowship grant number PF5-60037, issued by the Chandra X-Ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060. The VLA and VLBA are facilities of the National Radio Astronomy Observatory which is operated by Associated Universities, Inc., under cooperative agreement with the National Science Foundation. The European VLBI Network is a joint facility of European, Chinese, South African and other radio astronomy institutes funded by their national research councils. ParselTongue was developed in the context of the ALBUS project, which has benefited from research funding from the European Community’s sixth Framework Programme under RadioNet R113CT 2003 5058187. This research has made use of NASA’s Astrophysics Data System.
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---
abstract: 'Motivated by a recent surge of interest for Dynkin operators in mathematical physics and by problems in the combinatorial theory of dynamical systems, we propose here a systematic study of logarithmic derivatives in various contexts. In particular, we introduce and investigate generalizations of the Dynkin operator for which we obtain Magnus-type formulas.'
author:
- Frédéric Menous
- Frédéric Patras
date: 'June 6, 2012'
title: 'Logarithmic derivatives and generalized Dynkin operators.'
---
Introduction {#sect:intro .unnumbered}
============
Dynkin operators are usually defined as iterated brackettings. They are particularly popular in the framework of linear differential equations and the so-called continuous Baker-Campbell-Hausdorff problem (to compute the logarithm of an evolution operator). We refer to [@reutenauer] for details and an historical survey of the field. Dynkin operators can also be expressed as a particular type of logarithmic derivatives (see Corollary \[classical\] below). They have received increasingly more and more interest during the recent years, for various reasons.
1. In the theory of free Lie algebras and noncommutative symmetric functions, it was shown that Dynkin operators generate the descent algebra (the direct sum of Solomon’s algebras of type $A$, see [@gelfand; @reutenauer]) and play a crucial role in the theory of Lie idempotents.
2. Generalized Dynkin operators can be defined in the context of classical Hopf algebras [@patreu2002]. The properties of these operators generalize the classical ones. Among others (see also [@EGP; @EMP]), they can be used to derive fine properties of the renormalization process in perturbative quantum field theory (pQFT): the Dynkin operators can be shown to give rise to the infinitesimal generator of the differential equation satisfied by renormalized Feynman rules [@EFGP]. This phenomenon has attracted the attention on logarithmic derivatives in pQFT where generalized Dynkin operators are expected to lead to a renewed understanding of Dyson-Schwinger-type equations, see e.g. [@KY06].
The present article was however originated by different problems, namely problems in the combinatorial theory of dynamical systems (often referred to as Ecalle’s *mould calculus, see e.g. [@sauzin], also for further references on the subject) and in particular in the theory of normal forms of vector fields. It appeared very soon to us that the same machinery that had been successfully used in the above mentioned fields and problems was relevant for the study of dynamical systems. However, the particular form of logarithmic derivatives showing up in this field requires the generalization of the known results on logarithmic derivatives and Dynkin operators to a broader framework: we refer to the section \[exemple\] of the present article for more details on the subject.*
The purpose of the present article is therefore to develop further the algebraic and combinatorial theory of logarithmic derivatives, following various directions and point of views (free Lie algebras, Hopf algebras, Rota-Baxter algebras...), all of them known to be relevant for the study of dynamical systems but also to various other fields, running from the numerical analysis of differential equations to pQFT.
We limit the scope of the present article to the general theory and plan to use the results in forthcoming articles.
Twisted Dynkin operators on free Lie algebras
=============================================
Let $T(X)$ be the tensor algebra over an alphabet $X=\{x_1,...,x_n,...\}$. It is graded by the length of words: the degree $n$ component $T_n(X)$ of $T(X)$ is the linear span (say over the rationals) of the words $y_1...y_n,\ y_i\in X$. The length, $n$, of $y_1...y_n$ is written $l(y_1...y_n)$. It is equipped with the structure of a graded connected cocommutative Hopf algebra by the concatenation product: $$\mu(y_1...y_n\otimes z_1...z_m)=y_1...y_n\cdot z_1...z_m:=y_1...y_nz_1...z_m$$ and the unshuffling coproduct: $$\Delta(y_1...y_n):=\sum\limits_{p=0}^n\sum\limits_{I\coprod J=[n]}y_{i_1}...y_{i_p}\otimes y_{j_1}...y_{j_{n-p}},$$ where $1\leq i_1<i_2<...<i_p\leq n$, $1\leq j_1<j_2<...<j_{n-p}\leq n$ and $I=\{i_1,...,i_p\}$, $J=\{j_1,...,j_{n-p}\}$. The antipode is given by: $S(y_1...y_n)=(-1)^ny_n...y_1$, where $y_i\in X$. We refer e.g. to [@reutenauer] for further details on the subject and general properties of the tensor algebra viewed as a Hopf algebra and recall simply that the antipode is the convolution inverse of the identity of $T(X)$: $S\ast Id=Id\ast S=\varepsilon$, where $\varepsilon$ is the projection on the scalars $T_0(X)$ orthogonally to the higher degree components $T_n(X),\ n>0$, and the convolution product $f\ast g$ of two linear endomorphisms of $T(X)$ (and, more generally of a Hopf algebra with coproduct $\Delta$ and product $\mu$) is given by: $f\ast g:=\mu\circ (f\otimes g)\circ\Delta$. In particular, $S\ast Id$ is the null map on $T_n(X)$ for $n>0$.
We write $\delta$ an arbitrary derivation of $T(X)$ (in particular $\delta$ acts as the null application on the scalars, $T_0(X)$). The simplest and most common derivations are induced by maps $f$ from $X$ to its linear span: the associated derivation, written $\tilde f$, is then defined by $f(y_1...y_n)=\sum_{i=0}^ny_1...y_{i-1}f(y_i)y_{i+1}...y_n$. Since for $a,b\in T(X),\ f([a,b])=[f(a),b]+[a,f(b)]$, where $[a,b]:= ab-ba$, these particular derivations are also derivations of the free Lie algebra over $X$. These are the derivations we will be most interested in in practice (the ones that map the free Lie algebra over $X$ to itself), we call them *Lie derivations.*
In the particular case $f=Id$, we also write $Y$ for $\tilde{Id}$: $Y$ is the graduation operator, $Y(y_1...y_n)=n\cdot y_1...y_n$. When $f=\delta_{x_i}$ ($f(x_i)=x_i, \ f(x_j)=0$ for $j\not= i$), $\tilde f$ counts the multiplicity of the letter $x_i$ in words and is the noncommutative analog of the derivative with respect to $x_i$ of a monomial in the letters in $X$.
For arbitrary letters $y_1,...,y_n$ of $X$, we have, for $D_{\delta}:=S\ast \delta$: $$D_\delta (y_1...y_n)=[...[[\delta(y_1),y_2],y_3]...,y_n]$$
Let us assume, by induction, that the identity holds in degrees $<n$. Then, with the same notation as the ones used to define the coproduct $\Delta$: $$S\ast \delta (y_1...y_n) = \sum\limits_{p=0}^n(-1)^p y_{i_p}...y_{i_1}\delta(y_{j_1}...y_{j_{n-p}})$$ where $I\coprod J=[n]$. Notice that, if $i_p\not=n$, $j_{n-p}=n$ (and conversely). Therefore, since $\delta$ is a derivation: $$S\ast \delta (y_1...y_n) = \sum\limits_{p=0}^{n-1}(-1)^p (y_{i_p}...y_{i_1}\delta(y_{j_1}...y_{j_{n-p-1}})y_n$$ $$+ y_{i_p}...y_{i_1}y_{j_1}...y_{j_{n-p-1}}\delta(y_n))$$ $$+\sum\limits_{p=0}^{n-1}(-1)^{p+1}y_ny_{i_p}...y_{i_1}\delta(y_{j_1}...y_{j_{n-p-1}}),$$ where the sums run over the partitions $I\coprod J=[n-1]$.
The first and third term of the summation sum up to $[S\ast \delta (y_1...y_{n-1}),y_n]$ which is, by induction, equal to $[...[[\delta(y_1),y_2],y_3]...,y_n]$. The second computes $S\ast Id(y_1...y_{n-1}) \delta(y_n)$, which is equal to $0$ for $n>1$. The Proposition follows.
For a Lie derivation $\delta$, the map $D_\delta$ maps $T(X)$ to $Lie(X)$, the free Lie algebra over $X$. Moreover, we have, for $l\in Lie(X)$, $$D_\delta(l)=\delta(l).$$
The first part of the corollary follows from the previous proposition. To prove the second part, recall that $l\in Lie(X)$ if and only if $\Delta(l)=l\otimes 1+1\otimes l$. Notice furthermore that, since $D_\delta=S\ast \delta$, we have $\delta=Id\ast D_\delta$. Therefore: $$\delta(l)=(Id\ast D_\delta)(l)=D_\delta(1)\cdot l+D_\delta(l).$$ The proof follows since $D_\delta(1)=0$.
We recover in particular the theorem of Dynkin [@dynkin], Specht [@specht], Wever [@wever] (case $f=Id$) and obtain an extension thereof to the case $f=\delta_{x_i}$. We let the reader derive similar results for other families of Lie derivations.
\[classical\] We have, for the classical Dynkin operator $D=S\ast Y$ and an arbitrary element $l$ in $T_n(X)\cap Lie(X)$: $$D(l)=n\cdot l.$$ In particular, the operator $\frac D n$ is a projection from $T_n(X)$ onto $T_n(X)\cap Lie(X)$.
The definition $D=S*Y$ of the Dynkin operator seems due to von Waldenfels, see [@reutenauer].
Let us write $T_n^{i}(X)$ for the linear span of words over $X$ such that the letter $x_i$ appears exactly $n$ times. The derivation $\tilde \delta_{x_i}$ acts as the multiplication by $n$ on $T_n^{i}(X)$.
We have for $D_{x_i}:=S\ast \tilde\delta_{x_i}$ and an arbitrary element $l$ in $T_n^i(X)\cap Lie(X)$: $$D_{x_i}(l)=n\cdot l.$$ In particular, the operator $\frac D n$ is a projection from $T_n^i(X)$ onto $T_n^i(X)\cap Lie(X)$.
Abstract logarithmic derivatives
================================
Quite often, the logarithmic derivatives one is interested in arise from dynamical systems and geometry. We explain briefly why on a fundamental example, the classification of singular vector fields (section \[exemple\]). Although we settle our later computations in the general framework of Lie and enveloping algebras, the reader may keep that motivation in mind.
The second section (\[henv\]) shows briefly how to extend the results on generalized Dynkin operators obtained previously in the tensor algebra to the general setting of enveloping algebras.
We show at last (section (\[idc\]) how these results connect to the theory of Rota Baxter algebras, which is known to be the right framework to investigate the formal properties of derivations. Indeed, as we will recall below, Rota-Baxter algebra structures show up naturally when derivations have to be inverted. See also [@EGP; @EMP; @BCEP] for further details on the subject of Rota-Baxter algebras and their applications.
An example from the theory of dynamical systems {#exemple}
-----------------------------------------------
Derivations on graded complete Lie algebras appear naturally in the framework of dynamical systems, especially when dealing with the formal classification (up to formal change of coordinates) of singular vector fields.
The reader can refer to [@Ilya] for an overview and further details on the objects we consider (such as identity-tangent diffeomorphisms or substitution automorphisms) -let us also mention that the reader who is interested only in formal aspects of logarithmic derivatives may skip that section.
A formal singular vector field in dimension $\nu$ is an operator: $$X=\sum_{i=1}^{\nu} f_i(x_1,\dots,x_{\nu})\frac{\partial}{\partial x_i}$$ such that $f_i(0)=0$ for all $i$ (that is $f_i\in \mathbb{C}_{\geq 1}[[x_1,\dots,x_{\nu}]]$). Such operators act on the algebra of formal series in $\nu$ variables. In practice, a vector field is given by a series of operators such as $x_1^{n_1} \ldots
x_{\nu}^{n_{\nu}} \frac{\partial}{\partial x_i}$ with $n_1 + \ldots + n_{\nu}
\geqslant 1$ that acts on monomials $x_1^{m_1} \ldots x_{\nu}^{m_{\nu}}$: $$\left( \left. x_1^{n_1} \ldots x_{\nu}^{n_{\nu}} \frac{\partial}{\partial
x_i} \right) . x_1^{m_1} \ldots x_{\nu}^{m_{\nu}} = m_i\cdot x_1^{m_1 + n_1} \ldots
x_i^{m_i + n_i - 1} \ldots x_{\nu}^{m_{\nu} + n_{\nu}} \right.$$ so that the total degree goes from $m_1 + \ldots + m_{\nu}$ to $m_1 + \ldots +
m_{\nu} + n_1 + \ldots + n_{\nu} - 1$ and the graduation for such an operator is then $n_1 + \ldots + n_{\nu} - 1$.
The $0$ graded component of a vector field $X$ is called the linear part since it can be written $X_0=\sum A_{ij}x_i\frac{\partial}{\partial x_j}$ and a fundamental question in dynamical systems is to decide if $X$ is conjugate, up to a change of coordinates, to its linear part $X_0$. Notice that:
- The vector space $L$ of vector fields without linear part (or without component of graduation 0) is a graded complete Lie algebra.
- The exponential of a vector field in $L$ gives a one to one correspondence between vector fields and substitution automorphisms on formal power series, that is operators $F$ such that $$F(A(x_1,\dots,x_\nu))=A(F(x_1),\dots,F(x_{\nu}))$$ where $(F(x_1),\dots,F(x_{\nu}))$ is a formal identity-tangent diffeomorphism.
- The previous equation also determines an isomorphism between the Lie group of $L$ and the group of formal identity-tangent diffeomorphism $G_1$.
These are essentially the framework (the one of graded complete Lie algebras) and the objects (elements of the corresponding formal Lie groups) that we will consider and investigate in our forthcoming developments.
Consider now a vector field $X=X_0 +Y\in L_0\oplus L$ and suppose that it can be linearized by a change of coordinates in $G_1$, or rather by a substitution automorphism $F$ in the Lie group of $L$. It is a matter of fact (see [@Ilya], [@SNAG]) to check that the corresponding conjugacy equation reads: $$X_0 F=F(X_0 +Y) \Longleftrightarrow [X_0,F]=FY \Longleftrightarrow ad_{X_0}(F)=FY$$ This equation, called the homological equation, delivers a derivation $\delta=ad_{X_0}$ on $L$ that is compatible with the graduation. The linearization problem is then obviously related to the inversion of the logarithmic derivation $D_{\delta}(F):=F^{-1}\delta(F)$.
In the framework of dynamical systems, the forthcoming theorem \[invert\] ensures that if the derivation $ad_{X_0}$ is invertible on $L$, any vector field $X_0+Y$ can be linearized. This is the kind of problems that can be addressed using the general theory of logarithmic derivatives to be developed in the next sections.
Hopf and enveloping algebras {#henv}
----------------------------
We use freely in this section the results in [@patreu2002] to which we refer for further details and proofs. The purpose of this section is to extend the results in [@patreu2002] on the Dynkin operator to more general logarithmic derivatives.
Let $L=\bigoplus\limits_{n\in\NM^\ast}L_n$ be a graded Lie algebra, $\hat L=\prod\limits_{n\in\NM^\ast}L_n$ its completion, $U(L)=\bigoplus\limits_{n\in\NM^\ast}U(L)_n$ the (graded) enveloping algebra of $L$ and $\hat U(L)=\prod\limits_{n\in\NM^\ast}U(L)_n$ the completion of $U(L)$ with respect to the graduation.
The ground field is chosen to be $\QM$ (but the results in the article would hold for an arbitrary ground field of characteristic zero and, due to the Cartier-Milnor-Moore theorem [@mm; @patras], for arbitrary graded connected cocommutative Hopf algebras).
The enveloping algebra $U(L)$ is naturally provided with the structure of a Hopf algebra. We denote by $\e: \QM=U(L)_0\ra U(L)$ the unit of $U(L)$, by $\eta: U(L)\ra \QM$ the counit, by $\Delta :U(L)\ra U(L)\ot U(L)$ the coproduct and by $\mu :U(L)\ot U(L) \ra U(L)$ the product. An element $l$ of $U(L)$ is [*primitive*]{} if $\Delta (l)=l\ot 1+1\ot l$; the set of primitive elements identifies canonically with $L$. Recall that the [*convolution*]{} product $*$ of linear endomorphisms of $U(L)$ is defined by $f*g=\mu \circ (f\ot g) \circ \Delta$; $\n:=\e\circ\eta$ is the neutral element of $\ast$. The antipode is written $S$, as usual.
An element $f$ of $End(U(L))$ admits $F\in End(U(L))\ot End(U(L))$ as a [*pseudo-coproduct*]{} if $F \circ \Delta =
\Delta \circ f$. If $f$ admits the pseudo-coproduct $f\ot \nu +\nu\ot f$, we say that $f$ is [*pseudo-primitive*]{}.
In general, an element of $End(U(L))$ may admit several pseudo-coproducts. However, this concept is very flexible, as shows the following result [@patreu2002 Thm. 2].
If $f,g$ admit the pseudo-coproducts $F,G$ and $\a\in \F$, then $f+g, \a f, f*g, f\circ g$ admit respectively the pseudo-coproducts $F+G, \a F, F*G, F\circ G$, where the products $*$ and $\circ$ are naturally extended to $End(U(L))\ot End(U(L))$.
An element $f \in End(U(L))$ takes values in $Prim(U(L))$ if and only if it is pseudo-primitive.
Let $\delta$ be an arbitrary derivation of $L$ ($\forall l,l'\in L, \ \delta([l,l'])=[\delta(l),l']+[l,\delta(l')]$). We also write $\delta$ for its unique extension to a derivation of $U(L)$ and write $D_\delta:=S\ast \delta$. For an element $l\in L$, $\exp(l)$ is group-like ($\Delta(\exp(l))=\exp(l)\otimes \exp(l)$), from which it follows that: $$D_\delta(\exp(l))=S(\exp(l))\delta(\exp(l))=\exp(-l)\delta(\exp(l)),$$ the (noncommutative) logarithmic derivative of $\exp(l)$ with respect to $\delta$. We call therefore $D_\delta$ the *logarithmic derivative of $\delta$.*
The logarithmic derivative $D_\delta$ is a pseudo-primitive: it maps $U(L)$ to $L$.
Indeed, $S\otimes S$ is a pseudo-coproduct for $S$ (see [@patreu2002]). On the other hand $U(L)$ is spanned by products $l_1...l_n$ of elements of $L$. Since $\delta$ is a derivation, we get: $$\Delta\circ\delta (l_1...l_n)=\Delta(\sum_{i=1}^nl_1...\delta(l_i)...l_n)=(\delta\otimes Id+Id\otimes\delta)\circ\Delta(l_1...l_n),$$ where the last identity follows directly from the fact that the $l_i$ are primitive, which implies that $\Delta(l_1...l_n)$ can be computed by the same formula as the one for the coproduct in the tensor algebra. In particular, $\delta\otimes Id+Id\otimes\delta$ is a coproduct for $\delta$. We get finally: $$\Delta\circ D_\delta=\Delta\circ(S\ast\delta)=(S\otimes S)\ast(\delta\otimes Id+Id\otimes\delta)\circ\Delta$$ $$=(D_\delta\otimes \n+\n\otimes D_\delta)\circ\Delta ,$$ from which the proof follows.
\[primit\] For $l\in L$, we have $\delta(l)=D_\delta(l)$. In particular, when $\delta$ is invertible on $L$, $D_\delta(l)$ is a projection from $U(L)$ onto $L$.
The proof is similar to the one in the free Lie algebra. We have $D_\delta(l)=(S*\delta)(l)=\pi\circ (S\ot \delta)\circ \Delta (l)=\pi\circ (S\ot \delta)(l\ot 1+1\ot l)$ $=\pi (S(l)\ot \delta(1)+S(1)\ot \delta(l))=\delta(l)$, since $\delta(1)=0$ and $S(1)=1$.
Integro-differential calculus {#idc}
-----------------------------
The notation are the same as in the previous section, but we assume now that the derivation $\delta$ is invertible on $L$ and extends to an invertible derivation on $U(L)^+:=\bigoplus\limits_{n\geq 1}U(L)_n$. The simplest example is provided by the graduation operator $Y(l)=n\cdot l$ for $l\in L_n$ (resp. $Y(x)=n\cdot x$ for $x\in U(L)_n$). This includes the particular case, generic for various applications to the study of dynamical systems, where $L$ is the graded Lie algebra of polynomial vector fields spanned linearly by the $x^n\partial_x$ and $\delta :=x\partial_x$ acting on $P(x)\partial_x$ as $\delta(P(x)\partial_x):=xP'(x)\partial_x$.
We are interested in the linear differential equation $$\label{eqdif}
\delta \phi=\phi\cdot x,\ x\in L,\ \phi\in 1+U(L)^+.$$ The inverse of $\delta$ is written $R$ and satisfies, on $U(L)^+$, the identity: $$R(x)R(y)=R(R(x)y)+R(xR(y)),$$ that follows immediately from the fact that $\delta$ is an invertible derivation. In other terms, $U(L)^+$ is provided by $R$ with the structure of a weight $0$ Rota-Baxter algebra and solving (\[eqdif\]) amounts to solve the so-called Atkinson recursion: $$\phi =1+R(\phi\cdot x).$$ We refer to [@EGP; @EMP] for a detailed study of the solutions to the Atkinson recursion and further references on the subject. Perturbatively, the solution is given by the Picard (or Chen, or Dyson... the name given to the expansion depending on the application field) series: $$\phi = 1+\sum\limits_{n\geq 1}R^{[n]}(x),$$ where $R^{[1]}(x)=R(x)$ and $R^{[n]}(x):=R(R^{[n-1]}(x)x)$.
Since the restriction to the weight $0$ is not necessary for our forthcoming computations, we restate the problem in a more general setting and assume from now on that $R$ is a weight $\theta$ Rota-Baxter (RB) map on $U(L)^+$, the enveloping algebra of a graded Lie algebra. That is, we assume that: $$R(x)R(y)=R(R(x)y)+R(xR(y))-\theta R(xy).$$ This assumption allows to extend vastly the scope of our forthcoming results since the setting of Rota-Baxter algebras of arbitrary weight includes, among others, renormalization in perturbative quantum field theory and difference calculus, the later setting being relevant to the study of diffeomorphisms in the field of dynamical systems. We refer in particular to the various works of K. Ebrahimi-Fard on the subject (see e.g. [@EMP; @BCEP] for various examples of RB structures and further references). We assume furthermore that $R$ respects the graduation and restricts to a linear endomorphism of $L$.
The solution to the Atkinson recursion is a group-like element in $1+U(L)^+$. In particular, $S(\phi)=\phi^{-1}$.
Recall from [@patreu2002] and [@EFGP] that the generalized Dynkin operator $D:=S\ast Y$ (the convolution of the antipode with the graduation map in $U(L)$) maps $U(L)$ to $L$ and, more specifically, defines a bijection between the set of group-like elements in $\hat U(L)$ and $\hat L$. The inverse is given explicitly by ([@EFGP Thm. 4.1]): $$D^{-1}(l)=1+\sum_{n\in\NM^\ast}\sum_{k_1+...+k_l=n}\frac{l_{k_1}\cdot ...\cdot l_{k_l}}{k_1(k_1+k_2)...(k_1+...+k_l)},$$ where $l_n$ is the component of $l\in L$ in $L_n$. According to [@EGP Thm. 4.3], when $l=D(1+\sum\limits_{n\geq 1}R^{[n]}(x))$ we also have: $$1+\sum\limits_{n\geq 1}R^{[n]}(x)=1+\sum_{n\in\NM^\ast}\sum_{k_1+...+k_l=n}\frac{l_{k_1}\cdot ...\cdot l_{k_l}}{k_1(k_1+k_2)...(k_1+...+k_l)},$$ that is, since $l\in L$ by Prop. \[primit\], $1+\sum\limits_{n\geq 1}R^{[n]}(x)$ is a group-like element in $U(L)$. The last part of the Lemma is a general property of the antipode acting on a group-like element; the Lemma follows.
We are interested now in the situation where another Lie derivation $d$ acts on $U(L)$ and commutes with $R$ (or equivalently with $\delta$ when $R$ is the inverse of a derivation). A typical situation is given by Schwarz commuting rules between two different differential operators associated to two independent variables.
Let $d$ be a graded derivation on $U(L)$ commuting with the weight $\theta$ Rota-Baxter operator $R$. Then, for $\phi$ a solution of the Atkinson recursion as above, we have: $$D_d(\phi)=\phi^{-1}\cdot d(\phi)=\sum\limits_{n\geq 1}R_d^{[n]}(x)$$ with $I_d^{[1]}=d(x)$, $R_d^{[1]}(x)=R(d(x))$, $I_d^{[n+1]}(x)=[R_d^{[n]}(x),x]+\theta x\cdot I_d^{[n]}(x)$ and $R_d^{[n+1]}(x)=R(I_d^{[n+1]}(x))$.
Notice, although we won’t make use of this property, that the operation $x\circ y:=[R(x),y]+\theta y\cdot x$ showing up implicitly in this recursion is a preLie product, see e.g. [@EGP]. In particular, for $y$ the solution to the preLie recursion: $$y=d(x)+y\circ x,$$ we have: $D_d(\phi)=R(y)$.
The first identity $D_d(\phi)=\phi^{-1}\cdot d(\phi)$ follows from the definition of the logarithmic derivative $D_d:=S\ast d$ and from the previous Lemma.
The second is equivalent to $d(\phi)=\phi\cdot \sum\limits_{n\geq 1}R_d^{[n]}(x)$, that is: $$d(R^{[n]}(x))=R_d^{[n]}(x)+\sum_{i=1}^{n-1}R^{[i]}(x)R_d^{[n-i]}(x).$$ For $n=1$, the equation reads $d(R(x))=R(d(x))$ and expresses the commutation of $d$ and $R$.
The general case follows by induction. Let us assume that the identity holds for the components in degree $n<p$ of $D_d(\phi)$. We summarize in a technical Lemma the main ingredient of the proof. Notice that the Lemma follows directly from the Rota-Baxter relation and the definition of $R_d^{[n+1]}(x)$.
We have, for $n,m\geq 1$: $$R(R^{[m]}(x)\cdot ([R_d^{[n]}(x),x]+\theta x\cdot I_d^{[n]}(x)))=R^{[m]}(x)R_d^{[n+1]}(x)$$ $$-R(R^{[m-1]}(x)\cdot x\cdot R_d^{[n+1]}(x))+\theta R(R^{[m-1]}(x)\cdot x\cdot ([R_d^{[n]}(x),x]+\theta x\cdot I_d^{[n]}(x)).$$
We have, for the degree $p$ component of $D_d(\phi)$, using the Lemma to rewrite $R(R^{[m]}(x)\cdot R_d^{[n]}(x)\cdot x)$: $$d(R^{[p]}(x))=d(R(R^{[p-1]}(x)\cdot x))=R((dR^{[p-1]}(x))\cdot x+R^{[p-1]}(x)\cdot dx)$$ $$=R(R^{[p-1]}(x)\cdot dx)+R((R_d^{[p-1]}(x)+\sum_{k=1}^{p-2}R^{[p-1-k]}(x)\cdot R_d^{[k]}(x) )\cdot x)$$ $$=R(R^{[p-1]}(x)\cdot dx)+R(R_d^{[p-1]}(x)\cdot x)+\sum\limits_{k=1}^{p-2}[ R^{[p-1-k]}(x)\cdot R_d^{[k+1]}(x)$$ $$-R(R^{[p-2-k]}(x)\cdot x\cdot R_d^{[k+1]}(x))+\theta R(R^{[p-2-k]}(x)\cdot x\cdot ([R_d^{[k]}(x),x]+\theta x\cdot I_d^{[k]}(x))$$ $$+R(R^{[p-1-k]}(x)\cdot x\cdot R_d^{[k]}(x))-\theta R(R^{[p-1-k]}(x)\cdot x\cdot I_d^{[k]}(x))].$$ The fourth and sixth terms cancel partially and add up to $R(R^{[p-2]}(x)\cdot x\cdot R_d^{[1]}(x))-R(x\cdot R_d^{[p-1]}(x))$. The fifth and last terms cancel partially and add up to $\theta R(x\cdot I_d^{[p-1]}(x))-\theta R(R^{[p-2]}(x)\cdot x\cdot I_d^{[1]}(x))$. In the end, we get: $$d(R^{[p]}(x))=\sum\limits_{k=1}^{p-2}R^{[p-1-k]}(x)\cdot R_d^{[k+1]}(x)+[R(R^{[p-2]}(x)\cdot x\cdot R_d^{[1]}(x))+$$ $$R(R^{[p-1]}(x)\cdot dx)-\theta R(R^{[p-2]}(x)\cdot x\cdot I_d^{[1]}(x)]+[R(R_d^{[p-1]}(x)\cdot x)-R(x\cdot R_d^{[p-1]}(x))$$ $$+\theta R(x\cdot I_d^{[p-1]}(x))]$$ Using the RB identity for the expressions inside brackets, we get finally: $$d(R^{[p]}(x))=\sum\limits_{k=1}^{p-2}R^{[p-1-k]}(x)\cdot R_d^{[k+1]}(x)+R^{[p-1]}(x)R_d^{[1]}(x)+R_d^{[p]}(x),$$ from which the Theorem follows.
Magnus-type formulas
====================
The classical Magnus formula relates “logarithms and logarithmic derivatives” in the framework of linear differential equations. That is, it relates explicitly the logarithm $\log(X(t))=:\Omega(t)$ of the solution to an arbitrary matrix (or operator) differential equation $X'(t)=A(t)X(t),\ X(0)=1$ to the infinitesimal generator $A(t)$ of the differential equation: $$\Omega'(t)=\frac{ad_{\Omega(t)}}{\exp^{ad_{\Omega(t)}}-1}A(t)=A(t)+\sum\limits_{n>0}\frac{B_n}{n!}ad_{\Omega(t)}^n(A(t)),$$ where $ad$ stands for the adjoint representation and the $B_n$ for the Bernoulli numbers.
The Magnus formula is a useful tool for numerical applications (computing the logarithm of the solution improves the convergence at a given order of approximation). It has recently been investigated and generalized from various points of view, see e.g. [@EM2], where the formula has been extended to general dendriform algebras (i.e. noncommutative shuffle algebras such as the algebras of iterated integrals of operators), [@CP] where the algebraic structure of the equation was investigated from a purely preLie algebras point of view, or [@BCEP] where a generalization of the formula has been introduced to model the commutation of time-ordered products with time-derivations.
The link with preLie algebras follows from the observation that (under the hypothesis that the integrals and derivatives are well-defined), for arbitrary time-dependent operators, the preLie product $$M(t)\curvearrowleft N(t):=\int_0^t [N(u),M'(u)]du$$ satisfies $(M(t)\curvearrowleft N(t))'=ad_{N(t)}M'(t)$. The Magnus formula rewrites therefore: $$\Omega'(t)=\left(\int\limits_0^tA(x)dx\curvearrowleft\left( \frac{\Omega}{\exp(\Omega)-1} \right)\right)'$$ where $\frac{\Omega}{\exp(\Omega)-1}$ is computed in the enveloping algebra of the preLie algebra of time-dependent operators.
Here, we would like to go one step further and extend the formula to general logarithmic derivatives in the suitable framework in view of applications to dynamical systems and geometry, that is, the framework of enveloping algebras of graded Lie algebras and Lie derivations actions. Notations are as before, that is $L$ is a graded Lie algebra and $\delta$ a graded Lie derivation (notice that we do not assume its invertibility on $L$ or $U(L)$).
For $l\in L$, $k\geq 1$ and $x\in U(L)$, we have: $$x\cdot l^k=\sum\limits_{i=0}^k{{k}\choose{ i}}l^{k-i}\cdot (-ad_{l})^i(x).$$
The proof is by induction on $n$. The formula holds for $n=1$: $xl=-[l,x]+lx$. Let us assume that it holds for an arbitrary $n<p$. Then we have: $$x\cdot l^p= (x\cdot l^{p-1})\cdot l= (\sum\limits_{i=0}^{p-1}{{p-1}\choose{ i}}l^{p-1-i}\cdot (-ad_{l})^i(x))\cdot l$$ $$=\sum\limits_{i=0}^{p-1}{{p-1}\choose{ i}}l^{p-1-i}\cdot (-ad_{l})^{i+1}(x) +\sum\limits_{i=0}^{p-1}{{p-1}\choose{ i}} l^{p-i}\cdot(-ad_{l})^{i}(x).$$ The identity follows then from Pascal’s triangular computation of the binomial coefficients.
\[magnus\] For $l\in L$, we have: $$D_\delta (\exp(l))=\frac{\exp(-ad_l)-1}{-ad_l}\delta(l).$$
Indeed, from the previous formula we get, $$d(\exp(l))=\sum\limits_{n\geq 1}\frac{1}{n!}d(l^n)= \sum\limits_{n\geq 1}\frac{1}{n!}\sum_{k=0}^{n-1}l^{n-1-k}d(l)l^k$$ $$=\sum\limits_{n\geq 1}\sum\limits_{i=0}^{n-1}\frac{1}{n!}(\sum\limits_{k=i}^{n-1}{k\choose i})l^{n-1-i}(-ad_l)^i(d(l))$$ $$=\sum\limits_{n\geq 1}\sum\limits_{i=0}^{n-1}\frac{1}{n!}{n\choose i+1}l^{n-1-i}(-ad_l)^i(d(l))$$ $$=\sum\limits_{i\geq 0}\sum\limits_{n\geq i+1}\frac{1}{(i+1)!(n-1-i)!}l^{n-1-i}(-ad_l)^i(d(l))$$ $$=\exp(l)(\sum\limits_{i\geq 0}\frac{1}{(i+1)!}(-ad_l)^i)d(v).$$ Since $\exp(l)$, being the exponential of a Lie element is group-like, $D_\delta(\exp(l))= \exp({-l})d(\exp(l))$ and the theorem follows.
Let us show as a direct application of the Thm (\[magnus\]) how to recover the classical Magnus theorem (other applications to mould calculus and to the formal and analytic classification of vector fields are postponed to later works).
Let us consider once again an operator-valued linear differential equation $X'(t)=X(t)\lambda A(t),\ X(0)=1$. Notice that the generator $A(t)$ is written to the right, for consistency with our definition of logarithmic derivatives $D_\delta =S\ast d$. All our results can of course be easily adapted to the case $d\ast S$ (in the case of linear differential equations this amounts to consider instead $X'(t)=A(t)X(t)$), this easy task is left to the reader. Notice also that we introduce an extra parameter $\lambda$, so that the perturbative expansion of $X(t)$ is a formal power series in $\lambda$.
Consider then the Lie algebra $O$ of operators $M(t)$ (equipped with the bracket of operators) and the graded Lie algebra $L=\bigoplus_{n\in \NM^\ast}\lambda^n O=O\otimes \lambda \CM[\lambda]$. Applying the Thm (\[magnus\]), we recover the classical Magnus formula.
Although a direct consequence of the previous theorem (recall that any group-like element in the enveloping algebra $U(L)$ can be written as an exponential), the following proposition is important and we state it also as a theorem.
\[invert\] When $\delta$ is invertible on $L$, the logarithmic derivative $D_\delta$ is a bijection between the set of group-like elements in $U(L)$ and $L$.
Indeed, for $h\in L$ the Magnus-type equation in $L$ $$l=\delta^{-1}(\frac{-ad_l}{\exp (-ad_l)-1}(h))$$ has a unique recursive solution $ l\in L$ such that $exp(l)=D_\delta^{-1}(h)$.
[99]{}
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|
---
abstract: 'We investigate the microwave absorption arising from inhomogeneity in the superfluid density of a model high-T$_c$ superconductor. Such inhomogeneities may arise from a wide variety of sources, including quenched random disorder and static charge density waves such as stripes. We show that both mechanisms will inevitably produce additional absorption at finite frequencies. We present simple model calculations for this extra absorption, and discuss applications to other transport properties in high-T$_c$ materials. Finally, we discuss the connection of these predictions to recent measurements by Corson[@corson] of absorption by the high-temperature superconductor [[Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$]{} ]{}in the THz frequency regime.'
address: 'Department of Physics, The Ohio State University, Columbus, Ohio 43210'
author:
- 'Sergey V. Barabash, David Stroud'
title: 'Superfluid Inhomogeneity and Microwave Absorption in Model High-T$_c$ Superconductors'
---
Introduction
============
The high-T$_c$ superconductor [[Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$]{} ]{}shows a remarkably strong absorption in the microwave regime, even far below $T_c$[@corson; @lee]. In conventional, low-T$_c$, s-wave superconductors, there is no such background, because there is no absorption below the energy gap for pair excitations, $2\Delta$. But in high-T$_c$ materials, which are thought to have a $d_{x^2-y^2}$ order parameter, the gap vanishes in certain nodal $k$ directions. Hence, gapless nodal quasiparticles can be excited, and hence can absorb microwave radiation, at arbitrarily low temperatures. However, experiment suggests that this absorption is much stronger than expected from the quasiparticles alone[@corson2].
In this paper, we show that such extra absorption can be produced if the superfluid density $n({\bf x})$ within the $ab$ plane is a function of position ${\bf x}$. Such spatial variation can be produced, e. g., by quenched disorder, or by charge density waves (e. g. stripes). The extra absorption is most likely in the microwave frequency range. A similar absorption was found by us in a previous paper[@barabash], in which the inhomogeneity was described as static fluctuations in the Josephson coupling between superconducting grains.
Microwave Absorption in an Inhomogeneous Two-Dimensional Superfluid
===================================================================
We consider a two-fluid model of a superconductor in two dimensions, with local conductivity $$\sigma({\bf x}, \omega) = \sigma_{qp}(\omega) +
\frac{iq^2n_s( \omega, {\bf x})}{m^*\omega},$$ where $q = 2|e|$ and $m^*$ is twice the electron mass $m_e$. $n_s({\bf x}, \omega)$ is assumed spatially varying, while $\sigma_{qp}$ is taken to be spatially uniform. We wish to calculate the complex effective conductivity $\sigma_e(\omega)$. $\sigma_{qp}$ might be a contribution from the nodal quasiparticles, while $n_s$ represents the perfect-conductivity response of the superconductor. It can be spatially varying because of the very short in-plane coherence length.
We employ the Kramers-Kronig relations satisfied by $\sigma_e(\omega)$, namely $\sigma_{e1}(\omega) = \frac{2}{\pi}P\int_0^\infty
\frac{\omega^\prime \sigma_{e2}(\omega^\prime)}
{\omega^{\prime 2} - \omega^2}d\omega^\prime + \sigma_\infty$, and $\sigma_{e2} = \frac{q^2n_{s,e}}{m^*\omega} - \frac{2\omega}{\pi}
P\int_0^\infty\frac{\sigma_{e1}(\omega^\prime)-\sigma_\infty}
{\omega^{\prime 2} - \omega^2}d\omega^\prime$, where P means “principal part of,” $\sigma_e = \sigma_{e1} + i\sigma_{e2}$, and $n_{s,e}$ is the effective superfluid density \[the value of $\frac{q^2n_{s,e}}{m^*}$ can be found from the residue of $\sigma_e(\omega)$ at $\omega=0$\]. At very large frequencies $\omega$, the equation for $\sigma_{e2}$ becomes $\sigma_{e2} \rightarrow \frac{q^2n_{s,e}}{m^*\omega} +
\frac{2}{\pi\omega}\int_0^\infty
[\sigma_{e1}(\omega^\prime) - \sigma_\infty]d\omega^\prime$.
Frequency-Independent $\sigma_{qp}$
-----------------------------------
If $\sigma_{qp}(\omega)$ is real and frequency-independent, then $\sigma_\infty = \sigma_{qp}$. Then at high frequencies, the [*local*]{} complex conductivity, $\sigma_\infty + iq^2n_s({\bf x})/(m^*\omega)$, has only small spatial fluctuations. Then[@bergman] $$\sigma_e \approx \sigma_{av} -\frac{1}{2}\frac{\langle (\delta\sigma)^2\rangle}
{\sigma_{av}},
\label{sigma_e}$$ $\delta\sigma(\omega, {\bf x}) \equiv \sigma(\omega, {\bf x}) -
\sigma_{av}(\omega)$, where $\sigma_{av}(\omega) = \sigma_\infty + iq^2n_{s,av}/(m^*\omega)$, $\delta\sigma(\omega, {\bf x}) = \sigma(\omega, {\bf x}) -
\sigma_{av}(\omega)$, and $\langle...\rangle$ is a space average. Since only $n_s$, and not $\sigma_{qp}$, is fluctuating, this expression simplifies to \_e \_+ + . \[eq:sigma\_e\_const\] At large $\omega$, the imaginary part of $\sigma_e$ to leading order in $1/\omega$ is simply $\sigma_{e2} \sim q^2n_{s,av}/(m^*\omega)$. Equating this expression to the right-hand side of the Kramers-Kronig expression for $\sigma_{e2}$, we finally obtain $$\int_0^\infty
\left[\sigma_{e1}(\omega^\prime)-\sigma_\infty\right]d\omega^\prime
= \frac{q^2\pi}{2m^*}\left(n_{s,av} - n_{s,e}\right)
\label{eq:sige1}$$ If $n_s({\bf x})$ is spatially varying, the right-hand side is always positive, whence [*there will be an additional contribution to $\sigma_{e1}(\omega)$, beyond $\sigma_{qp}$*]{}.
At small $\omega$, eq. (\[eq:sigma\_e\_const\]) implies $\sigma_{e2} \sim \frac{q^2n_{s,av}}{m^*\omega} +
\frac{1}{2}\left(\frac{q^2/m^*\langle(\delta n_s)^2\rangle}
{\omega n_{s,av}}\right)$, and thus $n_{s,av} - n_{s,e} \approx \frac{q^2n_{s,av}}{m^*}
\langle(\delta n_s)^2\rangle/(2n_{s,av}^2)$. Then eq. (\[eq:sige1\]) becomes $$\int_0^\infty\left[\sigma_{e1}(\omega^\prime)-\sigma_\infty\right]
d\omega^\prime
\approx \frac{q^2\pi n_{s,av}}{4m^*}\frac{(\delta n_s)^2}{n_{s,av}^2}.
\label{eq:sige1p}$$ For fixed $\langle (\delta n_s)^2\rangle/n_{s,av}^2$, this integral is proportional to $n_{s,av}$. That is, the extra integrated fluctuation contribution to $\sigma_{e1}$, is proportional to the average superfluid density. A similar result has been reported in experiments[@corson; @corson2].
Drude $\sigma_{qp}(\omega)$
---------------------------
For a Drude $\sigma_{qp}(\omega) = \sigma_0/(1-i\omega\tau)$, we can carry out a similar analysis[@barabash1]. In the case of weak fluctuations in $n_s$, the result is[@barabash1] $$\begin{aligned}
\int_0^\infty &&\!\!\!\!\!\!\!\!\!\!\!\!
\left[\sigma_{e,1}(\omega^\prime)-\sigma_{qp,1}(\omega^\prime)
\right]d\omega^\prime \nonumber \\
&& \!\!\!\!\!\!\!\!\!\!\! \approx
\frac{\pi}{4}\left((q^2/m^*)^2\langle(\delta n_s)^2\rangle\right) \\
&& \!\!\times
\left(\frac{1}{q^2n_{s,av}/m^*}-\frac{1}{q^2n_{s,av}/m^*
+\sigma_0/\tau}\right). \nonumber\end{aligned}$$
In the limit $\sigma_0/\tau \gg q^2n_{s,av}/m^*$, this expression reduces to the right-hand side of eq. (\[eq:sige1p\]). Thus, in this regime, the extra spectral weight is indeed proportional to the average superfluid density $n_{s,av}$.
Tensor $n_s$
------------
Next, we consider a [*tensor*]{} superfluid density, as would be expected in a superconducting layer containing quenched charge density waves such as a charge stripes[@kivelson]. In this case, the ($2 \times 2$) superfluid density tensor should have the form $n_s^{\alpha\beta}({\bf x}) = R^{-1}({\bf x})n_s^dR({\bf x})$, where $n_s^d$ is a diagonal $2 \times 2$ matrix with diagonal components $n_{s,A}$, $n_{s,B}$, and $R({\bf x})$ is a position-dependent $2\times 2$ rotation matrix. describing the relative orientation of the charge density wave or stripes. If the stripes have either of two orientations along the crystal axes of the layer, with equal probability, then the [*effective*]{} superfluid density $n_{s,e}$ will be a scalar.
The arguments of the previous subsections can readily be transferred to the tensor case. For a two-fluid model with a frequency-independent quasiparticle conductivity which has the same value $\sigma_{\infty}$ in both the $A$ and $B$ directions, the effective scalar conductivity $\sigma_e(\omega)$ again satisfies Kramers-Kronig relations, and one again obtains the sum rule (\[eq:sige1\]), where $n_{s,av}$ is now the rotational average of a diagonal element of $n_{s;\alpha\beta}$. If the principal axes of $n_{s;\alpha\beta}$ point with equal probability along the two symmetry directions of the CuO$_2$ plane, as mentioned above, $n_{s,av} = (n_{s,A} + n_{s,B})/2$. Likewise, if the principal axes were to point in any direction in the plane with equal probability (a circumstance which seems unlikely for a stripe phase), then it can be shown[@kazaryan] that once again $n_{s,av} = (n_{s,A} + n_{s,B})/2$.
If the quasiparticle conductivity is frequency-dependent, then the analogous scalar results of the previous section continue to hold. For example, if $\sigma_{qp,A}(\omega)=\sigma_{qp,B}(\omega)$, then the extra spectral weight due to the superfluid inhomogeneity is again given by eq. (5) in the weak-inhomogeneity regime.
In the case of the stripe geometry, where the principal axes of the conductivity tensor take either of two perpendicular orientations with equal probability, $n_{s,e}$ is given by the duality result[@bergman; @kivelson] $$n_{s,e} = \sqrt{n_{s,A}n_{s,B}}.$$ This form allows the extra spectral weight to be evaluated straightforwardly, given $\sigma_{qp}(\omega)$, without making the small fluctuation approximation.
Numerical Example
=================
As a simple example, we consider a superconducting layer in which the conductivity has one of two possible values, $\sigma_A(\omega)$ or $\sigma_B(\omega)$ with equal probability, and we assume $\sigma_{A,B}(\omega) = \sigma_{qp}(\omega) + \frac{q^2n_{s;A,B}}
{m^*\omega}$, where we take $n_{s,A} > n_{s,B}$. This model would be suitable either for a layer with static scalar disorder, or for a model of stripe domains, as discussed above. We also write $\sigma_{qp} = \sigma_0/(1-i\omega\tau_{qp})$, with $\sigma_0 = n_{qp}q^2\tau_{qp}/m^*$. We also assume $n_{qp} = \alpha T$ for $T<T_c$ or $n_{qp} = \alpha T_c$ for $T>T_c$, $\tau_{qp}^{-1} = \beta T$, $n_{s,A} = \gamma n_{s,0}$, $n_{s,B} = \gamma^{-1} n_{s,0}$, where $n_{s,0}(T) = n_{s,0}(0)\sqrt{1-2\alpha T/n_{s,0}(0)}$, and $\gamma$ is a parameter describing the superfluid inhomogeneity (this form ensures that $n_{s,av}\equiv\sqrt{n_{s,A}n_{s,B}}=n_{s,0}$). Our choice of temperature dependence for $n_{s,0}(T)$ ensures that $n_{s,0}$ (i) decreases linearly with increasing temperature $T$ at small $T$, as observed experimentally[@linear]; and (ii) vanishes at a critical temperature $T_c$ as $\sqrt{T_c - T}$. The ingredients of this model are very similar to those of Ref. [@corson2], and have a straightforward interpretation. First, $n_{s,qp} $ should be proportional to $T$ in a gapless $d$-wave superconductor[@gap], and hence $n_{s,e}$ should be depleted by the same amount and fall off linearly in $T$. We also assume that $n_{s,A}$ and $n_{s,B}$ individually are linear in $T$. The form $1/\tau_{qp} = \beta T$, where $\beta$ is another constant[@corson2], has been observed for nodal quasiparticles in the superconducting states[@qp].
We compute $\sigma_e$ using Bruggeman effective-medium approximation (EMA)[@bergman; @bruggeman], which gives $(\sigma_A-\sigma_e)/(\sigma_A+\sigma_e)
+ (\sigma_B-\sigma_e)/(\sigma_B+\sigma_e) = 0$. The solution to this equation is simply $\sigma_e = \sqrt{\sigma_A\sigma_B}$. Fig. 1 shows the resulting $\sigma_{e,1}(\omega, T)$ for several frequencies ranging from 0.2 to 0.8 THz, the range measured in Ref.[@corson2]. The parameters $T_c$, $\sigma_0$, $n_{s,0}(0)$, $\alpha$ and $\beta$ were taken from Ref.[@corson2], and we assumed $\gamma=3$. Also shown are $\int_{\omega_{min}}^{\omega_{max}}\sigma_{e,1}(\omega,T)d\omega$ for $\omega_{min}/(2\pi) = 0.2$THz and $\omega_{max}/(2\pi) =
0.8$THz. Finally, we plot $\sigma_{qp,1}(\omega, T)$ for these frequencies, as well as $\int_{\omega_{min}}^{\omega_{max}}\sigma_{qp,1}d\omega$. Clearly, $\sigma_{e,1}(\omega)$ is considerably increased beyond the quasiparticle contribution, because of spatial fluctuations in the superfluid density.
Discussion
==========
We have shown that a superconducting layer with an inhomogeneous superfluid density will have an extra absorption not present in a homogeneous superconductor. The frequency integral of $\sigma_{e,1}(\omega)$ associated with this absorption is proportional to the superfluid density, in general agreement with experiments[@corson; @corson2]. We have also shown that this inhomogeneity, and hence the extra absorption seen in experiments, can arise from a stripe domain structure, as well as from random but isotropic disorder. Thus, we speculate that no such extra absorption should be observed in a high-T$_c$ superconductor unless one of these two types of inhomogeneities are present (beyond that expected purely from nodal quasiparticle absorption).
Acknowledgments.
================
This work has been supported by NSF Grant DMR01-04987, and by the U.-S./Israel Binational Science Foundation.
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|
---
abstract: 'In this paper we are interested in the prediction of preterm birth based on diagnosis codes from longitudinal EHR. We formulate the prediction problem as a supervised classification with noisy labels. Our base classifier is a Recurrent Neural Network with an attention mechanism. We assume the availability of a data subset with both noisy and clean labels. For the cohort definition, most of the diagnosis codes on mothers’ records related to pregnancy are ambiguous for the definition of full-term and preterm classes. On the other hand, diagnosis codes on babies’ records provide fine-grained information on prematurity. Due to data de-identification, the links between mothers and babies are not available. We developed a heuristic based on admission and discharge times to match babies to their mothers and hence enrich mothers’ records with additional information on delivery status. The obtained additional dataset from the matching heuristic has noisy labels and was used to leverage the training of the deep learning model. We propose an Alternating Loss Correction (ALC) method to train deep models with both clean and noisy labels. First, the label corruption matrix is estimated using the data subset with both noisy and clean labels. Then it is used in the model as a dense output layer to correct for the label noise. The network is alternately trained on epochs with the clean dataset with a simple cross-entropy loss and on next epoch with the noisy dataset and a loss corrected with the estimated corruption matrix. The experiments for the prediction of preterm birth at 90 days before delivery showed an improvement in performance compared with baseline and state of-the-art methods.'
author:
- |
Sabri Boughorbel\
Systems Biology\
Sidra Medicine\
Doha, P.O. Box 26999, Qatar\
`sboughorbel@sidra.org`\
Fethi Jarray\
Higher Institute of Computer\
Science, Medenine, Tunisia\
`fjarray@gmail.com`\
Neethu Venugopal\
Systems Biology\
Sidra Medicine\
Doha, P.O. Box 26999, Qatar\
`nvenugopal@sidra.org`\
Haithum Elhadi\
Medical Informatics\
Sidra Medicine\
Doha, P.O. Box 26999, Qatar\
`helhadi@sidra.org`\
bibliography:
- 'bib\_ml4h.bib'
title: 'Alternating Loss Correction for Preterm-Birth Prediction from EHR Data with Noisy Labels'
---
Introduction
============
The digitization of hospitals, by the adoption of Electronic Health Record systems, promises to revolutionize the future of healthcare. Several countries have achieved nearly 100% adoption. Therefore the complexity and size of EHR data is drastically increasing. This is creating new challenges and opportunities to the research community of machine learning for healthcare. In this paper we consider the clinical application of predicting preterm birth from EHR based on deep learning models. Between 10% to 15% of babies are born before 37 weeks of gestation [@barros2015distribution]. Preterm birth is the leading cause of mortality and long-term disabilities in neonates. It is also an important cause of developmental retardation. The cost of preterm deliveries and care exceed 26 billion dollars in the US [@behrman2006preterm]. The goal of our application is to predict in advance the risk for a preterm delivery [@tran2016preterm; @vovsha2016using]. Developing such predictive model can be of high value for obstetricians. The availability of large clinical EHR data should help in building accurate and interpretable models.
Related Work
============
In supervised learning, it is usually assumed that the data is accurately labeled. However in many real-life applications and especially with massive data, either data or labels may be corrupted by noise. In the case of noisy label, the noise might be class-dependent or instance-dependent. In the former case each label is flipped with some fixed probability independently of the instance itself. However in the later case, the label noise depends on the correct hidden label and the instance.
In this work, we are interested in developing deep learning methods that are tolerant to noisy labels. We suppose that the noise affects only the labels, i.e., preterm and full-term births, independently of the EHR mother records. By surveying the literature, we distinguish four approaches in learning with noisy labels [@Frenay2014]: 1) Data Cleansing by detecting and pre-processing (correcting or removing) the corrupted labels [@Pang2017influence], 2) Loss function reformulation by incorporating the noise in the learning criteria [@Natarajan2013; @Scott2014; @zhang2018generalized], 3) Noise robust approaches by using a noise robust loss function such as in SVM, kNNs or logistic regression [@Bootkrajang2012; @Chen2015] and 4) Noise tolerant approaches by using datasets with clean labels to approximate the corruption matrix between the clean and the noisy labels and then design a model to predict the correct labels [@patrini2017making; @Joulin2016; @li2017learning; @xiao2015learning; @han2018co; @goldberger2016training; @zhang2018generalized; @sukhbaatar2014training; @ma2018d2l; @wang2018iterative].
The majority of noise tolerant approaches are based on deep learning models. Sukhbataar et al. [@sukhbaatar2014training] integrated a constrained linear layer at the top of the softmax layer of the base model. Goldberger et al. [@goldberger2016training] considered the correct labels as latent variables. Instead of using EM algorithm, they used a neural network model where the noise is modelled by an additional softmax layer placed between the clean and the noisy output. In the context of loss correction, Jiang et al. [@Jiang2015] and Jindal et al. [@Jindal2016] added regularizers at the end of the network to fine-tune the adaptation layer. Azadi et al. [@Azadi2015] proposed a pre-trained model of AlexNet to extract information from the clean data and fine-tune the last layers with a noisy dataset. Yuncheng et al. [@Yuncheng2017] developed a distillation framework based on knowledge graph to correct the noisy label. Recently Hendrycks et al. [@hendrycks2018using] proposed a label noise correction method called gold loss correction (GLC). Given a small trusted dataset $( \mathcal{D}^*)$ and a large untrusted dataset $(\widetilde{\mathcal{D}})$, they first learn a classifier on the untrusted dataset and determine a label corruption matrix. Then they train the network on the noisy dataset with a loss corrected by the corruption matrix and on the clean dataset with a simple cross-entropy loss. Han et al. [@han2018co] proposed a co-teaching approach by simultaneously training two networks. At each epoch the weights of one network is updated through the gradient of the other network.
Preterm Dataset with Noisy Labels {#preterm:data}
=================================
[r]{}[2.7cm]{}
Pre Full
------ ------ ------
Pre 0.68 0.32
Full 0.2 0.8
captype[table]{}
We used Health Facts$^\copyright$ EMR Data (HF) to extract our pregnancy dataset. The cohort contained mothers with full-term and preterm deliveries from 164 hospitals. The complete diagnosis histories of these subjects were retrieved. Full-term delivery encounters were identified using the following ICD-9 diagnosis codes: (650, 645\*, 649.8, 652.5). Preterm deliveries are identified as encounters with one of the following diagnosis codes (644.2\*, 640.01). Ambiguous codes related to preterm deliveries were excluded from the criteria. Since the gestational age was not available in the data we used the delivery time as a reference time point. We introduced a prediction period which defines the time gap between the delivery event and the time when the model is asked to compute a prediction score. In this paper, we chose a prediction period of 90 days before delivery. In order to create a realistic prediction scenario and prevent data leakage, all future data with respect to the prediction time point are removed for each subject and made unavailable during training. We excluded subjects with less two visits left for model training. The obtained dataset $\mathcal{D}^*$ has clean labels and gathers in total 23,172 subjects.
ICD-9 codes on the newborn records are rich in information about the prematurity status. For example in codes 765.2\* and 765.1\*, the fifth-digit sub-classifications indicates respectively the gestational age at delivery and the baby weight. Since HF is a de-identified EHR database, it is not possible to link mothers to their babies records and thus we cannot augment mother records with additional information on the deliveries. We propose a simple and fast algorithm to re-link babies to mothers based on admission and discharge times. We note that the timestamps in HF has not been manipulated during de-identification. For each hospital, mother deliveries and newborn events are identified based on ICD codes. We have extracted 739,000 deliveries and 221,000 newborns. We have restricted the selection of the newborns to the ones that can be categorized into full-term or preterm newborns. We defined the time vector $t=[t_{adm},\ t_{dis}]$, where $t_{adm}$ and $t_{dis}$ are respectively the admission and discharge times. We used $t$ to find, for each hospital, the nearest mother in time to each newborn baby. As one mother could be the nearest to multiple babies, we applied a threshold of 3 (in case of multiple gestations) and an $L_1$ time distance of 24 hours to exclude the unlikely candidate mothers. The time threshold is justified by the fact that there might delays in the EHR system in capturing mothers and babies admission and discharge times. After applying the filtering criteria, the matching algorithm resulted in 23,578 mother-baby links allowing to extend the labeling of the mother deliveries as preterm or full-term. This constitutes our dataset $\widetilde{\mathcal{D}}$ with noisy labels. The obtained labels are considered noisy because the linkage algorithm is based solely on temporal information and hence can lead to erroneous matching.
Alternating Loss Correction {#gen_inst}
===========================
As explained in Section \[preterm:data\], we have two datasets available for the prediction of preterm births: a dataset $\mathcal{D}^*$ with clean labels and a dataset $\widetilde{\mathcal{D}}$ with noisy labels. The label corruption is specified by a distribution $p\left(\tilde{y}\mid y^*\right)$ independent of the input $x$. Since we have a binary classification problem, we can estimate the $2\times 2$ corruption matrix defined by $C_{ij}=p\left(\tilde{y}=j,\ y^*=i\right)$ using the subset $\mathcal{D}^\prime$ with both clean and noisy labels. $\mathcal{D}^\prime$ has 2,133 subjects. We used this subset to estimate the matrix $\widehat{C}$ of $C$ using equation (\[eq:C\]). Pre-training methods for noisy labels suffer from memorizing data with noisy labels. On the other hands Pre-training with clean data and fine tuning with noisy data tend to have poor performance. In order to circumvent this issue, we propose to alternate the use of both datasets during training. A loss correction is applied while training with $\widetilde{\mathcal{D}}$. A simple cross entropy loss is used when the model is trainined with $\mathcal{D}^*$.
$$\widehat{C}_{ij}=\frac{1}{|A_i|}{\sum_{x \in A_i} \mathbbm{1}{\left\{\tilde{y} = j\mid x\right\}}}
\label{eq:C}$$
where $A_i$ is the subset of $\mathcal{D}^\prime$ with clean label $i$. The classifier $f$ is a deep neural network similar to RETAIN [@choi2016retain]. The output layer of $f$ is a dense layer with a softmax activation having $2\times2$ parameters. The loss correction can be viewed as a dense layer of constant weights $\widehat{C}_{ij}$ and zero bias. The network $f$ is alternately trained on an epoch using $\mathcal{D}^*$ using a simple cross-entropy loss and on the next epoch using $\widetilde{\mathcal{D}}$ with a loss correction.
Experiments {#headings}
===========
[r]{}[7cm]{} captype[table]{}
Method AUC PR-UC
------------------------------------------------------ ------------------------ ------------------------
ALC - $\mathcal{D}^*\sim \widetilde{\mathcal{D}}$ [**82.79$\pm$0.72**]{} [**73.05$\pm$1.37**]{}
GLC - $\widetilde{\mathcal{D}}$ then $\mathcal{D}^*$ 80.77$\pm$1.44 71.07$\pm$2.05
GLC - $\mathcal{D}^*$ then $\widetilde{\mathcal{D}}$ 64.13$\pm$1.59 49.48$\pm$3.03
No-LC - $\mathcal{D}^*+\widetilde{\mathcal{D}}$ 77.87$\pm$0.87 66.65$\pm$1.76
No-LC - $\mathcal{D}^*$ 80.71$\pm$1.49 72.11$\pm$2.04
No-LC - $\widetilde{\mathcal{D}}$ 71.20$\pm$1.41 59.72$\pm$1.68
The sequence of diagnosis codes are ordered in the patient-visit timeline. Patient encounters of the same day are merged to reduce the temporal frequency in the data. The discrete ICD codes, represented by 17629-dimensional hot vector, are then passed into a 200-dimensional floating-point embedding. For each batch, the sequences are padded with zeros to have the same number of visits. The embedding vectors of the different codes within the same visit are summed up. Then they are passed to a Recurrent Neural Network with an attention mechanism. The network architecture is a modified version of RETAIN model [@choi2016retain]. We benchmarked the proposed ALC algorithm with 1) Gold Loss Correction (GLC) method [@hendrycks2018using] and 2) training without loss correction. All evaluations reported in this work are based on validation and test sets from the dataset with clean labels $\mathcal{D}^*$. We considered few training scenarios in our benchmark: GLC is trained first with the noisy label dataset $\widetilde{\mathcal{D}}$ using loss correction then the model further trained with the clean-labels dataset $\mathcal{D}^*$. This configuration is the one implemented in GLC. We evaluated GLC by flipping the order of the datasets, i.e., we first trained with $\mathcal{D}^*$ without loss correction and further trained the model on $\widetilde{\mathcal{D}}$ with loss correction. We evaluated also the baseline scenario where only 1) clean labels are used and samples of clean and noisy labels are mixed. In both cases, no loss correction is applied. This is denoted by No-LC in table \[tab:result\]. The models have been first evaluated with few different number of epochs $N_e$. Training with 10 epochs was enough to obtain result stabilization. The models were implemented using Keras 2.1.6 with Tensorflow 1.8.0 back-end. We trained the model in parallel using two Tesla V100 GPUs. It took approximately 2-3 mins to train a model on 10 epochs. Table \[tab:result\] summarizes the results in terms of Area under ROC (AUC) and Area under Precision Recall curve (PR-UC). The splitting of data into training, validation and test sets was randomly repeated 20 times to obtain performance means and standard deviations. The proposed ALC achieved the best results in the benchmark. By alternating through both datasets, ALC was able to leverage on sample from noisy labels and achieve a gain in performance compared with the baseline using only clean labels (No-LC - $\mathcal{D}^*$). ALC is followed by GLC - $\widetilde{\mathcal{D}}$ then $\mathcal{D}^*$ when it is first trained on noisy labels with corrected loss then on clean labels without correction. GLC - $\mathcal{D}^*$ then $\widetilde{\mathcal{D}}$ gave the worst result and seems to under-fit the data. This could be explained by the fact that the model tends to memorize the latest examples seen during training. The baseline results, No-LC - $\mathcal{D}^*$, solely on clean labels is 80.71 (AUC) and 72.11 (PR-UC). The data augmentation (No-LC - $\mathcal{D}^*+\widetilde{\mathcal{D}}$) by merging both datasets $\mathcal{D}^*$ and $\widetilde{\mathcal{D}}$ resulted in a performance degradation. \[others\]
Discussion and Future Work
==========================
The problem of matching of mothers and babies is relevant in general for massive de-identified EHR datasets and can be useful for studying pregnancy outcomes beyond the scope our application of predicting preterm births. Our simple matching heuristic has achieved an accuracy of 72% on the labels measured on $\mathcal{D}^\prime$. This could be further improved by considering a more global matching method based on linear optimization. The ambiguous ICD-9 codes that were excluded from the cohort definition could be used to add subjects with noisy labels. In a future work, we will consider the use of probabilistic noise model where the label values continuously range between 0 and 1 for binary classification. The values will reflect the noise level on the labels. Values of 0 or 1 indicate the clean labels and the rest are noisy with different levels of noise. A label of 0.5 has the highest uncertainty on the label assignment. This can help increasing the size of the training set. In this context, we plan to combine a regression problem on the continuous noisy labels and a classification problem on the clean labels. We mention that in the case where $\mathcal{D}^\prime$ is very small, we can use a similar approach proposed in [@hendrycks2018using] to estimate the corruption matrix $C$.
Conclusion
==========
In this paper, we were interested by the prediction of preterm birth using diagnosis information from de-identifed EHR data. We have devised a heuristic to match mothers and babies and hence augment our cohort with additional data examples with noisy labels. Then we have introduced a new learning algorithm called Alternating Loss Correction (ALC) to robustly train deep neural networks with noisy labels. The idea behind ALC is to involve both clean and noisy labels in an alternating fashion during training to avoid over-fitting and increase the generalization capability of the model. In ALC algorithm, we first estimate the corruption matrix on the data subset that have both clean and noisy labels, then we train the model on the clean and noisy datasets by alternating the losses. ALC achieved an improvement in prediction performance compared with baseline and state-of-the-art methods. It could be generalized to the training of deep learning models with multiple datasets.
|
---
abstract: 'We study the path realization of Demazure crystals related to solvable lattice models in statistical mechanics. Various characters are represented in a unified way as the sums over one dimensional configurations which we call unrestricted, classically restricted and restricted paths. As an application characters of Demazure modules are obtained in terms of $q$-multinomial coefficients for several level 1 modules of classical affine algebras.'
author:
- |
Atsuo Kuniba$^1$, Kailash C. Misra$^2$, Masato Okado$^3$,\
Taichiro Takagi$^4$ and Jun Uchiyama$^5$
date: |
*$^1$Institute of Physics,\
*University of Tokyo,\
*Komaba, Tokyo 153, Japan\
*$^2$Department of Mathematics,\
*North Carolina State University,\
*Raleigh, NC 27695-8205, USA\
*$^3$Department of Mathematical Science,\
*Faculty of Engineering Science,\
*Osaka University, Toyonaka, Osaka 560, Japan\
*$^4$Department of Mathematics and Physics,\
*National Defense Academy, Yokosuka 239, Japan\
*$^5$Department of Physics,\
*Rikkyo University,\
*Nishi-Ikebukuro, Tokyo 171, Japan**************
title: |
Characters of Demazure Modules\
and Solvable Lattice Models\
---
addtoreset[equation]{}[section]{}
\[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\]
\#1[\_[\#1]{}]{} \#1[\_[\#1]{}]{} \#1 Ł[[L]{}]{} ¶[[P]{}]{}
Introduction
============
Let $\Uq$ be a quantum affine algebra and $V(\la)$ be the integrable $\Uq$-module with highest weight $\la \in P^+$. Given a Weyl group element $w$, the associated Demazure module $V_w(\la)$ is the finite dimensional subspace of $V(\la)$ generated from the extermal weight space $V(\la)_{w \la}$ by the $e_i$ generators. In [@Ka], Kashiwara introduced its crystal $\B_w(\la)$, which is a finite subset of the crystal $\B(\la)$ for $V(\la)$. With this aid the character of the Demazure module $V_w(\la)$ is expressed as $$ch V_w(\la) = \sum_{p \in \B_w(\la)} e^{\wts p}.
\label{char}$$ The subject of this paper is to calculate (\[char\]) systematically by using the path realization of the Demazure crystal $\B_w(\la)$ studied in [@KMOU], [@KMOTU1]. The realization is based on the earlier one for $\B(\la)$ [@KMN1], [@KMN2] and has an origin in the analyses of solvable lattice models [@ABF], [@DJKMO1], [@DJKMO2]. In these works the object called one dimensional configuration sums (1dsums) played an essential role and were studied extensively either in their “infinite lattice limits $j \rightarrow \infty$” or “finite truncations $j < \infty$”. In this paper we consider three kinds of 1dsums $g_j, \overline{X}_j$ and $X_j$, which we call unrestricted, classically restricted and restricted 1dsums, respectively. The unrestricted $g_j$ is relevant to vertex models and so is $X_j$ to the restricted solid-on-solid (RSOS) type models. In section 2 we apply the main theorem in [@KMOU] to relate the Demazure character with the 1dsum $g_j$ for finite $j$. We shall also clarify the relations among the three kinds of 1dsums and thereby give a unified picture to understand the Demazure characters and various branching functions. (See Table 1.) This enables us to evaluate these quantities explicitly from several known results on the 1dsums. As an application, in section 3 we shall give $q$-multinomial formulae for $ch V_w(\la)$ for many level 1 modules $V(\la)$ over $\Uq$ for $\geh$ of classical types $\geh = A^{(1)}_n, B^{(1)}_n, D^{(1)}_n,
A^{(2)}_{2n-1}, A^{(2)}_{2n}$ and $D^{(2)}_{n+1}$. We hope to report on higher level cases in a future publication.
Path realization of Demazure crystals
=====================================
Perfect crystals
----------------
Let us recall relevant facts and notations from [@KMN1],[@KMN2],[@KMOU],[@KMOTU1]. $\alpha_i, \, h_i, \, \Lambda_i\, (i \in I)$ are the simple roots, coroots and fundamental weights, respectively. We put $\rho = \sum_{i \in I} \Lambda_i$ and let $\delta$ denote the null root. $P = \oplus_i {\bf Z} \Lambda_i \oplus {\bf Z} \delta$ and $\Pcl = \oplus_i {\bf Z} \Lambda_i \subset P$ are the weight and the classical weight lattices, respectively. Set further $P^+ (\Pcl^+) =\{\la\in P (\Pcl) \mid \langle\la,h_i\rangle\ge0
\mbox{ for any }i\}$ and $(\Pcl^+)_l=\{\la\in\Pcl^+\mid \langle\la,c\rangle=l\}$, where $c$ is the canonical central element. For $\la \in P^+$ we let $(\L(\la),\B(\la))$ denote the crystal base of the irreducible $\Uq$-module $V(\la)$ with highest weight $\la$. For a crystal base of a finite dimensional $\Uq$-module we use the symbol $(L,B)$. Let $B$ be a perfect crystal of level $l$. See Definition 4.6.1 in [@KMN1] for its definition. Then for any $\la\in(\Pcl^+)_l$, there uniquely exists $b(\la) \in B$ such that $\vphi(b(\la)) = \la$. Here we recall that $\veps_i(b) = \hbox{max }\{ k \mid \et{i}^k b \neq 0 \},
\vphi_i(b) = \hbox{max }\{ k \mid \ft{i}^k b \neq 0 \},
\veps(b) = \sum_{i \in I} \veps_i(b) \Lambda_i$ and $\vphi(b) = \sum_{i \in I} \vphi_i(b) \Lambda_i$. Let $\sigma$ be the automorphism of $(\Pcl^+)_l$ given by $\sigma\la=\veps(b(\la))$. We put $\bbar_k=b(\sigma^{k-1}\la)$ and $\la_k=\sigma^k\la$. Then perfectness assures that we have the isomorphism of crystals $$\B(\la_{k-1})\simeq \B(\la_k)\ot B.
\label{iso1}$$ Define the set of paths $\P(\la,B)$ by $$\P(\la,B)=\{p=\cdots\ot p(2)\ot p(1)\mid p(j)\in B,p(k)=
\bbar_k\mbox{ for }k\gg0\},$$ By iterating (\[iso1\]) we have an isomorphism of crystals $$\B(\la) \simeq \P(\la,B).
\label{iso2}$$ In particular, the image of the highest weight vector $u_\la \in \B(\la)$ is given by $\pbar = \cd\ot\bbar_k\ot\cd\ot\bbar_2\ot
\bbar_1$. We call $\pbar$ the [*ground-state*]{} path. The actions of $\et{i}$ and $\ft{i}$ on $\P(\la,B)$ are determined explicitly by the signature rule. See section 1.3 of [@KMOU].
To describe the weights on $\P(\la, B)$ it is necessary to introduce the energy function $ H:B\ot B\rightarrow\Z$. Up to an additive constant it is determined by requiring the following for any $b,b'\in B$ and $i\in I$ such that $\et{i}(b\ot b')\ne0$. $$H(\et{i}(b\ot b'))=\left\{\begin{array}{ll}
H(b\ot b')&\quad\mbox{if }i\ne0\\
H(b\ot b')+1&\quad\mbox{if }i=0\mbox{ and }\vphi_0(b)\ge\veps_0(b')\\
H(b\ot b')-1&\quad\mbox{if }i=0\mbox{ and }\vphi_0(b)<\veps_0(b').
\end{array}\right.
\label{defh}$$ Under the isomorphism (\[iso2\]), the weight of a path $p=\cdots\ot p(2)\ot p(1)$ is given by (Proposition 4.5.4 in [@KMN1]) $$\begin{aligned}
\wt p&=&\la+\sum_{i=1}^\infty
\left(\wt p(i) - \wt \bbar_i \right)\nonumber\\
&&\quad-\left(\sum_{i=1}^\infty
i(H(p(i+1)\ot p(i))-H(\bbar_{i+1}\ot \bbar_i))\right)\delta.
\label{eq:weight}\end{aligned}$$ We remark the weight relation $$\la_j = \la - \sum_{i=1}^j \wt \bbar_i,
\label{weightrelation}$$ which is valid for any $j \ge 0$.
Demazure modules
----------------
Let $\{r_i\}_{i\in I}$ be the set of simple reflections, and let $W$ be the Weyl group. For $\la \in (\Pcl^+)_l$ we consider the Demazure module $V_w(\la)$ generated from the extremal weight space $V(\la)_{w\la}$. By definition its character is given by $ch V_w(\la) = \sum_\mu {\rm dim }(V_w(\la))_\mu e^\mu$. For $\mu \in P, \, i \in I$ define the operator $D_i: {\bf Z}[P] \rightarrow {\bf Z}[P]$ by $$D_i(e^\mu) = {e^{\mu + \rho} - e^{r_i(\mu + \rho)} \over
1 - e^{-\alpha_i}}e^{-\rho}.$$ Let $w = r_{i_k} \cdots r_{i_1} \in W$ be a reduced expression. Then the following character formula is well known [@Dem], [@Ku], [@Mat]. $$ch V_w(\la) = D_{i_k} \cdots D_{i_2} D_{i_1}(e^\mu).
\label{kumar}$$ From this one has a recursion relation $$ch V_{r_i w}(\la) = D_i\left( ch V_w(\la) \right)
\quad{\rm if }\, r_i w \succ w.
\label{eq:demazurerec}$$
Let $(\L(\la),\B(\la))$ be the crystal base of $V(\la)$. In [@Ka] Kashiwara showed that for each $w\in W$, there exists a subset $\B_w(\la)$ of $\B(\la)$ such that $$\frac{V_w(\la)\cap\L(\la)}{V_w(\la)\cap q\L(\la)}
=\bigoplus_{b\in\B_w(\la)}\Q b.$$ Furthermore, $\B_w(\la)$ has the following recursive property. $$\begin{aligned}
&&\mbox{If }r_iw\succ w,\mbox{ then}\nonumber\\
&&\B_{r_iw}(\la)=\bigcup_{n\ge0}\ft{i}^n\B_w(\la)\setminus\{0\},
\label{recursive}\end{aligned}$$ which is analogous to (\[eq:demazurerec\]). We call $\B_w(\la)$ a [*Demazure crystal*]{}. One can now express $ch V_w(\la)$ as in (\[char\]). It affords an effecient way to calculate the Demazure character through the path realization of $\B_w(\la)$ given in the next subsection and (\[eq:weight\]). Relations with the formula (\[kumar\]) and the 1dsums will also be explained in section 2.
Path realization
----------------
In [@KMOU] the image of $\B_w(\la)$ under the isomorphism (\[iso2\]) is determined for a suitably chosen Weyl group element $w$. Let us recall the main theorem therein, which gives a path realization of the Demazure crystal. There appears the [*mixing index*]{} $\kappa$ specified from $\la$ and $B$. (See section 2.3 of [@KMOU].) In this paper we shall only consider the case $\kappa=1$. Let $\la$ be an element of $\Pcll$, and let $B$ be a classical crystal. For the theorem, we need to assume four conditions (I-IV).
(I)
: $B$ is perfect of level $l$.
Thus, we can assume an isomorphism between $\B(\la)$ and the set of paths $\P(\la,B)$. Let $\pbar=\cd\ot\bbar_2\ot\bbar_1$ denote the ground state path. Fix a positive integer $d$. For a set of elements $i_a^{(j)}$ ($j\ge1,1\le a\le d$) in $I$, we define $B_a^{(j)}$ ($j\ge1,0\le a\le d$) by $$B^{(j)}_0=\{\bbar_j\},\hspace{1cm}
B_a^{(j)}=\bigcup_{n\ge0}\ft{i_a^{(j)}}^n B_{a-1}^{(j)}\setminus\{0\}
\quad(a=1,\cdots,d).$$
(II)
: For any $j\ge1$, $B_d^{(j)}=B$.
(III)
: For any $j\ge1$ and $1\le a\le d$, $\langle\la_j,h_{i^{(j)}_a}\rangle\le\veps_{i^{(j)}_a}(b)$ for all $b\in B^{(j)}_{a-1}$.
We now define an element $w^{(k)}$ of the Weyl group $W$ by $$w^{(0)}=1,\hspace{1cm}
w^{(k)}=r_{i^{(j)}_a}w^{(k-1)}\quad\mbox{for }k>0,$$ where $j$ and $a$ are fixed from $k$ by $k=(j-1)d+a,j\ge1,1\le a\le d$.
(IV)
: $w^{(0)}\prec w^{(1)}\prec\cd\prec w^{(k)}\prec\cd$.
See [@KMOU], [@KMOTU1] on how to check the last condition.
Finally we define a subset $\P^{(k)}(\la,B)$ of $\P(\la,B)$ as follows. We set $\P^{(0)}(\la,B)=\{\overline{p}\}$. For $k>0$, $$\label{eq:def_P}
\P^{(k)}(\la,B) =
\cdots\ot B^{(j+2)}_0\ot B^{(j+1)}_0\ot B^{(j)}_a
\ot B^{\ot(j-1)},$$ where $j \ge 1$ and $1 \le a \le d$ are uniquely specified by $k=(j-1)d+a$.
Now we have
\[iso3\] Under the assumptions (I-IV), we have $$\B_{w^{(k)}}(\la)\simeq\P^{(k)}(\la,B).$$
The proof is done by showing the recursion relation (\[recursive\]) in the path realization.
One dimensional sums
====================
Here we first introduce the unrestricted 1 dimensional sum (1dsum) $g_j$ and express the Demazure characters in terms of it. After establishing its fundamental properties we then introduce classically restricted 1dsums $\overline{X}_j$ and restricted 1dsums $X_j$ and study their relations. In the working below we shall use the variable $$q = e^{-\delta}.$$
Unrestricted 1dsum
------------------
For $j \in {\bf Z}_{\ge 0}, \, b \in B$ and $\mu \in P$, put $$\P_j(b, \mu) = \{ b \ot b_j \ot \cdots \ot b_1 \in B^{\ot (j+1)} \vert
wt(b_j \ot \cdots \ot b_1) = cl(\mu) \}.
\label{defpj}$$ In the sequel we always assume the relation $k = (j-1)d + a$. Comparing (\[defpj\]) with (\[eq:def\_P\]) we have $$\P^{(k)}(\la, B) = \sqcup_{\mu \in P_{cl}}
\sqcup_{b \in B^{(j)}_a}
\P_{j-1}(b,\mu).$$ For $j \in {\bf Z}_{\ge 0}, \, b \in B$ and $\mu \in P$ we define the (unrestricted) 1dsum as follows. $$g_j(b,\mu) = q^{(\Lambda_0, \mu)}
\sum_{ b_{j+1} \ot \cdots \otimes b_1 \in \P_j(b,\mu)}
q^{\sum_{i=1}^jiH(b_{i+1} \ot b_i)}.
\label{def1dsum}$$ Defining a map $E: B^{\ot (j+1)} \rightarrow {\bf Z}$ by $$E(b_{j+1}\ot \cdots \ot b_1) = \sum_{i=1}^j i H(b_{i+1}\ot b_i),
\label{eq:energy}$$ one can write (\[def1dsum\]) as $$g_j(b, \mu) = q^{(\Lambda_0, \mu)}
\sum_{p \in \P_j(b,\mu)} q^{E(p)}.
\label{eq:def2}$$ Note that $$\begin{aligned}
&&g_j(b, \mu) = 0 \, \, \mbox{ unless } \langle\mu ,c\rangle = 0 \nonumber \\
&&g_j(b,\mu+m\delta) = q^m g_j(b,\mu) \,\, \mbox{ for any } \mu \in P
\mbox{ and } m \in {\bf Z}. \nonumber \\
\label{1dsumproperty}\end{aligned}$$ For the Weyl group element $w^{(k)}$ in ([**III**]{}), the Demazure character (\[char\]) is expressed in terms of the 1dsum.
\[pr:by1dsum\] $$\begin{aligned}
&&ch V_{w^{(k)}}(\la) = q^{-c_j}
\sum_{\mu \in P_{cl}} e^{\la_j + \mu}
\sum_{b \in B^{(j)}_a} q^{jH(\bbar_{j+1} \ot b)} g_{j-1}(b, \mu-wt(b)),
\nonumber \\
&&c_j = \sum_{i=1}^j i H(\bbar_{i+1} \ot \bbar_i).
\label{by1dsum}\end{aligned}$$
Substitute (\[weightrelation\]) and (\[def1dsum\]) into the rhs. Setting $p = \cdots \ot \bbar_{j+1} \ot b_j \ot \cdots \ot b_1$ and noting (\[eq:weight\]) one finds that the result is equal to $\sum_{p \in \P^{(k)}(\la, B)} e^{\wts p}$. Thus the assertion follows from Theorem \[iso3\].
The 1dsums are uniquely characterized also from the recursion relation and the initial condition as follows.
\[pr:rec1dsum\] $$\begin{aligned}
&& g_j(b,\mu) = \sum_{b^\prime \in B} q^{j H(b \ot b^\prime)}
g_{j-1}(b^\prime,\mu-wt(b^\prime)) \nonumber\\
&& g_0(b,\mu) = \delta_{0 \mu}.
\label{rec1dsum}\end{aligned}$$
In the definition (\[def1dsum\]) put $b^\prime = b_j$ and notice $b_{j+1} = b$ due to (\[defpj\]).
When $k =jd \, (a=d), B^{(j)}_a = B$ due to [**(II)**]{}. Thus Proposition \[pr:rec1dsum\] simplifies the sum in (\[by1dsum\]) into $$\label{simplecase}
ch V_{w^{(jd)}}(\la) = q^{-c_j}
\sum_{\mu \in P_{cl}} e^{\la_j + \mu}
g_j(\bbar_{j+1}, \mu).$$
The following relation is of primary importance in later discussion. See Remark \[fundamental\] and the proof of Proposition \[pr:xbyg\].
\[pr:2mrelation\] For $b \in B$, let $m = \vphi_i(b)$. Then we have $$\sum_{t=0}^m g_j(\ft{i}^t b,\mu+t\alpha_i)q^{tj\delta_{0 i}}=
\sum_{t=0}^m g_j\bigl(\ft{i}^t b,r_i(\mu+(m-t)\alpha_i)\bigr)
q^{tj\delta_{0 i}}.
\label{eq:2mrelation}$$
For the proof we need a few Lemmas. The following is an immediate consequence of the signature rule.
\[le:inequality\] For any $b_1, b_2 \in B$ and $i \in I$ we have $$\begin{aligned}
&&\vphi_i(b_1\ot b_2) \ge \vphi_i(b_1) + \langle h_i, \wt b_2 \rangle
\nonumber \\
&&\veps_i(b_1\ot b_2) \ge - \langle h_i, \wt(b_1\ot b_2) \rangle.\end{aligned}$$
\[le:bijection\] Let $n = \langle h_i, \mu \rangle + m$ and assume $n \ge 0$. Then the map $$\ft{i}^n : \sqcup_{t=0}^m
\P_j(\ft{i}^t b, \mu + t\alpha_i) \rightarrow
\sqcup_{t=0}^m
\P_j\left(\ft{i}^t b, r_i(\mu + (m-t)\alpha_i)\right)
\label{eq:bijection}$$ is a bijection.
The image is certainly within the rhs by the weight reason unless it is zero. Since $\ft{i}^n p = p^\prime$ is equivalent to $\et{i}^n p^\prime = p$, it suffices to show for $0 \le t \le m$ that $\vphi_i(p) \ge n$ for any $p \in \P_j(\ft{i}^t b, \mu + t\alpha_i)$ and $\veps_i(p^\prime) \ge n$ for any $p^\prime \in \P_j\left(\ft{i}^t b, r_i(\mu + (m-t)\alpha_i)\right)$. Applying Lemma \[le:inequality\], one has $$\begin{aligned}
&\vphi_i(p) & \ge m-t + \langle h_i, cl(\mu + t\alpha_i) \rangle = n + t, \\
&\veps_i(p^\prime) & \ge
\veps_i(\ft{i}^t b) - \vphi_i(\ft{i}^t b) -
\langle h_i, cl\left(r_i(\mu + (m-t)\alpha_i)\right) \rangle \\
&& \ge 2t-m + \langle h_i, \mu + (m-t)\alpha_i \rangle = n,\end{aligned}$$ which completes the proof.
\[le:energy\] Let $b \in B, \, m = \vphi_i(b), \, \xi \in P$. For $0 \le t \le s \le m$, assume that $p \in \P_j(\ft{i}^s b, \xi)$ and $\et{i}^n p \in \P_j(\ft{i}^t b, \xi+(n+t-s)\alpha_i)$. Then we have $$E(\et{i}^n p) = E(p) + \left( (j+1)(s-t) - n \right)\delta_{0 i}.$$
Use (\[defh\]) in (\[eq:energy\]). 0.2cm [*Proof of Proposition \[pr:2mrelation\]*]{}. Put $n = \langle h_i, \mu \rangle + m$. Since (\[eq:2mrelation\]) is invariant under the change $\mu \rightarrow r_i(\mu + m\alpha_i)$, we may assume $n \ge 0$ with no loss of generality. The lhs of (\[eq:2mrelation\]) is written as $$\sum_{0 \le t \le s \le m}
q^{(\Lambda_0, \mu + t\alpha_i)+ tj\delta_{0 i}}
\sum^{}_{p^\prime}{}^\prime q^{E(p^\prime)},
\label{eq:1}$$ where $\sum^{}_{p^\prime}{}^\prime$ extends over those $p^\prime \in \P_j(\ft{i}^t b, \mu + t\alpha_i)$ such that $\ft{i}^n p^\prime \in \P_j(\ft{i}^s b, r_i(\mu + (m-s)\alpha_i))$. The sum $\sum_{0 \le t \le s \le m}
\sum^{}_{p^\prime}{}^\prime$ in total ranges over the set appearing in the lhs of (\[eq:bijection\]). Thus Lemma \[le:bijection\] allows a change of the summation variable into $p = \ft{i}^n p^\prime$, and thereby transforms (\[eq:1\]) into $$\sum_{0 \le t \le s \le m}
q^{(\Lambda_0, \mu + t\alpha_i)+ tj\delta_{0 i}}
\sum^{}_p{}^{\prime\prime} q^{E(\et{i}^n p)}.
\label{eq:2}$$ Here $\sum^{}_{p}{}^{\prime\prime}$ extends over those $p \in \P_j(\ft{i}^s b, r_i(\mu + (m-s)\alpha_i))$ such that $\et{i}^n p \in \P_j(\ft{i}^t b, \mu + t\alpha_i)$. Setting $\xi = r_i(\mu + (m-s)\alpha_i)$, one can apply Lemma \[le:energy\] to rewrite (\[eq:2\]) as $$\begin{aligned}
&&\sum_{0 \le t \le s \le m}
q^{(\Lambda_0, r_i(\mu + (m-s)\alpha_i))+ sj\delta_{0 i}}
\sum^{}_p{}^{\prime\prime} q^{E(p)}\nonumber \\
&&=\sum_{0 \le s \le m}
q^{(\Lambda_0, r_i(\mu + (m-s)\alpha_i))+ sj\delta_{0 i}}
\sum^{}_{p \in \P_j(\ft{i}^sb, r_i(\mu+(m-s)\alpha_i))}
q^{E(p)}.\nonumber\end{aligned}$$ By (\[eq:def2\]) the last expression is the rhs of (\[eq:2mrelation\]).
The relation (\[eq:demazurerec\]) for the Demazure character (\[by1dsum\]) implies yet another recursion relation for the 1dsums than Proposition \[pr:rec1dsum\].
\[pr:1dsumdemarec\] For any $0 \le a \le d-1$ and $\mu \in P_{cl}$ one has $$\begin{aligned}
&&\sum_{b \in B^{(j)}_{a+1} \setminus B^{(j)}_a}
q^{jH(\bbar_{j+1} \ot b)} g_{j-1}(b, \mu-wt(b)) \nonumber \\
&& =
\sum_{b \in B^{(j)}_{a+1}}
q^{jH(\bbar_{j+1} \ot b)} g_{j-1}(b, \mu+\alpha_{i^{(j)}_{a+1}}-wt(b))
\nonumber \\
&& - \sum_{b \in B^{(j)}_a}
q^{jH(\bbar_{j+1} \ot b)}
g_{j-1}(b, r_{i^{(j)}_{a+1}}(\mu+\rho+\la_j)-\la_j-\rho-wt(b)).\end{aligned}$$
Suppose $1 \le a \le d-1$. Substitute (\[by1dsum\]) into (\[eq:demazurerec\]) with $w=w^{(k)}$ and $i = i^{(j)}_{a+1}$. Note that $D_i(e^{\mu + z \delta}) = e^{z \delta} D_i(e^\mu)$ holds for any $z \in {\bf Z}$ and $\mu \in P_{cl}$ because of $r_i(\delta) = \delta$. After multiplying both sides by $1 - e^{-\alpha_i}$, comparison of the coefficients of $e^{\lambda_j + \mu}$ leads to the above relation. The case $a=0$ is similar.
\[fundamental\] It is possible to prove Proposition \[pr:1dsumdemarec\] without using (\[eq:demazurerec\]) and (\[by1dsum\]) but only from (\[eq:2mrelation\]). In this sense, (\[eq:2mrelation\]) is the most fundamental relation of the 1dsums implied from the Demazure recursion relation (\[eq:demazurerec\]).
Classically restricted and restricted 1dsum
-------------------------------------------
The 1dsums discussed so far is related to vertex models. Now we proceed to the two variants of them related to restricted solid-on-solid (RSOS) type models (cf. [@ABF], [@DJKMO1]) and Kostka-type polynomials (cf.[@Ki], [@NY]). Here they shall be called the restricted 1dsums and the classically restricted 1dsums, respectively. Let $\overline{\Lambda}_i =
\Lambda_i - \langle c , \Lambda_i \rangle \Lambda_0$ and put $\overline{P}_{cl}^+ = \oplus_{i \in I \setminus \{0 \}}
{\bf Z}_{\ge 0} \overline{\Lambda}_i$. Fix a non-negative integer $l'$. Given $\xi \in (P^+_{cl})_{l + l'}$ (resp. $\overline{\xi} \in \overline{P}_{cl}^+$) and $b \in B$ we define $$\begin{aligned}
&&(\xi, b) \,\hbox{ is {\em admissible} } \Leftrightarrow
\et{i}^{\langle h_i, \xi \rangle + 1} b = 0 \, \,
\forall i \in I,\\
&&(\overline{\xi}, b) \,\hbox{ is {\em classically admissible} }
\Leftrightarrow
\et{i}^{\langle h_i, \overline{\xi} \rangle + 1} b = 0 \,\,
\forall i \in I \setminus \{ 0 \}.\\\end{aligned}$$ Recall that $l$ is the level of the perfect crystal $B$. It is easy to see that if $(\xi, b)$ is admissible (resp. $(\overline{\xi}, b)$ is classically admissible) then $\xi + wt(b) \in (P^+_{cl})_{l + l'}$ (resp. $\overline{\xi} + wt(b) \in \overline{P}_{cl}^+$). The admissibility condition has been introduced by [@DJO] in the study of $q$-vertex operators by the crystal base theory. For $j \in {\bf Z}_{\ge 0}, b \in B$ and $\xi, \eta \in (P^+_{cl})_{l + l'}$ such that $(\xi-wt(b), b)$ is admissible, (resp. $\overline{\xi}, \overline{\eta} \in \overline{P}_{cl}^+$ such that $(\overline{\xi}-wt(b), b)$ is classically admissible), we define $q$-polynomials $X_j(b,\xi, \eta)$ and $\overline{X}_j(b,\overline{\xi},\overline{\eta})$ to be the sum $$\sum_{ b_j, \ldots, b_1 \in B, b_{j+1} = b}
q^{\sum_{i=1}^jiH(b_{i+1} \ot b_i)}.$$ Here the outer sum $\sum$ is taken over $b_j, \ldots, b_1 \in B$ under the following conditions for each case. $$\begin{aligned}
X_j(b, \xi, \eta)\, {\rm case}:
&&\xi_i + wt(b_i) = \xi_{i-1} \, \mbox{ for } 1 \le i \le j, \quad
\xi_j = \xi, \, \xi_0 = \eta,\\
&&(\xi_i, b_i) \mbox{ is admissible for } 1 \le i \le j.\\
\overline{X}_j(b, \overline{\xi}, \overline{\eta})\, {\rm case}:
&&\overline{\xi}_i + wt(b_i) = \overline{\xi}_{i-1} \,
\mbox{ for } 1 \le i \le j, \quad
\overline{\xi}_j = \overline{\xi}, \,
\overline{\xi}_0 = \overline{\eta},\\
&&(\overline{\xi_i}, b_i) \mbox{ is classically admissible for }
1 \le i \le j.\\\end{aligned}$$ We define $X_j(b, \xi, \eta)$ (resp. $\overline{X}_j(b,\overline{\xi},\overline{\eta})$) to be zero if $(\xi-wt(b),b)$ is not admissible (resp. if $(\overline{\xi}-wt(b),b)$ is not classically admissible). This implies that $\varphi_i(b) \le \langle h_i, \xi \rangle$ for all $i \in I$ (resp. $\varphi_i(b) \le \langle h_i, \overline{\xi} \rangle$ for all $i \in I\setminus \{ 0 \}$). We shall call $X_j(b, \xi, \eta)$ and $\overline{X}_j(b, \overline{\xi}, \overline{\eta})$ the (level $l + l'$) restricted 1dsums and the classically restricted 1dsums, respectively.
For any $\overline{\xi} \in \overline{P}_{cl}^+$, $\overline{\xi} + (l + l')\Lambda_0$ belongs to $(P^+_{cl})_{l+l'}$ for sufficiently large $l'$. Moreover the pair $(\overline{\xi},b)$ is classically admissible if and only if $(\overline{\xi} + (l + l')\Lambda_0, b)$ is admissible in the limit $l' \rightarrow \infty$. Therefore we have
\[pr:xandxbar\] For any $\overline{\xi}, \overline{\eta} \in \overline{P}_{cl}^+$, $$\begin{aligned}
\overline{X}_j(b, \overline{\xi}, \overline{\eta}) =
\lim_{l' \rightarrow \infty} X_j(b, \overline{\xi} + (l+l')\Lambda_0,
\overline{\eta} + (l+l')\Lambda_0). \nonumber \\\end{aligned}$$
As Proposition \[pr:rec1dsum\], the 1dsums $X_j(b, \xi, \eta)$ and $\overline{X}_j(b, \overline{\xi}, \overline{\eta})$ are characterized by the recursion relation and the initial condition as follows.
\[pr:recr1dsum\] $$\begin{aligned}
&& X_j(b, \xi, \eta) =
\sum_{b^\prime \in B, \, (\xi, b'): \hbox{\scriptsize admissible}}
q^{j H(b \ot b^\prime)}
X_{j-1}(b^\prime,\xi + wt(b^\prime), \eta),\nonumber \\
&& X_0(b, \xi, \eta) = \delta_{\xi \eta}, \nonumber \\
&& \overline{X}_j(b, \overline{\xi}, \overline{\eta}) =
\sum_{b^\prime \in B, \, (\overline{\xi}, b'):
\hbox{\scriptsize classically admissible}}
q^{j H(b \ot b^\prime)}
\overline{X}_{j-1}
(b^\prime,\overline{\xi} + wt(b^\prime), \overline{\eta}),\nonumber \\
&& \overline{X}_0(b, \overline{\xi}, \overline{\eta}) =
\delta_{\overline{\xi} \overline{\eta}}.\end{aligned}$$
In order to express $X_j$ and $\overline{X}_j$ in terms of $g_j$, we need to assume
\[con:disjoint\] For any $\xi \in (P_{cl}^+)_l$, there exists a disjoint union decomposition (not necessarily unique) as $$\begin{aligned}
&&\{ b \in B \mid \, (\xi,b) \hbox{ is not admissible} \} \nonumber \\
&&= \bigsqcup_{b' \in B,
\varepsilon_i(b') = \langle h_i, \xi \rangle + 1\,
\hbox{\scriptsize for some } i \in I}
\{ \ft{i}^t b' \mid 0 \le t \le \varphi_i(b') \}.
\label{affinedisjoint}\end{aligned}$$
We have proved this for $U_q(A^{(1)}_n), \, B = l$-fold symmetric tesnor case and have checked several other cases. This property seems reflecting an intrinsic combinatorial nature of perfect crystals. If the conjecture holds, it follows by the same argument as for Proposition \[pr:xandxbar\] that for any $\overline{\xi} \in \overline{P}_{cl}^+$, there exists a disjoint union decomposition (not necessarily unique) as $$\begin{aligned}
&&\{ b \in B \mid \, (\overline{\xi},b)
\hbox{ is not classically admissible} \} \nonumber \\
&&= \bigsqcup_{b' \in B,
\varepsilon_i(b') = \langle h_i, \overline{\xi} \rangle + 1\,
\hbox{\scriptsize for some } i \in I\setminus \{0 \}}
\{ \ft{i}^t b' \mid 0 \le t \le \varphi_i(b') \}.
\label{classicaldisjoint}\end{aligned}$$
\[pr:xbyg\] If Conjecture \[con:disjoint\] holds, the restricted and classically restricted 1dsums are expressed as linear superpositions of the 1dsum $g_j$ over the affine Weyl group $W$ and the classical Weyl group $\overline{W}$, respectively as $$\begin{aligned}
&&X_j(b, \xi, \eta) =
\sum_{w \in W} \mbox{ det } w\,
g_j(b, w(\eta + \rho) - \xi - \rho),
\label{xbyg}\\
&&\overline{X}_j(b, \overline{\xi}, \overline{\eta}) =
\sum_{w \in \overline{W} } \mbox{ det } w\,
g_j(b, w(\overline{\eta} + \overline{\rho})
- \overline{\xi} - \overline{\rho}).
\label{xbarbyg}\end{aligned}$$
We show (\[xbyg\]) from (\[affinedisjoint\]). (\[xbarbyg\]) can be verified from (\[classicaldisjoint\]) analogously. Let $F_j(b, \xi, \eta)$ denote the rhs of (\[xbyg\]). We are to show that $F_j(b, \xi, \eta)$ fulfills the properties in Proposition \[pr:recr1dsum\]. To check the initial condition is easy. By using (\[rec1dsum\]) one has $$F_j(b, \xi, \eta) = \sum_{b' \in B} q^{jH(b \ot b')}
F_{j-1}(b', \xi + wt(b'), \eta).$$ The sum here is similar to the one in Proposition \[pr:recr1dsum\] but without the constraint $(\xi, b'):\hbox{ admissible}$. Thus it is enough to show the cancellation of those unwanted contributions from non-admissible $b'$, namely, $$0 = \sum_{b' \in B, (\xi,b'): \hbox{non-admissible}}
q^{jH(b \ot b')}
F_{j-1}(b', \xi + wt(b'), \eta).$$ Under the assumption (\[affinedisjoint\]) it suffices to show the further decomposed form of this as $$0 = \sum_{t=0}^m
q^{jH(b \ot \ft{i}^t b')}
F_{j-1}(\ft{i}^t b', \xi + wt(\ft{i}^t b'), \eta)$$ for each $b' \in B$ such that $\varepsilon_i(b') = \langle h_i, \xi \rangle + 1$. Here $m = \varphi_i(b') = \langle h_i, \xi + \rho + wt(b') \rangle$. From $\varphi_i(b) \le \langle h_i, \xi \rangle$ one has $H(b \ot \ft{i}^t b') = H(b \ot b') + t\delta_{0 i}$. By using $wt(\ft{i}^t b') = wt(b') - t \alpha_i + t \delta_{0 i} \delta$ and (\[1dsumproperty\]) further, the rhs of the above is expressed as $$q^{jH(b \ot b')} \sum_{w \in W} \mbox{ det }w
\sum_{t=0}^m g_{j-1}(\ft{i}^tb',
w(\eta+\rho) - \xi - \rho - wt(b') + t\alpha_i)
q^{t(j-1)\delta_{0 i}}.$$ Upon applying (\[eq:2mrelation\]), one finds that this quantity is precisely equal to itself with $w(\eta+\rho)$ replaced by $r_iw(\eta+\rho)$. Thus it vanishes because of $\mbox{ det } r_i w = - \mbox{ det } w$.
Relation with affine Lie algebra and coset characters
-----------------------------------------------------
Given a $\Uq$ module $M$ and $\mu \in P$, let $[M : \mu ]$ (resp. $[M : \mu ]_{cl}$) denote the dimension of the linear space $\{ v \in M \mid wt(v) = \mu, \,
e_i v = 0
\hbox{ for all } i \in I (\hbox{resp. } i \in I \setminus \{ 0 \}) \}$. Let $c_j$ be as in Proposition \[pr:by1dsum\], $\lambda, \lambda_j \in (P_{cl}^+)_l,
V(\lambda), V_w(\lambda), \bbar_j, b(\lambda), d$ and $\sigma$ be as in section 1. Put $\overline{\lambda}_j = \lambda_j - l \Lambda_0$. The $j \rightarrow \infty$ limits of $g_j, X_j$ and $\overline{X}_j$ give rise to various branching functions. We summarize them in
\[pr:branching\] For $\mu \in P,
\xi \in (P_{cl}^+)_{l'}, \eta \in (P_{cl}^+)_{l+l'}$ and $\overline{\eta} \in \overline{P}_{cl}^+$ we have $$\begin{aligned}
&&\lim_{j \rightarrow \infty} q^{-c_j} g_j(\bbar_{j+1}, \mu) =
\sum_i \left( \dim V(\lambda)_{\mu - i \delta} \right) q^i,
\label{glimit}\\
&&\lim_{j \rightarrow \infty} q^{-c_j}
X_j(\bbar_{j+1}, \xi + \lambda_j, \eta) =
\sum_i [V(\xi) \otimes V(\lambda) : \eta - i \delta] \, q^i,
\label{xlimit}\\
&&\lim_{j \rightarrow \infty} q^{-c_{j}}
\overline{X}_{j}(\bbar_{j+1}, \overline{\lambda}_j, \overline{\eta}) =
\sum_i [V(\lambda) :
\overline{\eta} + l\Lambda_0 - i \delta]_{cl}\, q^i.
\label{xbarlimit}\\
\nonumber\end{aligned}$$
(\[glimit\]) is due to [@KMN1]. (\[xlimit\]) is due to [@DJO]. To show (\[xbarlimit\]) recall that any path can be written in the form $p = u_{\lambda_j} \otimes b_j \otimes \cdots \otimes b_1$ $(b_1, \ldots, b_j \in B)$ for sufficiently large $j$, where one may identify $u_{\lambda_j} = \cdots \otimes \bbar_{j+2} \otimes \bbar_{j+1}$. For a path $p = u_{\lambda_j} \otimes b_j \otimes \cdots \otimes b_1$, the condition $\et{i}p = 0\, \forall i \in I \setminus \{ 0 \}$ is equivalent to the requirement that $( \overline{\lambda}_j + wt(b_j) + \cdots + wt(b_{i+1}), b_i)$ is classically admissible for $1 \le i \le j$. Since the weight of the path $p$ is given by $\overline{\eta} + l \Lambda_0 - \left(
E(\bbar_{j+1} \otimes b_j \otimes \cdots \otimes b_1) - c_j \right) \delta$, this completes the proof of (\[xbarlimit\]).
Up to an overall power of $q$, (\[glimit\]) is a string function [@KP], (\[xlimit\]) is a branching coefficient of the module $V(\eta)$ in the tensor product $V(\xi) \otimes V(\lambda)$, (\[xbarlimit\]) is the branching coefficient of the irreducible $U_q(\overline{\geh})$ module with highest weight $\overline{\eta}$ within the integrable highest weight module $V(\lambda)$, where $U_q(\overline{\geh})$ stands for the subalgebra of $\Uq$ generated by $e_i, f_i, t_i ( i \in I\setminus \{ 0 \})$.
\[kacwakimoto\] Multiply $q^{-c_j}$ on both sides of (\[xbyg\]) and take $j \rightarrow \infty$ limit. From (\[glimit\]) and (\[xlimit\]), the result turns out to be equivalent with Theorem 3.1 in [@KW] when $\mu$ there is dominant integral. In this sense (\[xbyg\]) is a finite $j$ analogue of it.
One can interpret the Kostka-Foulkes polynomial $K_{\xi \mu}(q)$ [@Ma] as a classically restricted 1dsum for $A^{(1)}_n$. Consider the level $l$ perfect crystal $B$ corresponding to the $l$-fold symmetric tensor representation [@KMN2]. It is parametrized by semistandard tableaux of shape $(l)$ and entries from $\{0, 1, \ldots, n \}$. In particular $b(l\Lambda_0)$ is the one with all entries being $n$. Let $0 \le x_1 \le \cdots \le x_l \le n$ and $0 \le y_1 \le \cdots \le y_l \le n$ stand for the semistandard tableaux for $b$ and $b' \in B$, respectively. Then the $H$-function (\[defh\]) is given by $H(b \otimes b') = \hbox{ min }_\tau (\sum_{i=1}^l
\theta(x_i \ge y_{\tau(i)}))$. Here $\theta(\hbox{true}) = 1, \theta(\hbox{false}) = 0$ and the minimum extends over the degree $l$ symmetric group. Let $\xi = (\xi_1, \xi_2, \ldots, \xi_{n+1})$ be any partition of $lj$ (depth $l(\xi) \le n+1$) and identify it with $\sum_{i=1}^n(\xi_i - \xi_{i+1})\overline{\Lambda}_i
\in \overline{P}_{cl}^+$. Then we have $$K_{\xi (l^j)}(q) = q^{-lj}
\overline{X}_j(b(l\Lambda_0),0,\xi).
\label{kostka}$$ This is just an interpretation of a special case of a theorem in [@NY] via the classically restricted 1dsums. See also [@KMOTU2] for another extension. Our picture can be summarized roughly in the following table.
------------------------ ---------------------------------------------------------------- -------------------------------------- ---------------------------------------------------------
[1dsum]{} [$g_j$]{} [$\overline{X}_j$]{} [$X_j$]{}
path unrestricted classically restricted restricted
$j < \infty$ ${{\hbox{string function of}} \atop {\hbox{Demazure module}}}$ $\geh$-Kostka restricted $\geh$-Kostka
$j \rightarrow \infty$ string function ${\goth{g}_l}/{\overline{\goth{g}}}$ $({\goth{g}_{l'} \oplus \goth{g}_l})/{\goth{g}_{l+l'}}$
------------------------ ---------------------------------------------------------------- -------------------------------------- ---------------------------------------------------------
Here $\goth{g}_l$ denotes the affine Lie algebra $\goth{g}$ at level $l$. By $\geh$-Kostka we generally mean the branching coefficients of the irreducible $U_q(\overline{\geh})$-modules in the Demazure module $V_{w^{(jd)}}(\sigma^{-j}(l\Lambda_0))$. See [@KMOTU1], [@KMOTU2] for the $U_q(\overline{\geh})$ invariance of the Demazure modules.
\[kostkadifference\] Combining (\[kostka\]) and (\[xbarbyg\]) one can express $K_{\xi (l^j)}(q)$ as an alternating sum over $\overline{W}$. However the resulting formula is different from the one on p244 in [@Ma].
$q$-multinomial formula for $g_j(b,\mu)$
========================================
In this section we present explict formulae for the 1dsums $g_j(b,\mu)$ in terms of $q$-multinomial coefficients. We shall also attach the data $B, d, i^{(j)}_a, B^{(j)}_a$, etc from [@KMOTU1], which satisfy ([**I**]{}) - ([**IV**]{}) in section 1. Combined with Proposition \[pr:by1dsum\] or (\[simplecase\]) they yield a character formula for the Demazure module $V_{w^{(k)}}(\lambda)$. We shall only consider level 1 cases of $U_{q}(\geh)$ with $\geh$ being classical types: $A^{(1)}_{n}$, $B^{(1)}_{n}$, $D^{(1)}_{n}$, $A^{(2)}_{2n-1}$, $A^{(2)}_{2n}$ and $D^{(2)}_{n+1}$. Other cases, especially higher level cases will be treated elsewhere. Except the $D^{(2)}_{n+1}$ case, the $q$-multinomial formulae for $g_j(b,\mu)$ have been effectively known in earlier works [@JMO], [@DJKMO1], [@Kun] on solvable lattice models. They can be proved by establishing the recursion relation (\[rec1dsum\]).
Given a crystal $B$ and an integer vector with $\sharp B$-components $\gamma = (\gamma_b)_{b \in B}$ we shall employ the notations
$$\begin{aligned}
&\left[ \begin{array}{c} j \\ \gamma \end{array} \right]_q &=
\left \{\begin{array}{cl}
{(q)_j \over \prod_{b \in B}(q)_{\gamma_b}} & \quad
\mbox{if \, $j = \sum_{b \in B} \gamma_b$ \,
and $\gamma_b \in {\bf Z}_{\ge 0}$}\\
0& \quad \mbox{otherwise}
\end{array}\right. \\
&(q)_m &= \prod_{i=1}^m(1-q^i) \quad {\rm for }\, m \in {\bf Z}_{\ge 0}.\end{aligned}$$
We shall also use $$\epsilon_s = \left\{
\begin{array}{ll}
0& {\rm if} \; s \mbox{ is even } \\
1& {\rm if} \; s \mbox{ is odd }. \\
\end{array}\right.$$ In view of (\[1dsumproperty\]) we shall assume $\mu$ is a level 0 integral weight, i.e., $\mu \in P_{cl},\, \langle \mu, c \rangle = 0$ in the rest of the paper.
$(A^{(1)}_n, B(\Lambda_1))$ case
--------------------------------
The level 1 perfect crystal $B = B(\Lambda_1) = \{0, 1, \ldots, n \}$ can be depicted in the crystal graph
(300,80)(0,0) (10,10)[ (10,20)(147.5,60)(285,20) (10,20)[(-3,-1)[1]{}]{} (20,10)(50,0)[3]{}[(1,0)[30]{}]{} (5,5)[(10,10)[0]{}]{} (55,5)[(10,10)[1]{}]{} (105,5)[(10,10)[2]{}]{} (32,10)[(10,10)[1]{}]{} (81,10)[(10,10)[2]{}]{} (130,10)[(10,10)[3]{}]{} (157,10)(5,0)[6]{} (190,10)(55,0)[2]{}[(1,0)[30]{}]{} (225,5)[(15,10)[n-1]{}]{} (280,5)[(10,10)[n]{}]{} (200,10)[(15,10)[n-1]{}]{} (257,10)[(10,10)[n]{}]{} (147.5,40)[(10,10)[0]{}]{} ]{}
Elements of $B$ have the weights $$wt(b) = \Lambda_{\overline{b+1}} - \La_b\,\, \mbox{ for } b \in B,$$ where $\overline{x}$ is uniquely specified from $x$ by $\overline{x} \equiv x$ mod $n+1$ and $0 \le \overline{x} \le n$. The energy function is given by $$H(b\otimes b') = \left\{\begin{array}{cc}
0& {\rm if} \; b < b' \\
1& {\rm if} \; b \ge b'.
\end{array}\right.$$ Due to the Dynkin diagram symmetry it suffices to consider the case $\la = \Lambda_0$. Then we have the result [@KMOTU1]: $$\begin{aligned}
&&d = n,\quad \lambda_j = \Lambda_{\overline{ -j}},\quad
\bbar_j = \overline{ -j}, \\
&&B^{(j)}_a = \{ \overline{-j}, \overline{-j+1}, \ldots, \overline{-j+a} \}
\quad 1 \le a \le n, \\
&&i^{(j)}_a = \overline{-j+a}. \\\end{aligned}$$
For $j \in {\bf Z}_{\ge 0}, \, b \in B$ and $\mu = (\mu_i)_{i \in B} =
(\mu_n - \mu_0)\Lambda_0 + (\mu_0 - \mu_1) \Lambda_1 + \cdots +
(\mu_{n-1} - \mu_n)\Lambda_n \in P_{cl}$, we have $$g_j(b,\mu) = q^{\frac{1}{2} \sum_{i\in B}\mu_{i}(\mu_{i}-1)+
\sum_{i\in B}H(b \ot i)\mu_i}{j \brack \mu}_q.$$
$(B^{(1)}_n, B(\Lambda_1))$ Case
--------------------------------
The level 1 perfect crystal $B = B(\Lambda_1) =
\{1,2, \ldots, n, 0, \overline{n}, \ldots, \overline{1} \}$ is depicted by the crystal graph
(300,100)(0,0) (0,15)[ (20,60)(60,0)[2]{}[(1,0)[40]{}]{} (5,55)[(10,10)[1]{}]{} (65,55)[(10,10)[2]{}]{} (34,60)[(10,10)[1]{}]{} (92,60)[(10,10)[2]{}]{} (127,60)(5,0)[6]{} (160,60)(65,0)[2]{}[(1,0)[40]{}]{} (205,55)[(15,10)[n-1]{}]{} (270,55)[(10,10)[n]{}]{} (172,60)[(15,10)[n-2]{}]{} (240,60)[(15,10)[n-1]{}]{} (60,10)(60,0)[2]{}[(-1,0)[40]{}]{} (5,5)[(10,10)[$\overline{1}$]{}]{} (65,5)[(10,10)[$\overline{2}$]{}]{} (34,10)[(10,10)[1]{}]{} (92,10)[(10,10)[2]{}]{} (127,10)(5,0)[6]{} (200,10)(65,0)[2]{}[(-1,0)[40]{}]{} (205,5)[(15,10)[$\overline{\mbox{n-1}}$]{}]{} (270,5)[(10,10)[$\overline{\mbox{n}}$]{}]{} (172,10)[(15,10)[n-2]{}]{} (240,10)[(15,10)[n-1]{}]{} (320,30)[(10,10)[0]{}]{} (285,60)[(2,-1)[33]{}]{} (319,28)[(-2,-1)[33]{}]{} (298,52)[(10,10)[n]{}]{} (298,21)[(10,10)[n]{}]{} (57,20)[(-1,1)[33]{}]{} (24,20)[(1,1)[33]{}]{} (27,30)[(10,10)[0]{}]{} (43,30)[(10,10)[0]{}]{} ]{}
Elements of $B$ have the weights $$\begin{aligned}
&&wt(b) = - wt(\overline{b}) = \left\{
\begin{array}{ll}
\Lambda_b - \Lambda_{b-1} & b = 1 \mbox{ or } 3 \le b \le n-1 \\
\La_2 - \La_1 - \La_0 & b = 2 \\
2\La_n - \La_{n-1} & b = n \\
\end{array}\right. \\
&&wt(0) = 0.\\\end{aligned}$$ We introduce an order $\prec$ on $B$ by $$1 \prec \cdots \prec n \prec 0 \prec \overline{n} \prec \cdots \prec
\overline{1}.$$ This and similar $\prec$ will be used in the subsequent subsections just for convenience and should not be confused with the Bruhat order. The energy function is given by $$H(b\otimes b') = \left\{\begin{array}{cc}
0& {\rm if} \; b \prec b' \\
1& {\rm if} \; b \succeq b'.
\end{array}\right.$$ with the exceptions: $$H(0\otimes 0) =0, \quad H(1\otimes \overline{1}) = -1.$$ There are 3 level 1 dominant integral weights $(P^+_{cl})_1 = \{ \Lambda_0, \Lambda_1, \Lambda_n \}$. Due to the Dynkin diagram symmetry it suffices to consider $\la = \Lambda_0$ and $\Lambda_n$. In both cases we have $d = 2n-1$. The other data given in [@KMOTU1] reads $$\begin{aligned}
&\la = \Lambda_0 &{\rm case}:\\
&&\la_j = \La_{\epsilon_j},
\quad
\bbar_j = \left\{
\begin{array}{ll}
1& \; j: {\rm even} \\
\overline{1}& \; j: {\rm odd} \\
\end{array}\right.,\\
&&B^{(j)}_a = \left\{
\begin{array}{ll}
\{\bbar_j, 2, \ldots, a+1 \} & 1 \le a \le n-1\\
\{\bbar_j, 2, \ldots, n, 0, \overline{n}, \overline{n-1}, \ldots,
\overline{2n-a} \} & n \le a \le 2n-2\\
\end{array}\right., \\
&&i^{(j)}_a = \left\{
\begin{array}{ll}
\epsilon_{1-j} & a=1 \mbox{ or } a = 2n-1\\
{\rm min}(a,2n-a)& 2 \le a \le 2n-2.\\
\end{array}\right.\end{aligned}$$ $$\begin{aligned}
&\la = \Lambda_n &{\rm case}:\\
&&\la_j = \La_n,
\quad
\bbar_j = 0,\\
&&B^{(j)}_a = \left\{
\begin{array}{ll}
\{0, \overline{n}, \overline{n-1}, \ldots, \overline{n+1-a} \} & 1 \le a \le n-1\\
\{0, \overline{n}, \ldots, \overline{2}, 1 \} & a = n\\
\{0, \overline{n}, \overline{n-1}, \ldots, \overline{1}, 1, 2, \ldots, a-n+1 \}
& n+1 \le a \le 2n-1\\
\end{array}\right., \\
&&i^{(j)}_a = \left\{
\begin{array}{ll}
n+1-a & 1 \le a \le n-1 \\
a-n & n \le a \le 2n-1. \\
\end{array}\right.\end{aligned}$$ For $\la = \La_n$ one can also make another choice of $B^{(j)}_a$ and $i^{(j)}_a$ as $$\begin{aligned}
&&B^{(j)}_a = \left\{
\begin{array}{ll}
\{0, \overline{n}, \overline{n-1}, \ldots, \overline{n+1-a} \} & 0 \le a \le n\\
\{0, \overline{n}, \overline{n-1}, \ldots, \overline{1}, 1, 2, \ldots, a-n+1 \}
& n+1 \le a \le 2n-1\\
\end{array}\right., \\
&&i^{(j)}_a = \left\{
\begin{array}{ll}
n+1-a & 1 \le a \le n+1 \\
a-n & n+2 \le a \le 2n-1. \\
\end{array}\right.\end{aligned}$$ In any case, $B^{(j)}_0 = \{ \bbar_j \}$ and $B^{(j)}_d = B$ hold. Let us parametrize the level 0 elements $\mu \in P_{cl}$ by $(\mu_i)_{i=1}^n \in {\bf Z}^n$ as $$\mu = (- \mu_1 - \mu_2)\La_0 + (\mu_1 - \mu_2)\La_1 + \cdots +
(\mu_{n-1} - \mu_n) \Lambda_{n-1} + 2\mu_n \La_n.$$
For $j \in {\bf Z}_{\ge 0}, b \in B$ and the above $\mu \in P_{cl}$, we have $$\begin{aligned}
g_j(b,\mu) &=& \mathop{{\sum}^{*}}_\gamma q^{\frac{1}{2}
\sum_{i\in B}\gamma_{i}(\gamma_{i}-1)-\gamma_s\gamma_{\overline{s}}+
\sum_{i \in B}H(b \ot i)\gamma_{i}}{j\brack \gamma}_q,\\\end{aligned}$$ where $s = n$ if $b \succ 0$ and $s = 1$ if $b \prec 0$. When $b = 0$, either choice $s=n$ or $s=1$ is valid. The sum ${\sum}^{*}_\gamma$ extends over $\gamma = (\gamma_i)_{i \in B} \in ({\bf Z}_{\ge 0})^{2n+1}$ such that $$\gamma_{i} - \gamma_{\overline{i}} = \mu_{i}\;\;
for\; i = 1, \cdots, n, \quad \sum_{i\in B} \gamma_{i} = j.$$
$(D^{(1)}_{n}, B(\Lambda_{1}))$ case
------------------------------------
The level 1 perfect crystal $B = B(\Lambda_1) =
\{1,2, \ldots, n, \overline{n}, \ldots, \overline{1} \}$ is depicted by the crystal graph
(300,100)(0,0) (10,15)[ (20,60)(60,0)[2]{}[(1,0)[40]{}]{} (5,55)[(10,10)[1]{}]{} (65,55)[(10,10)[2]{}]{} (34,60)[(10,10)[1]{}]{} (92,60)[(10,10)[2]{}]{} (127,60)(5,0)[6]{} (160,60)(65,0)[2]{}[(1,0)[40]{}]{} (205,55)[(15,10)[n-1]{}]{} (270,55)[(10,10)[n]{}]{} (172,60)[(15,10)[n-2]{}]{} (240,60)[(15,10)[n-1]{}]{} (60,10)(60,0)[2]{}[(-1,0)[40]{}]{} (5,5)[(10,10)[$\overline{1}$]{}]{} (65,5)[(10,10)[$\overline{2}$]{}]{} (34,10)[(10,10)[1]{}]{} (92,10)[(10,10)[2]{}]{} (127,10)(5,0)[6]{} (200,10)(65,0)[2]{}[(-1,0)[40]{}]{} (205,5)[(15,10)[$\overline{\mbox{n-1}}$]{}]{} (270,5)[(10,10)[$\overline{\mbox{n}}$]{}]{} (172,10)[(15,10)[n-2]{}]{} (240,10)[(15,10)[n-1]{}]{} (260,52)[(-1,-1)[33]{}]{} (227,52)[(1,-1)[33]{}]{} (246,30)[(10,10)[n]{}]{} (231,30)[(10,10)[n]{}]{} (57,20)[(-1,1)[33]{}]{} (24,20)[(1,1)[33]{}]{} (27,30)[(10,10)[0]{}]{} (43,30)[(10,10)[0]{}]{} ]{}
Elements of $B$ have the weights $$wt(b) = - wt(\overline{b}) = \left\{
\begin{array}{ll}
\Lambda_b - \Lambda_{b-1} & b \ne 2, n-1 \\
\La_2 - \La_1 - \La_0 & b = 2 \\
\La_n + \La_{n-1} - \La_{n-2} & b = n-1. \\
\end{array}\right.$$ We introduce an order on $B$ by $$1 \prec \cdots \prec n-1 \prec {n \atop\overline{n}} \prec
\overline{n-1} \prec \cdots \prec \overline{1}.$$ There is no order between $n$ and $\overline{n}$. The energy function is given by $$H(b\otimes b') = \left\{\begin{array}{cc}
0& {\rm if} \; b \prec b', \\
1& {\rm if} \; b \succeq b'.
\end{array}\right.$$ with the exceptions: $$H(n \otimes \overline{n}) = H(\overline{n}\otimes n) =0, \quad
H(1\otimes \overline{1}) = -1.$$ There are 4 level 1 dominant integral weights $(P^+_{cl})_1 = \{ \Lambda_0, \Lambda_1, \La_{n-1}, \Lambda_n \}$. Due to the Dynkin diagram symmetry it suffices to consider $\la = \Lambda_0$. Then the result in [@KMOTU1] reads $$\begin{aligned}
&&d = 2n-2,\quad \la_j = \La_{\epsilon_j},
\quad
\bbar_j = \left\{
\begin{array}{ll}
1& \; j: {\rm even} \\
\overline{1}& \; j: {\rm odd} \\
\end{array}\right.,\\
&&B^{(j)}_a = \left\{
\begin{array}{ll}
\{\bbar_j, 2, \ldots, a+1 \} & 1 \le a \le n-2\\
\{\bbar_j, 2, \ldots, n-1, \overline{n} \} & a = n-1\\
\{\bbar_j, 2, \ldots, n, \overline{n}, \overline{n-1}, \ldots,
\overline{2n-1-a} \} & n \le a \le 2n-3\\
\end{array}\right., \\
&&i^{(j)}_a = \left\{
\begin{array}{ll}
\epsilon_{1-j} & a=1, 2n-2\\
{\rm min}(a,2n-1-a)& a \neq 1, n-1, 2n-2\\
2n-1-a & a=n-1.\\
\end{array}\right.\end{aligned}$$ One can also make another choice of $B^{(j)}_a$ and $i^{(j)}_a$ as $$\begin{aligned}
&&B^{(j)}_a = \left\{
\begin{array}{ll}
\{\bbar_j, 2, \ldots, a+1 \} & 1 \le a \le n-1\\
\{\bbar_j, 2, \ldots, n, \overline{n}, \overline{n-1}, \ldots,
\overline{2n-1-a} \} & n \le a \le 2n-3\\
\end{array}\right., \\
&&i^{(j)}_a = \left\{
\begin{array}{ll}
\epsilon_{1-j} & a=1, 2n-2\\
{\rm min}(a,2n-1-a)& a \neq 1, n, 2n-2\\
n & a=n.\\
\end{array}\right. \end{aligned}$$ In any case, $B^{(j)}_0 = \{ \bbar_j \}$ and $B^{(j)}_d = B$ hold. Let us parametrize the level 0 elements $\mu \in P_{cl}$ by $(\mu_i)_{i=1}^n \in {\bf Z}^n$ as $$\mu = (- \mu_1 - \mu_2)\La_0 + (\mu_1 - \mu_2)\La_1 + \cdots +
(\mu_{n-1} - \mu_n) \Lambda_{n-1} + (\mu_{n-1} + \mu_n) \La_n.$$
For $j \in {\bf Z}_{\ge 0}, b \in B$ and the above $\mu \in P_{cl}$, we have $$\begin{aligned}
g_j(b,\mu) &=& \mathop{{\sum}^{*}}_\gamma q^{\frac{1}{2}
\sum_{i\in B}\gamma_{i}(\gamma_{i}-1)-\gamma_s\gamma_{\overline{s}}+
\sum_{i \in B}H(b \ot i)\gamma_{i}}{j\brack \gamma}_q,\\\end{aligned}$$ where $s = n$ if $b \in \{\overline{n}, \ldots, \overline{1} \}$ and $s = 1$ if $b \in \{ 1, \ldots, n \}$. The sum ${\sum}^{*}_\gamma$ extends over $\gamma = (\gamma_i)_{i \in B} \in ({\bf Z}_{\ge 0})^{2n}$ such that $$\gamma_{i} - \gamma_{\overline{i}} = \mu_{i}\;\;
for\; i = 1, \cdots, n, \quad \sum_{i\in B} \gamma_{i} = j.$$
This is a very similar form to the $B^{(1)}_n$ case.
$(A^{(2)}_{2n-1}, B(\Lambda_{1}))$ case
---------------------------------------
The level 1 perfect crystal $B = B(\Lambda_1) =
\{1,2, \ldots, n, \overline{n}, \ldots, \overline{1} \}$ is depicted by the crystal graph
(300,100)(0,0) (10,15)[ (20,60)(60,0)[2]{}[(1,0)[40]{}]{} (5,55)[(10,10)[1]{}]{} (65,55)[(10,10)[2]{}]{} (34,60)[(10,10)[1]{}]{} (92,60)[(10,10)[2]{}]{} (127,60)(5,0)[6]{} (160,60)(65,0)[2]{}[(1,0)[40]{}]{} (205,55)[(15,10)[n-1]{}]{} (270,55)[(10,10)[n]{}]{} (172,60)[(15,10)[n-2]{}]{} (240,60)[(15,10)[n-1]{}]{} (60,10)(60,0)[2]{}[(-1,0)[40]{}]{} (5,5)[(10,10)[$\overline{1}$]{}]{} (65,5)[(10,10)[$\overline{2}$]{}]{} (34,10)[(10,10)[1]{}]{} (92,10)[(10,10)[2]{}]{} (127,10)(5,0)[6]{} (200,10)(65,0)[2]{}[(-1,0)[40]{}]{} (205,5)[(15,10)[$\overline{\mbox{n-1}}$]{}]{} (270,5)[(10,10)[$\overline{\mbox{n}}$]{}]{} (172,10)[(15,10)[n-2]{}]{} (240,10)[(15,10)[n-1]{}]{} (275,51)[(0,-1)[32]{}]{} (275,30)[(10,10)[n]{}]{} (57,20)[(-1,1)[33]{}]{} (24,20)[(1,1)[33]{}]{} (27,30)[(10,10)[0]{}]{} (43,30)[(10,10)[0]{}]{} ]{}
Elements of $B$ have the weights $$wt(b) = - wt(\overline{b}) = \left\{
\begin{array}{ll}
\Lambda_b - \Lambda_{b-1} & b \ne 2\\
\La_2 - \La_1 - \La_0 & b = 2. \\
\end{array}\right.$$ We introduce an order on $B$ by $$1 \prec \cdots \prec n \prec \overline{n} \prec \cdots \prec
\overline{1}.$$ The energy function is given by $$H(b\otimes b') = \left\{\begin{array}{cc}
0& {\rm if} \; b \prec b' \\
1& {\rm if} \; b \succeq b',
\end{array}\right.$$ with the exception: $$H(1\otimes \overline{1}) = -1.$$ There are 2 level 1 dominant integral weights $(P^+_{cl})_1 = \{ \Lambda_0, \Lambda_1 \}$. Due to the Dynkin diagram symmetry it suffices to consider $\la = \Lambda_0$. Then the result in [@KMOTU1] reads $$\begin{aligned}
&&d = 2n-1,\quad \la_j = \La_{\epsilon_j},
\quad
\bbar_j = \left\{
\begin{array}{ll}
1& \; j: {\rm even} \\
\overline{1}& \; j: {\rm odd} \\
\end{array}\right.,\\
&&B^{(j)}_a = \left\{
\begin{array}{ll}
\{\bbar_j, 2, \ldots, a+1 \} & 1 \le a \le n-1\\
\{\bbar_j, 2, \ldots, n, \overline{n}, \overline{n-1}, \ldots,
\overline{2n-a} \} & n \le a \le 2n-2\\
\end{array}\right., \\
&&i^{(j)}_a = \left\{
\begin{array}{ll}
\epsilon_{1-j} & a=1, 2n-1\\
{\rm min}(a,2n-a)& 2 \le a \le 2n-2.\\
\end{array}\right.\end{aligned}$$ $B^{(j)}_0 = \{ \bbar_j \}$ and $B^{(j)}_d = B$ hold. Let us parametrize the level 0 elements $\mu \in P_{cl}$ by $(\mu_i)_{i=1}^n \in {\bf Z}^n$ as $$\mu = (- \mu_1 - \mu_2)\La_0 + (\mu_1 - \mu_2)\La_1 + \cdots +
(\mu_{n-1} - \mu_n) \Lambda_{n-1} + \mu_n \La_n.$$
For $j \in {\bf Z}_{\ge 0}, b \in B$ and the above $\mu \in P_{cl}$, we have $$\begin{aligned}
g_j(b,\mu) &=& \mathop{{\sum}^{*}}_\gamma q^{Q}
\frac{(q^2)_{\gamma_1 + \gamma_{\overline{1}}}(q)_j}
{(q^2)_{\gamma_1}(q^2)_{\gamma_{\overline{1}}}
(q)_{\gamma_1 + \gamma_{\overline{1}}}
\prod_{i=2}^n(q)_{\gamma_i}(q)_{\gamma_{\overline{i}}}}
G(b,\mu,\gamma),\\
Q &=& \frac{1}{2} \sum_{i\in B}\gamma_{i}(\gamma_{i}-1)-
\gamma_{1}\gamma_{\overline{1}}+
\sum_{i\in B}H(b \ot i)\gamma_{i}, \\
G(b,\mu,\gamma) &=& \left\{ \begin{array}{cc}
1 & if\;\; b = 1\;or\;\overline{1} \\
\frac{ q^{\frac{1}{2}\mu_{1}} + q^{-\frac{1}{2}\mu_{1}} }
{ q^{\frac{1}{2}(\gamma_{1}+\gamma_{\overline{1}})} + q^{-\frac{1}{2}
(\gamma_{1}+\gamma_{\overline{1}}) }} & otherwise.
\end{array}\right.,\end{aligned}$$ The sum ${\sum}^{*}_\gamma$ extends over $\gamma = (\gamma_i)_{i \in B} \in ({\bf Z}_{\ge 0})^{2n}$ such that $$\gamma_{i} - \gamma_{\overline{i}} = \mu_{i}\;\;
for\; i = 1, \cdots, n, \quad \sum_{i\in B} \gamma_{i} = j.$$
$(A^{(2)}_{2n}, B(0) \oplus B(\Lambda_{1}))$ case
-------------------------------------------------
For a technical reason, we take the opposite ordering for the labeling of vertices of the Dynkin diagram from [@KMOTU1]. The level 1 perfect crystal $B = B(0) \oplus B(\Lambda_1) =
\{1,2, \ldots, n, 0, \overline{n}, \ldots, \overline{1} \}$ is depicted by the crystal graph
(300,100)(0,0) (0,15)[ (20,60)(60,0)[2]{}[(1,0)[40]{}]{} (5,55)[(10,10)[1]{}]{} (65,55)[(10,10)[2]{}]{} (34,60)[(10,10)[1]{}]{} (92,60)[(10,10)[2]{}]{} (127,60)(5,0)[6]{} (160,60)(65,0)[2]{}[(1,0)[40]{}]{} (205,55)[(15,10)[n-1]{}]{} (270,55)[(10,10)[n]{}]{} (172,60)[(15,10)[n-2]{}]{} (240,60)[(15,10)[n-1]{}]{} (60,10)(60,0)[2]{}[(-1,0)[40]{}]{} (5,5)[(10,10)[$\overline{1}$]{}]{} (65,5)[(10,10)[$\overline{2}$]{}]{} (34,10)[(10,10)[1]{}]{} (92,10)[(10,10)[2]{}]{} (127,10)(5,0)[6]{} (200,10)(65,0)[2]{}[(-1,0)[40]{}]{} (205,5)[(15,10)[$\overline{\mbox{n-1}}$]{}]{} (270,5)[(10,10)[$\overline{\mbox{n}}$]{}]{} (172,10)[(15,10)[n-2]{}]{} (240,10)[(15,10)[n-1]{}]{} (320,30)[(10,10)[0]{}]{} (285,60)[(2,-1)[33]{}]{} (319,28)[(-2,-1)[33]{}]{} (298,52)[(10,10)[n]{}]{} (298,21)[(10,10)[n]{}]{} (10,19)[(0,1)[32]{}]{} (10,30)[(10,10)[0]{}]{} ]{}
Elements of $B$ have the weights $$\begin{aligned}
&&wt(b) = - wt(\overline{b}) = \left\{
\begin{array}{ll}
\Lambda_b - \Lambda_{b-1} & 1 \le b \le n-1 \\
2\La_n - \La_{n-1} & b = n \\
\end{array}\right. \\
&&wt(0) = 0.\\\end{aligned}$$ We introduce an order $\prec$ on $B$ by $$1 \prec \cdots \prec n \prec 0 \prec \overline{n} \prec \cdots \prec
\overline{1}.$$ The energy function is given by $$H(b\otimes b') = \left\{\begin{array}{cc}
0& if \; b \prec b' \\
1& if \; b \succeq b',
\end{array}\right.$$ with the exception: $$H(0\otimes 0) =0.$$ There is a unique level 1 dominant integral weight $(P^+_{cl})_1 = \{ \Lambda_n \}$. Thus we set $\la = \La_n$, for which the result in [@KMOTU1] reads $$\begin{aligned}
&&d = 2n,\quad \la_j = \La_n,
\quad
\bbar_j = 0,\\
&&B^{(j)}_a = \left\{
\begin{array}{ll}
\{0, \overline{n}, \overline{n-1}, \ldots,
\overline{n+1-a} \} & 1 \le a \le n\\
\{0, \overline{n}, \overline{n-1}, \ldots, \overline{1},
1, 2, \ldots, a-n \} & n+1 \le a \le 2n\\
\end{array}\right., \\
&&i^{(j)}_a = \vert n+1-a \vert \quad 1 \le a \le 2n.\end{aligned}$$ Let us parametrize the level 0 elements $\mu \in P_{cl}$ by $(\mu_i)_{i=1}^n \in {\bf Z}^n$ as $$\mu = - \mu_1 \La_0 + (\mu_1 - \mu_2)\La_1 + \cdots +
(\mu_{n-1} - \mu_n) \Lambda_{n-1} + 2\mu_n \La_n.$$
For $j \in {\bf Z}_{\ge 0}, b \in B$ and the above $\mu \in P_{cl}$, we have $$\begin{aligned}
g_j(b,\mu) &=& \mathop{{\sum}^{*}}_\gamma q^{\frac{1}{2}
\sum_{i\in B, i \neq 0}\gamma_{i}(\gamma_{i}-1)+
\sum_{i \in B}H(b \ot i)\gamma_{i}}{j\brack \gamma}_q,\\\end{aligned}$$ where the sum ${\sum}^{*}_\gamma$ extends over $\gamma = (\gamma_i)_{i \in B} \in ({\bf Z}_{\ge 0})^{2n+1}$ such that $$\gamma_{i} - \gamma_{\overline{i}} = \mu_{i}\;\;
for\; i = 1, \cdots, n, \quad \sum_{i\in B} \gamma_{i} = j.$$
$(D^{(2)}_{n+1}, B(0) \oplus B(\Lambda_1))$ Case
------------------------------------------------
The level 1 perfect crystal $B = B(0) \oplus B(\Lambda_1) =
\{1,2, \ldots, n, 0, \overline{n}, \ldots, \overline{1}, \phi \}$ is depicted by the crystal graph
(300,130)(0,0) (10,10)[ (20,100)(60,0)[2]{}[(1,0)[40]{}]{} (5,95)[(10,10)[1]{}]{} (65,95)[(10,10)[2]{}]{} (34,100)[(10,10)[1]{}]{} (92,100)[(10,10)[2]{}]{} (127,100)(5,0)[6]{} (160,100)(65,0)[2]{}[(1,0)[40]{}]{} (205,95)[(15,10)[n-1]{}]{} (270,95)[(10,10)[n]{}]{} (172,100)[(15,10)[n-2]{}]{} (240,100)[(15,10)[n-1]{}]{} (60,10)(60,0)[2]{}[(-1,0)[40]{}]{} (5,5)[(10,10)[$\overline{1}$]{}]{} (65,5)[(10,10)[$\overline{2}$]{}]{} (34,10)[(10,10)[1]{}]{} (92,10)[(10,10)[2]{}]{} (127,10)(5,0)[6]{} (200,10)(65,0)[2]{}[(-1,0)[40]{}]{} (205,5)[(15,10)[$\overline{\mbox{n-1}}$]{}]{} (270,5)[(10,10)[$\overline{\mbox{n}}$]{}]{} (172,10)[(15,10)[n-2]{}]{} (240,10)[(15,10)[n-1]{}]{} (270,50)[(10,10)[0]{}]{} (275,47)(0,46)[2]{}[(0,-1)[30]{}]{} (275,27)(0,46)[2]{}[(10,10)[n]{}]{} (5,50)[(10,10)[$\phi$]{}]{} (10,17)(0,45)[2]{}[(0,1)[30]{}]{} (9,26)(0,45)[2]{}[(10,10)[0]{}]{} ]{}
Elements of $B$ have the weights $$\begin{aligned}
&&wt(b) = - wt(\overline{b}) = \left\{
\begin{array}{ll}
\La_1-2\La_0 & b = 1 \\
\Lambda_b - \Lambda_{b-1} & 2 \le b \le n-1 \\
2\La_n - \La_{n-1} & b = n \\
\end{array}\right. \\
&&wt(0) = wt(\phi) = 0.\\\end{aligned}$$ We introduce an order $\prec$ on $B\setminus \{ \phi \}$ by $$1 \prec \cdots \prec n \prec 0 \prec \overline{n} \prec \cdots \prec
\overline{1}.$$ The energy function is given by $$H(b\otimes b') = \left\{\begin{array}{ll}
0& {\rm if} \; b \prec b' \mbox{ or } (b, b') = (0,0), (\phi,\phi) \\
1& \mbox{if one and only one of } b \mbox{ and } b'\mbox{ is } \phi \\
2& {\rm if} \; b \succeq b' \mbox{ and }
(b, b') \neq (0,0), (\phi,\phi). \\
\end{array}\right.$$ There are 2 level 1 dominant integral weights $(P^+_{cl})_1 = \{ \Lambda_0, \Lambda_n \}$. Due to the Dynkin diagram symmetry it suffices to consider $\la = \Lambda_0$. Then the result in [@KMOTU1] reads $$\begin{aligned}
&&d = 2n,\quad \la_j = \La_0,
\quad
\bbar_j = \phi,\\
&&B^{(j)}_a = \left\{
\begin{array}{ll}
\{\phi, 1, 2, \ldots, a \} & 0 \le a \le n \\
\{\phi, 1, \ldots, n, 0, \overline{n}, \overline{n-1}, \ldots,
\overline{2n+1-a} \} & n+1 \le a \le 2n \\
\end{array}\right. , \\
&&i^{(j)}_a = {\rm min}(a-1,2n+1-a) \quad 1 \le a \le 2n.\\\end{aligned}$$ Let us parametrize the level 0 elements $\mu \in P_{cl}$ by $(\mu_i)_{i=1}^n \in {\bf Z}^n$ as $$\mu = - 2\mu_1 \La_0 + (\mu_1 - \mu_2)\La_1 + \cdots +
(\mu_{n-1} - \mu_n) \Lambda_{n-1} + 2\mu_n \La_n.$$
For $j \in {\bf Z}_{\ge 0}, b \in B$ and the above $\mu \in P_{cl}$, we have $$\begin{aligned}
g_j(b,\mu) &=& \mathop{{\sum}^{*}}_\gamma
q^{\sum_{i\in B, i \neq 0, \phi}\gamma_{i}(\gamma_{i}-1)+
\sum_{i \in B}H(b \ot i)\gamma_{i}}{j\brack \gamma}_{q^2},\\\end{aligned}$$ where the sum ${\sum}^{*}_\gamma$ extends over $\gamma = (\gamma_i)_{i \in B} \in ({\bf Z}_{\ge 0})^{2n+2}$ such that $$\gamma_{i} - \gamma_{\overline{i}} = \mu_{i}\;\;
for\; i = 1, \cdots, n, \quad \sum_{i\in B} \gamma_{i} = j.$$
Discussion
==========
We have shown that various characters can be viewed in a unified way as the 1dsums under the path realization of Demazure crystals. Our picture is summarized in Table 1 in the end of section 2. It is yet another task to actually evaluate these 1dsums. In this paper it has been done in section 3 for level 1 cases of the unrestricted 1dsum $g_j$. Substitution of them into Proposition \[pr:xbyg\] generates formulae also for $X_j$ and $\overline{X}_j$. As seen explicitly there, the results necessarily involve alternating signs from the Weyl group signature. Such formulae are sometimes called bosonic. In this respect it is interesting also to seek fermionic formulae. By this one roughly means those series or polynomials which are free of signs, admit a quasi-particle interpretation or have an origin in string hypotheses in the Bethe ansatz, etc. Formulae with such features have been explored extensively for several cases of $\overline{X}_j$ and $X_j$ in our Table 1 by many authors. See for example [@Ki] and references therein. On the other hand relatively fewer fermionic formulae seem known or even conjectured for $g_j$. A possible reason for this is that $g_j$ does [*not*]{} correspond to a counting of highest weight vectors as opposed to $\overline{X}_j$ and $X_j$. We hope to discuss this point further and higher level cases in near future.
[*Acknowledgement*]{}. A.K. would like to thank I. Cherednik, T. Nakanishi and A. Varchenko for their warm hospitality at University of North Carolina. K.C.M. is supported in part by NSA/MSP Grant No. 96-1-0013. K.C.M. and M.O. thank O. Foda for his hospitality at the University of Melbourne, and for collaboration in their earlier work [@FMO]. M.O. would like to thank M. Wakimoto for discussions and sending the paper [@KW].
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S-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki, Perfect crystals of quantum affine Lie algebras, Duke Math. J. [**68**]{}, 499-607 (1992).
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|
---
author:
- |
Sugumi <span style="font-variant:small-caps;">Kanno</span>$^{1,}
$[^1] , Misao <span style="font-variant:small-caps;">Sasaki</span>$^{2,}
$[^2] and Jiro <span style="font-variant:small-caps;">Soda</span>$^{3,}
$[^3]
title: 'Born-Again Braneworld '
---
Introduction
============
The inflationary universe scenario is a natural solution to fundamental problems of the big-bang model, such as the horizon problem[@inflation]. However, it is not a unique choice. For example, a universe with an era of contraction is also a possibility. The pre-big-bang scenario is a realization of such a case in the superstring context[@prebigbang]. Unfortunately, however, the pre-big-bang scenario suffers from the singularity problem, which cannot be solved without understanding the stringy non-perturbative effects.
One of the remarkable features of superstring theory is the existence of extra dimensions. Conventionally, the extra dimensions are considered to be compactified to form a small compact space of the Planck scale. However, recent revolutionary progress in string theory has lead to the brane-world picture[@braneworld]. In this paper, we consider a system of two branes having tensions of opposite sign, with the intermediate spacetime (bulk) described by an anti-de Sitter space (AdS$_5$)[@RS1; @radion1; @radion2; @cosmo]. One of the branes is assumed to be our universe, and there exists an inflaton field that leads to inflation. The other brane is assumed to be vacuum, but with a non-zero cosmological constant (see Fig. 1).
Assuming the slow roll of the inflaton field, we can regard both branes as vacuum (de Sitter) branes. Hence, we analyze this case in detail. To this time, mostly the static de-Sitter two-brane system has been considered in the cosmological context[@dsbranes]. However, it is now well-known that a static de-Sitter two-brane system is unstable[@gen]. We therefore investigate the non-trivial radion dynamics and focus on its cosmological consequences (See the previous work on the radion dynamics given in Ref. 10). As a result, we find a new scenario of the braneworld, which we call the “born-again braneworld scenario". We show that the two branes can collide without developing serious singularities, as seen from an observer on either brane, and emerge as reborn branes with the signs of the Lagrangians reversed. We find that our scenario has features common to both the conventional inflationary scenario and the pre-big-bang scenario. In a sense, we can regard it as a non-singular realization of the pre-big-bang model in the braneworld context. (See related works and criticism of them presented in Refs. 11) and 12).) In particular, a flat spectrum for the density perturbation is naturally produced, while background gravitational waves with a very blue spectrum are generated through the collision. It may be possible to detect this using future interferometric gravitational wave detectors[@detector].
Effective action
=================
We begin by reviewing the effective equations on a brane at low energies, which we derived in previous papers[@kanno1; @kanno2] (see also Refs. 16) and 17)). This effective theory is valid if the energy density on a brane is much smaller than the brane tension. Strictly speaking, we cannot use this action when two branes collide, because the junction conditions lose their meaning in this case. Indeed, the collision process is singular from the 5-dimensional point of view. However, this singularity is relatively mild, and the action is completely regular at the collision point. This leads us to assume that the collision process can be described by this effective action. This assumption is crucial for later analysis.
Our system is described by the action $$\begin{aligned}
S&=&{1\over 2\kappa^2}\int d^5 x \sqrt{-g}\left({\cal R}
+{12\over \ell^2}\right)
-\sum_{i=A,B}\sigma_i \int d^4 x \sqrt{-g^{i\mathrm{\hbox{-}brane}}}
\nonumber\\
&& +\sum_{i=A,B} \int d^4 x \sqrt{-g^{i\mathrm{\hbox{-}brane}}}
\,{\cal L}_{\rm matter}^i \ ,
\label{5D:action}\end{aligned}$$ where ${\cal R}$, $g^{i\mathrm{\hbox{-}brane}}_{\mu\nu}$ and $\kappa^2$ are the 5-dimensional scalar curvature, the induced metric on the $i$-brane, and the 5-dimensional gravitational constant, respectively. We consider an $S_1/Z_2$ orbifold spacetime with the two branes as the fixed points. In the first Randall-Sundrum (RS1) model[@RS1], the two flat 3-branes are embedded in AdS$_5$ with the curvature radius $\ell$ and the brane tensions given by $\sigma_A=6/(\kappa^2\ell)$ and $\sigma_B=-6/(\kappa^2\ell)$. Then we have $g^{A\hbox{-}\rm brane}_{\mu\nu}=e^{2d/\ell}g^{B\hbox{-}\rm brane}$, where $d$ is the distance between the two branes. We assume this model to be the ground state of our model.
Adding the energy momentum tensor to each of the two branes, and allowing deviations from the pure AdS$_5$ bulk, the effective (non-local) Einstein equations on the branes at low energies take the forms[@ShiMaSa; @kanno1; @kanno2] $$\begin{aligned}
G^{\mu}{}_{\nu} (h )
&=&{\kappa^2 \over\ell} T^{A\mu}{}_{\nu}
-{2\over\ell}\chi^{\mu}{}_{\nu} \,,
\label{A:einstein}
\\
G^{\mu}{}_{\nu}(f)
&=& -{\kappa^2 \over\ell} T^{B\mu}{}_{\nu}
-{2\over\ell} {\chi^{\mu}{}_{\nu} \over \Omega^4} \ .
\label{B:einstein}\end{aligned}$$ where $h_{\mu\nu}=g^{A\hbox{-}{\rm brane}}_{\mu\nu}$ and $\Omega$ is a conformal factor that relates the metric on the $A$-brane to that on the $B$-brane (specifically, $f_{\mu\nu}=g^{B\hbox{-}{\rm brane}}_{\mu\nu}=\Omega^2h_{\mu\nu}$), and the terms proportional to $\chi_{\mu\nu}$ are 5-dimensional Weyl tensor contributions, which describe the non-local 5-dimensional effect. Although Eqs. (\[A:einstein\]) and (\[B:einstein\]) are non-local individually, with undetermined $\chi_{\mu\nu}$, they can be combined so as to reduce them to local equations for each brane. Since $\chi_{\mu\nu}$ appears only algebraically, one can easily eliminate $\chi_{\mu\nu}$ from Eqs. (\[A:einstein\]) and (\[B:einstein\]).
$A$-brane
---------
First, consider the effective equations on the $A$-brane. Defining a new field $\Psi = 1-\Omega^2$, we find $$\begin{aligned}
G^{\mu}{}_{\nu}(h)&=&{\kappa^2 \over\ell \Psi } T^{A\mu}{}_{\nu}
+{\kappa^2 (1-\Psi )^2 \over\ell\Psi} T^{B\mu}{}_{\nu}
+{ 1 \over \Psi } \left( \Psi^{|\mu}{}_{|\nu}
-\delta^\mu_\nu \Psi^{|\alpha}{}_{|\alpha} \right) \nonumber\\
&& +{3 \over 2 \Psi (1-\Psi )} \left( \Psi^{|\mu} \Psi_{|\nu}
- {1\over 2} \delta^\mu_\nu \Psi^{|\alpha} \Psi_{|\alpha}
\right),
\label{A:STG1} \\
\Box\Psi&=&{\kappa^2 \over 3\ell}(1-\Psi )
\left\{ T^A + (1-\Psi)T^B \right\}
-{1 \over 2 (1-\Psi )} \Psi^{|\mu}\Psi_{|\mu} \ ,
\label{A:STG2} \end{aligned}$$ where “$|$" denotes the covariant derivative with respect to the metric $h_{\mu\nu}$. Since $\Omega$ (or equivalently $\Psi$) contains the information of the distance between the two branes, we call $\Omega$ (or $\Psi$) the “radion".
We can also determine $\chi^{\mu}{}_{\nu}$ by eliminating $G^{\mu}{}_{\nu}$ from Eqs. (\[A:einstein\]) and (\[B:einstein\]). Then, we have $$\begin{aligned}
\chi^{\mu}{}_{\nu}&=&-{\kappa^2(1-\Psi)\over 2 \Psi}
\left\{ T^{A\mu}{}_{\nu}
+ (1-\Psi)T^{B\mu}{}_{\nu}\right\}
-{\ell\over 2 \Psi} \biggl[ \left( \Psi^{|\mu}{}_{|\nu}
-\delta^\mu_\nu \Psi^{|\alpha}{}_{|\alpha} \right)
\biggr. \nonumber \\
&& \biggl.+{3 \over 2(1 -\Psi )} \left( \Psi^{|\mu} \Psi_{|\nu}
-{1\over 2} \delta^\mu_\nu \Psi^{|\alpha} \Psi_{|\alpha}
\right) \biggr] \ .
\label{A:chi}\end{aligned}$$ Note that the index of $T^{B\mu}{}_{\nu}$ is to be raised or lowered by the induced metric on the $B$-brane, $f_{\mu\nu}$.
The effective action for the $A$-brane that gives Eqs. (\[A:STG1\]) and (\[A:STG2\]) is $$\begin{aligned}
S_{\rm A}&=&{\ell\over 2 \kappa^2} \int d^4 x \sqrt{-h}
\left[ \Psi R (h) - {3 \over 2(1- \Psi )}
\Psi^{|\alpha} \Psi_{|\alpha} \right] \nonumber\\
&&\!\!\!\! + \int d^4 x \sqrt{-h} {\cal L}^A
+ \int d^4 x \sqrt{-h} \left(1-\Psi \right)^2 {\cal L}^B
\ .
\label{A:action} \end{aligned}$$
$B$-brane
---------
Using the same procedure as that above (but exchanging the roles of $h_{\mu\nu}$ and $f_{\mu\nu}$) also yields the effective equations on the $B$-brane. Defining $\Phi = \Omega^{-2} -1$, we obtain $$\begin{aligned}
G^\mu_{\ \nu}(f)&=&{\kappa^2 \over\ell\Phi }T^{B\mu}{}_{\nu}
+{\kappa^2 (1+\Phi )^2 \over\ell\Phi} T^{A\mu}{}_{\nu}
+{ 1 \over \Phi } \left( \Phi^{;\mu}{}_{;\nu}
-\delta^\mu_\nu \Phi^{;\alpha}{}_{;\alpha} \right) \nonumber\\
&& -{3 \over 2\Phi(1+\Phi)} \left( \Phi^{;\mu} \Phi_{;\nu}
- {1\over 2} \delta^\mu_\nu \Phi^{;\alpha} \Phi_{;\alpha}
\right) \ ,
\label{B:STG1} \\
\Box\Phi&=&{\kappa^2 \over 3\ell} (1+\Phi )
\left\{ T^B + (1+\Phi) T^A \right\}
+{1 \over 2 (1+\Phi )} \Phi^{;\mu}
\Phi_{;\mu} \ .
\label{B:STG2} \end{aligned}$$ Here, “$;$" denotes the covariant derivative with respect to the metric $f_{\mu\nu}$. Note that the index of $T^{A\mu}{}_{\nu}$ is raised or lowered by $h_{\mu\nu}$. Because $\Phi$ is equivalent to $\Omega$ or $\Psi$, we also call $\Phi$ the “radion".
We can also express $\chi^{\mu}{}_{\nu}$ in terms of quantities on the $B$-brane. We find $$\begin{aligned}
\chi^{\mu}{}_{\nu} &=& -{\kappa^2 \over 2 \Phi (1+\Phi)}
\left\{ T^{B\mu}{}_{\nu}
+ (1+ \Phi) T^{A\mu}{}_{\nu} \right\}
-{\ell\over 2 \Phi (1+\Phi)^2 } \biggl[ \biggl( \Phi^{;\mu}{}_{;\nu}
-\delta^\mu_\nu \Phi^{;\alpha}{}_{;\alpha} \biggr)
\biggr. \nonumber \\
&& \biggl.-{3 \over 2(1+\Phi )} \left( \Phi^{;\mu} \Phi_{;\nu}
- {1\over 2} \delta^\mu_\nu \Phi^{;\alpha} \Phi_{;\alpha}
\right) \biggr] \ .
\label{B:chi}\end{aligned}$$ The effective action for the $B$-brane is given by $$\begin{aligned}
S_{\rm B}&=&{\ell\over 2 \kappa^2} \int d^4 x \sqrt{-f}
\left[ \Phi R(f) + {3 \over 2(1+\Phi )}
\Phi^{;\alpha} \Phi_{;\alpha} \right] \nonumber\\
&&\!\!\!\! +\int d^4 x \sqrt{-f} {\cal L}^B
+\int d^4 x \sqrt{-f} {\cal L}^A(1+\Phi)^2
\ .
\label{B:action} \end{aligned}$$
Radion dynamics
===============
Most inflationary models are based on a slow-roll inflation that has a sufficiently flat potential. In this section, we consider the dynamics of branes with vacuum energy as a first-order approximation of a slow-roll inflation model. Qualitative features of the brane cosmology can be understood with this simplified vacuum brane model.
We take the matter Lagrangians to be ${\cal L}^A=-\delta\sigma^A$ and ${\cal L}^B=-\delta\sigma^B$ in our effective action (\[A:action\]) or (\[B:action\]). The effective action on the $A$-brane in this case reads $$\begin{aligned}
S_{\rm A} &=& {\ell\over 2 \kappa^2} \int d^4 x \sqrt{-h}
\left[
\Psi R
- {3\over2(1-\Psi)}
\Psi^{|\alpha}
\Psi_{|\alpha} \right]
-\delta\sigma^A \int d^4 x \sqrt{-h} \nonumber\\
&& -\delta\sigma^B \int d^4 x \sqrt{-h}
\left(1-\Psi\right)^2.
\label{A:action-vcm}\end{aligned}$$ Because our theory is a scalar-tensor-type theory, we call this original action the “Jordan-frame effective action". In order to study the dynamics of the radion, it is convenient to move to the Einstein frame, in which the action takes the canonical Einstein-scalar form[@Nojiri]. Applying the conformal transformation defined by $h_{\mu\nu}=\frac{1}{\Psi}g_{\mu\nu}$ and introducing the new field $$\begin{aligned}
\eta=-\log\left|\frac{\sqrt{1-\Psi}-1}{\sqrt{1-\Psi}+1}\right| \ ,
\label{A:field}\end{aligned}$$ we obtain the Einstein-frame effective action as $$\begin{aligned}
S_{\rm A}&=&{\ell\over 2 \kappa^2} \int d^4 x \sqrt{-g}
\left[R(g)-\frac{3}{2}
\nabla^\alpha\eta
\nabla_\alpha\eta\right]
-\int d^4x\sqrt{-g}~V(\eta) \ ,
\label{EH-action}\end{aligned}$$ where $\nabla$ denotes the covariant derivative with respect to the metric $g_{\mu\nu}$, and the radion potential now takes the form $$V(\eta) = \delta\sigma^A\left[~\cosh^4\frac{\eta}{2}
+\beta\sinh^4\frac{\eta}{2}~\right], \quad
\beta=\frac{\delta\sigma^B}{\delta\sigma^A} \ .
\label{potential}$$ We can also start from the effective action on the $B$-brane to obtain the same Einstein-frame effective action. By applying the conformal transformation defined by $f_{\mu\nu}=\frac{1}{\Phi}g_{\mu\nu}$ and introducing the new field $$\eta=-\log\left|\frac{\sqrt{\Phi+1}-1}{\sqrt{\Phi+1}+1}\right| \ ,
\label{B:field}$$ we also arrive at Eq. (\[EH-action\]).
We are now ready to examine the radion dynamics (see Fig. 2).
Notice that the two branes are infinitely separated when $\eta=0~(\Psi=1)$, and they collide when $\eta=\infty~(\Psi=0)$. For definiteness, let us assume $\delta\sigma_A>0$. If $\delta\sigma^A+\delta\sigma^B>0$, $\Psi$ will move towards unity; i.e., the branes will move away from each other. If $\delta\sigma^A+\delta\sigma^B<0$, the potential has a maximum at $\Psi_c=1+1/\beta$, and the behavior depends on whether $\Psi>\Psi_c$ or $\Psi<\Psi_c$. If $\Psi>\Psi_c$, the branes will become infinitely separated. If $\Psi<\Psi_c$, the branes will approach each other and eventually collide.
The static two de-Sitter brane solution corresponds to the unstable point $\Psi=\Psi_c$. In fact, considering the fluctuations around $\Psi_c$, we find an instability characterized by the equation $$\delta \ddot{\Psi} + 3 H \delta \dot{\Psi}
-4\left(H^2+\frac{K}{a^2}\right)\delta \Psi=0 \ .
\label{perturbation}$$ We see that the mass square, $-4(H^2+K/a^2)$, is negative, in accordance with the previous linear perturbation analysis[@gen].
As we mentioned above, in the case $\Psi<\Psi_c$, the two branes collide. From the 5-dimensional point of view, this is certainly a singularity, where the spacetime degenerates to 4 dimensions. However, as far as observers on the branes are concerned, nothing seems to go wrong. In fact, the action (\[A:action\]) is well-defined even in the limit $\Psi\to0$. Let us assume that $\Psi$ smoothly becomes negative after collision. Then replacing $\Psi$ as $\Psi\to -\tilde{\Psi}$ in the action (\[A:action\]), we find $$\begin{aligned}
-S_A &=& {\ell\over 2\kappa^2} \int d^4 x \sqrt{-h}
\left[~
\tilde{\Psi}R(h)
+\frac{3}{2}\frac{1}{1+\tilde{\Psi}}
\tilde{\Psi}^{|\alpha}\tilde{\Psi}_{|\alpha}~
\right]
+\int d^4 x \sqrt{-h}
(-{\cal L}^A) \nonumber \\
&& +\int d^4 x \sqrt{-h}
\left(1+\tilde{\Psi}\right)^2
({-\cal L}^B) \ .\end{aligned}$$ This is the same as the effective action on the $B$-brane, given in Eq. (\[B:action\]), except for the overall change of sign and the associated changes of sign of the matter Lagrangians. This fact can be interpreted as follows. After collision, the positive tension brane becomes a negative tension brane, together with the sign change of the matter Lagrangian, and vice versa for the initially negative tension brane. This implies that, if we live on either of the branes, our world transmutes into quite a different world, and so do we without much damage to the world. That is, we are born again!
This procedure might cause a serious problem when we consider quantum theory. A similar issue arises in string theory if there exists a negative tension brane. However, string theory has the potential ability to overcome this difficulty. For the time being, we can only hope that our prescription has an appropriate interpretation in the context of string theory.
Born-again braneworld
=====================
After the collision, if our world had initially been a positive tension brane, we would now be on the negative tension brane. However, the theory described by the action (\[B:action\]) with any value of $\Phi$ contradicts observation. Therefore we assume that we were initially on the negative tension brane ($B$-brane) before the collision.
Let us first investigate the cosmological evolution of the $B$-brane in the original Jordan frame. We consider the spatially isotropic and homogeneous metric on the brane $$\begin{aligned}
ds^2=-dt^2+a^2(t)\gamma_{ij}dx^idx^j\,,
\label{mtrc:cosmology}\end{aligned}$$ where $a(t)$ is the scale factor and $\gamma_{ij}$ is the metric of a maximally symmetric 3-space with comoving curvature $K=0$, $\pm1$. Using Eqs. (\[B:STG1\]) and (\[B:STG2\]), the field equations on the $B$-brane can be written $$\begin{aligned}
-3\left(H^2+\frac{K}{a^2}\right)
&=&-\frac{\kappa^2}{\ell}\frac{1}{\Phi}\delta\sigma^B
-\frac{\kappa^2}{\ell}\frac{(1+\Phi)^2}{\Phi}\delta\sigma^A
+3H\frac{\dot{\Phi}}{\Phi}
+\frac{3}{4}\frac{\dot{\Phi}^2}{\Phi(1+\Phi)} \ ,
\label{B:00} \\
-2\left(\dot{H}-\frac{K}{a^2}\right)
&-&3\left(H^2+\frac{K}{a^2}\right)
\nonumber\\
&&\hspace{-1cm}
=-\frac{\kappa^2}{\ell}\frac{1}{\Phi}\delta\sigma^B
-\frac{\kappa^2}{\ell}\frac{(1+\Phi)^2}{\Phi}\delta\sigma^A
+\frac{\ddot{\Phi}}{\Phi}
+2H \frac{\dot{\Phi}}{\Phi}
-\frac{3}{4}\frac{\dot{\Phi}^2}{\Phi(1+\Phi)}
\label{B:ij} \ , \\
\ddot{\Phi}+3H\dot{\Phi}
&=&\frac{4\kappa^2}{3\ell}(1+\Phi)
\left[\delta\sigma^B+(1+\Phi)\delta\sigma^A\right]
+\frac{1}{2}\frac{1}{1+\Phi}\dot{\Phi}^2 \ .
\label{B:radion}\end{aligned}$$ Note that Eq. (\[B:00\]) is the Hamiltonian constraint. Eliminating $\ddot{\Phi}$ from Eq. (\[B:ij\]) by using Eq. (\[B:radion\]) and combining the resulting equation with Eq. (\[B:00\]), we obtain $$\begin{aligned}
\dot{H}-\frac{K}{a^2}=-2\left(H^2+\frac{K}{a^2}\right)
-\frac{2\kappa^2}{3\ell}\delta\sigma^B \ .
\label{B:ij2} \end{aligned}$$ Integrating this equation, we obtain the Friedmann equation with dark radiation, $$\begin{aligned}
H^2+\frac{K}{a^2}
=-\frac{\kappa^2}{3\ell}\delta\sigma^B+\frac{C}{a^4} \ .
\label{friedmann}\end{aligned}$$ Note that this is just a cosmological version of the non-local Einstein equations on the brane, given in Eq. (\[B:einstein\]), in which the $\chi^{\mu}{}_\nu$ term gives the dark radiation $C/a^4$.
Comparing Eq. (\[B:00\]) with Eq. (\[friedmann\]), we find the following relation between the radion and the dark radiation: $$\begin{aligned}
&&\frac{\kappa^2\delta\sigma^B}{3\ell}\frac{1+\Phi}{\Phi}
\left[1+\frac{(1+\Phi)}{\beta}\right]
-H\frac{\dot{\Phi}}{\Phi}
-\frac{1}{4}\frac{1}{1+\Phi}\frac{\dot{\Phi}^2}{\Phi}
=\frac{C}{a^4} \ .
\label{relation}\end{aligned}$$ This gives, in particular, the relation between the initial conditions of the radion and the sign of the dark radiation.
Assuming $\delta\sigma_B<0$, the Friedmann equation (\[friedmann\]) yields the dependence of $H$ on the dark radiation. Setting $H_*^2=(\kappa^2/3\ell)(-\delta\sigma_B)$, we find $$\begin{aligned}
H_*^2K^2&-&4C>0\,:
\nonumber\\
H&=&H_*\frac{\sqrt{H_*^2K^2-4C}\sinh2H_* t}
{\sqrt{H_*^2K^2-4C}\cosh2H_* t+H_*^2K} \ ,
\\
~\nonumber\\
H_*^2K^2&-&4C=0\,:
\nonumber\\
H&=&H_*\frac{2\,e^{2H_*t}}{2\,e^{2H_*t}+H_*^2K} \ ,
\\
~\nonumber\\
H_*^2K^2&-&4C<0\,:
\nonumber\\
H&=&H_*\frac{\sqrt{4C-H_*^2K^2}\cosh2H_* t}
{\sqrt{4C-H_*^2K^2}\sinh2H_* t+H_*^2K} \ .\end{aligned}$$ To realize the born-again braneworld scenario, we consider the case of colliding branes. For simplicity, we assume $K=0$. A numerical solution of $\Phi$ is displayed in Fig. 3. We indeed see that $\Phi$ passes through zero smoothly and approaches $-1$; i.e., the reborn branes will eventually be infinitely separated.
Let us analyze this collision. We denote the Hubble constant at the time of collision $t=t_c$ by $H_c$. Applying Eq. (\[relation\]) to the vicinity of the time of collision, we find $$\begin{aligned}
\Phi=-2(1-\sqrt{\gamma})H_c(t-t_c)\,;
\quad\gamma=1-\frac{H_*^2}{H_c^2}\left(1+\frac{1}{\beta}\right)\,.\end{aligned}$$ As expected, $\Phi$ behaves perfectly smoothly around the time of collision. The brane geometry is, of course, perfectly regular as well. In fact, the Friedmann equation (\[friedmann\]) continues to hold without a hint of collision.
Now, we transform these quantities into the Einstein frame. Because $\Phi\to-1$ eventually, we can regard our present universe to be described by the Einstein frame. The relation between the Einstein frame and the Jordan frame is $$\begin{aligned}
ds^2_E&=&-dt_E^2 + b^2 (t_E) \delta_{ij} dx^i dx^j \nonumber\\
&=&|\Phi| \left[-dt_J^2+a(t_J)^2\delta_{ij}dx^idx^j\right],\end{aligned}$$ where we have attached the subscripts $E$ and $J$ to the time coordinates to denote the cosmic time in the Einstein frame and the Jordan frame, respectively. Thus we have $$\begin{aligned}
b=\sqrt{|\Phi|}\,a, \quad dt_E = \sqrt{|\Phi|}\,dt_J \ .
\label{conftrans}\end{aligned}$$ Therefore, the Hubble parameter in the Einstein frame behaves in the vicinity of collision as $$\begin{aligned}
\frac{\dot{b}(t_E)}{b(t_E)}
=\frac{1}{3t_E}
+\frac{H_c}{\left(3(1-\sqrt{\gamma})H_c|t_E|\right)^{1/3}}\,,
\label{Ehubble}\end{aligned}$$ where the collision time in the Einstein frame is set to be $t_E=0$.
We note that in the Einstein frame, the universe contracts rapidly just before the collision, and the Hubble parameter diverges to $-\infty$ at collision. Then, the universe is reborn with an infinitely large Hubble parameter, which looks like a big-bang singularity. Thus, because there exists no singularity in the Jordan frame, the pre-big-bang phase and the post-big-bang phase in the Einstein frame are successfully connected. That is, our scenario is indeed a successful realization of the pre-big-bang scenario in the context of the braneworld (see Fig. 4).
Observational implication
=========================
As we can see from Eq. (\[friedmann\]), the universe will rapidly converge to the quasi-de-Sitter regime, while the radion can vary, as long as the relation (\[relation\]) is satisfied. In the Jordan frame, because the metric couples with the radion, the non-trivial evolution of the radion field affects the perturbations. This possibility discriminates our model from the usual inflationary scenario. On the other hand, the inflaton does not couple directly with the radion field. Hence, the inflaton fluctuations are expected to give adiabatic fluctuations with a flat spectrum. This feature of our model is an advantage it has over the pre-big-bang model.
Radion fluctuations
-------------------
To study the behavior of the radion fluctuations, it is convenient to work in the Einstein frame. We express the metric perturbation in the Einstein frame as $$\begin{aligned}
ds_E^2
&=&b^2\left[-(1+2A)d\tau^2+2\partial_iBdx^id\tau\right.
\left.+\left((1+2{\cal R})\delta_{ij}+2\partial_i\partial_jE\right)
dx^idx^j\right]
\label{mtrc:sclr-ptb}\end{aligned}$$ The action for a curvature perturbation ${\cal R}$ on the $\delta\eta=0$ (i.e., radion-comoving) slice reads (for a concise review, see Appendix B of Ref. 20)) $$\begin{aligned}
S={1\over2}\int d\eta\, d^3x\,z^2
\left[{\cal R}_c'{}^2-{\cal R}_c^{\,|i}{\cal R}_{c\,|i}\right]\,,
\label{ptb:action}\end{aligned}$$ where ${\cal H}=b'/b$ and $$\begin{aligned}
{\cal R}_c={\cal R}-{\cal H}{\delta\eta\over\eta'}\,,
\quad
z=\sqrt{3\ell\over2\kappa^2}\,{b\eta'\over{\cal H}}\,.
\label{Rcdef}\end{aligned}$$ The equation of motion for ${\cal R}_c$ is $${\cal R}_c''+ 2{z'\over z}{\cal R}_c'
-\mathop\Delta^{(3)}{\cal R}_c = 0 \ .
\label{sclr:schrodinger-typ}$$ Because the background behaves as $b \sim (-\tau)^{1/2} $, ${\cal H}\sim(2\tau)^{-1}$ and $\eta' \sim (-\tau)^{-1}$, we have $z\propto b$, and the positive frequency modes for the adiabatic vacuum are given by $${\cal R}_{c,k}\sim\sqrt{\pi\kappa^2\over6H_*\ell}\,
H_0^{(1)} (-k\tau ) \,,
\label{bessel}$$ where we have normalized $b$ as $b=|H_*\tau|^{1/2}$. Then we have $$\begin{aligned}
\left\langle{\cal R}_c^2\right\rangle_k
\equiv{k^3\over 2\pi^2}P(k)
={k^3\over 2\pi^2}|{\cal R}_{c,k}|^2
\sim {k^3\over H_*M_{pl}^2}\,,
\label{curvamp}\end{aligned}$$ where $M_{pl}^2=\kappa^2/\ell$. Thus the spectrum is very blue. If we define the spectral index by $P(k)\propto k^{n-4}$, this implies $n=4$.
There are, however, a couple of points that should be mentioned. First, the curvature perturbation ${\cal R}_c$ is logarithmically divergent at the instant of collision. This divergence does not disappear even in the Jordan frame.[^4]. Note that ${\cal R}_c$ is defined on the hypersurface on which the radion is uniform, and hence is invariant under the conformal transformation (\[conftrans\]) This suggests the marginal instability of our system. Nevertheless, if we cut off an infinitesimally small time interval around the singularity, say $[-\epsilon,\epsilon]$, and smoothly match ${\cal R}_c$ at $\tau=\pm\epsilon$, the result is quite insensitive to the choice of $\epsilon$, as long as $k\epsilon\ll1$, i.e., the scale is outside the effective Hubble horizon ${\cal H}^{-1}$. Thus, we expect the result (\[curvamp\]) to be valid for all scales of cosmological interest.
The second point, which is a possible drawback, is the following. If inflation (in the Jordan frame) continues for a time of $O(H_*^{-1})$ after collision, the blue spectrum given above should not be called “blue" after all. As can be seen from Fig. 4, or from Eq. (\[Ehubble\]), $|\cal H|$ is quite symmetric around $\tau=0$, at least for $|\tau|\lesssim H_*^{-1}$. This implies that all the modes with $k>H_*$ that were once outside the horizon in the pre-big-bang phase came inside the horizon again in the post-big-bang phase by the time $\tau\sim H_*^{-1}$. (It should be noted that the conformal time is approximately equal to the cosmic time in the Jordan frame near the collision, with our normalization such that $b=|H_*\tau|^{1/2}$.) Because the evolution near the collision time is approximately time symmetric, the standard vacuum state will re-emerge for the modes that re-enter the horizon in the Einstein frame. These modes will then come outside the horizon again in the inflationary phase, and their spectrum will be a normal scale-invariant one. Therefore, we would obtain a spectrum that is scale-invariant for $k>H_*$ and blue spectrum at $k<H_*$, with the maximum amplitude given by $$\begin{aligned}
\left\langle{\cal R}_c^2\right\rangle_{k=H_*}
\sim {H_*^2\over M_{pl}^2}\,.\end{aligned}$$ For the values of $H_*$ predicted by the standard inflationary models, this is not really a blue spectrum in the observational sense.
This problem can be avoided only if the inflation ends right after the collision, when $\tau\ll H_*^{-1}$. One possibility is to resort to the marginal instability mentioned above. There may be a model in which the marginally divergent spectrum at high frequencies triggers a phase transition to end inflation. It is not clear if it is possible to construct such a model in a natural way. We leave investigation of this point as a future problem.
Gravitational waves
-------------------
Next, consider the tensor perturbations $$ds^2 = b^2 (\tau) \left[
-d\tau^2 + (\delta_{ij} + h_{ij}) dx^i dx^j \right] \ ,
\label{mtrc:tnsr-ptb}$$ where $h_{ij}$ satisfy the transverse-traceless conditions $h_{ij}{}^{,j} = h^{i}{}_i =0$. For the gravitational tensor perturbations, we have $$h_k^{\prime\prime} + 2{\cal H}h_k'+k^2h_k
=0 \ ,
\label{tnsr:schrodinger-typ}$$ where $h_k$ is the amplitude of $h_{ij}$. Since ${\cal H}\sim (2\tau)^{-1}$, $h_k$ has approximately the same spectrum as ${\cal R}_c$, including the magnitude. In particular, the spectral index for the gravitational waves is also $n=4$. (Here, the spectral index is defined by $P_h(k)\propto k^{n-4}$, as in the case of the scalar curvature perturbation. For the tensor perturbation, the conventional definition is $n_T=n-1$.) Provided that inflation ends right after collision, as discussed in the previous subsection, this gives a sufficiently blue spectrum that can amplify $\Omega_g$ by several orders of magnitude or more on small scales as compared to conventional inflation models. Thus, there is the possibility that it may be detected by a space laser interferometer for low frequency gravitational waves, such as LISA [@detector].
Inflaton perturbation
---------------------
To investigate the inflaton perturbation rigorously, one needs to introduce an inflaton field explicitly and consider a system of equations fully coupled with the radion and the metric perturbation. However, since this is beyond the scope of the present paper, let us just estimate the effect of the metric perturbation induced by radion fluctuations on the inflaton perturbation.
The field equation for the inflaton, $\varphi$, in the Jordan frame is $$\begin{aligned}
&&a^{-2}(a^2\delta\varphi')'
-\Delta\delta\varphi+a^2\partial_\varphi^2V\delta\varphi
\nonumber\\
&&\quad\qquad=-3\varphi'{\cal R}_J'+a^{-2}(a^2\varphi'A_J)'
-a^2\partial_\varphi VA_J
-\Delta(E_J'-B_J)\varphi'\,,
\label{Feq}\end{aligned}$$ where the suffix $J$ indicates a quantity in the Jordan frame. Again, let us consider the radion-comoving slice. Then, all the metric perturbation variables in the Jordan frame coincide with their respective counterparts in the Einstein frame.
Ignoring the effect of the inflaton perturbation, the Hamiltonian and momentum constraints on the radion-comoving slice are [@KS] $$\begin{aligned}
\Delta({\cal R}_c-{\cal H}(E'-B)_c)
&=&{3\over4}\eta'{}^2A_c \ ,
\nonumber\\
{\cal R}_c'-{\cal H}A_c
&=&0 \ .
\label{Econstraints}\end{aligned}$$ Using these equations, Eq. (\[Feq\]) reduces to $$\begin{aligned}
&&a^{-2}(a^2\delta\varphi')'
-\Delta\delta\varphi+a^2\partial_\varphi^2V\delta\varphi
\nonumber\\
&&\qquad
=-{2{\cal H}^2+{\cal H}'\over{\cal H}^2}{\cal R}_c'\varphi'
+a^{-2}\left(a^2{{\cal R}_c'\over{\cal H}}\varphi'\right)'
%+2{\cal H}_J{{\cal R}_c'\over{\cal H}}\varphi'
-{{\cal R}_c'\over{\cal H}}a^2\partial_\varphi V
-{\Delta{\cal R}_c\over{\cal H}}{\varphi'}\,.\end{aligned}$$ Since ${\cal R}_c'/{\cal H}$ is finite, we see that the right-hand side of the above is regular and small for the slow-roll inflation. Hence, the inflaton fluctuations are not strongly affected by the radion fluctuations. Thus, the inflaton fluctuations should have a standard scale-invariant spectrum.
Conclusion
==========
In this paper, we proposed a scenario in which two branes collide and are reborn as new branes, called the “born-again braneworld scenario". Our model has the features of both inflationary and pre-big-bang scenarios. In the original frame, which we call the Jordan frame, because gravity on the brane is described by a scalar-tensor-type theory, the brane universe is assumed to be inflating due to an inflaton potential. From the 5-dimensional point of view, the radion, which represents the distance between the branes and which acts as a gravitational scalar on the branes, has non-trivial dynamics and these vacuum branes can collide and pass through smoothly. After collision, it is found that the positive tension and the negative tension branes exchange their role. Then, they move away from each other, and the radion becomes trivial after a sufficient lapse of time. The gravity on the originally negative tension brane (whose tension becomes positive after collision) then approaches that of the conventional Einstein theory, except for tiny Kaluza-Klein corrections.
We can also consider the cosmological evolution of the branes in the Einstein frame. Note that the two frames are indistinguishable at present if our universe is on the positive tension brane after collision. In the Einstein frame, the brane universe is contracting before the collision and a singularity is encountered at the collision point. This resembles the pre-big-bang scenario. Thus our scenario may be regarded as a non-singular realization of the pre-big-bang scenario in the braneworld context.
Because our braneworld is inflating, and the inflaton has essentially no coupling with the radion field, an adiabatic density perturbation with a flat spectrum is naturally realized. On the other hand, because the collision of branes mimics the pre-big-bang scenario, the primordial background gravitational waves with a very blue spectrum may be produced. This suggests the possibility that we may be able to observe the collision epoch using a future gravitational wave detector, such as LISA.
Admittedly, the collision process must be treated with a more fundamental theory. However, because the singularity at the collision point is very mild, we expect that the qualitative features of our scenario will remain unchanged, even if we include the effect of a (yet unknown) fundamental theory. The born-again scenario surely deserves further investigation.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported in part by Monbukagaku-sho Grant-in-Aid for Scientific Research, Nos. 14540258, 1047214 and 12640269. We would like to thank D. Langlois and C. van de Bruck for discussions and comments. A part of this work was done while one of us (MS) was visiting the gravitation and cosmology group (GRECO) at IAP in Paris. MS would like to thank the GRECO members of IAP for warm hospitality.
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[^1]: E-mail: kanno@phys.h.kyoto-u.ac.jp
[^2]: E-mail: misao@vega.ess.sci.osaka-u.ac.jp
[^3]: E-mail: jiro@phys.h.kyoto-u.ac.jp
[^4]: The fact that the perturbation is generically singular when the effective gravitational constant changes sign was first pointed out by Starobinsky[@star].
|
---
abstract: 'We experimentally demonstrate that the origin of multiply reversed rectified vortex motion in an asymmetric pinning landscape is a consequence not only of the vortex-vortex interactions but also essentially depends on the ratio between the characteristic interaction distance and the period of the asymmetric pinning potential. Our system consists of an Al film deposited on top of a square array of [*size-graded*]{} magnetic dots with a constant lattice period $a=2~\mu$m. Four samples with different periods of the size gradient $d$ were investigated. For large $d$ the dc voltage $V_{dc}$ recorded under a sinusoidal ac excitation indicates that the average vortex drift is from bigger to smaller dots for all explored positive fields. As $d$ is reduced a series of sign reversals in the dc response are observed as a function of field. We show that the number of sign reversals increases as $d$ decreases. These findings are in agreement with recent computer simulations and illustrate the relevance of the different characteristic lengths for the vortex rectification effects.'
author:
- 'W. Gillijns, A.V. Silhanek, and V. V. Moshchalkov'
- 'C.J. Olson Reichhardt and C. Reichhardt'
title: On the origin of the reversed vortex ratchet motion
---
When independent overdamped particles in an asymmetric potential are subject to an oscillatory driving force with zero mean, a stationary flux of particles can be achieved. This rectified motion of particles, known as a rocked ratchet, is basically the result of the broken spatial symmetry of the potential energy[@reviews]. It has been recently shown that this effect can be used to manipulate the motion of flux quanta in superconductors with asymmetric pinning landscapes[@lee; @villegas; @clecionature; @clecioprl; @silhanekapl]. In this case, the fact that the particles (flux lines) cannot be regarded as independent entities leads to a far richer ratchet motion which can even reverse the direction of the average motion of vortices.
The relevance of the vortex-vortex interaction has already been pointed out by Souza Silva [*et al.*]{}[@clecionature] who describe the sign reversal of the ratchet signal as a result of trapping several vortices in each pinning site. Lu [*et al.*]{}[@lu] predicted that reversed ratchet motion should also be present in a simpler system consisting of one dimensional asymmetric pinning potential. In the former description the penetration depth $\lambda$ was larger than the period of the ratchet potential $d$, whereas in the latter model $\lambda = d$. Similarly, this effect has been observed experimentally [@shalom] and modelled theoretically [@marconi] for arrays of Josephson junctions with asymmetric periodic pinning, where $\lambda = \infty$. These studies have unambiguously identified a strong vortex-vortex interaction as the basic ingredient for producing ratchet reversals. This situation is achieved when the intervortex distance $a_0$ becomes smaller than $\lambda$. Although the inequality $a_0 < \lambda$ seems to represent a [*necessary*]{} condition, it cannot be sufficient since the natural scale where the inversion symmetry is broken, $d$, is not explicitly taken into account. An important question is what the ultimate conditions are for obtaining a reversed ratchet motion and how the number of sign reversals depends on the characteristic scales of the system.
In this work we directly address these questions by investigating the vortex ratchet motion in samples with different periods of the asymmetric pinning potential $d$. For $d>\lambda$ no sign reversal is detected. This finding demonstrates that $\lambda > d$ represents a prerequisite to obtain reversed ratchet motion. In addition we show that as $d$ is decreased the number of sign reversals increases. We argue that this effect can be qualitatively described within a collective pinning scenario.
The pinning potential is created by a square array (period $a=2 \mu$m) of ferromagnetic dots of a size that changes linearly from $1.2 \mu$m to $0.4 \mu$m over a distance $d$ (see Fig. \[fig1\]). Four different periods $d = 10$, 18, $34$, and 162 $\mu$m were used. The magnetic dots are \[0.4 nm Co/1.0 nm Pt\]$_{n}$ multilayers with $n=10$ grown on top of a 2.5 nm Pt buffer layer. The dot arrays are then covered with a 5 nm thick Si layer followed by an Al bridge of thickness $t=50$ nm. In this way the Al is insulated from the ferromagnetic template and proximity effects are suppressed.
Since the typical field excursion used to explore the superconducting state is much smaller than the coercive field of the Co/Pt multilayers, the magnetic dots remain in the as-grown state during our measurements. The virgin state of the dots actually consists of a multidomain structure with zero net magnetization [@gillijns-PRB; @silhanek-milosevic]. Still the stray field generated by these domains is enough to deplete the superconducting condensate locally on top of the dots, making the dots behave like pinning centers
The electrical transport measurements were carried out in a He4 cryostat with a base temperature of 1 K and a stability better than 100 nK, which is mounted with standard electronics. The normal/superconductor (N/S) phase transition (not shown) for all studied samples shows a featureless and nearly linear $T_{c}(H)$ phase boundary. The slope of these N/S boundaries corresponds to a superconducting coherence length at zero temperature $\xi(0) \sim 114 \pm 5$ nm. From the dirty limit expression $\xi(0) \sim 0.855 \sqrt{\xi_0\ell}$, where $\xi_0 \sim 1400$ nm is the BCS coherence length for Al and $\ell$ the electronic mean free path, we estimate $\ell \sim 13$ nm. Taking into account the finite thickness of the film and the reduced mean free path we obtain an effective penetration depth $\lambda(0) \sim 1.46~\mu$m. On the right axes of Fig. \[fig2\] we have indicated the variation of $\lambda$ with temperature. Within the temperature window of our measurements $\lambda$ is always bigger than the vortex-vortex separation $a_0=\sqrt{\phi_0/H}$, where $\phi_0$ is the flux quantum[@comment]. This indicates that the [*vortex-vortex interaction is in all cases significant in our experimental conditions*]{}.
Rectification effects were investigated by applying a sinusoidal excitation of $500~\mu$A amplitude and frequency $f=1$ kHz while simultaneously recording the average dc voltage with a nanovoltmeter. The acquired dc signal, when the current is parallel to the gradient, i.e. Lorentz force $F_L$ parallel to the iso-size lines ($x$-axis in Fig. \[fig1\]), was below our experimental resolution (not shown). This is consistent with the fact that for this direction of the Lorentz force the system is fully symmetric. A different situation is expected when the external ac current shakes the vortex lattice with a $F_L$ in the same direction as the gradient ($y$-axis in Fig. \[fig1\]). In this case a flux line feels an effective pinning potential $U_p$ schematically shown in Fig. \[fig3\], with asymmetric forces $f=-\partial U_p/\partial y$ for motion along $+y$ and $-y$ orientations. The lack of the inversion symmetry of $U_p(y)$ is a natural consequence of the increasing pinning strength as the radius of the dots becomes larger [@gillijns-PRB; @gillijns-unp]. Under these circumstances an external ac excitation should induce a net motion of vortices in the direction of the smaller slope of $U_p$, i.e. towards $+y$. This is indeed confirmed by the positive dc voltage $V_{dc}$ at positive fields obtained for the largest period of size-gradient ($d/a=81$), shown in Fig. \[fig2\](a). For this sample, the ratchet signal is strong only when the density of dots outnumbers the density of vortices, i.e. for fields below the first matching field of the square array $H_1^{2D}=\phi_0/a^2=0.51$ mT. Near $H_1^{2D}$, the hopping of vortices is suppressed since most of the pinning centers are occupied, and therefore the ratchet signal decreases. The most important feature of this figure is the lack of reversal ratchet in the whole range of fields studied. We argue that this behavior, present even though the vortex-vortex interaction is important, is due to the large period of the asymmetric pinning landscape $d$ in comparison with the penetration depth $\lambda$. For the sake of clarity, in Fig. \[fig2\] we added horizontal dashed lines at the temperatures where $\lambda \sim d$.
The most conclusive evidence that the ratio $d/\lambda$ is indeed the relevant parameter which properly accounts for the appearance of reversed ratchet motion comes from Fig. \[fig2\](b),(c), and (d) where the ratchet period is progressively reduced to meet the condition $d<\lambda$. In contrast to the previous sample, where $d/\lambda>1$, for all these samples clear multiple sign reversals of the ratchet signal are observed. [*This finding shows that the existence of a strong vortex-vortex interaction is not a sufficient condition to reverse the easy ratchet direction, but instead is the relative size of $\lambda$ in comparison with the period of the ratchet potential $d$ which determines the occurrence of this effect*]{}. Furthermore, the number of sign reversals in the same window of temperature and field ranges from none for $d/a=81$, 2 for $d/a=17$, 3 for $d/a=9$, and to 5 for $d/a=5$, illustrating the influence of this ratio on the number of observed reversals of sign of the vortex rectification effect.
Lu [*et al.*]{}[@lu] recently showed that a two dimensional vortex lattice interacting with a one-dimensional ratchet potential produces a conventional ratchet when the vortex lattice is highly ordered in such a way that the vortex-vortex interactions cancel. In contrast, disordered configurations of the vortex lattice lead to reversed ratchet motion. The vortex lattice disordering occurred due to buckling transitions of the vortices confined in each row of the ratchet over the field range considered in Ref. [@lu], but additional reversals are expected at lower fields. As shown in Ref. [@Levitov], a two-dimensional vortex lattice interacting with a one-dimensional periodic pinning array undergoes a series of continuous phase transitions as the magnetic field is decreased provided that the vortex-vortex interaction remains strong enough to produce a triangular ordering of the vortex lattice. The number of possible vortex lattice orderings that appear for decreasing vortex density increases when the lattice constant of the pinning array decreases, so we expect to find more low-field order-disorder transitions, and thus more low-field ratchet reversals, for smaller lattice periods $d$. To illustrate this, we have performed new simulations with the system described in Ref. [@lu] for pinning strength $A_p=0.15f_0$, ac field $F_{ac}=0.065f_0$ and ratchet periods $d=8\lambda$, $3\lambda$, $\lambda$, and $0.5\lambda$, where $f_0=6\phi_0^2/2\pi\mu_0\lambda^3$. In Fig. \[fig4\] we plot the net velocity $\langle V\rangle$ versus vortex density $n_v$, showing that the number of ratchet reversals for $n_v<1.1$ changes from no reversals at $d=8\lambda$ to two reversals for $d=3\lambda$, four at $d=\lambda$, and more than eight reversals at $d=0.5\lambda$. This demonstrates that as the ratchet period $d$ decreases, more ratchet reversals occur, in agreement with the experimental results. We note that our experimental system represents, for fields $H<H_1^{2D}$, a discrete two dimensional version of the situation described by the simulations. As a last remark we would like to point out that strictly speaking it is unlikely that the single parameter $d/\lambda$ properly describes the crossover from no-reversal to reversed ratchet. For instance a correlation length similar to the Larkin-Ovchinnikov[@LO; @garten] (LO) correlation length $R_c=\sqrt{c_{66}a_0/F_p}$, where $c_{66}$ is the shear elastic modulus of the vortex lattice and $F_p$ is the pinning force density, should be taken into account instead of $\lambda$. Unfortunately the previous scale is a theoretical estimate for thick samples with weak random pinning and to our knowledge an extension of this equation to include thin film geometries with periodic array of pinning centers is still not available in the literature.
Concluding, we have experimentally demonstrated that when the characteristic vortex-vortex interaction scale length $\lambda$ is of the order of the period of the asymmetric pinning landscape $d$ an inversion of the effective ratchet potential for vortex motion occurs. This implies that the inequality $d < \lambda$ which represents a sufficient condition for the occurrence of multiple reversal in vortex systems should remain valid for any ratchet system with interacting particles such as colloids and granular materials.
This work was supported by the K.U.Leuven Research Fund GOA/2004/02 program, the Belgian IAP, the Fund for Scientific Research – Flanders (F.W.O.–Vlaanderen), by the F.W.O. fellowship (A.V.S.), and by the ESF-NES Programme. Work by C.J.O.R. and C.R. was carried out under the NNSA of the US DoE at LANL under Contract No. DE-AC52-06NA25396.
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|
---
abstract: 'We present an algorithm to decide the intruder deduction problem (IDP) for a class of locally stable theories enriched with normal forms. Our result relies on a new and efficient algorithm to solve a restricted case of higher-order associative-commutative matching, obtained by combining the *Distinct Occurrences of AC-matching* algorithm and a standard algorithm to solve systems of linear Diophantine equations. A translation between natural deduction and sequent calculus allows us to use the same approach to decide the *elementary deduction problem* for locally stable theories. As an application, we model the theory of blind signatures and derive an algorithm to decide IDP in this context, extending previous decidability results.'
author:
- 'Mauricio Ayala-Rincón [^1]'
- Maribel Fernández
- 'Daniele Nantes-Sobrinho[^2]'
nocite: '[@*]'
title: 'Elementary Deduction Problem for Locally Stable Theories with Normal Forms[^3]'
---
Introduction {#introduction .unnumbered}
============
There are different approaches to model cryptographic protocols and to analyse their security properties [@survey]. One technique consists of proving that an attack requires solving an algorithmically hard problem; another consists of using a process calculus, such as the spi-calculus [@spi], to represent the operations performed by the participants and the attacker. In recent years, the deductive approach of Dolev and Yao [@dolev], which abstracts from algorithmic details and models an attacker by a deduction system, has successfully shown the existence of flaws in well-known protocols. A deduction system under Dolev-Yao’s approach specifies how the attacker can obtain new information from previous knowledge obtained either by eavesdropping the communication between honest protocol participants (in the case of a passive attacker), or by eavesdropping and fraudulently emitting messages (in the case of an active attacker). The *intruder deduction problem* (IDP) is the question of whether a passive eavesdropper can obtain a certain information from messages observed on the network.
Abadi and Cortier’s approach [@AbadiCo2006] proposes conditions for analysing message deducibility and indistinguishability relations for security protocols modelled in the applied pi-calculus [@appliedpicalculus]. In particular, [@AbadiCo2006] shows that IDP is decidable for *locally stable* theories. However, to ensure the soundness of this approach, the definition of locally stable theories given in [@AbadiCo2006] needs to be modified (as confirmed via personal communication with the second author of [@AbadiCo2006]). In this work, we made the necessary modifications and propose a new approach to solve IDP in the context of locally stable theories.
Our notion of locally stable theory is based on the existence of a finite and computable saturated set, but, unlike [@AbadiCo2006], our saturated sets include normal forms[^4]. The new approach we propose in order to prove the decidability of IDP is based on an algorithm to solve a restricted case of higher-order associative-commutative matching (AC-matching). To design this algorithm we use well-known results for solving systems of linear Diophantine equations (SLDE) [@BouConDe; @ClauFor; @frumkin; @papadimitriou], which we combine with a polynomial algorithm to solve the DO-ACM problem (Distinct Occurrences of AC-Matching) [@narendran].
In the case where the signature of the equational theory contains, for each AC function symbol $\oplus$, its corresponding inverse $i_{\oplus}$, we obtain a decidability result which is polynomial with relation to the size of the saturated set (built from the initial knowledge of the intruder). Thanks to the use of the algorithm for solving SLDE over $\mathbb{Z}$, we avoid an exponential time search over the solution space in the case of AC symbols (improving over [@AbadiCo2006], where an exponential number of possible combinations have to be considered). For more details we refer the reader to the extended version of this paper [@AFSextended]. After introducing the class of locally stable theories and proving the decidability of the IDP for protocols in this class, we show that the Elementary Deduction Problem (EDP) introduced in [@TiGo2009] is also decidable in polynomial time with relation to the size of a saturated set of terms. EDP is stated as follows: given a set $\Gamma$ of messages and a message $M$, is there an $E$-context $C[\ldots]$ and messages $M_1,\ldots, M_k\in \Gamma$ such that $C[M_1,\ldots, M_k]\approx_{E}M$? Here, $E$ is the equational theory modelling the protocol. We use this approach to model theories with blind signatures. As an application, using a previous result that links the decidability of the EDP to the decidability of the IDP when the theory $E$ satisfies certain conditions, we obtain decidability of IDP for a subclass of locally stable theories combined with the theory $B$ of blind signatures. In this way, we generalise a result from [@AbadiCo2006] (Section 5.2.4): it is not necessary to prove that the combination of the theories $E$ and $B$ is locally stable.
**Related Work.** The analysis of cryptographic protocols has attracted a lot of attention in the last years and several tools are available to try to identify possible attacks, see Maude-NPA [@maudeNPA07], ProVerif [@proverif], CryptoVerif [@cryptoverif], Avispa [@avispa], Yapa [@YAPA].
Sequent calculus formulations of Dolev Yao intruders [@Tiu2007] have been used in a formulation of open bisimulation for the spi-calculus. In [@TiGo2009], deductive techniques for dealing with a protocol with blind signatures in mutually disjoint AC-convergent equational theories, containing a unique AC operator each, are considered. As an alternative approach, the intruder’s deduction capability is modelled inside a sequent calculus modulo a rewriting system, following the approach of [@BeC06]. Then, the IDP is reduced in polynomial time to EDP.
By combining the techniques in [@TiGo2009] and [@Bursucconstraints], the IDP formulation for an Electronic Purse Protocol with blind signatures was proved to reduce in polynomial time to EDP for an AC-convergent theory containing three different $AC$ operators and rules for exponentiation [@nantesayala], extending the previous results. However, no algorithm was provided to decide EDP. More precisely, assuming that EDP is solved in time $O(f(n))$, it was proved that IDP reduces polynomially to EDP with complexity $O(n^k \times f(n))$, for some constant $k$. Thus, whenever the former problem is polynomial, the IDP is also polynomial.
**Contributions.** We present a technique to decide EDP or IDP in AC-convergent equational theories. Our approach is based on a “local stability” property inspired by [@AbadiCo2006], instead of proving that the deduction rules are “local” in the sense of [@mcallester] as done in many previous works [@CoLuSh03; @De2006; @Laf07; @Bursucconstraints]. More precisely, the main contributions of this paper are:
- We adapt and refine the technique proposed in [@AbadiCo2006], where deducibility and indistinguishability relations are claimed to be decidable in polynomial time for locally stable theories. First, we changed the definition of locally stable theories, adding normal forms, which are needed to carry out the decidability proofs. Second, we designed a new algorithm to decide IDP in locally stable theories. The algorithm provided in [@AbadiCo2006] is polynomial for the class of subterm theories (Proposition 10 in [@AbadiCo2006]), but the proof does not extend directly to locally stable theories (despite the statement in Proposition 16). Our algorithm relies on solving a restricted case of higher-order AC-matching problem that is used to decide the deduction relation. It is a combination of two standard algorithms: one for solving the DO-ACM problem [@narendran] which has a polynomial bound in our case; and one for solving systems of Linear Diophantine Equations(SLDE), which is polynomial in $\mathbb{Z}$ [@BouConDe; @ClauFor; @frumkin; @papadimitriou]. Using this algorithm we prove that IDP is decidable in polynomial time with respect to the saturated set of terms, for locally stable theories with inverses.
- A decidability result for the EDP for locally stable theories, which extends the work of Tiu and Goré [@TiGo2009]. As an application, we present a strategy to decide IDP for locally stable theories combined with blind signatures. Here, the combination of theories does not need to be locally stable.
In order to get the polynomial decidability result claimed in [@AbadiCo2006] for locally stable theories, we had to restrict to theories that contain, for each $AC$ symbol in the signature, the corresponding inverse. The inverses are necessary when we interpret our term algebra inside the integers $\mathbb{Z}$ to solve SLDE (terms headed by the inverse function will be seen as negative integers). If the theory does not contain inverses, we would have to solve the SLDE for $\mathbb{N}$ which is a well known NP-complete problem.
Preliminaries
=============
Standard rewriting notation and notions are used (e.g. [@baader]). We assume the following sets: a countably infinite set $N$ of *names* (we use $a,b,c, m$ to denote names); a countably infinite set $X$ of *variables* (we use $x,y,z$ to denote variables); and a finite *signature* $\Sigma$, consisting of function names and their arities. We write $arity(f)$ for the arity of a function $f$, and let $ar(\Sigma)$ be the maximal arity of a function symbol in $\Sigma$.
The set of *terms* is generated by the following grammar: $$M,N:= a\,|\, x\,|\, f(M_1,\ldots, M_n)$$ where $f$ ranges over the function symbols of $\Sigma$ and $n$ matches the arity of $f$, $a$ denotes a name in $N$ (representing principal names, nonces, keys, constants involved in the protocol, etc) and $x$ a variable. We denote by $V(M)$ the set of variables occurring in $M$. A message $M$ is *ground* if $V(M)=\emptyset$. The *size* $|M|$ of a term $M$ is defined by $|u|=1$, if $u$ is a name or a variable; and $|f(M_1,\ldots, M_n)|=1+\sum_{i=1}^n|M_i|$.
The set of *positions* of a term $M$, denoted by $\mathcal{P}os(M)$, is defined by $\mathcal{P}os(M):= \{\epsilon\}$, if $M$ is a name or a variable; and $\mathcal{P}os(M):= \{\epsilon\} \cup \bigcup_{i=1}^n\{ip \,|\, p \in \mathcal{P}os(M_i)\}$, if $M=f(M_1, \ldots, M_n)$ where $f\in \Sigma$. The position $\epsilon$ is called the *root* position. The size of $|M|$ coincides with the cardinality of $\mathcal{P}os(M)$. The set of *subterms* of $M$ is defined as $st(M)=\{M|_p \,|\, p \in \mathcal{P}os(M)\}$, where $M|_p$ denotes the subterm of $M$ at position $p$. For a set $\Gamma$ of terms, the notion of subterm can be extended as usual: $st(\Gamma):= \bigcup_{M\in \Gamma}st(M)$. For $p \in \mathcal{P}os(M)$, we denote by $M[t]_p$ the term that is obtained from $M$ by replacing the subterm at position $p$ by $t$.
A term rewriting system (TRS) is a set $\mathcal{R}$ of oriented equations over terms in a given signature. For terms $s$ and $t$, $s\rightarrow_{\mathcal{R}} t$ denotes that $s$ rewrites to $t$ using an instance of a rewriting rule in $\mathcal{R}$. The transitive, reflexive-transitive and equivalence closures of $\rightarrow_{\mathcal{R}}$ are denoted by $\stackrel{+}{\rightarrow}_{\mathcal{R}}, \stackrel{*}{\rightarrow}_{\mathcal{R}}$ and $\stackrel{*}{\leftrightarrow}_{\mathcal{R}}$, respectively. The equivalence closure of the rewriting relation, $\stackrel{*}{\leftrightarrow}_{\mathcal{R}}$, is denoted by $\approx_{\mathcal{R}}$. Given a TRS $\mathcal{R}$ in which some function symbols are assumed to be AC, and two terms $s$ and $t$, $s\rightarrow_{\mathcal{R}\cup AC}t$ if there exists $w$ such that $s=_{AC}w$ and $w\rightarrow_{\mathcal{R}} t$, where $=_{AC}$ denotes equality modulo AC (according to the AC assumption on function symbols). For every term $s$, the set of normal forms $s\downarrow_{\mathcal{R}}$ (closed modulo AC) of $s$ is the set of terms $t$ such that $s\stackrel{*}{\rightarrow}_{\mathcal{R}\cup AC}t$ and $t$ is irreducible for $\rightarrow_{\mathcal{R}\cup AC}$. $\mathcal{R}$ is said to be AC-convergent whenever it is AC-terminating and AC-confluent.
We equip the signature $\Sigma$ with an equational theory $\approx_E$ induced by a set of $\Sigma$-equations $E$, that is, $\approx_E $ is the smallest equivalence relation that contains $E$ and is closed under substitutions and compatible with $\Sigma$-contexts. An equational theory $\approx_E$ is said to be equivalent to a TRS $\mathcal{R}$ whenever $\approx_{\mathcal{R}}\; =\; \approx_E$. An equational theory $\approx_E$ is AC-convergent when it has an equivalent rewrite system $\mathcal{R}$ which is AC-convergent. In the next sections, given an AC-convergent equational theory $\approx_E$, normal forms of terms are computed with respect to the TRS $\mathcal{R}$ associated to $\approx_E$, unless otherwise specified. To simplify the notation we will denote by $E$ the equational theory induced by the set of $\Sigma$-equations $E$. We will denote by $\Sigma_E$ the signature used in the set of equations $E$. The *size* $c_E$ of an equational theory $E$ with an associated TRS $\mathcal{R}$ consisting of rules $\bigcup_{i=1}^k\{l_i\rightarrow r_i\}$ is defined as $c_E=max_{1\leq i\leq k}\{|l_i|,|r_i|, ar(\Sigma)+1\}$. For $\mathcal{R}= \emptyset$, define $ c_E= ar(\Sigma)+1$.
Let $\square$ be a new symbol which does not yet occur in $\Sigma\cup X$. A $\Sigma$-*context* is a term $t\in T(\Sigma, X\cup \{ \square\})$ and can be seen as a term with “holes”, represented by $\square$, in it. Contexts are denoted by $C$. If $\{p_1,\ldots,p_n\}=\{p\in\mathcal{P}os(C) \,|\, C|_p = \square\}$, where $p_i$ is to the left of $p_{i+1}$ in the tree representation of $C$, then $C[T_1\ldots, T_n]:= C[T_1]_{p_1}\ldots [T_n]_{p_n}$. In what follows a context formed using only function symbols in $\Sigma_{E}$ will be called an $E$-*context* to emphasize the equational theory $E$.
A term $M$ is said to be an $E$-*alien* if $M$ is headed by a symbol $f\notin \Sigma_{E}$ or a private name/constant. We write $M==N$ to denote syntactic equality of ground terms.
In the rest of the paper, we use signatures, terms and equational theories to model protocols. *Messages* exchanged between participants of a protocol during its execution are represented by terms. Equational theories and rewriting systems are used to model the cryptographic primitives in the protocol and the algebraic capabilities of an intruder.
Deduction Problem {#sec:locallystable}
=================
Given a set $\Gamma$ that represents the information available to an attacker, we may ask whether a given ground term $M$ may be deduced from $\Gamma$ using equational reasoning. This relation is written $\Gamma \vdash M$ and axiomatised in a natural deduction like system of inference rules.
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Locally Stable Theories
-----------------------
Let $\oplus$ be an arbitrary function symbol in $\Sigma_E$ for an equational theory $E$. We write $\alpha \cdot_{\oplus} M$ for the term $M\oplus \ldots \oplus M$, $\alpha$ times ($\alpha \in \mathbb{N}$). Given a set $S$ of terms, we write $sum_{\oplus}(S)$ for the set of arbitrary sums of terms in $S$, closed modulo $AC$: $$sum_{\oplus}(S)=\{(\alpha_1 \cdot_{\oplus}T_1)\oplus \ldots\oplus(\alpha_n \cdot_{\oplus}T_n)\,|\, \alpha_i \geq 0, T_i\in S\}$$ Define $sum(S)= \bigcup_{i=1}^k sum_{\oplus_i}(S)$, where $\oplus_1,\ldots, \oplus_k$ are the AC-symbols of the theory.
For a rule $l\rightarrow r\in \mathcal{R}$ and a substitution $\theta$ such that
- either there exists a term $s_1$ such that $s=_{AC}s_1$, $s_1=_{AC}l\theta$ and $t=r\theta$;
- or there exist terms $s_1$ and $s_2$ such that $s=_{AC}s_1 \oplus s_2$, $s_1=_{AC}l\theta$ and $t=_{AC}r\theta \oplus s_2$.
we write $s\stackrel{h}{\rightarrow}t$ and say that the reduction occurs in the head.
As in [@AbadiCo2006] we associate with each set $\Gamma$ of messages, a set of subterms in $\Gamma$ that may be deduced from $\Gamma$ by applying only “small” contexts. The concept of small is arbitrary — in the definition below, we have bound the size of an $E$-context $C$ by $c_E$ and the size of $C'$ by $c_E^2$, but other bounds may be suitable. Notice that limiting the size of an $E$-context by $c_E$ makes the context big enough to be an instance of any of the rules in the TRS $\mathcal{R}$ associated to $E$.
\[locallystable\] An AC-convergent equational theory $E$ is *locally stable* if, for every finite set $\Gamma=\{M_1, \ldots,M_n\}$, where the terms $M_i$ are ground and in normal form, there exists a finite and computable set $sat(\Gamma)$, closed modulo $AC$, such that
1. $M_1, \ldots, M_n \in sat(\Gamma)$;\[rule1\]
2. if $M_1,\ldots,M_k \in sat(\Gamma)$ and $f(M_1,\ldots,M_k)\in st(sat(\Gamma))$ then $f(M_1,\ldots,M_k)\in sat(\Gamma)$, for $f\in \Sigma_E$;\[rule2\]
3. if $C[S_1, \ldots,S_l]\stackrel{h}{\rightarrow}M$, where $C$ is an $E$-context such that $|C|\leq c_{E}$, and $S_1, \ldots, S_l \in$ $sum_{\oplus}(sat(\Gamma))$, for some $AC$ symbol $\oplus$, then there exist an $ E$-context $C'$, a term $M'$, and terms $S_1', \ldots, S_k' \in sum_{\oplus}(sat(\Gamma))$, such that $|C'|\leq c_{E}^2$, and $M\stackrel{*}{\rightarrow}_{\mathcal{R}\cup AC}M'=_{AC}C'[S_1', \ldots, S_k']$;\[rule3\]
4. if $M\in sat(\Gamma)$ then $M\downarrow\in sat(\Gamma)$.\[rule4\]
5. if $M\in sat(\Gamma)$ then $\Gamma \vdash M$.\[rule5\]
Notice that the set $sat(\Gamma)$ may not be unique. Any set $sat(\Gamma)$ satisfying the five conditions is adequate for the results.
The addition of rule 4 in the Definition \[locallystable\] is necessary to prove case 1b of Lemma \[lemma:epcloseness\], where the rewriting reduction occurs in a term $M_i\in sat(\Gamma)$ in a position different from the “head”. Normal forms are strictly necessary in the set $sat(\Gamma)$, they are essential to lift the applications of rewriting rules in the head of “small” contexts to applications of rewriting rules in arbitrary positions of “small” contexts. With this additional condition, Lemma 11 in [@AbadiCo2006] can also be proved. This fact was confirmed via personal communication with the second author of [@AbadiCo2006].
The lemma and the corollary below, adapted from [@AbadiCo2006], are used in the proof of Theorem \[theorem:epdecidability\].
\[lemma:epcloseness\] Let $E$ be a locally stable theory and $\Gamma=\{M_1,\ldots,M_n\}$ a set of ground terms in normal form. For every $E$-context $C_1$, for every $M_i \in sat(\Gamma)$, for every term $T$ such that $C_1[M_1,\ldots,M_k]\rightarrow_{\mathcal{R}\cup AC} T$, there exist an $E$-context $C_2$, and terms $M_i' \in sat(\Gamma)$, such that $T \stackrel{*}{\rightarrow}_{\mathcal{R}\cup AC} C_2[M_1', \ldots, M_l']$.
Suppose that $C_1[M_1,\ldots,M_k]\rightarrow_{AC}T$, for an $E$-context $C_1$ and $M_i\in\,sat(\Gamma)$. The proof is divided in two cases:
1. The reduction happens inside one of the terms $M_i$:
1. if $M_i\stackrel{h}{\rightarrow}M_i'$ then by definition of $sat(\Gamma)$ (since $E$ is locally stable), there exist an $E$-context $C$ such that $|C|\leq c_E^2$ and $M_i'\stackrel{*}{\rightarrow}C[S_1,\ldots,S_l]$ where $S_j\in sum_{\oplus}(sat(\Gamma))$.
Each $S_j\in sum_{\oplus}(sat(\Gamma))$ is of the form $S_j=(\alpha_1\cdot_{\oplus}M_{j_1})\oplus \ldots\oplus (\alpha_n \cdot_{\oplus}M_{j_n}),$ for $M_{j_k}\in sat(\Gamma)$. That is, $S_j=C_j[M_{j_1},\ldots, M_{j_k}]$, for $1\leq j\leq l$. Therefore, $$\begin{split}
C_1[M_1,\ldots, M_i, \ldots,M_k]\stackrel{h}{\rightarrow}C_1[M_1,\ldots, M'_i, \ldots,M_k]&\stackrel{*}{\rightarrow}_{AC}C_1[M_1,\ldots, C[S_1,\ldots, S_l], \ldots,M_k]\\
&=_{AC}C_2[M_1^{''},\ldots, M_s^{''}],
\end{split}$$ where $M_t^{''} \in sat(\Gamma)$, for $1\leq t\leq s$.
2. if $M_i\rightarrow_{AC}M_i'$ in a position different from “head”, then \[case:correction\] $$C_1[M_1,\ldots, M_i, \ldots,M_k]\rightarrow C_1[M_1,\ldots, M'_i, \ldots,M_k]\stackrel{*}{\rightarrow}_{AC}C_1[M_1,\ldots, M_i\downarrow, \ldots,M_k].$$ By case 4 in Definition \[locallystable\], $M_i\downarrow \in sat(\Gamma)$.
2. The case where the reduction does not occur inside the terms $M_i$: this case if very technical and will be omitted here. The complete proof can be found in the extended version of this paper.
As a consequence we obtain the following Corollary:
\[corollary:epcloseness\] Let $E$ be a locally stable theory. Let $\Gamma=\{M_1,\ldots,M_n\}$ be a set of ground terms in normal form. For every $E$-context $C_1$, for every $M'_i\in sat(\Gamma)$, for every $T$ in normal form such that $C_1[M'_1,\ldots,M'_k]\stackrel{*}{\rightarrow}_{\mathcal{R}\cup AC}T$, there exist an $E$-context $C_2$ and terms $M_j{''}\in sat(\Gamma)$ such that $T=_{AC}C_2[M^{''}_1,\ldots,M_l^{''}]$.
The proof is the same as in [@AbadiCo2006].
In the following we show that any term $M$ deducible from $\Gamma$ is equal modulo AC to an $E$-context over terms in $sat(\Gamma)$.
\[lemma:deductioncontext\] Let $E$ be a locally stable theory. Let $\Gamma =\{M_1,\ldots,M_n\}$ be a finite set of ground terms in normal form, and $M$ be a ground term in normal form. Then $\Gamma \vdash M$ if and only if there exist an $E$-context $C$ and terms $M'_1, \ldots,M'_k\in sat(\Gamma)$ such that $M=_{AC}C[M^{'}_1,\ldots,M^{'}_n]$.
The proof is the same as in [@AbadiCo2006].
As a consequence of the previous results decidability of IDP for locally stable theories is obtained:
\[theo:IDPdecidable\] The Intruder Deduction Problem is decidable for locally stable theories.
In the next section we will provide a complexity bound for the decidability of the intruder deduction problem for a restricted case of locally stable theories.
Locally Stable Theories with Inverses
=====================================
In order to obtain the polynomial complexity bound of our decidability algorithm we will need to consider the existence of inverses for each $AC$ symbol in the signature of our equational theory. Our algorithm will rely on solving systems of linear Diophantine equations over $\mathbb{Z}$ and the inverses will be interpreted as *negative integers*.
(\*) *In the following results, let $E$ be a locally stable theory whose signature $\Sigma_E$ contains, for each $AC$ function symbol $\oplus$, its corresponding *inverse* $i_{\oplus}$.*
That is, the following results are related to equational theories $E$ containing the following equation: $$x\oplus i_{\oplus} (x) = e_{\oplus}$$ for each AC-symbol $\oplus$ in $\Sigma_E$, where $i_{\oplus}$ is the unary function symbol representing the inverse of $\oplus$ and $e_{\oplus}$ is the corresponding neutral element.
\[def:locallystableinverses\] An AC-convergent equational theory $E$ satisfying (\*) is *locally stable* if, for every finite set $\Gamma=\{M_1, \ldots,M_n\}$, where the terms $M_i$ are ground and in normal form, there exists a finite and computable set $sat(\Gamma)$, closed modulo $AC$, such that
1. $M_1, \ldots, M_n \in sat(\Gamma)$, $e_{\oplus} \in sat(\Gamma)$ for each $\oplus \in \Sigma_{E}$;
2. if $M_1,\ldots,M_k \in sat(\Gamma)$ and $f(M_1,\ldots,M_k)\in st(sat(\Gamma))$ then $f(M_1,\ldots,M_k)\in sat(\Gamma)$, for $f\in \Sigma_E$;
3. if $C[S_1, \ldots,S_l]\stackrel{h}{\rightarrow}M$, where $C$ is an $E$-context such that $|C|\leq c_{E}$, and $S_1, \ldots, S_l \in sum_{\oplus}(sat(\Gamma))$, for some $AC$ symbol $\oplus$, then there exist an $ E$-context $C'$, a term $M'$, and terms $S_1', \ldots, S_k' \in sum_{\oplus}(sat(\Gamma))$, such that $|C'|\leq c_{E}^2$, and $M\stackrel{*}{\rightarrow}_{\mathcal{R}\cup AC}M'=_{AC}C'[S_1', \ldots, S_k']$;
4. if $M\in sat(\Gamma)$ then $M\downarrow\in sat(\Gamma)$.
5. if $M\in sat(\Gamma)$ then $i_{\oplus}(M)\downarrow\in sat(\Gamma)$ for each AC symbol $\oplus$ in $E$.
6. if $M\in sat(\Gamma)$ then $\Gamma \vdash M$.
Based on a well-founded ordering over the symbols in the language, we prove that a restricted case of higher-order AC-matching (“is there an $E$-context $C$ such that $M=_{AC}C[M_1,\ldots,M_k]$ for some $M_1,\ldots,M_k\in sat(\Gamma)$?”) can be solved in polynomial time in $|sat(\Gamma)|$ and $|M|$. This AC-matching problem is solved using the DO-ACM (Distinct-Occurrences of AC-matching) [@narendran], where every variable in the term being matched occurs only once. In addition, we also use a standard and polynomial time algorithm for solving SLDE over $\mathbb{Z}$ [@BouConDe; @ClauFor; @frumkin; @papadimitriou].
To facilitate the description of the algorithm below we have considered only one AC-symbol $\oplus$ whose corresponding inverse will be denoted by $i$. The proof can be extended similarly for theories with multiple AC-symbols each one with its corresponding inverse.
\[lemma:acmatching\] Let $E$ be a locally stable theory satisfying (\*), $\Gamma=\{M_1,\ldots,M_n\}$ a finite set of ground messages in normal form and $M$ a ground term in normal form. Then the question of whether there exists an $E$-context $C$ and $T_1,\ldots,T_k\in sat(\Gamma)$ such that $M=_{AC}C[T_1,\ldots,T_k]$ is decidable in polynomial time in $|M|$ and $|sat(\Gamma)|$.
Given $\Gamma$, we construct the set $sat(\Gamma)=\{T_1,\ldots, T_s\}$, which is computable and finite by Definition \[locallystable\]. We can then check whether $M=^?_{AC}C[T_1,\ldots,T_k]$ for some $E$-context $C$ and terms $T_1,\ldots,T_k\in sat(\Gamma)$ using the following algorithm which is divided in its main component A), and procedures B) and C) for reducing linear Diophantine equations and selecting $T_i$’s from $sat(\Gamma)$, respectively.
A\) **Algorithm 1.**
1. For all positions $p$ in $M$ headed by $\oplus$ starting from the longest positions in decreasing order (positions seen as sequences) solve the *system of linear Diophantine equations* (see part B below) for $M|_p$ with $sat(\Gamma)\cup S$, where $S$ is built incrementally from $sat(\Gamma)$, starting with $S_0=\emptyset$, including all $M|_p$ that have solutions. In other words:
Let $\mathcal{P}'=\{p_1,\ldots,p_t\}$ be the set of positions of $M$ such that $M|_p$ is headed with $\oplus$, organised in decreasing order. For each $p_j \in \mathcal{P}'$ let $M|_{p_j}$ be the subterm of $M$ such that $$M|_{p_j}=n_{j_1}\oplus \ldots \oplus n_{j_{kj}}\,\, (j=1,\ldots, t)$$
Recursively find, but suppressing step 1 in this recursive call, solutions for the arguments $n_{j_{i_1}},\ldots, n_{j_{i_l}}$ of $M|_{p_j}$ with $n_{j_{im}} \in \{n_{j_1},\ldots,n_{j_{k_j}} \}$ with respective $E$-contexts $C_{j_{i_1}},\ldots, C_{j_{i_l}}$ such that $$n_{j_{i_m}}=C_{j_{i_m}}[T_1,\ldots,T_{s_{i_m}}]$$ where $T_q \in sat(\Gamma)\cup S_{j-1}$, $q = 1,\ldots, s_{i_m}$.
Then one checks satisfiability of the SLDE generated from $M|_{p_j}$ and $sat(\Gamma)\cup S_{j-1} \cup \{n_{j_{i_1}},\ldots, n_{j_{k_l}}\}$ (see steps B and C).
If there is a solution then $S_j:= S_{j-1} \cup \{n_{j_{i_1}},\ldots, n_{j_{k_l}}\}\cup \{M|_{p_j}\}$
2. Let $S:= S_t$. Classify the terms in $sat(\Gamma)\cup S$ by size.
3. For each term $T_i \in sat(\Gamma) \cup S$ (from terms of maximal size to terms of minimal size) check:
- For each position $q\in \mathcal{P}os(M)$ such that $T_i=_{AC}M|_q$ do
Check whether the path between $T_i$ and the root of $M$ contains a $\oplus$:
- if NOT, then delete $M|_q$ from $M $ and move to $T_{i+1}$.
- if YES (there is a $\oplus$) then $M$ has a subterm $ N$ such that $N=n_1\oplus \ldots \oplus n_j[T_i]\oplus \ldots \oplus n_k$ and $N$ cannot be constructed from $sat(\Gamma)\cup S$. Therefore, $M$ cannot be written as an $E$-context with terms from $sat(\Gamma)$.
4. Check whether the remaining part of $M$ still contains $E$-aliens. If it is not the case, we have found an $E$-context $C$ and terms $M_1,\ldots,M_k \in sat(\Gamma)$ and $M=_{AC}C[M_1,\ldots,M_k]$; otherwise such an $E$-context does not exist.
B\) **Reduction to linear Diophantine equations.**
First, notice that, for each position $p$ such that $M|_p$ is headed with $\oplus$ we have $$\label{Eq:eq1}
M|_p=\alpha_1 m_1\oplus \ldots \oplus \alpha_r m_r\, , \, \alpha_j \in \mathbb{N}$$ where $m_j$ is not headed with $\oplus$ and $\alpha_jm_j$ counts for $\underbrace{m_j\oplus \ldots \oplus m_j}_{\alpha_j-times}$.
We want to prove that there are $\beta_1,\ldots,\beta_q \in \mathbb{N}$ such that $$\label{Eq:eq2}
\beta_1T_1\oplus \ldots \oplus \beta_qT_q =_{AC} M|_p=\alpha_1 m_1\oplus \ldots \oplus \alpha_r m_r$$ This AC-equality is only possible when $T_i=\gamma_{1i}m_1\oplus \ldots\oplus \gamma_{ri}m_r $ for each $i$, $1\leq i \leq q\leq s$ and $\gamma_{j_i}\in \mathbb{N}$.
That is, $\beta_1T_1\oplus \ldots \oplus \beta_qT_q=_{AC}\alpha_1 m_1\oplus \ldots \oplus \alpha_r m_r$ if and only if $$\begin{split}
&\beta_1(\gamma_{1_1}m_1\oplus \ldots\oplus \gamma_{r_1}m_r)\oplus \beta_2(\gamma_{1_2}m_1\oplus \ldots\oplus \gamma_{r_2}m_r)\oplus\ldots \\
&\ldots\oplus \beta_q(\gamma_{1_q}m_1\oplus \ldots\oplus \gamma_{r_q}m_r)=\alpha_1 m_1\oplus \ldots \oplus \alpha_r m_r
\end{split}$$
if and only if $$\begin{split}
&(\gamma_{1_1}\beta_1 \oplus \gamma_{1_2}\beta_2 \ldots\oplus \gamma_{1_q}\beta_q)m_1\oplus (\gamma_{2_1}\beta_1 \oplus \gamma_{2_2}\beta_2 \ldots\oplus \gamma_{2_q}\beta_q)m_2\oplus \ldots\\
&\ldots(\gamma_{r_1}\beta_1 \oplus \gamma_{r_2}\beta_2 \ldots\oplus \gamma_{r_q}\beta_q)m_r=\alpha_1 m_1\oplus \ldots \oplus \alpha_r m_r
\end{split}$$ if and only if
$$S=\left\{
\begin{split}
\gamma_{1_1}\beta_1 \oplus \gamma_{1_2}\beta_2 \ldots\oplus \gamma_{1_q}\beta_q&=\alpha_1\\
\gamma_{2_1}\beta_1 \oplus \gamma_{2_2}\beta_2 \ldots\oplus \gamma_{2_q}\beta_q&=\alpha_2\\
\vdots \hspace{1cm} &\\
\gamma_{r_1}\beta_1 \oplus \gamma_{r_2}\beta_2 \ldots\oplus \gamma_{r_q}\beta_q&=\alpha_r\\
\end{split}
\right.$$
where $S$ is a system of linear Diophantine equations over $\mathbb{Z}$ which can be solved in polynomial time [@BouConDe; @ClauFor; @frumkin; @papadimitriou].
We will interpret the equations \[Eq:eq1\] and \[Eq:eq2\] inside integer arithmetic. If there exists an index $j$ such that $m_j= i(m_j')$ and $m_j' $ is not headed with $i$ then $\alpha_j m_j=\alpha_j (i(m_j'))$ and we will take it as $(-\alpha_j)m_j'$. Therefore, we can take $\alpha_j \in \mathbb{Z}$, for all $j$. We can use the same reasoning to conclude that $\beta_j \in \mathbb{Z}$, for all $1\leq j\leq q$ and $\gamma_{j_i} \in \mathbb{Z}$, for all $i$ and $j$.
C\) **Selecting the $T_j's$ from $sat(\Gamma)$.**
For each $T_i \in sat(\Gamma)$, $1\leq i\leq s$ we want to check if $T_i=\gamma_{1_i}m_1\oplus \ldots\oplus \gamma_{r_i}m_r $.
**Algorithm 2:**
For each $T_i \in sat(\Gamma)$, $1\leq i \leq s$, solve the equation $T_i \oplus x_i =_{AC} \alpha_{1}m_1 \oplus \ldots\oplus \alpha_{r}m_r $ where $x_i$ is a fresh variable.
Since the $T_i's$ and $M$ are ground terms, this equation can be seen as an instance of the DO-ACM matching problem which can be solved in time $\mathcal{O}(|T_i\oplus x_i|.|M|_p|)$ [@narendran]. If there exists $T_i \in sat(\Gamma)$ such that $T_i=\gamma_{1_i}^*m_1\oplus \ldots\oplus \gamma_{r_i}^*m_r\oplus u $, where $u$ is not empty, $\gamma_{i_j}^*\in \mathbb{N} $ and the **Algorithm 2** can no longer be applied then $T_i$ will not be selected.
Notice that each step of the algorithm can be done in polynomial time in $|M|$ and $|sat(\Gamma)|$. Therefore, the whole procedure is polynomial in $|M|$ and $sat(\Gamma)$.
For the proof we can adopt an ordering in which, for instance, variables are smaller than constants, constants smaller than function symbols, and function symbols are also ordered, but other suitable order can be used. Terms are compared by the associated lexicographical ordering built from this ordering on symbols.
We consider the theory of Abelian Groups where the signature is $\Sigma_{AG}=\{+,0,i\}$ for $i$ the inverse function and $+$ the AC group operator. The equational theory $E_{AG}$ is: $$E_{AG}=\left\{
\begin{array}{l@{\hspace{1cm}}c @{\hspace{1cm}}r}
\begin{array}{rcl}
x+(y+z)&=&(x+y)+z\\
x+y&=&y+x\\
i(x+y)&=&i(y)+i(x)\\
\end{array}
&
\begin{array}{rcl}
x+0&=&x\\
x+i(x)&=&0
\end{array}
&
\begin{array}{rcl}
i(i(x))&=&x\\
i(0)&=&0
\end{array}
\end{array}
\right.$$ We define $\mathcal{R}_{AG} $ by orienting the equations from left to right (excluding the equations for associativity and commutativity). $\mathcal{R}_{AG}$ is AC-convergent. The size $c_{E_{AG}}$ of the theory is at least 5. In the following prove that $E_{AG}$ is locally stable with inverses for finite models, i.e., we define a set $sat(\Gamma)$ satisfying the properties in the Definition \[locallystable\]. For a given set $\Gamma=\{M_1,\ldots,M_k\}$ of ground terms in normal form, $sat(\Gamma)$ is the smallest set such that:
1. $M_1,\ldots,M_k\in sat(\Gamma)$;
2. $M_1,\ldots,M_k\in sat(\Gamma)$ and $f(M_1,\ldots,M_k)\in st(sat(\Gamma))$ then $f(M_1,\ldots,M_k)\in sat(\Gamma)$, $f\in \Sigma_{AG}$;
3. if $M_i,M_j \in sat(\Gamma)$ and $M_i+M_j\stackrel{h}{\rightarrow}M$ via rule $x+i(x)\rightarrow 0$ then $M\downarrow\in sat(\Gamma)$;
4. if $M_j \in sat(\Gamma)$ then $i(M_j)\downarrow \in sat(\Gamma)$;
5. if $M_i=_{AC}M_j$ and $M_i \in sat(\Gamma)$ then $M_j \in sat(\Gamma).$
The set $sat(\Gamma)$ defined for Finite Abelian Groups is finite.
Although it was said in [@AbadiCo2006] that the theory of Abelian Groups is locally stable, no proof of such fact was found in the literature. With the proviso that the Abelian Group under consideration is finite, we have demonstrated that $|sat(\Gamma)|$ is exponential in the size of $|\Gamma|$.
These results give rise to the decidability of deduction for locally stable theories. Notice that polynomiality on $|sat(\Gamma)|$ relies on the use of the AC-matching algorithm proposed in Lemma \[lemma:acmatching\]. Unlike [@AbadiCo2006], we do not need to compute of the congruence class modulo AC of $M$ (which may be exponential). This gives us a slightly different version of the decidability theorem:
\[theorem:epdecidability\] Let $E$ be a locally stable theory satisfying (\*). If $\Gamma=\{M_1,\ldots,M_n\}$ is a finite set of ground terms in normal form and $M$ is a ground term in normal form, then $\Gamma\vdash M$ is decidable in polynomial time in $|M|$ and $|sat(\Gamma)|$.
The result follows directly from Lemmas \[lemma:acmatching\] and \[lemma:deductioncontext\].
In the following example we consider the *Pure AC-theory* which can be proven to be locally stable but does not contain the inverse of the AC-symbol $+$.
$\Sigma_{AC}$ contains only constant symbols, the AC-symbol $\oplus$ and the equational theory contains only the $AC$ equations for $\oplus$: $$AC=\left\{
\begin{array}{l@{\hspace{3cm}}r}
x\oplus y= y\oplus x
&
x\oplus(y\oplus z) =(x\oplus y)\oplus z
\end{array}
\right\}$$ In this case, $E=AC$ and $ \mathcal{R}=\emptyset$ is the AC-convergent TRS associated to $E$. Let $\Gamma=\{M_1,\ldots,M_k\}$ be a finite set of ground terms in normal form. Let us define $sat(\Gamma)$ for the pure $AC$ theory as the smallest set such that
1. $M_1,\ldots, M_k\in sat(\Gamma)$;
2. if $M_i,M_j\in sat(\Gamma)$ and $M_i\oplus M_j \in st(sat(\Gamma))$ then $M_i\oplus M_j\in sat(\Gamma)$.
3. if $M_i=_{AC}M_j$ and $M_i\in sat(\Gamma)$ then $M_j\in sat(\Gamma)$.
The set $sat(\Gamma)$ is finite since we add only terms whose size is smaller or equal than the maximal size of the terms in $\Gamma$. It is easy to see that the set $sat(\Gamma)$ satisfies the rules \[rule1\],\[rule2\], \[rule4\] and \[rule5\]. Since $\mathcal{R}=\emptyset$ it follows that \[rule3\] is also satisfied. Therefore, $AC$ is locally stable.
*The size of $sat(\Gamma)$:*
- Steps 1 and 2: only subterms in $sat(\Gamma)$ are added.
- Step 3: for each $M_i\in sat(\Gamma)$ add $M_j=_{AC}M_i\in sat(\Gamma)$. Notice that the number of terms added in $sat(\Gamma)$, in this case, depends on the number of occurrences of $\oplus$ in $M_i$. Suppose that $M_i$ contains $n$ occurrences of $\oplus$: $$M_i=M_{i_1}\oplus \ldots \oplus M_{i_{n+1}}.$$ There are $(n+1)!$ terms $M_j$ such that $M_1=_{AC}M_j$.
Suppose that each $M_i$ in $\Gamma$ contains $n_i$ occurrences of $\oplus$.Then, $|M_i|=\displaystyle\sum_{j=1}^{n_i+1}|M_{i_j}|+n_i.$ Let $n=\max_{1\leq i\leq k}\{n_i\}$. There exists an index $r$ such that $M_r$ contains $n_r=n$ occurrences of $\oplus$. Since $|\Gamma|=\displaystyle\sum_{i=1}^k|M_i|$ it follows that $ n \leq |M_r|-\displaystyle\sum_{j=1}^{n+1}|M_{r_j}|\leq |\Gamma|.$ Then the number of terms added in step 3 is $\displaystyle\sum_{i=1}^k (n_i+1)!\leq (n+1)! \cdot k \leq (|\Gamma|+1)!\cdot k .$
In this case one can adapt Lemma \[lemma:acmatching\] such that the algorithm would rely on solving systems of linear Diophantine equations over $\mathbb{N}$ which is NP-complete [@papadimitriou]. Therefore, the complexity of IDP for pure AC would be exponential, agreeing with previous results [@lafourcade].
Elementary Deduction Problem for Locally Stable Theories
========================================================
To establish necessary concepts for the next results, we recall the well-known translation between natural deduction and sequent calculus systems to model the IDP as a proof search in sequent calculus, whose properties (such as cut or subformula) facilitate the study of decidability of deductive systems. For an AC-convergent equational theory E, the System $\mathcal{N}$ in Table \[equationalreasoning\] is equivalent to the $(id)$-rule of the sequent calculus (Table \[DeductionRulesForIntruder\]) introduced in [@TiGo2009]:
Consequently, IDP for System $\mathcal{N}$ is equivalent to the *Elementary Deduction Problem*:
Given an AC-convergent equational theory $E$ and a sequent $\Gamma \vdash M$ ground and in normal form, the *elementary deduction problem* (EDP) for $E$, written $\Gamma \Vdash_{E}M$, is the problem of deciding whether the $(id)$-rule is applicable in $\Gamma\vdash M$.
The theorem below decides EDP for locally stable theories :
\[theorem:edpisptime\] Let $E$ be a locally stable equational theory satisfying (\*). Let $\Gamma \vdash M$ be a ground sequent in normal form. The *elementary deduction problem* for the theory $E$ ($\Gamma \Vdash_E M$) is decidable in polynomial time in $|sat(\Gamma)|$ and $|M|$.
By Lemma \[lemma:acmatching\], the problem whether $M=_{AC}C[M_1,\ldots,M_k]$ for an $E$-context $C$ and terms $M_1,\ldots,M_k\in sat(\Gamma)$ is decidable in polynomial time in $|sat(\Gamma)|$ and $|M|$. If $M=_{AC}C[M_1,\ldots,M_k]$ for an $E$-context $C$ and terms $M_1,\ldots,M_k\in sat(\Gamma)$ then there exist an $ E$-context $C'$ and terms $M'_1,\ldots,M'_n\in \Gamma$ such that $C[M'_1,\ldots,M'_n]\stackrel{*}{\rightarrow}_{\mathcal{R}\cup AC} M.$ It is enough to observe that for all $T\in sat(\Gamma)$, $T$ can be constructed from the terms in $\Gamma$.
If there is no $E$-context $C$ and terms $M_1,\ldots,M_k\in sat(\Gamma)$ such that $M=_{AC}C[M_1,\ldots,M_k]$ then, by Corollary \[corollary:epcloseness\], there are no $\textsf E$-context and terms $M'_1, \ldots, M'_t\in sat(\Gamma)$ such that $C[M'_1, \ldots, M'_t]\stackrel{*}{\rightarrow}_{\mathcal{R}\cup AC} M.$ Therefore, there is no $E$-context $C''$ and terms $M''_1,\ldots,M''_l\in \Gamma$ such that $C''[M''_1, \ldots, M''_l]\stackrel{*}{\rightarrow}_{\mathcal{R}\cup AC} M.$ Thus, the EDP for $E$ is decidable in polynomial time in $|sat(\Gamma)|$ and $|M|$.
Extension with Blind Signatures {#extension}
-------------------------------
Blind signature is a basic cryptographic primitive in e-cash. This concept was introduced by David Chaum in [@Chaum] to allow a bank (or anyone) sign messages without seeing them. David Chaum’s idea was to use this homomorphic property in such a way that Alice can multiply the original message with a random (encrypted) factor that will make the resulting image meaningless to the Bank. If the Bank agrees to sign this random-looking data and return it to Alice, she is able to divide out the blinding factor such that the Bank’s signature in the original message will appear.
Given a locally stable equational theory $E$, we extend the signature $\Sigma_{E}$ with $\Sigma_C$, a set containing function symbols for “constructors” for blind signatures, in order to obtain decidability results for the extension of the IDP for System $\mathcal{N}$ taking into account some rules for blind signatures.
### Extended Syntax {#extended-syntax .unnumbered}
The signature $\Sigma$ consists of function symbols and is defined by the union of two sets: $\Sigma =\Sigma_C \cup \Sigma_{E}$ ( with $\Sigma_{\textsf{E}}\cap \Sigma_C=\emptyset$), where $$\Sigma_C=\left\{\textsf{pub}(\_), \textsf{sign}(\_\; ,\_), \textsf{blind}(\_\; ,\_), \left\{\_\right\}_{\_}, <\_\; ,\_>\right\}$$ represents the *constructors*, whose interpretations are: $\textsf{pub}(M)$ gives the public key generated from a private key $M$; $\textsf{blind}(M,N)$ gives $M$ encrypted with $N$ using blinding encryption; $\textsf{sign}(M,N)$ gives $M$ signed with a private key $N$; $\left\{M\right\}_N$ gives $M$ encrypted with the key $N$ using Dolev-Yao symmetric encryption; $\langle M,N\rangle$ constructs a pair of terms from $M$ and $N$. Then the extended grammar of the set of *terms* or messages is given as $$M,N \;:= a \;|\; x \;| f(M_1,\ldots, M_n)|\textsf{pub}(M) | \textsf{sign}(M,N) | \textsf{blind}(M,N)|\left\{M\right\}_N|\langle M, N\rangle$$
Notice that, with the extension an $E$-alien term $M$ is a term headed with $f\in \Sigma_C$ or $M$ is a private name/constant. An $E$-alien subterm $M$ of $N$ is said to be an $E$-*factor* of $N$ if there is another subterm $F$ of $N$ such that $M$ is an immediate subterm of $F$ and $F$ is headed by a symbol $f\in\Sigma_{E}$. This notion can be extended to sets in the obvious way: a term $M$ is an $E$-factor of $\Gamma$ if it is an $E$-factor of a term in $\Gamma$. These notions were introduced in [@TiGo2009].
The operational meaning of each constructor will be defined by their corresponding inference rules in the sequent calculus to be described.
### Extending the EDP to Model Blind Signatures {#extending-the-edp-to-model-blind-signatures .unnumbered}
Following the approach proposed in [@TiGo2009], we extend EDP with blind signatures using the sequent calculus $\mathcal{S}$ described in Table \[DeductionRulesForIntruder\]. In this way, we can model intruder deduction for the combination of a locally stable theory $E$ with blind signatures in a modular way: the theory $E$ is used in the $id$ rule, while blind signatures are modelled with additional deduction rules. As shown below, this approach has the advantage that we can derive decidability results for the intruder deduction problem without needing to prove that the combined theory is locally stable (in contrast with the results in the previous section and in [@AbadiCo2006]).
------------------------------------------------------------------------
------------------------------------------------------------------------
Analysing the system $\mathcal{S}$ one can make the following observations:
1. The rules $p_L, e_L$,$ \textsf{sign}_L$,$ \textsf{blind}_{L1}$, $\textsf{blind}_{L2}$ and $acut$ are called *left rules* with $\langle M,N\rangle$, $\{M\}_K$, $\textsf{sign}(M,K)$, $\textsf{blind}(M,K)$, $\textsf{sign}(\textsf{blind}((M,R),K)$ and $A$ as *principal term*, respectively. The rules $p_R, e_R,\textsf{sign}_R$ and $ \textsf{blind}_{R}$ are called *right rules*.
2. The rule $(acut)$, called *analytic cut* is necessary to prove cut rule *admissibility*. A complete proof can be found in [@TiGo2009; @nantesayala].
Considerations about locally stable theories with blind signatures:
1. All the results proved on Section \[sec:locallystable\] are valid under this extension with blind signatures since the results depend only on the equational theory $E$ and on the symbols in $\Sigma_E$. Unlike example 5.2.4 [@AbadiCo2006], the theory of Blind Signatures is not considered as part of the equational theory, the functions are abstracted in the set of constructors with the operational meaning represented in the sequent calculus.
2. In [@TiGo2009] it is shown that the intruder deduction problem for $\mathcal{S}$ is *polynomially reducible* to the EDP for $E$: *if the EDP problem in $E$ has complexity $f(m)$ then the deduction problem $\Gamma\vdash M$ in $\mathcal{S}$ has complexity $O(n^k.f(n))$ for some constant $k$*[^5]. This result was proved for an AC-convergent equational theory $E$ containing only one $AC$ symbol and extended to finite a combination of disjoint AC-convergent equational theories each one containing only one AC-symbol.
3. In [@nantesayala], it was proved that deduction in $\mathcal{S}$ reduces polynomially to $EDP$ in the case of the AC-convergent equational theory ${\textsf{EP}}$, which contains three different AC-symbols and rules for exponentiation and cannot be split into disjoint parts.
As a consequence of the results mentioned in the above remark, we can state the following result:
\[theo:eppblind\] Let $E$ be a locally stable theory satisfying (\*) containing only one AC-symbol or formed by a finite and disjoint combination of AC-symbols. Let $\Gamma$ a finite set of ground terms in normal form and $M$ a ground term in normal form. The IDP for the theory $E$ combined with blind signatures ($\Gamma\vdash M$) is decidable in polynomial time in $|sat(\Gamma)|$ and $|M|$.
Conclusion
==========
We have shown that the IDP is decidable for locally stable theories. In order to obtain the polynomiality result, a restriction on the equational theory is necessary: the theory must contain inverses of all AC-symbols. We have proposed an algorithm to solve a restricted case of higher-order AC-matching by using the DO-ACM matching algorithm combined with an algorithm to solve linear Diophantine equations over $\mathbb{Z}$. Based on this algorithm, we obtain a polynomial decidability result for IDP for a class of locally stable theories with inverses. Our algorithm does not need to compute the set of normal forms modulo AC of a given term (which may be exponential). Therefore, we can conclude that the deducibility relation is decidable in polynomial time for a very restricted class of equational theories, it does not work for all locally stable theories as [@AbadiCo2006] has claimed. It also decides the IDP for the combination of locally stable theories with the theory of blind signatures, using a translation between natural deduction and sequent calculus.
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[^1]: Author partially supported by CNPq.
[^2]: Corresponding author. Author supported by CNPq
[^3]: Work supported by grants from the CNPq/CAPES *Science without Borders* programme and FAPDF PRONEX.
[^4]: With this simple modification, the correctness proof in [@AbadiCo2006] can also be carried out, fixing a gap in Lemma 11.
[^5]: Here, $m$ is the size of the input of EDP and $n$ is the cardinality of the set $St(\Gamma\cup \{M\})$ defined in [@TiGo2009]
|
---
abstract: 'Several BL Lac objects are confirmed sources of variable and strongly Doppler-boosted TeV emission produced in the nuclear portions of their relativistic jets. It is more than probable, that also many of the FR I radio galaxies, believed to be the parent population of BL Lacs, are TeV sources, for which Doppler-hidden nuclear $\gamma$-ray radiation may be only too weak to be directly observed. Here we show, however, that about one percent of the total time-averaged TeV radiation produced by the active nuclei of low-power FR I radio sources is inevitably absorbed and re-processed by photon-photon annihilation on the starlight photon field, and the following emission of the created and quickly isotropized electron-positron pairs. In the case of the radio galaxy Centaurus A, we found that the discussed mechanism can give a distinctive observable feature in the form of an isotropic $\gamma$-ray halo. It results from the electron-positron pairs injected to the interstellar medium of the inner parts of the elliptical host by the absorption process, and upscattering starlight radiation via the inverse-Compton process mainly to the GeV$-$TeV photon energy range. Such a galactic $\gamma$-ray halo is expected to possess a characteristic spectrum peaking at $\sim 0.1$ TeV photon energies, and the photon flux strong enough to be detected by modern Cherenkov Telescopes and, in the future, by GLAST. These findings should apply as well to the other nearby FR I sources.'
author:
- |
Ł. Stawarz$^{1, \, 2, \, 3,}$[^1], F. Aharonian$^2$, S. Wagner$^1$, and M. Ostrowski$^3$\
$^1$Landessternwarte Heidelberg, Königstuhl, D-69117 Heidelberg, Germany\
$^2$Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany\
$^3$Obserwatorium Astronomiczne, Uniwersytet Jagielloński, ul. Orla 171, 30-244 Kraków, Poland
title: 'Absorption of Nuclear Gamma-rays on the Starlight Radiation in FR I Sources: the Case of Centaurus A'
---
\[firstpage\]
radiation mechanisms: non-thermal — gamma-rays: theory — galaxies: active — galaxies: jets — galaxies: individual (Centaurus A)
Introduction
============
Centaurus A is the most nearby active galaxy[^2], hosted by a powerful elliptical [see a monograph by @isr98]. At low frequencies it reveals a giant ($8\deg \times 4\deg$ or $500$ kpc $\times$ $250$ kpc) and complex radio structure, with a total $5$ GHz energy flux of $3.4 \times 10^{-11}$ erg cm$^{-2}$ s$^{-1}$, containing a one-sided kpc-scale jet, few-kpc-long inner lobes, and extended outer lobes. The host galaxy is of $14' \times 18'$ optical size, has a blue luminosity of $7.5 \times 10^{10} \, L_{\odot}$, and contains a $\sim 10^8 \, M_{\odot}$ supermassive black hole in the center [@mar06]. The famous ‘dark lane’ pronounced within the galactic body, being in fact an edge-on disk of rotating gas, dust and young stars, is most probably a remnant of the merger with a spiral galaxy, which happened some $10^8 - 10^9$ years ago. The kpc-scale jet and the active nucleus are confirmed sources of X-ray emission [see most recent analysis by @har03; @kat06 and references therein].
Cen A source is classified as an FR I radio galaxy, although some of its morphological characteristics (double-double structure, one-sided jet) are not typical for this class of AGNs. FR I galaxies are believed to be a parent population of BL Lac objects [@urr95], some of which are confirmed sources of very high energy (VHE) $\gamma$-ray emission. In fact, the nuclear part of the Cen A jet was modeled in terms of a ‘misaligned’ (and therefore Doppler-hidden) BL Lac, since the jet viewing angle in this source is expected to be much larger than the typical inclination of the blazar jet [e.g., @bai86; @mor91; @bot93; @ste98; @chi01]. As such, Cen A was considered for some time as a potential source of TeV radiation, for which jet misalignment effects are compensated by the source’s proximity. This expectation was strengthened by the positive detection of the considered object at MeV$-$GeV energy range by all the instruments on board CGRO during the period 1991-1995 [@ste98 and references therein]. Nevertheless, the observations performed till now at the TeV energy range resulted in the upper limit for such an emission, only [@row99; @aha04; @aha05].
In this paper we investigate the emission resulting from re-procession of the potential $\gamma$-ray flux of the active nucleus in the Cen A radio galaxy via annihilation of the nuclear $\gamma$-rays on the starlight photon field due to the host galaxy, followed by the synchrotron and inverse-Compton cooling of the created electron-positron pairs. We show, that the analysis of this mechanism — which also applies to other FR I sources — can give us interesting constraints on the unknown parameters of the Doppler-hidden and heavily obscured Cen A nucleus, as well as of its host galaxy and outer parts of its radio outflow. We note that $\gamma$-ray opacity in blazar sources was considered before only in the context of nuclear target radiation fields, due to accretion discs, broad-line regions, or dusty tori [e.g., @sik94; @der94; @bla95; @wag95; @boe97; @wan00]. In the case of the FR I/BL Lac sources, however, such radiation fields are relatively weak, or even absent. Thus, the main source of the opacity for the TeV photons emitted from their active nuclei is the starlight radiation[^3]. This radiation can be modelled quite precisely, since all the low-power radio sources of the FR I/BL Lac type are hosted by giant elliptical galaxies [e.g., @col95; @urr00; @hei04], for which spectral and spatial distribution of the stellar emission is relatively well known [see in this context discussion in @sta05; @sta06a; @sta06b].
Below, in section 2, we present details regarding the calculation of the optical depth for photon-photon pair production by the Cen A nuclear $\gamma$-ray emission on the starlight photon field of the elliptical galaxy, and the energetics of the created electron-positron pairs. In section 3, we discuss further evolution of such particles injected into the interstellar medium of the elliptical host, calculating in particular the resulting synchrotron and inverse-Compton fluxes. Section 4 contains a final discussion and conclusions.
Calculations
============
Opacity
-------
In order to calculate the opacity for the $\gamma$-ray photon beam propagating through the galaxy, one has to specify the spectral and spatial distribution of the stellar photon field(s). Starlight surface brightness in elliptical galaxies, including those with active nuclei, are typicaly well fitted by an empirical Nuker law [@lau95]. In terms of monochromatic starlight intensity, this law can be expressed as $$I(r) = I_{\rm b} \, 2^{(b - d)/a} \, \left({r \over r_{\rm b}}\right)^{-d} \, \left[1+\left({r \over r_{\rm b}}\right)^{a}\right]^{-(b-d)/a} \, ,$$ where $r$ is the distance from the galactic nucleus, $r_{\rm b}$ is the ‘break’ radius, and $I_{\rm b} = I(r_{\rm b})$. Such profiles imply a power-law dependance $I(r) \propto r^{-d}$ for $r < r_{\rm b}$, and $I(r) \propto r^{-b}$ at larger distances. @rui05 found that in the case of weak radio galaxies (selected from the B2 sample with the criterium of not showing significant dust emission), the host galaxies are characterized by the typical values of $a = 1.9$, $b = 1.6$, and $d = 0.02$. In a specific case of the Cen A host galaxy, @cap05 give $a = 1.68$, $b = 1.3$, $d = 0.1$, and a break radius $r_{\rm b} = 2.56\arcsec = 41$ pc. These values are considered hereafter, allowing to approximate the starlight emissivity, $j(r) \propto r^{-1} \, I(r)$, in the NGC 5128 (Cen A host) galaxy, as $$\epsilon j_{\epsilon}(\xi) = j_{\rm V} \, g(\epsilon) \, h(\xi) \, ,$$ where $\xi \equiv r / r_{\rm b}$. Here $\epsilon \equiv \varepsilon / m_{\rm e} c^2$ is the starlight photon energy $\varepsilon$ in $m_{\rm e}c^2$ units, $j_{\rm V}$ is the total $V$-band galactic emissivity, $g(\epsilon)$ describes the $V$-band normalized spectral distribution of the stellar photon field, and $$h(\xi) = \left({r \over r_{\rm b}}\right)^{-1} \, {I(r) \over I_{\rm b}} = 1.64 \, \, \xi^{-1.1} \, \left(1+\xi^{1.68}\right)^{-0.7143}$$ is the radial depedance function. The considered galaxy is $14' \times 18'$ in optical size [@isr98], and hence we take the average terminal galactic radius of $16'$, or $\xi_{\rm t} \equiv r_{\rm t} / r_{\rm b} = 375$, leading to the galactic volume (${\cal V}$) integral $${\cal H} \equiv \int_{\cal V} h(\xi) \, d{\cal V} = 4 \pi \, r_{\rm b}^3 \int_0^{\xi_{\rm t}} \xi^2 \, h(\xi) \, d\xi = 1.83 \times 10^3 \, r_{\rm b}^3$$ (assuming spherical symmetry). For the spectral distribution of the NGC 5128 starlight, we assume the template spectrum of a powerful elliptical galaxy as provided by @sil98, restricted to the photon energy range between $\epsilon_{\rm min} = 10^{-7}$ and $\epsilon_{\rm max} = 10^{-5}$. This restriction implies that we consider only the direct stellar emission of the evolved red giants (constituting the main body of the elliptical host) and their winds, but not the far infrared emission resulting from the reprocession of the starlight photons by the cold galactic dust (see Appendix A). We normalize it to the $V$-band radiation, so that $g(\epsilon = h/m_{\rm e} c \lambda_{\rm V}) = 1$, where $\lambda_{\rm V} = 0.55$ $\mu$m. The resulting spectral function $g(\epsilon)$ is shown in Figure 1 (top-panel): values read directly from @sil98 are represented by crosses, and the broken power-law approximation considered hereafter, $$\begin{aligned}
g(\epsilon) = \left\{ \begin{array}{lll} 2.67 \times 10^{14} \, \epsilon^{2.44} & {\rm for} & 10^{-7.0} \leq \epsilon \leq 10^{-5.8} \\ 9.25 \times 10^{-4} \, \epsilon^{-0.57} & {\rm for} & 10^{-5.8} < \epsilon < 10^{-5.3} \\ 3.54 \times 10^{-31} \, \epsilon^{-5.74} & {\rm for} & 10^{-5.3} \leq \epsilon \leq 10^{-5.0} \end{array} \right. ,\end{aligned}$$ is shown as a solid line.
![[*Top:*]{} Spectral energy distribution of the starligh emission of a template giant elliptical; values read directly from @sil98 are represented by crosses, and the broken power-law approximation introduced in this paper is shown as a solid line. [*Bottom:*]{} Optical depth for annihilation of the nuclear $\gamma$-ray emission on the starlight photon field.](f1.eps)
The apparent $V$-band magnitude of NGC 5128 is $m_{\rm V} = 6.98$ [@isr98]. This gives the monochromatic $V$-band galactic luminosity $$\log \left({L_{\rm V} \over {\rm erg/s}}\right) = 50.078 + 2 \, \log \left({d_{\rm L} \over {\rm Mpc}}\right) -0.4 \, (m_{\rm V} - A_{\rm V}) - c_{\rm V} = 43.82 \quad ,$$ where the distance of Cen A is $d_{\rm L} = 3.4$ Mpc, the extinction is $A_{\rm V}=0.381$, and $c_{\rm V} = 4.68$. Since, by the definition, $$L_{\rm V} = 4 \pi \, \int_{\cal V} \left[\epsilon j_{\epsilon}(\xi)\right]_{\rm V} \, d{\cal V} = 4 \pi \, j_{\rm V} \, {\cal H} \quad ,$$ one obtains $j_{\rm V} = 1.42 \times 10^{-21}$ erg cm$^{-3}$ ster$^{-1}$ s$^{-1}$. Note, that by means of standard radiative transfer formulae, the spectral intensity within a solid angle $\Omega$, which can be obtained by integrating spectral emissivity along a ray as $I_{\nu} (\Omega) = \int \, j_{\nu}(\xi) \, dl$, is related to the differential photon number density, $n_{\epsilon}(\xi, \Omega)$, through the expression $I_{\nu}(\Omega) = c h \, \epsilon \, n_{\epsilon}(\xi, \Omega)$. This corresponds in fact to the neglegible absorption of the starlight along the light path, the approximation which may be considered as being in conflict with the presence of the dust lane in Cyg A host. It is however good enough for the purpose of the presented evaluation, since integration over the whole extended elliptical volume reduces the starlight absorption effects. As a result, $$n_{\epsilon}(\xi, \Omega) = {\epsilon^{-2} \over m_{\rm e} c^3} \, \int \, \left[\epsilon j_{\epsilon}(\xi)\right] \, dl \quad .$$
The optical depth for photon-photon annihilation, computed for the case of a monodirectional beam of $\gamma$-ray photons with a dimensionless energy $\epsilon_{\gamma}$, propagating through the stellar photon field of the host galaxy from the active center up to the terminal distance $r_{\rm t}$, is $$\tau(\epsilon_{\gamma}) = \int_0^{r_{\rm t}} dr \, \int \, dn \, (1-\varpi) \, \sigma_{\gamma \gamma} \quad ,$$ where $\varpi$ is the cos function of the angle between the $\gamma$-ray photon and the incident starlight photon, $dn = n_{\epsilon}(\xi, \Omega) \, d\epsilon \, d\Omega$ is the differential starlight photon number density, and $$\sigma_{\gamma \gamma}(\epsilon_{\gamma}, \epsilon, \varpi) = {3 \sigma_{\rm T} \over 16} \, (1 - \beta^2) \, \left[ (3 - \beta^4) \, \ln \left( {1 + \beta \over 1 - \beta} \right) - 2 \beta \, (2 - \beta^2) \right]$$ is the photon-photon annihilation cross section [@gou67], where $$\beta \equiv \left( 1 - {2 \over \epsilon_{\gamma} \epsilon \, ( 1- \varpi)} \right)^{1/2} \quad ,$$ is the velocity of the created electron/positron (computed via the conservation of four-momentum) in the appropriate center-of-momentum frame. By choosing $d\Omega = d\phi \, d\varpi$, the differential starlight photon number density (see equation 8) reads as $$n_{\epsilon}(\xi, \Omega) = {\epsilon^{-2} \, r_{\rm b} \over m_{\rm e} c^3} \, \int_0^{\eta_{\rm max}} \, \left[\epsilon j_{\epsilon}(\zeta)\right] \, d\eta \quad ,$$ where $\eta \equiv l / r_{\rm b}$, $\zeta = \sqrt{\xi^2 + \eta^2 - 2 \xi \eta \varpi}$, and the integration upper limit is $\eta_{\rm max} = \xi \varpi + \sqrt{\xi^2 \varpi^2 - \xi^2 + \xi_{\rm t}^2}$. This gives finaly $$\tau(\epsilon_{\gamma}) = {2 \pi \, j_{\rm V} \, r_{\rm b}^2 \over m_{\rm e} \, c^3} \, \int_{-1}^{+1} d\varpi \, (1 - \varpi) \, \int_{\max(\epsilon_{\rm min}, \epsilon_{\rm thre})}^{\epsilon_{\rm max}} d\epsilon \, \sigma_{\gamma \gamma}(\epsilon_{\gamma}, \epsilon, \varpi) \, { g(\epsilon) \over \epsilon^2 } \, \int_0^{\xi_{\rm t}} d \xi \int_0^{\eta_{\rm max}} d\eta \, h(\zeta) \, ,$$ where the threshold energy is $\epsilon_{\rm thre} = 2 / \epsilon_{\gamma} \, (1 - \varpi)$. This optical depth as a function of the $\gamma$-ray photon energy is shown in Figure 1 (bottom-panel). As can be seen, it possesses a broad maximum at $\varepsilon_{\gamma} \approx 1-10$ TeV, with the maximum value $\tau(2 \, {\rm TeV}) \approx 0.0145$.
Let us fix the $\gamma$-ray photon energy at $\epsilon_{\gamma} = 10^{6.6}$, and compute again the appropriate optical depth for photon-photon annihilation as a function of the upper bound in integration over the distance from the core, $\tau = \tau(\xi)$, replacing simply constant $\xi_{\rm t}$ in equation 13 with variable $\xi$. Such a profile is shown in Figure 2 (solid line): as can be seen, only about $40\%$ ($\tau \approx 0.006$) of the annihilated radiation is in fact absorbed within the innermost galactic volume of radius $r_{\rm b}$, and the remaining $60\%$ is absorbed at distances $r_{\rm b} < r < 100 \, r_{\rm b}$. The obtained profile for $\tau$ can be compared with the spatial distribution of the galactic radiation. Noting the definition for the photon field spectral energy density, $$U_{\nu} = {2 \pi \over c} \, \int_{-1}^{+1} I_{\nu}(\Omega) \, d\varpi \quad ,$$ one may find the appropriate bolometric energy density profile for the starlight emission as $$U_{\rm rad}(\xi) = f_{\rm bol} \, \left[\nu U_{\nu}\right]_{\rm V} = f_{\rm bol} \, {2 \pi \, r_{\rm b} \, j_{\rm V} \over c} \, \int_{-1}^{+1} d\varpi \, \int_0^{\eta_{\rm max}} d \eta \, \, h(\zeta) \quad ,$$ where $f_{\rm bol} = 2.5$ is the $V$-band bolometric correction for the adopted here stellar spectrum (equation 5). This profile is shown on Figure 2 as a dotted line.
![[*Left:*]{} Optical depth for $2$ TeV photons as a function of the distance from the active nucleus (solid line). [*Right:*]{} Profile of the energy denisty of the starlight photon field (dotted line).](f2.eps)
The spectral and spatial shapes of the optical depth for the photon-photon annihilation discussed above for the Cen A radio galaxy is universal, and should hold also for the other low-power radio sources hosted by elliptical galaxies. That is caused by the appropriate scaling $\tau \propto j_{\rm V} \, r_{\rm b}^2$ (equation 13), $j_{\rm V} \propto L_{\rm V} / {\cal H}$ (equation 7), and ${\cal H} = \kappa \, r_{\rm b}^3$ (equation 4), where $\kappa = \kappa(a, b, d)$ is a numerical factor depending on the parameters of the Nuker profile (for example, $\kappa = 1.83 \times 10^3$ in the case of NGC 5128, as given in the equation 3). These relations give $\tau \propto L_{\rm V} / \kappa \, r_{b}$. Since the parameters of the NGC 5128 Nuker profile [$a = 1.68$, $b = 1.3$, $d = 0.1$ @cap05] are similar to the average hosts of B2 radio sources [$a = 1.9$, $b = 1.6$, $d = 0.02$ @rui05], since the spectral energy distribution of the starlight emission considered here is based on a universal spectrum of the elliptical galaxy [@sil98], and since the break radius scales linearily with the $V$-band galactic luminosity, $$\left({r_{\rm b} \over {\rm kpc}}\right) \approx \left({L_{\rm V} \over 10^{45} \, {\rm erg/s}}\right) \, ,$$ [@rui05], the optical depth for the $\gamma$-ray photons shown on Figures 1 and 2 for the case of Cen A radio galaxy indeed should apply also to the other analogous sources. We note for example, that with the value of $L_{\rm V}$ found previously for the NGC 5128 galaxy, the simple scaling introduced in equation 16 implies the break radius $\approx 66$ pc, which is very close to the observed value of $r_{\rm b} \approx 41$ pc [@cap05]. On the other hand, the starlight energy density profile $U_{\rm rad} \propto j_{\rm V} \, r_{\rm b} \propto r_{\rm b}^{-1}$ is supposed to change for different sizes (and hence luminosities) of the elliptical host when compared with the particular profile shown on Figure 2 for the case of NGC 5128 parameters.
Energetics
----------
The obvious source of $\gamma$-ray photons in the Cen A system is the active nucleus (containing a relativistic jet), which is most probably responsible for production of the flux detected by OSSE ($0.05-4$ MeV), COMPTEL ($0.75-30$ MeV) and EGRET ($0.1-1.0$ GeV) instruments on board CGRO in the period 1991 – 1995, with the total observed $50$ keV – $1$ GeV luminosity $\sim 3 \times 10^{42}$ erg s$^{-1}$ [@ste98 and references therein]. This emission peaks at $\approx 0.1$ MeV, with the maximum energy flux about $\sim 10^{-9}$ erg cm$^{-2}$ s$^{-1}$. At a very high $\gamma$-ray energy range, a 3$\sigma$ detection of Cen A with the photon flux $F(>0.3 \, {\rm TeV}) = 4.4 \times 10^{-11}$ ph cm$^{-2}$ s$^{-1}$ was reported in the seventies [@gri75]. Subsequent CANGAROO observations resulted in the upper limit $F(>1.5 \, {\rm TeV}) < 5.45 \times 10^{-12}$ ph cm$^{-2}$ s$^{-1}$ for the point source centered at the Cen A nucleus, and $F(>1.5 \, {\rm TeV}) < 1.28 \times 10^{-11}$ ph cm$^{-2}$ s$^{-1}$ for the extended region with radius $14'$ [@row99]. The most recent HESS observations gave $F(>0.19 \, {\rm TeV}) < 5.68 \times 10^{-12}$ ph cm$^{-2}$ s$^{-1}$ for the point-like active nucleus [@aha05].
Here we are interested in the total, time-averaged and angle-averaged (i.e., ‘calorimetric’) flux of TeV photons produced by the active nucleus and ‘injected’ into the host galaxy as well as into the large-scale radio structure. With the most recent HESS results, we can only put some upper limits for it. Namely, assuming a standard apectral index $\alpha_{\gamma} = 1$ (equivalent to the photon index $\Gamma_{\gamma} = \alpha_{\gamma} + 1 = 2$) at the considered photon energy range, the relation between the integrated photon flux and the monochromatic flux energy density at any photon energy $\varepsilon \geq \varepsilon_0$ is simply $\left[\varepsilon S_{\varepsilon}\right] = \left[\varepsilon_0 \, F(>\varepsilon_0)\right]$. This flux is related to the emitting fluid (jet) intrinsic monochromatic power (assumed to be isotropic in the jet comoving frame) radiated in a given direction, $\partial L' / \partial \Omega' = L' / 4 \pi$, by the expression $$\left[\nu S_{\nu}\right] = {1 \over d_{\rm L}^2} \, {\delta_{\rm nuc}^3 \over \Gamma_{\rm nuc}} \, {\partial L' \over \partial \Omega'} = {1 \over d_{\rm L}^2} \, {\delta_{\rm nuc}^3 \over \Gamma_{\rm nuc}} \, {L' \over 4 \pi} \quad ,$$ where $\Gamma_{\rm nuc}$ and $\delta_{\rm nuc} = \Gamma_{\rm nuc}^{-1} \, \left(1 - \sqrt{1 - \Gamma_{\rm nuc}^{-2}} \cos \theta\right)^{-1}$ are, respectively, Lorentz and Doppler factors of the nuclear portion of the jet, and $\theta$ is the jet viewing angle [e.g., @sik97; @sta03]. On the other hand, the total power radiated into the ambient medium being of interest here, is $$L_{\rm inj} = \oint {\delta_{\rm nuc}^3 \over \Gamma_{\rm nuc}} \, {\partial L' \over \partial \Omega'} \, d\Omega = {1 \over 2} \, L' \, \Gamma_{\rm nuc}^{-1} \, \int_0^{\pi} \, \delta_{\rm nuc}^{-3} \, \sin \theta \, d\theta = L' \quad ,$$ and hence $$L_{\rm inj} = 4 \pi \, d_{\rm L}^2 \, \Gamma_{\rm nuc} \delta_{\rm nuc}^{-3} \, \left[\varepsilon_0 \, F(> \varepsilon_0)\right] \quad .$$ With the HESS photon flux $F(> 0.19 \, {\rm TeV}) < 5.68 \times 10^{-12}$ ph cm$^{-2}$ s$^{-1}$ (corresponding to the few-arcmin-integration area centered on the Cen A nucleus) one obtains the upper limit for the total injected monochromatic power $L_{\rm inj} < 2.4 \times 10^{39} \, \Gamma_{\rm nuc} \delta_{\rm nuc}^{-3}$ erg s$^{-1}$. With the prefered values $\theta \sim 50\deg-80\deg$ inferred from the VLBI radio observations [@jon96; @tin98], and $\Gamma_{\rm nuc} \sim 10$ widely considered as a typical value for the bulk Lorentz factor of sub-parsec scale AGN jets, this reads as $L_{\rm inj} < 10^{42} - 10^{43}$ erg s$^{-1}$. Below we take conservatively $L_{\rm inj} = 10^{42}$ erg s$^{-1}$ as an upper limit, noting that such a luminosity would correspond to less than $10 \%$ of the minimum total kinetic power of the Cen A jet, estimated to be $L_{\rm j} \gtrsim 10^{43}$ erg s$^{-1}$ from the radio/X-ray lobes’ energetic [see @cla92; @kra03]. As shown in the previous section, about $1\%$ of this power is absorbed on the starlight photon field within the central $100 \, r_{\rm b} \sim 4$ kpc region of the host galaxy, and is thus converted to an electron-positron population, mainly in the $0.1-1$ TeV energy range.
The nuclear $\gamma$-ray emission postulated here is expected to be Doppler-boosted within the narrow cone characterized by the opening angle $\Gamma_{\rm nuc}^{-1} \lesssim 6\deg$. Therefore, it is strongly Doppler hidden when viewed from $\theta \geq 50\deg$ (see equation 17). If the observer was located within the beaming cone of this emission however, he would detect a flux corresponding to the isotropic luminosity $L(0) = (\delta_{\rm nuc, \, \theta=0}^3 / \Gamma_{\rm nuc}) \, L' \approx \Gamma_{\rm nuc}^2 \, L' < 10^{44}$ erg s$^{-1}$. Such values are consistent with luminosities observed from the TeV-detected BL Lac objects [see, e.g., @katar06], which are believed to be beamed analogues of the low-power FR I radio galaxies like Cen A [@urr95].
![Different approximations for the pair injection energy spectrum (corresponding to the photon flux of the primary $\gamma$-rays $\propto \epsilon_{\gamma}^{-2}$) in normalized units: (i) $Q(\gamma) \propto \gamma^{-2} \, \tau( 2 \gamma)$ (open circles), (ii) $Q(\gamma)$ calculated using the pair production rate expression introduced by @aha83 [solid line], and (iii) broken power-law approximation introduced in this paper (dotted line).](f3.eps)
In order to find the energy spectrum of the created electrons and positrons, we note that the pair production rate for small values of the optical depth can be estimated as $$Q(\gamma, r) \propto \left. \left\{n_{\gamma}(\epsilon_{\gamma}) \times {d \tau (\epsilon_{\gamma}) \over dr}\right\}\right|_{\epsilon_{\gamma} = 2 \, \gamma}$$ [e.g., @cop90; @bot97], where $\gamma$ is the Lorentz factor of the created particles, and $n_{\gamma}(\epsilon_{\gamma})$ is the photon spectrum of the primary $\gamma$-ray photons. For the latter we assume power-law form $n_{\gamma}(\epsilon_{\gamma}) \propto \epsilon_{\gamma}^{-\Gamma_{\gamma}}$, where $\Gamma_{\gamma} = \alpha_{\gamma} + 1$ is the photon index. When averaged over the galactic radius, the pair production rate therefore scales as $$Q(\gamma) \propto \gamma^{-\Gamma_{\gamma}} \, \tau\left(2 \, \gamma\right) \quad .$$ This function is shown in Figure 3 in normalized units (open circles) for $\Gamma_{\gamma} = 2$, taken in this paper as a typical photon index of blazars’ TeV emission. As illustrated, $Q(\gamma)$ is peaked at $\gamma \sim 10^{5.5}$, and decreases as $\propto \gamma^{- (\Gamma_{\gamma}+0.5)}$ for $\gamma > 10^6$. Such behavior is in fact expected, since for $\epsilon_{\gamma} > 10^{6.6}$ the optical depth evaluated previously is roughly $\tau(\epsilon_{\gamma}) \propto \epsilon_{\gamma}^{-0.5}$. Figure 3 shows also the pair injection energy distribution calculated using the expression introduced by @aha83 [solid line], integrated over the spatial-averaged spectrum of the starlight photons (equation 12; see also equation 34 below) and the assumed photon spectrum of primary $\gamma$-rays, $\propto \epsilon_{\gamma}^{-2}$. We note that the approximation of @aha83 for the energy distribution of the secondary electrons provides accuracy better than $few \%$ [see detailed calculations in @bot97]. Small differences between the two presented estimates come from the fact that the function given by @aha83 corresponds to the isotropic distribution of the soft photons, while the optical depth calculated by us (i.e. equations 20-21) includes the anisotropy of the starlight radiation field. Finally, Figure 3 shows also the simplest broken power-law approximation for the injection pair spectrum considered hereafter for the purpose of the following calculations: $Q(\gamma) \propto const$ for $\gamma_0 \equiv 10^5 \leq \gamma \leq \gamma_{\rm br} \equiv 10^6$, and $Q(\gamma)\propto \gamma^{-2.5}$ for $\gamma > \gamma_{\rm br}$. We normalize it to the monochromatic $1$ TeV power $L_{\rm inj}$ specified above, to obtain $$Q(\gamma) = {\tau \, L_{\rm inj} \, {\cal{I}}(\gamma) \over \gamma_{\rm br}^2 \, m_{\rm e} c^2 \, \cal{V}_{\rm gal}} \quad {\rm where} \quad {\cal{I}}(\gamma) = \left\{ \begin{array}{lll} 1 & {\rm for} & \gamma_0 \leq \gamma \leq \gamma_{\rm br} \\ (\gamma_{\rm br} / \gamma)^{2.5} & {\rm for} & \gamma > \gamma_{\rm br} \end{array} \right.$$ and zero otherwise. Here $\tau \equiv \tau(\epsilon_{\gamma} = 1 \, {\rm TeV}) \approx 0.01$, and $\cal{V}_{\rm gal}$ is the galactic volume to which the pair injection is taking place.
Implications
============
The electrons and positrons produced by annihilation of the VHE $\gamma$-ray nuclear photons on starlight radiation are expected to have an initial spectrum peaked at $0.1-1$ TeV energies. After being injected into the galactic medium, they are quickly isotropized by the ambient magnetic field, and radiate via the synchrotron and inverse-Compton processes. The evolution of these electrons and details of their emission spectra depend therefore on the properties of the interstellar medium of the elliptical host.[^4]
Elliptical host
---------------
@mos96 argued, that elliptical galaxies have no ordered large-scale magnetic field, but only an unresolved random component. They further suggested, that the latter is due to a ‘fluctuation dynamo’ driven by the turbulent motions of the interstellar matter caused by type I supernovae and stellar motions. The latter ones are expected to be characterized by the Kolmogorov-like energy spectrum and an injection scale $\sim 3$ pc, while the former ones by a steeper (shock-like) spectrum and larger injection scale, $\sim 300$ pc. The resulting random magnetic field is expected to be characterized by the average galactic intensity $\sim 3$ $\mu$G (reaching $\sim 10$ $\mu$G in the central parts) and the correlation scale $\sim 100$ pc. @mos96 demonstrated that their model is consistent with all the observational constraints (in particular with the depolarization studies). In the case of Centaurus A, and in general all the ellipticals hosting radio-loud AGNs, a galactic magnetic field can be even higher than this, especially close to the galactic center, possessing even some regular component due to polution of the interstellar medium by the magnetized plasma transported from the active core in the form of the jets. However, for the purpose of order-of-magnitude evaluations, below we take the characteristic values for the NGC 5128 elliptical host’s magnetic field $B_{\rm gal} \approx 3-10$ $\mu$G, assuming that it consists solely of the (Alfvénic) turbulent component with the maximum wavelength $\lambda_{\rm max} \sim 100$ pc and Kolmogorov energy spectrum $W(k) \propto k^{-q}$, where $q = 5/3$.
With the interstellar medium parameters as discussed above, the mean free path of the created electron-positron pairs for resonant interactions with the turbulent Alfvèn modes, is [@sch89] $$\lambda_{\rm e} \approx r_{\rm g} \, \left({\lambda_{\rm max} \over r_{\rm g}}\right)^{q-1} = r_{\rm g}^{1/3} \, \lambda_{\rm max}^{2/3} \sim 0.82 \times \gamma_6^{1/3} \, B_{-5}^{-1/3} \, {\rm pc} \quad ,$$ where $\gamma_6 \equiv \gamma / 10^6$, $r_{\rm g} \equiv \gamma \, m_{\rm e} c^2 / e B_{\rm gal} \sim 5.5 \times 10^{-5} \ \gamma_6 \, B_{-5}^{-1}$ pc is the electrons’ gyroradius, and $B_{-5} \equiv B_{\rm gal} / 10$ $\mu$G [as a general reference for the particle-turbulent wave interactions see @sch02]. Such interactions lead to quick isotropisation of the injected electrons. In particular, the appropriate isotropisation time scale, $t_{\rm iso} \sim 3 \, \lambda_{\rm e} / c \sim 8 \, \gamma_6^{1/3} \, B_{-5}^{-1/3}$ yrs, is a few orders of magnitude shorter than the radiative cooling time scale (see below). The diffusive escape time scale from the central parts of the elliptical host ($R \sim 100 \, r_{\rm b} \sim 4$ kpc) is $t_{\rm esc} \sim 3 \, R^2 / \lambda_{\rm e} \, c \sim 2 \times 10^8 \, \gamma_6^{-1/3} \, B_{-5}^{1/3}$ yrs. We note, that the re-acceleration of the radiating particles by resonant scattering on the Alfvèn modes is not expected to be efficient enough to keep the electrons around $\gamma \sim 10^6$. In particular, the time scale for this process, $t_{\rm acc} \sim \beta_{\rm A}^{-2} \, t_{\rm iso} \sim 4.5 \times 10^6 \, \gamma_6^{1/3} \, B_{-5}^{-7/3}$ yrs, is much longer than the radiative cooling time scale. In the above, $v_{\rm A} \equiv \beta_{\rm A} c \approx B_{\rm gal} \, (4 \pi \, m_{\rm p} \, n_{\rm gas})^{-1/2} \sim 10^{-3} \, B_{-5} \, c$ is the Alfvèn velocity expected for the average number density of the cold gas within central ($< 4$ kpc) parts of the NGC 5128 galaxy $n_{\rm gas} \sim 3 \times 10^{-3}$ cm$^{-3}$. In this context we note that the most recent analysis presented by @kra03 indicates that the hot gaseous component of the galaxy discussed here is characterized by a tempreature $kT \sim 0.3$ keV and a central number density $\sim 4 \times 10^{-2}$ cm$^{-3}$, which is roughly constant within the radius $r \sim 0.5$ kpc and decreases further away as $\propto r^{-1.2}$. Such a behavior is consistent with general properties of the elliptical galaxies [@mat03]. Thus, one can conclude that the TeV energy electrons injected into the interstellar medium (via annihilation of the $\gamma$-ray emission of the active center) are effectively confined to the elliptical body and quickly isotropized by the galactic magnetic field, and therefore radiate all their energy there by inverse-Compton upscattering of the starlight photons and the synchrotron process (see below), before being re-accelerated by the turbulent processes. As a result, one should expect formation of a small-scale (galactic) version of the ‘isotropic pair halos’ discussed by @aha94, restricted however to the first generation of the secondary photons.
Below we investigate in more detail the emission spectrum of electron-positron pairs created within the centaral $100 \, r_{\rm b}$ parts of the elliptical host’s interstellar medium due to annihilation of the nuclear VHE $\gamma$-rays on the starlight photon field. First, we introduce the *averaged* energy density of the starlight photon field, assumed in this section to be isotropic within $100 \, r_{\rm b}$, $$\epsilon U_{\epsilon} \approx \langle\left[\nu U_{\nu}\right]_{\rm V}\rangle_{\xi} \times g(\epsilon)$$ (see equations 5 and 15), where $$\langle\left[\nu U_{\nu}\right]_{\rm V}\rangle_{\xi} \equiv {\int_0^{\xi_{\rm cr}} d\xi \, \left[\nu U_{\nu}\right]_{\rm V} \over \int_0^{\xi_{\rm cr}} d\xi} = {2 \pi \, r_{\rm b} \, j_{\rm V} \over c \, \xi_{\rm cr}} \, \int_0^{\xi_{\rm cr}} \, d\xi \, \int_{-1}^{+1} d\varpi \, \int_0^{\eta_{\rm max}} d \eta \, \, h(\zeta) \quad ,$$ and $r_{\rm cr} \equiv \xi_{\rm cr} \, r_{\rm b}$ is some particular critical radius over which the averaging is performed. The total averaged starlight energy density is $$\langle U_{\rm rad} \rangle = \int_{\epsilon_{\rm min}}^{\epsilon_{\rm max}} \, U_{\epsilon} \, d\epsilon = f_{\rm bol} \, \langle\left[\nu U_{\nu}\right]_{\rm V}\rangle_{\xi} \quad .$$ With $\xi_{\rm cr} = 100$ and other parameters as given above, one obtains $\langle\left[\nu U_{\nu}\right]_{\rm V}\rangle_{\xi} \approx 10^{-11}$ erg cm$^{-3}$, and $\langle U_{\rm rad} \rangle \approx 2.5 \times 10^{-11}$ erg cm$^{-3}$, which is still much higher than the energy density of the galactic magnetic field, $U_{\rm B} = B_{\rm gal}^2 / 8 \pi \approx 4 \times 10^{-12} \, B_{-5}^2$ erg cm$^{-3}$. The time-scale for the radiative losses of the electrons with Lorentz factor $\gamma$, including synchrotron and inverse-Compton losses (in both Thomson and Klein-Nishina regimes) can be then estimated as $$t_{\rm rad}(\gamma) = {3 \, m_{\rm e} c \over 4 \, \sigma_{\rm T} \, U_{\rm B} \, \gamma \, (1 + q \, F_{\rm KN})} \quad ,$$ where $q \equiv \langle U_{\rm rad} \rangle / U_{\rm B} \approx 7 \, B_{-5}^{-2}$, and $$F_{\rm KN} = {1 \over \langle U_{\rm rad} \rangle} \, \int {U_{\epsilon} \over \left(1+ 4 \, \gamma \, \epsilon\right)^{1.5} } \, d\epsilon$$ [@mod05]. The simplified form of the function $F_{\rm KN}$ introduced above is in fact a very accurate approximation for the values $4 \, \gamma \, \epsilon < 4 \, \gamma_{\rm inj} \, \epsilon_{\rm max} \lesssim 100$, as considered in this paper (with $\epsilon_{\rm max} = 10^{-5}$). The time scale $t_{\rm rad}(\gamma)$, together with the turbulent re-acceleration time scale and the escape time scale, is shown in Figue 4 as a function of the electron Lorentz factor $\gamma$ for the range of $B_{\rm gal} = 3-10$ $\mu$G (note that $t_{\rm iso}(\gamma)$, not shown in this figure, is much shorter than any other of the time scales plotted). As can be seen, $t_{\rm esc} \gg \max(t_{\rm acc}, \, t_{\rm rad})$ for all $\gamma \leq \gamma_{\rm br}$, and $t_{\rm acc} \gg t_{\rm rad}$ for $\gamma > \gamma_0$. All the electrons with Lorentz factors $\gamma < \gamma_{\rm KN} \equiv 1 / 4 \, \epsilon_{\rm max} \sim 10^{4.5}$ cool mainly via the inverse-Compton losses in the Thomson regime. Meanwhile, the electrons with $\gamma_{\rm KN} < \gamma < \gamma_{\rm cr}$, where $q \, F_{\rm KN}(\gamma_{\rm cr}) \equiv 1$, lose their energy mainly via the inverse-Compton emission in the Klein-Nishina regime. In the case of $B_{\rm gal} = 3$ $\mu$G, one has $\gamma_{\rm cr} \sim \gamma_{\rm br}$, while for $B_{\rm gal} = 10$ $\mu$G the critical electron Lorentz factor $\gamma_{\rm cr}$ is a factor of a few lower than that. Finally, the main process responsible for the cooling of the electrons with $\gamma > \gamma_{\rm cr}$ is synchrotron radiation.
The resulting electron energy distribution, $n_{\rm e}(\gamma)$, ignoring re-acceleration and escape effects, can be found from the continuity equation $${\partial n_{\rm e}(\gamma) \over \partial t} = {\partial \over \partial \gamma} \left\{ |\dot{\gamma}|_{\rm cool} \, n_{\rm e}(\gamma)\right\} + Q(\gamma) \quad ,$$ where $Q(\gamma)$ denotes injection of high-energy electrons through photon-photon annihilation, and $|\dot{\gamma}|_{\rm cool} = |\dot{\gamma}|_{\rm syn} + |\dot{\gamma}|_{\rm ic}$ is the total rate of the radiative cooling. The standard formulae give $$|\dot{\gamma}|_{\rm syn} = {4 \, c \sigma_{\rm T} \over 3 \, m_{\rm e} c^2} \, U_{\rm B} \, \gamma^2 \quad , \quad {\rm and} \quad |\dot{\gamma}|_{\rm ic} = {4 \, c \sigma_{\rm T} \over 3 \, m_{\rm e} c^2} \, U_{\rm B} \, \gamma^2 \, q \, F_{\rm KN}$$ [@mod05]. Note, that for the parameters considered here, and electron Lorentz factor $\gamma = 10^6$, the relative importance of the inverse-Compton and synchrotron energy losses, $|\dot{\gamma}|_{\rm ic} / |\dot{\gamma}|_{\rm syn} = q \, F_{\rm KN}$, is roughly $\sim 3$ for $B_{\rm gal} = 3$ $\mu$G, and $\sim 0.3$ for $10$ $\mu$G. Thus, the energy injected into the created pairs is re-radiated via their synchrotron and inverse-Compton (in the Klein-Nishina regime) processes in roughly comparable amounts. As for the function $Q(\gamma)$, we use approximation for the electrons freshly injected to the galactic volume $\cal{V}_{\rm gal}$ introduced by equation 22, to obtain the steady-state solution $$n_{\rm e}(\gamma) = {3 \, m_{\rm e} c \over 4 \, \sigma_{\rm T}} \, {\int_{\gamma} d\gamma' \, Q(\gamma')\over \gamma^2 \, U_{\rm B} \, (1 + q \, F_{\rm KN})} = {3 \, \tau \, L_{\rm inj} \over 4 \, c \, \sigma_{\rm T} \, \gamma_{\rm br}^2 \, {\cal{V}}_{\rm gal}} \, {\int_{\gamma} d\gamma' \, {\cal{I}}(\gamma') \over \gamma^2 \, U_{\rm B} \, (1 + q \, F_{\rm KN})} \quad .$$ This spectrum, in normalized units, is shown in Figure 5 for $B_{\rm gal} = 3$ $\mu$G (solid line) and $10$ $\mu$G (dashed line). The shaded region indicates the energy range for which the turbulent re-acceleration effects (neglected here) are expected to become important. As can be noted, at low electron energies $\gamma < \gamma_{\rm KN}$, for which the inverse-Compton cooling in the Thomson regime dominates, the energy spectrum has a standard form $n_{\rm e}(\gamma) \propto \gamma^{-p}$ with $p=2$, as expected in the case of the continuous injection of flat-spectrum ($Q(\gamma < \gamma_{\rm br}) \propto const$) particles, followed by the $|\dot{\gamma}|_{\rm ic/T} \propto \gamma^2$ energy losses. However, for $\gamma > \gamma_{\rm KN}$ the electron spectrum flattens, as a result of the dominant inverse-Compton/Klein-Nishina cooling [see the most recent discussion in @mod05 and references therein], although in the case of $B_{\rm gal} = 10$ $\mu$G this effect is relatively weak (the effective spectral index $p \sim 1.6$, to be compared with $p \sim 1.4$ for $B_{\rm gal} = 3$ $\mu$G). Meanwhile, at $\gamma > \gamma_{\rm cr}$ the electron sepectral index in both cases increases to $p = 3.5$, as expected for the dominant synchrotron cooling $|\dot{\gamma}|_{\rm syn} \propto \gamma^2$ of the continuously injected steep-spectrum ($Q(\gamma > \gamma_{\rm br}) \propto \gamma^{-2.5}$) electrons.
![Synchrotron and inverse-Compton spectra of the electrons injected into the interstellar medium of the elliptical host, for the galactic magnetic field $3$ $\mu$G (solid lines) and $10$ $\mu$G (dashed lines), and the injection luminosity $L_{\rm inj} = 10^{42}$ erg s$^{-1}$. Dotted lines indicates the $0.05-1000$ MeV spectrum of the Cen A source as constrained by different instruments on board CGRO [1991-1995; ‘low’ and ‘intermediate’ states; @ste98].](f6.eps)
With the evaluated electron energy distribution, one can find the energy spectrum of the resulting synchrotron and inverse-Compton emissions of the created pairs. In general, for the isotropic distribution of seed photons and particles, the respective luminosities can be simply written as $[\varepsilon L_{\varepsilon}]_{\rm syn/ic} = 4 \pi \, {\cal V}_{\rm gal} \, [\epsilon j_{\epsilon}]_{\rm syn/ic}$, where $$[\epsilon j_{\epsilon}]_{\rm syn} = {1 \over 2} \, \left.{n_{\rm e}(\gamma) \, \gamma \over 4 \pi} \, |\dot{\gamma}|_{\rm syn} \, m_{\rm e} c^2\right|_{\gamma = \sqrt{(3/4) \, \epsilon_{\rm syn} \, (B_{\rm cr} / B)}} \quad ,$$ [using $\delta$-approximation for the synchrotron emissivity; see @cru86], $B_{\rm cr} \equiv 2 \pi \, m_{\rm e}^2 \, c^3 / h \, e = 4.4 \times 10^{13}$ G, and $$[\epsilon j_{\epsilon}]_{\rm ic} = {3 \, \sigma_{\rm T} \, m_{\rm e} c^3 \over 16 \pi} \, \epsilon_{\rm ic}^2 \, \int d\gamma \, \int d\epsilon \, \, \, n_{\rm e}(\gamma) \, n_{\rm rad}(\epsilon) \, {{\cal F}_{\rm iso}(\gamma,\epsilon,\epsilon_{\rm ic}) \over \epsilon \, \gamma^{2}} \quad .$$ Here $\epsilon_{\rm syn/ic}$ denotes now the dimensionless energy of the synchrotron/inverse-Compton photons, electron energy distribution $n_{\rm e}(\gamma)$ is given by the equation 31, seed photons’ spectrum $$n_{\rm rad}(\epsilon) = {U_{\epsilon} \over \epsilon \, m_{\rm e} c^2} = {\langle U_{\rm rad}\rangle \over m_{\rm e} c^2 \, f_{\rm bol}} \, \epsilon^{-2} \, g(\epsilon)$$ is specified by the equation 5, and finally $${\cal F}_{\rm iso}(\gamma,\epsilon, \epsilon_{\rm ic}) = 2 \, {\cal P} \, \ln {\cal P} + {\cal P} + 1 - 2 {\cal P}^2 + {({\cal K} \, {\cal P})^2 \, (1 - {\cal P}) \over 2 \, (1 + {\cal K} \, {\cal P})}$$ with $${\cal K} \equiv 4 \, \epsilon \, \gamma \quad {\rm and} \quad {\cal P} \equiv {\epsilon_{\rm ic} \over 4 \, \epsilon \, \gamma \, ( \gamma - \epsilon_{\rm ic})} \quad ,$$ where $1/ 4 \gamma^2 \leq {\cal P} \leq 1$ [@blu70]. This leads to $$[\epsilon L_{\epsilon}]_{\rm syn} = {\tau \, L_{\rm inj} \over 2 \, \gamma_{\rm br}^2} \, \left.{\gamma \, \int_{\gamma} d\gamma' \, {\cal{I}}(\gamma') \over 1 + q \, F_{\rm KN}}\right|_{\gamma = \sqrt{(3/4) \, \epsilon_{\rm syn} \, (B_{\rm cr} / B)}} \quad ,$$ and $$[\epsilon L_{\epsilon}]_{\rm ic} = {9 \, q \, \tau \, L_{\rm inj} \over 16 \, f_{\rm bol} \, \gamma_{\rm br}^2} \, \, \epsilon_{\rm ic}^2 \, \int d\gamma \, \int_{\max(\epsilon_{\rm min}, \epsilon_{\rm low})}^{\min(\epsilon_{\rm max},\epsilon_{\rm up})} d\epsilon \, \, \, {{\cal F}_{\rm iso}(\gamma,\epsilon,\epsilon_{\rm ic}) \, g(\epsilon) \over \epsilon^3 \, \gamma^4 \, (1 + q \, F_{\rm KN})} \, \int_{\gamma} d\gamma' \, {\cal{I}}(\gamma') \quad ,$$ where $\epsilon_{\rm low} \equiv \epsilon_{\rm ic}/4 \gamma \, (\gamma - \epsilon_{\rm ic})$, and $\epsilon_{\rm up} \equiv \epsilon_{\rm ic} \gamma /(\gamma - \epsilon_{\rm ic})$. The evaluated luminosities are shown in Figure 6, for $L_{\rm inj} = 10^{42}$ erg s$^{-1}$ and the galactic magnetic fiel $B_{\rm gal} = 3$ $\mu$G (solid lines) and $10$ $\mu$G (dashed lines). In the case of the inverse-Compton emission, spectral index in the photon energy range $\epsilon_{\rm ic} = 10^{3} - 10^{6}$ (corresponding roughly to the electrons energies $\gamma_{\rm KN} < \gamma < \gamma_{\rm br}$), is $\alpha_{\rm ic/KN} \sim 0.36$ for $B_{\rm gal} = 3$ $\mu$G, and $\alpha_{\rm ic/KN} \sim 0.45$ for $10$ $\mu$G. These values are in agreement with the expected $\alpha_{\rm ic/KN} = p-1$ for the appropriate electron spectral index $p=1.4$ and $1.6$, respectively [see a wide discussion in @mod05]. At $\epsilon_{\rm ic} > 10^6$, the inverse-Compton emission breaks to $\propto \epsilon_{\rm ic}^{-2.6}$, again as expected in the case of a steep power law electron continuum $\propto \gamma^{-3.5}$ at $\gamma > \gamma_{\rm br}$. In addition, Figure 6 presents the $0.05-1000$ MeV spectrum of the Cen A source as constrained by different instruments on board CGRO [1991-1995; ‘low’ and ‘intermediate’ states; @ste98]. As shown, the low-energy ($1-100$ GeV) flat-spectrum part of the predicted halo, if detected by the GLAST instrument in the future, could be spectraly distinguished from the (presumably nuclear) component observed by CGRO.
Figure 6 indicates that almost all the energy injected to the elliptical host via annihilation of the nuclear $\gamma$-ray emission on the starlight photon field is re-emitted via the synchrotron emission at $\sim 10^{13} - 10^{14}$ Hz frequencies, and via the inverse-Compton emission at $\gtrsim 0.1$ TeV photon energies. Unfortunatelly, the secondary synchrotron photons have almost the same energy as the target starlight photons, and at the same time much lower total luminosity, and therefore are not likely to be directly observed. However, the secondary $\gamma$-ray emission, although also relatively weak, is more promising for the detection. In particular, the analysis presented above indicates that at high photon energies ($> 0.1$ TeV) one should expect the photon flux from the galactic pair halo $$F_{\rm iso} \sim {[\varepsilon_0 L_{\varepsilon_0}]_{\rm ic} \over 4 \pi \, d_{\rm L}^2 \, \varepsilon_0} \lesssim 10^{-11} \, \left({L_{\rm inj} \over 10^{42} \, {\rm erg/s}}\right) \quad {\rm ph \, cm^{-2} \, s^{-1}} \quad ,$$ with the Cen A distance $d_{\rm L} = 3.4$ Mpc and $\varepsilon_0 = 0.2$ TeV. Note, that although the starlight radiation absorbs and reprocesses only a tiny ($\sim 1\%$) fraction of the nuclear $\gamma$-ray emisison, the small size of the resulting isotropic galactic pair halos ($\sim 100 \, r_{\rm b}$) make it quite easy to detect such structures. In fact, in the the case of Cen A radio galaxy $100 \, r_{\rm b}$ corresponds to roughly $4$ kpc, or about $4$ arcmin, which is still within the active nucleus-centered flux integration region of the HESS telescope. Thus, the estimate given in equation 21 can be directly compared with the HESS upper limits given in @aha05. Such comparison implies that the time-averaged $\gamma$-ray output of the Cen A nucleus is indeed $L_{\rm inj} < 10^{42}$ erg s$^{-1}$, which is already a meaningful result. This demonstrates an important aspect of the presented analysis: observations of nearby AGNs at TeV photon energies can provide important constraints on the time-averaged VHE $\gamma$-ray fluxes produced in their active centers, even if these sources are not belonging to the blazar sub-class (i.e., even if the nuclear portions of their jets are not inclined at small angles to the line of sight, and therefore their direct $\gamma$-ray emission is Doppler-hidden). Figure 7 shows in more details the expected $0.19-5$ TeV and $0.1-300$ GeV photon fluxes of the inverse-Compton emission for the discussed Cen A galactic pair halo, together with the indicated HESS upper limit and the expected GLAST sensitivity [$\sim 1.5 \times 10^{-9}$ ph cm$^{-2}$ s$^{-1}$ for one-year all-sky survey; @blo96]. As is shown, future observations by HESS and GLAST will be able to put independent constraints on both the low- and high-energy spectral parts of the predicted halo.
![The expected photon fluxes of the Cen A halo for $\varepsilon_0 = 0.1$ GeV (thin lines) and $\varepsilon_0 = 0.2$ TeV (thick lines), corresponding to $B=3$ and $10$ $\mu$G (solid and dashed lines, respectively), as functions of the injection luminosity $L_{\rm inj}$. Horizontal dotted lines indicates HESS upper limit as given in @aha05, and the expected GLAST sensitivity [$\sim 1.5 \times 10^{-9}$ ph cm$^{-2}$ s$^{-1}$ for one-year all-sky survey; @blo96].](f7.eps)
Discussion and Conclusions
==========================
Several BL Lac objects are confirmed sources of variable and strongly Doppler-boosted TeV emission produced in the nuclear portions of their relativistic jets. It is more than probable, that also many of the FR I radio galaxies, believed to be the parent population of BL Lacs, are TeV sources, for which strongly Doppler-hidden nuclear $\gamma$-ray radiation may be only too weak to be directly observed [although see @aha03; @bei05 for the case of nearby FR I radio galaxy M 87 detected recently at TeV photon energies by HEGRA and HESS Cherenokov telescopes]. Here we show, however, that about one percent of the total, time-averaged TeV radiation produced by the active nuclei of low-power FR I radio sources is inevitably absorbed and re-processed by the photon-photon annihilation on the starlight photon field, and the following synchrotron/inverse-Compton emission of the created and quickly isotropized electron-positron pairs. Such a re-processed isotropic radiation could be detected in the cases of at least a few nearby FR I radio galaxies, providing interesting constraints on the unknown parameters of the active nucleus and the elliptical host.
In the case of the Cen A radio galaxy considered in this paper, we found that the discussed mechanism can give distinctive radiative features due to the isotropic $\gamma$-ray emission of the electron-positron pairs injected by the absorption process into the interstellar medium of the elliptical host (its inner parts in particular, roughly within the radius of $4$ kpc from the galactic center), and inverse-Compton upscattering thereby starlight radiation to the $\leq$ TeV photon energy range. The resulting $\gamma$-ray halo is expected to possess a spectral peak at $\sim 0.1$ TeV photon energies, preceded by a flat continuum due to the dominant Klein-Nishina cooling of the radiating electrons, and followed by a steep power-law $\propto \epsilon_{\rm ic}^{-(\Gamma_{\gamma}+0.5)}$, where $\Gamma_{\gamma}$ is the photon index of the primary (nuclear) $\gamma$-ray emission. Such a halo should be strong enough to be detected and mapped by stereoscopic systems of Cherenkov telescopes like HESS, and, at lower photon energies, by GLAST.
All of the above findings should apply as well to the other nearby FR I sources. We note in this context, that the kinetic power of the Cen A jet, $L_{\rm j} \sim 10^{43}$ erg s$^{-1}$, is rather low when compared to other FR I sources [e.g., $L_{\rm j} \gtrsim 10^{44}$ erg s$^{-1}$ in the case of M 87 radio galaxy; @sta06b and references therein]. Thus, other (though more distant) objects of the FR I type may posses more luminous isotropic halos than Cen A analyzed here. Indeed, taking the time- and angle average nuclear $\gamma$-ray ($\sim$ TeV) luminosity $L_{\rm inj} \sim \eta_{\rm rad} \, L_{\rm j}$, where $\eta_{\rm rad}$ is the radiative efficiency, and the re-processed ($\gtrsim 0.1$ TeV) luminosity $L_{\rm iso} \sim \tau \, L_{\rm inj}$ with $\tau \sim 0.01$ as estimated in this paper, one can find that modern Cherenkov telescopes with the available sensitivity limit $10^{-13}$ erg cm$^{-2}$ s$^{-1}$ will be able to detect the discussed halos from the objects located within the radius rougly $d_{\rm L} \leq 100 \, \sqrt{(\eta_{\rm rad} / 0.1) \, (L_{\rm j} / 10^{44} \, {\rm erg \, s^{-1}})}$ Mpc. Is it therefore possible to attribute the $\gamma$-ray flux detected recently from M 87 system [@aha04] to the isotropic halo of its host galaxy? The answer is negative, since variability of this emission established on the time-scale of months and years [@bei05] excludes any extended $\gamma$-ray emission sites.
Opacity due to the ‘Dust Lane’
==============================
The continuum far-infrared emission of NGC 5128, produced most likely by massive young stars and diffuse cirrus clouds [@joy88; @eck90], is concentrated within the ‘dust lane’. We model this feature as a thin disc, perpendicular to the jet axis, centered on the active nucleus, and extending up to radii $R_{\rm fir} \sim 4$ kpc [see @isr98]. We also restrict the analysis to $\lambda_{\rm fir} = 100$ $\mu$m radiation, for which we take the total observed (IRAS) flux $\sim 400$ Jy [@gol88], corresponding to the luminosity $L_{\rm fir} \sim 1.6 \times 10^{43}$ erg s$^{-1}$. The number density of the far-infrared photons, assumed to be uniformly distributed within the ‘dust-lane’, is then $$n_{\rm fir, \epsilon}(\xi, \Omega) = {L_{\rm fir} \over 8 \, \pi^2 \, R_{\rm fir}^2 \, \epsilon_{\rm fir} \, m_{\rm e} c^3} \, \delta(\epsilon - \epsilon_{\rm fir}) \quad .$$ The appropriate optical depth for the photon-photon annihilation, analogous to the one given by equation 13, can be then evaluated as $$\tau(\epsilon_{\gamma}) = {L_{\rm fir} \, r_{\rm b} \over 8 \, \pi^2 \, R_{\rm fir}^2 \, \epsilon_{\rm fir} \, m_{\rm e} c^3} \, \int_0^{\xi_{\rm t}} d \xi \, \int_{\cos[{\rm arccot}(\xi/100)]}^{+1} d\varpi \, \, \, (1 - \varpi) \, \, \, \sigma_{\gamma \gamma}(\epsilon_{\gamma}, \epsilon_{\rm fir}, \varpi) \quad .$$ This optical depth is shown on Figure 8 (solid line), together with the optical depth due to the starlight emission of the elliptical host evaluated peviously (dotted line). As shown, opacity due to the dust lane is much smaller than the opacity due to the elliptical host (because of the different spatial distribution of the target photons), and in addition regards only very high energies ($>100$ TeV) of the primary $\gamma$-rays.
Acknowledgments {#acknowledgments .unnumbered}
===============
Ł.S. and M.O. were supported by MEiN through the research project 1-P03D-003-29 in years 2005-2008. Ł.S. acknowledges also the ENIGMA Network through the grant HPRN-CT-2002-00321, and thanks Wystan Benbow as well as the referee, Marcus Böttcher, for helpful comments.
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\[lastpage\]
[^1]: E-mail: Lukasz.Stawarz@mpi-hd.mpg.de
[^2]: Distance $d = 3.4$ Mpc; $1$ arcsec corresponds to $16$ pc.
[^3]: We note, that ellipticals hosting radio sources may contain relatively large amounts of cold dust, prounounced at far infrared frequencies [@gol88; @kna90], for which the exact location is unknown. As mentioned before, the Cen A system itself is bright in infrared frequencies due to the dust and young stars emission, restricted however to the ‘dust lane’, i.e. relatively thin but extended disc-like feature (roughly perpendicular to the jet axis), being a remnant of the spiral which merged recently with the elliptical host. A role of this feature in absorbing and reprocessing the nuclear $\gamma$-rays is briefly discussed in Appendix A.
[^4]: We also note, that since the nuclear $\gamma$-ray emission is expected to be beamed within the narrow cone with an opening angle of less then $10\deg$, as discussed in the previous section, some part of it illuminates also the large-scale radio outflow, i.e. the kpc-scale jet (extending in the case of Cen A from $\sim 0.2$ kpc up to $\sim 4$ kpc from the center), and the ‘inner’ lobe [$\sim 4-5$ kpc; see @bur83; @cla92]. The evolution of the electron-positron pairs injected via the photon-photon annihilation into these regions can differ substantially from the evolution of the particles injected into the body of the elliptical host as considered below. This problem will be investigated in a subsequent paper.
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abstract: 'Let $\Omega\subset M$ be an open subset of a Riemannian manifold $M$ and let $V:M\to {\mathbb{R}}$ be a Kato decomposable potential. With $W^{1,2}_{0}(M;V)$ the natural form domain of the Schrödinger operator $-\Delta+V$ in $L^2(M)$, in this paper we study systematically the following question: Under which assumption on $\Omega$ is the statement $$\text{ for all $f\in W^{1,2}_{0}(M;V)$ with $f=0$ a.e. in $M\setminus \Omega$ one has $f|_\Omega\in W^{1,2}_{0}(\Omega;V)$}$$ true for every such $V$? We prove that without any further assumptions on $V$, the above property is satisfied, if $\Omega$ is Kac regular, a probabilistic property which means that the first exit time of Brownian motion on $M$ from $\Omega$ is equal to its first penetration time to $M\setminus \Omega$. In fact, we treat more general covariant Schrödinger operators acting on sections in metric vector bundles, allowing new results concerning the harmonicity of Dirac spinors on singular subsets. Finally, we prove that Lipschitz regular $\Omega$’s are Kac regular.'
address:
- 'Francesco Bei, Dipartimento di matematica, universit‘a degli studi di Padova, Italy'
- 'Batu Güneysu, Mathematisches Institut, Universität Bonn, Germany'
author:
- Francesco Bei
- Batu Güneysu
title: 'Kac regular sets and Sobolev spaces in geometry, probability and quantum physics'
---
Introduction
============
Consider the following three properties that an open subset $\Omega$ of a noncompact Riemannian manifold $M$ may or may not have:\
(A) Let $V$ be a Kato decomposable potential on $M$ and let $H_M(V)$ denote the natural quadratic form realization of $-\Delta+V$ in $L^2(M)$ and let $H_{\Omega}(V)$ denote the self-adjoint realization of $-\Delta+V$ in $L^2(\Omega)$ subject to Dirichlet boundary conditions (also the Friedrichs realization). *For every such $V$ one has $$\begin{aligned}
\exp\big(-t(H_M(V)+\infty\cdot 1_{M\setminus \Omega})\big)=\exp(-t H_{\Omega}(V))P_\Omega\quad\text{ for all $t>0$,}\end{aligned}$$ where $\exp\big(-t(H_M(V)+\infty\cdot 1_{M\setminus \Omega})\big)$ is understood as the strong limit of $\exp\big(-t(H_M(V)+n\cdot 1_{M\setminus \Omega})\big)$ as $n\to\infty$, and where $P_\Omega:L^2(M)\to L^2(\Omega)$ denotes the natural projection.*
\(B) Assume $M$ is a geodesically complete Riemannian spin manifold. *For every spin bundle ${\mathscr{S}}\to M$ with corresponding Dirac operator $D$ acting on sections of ${\mathscr{S}}\to M$, and every spinor $\Psi \in \Gamma_{L^2}(M,{\mathscr{S}})$ with $D\Psi \in \Gamma_{L^2}(M,{\mathscr{S}})$ and $\Psi=0$ almost everywhere in $M\setminus \Omega$, one has the implication* $$\text{\emph{$D^2\Psi=0$ in $\Omega$ $\Rightarrow$ $D\Psi=0$ in $\Omega$ .}}$$ (C) *The first exit time of Brownian motion from $\Omega$ equals the first penetration time of Brownian motion to $M\setminus\Omega$*. This property in particular implies[^1] that with $\mathbb{P}^x$ the Riemannian Brownian motion measure with initial point $x$ and ${\mathbb{X}}$ the coordinate process on the path space of $M$ one has $$\begin{aligned}
& \{ {\mathbb{X}}_s\in \Omega\text{\emph{ for a.e. $s\in [0,t]$ and ${\mathbb{X}}_s\in M$ for all $s\in [0,t]$}}\}\\
&=_{\mathbb{P}^x} \{ {\mathbb{X}}_s\in \Omega\text{ \emph{for all $s\in [0,t]$}}\}\quad\text{\emph{ for all $x\in \Omega$, $t>0$}}.\end{aligned}$$ To the best of our knowledge, property (C) has been introduced by D. Stroock in the Euclidean space ${\mathbb{R}}^m$, and is usually referred to as the *Kac regularity* of $\Omega$.
While (A), (B) and (C) seem unrelated at first glance, the main results of this paper show that the above three problems are much more correlated than one might expect: Indeed we prove that given an arbitrary open subset $\Omega\subset M$,
- \(A) is equivalent to $$\quad\quad\quad\text{(A')}\>\>\text{ for all $f\in W^{1,2}_{0}(M;V)$ with $f=0$ a.e. in $M\setminus \Omega$ one has $f|_\Omega\in W^{1,2}_{0}(\Omega;V)$}$$ for every $V$ as in (A); here $W^{1,2}_{0}(M;V)$ and $W^{1,2}_{0}(\Omega;V)$ denote, respectively, the form domain of $H_M(V)$ and $H_{\Omega}(V)$,
- \(C) is equivalent to (A),
- \(C) implies (B),
- If $ \Omega$ is Lipschitz regular (cf. Definition \[lipreg\]), then one has (C).
While (i), (ii) and a variant of (iv) have been obtained in the Euclidean space ${\mathbb{R}}^m$ [@herbst] for $V=0$ (cf. Remark \[ende\]), our analysis is the first systematic treatment of these questions on manifolds, allowing in addition potentials and in fact covariant Schrödinger operators. As we only require a very weak regularity on the potential $V$, we can also treat Schrödinger operators that appear naturally in quantum mechanics, having potentials with Coulomb type singularities. In fact, we are going to treat these problems within the much more general class of covariant Schrödinger operators that act on sections of metric vector bundles over $M$, allowing to treat magnetic fields or squares of geometric Dirac operators simulatinously, making contact with (B). Our proofs rely on various (partially new) covariant Feynman-Kac formulas.
Main result
===========
In the sequel, we work in the smooth category, that is, all differential, topological and geometric data (like manifolds, bundles, metrics and covariant derivatives) are understood to be smooth. In addition, any manifold is understood to be without boundary, unless otherwise stated. Without loss of generality, we will consider only complex function spaces. Furthermore, for every manifold $M$ and every complex vector bundle $E\to M$ its fibers will be denoted with $E_x$, $x\in M$, and the corresponding space of sections of $E\to M$ having a certain regularity $\mathscr{C}$ like $\mathscr{C}=C^{\infty}_c$ etc. will be denoted with $\Gamma_{\mathscr{C}}(M,E)$. The smooth linear partial differential operator $$d_{\bullet}:\Gamma_{C^{\infty}}(M,\wedge^{\bullet} T^*_{{\mathbb{C}}}M)\longrightarrow \Gamma_{C^{\infty}}(M,\wedge^{\bullet+1} T^*_{{\mathbb{C}}}M)$$ from $\wedge^{\bullet} T^*_{{\mathbb{C}}}M\to M$ to itself denotes the exterior derivative acting on complexified differential forms.
Let $M$ be an arbitrary connected Riemannian $m$-manifold. As such, $M$ is equipped with its Riemannian volume measure $\mu$. Given a complex metric vector bundle ${E}\to M$, the scalar product on the complex Hilbert space $\Gamma_{L^2}(M,{E})$ of Borel equivalence classes of square integrable sections of ${E}\to M$ is simply denoted with $$\left\langle f_1,f_2\right\rangle =\int (f_1,f_2) { d}\mu,$$ with $$\left\|f\right\|^2=\int |f|^2 { d}\mu$$ the induced norm. Given two such bundles ${E}_1\to M$, ${E}_2\to M$, the formal adjoint of a smooth linear partial differential operator $D$ mapping sections of ${E}_1\to M$ to sections of ${E}_2\to M$ with respect to $\left\langle \cdot,\cdot\right\rangle $ is simply denoted with $D^{\dagger}$. The reader may find the basics of linear partial differential operators on vector bundles over Riemannian manifolds in [@gunbook]. The following conventions will be very convenient in the sequel:
If $\Omega\subset M$ is an open subset and ${E}\to M$ is a complex vector bundle, we define $$\begin{aligned}
\Gamma_{C^{\infty}_c}(\Omega,{E}):=\left\{f\in \Gamma_{C^{\infty}_c}(M,{E}):\text{ $f$ is compactly supported in $\Omega$} \right\}\subset \Gamma_{C^{\infty}_c}(M,{E}),\end{aligned}$$ and if ${E}\to M$ is equipped with a metric, then $$\begin{aligned}
\Gamma_{L^2}(\Omega,{E}):=\left\{f\in \Gamma_{L^2}(M,{E}):\text{ $f=0$ $\mu$-a.e. in $M\setminus \Omega$} \right\}\subset \Gamma_{L^2}(M,{E}).\end{aligned}$$
Then $\Gamma_{L^2}(\Omega,{E})$ is a closed subspace of $\Gamma_{L^2}(M,{E})$, thus a Hilbert space in itself. We denote with $$P_{\Omega}: \Gamma_{L^2}(M,{E})\longrightarrow \Gamma_{L^2}(\Omega,{E})$$ the orthogonal projection onto $\Gamma_{L^2}(\Omega,{E})$. Note that $P_{\Omega}$ is nothing but the restriction map $f\mapsto f|_\Omega$.\
Let $(t,y)\mapsto p_{\Omega}(t,x,y)$ denote the pointwise minimal solution of the heat equation on the open subset $\Omega\subset M$ with $\lim_{t\to 0+}p_{\Omega}(t,x,\cdot)=\delta_x$. We recall that the Kato class $\mathcal{K}(M)$ of $M$ is defined by all Borel functions $w$ on $M$ such that $$\begin{aligned}
\label{dre}
\lim_{t\to 0+}\sup_{x\in M}\int p_M(t,x,y) |w(y)| d\mu(y)=0.\end{aligned}$$
Based on the results from [@gunbook] (cf. Chapter VII), we propose:
\[popo\] By a *Kato-Schrödinger bundle over $M$*, we will understand a datum $$({E},\nabla, V)\longrightarrow M$$ with
- ${E}\to M$ a complex metric vector bundle
- $\nabla$ a metric covariant derivative on ${E}\to M$,
- $V:M\to \mathrm{End}({E})$ is a Kato decomposable potential, that is, there exist pointwise self-adjoint Borel sections $V_{\pm}$ of $\mathrm{End}(E)\to M$ with $V_{\pm}(x)\geq 0$ for all $x\in M$ and $$\begin{aligned}
\label{pqyss}
V=V_+-V_-, \quad |V_+|\in L^1_{{\mathrm{loc}}}(M) , \quad |V_-|\in \mathcal{K}(M),\end{aligned}$$ where $|\cdot|$ denotes the fiberwise operatornorm.
It follows in the above situation that $|V_-|\in L^1_{{\mathrm{loc}}}(M)$ (cf. Lemma VI.3 in [@gunbook]) and so $|V|\in L^1_{{\mathrm{loc}}}(M)$. In view of [@gri] $$\int p_M(t,x,y) d\mu (y) \leq 1\quad \text{ for all $(t,x)\in (0,\infty)\times M$, }$$ it follows that every bounded Borel function $w$ on $M$ satisfies (\[dre\]), and the reader may find various (weighted) $L^q$-assumptions on $w$ that imply (\[dre\]) in [@kw1; @kw2; @guenkat] (see also Chapter VI from [@gunbook]). In particular, every pointwise self-adjoint $L^1_{\mathrm{loc}}$-section $V$ of $\mathrm{End}(E)\to M$ which is bounded from below, in the sense that for some constant $C\in {\mathbb{R}}$ one has $V(x)\geq C$ for all $x\in M$, satisfies (\[pqyss\]).
In the situation of Definition \[popo\], let $\Omega\subset M$ be an open subset. Then $\Gamma_{W^{1,2}_0}(\Omega,{E};\nabla,V)$ is defined to be the space of all $f\in \Gamma_{L^2}(\Omega,{E};\nabla,V)$ which admit a sequence $(f_n)\subset \Gamma_{C^{\infty}_c}(\Omega,{E};\nabla,V)$ such that $$\begin{aligned}
&\left\|f_n-f\right\|\to 0\quad\text{as $n\to \infty$},\\
&\int\big(\nabla^{\dagger}\nabla (f_n-f_m),(f_n-f_m) \big)d\mu+\int \big(V(f_n-f_m),(f_n-f_m)\big) d\mu \to 0\quad\text{as $n,m\to \infty$}.\end{aligned}$$ By Corollary XIII.3 in [@gunbook] the symmetric densely defined sesquilinear form $$\Gamma_{C^{\infty}_c}(\Omega,{E})\times \Gamma_{C^{\infty}_c}(\Omega,{E})\ni (f_1,f_2)\longmapsto \int( \nabla^{\dagger}\nabla f_1,f_2 )d\mu+\int (Vf_1,f_2 ) d\mu \in {\mathbb{C}}$$ is closable and semibounded from below in $\Gamma_{L^2}(\Omega,{E})$. Its closure $\langle \cdot, \cdot \rangle_{\nabla,V,*}$ has the domain of definition $\Gamma_{W^{1,2}_0}(\Omega,{E};\nabla,V)$ and is explicitly given as follows: For $f, \tilde{f} \in \Gamma_{W^{1,2}_0}(\Omega,{E};\nabla,V)$ with defining sequences $(f_n), (\tilde{f}_n)\subset \Gamma_{C^{\infty}_c}(\Omega,{E};\nabla,V)$, one has $$\langle f, \tilde{f} \rangle_{\nabla,V,*}=\lim_{n\to \infty} \left( \int( \nabla^{\dagger}\nabla f_n,\tilde{f}_n )d\mu +\int (Vf_n,\tilde{f}_n ) d\mu\right),$$ and this number does not depend on the defining sequences. The semibounded from below and self-adjoint operator in $\Gamma_{L^2}(\Omega,{E})$ induced by $\langle \cdot, \cdot \rangle_{\nabla,V,*}$ is denoted with $H_\Omega(\nabla,V)$.\
In addition, $\Gamma_{W^{1,2}_0}(\Omega,{E};\nabla,V)$ becomes a Hilbert space with respect to the scalar product $$\begin{aligned}
\langle f_1,f_2 \rangle_{\nabla,V}:= (1+C)
\langle f_1,f_2 \rangle + \left\langle f_1,f_2 \right\rangle_{\nabla,V,*},\end{aligned}$$ where $C\geq 0$ is any constant satisfying $$\begin{aligned}
\label{swq}
\int( \nabla^{\dagger}\nabla f, f ) d\mu+ \int (V f, f)d\mu \geq -C\left\|f\right\|^2\quad\text{ for all $ f\in \Gamma_{C^{\infty}_c}(M,{E})$},\end{aligned}$$ noting that any two such constants produce equivalent scalar products.\
Note also that $H_\Omega(\nabla,V)$ is nothing but the Friedrichs realization of $\nabla^{\dagger}\nabla +V$ in $\Gamma_{L^2}(\Omega,{E})$. Traditionally, if $\Omega\ne M$, one says that $H_\Omega(\nabla,V)$ is the self-adjoint realization of $\nabla^{\dagger}\nabla +V$ in $\Gamma_{L^2}(\Omega,{E})$ subject to Dirichlet boundary conditions.
To simplify the notation in some special cases, we add:
\[bemee\] If ${E}\to M$ is the trivial complex line bundle and $\nabla$ the covariant derivative which is induced by the exterior differential, then we simply omit $\nabla$ and ${E}$ everywhere in the notation. Note that in this case $\nabla^{\dagger}\nabla=:-\Delta$ is by definition the usual Laplace-Beltrami operator on functions. In case $V=0$ we will in addition omit $V$ everywhere in the notation. These conventions lead to natural notations such as $W^{1,2}_0(\Omega;V)$, $H_{\Omega}(V)$, $W^{1,2}_0(\Omega)$, $L^2(\Omega)$, $H_{\Omega}$. In particular, the self-adjoint operator $H_{\Omega}$ in $L^2(\Omega)$ is $-\Delta$ with Dirichlet boundary conditions and the integral kernel $\exp(-tH_\Omega)(x,y)$ of $\exp(-tH_{\Omega})$ is precisely $p_{\Omega}(t,x,y)$ [@gri].
Here is an example from quantum mechanics:
If $M$ is the Euclidean space ${\mathbb{R}}^3$ and $\eta=\sum^3_{j=1}\eta_j dx^j$ is a smooth real-valued $1$-form on ${\mathbb{R}}^3$ (considered as a magnetic potential), $Z\in{\mathbb{N}}$, and $V:{\mathbb{R}}^3\to {\mathbb{R}}$ is the Coulomb potential $V(x):=-Z/|x-x_0|$ with mass at $x_0\in{\mathbb{R}}^3$, then with $$\nabla^{\eta}: C^{\infty}({\mathbb{R}}^3)\longrightarrow \Gamma_{C^{\infty}}({\mathbb{R}}^3,T^*_{{\mathbb{C}}}{\mathbb{R}}^3),\quad
\nabla^{\eta}f:=\sum^3_{j=1}(\partial_j f+\sqrt{-1}f\eta_j) dx^j ,$$ the datum $$({\mathbb{R}}^3\times {\mathbb{C}}, \nabla^{\eta}, V)\longrightarrow {\mathbb{R}}^3$$ is a Kato-Schrödinger bundle (cf. Proposition VI.14 in [@gunbook]). In this case, identifying sections in ${\mathbb{R}}^3\times {\mathbb{C}}\to {\mathbb{R}}^3$ with functions on $M$, the semibounded self-adjoint operator $H_{{\mathbb{R}}^3}(\nabla^{\eta},V)$ in $L^2({\mathbb{R}}^3)$ is the Hamilton operator of an atom having one electron and a nucleus having $Z$ protons, in the magnetic field given by $$d\eta=\sum_{i<j} (\partial_i \eta_j-\partial_j \eta_i)dx^i \wedge dx^j.$$ More generally, one could replace in this example ${\mathbb{R}}^3$ with a Riemannian $3$-fold $M$ such that $p_M(t,x,y)$ satisfies a Gaussian upper bound of the form $$p_M(t,x,y)\leq C_1t^{-3/2}\exp\left(-\frac{d(x,y)^2}{C_2t}\right),\quad (t,x,y)\in (0,\infty)\times M\times M,$$ taking $$V(x):=-Z\int^{\infty}_0 p_M(t,x,x_0)d t$$ to be the Coulomb potential with mass at $x_0$. One could even take the electrons spin into account, as is shown in [@guneysu2].
Here is an example from geometry:
\[diracc\] 1. A *geometric Dirac bundle over $M$* is understood to be a datum $$({E};c,\nabla)\longrightarrow M$$ such that
- ${E}\to M$ is a complex metric vector bundle
- $c$ is a Clifford multiplication on ${E}\to M$, in the sense that $c$ is a homomorphism of real (!) vector bundles $ c: TM\to\mathrm{End}({E})$, such that for all $X\in \Gamma_{C^{\infty}}(M,TM)$ one has $
c(X)=-c(X)^*$ and $c(X)^{*}c(X)=\left|X\right|^2$
- $\nabla$ is a Clifford connection on $({E};c)\to M$, that is, $\nabla$ is a metric covariant derivative on ${E}\to M$ such that for all $X,Y\in \Gamma_{C^{\infty}}(M,TM)$, $\Psi\in\Gamma_{C^{\infty}}(M,{E})$ one has $$\nabla_X(c(Y)\Psi)=c(\nabla^{T M}_X Y)\Psi+ c(Y)\nabla_X \Psi.$$
Then the associated geometric Dirac operator is the first order linear partial differential operator on ${E}\to M$ defined by $D(c,\nabla):=\sum^m_{j=1} c(e_j)\nabla_{e_j}$, where $e_j$ is any local orthonormal-frame for $TM\to M$ and $m=\dim M$. The operator $D(c,\nabla)$ is formally self-adjoint with symbol $c$, in particular elliptic. By the abstract Lichnerowicz formula [@nico; @lawson], $$V(c,\nabla):= D(c,\nabla)^2-\nabla^{\dagger}\nabla:M\to \mathrm{End}({E})$$ is a smooth potential. In case $V(c,\nabla)$ is Kato decomposable (which is the case, e.g., if $V(c,\nabla)$ is bounded from below), we call $$({E};c,\nabla)\longrightarrow M$$ a *geometric Kato-Dirac bundle*, and we get the Kato-Schrödinger bundle $$\big({E}, \nabla,V(c,\nabla) \big)\longrightarrow M.$$ In this case, $\Gamma_{W^{1,2}_0}(\Omega,{E};\nabla,V(c,\nabla))$ is given by all $f\in \Gamma_{L^2}(\Omega,{E})$ with $D(c,\nabla)f\in \Gamma_{L^2}(\Omega,{E})$ in the sense of distributions, which admit a sequence $f_n\in \Gamma_{C^{\infty}_c}(\Omega,{E})$, $n\in{\mathbb{N}}$, with $$\left\|f_n-f\right\|+\left\|D(c,\nabla)f_n-D(c,\nabla)f\right\|\to 0.$$ We have $$\left\langle f_1,f_2 \right\rangle_{ \nabla,V(c,\nabla),*}= \left\langle D(c,\nabla)f_1,D(c,\nabla)f_2 \right\rangle ,$$ so that in particular the scalar product on $\Gamma_{W^{1,2}_0}(\Omega,{E};\nabla,V(c,\nabla))$ is given by $$\left\langle f_1,f_2 \right\rangle_{ \nabla,V(c,\nabla)}= \left\langle f_1,f_2\right\rangle+ \left\langle D(c,\nabla)f_1,D(c,\nabla)f_2 \right\rangle .$$ 2. For example, if $M$ is a spin manifold, then every spin structure on $M$ canonically induces a geometric Dirac bundle over $M$ [@lawson; @nico], which is a geometric Kato-Dirac bundle, if the scalar curvature of $M$ is Kato decomposable.\
3. Another example of a geometric Dirac bundle over $M$ (which does not require any topological assumptions on $M$) is given by taking ${E}=\wedge T^*_{{\mathbb{C}}}M$ with the metric induced by the Riemannian metric on $M$ and $\nabla$ the covariant derivative which is induced by the Levi-Civita connection on $M$, and $$c(\alpha)\beta:=\alpha\wedge\beta - \iota_{\alpha^{\sharp}}\beta, \quad \alpha\in \Gamma_{C^{\infty}}(M,T^*_{{\mathbb{C}}}M),\beta\in \Gamma_{C^{\infty}}(M,\wedge T^*_{{\mathbb{C}}}M),$$ where $\iota_{\alpha^{\sharp}}$ denotes the contraction with the vector field $\alpha^{\sharp}$ which is induced by $\alpha$ via the Riemannian metric on $M$. In this case, one can calculate that $D(c,\nabla)=d+d^{\dagger}$, so that $D(c,\nabla)^2$ is by definition the Laplace-Beltrami operator acting on differential forms. Note that the restriction of $D(c,\nabla)^2$ to $0$-forms is precisely the usual Laplace-Beltrami operator $-\Delta$ on functions, and that the restriction of $D(c,\nabla)$ to $0$-forms can be identified with the gradient (cf. Proposition 11.2.1 in [@nico] for detailed proofs of these facts).
Given a Kato-Schrödinger bundle $$({E},\nabla, V)\longrightarrow M,$$ the inclusion $$\Gamma_{W^{1,2}_0}(\Omega,{E};\nabla ,V )\subset\big\{ f\in \Gamma_{W^{1,2}_0}(M,{E};\nabla,V ): f|_{M\setminus \Omega}=0\>\>\text{\rm $\mu$-a.e. }\big\}$$ is easily seen to be satisfied without any further assumptions on the underlying data: Indeed, given $\psi\in\Gamma_{W^{1,2}_0}(\Omega,{E},\nabla , V )$ we can per definitionem pick a sequence $\psi_{n}\in\Gamma_{C^{\infty}_c}(\Omega,{E})$ with $\left\|\psi_n- \psi\right\|_{ \nabla ,V }\to 0$, so that $\left\|\psi_n- \psi\right\|\to 0$ and so $\psi=0$ $\mu$-a.e. in $M\setminus \Omega$.
*The question we address in this paper is: Under which condition on $\Omega$ does the reverse inclusion $$\begin{aligned}
\label{poa44}
\Gamma_{W^{1,2}_0}(\Omega,{E};\nabla,V )\supset\big\{ f\in \Gamma_{W^{1,2}_0}(M,{E};\nabla,V ): f|_{M\setminus \Omega}=0\>\>\text{\rm $\mu$-a.e. }\big\}\end{aligned}$$ hold true?*
By the above, (\[poa44\]) is equivalent to
$$\begin{aligned}
\label{poa}
\Gamma_{W^{1,2}_0}(\Omega,{E};\nabla,V )=\big\{ f\in \Gamma_{W^{1,2}_0}(M,{E};\nabla,V ): f|_{M\setminus \Omega}=0\>\>\text{\rm $\mu$-a.e. }\big\}.\end{aligned}$$
The validity of (\[poa\]) turns out to be equivalent to the corresponding Dirichlet heat-flow of $H_{\Omega}(\nabla,V)$ on $\Omega$ being realized as the heat-flow of $H_M(\nabla,V)$ on $M$ perturbed by the potential $\infty\cdot 1_{M\setminus \Omega}$, for one has:
\[main1\] Let $\Omega\subset M$ be an arbitrary open subset and let $({E},\nabla,V)\to M$ be a Kato-Schrödinger bundle over $M$. Then one has (\[poa\]), if and only if for all $t\geq 0$ one has $$\begin{aligned}
\label{aposssay}
\lim_{n\to\infty} \mathrm{exp}\big(-tH_M(\nabla,V+n1_{M\setminus \Omega})\big) = \mathrm{exp}\big(-t H_\Omega(\nabla, V)\big)P_{\Omega}\end{aligned}$$ strongly as bounded operators in $\Gamma_{L^2}(M,{E})$.
The proof of Proposition \[main1\] will be given in Section \[pon\]. It is a consequence of monotone convergence results for sesquilinear forms.\
As one might guess, the validity of (\[poa\]) or (\[aposssay\]) should depend on the local regularity of $\partial \Omega$. In order to make precise how (\[poa\]) or (\[aposssay\]) depend on the local regularity of $\partial \Omega$, the probabilistic Definition \[paa\] below will turn out to be crucial. To this end, let $W(\hat{M})$ denote the Wiener space of continuous paths $\omega:[0,\infty)\to \hat{M}$ with its Borel-sigma-algebra $\mathcal{F}$, where $ \hat{M}=M\cup\{\infty_M\}$ is defined to be the (essentially uniquely determined) Alexandroff-compactification of $M$ in case $M$ is noncompact, and $\hat{M}:=M$ if $M$ is compact.[^2] In any case, $W(\hat{M})$ is equipped with the topology of locally uniform convergence. Let $${\mathbb{X}}: [0,\infty)\times W(\hat{M})\longrightarrow \hat{M},\quad {\mathbb{X}}_t(\omega):= \omega(t)$$ denote the coordinate process, and for every $x\in M$, the symbol $\mathbb{P}^x $ stands for the Riemannian Brownian motion measure on $\mathcal{F}$ with $$\mathbb{P}^x\{{\mathbb{X}}_0=x\}=1.$$ In other words, the transition density of $\mathbb{P}^x$ with respect to $\mu$ is given by the natural extension of $ \exp(-t H_M)(x,y)$ to $(0,\infty)\times \hat{M}\times \hat{M}$. We refer the reader to Section 8 in [@grio] and the references therein for the definition and the existence of $\mathbb{P}^x$.\
With $\mathcal{F}_*$ the filtration of $\mathcal{F}$ which is generated by ${\mathbb{X}}$, the family $\mathbb{P}^{x}$, $x\in M$, has the strong Markov property for stopping times in $\mathcal{F}_*$. If $\Omega\subset \hat{M}$ is an open set, then[^3] $$\begin{aligned}
\alpha_\Omega:W(\hat{M})\longrightarrow [0,\infty],\quad \alpha_\Omega:= \inf\big\{t>0: {\mathbb{X}}_t\in \hat{M}\setminus \Omega\big\}\end{aligned}$$ denotes the first exit time of ${\mathbb{X}}$ from $\Omega$. It is well-known that $\alpha_\Omega$ is an $\mathcal{F}_*$-stopping time. In particular, if $M$ is noncompact, then the stopping time $\alpha_M$ is the explosion time of $\mathbb{X}$, and in this case the point at infinity $\infty_M$ of $\hat{M}$ is $\mathbb{P}^x$-almost surely (a.s.) absorbing for all $x\in M$, in the sense that $$\mathbb{P}^x\left(\{\alpha_M=\infty\}\bigcup \{\alpha_M<\infty\>\text{ and $\mathbb{X}_t=\infty_M$ for all $t\in [\alpha_M,\infty)$} \}\right)=1\quad\text{ for all $x\in M$}.$$
Likewise, $$\begin{aligned}
\beta_\Omega:= \inf\big\{t>0: \int^t_0 1_{ \hat{M}\setminus \Omega}({\mathbb{X}}_s) { d}s >0\big\}:W(\hat{M})\longrightarrow [0,\infty]\end{aligned}$$ denotes the first penetration time of ${\mathbb{X}}$ into $ M\setminus \Omega$. One trivially has $$\begin{aligned}
\label{didaa}
\alpha_\Omega\leq \beta_\Omega,\end{aligned}$$ and again $\beta_\Omega$ induces an $\mathcal{F}_*$-stopping time. Note that both $\alpha$ and $\beta$ are pathwise monotonely increasing in $\Omega$, and that one pathwise has $$\beta_\Omega= \inf\big\{t\geq 0: \int^t_0 1_{ \hat{M}\setminus \Omega}({\mathbb{X}}_s) { d}s >0\big\}= \inf\big\{t\geq 0: {\mathbb{X}}_s\in \Omega\text{ for a.e. $s\in [0,t]$}\big\}.$$
If $\omega\in W(\hat{M})$ is such that $\omega(0)\in \Omega$, then one also gets $$\beta_\Omega(\omega) \geq \alpha_\Omega(\omega)= \inf\big\{t\geq 0: \omega(t)\in \hat{M}\setminus \Omega\big\}>0.$$
The following regularity result will be very useful in the sequel:
\[dnaarrr\] Let $\omega\in W(\hat{M})$ be such that in case $\alpha_M(\omega)<\infty$ one has $\omega(t)=\infty_M$ for all $t\in [\alpha_M(\omega),\infty)$, let $\Omega$ be an open subset of $M$, and let $\Omega_n\subset \Omega$, $n\in{\mathbb{N}}$, be open with $\Omega_n\subset \Omega_{n+1}$ for all $n\in{\mathbb{N}}$, $\omega(0)\in \Omega_1$ and $\bigcup_{n\in{\mathbb{N}}}\Omega_n=\Omega$.\
*a)* One has $\alpha_{\Omega_n}(\omega)\nearrow\alpha_{\Omega }(\omega) $ as $n \to\infty$.\
*b)* Assume in addition that for each $n\in \mathbb{N}$ there exists an open subset $\Upsilon_n\subset M$ such that $\Upsilon_n\cap \Omega=\Omega_n$ and $\overline{\Omega}=\bigcup_{n\in{\mathbb{N}}}(\overline{\Omega}\cap \Upsilon_n)$. Then one has $\beta_{\Omega_n}(\omega)\nearrow\beta_{\Omega }(\omega) $ as $n \to\infty$.
While Lemma \[dnaarrr\] a) is well-known, a proof of Lemma \[dnaarrr\] b) will be given in Section \[weqyx\].
\[lop\] If $\{\Upsilon_n:n\in{\mathbb{N}}\}$ is a family of open subsets of $M$ such that $\Upsilon_n\subset \Upsilon_{n+1}$ for all $n\in{\mathbb{N}}$ and $\bigcup_{n\in{\mathbb{N}}}\Upsilon_n=M$, then the family $\Omega_n:=\Omega\cap \Upsilon_n$, $n\in{\mathbb{N}}$, satisfies the hypothesis of Lemma \[dnaarrr\] part b).
The following definition is motivated by [@herbst] [@stroock], who treat the Euclidean space ${\mathbb{R}}^m$:
\[paa\] An open set $\Omega\subset M$ is called *Kac regular*, if one has $$\begin{aligned}
\label{apossaq}
\mathbb{P}^x \{ \alpha_\Omega =\min(\beta_\Omega,\alpha_M) \}=1\quad\text{ for all $x\in \Omega$}.\end{aligned}$$
Note that we do not assume $M$ to be stochastically complete, that is $\alpha_M=\infty$ $\mathbb{P}^x$-a.s. for all $x\in M$, noting that in the latter case one has $$\mathbb{P}^x \{ \alpha_\Omega =\min(\beta_\Omega,\alpha_M) \}=\mathbb{P}^x \{ \alpha_\Omega =\beta_\Omega \},$$ and some technical problems are not present, that we will have to deal with. The connection between problems such as (\[poa\]) and Kac regularity is clarified in the following result, which has been established in the Euclidean space ${\mathbb{R}}^m$ in [@herbst]:
\[scal1\] Let $\Omega\subset M$ be an arbitrary open subset. The following properties are equivalent:
- $\Omega$ is Kac regular.
- One has $$\begin{aligned}
\label{alkdd}
W^{1,2}_{0}(\Omega )=\big\{f\in W^{1,2}_0 (M ): f|_{M\setminus \Omega}=0\>\>\text{\rm $\mu$-a.e. }\big\}.\end{aligned}$$
- For all $ t\geq 0$ one has $$\begin{aligned}
\label{apossq}
\lim_{n\to\infty}\mathrm{exp}\big(-t H_M( n 1_{M\setminus \Omega})\big) = \mathrm{exp}\big(-t H_\Omega \big)P_\Omega\end{aligned}$$ strongly as bounded operators in $L^2(M)$.
The proof of Proposition \[scal1\] will be given in Section \[pon\]. Note that in our notation $H_M( n 1_{M\setminus \Omega})$ is the Friedrichs realization of $-\Delta+n 1_{M\setminus \Omega}$ in $L^2(M)$, and $H_\Omega $ is the Dirichlet realization of $-\Delta$ in $L^2(\Omega)$. The equivalence of (\[alkdd\]) and (\[apossq\]) follows immediately from specializing Proposition \[main1\] to the scalar case. The equivalence of (\[apossq\]) and Kac-regularity will be established by using a Feynman-Kac formula.
\[scal2\]1. The potential theoretic variant of (\[alkdd\]) does not see anything from the geometry of $\Omega$, namely [@fuku], for *every open $\Omega\subset M$* one has $$W^{1,2}_0(\Omega )=\big\{ f\in W^{1,2}_0(M ): \breve{f}|_{M\setminus \Omega}=0\>\> \text{\rm q.e.}\big\},$$ where q.e. (quasi everywhere) is understood with respect to the capacity associated with the regular strongly local Dirichlet form $$\left\langle f_1,f_2 \right\rangle_{ * }= \int (df_1,df_2) d\mu$$ in $L^2(M)$ with domain of definition $W^{1,2}_0(M )$, and where $\breve{f}$ denotes the quasi-continuous representative of $f$. This again reflects the subtlety of the questions under investigation.\
2. It was conjectured in [@simon] by B. Simon that open subsets $\Omega\subset {\mathbb{R}}^m$ of the Euclidean space with ${\mathbb{R}}^m\setminus \Omega$ perfect satisfy (\[apossq\]). A counterexample to this conjecture was given in [@hedberg] and [@stollmann1] (cf. [@stollmann2], p.127), who even show that there exists $\Omega\subset {\mathbb{R}}^m$ open with ${\mathbb{R}}^m\setminus \Omega$ compact and $${\mathbb{R}}^m\setminus \Omega= \overline{\mathring{{\mathbb{R}}^m\setminus \Omega }}$$ such that (\[alkdd\]) and thus (\[apossq\]) fails. Note that this counterexample also entails that even for quasi-continuous $f\rq{}s$ the assumption $f|_{M\setminus \Omega}=0$ a.e. does not imply that $f|_{M\setminus \Omega}=0$ q.e.
In order to formulate our main result, we add:
\[lipreg\] An open subset $\Omega\subset M$ is called *Lipschitz regular*, if there exists a family $\{\Omega_n: n\in{\mathbb{N}}\}$ of open subsets of $\Omega$ having Lipschitz boundary and compact closure in $M$, such that for all $n\in {\mathbb{N}}$ there exists an open subset $\Upsilon \subset M$ such that $\Upsilon_n\cap \Omega = \Omega_n$ and $\overline{\Omega}=\bigcup_{n\in{\mathbb{N}}}(\overline{\Omega}\cap \Upsilon_n)$.
Here comes our main result:
\[main2\] Let $\Omega$ be an arbitrary open subset of $M$.\
[a)]{} If $\Omega$ is Kac regular, then for every Kato-Schrödinger bundle $({E},\nabla,V)\to M$ one has (\[poa\]) and (\[aposssay\]).\
[b)]{} If $\Omega$ is Lipschitz regular, then $\Omega$ is Kac regular.
Using transversality theory one finds that every open subset $\Omega\subset M$ having a smooth boundary is Lipschitz regular, implying:
Every open subset $\Omega\subset M$ having a smooth boundary is Lipschitz regular and thus Kac regular.
The proof of the fact that $\Omega$ is Lipschitz regular follows from Lemma 18 in [@pigo]. For the sake of completeness, we provide a sketch of proof: pick a family $\{W_n:n\in{\mathbb{N}}\}$ of relatively compact open subsets of $M$ with smooth boundary such that $W_n\subset W_{n+1}$ for all $n\in{\mathbb{N}}$ and $\bigcup_{n\in{\mathbb{N}}}W_n=M$. Let $i_n:W_n\rightarrow M$ be the natural inclusion of $W_n$ in $M$. Using classical results of transversality theory in differential topology we can find for each $n\in \mathbb{N}$ a smooth embedding $j_n:\overline{W_n}\rightarrow M$ such that $j_n$ and $i_n$ are arbitrarily close in the Whitney strong topology and $j_n(\partial\overline{W_n})$ is transverse to $\partial\overline{\Omega}$. Altogether this tells us that $\{j_n(W_n): n\in{\mathbb{N}}\}$ is a family of open subsets of $M$ such that $j_n(W_n)\subset j(W_{n+1})$ for all $n\in{\mathbb{N}}$ and $\bigcup_{n\in{\mathbb{N}}}j(W_n)=M$ in a way that $\Omega_n:=j_n(W_n)\cap \Omega$ has a Lipschitz boundary. It follows from Remark \[lop\] that the set $\Omega$ becomes a Lipschitz regular set. We refer to [@GuPo] and [@MoHi] for classical results about transversality theory and for the definition of Whitney topology.
We [**conjecture**]{} that every open subset $\Omega\subset M$ having a locally Lipschitz boundary is Lipschitz regular.
Theorem \[main2\] a) is new even in the Euclidean case, where so far only $-\Delta$ has been treated, and not even Schrödinger operators $-\Delta+V$. The proof of Theorem \[main2\] will be given in Section \[pon\]. We continue with remarks on this proof: The proof of Theorem \[main2\] a) relies on Proposition \[main1\] and Proposition \[scal1\], while both worlds are linked through various partially completely new covariant Feynman-Kac formula. We believe it is a very surprising fact, that no conditions on the negative part of $V$ is needed in Theorem \[main2\] a), as such a condition is certainly required for the covariant Feynman-Kac formula to hold (roughly speaking, because Brownian paths have an infinite speed). The point here is that using rather subtle approximation arguments, one can reduce everything to the case of $V$’s that are bounded from below by a constant.\
The proof of Theorem \[main2\] b) relies again on Proposition \[scal1\]: In a first step , we will give an analytic proof of the stronger statement $$\begin{aligned}
\label{asoiss}
W^{1,2}_0(\Omega )=\big\{ f\in W^{1,2}(M ): f|_{M\setminus \Omega}=0 \>\>\text{\rm $\mu$-a.e. } \big\},\end{aligned}$$ under the assumption that $\Omega$ is relatively compact with $\partial \Omega$ Lipschitz, directly verifying (\[alkdd\]). As usual, $W^{1,2}(M )$ is defined to be the space of all $f\in L^2(M)$ such that $df\in\Gamma_{L^2}(M,T^*M)$ in the sense of distributions. In a second step we will then combine this local result with a probabilistic approximation argument, directly confirming (\[apossaq\]).
\[ende\] 1. It is false that every open subset $\Omega\subset M$ with smooth boundary satisfies[^4] (\[asoiss\]). As a counterexample, one can consider the Euclidean ball and its open upper half $$M:= \{(x,y,z)\in {\mathbb{R}}^3: x^2+y^2+z^2<1\} ,\quad \Omega:=\{(x,y,z)\in M: z>0\},$$ with the Euclidean metric.\
2. It is shown in [@herbst] that in the Euclidean case $\Omega$’s with $\partial \Omega$ having the segment property are Kac regular. In fact, many comparable Euclidean results can be found in [@stollmann1]. The segment property is more general then being locally Lipschitz, but is also a concept that does not apply to manifolds.
In view of Example \[diracc\].3, the following result includes (when applied to $0$-forms) a criterion for a harmonic function $f$ on an open subset $\Omega\subset M$ with $f=0$ $\mu$-a.e. in $M\setminus \Omega$ to have vanishing gradient in $\Omega$ (cf. [@herbst] for the Euclidean case):
\[pw\] Let $M$ be geodesically complete and let $({E};c,\nabla)\to M$ be a geometric Kato-Dirac bundle. Then for every open subset $\Omega\subset M$ which is Kac regular, and every $\Psi$ with $$\begin{aligned}
&\Psi\in \Gamma_{L^2}(M, {E}),\quad D (c,\nabla)\Psi\in \Gamma_{L^2}(M, {E}),\quad\text{$\Psi=0$ $\mu$-a.e. in $M\setminus \Omega$,}\end{aligned}$$ one has the implication $$D (c,\nabla)^2\Psi=0\>\>\>\text{in $\Omega$}\quad\Rightarrow \quad\text{$D(c,\nabla)\Psi=0$ in $\Omega$},$$ where above the action of $D (c,\nabla)^2$ and $D(c,\nabla)$ is understood in the distributional sense.
Recall that $({E};c,\nabla)\to M$ induces the Kato-Schrödinger bundle $$({E},\nabla, V(c,\nabla))\longrightarrow M,$$ As $M$ is geodesically complete one has $$\begin{aligned}
\label{helpme}
\Psi\in \Gamma_{W^{1,2}_0}\big(M, {E}; \nabla,V(c,\nabla)\big),\end{aligned}$$ which in view of the product rule for $D (c,\nabla)$ (p. 171 in [@gunbook]) and a manifold version of the Meyers-Serrin Theorem (Theorem 2.9 in [@guidetti]) can be seen as in the case of functions [@aubin]. Now (\[helpme\]) and the Kac regularity of $\Omega$ imply $$\Psi \in\Gamma_{W^{1,2}_0}\big(\Omega, {E},\nabla , V(c,\nabla) \big),$$ and we can pick a sequence $\Psi_n\in \Gamma_{C^{\infty}_c}(\Omega,{E})$ with $$\lim_{n\to\infty}\|\Psi_n-\Psi \|_{ \nabla ,V(c,\nabla) }\>=0.$$ It follows that $$\begin{aligned}
&\big\| D(c,\nabla ) \Psi \big\|^2 \>= \lim_{n\to\infty} \big\langle D (c,\nabla) \Psi_n, D (c,\nabla) \Psi \big\rangle\\
&= \int_\Omega ( D (c,\nabla)\Psi_n, D (c,\nabla) \Psi ){ d}\mu =\int_\Omega ( \Psi_n, D (c,\nabla)^2 \Psi ){ d}\mu =0,\end{aligned}$$ where the integration by parts is justified as $\Psi$ is smooth in $\Omega$ by local elliptic regularity, and $\Psi_n$ is smooth and compactly supported in $\Omega$.
Note that in Corollary \[pw\] the geodesic completeness of $M$ was only assumed to guarantee $$\Psi\in \Gamma_{W^{1,2}_0}(M, {E};\nabla, V(c,\nabla)),$$ which one could assume instead.
Proof of Lemma \[dnaarrr\] {#weqyx}
==========================
a\) This statement is well-known and elementary to check.\
b) As this observation seems to be new, we have decided to add its proof (which is rather technical): Assume first that there exists $t_0\in (0,\infty)$ such that $\beta_\Omega(\omega) =t_0$. This means that the Lebesgue measure of the set $\{s\in [0,t_0]: \omega(s)\notin \Omega\}$ is zero, briefly $$|\{s\in [0,t_0]: \omega(s)\notin \Omega\}|=0$$ and moreover that for each $\epsilon>0$ we have $$|\{s\in [0,t_0+\epsilon]: \omega(s)\notin \Omega\}|>0.$$ Let $n_0>0$ be such that $$\omega([0,t_0])\cap \Omega= \omega([0,t_0])\cap \Omega_{n_0}.$$ In this way we have $$|\{s\in [0,t_0]: \omega(s)\notin \Omega_n\}|=0$$ for each $n\geq n_0$, because $$\{s\in [0,t_0]: \omega(s)\notin \Omega_n\}=\{s\in [0,t_0]: \omega(s)\notin \Omega_{n_0}\}=\{s\in [0,t_0]: \omega(s)\notin \Omega\}$$ and the latter has Lebesgue measure $0$. Moreover we have $$|\{s\in [0,t_0+\delta]: \omega(s)\notin \Omega_n\}|>0$$ for each $\delta>0$ and for each $n\geq n_0$, because $$\{s\in [0,t_0+\delta]: \omega(s)\notin \Omega\}\subset \{s\in [0,t_0+\delta]: \omega(s)\notin \Omega_n\}\subset \{s\in [0,t_0+\delta]: \omega(s)\notin \Omega_{n_0}\}$$ and $$|\{s\in [0,t_0+\delta]: \omega(s)\notin \Omega\}|)>0.$$ Therefore $\beta_{\Omega_n}(\omega)=t_0$ for each $n\geq n_0$ and hence we can conclude that $\lim \beta_{\Omega_n}(\omega)=\beta_\Omega(\omega)$ as $n\rightarrow \infty$.\
Assume now that there is no $t_0\in (0,\infty)$ such that $\beta_\Omega(\omega)=t_0$. Therefore $$|\{s\in [0,\infty): \omega(s)\notin \Omega\}|=0.$$ In this case we set $\beta_\Omega(\omega):=\infty$. Consider first the case where $\overline{\omega([0,\infty))}$ is compact. Let $$A:=\{s\in [0,\infty): \omega(s)\notin \Omega\}$$ and let $B:=[0,\infty)\setminus A$. Let $n_0$ be such that $\omega(B)\subset \Omega_{n_0}$. Since $\omega(B)\subset \Omega_{n_0}$ we have $$\omega([0,\infty))\cap \Omega=\omega([0,\infty))\cap \Omega_n$$ for each $n\geq n_0$. Therefore $$|\{s\in [0,\infty): \omega(s)\notin \Omega_n\}|=0$$ for any $n\geq n_0$ and hence we can conclude that $\lim \beta_{\Omega_n}(\omega)=\beta_\Omega(\omega)$ as $n\rightarrow \infty$. Finally consider the case where $\overline{\omega([0,\infty))}$ is not compact. The sequence $\beta_{\Omega_n}(\omega)$ is increasing and unbounded, because given an arbitrarily big $t_0$ we can find an integer $n_0$ such that $\omega([0,t_0])\cap \Omega=\omega([0,t_0])\cap \Omega_{n_0}$ and $\omega(t_1)\notin F_{n_0}$ for some $t_1>t_0$. Hence $\beta_{\Omega_{n_0}}(\omega)\geq t_0$ and therefore $\lim \beta_{\Omega_n}(\omega)=\infty$ as $n\rightarrow \infty$.
Proofs of main results {#pon}
======================
We first introduce some notation that will be relevant in the sequel. Let $$({E},V,\nabla)\longrightarrow M$$ be a Kato-Schrödinger bundle.
Given an open subset $\Omega\subset M$ let $$\begin{aligned}
\label{aqya}
\Gamma_{\widetilde{W}^{1,2}_0}(\Omega,{E};\nabla,V):=\big\{ f\in \Gamma_{W^{1,2}_0}(M,{E};\nabla,V): f|_{M\setminus \Omega}=0\>\>\text{ \rm $\mu$-a.e. }\big\}\subset \Gamma_{W^{1,2}_0}(M,{E};\nabla,V),\end{aligned}$$ and let $\widetilde{H}_\Omega(\nabla,V)$ denote the self-adjoint nonnegative operator in $\Gamma_{L^2}(\Omega,{E})$ which corresponds to $\left\langle \cdot,\cdot\right\rangle_{\nabla,V,*}$ with domain of definition $\Gamma_{\widetilde{W}^{1,2}_0}(\Omega,{E};\nabla,V)$, a closed densely defined symmetric sesquilinear form in $\Gamma_{L^2}(\Omega,{E})$. Again we will follow the conventions from (\[bemee\]) analogously.
The above auxiliary operator $\widetilde{H}_\Omega(\nabla,V)$ satisfies the following two important results, without any further assumptions on $\Omega$:
\[zbl\] For every open subset $\Omega$ of $M$ and every $t\geq 0$ one has $$\lim_{n\to\infty}\mathrm{exp}\big(-t H_M(\nabla,V+n1_{M\setminus \Omega})\big)= \mathrm{exp}\big(-t \widetilde{H}_{\Omega}(\nabla,V)\big)P_{\Omega}$$ strongly as bounded operators in $\Gamma_{L^2}(M,{E})$.
This is a simple application of Theorem \[monn\]: Regarding $n1_{M\setminus \Omega}$ as a bounded multiplication operator it follows that with $$Q_n:=\left\langle \cdot,\cdot\right\rangle_{\nabla,V+n1_{M\setminus \Omega},*}=\left\langle \cdot,\cdot\right\rangle_{\nabla,V,*}+ n\int_{M\setminus \Omega } (\cdot,\cdot) d\mu,\quad
{\mathrm{Dom}}(Q_n):= \Gamma_{ W^{1,2}_0 }(M,{E};\nabla,V),$$ one has $$S_{Q_n}=H_M(\nabla,V+n1_{M\setminus \Omega}).$$ It follows that $f\in \Gamma_{W^{1,2}_0 }(M,{E};\nabla,V)$ is in ${\mathrm{Dom}}(Q_{\infty})$, if and only if $$\sup_{n\in{\mathbb{N}}}n\int_{M\setminus \Omega} |f|^2 d\mu<\infty,$$ which again is equivalent to $f=0$ in $M\setminus \Omega$ $\mu$-a.e., and for such $f$s one has $$Q_{\infty}(f,f)=\lim_{n\to \infty}Q_n(f,f)=\left\langle f,f\right\rangle_{\nabla,V,*},$$ proving $S_{Q_\infty}=\widetilde{H}_{\Omega}(\nabla,V)$.
Let us now prepare the formulation of various covariant Feynman-Kac formulae, that is, path integral representations of the semigroups corresponding to covariant Schrödinger operators. Note that the usual Feynman-Kac formula [@guneysu; @hsu] is for semigroups of the form $\mathrm{exp}\big(-t H_M(\nabla,V)\big)$, and it is the aim of this section to derive formulae for the semigroups $\mathrm{exp}\big(-t \widetilde{H}_{\Omega}(\nabla,V) \big)$ and $\mathrm{exp}\big(-t H_{\Omega}(\nabla,V)\big)$. As these formulae rely on semimartingale theory and stochastic integrals (through the definition of the Stratonovic parallel transport), one needs to work here with a filtered probability, which allows to pick continuous versions of these semimartingales. We refer the reader to [@hsu; @guneysu] for the basics of stochastic analysis on manifolds. To set the stage, suppose that for every $x\in M$ we are given an adapted Riemannian Brownian motion $\mathbb{X}(x)$ on $M$ with $${\mathbb{P}}\{\mathbb{X}_0(x)=x\}=1,$$ which is defined on some fixed filtered probability space which satisfies the usual assumptions of completeness and right-continuity. In other words, $\mathbb{X}(x)$ is an $M$-valued continuous and adapted process defined up to its explosion time, such that the law of ${\mathbb{X}}(x)$ is $\mathbb{P}^x$. Given an open subset $\Omega\subset M$, let $\alpha_\Omega(x):=\alpha_\Omega({\mathbb{X}}(x))$ denote the first exit time of $\mathbb{X}(x)$ from $\Omega$, and let $\beta_\Omega(x):=\beta_\Omega({\mathbb{X}}(x))$ denote the penetration time of $\mathbb{X}(x)$ to $M\setminus \Omega$. Let ${\slash\slash}^{\nabla}_t(x)$, $t<\alpha_M(x)$, be the Stratonovic parallel transport with respect to $\nabla$ along the paths of $\mathbb{X}(x)$, which, $\nabla$ being metric, is a random variable taking values in the unitary maps ${E}_{x} \to {E}_{{\mathbb{X}}(t)}$. So far $x\in M$ was arbitrary. On the other hand, for $\mu$-a.e. $x\in M$, we can define pathwise for $t<\alpha_M(x)$ a random variable $\mathscr{A}^{\nabla}_V(x,t)$ taking values in $\mathrm{End}( {E}_x)$, to be the unique locally absolutely continuous solution of $$\begin{aligned}
&({ d}/{ d}t)\mathscr{A}^{\nabla}_V(x,t)=-\mathscr{A}^{\nabla}_V(x,t)\big({\slash\slash}^{\nabla}_t(x)^{-1}V({\mathbb{X}}_t(x)){\slash\slash}^{\nabla}_t(x)\big),\\
&\mathscr{A}^{\nabla}_V(x,0)=1_{\mathrm{End}( {E}_{x})}.\end{aligned}$$
The existence and uniqueness of $\mathscr{A}^{\nabla}_V(x,t)$ is guaranteed by the $L^1_{{\mathrm{loc}}}$-assumption on $V$, which implies $$\mathbb{P} \left\{|V({\mathbb{X}}(x))|\in L^1_{{\mathrm{loc}}}\big[0,\alpha_M(x)\big)\right\}=1\quad\text{ for $\mu$-a.e. $x\in M$,}$$ and so $$\mathbb{P}\left\{\left|{\slash\slash}^{\nabla}_t(x)^{-1}V({\mathbb{X}}_t(x)){\slash\slash}^{\nabla}_t(x)\right|\in L^1_{{\mathrm{loc}}}\big[0,\alpha_M(x)\big)\right\}=1\quad\text{ for $\mu$-a.e. $x\in M$}.$$ We refer the reader to [@guneysu] for detailed proofs of these facts.
\[aposssy\] For every $\Omega\subset M$ open, $t\geq 0$, $f\in \Gamma_{L^2}(\Omega,{E})$, and $\mu$-a.e. $x\in M$ one has $$\begin{aligned}
\label{feyn3}
\mathrm{exp}\big(-t \widetilde{H}_{\Omega}(\nabla,V) \big)f(x) =\int_{\{t<\min(\beta_\Omega(x), \alpha_M(x) )\}} \mathscr{A}^{\nabla}_V(x,t){\slash\slash}^{\nabla}_t(x)^{-1}f({\mathbb{X}}_t(x)) { d}\mathbb{P}. \end{aligned}$$
Note that the expectation in (\[feyn3\]) is on the fiber ${E}_x$, as $f({\mathbb{X}}_t(x))\in {E}_{{\mathbb{X}}_t(x)}$ in $\{t<\alpha_M(x)\}$. In the scalar case we get
$$\begin{aligned}
\label{pqvr3}
\mathrm{exp}\big(-t \widetilde{H}_{\Omega}(w)\big)f(x)=\int_{\{t<\min(\beta_\Omega(x),\alpha_M(x) )\}} \mathrm{exp}\big(-\int^t_0 w({\mathbb{X}}_s)\big)f({\mathbb{X}}_t){ d}\mathbb{P} .\end{aligned}$$
We start by recalling the usual covariant Feynman-Kac formula [@guneysu] $$\mathrm{exp}\big(-t H_M(\nabla,W)\big)f(x)=\int_{\{t< \alpha_M(x)\}} \mathscr{A}^{\nabla}_W(x,t){\slash\slash}^{\nabla}_t(x)^{-1}f({\mathbb{X}}_t(x)) { d}\mathbb{P},$$ which is valid, for example if the potential $W$ is in $L^2_{{\mathrm{loc}}}$ is such that for some constant $A\in{\mathbb{R}}$ and all $x\in M$, the eigenvalues of $W(x)\in\mathrm{End}({E}_x)$ are $\geq A$. In particular, this statement includes that $$\begin{aligned}
\label{domii}
\int_{\{t< \alpha_M(x)\}} \left| \mathscr{A}^{\nabla}_W(x,t){\slash\slash}^{\nabla}_t(x)^{-1}f({\mathbb{X}}_t(x))\right| { d}\mathbb{P}<\infty.\end{aligned}$$
Applying this with $W=V+n1_{M\setminus \Omega}$ and using Proposition \[zbl\] we have, possibly after taking a subsequence of $$\mathrm{exp}\big(-t H_M(\nabla,V+n1_{M\setminus \Omega})\big)f$$ if necessary, that for $\mu$-a.e. $x\in M$ it holds that $$\begin{aligned}
&\mathrm{exp}\big(-t \widetilde{H}_{\Omega}(\nabla,V)\big)f(x)=\lim_{n\to\infty}\mathrm{exp}\big(-t H_M(\nabla,V+n1_{M\setminus \Omega})\big)f(x)\\
&=\lim_{n\to\infty}\int_{\{t< \alpha_M(x)\}} \exp\Big(-n\int^t_01_{M\setminus \Omega}({\mathbb{X}}_s(x)) ds\Big)\mathscr{A}^{\nabla}_V(x,t){\slash\slash}^{\nabla}_t(x)^{-1}f({\mathbb{X}}_t(x)) { d}\mathbb{P}.\end{aligned}$$ For paths from the set $$\{t< \alpha_M(x)\}=_{{\mathbb{P}}}\{{\mathbb{X}}_s(x)\in M\text{ for all $s\in [0,t]$}\}$$ one has $$\int^t_01_{\hat{M}\setminus \Omega}({\mathbb{X}}_s(x)) ds=\int^t_01_{M\setminus \Omega}({\mathbb{X}}_s(x)) ds,$$ and $\int^t_01_{M\setminus \Omega}({\mathbb{X}}_s(x)) ds=0$ is equivalent to $t<\beta_\Omega(x)$. Thus, ${\mathbb{P}}$-a.s. in $\{t< \alpha_M(x)\}$ one has $$\lim_{n\to\infty}\exp\Big(-n\int^t_01_{M\setminus \Omega}({\mathbb{X}}_s(x)) ds\Big)=1_{\{t< \beta_\Omega(x)\}}$$ and $$\begin{aligned}
&\lim_{n\to\infty}\int_{\{t< \alpha_M(x)\}} \exp\Big(-n\int^t_01_{M\setminus \Omega}({\mathbb{X}}_s(x)) ds\Big)\mathscr{A}^{\nabla}_V(x,t){\slash\slash}^{\nabla}_t(x)^{-1}f({\mathbb{X}}_t(x)) { d}\mathbb{P}\\
&=\int_{\{t<\min(\beta_\Omega(x),\alpha_M(x) )\}} \mathscr{A}^{\nabla}_V(x,t){\slash\slash}^{\nabla}_t(x)^{-1}f({\mathbb{X}}_t(x)) { d}\mathbb{P}\end{aligned}$$ follows from dominated convergence, in view of (\[domii\]).
Proof of Proposition \[main1\]
------------------------------
With the preparations made, there is almost nothing left to prove: Note first that (\[poa\]) is equivalent to
$$\begin{aligned}
\label{hipo}
H_\Omega(\nabla,V)=\widetilde{H}_\Omega(\nabla,V).\end{aligned}$$
*(\[poa\]) $\Rightarrow$ (\[aposssay\]):* We have (\[hipo\]), so that (\[aposssay\]) follows from Proposition \[zbl\].\
*(\[aposssay\]) $\Rightarrow$ (\[poa\]):* (\[aposssay\]) and Proposition \[zbl\] imply $$\exp(-tH_\Omega(\nabla,V))=\exp(-t\tilde{H_\Omega}(\nabla,V))\quad\text{ for all $t>0$,}$$ showing (\[hipo\]).
Proof of Proposition \[scal1\]
------------------------------
Note first that one has the Feynman-Kac formula $$\exp(-tH_\Omega )f(x) = \int_{\{t<\alpha_\Omega\}}f({\mathbb{X}}_t) { d}\mathbb{P}^x$$ for all $f\in L^2(\Omega)$, $t>0$, $x\in \Omega$, meaning also that the right hand side is the smooth representative of $\exp(-tH_\Omega )f$. In case $\Omega$ is relatively compact with smooth boundary this is well-known. For the general case we can assume $f\geq 0$. Fix $x\in \Omega$ and exhaust $\Omega$ with a sequence of relatively compact smooth $\Omega_n$’s with $x\in \Omega_1$. Then the latter formula is valid with $\Omega$ replaced with $\Omega_n$, and we can take $n\to\infty$ using monotone convergence for integrals on the probabilistic side, noting that $\alpha_{\Omega_n}\nearrow \alpha_\Omega$ by Lemma \[dnaarrr\] a), and the fact that (cf. p.213 in [@gri]) $$\exp(-tH_{\Omega_n} )f(x)\nearrow \exp(-tH_{\Omega} )f(x).$$ Let us come to the actual proof of Proposition \[scal1\]. Being equipped with the previously established results, we can follow the Euclidean proof from [@herbst] from here on:\
As we have already remarked, the equivalence of (\[alkdd\]) and (\[apossq\]) follows immediately from Proposition \[main1\].
*(\[alkdd\]) $\Rightarrow$ Kac-regularity:* (\[alkdd\]) implies $$\begin{aligned}
\label{rtl2}
\widetilde{H}_\Omega=H_\Omega ,\end{aligned}$$ by comparing the associated forms. In particular, the semigroups coincide and the Feynman-Kac formulae for the semigroups imply the first identity in $$0=\int(1_{\{t< \min(\beta_\Omega,\alpha_M) \}} -1_{\{t<\alpha_\Omega\}})f({\mathbb{X}}_t) { d}\mathbb{P}^x= \int_{\{ \alpha_\Omega\leq t< \beta_\Omega \}} f({\mathbb{X}}_t) { d}\mathbb{P}^x$$ for all $f\in L^2(\Omega)$, $t>0$, $\mu$-a.e. $x\in \Omega$, where the second equality follows from $$1_{\{ \alpha_\Omega\leq t< \min(\beta_\Omega,\alpha_M) \}}=1_{\{t< \min(\beta_\Omega,\alpha_M) )\}}-1_{\{t<\alpha_\Omega\}\cap \{t< \min(\beta_\Omega,\alpha_M) \}}$$ and $\alpha_\Omega\leq\beta_\Omega$ and $\alpha_\Omega\leq \alpha_M$. Letting $f$ run through $f_n:=1_{A_n}$, where $\{A_n:n\in{\mathbb{N}}\}$ is a family of compact subsets of $M$ such that $A_n\subset A_{n+1}$ for all $n\in{\mathbb{N}}$ and $\bigcup_{n\in{\mathbb{N}}}A_n=M$, we get $$\mathbb{P}^x\{ \alpha_\Omega\leq t< \min(\beta_\Omega,\alpha_M) \}=0\quad\text{ for all $t>0$, $\mu$-a.e. $x\in \Omega$}.$$ Letting $t$ run through $\mathbb{Q}_{>0}$, $$\begin{aligned}
\label{apaaoaaws}
{\mathbb{P}}^{x}\{\alpha_\Omega< \min(\beta_\Omega,\alpha_M) \}=0\quad\text{ for $\mu$-a.e. $x\in \Omega$}.\end{aligned}$$ In view of $\alpha_\Omega\leq\beta_\Omega$, the proof of the Kac-regularity of $\Omega$ is complete, once we have shown that the bounded function $h(x):= {\mathbb{P}}^{x}\{\alpha_\Omega< \min(\beta_\Omega,\alpha_M) \}$ is continuous in $\Omega$. We are going to prove that for all open relatively compact subsets $D\subset \Omega$ with $\overline{D}\subset \Omega$ one has $$\begin{aligned}
\label{apoaas}
\int h(\mathbb{X}_{\alpha_D}) d {\mathbb{P}}^x =h(x)\text{ for all $x\in D$,}\end{aligned}$$ which implies that $x\mapsto {\mathbb{P}}^{x}\{\alpha_\Omega< \min(\beta_\Omega,\alpha_M) \}$ is harmonic and thus smooth in $\Omega$. To see (\[apoaas\]), note first that $\alpha_D<\alpha_{\Omega}\leq \alpha_M$ ${\mathbb{P}}^x$-a.s., as the transition density of Brownian motion is continuous. Now we can calculate $$\begin{aligned}
&\int h(\mathbb{X}_{\alpha_D}) d {\mathbb{P}}^x\\
&=\int {\mathbb{P}}^{{\mathbb{X}}_{\alpha_D} }\{\alpha_\Omega< \min(\beta_\Omega,\alpha_M) \}d{\mathbb{P}}^{x}\\
&= \int 1_{\{\alpha_\Omega< \min(\beta_\Omega,\alpha_M) \}}\big(\omega(\alpha_D(\omega)+\bullet)\big)d{\mathbb{P}}^{x}(\omega)\\
&= \int1_{\{\alpha_\Omega+\alpha_D< \min(\beta_\Omega,\alpha_M)+\alpha_D \}}(\omega)d{\mathbb{P}}^{x}(\omega)\\
&=\int1_{\{\alpha_\Omega< \min(\beta_\Omega,\alpha_M) \}}(\omega)d{\mathbb{P}}^{x}(\omega)\\
&=h(x),\end{aligned}$$ where the first inequality holds by definition, the second equality holds by the strong Markov property of Brownian motion, the third equality holds by elementary considerations, the fourth equality is trivial and and the last equality again holds by definition.
*Kac-regularity $\Rightarrow$ (\[alkdd\]):* By the corresponding Feynman-Kac formulae we immediately find that Kac-regularity implies $$\exp(-t\widetilde{H}_\Omega)=\exp(-tH_\Omega )\quad\text{ for all $t>0$},$$ so that $\widetilde{H}_\Omega=H_\Omega$.
Proof of Theorem \[main2\] a)
-----------------------------
We will need the following covariant-Feynman-Kac formula:
\[feyn1\] For every $\Omega\subset M$ open, $t\geq 0$, $f\in \Gamma_{L^2}(\Omega,{E})$, and $\mu$-a.e. $x\in M$ one has the covariant Feynman-Kac formula $$\begin{aligned}
\label{feyn2}
\mathrm{exp}\big(-t H_\Omega(\nabla,V)\big)f(x) =\int_{\{t<\alpha_\Omega(x) \}} \mathscr{A}^{\nabla}_V(x,t){\slash\slash}^{\nabla}_t(x)^{-1}f({\mathbb{X}}_t(x)) { d}\mathbb{P}. \end{aligned}$$
In case $\Omega$ is relatively compact with smooth boundary and $V$ is smooth, formula (\[feyn2\]) is implicitly included in [@driver]. For general $\Omega$’s and $V$’s (\[feyn2\]) seems to be new.
It is enough to prove the formula in case $\Omega$ is relatively compact with a smooth boundary. Indeed, in the general case we can then pick a family of open subsets $\{\Omega_n:n\in{\mathbb{N}}\}$ of $\Omega$ having a smooth boundary, such that $\Omega_n\subset \Omega_{n+1}$ for all $n\in{\mathbb{N}}$ and $\bigcup_{n\in{\mathbb{N}}}\Omega_n=\Omega$. Then we have $\alpha_{\Omega_n}(x)\nearrow\alpha_\Omega(x) $ and $$\begin{aligned}
\label{feyn4}
\int_{\{t<\alpha_{\Omega_n}(x) \}} \mathscr{A}^{\nabla}_V(x,t){\slash\slash}^{\nabla}_t(x)^{-1}f({\mathbb{X}}_t(x)) { d}\mathbb{P}\to\int_{\{t<\alpha_{\Omega}(x) \}} \mathscr{A}^{\nabla}_V(x,t){\slash\slash}^{\nabla}_t(x)^{-1}f({\mathbb{X}}_t(x)) { d}\mathbb{P}. \end{aligned}$$ follows from dominated convergence. Likewise, $$\begin{aligned}
\label{feyn5}
\mathrm{exp}\big(-t H_{\Omega_n}(\nabla,V)\big)f(x) \to \mathrm{exp}\big(-t H_\Omega(\nabla,V)\big)f(x) \end{aligned}$$ follows from applying Theorem \[monn2\] and Remark \[aposv\] thereafter: indeed, define thee form $Q$ by $${\mathrm{Dom}}(Q):=\Gamma_{W^{1,2}_0}(\Omega,E;\nabla,V),\quad Q(f_1,f_2)=\left\langle f_1,f_2\right\rangle_{\nabla,V,*},\quad f_1,f_2\in {\mathrm{Dom}}(Q).$$ Then the form $Q'$ from Theorem \[monn2\] is closable (it has the closed extension $Q$), so that it remains to prove $\overline{Q'}=Q$, which is easily checked from the definitions.\
Thus we can and we will assume $\Omega$ is relatively compact and has a smooth boundary. In case $V$ is smooth and bounded, the asserted formula is a simple consequence of Itos formula and parabolic regularity up to the boundary (cf. p. 102 in [@driver]). In case $V$ is bounded, we can use Friedrichs mollifiers to pick a sequence of smooth potentials $V_n:M\to \mathrm{End}(E)$ with $|V_n|\leq |V|$ and $|V_n-V|\to 0$ $\mu$-a.e. in $M$. Applying (\[feyn2\]) with $V$ replaced with $V_n$ and taking $n\to \infty$ gives the result in this case. Finally, if $V$ is bounded from below, we can use the spectral calculus on the fibers of $E\to M$ to define a sequence of potentials $$V_n:M\longrightarrow \mathrm{End}(E), \quad V_n(x):=\min(n,V(x))$$ then apply (\[feyn2\]) with $V$ replaced with $V_n$ and take $n\to \infty$. Each approximation argument can be justified precisely as in the proof of Theorem 2.11 in [@guneysu], which treats the case $M=\Omega$: In each situation one uses convergence results for sesqulinear forms to control the left-hand site of (\[feyn2\]) and convergence theorems for integrals to control the right-hand-side.
There is notihing left to prove: We know from Proposition \[zbl\] that $$\lim_{n\to\infty}\mathrm{exp}\big(-t H_M(\nabla,V+n1_{M\setminus \Omega})\big)= \mathrm{exp}\big(-t \widetilde{H}_{\Omega}(\nabla,V)\big)P_{\Omega}$$ strongly (without any further assumptions on $\Omega$), and $$\widetilde{H}_{\Omega}(\nabla,V)=H_{\Omega}(\nabla,V)$$ follows from a comparison of the covariant Feynman-Kac formulae for $\exp(-t\widetilde{H_\Omega}(\nabla,V_n))$ and $\exp(-tH_\Omega(\nabla,V_n))$ (cf. Proposition \[aposssy\] and Proposition \[feyn1\]), using that $\Omega$ is Kac regular.
Proof of Theorem \[main2\] b)
-----------------------------
We recall that given an open subset $\Omega\subset M$, the space $W^{1,2}(\Omega;d) $ is defined to be the space of all $f\in L^2(\Omega)$ such that $df\in \Gamma_{L^2}(\Omega;T^*M)$ in the sense of distributions. We are going to need a special case of the following result for the proof of Theorem \[main2\] b):
\[bkjs\] \[identification\] Let $\Omega\subset M$ be a relatively compact open subset with a Lipschitz boundary. Then one has $$\begin{aligned}
W^{1,2}_0(\Omega ) = \big\{ f\in W^{1,2}(M ): f|_{M\setminus \Omega}=0 \>\>\text{\rm $\mu$-a.e.} \big\},\end{aligned}$$ in particular, $\Omega$ is Kac-regular.
Set $m:=\dim M$. It remains to prove $$W^{1,2}_0(\Omega)\supset \{f\in W^{1,2}(M): f|_{M\setminus \Omega}= 0\>\text{$\mu$-a.e.}\}.$$ To this end, we record that for every $p\in \overline{\Omega}\setminus \Omega=\partial\Omega $ there exists a pair of open neighborhoods $V$, $X$ of $p$, and a smooth diffeomorphism $\alpha: X\rightarrow A\subset \mathbb{R}^m$ such that:
1. $\overline{V}\subset X$,
2. $\overline{V}$ is compact,
3. $\alpha(V\cap \Omega)$ is a bounded Lipschitz set of $\mathbb{R}^m$.
Let us consider a finite collection of open subsets of $M$, $\mathcal{W}:=\{V_1,...,V_q,W_1,...,W_n\}$, such that
- for each $j\in \{1,...,n\}$ we have $\overline{W_j}\subset \Omega$,
- $\overline{\Omega}\subset V_1\cup...\cup V_q\cup W_1\cup...\cup W_n$,
- for each $i\in \{1,...,q\}$ there exists another open subset $X_i$ such that the pair given by $V_i$ and $X_i$ satisfies the properties $(1)$–$(3)$ required in the statement.
Clearly such a finite collection of open subsets exists. Let $I$ be an open subset of $M$ such that $I\cap \Omega=\emptyset$ and $$M=V_1\cup...\cup V_q\cup W_1\cup...\cup W_n\cup I.$$ Let $$\mathcal{W'}:=\{V_1,...,V_q,W_1,...,W_n,I\}$$ and let $$\{\phi_1,...,\phi_q,\psi_1,...,\psi_n,\tau\}$$ be a partition of unity subordinated to $\mathcal{W}$. Given now $f\in W^{1,2}(M)$ with $f|_{M\setminus \Omega}= 0$ $\mu$-a.e., for each $i\in \{1,...,n\}$ we have $\operatorname{supp}(\psi_if)\subset W_i$. Therefore there exists a sequence $\{\beta^i_p\}_{p\in \mathbb{N}}\subset C^{\infty}_c(W_i)$ such that $\beta^i_p\rightarrow \psi_if$ in $W^{1,2}_0(W_i)$ as $p\rightarrow \infty$. Indeed, by a Meyers-Serrin type theorem [@guidetti], we can pick a sequence $\{\overline{\beta}^i_p\}_{p\in \mathbb{N}}$ in $C^{\infty}(W_i)\cap W^{1,2}(W_i)$ such that $\overline{\beta}^i_p\rightarrow f$ in $W^{1,2}(W_i)$. Then, by defining $\beta^i_p:= \psi_i\overline{\beta}^i_p$, it is clear that $\{\beta^i_p\}_{p\in \mathbb{N}}\subset C^{\infty}_c(W_i)$ and that $\beta^i_p\rightarrow \psi_if$ in $W^{1,2}_0(W_i)$ as $p\rightarrow \infty$. Let us now consider the other case. For each $i\in \{1,...,q\}$ let $Y_i:=V_i\cap \Omega$. Then we have $\operatorname{supp}(\phi_if)\subset (\overline{Y_i}\cap V_i)$. Let $A_i:=\alpha_i(X_i)$, $E_i:=\alpha_i(V_i)$ and $B_i:=\alpha_i(Y_i)$. Since we assumed that $\overline{V_i}\subset X_i$ we have that $(\alpha_i^*g_e)|_{V_i}$ is quasi-isometric to $g|_{V_i}$ where $g_e$ is the standard Euclidean metric on $\mathbb{R}^m$ and g the metric on $M$. In this way we can conclude that $(\phi_if)\circ (\alpha|_{V_i})^{-1}\in W_{0,e}^{1,2}(E_i)$ and that $(\phi_if)\circ (\alpha|_{Y_i})^{-1}\in W^{1,2}_e(B_i)$, where of course $W^{1,2}_e$ stands for the various Sobolev spaces that are defined with respect to $g_e$. Furthermore we know that $(\phi_if)\circ (\alpha|_{Y_i})^{-1}|_{A_i\setminus B_i}\equiv 0$ because $(\phi_if)|_{X_i\setminus Y_i}\equiv0$. Therefore, by extending $(\phi_if)\circ (\alpha|_{Y_i})^{-1}$ outside its support as the identically zero function, we can say, with a little abuse of notation, that $(\phi_if)\circ (\alpha|_{Y_i})^{-1}\in W^{1,2}_e(\mathbb{R}^m)$ and that $(\phi_if)\circ (\alpha|_{Y_i})^{-1}|_{\mathbb{R}^m\setminus B_i}\equiv 0$. As $B_i$ is a bounded Lipschitz set in $\mathbb{R}^m$, it follows from Remark \[ende\].2 that there exists a sequence $\{\upsilon^i_p\}_{p\in \mathbb{N}}\subset C_c^{\infty}(B_i)$ such that $\upsilon^i_p \rightarrow (\phi_if)\circ (\alpha|_{Y_i})^{-1}$ in $W^{1,2}_{0,e}(B_i)$ as $p\rightarrow \infty$. Finally, using the fact that $(\alpha_i^*g_e)|_{V_i}$ is quasi-isometric to $g|_{V_i}$, we can conclude that $\gamma^i_p\rightarrow \phi_if$ in $W^{1,2}_0(Y_i)$ as $p\rightarrow \infty$ where $\gamma^i_p:=\upsilon^i_p\circ (\alpha_i|_{Y_i})$ and clearly $\{\gamma^i_p\}_{p\in \mathbb{N}}\subset C^{\infty}_c(Y_i)$ by construction. Let us define now the following sequence of functions $\eta_p:=\gamma^1_p+...+\gamma_p^q+\beta_p^1+...+\beta_p^n$. It is clear by construction that $\{\eta_p\}_{p\in \mathbb{N}}\subset C^{\infty}_c(\Omega)$. Moreover we have $$\begin{aligned}
& \nonumber \|\eta_p-f\| =\|\gamma^1_p+...+\gamma_p^q+\beta_j^1+...+\beta_p^n-\sum_{i=1}^q\phi_if-\sum_{i=1}^n\psi_if\| \\
& \nonumber \leq \|\gamma^1_p-\phi_1f\| +...+\|\gamma^q_p-\phi_qf\| +\|\beta^1_p-\psi_1f\| +...+\|\beta^n_p-\psi_nf\| \end{aligned}$$ and by construction all the terms in the second line tend to zero as $p\rightarrow \infty$. Hence we have shown that $\eta_j\rightarrow f$ in $\Gamma_{L^2}(\Omega,T^*_{{\mathbb{C}}}M)$ as $p\rightarrow \infty$. Similarly we have $$\begin{aligned}
& \nonumber \|d\eta_p-df\| =\|d\gamma^1_p+...+d\gamma_p^q+d\beta_p^1+...+d\beta_p^n-d(\sum_{i=1}^q\phi_if-\sum_{i=1}^n\psi_if)\| \\
& \nonumber \leq \|d\gamma^1_p-d(\phi_1f)\| +...+\|d\gamma^q_p-d(\phi_qf)\| +\|d\beta^1_p-d(\psi_1f)\| +...\\
& \nonumber+\|d\beta^n_p-d(\psi_nf)\| \end{aligned}$$ and again by construction we know that all the terms on the right hand side of the inequality tend to zero as $p\rightarrow \infty$. This tells us that $d\eta_p\rightarrow df$ in $\Gamma_{L^2}(\Omega,T^*_{{\mathbb{C}}}M)$. In conclusion we have shown that $\eta_p\rightarrow f$ in $W^{1,2}(\Omega)$ and thereby we can conclude that $f\in W^{1,2}_0(\Omega)$ as desired.
Fix $x\in \Omega$. Pick a sequence $\Omega_n\subset \Omega$, $n\in{\mathbb{N}}$ of relatively compact open subsets of $M$ with Lipschitz boundary, such that $x\in \Omega_1$, $\Omega_n\subset \Omega_{n+1}$ for all $n\in{\mathbb{N}}$, $\bigcup_{n\in{\mathbb{N}}} \Omega_n=\Omega$ and such that for all $n\in{\mathbb{N}}$ there exists an open subset $\Upsilon\subset M$ such that $\Upsilon_n\cap \Omega=\Omega_n$ for all $n\in{\mathbb{N}}$ and $\bigcup_{n\in{\mathbb{N}}} (\Upsilon_n\cap \overline{\Omega})= \overline{\Omega}$. Then by Proposition \[main2\] and Proposition \[bkjs\] we have $$\mathbb{P}^x \{ \alpha_{\Omega_n} =\min(\beta_{\Omega_n},\alpha_M) \}=1\quad\text{for all $n$},$$ so that using $\alpha_{\Omega_n}\to \alpha_{\Omega}$ and $\beta_{\Omega_n}\to \beta_{\Omega}$ $\mathbb{P}^x$-a.s. as $n\to \infty$, a consequence of Lemma \[dnaarrr\], we arrive at $$\begin{aligned}
\mathbb{P}^x \{ \alpha_{\Omega} \ne \min(\beta_{\Omega },\alpha_M) \}\subset \mathbb{P}^x\bigcup_{n\in{\mathbb{N}}}\{ \alpha_{\Omega_n} \ne \min(\beta_{\Omega_n},\alpha_M) \}\leq \sum_{n\in{\mathbb{N}}} \mathbb{P}^x \{ \alpha_{\Omega_n} \ne \min(\beta_{\Omega_n},\alpha_M) \}=0,\end{aligned}$$ completing the proof.
Appendix: Some functional analytic facts
========================================
In this section we collect some facts about the monotone convergence of nonnegative closed sesquilinear forms which are possibly not densely defined.
Let ${\mathscr{H}}$ be a complex seperable Hilbert space and assume that $Q$ is a semibounded from below closed sesquilinear forms such a form on ${\mathscr{H}}$. Then $Q$ is a densely defined semibounded from below closed sesquilinear form on the Hilbert subspace ${\mathscr{H}}_{Q}:= \overline{{\mathrm{Dom}}(Q)}^{\left\|\cdot\right\|_{{\mathscr{H}}}}\subset {\mathscr{H}}$ and thus there exists a unique semibounded from below self-adjoint operator $S_{Q}$ on ${\mathscr{H}}_{Q}$ such that $${\mathrm{Dom}}(\sqrt{S_{Q}})={\mathrm{Dom}}(Q),\quad Q(f_1,f_2)=\left\langle \sqrt{S_{Q}}f_1,\sqrt{S_{Q}}f_2\right\rangle_{{\mathscr{H}}}.$$ Let then $P_{Q}:{\mathscr{H}}\to {\mathscr{H}}_{Q}$ denote the orthogonal projection. Recall also that for semibounded forms $Q_1$ and $Q_2$ in ${\mathscr{H}}$ one per definitionem has $Q_1\leq Q_2$ if and only if ${\mathrm{Dom}}(Q_1)\supset {\mathrm{Dom}}(Q_2)$ and $Q_1(f,f)\leq Q_2(f,f)$ for all $f\in {\mathrm{Dom}}(Q_2)$. Given a semibounded sesquilinear form $Q$ on ${\mathscr{H}}$ there exists a largest (with respect to ’$\leq$’) closable semibounded sesquilinear form $\mathrm{ref}(Q)$ on ${\mathscr{H}}$, such that $\mathrm{ref}(Q)\leq Q$.\
The following results follow from Theorem 4.2 in [@simon]:
\[monn\] Let $ Q_1\leq Q_2\leq\dots$ be a sequence of closed semibounded from below sesquilinear forms on ${\mathscr{H}}$. Then $$Q_{\infty}(f_1,f_2):=\lim_{n\to\infty} Q_n(f_1,f_2)$$ with $${\mathrm{Dom}}(Q_{\infty}):=\Big\{ f\in\bigcap_{n\in{\mathbb{N}}}{\mathrm{Dom}}(Q_n):\sup_{n\in{\mathbb{N}}} Q_n(f,f)<\infty\Big\}$$ is a closed semibounded from below sesquilinear form $Q_{\infty}$ in ${\mathscr{H}}$, and one has $$\mathrm{e}^{-t S_{Q_n}}P_{Q_n} \to \mathrm{e}^{ -t S_{Q_{\infty}}}P_{Q_{\infty}} \>\>\text{ strongly in ${\mathscr{L}}({\mathscr{H}})$ as $n\to\infty$, for all $t\geq 0$.}$$
\[monn2\] Let $Q_1\geq Q_2\geq\dots$ be a sequence of closed semibounded sesquilinear forms on ${\mathscr{H}}$. Define a nonnegative sesquilinear form in ${\mathscr{H}}$ given by $$Q'(f_1,f_2):=\lim_{n\to\infty} Q_n(f_1,f_2),\quad{\mathrm{Dom}}(Q'):=\bigcup_{n\in{\mathbb{N}}} {\mathrm{Dom}}(Q_n)$$ and let $Q_{\infty}$ denote the form on ${\mathscr{H}}$ given by the closure of $\mathrm{reg}(Q\rq{})$ in ${\mathscr{H}}$. Then one has $$\mathrm{e}^{-t S_{Q_n}}P_{Q_n}\to \mathrm{e}^{-t S_{Q_{\infty}}}P_{Q_{\infty}}\text{ strongly in ${\mathscr{L}}({\mathscr{H}})$ as $n\to\infty$, for all $t\geq 0$ .}$$
\[aposv\] Note that in the above situation $Q_{\infty}$ is the closure of $Q\rq{}$ if $Q\rq{}$ is closable, and $Q_{\infty}=Q\rq{}$ if $Q\rq{}$ is even closed.
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[^1]: We will not assume $M$ to be stochastically complete
[^2]: Note that every open subset of $M$ is also open in $\hat{M}$. In addition, the closure of a subset $\Omega$ of $M$ is always understood with respect to the topology of $M$ and not the topology of $\hat{M}$, noting that both closures coincide if and only if the closure of $\Omega$ in $M$ is a compact subset of $M$.
[^3]: In the sequel the infimum of an empty set is understood to be $\infty$.
[^4]: But of course this statement is true by Proposition \[scal1\] and Theorem \[main2\] b), if $M$ is geodesically complete, as then $W^{1,2}(M )=W^{1,2}_0(M )$ [@aubin].
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abstract: 'Given a Dedekind incomplete ordered field, a pair of convergent nets of gaps which are respectively increasing or decreasing to the same point is used to obtain a further equivalent criterion for Dedekind completeness of ordered fields: Every continuous one-to-one function defined on a closed bounded interval maps interior of that interval to the interior of the image. Next, it is shown that over all closed bounded intervals in any monotone incomplete ordered field, there are continuous not uniformly continuous unbounded functions whose ranges are not closed, and continuous 1-1 functions which map every interior point to an interior point (of the image) but are not open. These are achieved using appropriate nets cofinal in gaps or coinitial in their complements. In our third main theorem, an ordered field is constructed which has parametrically definable regular gaps but no $\emptyset$-definable divergent Cauchy functions (while we show that, in either of the two cases where parameters are or are not allowed, any definable divergent Cauchy function gives rise to a definable regular gap). Our proof for the mentioned independence result uses existence of infinite primes in the subring of the ordered field of generalized power series with rational exponents and real coefficients consisting of series with no infinitesimal terms, as recently established by D. Pitteloud.'
address: 'Department of Mathematics, Tarbiat Modarres University, Tehran, Iran'
author:
- Mojtaba Moniri
- 'Jafar S. Eivazloo'
title: 'Using nets in Dedekind, monotone, or Scott incomplete ordered fields and definability issues'
---
(474,66)(0,0) (0,66)(1,0)[40]{}[(0,-1)[24]{}]{} (43,65)(1,-1)[24]{}[(0,-1)[40]{}]{} (1,39)(1,-1)[40]{}[(1,0)[24]{}]{} (70,2)(1,1)[24]{}[(0,1)[40]{}]{} (72,0)(1,1)[24]{}[(1,0)[40]{}]{} (97,66)(1,0)[40]{}[(0,-1)[40]{}]{} (143,66)[(0,0)\[tl\][Proceedings of the Ninth Prague Topological Symposium]{}]{} (143,50)[(0,0)\[tl\][Contributed papers from the symposium held in]{}]{} (143,34)[(0,0)\[tl\][Prague, Czech Republic, August 19–25, 2001]{}]{}
[^1]
A Dedekind Incompleteness Feature via Convergent Nets of Gaps
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A cut of an ordered field $F$ is a subset which is downward closed in $F$. By a nontrivial cut, we mean a nonempty proper cut. A nontrivial cut is a gap if it does not have a least upper bound in the field. An ordered field is Archimedean (has no infinitesimals) just in case it can be embedded in ${\mathbb R}$.
The following fact presents some of the well known characterizations of the ordered field of real numbers. A more delicate equivalent condition is presented in Theorem 1.2.
\[easy2nd\] The real ordered field ${\mathbb R}$ is, up to isomorphism, the unique ordered field which satisfies either of the following equivalent conditions:
- Dedekind Completeness, i.e. not having any gaps,
- connectedness,
- not being totally disconnected,
- every (nonempty) convex subset being an interval (of one of the usual kinds),
- all convex subsets being connected,
- all intervals being connected,
- all continuous functions on the field mapping any convex subset onto a convex subset,
- all continuous functions on the field satisfying the intermediate value property.
These are fairly well known, let us give the argument for (iii). Pick a Dedekind incomplete ordered field $F$. In the Archimedean case, $F$ is a proper subfield of ${\mathbb R}$ which therefore misses some points in any real interval. In the non-Archimedean case, and given any two points $a<b$ of a subset $A$ of $F$, we have the nontrivial clopen subset of those points $x$ in $A$ such that $\frac{x-a}{b-a}$ is a ${\mathbb Q}$-infinitesimal. Alternatively, one can use linear increasing functions between intervals to map a given gap (somewhere in the field) into a given interval.
Notice that if $F$ is an ordered field and $P(x)\in F[x]$, then $P(x)$ is continuous on $F$. To see this, it suffices to show that if $a\in F$ and $P(a)<0$, then $P$ is negative on a neighborhood of $a$ in $F$. As the same inequality also holds in $RC(F)$, where polynomials have factorizations into linear and irreducible quadratics, there are $b,c\in RC(F)$ with $b<a<c$ such that $\forall x\in (b,c)_{RC(F)}$, $P(x)<0$. Using cofinality of $F$ in $RC(F)$, one can now see that there are $d,e\in F$ such that $b<d<a<e<c$ (observe that there exists $g\in F$ such that $0<g<a-b$ and put $d=a-g$, similarly for $e$). A very mild use of this takes place in the claim within the next theorem.
\[involved2nd\] An ordered field is Dedekind complete if and only if all continuous 1-1 functions defined on some (equivalently all) non-degenerated closed bounded interval(s) of it map interior points \[of the interval(s)\] to interior points \[of their range(s)\].
We only need to prove the [*if*]{} part.
Given two convex subsets $A$ and $B$ of an ordered field $F$, with $A$ bounded and $B$ not a singleton, there exists a nonconstant linear (and so 1-1 continuous) function from $A$ into $B$.
Let $A\subseteq [a_{1},a_{2}]$, $[b_{1},b_{2}]\subseteq B$, with $a_{1}\not =a_{2}$ and $b_{1}\not =b_{2}$. Consider the function $$f(x)=b_{1}+\frac{b_{2}-b_{1}}{a_{2}-a_{1}}(x-a_{1}),$$ whose restriction to $A$ does the required job.
Let $F$ be a Dedekind incomplete ordered field of cofinality $\lambda$. Consider an interval $[a,b]$ in $F$ with a point $c\in (a,b)$. Pick a strictly increasing net $(a_{\alpha})_{\alpha <\lambda}$ and a strictly decreasing one $(b_{\alpha})_{\alpha <\lambda}$ in $[a,b]$ with $a_{0}=a$ and $b_{0}=b$ which converge in $F$ to $c$. For each $\alpha$ less than $\lambda$, let $U_{\alpha}$ be a gap in $(a_{\alpha },a_{\alpha +1})$ and $V_{\alpha}$ a gap in $(b_{\alpha +1 },b_{\alpha })$. Put $$S_{0}=U_{0}\cap [a,b],\
T_{0}=[a,b]\setminus V_{0},$$ $$S_{\alpha+1}=U_{\alpha+1}\setminus U_{\alpha},\
T_{\alpha+1}=V_{\alpha}\setminus V_{\alpha +1}$$ for $\alpha < \lambda$, and for limit $\beta<\lambda$, $$S_{\beta}=U_{\beta}\setminus\cup_{\gamma<\beta}U_{\gamma},\
T_{\beta}=(\cap_{\gamma<\beta}V_{\gamma})\setminus V_{\beta}.$$ Note that $[a,c)=\cup_{\alpha<\lambda}S_{\alpha}$, and $(c,b]=\cup_{\alpha<\lambda}T_{\alpha}$, and that both unions are disjoint. Using the claim, consider a function $f$ on $[a,b]$ which linearly injects, for $\alpha <\lambda$, the set $S_{\alpha}$ into $T_{\alpha \cdot
2}$, $T_{\alpha}$ into $T_{(\alpha\cdot 2 )+1}$, and finally $c$ to $c$. Then $f$ maps the interior point $c$ of $[a,b]$ to the boundary point $c$ of $f([a,b])$.
\[openmaps\] An ordered field is Dedekind complete if and only if all continuous 1-1 functions on some (equivalently all) non-degenerated closed bounded interval(s) are open.
For the non-trivial [*if*]{} part, notice that all open maps on any ordered field map interior to interior.
Some Consequences of Monotone Incompleteness via Nets Cofinal in Gaps
=====================================================================
An ordered field $F$ is Scott complete, see [@S], if it satisfies either of the following three equivalent conditions:
- It does not have any proper extensions to an ordered field in which it is dense,
- It does not have any regular gaps, where a gap $C\subset F$ is regular if $\forall\epsilon\in F^{>0}$, $C+\epsilon\not\subseteq C$,
- All functions $f:F\rightarrow F$ which are Cauchy at $\infty_{F}$, are convergent there.
We abbreviate the (equivalent non first order) axiomatizations obtained from the theory $OF$ of ordered fields by adding condition (ii) or (iii) as $S_{rc}COF$ and $S_{cf}COF$ respectively.
Observe that an ordered field of cofinality $\lambda$ is Scott complete if and only if it has no divergent Cauchy nets of length $\lambda$.
If an ordered field $F$ is not Scott complete, then for all $a<b\in F$, there is a regular gap $V$ of $F$ with $a\in V$ and $b\in F\setminus V$ (and so a divergent Cauchy net in $(a,b)$ of length equal to cofinality of the field). In fact, any regular gap of $F$ has an additive translation to a regular gap in $(a,b)$. To see this, let $U$ be a regular gap of $F$. We can pick $c\in U$ and $d\in F\setminus U$ such that $d-c<b-a$. Let $V=U+(a-c)$. It is straightforward to see that $V$ is a regular gap and $a<V<b$.
It was proved in [@S] (Theorem 1), that any ordered field $F$ has a (unique, up to an isomorphism of ordered fields which is identity on $F$) Scott completion. It is characterized by being Scott complete and having $F$ dense in it. Furthermore $F$ is dense in $RC(F)$ if and only if its Scott completion is real closed, see [@S] (Theorem 2).
As shown in [@EGH] (Lemma 2.3), any uncountable ordered field has a dense transcendence basis over the rationals. Therefore all Scott complete ordered fields necessarily have proper dense subfields and so can be obtained by Scott completion of a proper subfield (the field extension, inside the original ordered field, of rationals by the just mentioned basis minus a point does the job).
We will state and prove a number of results on Scott completeness and its two first order versions in section 3. In this section, we are aiming at a result dealing with the stronger notion of monotone completeness. Monotone complete ordered fields were introduced in [@K]. They are ordered fields with no bounded strictly increasing divergent functions. Scott complete ordered fields that are $\kappa$-Archimedean for some regular cardinal $\kappa$ are Monotone Complete (by $\kappa$-Archimedean, one means that there are subsets of cardinality $\kappa$ whose distinct elements are at least a unit apart and all such sets are unbounded), see [@CL].
As we have already mentioned, all open maps on any interval in an arbitrary ordered field map every interior point to an interior point of the image. However, the converse property, that of mapping interior to interior implying being open even when restricted to continuous 1-1 functions, is strong enough to imply monotone completeness as the next theorem shows.
Over all closed bounded intervals in arbitrary monotone incomplete ordered fields, there are
- continuous non uniformly continuous unbounded functions whose ranges are not closed,
- continuous 1-1 functions mapping all interior points to interior of the image which, nevertheless, are not open.
Let $F$ be a monotone incomplete ordered field of cofinality $\lambda$. We first observe the following.
Any nondegenerated interval of $F$ contains a strictly increasing divergent net of length $\lambda$ \[compare with [@K] (Proposition 1(a))\].
Given a divergent bounded strictly increasing net of length $\lambda$, consider an interval containing its range. Then for any other interval, one may apply the linear strictly increasing function mapping the former onto the latter interval, thereby producing a strictly increasing divergent net $\lambda$ in the latter. This concludes the claim.
(i). Given a closed bounded interval $[a,b]$ of $F$, using the claim, we can pick divergent $\lambda$-nets $(a_{\alpha})_{\alpha <\lambda}$ and $(b_{\alpha})_{\alpha <\lambda}$ with $a_{0}=a$, $b_{0}=b$, $a_{\alpha}<b_{\beta}$, for all $\alpha,\beta <\lambda$, which are respectively strictly increasing or strictly decreasing. Also, let $(u_{\alpha})_{\alpha <\lambda}$ and $(d_{\alpha})_{\alpha
<\lambda}$ be both strictly increasing, the former being unbounded and having consecutive terms which are at least a unit apart, the latter converging in $F$ to $0$. Now, for each $\alpha<\lambda$, let $f$ map $[a_{\alpha},a_{\alpha +1})$ linearly and increasingly onto $[u_{\alpha},u_{\alpha +1})$ and $(b_{\alpha +1},b_{\alpha}]$ linearly and in a decreasing manner onto $[d_{\alpha},d_{\alpha+1})$ and be equal to 1 on the rest of $[a,b]$. We may assume without loss of generality that for any $\delta>0$, there exists $\alpha_{0} <\lambda$ such that $a_{\alpha_{0}
+1}-a_{\alpha_{0}}<\delta$. To see that such a reduction indeed causes no loss of generality, take a net $(c_{\alpha})_{\alpha<\lambda}$ converging to $0$ such that $0<c_{\alpha}<a_{\alpha +1} - a_{\alpha}$ and consider the net obtained from $(a_{\alpha})_{\alpha<\lambda}$ by right-shifting those of its terms which have limit ordinal indices by the corresponding same-indexed terms in $(c_{\alpha})_{\alpha<\lambda}$. The function $f$ is a continuous non uniformly continuous function on $[a,b]$ whose values are $F$-unbounded on the downward closure of $(a_{\alpha})_{\alpha <\lambda}$ in $[a,b]$ and asymptotic to zero on the upward closure of $(b_{\alpha})_{\alpha <\lambda}$ in $[a,b]$ traversed backwards. To see that $f$ is not uniformly continuous, let $\epsilon =\frac{1}{2}$. Then for any $\delta >0$, by the assumption we made above, there exists $\alpha_{0} <\lambda$ such that $a_{\alpha_{0} +1}-a_{\alpha_{0}}<\delta$. Now by $u_{\alpha +1}-u_{\alpha} \geq 1$ and the way $f$ is constructed, we have $$f(a_{\alpha_{0} +1})-f(a_{\alpha_{0}})\geq 1 >\epsilon.$$
(ii). Consider the interval $[a,b]$ with a point $c\in (a,b)$. We now follow a construction similar to the one in Theorem \[involved2nd\]. Using the same notation as there, the change we make is the following. We linearly inject $T_{\alpha}$, for $1\leq \alpha <\lambda$, into $T_{(\alpha\cdot 2 )+1}$ but $T_{0}$ onto $[a,c)$. The latter can be done as in part (i) in a piece-wise manner. Here we may assume that $T_{0}$ is the upward closure of a strictly decreasing divergent $\lambda$-net in $(b_{1},b_{0}]$. Then $c$ will be a boundary point of the image of the open interval $(a_{1},b_{1})$ under $f$, so that image is not open. On the other hand, by construction, $f$ sends interior to interior.
P-Definable Regular Gaps Not Traversed by $\emptyset$-Definable Cauchy Functions
================================================================================
We now consider two notions of being definably (with or without parameters) Scott complete for ordered fields, those corresponding to $S_{rc}COF$ and $S_{cf}COF$. They are ordered fields with no definable regular gaps, respectively no definable functions which are Cauchy at infinity but divergent there. We denote the corresponding theories by $D_{p}S_{rc}COF$, $D_{p}S_{cf}COF$, $D_{\emptyset}S_{rc}COF$, and $D_{\emptyset}S_{cf}COF$.
It is easy to see that, as long as there are no definability concerns, all regular gaps can be traversed by suitable (divergent Cauchy) functions and vice versa, every divergent Cauchy function induces a regular gap. The latter converse is shown in Theorem \[conditional\](i) to be true in either of the p-definable or $\emptyset$-definable cases. For the former direction, however, we are only able in Theorem \[conditional\](ii) to prove an independence result in a mixed $\emptyset$-definable / p-definable case.
\[shepherdson\] If the ordered field $F$ is a proper dense sub-field of its real closure, then $F\nvDash D_{p}S_{rc}COF$.
Pick out an element $r\in RC(F)\setminus F$ and consider the set $$C=\{x\in F: \ RC(F)\vDash x<r\}.$$ It is obviously a gap of $F$. As $F$ is dense in $RC(F)$, $C$ is regular. Finally, $C$ is p-definable in $F$: consider the minimal polynomial of $r$ over $F$ and the number of roots of that polynomial in $RC(F)$ which are less than $r$.
For any ordered field $F$ and ordered abelian group $G$, the set $[[F^{G}]]$ of all functions $G\rightarrow F$ whose supports are well ordered in $G$ equipped with pointwise sum and Cauchy product $$(f_{1}f_{2})(g)=\sum_{i+j=g}f_{1}(i)f_{2}(j)$$ (a finite sum by the condition on the supports) forms a field. It can be ordered by comparison of values at the minimum of support of the difference. Elements of $[[F^{G}]]$ can also be thought of as those formal power series $\sum_{g\in G}f(g)t^{g}$ which have well ordered supports. The indeterminate $t$ is taken to be a positive $F$-infinitesimal. The field $[[F^{G}]]$ is real closed if and only if $F$ is so and $G$ is divisible, see [@R] (6.10).
In the proposition below, $\chi_{S}$ stands for the characteristic function of (a set) $S$.
\[scottfieldsofgps\] For any ordered abelian group $G$ and ordered field $F$, the generalized power series field $[[F^{G}]]$ is Scott complete if $F$ is so.
Let $\lambda$ be the cofinality of $G$ and consider a strictly decreasing $\lambda$-net $(a_{\theta})_{\theta <\lambda}$ in $G^{<0}$ which is unbounded below there. Assume $\mathcal{F}:[[F^{G}]]\rightarrow [[F^{G}]]$ is Cauchy. For each $g\in G$, define $f_{g}:\lambda\rightarrow F$ by $$f_{g}(\theta)=(\mathcal{F}(\chi_{\{a_{\theta}\}}))(g),
\quad
\mbox{for $\theta <\lambda$}.$$
\[c1\] For all $g\in G$, the net $(f_{g}(\theta))_{\theta <\lambda}$ is convergent in $F$.
Fix $g\in G$ and $\epsilon\in F^{>0}$. By the Cauchy condition for $\mathcal{F}$, there exists $\theta_{0}<\lambda$ such that $\forall \alpha,\beta\in[[F^{G}]]$ with $\alpha,\beta\geq\chi_{\{a_{\theta_{0}}\}}$, we have $|\mathcal{F}(\alpha)-\mathcal{F}(\beta)|<\epsilon\chi_{\{g\}}$. This shows that $|\mathcal{F}(\alpha)(g)-\mathcal{F}(\beta)(g)|<\epsilon$. Therefore $\forall\theta_{1},\theta_{2}\geq\theta_{0}$, $$|f_{g}(\theta_{1})-f_{g}(\theta_{2})| =
|(\mathcal{F}(\chi_{\{a_{\theta_{1}}\}}))(g) -
(\mathcal{F}(\chi_{\{a_{\theta_{2}}\}}))(g)|<\epsilon,$$ since $\chi_{\{a_{\theta_{1}}\}},\chi_{\{a_{\theta_{2}}\}}
\geq \chi_{\{a_{\theta_{0}}\}}$. Hence the net $(f_{g}(\theta))_{\theta <\lambda}$ is Cauchy in $F$ and so convergent there, since $F$ is Scott complete.
Let $\gamma:G\rightarrow F$ be defined by $\gamma (g)=\lim(f_{g}(\theta))_{\theta <\lambda}$.
\[c2\] $(\forall \eta<\lambda)
(\exists \theta_{0}<\lambda)
(\forall \theta\geq\theta_{0})
(\forall g\in G^{<-a_{\eta}})
\mathcal{F}(\chi_{\{a_{\theta}\}})(g) = \gamma(g)$.
For any $\epsilon=\chi_{\{-a_{\eta}\}}$, with $\eta <\lambda$, there exists $\theta_{0}<\lambda$ such that $\forall\theta_{1},\theta_{2}<\lambda$ with $\theta_{1},\theta_{2}\geq\theta_{0}$, we have $$|\mathcal{F}(\chi_{\{a_{\theta_{1}}\}}) -
\mathcal{F}(\chi_{\{a_{\theta_{2}}\}})|<\chi_{\{-a_{\eta}\}}.$$ This shows that for all $\theta_{1},\theta_{2}\geq\theta_{0}$ and $g<-a_{\eta}$, we have $$(\mathcal{F}(\chi_{\{a_{\theta_{1}}\}}))(g) =
(\mathcal{F}(\chi_{\{a_{\theta_{2}}\}}))(g)=\gamma (g).$$ Therefore, for all $\theta\geq\theta_{0}$ and $g\in G^{<-a_{\eta}}$, we have $\mathcal{F}(\chi_{\{a_{\theta}\}})(g)=\gamma(g)$.
\[c3\] $\gamma\in [[F^{G}]]$.
It suffices to show that the support of $\gamma$ is well ordered. For all $g\in G$, there exists $\eta <\lambda$ such that $g<-a_{\eta}$. Let $\theta_{0}$ be as in Claim \[c2\]. As $g$ can not be the initial term of any infinite strictly decreasing sequence in the support of $\mathcal{F}(\chi_{\{a_{\theta_{0}}\}})$, Claim \[c2\] shows that the same holds for the support of $\gamma$.
\[c4\] The function $\mathcal{F}$ on $[[F^{G}]]$ tends to $\gamma$ at infinity.
It is enough, by the Cauchy criterion for $\mathcal{F}$, to apply $\mathcal{F}$ on those $f$’s in $[[F^{G}]]$ that are of the form $\chi_{\{a_{\theta}\}}$, for $\theta <\lambda$ and let $\theta$ tend to $\lambda$. The result is then immediate from Claim \[c2\].
The above claims give the result.
\[ifthencf\] Suppose that $F$ is a Scott complete ordered field, $G$ is a 2-divisible ordered abelian group, and $K$ is a dense ordered sub-field of $[[F^{G}]]$ which contains $F$. Assume furthermore that $K$ is closed under the automorphism on $[[F^{G}]]$ sending $\chi_{\{g\}}$ to $\chi_{\{g+g\}}$ and its inverse. Then $K$ satisfies $D_{\emptyset}S_{cf}COF$.
If $K=[[F^{G}]]$, then the conclusion will trivially hold since $[[F^{G}]]$ is Scott complete. Suppose $K\subsetneq
[[F^{G}]]$. Assume for the purpose of a contradiction that $\mathcal{F}$ is a $\emptyset$-definable divergent Cauchy function on $K$. Fix a net $(k_{\alpha})_{\alpha<\lambda}$ of elements of $K$, where $\lambda$ is cofinality of $[[F^{G}]]$, which is cofinal in the latter. For any $f\in [[F^{G}]]$, let $\lambda (f)$ be the least ordinal less than $\lambda$ such that $f<k_{\lambda (f)}$. Consider the function $\tilde{\mathcal{F}}$ on $[[F^{G}]]$ with $\tilde{\mathcal{F}}(f)=\mathcal{F}(k_{\lambda (f)})$. As a Cauchy function on the Scott complete ordered field $[[F^{G}]]$, it will converge to some $f_{0}\in [[F^{G}]]$. Clearly $f_{0}\not\in K$ and therefore $f_{0}\not\in F$. Let $\mathcal{A}$ be the ordered field automorphism on $[[F^{G}]]$ described in the statement of the lemma. The only fixed points of $[[F^{G}]]$ under $\mathcal{A}$ are elements of $F$ (since if there is a leading $F$-infinitely large or a highest $F$-infinitely small term, then the result of applying $\mathcal{A}$ to such elements will have a different leading $F$-infinitely large, respectively highest $F$-infinitely small term and so must be different). Therefore, $\mathcal{A}(f_{0}) \not = f_{0}$. Let $$C =
\{f\in K : \mbox {$f$ is cofinally many times dominated in $K$ by
values of $\mathcal{F}$}\}.$$ Now since in $[[F^{G}]]$, $\mathcal{F}$ tends to $f_{0}$, while $\mathcal{A}(\mathcal{F})$ approaches $\mathcal{A}(f_{0})$ there (as $\mathcal{A}$ is continuous) and also $K$ is dense in $[[F^{G}]]$, we get $C\not = \mathcal{A}(C)$. This contradicts $\emptyset$-definability of $\mathcal{F}$, since $\mathcal{A}$ restricted to $K$ is an (onto) automorphism.
\[conditional\]
- $D_{p}S_{rc}COF\vdash D_{p}S_{cf}COF$, $D_{\emptyset}S_{rc}COF\vdash D_{\emptyset}S_{cf}COF$.
- $D_{\emptyset}S_{cf}COF\nvdash D_{p}S_{rc}COF$.
(i). Let $F\vDash D_{p}S_{rc}COF$ (respectively $F\vDash D_{\emptyset}S_{rc}COF$) and $f:F\rightarrow F$ be a p-definable (respectively $\emptyset$-definable) Cauchy function. Let $\psi(z)$ be the formula expressing, using $f$ as a shorthand, $(\forall t)(\exists x\geq t)(z\leq f(x))$. We claim that the set $C$ defined by $\psi$ in $F$ is a regular cut whose supremum is the limit of $f$.
To see regularity of $C$, fix $\epsilon\in F^{>0}$. From the Cauchy criterion for $f$, there exists $d\in F$ such that $\forall x,y\geq d$, $|f(x)-f(y)|<\frac{\epsilon}{2}$. Let $z=f(d)-\frac{\epsilon}{2}$. Then $F\vDash\psi (z)$ and $F\vDash\neg\psi (z+\epsilon)$.
Now let $\sup(C)=\alpha$. To show $\lim f(x)=\alpha$, as $x$ becomes arbitrarily large in $F$, let $\epsilon\in F^{>0}$ and $d$ be as before. By the definition of $\alpha$, there exists $z\in C$ with $z>\alpha-\frac{\epsilon}{2}$. From this, we get $\exists
x_{0}\geq d$ with $\alpha-\frac{\epsilon}{2}<z\leq f(x_{0})$. On the other hand, for any $t\in F$, the element $x=\max\{t,d\}$ satisfies $f(x_{0})-\frac{\epsilon}{2}<f(x)$ and so $f(x_{0})-\frac{\epsilon}{2}\leq\alpha$. Therefore, $|f(x_{0})-\alpha|\leq\frac{\epsilon}{2}$. Hence, $$\forall x\geq d,\
|f(x)-\alpha| \leq |f(x)-f(x_{0})|+|f(x_{0})-\alpha|
< \frac{\epsilon}{2}+\frac{\epsilon}{2}
= \epsilon.$$
(ii). Let $R$ be the ordered ring $[[{\mathbb R}^{{\mathbb Q}^{\leq 0}}]]$ and $K$ its fraction field. We claim that $K$ is a witness for the independence assertion at hand. By Lemmas 3.1 and 3.3 and $[[{\mathbb R}^{\mathbb Q}]]$ being real closed, it suffices to show that $K$ is a non real closed dense sub-field of $[[{\mathbb R}^{\mathbb Q}]]$ closed under the automorphism mentioned there and its inverse. The latter is immediate from the same property for $R$.
As to $K$ being proper in its real closure, we use a result which has recently been shown in [@P]. It states that in $R$, all elements $a+1$ where $a$ has a strictly increasing $\omega$-sequence support converging to $0$, are prime. The series $$t^{-1}+t^{-\frac{1}{2}}+t^{-\frac{1}{3}}+\cdots+1$$ is therefore an infinite prime in $R$. The square root of any (positive) infinite prime in $R$ (which has cofinal positive-integer powers there) witnesses that $K$ is not real closed.
To see that $K$ is dense in $[[{\mathbb R}^{\mathbb Q}]]$, notice that $R$ approximates within $1$ any element of $[[{\mathbb R}^{\mathbb Q}]]$.
It is interesting to note that, by Pitteloud’s results again, the same infinite prime remains prime in all $[[{\mathbb R}^{({\mathbb R}^{\alpha})^{\leq 0}}]]$ for any ordinal $\alpha$. The p-definable regular gap corresponding to its square root is therefore never realized in the fraction fields of such ordered extension fields and can not be traversed by $\emptyset$-definable functions over them.
Acknowledgement {#acknowledgement .unnumbered}
---------------
The first author thanks Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran, for partial support. This work will form part of second author’s PhD thesis under first author’s supervision at Tarbiat Modarres University, Tehran, Iran. The authors are grateful to D. Pitteloud for kindly sending his paper to them before it was published.
Note added in proof {#note-added-in-proof .unnumbered}
-------------------
The above proof for Proposition \[scottfieldsofgps\] will be supplemented elsewhere by an appropriate argument to show that $[[F^{G}]]$ is always Scott complete, no matter whether $F$ is so or not (of course, there $G$ will be a non-zero group). Lemma \[ifthencf\] therefore holds without assuming that $F$ is Scott complete. Theorem \[conditional\](ii) will also be improved to $D_{\emptyset}S_{rc}COF\nvdash D_{p}S_{rc}COF$ (with the same witness $K$ as above). Consequently, either $D_{\emptyset}S_{cf}COF\nvdash^{(?)} D_{p}S_{cf}COF$ or $D_{p}S_{cf}COF\nvdash^{(?)} D_{p}S_{rc}COF$ (or both). The parameter-free version of the latter remains open also.
[1]{}
John Cowles and Robert LaGrange, *Generalized [A]{}rchimedean fields*, Notre Dame J. Formal Logic **24** (1983), no. 1, 133–140. [MR ]{}[84m:03054]{}
P. Erd[ö]{}s, L. Gillman, and M. Henriksen, *An isomorphism theorem for real-closed fields*, Ann. of Math. (2) **61** (1955), 542–554. [MR ]{}[16,993e]{}
H. Jerome Keisler, *Monotone complete fields*, Victoria Symposium on Nonstandard Analysis (Univ. Victoria, Victoria, B.C., 1972), Springer, Berlin, 1974, pp. 113–115. Lecture Notes in Math., Vol. 369. [MR ]{}[58 \#21585]{}
Daniel Pitteloud, *Existence of prime elements in rings of generalized power series*, J. Symbolic Logic **66** (2001), no. 3, 1206–1216. [MR ]{}[1 856 737]{}
Paulo Ribenboim, *Fields: algebraically closed and others*, Manuscripta Math. **75** (1992), no. 2, 115–150. [MR ]{}[93f:13014]{}
Dana Scott, *On completing ordered fields*, Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967), Holt, Rinehart and Winston, New York, 1969, pp. 274–278. [MR ]{}[39 \#6866]{}
[^1]: Mojtaba Moniri and Jafar S. Eivazloo, [*Using nets in Dedekind, monotone, or Scott incomplete ordered fields and definability issues*]{}, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 195–203, Topology Atlas, Toronto, 2002
|
---
author:
- 'Arif Mahmood and Ajmal S. Mian'
title: |
Semi-supervised Spectral Clustering\
for Classification
---
Currently this article is under review and will be redirected to the organizational web page once accepted {#currently-this-article-is-under-review-and-will-be-redirected-to-the-organizational-web-page-once-accepted .unnumbered}
==========================================================================================================
Introduction {#sec:intro}
============
Clustering finds the intrinsic data structure by splitting the data into similar clusters whereas classification assigns labels based on prior knowledge. Thus, clustering follows the intrinsic data boundaries whereas classification follows externally imposed boundaries. The two boundaries are generally different resulting in clusters across multiple classes (Fig. \[fig:pcaPlot\]). Due to this difficulty, clustering has not been widely used for classification. We bridge this gap by proposing a classification algorithm based on semi-supervised clustering of labelled data combined with unlabelled data. The proposed approach is very powerful in the context of image-set classification [@chenimproved; @chen2012dictionary; @ImagesetAlignment2012; @harandi2011graph; @Hu_TPAMI_2012; @ProbElasticMatching; @Nishiyama_CVPR_2007; @wang2012manifold; @VidClust2013].
![Plot of top three eigen-coefficients of 6 facial image-sets of 5 subjects from the CMU Mobo data [@Gross_TR_2001]. (a) Ground-truth identities labeled with different shapes and colors. Test set is yellow triangles. Five clusters obtained with (b) K-means, (c) SSC [@ElhamPAMI] and (d) SKMS [@Anand]. In each case, clusters are across class boundaries. No class cluster correspondence exists. []{data-label="fig:pcaPlot"}](motFig2-2.pdf){width="8.4cm"}
{width="17.5cm"}
Unsupervised data clustering has been extensively studied for the analysis of complex data [@chen2009spectral; @ElhamPAMI; @Fitzgibbon_CVPR_2003; @fowlkes2004spectral; @Kwok; @von2007tutorial]. Despite significant advances, clustering and classification have remained two separate streams with no direct mapping. Figure \[fig:pcaPlot\] compares the ground truth labels with K-means clustering, Sparse Subspace Clustering (SSC) [@ElhamPAMI] and Semi-supervised Kernel Mean Shift (SKMS) clustering [@Anand]. Some of the basic questions that still remain unanswered include finding the appropriate number of clusters and establishing class-cluster correspondence when the clusters overlap with multiple classes. One can observe in Figure \[fig:pcaPlot\] that no well defined class-cluster correspondence exists and an optimal number of clusters cannot be defined in general. Even the latest algorithms such as Semi-supervised Kernel Mean Shift clustering [@Anand] does not make the problem easier. Both these ill-posed questions have long occluded the utilization of clustering in classification tasks. One of the main contributions of this paper is bridging the gap between clustering and classification. The proposed algorithm does not require a unique class-cluster mapping, and can efficiently handle clusters across class boundaries as well as multiple clusters within the same class.
Our first contribution is a Classification Via Clustering (CVC) algorithm where the final classification decision is based on semi-supervised clusters computed over all data. This ensures global classification decisions as opposed to the local decisions made in the existing classification literature [@harandi2011graph; @Hu_TPAMI_2012; @ProbElasticMatching; @Nishiyama_CVPR_2007; @wang2012manifold; @VidClust2013]. For this purpose, we apply semi-supervised clustering on the combined training and test data without enforcing class boundaries. We compute the clusters based on the data characteristics without using label information. The probability distribution of each class over the set of clusters is computed using the label information. This distribution can be thought of as a compact representation of the class. Classification is performed by measuring distances between the probability distribution of the test data from each training class. The proposed CVC algorithm is generic and applicable to any clustering algorithm.
Our second contribution is a Semi-supervised Hierarchical Clustering (SHC) algorithm where every parent cluster is partitioned into two child clusters in an unsupervised way while the labels are used only as a stopping criterion for partitioning. Note that the term semi-supervised is used in a different context compared to the existing semi-supervised clustering algorithms, such as SKMS [@Anand], where the label information forms a part of the main objective function to be minimized for clustering. In our case, the objective function is independent of the class labels. Although, SHC can be used with any partitioning criterion, we consider NCut graph partitioning where the two partitions are determined by the signs of the Fiedler vector of the Laplacian matrix. Our motiviation for this choice comes from the recent advances in spectral clustering [@chen2009spectral; @ElhamPAMI; @fowlkes2004spectral; @Kwok; @ngspectral].
Our third contribution is a Direct Fiedler Vector Computation (DFVC) algorithm which is based on the shifted inverse iteration method. Hierarchical spectral clustering was not previously considered a viable option because all the eigenvectors were required to be computed while only the second least significant (Fiedler Vector) was used. The proposed DFVC algorithm solves this problem. Moreover, existing spectral clustering research is based on off-the-shelf eigen-solvers which aim to accurately find the magnitudes of the eigenvector coefficients even though only their signs are required for partitioning. The convergence of our DFVC algorithm is based on the signs.
We apply the proposed algorithms to the problem of image-set classification where the training classes and the test class consist of a collection of images. The proposed algorithms can be directly applied to the raw data however, we represent each image-set with a Grassmannian manifold and perform clustering on the manifold basis. This strategy reduces computational complexity and increases discrimination and robustness to noise. The use of Grassmann manifolds enables us to make an ensemble of spectral classifiers, each based on a different dimensionality of the manifold. This significantly increases the accuracy and also gives reliability (confidence) of the label assignment. An overview of our approach is given in Fig. \[fig:blockDiagram\]. Our initial work in this direction appeared in [@MyCVPR14] where we computed all the eigenvectors and partitioned the data into two or more clusters. In this paper, we perform only binary partitioning and propose a Direct Fiedler Vector Computation (DFVC) algorithm thus avoiding the computation of all eigenvectors.
Experiments were performed on three standard face image-sets (Honda [@Lee_CVPR_2003], CMU Mobo [@Gross_TR_2001] and YouTube Celebrities [@Kim_CVPR_2008]), an object categorization (ETH 80) [@ETH80], and Cambridge hand gesture [@Camdatabase] datasets. Results were compared to seven state of the art algorithms. The proposed technique achieved higher accuracy on all datasets. The maximum improvement was observed on the most challenging You-Tube dataset where our algorithm achieved 11.03% higher accuracy than the previous best reported.
$\mathcal{G}$, [Probe Set:]{} $c_p$, [Gallery Labels:]{} $\ell_\mathcal{G}$ ${\ell_p}$ [{Probe Set Label}]{} $\mathcal{D}=[ \mathcal{G}~~c_p]$ $n_c \leftarrow$ unique($\ell_\mathcal{G}$) [{Number of Classes}]{} $\ell_{\mathcal{D}}\leftarrow [\ell_\mathcal{G}~~ \widehat{\ell}_{p}]$, [{$\widehat{\ell}_{p}=n_c+1$ is dummy probe label}]{} $[\ell_k, n_k] \leftarrow$ **SHC** $(\mathcal{D},\ell_{\mathcal{D}})$ [{Semi-sup. Hierar. Clust.}]{} $H_{cc} \leftarrow $zeros$(n_c+1,n_k)$ [{Class cluster histogram}]{} $H_{cc}(\ell_\mathcal{D}(i),\ell_k(i))=H_{cc}(\ell_\mathcal{D}(i),\ell_k(i))+1$ $H_{cc}(i,:)=H_{cc}(i,:)/\sum_{j=1}^{n_k}{H_{cc}(i,j)}$ $\textbf{p}_p\leftarrow H_{cc}(n_c+1,:)$ [{Probe cluster distribution}]{} $\textbf{p}_i\leftarrow H_{cc}(i,:)$ [{Class cluster distribution}]{} $\mathcal{B}(i)=-\ln\sum\limits_{j=1}^{n_k}\sqrt{p_{i}(j)p_{p}(j)}$ [{Bhattacharyya distance}]{} $\ell_p \leftarrow \ell_\mathcal{G}(i_{\min})$ such that $i_{\min}\equiv \min_i (\mathcal{B})$
\[algo:CVC\]
Related Work
============
Many existing image-set classification techniques are variants of the nearest neighbor (NN) algorithm where the NN distance is measured under some constraint such as representing sets with affine or convex hulls [@Cevikalp_CVPR_2010], regularized affine hull [@RegNearestPoints], or using the sparsity constraint to find the nearest points between image-sets [@Hu_TPAMI_2012]. Since NN techniques utilize only a small part of the available data, they are more vulnerable to outliers.
At the other end of the spectrum are algorithms that represent the holistic set structure, generally as a linear subspace, and compute similarity as canonical correlations or principle angles [@Kim_TPAMI_2007]. However, the global structure may be a non-linear complex manifold and representing it with a single subspace may lead to incorrect classification [@chenimproved]. Discriminant analysis has been used to force the class boundaries by finding a space where each class is more compact while different classes are apart. Due to multi-modal nature of the sets, such an optimization may not scale the inter class distances appropriately (Fig. \[fig:pcaPlot\]a). In the middle of the spectrum are algorithms that divide an image-set into multiple local clusters (local subspaces or manifolds) and measure cluster-to-cluster distance [@chenimproved; @Wang_CVPR_2009; @wang2012manifold]. In all these techniques, label assignment decision is dominated by either a few data points, only one cluster, one local subspace, or one basis of the global set structure, while the rest of the set data or local structure variations are ignored. In contrast, our approach uses all the samples and exploits the local and global structure of the image-sets.
Classification Via Clustering
=============================
Classification algorithms enforce the class boundaries in a supervised way which may not be optimal because training data label assignment is often manual and based on the real world semantics or prior information instead of the underlying data characteristics. For example, all training images of the same person are pre-assigned the same label despite that the intra-person distance may exceed inter-person distance. Therefore, the intrinsic data clusters will not align with the imposed class boundaries.
We propose to compute the class to class distance based on the probability distribution of each class over a finite set of clusters. In the proposed technique, clustering is performed on all data comprising the training and test sets combined without considering their labels. By doing so, we ensure natural clusters based on the inherent data structure. Clusters are allowed to be formed across two or more training classes. Once an appropriate number of clusters is obtained, we use the labels of training data to compute the probability distribution of each class over the set of clusters. Then we use distribution based distance measures to find which class is the nearest to the test set. Algorithm \[algo:CVC\] shows the basic steps of the proposed approach.
Let $\mathcal{G}=\{X_i\}_{i=1}^{g} \in \mathds{R}^{l \times n_g}$ be the gallery containing labeled training data, where $n_g=\sum_{i=1}^{g}{n_i}$ are the total number of data points in the gallery and $n_i$ be the data points in a set $X_i=\{x_j\}_{j=1}^{n_i} \in\mathds{R}^{l\times n_i}$. A data point $x_j\in \mathds{R}^l$ could be a feature vector (e.g. LBP or HoG) or simply the pixel values. The number of classes in the gallery $n_c$ are generally less than or equal to the number of sets $ n_c \le g$. Let $c_p=\{x_i\}_{i=1}^{n_p} \in \mathds{R}^{l \times n_p}$ be the probe-set with a dummy label $n_c+1$. We make a data matrix by appending all gallery sets and the probe set: $\mathcal{D}=[\mathcal{G}~~c_p]\in \mathds{R}^{l \times n_d}$, where $n_d=n_g+n_p$. Let $\ell_\mathcal{G}\in \mathds{R}^{ n_g}$ be the labels of the gallery data and $\widehat{\ell}_{k} \in \mathds{R}^{n_p}$ be the dummy labels of the probe set, selected as $n_c+1$. A label matrix is formed as $\ell_\mathcal{D}=[\ell_\mathcal{G}~~\widehat{\ell}_{k}]\in \mathds{R}^{ n_d}$.
For the purpose of clustering we propose Semi-supervised Hierarchical Clustering (SHC) due to better control on the quality of the clusters as discussed in Section \[SHC\]. The output of the SHC is a cluster label array $\ell_k$ and the number of clusters $n_k$. Note that for the case of existing unsupervised clustering algorithms such as SSC or K-means, $n_k$ is a user defined parameter while our proposed SHC Algorithm automatically finds $n_k$. Using the label arrays $\ell_\mathcal{D}$ and $\ell_k$ we compute a 2D class-cluster histogram $H_{cc}$. Each row of $H_{cc}$ corresponds to a specific class and each column corresponds to a specific cluster. For a class $c_i$ let $\textbf{p}_i=H_{cc}(i,:)/\sum_{k=1}^{n_k}{H_{cc}(i,k)} \in \mathds{R}^{n_k}$ be the distribution over all clusters, $\sum_{k=1}^{n_k}{\textbf{p}_i[k]}=1$ and $1 \ge \textbf{p}_i[k] \ge 0$. Since we set dummy probe label to be $n_c+1$, the last row of $H_{cc}$ is the probe set distribution over all clusters $\textbf{p}_p=H_{cc}(n_c+1,:)/\sum_{k=1}^{n_k}{H_{cc}(i,k)}$.
Distance between the two distributions $\textbf{p}_i$ and $\textbf{p}_p$ can be found by using an appropriate distance measure such as Bhattacharyya [@div1967] distance $$\label{bhat}
\mathcal{B}_{i,p}=-\ln \sum\limits_{k=1}^{n_k}\sqrt{\textbf{p}_{i}(k)\textbf{p}_{p}(k)}.$$ Bhattacharyya distance is based on the angle between the square-root of the two distributions: $\mathcal{B}_{i,p}=-\ln (\cos(\theta_{ip}))$. Another closely related measure is Hellinger distance [@beran1977minimum] which is the $\ell_2$ norm of the difference between the square-root of the two distributions $$\label{Hal}
\mathcal{H}_{i,p}= \frac{1}{\sqrt{2}} \sqrt{\sum_{k=1}^{n_k}{(\sqrt{\textbf{{p}}_i[k]}-\sqrt{{\textbf{p}}_p[k]})^2}}.$$
Fig. \[fig:Fig1Stat\] shows the class cluster distributions for the three clustering results shown in Fig. \[fig:pcaPlot\]. Distance $\mathcal{B}_{i,p}$ for each algorithm is shown in Fig. \[fig:Fig1Stat\]d. Minimum value of $\mathcal{B}_{i,p}$ is found to be {0.0680, 0.00423, 0.0342} for k-means, SSC and SKMS respectively. In all three cases, the label of the test class is correctly found. We consider the ratio of rank 2 to rank 1 distance as SNR. A larger value of SNR shows more robustness to noise. In this experiment SNR of k-means, SSC and SKMS is {2.63, 78.84, 5.20} respectively indicating that SSC based classification is more robust compared to the others.
![(a-c) Plots of class-cluster distributions ($H_{cc}$) of the five gallery classes and 6-th test class over five clusters obtained by k-means, SSC and SKMS shown in Fig. \[fig:pcaPlot\] (d) Bhattacharyya distance of the test set from each gallery class.[]{data-label="fig:Fig1Stat"}](ClassClusterDistr.pdf){width="8.5cm"}
Optimizing the Number of Clusters
==================================
Algorithm \[algo:CVC\] does not impose any constraint on the number of clusters however, in Lemma \[lam2.1\] we argue that an optimal number of clusters exists and can be found in the context of classification problem. The derivation of Lemma \[lam2.1\] is based on the notion of [*conditional orthogonality*]{} and [*indivisibility*]{} of clusters as defined below. We assume that classes $c_i$ and $c_j$ belong to the gallery ${\mathcal{G}}$ while $c_p$ is the probe set with unknown label.
Class cluster distribution of class $c_i$ is [*conditionally orthogonal*]{} to the distribution of a class $c_j$ w.r.t the distribution of probe set $c_p$ if $$\label{CO}
(\textbf{p}_i \perp_{c} \textbf{p}_j)_{\textbf{p}_p} {\mathrel{\mathop:}=}\llangle(\textbf{p}_i \land \textbf{p}_p), (\textbf{p}_j \land \textbf{p}_p)\rrangle=0 ~~\forall (c_i,c_j)\in \mathcal{G}.$$
A cluster $k$ is indivisible from the probe set label estimation perspective if $\forall (c_i,c_j)\in \mathcal{G}$ $$\label{indC}
p_i[k]\land p_j[k]\land p_p[k]=0.$$ $$p_i[k] \lor p_j[k] \lor p_p[k]=1.$$ A cluster is [*indivisible*]{} if at least one of the three probabilities, $p_i[k]$, $p_j[k]$ and $p_p[k]$ is zero. If all clusters become indivisible, class cluster distributions of all gallery classes will become [*conditionally orthogonal*]{}.
The optimal number of clusters for a set labeling problem are the minimum number of clusters such that all class-cluster distributions become [*conditionally orthogonal*]{}. $$\label{lam1}
n_k^* \triangleq \min_{n_k} (\textbf{p}_i \perp_{c} \textbf{p}_j)_{\textbf{p}_p}~~~~~ \forall c_i,c_j\in \mathcal{G}$$\[lam2.1\]
Consider an indivisible cluster with two non-zero probabilities $p_i$ and $p_p$ corresponding to the gallery class $c_i$ and the probe set. Suppose this cluster is divided into $n\ge 2$ child clusters with $q_i[1],q_i[2]\cdots q_i[n]$ and $q_p[1],q_p[2]\cdots q_p[n]$ probabilities such that $q_i[1]+q_i[2]\cdots +q_i[n]=p_i$ and $q_p[1]+q_p[2]\cdots +q_p[n]=p_p$. Then $$p_ip_p\ge \sum_{j=1}^n q_i[j]q_p[j].$$ Therefore, reduction in the cluster size will cause increase in the distances $\mathcal{B}_{i,p}$ and $\mathcal{H}_{i,p}$. Moreover, some of the probabilities $q_i[j]$ and $q_p[j]$ may become zero causing the loss of useful discriminating information.
Division of the clusters with $p_p=0$ will not have any effect on $\mathcal{B}_{i,p}$ while $\mathcal{H}_{i,p}$ will decrease.
Decreasing the number of clusters such that $n_k < n_k^*$, will result in some clusters with overlapped class-cluster distributions leading to an increased intra-class similarity and reduced discrimination. Therefore, $n_k^*$ are the optimal number of clusters for classification.
Existing unsupervised clustering algorithms do not ensure the clusters to be indivisible and require the number of clusters to be user defined. In the next section, we propose Semi-supervised Hierarchical Clustering (SHC) which efficiently solves this problem.
$A$, [class labels array:]{} $\ell_\mathcal{D}$, [cluster label matrix:]{} $\ell_k$, [current cluster ID:]{} $c$, [current recursion level:]{} $r$, [dummy probe label:]{} $\widehat{\ell}_p$ $\ell'_k$ [{Updated cluster labels matrix}]{} $\ell_b\leftarrow \ell_k(r,:)$ [{Cluster labels at current level}]{} $i_c \leftarrow \ell_b==c$ [{Current cluster indicator vector}]{} $\ell_c \leftarrow \ell_\mathcal{D}(i_c)$ [{Class labels in the current cluster}]{} **Return** [{Retain indivisible cluster}]{} $A_c \leftarrow A(i_c,i_c)$ [{Local proximity matrix: $n \times n$}]{} $D_c(j,j)=\sum_{i=1}^{n}{A_c(i,j)}$ [{Local degree matrix}]{} $L_s \leftarrow {D_c^{-1/2}}(D_c-A_c){D_c^{-1/2}}$ { $u_{n-1}\leftarrow$**DFVC**$(Ls, D_c)$ [{Direct Fiedler Vector Comp.}]{} $\hat{\ell}=u_{n-1} \ge 0$ [{Sign based partitioning}]{} $c \leftarrow c+1$ $\ell'_k \leftarrow$ Update-Cluster-Labels $(\ell_k,\hat{\ell},c,p,r)$ $\ell'_k \leftarrow\textbf{ SHC}(A,\ell_\mathcal{D},\ell'_k,c,r+1,t)$ [{Recursive Call}]{}
\[algo:SHC\]
SHC: Semi-supervised Hierarchical Clustering {#SHC}
=============================================
The class cluster distributions shown in Fig. \[fig:Fig1Stat\] are not conditionally orthogonal because two clusters need further partitioning. However, if the number of clusters is blindly increased, other indivisible clusters may get partitioned. Such a break down of indivisible clusters will reduce the discrimination capability of the CVC algorithm. To this end, we propose a semi-supervised algorithm based on two-way hierarchical clustering which can identify indivisible clusters and hence avoid further partitioning of such clusters. A parent cluster is partitioned into two child clusters only if it is divisible. Once all clusters become [*indivisible*]{}, the algorithm stops and hence the optimal number of clusters is automatically determined.
Algorithm \[algo:SHC\] recursively implements the proposed Semi-supervised Hierarchical Clustering (SHC). For the purpose of dividing data into two clusters, we consider NCut based graph partitioning. This choice is motivated by the robustness of spectral clustering [@ElhamPAMI]. In NCut, each data point in the data matrix $\mathcal{D}$ is mapped to a vertex of a weighted undirected graph $G=(V,E)$, where the edge weights correspond to the similarity between the two vertices [@ngspectral; @von2007tutorial]. The adjacency matrix $A \in \mathds{R}^{n_d \times n_d}$ is computed as $$\label{eq:adj} A_{i,j}=\begin{cases} &\exp{\bigl(-\frac{1}{2}|x_i-x_j|^\top \Sigma^{-1}|x_i-x_j|\bigr)} \text{ if } i\ne j\\ & 0 \text{ if } i = j,\end{cases}$$where the parameter $\Sigma$ controls the connectivity. The corresponding degree matrix is defined as $$D(i,j)=\begin{cases}&\sum_{i=1}^{n_d} {A(i,j)}\text{ if } i = j\\& 0 \text{ if } i \ne j.\end{cases}$$ If $p_r$ and $p_l$ are the two partitions, the normalized cut (NCut) objective function [@fowlkes2004spectral; @NormCut] is given by $$J_{nc}=\sum_{i\in |p_r|, j \in |p_l|}{A(i,j) }(\frac{1}{v_{r}}+\frac{1}{v_{l}}), \label{J_nc}$$ where $v_{r}$ and $v_{l}$ are the volumes of $p_r$ and $p_l$, given by the sum of all edge weights attached to the vertices in that partition $$v_{r}=\sum\limits_{i\in p_r}{D(i,i)}, v_{l}=\sum\limits_{i\in p_l}{D(i,i)}.$$ Shi and Malik [@NormCut] have shown that the NCut objective function is equivalent to $$J_{nc}=\frac{y_{r}^\top(D-A)y_{r}}{y_{r}^\top Dy_{r}},$$ where $y_{r}$ is an indicator vector $$y_{r}(i)=\begin{cases}&v_l \text{ if } i \in p_r\\&- {v_{r}}{} \text{ if } i \in p_l.\end{cases}$$
An exhaustive search for a $y_r$, such that $J_{nc}$ is minimized, is NP complete. However, if $y_r$ is relaxed to have real values then an approximate solution can be obtained by the generalized eigenvalue problem $$(D-A)\hat{y}_{r}=\lambda D \hat{y}_{r}.$$ Transforming to the standard eigen-system $$D^{-\frac{1}{2}}(D-A)D^{-\frac{1}{2}}\hat{q}_{r}=\lambda \hat{q}_{r},$$ where $\hat{q}_{r}=D^{\frac{1}{2}}\hat{y}_{r}$ is an approximation to $q_r$ and $$L_{s}=D^{-\frac{1}{2}}(D-A)D^{-\frac{1}{2}}$$ is a symmetric normalized Laplacian matrix. For a connected graph, only one eigenvalue of $L_s$ is zero which corresponds to the eigenvector given by $\textbf{u}_n=\sqrt{\text{diag}(D)}$ and $\lambda_n=\textbf{u}_n^\top L_s \textbf{u}_n={0}$.
The smallest non-zero eigenvalue $\lambda_{n-1}$ of $L_s$ represents the algebraic connectivity of the graph [@fiedler1973algebraic] and the corresponding eigenvector is known as the Fiedler vector $$\textbf{u}_{n-1}=\min_{\textbf{u} \perp \textbf{u}_n} (\textbf{u}^\top L_s \textbf{u}).$$ The signs of the Fiedler vector can be used to divide the graph into two spectral partitions $$\ell(x_i)=\begin{cases}& p_r \text{ if } \textbf{u}_{n-1}(i)\ge0\\&p_l \text{ if } \textbf{u}_{n-1}(i)<0.\end{cases}$$
Algorithm \[algo:SHC\] combines the NCut based graph partitioning with semi-supervised hierarchical clustering. It takes the proximity matrix $A$ defined in , class labels $\ell_\mathcal{D}$, cluster labels matrix $\ell_k$, current cluster ID $c$, and the current recursion level $r$. The dummy label given to the test set $\widehat{\ell}_p$ is unique from the class labels. Initially $\ell_\mathcal{k}$ is set to all ones, $c=1$, $r=1$ and at each iteration, the algorithm updates these values. The algorithm stops when all clusters become [*indivisible*]{}.
We perform an experiment on synthetic data comprising two random 3D Gaussian clusters with means $\mu_1=[0~ 0~ 0]^\top$ and $\mu_2=[d~ 0~ 0]^\top$, where $d$ is the distance between the cluster centers which is varied from $2$ to $7$ (Fig. \[fig:twoClusters\]a). Both clusters have the same variance $\sigma I$, where $\sigma=2$ and $I$ is a $3\times 3$ identity matrix. The size of each cluster grows from $100$ to $500$ data points in steps of $25$ points. NCut is applied to the data and the relationship between different parameters is analyzed in Figure \[fig:twoClusters\]. Notice how the value of $\lambda_{n-1}$ remains fairly stable, given a fixed $d$, for increasing number of data points. This empirically varifies that the value of $\lambda_{n-1}$ is a close approximation of the NCut objective function [@NormCut]. Comparing Fig. \[fig:twoClusters\]b and \[fig:twoClusters\]c, we notice that the magnitude of $\lambda_{n-1}$ is also proportional to the classification error.
A sign change in the Fiedler vector $\textbf{u}_{n-1}$ basically means that the corresponding data point has moved from one cluster to the other. Since Fiedler vector is computed iteratively, we can count the number of sign changes till convergence (Section \[sec:DFV\]). Hence, we can find the number of points switching partitions and the number of times each point switches partitions. We observe, in Fig. \[fig:twoClusters\]d, that the average number of sign changes (points switching partitions) reduces as $\lambda_{n-1}$ reduces (i.e. $d$ increases). Figure \[fig:twoClusters\]e shows the classification error only for those points that swith partitions 0, 1 and 2 times. We observe that those data points which change signs (switch partitions) more frequently have higher error rates.
If a parent cluster is divided into two balanced child clusters, the size of each child’s local proximity matrix is four times smaller than the parent’s matrix. Since the complexity of eigenvector solvers is $O(n^3)$, the complexity reduces in the next iteration by $O((\frac{n}{2})^3)$ for each sub-problem. The depth of the recursive tree is $\log_2(n_d)$, however, the proposed supervised stopping criterion does not let the iterations to continue until the very end. The process stops as soon as all clusters are indivisible. A computational complexity analysis of the recursion tree reveals that the overall complexity of the eigenvector computations remains the same, $O(n_d^3)$.
![(a) 3D Gaussian clusters with center to center distance varying from $d=2$ to $d=7$ units. (b) Variation of algebraic connectivity $\lambda_{n-1}$ with graph size and distance $d$. (c) Classification error versus the graph size and $d$ (d) Average number of sign switches of Fiedler vector coefficients versus the graph size and $d$ (e) Classification error is more in data points which change signs (switch partitions) more frequently.[]{data-label="fig:twoClusters"}](Plot2ndEigClusterDistance1.pdf){width="8.5cm"}
Proximity Matrix Computation through Regularized Linear Regression
------------------------------------------------------------------
Often high dimensional data sets lie on low dimensional manifolds. In such cases, the Euclidean distance based adjacency matrix may not be an effective way to represent the geometric relationships among the data points. A more viable option is the sparse representation of data which has been used for many tasks including label propagation [@SparseLabel], dimensionality reduction, image segmentation and face recognition [@Wright_TPAMI_2009]. Recently, Elhamifar and Vidal [@ElhamPAMI] proposed sparse subspace clustering which can discriminate data lying on independent and disjoint subspaces.
A vector can only be represented as a linear combination of other vectors spanning the same subspace. Therefore, the proximity matrices based on linear decomposition of data lead to subspace based clustering. Representing a data point $x_i$ as a linear combination of the remaining data points $ (\mathcal{\hat{D}})$ ensures that zero coefficients will only correspond the points spanning subspaces independent to the subspace spanned by $x_i$. Such a decomposition may be computed with least squares: $\alpha_i=(\mathcal{\hat{D}}^\top \mathcal{\hat{D}})^{-1}\mathcal{\hat{D}}^\top x_i$, where $\alpha$ are the linear coefficients. For high dimensional data, $\mathcal{\hat{D}}^\top \mathcal{\hat{D}}$ can be rank deficient. Inverse may be computed through eigen decomposition: $D^\top D=USU^\top$, where $U$ are the eigenvectors of $D^\top D$ and $S$ is the diagonal matrix of singular values. By selecting non-zero singular values and the corresponding eigenvectors, $D^\top D=\hat{U}\hat{S}\hat{U}^\top$ and $\alpha_i=\hat{U}\hat{S}^{-1}\hat{U}^\top\mathcal{\hat{D}}^\top x_i$.
One may introduce sparsity by only considering the $w$ largest coefficients in $\alpha_i$ and forcing the rest to zero. An alternate approach is to use $\ell_1$ regularized linear regression also known as Lasso [@Tibshirani94] $$\label{eqn:lasso2}\alpha_i^* {\mathrel{\mathop:}=}\min_{\alpha_i} \left(\frac{1}{2}||x_i-\mathcal{\hat{D}}\alpha_i||^2_2 + w||\alpha_i||_1 \right)~,$$ where $||\alpha_i||_1$ approximates the sparsity induction term and $w_i>0$ is the relative importance of the sparsity term. In , $\ell_1$ is a soft constraint on sparsity and the actual sparsity (number of non-zero coefficients) may vary. To ensure a fixed sparsity linear regression $$\label{eqn:womp}\alpha_i {\mathrel{\mathop:}=}\min_{\alpha_i} \left(\frac{1}{2}||x_i-\mathcal{\hat{D}}\alpha_i||^2_2\right) \text{such that} ||\alpha_i||_o \le w,$$ can be used. We observed that while this approach is faster, it yeilds less accuracy compared to the $\ell_1$ formulation.
To complete the proximity matrix $S$, the same process is repeated for all $x_i$ and the corresponding $\alpha_i$ are appended as columns $S=\{\alpha_i\}_{i=1}^{n_d}\in\mathds{R}^{{n_d}\times
{n_d}}$. Some of the $\alpha$ coefficients may be negative and in general $S(i,j) \ne S(j,i)$. Therefore, a symmetric sparse LS proximity matrix is computed as $A=|S|+|S^\top|$ for spectral clustering.
Spectral Clustering on Grassmannian Manifolds
=============================================
Eigenvectors computation of large Laplacian matrices $L_{s}\in\mathds{R}^{n_d\times n_ d}$ incurs high computational cost. A naive approach to reduce the cost is to compute eigenvectors for randomly sampled columns of $L_{s}$ and extrapolate to the rest of the data [@fowlkes2004spectral; @pengscalable; @Kwok]. Instead, we replace each image-set $X_i=\{x_j\}_{j=1}^{n_i} \in\mathds{R}^{l\times n_i}$ by a compact representation and perform clustering on the representation. Our choice of compact representation is motivated from linear subspace based image-set representations [@wang2012manifold; @Kim_TPAMI_2007]. These subspaces can be considered as points on Grassmannian manifolds [@hamm2008grassmann; @harandi2011graph]. While others perform discriminant analysis on Grassmannian manifolds or compute manifold to manifold distances, we propose sparse spectral clustering on Grassmannian manifolds.
A set of $d$-dimensional linear subspaces of $\mathds{R}^{n}$, $n=\min(l,n_i)$ and $d \le n$, is termed the Grassmann manifold $Grass(d, n)$ [@grassmann]. An element $\mathcal{Y}$ of $Grass(d, n)$ is a $d$-dimensional subspace which can be specified by a set of $d$ vectors: $Y=\{y_1, . . . , y_d\}\in \mathds{R}^{l\times d}$ and $\mathcal{Y}$ is the set of all linear combinations. For each data element of the image-set, we compute a set of basis ${Y}$ and the set of all such ${Y}$ matrices is termed as a non-compact Stiefel manifold $ST(d,n){\mathrel{\mathop:}=}\{{Y} \in R^{l \times d}: \text{ rank }(\mathcal{Y} ) = d\}$. We arrange all the $Y$ matrices in a basis matrix $B$ which is capable of representing each data point in the image-set by using only $d$ of its columns. For the $i^{th}$ data point in the $j^{th}$ image-set $x_j^i \in X_j$, having $B_j$ as the basis matrix, $x_j^i=B_j\alpha_j^i$, where $\alpha_j^i$ is the set of linear parameters with $|\alpha_j^i|_o=d$. For the case of known $B_j$, we can find a matrix $\alpha_j =\{\alpha_j^1, \alpha_j^2, \cdots \alpha_j^{n_i}\}$ such that the residue is minimized $$\label{eqn:lasso} \min_{\alpha_j}(\sum\limits_{i=1}^{n_i}||x_j^i-B_j\alpha_i^j||^2_2) \text{ ~~s.t. } ||\alpha_j||_o \le d ~.$$ Using the fact that $\ell_o$ norm can be approximated by $\ell_1$ norm, we can estimate both $\alpha_j$ and $B_j$ iteratively by using the following objective function [@Lee07efficientsparse]$$\label{eqn:loss}\min_{\alpha_j,B_j}\left(\frac{1}{n_j}\sum_{i=1}^{n_j} {\frac{1}{2}||x_j^i-B_j\alpha_j^i||^2_2 \text{ ~~s.t. }||\alpha_j^i||_1 \le d}\right)~.$$ The solution is obtained by randomly initializing $B_j$ and computing $\alpha_j$, then fixing $\alpha_j$ and recomputing $B_j$ until convergence. The columns of $B_j$ are significantly smaller than the number of data points $n_j$ in the corresponding image-set leading to computational cost reduction. Interestingly, this compact representation also increases the classification accuracy of the proposed CVC algorithm because the underlying subspaces of each class are robustly captured in $B$ while the outliers are discarded.
Ensemble of Spectral Classifiers {#sec:ensemble}
--------------------------------
Representing image-sets with Grassmannian manifolds facilitates the formation of an ensemble of spectral classifiers. For an image-set $X_i$ varying the random initializations and dimensionality of $B_i$ in will converge to different solutions. During off line training, we compute a set of manifolds of varying dimensionality for each image-set $X_i \equiv$ {$B_i^1,B_i^2,...B_i^\kappa$}. Spectral clustering is independently performed on each set of manifolds of the same dimensionality $\{B_1^j,B_2^j,...B_{g}^j,B_{g+1}^j\}$, where $B_{g+1}^j$ is the probe set representation.
The distance of each training set is computed from the probe set. For example, Bhattacharyya distance of image-set $X_i$ with representation $B_j$ from the probe set $X_p$ is given by $\mathcal{B}_{i,j,p}$ . For the $j^{th}$ manifold representation, we get a set of distances $\mathcal{B}_{j,p}\in \mathds{R}^{g}$. We consider two simple fusion strategies, sum rule and mode or maximum frequency rule. In sum rule, we sum all the distance vectors $\mathcal{B}_p=\mathcal{B}_{1,p}+\mathcal{B}_{2,p}+...\mathcal{B}_{\kappa,p}$ and the probe set label corresponds to the gallery set with minimum overall distance $L_p\equiv\min_i(\mathcal{B}_p(i))$.
In mode based fusion, for the $j^{th}$ classifier, probe set label is independently estimated as the label of $i^{th}$ gallery set with minimum distance: $L_{j,p}^i\equiv\min_i (\mathcal{B}_{j,p}(i))$, and the final label is the mode of all labels: $L_p \equiv$ mode($\{L_{j,p}^i\}_{j=1}^\kappa$). We empirically observe that the mode based fusion is more robust in case of noisy image-sets and often generates more accuracy than the sum rule based fusion.
Direct Fiedler Vector Computation {#sec:DFV}
=================================
Spectral clustering research is based on generic eigensolvers. As discussed in Section \[SHC\], the signs of the eigenvector coefficients are more important than the numerical accuracy of their magnitudes. Therefore, while iteratively computing eigenvectors, we enforce the convergence of the signs of eigenvector coefficients instead of the eigenvalues. The iterations terminate when the number of sign changes reduces below a threshold.
We consider power iterations, a fundamental algorithm for eigen computation [@Matrix]. To find an eigenvector of $L_s$, a random vector $v^{(0)}_1$ is repeatedly multiplied with $L_s$ and normalized until after $k$ iterations $L_sv^{(k)}_1=\lambda_1 v^{(k)}_1$, where $\lambda_1$ is the maximum eigenvalue and $u_1=v^{(k)}_1$ is the most dominant eigenvector of $L_s$. The convergence of the power iterations algorithm depends on the ratio of first two eigenvalues $\lambda_2/ \lambda_1$. If this ratio is very small, the algorithm will converge in few iterations. Once $u_1=v^{(k)}$ is found, $L_s$ is deflated to find the next dominant eigenvector $$L_s^{(i+1)}=L_s^{(i)} - u_1 u_1^{\top} L_s^{(i)},$$ and the same process is repeated. Thus the computation of eigenvectors proceeds from the most significant to the least significant without skipping any intermediate vector. Such a computational order is very inefficient from the spectral clustering perspective where only the least significant eigenvectors are required. For the specific case of 2-way hierarchical clustering, only the eigenvector corresponding to the second smallest eigenvalue is required. This is the main reason why hierarchical spectral clustering was not previously considered a viable option [@NormCut]. We solve this problem by proposing Algorithm \[algo:Fiedler\] for the direct computation of the Fiedler vector using an inverse iteration method [@Matrix]. Moreover, since the signs are more important than the numerical values, our proposed algorithm stops when most of the signs of the Fiedler vector stabilize, further reducing the computational cost.
$L_s$, $u_n$, $\eta$, $\epsilon_s$ $u_{n-1}, \lambda_{n-1}$ {Fiedler Vector and Value } $\hat{L}_s\leftarrow L_s-u_n \textbf{1}^{1\times n}-\eta I$ {Over Deflation and Eigen-shift} $Q_sR_s \leftarrow \hat{L}_s$ {QR Decomposition} $v^{(0)}\leftarrow\sum_{i=1}^{n-1}{q_i}-u_n$, $v^{(0)}\leftarrow{v^{(0)}}/{||v^{(0)}||_2}$ $\Delta_s\leftarrow \epsilon_s+1$, $k\leftarrow1$ $v^{(k')}\leftarrow Q_s^\top v^{(k-1)}$ $w\leftarrow\textbf{0}^{n\times 1}$ $w(i)\leftarrow(v^{(k')}(i)-R_s(i,:)w)/R(i,i)$ $v^{(k)}\leftarrow{w}/{||w||_2}$ $\Delta_s \leftarrow \sum{(v^{(k)}>0)\oplus (v^{(k-1)}>0)}$ $k\leftarrow k+1$ $u_{n-1}\leftarrow v^{(k)}$, $\lambda_{n-1}\leftarrow u_{n-1}^\top L_s u_{n-1}$
\[algo:Fiedler\]
Graph Laplacian is a symmetric matrix, therefore its eigen decomposition is $L_s=U^\top \Lambda U$ and $L_s^{-1}=U^\top \Lambda^{-1} U$. For a scalar $\eta$ and identity matrix $I$ one can show that $$(L_s-\eta I)^{-1}=U^\top(\Lambda-\eta I)^{-1}U.$$ That is, the eigenvectors of $(L_s-\eta I)^{-1}$ are the same as that of $L_s$ while the eigenvalues are $1/(\lambda_i-\eta)$. If $\eta$ is the same as the k$^{th}$ eigenvalue, then $1/(\lambda_k-\eta)\rightarrow \infty$ and will become significantly larger than all other eigenvalues. If power iteration method is applied to $(L_s-\eta I)^{-1}$, it will converge in only one iteration. However, if the difference between $\eta$ and $\lambda_k$ is relatively large, power iteration will converge depending on the ratio of $(\lambda_j-\eta)/(\lambda_k-\eta)$, where $\lambda_j$ is the next closest eigenvalue.
To avoid matrix inversion, $(L_s-\eta I)^{-1}$, the matrix vector multiplication step in the power iteration method $$v^{(k)}=(L_s-\eta I)^{-1}v^{(k-1)}$$ is modified as $$(L_s-\eta I)v^{(k)}=v^{(k-1)}.$$ To find $v^{(k)}$ we solve a set of linear equations. For this purpose, we use QR decomposition $Q_sR_s=(L_s-\eta I)$, where $Q_s$ is an orthonormal matrix and $R_s$ is an upper triangular matrix. Therefore, the following system of equations $$R_{s} v^{(k)}=Q_{s}^\top v^{(k-1)}$$ is upper triangular and we use back substitution to efficiently solve this system.
If $\lambda_k$ is known, we can directly compute the $k^{th}$ eigenvector by setting $\eta=\lambda_k$. Otherwise, for a given $\eta$, the algorithm will find an eigenvector with eigenvalue closest to $\eta$. For direct computation of the Fiedler vector, we need an estimate of $\lambda_{n-1}$. Unfortunately, in the spectral clustering literature [@chung1997spectral], sufficiently tight and easy to compute bounds on $\lambda_{n-1}$ have not been found for generic graphs. The existing bounds are either loose or require significant computational complexity.
For a connected graph, $\lambda_{n}=0$ and $\lambda_{n-1}>\lambda_n$, for a very small value of $\eta$, the inverse power iteration algorithm may converge to the zero eigenvalue, while setting a relatively larger value of $\eta$, the algorithm may converge to another eigenvalue $\lambda>\lambda_{n-1}$. We solve this problem by over-deflating $L_s$ with $u_n=\sqrt{diag(D)}$, where $D$ is the degree matrix $$\hat{L}_s=L_s-u_n 1^{1\times n}.$$ Note that a conventional deflation approach cannot be used because $u_n$ is a null vector of $L_s$. Over-deflation ensures that the proposed Algorithm 3 does not converge towards $u_n$ no matter how small the value of $\eta>0$ is used.
To find a rough estimate of $\eta$ we use the bound $ \lambda_{n-1}\ge{1}/{\varphi v_g}$ [@chung1997spectral], where $\varphi$ is the diameter of the graph and $v_g$ is the volume of the graph. Note that $v_g$ can be easily computed from the degree matrix while no simple method exists for the computation of $\varphi$, the weight of the longest path in the graph. We use the fact that $\varphi \le v_g$, therefore $ \lambda_{n-1}\ge{1}/{ v_g^2}$. Which is a loose bound but readily known.
After convergence, Algorithm 3 can find the exact value of $\lambda_{n-1}$ however, we are not interested in the exact solution for the purpose of spectral clustering. Rather we are interested in the signs which represent the cluster occupancy. We compute the cluster indicator vector $q^{k}=v^{(k)}>0$. Instead of setting the convergence criteria on $|\lambda_{n-1}^k-\lambda_{n-1}^{k-1}|$, the iterations are stopped when $\Delta_s=\sum (q^{(k)} \oplus q^{k-1})<\epsilon_s$, where $\oplus$ is XOR operation and $\epsilon_s$ is a user defined threshold. The value of $\Delta_s$ also represents the quality of the NCut. A larger value shows that there are many data points which are switching across the partition in consecutive iterations and a good cut is not found.
Experimental Evaluation
=======================
The proposed algorithms were evaluated for image-set based face recognition, object categorization and gesture recognition. Comparisons are performed with seven existing state of the art image-set classification algorithms including Discriminant Canonical Correlation (DCC) [@Kim_TPAMI_2007], Manifold-Manifold Distance (MMD) [@wang2012manifold], Manifold Discriminant Analysis (MDA) [@Wang_CVPR_2009], linear Affine and Convex Hull based image-set Distance (AHISD, CHISD) [@Cevikalp_CVPR_2010], Sparse Approximated Nearest Points [@Hu_TPAMI_2012], and Covariance Discriminative Learning (CDL) [@CDL]. The same experimental protocol was replicated for all algorithms. The implementations of [@Kim_TPAMI_2007; @wang2012manifold; @Cevikalp_CVPR_2010; @Hu_TPAMI_2012] were provided by the original authors whereas the implementation of [@Wang_CVPR_2009] was provided by the authors of Hu et al. [@Hu_TPAMI_2012]. We carefully implemented CDL which was verified by the original authors of the algorithm [@CDL]. The parameters of all methods were carefully optimized to maximize their accuracy. For DCC, the embedding space dimension was set to 100, the subspace dimensionality was set to 10 and set similarity was computed from the 10 maximum correlations. For MMD and MDA, the parameters were selected as suggested in [@Wang_CVPR_2008] and [@Wang_CVPR_2009].
Each image-set was represented by $\nu$ manifolds with dimensionality increasing from 1 to $\nu$. Each set of equal dimensionality manifolds was used to make an independent classifier. The results of $\nu$ classifiers were fused using sum and mode rules. Table \[tab:CVC\] shows the results of the proposed CVC algorithm using Semi-supervised Hierarchical Clustering (SHC) for the two fusion schemes over all datasets. The proximity matrix was based on the $\ell_1$ norm regularized linear regression and solved using the SPAMS [@Mairal1] library with $w=0.01$ [@MyBMVC12]. For eigenvector computation, the proposed DFVC Algorithm \[algo:Fiedler\] was used. DFVC was taken to be converged when the sign changes reduced to $ \le 1.00$%. The code, manifold basis and clustering results will be made available at <http://www.csse.uwa.edu.au/~arifm/CVC.htm>.
-------------------------------------------------------------------------------------------------------------------------------------------- ------- ------- ------- ------- ------- ------- ------ ------- ------- ------- ------- ------- ------- ------- -------
Mod Sum NFus Mod Sum NFus Mod Sum NFus Mod Sum NFus Mod Sum NFus
(r[5pt]{}l[5pt]{})[2-4]{} (r[5pt]{}l[5pt]{})[5-7]{} (r[5pt]{}l[5pt]{})[8-10]{} (r[5pt]{}l[5pt]{})[11-13]{} (r[5pt]{}l[5pt]{})[14-16]{} Avg 76.03 70.28 62.27 98.33 96.39 96.53 100 95.13 94.36 91.75 88.75 84.75 - - -
Min 70.56 64.89 53.19 97.22 93.06 94.44 100 89.74 89.74 87.5 75.0 77.5 - - -
Max 81.91 76.59 66.31 100.0 98.61 100 100 100 100 100 97.50 92.50 83.10 76.53 74.58
STD 4.69 5.38 5.33 0.88 2.09 1.78 0.00 2.91 3.38 4.42 6.80 5.33 - - -
-------------------------------------------------------------------------------------------------------------------------------------------- ------- ------- ------- ------- ------- ------- ------ ------- ------- ------- ------- ------- ------- ------- -------
\[tab:CVC\]
![Two example image-sets from (a) Honda, (b) CMU Mobo and (c) You-tube Celebrities datasets. (d) One object category from the ETH 80 dataset.[]{data-label="fig:dataset"}](FigureALLData1.pdf){width="8.5cm"}
Face Recognition using Image-sets
---------------------------------
Our first dataset is the You-tube Celebrities [@Kim_CVPR_2008] which is very challenging and includes 1910 very low resolution videos (of 47 subjects) containing motion blur, high compression, pose and expression variations (Fig. \[fig:dataset\]c). Faces were automatically detected, tracked and cropped to 30$\times$30 gray-scale images. Due to tracking failures our sets contained fewer images (8 to 400 per set) than the total number of video frames. Experiments were performed on both intensity based features and HOG features. The proposed algorithm performed better on the HOG features. Five-fold cross validation experiments were performed where 3 image-sets were selected for training and the remaining 6 for testing.
Each image-set was represented by two Grassmannian manifold-sets for $\lambda=\{1, 2\}$ in . Each set had 8 classifiers with dimensionality increasing from 1 to 8. A different classifier was made for each dimension. Fig. \[fig:YoutubeMobo\]a shows that the accuracy increases as the dimensionality increases from 1 to 8 for different fusion options. The average accuracies obtained by mode fusion, sum fusion and no-fusion were {76.03$\pm$4.69, 70.28$\pm$5.38, 62.27$\pm$5.33}. Due to noise in the dataset the mode fusion performed better than the sum fusion. The mode fusion is more robust due to independent decisions at each level while the errors may get accumulated in sum fusion. Both mode and sum fusions outperformed the no-fusion case. This shows that fusion can be used to compensate for the cases where good quality cuts are not found for graph partitioning. The high dimensional manifolds are more separable in some dimensions and less in the others. The ensemble exploits the more separable dimensions to obtain better accuracy. On this dataset Bhattacharyya and Hellinger distance measures achieved very similar accuracies. Table \[tab:Comp\] shows a comparison with the existing state of the art image-set classification algorithms. Note that CVC algorithm with all combinations is more accurate than the previous best reported accuracy. The average accuracy of CVC algorithm is 76.03$\pm$4.69% outperforming the existing methods by 11.03%.
![CVC Algorithm performance on (a) YouTube and (b) CMU Mobo datasets.[]{data-label="fig:YoutubeMobo"}](youtubeMobo.pdf){width="8.6cm"}
![CVC algorithm performance on (a) Honda and (b) ETH80 datasets. []{data-label="fig:MoboG"}](HondaETH.pdf){width="8.6cm"}
The second dataset we used is CMU Mobo [@Gross_TR_2001] containing 96 videos of 24 subjects. Face images were resized to $40\times 40$ and LBP features were computed using circular (8, 1) neighborhoods extracted from $8\times 8$ gray scale patches similar to [@Cevikalp_CVPR_2010]. We performed 10-fold experiments by randomly selecting one image-set per subject as training and the remaining 3 as probes. This experiment was repeated by varying the manifold dimensionality from 1 to 12. The number of classifiers in the ensemble also vary from 1 to 12. Fig. \[fig:YoutubeMobo\]b shows the accuracy of CVC algorithm versus the number of classifiers. Most of the accuracy is gained by the first eight classifiers and any further increase provides minor improvement.
For the Mobo dataset, the mode fusion again performed better than the sum fusion (Table \[tab:CVC\]). The improvement over sum rule is relatively smaller compared to the Youtube dataset because Mobo dataset is less noisy. Thus, the no-fusion case also has very similar average accuracy to the sum fusion but with a higher standard deviation. On this dataset, CVC algorithm achieved a maximum of 100% and average 98.33$\pm$0.88% accuracy which is again the highest (Table \[tab:Comp\]).
Our final face recognition dataset is Honda/UCSD [@Lee_CVPR_2003] containing 59 videos of 20 subjects with varying poses and expressions. Histogram equalized 20$\times$20 gray scale face image pixel values were used as features [@wang2012manifold]. We performed 10-fold experiments by randomly selecting one set per subject as gallery and the remaining 39 as probes. In this dataset, each image-set was represented by 12 manifolds of dimensionality 1 to 12. The CVC algorithm achieved 100% accuracy with mode fusion (Table \[tab:CVC\]). Note that the accuracies of other algorithms in Table \[tab:CVC\] for the Honda dataset are different than those reported by their original authors because they are either used a single fold [@Hu_TPAMI_2012] or different folds.
\[tab:Comp\]
Object Categorization & Gesture Recognition
-------------------------------------------
For object categorization, we used the ETH-80 dataset [@ETH80] containing images of 8 object categories each with 10 different objects. Each object has 41 images taken at different views forming an image-set. We used 20$\times$20 intensity images for classifying an image-set of an object into a category. ETH-80 is a challenging database because it has fewer images per set and significant appearance variations across objects of the same class. Experiments were repeated 10 folds. Each time for each class, 5 random image-sets were used for training and the remaining 5 for testing. Each image-set was represented with manifolds of dimensionality varying from 1 to 10.
A comparison of mode, sum and no-fusion schemes is shown in Fig. \[fig:MoboG\]b. As the dimensionality of the manifold increases, the accuracy of all fusion schemes increases. The mode fusion decreases at $\nu=3$ showing that the third dimension of the manifold was less discriminative. Despite variations, mode fusion obtained better accuracy for $\nu>1$. Table \[tab:Comp\] shows that the CVC algorithm outperformed all other methods on the ETH-80 dataset giving a significant margin from algorithms that performed quite well on the face datasets such as SANP.
![(a) Cambridge Hand Gesture dataset. (b) CVC algorithm performance on Cambridge HG dataset. []{data-label="fig:ETHG"}](CambHG.pdf){width="8.6cm"}
SHC K-means SSC SKMS
-------------- ------- --------- ------- -------
Youtube 68.44 40.85 63.83 -
Mobo 94.44 61.53 93.60 72.23
Honda 97.44 29.50 88.33 69.23
ETH80 90.00 73.00 87.50 -
Cambridge HG 74.58 73.15 79.58 -
: Comparison of CVC algorithm (no-fusion) accuracy by using SHC, K-Means, SSC [@ElhamPAMI] and SKMS [@Anand].
\[tab:cmpSHC\]
SHC K-means SSC SKMS
-------------- ------- --------- -------- ---------
Youtube 7.30 3.17 20.57 109.34
Mobo 0.43 0.24 1.08 2.97
Honda 0.41 0.19 0.83 1.95
ETH80 1.21 1.71 3.03 10.63
Cambridge HG 22.75 85.72 160.00 1069.00
: Execution time (sec) comparison of SHC algorithm with K-means, SSC and SKMS clustering algorithms when matching a single probe set to the gallery.
\[tab:time\]
Our last dataset is the Cambridge Hand Gesture dataset [@Camdatabase] which contains 900 image-sets of 9 gesture classes with large intra-class variations (Fig. \[fig:ETHG\]a). Each class has 100 image-sets, divided into two parts, 81-100 were used as gallery and 1-80 as probes [@KimTensor]. Pixel values of $20 \times 20$ gray scale images were used as feature vectors. Each image-set was represented by a set of manifolds with dimensionality increasing from 1 to 7.
The accuracy of CVC algorithm using different fusion schemes versus the dimensionality of Grassmannian manifolds is shown in Fig. \[fig:ETHG\]b. The accuracy of all fusion schemes increased with increasing the number of classifiers. The maximum accuracy of the proposed CVC algorithm was 83.1% obtained by mode fusion (Table \[tab:CVC\]). Table \[tab:Comp\]) shows that the CVC algorithm outperformed the state of the art image-set classification algorithms by a significant margin. Notice that some of the image-set classification algorithms did not generalize to the hand gesture recognition problem. On the other hand, the proposed algorithm consistantly gives good performance for face recognition, object categorization and gesture recognition.
Comparison of CVC when combined with Different Clustering Algorithms
--------------------------------------------------------------------
We compare the performance of CVC algorithm by combining it with different existing clustering algorithms including k-means clustering, Sparse Subspace Clustering (SSC) [@ElhamifarV09] and Semi-supervised Kernel Mean Shift (SKMS) clustering [@Anand] (Table \[tab:cmpSHC\]). Comparison was performed in the no-fusion setting using Bhattacharyya measure. For each dataset, a single classifier based on the maximum manifold dimensionality was used. For the case of k-means and SSC the number of clusters were the same as the number of classes in the gallery. For SKMS, the algorithm automatically finds the optimal number of clusters.
For the SSC and the SKMS algorithms, the implementations of the original authors were used. The proximity matrices proposed by SSC [@ElhamPAMI] was based on the $\ell_1$ norm regularized linear regression which was computed by using ADMM [@ADMM] with default parameters recommended by the original authors [@ElhamPAMI]. The SHC proximity matrix was computed with the SPAMS [@Mairal1] library using $w=.01$, Mode=2. The SHC proximity matrix computaion time was significantly less than that required by SSC (Table \[tab:time\]).
The number of constraints in SKMS were repeated as {5%, 10%, 20%, 50%, 100%} of the data points in the gallery. For a randomly selected gallery point, another random gallery point was selected and a constraint was defined to show if both points belong to the same class (and hence the same cluster) or not. No constraint was specified for the data points in the test set. For other parameters of the SKMS algorithm, default values were used. We report the best performance of SKMS over these constraints.
In the SKMS algorithm, the constraints become part of the objective function. This type of supervision causes lack of generalization and hence the accuracy reduced. The supervision used in the proposed SHC algorithm keeps the partitioning itself unsupervised while using the labels to decide whether partitioning is required or not. Therefore, the generalization of the SHC algorithm is equivalent to that of the unsupervised clustering algorithms.
In Table \[tab:cmpSHC\], CVC combined with SHC refers to the proposed algorithm wich outperforms others on the first four datasets and has the second best performance on the last dataset. This is because the class-cluster distributions obtained by the proposed SHC algorithm are conditionally orthogonal reducing the within Gallery similarities and increasing the discrimination across the gallery classes. CVC algorithm combined with SSC clustering has the highest accuracy on Cambridge dataset and second highest on the remaining datasets.
CVC combined with SKMS does not perform as good as SHC or SSC because SKMS incorporates the class labels into the objective function. While SKMS may achieve state of the art results for clustering, it is not suitable for Classification Via Clustering (CVC) as it does not expoit the main strength of CVC i.e. to find the label independant internal data structure. Note that we do not report the performance of CVC combined with SKMS on three datasets as these datasets are large and SKMS is computationally demanding (see Table \[tab:time\]) and we could not fine tune the SKMS parameters.
The performance of CVC combined with k-means is reasonable given the simplicity and fast execution time of k-means (see next section). These results show that CVC is a generic classification algorithm and has good generalization capability when combined with off-the-shelf clustering algorithms especially when the clustering algorithms do not use labels in their objective function.
Execution Time Comparisons
--------------------------
Table \[tab:time\] compares the execution time of the proposed clustering algorithm SHC with the existing algorithms including k-means, SSC and SKMS on all five datasets. The experiment was performed in no-fusion mode on the maximum dimensionality of the manifold for each dataset. On the average the SHC algorithm is {1.33, 3.38, 16.47} times faster than the k-means, SSC and SKMS respectively. Maximum speedups are {3.77, 7.03, 47.0} and the minimum speedups are {0.434, 2.03, 4.74} respectively. Maximum speedup is obtained for the Cambridge dataset. The computational advantage of SHC increases with the gallery size. For SSC, the proximity matrix computation is significantly slower than that of the proposed SHC algorithm. In SSC, once the proximity matrix is computed, then k-means clustering is used to cluster the rows of the eigenvectors into $n_k$ clusters. In contrast, in the proposed SHC algorithm, the Fiedler vector computation and partitioning are done alternatively. The partitioning step is a binary decision using the signs of the Fiedler vector coefficients. The computational cost of the proposed DFVC algorithm reduces if a cut with low cost is present because the algorithm converges very quickly.
The computational cost of the proposed SHC algorithm increases as the dimensionality of the manifold increases. For mode fusion, a faster CVC algorithm can be designed by terminating computations once a decisive number of label agreements are obtained. For example, if there are $\nu$ classifiers corresponding to 1 to $\nu$ dimensions, then starting from the lowest dimensional classifier, if floor($\nu/2$)+1 classifiers get the same label, the remaining classifiers cannot change the mode and hence the classification decision. Therefore, computations of the remaining classifiers can be safely skipped making the algorithm significantly faster.
Robustness to Outliers
----------------------
We performed robustness experiments in a setting similar to [@Cevikalp_CVPR_2010]. The Honda dataset was modified to have 100 randomly selected images per set. In the first experiment, each [*gallery*]{} set was corrupted by adding 1 to 3 random images from each other gallery set resulting in 19%, 38% and 57% outliers respectively. The proposed CVC algorithm with Semi-supervised Hierarchical Clustering (SHC) achieved 100% accuracy for all three cases. In the second experiment, the [*probe*]{} set was corrupted by adding 1 to 3 random images from each gallery set. In this case, the CVC algorithm achieved {100%, 100%, 97.43%} recognition rates respectively. The proposed CVC algorithm outperformed all seven algorithms. Fig. \[fig:robustness\] compares our algorithm to the nearest two competitors in both experiments which are CDL [@CDL] and SANP [@Hu_TPAMI_2012].
![Robustness to outliers experiment: (a) Corrupted gallery case (b) Corrupted probe case. []{data-label="fig:robustness"}](robustness.pdf){width="8.6cm"}
Conclusion
==========
We presented a Classification Via Clustering (CVC) algorithm which bridges the gap between clustering algorithms and classification problems. The CVC algorithm performs classification by using unsupervised or semi-supervised clustering. Distribution based distance measures are used to match the class-cluster distributions of the test set and the gallery classes. A Semi-supervised Hierarchical Clustering (SHC) algorithm is proposed which optimizes the number of clusters using the class labels. An algorithm for Direct Fiedler Vector Computation (DFVC) is proposed to directly compute the second least-significant eigenvector of the Laplacian matrix. Image-sets are mapped on Grassmannian manifolds and clustering is performed on the manifold bases. By using multiple representations of each set, multiple classifiers are designed and results are combined by mode fusion. The proposed algorithm consistently showed better performance when compared with state of the art image-set classification algorithms on five standard datasets.
Acknowledgements
================
This research was supported by ARC Discovery Grants DP1096801 and DP110102399. We thank T. Kim and R. Wang for sharing the implementations of DCC and MMD and the cropped faces of Honda data. We thank H. Cevikalp for providing the LBP features of Mobo data and Y. Hu for the MDA implementation.
|
---
author:
- 'Matthew Buican$^{\diamondsuit, 1}$, Zoltan Laczko$^{\clubsuit, 1}$, and Takahiro Nishinaka$^{\heartsuit, 2}$'
bibliography:
- 'chetdocbib.bib'
date: June 2017
title: '${\mathcal{N}}=2$ $S$-duality Revisited'
---
Four-dimensional (4D) superconformal field theories (SCFTs) often admit exactly marginal deformations (the spaces of these deformations are typically called conformal manifolds"). In the context of theories with ${\mathcal{N}}\ge2$ supersymmetry (SUSY), one can easily obtain examples with exactly marginal deformations by coupling a gauge multiplet to precisely enough matter so that the one-loop beta function vanishes. A canonical example of this phenomenon occurs in $su(N)$ ${\mathcal{N}}=4$ Super Yang-Mills (SYM). At the level of the Lie algebra and the local operators, this theory is self-dual:[^1] as we vary the exactly marginal gauge coupling, $\tau$, towards a strong-coupling cusp on the conformal manifold, an $S$-dual weakly coupled $su(n)$ ${\mathcal{N}}=4$ SYM theory emerges. A similar story holds in $su(2)$ ${\mathcal{N}}=2$ gauge theory with four fundamental flavors [@Seiberg:1994aj].
On the other hand, the $S$-duality in $su(3)$ ${\mathcal{N}}=2$ gauge theory with six fundamental flavors is dramatically different [@Argyres:2007cn]. As one takes the gauge coupling to infinity, Argyres and Seiberg found that, instead of getting a weakly coupled $S$-dual description in terms of another $su(3)$ gauge theory with fundamental matter, one instead finds a dual consisting of an $su(2)$ theory coupled to a doublet of hypermultiplets and an $su(2)\subset\mathfrak{e}_6$ factor of the global symmetry of the Minahan-Nemeschansky $E_6$ SCFT [@Minahan:1996fg].
The message of [@Argyres:2007cn] is clear: sometimes, starting from vanilla building blocks, the matter" that appears via ${\mathcal{N}}=2$ $S$-duality is not standard matter (i.e., hypermultiplets) but is instead a strongly coupled isolated SCFT[^2] whose global symmetry (or a proper subgroup thereof) is weakly gauged.[^3] Moreover, $S$-duality can be a machine for generating exotic isolated theories.
This latter point was driven home in [@Gaiotto:2009we]. Indeed, Gaiotto generalized [@Argyres:2007cn] to higher-rank gauge theories and, in the process, found an infinite number of new isolated SCFTs—the so-called $T_N$ theories—at strong-coupling cusps on the resulting conformal manifolds.[^4] Since a $T_N$ theory has $SU(N)^3$ global symmetry[^5] and the following $SU(N)$ current two-point function (and hence 1-loop beta function contribution upon gauging) for each such factor $$\label{TNk}
k_{SU(N)_{i}}^{T_N}=2N~,\ \ \ i=1,2,3~,$$ one can always find a non-trivial conformal manifold by taking two $T_N$ theories and gauging a diagonal $SU(N)$. Indeed, the contributions from the $T_N$ theories in cancel those of the $SU(N)$ gauge fields $$\label{1loop}
\beta^{\rm 1-loop}_{SU(N)}=-4N+2N+2N=0~.$$ One can then proceed to construct a conformal manifold consisting only of arbitrarily many $T_N$ theories and conformal gauge fields.
While the above set of theories is quite vast, the $T_N$ theories (and their cousins) are somewhat special: their ${\mathcal{N}}=2$ chiral primaries have integer scaling dimensions.[^6] The underlying reason is that these theories emerge in a duality with a Lagrangian theory.[^7] On the other hand, the most generally allowed values for the scaling dimensions, $\Delta_i$, of ${\mathcal{N}}=2$ chiral operators are widely believed to be $\Delta_i\in\mathbb{Q}$, and non-integer rational values are indeed realized in so-called Argyres-Douglas (AD) theories [@Argyres:1995jj; @Argyres:1995xn; @Xie:2012hs].[^8] These theories cannot emerge in an ${\mathcal{N}}=2$ $S$-duality with a Lagrangian theory.
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\(1) at (1.5,0) [$\ (A_1, D_4)\ $]{}; (2) at (3,0) \[shape=circle\] [$3$]{} edge \[-\] node\[auto\] (1); (3) at (4.5,0) [$\ (A_1, D_4)\ $]{} edge \[-\] node\[auto\] (2); (9) at (3,-1.2) [$3$]{} edge\[-\] (2);
Motivated by a desire to understand ${\mathcal{N}}=2$ $S$-duality more broadly, it is then natural to ask what is the minimal (which we will define to be lowest rank[^9]) AD generalization of Argyres-Seiberg (i.e., non self-similar) duality [@Buican:2014hfa]. Since the starting point cannot be a Lagrangian theory, one must engineer such a conformal manifold from a weakly coupled gauging of a global symmetry of a collection of AD building blocks (potentially with additional hypermultiplets). An answer, using general consistency conditions and the class ${\mathcal{S}}$ Argyres-Douglas theories in [@Xie:2012hs], was given in [@Buican:2014hfa] and is reproduced in Fig. \[quiver1\] (there, this theory was referred to as the ${\mathcal{T}}_{3,2,{3\over2},{3\over2}}$" SCFT). This theory is constructed by gauging the diagonal $SU(3)$ symmetry of three fundamental flavors and a pair of $(A_1, D_4)$ SCFTs (the $(A_1, D_4)$ theory, originally discussed in [@Argyres:1995xn], has $SU(3)$ flavor symmetry and a single ${\mathcal{N}}=2$ chiral ring generator of dimension $3/2$). The resulting global symmetry is $U(3)$ and is furnished by the three fundamental flavors.
The $S$-dual frame of this theory is given in Fig. \[quiver2\] and consists of an $SU(2)$ gauge theory coupled to an $(A_1, D_4)$ factor and a more exotic AD theory called the ${\mathcal{T}}_{3,{3\over 2}}$ SCFT [@Buican:2014hfa] which has flavor symmetry $G\supset SU(3)\times SU(2)$.[^10] Therefore, in rough analogy with Argyres-Seiberg duality, the strongly coupled $(A_1, D_4)$ theory plays the role of the hypermultiplets on the $SU(2)$ side of the duality and the ${\mathcal{T}}_{3,{3\over2}}$ theory plays the role of the $E_6=T_3$ theory.
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\(1) at (1.8,0) [$\ {\mathcal{T}}_{3,{3\over2}}\ $]{}; (2) at (3,0) \[shape=circle\] [$2$]{} edge \[-\] node\[auto\] (1); (3) at (4.4,0) [$\ (A_1, D_4)\ $]{} edge \[-\] node\[auto\] (2);
However, upon closer inspection, the analogy with Argyres-Seiberg duality seems to break down. Indeed, the anomalies of the ${\mathcal{T}}_{3,{3\over2}}$ theory were computed in [@Buican:2014hfa] and found to be $$\label{T332anom}
k_{SU(2)}^{{\mathcal{T}}_{3,{3\over2}}}=5~, \ \ \ k_{SU(3)}^{{\mathcal{T}}_{3,{3\over2}}}=6~, \ \ \ c^{{\mathcal{T}}_{3,{3\over2}}}={9\over4}~, \ \ \ a^{{\mathcal{T}}_{3,{3\over2}}}=2~.$$ Using these symmetries, one cannot construct conformal manifolds built only out of arbitrary numbers of ${\mathcal{T}}_{3,{3\over2}}$ SCFTs and conformal gauge fields. The reason is that the contribution to the $SU(2)$ beta function in is too large and the required $SU(2)$ gauging would be infrared (IR) free. This state of affairs is quite unlike the $E_6=T_3$ case described above, where an arbitrary number of such theories can be concatenated by gauging enough diagonal symmetries.
Still, there are some puzzles in the above picture. To begin with, the flavor symmetry group of the ${\mathcal{T}}_{3,{3\over2}}$ theory is not obvious. One standard way to find such symmetries for SCFTs that, like the ${\mathcal{T}}_{3,{3\over2}}$ theory, can be derived from M5-branes wrapping a (punctured) Riemann surface, ${\mathcal{C}}$, (so-called class ${\mathcal{S}}$ theories) is to construct the Hitchin system corresponding to the theory [@Xie:2012hs; @Gaiotto:2009hg]. In particular, the Hitchin system has a meromorphic 1-form, $\varphi(z)dz$, with singularities at the punctures of ${\mathcal{C}}$. In the case of the ${\mathcal{T}}_{3,{3\over2}}$ SCFT, one can construct the corresponding $\varphi$ using the methods in [@Xie:2012hs] $$\label{Hfield}
\varphi(z)=z M_1+M_2+{1\over z}M_3+{\mathcal{O}}(z^{-2})~,$$ where we have expanded around a third-order pole at $z=\infty$ ($\varphi$ is non-singular at all other points $z\in{\mathcal{C}}=\mathbb{CP}^1$), and the $M_i$ are the following diagonal traceless matrices $$\begin{aligned}
\label{matrices}
M_1&=&{\rm diag}\left(\tilde a_1, \tilde a_1,\tilde a_2,\tilde a_2,\tilde a_3,\tilde a_3\right)~, \ \ \ M_2={\rm diag}\left(\tilde b_1, \tilde b_1,\tilde b_2,\tilde b_2,\tilde b_3,\tilde b_3\right)~, \cr M_3&=&{\rm diag}\left(\tilde m_1, \tilde m_1,\tilde m_2,\tilde m_2,\tilde m_3,\tilde m_4\right)~.\end{aligned}$$ The flavor symmetries are then read off by studying the independent parameters appearing as coefficients of the simple pole, i.e., the entries of $M_3$.[^11] This traceless matrix has three degrees of freedom which correspond to the Cartans of $SU(3)\times SU(2)$. Therefore, according to this description, $G_{{\mathcal{T}}_{3,{3\over2}}}=SU(3)\times SU(2)$. One reaches the same conclusion by constructing the Seiberg-Witten (SW) curve from this description via the spectral curve, $\det\left(xdz-\varphi(z)dz\right)=0$, and looking at the mass parameters (i.e., the simple poles in the SW 1-form, $\lambda=xdz$).
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On the other hand, one often computes flavor symmetries of strongly interacting 4D ${\mathcal{N}}=2$ theories by taking their $S^1$ reductions and studying the mirror theory (which may sometimes be described by a Lagrangian that flows to the same 3D ${\mathcal{N}}=4$ SCFT). Now, the ${\mathcal{T}}_{3,{3\over2}}$ theory has a proposed Lagrangian mirror for its $S^1$ reduction given in Fig. \[quiver3\] (following the rules in [@Xie:2012hs]) that predicts flavor symmetry $G^{3d}_{{\mathcal{T}}_{3,{3\over2}}}= SU(3)\times SU(2)^2$. Indeed, IR dimension-one monopole operators in this theory describe the enhancement of the manifest $U(1)^3$ topological symmetry to $SU(3)\times SU(2)^2$ [@Buican:2014hfa]. In particular, there is a free monopole operator in the IR that gives rise to an additional $SU(2)$ factor.[^12] By mirror symmetry [@Intriligator:1996ex], one expects, upon performing an $S^1$ reduction, the enhancement of $G_{{\mathcal{T}}_{3,{3\over2}}}\to SU(3)\times SU(2)^2$ with a decoupled hypermultiplet.
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A priori, there are various possible resolutions to the different predictions for $G_{{\mathcal{T}}_{3,{3\over2}}}$. First, it could be that the extra $SU(2)$ factor is an accidental symmetry at energies $E\ll R^{-1}$ (where $R$ is the radius of the compactification circle). Second, it could be that the 4D description around from the M5 brane simply misses some flavor symmetries.[^13] Finally, it could be that neither description gets the correct symmetries.
We claim the 3D quiver of Fig. \[quiver3\] captures the full flavor symmetry and the 4D description around does not. In particular, we will argue that the ${\mathcal{T}}_{3,{3\over2}}$ SCFT splits into a free hypermultiplet and an interacting theory, ${\mathcal{T}}_X$, as in Fig. \[factorization\] and that the $SU(2)$ symmetry detected around corresponds to a diagonal subgroup of the $SU(2)^2\subset G_{{\mathcal{T}}_{3,{3\over2}}}$ factor. Happily, the interacting ${\mathcal{T}}_X$ theory then has (${\mathcal{N}}=2$) flavor symmetry $G_{{\mathcal{T}}_X}=SU(3)\times SU(2)$ and the following anomalies[^14] $$\label{TXanom}
k_{SU(2)}^{{\mathcal{T}}_X}=4~, \ \ \ k_{SU(3)}^{{\mathcal{T}}_X}=6~, \ \ \ c^{{\mathcal{T}}_X}={13\over6}~, \ \ \ a^{{\mathcal{T}}_X}={47\over24}~.$$ In particular, we can now, in more direct analogy with the $E_6=T_3$ theory, construct conformal manifolds just from arbitrarily many ${\mathcal{T}}_X$ theories and conformal gauge fields.[^15] On the other hand, we need to be careful when constructing theories by gauging the $SU(2)$ factor since it has a $\mathbb{Z}_2$ Witten anomaly [@Witten:1982fp]! Indeed, as argued in [@Buican:2014hfa], the (diagonal) ${\mathcal{T}}_{3,{3\over2}}$ $SU(2)$ factor is anomaly free. However, since a single hypermultiplet has a Witten anomaly, the ${\mathcal{T}}_X$ theory must have a non-trivial compensating anomaly.
In order to substantiate our claim in Fig. \[factorization\] and also to further examine the analogy between the ${\mathcal{T}}_X$ theory and the $T_N$ theories, we must go beyond the simple description around . To that end, we will focus on the Schur" sector [@Gadde:2011uv] of the various component theories in our duality. This is a sector of operators that contains a wealth of information and is often exactly solvable, since it contains the (hidden) symmetries of a 2D chiral algebra [@Beem:2013sza].
In order to get a handle on the Schur sector, it is useful to first compute the limit of the superconformal index (i.e., the Schur“ index) that captures contributions only from operators in this sector (i.e., the Schur” operators). For our starting point in Fig. \[quiver1\], this computation can easily be carried out using the results of [@Buican:2015ina; @Cordova:2015nma]. Invariance of the Schur index under $S$-duality guarantees that we then also have the index for the theory in Fig. \[quiver2\].[^16]
Obtaining the index of the ${\mathcal{T}}_X$ theory itself is somewhat more delicate. However, using a recent conjecture in [@Xie:2016evu] (proven in [@kac2017remark] and reviewed in Appendix \[app:PEproof\]), we are able to find the Schur index of ${\mathcal{T}}_X$ from the index of the quiver in Fig. \[quiver2\] using the inversion theorem in [@spiridonov2006inversions]. Our use of the result in [@spiridonov2006inversions] is in the same spirit that it was used by the authors of [@Gadde:2010te] to determine the index of the $E_6$ SCFT (however, there are some technical differences, because our $SU(2)$ duality frame involves an additional strongly interacting factor).
In order to check our index computation and also to gain more insight into the ${\mathcal{T}}_X$ theory, we bootstrap its chiral algebra, $\chi({\mathcal{T}}_X)$, (and hence by the correspondence of [@Beem:2013sza], we find its Schur operators) using techniques described in [@Lemos:2014lua]. In particular, we show that there is a unique consistent chiral algebra with the (minimal) number of generators required, via the correspondence in [@Beem:2013sza], for compatibility with our inversion result and the anomalies in . Then, using arguments closely related to those in [@Lemos:2014lua], we argue for an exact expression for the vacuum character of $\chi({\mathcal{T}}_X)$ in terms of certain diagonal" $\widehat{su(2)}_{-2}\times\widehat{su(3)}_{-3}$ Affine Kac-Moody (AKM) characters. By the correspondence of [@Beem:2013sza], this gives us a simple closed-form expression for the Schur index of the ${\mathcal{T}}_X$ theory and allows us to recover the $S^3$ partition function of the proposed 3D mirror in Fig. \[quiver3\] by taking the $q\to1$ limit of this quantity.
As we will see, our expression for the Schur index in terms of AKM characters reveals a much deeper connection with the $T_N$ theories: the structure constants" that emerge are precisely those of the $T_2$ theory (although the AKM characters we sum over are different, they are in one-to-one correspondence with those we sum over in the $T_2$ case). We explore these connections in greater detail below and also comment on some consequences of the non-trivial Witten anomaly of the ${\mathcal{T}}_X$ theory for the 2D/4D correspondence of [@Beem:2013sza].
Before proceeding, let us discus the plan of the paper. In the next section, we review the basics of the Schur sector and its correspondence with 2D chiral algebras. With this formalism under our belts, we give a simple argument for the factorization in Fig. \[quiver3\]. We then move on to describe the Schur index of the ${\mathcal{T}}_X$ theory via the $S$-duality of [@Buican:2014hfa]. Using this result, we bootstrap the corresponding chiral algebra, construct its vacuum character, and make contact with Fig. \[quiver3\]. We then compute the Hall-Littlewood index of our theory using the data in Fig. \[quiver3\] and compare it with our Schur index in order to highlight some subtle aspects of the Schur sector. We conclude with a discussion of various open problems suggested by our work.
\[2D4Dcorr\] In this section we conduct a lightning review of Schur operators and the parts of the associated 4D/2D correspondence described in [@Beem:2013sza] that are useful for us below. These operators sit in short multiplets of the 4D ${\mathcal{N}}=2$ superconformal algebra and satisfy $$\label{Schurcond}
\left\{\tilde{\mathcal{Q}}_{2\dot-},{\mathcal{O}}\right]=\left\{{\mathcal{Q}}_{-}^1,{\mathcal{O}}\right]=0~,$$ along with corresponding equations for the conjugate charges acting on ${\mathcal{O}}(0)$. In , numerical indices denote spin-half $SU(2)_R\subset U(1)_R\times SU(2)_R$ quantum numbers, while the remaining indices are for spinors of the left and right parts of the Lorentz group. To simplify our notation, we have dropped any $SU(2)_R$ or Lorentz indices of ${\mathcal{O}}$, but the above definition guarantees that Schur operators are $SU(2)_R$ and Lorentz highest-weight states satisfying $$E({\mathcal{O}})=2R({\mathcal{O}})+j_1({\mathcal{O}})+j_2({\mathcal{O}})~, \ \ \ r({\mathcal{O}})=j_2({\mathcal{O}})-j_1({\mathcal{O}})~,$$ where $E$ is the scaling dimension, $R$ is the $SU(2)_R$ weight, $j_{1,2}$ are the Lorentz weights, and $r$ is the $U(1)_R\subset U(1)_R\times SU(2)_R$ charge.
The Schur operators also give the unique contributions to a simpler (but highly non-trivial) limit of the superconformal index called the Schur limit $${\mathcal{I}}(q, {\bf x})={\rm Tr}_{{\mathcal{H}}}(-1)^Fe^{-\beta\Delta}q^{E-R}\prod_i(x_i)^{f_i}~,$$ where the trace is over the Hilbert space of local operators, ${\mathcal{H}}$, $F$ is fermion number, $|q|<1$ is a superconformal fugacity, the $|x_i|=1$ are flavor fugacities, $f_i$ are flavor charges, and $\Delta=\left\{\tilde{\mathcal{Q}}_{2\dot-},\left(\tilde{\mathcal{Q}}_{2\dot-}\right)^{\dagger}\right\}$. Schur operators sit in the following multiplets $$\label{multtypes}
\hat{\mathcal{B}}_R~, \ \ \ {\mathcal{D}}_{R(0,j_2)}\oplus\bar{\mathcal{D}}_{R(j_1,0)}~, \ \ \ \hat{\mathcal{C}}_{R(j_1,j_2)}~,$$ where we have used the notation and conventions of [@Dolan:2002zh].[^17]
The $\hat{\mathcal{C}}_{R(j_1,j_2)}$ multiplets are semi-short multiplets, and the component Schur operators are obtained by acting on the highest-weight state with $\tilde Q_{2\dot+}Q^1_+$. The most important example of such multiplets for us below will be the stress tensor multiplet, $\hat{\mathcal{C}}_{0(0,0)}$. The associated Schur operator is the $SU(2)_R$ and Lorentz highest weight component of the $SU(2)_R$ current, $J^{11}_{+\dot+}$.
The $\hat{\mathcal{B}}_R$ multiplets will also play an important role below. The corresponding Schur operators are the highest $SU(2)_R$ weight components of the primaries and are annihilated by half the ${\mathcal{N}}=2$ superspace. These operators can parameterize the Higgs branch (when it exists). A particularly important example of a $\hat{\mathcal{B}}_R$ multiplet is the dimension two $\hat{\mathcal{B}}_1$ multiplet. It contains flavor symmetry currents and has as its Schur operator the holomorphic moment map, $\mu$.
The ${\mathcal{D}}_{R(0,j_2)}\oplus\bar{\mathcal{D}}_{R(j_1,0)}$ multiplets are somewhat less familiar (the component Schur operators are $\tilde Q_{2\dot+}$ and $Q^1_+$ highest-weight descendants),[^18] but, together with the $\hat{\mathcal{B}}_R$ multiplets, the $\bar{\mathcal{D}}_{R(j_1,0)}$ multiplets comprise an important subring of operators called the Hall-Littlewood (HL) chiral ring [@Gadde:2011uv]. It is an interesting general question to understand the class of theories whose HL ring includes ${\mathcal{D}}_{R(0,j_2)}\oplus\bar{\mathcal{D}}_{R(j_1,0)}$.[^19] As we will see below, the HL ring of the ${\mathcal{T}}_X$ theory is generated only by operators of type $\hat{\mathcal{B}}_R$.
The authors of [@Beem:2013sza] found a general organizing principle for all of the above operators: they are related to a 2D chiral algebra. More precisely, the Schur operators define non-trivial cohomology classes with respect to a nilpotent supercharge, $\mathbbmtt{Q}={\mathcal{Q}}^1+\tilde{\mathcal{S}}^2$ $$\left\{\mathbbmtt{Q},{\mathcal{O}}(0)\right]=0~, \ \ \ {\mathcal{O}}(0)\ne\left\{\mathbbmtt{Q},{\mathcal{O}}'(0)\right]~.$$ One then considers ${\mathcal{O}}$ to be fixed in a plane $\mathcal{P}\subset\mathbb{R}^4$ with coordinates $(z, \bar z)$. Translations (and the rest of the global conformal group) in the $\bar z$ direction are twisted with the $SU(2)_R$ symmetries. It then turns out that the quantum numbers of ${\mathcal{O}}$ are such that its twisted $\bar z$ translations are $\mathbbmtt{Q}$-exact. In general, this translation process introduces lower $SU(2)_R$ partner components of ${\mathcal{O}}$ when $\bar z\ne0$.[^20] However, these translations do not take one out of the cohomology class defined by ${\mathcal{O}}(z,0)$ and so the cohomology classes form an infinite dimensional chiral algebra with meromorphic correlators (translations out of the plane take one out of the cohomology).
While the precise details of the map between 4D and 2D are somewhat technical, the basic results are intuitive. For example, we have the following correspondences [@Beem:2013sza] $$\label{map}
\chi\left[J^{11}_{+\dot+}\right]=-{1\over2\pi^2}T~, \ \ \ \chi\left[\mu^I\right]={1\over2\sqrt{2}\pi}J^I~, \ \ \ \chi\left[\partial_{+\dot+}\right]=\partial_z\equiv\partial~,$$ where $\chi\left[\cdots\right]$ takes a 4D Schur operator to its 2D counterpart. As one might naturally expect, $T$ is the holomorphic stress tensor, $J^I$ is an AKM current ($I$ is an adjoint index), and $\partial$ is the holomorphic derivative in $\mathcal{P}$. Note that any local 4D ${\mathcal{N}}=2$ SCFT has a stress tensor and therefore, by ${\mathcal{N}}=2$ SUSY, a $J^{11}_{+\dot+}$ operator. As a result, tells us that the associated chiral algebra must contain at least a Virasoro sub-algebra. Moreover, 4D theories with flavor symmetries have an associated chiral algebra with an AKM subalgebra. Interestingly, there is a universal map between the corresponding anomalies in 4D and 2D for the universal currents we have just described [@Beem:2013sza] $$\label{anomalyMap}
k_{2d}=-{1\over2}k_{4d}~, \ \ \ c_{2d}=-12c_{4d}~.$$
More generally, the chiral algebras arising via this correspondence typically contain generators[^21] beyond the ones appearing in . However, all generators must satisfy basic consistency conditions in the form of Jacobi identities $$\label{JacobiDef}
\left[{\mathcal{O}}_1(z_1)\left[{\mathcal{O}}_2(z_2){\mathcal{O}}_3(z_3)\right]\right]-\left[{\mathcal{O}}_3(z_3)\left[{\mathcal{O}}_1(z_1){\mathcal{O}}_2(z_2)\right]\right]-\left[{\mathcal{O}}_2(z_2)\left[{\mathcal{O}}_3(z_3){\mathcal{O}}_1(z_1)\right]\right]=0~,$$ where we take $|z_2-z_3|<|z_1-z_3|$, $\left[\cdots\right]$ is the singular part of the OPE of the operators enclosed, and we have assumed the ${\mathcal{O}}_i$ are all bosonic (as we will see is the case for $\chi({\mathcal{T}}_X)$ below). These constraints are the basis of the chiral algebra bootstrap, and we will make heavy use of them in Sec. \[chiboot\].
Finally, we note that the holomorphic dimension in the chiral algebra, $h$, satisfies $$\label{dim}
h=E-R~.$$ Moreover, the torus partition function of the chiral algebra can be written as follows $$\label{T2fn}
Z(y,q,{\bf x})={\rm Tr}\ y^{M^{\perp}} q^{L_0}\prod(x_i)^{f_i}~,$$ where $M^{\perp}=j_1-j_2$, and the relation to the Schur index is $$\label{SchurT2rel}
Z(-1, q,{\bf x})={\mathcal{I}}(q,{\bf x})~.$$ This equation allows us to read off the vacuum character of the chiral algebra from the Schur index and is instrumental in allowing us to find the set of generators of $\chi({\mathcal{T}}_X)$ below.
Therefore, we see that the Schur sector of the theory contains a remarkably constrained—but still interesting—set of operators that are complementary to the Coulomb branch degrees of freedom characterizing the SW curve description discussed in the introduction.[^22] Since chiral algebras are such rigid objects, finding a unique chiral algebra with a particular set of generators and anomalies that satisfies Jacobi identities like those in is strong evidence for having found the Schur sector of a 4D theory exactly.
In the next section, we will apply our above discussion and argue for the factorization in Fig. \[factorization\]. Along the way, we also make use of the results in [@Buican:2015ina; @Cordova:2015nma].
\[factarg\] To understand why the ${\mathcal{T}}_{3,{3\over2}}$ theory factorizes, note that a simple consequence of the duality discussed in the introduction is that the spectrum of gauge invariant operators arising from the quiver in Fig. \[quiver1\] must match the spectrum of such operators arising from the quiver in the dual frame in Fig. \[quiver2\]. In particular, the $SU(3)$ side of the theory clearly has dimension three and $SU(2)_R$ weight ${3\over2}$ baryons $$B=\epsilon^{ijk}Q_i^aQ_j^bQ_k^c~, \ \ \ \tilde B=\epsilon_{ijk}\tilde Q^i_a\tilde Q^j_b\tilde Q^k_c~,$$ that are charged under the baryonic $U(1)\subset U(3)$ factor of the flavor symmetry. Moreover, we have $$\left[\tilde Q_{2\dot-},B\right]=\left[Q^1_{-},B\right]=\left[\tilde Q_{2\dot-},\tilde B\right]=\left[Q^1_{-},\tilde B\right]=0~,$$ and so these degrees of freedom are Schur operators of type $\hat{\mathcal{B}}_{3\over2}$ discussed around . By , Such operators are in turn related to 2D chiral algebra primaries ${\mathcal{B}}$ and $\tilde{\mathcal{B}}$ of holomorphic scaling dimension $h=E-R={3\over2}$.
As a result, the $SU(2)$ side of the duality must also have operators $B$ and $\tilde B$. Since the $(A_1, D_4)$ factor in this duality frame is responsible for the baryonic symmetry, ${\mathcal{B}}$ and $\tilde{\mathcal{B}}$ must either be Schur operators of the $(A_1, D_4)$ sector or composite gauge-invariant operators built from Schur operators of this sector and Schur operators of at least one other sector. However, we know the Schur sector of the $(A_1, D_4)$ theory exactly: it corresponds, via the map described in Sec. \[2D4Dcorr\], to the $\widehat{su(3)}_{-{3\over2}}$ AKM chiral algebra [@Beem:2014zpa; @Buican:2015ina; @Cordova:2015nma; @Buican:2015tda][^23] generated by the AKM current $J^I_{SU(3)}$ ($I=1,\cdots,8$ is an adjoint index of $SU(3)$).
Therefore, $\chi\left[(A_1, D_4)\right]$ has no operators with the quantum numbers of ${\mathcal{B}}$ and $\tilde{\mathcal{B}}$ (since $J^I_{SU(3)}$ has $h=1$, there are no operators with $h={3\over2}$ in the $\widehat{su(3)}_{-{3\over2}}$ vacuum module). As a result, we must construct $B$ and $\tilde B$ as composites of the holomorphic moment map of the $(A_1, D_4)$ theory, $\mu_{SU(3)}^I$, with a field of dimension one (and $h=1/2$).[^24] In other words, we must have a sector consisting of a hypermultiplet, $Q_i$ (with $i=1,2$), charged under the gauged $SU(2)$ (recall that the hypermultiplet has $Sp(1)\simeq SU(2)$ flavor symmetry) from which we can construct $$B=\mu_{SU(3)}^iQ_i~, \ \ \ \tilde B=\tilde\mu_{SU(3)}^iQ_i~,$$ where $\mu_{SU(3)}^i$ and $\tilde\mu_{SU(3)}^i$ are the two doublets descending from the eight $\mu_{SU(3)}^I$ moment maps under the decomposition of $SU(3)$ into representations of the $SU(2)$ gauge group (we have ${\bf 8}={\bf1}+2{\bf\times 2}+{\bf3}$). In particular, we see that the ${\mathcal{T}}_{3,{3\over2}}$ SCFT splits into a free hyper and another theory which we call ${\mathcal{T}}_X$ (as in Fig. \[factorization\]).[^25] Moreover, as discussed in the introduction, since the ${\mathcal{T}}_{3,{3\over2}}$ theory doesn’t have a Witten anomaly for its $SU(2)$ global symmetry subgroup but the free hypermultiplet does, the $SU(2)$ global symmetry subgroup of the ${\mathcal{T}}_X$ theory has a Witten anomaly. We will see an interesting consequence of this fact below. This discussion also derives the result in from .
In the next section, we begin a deeper exploration of the ${\mathcal{T}}_X$ theory. To do so, we first construct the Schur index of the theory. After finding this index, we will conjecture a chiral algebra, $\chi({\mathcal{T}}_X)$, that reproduces it and then use bootstrap techniques to confirm our conjecture.
\[inversion\] In order to get more detailed information about the ${\mathcal{T}}_X$ theory, we compute its Schur index using the $S$-duality described in Fig. \[quiver1\] and Fig. \[quiver2\]. Indeed, since the index is invariant under $S$-duality, the Schur indices of the theories in these two figures must agree. On the $SU(3)$ side of the duality, it is easy to compute the Schur index as follows $$\begin{aligned}
\label{su3side}
{\mathcal{I}}_{SU(3)}(q, s, z_1, z_2)&=&\oint d\mu_{SU(3)}(x_1,x_2)\times{\mathcal{I}}_{\rm vect}(q,x_1, x_2)\times{\mathcal{I}}_{\rm flavors}(q, x_1, x_2, s, z_1, z_2)\times\cr&\times&{\mathcal{I}}_{(A_1, D_4)}(q,x_1, x_2)^2~,\end{aligned}$$ where the measure of integration is the $SU(3)$ Haar measure, ${\mathcal{I}}_{\rm flavors}$ is the index of the three fundamental flavors, ${\mathcal{I}}_{(A_1, D_4)}$ is the index of the $(A_1, D_4)$ theory, and ${\mathcal{I}}_{\rm vect}$ is the vector multiplet index (see Appendix \[inversionap\] for detailed expressions). The fugacities, $s$ and $(z_1,z_2)$, are for $U(1)\subset U(3)$ and $SU(3)\subset U(3)$ flavor subgroups, respectively. All terms appearing in the integrand of have known closed-form expressions (${\mathcal{I}}_{(A_1, D_4)}$ was computed in [@Buican:2015ina; @Cordova:2015nma]). Now, on the $SU(2)$ side of the duality, we have $$\label{su2side}
{\mathcal{I}}_{SU(2)}(q,s,z_1,z_2)=\oint d\mu_{SU(2)}(e)\times{\mathcal{I}}_{\rm vect}(q,e)\times{\mathcal{I}}_{(A_1, D_4)}(q,e,s)\times {\mathcal{I}}_{{\mathcal{T}}_{3,{3\over2}}}(q,e,z_1,z_2)~,$$ where ${\mathcal{I}}_{{\mathcal{T}}_{3,{3\over2}}}$ is the Schur index of the ${\mathcal{T}}_{3,{3\over2}}$ theory. From the general discussion in the previous section and Fig. \[factorization\], we must have $$\label{Ifact}
{\mathcal{I}}_{{\mathcal{T}}_{3,{3\over2}}}(q,e,z_1,z_2)={\mathcal{I}}_{{\mathcal{T}}_{X}}(q,e,z_1,z_2)\times {\mathcal{I}}_{\rm hyper}(q,e)~,$$ where the second factor on the RHS is the Schur index of a free hypermultiplet, and the first factor is the index of the ${\mathcal{T}}_X$ SCFT.
In order to compute the index in , we will use an inversion procedure based on the theorem in [@spiridonov2006inversions] to extract it from the expression in . Roughly the same basic procedure was first used in [@Gadde:2010te] to extract the index of the $E_6$ SCFT from Argyres-Seiberg duality. However, there are some technical differences (due to the fact that our $SU(2)$ duality frame has an additional strongly interacting factor) in our use of [@spiridonov2006inversions] that are reviewed in Appendix \[inversionap\]. One important precondition for our inversion procedure involves the use of a conjectured form for ${\mathcal{I}}_{(A_1, D_4)}(q,x_1, x_2)$ due to Xie-Yan-Yau (XYY) [@Xie:2016evu] (recently proved in [@kac2017remark] and reviewed in Appendix \[app:PEproof\]) that is compatible with its known form in [@Buican:2015ina; @Cordova:2015nma] $$\label{XYYform}
{\mathcal{I}}_{(A_1, D_4)}(q,x_1,x_2)=P.E.\left[{q\over1-q^2}\chi_{\rm Adj}(x_1, x_2)\right]~,$$ where the plethystic exponential" is defined as $$\label{PEdef}
P.E.\left[G(a_1,\cdots,a_p)\right]\equiv\exp\left[\sum_{n=1}^{\infty}{1\over n}G(a_1^n,\cdots, a_p^n)\right]~,$$ for any function of the fugacities, $G$. Indeed, the surprising fact that the index of the strongly interacting $(A_1, D_4)$ SCFT in is related to the index of a free adjoint hypermultiplet by the rescaling $q\to \sqrt{q}$ allows us to use the inversion theorem of [@spiridonov2006inversions] (as in [@Gadde:2010te], we will a posteriori justify the assumptions used in applying this theorem by finding a consistent symmetry structure for our index). One surprising fact we will uncover later on is that, when appropriately re-written, ${\mathcal{I}}_{{\mathcal{T}}_X}$ will also be closely related to a Schur index for free fields.
Applying the procedure in Appendix \[inversionap\], we find that the Schur index of the ${\mathcal{T}}_{3,{3\over2}}$ theory can be written as $$\label{schurtotal}
{\mathcal{I}}_{{\mathcal{T}}_{3,{3\over2}}}(q,w,z_1,z_2)=\frac{1}{(w^{\pm 2} q;q)}\left[ \frac{1}{1-w^2}{\mathcal{I}}_{SU(3)}(q,wq,z_1,z_2)+\frac{w^2}{w^2-1}{\mathcal{I}}_{SU(3)}(q,\frac{q}{w},z_1,z_2)\right]~,$$ where $(a;q)$ denotes the $q$-Pochhammer symbol $$(a;q)=\prod_{n=0}^{\infty} (1-a q^n)~,$$ and we also use the condensed notation $$\label{repeat}
(a^{\pm};q)\equiv (a;q)(a^{-1};q)~.$$ Expanding perturbatively in $q$ we obtain $$\begin{aligned}
\label{schurT332}
{\mathcal{I}}_{{\mathcal{T}}_{3,{3\over2}}}(q, w, z_1, z_2)&=&1+\chi_{1}q^{1\over2}+(2 \chi_{2} + \chi_{1,1})q+2(\chi_{1} + \chi_{3} + \chi_{1} \chi_{1,1})q^{3\over2}+(4 + 3 \chi_{2} +\cr&+& 3 \chi_{4} + 3 \chi_{1,1} +
3 \chi_{2} \chi_{1,1} + \chi_{2,2})q^2+(8 \chi_{1} + 5 \chi_{3} + 3 \chi_{5} + 7 \chi_{1} \chi_{1,1} +\cr&+&
4 \chi_{3} \chi_{1,1} + \chi_{1} \chi_{3, 0} + \chi_{1} \chi_{0, 3} + 2 \chi_{1} \chi_{2,2})q^{5\over2}+(6 + 15 \chi_{2} + 6 \chi_{4} +\cr&+& 4 \chi_{6} + 10 \chi_{1,1} + 12 \chi_{2} \chi_{1,1} + 5 \chi_{4} \chi_{1,1} + 3 \chi_{3, 0} + \chi_{2} \chi_{3, 0} + 3 \chi_{0,3} +\cr&+& \chi_{2} \chi_{0, 3} + 3 \chi_{2,2} + 4 \chi_{2} \chi_{2,2} + \chi_{3,3})q^3+{\mathcal{O}}(q^{7\over2})~,\end{aligned}$$ where $\chi_{\lambda}\equiv\chi_{\lambda}(w)$ is the character of the spin ${\lambda\over2}$ representation of $SU(2)$ and $\chi_{\lambda_1,\lambda_2}\equiv\chi_{\lambda_1,\lambda_2}(z_1, z_2)$ is the character of the $SU(3)$ representation with Dynkin labels $\lambda_{1,2}\in\mathbb{Z}_{\ge0}$.
One check of and is that they are compatible with the factorization we argued for in Sec. \[factarg\] and explained at the level of the index in . In particular, we see a free hypermultiplet at ${\mathcal{O}}(q^{1\over2})$. Moreover, the total global symmetry of the ${\mathcal{T}}_{3,{3\over2}}$ theory is then, as explained in the introduction, $SU(2)^2\times SU(3)$ with one $SU(2)$ factor coming from the free hypermultiplet.[^26] Although this enhancement is not quite as dramatic as the $E_6$ enhancement of flavor symmetry observed in the example studied in [@Gadde:2010te], we will find a much deeper statement about the (hidden) symmetries of this theory (and hence the consistency of our picture) by bootstrapping the chiral algebra associated with ${\mathcal{T}}_X$ below.
As a first step towards this goal, we arrive at the index of the ${\mathcal{T}}_X$ theory by dividing both sides of by the free hypermultiplet contribution $$\begin{aligned}
\label{schurTX}
{\mathcal{I}}_{{\mathcal{T}}_X}(q,w,z_1,z_2)&=&1+(\chi_{2}+\chi_{1,1})q+\chi_{1} \chi_{1,1} q^{3\over2}+(2 + \chi_{2} + \chi_{4} + 2 \chi_{1,1} + \chi_{2} \chi_{1,1} + \cr&+&\chi_{2,2})q^2+(\chi_{1} + 2 \chi_{1} \chi_{1,1}+ \chi_{3} \chi_{1,1} + \chi_{1} \chi_{0,3} + \chi_{1} \chi_{3,0} +\cr&+& \chi_{1} \chi_{2,2})q^{5\over2}+(2 + 4 \chi_{2} + \chi_{4} + \chi_{6} + 5 \chi_{1,1} +3 \chi_{2} \chi_{1,1} + \chi_{4} \chi_{1,1} +\cr&+& 2 \chi_{3, 0} + 2 \chi_{0,3} + 2 \chi_{2,2} + 2 \chi_{2} \chi_{2,2} + \chi_{3,3})q^3+{\mathcal{O}}(q^{7\over2})~,\end{aligned}$$ which has, as promised, $SU(2)\times SU(3)$ global symmetry (we see currents in the adjoint of this symmetry group at $O(q)$, and the index organizes into characters of this symmetry).
In the next section, we use this expansion to conjecture the generators of the associated chiral algebra, $\chi({\mathcal{T}}_X)$. We then bootstrap this chiral algebra and show that it is consistent (in the sense that it obeys Jacobi identities of the form reviewed in ). Moreover, we will argue that it is the unique such chiral algebra with the generators we conjecture and the anomalies required from the discussion in the introduction and Sec. \[2D4Dcorr\].[^27]
From the simple expansion presented in , we can immediately conjecture the generators of the corresponding chiral algebra in the sense of [@Beem:2013sza] reviewed in Sec. \[2D4Dcorr\]. Indeed, using the map in , is also an expansion for the character of the vacuum module of the chiral algebra we want to find.
The only possible contributions in the vacuum module at ${\mathcal{O}}(q)$ must come from AKM currents, which, in this case, are for $\widehat{su(2)}_{-2}\times\widehat{su(3)}_{-3}$. We have used and to fix the levels of the AKM algebras to the so-called critical levels (these are $k=-h^{\vee}$, where $h^{\vee}$ is the dual Coxeter number). As in the case of the $T_N$ theories (with the exception of the $T_3=E_6$ theory which has enhanced $E_6\supset SU(3)^3$ flavor symmetry and the $T_2$ theory which has $Sp(4)\supset SU(2)^3$ symmetry and no AKM currents as generators), this discussion means that the holomorphic stress tensor of the chiral algebra must be an independent generator, since the Sugawara stress tensor is not normalizable (note that from and we have $c=-26$ for the Virasoro subalgebra[^28]). Looking at ${\mathcal{O}}(q^{3\over2})$, we see that there must be at least one operator, ${\mathcal{O}}_{aI}$, transforming in the ${\bf2}\times{\bf8}$ representation of the global symmetry (since all the other generators are integer dimensional).[^29] This operator is mapped to an AKM primary, $\chi[{\mathcal{O}}_{aI}]=W_{aI}$. Therefore, the minimal conjecture for $\chi\left[{\mathcal{T}}_X\right]$ is the following
\[conjecture\] [**Conjecture:**]{} The chiral algebra, $\chi\left[{\mathcal{T}}_X\right]$, is generated by a stress tensor, $T$ (with $c=-26$), AKM currents, $J_{su(2)}^A$ and $J_{su(3)}^I$ (with $A=1,\cdots,3$ and $I=1,\cdots,8$) for $\widehat{su(2)}_{-2}\times\widehat{su(3)}_{-3}$, and an $h={3\over2}$ AKM primary, $W_{aI}$ (with $a=1,2$ and $I=1,\cdots,8$), transforming in the ${\bf 2}\times{\bf8}$ representation of $su(2)\times su(3)$.
Note that this conjecture is consistent with the simplicity of AD theories: to get the chiral algebra of ${\mathcal{T}}_X$, one needs to add only a single additional generator (really 16 generators if one counts all the allowed $a,I$ pairs) beyond the universal ones required by 4D symmetries. Indeed, this algebra is considerably simpler than those of the interacting $T_N$ theories (even the $T_3=E_6$ theory has a larger number of generators by virtue of its large global symmetry).
We will give convincing evidence for this conjecture in Sec. \[chiboot\], where we will show there is a unique consistent chiral algebra satisfying this conjecture. For now, we also give some powerful circumstantial evidence in favor of our proposal. In particular, if this conjecture is correct, then all contributions appearing in can be generated by plethystic exponentials of our generators modulo constraints. Assuming our conjecture is correct, we find some natural operator relations at low order in $q$
- [A singlet relation at $O(q^2)$. As we will see in greater detail below, we expect that $$\label{flavornull2}
{\rm Tr}J_{SU(3)}^2\sim{\rm Tr}J_{SU(2)}^2~,$$ where we will fix the non-zero constant of proportionality in the next section. The motivation for this relation is that the $\widehat{su(2)}_{-2}\times\widehat{su(3)}_{-3}$ subalgebras of $\chi({\mathcal{T}}_X)$ are at the critical level. Therefore, in their respective modules, the LHS and RHS of separately vanish. However, it is natural to expect that, as in the case of the $T_N$ theories [@Lemos:2014lua], one linear combination of these operators becomes non-null in the full chiral algebra and therefore remains as a non-trivial operator.]{}
- [At ${\mathcal{O}}(q^{5\over2})$ we have two operator relations with quantum numbers ${\bf 2}\times {\bf 8}$]{}.
- [At ${\mathcal{O}}(q^3)$ we have many operator relations. One important set of relations are the singlets of the form $$\label{t3sing}
{\rm Tr}J_{SU(3)}^3={\rm Tr}J_{SU(2)}^3=W_{3\over2}^{aI}W_{{3\over2} aI}=0~.$$ The first relation again follows from the fact that the flavor symmetry is at the critical level and is a non-trivial statement, while the last two relations are a simple consequence of bosonic statistics.]{}
\[chiboot\] One strong piece of evidence in favor of our conjecture in the previous section is that there exists a (unique) set of operator product expansions (OPEs) among the generators described there that is consistent with Jacobi identities of the type described in . To understand this statement, let us consider the most general OPEs among the generators. The non-vanishing singular parts of the OPEs among the stress tensor and the $SU(2)\times SU(3)$ currents are completely fixed by Ward identities to take the form $$\begin{aligned}
T(z)T(0) &\sim \frac{c_{2d}}{2z^4} + \frac{2T}{z^2} + \frac{\partial T}{z}~,\nonumber
\\
T(z)J^A_{SU(2)}(0) &\sim \frac{J^A_{SU(2)}}{z^2} + \frac{\partial J^A_{SU(2)}}{z}~,\nonumber
\\
T(z) J^I_{SU(3)}(0) &\sim \frac{J^I_{SU(3)}}{z^2} + \frac{\partial J^I_{SU(3)}}{z}~,
\\
J_{SU(2)}^A(z) J_{SU(2)}^B(0) &\sim \frac{k_{2d}^{su(2)}\delta^{AB}}{2z^2} + \frac{i\epsilon^{ABC}J_{SU(2)}^C}{z}~,\nonumber
\\
J_{SU(3)}^I(z)J_{SU(3)}^J &\sim \frac{k_{2d}^{su(3)}\delta^{IJ}}{2z^2} + \frac{if^{IJK}J_{SU(3)}^K}{z}~,\nonumber\end{aligned}$$ where $f_{IJK}$ is the structure constant of $su(3)$ and, as discussed in the previous section, $c_{2d}=-26,\, k_{2d}^{su(2)} = -2$ and $k_{2d}^{su(3)}=-3$. Moreover, since there is no generator with $h=1/2$, $W_a{}^I$ has to be a primary of the Virasoro and $\widehat{su(2)}_{-2}\times \widehat{su(3)}_{-3}$ algebras. This fact implies the following singular parts of the OPEs: $$\begin{aligned}
T(z)W_a{}^I(0) &\sim \frac{3W_a{}^I}{2z^2} + \frac{\partial W_a{}^I}{z}~,\nonumber
\\
J^A_{SU(2)}(z)W_a{}^I(0) &\sim \frac{ \sigma^A_{ab} W^{bI}}{2z}~,
\\
J^I_{SU(3)}(z)W_a{}^J(0) &\sim \frac{f^{IJK}W_{aK}}{z}~,\nonumber\end{aligned}$$ where the $\sigma^A$ are Pauli matrices.
On the other hand, the OPE between $W_a{}^I$ and $W_b{}^J$ is not fixed by the symmetries. Therefore, we adopt the following general ansatz for the singular parts of this OPE: $$\begin{aligned}
W_a{}^I(z) W_b{}^J(0) &\sim \frac{\epsilon_{ab}\delta^{IJ}}{z^3} + \frac{1}{z^2}\bigg(\frac{a_1}{2} \delta^{IJ} \sigma^A_{ab}\, J_{SU(2)A} + \epsilon_{ab}\,(a_2\, f^{IJK} + a_3\, d^{IJK}) J_{SU(3)K}\bigg)
\nonumber\\
& + \frac{1}{z}\Bigg[ \epsilon_{ab}\delta^{IJ}\bigg(a_4\, T+a_5\, J^A_{SU(2)}J_{SU(2)A} + a_6\, J^K_{SU(3)}J_{SU(3)K} \bigg)
\nonumber\\
&\qquad + \frac{a_7}{2}\,\delta^{IJ}\sigma^A_{ab}\,J_{SU(2)A}' + a_8\, \epsilon_{ab}\,f^{IJK}\, J_{SU(3)K}'+\frac{a_9}{2} \,\sigma^A_{ab}\,f^{IJK}J_{SU(2)A} J_{SU(3)K}
\nonumber\\
&\qquad + \epsilon_{ab}(a_{10}\, f^{IJK}+a_{11}\,d^{IJK})d_{KLM}J^{L}_{SU(3)}J^M_{SU(3)} + 2a_{12}\,\epsilon_{ab}J^{(I}_{SU(3)}J^{J)}_{SU(3)}\Bigg]~,
\label{eq:WW-OPE}\end{aligned}$$ where $d^{IJK}$ is the totally symmetric tensor of $su(3)$ normalized so that $d^{IJK}d_{IJK}=\frac{40}{3}$, and the $W_a{}^I$ are normalized so that the coefficient of $\epsilon_{ab}\delta^{IJ}/z^3$ is one.[^30] The twelve coefficients, $a_1,\cdots,a_{12}$, are free parameters to be fixed in such a way that the Jacobi identities are satisfied. Note that is the most general OPE written in terms of the generators, $T,\, J^A_{SU(2)},\, J^I_{SU(3)}$ and $W_a{}^I$.[^31]
To fix the above constants and test the consistency of what we have written, we impose the various Jacobi identities among the generators. In particular, the Jacobi identities among $\mathcal{O},\,W_a{}^I$, and $W_b{}^J$ for $\mathcal{O}\in\left\{T,\, J_{SU(2)}^A,\,J_{SU(3)}^I\right\}$ imply that $$\begin{aligned}
a_1 = 1~,\quad a_2 =a_9&= -\frac{2i}{3}~,\quad a_3=a_{10}=0,\quad a_4=-\frac{1}{4}~,\quad a_6=\frac{2-3a_5}{12}~,
\nonumber\\[2mm]
& a_7=a_{11}=\frac{1}{2}~,\quad a_8=-\frac{i}{3}~, \quad a_{12}=-\frac{1}{12}~.
\label{eq:solution}\end{aligned}$$ Note that this condition fixes all the OPE coefficients except for $a_5$. Moreover, it turns out that, with $a_6=(2-3a_5)/12$ imposed, the undetermined parameter $a_5$ is only coupled to a null operator. Indeed, under the condition $a_6=(2-3a_5)/12$, the only $a_5$-dependent term in is $$\begin{aligned}
a_5 \left(J_{SU(2)}^A J_{SU(2)A} - \frac{1}{4}J_{SU(3)}^K J_{SU(3)K}\right)~.
\label{eq:null1}\end{aligned}$$ Since the OPEs of this operator with the generators only involve operators of holomorphic dimension larger than or equal to its own dimension, is a null operator. Therefore, we set $a_5=0$ in the rest of this section.
Let us now look at the Jacobi identities among $W_a{}^I,\,W_b{}^J,$ and $W_c{}^K$. With the condition , they are automatically satisfied up to the following operators: $$\begin{aligned}
\sigma^A_{ab}J_{SU(2)A}W^{bI} + \frac{if^{IJK}}{2}J_{SU(3)J}W_{aK}~,\ \ \ d^{IJK}J_{SU(3)J}W_{aK}~.
\label{eq:null2}\end{aligned}$$ Since the OPEs of these operators with the generators of the chiral algebra only involve operators of holomorphic dimensions larger than or equal to their own dimensions, the above two operators are both null. This means that is consistent with all the Jacobi identities among the generators. The existence of such a consistent $WW$ OPE is strong evidence for our chiral algebra conjecture in the previous section.
Another interesting observation is that the chiral algebra generated by $T,\,W_a{}^I$, and $J_{SU(2)}^A,\, J_{SU(3)}^I$ at the critical levels exist if and only if the Virasoro central charge is $c_{2d}=-26$. Indeed, when we do the above analysis with $c_{2d}$ unfixed, we see that the Jacobi identities among the generators imply $c_{2d}=-26$. Similarly, if we take $c_{2d}=-26$ with the levels of the AKM algebras unfixed, we can show that the Jacobi identities imply that $k_{SU(2)}=-2$ and $k_{SU(3)}=-3$.[^32]
We have seen there are at least three null operators up to $h=\frac{5}{2}$. The first one is shown in and is a singlet of $SU(2)\times SU(3)$ with $h=2$. The second and third null operators are shown in and are in the ${\bf 2}\times {\bf 8}$ representation of $SU(2)\times SU(3)$ with $h=\frac{5}{2}$. These three null operators are perfectly consistent with the 4D operator relations discussed in Sec. \[conjecture\].
Finally, we note that the following normal-ordered product $$\label{JWnon0}
J^I_{SU(3)}W^a_I\ne0~,$$ does not vanish. On the other hand, as we will see below when we discuss the HL chiral ring, there is a non-trivial operator relation for the 4D $\hat{\mathcal{B}}_R$ ancestors of these operators. However, as we will explain in greater detail below, this statement is consistent with because of the $SU(2)_R$ mixing described in Footnote \[ttfootnote\] which induces a non-trivial $\hat{\mathcal{C}}_{{1\over2}(0,0)}$ component for the chiral algebra normal-ordered product.[^33]
Given this chiral algebra, we will argue that its vacuum character has a surprisingly simple exact expression in terms of certain $\widehat{su(2)}_{-2}\times\widehat{su(3)}_{-3}$ characters. This expansion will turn out to be remarkably similar to the expansion one finds for the $T_2$ theory (although the precise characters we sum over are different). In addition to pointing to some mysterious connections between AD theories and $T_N$ SCFTs, we are able to use this formula to take the $q\to1$ limit and make contact with the $S^3$ partition function of the 3D quiver appearing in Fig. \[quiver3\].
Since $\chi\left[{\mathcal{T}}_X\right]$ has AKM symmetry, it is reasonable to organize the index in terms of AKM representations. In particular, we claim that can be re-written as follows $$\begin{aligned}
\label{2d partition}
I_{\mathrm{{\mathcal{T}}_X}}(q,w,z_1,z_2)=\sum_{\lambda=0}^\infty q^{\frac{3}{2} \lambda} P.E.\left[\frac{2q^2}{1-q}+2q-2q^{\lambda+1}\right]\mathrm{ch}^{SU(2)}_{R_\lambda}(q,w)\mathrm{ch}^{SU(3)}_{R_{\lambda,\lambda}}(q,z_1,z_2)~, \end{aligned}$$ where ${\rm ch}_{R_{\lambda}}^{SU(2)}$ and ${\rm ch}_{R_{\lambda,\lambda}}^{SU(3)}$ are AKM characters with highest-weight states transforming in representations of $SU(2)$ and $SU(3)$ characterized by Dynkin labels $\lambda$ and $\lambda_1=\lambda_2=\lambda$ respectively.
In fact, is a completely explicit formula, since AKM characters of $\widehat{su}(N)$ at the critical level have the following simple closed-form expression (e.g., see [@Lemos:2014lua]) $$\begin{aligned}
\mathrm{ch}_{R_{\vec{\lambda}}}(\boldsymbol{x})=\frac{\mathrm{P.E.}[\frac{q}{1-q}\chi_{adj}(\boldsymbol{x})]\chi_{R_{\vec{\lambda}}}(\boldsymbol{x})}{q^{\braket{\vec{\lambda},\rho}}\mathrm{P.E.}[\sum_{j=1}^{N-1}\frac{q^{j+1}}{1-q}]\dim_q R_{\vec{\lambda}}}~,\end{aligned}$$ where $\vec{\lambda}$ is a vector containing the $N-1$ Dynkin labels characterizing the $su(N)$ quantum numbers of the highest-weight state, $\rho$ is the Weyl vector, $\langle\cdot,\cdot\rangle$ is the standard inner product,[^34] and the $q$-dimension is defined as $$\dim_q R_{\vec{\lambda}}=\prod_{\alpha \in \Delta_+}\frac{\left[\braket{\vec{\lambda}+\rho,\alpha} \right]_q}{\left[\braket{\rho,\alpha} \right]_q}~,$$ where $\Delta_+$ denotes the set of positive roots, and the $q$-deformed number is given by $$[x]_q=\frac{q^{-\frac{x}{2}}-q^{\frac{x}{2}}}{q^{-\frac{1}{2}}-q^{\frac{1}{2}}}~.$$
Amusingly, we can give an argument in favor of that parallels the discussion in [@Lemos:2014lua] for the $T_N$ case. The first term, $q^{{3\over2}\lambda}$, is related to the dimension of the non-trivial AKM primary, $W^a_I$, and the dimensions of its products. The plethystic exponential structure constants" $$\label{PEakm}
P.E.\left[{2q^2\over1-q}+2q-2q^{\lambda+1}\right]~,$$ have a simple interpretation as well. Indeed, the first term adds in normal-ordered products of the stress tensor and its derivatives with the other operators in the theory (note that these operators vanish in the AKM modules at the critical level) and also adds in normal-ordered products of the $h=2$ state built out of Casimirs of currents orthogonal to with other operators in the theory (since this linear combination should not be null in the full chiral algebra). The second term in adds back in the level one modes of these two operators, and the final term subtracts relations (for $\lambda=0$, this relation is required by the invariance of the vacuum under these modes).
We have also conducted many highly non-trivial checks of . For example, we have checked that, perturbatively in $q$, coincides with the expression in to very high order. Non-perturbatively in $q=e^{-\beta}$ we have also performed various checks. For example, it is straightforward to see that $$\label{cardyTX}
\lim_{\beta\to0}\, \log\, {\mathcal{I}}_{{\mathcal{T}}_X}(q, w, z_1, z_2)={5\pi^2\over3\beta}+\cdots~.$$ This behavior is consistent with the expected Cardy-like scaling discussed in [@DiPietro:2014bca; @Ardehali:2015bla; @DiPietro:2016ond][^35] $$\lim_{\beta\to0}\,\log\, {\mathcal{I}}(q,{\bf x})=-{8\pi^2\over3\beta}(a-c)+\cdots={\pi^2\over3\beta}{\rm dim}_{\mathbb{Q}}{\mathcal{M}}_H+\cdots~,$$ where, the last equality holds by $U(1)_R$ ’t Hooft anomaly matching in theories with genuine Higgs branches (i.e., moduli spaces where, at generic points, the theory just has free hypermultiplets). In the case of the ${\mathcal{T}}_X$ theory, we expect there to be a genuine Higgs branch since the mirror of the $S^1$ reduction of the ${\mathcal{T}}_{3,{3\over2}}$ theory in Fig. \[quiver3\] has a genuinie Coulomb branch (the result in can also be taken as further evidence for the proposal in Fig. \[quiver3\]).
An even more interesting non-perturbative in $q$ check of our above discussion is to take the $\beta\to0$ limit of , drop the divergent piece in , and study the resulting $S^3$ partition function, $Z_{S^3}$. As we review in greater detail in Appendix \[S3partfn\], using the prescription in [@Nishioka:2011dq] we obtain $$\begin{aligned}
\label{beta0TX}
\lim_{\beta\to0}{\mathcal{I}}_{{\mathcal{T}}_X}(q, w, z_1, z_2)&=&{\rm Div.}\times\int_{-\infty}^{\infty} \frac{d m}{\sinh 2 \pi m \sinh \pi m}{\sin\pi m(\zeta_1-\zeta_2)\sin\pi m(2\zeta_1+\zeta_2)\over\sinh\pi(\zeta_1-\zeta_2)\sinh\pi(2\zeta_1+\zeta_2)}\cr&\times&{\sin\pi m(2\zeta_2+\zeta_1)\sin2\pi m\zeta\over\sinh\pi(2\zeta_2+\zeta_1)\sinh2\pi\zeta}~,\end{aligned}$$ where the Div." factor is the flavor-independent divergent piece in , $w=e^{-i\beta\zeta}, \ z_k=e^{-i\beta\zeta_k}$, and the summation over $\lambda$ in becomes an integral over $m$. On the other hand, we can compute the partition function of the mirror of the quiver in Fig. \[quiver3\], given in Fig. \[quiver5\] of Appendix \[S3partfn\], (or of the original quiver in Fig. \[quiver3\] itself) and divide out by the contribution of a decoupled hypermultiplet to obtain $$\begin{aligned}
Z_{S^3}^{\rm quiver}&=&{\rm Div.}\times{1\over2}\int_{-\infty}^{\infty} dx_1 dx_2 \frac{\sinh^2(\pi(x_1 - x_2)) e^{2\pi i \eta(x_1 + x_2)}}{\cosh\pi(x_1 - x_2 - m’) \cosh\pi(x_2 - x_1 - m’)} \cr&\times&\frac{1}{\cosh\pi m'\cosh\pi(x_1 - m_1) \cosh\pi(x_2 - m_1) \cosh\pi(x_1 - m_2)}\cr&\times&\frac{1}{\cosh\pi(x_2 - m_2) \cosh\pi(x_1 + m_1 + m_2) \cosh\pi(x_2 + m_1 + m_2)}~.\end{aligned}$$ A direct calculation carried out in further detail in Appendix \[S3partfn\] reveals that (up to an unimportant overall constant) $$\lim_{\beta\to0}\left({\rm Div.}^{-1}\times {\mathcal{I}}_{{\mathcal{T}}_X}\right)=Z_{S^3}^{\rm quiver}~,$$ when we identify $m_i\leftrightarrow\zeta_i$ and $m'\leftrightarrow\zeta$.[^36] This result is a strong check of our discussion and also of the proposal in [@Xie:2012hs; @Xie:2013jc].
In the next section we move on and discuss the HL limit of the index and some additional predictions for the Schur sector of ${\mathcal{T}}_X$. Before doing so, let us make a few brief comments on what we have found in this section
- [The structure constants given in that multiply the AKM characters in are precisely those of the free $T_2$ theory [@Lemos:2014lua]. While the set of modules we sum over is diagonal," it is not the same set of modules we sum over for the $T_2$ theory (although the modules are in one-to-one correspondence). It is quite remarkable that all the component Schur indices in our duality described in Fig. \[quiver1\] and \[quiver2\] are so closely related to those of free fields. Moreover, the form of the partition function in suggests simple generalizations to other (hypothetical) SCFTs.]{}
- [We have found strong evidence in favor of the quiver given in Fig. \[quiver3\] for the mirror of the $S^1$ reduction of the ${\mathcal{T}}_{3,{3\over2}}$ theory. Note, however, that the corresponding mirror for the $S^1$ reduction of the ${\mathcal{T}}_X$ theory contains 3D monopole mass terms[^37] $$\label{N2massdef}
\delta W_{{\mathcal{N}}=2}=m\varphi_+{\mathcal{O}}_++m\varphi_-{\mathcal{O}}_-~,$$ where ${\mathcal{O}}_{\pm}$ are the monopoles in the UV theory that map to the free (twisted) hypermultiplet according to the discussion in Footnote \[topology\], and $\varphi_{\pm}$ are fields we add by hand in order to reproduce the IR SCFT that the ${\mathcal{T}}_X$ theory reduced on a circle flows to. This situation is quite unlike what happens for the mirrors of many of the dimensional reductions of the AD theories discussed in [@Xie:2012hs; @Xie:2013jc]]{} (see also the discussions in [@Buican:2015hsa; @Buican:2017uka; @Benvenuti:2017lle]).
In this section, we briefly discuss the Hall-Littlewood (HL) chiral ring of the ${\mathcal{T}}_X$ theory in order to tease out some additional information about the Schur sector of the ${\mathcal{T}}_X$ SCFT. Based on our discussion above, the HL ring is generated by the following 4D Schur operators $$\mu^A_{SU(2)}\in\hat{\mathcal{B}}_1~, \ \ \ \mu^I_{SU(3)}\in\hat{\mathcal{B}}_1~, \ \ \ {\mathcal{O}}^a_I\in\hat{\mathcal{B}}_{3\over2}~,$$ where $A$ and $I$ are adjoint indices of $SU(2)$ and $SU(3)$ respectively, and $a$ is a fundamental index of $SU(2)$.
In , we saw that $W^a_I=\chi\left[{\mathcal{O}}^a_I\right]$ and $J^I_{SU(3)}=\chi\left[\mu^I_{SU(3)}\right]$ had a non-trivial normal-ordered product in the ${\bf2}\times{\bf1}$ channel of $SU(2)\times SU(3)$. On the other hand, as we show in Appendix \[HLlim\], the HL limit of the ${\mathcal{T}}_X$ index has the following expansion $$\begin{aligned}
\label{TXHLpert}
{\mathcal{I}}_{HL}^{{\mathcal{T}}_X}(t,w,z_1,z_2)&=&1+(\chi_{2}+\chi_{1,1})t+\chi_{1}\chi_{1,1}t^{3\over2}+(1+\chi_{4}+\chi_{1,1}+\chi_{2}\chi_{1,1}+\chi_{2,2})t^2+\cr&+&(\chi_{1} \chi_{1,1} + \chi_{3}\chi_{1,1}+\chi_{1}\chi_{3,0} +
\chi_{1} \chi_{0,3} + \chi_{1}\chi_{2, 2})t^{5\over2}+{\mathcal{O}}(t^3)~.\ \ \ \ \ \ \end{aligned}$$ Note that, compared with the Schur index in , the HL index is missing a contribution of the form $\chi_1$ at ${\mathcal{O}}(t^{5\over2})\sim{\mathcal{O}}(q^{5\over2})$ (recall that the power of the fugacity in the HL limit of the index is also given by $h=E-R$). The only apparent explanation, given our generators and the above discussion, is that there is a relation in the HL ring of the form $$\label{HL52rel}
\mu_{SU(3)}^I{\mathcal{O}}^{a}_{{3\over2}\ I}=0~.$$ In order to reconcile this relation with , we conjecture that the theory has a $\hat{\mathcal{C}}_{{1\over2}(0,0)}$ multiplet with Schur operator, $\hat{\mathcal{O}}^{111}_{+\dot+}$, and that this operator appears in the $SU(2)_R$ twisted OPE of the $\mu^I_{SU(3)}$ and ${\mathcal{O}}^a_I$ operators (in the sense described in Footnote \[ttfootnote\]) so that $$\mu^I_{SU(3)}(z,\bar z){\mathcal{O}}^a_I(0)\supset\hat{\mathcal{O}}_{+\dot+}^{111}(0)~.$$ At the level of component (untwisted OPEs), we have $$\label{chirOPE12}
J^{4d,I}_{SU(3)}(x){\mathcal{O}}_I^a(0)\supset {x_{-\dot-}\over x^2}\hat{\mathcal{O}}_{+\dot+}^{111}(0)~,$$ where the operator on the far left of this inclusion is the $R=0$ partner of the holomorphic moment map, $\mu^I_{SU(3)}$. It is straightforward to check that such mixing is compatible with ${\mathcal{N}}=2$ superconformal Ward identities and that therefore $\hat{\mathcal{O}}_{+\dot+}^{111}$ maps to a normal ordered product of generators of $\chi({\mathcal{T}}_X)$.[^38] This discussion is analogous to what happens in the OPE of moment maps in the rank one theories discussed in [@Beem:2013sza] (there the 2D interpretation of the corresponding OPE is that the stress tensor is a Sugawara stress tensor; in the case of the ${\mathcal{T}}_X$ theory, the conclusion is quite different).
In the next section we will switch gears and focus on the implication of the non-vanishing Witten anomaly of $SU(2)\supset G_{{\mathcal{T}}_X}$ for the 2D/4D correspondence of [@Beem:2013sza].
\[wittenanom\] One of the deepest questions in the 4D/2D correspondence of [@Beem:2013sza] is to understand which chiral algebras in 2D are part of a swampland" of theories that cannot be related to consistent (and unitary) 4D ${\mathcal{N}}=2$ SCFTs. One example of a constraint all chiral algebras that are not part of this swampland must obey (unless they are part of the special set of chiral algebras related to a finite subset of free SCFTs in 4D with sufficiently few fields) follows from the analysis in [@Liendo:2015ofa] $$c_{2d}\le-{22\over5}~.$$
.5cm
\(1) at (1.8,0) [$\;{\mathcal{T}}_{X}\;$]{}; (2) at (3,0) \[shape=circle\] [$2$]{} edge \[-\] node\[auto\] (1); (3) at (4.1,0) [$2$]{} edge \[-\] node\[auto\] (2);
We would like to point out that another constraint chiral algebras outside the swampland must obey is that they are not related to 4D ${\mathcal{N}}=2$ SCFTs that have a gauge symmetry with a Witten anomaly [@Witten:1982fp].[^39] Indeed, the corresponding 4D theory is inconsistent. Interestingly, our ${\mathcal{T}}_X$ theory allows us to construct an infinite number of pathological SCFTs by gauging the $SU(2)$ global symmetry (of course, we can also construct infinitely many conformal manifolds that are consistent and have no Witten anomaly; note that the ${\mathcal{T}}_X$ theory on its own is also perfectly consistent since the $SU(2)$ symmetry is global).
A simple example of such a pathological theory is given in Fig. \[anom\]. To construct this SCFT, we gauge a diagonal $SU(2)$ flavor symmetry of the $T_2$ and ${\mathcal{T}}_X$ theories (where the ${\mathcal{T}}_X$ contribution is the anomalous $SU(2)$ factor and not a subgroup of $SU(3)$). Using the expression for the $T_2$ index given in [@Lemos:2014lua] and our expression in , it is straightforward to verify that the naive index of the pathological theory is[^40] $$\begin{aligned}
\label{wittenanomI}
{\mathcal{I}}(q,y_1,y_2,z_1,z_2)&=&\sum_{\lambda}q^{2\lambda}P.E.\left[{2q^2\over1-q}+2q-2q^{1+\lambda}\right]{\rm ch}_{R_{\lambda}}^{SU(2)}(q,y_1){\rm ch}_{R_{\lambda}}^{SU(2)}(q,y_2)\times\cr&\times&{\rm ch}_{R_{\lambda,\lambda}}^{SU(3)}(q,z_1,z_2)~,\end{aligned}$$ where $y_{1,2}$ are $SO(4)$ fugacties, and $z_{1,2}$ are the $SU(3)$ fugacities introduced above.
It would be interesting to understand how (or even if!) this pathology is manifested in the 2D setting. One possibility is that such chiral algebras (like the one whose vacuum character is given in ) are somehow pathological (or perhaps the non-trivial representations of these chiral algebras are pathological). Another possibility is that the chiral algebras and their modules are perfectly consistent at the level of 2D QFT but still detect the pathology of the 4D theory. While we have not fully investigated this question, we suspect the latter possibility holds (we should also note that, in principle, it could be that the chiral algebra and its representations are perfectly consistent and also do not detect the 4D pathology). We hope to return to this question soon.[^41]
Using very little data, we found the Schur index and chiral algebra of the exotic isolated irreducible SCFT, ${\mathcal{T}}_X$,[^42] that emerges in the simplest AD generalization of Argyres-Seiberg duality. Moreover, we saw this theory has a remarkable resemblance to its cousin $T_N$ theories (although its chiral algebra is even simpler) and that, like the other component theories of the duality described in [@Buican:2014hfa], the ${\mathcal{T}}_X$ Schur index is intimately related to the index of free fields (even though the theory itself is strongly interacting). As a result of this study, we found a more pleasing place for the duality described in [@Buican:2014hfa] in the landscape of ${\mathcal{N}}=2$ dualities.
Our work raises many open questions. Among them are the following:
- [Is there a deeper relation between the ${\mathcal{T}}_X$ SCFT and the $T_N$ theories? We saw the Schur indices were closely related. What about more general limits of the index? Is there a family of ${\mathcal{T}}_X$ theories arising from ${\mathcal{N}}=2$ $S$-dualities that are close cousins of the $T_N$ theories?]{}
- [Is there an explanation for why all the component theories in the duality we considered have Schur indices that are so closely related to those of free fields (perhaps generalizing the reasoning in [@kac2017remark])? Could this be some interesting manifestation of modularity in disguise?]{}
- [We saw that our indices are naturally written in terms of AKM characters. Is there a form of the index that is more natural from a TFT perspective (perhaps generalizing [@Buican:2015ina; @Song:2015wta; @Buican:2017uka; @Song:2017oew])?]{}
- [We know that the ${\mathcal{T}}_{3,{3\over2}}$ theory has a class ${\mathcal{S}}$ description (using the results in [@Xie:2012hs]). Does the ${\mathcal{T}}_X$ theory have such a description? Could the TFT description of the index shed some light on this question?]{}
- [If the ${\mathcal{T}}_X$ theory has a class ${\mathcal{S}}$ description, is there a geometrical way to encode the presence of the Witten anomaly in a puncture?]{}
- [This theory lacks ${\mathcal{D}}\oplus\bar{\mathcal{D}}$ operators in its HL ring. Is this absence a clue for the appropriate way to think about the topology of the Riemann surface in this case (again, assuming the theory is class ${\mathcal{S}}$)? See [@Xie:2017vaf] for some recent ideas on the topology that is naturally associated with AD theories.]{}
- [The ${\mathcal{T}}_X$ chiral algebra has only bosonic operators. Is this part of some larger pattern for isolated $1<{\mathcal{N}}<3$ SCFTs?]{}
- [Our theory has $SU(3)\times SU(2)\times U(1)$ flavor symmetry (when viewed as an ${\mathcal{N}}=1$ theory). We are not aware of another way to find this symmetry group in string or field theory from a minimality condition (recall that in our case, this symmetry emerges from requiring that we study the minimal generalization of Argyres-Seiberg duality to ${\mathcal{N}}=2$ SCFTs with non-integer chiral primaries). Can the minimality we are discussing be made more precise so that one can find this SCFT using the conformal bootstrap (perhaps, in light of and , it will be useful to study the $\langle J^I W^a_K J^L W^b_M\rangle$ four-point function)? What if we gauge the flavor symmetry–can this SCFT act as a hidden sector for beyond the standard model physics (since the $U(1)$ is not asymptotically free, this gauged theory can, at best, be part of an effective field theory)?]{}
- [Is it possible to make contact with a generalization of [@Gadde:2015xta; @Maruyoshi:2016tqk] to the case at hand?]{}
- [Can we find a manifestation of the 4D Witten anomaly for the (inconsistent) SCFT in Fig. \[anom\] in the corresponding 2D chiral algebra (as discussed in Sec. \[wittenanom\])?]{}
- [As a final amusing note, it is interesting to observe that the expression in makes it rather trivial to write down simple formulae for the indices of conformal manifolds built out of ${\mathcal{T}}_X$ theories (as in the case of the $T_N$ theories). For typical conformal manifolds built out of AD theories (e.g., as in the case of the $(A_N, A_M)$ conformal manifolds studied in [@Buican:2017uka]), this procedure is considerably more complicated.]{}
Proof of the XYY formula {#app:PEproof}
========================
In this appendix we review the fact that the conjectured XYY formula for the Schur index of the $(A_1,D_4)$ theory [@Xie:2016evu] reproduced in can be proven using Theorem 5.5 of [@Creutzig:2017qyf] (in fact, this result follows directly from (11) of [@kac2017remark]).[^43]
To that end, we start with the XYY formula $$\begin{aligned}
\label{XYYform2}
{\mathcal{I}}_{(A_1, D_4)}(q,a,b)&=&\mathrm{P.E.}\left[\frac{q}{1-q^{2}}\,\chi_{\text{Adj}}^{SU(3)}(a,b)\right]\cr&=&\mathrm{P.E.}\left[\frac{q}{1-q^{2}}(2+\frac{1}{a^{2}b}+\frac{1}{ab^{2}}+\frac{a}{b}+\frac{b}{a}+a^{2}b+ab^{2})\right]~.\end{aligned}$$ Expanding the plethystic exponentials, we obtain $$\label{PEconv}
\mathrm{P.E.}\left[\frac{a}{1-b}\right]=\prod_{i=0}^{\infty}{1\over1-ab^i}~,$$ and we can then rewrite as $$\begin{aligned}
\label{rewriteXYY}
\mathrm{P.E.}\left[\frac{q}{1-q^{2}}\,\chi_{\text{Adj}}^{SU(3)}(a,b)\right]=&\prod_{n=0}^{\infty}\frac{1}{(1-q^{2n+1})^{2}}\frac{1}{(1-\frac{1}{a^2b}q^{2n+1})}\frac{1}{(1-\frac{1}{ab^2}q^{2n+1})}\frac{1}{(1-\frac{a}{b}q^{2n+1})}\times\nonumber\\ \times&\frac{1}{(1-\frac{b}{a}q^{2n+1})}\frac{1}{(1-a^{2}bq^{2n+1})}\frac{1}{(1-ab^{2}q^{2n+1})}~.\end{aligned}$$ It is then straightforward to show that becomes the Schur index of the $(A_1,D_4)$ SCFT given by Theorem 5.5 of [@Creutzig:2017qyf] (setting $p=2$ and with the $q^{1/3}$ prefactor stripped off) $$\begin{aligned}
\label{CA1D4}
{\mathcal{I}}_{(A_1,D_4)}(q,x,y)=&\prod_{n=0}^{\infty}\frac{\left(1-y^{2}q^{2(n+1)}\right)\left(1-q^{2(n+1)}\right)^{2}\left(1-y^{-2}q^{2(n+1)}\right)}{\left(1-y^{2}q^{n+1}\right)\left(1-q^{n+1}\right)^{2}\left(1-y^{-2}q^{n+1}\right)\left(1-x y q^{2\left(n+\frac{1}{2}\right)}\right)}\times \nonumber \\
&\times\frac{1}{\left(1-x^{-1} y q^{2\left(n+\frac{1}{2}\right)}\right)\left(1-x y^{-1} q^{2\left(n+\frac{1}{2}\right)}\right)\left(1-x^{-1} y^{-1} q^{2\left(n+\frac{1}{2}\right)}\right)}\nonumber\\&=\prod_{n=0}^{\infty}\frac{1}{\left(1-q^{2n+1}\right)^{2}\left(1-y^{\pm 2}q^{2n+1}\right)\left(1-x^{\pm} y^{\pm} q^{2n+1}\right)}~,\end{aligned}$$ under the fugacity map $$\label{fugacitymap}
a=y\,x^{1/3} \qquad b=y^{-1}\,x^{1/3}~.$$ The relation in corresponds to the decomposition of the $SU(3)$ fugacities into fugacities of $SU(2)\times U(1)$. Before concluding, note that, as in , the $\pm$" superscripts in are understood as a product over each sign, e.g. $${1\over1-y^{\pm2}q^{2n+1}}\equiv {1\over1-y^{2}q^{2n+1}}{1\over1-y^{-2}q^{2n+1}}~.$$
Details of the Inversion Formula {#inversionap}
================================
In this appendix we find an integral expression for the superconformal index of the ${\mathcal{T}}_{3,{3\over2}}$ theory in the Schur limit by employing the inversion theorem proved in [@spiridonov2006inversions]. Our use of the inversion theorem is similar to its use in the case of the $E_6$ SCFT by the authors of [@Gadde:2010te], but there are some technical differences here since our $SU(2)$ duality frame in Fig. \[quiver2\] has, in addition to the ${\mathcal{T}}_{3,{3\over2}}$ theory, a strongly interacting $(A_1, D_4)$ SCFT instead of a pair of hypermultiplets as in the $E_6$ case. Nonetheless, we will argue that, using the results reviewed in Appendix \[app:PEproof\] and an argument about analytic properties of the index, we can invert the gauge integral of the index in the $SU(2)$ duality frame.
In order to find the index in the two duality frames we need the index of the basic building blocks in Figs. \[quiver1\] and \[quiver2\]. To that end, the single letter index of the $\mathcal{N}=2$ vector multiplet (transforming in the adjoint of the gauge group) and half-hypermultiplet (transforming in representation $R$ of the combined gauge and flavor groups) can be found in [@Gadde:2011uv] (whose labelling conventions for fugacities we follow). Here we reproduce these indices in the Schur limit $$\begin{aligned}
{\mathcal{I}}_{\rm vect}(q,{\bf x})&=&-\frac{2q}{1-q}\chi_{\text{adj}}({\bf x})~,\cr
{\mathcal{I}}_{\frac{1}{2}H}(q,{\bf x},{\bf z})&=&\frac{\sqrt{q}}{1-q}\,\chi_{R}({\bf x},{\bf z})~.\end{aligned}$$ We can glue" these indices along with the index of the $(A_1,D_4)$ SCFT given in by integrating their product over the Haar measure of the diagonal subgroup we are gauging.
We start with the $SU(3)$ side of the duality where we are gauging the diagonal part of the $SU(3)$ flavor symmetries of the two $(A_1,D_4)$ theories along with 3 fundamental hypermultiplets as in Fig. \[quiver1\]. The latter degrees of freedom supply the $U(3)$ symmetry, which is decomposed as $U(3)=SU(3)_z\otimes U(1)_s$. The index on this side of the duality is then given by $$\begin{aligned}
\label{su3sideapp}
&{\mathcal{I}}_{SU(3)}(q,s,z_1,z_2)=\frac{(q;q)^4}{6}\oint_{\mathbb{T}^2} \prod_{k=1}^{2}\frac{dx_k}{2 \pi i x_k} \prod_{i\neq j}(x_i-x_j) \left(q \frac{x_i}{x_j};q\right)^2 \times \nonumber\\
& \times \mathrm{P.E.}\left[\frac{q}{1-q^{2}}\,\chi_{\text{Adj}}^{SU(3)}(x_1,x_2)\right]^2 \prod_{i,j} \left( \sqrt{q} \left(\frac{z_{j}s^{1/3}}{x_{i}} \right)^{\pm};q \right)^{-1}~,\end{aligned}$$ where $\mathbb{T}$ is the positively oriented unit circle, $\prod_{k=1}^{2}\frac{dx_k}{2 \pi i x_k} \frac{1}{3!}\prod_{i\neq j}(x_i-x_j)$ is the Haar measure of $SU(3)$, and the $x_i$ ($i = 1,2,3$) satisfy the constraint $\prod_{i=1}^3x_i=1$. We can rewrite slightly using elementary computations described in appendix \[app:PEproof\] $$\label{rew}
\mathrm{P.E.}\left[\frac{q}{1-q^{2}}\,\chi_{\text{adj}}^{SU(3)}(x_1,x_2)\right]= (q;q^2)^{-2}\prod_{i\neq j}\left(q\frac{x_i}{x_j};q^2\right)^{-1}~, \ \ \ x_3=x_1^{-1}x_2^{-1}~.$$ Substituting into and performing some simplifications yields the following explicit formula $$\begin{aligned}
\label{SU(3)side}
{\mathcal{I}}_{SU(3)}(q,s,z_1,z_2)=\frac{(q^2;q^2)^4}{6}\oint_{\mathbb{T}^2} \prod_{i=1}^{2}\frac{dx_i}{2 \pi i x_i} \prod_{i\neq j}(x_i-x_j) \left(q^2 \frac{x_i}{x_j};q^2\right)^2 \prod_{i,j} \left( \sqrt{q} \left(\frac{z_{j}s^{1/3}}{x_{i}} \right)^{\pm};q \right)^{-1}~.\end{aligned}$$ Since the index is invariant under duality transformations, has to equal the index on the $SU(2)$ side of the duality where we are gauging the diagonal $SU(2)_e$ of the $(A_1,D_4)$ and ${\mathcal{T}}_{3,{3\over2}}$ theories as in Fig. \[quiver2\]. We can write the index in this duality frame as $$\begin{aligned}
{\mathcal{I}}_{SU(2)}(q,s,z_1,z_2)=\frac{(q;q)^2}{2} \oint_\mathbb{T} \frac{de}{2 \pi i e}(e^{\pm 2}q;q)^2 &(1-e^{\pm 2})\times\nonumber\\ \times\mathrm{P.E.}\left[\frac{q}{1-q^2}\,\chi_{\text{adj}}^{SU(3)}\left(es^{\frac{1}{3}},e^{-1}s^{\frac{1}{3}},s^{-\frac{2}{3}}\right)\right] &{\mathcal{I}}_{\mathrm{{\mathcal{T}}_{3,{3\over2}}}}(q,e,z_1,z_2)~,\end{aligned}$$ where $\frac{de}{2 \pi i e}\frac{1}{2}(e-e^{-1})(e^{-1}-e)$ is the Haar measure of $SU(2)$. Rewriting the plethystic exponential as in and performing some simplifications leads to $$\begin{aligned}
{\mathcal{I}}_{SU(2)}(q,s,z_1,z_2)=\frac{(q^2;q^2)^2}{2} \oint_\mathbb{T} \frac{de}{2 \pi i e}\frac{(e^{\pm 2}q;q)(e^{\pm 2};q^2) {\mathcal{I}}_{\mathrm{{\mathcal{T}}_{3,{3\over2}}}}(q,e,z_1,z_2)}{(qs^{\pm} e^{\pm};q^2)} ~.\end{aligned}$$ Finally, to make contact with the inversion theorem, we replace $q\rightarrow \sqrt{q}$ $$\begin{aligned}
\label{comp}
{\mathcal{I}}_{SU(2)}(q,s,z_1,z_2)|_{q\rightarrow \sqrt{q}}=\frac{(q;q)^2}{2} \oint_\mathbb{T} \frac{de}{2 \pi i e}\frac{(e^{\pm 2};q) }{(\sqrt{q}s^{\pm} e^{\pm};q)} (e^{\pm 2}\sqrt{q};\sqrt{q}){\mathcal{I}}_{\mathrm{{\mathcal{T}}_{3,{3\over2}}}}(q,e,z_1,z_2)|_{q\rightarrow \sqrt{q}}~.\ \\end{aligned}$$ Now we will explain how to use the inversion theorem in order to extract ${\mathcal{I}}_{{\mathcal{T}}_{3,{3\over2}}}$ from this equation. Extracting ${\mathcal{I}}_{{\mathcal{T}}_{3,{3\over2}}}$ is highly non-trivial since it is not at all obvious why preserves all the information about this quantity.
Inversion Theorem
-----------------
This subsection closely follows Appendix B of [@Gadde:2010te]. The input to the inversion theorem of [@spiridonov2006inversions] is the following type of contour integral $$\begin{aligned}
\label{invin}
\hat{f}(w)=\kappa \oint_{C_{w}}\frac{ds}{2 \pi i s}\delta(s,w;T^{-1},p,q)f(s)~,\end{aligned}$$ where $\kappa=\frac{1}{2}(p;p)(q;q)$, $w$ is on the unit circle, and the integral kernel is defined as $$\label{delta}
\delta(s,w;T,p,q) \equiv \frac{\Gamma(T s^{\pm 1}w^{\pm 1};p,q)}{\Gamma(T^{2};p,q)\Gamma (s^{\pm 2};p,q)}~.$$ In , $T$ is a function of $p,q,t\in\mathbb{C}$ satisfying $$\label{eq:cond}
\mathrm{max}\,\left(|p|,|q|\right)<|T|^2<1~,$$ $\Gamma(z;p,q)$ is defined as $$\label{Gammadef}
\Gamma(z;p,q)\equiv\prod_{j,k\ge0}{1-z^{-1}p^{j+1}q^{k+1}\over1-zp^jq^k}~,$$ and $f(s)\equiv f(s,p,q,t)$ is a function that is holomorphic in the annulus $$\mathbb{A}=\{|T|-\varepsilon<|s|<|T|^{-1}+\varepsilon\}~,$$ for small but finite $\varepsilon>0$ and also satisfies $$\label{scond}
f(s)=f(s^{-1})~.$$ The contour $C_{w}=C_{w}^{-1}$ lies in the annulus $\mathbb{A}$ with the points $T^{-1}w^{\pm}$ in its interior (and therefore the points $Tw^{\pm}$ in its exterior). If these conditions are all satisfied, then the inversion theorem states that $f$ can be recovered from the contour integral $$\begin{aligned}
\label{invout}
f(s)=\kappa \oint_{\mathbb{T}}\frac{de}{2 \pi i e} \delta(e,s;T,p,q)\hat{f}(e)~.\end{aligned}$$
As first applied to the index in [@Gadde:2010te], this inversion theorem is used as follows. First, one finds a representation of the conformal manifold index that is of the form of the RHS of . In particular, $\hat f(e)$ should contain the index of the isolated SCFT (the $E_6$ theory in [@Gadde:2010te] or the ${\mathcal{T}}_{3,{3\over2}}$ SCFT in the case at hand) we wish to determine. One then makes an analytic assumption that $\hat f(e)$ can be written as in for some function $f(s)$ satisfying while being analytic in the annulus, $\mathbb{A}$. Then, the inversion theorem implies that $f(s)$ is the index of the conformal manifold. However, in general, one is not guaranteed that the analytic assumption described above holds.[^44]
As a result, to apply this theorem in our case, we first need to choose $\hat f(e)$ in so that coincides with . To that end, using $$\Gamma(z;p,q)=\mathrm{P.E.}\left[ \frac{z-pq/z}{(1-p)(1-q)}\right]~,$$ and one finds that the delta function" in satisfies (for our choice of $T$ discussed below) $$\begin{aligned}
\label{deltadefn}
\delta(e,s;T,p,q)= \frac{(T^2;q)(e^{\pm 2};q)}{(T e^{\pm} s^{\pm};q)}\tilde\delta(e;T,p,q)~,\end{aligned}$$ where $\tilde\delta(e;T,p,q)$ contains $p$-dependent terms. By comparing with , one can see that if we choose $T=\sqrt{q}$ and $$\label{sep}
\hat{f}(e)=(e^{\pm 2}\sqrt{q};\sqrt{q})\times (e^{\pm2}p;p)^{-1}\times{\mathcal{I}}_{\mathrm{{\mathcal{T}}_{3,{3\over2}}}}(q,e,z_1,z_2)|_{q\rightarrow \sqrt{q}}~,$$ the two expressions coincide.
However, there is an additional wrinkle in our application of the inversion theorem relative to the $E_6$ case in [@Gadde:2010te]. Indeed, under the analytic assumption described in the paragraph below , we have $$\begin{aligned}
\label{inversion2}
(w^{\pm 2} \sqrt{q};\sqrt{q})\times(w^{\pm2}p;p)^{-1}&\times&{\mathcal{I}}_{\mathrm{{\mathcal{T}}_{3,{3\over2}}}}(q,w,z_1,z_2)|_{q\rightarrow\sqrt{q}}=\frac{(q;q)(p;p)}{2}\times\oint_{C_w}\frac{ds}{2 \pi i s} \frac{(\frac{1}{q};q)(s^{\pm 2};q)}{(\frac{1}{\sqrt{q}}s^{\pm}w^{\pm};q)}\times\cr&\times&\tilde\delta(s,w;{1\over\sqrt{q}},p,q)\times{\mathcal{I}}_{SU(3)}(q,s,z_1,z_2)|_{q\rightarrow \sqrt{q}}~,\end{aligned}$$ where, as in , we have separated $\delta$ into a $p$-independent part and a $p$-dependent part, $\tilde\delta$. While the $p$-dependence in can be cancelled so that $$\begin{aligned}
\label{inversion3}
(w^{\pm 2} \sqrt{q};\sqrt{q})&\times&{\mathcal{I}}_{\mathrm{{\mathcal{T}}_{3,{3\over2}}}}(q,w,z_1,z_2)|_{q\rightarrow\sqrt{q}}=\frac{(q;q)}{2}\times\oint_{C_w}\frac{ds}{2 \pi i s} \frac{(\frac{1}{q};q)(s^{\pm 2};q)}{(\frac{1}{\sqrt{q}}s^{\pm}w^{\pm};q)}\times\cr&\times&{\mathcal{I}}_{SU(3)}(q,s,z_1,z_2)|_{q\rightarrow \sqrt{q}}~,\end{aligned}$$ the condition fails for $T=\sqrt{q}$, and $\delta(s,w;{1\over\sqrt{q}};p,q)=0$ (since $({1\over q};q)=0$). Therefore, the RHS of vanishes.[^45]
To get a more sensible answer, we can consider taking $T=\sqrt{q}(1+\varepsilon')$ for $\varepsilon'\ll1$. In this case, we have $$\label{zerofact}
\left({1\over q};q\right)\to\left({1-2\varepsilon'\over q};q\right)\ne0~,$$ and the expression on the RHS of is non-vanishing since it becomes $$\label{inversion4}
\frac{(q;q)}{2}\times\oint_{C_w}\frac{ds}{2 \pi i s} \frac{(\frac{1-2\varepsilon'}{q};q)(s^{\pm 2};q)}{(\frac{1-\varepsilon'}{\sqrt{q}}s^{\pm}w^{\pm};q)}\times{\mathcal{I}}_{SU(3)}(q,s,z_1,z_2)|_{q\rightarrow \sqrt{q}}~.$$ In particular, note that the double poles at $s=T^{-1}w^{\pm1}$ and $s=qT^{-1}w^{\pm1}$ in are resolved into eight single poles in with one of each pair still taken to be in the integration contour (for a total of four) and a factor of $\varepsilon'^{-1}$ from the residues that cancels the factor of $\varepsilon'$ arising from (all other contributions will be parametrically smaller in $\varepsilon'$). Taking the $\varepsilon'\to0$ limit then gives us a prescription for computing the Schur index with $$\begin{aligned}
\label{prescription}
(w^{\pm 2} \sqrt{q};\sqrt{q})\times{\mathcal{I}}_{\mathrm{{\mathcal{T}}_{3,{3\over2}}}}(q,w,z_1,z_2)|_{q\rightarrow\sqrt{q}}&=&\lim_{\varepsilon' \rightarrow 0}\frac{(q;q)}{2}\times\oint_{C_w}\frac{ds}{2 \pi i s} \frac{(\frac{1-2\varepsilon'}{q};q)(s^{\pm 2};q)}{(\frac{1-\varepsilon'}{\sqrt{q}}s^{\pm}w^{\pm};q)}\times\cr&\times&{\mathcal{I}}_{SU(3)}(q,s,z_1,z_2)|_{q\rightarrow \sqrt{q}}~.\end{aligned}$$ The contour integration around an infinite number of poles thus reduces to the residues of just four poles whose contribution gives us the simple expression $${\mathcal{I}}_{{\mathcal{T}}_{3,{3\over2}}}(q,w,z_1,z_2)=\frac{1}{(w^{\pm 2} q;q)}\left[ \frac{1}{1-w^2}{\mathcal{I}}_{SU(3)}(q,wq,z_1,z_2)+\frac{w^2}{w^2-1}{\mathcal{I}}_{SU(3)}(q,\frac{q}{w},z_1,z_2)\right]~.$$
We can justify the above discussion a posteriori by noting that the non-trivial checks in the main text strongly suggest that is a consistent prescription. While a similar procedure works for the Schur index of the $E_6$ SCFT discussed in [@Gadde:2010te], our case at hand is somewhat more special. Indeed, we used the fact that the $(A_1, D_4)$ SCFT has a Schur index whose $s$ dependence (after taking $q\to\sqrt{q}$) in is the same as for $\delta(e,s;\sqrt{q},p,q)$. On the other hand, when we take $T\to\sqrt{q}(1+\varepsilon')$, we do not necessarily expect that the $(A_1, D_4)$ SCFT has a limit of the index whose $s$ dependence matches the $s$ dependence in $\delta(e,s;\sqrt{q}(1+\varepsilon'),p,q)$ to all orders in $\varepsilon'$. However, the ${\mathcal{O}}(\epsilon')$ resolution of the double poles into single poles described above should correspond to a shift in the fugacities of the index so that previously degenerate contributions from sets of operators are no longer degenerate (this statement is quite natural since generic single letter contributions to the index will be shifted at ${\mathcal{O}}(\epsilon')$ if we identify $T$ with a fugacity) and that higher-order differences with respect to $\delta(e,s;\sqrt{q}(1+\varepsilon'),p,q)$ do not affect the validity of our computation in the limit of small $\varepsilon'$.
$q\rightarrow1$ and $S^3$ partition function {#S3partfn}
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(3,.2) arc \[radius=0.5, start angle=20, end angle= 340\]; (2) at (3,0) \[shape=circle\] [$U(2)$]{} edge \[-\] node\[auto\] (2); (3) at (4.4,0) [$3$]{} edge \[-\] node\[auto\] (2);
The superconformal index can alternatively be viewed as a partition function on $S^3\times S^1$. Moreover, the fugacity $q=e^{-\beta}$ introduced in the main text controls the relative radii of the $S^3$ and $S^1$ factors. In particular, in the $\beta\to0$ limit, the $S^1$ factor shrinks relative to the $S^3$ factor and, up to divergent terms, we expect the index to reduce to the $S^3$ partition function, $Z_{S^3}$.
In the limit of $\beta\to0$, our expression for the ${\mathcal{T}}_X$ index in can be described by the rules in [@Nishioka:2011dq]. In particular, the sum over $\lambda$ is replaced by an integral on $m$, where $$\lambda=-\frac{2\pi m}{\beta}~,$$ and the group fugacities are $w=e^{-i\beta \zeta}$, $z_i=e^{-i\beta \zeta_i}$. We drop group fugacity independent factors in and only work to leading order in $\beta$. The $\beta \rightarrow 0$ limit of the remaining quantities are given by the following dictionary [@Nishioka:2011dq] $$\begin{aligned}
\label{dictionary}
\mathrm{P.E.}[-2q^{\lambda+1}]&\rightarrow& (1-e^{2 \pi m})^2~,\cr
\dim_q R_{\lambda}^{SU(2)}&\rightarrow& \sinh(\pi m)~,\cr
\dim_q R_{\lambda,\lambda}^{SU(3)} &\rightarrow& \sinh(2 \pi m) \sinh^2(\pi m)~,\cr
\mathrm{P.E.}\left[\frac{q}{1-q}\chi_{adj}\right] &\rightarrow&\prod_{j<k}\frac{(\zeta_j-\zeta_k)}{\sinh \pi (\zeta_j-\zeta_k)}~,\cr
\chi_{R_\lambda}^{SU(2)}(w) &\rightarrow& \frac{\sin (2 \pi m \zeta)}{\zeta}~,\cr
\chi_{R_{\lambda,\lambda}}^{SU(3)}(z_1,z_2,z_3)&\rightarrow& \frac{\sin \pi m(\zeta_1-\zeta_2)\sin \pi m(2\zeta_1+\zeta_2)\sin \pi m(2\zeta_2+\zeta_1)}{(\zeta_1-\zeta_2)(2\zeta_1+\zeta_2)(2\zeta_2+\zeta_1)}~.\end{aligned}$$ Using and replacing the sum over $\lambda$ with an integral over $m$, the $\beta\rightarrow 0$ limit of becomes $$\begin{aligned}
\label{q to 1}
\int_{-\infty}^{\infty} \frac{d m}{\sinh 2 \pi m \sinh \pi m} \frac{\sin \pi m(\zeta_1-\zeta_2)\sin \pi m(2\zeta_1+\zeta_2)\sin \pi m(2\zeta_2+\zeta_1)}{\sinh \pi (\zeta_1-\zeta_2)\sinh \pi (2\zeta_1+\zeta_2)\sinh \pi (2\zeta_2+\zeta_1)} \frac{\sin 2\pi m \zeta}{\sinh 2 \pi \zeta}~.\end{aligned}$$ One can integrate by turning it into a contour integral and using the residue theorem. The result is the following $$\begin{aligned}
\label{intresult}
& \frac{1}{32}\mathrm{sech\,\pi \zeta}\,(2\mathrm{csch}\,\pi (\zeta_1-\zeta_2)\, \mathrm{csch}\,\pi (\zeta_1+2 \zeta_2)\, \mathrm{sech}\,\pi (\zeta -2 \zeta_1-\zeta_2)\, \mathrm{sech}\,\pi (\zeta +2 \zeta_1+\zeta_2)\nonumber\\
&\qquad-\mathrm{csch}\,\pi (2 \zeta_1+\zeta_2) \,\mathrm{csch}\,\pi (\zeta_1+2 \zeta_2)\, \mathrm{sech}\,\pi (\zeta +\zeta_1-\zeta_2)\, \mathrm{sech}\,\pi (\zeta -\zeta_1+\zeta_2) \nonumber \\
&\qquad-\text{csch}\,\pi \zeta \,\text{csch}\, \pi \left(\zeta _1-\zeta _2\right)\, \text{csch}\,\pi \left(\zeta _1+2 \zeta _2\right)\nonumber \\
&\qquad \qquad\times (\left(2 \zeta +3 \zeta _1+5 \zeta _2\right) \text{sech}\,\pi \left(\zeta -\zeta _1-2 \zeta _2\right)\, \text{sech}\,\pi \left(\zeta +\zeta _1-\zeta _2\right)\nonumber\\
&\qquad\qquad-\left(4 \zeta +3 \zeta _1+5 \zeta _2\right) \,\text{sech}\,\pi \left(\zeta -\zeta _1+\zeta _2\right)\, \text{sech}\,\pi \left(\zeta +\zeta _1+2 \zeta _2\right))\nonumber\\
&\qquad-\frac{1}{2}\text{csch}\,\pi \zeta \,\text{csch}\,\pi \left(2 \zeta _1+\zeta _2\right)\,\text{csch}\,\pi \left(\zeta _1+2 \zeta _2\right)\nonumber \\
&\qquad\qquad\times (\left(4 \zeta +\zeta _1+\zeta _2\right) \text{sech}\,\pi \left(\zeta -\zeta _1-2 \zeta _2\right) \text{sech}\,\pi \left(\zeta -2 \zeta _1-\zeta _2\right)\nonumber\\
&\qquad\qquad-\left(2 \zeta +\zeta _1+\zeta _2\right) \text{sech}\, \pi \left(\zeta +2 \zeta _1+\zeta _2\right) \text{sech}\,\pi \left(\zeta +\zeta _1+2 \zeta _2\right))\nonumber \\
&+(\zeta_1 \leftrightarrow \zeta_2)~.\end{aligned}$$ This answer can then be compared with the partition function of the $S^1$ reduction of ${\mathcal{T}}_{X}$ or of the mirror theory in Fig. \[quiver3\]. The direct $S^1$ reduction of ${\mathcal{T}}_{3,{3\over2}}$ is described by an ${\mathcal{N}}=4$ $U(2)$ gauge theory whose Lagrangian quiver is illustrated in Fig. \[quiver5\] [@Cremonesi:2014xha]. Once we decouple the contribution of the $SU(2)$ gauge singlet part of the adjoint hypermultiplet, $\frac{1}{\cosh \pi m’}$, which is the 3D descendant of the decoupled hyper of ${\mathcal{T}}_{3,{3\over2}}$ we can write down the partition function of the 3D reduction of ${\mathcal{T}}_{X}$ [@Hama:2010av] [@Benvenuti:2011ga] $$\begin{aligned}
\label{S1 reduction}
Z_{S^3}^{\rm quiver}&=&{1\over2}\int_{-\infty}^{\infty} dx_1 dx_2 \frac{\sinh^2(\pi(x_1 - x_2)) e^{2\pi i \eta(x_1 + x_2)}}{\cosh\pi(x_1 - x_2 - m’) \cosh\pi(x_2 - x_1 - m’)} \cr&\times&\frac{1}{\cosh\pi m'\cosh\pi(x_1 - m_1) \cosh\pi(x_2 - m_1) \cosh\pi(x_1 - m_2)}\cr&\times&\frac{1}{\cosh\pi(x_2 - m_2) \cosh\pi(x_1 + m_1 + m_2) \cosh\pi(x_2 + m_1 + m_2)}~.\end{aligned}$$ This integral can be evaluated similary to with the same result (up to an unimportant overall constant and after using the map $\zeta\to m'$, $\zeta_i\to m_i$) as in (again, a similar statement holds for the partition function of the mirror in Fig. \[quiver3\], which involves six integrations and for which one should use the fugacity map in ).
The Hall-Littlewood index of ${\mathcal{T}}_X$ {#HLlim}
==============================================
In this appendix, we derive the HL index in . In the language of [@Dolan:2002zh], the HL operators are a subset of the Shur operators described around and are of type $\hat{\mathcal{B}}_R$ and ${\mathcal{D}}_{R(0,j_2)}\oplus\bar{\mathcal{D}}_{R(j_1,0)}$ (see Sec. \[2D4Dcorr\] for more details). In this section we merely note that they contribute to a limit of the superconformal index described in [@Gadde:2011uv] where their contributions are of the form $t^{E-R}$ where $t$ is the HL superconformal fugacity (this limit of the index also detects flavor symmetries).
When a 4D ${\mathcal{N}}=2$ theory is put on a circle, we can often compute the HL limit of the index from the 3D ${\mathcal{N}}=4$ Higgs branch Hilbert series provided the compactification is sufficiently well-behaved. Equivalently, mirror symmetry allows us to compute the HL limit of the 4D theory from the Coulomb branch Hilbert series of the mirror theory.
Indeed, we can try to compute ${\mathcal{I}}_{HL}^{{\mathcal{T}}_X}$ by first computing ${\mathcal{I}}_{HL}^{{\mathcal{T}}_{3,{3\over2}}}$ from the 3D mirror gauge theory that follows from the rules reproduced in Fig. \[quiver3\] and described in [@Xie:2012hs].[^46] Using the results in [@Cremonesi:2013lqa], we can write this index as follows $$\label{3DT332}
{\mathcal{I}}_{HL}^{{\mathcal{T}}_{3,{3\over2}}}(t)={1\over(1-t)^3}\sum_{a_1, a_{A,i},a_{B,i}\in\Gamma^*_{\hat G}/\mathcal{W}_{\hat G}}\zeta_A^{a_{A,1}+a_{A,2}}\zeta_B^{a_{B,1}+a_{B,2}}\zeta_C^{a_{C,1}+a_{C,2}}\cdot P(a_{A,i},a_{B,i},a_{C,i})\cdot t^{\Delta}~,$$ where the arguments of $P$ denote integral GNO flux (restricted to a Weyl chamber of the weight lattice of the GNO dual gauge group as described in [@Cremonesi:2013lqa]), $\zeta_{A,B,C}$ are fugacities for the $U(1)^3$ topological symmetry, $\Delta$ is a monopole scaling dimension for operators charged under the GNO flux, and $$\begin{aligned}
\label{PHLii}
P(a_{A,1}=a_{A,1},a_{B,1}=a_{B,2},a_{C,1}=a_{C,2})&=&{1\over(1-t^2)^3}~, \cr P(a_{A,1}>a_{A,1},a_{B,1}=a_{B,2},a_{C,1}=a_{C,2})&=&P(a_{A,1}=a_{A,1},a_{B,1}>a_{B,2},a_{C,1}=a_{C,2})=\cr P(a_{A,1}=a_{A,1},a_{B,1}=a_{B,2},a_{C,1}>a_{C,2})&=&{1\over(1-t)(1-t^2)^2}~,\cr P(a_{A,1}>a_{A,1},a_{B,1}>a_{B,2},a_{C,1}=a_{C,2})&=&P(a_{A,1}>a_{A,1},a_{B,1}=a_{B,2},a_{C,1}>a_{C,2})=\cr P(a_{A,1}=a_{A,1},a_{B,1}>a_{B,2},a_{C,1}>a_{C,2})&=&{1\over(1-t)^2(1-t^2)}~,\cr P(a_{A,1}>a_{A,1},a_{B,1}>a_{B,2},a_{C,1}>a_{C,2})&=&{1\over(1-t)^3}~.\end{aligned}$$ The monopole scaling dimension in is given by [@Buican:2014hfa] $$\begin{aligned}
\label{monodim}
\Delta&=&{1\over2}\Big(|a_{A,1}|+|a_{A,2}|\Big)+{1\over2}\Big(|a_{A,1}-a_{B,1}|+|a_{A,2}-a_{B,1}|+|a_{A,1}-a_{B,2}|+|a_{A,2}-a_{B,2}|\cr&+&|a_{A,1}-a_{C,1}|+|a_{A,2}-a_{C,1}|+|a_{A,1}-a_{C,2}|+|a_{A,2}-a_{C,2}|+|a_{B,1}-a_{C,1}|\cr&+&|a_{B,2}-a_{C,1}|+|a_{B,1}-a_{C,2}|+|a_{B,2}-a_{C,2}|\Big)-\Big(|a_{A,1}-a_{A,2}|+|a_{B,1}-a_{B,2}|\cr&+&|a_{C,1}-a_{C,2}|\Big)~.\end{aligned}$$ After identifying fugacities according to $$\label{fmap2}
\zeta_A=wz_1^{-2}z_2^{-1}~, \ \ \ \zeta_B=z_1z_2^2~, \ \ \ \zeta_C=z_1z_2^{-1}~,$$ we can then expand the HL index in $t$ to find $$\begin{aligned}
\label{T332HLpert}
{\mathcal{I}}_{HL}^{{\mathcal{T}}_{3,{3\over2}}}(t)&=&1+\chi_{1}t^{1\over2}+(2\chi_{2}+\chi_{1,1})t+(\chi_{1}+2\chi_{3}+2\chi_{1}\chi_{1,1})t^{3\over2}+(2+\chi_{2}+3\chi_{4}+\cr&+&2\chi_{1,1}+3\chi_{2}\chi_{1,1}+\chi_{2,2})t^2+(3\chi_{5}+\chi_{3}(2+4\chi_{1,1})+\chi_{1}(2+\chi_{1,1}+\cr&+&\chi_{3,0}+\chi_{0,3}+2\chi_{2,2}))t^{5\over2}+{\mathcal{O}}(t^3) \end{aligned}$$ We immediately see a free hypermultiplet at ${\mathcal{O}}(t^{1\over2})$ as expected from our discussion in the main text. Stripping off this free hypermultiplet, we get the putative HL index of the ${\mathcal{T}}_X$ theory $$\begin{aligned}
\label{TXHLpert2}
{\mathcal{I}}_{HL}^{{\mathcal{T}}_X}(t,w,z_1,z_2)&=&1+(\chi_{2}+\chi_{1,1})t+\chi_{1}\chi_{1,1}t^{3\over2}+(1+\chi_{4}+\chi_{1,1}+\chi_{2}\chi_{1,1}+\chi_{2,2})t^2+\cr&+&(\chi_{1} \chi_{1,1} + \chi_{3}\chi_{1,1}+\chi_{1}\chi_{3,0} +
\chi_{1} \chi_{0,3} + \chi_{1}\chi_{2, 2})t^{5\over2}+{\mathcal{O}}(t^3)~,\ \ \ \ \ \ \end{aligned}$$ described around .
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[^1]: See the recent analysis in [@Aharony:2013hda] for a discussion of subtleties at the level of the gauge group and the line operators.
[^2]: By isolated," we mean a theory that lacks an exactly marginal deformation.
[^3]: The corresponding contribution to the beta function—the current two point function coefficient, $k$—is often exactly computable since it is given by a contact term in the correlator of the superconformal $U(1)_R$ current with two flavor currents.
[^4]: The $T_3$ case is just the $E_6$ SCFT of [@Minahan:1996fg], and the $T_2$ case is eight free half-hypermultiplets. However, the $T_N$ SCFTs with $N\ge4$ are new isolated theories.
[^5]: The $T_3$ case has an enhanced $E_6\supset SU(3)^3$ global symmetry, but the discussion below applies to this theory as well. A similar discussion holds for the $T_2$ theory, which has $Sp(4)\supset SU(2)^3$ global symmetry.
[^6]: By ${\mathcal{N}}=2$ chiral primaries, we mean superconformal primaries that are annihilated by all the anti-chiral Poincaré supercharges of ${\mathcal{N}}=2$ SUSY.
[^7]: By the rules of [@Dolan:2002zh], ${\mathcal{N}}=2$ chiral operators cannot disappear from the spectrum or, by the discussion in [@Papadodimas:2009eu], have their dimensions renormalized as we vary $\tau$, so the $T_N$ ${\mathcal{N}}=2$ chiral ring generators must correspond to some subset of the gauge Casimirs of a Lagrangian theory.
[^8]: We define any ${\mathcal{N}}=2$ SCFT with non-integer scaling dimension chiral primaries to be of AD type.
[^9]: By rank, we mean the complex dimension of the Coulomb branch.
[^10]: This latter theory first appeared in the classification of [@Xie:2012hs] (using the nomenclature of this paper, ${\mathcal{T}}_{3,{3\over2}}$ is a Type III" theory with Young diagrams $[2,2,2],[2,2,2],[2,2,1,1]$).
[^11]: This data gives us the Cartans of the flavor symmetry. By studying various limits of the Hitchin system, we can often identify the full flavor symmetry by matching onto Hitchin sub-systems with known flavor symmetries.
[^12]: This result is somewhat counterintuitive since the rules derived in [@Gaiotto:2008ak] for the case of linear quivers suggest that the presence of a free monopole operator can be detected by looking at each gauge node in the quiver and counting the number of local flavors. If this number reaches a certain threshold, then the theory produces a free monopole after one turns on the corresponding gauge coupling(s) and flows to the IR (the theory is then referred to as ugly" in the nomenclature of [@Gaiotto:2008ak]). However, it is straightforward to check that the quiver in Fig. \[quiver3\] should have no free monopoles by these tests and no accidental superconformal $R$ symmetries. The resolution to this puzzle is that the free monopole depends on the global topology of the quiver—it has non-trivial flux through each gauge node—and so the linear quiver tests of [@Gaiotto:2008ak] do not apply. \[topology\]
[^13]: A similar phenomenon occurs in some theories with only regular punctures.
[^14]: Somewhat intriguingly, as an ${\mathcal{N}}=1$ theory, the flavor symmetry is $SU(3)\times SU(2)\times U(1)$. Note that since the $U(1)$ symmetry comes from the ${\mathcal{N}}=2$ $U(1)_R\times SU(2)_R$ symmetry, it is chiral (although the $SU(3)\times SU(2)$ factors are not by the general analysis of [@Buican:2013ica]). We are not aware of another method in field or string theory to impose a minimality condition and find $SU(3)\times SU(2)\times U(1)$ as a set of symmetries. However, note that these are genuine (global) symmetries and not gauge symmetries as in the Standard Model.
[^15]: Since now we can build an infinite linear quiver of ${\mathcal{T}}_X$ theories where we alternate gauging $SU(2)$ and $SU(3)$ flavor symmetry factors.
[^16]: Moreover, the consistency of the resulting picture we will find below bolsters the claimed duality in Fig. \[quiver1\] and Fig. \[quiver2\] beyond the checks that were performed in [@Buican:2014hfa] at the level of the SW curves and dimensional reductions.
[^17]: See also [@Dobrev:1985qv; @Minwalla:1997ka].
[^18]: Although the case with $R=j_1=j_2=0$ is just the free abelian vector multiplet, and the Schur operators are highest weight gauginos.
[^19]: In the class ${\mathcal{S}}$ construction, the existence of these operators can sometimes be related to the topology of the compactification surface, ${\mathcal{C}}$ [@Gadde:2011uv].
[^20]: In the notation of [@Beem:2013sza], the twisted-translated Schur operators are written as ${\mathcal{O}}(z, \bar z)\equiv u_{i_1}(\bar z)\cdots u_{i_{2N}}(\bar z){\mathcal{O}}^{i_{1}\cdots i_{2N}}$, where $i_k$ are $SU(2)_R$ spin-half indices and $u_{i}\equiv(1, \bar z)$.\[ttfootnote\]
[^21]: Generators are defined to be the operators whose normal-ordered products—along with their derivatives—span the chiral algebra.
[^22]: However, these operators are not independent of the Coulomb branch sector. Indeed, a study of the Schur index of AD theories reveals that the $q\to1$ limit of the index secretly encodes Coulomb branch physics [@Buican:2015hsa] (see also related work in [@Fredrickson:2017yka]). Moreover, the Schur index can be computed from particular sums over BPS states on the Coulomb branch [@Cordova:2015nma].
[^23]: See also the beautiful recent generalization in [@Creutzig:2017qyf].
[^24]: In fact, the baryons map to generators of the chiral algebra related to the theory in Figs. \[quiver1\] and \[quiver2\]. Note that, in accord with the bound in [@Buican:2016arp], this chiral algebra has at least three generators, since there are also multiple generators with $h=1$ as well.
[^25]: One may also derive this result using facts about the moduli spaces of vacua for the theories in our duality. However, our arguments at the level of the chiral algebra provide a stronger consistency check of the duality in [@Buican:2014hfa] as well as of the picture we propose in Fig. \[factorization\].
[^26]: Note that, on the $SU(2)$ side of our duality, we gauge the diagonal $SU(2)\subset SU(2)^2$ to construct the theory in Fig. \[quiver2\].
[^27]: We will also see that, for example, the central charge of the chiral algebra is fixed to be $c_{2d}=-26$ given our generators and AKM levels. Similarly, the AKM levels are fixed given our generators and $c_{2d}=-26$ (here we assume that the 2D chiral algebra is related to a unitary 4D SCFT by the correspondence of [@Beem:2013sza]).
[^28]: Amusingly, this value is the same as the $c$ anomaly for the $bc$ ghost system.
[^29]: This operator must be of type $\hat{\mathcal{B}}_{3\over2}$. The only other Schur multiplets (see ) of the appropriate statistics that can appear at ${\mathcal{O}}(q^{3\over2})$ are ${\mathcal{D}}_{0(0,{1\over2})}\oplus\bar{\mathcal{D}}_{0({1\over2},0)}$. However, these operators have the wrong multiplicity and, on general grounds, should not be present in this theory [@Buican:2014qla] (note that they also satisfy free field equations of motion and so presumably should not appear on such grounds as well).
[^30]: Note that the coefficient of $\epsilon_{ab}\delta^{IJ}/z^3$ is non-vanishing because otherwise $W_a{}^I$ is null. Therefore, this normalization is always possible.
[^31]: In particular, note that $J^{[I}_{SU(3)}J^{J]}_{SU(3)}$ is vanishing and therefore does not appear as an independent term.
[^32]: This last statement is true as long as the 2D chiral algebra is related to a unitary 4D SCFT by the correspondence discussed in [@Beem:2013sza].
[^33]: Therefore, the Schur operator sitting in this $\hat{\mathcal{C}}_{{1\over2}(0,0)}$ multiplet does not map to a generator of the chiral algebra. This situation is quite similar to what happens in, say, the chiral algebra of the $T_3=E_6$ theory, where the stress tensor is not a new generator of $\chi(E_6)$ due to the $SU(2)_R$ twisting of the moment maps and the mixing in of the $\hat{\mathcal{C}}_{0(0,0)}$ multiplet in the corresponding normal-ordered product.
[^34]: For $SU(N)$, we have $\langle\vec{\lambda},\rho\rangle=\sum_{i,j}\lambda_iF^{ij}\rho_j=\sum_{i,j}\lambda_iF^{ij}$ (where we have used that $\rho=(1,\cdots,1)$ in the last step) and $F^{ij}$ is the quadratic form matrix (i.e., the inverse of the Cartan matrix). In the cases of interest, this inner product reduces to $$\braket{\lambda,\rho}_{SU(2)}=\frac{1}{2}\lambda_1~,\ \ \ \braket{\vec{\lambda},\rho}_{SU(3)}=\lambda_1+\lambda_2~.$$
[^35]: Such behavior holds for theories whose $S^3$ partition function (upon performing an $S^1$ reduction) is finite. On the other hand, we are not aware of any ${\mathcal{N}}=2$ SCFT counterexamples to this behavior. Moreover, this scaling has been observed in many classes of strongly interacting ${\mathcal{N}}=2$ SCFTs [@Buican:2015ina; @Buican:2017uka].
[^36]: The fact that there are no imaginary FI parameters turned on is consistent with the 4D $U(1)_R$ symmetry flowing to the Cartan of the 3D $SU(2)_L\subset SO(4)_R$. This statement is also consistent (at least as far as the ${\mathcal{N}}=2$ chiral operators of the ${\mathcal{T}}_X$ theory are concerned) with the $SU(2)$ quantization condition discussed in [@Buican:2015hsa].
[^37]: We thank S. Benvenuti and S. Giacomelli for a discussion on this point.
[^38]: Often one must use highly non-trivial superspace techniques to determine which short multiplets are allowed by ${\mathcal{N}}=2$ superconformal symmetry to appear in the OPE of two short multiplets (e.g., see [@Liendo:2015ofa; @Ramirez:2016lyk]). However, in our case, a more pedestrian approach suffices to show that is allowed. Indeed, we can show that such terms exist in free SCFTs. To that end, consider a free hypermultiplet $$q^i=\begin{pmatrix}
Q \\
\tilde Q^{\dagger}
\end{pmatrix}~, \ \ \ q^{\dagger i}=\tilde q^i=\begin{pmatrix}
\tilde Q \\
-Q^{\dagger}
\end{pmatrix}~,$$ where $i$ is an $SU(2)_R$ spin-half index. Let us construct $\hat{\mathcal{B}}_1$ and $\hat{\mathcal{B}}_{3\over2}$ multiplets of the form $q^{(i}\tilde q^{j)}$ and $q^{(i}q^j\tilde q^{k)}$ respectively (where $(\cdots)$" denotes symmetrization of the enclosed indices). This theory has a $\hat{\mathcal{C}}_{{1\over2}(0,0)}$ multiplet with a primary of the form $\epsilon_{ij}q^i\tilde q^jq^k$. The associated Schur operator is (up to an overall normalization) $$\label{C12schur}
{\mathcal{O}}_{+\dot+}^{111}\sim(\tilde Q\partial_{+\dot+}Q-Q\partial_{+\dot+}\tilde Q)Q~.$$ We then see that is allowed by supersymmetry since a trivial computation in free field theory reveals that (at separated points) $$\label{3pt}
\langle(QQ^{\dagger}-\tilde Q\tilde Q^{\dagger})(x)QQ\tilde Q(y)(\tilde Q^{\dagger}\partial_{+\dot+}Q^{\dagger}-Q^{\dagger}\partial_{+\dot+}\tilde Q^{\dagger})Q^{\dagger})(0)\rangle\ne0~.$$
[^39]: However, it is conceivable that two different 4D ${\mathcal{N}}=2$ SCFTs might have the same chiral algebra (although we are not aware of any such examples). Therefore, we cannot immediately rule out the (perhaps remote) possibility that one might have a 2D chiral algebra that is related both to a well-defined 4D SCFT and a pathological one of the type described here.
[^40]: We are making this statement at the naive level of operator counting. Note that the $Z_{S^1\times S^3}$ partition function (which differs from the index by certain pre-factors) may have additional pathologies.
[^41]: It may be possible to use some of the theories described in [@Argyres:2007tq; @Argyres:2016xmc] to study this question as well.
[^42]: Note that this chiral algebra lies outside the classes of AD chiral algebras considered in the literature before (e.g., see [@Buican:2015ina; @Cordova:2015nma; @Cecotti:2015lab; @Buican:2016arp; @Creutzig:2017qyf; @Song:2017oew]).
[^43]: Note that the authors of [@kac2017remark] also demonstrate more general conjectures [@Xie:2016evu] for theories closely related to the $(A_1, D_4)$ SCFT.
[^44]: Therefore, the authors of [@Gadde:2010te] performed many non-trivial consistency checks of this procedure in the $E_6$ case. Our results in the main text can be viewed as highly non-trivial consistency checks of this procedure for the ${\mathcal{T}}_{3,{3\over2}}$ SCFT.
[^45]: A similar situation occurs in the $E_6$ example of [@Gadde:2010te] if one first takes the Schur limit and then performs the integration.
[^46]: Note that we found substantial evidence in favor of this proposed quiver in the main body of the text.
|
---
abstract: 'Dirac’s relativistic constraint dynamics have been successfully applied to obtain a covariant nonperturbative description of QED and QCD bound states. We use this formalism to describe a microscopic theory of meson-meson scattering as a relativistic generalization of the nonrelativistic quark-interchange model developed by Barnes and Swanson.'
address:
- '$^{1}$University of Tennessee Space Institute, Tullahoma, TN 37388'
- ' $^2$Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831'
- ' $^3$Department of Physics, University of Tennessee, Knoxville, TN 37996'
- 'E-mail: hcrater@utsi.edu & wongc@ornl.gov'
author:
- 'Horace W. Crater$^{1}$ & Cheuk-Yin Wong$^{2,3}$'
title: 'Meson-Meson Scattering in Relativistic Constraint Dynamics'
---
In order to rule out false signals for the onset of the formation of a quark-gluon plasma one needs a reliable relativistic meson-meson scattering formalism. For example the dissociative process $\pi+
J/\psi \rightarrow D+{\bar D}^*$ could, by rapidly taking the $J/\psi
$ out of the picture, mimic the suppression of $J/\psi $ production thought to occur in a quark-gluon plasma at high temperatures [@Mat86]-[@Won04]. A nonrelativistic formalism for such a process in the microscopic quark-interchange picture of meson-meson scattering was developed by Barnes and Swanson [@barnes] and later supplemented by a detailed quark model by Wong, Barnes, and Swanson [@wong]. Here we show how to extend the quark-interchange model to the relativistic domain using constraint dynamics, which has been successfully applied to two-body bound state problems in QED [@crater1], QCD [@crater2], and to two-body nucleon-nucleon scattering [@crater3].
The two-body relativistic wave equations of constraint dynamics can be derived from the Bethe Salpeter Equation. They have their origins however in classical relativistic mechanics where one starts with two mass shell constraints and introduces interactions $\Phi_i$ (here world scalar interactions) $$\begin{aligned}
\fl ~~~~~~\mathcal{H}_{i}^{0} &=&p_{i}^{2}+m_{i}^{2}\rightarrow
p_{i}^{2}+M_{i}^{2}\equiv \mathcal{H}_{i}\equiv p_{i}^{2}+m_{i}^{2}+\Phi
_{i}(x_{1}-x_{2},p_{1},p_{2});~~~~i=1,2,\end{aligned}$$ in such a way that the constraints are compatible $\{H_{1},H_{2}\}\approx 0$. These constraints in turn imply that the interaction potentials satisfy a relativistic third-law condition $$\Phi _{1}=\Phi _{2}=\Phi (x_{12\perp },p_{1},p_{2})\equiv \Phi _{w},$$ and that they depend, as Fig. 1$a$ indicates, on its “perped” component $ x_{12\perp}=(x_1-x_2)_{\perp}$ perpendicular to the total momentum $P=p_1+p_2 $. The relative time is covariantly eliminated since in the CM system $r_{12} {\Large \equiv }\sqrt{x_{12\perp
}^{2}}{\Large =}\sqrt{\mathbf{r_{12}}^{2}} ;~t_{1}-t_{2}=0.$
For two particles with spins, one has two Dirac equations [@crater4] (here given for minimal scalar and vector interactions) instead of two generalized mass shell constraints, $$\mathcal{S}_{i}\psi \equiv \gamma _{5i}(\gamma _{i}\cdot (p_{i}-\tilde{A}
_{i})+m_{i}+\tilde{S}_{i})\psi =0,~~~~i=1,2.$$ Their compatibility ($[\mathcal{S}_{1},\mathcal{S}_{2}]\psi =0$) is guaranteed if supersymmetry is added to the conditions that applied in the two body spinless case. The vector and scalar interactions each depend on underlying invariant functions $A(r)$ and $S(r)$. The compatibility condition leads to an automatic incorporation of correct spin-dependent recoil terms,
$$\fl ~~~~~~\tilde{A}_{i}^{\mu } =\tilde{A}_{i}^{\mu }(A(r),x_{\perp
},p_{1},p_{2},\gamma _{1},\gamma _{2});~\ \tilde{S}=\tilde{S}
_{1}(S(r),A(r),x_{\perp },p_{1},p_{2},\gamma _{1},\gamma _{2}).$$
The two-body Dirac equations can be put into a simple and local 4-component Schrödinger-like form. In the case of lowest order QED, they have an exact solution [@crater5] for singlet positronium that agrees with standard perturbative results. Thus, they are less likely to produce spurious results when applied to QCD.
Using such a formalism, we obtain very good results for the entire meson spectrum from the light pion to the heavy upsilon states [@crater2]. The nonperturbative structures in our equations provide for chiral symmetry in the sense that the pion (although not its excited states or the $\rho $) behave like a Goldstone boson.
The compatibility conditions for four spinless particles, $
\{H_{i},H_{j}\}\approx 0$, unlike their two body counterpart, are not tractable as the set of two-body momenta are not separately conserved and three- and four-body interactions are needed for full compatability, $$\fl \mathcal{H}_{i0}=p_{i}^{2}+m_{i}^{2}\rightarrow \mathcal{H}_{i}\equiv
p_{i}^{2}+m_{i}^{2}+\sum_{j\neq i}\Phi _{ij}(x_{ij\perp })+\sum_{j\neq k\neq
i}\Phi _{ijk}+\Phi _{1234}, i=1,2,3,4$$
Previously, we used Dirac’s constraint dynamics to obtain a Hamiltonian formulation of the relativistic $N$-body problem in a separable two-body basis in which the particles interact pair-wise through scalar and vector interactions by neglecting the many-body interactions [@Won01]. The resultant $N$-body Hamiltonian is relativistically covariant and can be separated in terms of the center-of-mass and the relative motion of any two-body subsystem. The two-body wave functions can be used as basis states to evaluate reaction matrix elements in the general $N$-body problem. In such a formalism, there is however the difficulty of determining the commutation relations involving the creation or annihilation operators of particles that belong to different composites.
Sazdjian [@saz1] has found alternatively that compatibility can be obtained if one demands that the two-body interactions depend on the component of the relative coordinates transverse to the total momentum of the four-body system instead of the two-body system. In this formalism, one introduces the transverse ($T$) component $$\begin{aligned}
x_{ijT}=x_{ij}+ (x_{ij}\cdot P) \,P/\sqrt{
-P^{2}},\end{aligned}$$ where $P=p_{1}+p_{2}+p_{3}+p_{4}$, $x_{ij}=x_{i}-x_{j}$, and one assumes $\Phi _{ij}=\Phi _{ij}(x_{ijT})$. This formalism is suited for bound systems. Below we shows its adaptation to the scattering problem.
We review the nonrelativistic approach, starting with the orthogonality and completeness conditions $$\langle \mathbf{p}_{1}^{\prime }\mathbf{,p}_{2}^{\prime }\mathbf{|p}
_{1}^{\prime \prime }\mathbf{,p}_{2}^{\prime \prime }\mathbf{\rangle }
\mathbf{=}\delta ^{3}\mathbf{(p}_{1}^{\prime }\mathbf{-p}_{1}^{\prime \prime
}\mathbf{)\delta }^{3}\mathbf{(p}_{2}^{\prime }\mathbf{-p}_{2}^{\prime
\prime }),$$ $$~1_{12p}=\int d^{3}p_{1}^{\prime }d^{3}p_{2}^{\prime }|\mathbf{p}
_{1}^{\prime }\mathbf{,p}_{2}^{\prime }\mathbf{\rangle \langle p}
_{1}^{\prime }\mathbf{,p}_{2}^{\prime }|,$$ so that with the wave function defined by $\langle \mathbf{p}_{1}^{\prime }
\mathbf{,p}_{2}^{\prime }|M(\mathbf{P})\rangle =\delta ^{3}(\mathbf{P}
_{12}^{\prime }\mathbf{-P}_{12})\tilde{\psi}_{P}(\mathbf{p}_{12}^{\prime }),$ the scalar products of meson wave functions is simply given by[ ]{} $$\langle M(\mathbf{Q})|M(\mathbf{P})\rangle =\delta ^{3}(\mathbf{P}^{\prime }
\mathbf{-Q})\int d^{3}p_{12}\tilde{\psi}_{Q}^{\ast }(\mathbf{p}_{12}^{\prime
})\tilde{\psi}_{P}(\mathbf{p}_{12}^{\prime }).$$ Using the above orthogonality, completeness conditions, and wave function, we can construct the meson scattering amplitude for the reaction $\mathbf{P}
_{12}\mathbf{+P}_{34}\mathbf{\rightarrow Q}_{14}\mathbf{+Q}_{32}$ shown in Fig. 1($b$) in terms of the momentum matrix elements of the interaction potential, $$\begin{aligned}
&&\langle M(\mathbf{Q}_{14})M(\mathbf{Q}_{23});\mathbf{Q}|V(\mathbf{x}
_{13})|M(\mathbf{P}_{12})M(\mathbf{P}_{34});\mathbf{P}\rangle \nonumber \\
&=&\int
d^{3}q_{1}^{\prime }d^{3}q_{3}^{\prime }d^{3}p_{1}^{\prime
}d^{3}p_{3}^{\prime }
\langle \mathbf{q}_{1}^{\prime }\mathbf{,q}_{2}^{\prime }\mathbf{,q}
_{3}^{\prime }\mathbf{,q}_{4}^{\prime }|V(\mathbf{x}_{13})|\mathbf{p}
_{1}^{\prime }\mathbf{,p}_{2}^{\prime }\mathbf{,p}_{3}^{\prime }\mathbf{,p}
_{4}^{\prime }\rangle \nonumber \\
&&\times \tilde{\psi}_{\mathbf{Q}_{14}}(\mathbf{q}_{14}^{\prime })\tilde{\psi
}_{\mathbf{Q}_{32}}(\mathbf{q}_{32}^{\prime })\tilde{\psi}_{\mathbf{P}_{12}}(
\mathbf{p}_{12}^{\prime })\tilde{\psi}_{\mathbf{P}_{34}}(\mathbf{p}
_{34}^{\prime })\end{aligned}$$ The coordinate completeness condition then leads finally to the results of Barnes, Swanson and Wong in [@barnes] and [@wong], $$\begin{aligned}
\fl
&&\langle M(\mathbf{Q}_{14})M(\mathbf{Q}_{23});\mathbf{Q}|V(\mathbf{x}
_{13})|M(\mathbf{P}_{12})M(\mathbf{P}_{34});\mathbf{P}\rangle \nonumber \\
&\mathbf{=}&\mathbf{\delta }^{3}\mathbf{(Q}_{14}\mathbf{+Q}_{32}\mathbf{-P}
_{12}\mathbf{-P}_{34}\mathbf{)}\int d^{3}p_{1}^{\prime }d^{3}q_{1}^{\prime }
\tilde{V}\mathbf{(p}_{1}^{\prime }\mathbf{-q}_{1}^{\prime }\mathbf{)}\tilde{
\psi}_{\mathbf{Q}_{14}}\mathbf{(q}_{1}^{\prime }\mathbf{-}\frac{m_{1}}{M_{14}
}\mathbf{Q}_{14}\mathbf{)} \nonumber \\
&&\mathbf{\times }\tilde{\psi}_{\mathbf{Q}_{32}}\mathbf{(}\frac{m_{2}}{M_{32}
}\mathbf{Q}_{32}\mathbf{-P}_{12}\mathbf{+p}_{1}^{\prime }\mathbf{)}\tilde{
\psi}_{\mathbf{P}_{12}}\mathbf{(p}_{1}^{\prime }\mathbf{-}\frac{m_{1}}{M_{12}
}\mathbf{P}_{12}\mathbf{)}\tilde{\psi}_{\mathbf{P}_{34}}\mathbf{(q}
_{1}^{\prime }\mathbf{-Q}_{14}\mathbf{+}\frac{m_{4}}{M_{34}}\mathbf{P}_{34}
\mathbf{)}. \label{a}\end{aligned}$$
We now display an analogous formalism in the relativistic case in which, as in the nonrelativistic case, the key ingredients are the scalar product, orthogonality, and completeness conditions.
Scalar products [@saz2] in the relativistic case are complicated by the fact that the relativistic effective potentials are energy dependent (as occurs for example in the one-body Klein-Gordon equation). In a general case this may lead to important contributions but in the Born approximation that we follow here it can be ignored.
We introduce here the completeness and orthogonality conditions, $$\begin{aligned}
1_{12p} &=&\int d^{4}p_{12}^{\prime }d^{4}P_{12}^{\prime }|p_{1}^{\prime
},p_{2}^{\prime }\rangle \delta (P_{12}^{\prime }\cdot \hat{P}+w)\delta
(p^{\prime }\cdot \hat{P})\langle p_{1}^{\prime },p_{2}^{\prime }|,
\nonumber \\
\langle p_{1}^{\prime \prime },p_{2}^{\prime \prime }|p_{1}^{\prime
},p_{2}^{\prime }\rangle &=&\tilde{\delta}^{4}(p_{1}^{\prime \prime
}-p_{1}^{\prime },\tau )\tilde{\delta}^{4}(p_{2}^{\prime \prime
}-p_{1}^{\prime },\tau ), \nonumber \\
\tilde{\delta}^{4}(p_{i}^{\prime \prime }-p_{i}^{\prime },\tau ) &\equiv
&\int dr_{i}\delta ^{4}(p_{i}^{\prime \prime }-p_{i}^{\prime }+r\hat{P}
_{12})\exp (-ir_{i}\tau {\Large )},\end{aligned}$$ where $p^{\prime }=(\varepsilon _{2}p_{1}^{\prime }-\varepsilon
_{1}p_{2}^{\prime })/w$, and $P_{12}^{\prime }=p_{1}^{\prime
}+p_{2}^{\prime }$. Unlike the nonrelativistic case there is here a $\hat{P},$ showing the dependence on the particular two-body state.
Using the wave function ($\hat{n}$ an arbitrary time-like unit vector) $$\langle x_{1}^{\prime },x_{2}^{\prime }|M(\hat{P}_{12})\rangle \equiv \sqrt{
\frac{\left( {\normalsize -}\hat{P}_{12}{\normalsize \cdot }\hat{n}\right) }{
(2\pi )^{3}}}\exp i(P_{12}\cdot X_{12}^{\prime })\psi _{P_{12}}(x_{12\perp
}^{\prime })$$ and completeness conditions, we obtain the scalar product $$\begin{aligned}
& &\langle M(Q_{12}|M(P_{12})\rangle
\nonumber\\
&=&\tilde{\delta}^{4}(P_{12}-Q_{12},\tau
)(-\hat{P}_{12}\cdot \hat{n})\int d^{4}x_{12}^{\prime }\delta
(x_{12}^{\prime }\cdot \hat{P}_{12})\psi _{Q_{12}}^{\ast }(x_{12\perp
}^{\prime })\psi _{P_{12}}(x_{12\perp }^{\prime }).\end{aligned}$$ The derivation of the meson-meson scattering amplitude parallels its nonrelativistic counterpart until one gets to the the momentum space matrix element of the potential $\langle q_{1}^{\prime
},q_{2}^{\prime },q_{3}^{\prime },q_{4}^{\prime }|\Phi (x_{13
T})|p_{1}^{\prime },p_{2}^{\prime },p_{3}^{\prime },p_{4}^{\prime
}\rangle $ that is analogue of the nonrelativistic matrix element $\mathbf{\langle q}_{1}^{\prime } \mathbf{,q}_{2}^{\prime
}\mathbf{,q}_{3}^{\prime }\mathbf{,q}_{4}^{\prime }
\mathbf{|}V\mathbf{(x}_{13}\mathbf{)|p}_{1}^{\prime
}\mathbf{,p}_{2}^{\prime }\mathbf{,p}_{3}^{\prime
}\mathbf{,p}_{4}^{\prime }\mathbf{\rangle }$. The problem is that the bra and ket momentum states in the relativistic expression belong to different mesons.
The initial state orthogonality condition below (final state condition is similar) shows the explicit dependence on meson momenta $P_{12}$ and $P_{34}$, $$\begin{aligned}
& & \!\!\!\!
\langle p_{1}^{\prime \prime },p_{2}^{\prime \prime
},p_{3}^{\prime \prime },p_{4}^{\prime \prime };\hat{P}_{12},\hat{P}
_{34}|p_{1}^{\prime },p_{2}^{\prime },p_{3}^{\prime },p_{4}^{\prime };\hat{P}
_{12},\hat{P}_{34}\rangle \nonumber \\
& &\!\!\!\!
=\int dr_{1}dr_{2}dr_{3}dr_{4}\delta ^{4}(p_{1}^{\prime \prime
}-p_{1}^{\prime }+r_{1}\hat{P}_{12})\delta ^{4}(p_{2}^{\prime \prime
}-p_{2}^{\prime }+r_{2}\hat{P}_{12}) \nonumber \\
& & \!\!\!\!\times
\delta ^{4}(p_{3}^{\prime \prime }-p_{3}^{\prime }+r_{3}\hat{P}
_{34})\delta ^{4}(p_{4}^{\prime \prime }-p_{4}^{\prime }+r_{4}\hat{P}
_{34})\exp (-i(r_{1}+r_{2}+r_{3}+r_{4})\tau ).\end{aligned}$$ Our postulate for different sets of mesons in the bra and ket states is one that uses the total four momentum unit vector $\hat{n}$ of the four quark system in place of the constituent four momenta, $$\begin{aligned}
&&\!\!\!\!\!\!\!\!\!\!\!
\langle q_{1}^{\prime },q_{2}^{\prime },q_{3}^{\prime },q_{4}^{\prime
}|p_{1}^{\prime },p_{2}^{\prime },p_{3}^{\prime },p_{4}^{\prime }\rangle
\equiv \langle q_{1}^{\prime },q_{2}^{\prime },q_{3}^{\prime },q_{4}^{\prime
};\hat{Q}_{14},\hat{Q}_{32}|p_{1}^{\prime },p_{2}^{\prime },p_{3}^{\prime
},p_{4}^{\prime };\hat{P}_{12},\hat{P}_{34}\rangle \nonumber \\
&=&\int dr_{1}dr_{2}dr_{3}dr_{4}\delta ^{4}(q_{1}^{\prime }-p_{1}^{\prime
}+r_{1}\hat{n})\delta ^{4}(q_{3}^{\prime }-p_{3}^{\prime }+r_{3}\hat{n})
\nonumber \\
&&\times \delta ^{4}(q_{2}^{\prime }-p_{2}^{\prime }+r_{2}\hat{n})\delta
^{4}(q_{4}^{\prime }-p_{4}^{\prime }+r_{4}\hat{n})\exp
(-i(r_{1}+r_{2}+r_{3}+r_{4})\tau ).\end{aligned}$$
The physical assumption this reflects is that in the collision process the individual mesons lose their identity and momentarily we have a four body system as described by the Sazdjian formalism. In a like manner the momentum matrix element of the potential $$\begin{aligned}
&&\langle q_{1}^{\prime },q_{2}^{\prime },q_{3}^{\prime },q_{4}^{\prime
}|\Phi (x_{13 T})|p_{1}^{\prime },p_{2}^{\prime },p_{3}^{\prime
},p_{4}^{\prime }\rangle \nonumber \\
&=&\langle q_{1}^{\prime },q_{2}^{\prime },q_{3}^{\prime },q_{4}^{\prime
}|\Phi (x_{13}\cdot (1+\hat{n}\hat{n}))|p_{1}^{\prime },p_{2}^{\prime
},p_{3}^{\prime },p_{4}^{\prime }\rangle \end{aligned}$$ is one that reflects the Sazdjian hypothesis of coordinate dependence only through its component perpendicular to the total momentum. We thus have a hybrid model in which the meson wave functions have the usual two-body perped variable ($\perp$) dependence but the potential has the four-body transversality ($T$) dependence. The diagram in Fig. (1b) details the hybrid nature of the two combined constraint formalisms.
We obtain finally an expression that is a relatively simple three-dimensional but covariant generalization of the nonrelativistic expression given in Eq.(\[a\]) $$\begin{aligned}
&&\langle M(Q_{14})M(Q_{23});Q|\Phi (x_{13 T})
|M(P_{12})M(P_{34});P\rangle \nonumber \\
&{\Large =}&\sqrt{\frac{\hat{n}\cdot \hat{P}_{12}\hat{n}\cdot \hat{Q}_{14}}{
\hat{n}\cdot \hat{P}_{34}\hat{n}\cdot \hat{Q}_{32}}}\tilde{\delta}
^{4}(P-Q,\tau )\int \delta (\hat{P}_{12}\cdot p_{12}^{\prime })\delta
(q_{14}^{\prime }\cdot \hat{Q}_{14}) \nonumber \\
&&\times \tilde{\psi}_{Q_{14}}(q_{14}^{\prime })\tilde{\psi}
_{Q_{32}}(q_{32}^{\prime })\tilde{\psi}_{P_{12}}(p_{12}^{\prime })\tilde{\psi
}_{P_{34}}(p_{34}^{\prime })\Phi \lbrack (p_{13}^{\prime }-q_{13}^{\prime
})_{T}]d^{4}q_{1}^{\prime }d^{4}p_{1}^{\prime }.\end{aligned}$$ Our aim now is to apply the relativistic quark model wave functions we have developed to compute meson-meson cross sections.
**Acknowledgment**
The authors would like to thank T. Barnes and E. S. Swanson for helpful discussions. This research was supported by the NSF under Contract No. NSF-PHY-0244819.
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H. Sazdjian, Annals of Physics, **191**, 82 (1989).
H. Sazdjian, J. Math. Phys., **29**, 1620, (1988)
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---
abstract: 'We report a large tunneling anisotropic magnetoresistance (TAMR) in a thin (Ga,Mn)As epilayer with lateral nanoconstrictions. The observation establishes the generic nature of this effect, which originates from the spin-orbit coupling in a ferromagnet and is not specific to a particular tunnel device design. The lateral geometry allows us to link directly normal anisotropic magnetoresistance (AMR) and TAMR. This indicates that TAMR may be observable in other materials showing a comparable AMR at room temperature, such as transition metal alloys.'
author:
- 'A.D. Giddings'
- 'M.N. Khalid'
- 'J. Wunderlich'
- 'S. Yasin'
- 'R.P. Campion'
- 'K.W. Edmonds'
- 'J. Sinova'
- 'T. Jungwirth'
- 'K. Ito'
- 'K. Y. Wang'
- 'D. Williams'
- 'B.L. Gallagher'
- 'C.T. Foxon'
title: 'Large tunneling anisotropic magnetoresistance in (Ga,Mn)As nanoconstrictions'
---
The family of (III,Mn)V ferromagnetic semiconductors offers unique opportunities for exploring the integration of two frontier areas in information technologies: spintronics and nanoelectronics. A striking example of the synergy of the two fields is the very large magnetoresistance (MR) effect recently observed in lithographically defined (Ga,Mn)As nanostructures in which tunnel barriers are formed in sub-10 nm lateral constrictions [@Ruester:2003_a]. The structure studied in Ref. [@Ruester:2003_a] consists of two such constrictions dividing a lithographically defined (Ga,Mn)As wire into contact leads and a narrower central region. The observed $\sim 2000\%$ spin-valve like signal was interpreted as a type of tunneling MR (TMR) effect arising from the relative alignment of the magnetizations in the regions on either side of the constriction, and in which the barrier shape was spin dependent. This experiment is clearly of great importance as the size of the effect indicates that nanospintronic structures may provide a new route to memory and sensor devices.
Recently, seemingly unrelated strongly anisotropic hysteretic MR of magnitude $\sim 3\%$ was reported [@Gould:2004_cond-mat/0407735] in a (Ga,Mn)As/AlOx/Au tunneling device. The effect is not due to the normal TMR as only a single ferromagnetic layer is present. It is a manifestation of a novel tunneling anisotropic MR (TAMR) effect that had been previously overlooked. The TAMR arises directly from the spin-orbit (SO) coupling induced dependence of the tunneling density of states of the ferromagnetic layer on the orientation of the magnetization with respect to the crystallographic axes [@Gould:2004_cond-mat/0407735].
In this paper we report that TAMR effects can also dominate the MR response of (Ga,Mn)As nanoconstrictions. It establishes that TAMR is a generic phenomenon whose occurrence is not dependent upon a particular device structure. The TAMR signals we observe are of order 100%. We note that very recent low-temperature studies of (Ga,Mn)As/GaAs/(Ga,Mn)As vertical tunnel structures find that the TAMR can be much larger than typical TMR signals in metallic magnetic tunnel junctions and astonishingly can even lead to the realization of a full MR current switch [@Ruester:2004_cond-mat/0408532]. Our lateral microstructures make it possible to study the link between the normal anisotropic MR (AMR) [@Baxter:2002_a; @Jungwirth:2003_b] in devices without constrictions, which also originates from the SO coupled band structure and is present in many metallic ferromagnets [@Jaoul:1977_a], and TAMR measured across a tunnel junction.
![(a) Schematic of an unstructured bar and SEM image of a double constricted nanodevice. (b) Magnetotransport measurements for unconstricted and constricted devices with applied field parallel to current at a temperature of 4.2 K (c) I-V characteristics for the 30 nm constriction device and the 50 nm device (inset)[]{data-label="figure1"}](Figure1a.eps){width=".48\textwidth"}
![(a) Schematic of an unstructured bar and SEM image of a double constricted nanodevice. (b) Magnetotransport measurements for unconstricted and constricted devices with applied field parallel to current at a temperature of 4.2 K (c) I-V characteristics for the 30 nm constriction device and the 50 nm device (inset)[]{data-label="figure1"}](Figure1bc.eps){width=".41\textwidth"}
The lateral geometry of the devices is shown in Fig. \[figure1\](a). All microstructures discussed in this paper were fabricated on a single Ga$_{0.98}$Mn$_{0.02}$As epilayer grown along the \[001\] crystal axis by low-temperature molecular beam epitaxy [@Campion:2003_a]. Despite being only 5 nm thick the layer has a Curie temperature of 40 K and conductivity of 130 $\Omega^{-1}\,$cm$^{-1}$ at room temperature: values which are comparable with those achieved in high quality thicker layers for 2% Mn. Device fabrication was carried out by e-beam lithography using PMMA positive resist and reactive ion etching.
The 3 $\mu$m wide Hall bar, aligned along the \[110\] direction, has pairs of constrictions from 30 nm to 400 nm wide separated by a distance of 9 $\mu$m. For reference AMR experiments, a separate unstructured bar was fabricated in parallel to the stripe without constrictions. Four point I-V curves and resistances were measured for both the unstructured Hall bars and across the constrictions (see Fig.\[figure1\](a)). A standard low frequency lock-in technique was used.
The comparison of MR characteristics of different devices is presented in Fig. \[figure1\](b) for external magnetic field applied parallel to the stripe (parallel to current). The unstructured bar and the 100 nm constriction show MRs typical of the bulk (Ga,Mn)As epilayers [@Baxter:2002_a; @Wang:2002_a]. The overall isotropic (independent of applied field orientation) negative MR in these traces is attributed to the suppression of magnetic disorder at large fields [@Baxter:2002_a]. The hysteretic low field effect is associated with the magnetization reversal and since its magnitude and sense change with applied field orientation it is a manifestation of the AMR. The shape of the 50 nm constriction MR partly deviates from this normal bulk (Ga,Mn)As behavior and a dramatic change is observed in the 30 nm constriction, both in the size and the sign of the low-field effect. The marked increase of the overall resistance of the 30 nm constriction device suggests that the anomalies occur due to the formation of a tunnel junction. This is confirmed by the measured temperature dependence of the I-V curves. Constrictions greater than 100 nm have Ohmic behavior. As shown in Fig. \[figure1\](c), deviations become more pronounced as the constriction size and temperature is reduced. At low temperature and bias, conduction through the 30 nm constrictions is by tunneling. The occurrence of tunneling in such a wide constriction suggests that disorder in the very thin, low Mn density (Ga,Mn)As material leads to local depletion and a tunnel barrier of lateral width considerably smaller than the nominal physical width.
![Low field magnetotransport measurements for the unstructured bar with applied field in three orthogonal orientations at a temperature of 4.2 K.[]{data-label="figure2"}](Figure2.eps){width=".41\textwidth"}
The negative sign of the hysteretic effect in our tunneling device is incompatible with TMR, for which antiparallel alignment on either side of the constriction at intermediate fields would lead to a positive hysteretic effect in the present geometry. Instead, we interpret the data as the TAMR which can show both the normal and inverted spin-valve like signals depending on the applied field orientation [@Gould:2004_cond-mat/0407735; @Ruester:2004_cond-mat/0408532]. This interpretation is also consistent with the geometry of our lateral device in which the central region between constrictions and leads have the same, relatively large, width and are therefore expected to reverse simultaneously.
We now present a detailed analysis of the anisotropic magnetotransport characteristics of our devices. In Fig. \[figure2\] we plot the low-field AMR characteristics of the unstructured bar for magnetic fields applied parallel to the stripe (${\bf B}||{\bf x}$), perpendicular to the stripe in-plane (${\bf B}||{\bf y}$), and perpendicular to the stripe out-of-plane (${\bf B}||{\bf z}$). The three curves in the figure are offset for clarity and also because the absolute comparison between resistances for different field orientations is impossible due to our experimental set up which does not allow us to rotate the sample in the cryostat during the measurement. Each thermal cycling of the sample leads to overall resistance shifts comparable to the size of the anisotropic magnetotransport effects. Apart from this constant offset the MR traces are reproducible which allows us to analyze the magnetotransport anisotropies based on the low-field parts of individual MRs.
We associate the hysteretic steps in the two lower curves in Fig. \[figure2\] with in-plane magnetization reversal precesses. A much stronger MR response is observed in the upper curve with the resistance increasing as the magnetization is rotated from the epilayer plane towards the vertical $z$-direction. In previously studied 50 nm thin Ga$_{0.98}$Mn$_{0.02}$As epilayers there was virtually no difference in the magnitude of the AMR for the two perpendicular-to-current orientations. The large (8%) out-of-plane AMR we observe is therefore attributed to the strong vertical confinement of the carriers in our ultra-thin Ga$_{0.98}$Mn$_{0.02}$As epilayer which breaks the symmetry between states with magnetization ${\bf M}||{\bf y}$ and ${\bf M}||{\bf z}$. Another indication of confinement effects is the presence of hysteresis in the ${\bf B}||{\bf z}$ MR. In thicker Ga$_{0.98}$Mn$_{0.02}$As epilayers the growth direction is magnetically hard with zero remanence due to a small compressive strain induced by the GaAs substrate and due to the shape anisotropy [@Wang:2002_a; @Konig:2003_a]. These effects compete in our epilayer with an increase in the relative population of the heavy hole states due to the confinement, which tends to favor spin polarization along the growth direction [@Lee:2002_a] and therefore changes the magnetic anisotropy energy landscape.
![ Detail of the TAMR measured in the 30 nm constrictions with applied field in the three orthogonal directions. Left inset: comparison of the perpendicular to plane AMR of the unconstricted stripe with the TAMR of the 30 nm constriction. The graph for the bar has been scaled up 300 times. Right inset: the temperature dependence of the TAMR for three different voltages, with ${\bf B}||{\bf x}$.[]{data-label="figure3"}](Figure3.eps){width=".41\textwidth"}
The dominance of the TAMR effect in the tunneling regime is clearly demonstrated in Fig. \[figure3\]. This shows that the measured MR is quite different for the three orthogonal applied field directions. The comparable magnitude but opposite sign of the TAMR for ${\bf B}||{\bf x}$ and ${\bf B}||{\bf y}$, indicates that the low resistance tunneling state is for ${\bf M}||{\bf y}$ and the high resistance state for ${\bf M}||{\bf x}$, and that the in-plane reversal process involves 90$^{\circ}$ switching through the two axes. In the right inset of Fig. \[figure3\] we plot temperature dependence of the TAMR for several excitation voltages. For the lowest temperature in the figure, $T=2$ K, and lowest voltage, $V=2.3$ mV, we obtain a 65% in-plane TAMR and the curves show no signs of saturation at these values. Even larger TAMR signals are recorded when ${\bf M}$ is rotated out of the (Ga,Mn)As epilayer plane. For $V=4.3$ mV and $T=4.2$ K we obtained a 110% TAMR for ${\bf M}||{\bf z}$ which compares to only 31% for ${\bf M}||{\bf x}$ at the same temperature and excitation voltage.
The close correspondence between the AMR results of Fig. \[figure2\] and the TAMR results of Fig. \[figure3\] is evident. The switching events in the in-plane MR traces occur at comparable magnetic fields for the two devices. In both the AMR and the TAMR experiments, the effects at ${\bf B}||{\bf x}$ and ${\bf B}||{\bf y}$ have a similar magnitude and the opposite sign. (Note that the high and low resistance states switch places in the AMR and TAMR traces which is not surprising given the different transport regimes of the two devices.) The most important comparison is between the ${\bf B}||{\bf z}$ AMR and TAMR as we expect the hysteretic magnetization to be unaffected by the constriction as it approaches saturation. The inset of Fig. \[figure3\] shows the expected similarity in general form and field scale of the AMR and TAMR in this geometry. The observation that the magnitude of the TAMR is considerably larger for ${\bf B}||{\bf z}$ than for the in-plane fields as is the case for the AMR, is another manifestation of the direct link between the AMR and TAMR effects. The fact that the observed TAMR effects are all much larger than the AMR effects is a manifestation of the general high sensitivity of tunneling probabilities compared to ohmic transport coefficients.
The AMR in (Ga,Mn)As was successfully modeled [@Jungwirth:2003_b] within the Boltzmann transport theory that accounts for the SO induced anisotropies with respect to the magnetization orientation in the hole group velocities and scattering rates. The TAMR has been analyzed in terms of tunneling density of states anisotropies [@Gould:2004_cond-mat/0407735; @Ruester:2004_cond-mat/0408532] or by calculating the transmission coefficient anisotropies using the Landauer formalism [@Brey:2004_cond-mat/0405473; @Petukhov:2002_a]. Both approaches confirmed the presence of the TAMR effects. The density of states calculations also provided additional qualitative interpretation of the measured field-angle and temperature dependence of TAMR in the vertical tunnel structures [@Gould:2004_cond-mat/0407735; @Ruester:2004_cond-mat/0408532]. The (Ga,Mn)As band structure in these calculations is obtained using the ${\bf
k}\cdot{\bf p}$ envelope function description of the host semiconductor valence bands in the presence of an effective kinetic-exchange field produced by the polarized local Mn moments [@Konig:2003_a].
![Color plot of the calculated tunneling transmission probabilities vs. conserved in-plane momenta at the Fermi energy. The carrier densities are $0.01$ nm$^{-3}$ (a,b), $0.05$ nm$^{-3}$ (c,d), and $0.1$ nm$^{-3}$ (e,f). The barrier height and width are 1 eV and 2 nm, respectively. Red is the highest probability for a given density and blue is zero. The tunneling current is along the x-direction and the magnetization is oriented along the z-direction for the first row and along the x-direction for the second row.[]{data-label="figure4"}](Figure4.eps){width="8cm" height="5.33cm"}
In Fig. \[figure4\] we plot illustrative Landauer transmission probabilities at the Fermi energy as a function of conserved momenta in the $(k_z,k_y)$-plane for two semi-infinite 3D (Ga,Mn)As regions separated by a tunnel barrier. The tunnel current is along the $x$-direction. In both ferromagnetic semiconductor contacts we consider substitutional Mn doping of 2% and a growth direction strain of 0.2%. Details of such calculations can be found in Ref. . The additional component of the strain, which was not considered in previous Landauer transport studies, allows us to model the broken cubic symmetry effects observed in experimental TAMR [@Gould:2004_cond-mat/0407735; @Ruester:2004_cond-mat/0408532]. The bulk 3D hole densities in our (Ga,Mn)As epilayer are of order 1$\times
10^{20}$ cm$^{-3}$ and a gradual depletion of the carriers is expected near the tunnel constriction. Data in panels (a) and (b) correspond to hole density 0.1$\times 10^{20}$ cm$^{-3}$, in (c) and (d) to density 0.5$\times 10^{20}$ cm$^{-3}$, and in (e) and (f) to 1$\times
10^{20}$ cm$^{-3}$.
The diagrams in Fig. \[figure4\] show an intricate dependence of the theoretical TAMR on the position in the $(k_z,k_y)$-plane. When integrated over all states at the Fermi energy, the TAMR ranges between $\sim 50\%$ and $\sim 1\%$ for the studied hole densities 0.1–1$\times 10^{20}$ cm$^{-3}$. In the experimental structure, however, the (Ga,Mn)As is strongly confined in the growth direction which leads to depopulation of high $k_z$ momenta states. The tunnel constriction further reduces the number of $k_y$-states contributing to the signal. Classically, the current is carried only by particles with small momenta in the $x$ and $y$-directions and wave-mechanics adds a condition $k_y=\pm\pi/w$, where $w$ is the effective width of the constriction. Fig. \[figure4\] illustrates that the theoretical TAMR can change significantly depending on the $k_z$ and $k_y$ values selected by the confinements which suggests that both the magnitude and sign of the effect are strongly sensitive to the detailed parameters of the tunnel barrier and of the ferromagnetic semiconductor epilayer.
To conclude, we have established the TAMR as a generic effect in tunnel devices with SO coupled ferromagnetic contacts. The anisotropic transport nature of the large MR signal in our lateral device was demonstrated by directly comparing the TAMR with the AMR effects in the contact leads. Our measurements open a new avenue for integration of spintronics through the TAMR with semiconductor nanoelectronics and motivate studies of the effect in other materials showing the AMR, including high Curie temperature ferromagnetic metals.
The authors thank L. Eaves, C. Gould, A.H. MacDonald, L. Molenkamp, and P. Novák for useful discussions and acknowledge financial support from the Grant Agency of the Czech Republic through grant 202/02/0912, from the EU FENIKS project EC:G5RD-CT-2001-00535, and from the UK EPSRC through grant GR/S81407/01.
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abstract: 'An important approach to studying high-energy cosmic rays is the investigation of the properties of extensive air showers; however, the lateral distribution of particles in simulations of such showers strongly depends on the applied model of low-energy hadronic interactions. It has been shown that many constraints to be applied to these models can be obtained by studying identified-particle spectra from accelerator collisions, in the energy range of the CERN Super Proton Synchrotron. Here we present measurements of the pion production cross-section obtained by the NA61/SHINE experiment at the SPS, in proton-carbon collisions at the beam energy of 31 GeV from the year 2007. Further analyses of identified-particle yields in SHINE, in particular with a pion beam, are in preparation.'
author:
- 'Marek Szuba for the NA61/SHINE Collaboration'
bibliography:
- 'bibl.bib'
title: 'Pion Production Cross-section Measurements in p+C Collisions at the CERN SPS for Understanding Extensive Air Showers'
---
\[sec:introduction\]Introduction
================================
The most common way employed nowadays to study high-energy cosmic rays is to examine the properties of extensive air showers (EAS) they induce in the atmosphere, using several different observables and detection techniques. The latter include measurements of shower size and composition at ground level using surface arrays of scintillation or Cherenkov-light detectors, observation of energy losses of shower particles as they traverse the atmosphere using fluorescence detectors, detection of radio emission from the shower using appropriate antennae, and others. Examples of experiments measuring EAS include the surface arrays KASCADE and KASCADE-Grande as well as the hybrid surface-and-fluorescence Pierre Auger Observatory [@Antoni:2003gd; @Navarra:2004hi; @Abraham:2004dt].
Unfortunately, determination of properties of the initial particle from such observables strongly depends on models, especially those of hadronic interactions occurring during shower development. Many such models exist yet they frequently fail to consistently and accurately reproduce experimental data, for instance underestimating the yield of shower muons at ground level or showing non-smooth transition from low- to high-energy hadronic interactions [@Bluemer:2009zf; @Meurer:2005dt; @Antoni:2001pw; @Abraham:2009ds].
In light of the above, additional data is required in order to appropriately tune models of hadronic interactions in EAS. Such data can be provided by accelerator experiments observing collisions of hadrons such as protons or pions with light ions such as carbon nuclei. In particular, it has been demonstrated that the energy range of the Super Proton Synchrotron (SPS) at CERN makes it highly suitable for reproducing final hadronic interactions[^1] in showers of energies studied by KASCADE, KASCADE-Grande and the Pierre Auger Observatory [@Meurer:2005dt].
This article presents the first results by the NA61/SHINE experiment at the SPS obtained for the purpose of measuring the particle yield in the long-baseline neutrino experiment T2K and tuning EAS models — pion spectra from *p+C* collisions at 31 GeV.
\[sec:shine\]The NA61/SHINE Experiment
======================================
NA61/SHINE is an experiment at the CERN SPS using the upgraded NA49 hadron spectrometer to accomplish a number of physics goals. In addition to providing reference data for cosmic-ray experiments it shall also provide model-tuning information for the neutrino experiment T2K, produce for the first time in the SPS energy range large-statistics proton-proton data sets for high-$p_{T}$ studies and, last but not least, perform a comprehensive energy and system-size scan in search for the QCD critical point. Its large acceptance (around 50 % for $p_{T} \le$ 2.5 GeV/c), high momentum resolution ($\sigma(p)/p^{2} \approx 10^{-4}~(GeV/c)^{-1}$) and tracking efficiency (over 95 %), and excellent particle-identification capabilities ($\sigma(\frac{\mathrm{d}E}{\mathrm{d}x})
/ \frac{\mathrm{d}E}{\mathrm{d}x} \approx 4~\%, \sigma(t_{ToF}) \approx 100~ps$) make it an excellent tool for investigating hadron spectra. Moreover, its kinematic range covers well that of KASCADE, KASCADE-Grande and the Pierre Auger Observatory (see Figure \[fig:na61coverage\]).
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"}
The following are the main features of the NA61 detector as shown in Figure \[fig:na61diagram\] [@Afanasev:1999iu; @Antoniou:2006mh]:
- tracking plus momentum, charge and $\mathrm{d}E/\mathrm{d}x$ measurement with five Time-Projection Chambers;
- three Time-of-Flight walls for additional identification information;
- high-precision downstream Projectile Spectator Detector;
- a number of beam and triggering detectors.
{width="\textwidth"}
For the purpose of cosmic-ray studies, SHINE acquired in the years 2007 and 2009 5.4 million *p+C* events at 31 GeV, 3.6 million *$\pi^{-}$+C* events at 158 GeV and 4.7 million *$\pi^{-}$+C* events at 350 GeV. The analysis of these data sets is currently in progress. The results presented here have been produced from the 0.6 million *p+C*-at-31 GeV events registered during the 2007 pilot run.
\[sec:method\]The Method
========================
The pion spectra presented here have been obtained using three independent analysis techniques:
- The $h^{-}$ method, in which all negative hadrons produced in a collision are assumed to be pions and the contribution of other species is corrected for using simulations. Pros: simple, high statistics. Cons: stronger model dependence, doesn’t work for positive pions;
- $\mathrm{d}E/\mathrm{d}x$ identification of $\pi^{\pm}$. Pros: explicit identification, still high statistics thanks to NA61 design. Cons: only works in the momentum regions where Bethe-Bloch bands do not overlap;
- $\mathrm{d}E/\mathrm{d}x$-plus-ToF identification of $\pi^{\pm}$. Pros: explicit identification over a wide momentum range. Cons: limited acceptance.
For all three approaches, particles passing all the cuts are divided into $(p,~\theta)$ bins, where $p$ is the total momentum and $\theta$ is the polar angle, to account for changing detection and identification properties.
Last but not least SHINE has estimated the systematic uncertainty of results obtained from all three analyses. At present this has been found to be less than on equal to 20 %; we are now working on reducing the systematic uncertainty further.
\[sec:results\]Results
======================
Figures \[fig:xsecMinus\] and \[fig:xsecPlus\] show the production cross-section ($\sigma_{prod} = \sigma_{inel} - \sigma_{qel}$, where $\sigma_{inel}$ and $\sigma_{qel}$ are inelastic and quasi-elastic cross-section, respectively) of negative and positive pions in different $\theta$ bins, compared to air-shower simulations based on the CORSIKA package, using three different interaction models: GHEISHA, FLUKA and UrQMD [@Heck:1998aa; @Fesefeldt:1985yw; @Fasso:2000hd; @Bass:1998ca]. Please see the summary for a discussion of these results.
{width="87.00000%"}
{width="58.00000%"}
\[sec:summary\]Summary and Outlook
==================================
NA61/SHINE has produced its first results relevant to tuning models of hadron production in extensive air showers: preliminary $\pi^{\pm}$ spectra from *p+C* collisions at 31 GeV. The spectra were obtained using three different methods, with very good agreement observed between them. Systematic uncertainties are at the moment no greater than 20 %, with work ongoing to reduce them further. Last but not least, preliminary comparisons with simulations show good agreement with FLUKA for polar angles below 180 mrad and with UrQMD above that threshold.
We are now working on finalising and publishing these results, as well getting ready to analyse the large-statistics *p+C* and *$\pi$+C* runs from 2009.
This work has been supported by the Hungarian Scientific Research Fund (OTKA 68506), the Polish Ministry of Science and Higher Education (N N202 3956 33), the Federal Agency of Education of the Ministry of Education and Science of the Russian Federation (grant RNP 2.2.2.2.1547) and the Russian Foundation for Basic Research (grants 08-02-00018 and 09-02-00664), the Ministry of Education, Culture, Sports, Science and Technology, Japan, Grant-in-Aid for Scientific Research (18071005, 19034011, 19740162), Swiss Nationalfonds Foundation 200020-117913/1 and ETH Research Grant TH-01 07-3.
[^1]: That is, interactions producing hadrons which do not interact further but decay into leptons instead.
|
---
abstract: 'It has been suggested that the long-lived residual radial velocity variations observed in the precision radial velocity measurements of the primary of $\gamma$ Cephei (HR8974, HD222404, HIP116727) are likely due to a Jupiter-like planet around this star [@Hatzes03]. In this paper, the orbital dynamics of this plant is studied and also the possibility of the existence of a hypothetical Earth-like planet in the habitable zone of its central star is discussed. Simulations, which have been carried out for different values of the eccentricity and semimajor axis of the binary, as well as the orbital inclination of its Jupiter-like planet, expand on previous studies of this system and indicate that, for the values of the binary eccentricity smaller than 0.5, and for all values of the orbital inclination of the Jupiter-like planet ranging from 0$^\circ$ to 40$^\circ$, the orbit of this planet is stable. For larger values of the binary eccentricity, the system becomes gradually unstable. Integrations also indicate that, within this range of orbital parameters, a hypothetical Earth-like planet can have a long-term stable orbit only at distances of 0.3 to 0.8 AU from the primary star. The habitable zone of the primary, at a range of approximately 3.1 to 3.8 AU, is, however, unstable.'
author:
- Nader Haghighipour
title: |
Dynamical Stability and Habitability of $\gamma$ Cephei\
Binary-Planetary System
---
Introduction
============
Among the currently known extrasolar planet-hosting stars, approximately 20$\%$ are members of binaries or multistar systems (Table 1)[^1]. With the exception of the pulsar-planetary system PSR B1620-26 [@sigurdsson03; @richer03; @beer04], and possibly the newly discovered system HD202206 [@correia05], the planets in these systems revolve only around one of the stars. These systems are mostly wide with separations between 250 to 6500 AU. At such large distances, the gravitational influence of the farther companion on the dynamics of planets around the other star is un-substantial. Simulations of the orbital stability of a Jupiter-like planet around a star of a binary system have shown that the existence of the farther companion will have considerable effect if the separation of the binary is less then 100 AU [@Jim02]. At the present, there are three planet-hosting binary/multistar systems with such a separation; $\gamma$ Cephei [@Hatzes03], GJ 86 [@Els01], and HD188753 [@Konacki05]. This paper focuses on the dynamics, long-term stability, and the habitability of $\gamma$ Cephei.
To many observers, the discovery of a planet in a dual-star system is of no surprise. There are many observational evidence that indicate the most common outcome of the star formation process is a binary system [@Math94; @White01]. There is also substantial evidence for the existence of potential planet-forming circumstellar disks in multiple star systems [@Math94; @Akeson98; @Rodriguez98; @White99; @Silbert00; @Math00]. To dynamicists, on the other hand, the discovery of a planet in a binary star system marks the beginning of a new era of more challenging questions. Three decades ago, models of planet formation in binary systems did not permit planet growth in binaries with separation comparable to those of $\gamma$ Cephei, GJ 86, and HD188753 [@Hep74; @Hep78]. Results of recent simulations by @Nelson00 also agree with those studies and imply that planets cannot grow via either core-accretion or disk instability mechanisms in binaries with separation of approximately 50 AU. Recent discoveries of planets in dual star systems, however, have cast doubt in the validity of those theories, and have now confronted astrodynamicists with new challenges. Questions such as, how planets are formed in binary star systems, what are the criteria for their long-term stability, can such planet-harboring systems be habitable, and how are habitable planets formed in binary star systems, are no longer within the context of hypothetical systems and have now found real applications.
Theorizing the formation of a planet in a dual star system requires a detailed analysis of planet formation at the presence of a stellar companion. Such a study is beyond the scope of this paper. However, for the sake of completeness, papers by @Boss98 [@Boss04; @Boss05], @nelson03, and @mayer04 on the effect of a stellar companion on the dynamics of a planet-forming nebula, and articles by @Marzari00, @Barbieri02, @Quintana02, and @Lissauer04 on the formation of Jupiter-like and terrestrial planets in and around binary star systems are cited.
Prior to constructing a theory for the formation of planets in binary star systems, it proves useful to develop a detailed understanding of the dynamics of planets in such environments. This is a topic that despite the lack of observational evidence, has always been of particular interest to dynamicists. For instance, in 1977, in search of criteria for the stability of planets in binary star systems, Harrington carried out a study of the orbital stability of Jupiter- and Earth-like planets around the components of a binary. In his simulations, Harrington considered two equal-mass stars and numerically integrated the equations of motion of a planet on a circular orbit in and around a binary with an eccentricity of 0, and 0.5. As expected, Harrington’s results indicated that planets can, indeed, have stable orbits in binary star systems provided they are either sufficiently close to their host stars, or sufficiently far from the entire binary system [@Har77].
Stability of planets in binary star systems has also been studied by @Black81, @Black82, and @Black83. In an effort to establish criteria for the orbital stability of three-body systems, these authors studied the survival time of a planet in the gravitational field of two massive bodies and mapped the parameter-space of the system for planetary orbits around each of the stars, as well as the entire binary system. Their results indicated that, in binary systems where the stellar components have comparable masses, orbital inclination of the planet will not have significant effect on its stability. A result that had also been reported by @Har77. However, when the mass of one of the components of the binary is comparable to the mass of Jupiter, planetary orbits with inclinations higher than 50$^\circ$ tend to be unstable.
To study the orbital stability of planetary bodies, different authors have used different stability criteria. For instance, the notion of stability as introduced by @Har77 implied no secular changes in the semimajor axis and orbital eccentricity of a planet during the time of integration. @Szeb80, and @Szeb81, on the other hand, used the integrals of motion and curves of zero velocity to establish orbital stability. These authors considered a restricted, planar and circular three-body system with a small planet (with negligible mass) orbiting either of the stars, or the entire binary system. Allowing arbitrary perturbations in the equation of motion of the planet, they mapped the parameter-space of the system (i.e., orbital radius, vs. ratio of the mass of the smaller component to the total mass of the binary) and identified regions where the orbit of the planet could be Hill stable. In the present paper, the orbital eccentricity of an object and its distance to other bodies of the system are used to set the criteria for stability. The orbit of an object is considered stable if, for the entire duration of integration, it’s orbital eccentricity stays below unity, it doesn’t collide with other bodies, and it doesn’t leave the gravitational field of the system.
Orbits of planets in binary star systems can be divided into different categories. In an article in 1980, Szebehely distinguished these categories as: Inner orbits, where a planet revolves around the primary star, satellite orbits, where a planet revolves around the secondary star, and outer orbits, where a planet revolves around the entire binary system [@Szeb80]. Another classification has also been reported by @Dvorak83. As noted by this author, the systematic study of the stability of resonant periodic orbits in a restricted, circular, three-body system by @Henon70 implies three types of planetary orbits in a binary system; the S-type where the planet revolves around one of the stars, the P-type where the planet orbits the entire binary systems, and the L-type where the planet has a stable librating orbit around $L_4$ and $L_5$ Lagrangian points. According to this classification, the dual star system of $\gamma$ Cephei is an S-type binary-planetary system.
Extensive studies have been done on the dynamical stability of S-type binary-planetary systems [@Dvorak88a; @Benest88; @Benest89; @Benest93; @Benest96; @Holman97; @Holman99; @Pilat02; @Dvorak03; @Dvorak04; @Pilat04; @Musielak05]. Although in these articles, the stability of S-type systems has been studied for different values of the binary’s mass-ratio and orbital parameters, simulations have been limited to restricted cases such as co-planar orbits, similar-mass binary components, and circular planetary orbits, and/or the durations of simulations have been no more than tens of thousands of the binary’s orbital period. A more detailed analysis of the stability of binary-planetary systems, particularly within the context of habitability, however, requires simulating the orbital dynamics of these systems for longer times. This paper extends previous studies by focusing on (1) the study of the long-term stability of $\gamma$ Cephei binary-planetary system, and (2) identifying regions of its parameter-space where, in the habitable zone of its primary star and at the presence of its Jupiter-like planet, an Earth-like object can have a long-term stable orbit. In this study, simulations are extended to a larger parameter-space where the orbital elements of the binary and the inclination of the planets’ orbits are included, and the stability of the system is studied for ten to hundred million years.
The outline of this paper is as follows. In $\S$ 2, the initial set up for the numerical integration of the system is presented. The results of the numerical simulations are given in $\S$ 3, and in $\S$ 4 the habitability of the system is discussed. Section 5 concludes this study by reviewing the results and comparing them with previous studies.
Initial Set Up
==============
The dual-star system of $\gamma$ Cephei is a spectroscopic binary with a 1.59 solar-mass K1 IV subgiant as its primary [@Fuhr03] and a probable red M dwarf, with a mass-range of 0.34 to 0.78 solar-mass [@Endl05], as its secondary. The semimajor axis and eccentricity of this system are, respectively, $18.5 \pm 1.1$ AU and $0.361 \pm 0.023$, as reported by @Hatzes03, and $20.3 \pm 0.7$ AU and $0.389 \pm 0.017$, as reported by @Griffin02. The primary star of this system has been suggested to be the host to a planet with a minimum mass of 1.7 Jupiter-mass, on an orbit with semimajor axis of $2.13 \pm 0.05$ AU, and eccentricity of $0.12 \pm 0.05$ [@Hatzes03].
The existence of two sets of reported values for the orbital semimajor axis and eccentricity of this binary, and also a mass-range for its secondary component, have caused $\gamma$ Cephei to have a large parameter-space. This parameter-space that consists of the binary’s semimajor axis $(a_b)$ and eccentricity $(e_b)$, the planet’s orbital inclination with respect to the plane of the binary $(i_p)$, and the binary’s mass-ratio $\mu={m_2}/({m_1}+{m_2})$ with $m_1$ and $m_2$ being the masses of the primary and secondary stars, respectively, is the space of the initial conditions for numerical integrations of the system. The first goal of this study is to identify regions of this parameter-space where the Jupiter-like planet of the system can have long-term stable orbits.
An important quantity in determining the stability of a planet in a binary star system is the planet’s semimajor axis. @Dvorak88a, and @Holman99 have obtained an empirical formula for the maximum value of the semimajor axis of a stable planetary orbit ([*critical*]{} semimajor axis, $a_c$) in terms of the binary mass-ratio and orbital eccentricity in a co-planar S-type binary-planetary system. As shown by these authors, $$\begin{aligned}
&{{a_c}/{a_b}}=(0.464\pm 0.006)+
(-0.380 \pm 0.010)\mu + (-0.631\pm0.034) {e_b}\nonumber\\
&\qquad\qquad\qquad
+ (0.586 \pm 0.061) \mu {e_b} + (0.150 \pm 0.041) {e_b^2}
+(-0.198 \pm 0.047)\mu {e_b^2}\>.\end{aligned}$$ Since, due to their uncertainties, the reported values of $a_b$, $e_b$, and $\mu$ for $\gamma$ Cephei binary system vary within certain ranges, the value of the planet’s critical semimajor axis will also vary within a range of values. Figure 1 shows the graph of $a_c$ in terms of the binary eccentricity for different reported values of $a_b$ (including its corresponding uncertainties), and also for all permutations of $\pm$ sings of Eq. (1). The value of $\mu$ in this graph is 0.2 corresponding to a mass of 0.4 solar-mass for the farther companion. Figure 1 shows that, in a co-planar system, for any given value of the binary eccentricity, the orbit of the Jupiter-like planet will be stable as long as the value of its semimajor axis stays below the minimum value of its corresponding range of $a_c$. In fact, the lower boundary of this graph makes the upper limit of the admissible values of the planet’s semimajor axis for which the planet’s orbit will be stable.
Although Fig. 1 presents a general idea of the stability of the $\gamma$ Cephei’s planet for low or zero orbital inclination, in order to portray a more detailed picture of the dynamical state of this body, and for the purpose of extending the analysis to more general cases which include inclined orbits as well, and also, to better understand the dynamical effects of this planet on the long-term stability of a habitable planet in this system, numerical simulations were carried out for different values of the semimajor axis and eccentricity of the binary, as well as the orbital inclination of the planet. The initial value of $e_b$ was chosen from the range of 0.2 to 0.65 in increments of 0.05, and the initial orbital inclination of the planet was chosen from the values of ${i_p}=$0, 2$^\circ$, 5$^\circ$, 10$^\circ$, 20$^\circ$, 40$^\circ$, 60$^\circ$, and 80$^\circ$. Numerical simulations were carried out for different values of ${a_b}$ ranging from 18 to 22 AU.
Stability of the Jupiter-like Planet
====================================
The three-body system of $\gamma$ Cephei binary-planetary system was integrated numerically using a conventional Bulirsch-Stoer integrator. Integrations were carried out for different values of ${a_b}, \, {e_b}$ and $i_p$, as indicated in the previous section. Table 1 shows the initial orbital parameters of the system. For the future purpose of integrating the equation of motion of an Earth-like planet within a large range of distances from the primary star, the timestep of each simulation was set to 1.88 days, equal to 1/20 of the orbital period of a planet at a distance of 0.3 AU from the primary star. This timestep was used in all orbital integrations of the system.
Figure 2 shows the results of integrations for a co-planar system with $\mu=0.2$ and for different values of the binary eccentricity. The initial value of the semimajor axis of the binary is 21.5 AU. As shown here, the system is stable for $0.2 \leq {e_b} \leq 0.45$. This result is consistent with the stability condition depicted by Fig. 1. Integrations also indicate that the system becomes unstable in less than a few thousand years when the initial value of the binary eccentricity exceeds 0.5.
To investigate the effect of planet’s orbital inclination $({i_p})$ on its stability, the system was also integrated for different values of $i_p$. The results indicate that for $0.20 \leq {e_b} \leq 0.45$, the system is stable for all values of planet’s orbital inclination less than 40$^\circ$. Figure 3 shows the semimajor axes and orbital eccentricities of the system for ${e_b}=0.2$ and for $i_p$=5$^\circ$, 10$^\circ$, and 20$^\circ$. For orbital inclinations larger than 40$^\circ$, the system becomes unstable in a few thousand years.
Kozai Resonance
---------------
An exception to the above-mentioned instability condition was observed for the planet’s inclination equal to 60$^\circ$. At this orbital inclination and for the initial eccentricities of 0.25 and 0.17 for the binary and planet, respectively, the system showed the signs of a Kozai resonance. Figure 4 shows the semimajor axes and eccentricities of the binary and planet in this case.
As demonstrated by @Kozai62, the exchange of angular momentum between the small body of the system (here, the Jupiter-like planet) and the binary, can cause the orbital eccentricity of the planet to reach high values at large inclinations (Fig. 4). Averaging the equations of motion of the system over mean anomalies, one can show that in this case, the averaged system is integrable when the ratio of distances are large [the Hill’s approximation, @Kozai62]. The Lagrange equations of motion in this case, indicate that, to the first order of planet’s eccentricity, the longitude of the periastron of the planet, $\omega_p$, librates around a fix value. Figure 4 also shows $\omega_p$ for the duration of integration. As shown here, this quantity librates around 90$^\circ$.
In a Kozai resonance, the longitude of periastron and the orbital eccentricity of the small body are related to its orbital inclination as [@Innanen97] $${\sin^2}{\omega_p}=\,0.4\,{\csc^2}{i_p},
\eqno (2)$$ and $${(e_p^2)_{\rm max}}={1\over 6}\,\Bigl[1-5\cos (2{i_p})\Bigr].
\eqno(3)$$ From Eq. (2), one can show that the Kozai resonance may occur if the orbital inclination of the small body is larger than 39.23$^\circ$. As mentioned above, in this study, the Kozai resonance occurred for ${i_p}={60^\circ}$. For the minimum value of ${i_p}$, the maximum value of the planet’s orbital eccentricity, as given by Eq. (3), is equal to 0.764. Figure (4) also shows that $e_p$ stays below this limiting value at all times.
As shown by @Kozai62 and @Innanen97, in a Kozai resonance, the disturbing function of the system, averaged over the mean anomalies, is independent of the longitudes of ascending nodes of the small object and perturbing body. As a result, the quantity ${\sqrt{{a}(1-{e^2})}}\,\cos {i}$ (shown as the “Reduced Delaunay Momentum” in Fig. 4) becomes a constant of motion. Figure 4 shows this quantity for the Jupiter-like planet of the $\gamma$ Cephei system. Since the eccentricity and inclination of the planet vary with time, the fact that the quantity above is a constant of motion implies that the time-variations of these two quantities have the same periods and they vary in such a way that when $i_p$ reaches its maximum, $e_p$ reaches its minimum and vice versa. Figure 5 shows this clearly.
Habitability
============
To study the habitability of $\gamma$ Cephei, one has to investigate the long-term stability of a habitable planet in the habitable zone of this system. A habitable zone is commonly referred to a region around a star where an Earth-like planet can maintain liquid water on its surface. The capability of maintaining liquid water depends on several factors such as the amount of radiation that such a planet receives from the star. This radiation itself depends on the star’s luminosity, and vary with its radius $R$ and surface temperature $T$ as, $$F(r)={1\over {4\pi}}L(R,T){r^{-2}}=\sigma{T^4}{R^2}{r^{-2}}.
\eqno (4)$$ In this equation, $L(R,T)$ is the luminosity of the star, $\sigma$ is the Boltzmann constant, and $F(r)$ is the apparent brightness of the star denoting the amount of stellar radiation that, in a unit of time, is distributed over the unit area of a sphere with radius $r$.
Equation (4) indicates that the width of a habitable zone and the locations of its inner and outer boundaries vary with the physical properties of the star. The inner edge of a habitable zone is defined as the largest distance from a star where, due to photodissociation and runaway greenhouse effect, the planet loses all its water. The outer edge of a habitable zone, on the other hand, corresponds to the shortest distance from a star where water can no longer exist in liquid phase and begins to freeze. In other words, the outer edge of a habitable zone corresponds to the largest distance from a star where an Earth-like planet with a carbon-dioxide atmosphere can still, in average, maintain a temperature of 273 K on its surface [@Kasting93]. As noted by @Jones05, such a definition for the boundaries of a habitable zone is somewhat conservative, and the outer edge of this zone may, in fact, be farther away [@Forget97; @Williams97; @Mischna00].
Since the above-mentioned definition of a habitable zone is based on the notion of habitability and life on Earth, one can use Eq. (4) to determine habitable regions around other stars by comparing their luminosities with that of the Sun. That is, a habitable zone can be defined as a region around a star where an Earth-like planet can receive the same amount of radiation as Earth receives from the Sun, so that it can develop and maintain similar habitable conditions as those on Earth. From Eq. (4), the statement above implies $${{F(r)}}\,=\,
{\Bigl({T\over {T_{\rm Sun}}}\Bigr)^4}\,
{\Bigl({R\over {R_{\rm Sun}}}\Bigr)^2}\,
{\Bigl({r\over{r_{\rm Earth}}}\Bigr)^{-2}}\,
{{F_{\rm Sun}}}({r_{\rm Earth}})
\eqno (5)$$ where now $F(r)$ represents the apparent brightness of a star with a luminosity of $L(R,T)$ as observed from an Earth-like planet at a distance $r$ from the star, $r_{\rm Earth}$ is the distance of Earth from the Sun, and ${F_{\rm Sun}}({r_{\rm Earth}})$ represents the brightness of the Sun at the location of Earth.
The primary star of $\gamma$ Cephei binary system has a temperature of 4900 K and a radius of 4.66 solar-radii [@Hatzes03]. Considering 5900 K as the surface temperature of the Sun, Eq. (5) indicates that in order for Earth to receive the same amount of radiation as it receives from the Sun, it has to be at a distance of $r\sim$ 3.2 AU from the $\gamma$ Cephei’s primary star. The habitable zone of the Sun, on the other hand, is considered to be between 0.95 to 1.37 AU [@Kasting93]. This range corresponds to a range of apparent solar brightness between 1.1 and 0.53, and implies that a similar region around the primary of $\gamma$ Cephei extends from 3.13 to 3.76 AU from this star.
The width and distance of the habitable zone of $\gamma$ Cephei have also been reported in papers by @Dvorak03, and @Jones05. In a study of the stability of planets in the $\gamma$ Cephei binary system, @Dvorak03 have considered a range of 1 to 2.2 AU as the habitable zone of the system’s primary star. From Eq. (5), at these distances, the apparent brightness of this star varies between 10.3 to 2.1 times the brightness of the Sun at 1 AU. With such brightness, it is unlikely that an Earth-like planet around the primary of $\gamma$ Cephei, at the range of distances reported by these authors, can maintain similar habitable conditions as those on Earth. More recently, @Jones05 studied the habitability of extrasolar planetary systems and tabulated the habitable regions of a large number of planet-hosting stars. According to these authors, the habitable zone of the primary of $\gamma$ Cephei extends from 2.07 to 4.17 AU from this star [@Jones05]. These authors consider this range to be conservative and mention that the outer boundary of the actual habitable zone may be somewhat larger. In the present study, however, the habitable zone of the primary of $\gamma$ Cephei is considered to be narrower and between 3.1 to 3.8 AU from this star.
Numerical integrations were carried out to study the stability of an Earth-like planet in $\gamma$ Cephei system. Although the habitable zone of the system was considered to be from 3.1 to 3.8 AU, stability of an Earth-like planet was studied at different locations, ranging from 0.3 to 4.0 AU from the primary star. Table 3 shows the ranges of the orbital parameters of this object, as well as those of the binary and its Jupiter-like planet. As shown in this table, numerical simulations were also carried out for different values of the orbital inclinations of Earth- and Jupiter-like planets. For each arrangement of these bodies, the equations of motion of a full four-body system consisting of the binary, its Jupiter-like planet, and an Earth-like object were numerically integrated. Figure 6 shows the survival times of Earth-like planets in terms of their initial positions for a co-planar arrangement with ${e_b}=0.3$. As shown here, an Earth-like planet will not be able to sustain a stable orbit in the habitable zone of the primary star. Results of numerical simulations indicate that the orbit of an Earth-like planet is stable when $0.3 \leq {a_E} \leq 0.8$ AU, ${0^\circ} \leq {i_E}=
{i_p} \leq {10^\circ}$, and ${e_b}\leq 0.4$, where $a_E$ and $i_E$ represent the semimajor axis and orbital inclination of the Earth-like planet, respectively. Figures 7 shows the time-variations of the semimajor axes, eccentricities, and orbital inclinations of one of such four-body systems for ${i_p}=5^\circ$, and for 100 million years.
Conclusion
==========
The results of a study of the orbital stability of the binary-planetary system of $\gamma$ Cephei have been presented. Numerical integrations of the full three-body system of the binary and its Jupiter-like planet indicate that the orbit of this planet is stable for the values of the binary eccentricity less than 0.5 \[see Fig. 5 of @Musielak05 for simulations of the system for $e_b$ close to zero\] and for the planet’s orbital inclination up to 40$^\circ$. For larger values of the inclination, the system becomes unstable except at 60$^\circ$ where the planet may be in a Kozai resonance.
The focus of the first part of this study was on the effects of the variations of the binary eccentricity and planet’s orbital inclination on the stability of the system. For that reason, numerical simulations were carried out for only one value of the semimajor axis of the Jupiter-like planet. However, simulations by @Pilat02, @Dvorak03, and @Pilat04 have indicated that the region of the stability of this planet extends to larger distances beyond its current location. @Dvorak03 studied the stability of this planet by numerically integrating the equation of motion of a massless object in a restricted elliptical three-body system and showed that its stable region extends to 4 AU from the primary star. @Pilat04, on the other hand, have shown that, allowing a range of 0.1 to 0.9 for the binary’s mass-ratio $(\mu)$ will limit this stable reigon to only 3.6 AU from the primary star.
In addition to the dynamics of its Jupiter-like planet, the binary-planetary system of $\gamma$ Cephei was also studied as a possible system for harboring habitable planets. The habitability of this system was studied by including a hypothetical Earth-like planet at different locations from its primary star and integrating the equations of motion of a complete elliptical four-body system. Simulations indicated that an Earth-like planet, initially on a circular orbit, would have an unstable orbit for the values of its semimajor axis larger than 0.8 AU. The habitable zone of the system, between 3.1 to 3.8 AU, is within this unstable region.
A report of the instability of an Earth-like planet for ${a_E}>0.8$ AU can also be found in the work of @Dvorak04. In a study of the stability of a fictitious massless planet in the vicinity of the Jupiter-like planet of $\gamma$ Cephei, these authors extended two previous studies by @Dvorak03 and @Pilat04, and showed that such an object, when initially on a circular orbit, cannot maintain its stability between 1.7 to 2.6 AU from the primary star. Considering an Earth-like planet as a massless object, and simulating its dynamics in the $\gamma$ Cephei system, @Dvorak03 and @Pilat04 had already shown that, when the inclination of the Jupiter-like planet of the system varies from 0$^\circ$ to 40$^\circ$, such an Earth-like object could have a stable orbit only in a region between 0.6 to 0.8 AU from the primary star. Their results had also indicated an island of stability at 1 AU from this star.
It is important to emphasize that, despite of some similarities between the results presented in this paper and those of @Dvorak03 and @Pilat04, the latter two studies do not fully represent the dynamical state of the $\gamma$ Caphei system. Studies of the stability of this system, as presented by these authors, are limited to elliptic restricted three-body cases. Integrations of the equation of motion of an Earth-like planet, in those articles, were also carried out within the context of the motion of a massless particle in an elliptic restricted system. These restrictions limit the generalization of the results obtained by these authors, particularly, when studying the habitability of the system. In the present paper, however, these limitations were overcome by carrying out simulations for a complete elliptical four-body system. The results of these simulations indicated that the range of the orbital stability of an Earth-like planet does not extend beyond 0.8 AU from the primary star, and the island of stability at 1 AU, as reported by @Dvorak03 and @Pilat04 is indeed unstable. Simulations also indicated that, although the region of the stability of an Earth-like planet as reported by these authors (0.6 to 0.8 AU) is within the region of stability indicated in this paper (0.3 to 0.8 AU), unlike what they report, the orbital inclination of the Jupiter-like planet of the system cannot exceed 10$^\circ$.
In closing, it is necessary to mention that the study of the habitability of the primary star of $\gamma$ Cephei, as presented in this paper, has one limitation. The evolution of this star during the time of integration has not been taken into consideration. This star is a 3 billion years old K1 IV subgiant that is still in the process of approaching the giants’ region of the HR diagram. While this star expands, its luminosity increases, and as a result, its habitable zone will move toward larger distances. Although the results of numerical simulations indicate that an Earth-like planet will have an unstable orbit at distances beyond the current habitable zone of the primary of $\gamma$ Cephei, it is necessary to extend these simulations to even larger distances, particularly when the stability of Earth-like planets is studied for several hundred million years. Additionally, it is important to investigate how different values of the mass of the farther companion would affect the dynamical stability and habitability of the system. The results of such simulations are currently in preparation for publication.
I am thankful to the Department of Terrestrial Magnetism at the Carnegie Institution of Washington for access to their computational facilities where the numerical simulations of this work were performed. This work has been supported by the NASA Astrobiology Institute under Cooperative Agreement NNA04CC08A at the Institute for Astronomy at the University of Hawaii-Manoa.
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[llll]{} HD142 (GJ 9002) & HD3651 & HD9826 ($\upsilon$ And) & HD13445 (GJ 86)\
HD19994 & HD22049 ($\epsilon$ Eri) & HD27442 & HD40979\
HD41004 & HD75732 (55 Cnc) & HD80606 & HD89744\
HD114762 & HD117176 (70 Vir) & HD120136 ($\tau$ Boo) & HD121504\
HD137759 & HD143761 ($\rho$ Crb) & HD178911 & HD186472 (16 Cyg)\
HD190360 (GJ 777A) & HD192263 & HD195019 & HD213240\
HD217107 & HD219449 & HD219542 & HD222404 ($\gamma$ Cephei)\
HD178911 & HD202206 & HD188753\
PSR B1257-20 & PSR B1620-26 &
[lcc]{} $a$(AU) & 2.13 & 18-22\
$e$ & 0.17 & 0.20-0.65\
$i$(deg) & 0-80 & 0\
$\Omega$(deg) & 0 & 0\
$\omega$(deg) & 74 & 160\
$M$(deg) & 104 & 353\
[lccc]{} $a$(AU) & 0.3-4.0 & 2.13 & 18,19,20\
$e$ & 0 & 0.17 & 0.2,0.3,0.4\
$i$(deg) & 0,2,5,10 & 2,5,10,20,40 & 0\
$\Omega$(deg) &0 & 0 & 0\
$\omega$(deg) &0 & 74 & 160\
$M$(deg) &0 & 104 & 353\
[^1]: See http://www.obspm.fr/planets for a complete and up-to-date list of extrasolar planets with their corresponding references.
|
---
abstract: 'Maximum likelihood estimators are used extensively to estimate unknown parameters of stochastic trait evolution models on phylogenetic trees. Although the MLE has been proven to converge to the true value in the independent-sample case, we cannot appeal to this result because trait values of different species are correlated due to shared evolutionary history. In this paper, we consider a $2$-state symmetric model for a single binary trait and investigate the theoretical properties of the MLE for the transition rate in the large-tree limit. Here, the large-tree limit is a theoretical scenario where the number of taxa increases to infinity and we can observe the trait values for all species. Specifically, we prove that the MLE converges to the true value under some regularity conditions. These conditions ensure that the tree shape is not too irregular, and holds for many practical scenarios such as trees with bounded edges, trees generated from the Yule (pure birth) process, and trees generated from the coalescent point process. Our result also provides an upper bound for the distance between the MLE and the true value.'
author:
- |
Lam Si Tung Ho[^1]\
Department of Mathematics and Statistics\
Dalhousie University, Halifax, Nova Scotia, Canada
- |
Vu Dinh$^{*}$\
Department of Mathematical Sciences\
University of Delaware
- |
Frederick A. Matsen IV\
Program in Computational Biology\
Fred Hutchinson Cancer Research Center
- |
Marc A. Suchard\
Departments of Biomathematics, Biostatistics and Human Genetics\
University of California, Los Angeles
bibliography:
- 'ms.bib'
title: 'On the convergence of the maximum likelihood estimator for the transition rate under a $2$-state symmetric model'
---
Introduction
============
The *maximum likelihood estimator* (MLE) is frequently used to estimate unknown parameters in trait evolution models on phylogenetic trees. To safely use this machinery, it is important to know that the MLE is *consistent*: that is, the estimate converges to the true value as we observe trait values for more species. However, traditional statistical theory only guarantees the consistency of the MLE when observations are independent and identically distributed. In contrast, trait values of biological species are not independent because species are related to each other according to a phylogenetic tree. Therefore, traditional consistency results are not directly applicable to trait evolution studies. For this paper we consider the scenario where only a single trait is observed and the number of species increases to infinity. This is different from the setting where many traits/characters are observed for the same set of species; this alternate setting typically assumes independence between traits.
Binary traits studied by comparative biologists come in various types, including morphological traits (whether a fruit fly has curly wings), behavioral traits (whether an antelope hides from predators), and geographical traits (whether a species is aquatic). Here we study the consistency property of the MLE for estimating the transition rate of a binary trait evolved along a phylogenetic tree according to a $2$-state symmetric Markov process. Under this $2$-state symmetric model, there is an unique rate of switching back and forth between the two states of a phenotype, which is the parameter of interest. We show that under two mild conditions, the MLE of this transition rate is consistent, that is, the estimate converges to the correct value as we observe the trait values of more species. These conditions ensure that edge lengths of the tree are not too small and the pairwise distances between leaves are neither too small nor too large. We verify that these conditions holds for many practical cases including trees with bounded edges, trees generated from the Yule process [@yule1925mathematical], and trees generated from the coalescent point process [@lambert2013birth].
The consistency of the MLE does not always hold under trait evolution models. Indeed, for estimating the ancestral state of the $2$-state symmetric model, @li2008more point out that the MLE can be inconsistent. As a consequence, the estimate of the ancestral state may not be close to the true value no matter how many species have been sampled. Several efforts have also been made to investigate the consistency of the MLE under evolution models of a continuous trait. @ane2008analysis points out that unlike traditional linear regression, the MLE of the coefficients of phylogenetic linear regression under the Brownian motion model can be inconsistent. Additionally, @sagitov2012interspecies show that the sample mean is an inconsistent estimator for the ancestral state under this model when the tree is generated from the Yule process. For the Ornstein-Uhlenbeck model, @ho2013asymptotic show that if the height of the phylogenetic tree is bounded as more observations are collected, then the MLE of the selective optimum is not consistent. Moreover, they discovered that in this scenario, no consistent estimator for the selective optimum exists. Recently, @ane2016phase provide a necessary and sufficient condition for consistency of the MLE under the Ornstein-Uhlenbeck model. Although the problem of reconstructing the ancestral state under the $2$-state symmetric model has been studied extensively [@tuffley1997links; @mooers1999reconstructing; @li2008more; @mossel2014majority], it remains unknown whether the transition rate can be estimated consistently.
In this paper, we show that the MLE is a consistent estimator for the transition rate of the $2$-state symmetric model under simple conditions. We start with introducing the $2$-state symmetric model for binary traits and derive several statistical properties of this model in Section \[sec:2state\]. In Section \[sec:consistency\], we state two necessary conditions for the consistency of the MLE of the transition rate and provide a detailed proof for this result. Section \[sec:app\] verifies these conditions for several practical scenarios and illustrates our result through a simulation.
Properties of the $2$-state symmetric model {#sec:2state}
===========================================
Let $\mu$ be the transition rate of a $2$-state symmetric Markov process. Then, the transition probability matrix has an analytical form: $${\mathbf}{P}_\mu(t) = \begin{pmatrix} \frac{1}{2} + \frac{1}{2}e^{-2 \mu t} & \frac{1}{2} - \frac{1}{2}e^{-2 \mu t} \\ \frac{1}{2} - \frac{1}{2}e^{- 2 \mu t} & \frac{1}{2} + \frac{1}{2}e^{-2 \mu t} \end{pmatrix}.
\label{eqn:trans}$$ Hence, the probability that the process switches state after $t$ unit of time is $\frac{1}{2} - \frac{1}{2}e^{- 2 \mu t}$.
In this paper, the term phylogenetic tree (or phylogeny) refers to a bifurcating rooted tree with leaves labeled by a set of taxon (species) names. We can reroot a phylogenetic tree by moving the root to another location along the tree (see Figure \[fig:reroot\]). The evolution of a binary trait along a tree is modeled using the $2$-state symmetric Markov process as follows. At each node in the phylogeny, the children inherit the trait value of their parent and the trait of each child evolves independently of one another.
![Example of rerooting a 4-taxa tree.[]{data-label="fig:reroot"}](reroot){width="80.00000%"}
Let ${\mathbb{T}}$ be a phylogenetic tree with $n$ leaves, and ${\mathbf}{Y}$ be the trait values at the leaves of ${\mathbb{T}}$. We assume that the ancestral state at the root $\rho$ of ${\mathbb{T}}$ follows a stationary distribution, which is a Bernoulli distribution with success probability $1/2$. Let $E$ be the set of all edges of ${\mathbb{T}}$. The joint probability distribution of ${\mathbf}{Y}$ is $$P_{{\mathbb{T}},\mu}({\mathbf}{Y}) := {{\mathrm{I}\!\mathrm{P}}}({\mathbf}{Y}~|~{\mathbb{T}},\mu) = \frac{1}{2} \sum_{y}{\left ( \prod_{(u,v)\in E}{[{\mathbf}{P}_\mu( d_{uv})}]_{y_u y_v} \right )}$$ where $y$ ranges over all extensions of ${\mathbf}{Y}$ to the internal nodes of the tree, $y_u$ denotes the assigned state of node $u$ by $y$, $d_{uv}$ is the edge length of $(u,v)$, and $[{\mathbf}{P}_\mu(t)]_{kl}$ is the element at $k$-th row and $l$-th column of matrix ${\mathbf}{P}_\mu(t)$. We define the log-likelihood function as $\ell_{{\mathbb{T}},\mu}({\mathbf}{Y}) = \log P_{{\mathbb{T}},\mu}({\mathbf}{Y})$. Throughout this paper, we denote the true transition rate with $\mu^*$ and assume that $\mu^* \in [{\underline}\mu, {\overline}\mu]$ where ${\underline}\mu, {\overline}\mu$ are two known positive numbers. Define $$R_{{\mathbb{T}},\mu}({\mathbf}{Y}) = \ell_{{\mathbb{T}},\mu^*}({\mathbf}{Y}) - \ell_{{\mathbb{T}},\mu}({\mathbf}{Y}).$$ Then the Kullback-Leibler divergence from $P_{{\mathbb{T}},\mu}$ to $P_{{\mathbb{T}},\mu^*}$ is $$\text{KL}(P_{{\mathbb{T}},\mu^*} \| P_{{\mathbb{T}},\mu}) = {\mathbb{E}}[\ell_{{\mathbb{T}},\mu^*}({\mathbf}{Y}) - \ell_{{\mathbb{T}},\mu}({\mathbf}{Y})] = {\mathbb{E}}[R_{{\mathbb{T}},\mu}({\mathbf}{Y})]$$ where ${\mathbb{E}}$ is the expectation with respect to $P_{{\mathbb{T}},\mu^*}$.
Let ${\mathbb{T}}'$ be the phylogenetic tree obtained by rerooting a tree ${\mathbb{T}}$, then $$\ell_{{\mathbb{T}}',\mu}({\mathbf}{Y}) = \ell_{{\mathbb{T}},\mu}({\mathbf}{Y})$$ for any trait values ${\mathbf}{Y}$ at the leaves. \[lem:reroot\]
This lemma, sometimes called the “pulley principle,” is a direct consequence of the fact that the ancestral state follows a stationary distribution [@felsenstein1981evolutionary]. We also have the following Lemmas:
Let ${\mathbb{T}}$ be a rooted tree with root $\rho$, and ${\mathbf}{Y}$ be the trait values at the leaves of ${\mathbb{T}}$ generated under the $2$-state symmetric model. Let $h$ be a function such that $h({\mathbf}{Y}) = h({\mathbf}{1} - {\mathbf}{Y})$, where ${\mathbf}{1}$ denotes the all-ones vector. Then, $h({\mathbf}{Y})$ and the trait value at $\rho$ are independent. \[lem:indep\]
The following two lemmas concern the regularity of the log-likelihood function. In particular, Lemma \[lem:subtrees\] shows that the log-likelihood function of a binary tree can be bounded by the sum of the log-likelihood functions of its two subtrees.
Let $\rho_1$ and $\rho_2$ be two direct descendants of $\rho$, and ${\mathbb{T}}_1$ and ${\mathbb{T}}_2$ the two subtrees descending from them. Let ${\mathbf}{Y}_1$, ${\mathbf}{Y}_2$ be the observations at the leaves of ${\mathbb{T}}_1$ and ${\mathbb{T}}_2$. We have $$|\ell_{{\mathbb{T}},\mu}({\mathbf}{Y}) - \ell_{{\mathbb{T}}_1,\mu}({\mathbf}{Y}_1) - \ell_{{\mathbb{T}}_2,\mu}({\mathbf}{Y}_2)| \leq \max \left\{\log 2, \log \frac{1}{1 - e^{-2 \mu d}} \right\}.$$ where $d$ is the tree distance between $\rho_1$ and $\rho_2$. \[lem:subtrees\]
There exists $C>0$ such that $$\left|\frac{1}{n}\ell_{{\mathbb{T}}, \mu_1}({\mathbf}{Y}) - \frac{1}{n}\ell_{{\mathbb{T}}, \mu_2}({\mathbf}{Y}) \right| \le C |\mu_1 - \mu_2| \qquad \forall \mu_1, \mu_2 \in [\underline{\mu}, \overline{\mu}], \forall {\mathbb{T}}$$ where $n$ denotes the number of leaves of ${\mathbb{T}}$. \[lem:Lipschitz\]
The proofs of Lemmas \[lem:indep\], \[lem:subtrees\], and \[lem:Lipschitz\] are provided in the Appendix \[sec:proof\]. Henceforth we will also use $\rho$ to denote the trait value at the root by an abuse of notation.
Convergence of the MLE of the transition rate {#sec:consistency}
=============================================
The MLE of $\mu$ is defined as follows: $$\hat \mu = \operatorname*{argmax}_{\mu \in [{\underline}\mu, {\overline}\mu]} \ell_{{\mathbb{T}},\mu}({\mathbf}{Y}).$$ In this section, we will state our main result regarding the consistency of $\hat \mu$. We will need to make two assumptions to ensure that the shape of ${\mathbb{T}}$ is not too irregular. Let us define $$f(x) = \max \left\{\log 2,\log \frac{1}{1-e^{-2x}}\right\} \text{ and } S_{{\mathbb{T}}, \mu}= \sum_{e \in E({\mathbb{T}})}{f(\mu t_e)^2}.$$ where $t_e$ denotes the edge length of edge $e$.
$S_{{\mathbb{T}}, \mu} = {\mathcal}{O}(n^{\gamma}$) for some $1\leq \gamma < 2$. That is, there exists a universal constant $c < \infty$ such that $S_{{\mathbb{T}}, \mu}< c n^{\gamma} $. \[assump:length\]
There exist $\Omega(n)$ pairs of leaves such that the paths connecting each pair are pairwise disconnected and their lengths are bounded in some fixed range $[{\underline}d, {\overline}d]$. Here, $\Omega(n)$ denotes a quantity that is greater than $c n$ for a positive constant $c$ and all $n$. \[assump:contrast\]
Assumption \[assump:length\] makes sure that the edge lengths of ${\mathbb{T}}$ are not too small. It is worth noticing that $1/(1 - e^{-2x}) \leq 1/x$ when $x$ is small enough. Therefore, this assumption holds when the smallest edge length is $\Omega(e^{-n (\gamma - 1)/2})$. On the other hand, Assumption \[assump:contrast\] guarantees that the pairwise distances between leaves of ${\mathbb{T}}$ do not vary too extremely. In Section \[sec:app\], we will verify these assumptions for several common tree models. Although we employ rerooting in proofs below, Assumption \[assump:length\] is for the original root of the tree.
Under Assumptions \[assump:length\] and \[assump:contrast\], for any $\delta > 0$, $\forall \alpha \in \left ( \max \left \{ \frac{\log(2)}{\log(3/2)}, \gamma \right \}, 2 \right )$, there exists a constant $ C_{\delta,\alpha,{\underline}d, {\overline}d, {\underline}\mu, {\overline}\mu} > 0$ such that $$|\hat \mu - \mu^*| \leq C_{\delta,\alpha,{\underline}d, {\overline}d, {\underline}\mu, {\overline}\mu} \left( \frac{\sqrt{\log n}}{n^{(2 - \alpha)/6}}\right)$$ with probability $1-\delta$. \[thm:consistency\]
From now on, we will use $\ell_{{\mathbb{T}}, \mu}$ and $R_{{\mathbb{T}},\mu}$ as short for $\ell_{{\mathbb{T}}, \mu}({\mathbf}{Y})$ and $R_{{\mathbb{T}},\mu}({\mathbf}{Y})$. Recall that $\ell_{{\mathbb{T}}, \mu}$ is the log-likelihood function and $R_{{\mathbb{T}},\mu} = \ell_{{\mathbb{T}}, \mu^*} - \ell_{{\mathbb{T}}, \mu}$ where $\mu^*$ is the true value of the transition rate. The main ideas of the proof of Theorem \[thm:consistency\] can be outlined as follows. For a fixed value of $\mu$, we can view the function $R_{{\mathbb{T}}, \mu}$ as the evidence to distinguish between $\mu$ and $\mu^*$. We will prove that as the number of leaves approaches infinity, the information to distinguish $\mu$ and $\mu^*$, characterized by the KL divergence between $P_{{\mathbb{T}},\mu}$ to $P_{{\mathbb{T}},\mu^*}$ (which is equal to ${\mathbb{E}}(R_{{\mathbb{T}}, \mu})$) increases linearly (Lemma \[lem:lowerbound\]), while the associated uncertainty, characterized by the variance of $R_{T, \mu}$, only increases sub-quadratically (Lemma \[lem:boundvar\]). It is worth noting that while the results of Lemma \[lem:lowerbound\] and Lemma \[lem:boundvar\] are straightforward for independent and identically distributed data, the analyses for phylogenetic traits are more complicated due to the correlations between trait values at the leaves. To overcome this issue, we use the independent phylogenetic contrasts, introduced in the next section, to obtain a lower bound on the information. On the other hand, Lemma \[lem:subtrees\] shows that the log-likelihood function of a binary tree can be bounded by the sum of the log-likelihood functions of its two subtrees, which allows us to exploit the sparse structure of the tree to derive an upper bound on uncertainty through an induction argument. Finally, we obtain a uniform bound on the difference between the log-likelihood functions and its expected values (Lemma \[lem:concentration\]), which enables us to derive an analysis of convergence of the MLE.
Lower bound on information {#sec:lower}
--------------------------
#### Phylogenetic contrasts
Letting $i$ and $j$ be two different species, we define ${\mathcal}{C}_{ij} =Y_i - Y_j$ to be a contrast between the two species. This is a popular notion introduced by @felsenstein1985phylogenies for computing the likelihood function under the Brownian motion model. @ane2016phase use independent contrasts to construct consistent estimators for the covariance parameters of the Ornstein-Uhlenbeck model. Here, we will introduce a notion of a squared-contrast ${\mathcal}{C}_{ij}^2$, which is simply the square of a contrast ${\mathcal}{C}_{ij}$, and show that squared-contrasts have the same independence properties as independent contrasts. Let ${\mathcal}{A}_{ij}$ be the state of the most recent common ancestor of $i$ and $j$, $d_{ij}$ the tree distance from $i$ to $j$, and $p_{ij}$ the path on the tree connecting $i$ and $j$. By symmetry of the model, we have $${{\mathrm{I}\!\mathrm{P}}}({\mathcal}{C}_{ij}^2 = y~|~{\mathcal}{A}_{ij} = 0) = {{\mathrm{I}\!\mathrm{P}}}({\mathcal}{C}_{ij}^2 = y~|~{\mathcal}{A}_{ij} = 1) = \begin{cases}
\frac{1}{2}(1 + e^{-2 \mu d_{ij}}), & \mbox{if } y = 0 \\
\frac{1}{2}(1 - e^{-2 \mu d_{ij}}), & \mbox{if } y= 1. \end{cases}$$ Therefore, ${\mathcal}{C}_{ij}^2$ and ${\mathcal}{A}_{ij}$ are independent.
Let $\{{\mathcal}{C}_{i_k j_k}\}_{k=1}^m$ be a sequence of contrasts such that any two paths in $\{p_{i_k j_k}\}_{k=1}^m$ have no common node. Then $\{{\mathcal}{C}_{i_k j_k}^2\}_{k=1}^m$ are independent. \[lem:contrast\]
We have, using ${{\mathbb{E}}\!\left(\cdot\right)}$ here to denote expectation over ancestral states, $$\begin{aligned}
&{{\mathrm{I}\!\mathrm{P}}}({\mathcal}{C}_{i_1 j_1}^2 = y_1, {\mathcal}{C}_{i_2 j_2}^2 = y_2, \ldots, {\mathcal}{C}_{i_m j_m}^2 = y_m) \\
& = {{\mathbb{E}}\!\left({{\mathrm{I}\!\mathrm{P}}}({\mathcal}{C}_{i_1 j_1}^2 = y_1, {\mathcal}{C}_{i_2 j_2}^2 = y_2, \ldots, {\mathcal}{C}_{i_m j_m}^2 = y_m | \{{\mathcal}{A}_{i_k j_k}\}_{k=1}^m)\right)} \\
& = {{\mathbb{E}}\!\left(\prod_{k=1}^m {{{\mathrm{I}\!\mathrm{P}}}\left({\mathcal}{C}_{i_k j_k}^2 = y_k | \{{\mathcal}{A}_{i_k j_k}\}_{k=1}^m \right)}\right)} = {{\mathbb{E}}\!\left(\prod_{k=1}^m {{{\mathrm{I}\!\mathrm{P}}}\left({\mathcal}{C}_{i_k j_k}^2 = y_k | {\mathcal}{A}_{i_k j_k}\right)}\right)} \\
& = {{\mathbb{E}}\!\left(\prod_{k=1}^m {{{\mathrm{I}\!\mathrm{P}}}\left({\mathcal}{C}_{i_k j_k}^2 = y_k \right)}\right)} = \prod_{k=1}^m {{{\mathrm{I}\!\mathrm{P}}}\left({\mathcal}{C}_{i_k j_k}^2 = y_k \right)},\end{aligned}$$ where the third equality comes from the Markov property. This completes the proof.
The independent squared-contrasts allow us to derive the following lower bound on the information.
Under Assumption \[assump:contrast\], there exists $C_{{\underline}d, {\overline}d, {\overline}\mu}>0$ such that $${\mathbb{E}}[R_{{\mathbb{T}},\mu}] \ge C_{{\underline}d, {\overline}d, {\overline}\mu} n |\mu - \mu^*|^2$$ for all $\mu \in [\underline{\mu}, \overline{\mu}]$. \[lem:lowerbound\]
Under Assumption \[assump:contrast\], there exists a set of $m = \Omega(n)$ pairwise disjoint pairs (that is, the paths connecting each pair are pairwise disconnected) $(Y_{k,1}, Y_{k,2})_{k=1}^m$. Consider the corresponding set of squared-contrasts ${\mathcal}{C}^2_{i_k j_k} = (Y_{k,1} - Y_{k,2})^2$. By Lemma \[lem:contrast\], $({\mathcal}{C}^2_{i_k j_k})_{k=1}^m$ are independent. Let $Q_{k,\mu}$ be the distribution of ${\mathcal}{C}^2_{i_k j_k}$ corresponding to parameter $\mu$. The total variation from $Q_{k,\mu^*}$ to $Q_{k,\mu}$ is $$\begin{aligned}
\text{TV}(Q_{k, \mu^*} \| Q_{k, \mu}) &= \frac{1}{2}\sum_{x \in \{ 0,1 \}} |Q_{k,\mu^*}(x) - Q_{k,\mu}(x)| \\
&= | e^{- 2 \mu^* d_{i_k j_k}} - e^{- 2 \mu d_{i_k j_k}} | \geq C_{{\underline}d, {\overline}d,{\overline}\mu} | \mu^* - \mu |.\end{aligned}$$ By the data processing inequality [Theorem 9 in @van2014renyi], the fact that $({\mathcal}{C}^2_{i_k j_k})_{k=1}^m$ are independent, and Pinsker’s inequalities, we have $$\begin{aligned}
{\mathbb{E}}[R_{{\mathbb{T}},\mu}] &= \text{KL}(P_{{\mathbb{T}},\mu^*} \| P_{{\mathbb{T}},\mu}) \geq \sum_{k=1}^m{\text{KL}(Q_{k,\mu^*} \| Q_{k,\mu})} \\
& \geq 2\sum_{k=1}^m{[\text{TV}(Q_{k,\mu^*} \| Q_{k,\mu})]^2} \geq m C_{{\underline}d, {\overline}d, {\overline}\mu} |\mu^* - \mu|^2.\end{aligned}$$ Note that $m = \Omega(n)$ which completes the proof.
Upper bound on the uncertainty
------------------------------
The result of Section \[sec:lower\] indicates that as the number of leaves increases, the information to distinguish $\mu$ and $\mu^*$ also increases linearly. In order to prove that the MLE can successfully reconstruct the true parameter $\mu$, we need to ensure that such information is not confounded by the uncertainty associated with the log-likelihood function due to randomness in the data. In other words, we want to guarantee that as $n \to \infty$, the variance of $R_{{\mathbb{T}}, \mu}$ only grows sub-quadratically, that is, there exist $\alpha<2$ and $C>0$ such that ${\mathrm{Var}}R_{{\mathbb{T}},\mu} \le C n^\alpha$. To obtain the bound, we need the following Lemma, the proof of which appears in Appendix \[sec:proof\].
If two functions $u$ and $v$ satisfy $\|u - v\|_{\infty} \le c$, then for all $\omega>1$ and random variables $X$, $${\mathrm{Var}}[u(X)] \le \omega {\mathrm{Var}}[v(X)] + \frac{2\omega}{\omega -1}c^2.$$ \[lem:approx\]
Recalling that, $$f(x) = \max \left\{\log 2,\log \frac{1}{1-e^{-2x}}\right\},~~ S_{{\mathbb{T}}, \mu}= \sum_{e \in E({\mathbb{T}})}{f(\mu t_e)^2},$$ we have the following estimate that does not depend on Assumption \[assump:length\].
For all $\alpha \in \left ( \frac{\log(2)}{\log(3/2)}, 2 \right )$ and $\beta \in (0,1)$, there exist $C_{\alpha, \underline{\mu}, \overline{\mu}} , \omega_{\alpha,\beta} > 1$ such that $${\mathrm{Var}}R_{{\mathbb{T}},\mu} \le C_{\alpha, \underline{\mu}, \overline{\mu}} n^\alpha + \frac{8\omega_{\alpha,\beta}}{\omega_{\alpha,\beta} -1} n^{\beta} S_{{\mathbb{T}}, \mu}$$ for all trees ${\mathbb{T}}$ with $n \ge 2$ taxa and $\mu \in [\underline{\mu}, \overline{\mu}]$. \[lem:boundvar0\]
We will prove the result by induction in the number of taxa $n$. For $n = 2$, we apply Lemma \[lem:Lipschitz\] to obtain $$R_{{\mathbb{T}},\mu} \leq 2 C | \mu^* - \mu | \leq 2 C | \overline{\mu} - \underline{\mu} |$$ where $C$ is the constant in Lemma \[lem:Lipschitz\]. Thus $ {\mathrm{Var}}R_{{\mathbb{T}},\mu} \leq 4 C^2 | \overline{\mu} - \underline{\mu} |^2$. Assume that the statement is valid for all $k < n$. We will prove that it is also valid for $k = n$. Now, let ${\mathbb{T}}$ be a bifurcating tree with $n$ taxa. @lipton1979separator show that there exists an edge $e = (I_1, I_2)$ of ${\mathbb{T}}$ such that if we reroot ${\mathbb{T}}$ at the middle of this edge, the two subtrees ${\mathbb{T}}_1$ and ${\mathbb{T}}_2$ stemming from $I_1$ and $I_2$ have no more than $2n/3$ leaves. Let $I$ be the middle point of edge $e$ and ${\mathbb{T}}'$ be the tree obtained by rerooting ${\mathbb{T}}$ to $I$. Denote by $n_1$ and $n_2$ the number of leaves of ${\mathbb{T}}_1$ and ${\mathbb{T}}_2$. According to Lemma \[lem:reroot\] and Lemma \[lem:indep\], $${\mathrm{Var}}R_{{\mathbb{T}},\mu} = {\mathrm{Var}}R_{{\mathbb{T}}',\mu} = {\mathrm{Var}}(R_{{\mathbb{T}}',\mu}~|~I).$$ By Lemma \[lem:subtrees\], we have $| R_{{\mathbb{T}}',\mu} - R_{{\mathbb{T}}_1,\mu} - R_{{\mathbb{T}}_2,\mu} | \leq 2f(\mu t_e)$. We apply Lemma \[lem:approx\] to obtain $${\mathrm{Var}}R_{{\mathbb{T}},\mu} = {\mathrm{Var}}(R_{{\mathbb{T}}',\mu}~|~I)\leq \omega {\mathrm{Var}}(R_{{\mathbb{T}}_1,\mu} + R_{{\mathbb{T}}_2,\mu} ~|~I) + \frac{8\omega}{\omega -1}f(\mu t_e)^2. \\$$ Let ${\mathbf}{Y}_1$ and ${\mathbf}{Y}_2$ be the observations at the leaves of ${\mathbb{T}}_1$ and ${\mathbb{T}}_2$ respectively. Since ${\mathbf}{Y}_1$ and ${\mathbf}{Y}_2$ are independent conditional on $I$, we have ${\mathrm{Var}}(R_{{\mathbb{T}}_1,\mu} + R_{{\mathbb{T}}_2,\mu} ~|~I) = {\mathrm{Var}}(R_{{\mathbb{T}}_1,\mu} ~|~I) + {\mathrm{Var}}(R_{{\mathbb{T}}_2,\mu} ~|~I)$. By Lemma \[lem:indep\], we deduce that $$\begin{aligned}
{\mathrm{Var}}R_{{\mathbb{T}},\mu} & \leq \omega [{\mathrm{Var}}R_{{\mathbb{T}}_1,\mu} + {\mathrm{Var}}R_{{\mathbb{T}}_2,\mu}] + \frac{8\omega}{\omega -1}f(\mu t_e)^2.\end{aligned}$$ Recalling that the trees $I_1$ and $I_2$ have no more than $2n/3$ leaves, using the induction hypothesis for ${\mathbb{T}}_1$ and ${\mathbb{T}}_2$, we have $$\begin{aligned}
{\mathrm{Var}}R_{{\mathbb{T}},\mu} & \leq \omega C_\alpha [n_1^\alpha + n_2^\alpha] + \frac{8\omega^2}{\omega -1} [n_1^\beta S_{{\mathbb{T}}_1} + n_2^\beta S_{{\mathbb{T}}_2}] + \frac{8\omega}{\omega -1} f(\mu t_e)^2 \\
& \leq \omega C_\alpha \frac{2^{\alpha + 1}}{3^\alpha} n^\alpha + \frac{8\omega}{\omega -1} \left [ \omega \frac{2^\beta}{3^\beta} (S_{{\mathbb{T}}_1} + S_{{\mathbb{T}}_2}) + f(\mu t_e)^2 \right ]n^\beta.\end{aligned}$$
Let $e_1$ and $e_2$ be the edges stemming from the root of ${\mathbb{T}}$. By definition, we have $$S_{{\mathbb{T}}} - [S_{{\mathbb{T}}_1} + S_{{\mathbb{T}}_2} + f(\mu t_e)^2] = f(\mu t_{e_1})^2 + f(\mu t_{e_2})^2 - f[\mu (t_{e_1} + t_{e_2})]^2 \geq 0$$ where the last inequality comes from the fact that $f$ is a non-increasing function. Thus, if we choose $\omega$ such that $$1<\omega \leq \min \left \{ \left(\frac{3}{2}\right)^{\beta}, \frac{1}{2}\left(\frac{3}{2}\right)^{\alpha} \right \},$$ then $${\mathrm{Var}}R_{{\mathbb{T}},\mu} \leq C_\alpha n^\alpha + \frac{8\omega}{\omega -1} n^{\beta} S_{{\mathbb{T}}},$$ which completes the proof.
A combination of Lemma $\ref{lem:boundvar0}$ and Assumption \[assump:length\] gives rise to the desired bound:
Under Assumption \[assump:length\], $\forall \alpha \in \left ( \max \left \{ \frac{\log(2)}{\log(3/2)}, \gamma \right \}, 2 \right )$, there exists $C_{\alpha, \gamma} > 0$ such that $${\mathrm{Var}}R_{{\mathbb{T}},\mu} \le C_{\alpha, \gamma} n^\alpha$$ for all $n \ge 2$ and $\mu \in [\underline{\mu}, \overline{\mu}]$. \[lem:boundvar\]
By Lemma \[lem:boundvar0\] and Assumption \[assump:length\], we have $${\mathrm{Var}}R_{{\mathbb{T}},\mu} \le C_\alpha n^\alpha + C_{\alpha, \beta, \gamma} n^{\beta + \gamma},~~\forall \alpha \in \left ( \frac{\log(2)}{\log(3/2)}, 2 \right ), \beta \in (0,1).$$ Note that $1 \leq \gamma < 2$. So, $\forall \alpha \in \left ( \max \left \{ \frac{\log(2)}{\log(3/2)}, \gamma \right \}, 2 \right )$, we can choose $\beta = \alpha -\gamma$.
Concentration bound
-------------------
We are now ready to prove the concentration inequality for the 2-state symmetric model.
Under Assumption \[assump:length\], for any $\delta>0$ and $\forall \alpha \in \left ( \max \left \{ \frac{\log(2)}{\log(3/2)}, \gamma \right \}, 2 \right )$, there exists $C_{\delta, \alpha,{\underline}\mu, {\overline}\mu}$ such that $$\left|\frac{1}{n}R_{{\mathbb{T}},\mu}- \mathbb{E} \left[\frac{1}{n}R_{{\mathbb{T}},\mu} \right] \right| \le \frac{C_{\delta,\alpha,{\underline}\mu, {\overline}\mu, \gamma}}{n^{(2-\alpha)/3}} \qquad \forall \mu \in [{\underline}\mu, {\overline}\mu]$$ with probability at least $1-\delta$. \[lem:concentration\]
Applying Chebyshev’s inequality, we obtain $$\mathbb{P}\left[ \left|\frac{1}{n}R_{{\mathbb{T}}, \mu}- \mathbb{E} \left[\frac{1}{n}R_{{\mathbb{T}}, \mu} \right] \right| \ge \eta \right] \le \frac{{\mathrm{Var}}(R_{{\mathbb{T}}, \mu})}{n^2 \eta^2}$$ for any $\mu \in [{\underline}\mu, {\overline}\mu]$. On the other hand, by Lemma \[lem:Lipschitz\], we have $$\left|\frac{1}{n}R_{{\mathbb{T}}, \mu_1} - \frac{1}{n}R_{{\mathbb{T}}, \mu_2} \right| = \left|\frac{1}{n}\ell_{{\mathbb{T}}, \mu_1} - \frac{1}{n}\ell_{{\mathbb{T}}, \mu_2} \right| \le C_{{\underline}\mu, {\overline}\mu} |\mu_1 - \mu_2|, ~~ \forall \mu_1, \mu_2 \in [{\underline}\mu, {\overline}\mu].$$ Therefore, if $$\left|\frac{1}{n}R_{{\mathbb{T}}, \mu_0}- \mathbb{E} \left[\frac{1}{n}R_{{\mathbb{T}}, \mu_0} \right] \right| \ge \eta,$$ then $$\left|\frac{1}{n}R_{{\mathbb{T}}, \mu}- \mathbb{E} \left[\frac{1}{n}R_{{\mathbb{T}}, \mu} \right] \right| \ge \frac{\eta}{2},$$ for all $$\mu \in \left [\mu_0 - \frac{\eta}{4 C_{{\underline}\mu, {\overline}\mu}}, \mu_0 + \frac{\eta}{4 C_{{\underline}\mu, {\overline}\mu}} \right ] \bigcap [{\underline}\mu, {\overline}\mu].$$ Define $$\mu_k = {\underline}\mu + k\frac{\eta}{4 C_{{\underline}\mu, {\overline}\mu}},~~ k = 1, 2, \ldots, \left \lfloor \frac{4 C_{{\underline}\mu, {\overline}\mu} ({\overline}\mu - {\underline}\mu)}{\eta} \right \rfloor.$$ We have $$\begin{aligned}
& {{\mathrm{I}\!\mathrm{P}}}\left[ \exists \mu \in [{\underline}\mu, {\overline}\mu]: \left|\frac{1}{n}R_{{\mathbb{T}}, \mu}- \mathbb{E} \left[\frac{1}{n}R_{{\mathbb{T}}, \mu} \right ] \right| \ge \eta \right ] \\
& \qquad \leq {{\mathrm{I}\!\mathrm{P}}}\left [ \bigcup_k \left \{ \left|\frac{1}{n}R_{{\mathbb{T}}, \mu_k}- \mathbb{E} \left[\frac{1}{n}R_{{\mathbb{T}}, \mu_k} \right ] \right | \ge \frac{\eta}{2} \right \} \right ]\\
& \qquad \leq \sum_k {{{\mathrm{I}\!\mathrm{P}}}\left [ \left|\frac{1}{n}R_{{\mathbb{T}}, \mu_k}- \mathbb{E} \left[\frac{1}{n}R_{{\mathbb{T}}, \mu_k} \right ] \right | \ge \frac{\eta}{2} \right ]} \\
& \qquad \le \sum_k{\frac{{\mathrm{Var}}(R_{{\mathbb{T}}, \mu_k})}{n^2 \eta^2}} \\
& \qquad \leq \frac{8 C_{{\underline}\mu, {\overline}\mu} ({\overline}\mu - {\underline}\mu) C_{\alpha, \gamma} n^{\alpha-2}}{\eta^3}.\end{aligned}$$ The last inequality comes from Lemma \[lem:boundvar\]. We complete the proof by picking $$\eta = \left(\frac{8 C_{{\underline}\mu, {\overline}\mu} ({\overline}\mu - {\underline}\mu) C_{\alpha, \gamma}}{\delta n^{2-\alpha}} \right)^{1/3}.$$
Proof of Theorem \[thm:consistency\]
------------------------------------
From Lemma \[lem:lowerbound\], we have $$C_{{\underline}d, {\overline}d, {\overline}\mu} n |\hat \mu - \mu^*|^2 \leq {\mathbb{E}}_{\mu^*} [R_{{\mathbb{T}}, \hat \mu}] = {\mathbb{E}}_{\mu^*} \left[\ell_{{\mathbb{T}},\mu^*} \right] - {\mathbb{E}}_{\mu^*} \left[\ell_{{\mathbb{T}},\hat \mu} \right]$$ where $E_{\mu^*}$ is the expectation with respect to $\bf P_{{\mathbb{T}}, \mu^*}$.
Note that $$\frac{1}{n} \ell_{{\mathbb{T}}, \hat \mu} - \frac{1}{n} \ell_{{\mathbb{T}}, \mu^*} \ge 0.$$ By Lemma \[lem:concentration\], with probability $1 - \delta$, we obtain $$\begin{aligned}
C_{{\underline}d, {\overline}d, {\overline}\mu} |\hat \mu - \mu^*|^2 & \leq \left ( \frac{1}{n} \ell_{{\mathbb{T}},\hat \mu} - {\mathbb{E}}_{\mu^*} \left[\frac{1}{n} \ell_{{\mathbb{T}},\hat \mu} \right] \right) - \left ( \frac{1}{n} \ell_{{\mathbb{T}},\mu^*} - {\mathbb{E}}_{\mu^*} \left[\frac{1}{n} \ell_{{\mathbb{T}}, \mu^*} \right] \right) \\
& \le \frac{C_{\delta,\alpha,{\underline}\mu, {\overline}\mu, \gamma}}{n^{(2-\alpha)/3}},\end{aligned}$$ which completes the proof.
Applications {#sec:app}
============
In this section, we discuss applications of Theorem \[thm:consistency\] for several practical scenarios. We first consider trees with edges of bounded length.
If edge lengths of ${\mathbb{T}}$ are bounded from below and above, then the MLE is consistent. \[thm:boundededges\]
Since all edge lengths are bounded from below, Assumption \[assump:length\] holds with $\gamma = 1$. Next, we will show that Assumption \[assump:contrast\] is satisfied. We select ${\lfloor n/2 \rfloor}$ pairs of leaves using the following procedure:
1. Pick a cherry as a pair. \[step1\]
2. Remove the cherry from the tree as well as the edge immediately above it. \[step2\]
3. Repeat step \[step1\] and \[step2\] until the tree has $0$ or $1$ leaves.
Denote ${\mathcal}{C} = (i_k,j_k)_{k=1}^m$ be the set of pairs returned by the procedure where $m = {\lfloor n/2 \rfloor}$. It is obvious that the paths connecting each pair are pairwise disjoint. Let ${\underline}e$ and ${\overline}e$ be the lower and upper bound of edge lengths, so we have a lower bound for distances between each pair $d_{i_k j_k} \geq 2 {\underline}e$. Moreover, $$\sum_{k=1}^m{d_{i_k j_k}} \leq (2n - 2) {\overline}e.
\label{eqn:lower}$$ We will prove by contradiction that the set ${\mathcal}{B} = \{ k: d_{i_k j_k} \leq 8 {\overline}e \}$ has at least $n/8$ elements. Assume that $|{\mathcal}{B}| < n/8$. Note that $$|{\mathcal}{C}| = {\lfloor n/2 \rfloor} \geq \frac{n-1}{2}.$$ We deduce that $$|{\mathcal}{C} \setminus {\mathcal}{B}| > \frac{n-1}{2} - \frac{n}{8} = \frac{3n-4}{8}.$$ Therefore $$\sum_{k=1}^m{d_{i_k j_k}} \geq \sum_{k \in {\mathcal}{C} \setminus {\mathcal}{B}}{d_{i_k j_k}} > 8 {\overline}e \frac{3n-4}{8} = (3n-4) {\overline}e \geq (2n - 2) {\overline}e, ~~ \forall n \geq 2$$ which contradicts . Therefore, $|{\mathcal}{B}| \geq n/8$.
Next, we apply our result to trees generated from a pure-birth process [@yule1925mathematical]. This is a classical tree-generating process which assumes that lineages give birth independently to one another with the same birth rate.
If ${\mathbb{T}}$ is generated from a Yule process, then the MLE is consistent. \[thm:Yule\]
Again, we only need to check Assumptions \[assump:length\] and \[assump:contrast\]. Let $(t_k)_{k=1}^n$ be the amount of time during which ${\mathbb{T}}$ has $k$ lineages. Then, $(t_k)_{k=1}^n$ are independent exponential random variables with rate $k\lambda$ where $\lambda$ is the birth rate of the Yule process. Let $e_{\min}$ be the minimum edge length of ${\mathbb{T}}$. It is sufficient for us to show that $e_{\min} \geq e^{-n (\gamma - 1)/2}$. Indeed, we have $$\begin{aligned}
{{\mathrm{I}\!\mathrm{P}}}\left(e_{\min} \geq e^{-n (\gamma - 1)/2}\right) & = {{\mathrm{I}\!\mathrm{P}}}\left(\min_k {t_k} \geq e^{-n (\gamma - 1)/2}\right) \\
&= \exp \left (- \frac{n(n+1)}{2} \lambda e^{-n (\gamma - 1)/2} \right )\end{aligned}$$ which converges to $1$ as $n \to \infty$. Hence, Assumption \[assump:length\] is satisfied asymptotically with $\gamma > 1$.
Define the age of an internal node to be its distance to the leaves. By Corollary 4 of @ane2016phase, there exist $0 < c_1 < c_2 < + \infty$ such that the probability of there being $\Omega(n)$ internal nodes with age between $c_1$ and $c_2$ is $1 - {\mathcal}{O}(n^{-1})$. Let ${\mathcal}{I}$ be the set of internal nodes which have age between $c_1$ and $c_2$, we select $\Omega(n)$ pairs of leaves as follows:
1. Start with an internal node $i \in {\mathcal}{I}$ that has the smallest age. Pick two of its descendants such that $i$ is their most recent common ancestor to form a pair. \[step1b\]
2. Remove $i$ and its parent node from ${\mathcal}{I}$. \[step2b\]
3. Repeat step \[step1b\] and \[step2b\] until ${\mathcal}{I}$ is empty.
This procedure is similar to the procedure for selecting contrasts in @ane2016phase [Lemma 3]. Hence, it selects at least $|{\mathcal}{I}|/2$ pairs, which is of order $\Omega(n)$. Since these internal nodes have age between $c_1$ and $c_2$, Assumption \[assump:contrast\] is satisfied.
Finally, we consider the coalescent point process (CPP), which generates a random ultrametric tree with a given height $T$ [see @lambert2013birth for more details]. Conditioning on the number of species $n$, the internal node ages $(t_k)_{k=1}^{n-1}$ of the tree generated from the CPP are independent and identically distributed according to a probability distribution in $[0,T]$. The probability density function $\phi$ of this common distribution is called the coalescent density. We say that a CPP is *regular* if its common distribution is not a point mass at $0$ and the coalescent density is bounded from above.
If ${\mathbb{T}}$ is generated from a regular coalescent point process, then the MLE is consistent. \[thm:CPP\]
Denote the minimum edge length of ${\mathbb{T}}$ by $e_{\min}$. We have $$e_{\min} \geq \min \left \{ \min_{1 \leq i < j \leq n-1}{|t_i - t_j|}, \min_{1 \leq i \leq n-1}{t_i}, \min_{1 \leq i \leq n-1}{(T - t_i)} \right \} .$$ Therefore, $$\begin{gathered}
{{\mathrm{I}\!\mathrm{P}}}\left (e_{\min} \leq \frac{1}{n^3} \right ) \leq {{\mathrm{I}\!\mathrm{P}}}\left (\min_{1 \leq i < j \leq n-1}{|t_i - t_j|} \leq \frac{1}{n^3} \right ) \\
+ {{\mathrm{I}\!\mathrm{P}}}\left ( \min_{1 \leq i \leq n-1}{t_i} \leq \frac{1}{n^3} \right ) + {{\mathrm{I}\!\mathrm{P}}}\left ( \min_{1 \leq i \leq n-1}(T - t_i) \leq \frac{1}{n^3} \right ).\end{gathered}$$ By the results of Section 4 in @jammalamadaka1986limit (for which the details will be provided in the Appendix), we have $${{\mathrm{I}\!\mathrm{P}}}\left (n^2 \min_{1 \leq i < j \leq n-1}{|t_i - t_j|} \leq \epsilon \right ) \to 1 - \exp \left (-c \epsilon \int_0^T{\phi^2} \right )
\label{eqn:jamma}$$ for any $\epsilon > 0$. We deduce that $$\begin{aligned}
\lim_{n \to \infty} {{\mathrm{I}\!\mathrm{P}}}\left (n^2 \min_{1 \leq i < j \leq n-1}{|t_i - t_j|} \leq \frac{1}{n^3} \right ) & \leq \lim_{n \to \infty} {{\mathrm{I}\!\mathrm{P}}}\left (n^2 \min_{1 \leq i < j \leq n-1}{|t_i - t_j|} \leq \epsilon \right )\\
& = 1 - \exp \left (-c \epsilon \int_0^T{\phi^2} \right ).\end{aligned}$$ Since $\epsilon$ can be arbitrary small, we conclude $$\lim_{n \to \infty} {{\mathrm{I}\!\mathrm{P}}}\left (\min_{1 \leq i < j \leq n-1}{|t_i - t_j|} \leq \frac{1}{n^3} \right ) = 0.$$ On the other hand $${{\mathrm{I}\!\mathrm{P}}}\left ( \min_{1 \leq i \leq n-1}{t_i} \leq \frac{1}{n^3} \right ) = 1 - {{\mathrm{I}\!\mathrm{P}}}\left ( \min_{1 \leq i \leq n-1}{t_i} \geq \frac{1}{n^3} \right ) = 1 - \left ( \int_{1/n^3}^T{\phi} \right )^n.$$ Note that $\phi$ is bounded from above by $M$ in $[0,T]$, thus $$\left ( \int_{1/n^3}^T{\phi} \right )^n = \left ( 1 - \int_{0}^{1/n^3}{\phi} \right )^n \geq \left (1 - \frac{M}{n^3} \right )^n \to 1.$$ where $M$ is the upper bound of $\phi$. Hence, $$\lim_{n \to \infty} {{\mathrm{I}\!\mathrm{P}}}\left ( \min_{1 \leq i \leq n-1}{t_i} \leq \frac{1}{n^3} \right ) = 0.$$ Similarly, $$\lim_{n \to \infty} {{\mathrm{I}\!\mathrm{P}}}\left ( \min_{1 \leq i \leq n-1}(T - t_i) \leq t \right ) = 0.$$ Therefore, $${{\mathrm{I}\!\mathrm{P}}}\left (e_{\min} \geq \frac{1}{n^3} \right ) \to 1.$$ Hence, Assumption \[assump:length\] is satisfied.
Since the common distribution is not a point mass at $0$, there exist a constant $c > 0$ such that $\int_c^T{\phi} > 0$. Let ${\mathcal}{I}$ be the set of internal nodes of ${\mathbb{T}}$ which have age between $c$ and $T$. Then, Assumption \[assump:contrast\] holds if we can prove that $|{\mathcal}{I}|$ is of order $\Omega(n)$ because we can use the same argument as in the proof of Theorem \[thm:Yule\]. By strong law of large numbers, we have $$\frac{|{\mathcal}{I}|}{n-1} \to \int_c^T{\phi} > 0,$$ which completes the proof.
Practical implication
=====================
The consistency property proved in Theorem \[thm:consistency\] suggests that adding more taxa helps to significantly improve the accuracy of the MLE of the transition rate. It is worth noticing that this property does not hold in many scenarios [see @li2008more; @ane2008analysis; @ho2013asymptotic; @ho2014intrinsic]. To illustrate our theoretical results, we perform the following simulation using the `R` package `geiger` [@harmon2007geiger; @pennell2014geiger]. We simulate $100$ trees (the number of taxa varies from $50$ to $2000$) according to the Yule process with birth rate $\lambda = 1$. For each tree, we simulate $100$ traits under the $2$-state symmetric model with the transition rate $\mu = 0.5$. For each of these 100 traits, the MLE for the transition rate is computed separately. The result is summarized in Figure \[fig:MLE\]. We can see that the MLEs concentrate more and more around the true transition rate as the number of species increases.
Discussion
==========
In this paper, we investigate the convergence of the MLE for the transition rate of a $2$-state symmetric model under regularity conditions on tree shape. These conditions ensure that edge lengths of the tree are not too small and the pairwise distances between leaves are not too extreme. For example, these conditions are satisfied when the edge lengths of the tree are bounded from above and below. We have also verified that trees generated from pure-birth process and coalescent point process satisfy these conditions. On the other hand, whether these sufficient conditions are also necessary conditions remains open.
Our results suggest that adding data in terms of the number of tips tends to improve the accuracy of the MLE of the transition rate in general. To demonstrate this result, we use simulations to confirm that the MLE of the transition rate is consistent when the tree is generated from the Yule process. It is worth noticing that for evolutionary data, the MLE for other quantities of interest in evolutionary studies can be inconsistent [@li2008more; @ane2008analysis; @ho2013asymptotic; @ho2014intrinsic]. In these situations, adding more data is a waste of resources because it does not significantly improve the precision of the MLE. Therefore, it is important to study the consistency of the MLE in the context of trait evolution models.
Beside its practical biological application, the paper also investigates many theoretical properties of tree-generating processes. Specifically, in Theorem $\ref{thm:Yule}$ and Theorem $\ref{thm:CPP}$, we consider the problem of bounding the minimum edge length $e_{min}$ of a tree generated by Yule and coalescent point processes. The theorems show that generically, $e_{min}$ is bounded from below by a function of order $n^{-2}$ for the Yule process and $n^{-3}$ for the coalescent point process. Since the minimum edge length plays an important role in many phylogenetic problems, these results may be of independent interest.
Finally, we remark that if we have multiple independent traits which evolve according to the same 2-state Markov process, the distance from the MLE to the true value of the transition rate will decrease linearly with respect to the number of traits. This result comes from the fact that traditional properties of MLE can be applied because these traits are independent. Therefore, incorporating additional traits to the analysis is another way to improve the precision of the MLE.
Proofs {#sec:proof}
======
Proof of lemma \[lem:indep\]
----------------------------
Note that by symmetry, we have ${{\mathrm{I}\!\mathrm{P}}}({\mathbf}{Y} = {\mathbf}{y}~|~\rho = 0) = {{\mathrm{I}\!\mathrm{P}}}(1 - {\mathbf}{Y} = {\mathbf}{y}~|~ \rho = 1)$. We deduce that $$\begin{aligned}
{{\mathrm{I}\!\mathrm{P}}}(h({\mathbf}{Y}) = x~|~\rho = 0) &= {{\mathrm{I}\!\mathrm{P}}}({\mathbf}{Y} \in h^{-1}(x)~|~\rho = 0) \\
&= {{\mathrm{I}\!\mathrm{P}}}({\mathbf}{1} - {\mathbf}{Y} \in h^{-1}(x)~|~\rho = 1) \\
&= {{\mathrm{I}\!\mathrm{P}}}(h({\mathbf}{1} - {\mathbf}{Y}) = x~|~\rho = 1) \\
&= {{\mathrm{I}\!\mathrm{P}}}(h({\mathbf}{Y}) = x~|~\rho = 1)\end{aligned}$$ which completes the proof.
Proof of lemma \[lem:subtrees\]
-------------------------------
Denote $P^{(u)}_v = {{\mathrm{I}\!\mathrm{P}}}({\mathbf}{Y_u} ~|~ {\mathbb{T}}_u, \mu, \rho_u = v)$ for $u \in \{ 0, 1\}$, $v \in \{ 0,1 \}$. We have $$P_{{\mathbb{T}}_1,\mu}({\mathbf}{Y_1}) P_{{\mathbb{T}}_2,\mu}({\mathbf}{Y_2}) = \frac{1}{4} \sum_{u,v \in \{ 0, 1\}}{P^{(1)}_u P^{(2)}_v}.$$ Moreover $$P_{{\mathbb{T}},\mu}({\mathbf}{Y}) = \frac{1 + e^{-2 \mu d}}{4} \sum_{u \in \{ 0, 1 \}}{P^{(1)}_u P^{(2)}_u} + \frac{1 - e^{-2 \mu d}}{4} \sum_{u \in \{ 0, 1 \}}{P^{(1)}_u P^{(2)}_{1-u}}.$$ Therefore $$\frac{1}{1 - e^{-2 \mu d}} P_{{\mathbb{T}}_1,\mu}({\mathbf}{Y_1}) P_{{\mathbb{T}}_2,\mu}({\mathbf}{Y_2}) \leq P_{{\mathbb{T}},\mu}({\mathbf}{Y}) \leq 2 P_{{\mathbb{T}}_1,\mu}({\mathbf}{Y_1}) P_{{\mathbb{T}}_2,\mu}({\mathbf}{Y_2}).$$
Proof of lemma \[lem:Lipschitz\]
--------------------------------
Without loss of generality, we assume that $\mu_1 < \mu_2$. By the mean value theorem, there exists $\tilde \mu_{uv} \in (\mu_1, \mu_2)$ for any $u, v \in \{ 0, 1\}$ such that $$\left| \log [{\mathbf}{P}_{\mu_1}(t)]_{uv} - \log [{\mathbf}{P}_{\mu_2}(t)]_{uv} \right| = \frac{ t e^{- 2 \tilde \mu_{uv} t}}{[{\mathbf}{P}_{\tilde \mu_{uv}}(t)]_{uv}} |\mu_1 - \mu_2| \leq \frac{ t e^{- 2 \tilde \mu_{uv} t}}{1 - e^{- 2 \tilde \mu_{uv} t}} |\mu_1 - \mu_2|.$$ We observe that there exists a $C_{\underline{\mu}, \overline{\mu}}>0$ such that $$\sup_{t \ge 0; \tilde \mu_{uv} \in (\underline{\mu}, \overline{\mu})} {\frac{ t e^{- 2 \tilde \mu_{uv} t}}{1 - e^{- 2 \tilde \mu_{uv} t}}} \le C_{\underline{\mu}, \overline{\mu}}.$$ Therefore, $$| \log [{\mathbf}{P}_{\mu_1}(t)]_{uv} - \log [{\mathbf}{P}_{\mu_2}(t)]_{uv} | \leq C_{\underline{\mu}, \overline{\mu}} |\mu_1 - \mu_2|.$$ This implies that $$[{\mathbf}{P}_{\mu_1}(t)]_{uv} \leq e^{C_{\underline{\mu}, \overline{\mu}} |\mu_1 - \mu_2|} [{\mathbf}{P}_{\mu_2}(t)]_{uv}.
\label{eqn:Pbound}$$ Note that $$P_{{\mathbb{T}},\mu}({\mathbf}{Y}) = \frac{1}{2} \sum_{y}{\left ( \prod_{(u,v)\in E}{[{\mathbf}{P}_\mu( d_{uv})}]_{y_u y_v} \right )}.$$ By applying for all $2n-3$ edges on the tree, we deduce that $$P_{{\mathbb{T}},\mu_1}({\mathbf}{Y}) \leq e^{(2n-3) C_{\underline{\mu}, \overline{\mu}} |\mu_1 - \mu_2| } P_{{\mathbb{T}},\mu_2}({\mathbf}{Y}) .$$ Hence, $$|\ell_{{\mathbb{T}},\mu_1}({\mathbf}{Y}) - \ell_{{\mathbb{T}},\mu_2}({\mathbf}{Y})| \leq (2n-3) C_{\underline{\mu}, \overline{\mu}} |\mu_1 - \mu_2|,$$ which validates the lemma.
Proof of lemma \[lem:approx\]
-----------------------------
For all $x, y$, we have $|u(x) - u(y)| \le |v(x)-v(y)| + 2c$. Let $Y$ be an independent and identically distributed copy of $X$, we have $$\begin{aligned}
2{\mathrm{Var}}[u(X)] &= {\mathbb{E}}_{X}[u(X)^2] + {\mathbb{E}}_{Y}[u(Y)^2] - 2 {\mathbb{E}}_{X}[u(X)] {\mathbb{E}}_{Y}[u(Y)]\\
&= {\mathbb{E}}_{X, Y}\left(u(X)^2 + u(Y)^2 - 2 u(X) u(Y)\right)\\
&= {\mathbb{E}}_{X, Y}\left([u(X)-u(Y)]^2\right) \\
&\le {\mathbb{E}}_{X, Y}\left( [|v(X)-v(Y)| + 2c]^2 \right).\end{aligned}$$ Note that for all $z, c \in \mathbb{R}$ and $\omega>1$, $$(z + 2c)^2 \le \omega z^2 + \frac{4\omega}{\omega -1} c^2.$$ Therefore, $$\begin{aligned}
2{\mathrm{Var}}[u(X)] & \le \omega {\mathbb{E}}_{X, Y}\left([v(X)-v(Y)]^2 \right) +\frac{4\omega}{\omega -1}c^2\\
& = 2 \omega {\mathrm{Var}}[v(X)] +\frac{4\omega}{\omega -1}c^2.\end{aligned}$$
Proof of Equation
------------------
In order to establish Equation , we use the following Lemma.
Let $X_1, X_2, \ldots, X_n$ be an i.i.d. sequence of random variables and $f_n(x, y)$ be an indicator function on $\mathbb{R}^2$ such that $$n^3 E[f_n(X_1, X_2) f_n(X_1, X_3)] \to 0 ~~~ \text{and} ~~~\frac{1}{2}n^2E[f_n(X_1, X_2)] \to \lambda$$ for some constant $\lambda>0$. Define $U_n =\sum_{1 \le i< j \le n}{f_n(X_i, X_j)}$.
Then $U_n \to_d \text{Poisson}(\lambda)$.
We apply this Lemma with $f_n(x, y) = I\{|x-y| < r_n\}$ where $r_n = \epsilon/n^2$ for the sequence $t_1, t_2, \ldots, t_n$ of the coalescent point process. Note that by Equation (4.3) in @jammalamadaka1986limit, $$\frac{1}{2}n^2E[f_n(t_1, t_2)] \to c \epsilon \int_{0}^T{\phi(x)^2 dx}$$ for some constant $c>0$. On the other hand, we have $$\begin{aligned}
E[f_n(t_1, t_2) f_n(t_1, t_3)] &= E[(E[f_n(t_1, t_2) \mid t_1] )^2]\\
&= \int_{0}^T{\left(\int_{t - r_n}^{t+r_n}{\phi(\tau)d\tau}\right)^2\phi(t) dt}\le \frac{4 \| \phi \|_\infty^2 \epsilon^2}{n^4}.\end{aligned}$$ Therefore, $n^3 E[f_n(t_1, t_2) f_n(t_1, t_3)] \to 0$. Hence, $${{\mathrm{I}\!\mathrm{P}}}\left (n^2 \min_{1 \leq i < j \leq n-1}{|t_i - t_j|} \leq \epsilon \right ) = P(U_n = 0) \to 1 - \exp \left (-c \epsilon \int_0^T{\phi^2} \right ).$$
[^1]: These authors contributed equally to this work.
|
---
abstract: '[*Generic higher character Lifshitz critical behaviors are described using field theory and $\epsilon_{L}$-expansion renormalization group methods. These critical behaviors describe systems with arbitrary competing interactions. We derive the scaling relations and the critical exponents at the two-loop level for anisotropic and isotropic points of arbitrary higher character. The framework is illustrated for the $N$-vector $\phi^{4}$ model describing a $d$-dimensional system. The anisotropic behaviors are derived in terms of many independent renormalization group transformations, each one characterized by independent correlation lengths. The isotropic behaviors can be understood using only one renormalization group transformation. Feynman diagrams are solved for the anisotropic behaviors using a new dimensional regularization associated to a generalized orthogonal approximation. The isotropic diagrams are treated using this approximation as well as with a new exact technique to compute the integrals. The entire procedure leads to the analytical solution of generic loop order integrals with arbitrary external momenta. The property of universality class reduction is also satisfied when the competing interactions are turned off. We show how the results presented here reduce to the usual $m$-fold Lifshitz critical behaviors for both isotropic and anisotropic criticalities.*]{}'
address: |
[*Laboratório de Física Teórica e Computacional, Departamento de Física,\
Universidade Federal de Pernambuco,\
50670-901, Recife, PE, Brazil*]{}
author:
- 'Marcelo M. Leite[^1]'
title: '**Critical behavior of generic competing systems**'
---
Introduction
============
Field theoretic renormalization group techniques are invaluable tools for studying usual critical phenomena as well as the critical behavior associated to the physics of systems presenting arbitrary short range competing interactions. The universality classes of the ordinary critical behavior are characterized by the space dimension of the system $d$ and the number of components of the (field) order parameter $N$ [@Am; @BLZ]. Competing systems, on the other hand, possess different types of space directions known as competition axes.
The simplest type of competition directions can be most easily visualized using the terminology of magnetic systems via a generalized Ising model. One permits exchange ferromagnetic couplings between nearest neighbors $(J_{1}>0)$ [*and*]{} antiferromagnetic interactions between second neighbors $(J_{2}<0)$ occurring along $m_{2}$ dimensions. Whenever $m_{2}<d$ the system presents a (usual) second character anisotropic Lifshitz critical behavior whose universality classes are characterized by $(N,d,m_{2})$, whereas the isotropic behavior characterized by $d=m_{2}$ close to 8 was formerly described at the same time [@Ho-Lu-Sh]. The phenomenological model corresponding to a uniaxial anisotropy ($m_{2}=1$) in a cubic lattice is known as ANNNI model [@Selke]. In the critical region, this sort of system is characterized by a disordered, a uniformly ordered [*and*]{} a modulated ordered phase which meet in a uniaxial Lifshitz multicritical point, where the ratio $\frac{J_{2}}{J_{1}}$ is fixed at the corresponding Lifshitz temperature $T_{L}$. High-precision numerical Monte Carlo simulations were carried out for the critical exponents of this model [@Pleim-Hen] and checked using two different two-loop analytical calculations [@AL1; @Leite1; @Leite2]. From the renormalization group perspective there is an important difference between these Lifshitz critical behaviors. The anisotropic behaviors have two independent correlation lengths, $\xi_{L2}$ perpendicular to the competing axes as well as $\xi_{L4}$ parallel to the $m_{2}$ competing axes. The isotropic behavior has only one correlation length $\xi_{L4}$.[^2]
If we go on to include ferromagnetic couplings up to the third neighbors $(J_{3}>0)$ along a single axis, the system will present a uniaxial third character Lifshitz point whenever $\frac{J_{2}}{J_{1}}$ and $\frac{J_{3}}{J_{1}}$ take certain fixed values at the corresponding Lifshitz temperature [@Se1]. When this sort of competition takes place along $m_{3}$ spatial directions, the system presents a $m_{3}$-fold third character Lifshitz point. On the other hand, if simultaneous and independent competing interactions take place between second neighbors along $m_{2}$ space directions and third neighbors along $m_{3}$ space dimensions, the system presents a [*generic*]{} third character $m_{3}$-fold Lifshitz critical behavior. The generic third character universality classes are defined by $(N,d,m_{2},m_{3})$, thus describing a wider sort of critical behavior when compared with the thrid character universality classes $(N,d,m_{3})$.
This idea can be extended in order to define the $m_{L}$-fold Lifshitz point of character $L$, when further alternate couplings are permitted up to the $L$th neighbors along $m_{L}$ directions, provided the ratios $\frac{J_{L}}{J_{1}}$, $\frac{J_{L-1}}{J_{1}}$ ,..., $\frac{J_{2}}{J_{1}}$ take especial values at the associated Lifshitz temperature [@Se2; @NCS; @NTCS]. However, the most general anisotropic situation is to consider [*several*]{} types of competing axes occurring simultaneously in the system such that second neighbors interact along $m_{2}$ space directions, $m_{3}$ directions couple third neighbors, etc., up to the interactions of $L$ neighbors along $m_{L}$ dimensions, with all competing axes perpendicular among each other. In that case, the corresponding critical behavior is called a generic $Lth$ character Lifshitz critical behavior [@Leite4].
In this work we shall undertake an exploration of the field theoretical renormalization group structure of the most general competing system using $\epsilon_{L}$-expansion techniques for anisotropic and isotropic higher character Lifshitz critical behaviors. A rather brief description of this structure was set forth in a previous letter [@Leite4] where it was first described; here we shall present the details and extend the formalism in order to incorporate the exact two-loop calculation for arbitrary isotropic higher character criticalities. Renormalization group (RG) arguments are constructed in order to find out the scaling relations for the anisotropic as well as the isotropic critical behaviors. The discussion parallels that for the usual second character Lifshitz points [@Leite2].
The arbitrary competing exchange coupling Ising model (CECI model) is the lattice model associated to this new critical behavior. It has a general renormalization group structure which contains many independent length scales, and its construction can be utilized for both anisotropic and isotropic cases. In the anisotropic cases, the system has only nearest neighbor interactions along $(d-m_{2}-...-m_{L})$, second neighbors competing interactions along $m_{2}$ dimensions, and so on, up to $L$th neighbors competing interactions along $m_{L}$ spatial directions. The distinct competing axes originate several types of independent correlation lengths, namely $\xi_{1}$ for directions parallel to the $(d-m_{2}-...-m_{L})$-dimensional noncompeting subspace, $\xi_{2}$ for directions parallel to the $m_{2}$-dimensional competing subspace, etc., and $\xi_{L}$ characterizing the $m_{L}$-dimensional subspace. The simplest representative of the CECI model is better understood with the help of Fig.\[figCECI1\], which is the particular case $m_{2}=m_{3}=1$, $m_{4}=...=m_{L}=0$. There are two competing subspaces and three types of correlation lengths which define three independent renormalization group transformations. It defines a particular generic third character anisotropic Lifshitz critical behavior.
In the phase diagram of the ANNNI model, the parameters which are varied are the temperature $T$ and $p=\frac{J_{2}}{J_{1}}$ which take a particular value at the uniaxial second character Lifshitz multicritical point as depicted in Fig.\[figCECI2\]. It is a particular case of the CECI model whenever $m_{2}=1$, with [$m_{3}=...=m_{L}=0$. Although the ANNNI model has applications in several real physical systems (see for example [@Selke]), the prototype of second character Lifshitz points in magnetic materials is manganese phosphide ($MnP$). Experimental as well as theoretical investigations have determined that $MnP$ presents a pure uniaxial Lifshitz point ($m_{2}=1, d=3, N=1$)[@Becerra; @Yokoi]]{}.
When adding further competing interactions to the ANNNI model, the number of parameters in the phase diagram increases [@Se2]. For instance, in the phase diagram of the model including uniaxial competing interactions up to third neighbors the parameters to be varied are the temperature $T$, $p_{1}=\frac{J_{2}}{J_{1}}$ and $p_{2}=\frac{J_{3}}{J_{1}}$. One can locate the third character Lifshitz point by looking at the projection of the phase diagram in the plane $(p_{1}, p_{2})$, as was demonstrated using numerical means [@Se2]. In the example of the CECI model displayed in Fig.\[figCECI1\], let the competing exchange be completely independent along the different competing axes. In that case, the phase diagram can be described by $T$, $p_{z} = \frac{J_{2 z}}{J_{1 z}}$, $p_{1y}=\frac{J_{2y}}{J_{1y}}$, $p_{2y}=\frac{J_{3y}}{J_{1y}}$. A useful two-dimensional representation can be obtained by separating the phase diagrams in two parts. The diagram $(T,p_{z})$ characterizing the second character behavior (with $p_{1y},p_{2y}$ fixed) and the diagram $(p_{1y}, p_{2y})$ (with $T,p_{z}$ fixed) corresponding to the third character behavior can be ploted independently. The superposition of the two diagrams at the generic third character Lifshitz point is indicated in Fig.\[figCECI3\]. As a consequence, there is a uniformly ordered phase and two modulated phases called $Helical_{2}$ and $Helical_{3}$ in Fig.\[figCECI3\] which meet at the uniaxial generic third character anisotropic Lifshitz point. Now there are two first order lines separating the ferromagnetic-$Helical_{2}$ and $Helical_{2}-Helical_{3}$ phases. Analogously, when there are arbitrary independent types of competing axes, we can consider several independent phase diagrams and each two-dimensional projection of them. The superposition of them in one two-dimensional diagram gives origin to a situation that resembles that illustrated in Fig.\[figCECI3\] . Instead, there are [*several*]{} modulated phases and one uniformly ordered phase which meet at the generich $Lth$ character anisotropic Lifshitz point. Each competing subspace has its own characteristic modulated phase along with its own independent correlation length. Although these objects go critical simultaneously at the Lifshitz critical temperature, they define independent renormalization group transformations in each different subspace.
Therefore, we find multiscale scaling laws as a consequence of this renormalization group flow independence in parameter space. This implies that we find several independent coupling constants, each one depending on a definite momenta scale characterizing the particular competition axes under consideration. Nevertheless, all coupling constants flow to the same fixed point. The universality classes of this system are characterized by the parameters $(N,d,m_{2},..., m_{L})$, therefore generalizing the usual Lifshitz behavior. It is important to mention that when we turn off all the competing interactions between third and more distant neighbors, the universality classes of the generic higher character Lifshitz point turn out to reduce to that associated to the second character behavior $(N,d,m_{2})$. Notice that these anisotropic behaviors generalize previous lattice models with competing interactions [@Fra-Hen] as it includes all types of competing axes. The isotropic critical behaviors $d=m_{n}$ have a distinct feature in which there is only one type of correlation length $\xi_{4n}$. Their universality classes are characterized by $(N,d,n)$ where $n$ is the number of neighbors coupled through competing interactions.
In addition, we compute the critical exponents at least at $O(\epsilon_{L}^{2})$ using dimensional regularization to resolving the Feynman diagrams and normalization conditions (and) or minimal subtraction as the renormalization procedures. The computation is realized in momentum space. For the anisotropic cases, the Feynman diagrams are performed with an approximation which is the most general one consistent with the homogeneity of these integrals in the external momenta scales. The isotropic situations are treated using this approximation as well, but we also present the exact calculation at the same loop order and make a comparison with the above mentioned approximation.
We present the functional integral representation of the model in terms of a $\lambda\phi^{4}$ setting and define the normalization conditions for this higher character Lifshitz critical behavior in section II. We show that many sets of normalization conditions, each one corresponding to a specific type of competition axes, are convenient to have a satisfactory description of the problem in its maximal generality.
In Section III we present the renormalization group analysis for the anisotropic critical behaviors. We construct the several renormalization functions appropriate to each competing subspace and study their flow with the various renormalization group transformations. We find the proper scaling relations to each competition subspace.
Section IV discusses the renormalization group treatment for the various isotropic behaviors. We obtain the scaling relations and show that they reduce to the usual $\lambda\phi^{4}$ case when the interactions beyond the first neighbors are switched off.
Section V is an exposition of the calculation of the critical exponents for the anisotropic cases using the scaling relations obtained in section III. We describe two different ways of calculating the critical exponents either in normalization conditions or in minimal subtraction. We discuss the limit $L \rightarrow \infty$ and some of its implications. We point out the analogy of the Lifshitz critical region with an effective field theory which arises from a recent cosmological model including modifications of gravity in the long distance limit [@AH].
Sections VI and VII are an in-depth analysis of the critical exponents for the isotropic cases. The critical exponents for the isotropic cases are calculated using the orthogonal approximation and the scaling relations in section VI. The exact calculation of the critical exponents and the comparison with those obtained from the orthogonal approximation are carried out in section VII.
The particular case corresponding to the second character isotropic Lifshitz critical behavior is discussed explicitly in Section VIII. The resulting critical exponents are shown to generalize those obtained previously [@Ho-Lu-Sh]. We discuss their relationship with those coming from the orthogonal approximation.
Section IX concludes this paper, a discussion of the ideas is summarized and some possible applications will be proposed. We calculate the Feynman integrals in the appendices. We describe in detail the generalized orthogonal approximation for the calculation of higher loop integrals of the anisotropic behaviors in Appendix A. It will be shown there that one can obtain the answer in a simple analytical form for arbitrary external momenta scales. The property of homogeneity of these integrals along arbitrary external momenta scales is preserved. Then, we use the same approximation to compute diagrams for the isotropic behaviors in Appendix B. In addition, we perform the exact calculation for arbitrary isotropic cases in Appendix C. We also analyse the simple particular case associated to the usual second character isotropic behavior.
Field theory and normalization conditions for higher character Lifshitz points
==============================================================================
The field theoretical representation can be expressed in terms of a modified $\lambda\phi^{4}$ field theory presenting arbitrary higher derivative terms due to the effect of competition along the different kinds of competing axes. The type of competing axes are defined by the number of neighbors that interact among each other via exchange competing couplings. Let $m_{n}$ be the number of space directions whose competing interactions extend to the $n$th neighbor. Thus, the $m_{n}$-dimensional competition subspace will be represented in the Lagrangian with even powers (up to the $2n$th) of the gradient acting on the order parameter scalar field. Thus, the effect of the competition resides in the higher derivatives of the field.
The corresponding bare Lagrangian density can be written as [@Leite4] : $$\begin{aligned}
L &=& \frac{1}{2}
|\bigtriangledown_{(d- \sum_{n=2}^{L} m_{n})} \phi_0\,|^{2} +
\sum_{n=2}^{L} \frac{\sigma_{n}}{2}
|\bigtriangledown_{m_{n}}^{n} \phi_0\,|^{2} \\ \nonumber
&& + \sum_{n=2}^{L} \delta_{0n} \frac{1}{2}
|\bigtriangledown_{m_{n}} \phi_0\,|^{2}
+ \sum_{n=3}^{L-1} \sum_{n'=2}^{n-1}\frac{1}{2} \tau_{nn'}
|\bigtriangledown_{m_{n}}^{n'} \phi_0\,|^{2} \\ \nonumber
&&+ \frac{1}{2} t_{0}\phi_0^{2} + \frac{1}{4!}\lambda_0\phi_0^{4} .\end{aligned}$$ At the Lifshitz point, the fixed ratios among the exchange couplings explained above translate into this field-theoretic version though the conditions $\delta_{0n} = \tau_{n n'} =0$. All even momentum powers up to $2L$ become relevant in the free propagator [@Wilson]. This condition simplifies the treatment of the system since it allows the decoupling of the several competing subspaces of Feynman integrals in momentum space. It indeed makes possible to solve these diagrams to any desired order in a perturbative approach. Within the loop order chosen, we can set a perturbative regime with maximal generality as far as critical behavior of competing systems are concerned. So we need the small loop parameter, which is intimately connected to the critical dimension of the theory.
It is instructive to find the critical dimension of this field theory through the use of the Ginzburg criterion [@Lev; @Ginz; @Amit1]. From the perspective of magnetic systems in the above Lagrangian density, $t_{0}= t_{0L} + (T - T_{L})$ measures the temperature difference from the critical temperature $T_{L}$. In the mean-field approximation the inverse susceptibility is proportional to $(T - T_{L})$, but has no longer this behavior when fluctuations get bigger due to the closenesss of the critical point and the mean-field argument breaks down. This immediately leads to the critical dimension $d_{c} = 4 + \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n}$. Above this critical dimension the mean-field behavior dominates the system. Consequently, the small parameter for a consistent perturbative expansion is $\epsilon_{L} = 4 + \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n} - d$. The same type of argument can be constructed for the isotropic behaviors. When $d=m_{n}$, the critical dimension is $d_{c}= 4n -\epsilon_{4n}$.
The renormalized theory can be defined starting from the bare Lagrangian (1). The approach we shall follow here is completely analogous to that exposed in [@Leite2] (see also [@Am]) and the reader is invited to consult that reference in order to be familiarized with the notation employed. The renormalization functions are determined in terms of the renormalized reduced temperature and order parameter (magnetization in the context of magnetic systems) as $t = Z_{\phi^2}^{-1} t_0$, $M = Z_{\phi}^{\frac{-1}{2}} \phi_0$ and depend on Feynman graphs. When the theory is renormalized at the critical temperature $(t=0)$, a nonvanishing external momenta must be used to define the renormalized theory. Consequently, the renormalization constants at the critical temperatute $T_{L}$ depend on the external momenta scales involved in the renormalization algorithm.
Let us analyse the anisotropic behaviors. The Feynman integrals depend on various external momenta scales, namely that characterizing the $(d-m_{2}-...-m_{L})$-dimensional noncompeting subspace, a momentum scale associated to the $m_{2}$ space directions, etc., up to the momentum scale corresponding to the $m_{L}$ competing axes. Thus, it is appropriate to define $L$ sets of normalization conditions in order to compute the critical exponents associated to correlations either perpendicular to or along the several types of competition axes. We define $\kappa_{1}$ to be the external momenta scale asssociated to the $(d-m_{2}-...-m_{L})$-dimensional noncompeting directions. If we define the noncompeting directions to be along the $m_{1}$-dimensional subspace, where $m_{1}=d-m_{2}-...-m_{L}$, we can unify the language by stating that $\kappa_{n}$ is the typical external momenta scale characterizing the $m_{n}$ competing axes ($n=1,...,L$).
We now turn our attention to the definition of the symmetry points (SP). In case we wish to evaluate the critical exponents along the $j$th type of competing axes, we set $\kappa_{n}=0$ for $n \neq j$ keeping, however, $\kappa_{j} \neq 0$. The proper normalization conditions to evaluating exponents along the various competition axes can be defined as follows. If $k'_{i(n)}$ is the external momenta along the competition axes associated to a generic 1PI vertex part, the external momenta along the $n$th type of competing directions are chosen as $ k'_{i(n)}. k'_{j(n)} = \frac{\kappa_{n}^{2}}{4} (4\delta_{ij} - 1)$. This implies that $(k'_{i(n)} + k'_{j(n)})^{2} = \kappa_{n}^{2}$ for $i \neq j$. The momentum scale of the two-point function is defined by $k_{(n)}^{' 2} = \kappa_{n}^{2} = 1$. The set of renormalized 1PI vertex parts is given by:
$$\begin{aligned}
&& \Gamma_{R(n)}^{(2)}(0,g_{n}) = 0, \\
&& \frac{\partial\Gamma_{R(n)}^{(2)}(k'_{(n)}, g_{n})}
{\partial k_{(n)}^{' 2n}}|_{k_{(n)}^{' 2n}=\kappa_{n}^{2n}}
= 1, \\
&& \Gamma_{R(n)}^{(4)}(k'_{i(n)}, g_{n})|_{SP_{n}} = g_{n} , \\
&& \Gamma_{R(n)}^{(2,1)}(k'_{1(n)}, k'_{2(n)}, k', g_{n})|_{\bar{SP_{n}}} = 1 , \\
&& \Gamma_{R(n)}^{(0,2)}(k'_{(n)}, g_{n})|_{k_{(n)}^{' 2n}}=\kappa_{n}^{2n}= 0 .\end{aligned}$$
These $L$ systems of normalization conditions seem to provide $L$ renormalized coupling constants. The origin of this overcounting is a consequence of the $L$ independent flow in the renormalization momenta scales $\kappa_{1}$,...,$\kappa_{L}$. The analysis works with $L$ coupling constants, namely $g_{n} = u_{n} (\kappa_{n}^{2n})^{\frac{\epsilon_{L}}{2}}$ (and $ \lambda_{n} = u_{0n} (\kappa_{n}^{2n})^{\frac{\epsilon_{L}}{2}}$) characterizing the flow along the momenta components parallel to each $m_{n}$-dimensional competing subspace. This is really a disguise since the situation becomes simpler at the fixed point: the many couplings will flow to the same fixed point, at two-loop level, giving a clear indication that this property is kept in higher-loop calculations. The explicit demonstration of this fact will be tackled in Sec. V.[^3]
The normalization conditions for the isotropic case ($m_{n}=d$ near $4n$) can be defined in a close analogy to its second character isotropic particular case [@Leite2]. If $k'_{i}$ is the external momenta along the $m_{n}$ competition axes, the external momenta along the $4n$ directions are chosen as $ k'_{i}. k'_{j} = \frac{\kappa_{n}^{2n}}{4}
(4\delta_{ij} - 1)$. This implies that $(k'_{i} + k'_{j})^{2} = \kappa_{n}^{2}$ for $i \neq j$. The momentum scale of the two-point function is fixed through $k'^{2n} = \kappa_{n}^{2n} = 1$. Then we have the same normalization conditions Eqs.(2), but now there is solely one type of external momenta scale. The others are absent in this situation as an effect of the Lifshitz condition $\delta_{0n}=\tau_{nn'}=0$.
We can express all the renormalization functions and bare coupling constants in terms of the dimensionless couplings in a unified perspective for both anisotropic and isotropic behaviors. The subscript $n = 1,2,3,...,L$ labels the different external momenta scales belonging to the general Lifshitz critical behavior, as defined above for the anisotropic and isotropic cases. Expansion of the dimensionless bare coupling constants $u_{o n}$ and the normalization constants $Z_{\phi (n)}$, $\bar{Z}_{\phi^{2} (n)} = Z_{\phi (n)} Z_{\phi^{2} (n)}$ as functions of the dimensionless renormalized couplings $u_{n}$ up to two-loop order as
$$\begin{aligned}
&& u_{o n} = u_{n} (1 + a_{1 n} u_{n} + a_{2 n} u_{n}^{2}) ,\\
&& Z_{\phi (n)} = 1 + b_{2 n} u_{n}^{2} + b_{3 n} u_{n}^{3} ,\\
&& \bar{Z}_{\phi^{2} (n)} = 1 + c_{1 n} u_{n} + c_{2 n} u_{n}^{2} ,\end{aligned}$$
along with dimensional regularization will be sufficient to find out all critical exponents.
Scaling theory for the anisotropic cases
========================================
The anisotropic behaviors are characterized by correlation lengths $\xi_{1},..., \xi_{L}$. When considered independently they define independent renormalization group transformations along the several competing directions. In momentum space, they induce independent flows in each external momenta scale $\kappa_{1},..., \kappa_{L}$.
In order to define the renormalized vertex parts we consider a set of cutoffs $\Lambda_{j}$ ($j=1,...,L$), each of them characterizing a different competing subspace. As functions of the bare vertices and normalization constants they read $$\begin{aligned}
\Gamma_{R(n)}^{(N,M)} (p_{i (n)}, Q_{i(n)}, g_{n}, \kappa_{n})
&=& Z_{\phi (n)}^{\frac{N}{2}} Z_{\phi^{2} (n)}^{M}
(\Gamma^{(N,L)} (p_{i (n)}, Q_{i (n)}, \lambda_{n}, \Lambda_{n})\\ \nonumber
&& - \delta_{N,0} \delta_{L,2}
\Gamma^{(0,2)}_{(n)} (Q_{(n)}, Q_{(n)}, \lambda_{n}, \Lambda_{n})|_{Q^{2}_{(n)} = \kappa_{n}^{2}})\end{aligned}$$ where $p_{i (n)}$ ($i=1,...,N$) are the external momenta associated to the vertex functions $\Gamma_{R(n)}^{(N,L)}$ with $N$ external legs and $Q_{i (n)}$ ($i=1,...,M$) are the external momenta associated to the $M$ insertions of $\phi^{2}$ operators. From the last section, $u_{0 n}$, $Z_{\phi (n)}$ and $Z_{\phi^{2} (n)}$ are represented as power series in $u_{n}$. In order to write the renormalization group equations in terms of dimensionless bare and renormalized coupling constants, we shall discuss the central idea which underlies the subsequent scaling theory.
Consider the volume element in momentum space for calculating an arbitrary Feynman integral. It is given by $d^{d - \sum_{i=2}^{L} m_{i}}q
\Pi_{i=2}^{L} d^{m_{i}}k_{(i)}$. Recall that $\vec{q}$ represents a $(d- \sum_{i=2}^{L}m_{i})$-dimensional vector perpendicular to the competing axes and $\vec{k_{(i)}}$ denotes an $m_{i}$-dimensional vector along the ith competing subspace, respectively. The Lifshitz condition $\delta_{0n}= \tau_{nn'}=0$ suppresses the quadratic part of the momentum along the $m_{2}$ competition axes, the quadratic and quartic part of the momentum along the $m_{3}$ competing directions, and so on, such that the $m_{L}$ competing subspace is represented by a $2L$th power of momentum in the inverse free critical $(t=0)$ propagator, i.e., $G_{0}^{(2) -1}(q,k) = q^{2} + \sum_{n=2}^{L} \sigma_{n}(k_{(n)}^{2})^{n} $. In order to be dimensionally consistent, the canonical dimension in mass units of the various terms in the propagator should be equal.
Our normalization conditions give us a hint that we can get rid of the $\sigma_{n}$ parameters provided we make simultaneously dimensional redefinitions of the momenta components along each type of competition subspace in a complete analogy to the second character case. Let $[\vec{q}] = M$ be the mass dimension of the quadratic momenta. Since all momentum terms in the propagator should have the same canonical dimension, this requires that $[\vec{k_{(i)}}] = M^{\frac{1}{i}}$. These simultaneous dimensional redefinitions of the momenta along the competing axes is only possible due to the Lifshitz condition. The volume element in momentum space $d^{d - \sum_{i=2}^{L} m_{i}}q
\Pi_{i=2}^{L} d^{m_{i}}k_{(i)}$ has mass dimension $M^{d - \sum_{i=2}^{L}\frac{(i-1)m_{i}}{i}}$. The dimension of the field can be found from the requirement that the volume integral of the Lagrangian density (1) is dimensionless in mass units. In other words, one obtains $[\phi] = M^{\frac{1}{2}(d-\sum_{i=2}^{L}\frac{(i-1)m_{i}}{i})-1}$. In momentum space the one particle irreducible (1PI) vertex functions have canonical dimension $[\Gamma^{(N)}(k_{i})]= M^{N + (d - \sum_{i=2}^{L}\frac{(i-1)m_{i}}{i}) -
\frac{N (d -\sum_{i=2}^{L}\frac{(i-1)m_{i}}{i})}{2}}$.
Let us describe the theory in terms of dimensionless parameters. As the coupling constants are associated to $\Gamma^{(4)}$, we can write $g_{n} = u_{n} (\kappa_{n}^{2 n})^{\frac{\epsilon_{L}}{2}}$, and $ \lambda_{n} = u_{0 n}
(\kappa_{\tau}^{2 n})^{\frac{\epsilon_{L}}{2}}$, where $\epsilon_{L} = 4 + \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n} - d$. Expressed in terms of these dimensionless coupling constants, the renormalization group equation can be cast in the form: $$(\kappa_{n} \frac{\partial}{\partial \kappa_{n}} +
\beta_{n}\frac{\partial}{\partial u_{n}}
- \frac{1}{2} N \gamma_{\phi (n)}(u_{n}) + L \gamma_{\phi^{2} (n)}(u_{n}))
\Gamma_{R(n)}^{(N,L)} = \delta_{N,0} \delta_{L,2} (\kappa_{n}^{-2 n})^\frac{\epsilon_{L}}{2} B_{n}(u_{n}) .$$ The functions
$$\begin{aligned}
&& \beta_{n} = (\kappa_{n}\frac{\partial u_{n}}{\partial \kappa_{n}}), \\
&& \gamma_{\phi (n)}(u_{n}) = \beta_{n}
\frac{\partial ln Z_{\phi (n)}}{\partial u_{n}}\\
&& \gamma_{\phi^{2} (n)}(u_{n}) = - \beta_{n}
\frac{\partial ln Z_{\phi^{2} (n)}}{\partial u_{n}}\end{aligned}$$
are calculated at fixed bare coupling $\lambda_{n}$. The $\beta_{n}$-functions can be rewitten in terms of dimensionless quantities as
$$\beta_{n} = - n \epsilon_{L}(\frac{\partial ln u_{0 n}}{\partial u_{n}})^{-1}.$$
Note that the beta function corresponding to the flow in $\kappa_{n}$ has a factor of $n$ compared to that associated to the flow in $\kappa_{1}$.
For the anisotropic case, the multi-parameters group of invariance is manifest in the solution of the renormalization group equation, which is given by $$\Gamma_{R (n)}^{(N)} (k_{i (n)}, u_{n}, \kappa_{n}) =
exp[-\frac{N}{2} \int_{1}^{\rho_{n}} \gamma_{\phi (n)}(u_{n}(\rho_{n}))
\frac{{d x_{n}}}{x_{n}}]
\;\Gamma_{R (n)}^{(N)} (k_{i (n)}, u_{n}(\rho_{n}),
\kappa_{n} \rho_{n}).$$
From the above analysis, the dimensional redefinitions of the momenta along the distinct competing axes result in an effective space dimension for the anisotropic case, namely, $(d - \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n})$ . We discover the following behavior for the 1PI vertex parts $\Gamma_{R (n)}^{(N)}$ under flows in the external momenta: $$\begin{aligned}
\Gamma_{R (n)}^{(N)} (\rho_{n} k_{i (n)}, u_{n}, \kappa_{n})&=&
\rho_{n}^{n (N + (d- \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n}) -
\frac{N(d-\sum_{n=2}^{L} \frac{(n-1)}{n} m_{n})}{2})}\\
&& exp[-\frac{N}{2} \int_{1}^{\rho_{n}} \gamma_{\phi (n)}(u_{n}(x_{n}))
\frac{{d x_{n}}}{x_{n}}]\nonumber\\
&&\Gamma_{R (n)}^{(N)} (k_{i (n)}, u_{n}(\rho_{n}),
\kappa_{n})\nonumber.\end{aligned}$$
The behavior of the vertex functions at the infrared regime is worthwhile, since their fixed point structure will determine the scaling laws and the critical exponents for arbitrary $m_{n}$-dimensional competing subspace. These $L$ independent fixed points are defined by $\beta_{n}(u_{n}^{*}) = 0$. At the fixed points the simple scaling property holds $$\begin{aligned}
\Gamma_{R (n)}^{(N)} (\rho_{n} k_{i (n)}, u_{n}^{*}, \kappa_{n})&=&
\rho_{n}^{n (N + (d-\sum_{n=2}^{L} \frac{(n-1)}{n} m_{n}) -
\frac{N(d-\sum_{n=2}^{L} \frac{(n-1)}{n} m_{n})}{2}) -\frac{N}{2} \gamma_{\phi (n)}(u_{n}^{*})}\\
&&\Gamma_{R (n)}^{(N)} (k_{i (n)}, u_{n}^{*},\kappa_{n})\nonumber.\end{aligned}$$
For $N=2$, we have $$\Gamma_{R (n)}^{(2)} (\rho_{n} k_{(n)}, u_{n}^{*}, \kappa_{n}) =
\rho_{n}^{2 n - \gamma_{\phi (n)}(u_{n}^{*})}\Gamma_{R (n)}^{(2)} (k_{(n)},u_{n}^{*}, \kappa_{n}).$$ The quantity $\gamma_{\phi (n)}(u_{n}^{*})$ can be identified as the anomalous dimension of the competing subspace under consideration.
This can be readily generalized to include $L$ insertions of $\phi^{2}$ operators such that the RG equations at the fixed point lead to the solution ($(N,M) \neq (0,2)$) :
$$\begin{aligned}
&\Gamma_{R (n)}^{(N,M)} (\rho k_{i (n)}, \rho p_{i (n)}, u_{n}^{*},
\kappa_{n}) = \rho_{n}^{n [N + (d -\sum_{n=2}^{L} \frac{(n-1)}{n} m_{n}) - \frac{N(d -\sum_{n=2}^{L} \frac{(n-1)}{n} m_{n})}{2} -2M] -
\frac{N \gamma_{\phi (n)}^{*}}{2} + M \gamma_{\phi^{2} (n)}^{*}}
\nonumber\\
& \qquad \times \Gamma_{R (n)}^{(N,M)} (k_{i (n)}, p_{i (n)}, u_{n}^{*}, \kappa_{n}).\end{aligned}$$
Writing this at the fixed point as $$\Gamma_{R (n)}^{(N,M)} (\rho k_{i (n)}, \rho p_{i (n)}, u_{n}^{*},
\kappa_{n}) = \rho_{n}^{n [(d -\sum_{n=2}^{L} \frac{(n-1)}{n} m_{n}) - N d_{\phi}] + M d_{\phi^{2}}}
\Gamma_{R (n)}^{(N,M)} (k_{i (n)}, p_{i (n)}, u_{n}^{*}, \kappa_{n}),$$ the anomalous dimensions of the insertions of $\phi^{2}$ operators are $d_{\phi^{2}} = -2 n + \gamma_{\phi^{2} (n)}(u_{n}^{*})$.
The scaling relations can be found by going away from the Lifshitz critical temperature ($t \neq 0$) staying, however, at the critical region $\delta_{0n} = \tau_{nn'}=0$, which is the generalization of that from the ordinary second character Lifshitz behavior. Above the Lifshitz critical temperature, the renormalized vertices for $t\neq 0$ can be expressed as a power series in $t$ around the renormalized vertex parts at $t=0$, as long as $N\neq 0$. Then one can show that the RGE for the vertex parts when $t\neq 0$ are given by $$[\kappa_{n} \frac{\partial}{\partial \kappa_{n}} +
\beta_{n}\frac{\partial}{\partial u_{n}}
- \frac{1}{2} N \gamma_{\phi (n)}(u_{n}) +
\gamma_{\phi^{2} (n)}(u_{n}) t \frac{\partial}{\partial t}]
\Gamma_{R (n)}^{(N)} (k_{i (n)}, t, u_{n}, \kappa_{n}) = 0.$$ The key property of the solution is that it is a homogeneous function of the product of $k_{i (n)}$ (to some power) and $t$ only at the fixed point $u_{n}^{*}$. As the value of $u_{n}$ is fixed at $u_{n}^{*}$, we shall omit it from the notation of this section henceforth. Thus, at the fixed point the solution of the RGE reads $$\Gamma_{R (n)}^{(N)} (k_{i (n)}, t, \kappa_{n})=
\kappa_{n}^{\frac{N \gamma_{\phi (n)}^{*}}{2}}
F_{(n)}^{(N)}(k_{i (n)},\kappa_{n} t^{\frac{-1}{\gamma_{\phi^{2} (n)}^{*}}}) .$$ Defining $\theta_{n} = -\gamma_{\phi^{2} (n)}^{*}$, and using dimensional analysis, it is easy to show that $$\begin{aligned}
\Gamma_{R (n)}^{(N)} (k_{i (n)}, t, \kappa_{n}) =&&
\rho_{n}^{n [N + (d - \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n}) - \frac{N}{2}(d - \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n})] -\frac{N}{2}
\eta_{n}}
\kappa_{n}^{\frac{N}{2} \eta_{n}} \nonumber\\
&&F_{(n)}^{(N)}(\rho_{n}^{-1} k_{i (n)},(\rho_{n}^{-1}\kappa_{n})
(\rho_{n}^{-2 n}t)^{\frac{-1}{\theta_{n}}} ) .\end{aligned}$$
The choice $\rho_{n} = \kappa_{n} (\frac{t}{\kappa_{n}^{2 n}})^{\frac{1}{\theta_{n} +
2 n}}$, can be substituted back in (16), implying that the vertex function depends only on the combination $k_{i (n)} \xi_{n}$ apart from a power of $t$. Since the correlation lengths $\xi_{n}$ are proportional to $t^{- \nu_{n}}$, it implies that the critical exponents $\nu_{n}$ satisfy the identity $$\nu_{n}^{-1} = 2 n + \theta_{n}^{*} = 2 n - \gamma_{\phi^{2} (n)}^{*} .$$ For convenience we could have defined the function $$\bar{\gamma}_{\phi^{2} (n)}(u_{n}) = - \beta_{n}
\frac{\partial ln (Z_{\phi^{2} (n)}Z_{\phi (n)}) }{\partial u_{n}}.$$ In that case we would have discovered the equivalent relations $$\begin{aligned}
\nu_{n}^{-1} &=& 2n - \eta_{n} - \bar{\gamma}_{\phi^{2} (n)}(u_{n}^{*}).\end{aligned}$$ For $N=2$ we choose $\rho_{n} = k_{(n)}$, the external momenta. Then $\Gamma_{R(n)}^{(2)}(k_{(n)}, t, \kappa_{n}) =
k^{2 n - \eta_{n}}
\kappa_{n}^{\eta_{n}} f(k_{(n)} \xi_{n})$. The infrared regime corresponds to $\xi_{n} \rightarrow \infty$ and $k_{(n)} \rightarrow 0$ such that $f(k_{(n)} \xi_{n}) \rightarrow Constant$. The definition $f_{n} = (k_{(n)} \xi_{n})^{2 n - \eta_{n}}
f(k_{(n)} \xi_{n})$, leads to $$\Gamma_{R(n)}^{(2)}(k_{(n)}, t, \kappa_{n}) = (k_{(n)} \xi_{n})^{2 n - \eta_{n}}
\kappa_{n}^{\eta_{n}} f_{n}(k_{(n)} \xi_{n}).$$ Since the susceptibility is proportional to $ t^{-\gamma_{n}}$ as $k_{(n)} \rightarrow 0$, and $\Gamma_{R (n)}^{(2)} = \chi_{(n)}^{-1}$, the susceptibility critical exponents are given by $$\gamma_{n} = \nu_{n} (2 n - \eta_{n}).$$
We now discuss the scaling law appropriate to relate the specific heat critical exponent to the others critical indices. The analysis of the RG equation for $\Gamma_{R (n)}^{(0,2)}$ above $T_{L}$ at the fixed point yields information about the specific heat exponents. In that case it reads $$(\kappa_{n} \frac{\partial}{\partial \kappa_{n}} +
\gamma_{\phi^2 (n)}^{*} (2 + t\frac {\partial}{\partial t}))
\Gamma_{R (n)}^{(0,2)} =
(\kappa_{n}^{-2 n})^{\frac{\epsilon_{L}}{2}} B_{n}(u_{n}^{*}) ,$$ where $B_{n}(u_{n}^{*})$ is given by $$(\kappa_{n}^{-2 n})^{\frac{\epsilon_{L}}{2}}
B_{n}(u_{n}^{*}) = - Z_{\phi^{2}(n)}^{2}
\kappa_{n} \frac{\partial}{\partial \kappa_{n}}
\Gamma_{(n)}^{(0,2)}(Q_{(n)}; -Q_{(n)}, \lambda_{n})
|_{Q_{(n)}^{2}=\kappa_{n}^{2}}.$$ The general discussion given up to now for the vertex part $\Gamma_{R (n)}^{(N,M)}$ will be useful to uncover the homogeneous part of the solution. In fact, at the fixed point the generalization of the solution for $\Gamma_{R(n)}^{(N,M)}$ is written as $$\Gamma_{R (n)}^{(N,M)} (p_{i (n)}, Q_{i (n)}, t, \kappa_{n}) =
\kappa_{n}^{\frac{1}{2} N \gamma_{\phi(n)}^{*} -
M \gamma_{\phi^{2}(n)}^{*}} F_{n}^{(N,M)}(p_{i (n)}, Q_{i (n)},
\kappa_{n} t^{\frac{-1}{\gamma_{\phi^{2} (n)}^{*}}}) .$$
At the fixed point, the temperature dependent homogeneous part for $\Gamma_{R(n),h}^{(0,2)}$ has the following property $$\Gamma_{R (n),h}^{(0,2)}(Q_{(n)}, -Q_{(n)}, t, \kappa_{n}) =
\kappa_{n}^{- 2 \gamma_{\phi^{2}(n)}^{*}} F_{n}^{(0,2)}
(Q_{(n)},- Q_{(n)},
\kappa_{n} t^{\frac{-1}{\gamma_{\phi^{2} (n)}^{*}}}) .$$ This is going to be identified with the specific heat at zero external momentum insertion $Q_{(n)}=0$. Using the dimensional analysis results, one can show that $$\begin{aligned}
&&\Gamma_{R (n),h}^{(0,2)}(Q_{(n)}, -Q_{(n)}, t, \kappa_{n}) =
\rho_{n}^{n [(d - \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n}) - 4] + 2\gamma_{\phi^{2} (n)}^{*}}\\
&& \qquad \times \;\; \Gamma_{R (n),h}^{(0,2)}(\rho_{n}^{-1}Q_{(n)},
- \rho_{n}^{-1}Q_{(n)},
\rho_{n}^{-2 n} t,\rho_{n}^{-1} \kappa_{n}) ,\nonumber\end{aligned}$$ and substituting this into the solution at the fixed point, it yields $$\begin{aligned}
&&\Gamma_{R (n),h}^{(0,2)}(Q_{(n)}, -Q_{(n)}, t, \kappa_{n}) =
\rho_{n}^{n [(d - \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n}) - 4] + 2\gamma_{\phi^{2} (n)}^{*}}
\kappa_{n}^{- 2 \gamma_{\phi^{2}(n)}^{*}}\\
&& \qquad \times \;\; F_{n}^{(0,2)}(\rho_{n}^{-1} Q_{(n)},
-\rho_{n}^{-1} Q_{(n)},
\rho_{n}^{-1} \kappa_{n}(\rho_{n}^{-2 n} t)^{\frac{-1}{\gamma_{\phi^{2} (n)}^{*}}}) .\nonumber\end{aligned}$$ Once more, choose $\rho_{n} = \kappa_{n} (\frac{t}{\kappa_{n}^{2 n}})^{\frac{1}{\theta_{n} + 2 n}}$. Replace this in last equation, take the limit $Q_{(n)} \rightarrow 0$ and identify the power of $t$ with the specific heat exponent $\alpha_{n}$. The result is $$\alpha_{n} = 2 - n (d - \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n})\nu_{n} .$$ The inhomogeneous part can now be discussed. Take $Q_{(n)}=0$ and choose a particular solution of the form: $$C_{p}(u_{n}) = (\kappa_{n}^{2 n})^{\frac{- \epsilon_{L}}{2}}
\tilde{C}_{p}(u_{n}).$$ When this is replaced into the RG equation for $\Gamma_{R (n)}^{(0,2)}$ at the fixed point, we learn that $$C_{p}(u_{n}^{*}) = (\kappa_{n}^{2 n})^{\frac{- \epsilon_{L}}{2}}
\frac{\nu_{n}}{\nu_{n} n (d - \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n}) -2} B_{n}(u_{n}^{*}).$$ Summing up both terms gives the following general solution at the fixed point: $$\Gamma_{R (n)}^{(0,2)} = (\kappa_{n}^{-2 n})^{\frac{\epsilon_{L}}{2}}
(C_{n} (\frac{t}{\kappa_{n}^{2 n}})^{- \alpha_{n}} +
\frac{\nu_{n}}{\nu_{n} n (d - \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n}) -2} B_{n}(u_{n}^{*})).$$
The situation for $T<T_{L}$ is as follows. For simplicity consider the case of magnetic systems. The renormalized equation of state furnishes a relation between the renormalized magnetic field and the renormalized vertex parts for $t<0$ via a power series in the magnetization $M$, i.e., $$H_{(n)}(t, M, u_{n}, \kappa_{n}) = \sum_{N=1}^{\infty} \frac{1}{N!} M^{N}
\Gamma_{R (n)}^{(1+N)}(k_{i (n)} = 0; t, u_{n}, \kappa_{n}),$$ where the zero momentum limit must be taken after performing the summation. The magnetic field satisfies the following RG equation: $$(\kappa_{n} \frac{\partial}{\partial \kappa_{n}} +
\beta_{n}\frac{\partial}{\partial u_{n}}
- \frac{1}{2} N \gamma_{\phi (n)}(N + M \frac{\partial}{\partial M}) +
\gamma_{\phi^{2} (n)} t \frac{\partial}{\partial t})
H_{(n)}(t, M, u_{n}, \kappa_{n}) = 0 .$$ The equation of state has the following form at the fixed point: $$H_{(n)}(t, M, \kappa_{n}) = \kappa_{n}^{\frac{\eta_{n}}{2}}
h_{1 n}(\kappa_{n} M^{\frac{2}{\eta_{n}}}, \kappa_{n} t^{\frac{-1}
{\gamma_{\phi^{2} (n)}}}).$$ Dimensional analysis arguments can be used to determine how a flow in the external momenta affects the renormalized magnetic field. The flow produces the following expression: $$H_{(n)}(t, M, \kappa_{n}) = \rho_{n}^{n [\frac{(d - \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n})}{2} + 1]} H_{(n)}(\frac{t}{\rho_{n}^{2 n}},
\frac{M}{\rho_{n}^{n[\frac{(d - \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n})}{2} - 1]}},
\frac{\kappa_{n}}{\rho_{n}}) .$$ The standard choice corresponds to $\rho_{n}$ being a power of $M$ $$\rho_{n} = \kappa_{n} [\frac{M}{\kappa_{n}^{\frac{n}{2}[(d - \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n}) - 2]}}]^{\frac{2}
{n [(d - \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n}) - 2] + \eta_{n}}} .$$ Replacing this into (35) and from the scaling form of the equation of state $H_{(n)}(t, M) =
M^{\delta_{n}} f(\frac{t}{M^{\frac{1}{\beta_{n}}}})$, we obtain the remaining scaling laws
$$\begin{aligned}
&&\beta_{n} = \frac{1}{2} \nu_{n} (n(d-\sum_{i=2}^{L} \frac{(i-1)}{i} m_{i}) - 2n + \eta_{n}),\\
&&\delta_{n} = \frac{n(d-\sum_{i=2}^{L} \frac{(i-1)}{i} m_{i}) + 2n -
\eta_{n}}{n(d-\sum_{i=2}^{L} \frac{(i-1)}{i} m_{i}) - 2n + \eta_{n}},\end{aligned}$$
which imply the Widom $\gamma_{n} = \beta_{n} (\delta_{n} -1)$ and Rushbrooke $\alpha_{n} + 2 \beta_{n} + \gamma_{n} = 2$ relations for [*arbitrary*]{} competing or noncompeting subspaces, since $n=1,...,L$. We note that there is one set of scaling relations for each competing subspace. This suggests that all the critical exponents take different values in distinct subspaces. We are going to see that this is not necessarily true, since the fixed point structure restricts the values of most critical exponents to be the same in different competing subspaces. It is a direct consequence that there is only one fixed point independent of the space directions under consideration.
The perturbative calculation of the critical exponents and other universal quantities follows from a diagrammatic expansion whose basic objects are Feynman diagrams. We shall use the loop expansion for the anisotropic integrals with the perturbation parameter $\epsilon_{L} = 4 + \sum_{n=2}^{L} \frac{(n-1)}{n} m_{n} - d$. The solution of the Feynman diagrams in terms of $\epsilon_{L}$ results in the $\epsilon_{L}$-expansion for the universal critical ammounts of the anisotropic criticalities. The anisotropic integrals are described using the generalized orthogonal approximation in Appendix A. This approximation yields a solution which is the most general one compatible with the homogeneity of the Feynman integrals for [*arbitrary*]{} external momenta scales. With this technique all the critical exponents in the anisotropic cases can be obtained as will be shown in Section V.
Scaling theory for the isotropic behaviors
==========================================
To begin with let us promote a slight change of notation with respect to the conventions presented in our previous letter [@Leite4]. There, the subscript associated to each type of $m_{n}$ competing axes was chosen as $4n$. Here, we choose the subscript $n$ to express the same thing. This will cause no confusion to the reader since the anisotropic and isotropic cases are considered separately in this work. Then an arbitrary ammount $A_{4n}$, should be changed to $A_{n}$. In particular, the perturbative parameter discussed in section II is now represented as $\epsilon_{n}$. Obviously, whenever $d=m_{n}$, the volume element in momentum space is given by $d^{m_{n}}k$. Setting $\sigma_{n}=1$ we perform a dimensional redefinition of the momenta such that $[k] = M^{\frac{1}{n}}$. Accordingly, the volume element has dimension $[d^{m_{n}}k] = M^{\frac{m_{n}}{n}}$. The dimension of the field in mass units is $[\phi] = M^{\frac{m_{n}}{2n} - 1}$. The 1PI vertex parts have dimensions $[\Gamma^{(N)}(k_{n})] = M^{N + \frac{m_{n}}{n} - N \frac{m_{n}}{2n}}$. Then, make the continuation $m_{n}=4n-\epsilon_{n}$. The coupling constant has dimension $\lambda_{4n} = M^{\frac{4n-m_{n}}{n}}= M^{\frac{\epsilon_{n}}{n}}$. In terms of dimensionless quantities, one has the renormalized $g_{n} = u_{n} (\kappa_{n}^{2n})^{\frac{\epsilon_{n}}{2n}}$ and bare $\lambda_{n} = u_{0n} (\kappa_{n}^{2n})^{\frac{\epsilon_{n}}{2n}}$ coupling constants, respectively. Again, the functions
$$\begin{aligned}
\beta_{n} &=& (\kappa_{n}\frac{\partial u_{n}}{\partial \kappa_{n}})\\
\gamma_{\phi (n)}(u_{n}) &=& \beta_{n}
\frac{\partial ln Z_{\phi (n)}}{\partial u_{n}}\\
\gamma_{\phi^{2} (n)}(u_{n}) &=& - \beta_{n}
\frac{\partial ln Z_{\phi^{2} (n)}}{\partial u_{n}}\end{aligned}$$
are computed at fixed bare coupling constant $\lambda_{n}$. The beta functions in terms of dimensionless quantities are given by $\beta_{n} = - \epsilon_{n}(\frac{\partial ln u_{0n}}{\partial u_{n}})^{-1}$. Notice that the beta function for the isotropic case does not possess the overall factor of $n$ present in the anisotropic beta function $\beta_{n}$ obtained from renormalization group transformations over the $m_{n}$-dimensional competing subspace. This is a very close analogy to the second character behaviors and a general property of Lifshitz critical behaviors.
The dimensional redefinition of the momenta along the $m_{n}$ competing axes leads to an effective space dimension for the isotropic case, i.e., $(\frac{m_{n}}{n})$. Under a flow in the external momenta we find the following behavior for the 1PI vertex parts $\Gamma_{R (n)}^{(N)}$: $$\begin{aligned}
&\Gamma_{R (n)}^{(N)} (\rho_{n} k_{i}, u_{n}, \kappa_{n}) =
\rho_{n}^{n [N + \frac{m_{n}}{n} - N \frac{m_{n}}{2n}]}
exp[-\frac{N}{2} \int_{1}^{\rho_{n}} \gamma_{\phi (n)}(u_{n}(x_{n}))
\frac{{d x_{n}}}{x_{n}}] \\
& \times \;\Gamma_{R (n)}^{(N)} (k_{i}, u_{n}(\rho_{n}),
\kappa_{n})\nonumber.\end{aligned}$$ Note that since there is only one type of space directions in the isotropic behaviors, we do not need to use a label in the external momenta specifying the type of competing axes considered as we did in the anisotropic cases. At the fixed point, the simple scaling property for the vertex parts $\Gamma_{R (n)}^{(N)}$ follows:
$$\begin{aligned}
&\Gamma_{R (n)}^{(N)} (\rho_{n} k_{i}, u_{n}^{*}, \kappa_{n}) =
\rho_{n}^{n [N + \frac{m_{n}}{n} - N \frac{m_{n}}{2n}] -
\frac{N}{2} \gamma_{\phi (n)}(u_{n}^{*})}\\
& \times \Gamma_{R (n)}^{(N)} (k_{i}, u_{n}^{*},\kappa_{n})\nonumber.\end{aligned}$$
For $N=2$, we have $$\Gamma_{R (n)}^{(2)} (\rho_{n} k, u_{n}^{*}, \kappa_{n}) =
\rho_{n}^{2n - \gamma_{\phi (n)}(u_{n}^{*})}\Gamma_{R (n)}^{(2)} (k, u_{n}^{*}, \kappa_{n} ).$$ In the noninteracting theory $d_{\phi}^{0} = \frac{\frac{m_{n}}{n}}{2} - 1$ is the naive dimension of the field. At the isotropic fixed point, the presence of interactions modify it such that $d_{\phi}= \frac{\frac{m}{n}}{2} - 1 + \frac{\eta_{n}}{2n}$. The generalization to include $L$ insertions of $\phi^{2}$ operators can be written at the fixed point as ($(N,L) \neq (0,2)$) :
$$\Gamma_{R (n)}^{(N,L)} (\rho_{n} k_{i}, \rho_{n} p_{i}, u_{n}^{*},
\kappa_{n}) = \rho_{n}^{n [N + \frac{m_{n}}{n} - \frac{N(\frac{m_{n}}{n})}{2} -2L] -
\frac{N \gamma_{\phi (n)}^{*}}{2} + L \gamma_{\phi^{2} (n)}^{*}}
\Gamma_{R (n)}^{(N,L)} (k_{i}, p_{i}, u_{n}^{*}, \kappa_{n}).$$
Writing at the fixed point $$\Gamma_{R (n)}^{(N,L)} (\rho_{n} k_{i}, \rho_{n} p_{i}, u_{n}^{*},
\kappa_{n}) = \rho_{n}^{ m_{n} - N d_{\phi} + L d_{\phi^{2}}}
\Gamma_{R (n)}^{(N,L)} (k_{i}, p_{i}, u_{n}^{*}, \kappa_{n}),$$ the anomalous dimension of the insertions of $\phi^{2}$ operators is given by $d_{\phi^{2}} = -2n + \gamma_{\phi^{2} (n)}(u_{n}^{*})$.
Above the Lifshitz critical temperature we find the following RGE $$[\kappa_{n} \frac{\partial}{\partial \kappa_{n}} +
\beta_{n}\frac{\partial}{\partial u_{n}}
- \frac{1}{2} N \gamma_{\phi (n)}(u_{n}) +
\gamma_{\phi^{2} (n)}(u_{n}) t \frac{\partial}{\partial t}]
\Gamma_{R (n)}^{(N)} (k_{i}, t, u_{n}, \kappa_{n}) = 0.$$ The solution at the fixed point is given by $$\Gamma_{R (n)}^{(N)} (k_{i (n)}, t, u_{n}^{*}, \kappa_{n})=
\kappa_{n}^{\frac{N \gamma_{\phi (n)}^{*}}{2}}
F_{(n)}^{(N)}(k_{i},\kappa_{n} t^{\frac{-1}{\gamma_{\phi^{2} (n)}^{*}}}) .$$ If we define $\theta_{n} = -\gamma_{\phi^{2} (n)}^{*}$, we can use dimensional analysis to obtain $$\begin{aligned}
&\Gamma_{R (n)}^{(N)} (k_{i}, t, \kappa_{n}) =
\rho_{n}^{n [N + \frac{m_{n}}{n} - \frac{N}{2} \frac{m_{n}}{n}] -\frac{N}{2}
\eta_{n}}
\kappa_{n}^{\frac{N}{2} \eta_{n}} \nonumber\\
& \times \; F_{(n)}^{(N)}(\rho_{n}^{-1} k_{i},(\rho_{n}^{-1}\kappa_{n})
(\rho_{n}^{-4}t)^{\frac{1}{\theta_{n}}} ) .\end{aligned}$$
We can choose $\rho_{n} = \kappa_{n} (\frac{t}{\kappa_{n}^{2n}})^{\frac{1}{\theta_{n} + 2n}}$, and replacing it in the last two equations, the vertex parts depend only on the combination $k_{i} \xi_{n}$ apart from a power of $t$. As $\xi_{n}$ is proportional to $t^{-\nu_{n}}$ we can identify the critical exponent $\nu_{n}$ as $$\nu_{n}^{-1} = 2n + \theta_{n}^{*} = 2n - \gamma_{\phi^{2} (n)}^{*} .$$
For the sake of convenience we define the function $$\bar{\gamma}_{\phi^{2} (n)}(u_{n}) = - \beta_{n}
\frac{\partial ln (Z_{\phi^{2} (n)}Z_{\phi (n)}) }{\partial u_{n}}.$$ In terms of this function the last equation turns into the following relation $$\nu_{n}^{-1} = 2n - \eta_{n} - \bar{\gamma}_{\phi^{2} (n)}(u_{n}^{*}).$$ For $N=2$ we choose $\rho_{n} = k$, the external momenta. When $\xi_{n} \rightarrow \infty$ and $k \rightarrow 0$, simultaneously, then $f(k \xi_{n}) \rightarrow Constant$. The susceptibility is proportional to $ t^{-\gamma_{n}}$ as $k_{i} \rightarrow 0$. As $\Gamma_{R}^{(2)} = \chi^{-1}$, the susceptibility critical exponent follows $$\gamma_{n} = \nu_{n} (2n - \eta_{n}).$$
From the RG equation for $\Gamma_{R (n)}^{(0,2)}$ above $T_{L}$ at the fixed point, the scaling relation for the specific heat exponent can be found. The RG equation is $$(\kappa_{n} \frac{\partial}{\partial \kappa_{n}} +
\gamma_{\phi^2 (n)}^{*} (2 + t\frac {\partial}{\partial t}))
\Gamma_{R (n)}^{(0,2)} = (\kappa_{n}^{-2 })^{\frac{\epsilon_{n}}{2}}
B_{n}(u_{n}^{*}) ,$$ where $$(\kappa_{n}^{-2n})^{\frac{\epsilon_{n}}{2n}}
B_{n}(u_{n}^{*}) = - Z_{\phi^{2}(n)}^{2}
\kappa_{n} \frac{\partial}{\partial \kappa_{n}}
[\Gamma_{(n)}^{(0,2)}(Q; -Q, \lambda_{n})|_{Q^{2}=\kappa_{n}^{2}}].$$ The $\Gamma_{R(n)}^{(N,L)}$ can be generalized to $$\Gamma_{R (n)}^{(N,L)}(p_{i}, Q_{i}, t, \kappa_{n}) =
\kappa_{n}^{\frac{1}{2} N \gamma_{\phi(n)}^{*} -
L \gamma_{\phi^{2}(n)}^{*}} F_{n}^{(N,L)}(p_{i}, Q_{i},
\kappa_{n} t^{\frac{-1}{\gamma_{\phi^{2} (n)}^{*}}}) .$$ The homogeneous part of the solution for $\Gamma_{R(n),h}^{(0,2)}$ is temperature dependent and scales at the fixed point as $$\Gamma_{R (n),h}^{(0,2)}(Q, -Q, t, \kappa_{n}) =
\kappa_{n}^{- 2 \gamma_{\phi^{2}(n)}^{*}} F_{n}^{(0,2)}(Q,- Q,
\kappa_{n} t^{\frac{-1}{\gamma_{\phi^{2} (n)}^{*}}}) .$$ This vertex function is going to be identified with the specific heat at zero external momentum insertion $Q=0$. Use of dimensional analysis yields the result: $$\Gamma_{R (n),h}^{(0,2)}(Q, -Q, t, \kappa_{n}) =
\rho_{n}^{n[\frac{m_{n}}{n} - 4] + 2\gamma_{\phi^{2} (n)}^{*}}
\Gamma_{R (n),h}^{(0,2)}(\rho_{n}^{-1}Q, - \rho_{n}^{-1}Q,
\rho_{n}^{-2n} t,\rho_{n}^{-1} \kappa_{n}) .$$ Substituting this equation in the solution at the fixed point leads to $$\Gamma_{R (n),h}^{(0,2)}(Q, -Q, t, \kappa_{n}) =
\rho_{n}^{n [\frac{m_{n}}{n} - 4] + 2\gamma_{\phi^{2} (n)}^{*}}
\kappa_{n}^{- 2 \gamma_{\phi^{2}(n)}^{*}} F_{n}^{(0,2)}(\rho_{n}^{-1} Q,
-\rho_{n}^{-1} Q,
\rho_{n}^{-1} \kappa_{n}(\rho_{n}^{-2n} t)^{\frac{-1}{\gamma_{\phi^{2} (n)}^{*}}}) .$$ The choice $\rho_{n} = \kappa_{n} (\frac{t}{\kappa_{n}^{2n}})^{\frac{1}{\theta_{n} + 2n}}$ can be made. Substitution of this choice into the last equation in the limit $Q \rightarrow 0$ and identifying the power of $t$ with the specific heat exponent $\alpha_{n}$, we obtain: $$\alpha_{n} = 2 - m_{n} \nu_{n} .$$ The inhomogeneous part can be found by taking $Q=0$ and choosing a particular solution in the standard way. Therefore, the general solution at the fixed point is given by $$\Gamma_{R (n)}^{(0,2)} = (\kappa_{n}^{-2 n})^{\frac{\epsilon_{n}}{2 n}}
(C_{n} (\frac{t}{\kappa_{n}^{2 n}})^{- \alpha_{n}} +
\frac{\nu_{n}}{\nu_{n} m -2} B_{n}(u_{n}^{*})).$$
Next, let us concentrate ourselves in the scaling relations when the system is below the Lifshitz critical temperature $T<T_{L}$. The relation among the renormalized magnetic field, the renormalized vertex parts for $t<0$ and the magnetization $M$ is given by $$H_{(n)}(t, M, u_{n}, \kappa_{n}) = \sum_{N=1}^{\infty} \frac{1}{N!} M^{N}
\Gamma_{R(n)}^{(1+N)}(k_{i} = 0; t, u_{n}, \kappa_{n}).$$ The RG equation satisfied by the magnetic field is: $$(\kappa_{n} \frac{\partial}{\partial \kappa_{n}} +
\beta_{n}\frac{\partial}{\partial u_{n}}
- \frac{1}{2} N \gamma_{\phi (n)}(u_{n})(N + M \frac{\partial}{\partial M}) +
\gamma_{\phi^{2} (n)} t \frac{\partial}{\partial t})
H_{(n)}(t, M, u_{n}, \kappa_{n}) = 0 .$$ The solution of the equation of state at the fixed point has the following property $$H_{(n)}(t, M, \kappa_{n}) = \kappa_{n}^{\frac{\eta_{n}}{2}}
h_{n}(\kappa_{n} M^{\frac{2}{\eta_{n}}}, \kappa_{n} t^{\frac{-1}
{\gamma_{\phi^{2} (n)}}}).$$ The scale change in the magnetic field followed by a flow in the external momenta can be written in the form: $$H_{(n)}(t, M, \kappa_{n}) = \rho_{n}^{n [\frac{m_{n}}{2n} + 1]}\nonumber\\
\;\; H_{n}(\frac{t}{\rho_{n}^{2n}}, \frac{M}{\rho_{n}^{n[\frac{m_{n}}{2n} - 1]}},
\frac{\kappa_{n}}{\rho_{n}}) .$$ The flow parameter $\rho_{n}$ is chosen as a power of $M$ such that: $$\rho_{n} = \kappa_{n} [\frac{M}{\kappa_{n}^{[\frac{m_{n}}{2} - n]}}]^{\frac{2}
{m_{n} - 2n + \eta_{n}}} ,$$ and from the scaling form of the equation of state $H_{(n)}(t, M) = M^{\delta_{n}} f_{(n)}(\frac{t}{M^{\frac{1}{\beta_{n}}}})$, we obtain the following scaling relations:
$$\begin{aligned}
\delta_{n} &=& \frac{m_{n} + 2n - \eta_{n}}{m_{n} - 2n + \eta_{n}}, \\
\beta_{n} &=& \frac{1}{2} \nu_{n} (m_{n} - 2n + \eta_{n}),\end{aligned}$$
which imply the Widom $\gamma_{n} = \beta_{n} (\delta_{n} -1)$ and Rushbrooke $\alpha_{n} + 2 \beta_{n} + \gamma_{n} = 2$ relations.
Except for some minor modifications, the renormalization group treatment of each isotropic behavior is equivalent to treat separately each competing subspace appearing in the most general anisotropic behavior. It is easy to see that all scaling laws reduce to those from the usual critical behavior described by a $\phi^{4}$ field theory for $n=1$. Then, the usual critical behavior is actually a first character isotropic Lifshitz critical behavior. For $n=2$, they easily reproduce those associated to the second character Lifshitz point [@Leite2]. Therefore, a new nomenclature emerges from the study of these higher character Lifshitz critical behaviors: the number of neighbors coupled through competing interactions is a fundamental parameter, generalizing the concept of universality class. Thus, the universality classes of isotropic behaviors $(d=m_{n})$ are characterized by $(N,d,n)$. These statements will be put on a firmer ground when we calculate the critical exponents, as we shall see in the next sections.
The isotropic behaviors are calculated using the generalized orthogonal approximation as well as exactly, without the resource of any approximation. The approximation is useful to complete the unified analytical description of the higher character Lifshitz critical behavior in its full generality, at least at the loop order considered here. On the other hand, the analytic exact (perturbative) solution is a conceptual step forward towards a better comprehension of this sort of system. At this point, the reader should consult the Appendix B in order to access the computation of the Feynman integrals using the generalized orthogonal approximation and Appendix C to see the exact computation required to finding the critical exponents.
Critical exponents for the anisotropic behaviors
================================================
In this section we compute the critical exponents using the generalized orthogonal approximation with the results derived in Appendix A. First, we attack the problem using normalization conditions. Second, the results are checked using a variant of the minimal subtraction scheme first developed in [@Leite2] for the anisotropic second character $m_{2}$-fold Lifshitz point.
Normalization conditions and critical exponents
-----------------------------------------------
The bare coupling constants and renormalization functions were defined in section II. They are given by
$$\begin{aligned}
&& u_{on} = u_{n} (1 + a_{1n} u_{n} + a_{2n} u_{n}^{2}) ,\\
&& Z_{\phi (n)} = 1 + b_{2n} u_{n}^{2} + b_{3n} u_{n}^{3} ,\\
&& \bar{Z}_{\phi^{2} (n)} = 1 + c_{1n} u_{n} + c_{2n} u_{n}^{2} ,\end{aligned}$$
where the constants $a_{in}, b_{in}, c_{in}$ depend on Feynman diagrams computed at suitable symmetry points. The critical exponents associated to correlations perpendicular or parallel to the arbitrary competing $m_{n}$-dimensional subspace can be calculated through the specification of the corresponding symmetry point.
In terms of the constants defined above, the beta functions and renormalization constants can be cast in the form:
$$\begin{aligned}
&& \beta_{n} = -n \epsilon_{L}u_{n}[1 - a_{1n} u_{n}
+2(a_{1n}^{2} -a_{2n}) u_{n}^{2}],\\
&& \gamma_{\phi (n)} = -n \epsilon_{L}u_{n}[2b_{2n} u_{n}
+ (3 b_{3n} - 2 b_{2n} a_{1n}) u_{n}^{2}],\\
&& \bar{\gamma}_{\phi^{2} (n)} = n \epsilon_{L}u_{n}[c_{1n}
+ (2 c_{2n} - c_{1n}^{2} - a_{1n} c_{1n})u_{n} ].\end{aligned}$$
The above coefficients can be figured out as functions of the integrals calculated at the symmetry points. We find
$$\begin{aligned}
&& a_{1n} = \frac{N+8}{6 \epsilon_{L}}[1 + h_{m_{L}} \epsilon_{L}] ,\\
&& a_{2n} = (\frac{N+8}{6 \epsilon_{L}})^{2}
+ [\frac{(N+8)^{2}}{18}h_{m_{L}} - \frac{(3N+14)}{24}]
\frac{1}{\epsilon_{L}} ,\\
&& b_{2n} = -\frac{(N+2)}{144 \epsilon_{L}}[1 +
(2 h_{m_{L}} + \frac{1}{4}) \epsilon_{L}], \\
&& b_{3n} = -\frac{(N+2)(N+8)}{1296 \epsilon_{L}^{2}} +
\frac{(N+2)(N+8)}{108 \epsilon_{L}}(-\frac{1}{4} h_{m_{L}} +
\frac{1}{48}), \\
&& c_{1n} = \frac{(N+2)}{6 \epsilon_{L}}[1 + h_{m_{L}} \epsilon_{L}], \\
&& c_{2n} = \frac{(N+2)(N+5)}{36 \epsilon_{L}^{2}}
+ \frac{(N+2)}{3 \epsilon_{L}}[\frac{(N+5)}{3} h_{m_{L}} - \frac{1}{4}].\end{aligned}$$
These expressions are sufficient to find out the fixed points at $O(\epsilon_{L}^{2})$, which are defined by $\beta_{n}(u_{n}^{*}) = 0$. As was seen in the last section, every integral computed at arbitrary symmetry points $SP_{1}, ..., SP_{L}$ gives the same result, irrespective of the considered subspace. The overall factor of $n=1,...,L$ in the $\beta_{n}$ functions drops out at the fixed points, such that the renormalization group transformations realized over $\kappa_{1}$,..., and $\kappa_{L}$ will flow to the same fixed point given by ($u_{1}^{*} = ... = u_{L}^{*} \equiv u^{*}$) $$u^{\ast}=\frac{6}{8 + N}\,\epsilon_L\Biggl\{1 + \epsilon_L
\,\Biggl[ - h_{m_{L}} + \frac{(9N + 42)}{(8 + N)^{2}}\Biggr]\Biggr\}\;\;.$$ It is instructive to separate explicitly the noncompeting and competing subspaces. The functions $\gamma_{\phi (1)}$ and $\bar{\gamma}_{\phi^{2} (1)}$ read $$\begin{aligned}
&& \gamma_{\phi (1)} = \frac{(N+2)}{72} [1 + (2 h_{m_{L}} + \frac{1}{4})
\epsilon_{L}]u_{1}^{2} - \frac{(N+2)(N+8)}{864} u_{1}^{3}, \\
&& \bar{\gamma}_{\phi^{2} (1)} = \frac{(N+2)}{6} u_{1}[1
+ h_{m_{L}} \epsilon_{L} - \frac{1}{2} u_{1}].\end{aligned}$$ When the value of the fixed point is substituted into these equations, using the relation among these functions and the critical exponents $\eta_{1}(\equiv \eta_{L2})$ and $\nu_{1}(\equiv \nu_{L2})$, we find: $$\begin{aligned}
&& \eta_{1}= \frac{1}{2} \epsilon_{L}^{2}\,\frac{N + 2}{(N+8)^2}
[1 + \epsilon_{L}(\frac{6(3N + 14)}{(N + 8)^{2}} - \frac{1}{4})] ,\\
&& \nu_{1} =\frac{1}{2} + \frac{(N + 2)}{4(N + 8)} \epsilon_{L}
+ \frac{1}{8}\frac{(N + 2)(N^{2} + 23N + 60)} {(N + 8)^3} \epsilon_{L}^{2}.\end{aligned}$$ The coefficient of each power of $\epsilon_{L}$ is the same as that coming from the second character Lifshitz behavior $m_{3}=...=m_{L}=0$. Consequently, the reduction to the Ising-like universality class $m_{2}=0$ case is warranted. The several beta functions corresponding to distinct competing axes satisfy the property $\beta_{n} = n \beta_{1}$. This implies that $\gamma_{\phi (n)}= n \gamma_{\phi (1)}$ and $\bar{\gamma}_{\phi^{2} (n)} = n \bar{\gamma}_{\phi^{2} (1)}$. Then, we have $$\begin{aligned}
&& \eta_{n}= \frac{n}{2}(\epsilon_{L}^{2}\,\frac{(N + 2)}{(N+8)^2}
[1 + \epsilon_{L}(\frac{6(3N + 14)}{(N + 8)^{2}} - \frac{1}{4})]) ,\\
&& \nu_{n} = \frac{1}{n} (\frac{1}{2} + \frac{(N + 2)}{4(N + 8)} \epsilon_{L}
+ \frac{1}{8}\frac{(N + 2)(N^{2} + 23N + 60)} {(N + 8)^3} \epsilon_{L}^{2}).\end{aligned}$$ At $O(\epsilon_{L}^{3})$, the relation $\eta_{n} = n \eta_{1}$ is satisfied, whereas at $O(\epsilon_{L}^{2})$, the relation $\nu_{n} = \frac{1}{n} \nu_{1}$ holds. Strong anisotropic scale invariance[@He] is [*exact*]{} to the perturbative order considered here, and within the generalized orthogonal approximation it is expected to hold at arbitrary higher loop order. Differently from the critical indices $\eta_{n}$ and $\nu_{n}$ which depend explicitly on the $m_{n}$-dimensional subspace under consideration, the other exponents take the same value in each subspace even though they are obtained through independent scaling relations along the distinct competing axes. They are given by $$\begin{aligned}
&& \gamma_{L} = 1 + \frac{(N+2)}{2(N + 8)} \epsilon_{L}
+\frac{(N + 2)(N^{2} + 22N + 52)}{4(N + 8)^{3}} \epsilon_{L}^{2}, \\
&& \alpha_{L} = \frac{(4 - N)}{2(N + 8)} \epsilon_{L}
- \frac{(N + 2)(N^{2} + 30N + 56)}{4(N + 8)^{3}} \epsilon_{L}^{2} ,\\
&& \beta_{L} = \frac{1}{2} - \frac{3}{2(N + 8)} \epsilon_{L}
+ \frac{(N + 2)(2N + 1)}{2(N + 8)^{3}} \epsilon_{L}^{2} ,\\
&& \delta_{L} = 3 + \epsilon_{L}
+ \frac{(N^{2} + 14N + 60)}{2(N + 8)^{2}} \epsilon_{L}^{2}.\end{aligned}$$ The exponents correctly reduce to those from the second character behavior [@Leite2; @Leite4], with a further reduction to the Ising-like case when $m_{2}=0$. In fact the universality classes reduction of generic higher character anisotropic Lifshitz points to that from Ising-like critical points is manifest in [*all*]{} critical exponents. Hence, this universality class reduction is a generic property of arbitrary competing systems.
To check the correctness of these exponents, it is convenient to calculate them in another renormalization procedure. Let us check these results using minimal subtraction of dimensional poles, as we are going to show next.
Minimal subtraction and critical exponents
------------------------------------------
In the minimal subtraction renormalization scheme, the common situation is to have just one momenta scale $\mu$[@Si] which is called $\kappa$ in the present paper. Nevertheless, the dimensional redefinitions performed over the external momenta characterizing arbitrary types of competing axes permit a picture of the anisotropic cases with $L$ independent momenta scales.
The calculation of the critical exponents along an arbitrary kind of competition subspace can be done, provided all external momenta not belonging to that subspace are set to zero. Then, we define $\kappa_{j}$ to be the typical scale parameter of the $j$th subspace, calculate the renormalization functions for arbitrary external momenta along the $m_{j}$th space directions and require minimal subtraction of dimensional poles. This procedure is inspired in the method formerly discussed in the second character anisotropic Lifshitz behaviors [@Leite2]. Thus, although the external momentum associated to the competing subspace under consideration is kept arbitrary in all stages of the computation, the same is not true for all other external momenta corresponding to distinct competing subspaces. This restriction on the values of all the external momenta is the price to be paid in order to describe independently the scale transformations of each inequivalent subspace. This is the main difference of this minimal subtraction scheme with several independent momentum scales from the conventional method with just one momentum scale.
Here we will content ourselves in showing that the diagrammatic procedure to calculate the fixed point using minimal subtraction results in the same functions $\gamma_{\phi (n)}$ and $\bar{\gamma}_{\phi^{2} (n)}$ at the fixed point as those coming from normalization conditions. This is equivalent to prove the renormalization scheme independence of all critical indices.
In minimal subtraction, the dimensionless bare coupling constants and the renormalization functions are defined by
$$\begin{aligned}
&& u_{0n} = u_{n}[1 + \sum_{i=1}^{\infty} a_{in}(\epsilon_{L})
u_{n}^{i}], \\
&& Z_{\phi (n)} = 1 + \sum_{i=1}^{\infty} b_{in}(\epsilon_{L})
u_{n}^{i},\\
&& \bar{Z}_{\phi^{2} (n)} = 1 + \sum_{i=1}^{\infty} c_{in}(\epsilon_{L})
u_{n}^{i}.\end{aligned}$$
The renormalized vertex parts
$$\begin{aligned}
&& \Gamma_{R (n)}^{(2)}(k_{(n)}, u_{n}, \kappa_{n}) = Z_{\phi (n)}
\Gamma_{(n)}^{(2)}(k_{n}, u_{0n}, \kappa_{n}), \\
&& \Gamma_{R (n)}^{(4)}(k_{i(n)}, u_{n}, \kappa_{n}) = Z_{\phi (n)}^{2} \Gamma_{(n)}^{(4)}(k_{i (n)}, u_{0n}, \kappa_{n}), \\
&& \Gamma_{R (n)}^{(2,1)}(k_{1(n)}, k_{2(n)}, p_{(n)};
u_{n}, \kappa_{n}) = \bar{Z}_{\phi^{2} (n)}
\Gamma_{(n)}^{(2,1)}(k_{1(n)}, k_{2(n)}, p_{(n)}, u_{0n}, \kappa_{n}),\end{aligned}$$
are finite by construction when $\epsilon_{L} \rightarrow 0$, order by order in $u_{n}$. One should bear in mind that the external momenta in the bare vertices are mutiplied by $\kappa_{n}^{-1}$. Since $k_{i(1)} = p_{i}$ are the external momenta perpendicular to the competing axes, whereas $k_{i(n)} = k'_{i(n)}$ are the external momenta parallel to the $m_{n}$-dimensional type of competing subspace, the coefficients $a_{in}(\epsilon_{L}),
b_{in}(\epsilon_{L})$ and $c_{in}(\epsilon_{L})$ are obtained by requiring that the poles in $\epsilon_{L}$ be minimally subtracted. The bare vertices are written in the form
$$\begin{aligned}
&& \Gamma_{(n)}^{(2)}(k_{(n)}, u_{0n}, \kappa_{n}) =
k_{(n)}^{2n}(1- B_{2n} u_{0n}^{2} + B_{3n}u_{0n}^{3}), \\
&& \Gamma_{(n)}^{(4)}(k_{i(n)}, u_{0n}, \kappa_{(n)}) =
\kappa_{n}^{n \epsilon} u_{0n}
[1- A_{1n} u_{0n}
+ (A_{2n}^{(1)} + A_{2n}^{(2)})u_{0n}^{2}], \\
&& \Gamma_{(n)}^{(2,1)}(k_{1(n)}, k_{2(n)}, p_{(n)};
u_{0n}, \kappa_{n}) = 1 - C_{1n} u_{0n}
+ (C_{2n}^{(1)} + C_{2n}^{(2)}) u_{0n}^{2}.\end{aligned}$$
Remember that $B_{2n}$ is proportional to the integral $I_{3}$ and $B_{3n}$ is proportional to $I_{5}$ which are calculated with all external momenta not belonging to the $m_{n}$-dimensional subspace set to zero.
The coefficients are expressed explicitly by the following integrals:
$$\begin{aligned}
&& A_{1n} = \frac{(N+8)}{18}[ I_{2}(\frac{k_{1(n)} + k_{2(n)}}
{\kappa_{n}}) + I_{2}(\frac{k_{1(n)} + k_{3(n)}}
{\kappa_{n}}) + I_{2}(\frac{k_{2(n)} + k_{3(n)}}
{\kappa_{n}})] ,\\
&& A_{2n}^{(1)} = \frac{(N^{2} + 6N + 20)}{108}
[I_{2}^{2}(\frac{k_{1(n)} + k_{2(n)}}{\kappa_{n}})
+ I_{2}^{2}(\frac{k_{1(n)} + k_{3(n)}}
{\kappa_{n}}) + I_{2}^{2}(\frac{k_{2(n)} + k_{3(n)}}
{\kappa_{n}})] ,\\
&& A_{2n}^{(2)} = \frac{(5N + 22)}{54}
[I_{4}( \frac{k_{i(n)}}{\kappa_{n}}) + 5 \;permutations] ,\\
&& B_{2n} = \frac{(N+2)}{18}I_{3}(\frac{k_{(n)}}{\kappa_{n}}) ,\\
&& B_{3n} =
\frac{(N+2)(N+8)}{108}I_{5}(\frac{k_{n}}{\kappa_{n}}) ,\\
&& C_{1n} = \frac{N+2}{18}[ I_{2}(\frac{k_{1(n)} + k_{2(n)}}
{\kappa_{n}}) + I_{2}(\frac{k_{1(n)} + k_{3(n)}}
{\kappa_{n}}) + I_{2}(\frac{k_{2(n)} + k_{3(n)}}
{\kappa_{n}})] ,\\
&& C_{2n}^{(1)} = \frac{(N+2)^{2}}{108}
[I_{2}^{2}(\frac{k_{1(n)} + k_{2(n)}}{\kappa_{n}})
+ I_{2}^{2}(\frac{k_{1(n)} + k_{3(n)}}
{\kappa_{n}}) + I_{2}^{2}(\frac{k_{2(n)} + k_{3(n)}}
{\kappa_{n}})] ,\\
&& C_{2n}^{(2)} = \frac{N+2}{36}[I_{4}(\frac{k_{i(n)}}{\kappa_{n}}) + 5 \;permutations].\end{aligned}$$
We have at hand all we need to determine the normalization constants at least at two-loop order. Requiring minimal subtraction for the renormalized vertex parts above listed, it can be verified that all the logarithmic integrals depending upon each arbitrary (nonvanishing) external momenta subspace appearing in $I_{2}, I_{3}, I_{4}$, and $I_{5}$ cancell out. This leads to the following expressions for the normalization functions and coupling constants:
$$\begin{aligned}
&& u_{0n} = u_{n}(1 + \frac{(N+8)}{6 \epsilon_{L}} u_{n}
+ [\frac{(N+8)^{2}}{36 \epsilon_{L}^{2}} - \frac{(3N+14)}{24
\epsilon_{L}}] u_{n}^{2}), \\
&& Z_{\phi (n)} = 1 - \frac{N+2}{144 \epsilon_L} u_{n}^{2}
+ [-\frac{(N+2)(N+8)}{1296 \epsilon_{L}^{2}} + \frac{(N+2)(N+8)}{5184
\epsilon_{L}}] u_{n}^{3}, \\
&& \bar{Z}_{\phi^{2} (n)} = 1 + \frac{N+2}{6 \epsilon_L} u_{n}
+ [\frac{(N+2)(N+5)}{36 \epsilon_{L}^{2}} - \frac{(N+2)}{24
\epsilon_{L}}] u_{n}^{2}).\end{aligned}$$
Using the renormalization constants we obtain: $$\begin{aligned}
&& \gamma_{\phi (n)} = n [\frac{(N+2)}{72}u_{n}^{2}
- \frac{(N+2)(N+8)}{1728}u_{n}^{3}],\\
&& \bar{\gamma}_{\phi^{2} (n)} = n \frac{(N+2)}{6} u_{n}
[ 1 - \frac{1}{2} u_{n}].\end{aligned}$$
The fixed points are defined by $\beta_{n}(u_{n}^{*}) =
0$. Then, we learn that the fixed points generated by renormalization group transformations over $\kappa_{1}$,..., and $\kappa_{L}$ are the same, namely $$u_{n}^{\ast}=\frac{6}{8 + N}\,\epsilon_L\Biggl\{1 + \epsilon_L
\,\Biggl[\frac{(9N + 42)}{(8 + N)^{2}}\Biggr]\Biggr\}\;\;.$$
When this result is replaced in the renormalization constants at the fixed point it yields $\gamma_{\phi (n)}^{*}= \eta_{n}$, where $\eta_{n}$ are given by Eqs. (71) and (73). In addition, we have $$\bar{\gamma^{*}}_{\phi^{2} (n)} = n \frac{(N+2)}{(N+8)} \epsilon_{L}
[ 1 + \frac{6(N+3)}{(N+8)^{2}} \epsilon_{L}].$$ The reader can verify that the exponents $\nu_{n}$ encountered by using the last equation are the same as those obtained via normalization conditions Eqs. (72) and (74). This proves the consistency of either renormalization scheme for the anisotropic Lifshitz critical behaviors.
Discussion
----------
The generic $L$th character Lifshitz critical behavior naturally extends the comprehension of the usual second character Lifshitz criticality. The latter is characterized by one noncompeting subspace and only one competing subspace, whereas the former is characterized by several types of competing subspaces. The expressions for the critical exponents can be analysed to extract further information concerning those systems. For instance, when $m_{3}\neq 0$ and $m_{2}=m_{4}=...=m_{L}=0$, the third character behavior is recovered and correctly reduces to the Ising-like behavior for $m_{3}=0$. The main characteristic of generic third character behavior is that there are $m_{2}, m_{3}\neq 0$ competing axes with $m_{4}=...=m_{L}=0$, and so on. From a phenomenological perspective in magnetic systems, may be it is worthy to assemble all magnetic materials presenting Lishitz critical behavior and analyse their critical exponents. Choosing those alloys in the same conjectured universality class, the greater the difference in their critical exponents the more likely they are in an alternative universality class contained in the CECI model analysed here.
It is interesting to note that exact strong anisotropic scale invariance [@He] is valid in the CECI model as a result of the generalized orthogonal approximation. Since the model describes the physics of short ranged competing systems, the limit $L \rightarrow \infty$ in the $Lth$ charater Lifshitz point should not be taken, since it would describe [*long range*]{} competing systems. Nevertheless, since no restriction on $L$ was made in the beginning of the discussion, the CECI model can be viewed as describing a particular type of long range competing interactions. If we go on and take this limit in the expression of the critical exponents we find that $\eta_{L}$ tends to infinity, whereas $\nu_{L}\rightarrow 0$ as a consequence of the strong anisotropic scale invariance. This implies that close to the Lifshitz critical temperature the correlation length $\xi_{L}$ does not diverge in that limit, instead of having a usual power law divergence. This fact is a nonperturbative result valid to all orders in perturbation theory within the context of the $\epsilon_{L}$-expansion.
Another feature emerges from this limit by looking at the critical dimension $d_{c}=4+m_{\infty}$. (Recall that in the $L$th anisotropic character critical behavior the system has $m_{L}$ competing axes and $d-m_{L}$ noncompeting space directions, i.e., $m_{2}=...=m_{L-1}=0$.) This limit yields a mechanism which is a natural new way to study extra dimensions without destroying the renormalizability of the corresponding field theory, while retaining the nontrivial aspect of the fixed point. In spite of the divergence of the anomalous dimension and the vanishing of the correlation length exponent along the $m_{L}$ competing directions in this limit, all other exponents are well behaved and have the same value of those corresponding to space directions without competition, as can be seen explicitly by the expressions for the exponents. This is so since the anisotropic scaling laws only contain the safe combination $L \nu_{L}$ which is always finite in the limit $L \rightarrow \infty$. Once more, note that $m_{\infty}\rightarrow 0$ reduces to the previous $\phi^{4}$ universality classes. Recall that the isotropic case is characterized by $d=m_{L}=4L$. Some care care must be taken in the interpretation of this case in the limit $L\rightarrow \infty$, for the approach from the anisotropic ($0\leq m_{L}<4L$) to the isotropic case takes infinite steps, which is not as simple as in the case when $L$ is finite. This point deserves its own analysis for future work.
On the other hand, the last few years have witnessed some ideas in quantum field theories that resemble very much issues contained in the analysis of Lifshitz critical phenomena. The closest analogy with the Lifshitz field theoretic tools presented here is the very recent idea of ghost condensation producing a consistent modification of gravity in the infrared (long distance limit) [@AH; @Tsu-Sa; @Amen]. In this framework, gravity is modified to have attractive [*as well as*]{} repulsive components. The latter mimics a kind of dark energy [@KNg], whose ghost condensate is a physical fluid arising from a theory where a real scalar field changes with a constant velocity. The ghost condensate appears as the physical scalar excitation around the background generated by the scalar field whose vacuum expectation value is defined by a constant value of its time derivative. Consequently, the effective action for the field representing the ghost condensate has kinetic terms with quadratic time derivatives and quartic space derivatives of the field, therefore breaking Lorentz invariance [@AH]. The instability in momentum space is manifest in the absence of kinetic terms quadratic in the space derivatives of the ghost condensate. This characteristic is a result of the competing nature between the attractive and repulsive components of the gravitational force. This is a precise analogy with the Lifshitz critical region in the case of the usual second character behavior included in the discussion given in the present paper. The utilization of an analogous reasoning leads us to conclude that the CECI model can be used to extract further insights from these new effective quantum field theories where Lorentz invariance is broken. It permits generalizations for the ghost condensate when higher powers in space derivatives of this field are present in its corresponding effective action. For example, at large distances the case of quadratic time derivatives and 6th space derivatives in the kinetic term would correspond to a gravitational interaction with attractive/repulsive/attractive competing components and so on. Further analysis of the model might be helpful to address the perturbative calculations regarding the ghost condensate in that scalar field background.
Numerical methods to probe the results obtained here are in their infancy for the CECI model. Earlier Monte Carlo simulations for a uniaxial third character behavior were performed [@Se1] and the existence of the corresponding Lifshitz point at nonzero temperature was established. Unfortunately, no critical exponent was determined for the third character Lifshitz point. Since then, perhaps due to the fact that these higher character criticalities were not well understood theoretically, these methods are still waiting for more investigations. Recently, a model with antiferromagnetic couplings between nearest neighbors as well as antiferromagnetic exchange interactions between third neighbors was studied using Monte Carlo simulations and some quantum properties were investigated [@Ca-Sa]. No ferromagnetic couplings appear among arbitrary neighbors. Even though there is a quantum Lifshitz point there, it pertains to a universality class which might be different from that representing third character critical points discussed in the present paper. Further numerical studies motivated by the CECI model are necessary to understand completely the classical and quantum properties of its critical regions.
Isotropic critical exponents in the orthogonal approximation
============================================================
We now address the calculation of critical exponents using the generalized orthogonal approximation from the results obtained in Appendix B. As we have seen in the calculation of Feynman integrals, this problem can also be tackled without performing any approximation during all steps of the calculation. This technique shall be postponed until next section. In a way or another, the approach is equivalent to treat each competing subspace separately. Though very similar, the framework of this section turns out to be more economical than that reported in the anisotropic cases.
Critical exponents in normalization conditions
----------------------------------------------
The basic definitions of the bare coupling constants and renormalization functions were encountered before and are given by
$$\begin{aligned}
&& u_{0n} = u_{n} (1 + a_{1n} u_{n} + a_{2n} u_{n}^{2}) ,\\
&& Z_{\phi (n)} = 1 + b_{2n} u_{n}^{2} + b_{3n} u_{n}^{3} ,\\
&& \bar{Z}_{\phi^{2} (n)} = 1 + c_{1n} u_{n} + c_{2n} u_{n}^{2} ,\end{aligned}$$
where the constants $a_{in}, b_{in}, c_{in}$ depend on Feynman integrals at the symmetry point called $SP_{n}$. Let $\kappa_{n}$ denote the competing $m_{n}$-dimensional subspace in the isotropic cases.
The beta-function and renormalization constants can be expressed in the form
$$\begin{aligned}
&& \beta_{n} = - \epsilon_{n}u_{n}[1 - a_{1n} u_{n}
+2(a_{1n}^{2} -a_{2n}) u_{n}^{2}],\\
&& \gamma_{\phi (n)} = - \epsilon_{n}u_{n}[2b_{2n} u_{n}
+ (3 b_{3n} - 2 b_{2n} a_{1n}) u_{n}^{2}],\\
&& \bar{\gamma}_{\phi^{2} (n)} = \epsilon_{n}u_{n}[c_{1n}
+ (2 c_{2n} - c_{1n}^{2} - a_{1n} c_{1n})u_{n} ].\end{aligned}$$
The coefficients above obtained as functions of the integrals calculated at the symmetry point read
$$\begin{aligned}
&& a_{1n} = \frac{N+8}{6 \epsilon_{n}}[1 + \frac{1}{2n} \epsilon_{n}] ,\\
&& a_{2n} = (\frac{N+8}{6 \epsilon_{n}})^{2}
+ [\frac{2N^{2} + 23N + 86}{72n \epsilon_{n}}] ,\\
&& b_{2n} = -\frac{(N+2)}{144n \epsilon_{n}}[1 + \frac{5}{4n} \epsilon_{n}], \\
&& b_{3n} = -\frac{(N+2)(N+8)}{1296n \epsilon_{n}^{2}} -
5 \frac{(N+2)(N+8)}{5184n^{2} \epsilon_{n}}, \\
&& c_{1n} = \frac{(N+2)}{6 \epsilon_{n}}[1 + \frac{1}{2n} \epsilon_{n}], \\
&& c_{2n} = \frac{(N+2)(N+5)}{36 \epsilon_{n}^{2}}
+ \frac{(N+2)(2N+7)}{72n \epsilon_{n}}.\end{aligned}$$
The equation $\beta_{n}(u_{n}^{*}) = 0$ defines the fixed point. Thus, we find $$u_{n}^{\ast}=\frac{6}{8 + N}\,\epsilon_{n}\Biggl\{1 + \epsilon_{n}
\frac{1}{n}\Biggl[ - \frac{1}{2} + \frac{(9N + 42)}{(8 + N)^{2}}\Biggr]\Biggr\}\;\;.$$ We stress that this fixed point is different from that arising in the anisotropic behavior and cannot be obtained from it within the $\epsilon_{L}$-expansion described above. The functions $\gamma_{\phi (n)}$ and $\bar{\gamma}_{\phi^{2} (n)}$ are found to be $$\begin{aligned}
&& \gamma_{\phi (n)} = \frac{(N+2)}{72n} [1 + \frac{5}{4n}
\epsilon_{n}]u_{n}^{2} - \frac{(N+2)(N+8)}{864 n^{2}} u_{n}^{3}, \\
&& \bar{\gamma}_{\phi^{2} (n)} = \frac{(N+2)}{6} u_{n}[1
+ \frac{1}{2n} \epsilon_{n} - \frac{1}{2n} u_{n}].\end{aligned}$$ When the fixed point is replaced inside these equations, using the relation among these functions and the critical exponents $\eta_{n}$ and $\nu_{n}$, we find: $$\begin{aligned}
&& \eta_{n}= \frac{1}{2n} \epsilon_{n}^{2}\,\frac{N + 2}{(N+8)^2}
[1 + \epsilon_{n}\frac{1}{n}(\frac{6(3N + 14)}{(N + 8)^{2}} - \frac{1}{4})] ,\\
&& \nu_{n} =\frac{1}{2n} + \frac{(N + 2)}{4 n^{2}(N + 8)} \epsilon_{n}
+ \frac{1}{8 n^{3}}\frac{(N + 2)(N^{2} + 23N + 60)} {(N + 8)^3} \epsilon_{n}^{2}.\end{aligned}$$ The coefficient of the $\epsilon_{n}^{2}$ term in the exponent $\eta_{n}$ is positive, consistent with its counterpart in the anisotropic cases as well as in the Ising-like case. In the generalized orthogonal approximation the competing momenta are not sufficient to induce its change of sign.
Now using the scaling relations derived for the isotropic case we obtain immediately $$\begin{aligned}
&& \gamma_{n} = 1 + \frac{(N+2)}{2n (N + 8)} \epsilon_{n}
+\frac{(N + 2)(N^{2} + 22N + 52)}{4 n^{2}(N + 8)^{3}} \epsilon_{n}^{2}, \\
&& \alpha_{n} = \frac{(4 - N)}{2n (N + 8)} \epsilon_{n}
- \frac{(N + 2)(N^{2} + 30N + 56)}{4 n^{2} (N + 8)^{3}} \epsilon_{n}^{2} ,\\
&& \beta_{n} = \frac{1}{2} - \frac{3}{2n (N + 8)} \epsilon_{n}
+ \frac{(N + 2)(2N + 1)}{2 n^{2} (N + 8)^{3}} \epsilon_{n}^{2} ,\\
&& \delta_{n} = 3 + \frac{1}{n} \epsilon_{n}
+ \frac{(N^{2} + 14N + 60)}{2 n^{2} (N + 8)^{2}} \epsilon_{n}^{2}.\end{aligned}$$ The explicit dependence on the number of neighbors coupled through competing interactions is manifest in the above exponents. Hence, the universality classes for an arbitrary isotropic ($d=m_{n}$) competing system are determined by $(N,d,n)$. The interesting fact is that the results for the ordinary critical behavior are correctly recovered in the limit $n \rightarrow 1$.
The orthogonal approximation is a good approximation even for isotropic higher character Lifshitz critical behaviors for at least three main reasons. It preserves the homogenity of the Feynman integrals in the external momenta scales. Second, it indicates which parameters are important to describe the universality classes of isotropic systems. And last, but not least, it manifests one of the most important properties of competing systems, namely, the reduction to the Ising-like universality classes in the limit when all interactions beyond first neighbors are turned off.
It is important to emphasize a technical detail concerning the approximation just employed. The leading singularities from the one- and two-loop diagrams contributing to the four-point 1PI vertex part do not get modified from the usual $\phi^{4}$. On the other hand, the leading singularities of the two- and three-loop for the two-point 1PI vertex function do get a factor of $\frac{1}{n}$ with respect to those from the usual critical behavior. These integrals also do not change signs under the generalized orthogonal approximations. The calculation performed without approximations shows that the leading singularities of diagrams contributing to the 1PI two-point function change in a more complicated way, also changing sign depending on the value of $n$. We shall compare the differences in the values of the exponents utilizing the orthogonal approximation and the exact treatment later on. In particular, we shall perform a numerical analysis for the isotropic second character behavior to understanding the deviations in both approaches.
In order to check these results, let us analyse the situation using the minimal subtraction scheme.
Critical exponents in minimal subtraction
-----------------------------------------
Minimal subtraction of dimensional poles in the renormalized vertex $\Gamma_{R (n)}^{(4)}$ can be used to show that all momentum-dependent logarithimic integrals are eliminated in the renormalization process leading the bare dimensionless coupling constant to be written as $$u_{0n} = u_{n}[1 + \frac{(N+8)}{6 \epsilon_{n}} u_{n} +
(\frac{(N+8)^{2}}{36 \epsilon_{n}^{2}} - \frac{(3N+14)}{24n \epsilon_{n}})
u_{n}^{2}].$$ The fixed point can be found out $$u_{n}^{*} = \frac{6}{(N+8)} \epsilon_{n} +
\frac{18(3N+14)}{n (N+8)^{3}} \epsilon_{n}^{2}.$$ In addition, the normalization constants are given by: $$\begin{aligned}
&& Z_{\phi (n)} = 1 - \frac{(N+2)}{144n \epsilon_{n}} u_{n}^{2} \nonumber\\
&& + [-\frac{(N+2)(N+8)}{1296n \epsilon_{n}^{2}} + \frac{(N+2)(N+8)}{5184 n^{2}
\epsilon_{n}}] u_{n}^{3},\\
&& \bar{Z}_{\phi^{2} (n)} = 1 + \frac{(N+2)}{6 \epsilon_{n}} u_{n} \nonumber\\
&& + [\frac{(N+2)(N+5)}{36 \epsilon_{n}^{2}} - \frac{(N+2)}{24n \epsilon_{n}}]
u_{n}^{2}.\end{aligned}$$
Consequently, the functions $\gamma_{\phi (n)}$ and $\bar{\gamma}_{\phi^{2} (n)}$ can be obtained directly
$$\begin{aligned}
&& \gamma_{\phi (n)} = \frac{(N+2)}{72n} u_{n}^{2} - \frac{(N+2)(N+8)}{1728 n^{2}} u_{n}^{3},\\
&& \gamma_{\phi^{2} (n)} = \frac{(N+2)}{6}(u_{n} - \frac{1}{2n} u_{n}^{2}).\end{aligned}$$
Substitution of these expressions in the function $\gamma_{\phi (n)}^{*}$ at the fixed point, one obtains the value of $\eta_{n}$ as obtained in (94). The function $\bar{\gamma}_{\phi^{2} (n)}^{*}$ at the fixed point reads $$\bar{\gamma}_{\phi^{2} (n)}^{*} = \frac{(N+2)}{(N+8)} \epsilon_{n} [1 +
\frac{6(N+3)}{n (N+8)^{2}} \epsilon_{n}],$$ that is the same as the one obtained in the fixed point using normalization conditions. Therefore, it yields the same critical exponent $\nu_{n}$ from (95) as can be easily cheched. This constitutes the equivalence between the two renormalization schemes.
Isotropic critical exponents in the exact calculation
=====================================================
We now turn our attention to the calculation of the isotropic critical exponents exactly, i.e., not using the orthogonal approximation. The algorithm we need to employ to obtain the critical indices is pretty much the same as that used in the orthogonal approximation, since the renormalization program and the scaling laws are approximation independent. The difference is that one should replace the Feynman integrals by their exact values already calculated in Appendix C. As we are going to discuss the case $n=2$ in the next section using normalization conditions and minimal subtraction, we shall take normalization conditions in this section. Nevertheless, with the resources furnished in the text, the reader should be able to check the exponents using minimal subtraction as well.
Using the definitions given for the power series of the bare dimensionless coupling constant and renormalization functions in terms of the dimensionless renormalized coupling constant, we find the following values for the coefficients of each power of $u_{n}$:
$$\begin{aligned}
&& a_{1n} = \frac{N+8}{6 \epsilon_{n}}[1 + D(n) \epsilon_{n}] ,\\
&& a_{2n} = (\frac{N+8}{6 \epsilon_{n}})^{2}
+ \frac{1}{\epsilon_{n}}[\frac{(N^{2} + 21N + 86) D(n)}{18}
- \frac{5N+22}{18}
- \frac{(N+2) (-1)^{n} \Gamma(2n)^{2}}{36 \Gamma(n+1) \Gamma(3n)}\nonumber\\
&& -\frac{5N+22}{18}(\sum_{p=1}^{2n-2} \frac{1}{2n-p}
- 2 \sum_{p=1}^{n-1} \frac{1}{n-p})],\\
&& b_{2n} = (-1)^{n} \frac{(N+2) \Gamma(2n)^{2}}
{72 \Gamma(n+1) \Gamma(3n) \epsilon_{n}}[1 + (D(n) + \frac{3}{4}
- \frac{3}{2} \sum_{p=2}^{2n-1} \frac{1}{p} \nonumber\\
&& + \frac{1}{2} \sum_{p=1}^{n-1} \frac{1}{n-p}
+ \frac{3}{2} \sum_{p=3}^{3n-1} \frac{1}{p})\epsilon_{n}], \\
&& b_{3n} = (-1)^{n}\frac{(N+2)(N+8)\Gamma(2n)^{2}}{108 \Gamma(n+1) \Gamma(3n)}[\frac{1}{6 \epsilon_{n}^{2}}
+ \frac{1}{\epsilon_{n}}(\frac{D(n)}{3} + \frac{1}{24}\nonumber\\
&& - \frac{1}{12} \sum_{p=2}^{2n-1} \frac{1}{p}
- \frac{1}{12} \sum_{p=1}^{n-1} \frac{1}{n-p}
+\frac{1}{12} \sum_{p=3}^{3n-1} \frac{1}{p})], \\
&& c_{1n} = \frac{(N+2)}{6 \epsilon_{n}}[1 + D(n) \epsilon_{n}], \\
&& c_{2n} = \frac{(N+2)(N+5)}{36 \epsilon_{n}^{2}}
+ \frac{(N+2)}{6\epsilon_{n}}[ \frac{(N+8)}{3} D(n) -\frac{1}{2}(1 + D(n)
+ \sum_{p=1}^{2n-2} \frac{1}{2n-p} - 2 \sum_{p=1}^{n-1} \frac{1}{n-p})].\end{aligned}$$
Here $D(n)= \frac{1}{2} \psi(2n) - \psi(n) + \frac{1}{2} \psi(1)$. The fixed point at two-loop level is defined as the zero of the $\beta$-function and it is given by the following expression: $$\begin{aligned}
&& u_{n}^{*} = \frac{6 \epsilon_{n}}{(N+8)}[1 +
\epsilon_{n}[\frac{1}{(N+8)^{2}}
(\frac{(-1)^{n} \Gamma(2n)^{2} (2N+4)}{\Gamma(n+1) \Gamma(3n)} +
(20N+88)(1-D(n)) - D(n) \nonumber\\
&& + \frac{(20N+88)}{(N+8)^{2}}(\sum_{p=1}^{2n-2} \frac{1}{2n-p}
- 2 \sum_{p=1}^{n-1} \frac{1}{2n-p})]].\end{aligned}$$ The renormalization functions $\gamma_{\phi (n)}(u_{n})$ and $\bar{\gamma}_{\phi^{2} (n)}(u_{n})$ can be expressed in the following simple manner in terms of $u_{n}$: $$\begin{aligned}
&&\gamma_{\phi(n)}(u_{n}) = (-1)^{n+1}
\frac{(N+2) \Gamma(2n)^{2}}{36 \Gamma(n+1) \Gamma(3n)}[1
+ (D(n) + \frac{3}{4} - \frac{3}{2} \sum_{p=2}^{2n-1} \frac{1}{p}
+ \frac{1}{2} \sum_{p=1}^{n-1} \frac{1}{n-p} \nonumber\\
&& + \frac{3}{2} \sum_{p=3}^{3n-1} \frac{1}{p})\epsilon_{n}]u_{n}^{2}
+ (-1)^{n+1} \frac{(N+2)(N+8)\Gamma(2n)^{2}}{216 \Gamma(n+1) \Gamma(3n)}
[ - \frac{1}{2} + \sum_{p=2}^{2n-1} \frac{1}{p}
- \sum_{p=1}^{n-1} \frac{1}{n-p} - \sum_{p=3}^{3n-1} \frac{1}{p}]u_{n}^{3};\end{aligned}$$ $$\begin{aligned}
\bar{\gamma}_{\phi^{2}(n)}(u_{n}) = \frac{(N+2)}{6}u_{n}[1 +
D(n) \epsilon_{n} -D(n) u_{n}].\end{aligned}$$ Substitution of the fixed point in the first equation gives directly the anomalous dimensions $\eta_{n}$. Using the scaling law relating the second expression with the exponents $\eta_{n}$ and $\nu_{n}$, one obtains the exponent $\nu_{n}$. Thus, we get $$\begin{aligned}
\eta_{n} = (-1)^{n+1}
\frac{(N+2)\Gamma(2n)^{2}}{(N+8)^{2} \Gamma(n+1) \Gamma(3n)} \epsilon_{n}^{2}
+(-1)^{n+1}\frac{(N+2)\Gamma(2n)^{2} F(N,n)}{(N+8)^{2} \Gamma(n+1) \Gamma(3n)}
\epsilon_{n}^{3};\end{aligned}$$ where $$\begin{aligned}
&&F(N,n)=[((-1)^{n} \frac{\Gamma(2n)^{2} (4N+8)}{\Gamma(n+1)
\Gamma(3n)} + (40N+176)D(n)) \frac{1}{(N+8)^{2}}\nonumber\\
&&- \frac{3}{4}
- \sum_{p=1}^{2n-1} \frac{1}{p} + \frac{1}{2} \sum_{p=1}^{n-1}
\frac{1}{p} + \frac{1}{2} \sum_{p=1}^{3n-1} \frac{1}{p}].\end{aligned}$$ Analogously, $$\begin{aligned}
&& \nu_{n} = \frac{1}{2n} + \frac{(N+2)}{4n^{2}(N+8)}\epsilon_{n}
+ \frac{(N+2)}{4n^{2}(N+8)^{3}} \epsilon_{n}^{2}
[(-1)^{n} (N-4) \frac{\Gamma(2n)^{2}}{\Gamma(n+1) \Gamma(3n)}
+ \frac{(N+2)(N+8)}{2n} \nonumber\\
&& + (14N+40)D(n)].\end{aligned}$$
Now, we use the scaling relations to obtain the remaining critical exponents. We find: $$\begin{aligned}
&&\gamma_{n}= 1 + \frac{(N+2)}{2n(N+8)}\epsilon_{n}
+ \frac{(N+2)}{4n^{2}(N+8)^{3}}\epsilon_{n}^{2}[(-1)^{n}\frac{2n(2N+4)
\Gamma(2n)^{2}}{\Gamma(n+1) \Gamma(3n)} + (N+2)(N+8)\nonumber\\
&& + 2n(14N+40)D(n)],\end{aligned}$$ $$\begin{aligned}
&&\alpha_{n}= \frac{(4-N)}{2n(N+8)}\epsilon_{n}
- \frac{(N+2)}{4n^{2}(N+8)^{3}}\epsilon_{n}^{2}[(-1)^{n}\frac{4n(N-4)
\Gamma(2n)^{2}}{\Gamma(n+1) \Gamma(3n)} + (N-4)(N+8)\nonumber\\
&& + 4n(14N+40)D(n)],\end{aligned}$$ $$\begin{aligned}
&&\beta_{n}= \frac{1}{2} - \frac{3}{2n(N+8)}\epsilon_{n}
+ \frac{(N+2)}{4n^{2}(N+8)^{3}}\epsilon_{n}^{2}[(-1)^{n+1}
\frac{12n \Gamma(2n)^{2}}{\Gamma(n+1) \Gamma(3n)} -3(N+8)\nonumber\\
&& + n(14N+40)D(n)],\end{aligned}$$ $$\begin{aligned}
&&\delta_{n} = 3 + \frac{\epsilon_{n}}{n}
+ \frac{\epsilon_{n}^{2}}{2 n^{2} (N+8)^{2}}[(N+8)^{2} + (-1)^{n} \frac{4n(N+2)\Gamma(2n)^{2}}{\Gamma(n+1) \Gamma(3n)}].\end{aligned}$$
Two important features emerge from this exact picture of the critical exponents shown above. It is instructive to discuss the exponent $\eta_{n}$. First, the sign of the lowest ($O(\epsilon_{n}^{2})$) contribution to the exponent $\eta_{n}$ is determined by the value of $n$. Remember that this fact merely reflects the change of sign of the two-point diagrams depending on the value of $n$. Second, instead of a global factor proportional to $n$ the exact solution has a dependence on $n$ coming from a product of $\Gamma$ functions having $n$ on their arguments as well as a finite sum with terms which depend on $n$. This property is valid for all critical exponents and appears explicitly at two- and higher-loop corrections. This is quite a remarkable new feature of arbitrary isotropic competing systems, going beyond the simple polynomial dependence on $n$ found using the orthogonal approximation.
The universality class reduction to that from the Ising-like one in the limit $n \rightarrow 1$ is obvious. Moreover, the case $n=2$ correctly reproduces the solution of an earlier two-loop calculation. Actually, the results given above extend the calculation of $\eta_{n}$ for arbitrary $n$ to include $O(\epsilon_{n}^{3})$ corrections. In addition, all critical exponents presented here at least up to two-loop order generalize all previous results for arbitrary higher character isotropic Lifshitz points. Therefore, at the loop order considered the results above represent the complete solution to the critical exponents of the CECI model for arbitrary types of isotropic competing interactions.
A numerical analysis for comparison between the results obtained either using the orthogonal approximation or in the exact calculation are worthwhile. The case $n=2$ will be analysed later. Here we display the numerical results for the cases $n=3,4,5,6$.
It would be appropriate to calculate the exponents in either approach in a particular case in order to see if the difference is meaningful. The usual $\epsilon$-expansion is good enough for the numerical estimation of critical exponents associated to three-dimensional critical systems even though the expansion parameter is not small. We can then ask ouselves if the same analogy is valid in order to extract concrete results from a fixed value of the space dimension and number of components of the order parameter field for arbitrary isotropic higher character Lifshitz points. We shall look at space dimensions which yields $\epsilon_{n}=1$ in analogy to the method used to extract numerical results from the ordinary $\epsilon-$expansion for three-dimensional systems.
For an eleven-dimensional lattice, take $N=1$ and $n=3$ for the isotropic third character behavior. The orthogonal approximation yields $\eta_{3}=0.006$, $\nu_{3}=0.177$, $\alpha_{3}=0.046$, $\beta_{3}=0.446$, $\gamma_{3}=1.064$ and $\delta_{3}=3.385$. The exact calculation produces the exponents $\eta_{3}=0.002$, $\nu_{3}=0.174$, $\alpha_{3}=0.085$, $\beta_{3}=0.435$, $\gamma_{3}=1.046$ and $\delta_{3}=3.390$. The maximal deviation ($4.1\%$) occurs for the exponent $\alpha_{3}$ followed by the deviation in the $\gamma_{3}$ exponent ($1.8\%$) and an error in the exponent $\beta_{3}$ around $1\%$. The other exponents have deviations smaller than $0.5\%$.
Consider the case $N=1$, $d=15$, $n=4$. The results from the orthogonal approximation are: $\eta_{4}=0.005$, $\nu_{4}=0.131$, $\alpha_{4}=0.036$, $\beta_{4}=0.459$, $\gamma_{4}=1.046$ and $\delta_{4}=3.279$. The exact calculation yields $\eta_{4}=-0.001$, $\nu_{4}=0.129$, $\alpha_{4}=0.058$, $\beta_{4}=0.449$, $\gamma_{4}=1.029$ and $\delta_{4}=3.282$. The maximal deviations takes place in $\alpha_{4} (2.2\%)$, $\gamma_{4} (1.7\%)$ and $\beta_{4} (1\%)$, while the other exponents have errors smaller than $0.6\%$.
Let us examine the case $N=1$, $d=19$ and $n=5$. The isotropic exponents in the orthogonal approximation are $\eta_{5}=0.004$, $\nu_{5}=0.104$, $\alpha_{5}=0.030$, $\beta_{5}=0.467$, $\gamma_{5}=1.036$ and $\delta_{5}=3.219$. The exact exponents are $\eta_{5}=0.001$, $\nu_{5}=0.102$, $\alpha_{5}=0.064$, $\beta_{5}=0.475$, $\gamma_{5}=1.020$ and $\delta_{5}=3.220$. The maximal deviations are in $\alpha_{5} (3.4\%)$, and $\gamma_{5} (1.6\%)$, whereas the remaining exponents have errors smaller than $0.8\%$[^4].
This analysis leads us to conclude that the orthogonal approximation is very precise to predict numerical values in specific situations, since the deviations are negligible when compared with the exact calculation. Moreover, the above data indicate that the deviations are under control no matter how the number of neighbors increases.
The extra insight from the field-theoretic viewpoint is that the more neighbors are introduced and coupled trough isotropic competing interactions the more the space dimensions seem to split. Then, one line with competing interactions up to second neighbors behaves for all practical purposes as having 2 dimensions. Pushing the argument further, a line with $n$ neighbors interacting through competing interactions seem to have $n$ dimensions. This is a striking general property of the field theory under consideration: in the massless limit presented here, when the free critical propagator is proportional to a $2n$th power of momenta each space direction “splits” in $n$ dimensions. This is a kind of degeneracy in the dimension, which can only be unveiled when more participants (neighbors) are allowed to take place in the isotropic competing system. This is a new aspect of systems whose Lagrangians have kinetic terms described by higher derivatives of the field. Further implications of this phenomenon remain to be investigated.
Now, let us show that these findings easily reproduce and extend the original calculation done by Hornreich, Luban and Shtrikman [@Ho-Lu-Sh] for the $n=2$ case.
Exact isotropic exponents for the second character case
=======================================================
In the last section we found the critical exponents for the isotropic case when competing interactions among arbitrary neighbors are allowed. We now discuss its reduction for the usual second character Lifshitz critical behavior. Since we want to compare our results with other which already appeared in the literature we shall derive the critical indices using normalization conditions and minimal subtraction of poles.
Critical exponents in normalization conditions
----------------------------------------------
The fixed point can be calculated by replacing $n=2$ in Eq.(107) in order to obtain: $$\begin{aligned}
&&u_{2}^{*} = \frac{6 \epsilon_{2}}{(N+8)}(1 - \frac{1}{3} \epsilon_{2}
[\frac{(41N+202)}{(N+8)^{2}} - \frac{1}{4}]).\end{aligned}$$ The renormalization functions $\gamma_{\phi (2)}(u_{2})$ and $\bar{\gamma}_{\phi^{2} (2)}(u_{2})$ can be expressed in the following simple manner in terms of $u_{2}$: $$\begin{aligned}
&&\gamma_{\phi(2)}(u_{2}) = - \frac{(N+2)}{240}[1 +
\frac{131}{120} \epsilon_{2}]u_{2}^{2}
+ \frac{29 (N+2)(N+8)}{28800}u_{2}^{3};\end{aligned}$$ $$\begin{aligned}
&&\bar{\gamma}_{\phi^{2}(2)}(u_{2}) = \frac{(N+2)}{6}u_{2}[1 -
\frac{1}{12} \epsilon_{2} + \frac{1}{12}u_{2}].\end{aligned}$$ Substitution of the fixed point in the first equation gives directly the anomalous dimensions $\eta_{2}$. Thus, we get $$\begin{aligned}
&&\eta_{2} = - \frac{3(N+2)}{20(N+8)^{2}} \epsilon_{2}^{2}
+ \frac{(N+2)}{10(N+8)^{2}}[\frac{(41N+202)}{10(N+8)^{2}} + \frac{23}{80}]
\epsilon_{2}^{3};\end{aligned}$$ From the scaling law relating $\eta_{2}$ with $\nu_{2}$ we obtain: $$\begin{aligned}
&&\nu_{2} = \frac{1}{4} + \frac{(N+2)}{16(N+8)}\epsilon_{2}
+ \frac{(N+2)(15N^{2}+89N+4)}{960(N+8)^{3}} \epsilon_{2}^{2}\end{aligned}$$ Using the scaling relations to obtain the remaining critical exponents, we find: $$\begin{aligned}
&&\gamma_{2}= 1 + \frac{(N+2)}{4(N+8)}\epsilon_{2}
+ \frac{(N+2)(15N^{2}+98N+76)}{240(N+8)^{3}}\epsilon_{2}^{2};\end{aligned}$$ $$\begin{aligned}
&&\alpha_{2}= \frac{(4-N)}{4(N+8)}\epsilon_{2}
+ \frac{(N+2)(-15N^{2}+62N+952)}{240(N+8)^{3}}\epsilon_{2}^{2};\end{aligned}$$ $$\begin{aligned}
&&\beta_{2}= \frac{1}{2} - \frac{3}{4(N+8)}\epsilon_{2}
- \frac{(N+2)(80N+514)}{240(N+8)^{3}}\epsilon_{2}^{2};\end{aligned}$$ $$\begin{aligned}
&&\delta_{2} = 3 + \frac{1}{2} \epsilon_{2}
+ \frac{(5N^{2}+86N+332)}{40(N+8)^{2}} \epsilon_{2}^{2}.\end{aligned}$$ Since the earlier calculations in the literature were performed using minimal subtraction, we shall discuss it next.
Critical exponents in minimal subtraction
-----------------------------------------
Now, let us obtain some renormalization functions at the fixed point using minimal subtraction. Since these objects are universal, finding them is equivalent to find the fixed point. Requiring minimal subtraction of the renormalized vertex $\Gamma_{R(2)}^{(4)}$, the cancellations among logarithmic integrals for arbitrary external momenta indeed take place as expected. It leads to the expression of the bare dimensionless coupling constant $u_{02}$ written in terms of the renormalized dimensionless coupling constant $u_{2}$, namely $$u_{02}= u_{2}[1 + \frac{(N+8)}{6 \epsilon_{2}}u_{2}
+ (\frac{(N+8)^{2}}{36 \epsilon_{2}^{2}} + \frac{(41N+202)}{2160 \epsilon_{2}})
u_{2}^{2}].$$ From the definitions of the normalization constants, we can write the following expressions $$\begin{aligned}
&& Z_{\phi (2)} = 1 + \frac{(N+2)}{480 \epsilon_{2}} u_{2}^{2} \nonumber\\
&& + \frac{(N+2)(N+8)}{4320}[\frac{1}{\epsilon_{2}^{2}} - \frac{23}{518400
\epsilon_{2}}] u_{2}^{3},\\
&& \bar{Z}_{\phi^{2} (2)} = 1 + \frac{(N+2)}{6 \epsilon_{2}} u_{2} \nonumber\\
&& + [\frac{(N+2)(N+5)}{36 \epsilon_{2}^{2}}
+ \frac{N+2}{144 \epsilon_{2}}]u_{2}^{2}.\end{aligned}$$ Consequently, the functions $\gamma_{\phi (2)}(u_{2})$ and $\bar{\gamma}_{\phi^{2} (2)}(u_{2})$ read $$\begin{aligned}
&&\gamma_{\phi(2)}(u_{2}) = - \frac{(N+2)}{240}u_{2}^{2}
+ \frac{23 (N+2)(N+8)}{172800}u_{2}^{3};\end{aligned}$$ $$\begin{aligned}
&&\bar{\gamma}_{\phi^{2}(2)}(u_{2}) = \frac{(N+2)}{6}u_{2}[1 +
\frac{1}{12}u_{2}].\end{aligned}$$ The fixed point can be calculated and shown to be $$u_{2}^{*} = \frac{6}{(N+8)} \epsilon_{2} -
\frac{(41N+202)}{5 (N+8)^{3}} \epsilon_{2}^{2}.$$ Replacement of this expression for the function $\gamma_{\phi (2)}(u_{2}^{*})$ gives precisely the exponent $\eta_{2}$ of the last section up to $O(\epsilon_{2}^{3})$. The corresponding expression for $\bar{\gamma}_{\phi^{2} (2)}(u_{2}^{*})$ is given by $$\bar{\gamma}_{\phi^{2} (2)}^{*} = \frac{(N+2)}{(N+8)} \epsilon_{2} [1 -
\frac{(N+2)(13N+41)}{15 (N+8)^{2}} \epsilon_{2}].$$ The last expression is actually the same as that coming from normalization conditions, therefore leading to the same exponent $\nu_{2}$ whereas the remaining exponents are obtained using the scaling laws. Thus, the equivalence between the two renormalization schemes is complete.
Discussion
----------
First of all, our results for the isotropic $n=2$ case generalizes those in the seminal work by Hornreich, Luban and Shtrikman [@Ho-Lu-Sh]. To see this, we make the identifications $\epsilon_{\alpha} \equiv \epsilon_{2}, \eta_{l4} \equiv \eta_{2}$ and $\nu_{l4} \equiv \nu_{2}$. The equations (10a,b) in [@Ho-Lu-Sh] are identical to our results for $\eta_{2}$ and $\nu_{2}$ Eqs.(120)-(121). The step forward within our method is the exponent $\eta_{2}$ which is obtained up to $O(\epsilon_{2}^{3})$ for the first time. Furthermore, using the scaling relations reported in our previous letter [@Leite4], we found all critical exponents (Eqs.(122)-(125)) exactly, at least up to two-loop order which constitutes another new result for the usual isotropic case.
Next, let us confront the results coming from the generalized orthogonal approximation with those from the exact solution at the same loop order. Both the exact and the approximated one-loop exponents are the same. This can be seen from the $n=2$ particular case or from the generic $n$ isotropic criticality by the direct examination of the explicit results shown in the present article. Two-loop deviations start at $n=2$ and for higher $n$.
Consider the case $n=2$. Take a magnetic system which has $N=1$ in a seven-dimensional lattice and analyse the exponents in each case. We shall restrict ourselves to three significative algarisms in our discussion. First use the orthogonal approximation. At two-loop order, the correlation length exponent yields the result $\nu_{2}=0.276$, and the anomalous dimension is given by $\eta_{2}=0.009$. The susceptibility, specific heat, magnetization and magnetic field exponents are given by $\gamma_{2}=1.103$, $\alpha_{2}=0.061$, $\beta_{2}=0.418$, $\delta_{2}=3.616$. Now use the exact two-loop calculation. We obtain the following numerical values for the critical indices: $\nu_{2}=0.271$, $\eta_{2}=-0.006$, $\gamma_{2}=1.087$, $\alpha_{2}=0.100$, $\beta_{2}=0.406$, $\delta_{2}=3.631$.
Therefore, for all the exponents the difference using either approach starts in the second significative algarisms. Specifically, the maximal error made by using the orthogonal approximation takes place for the specific heat exponent with a deviation of $3.9 \%$ , whereas the minimal error occurs in the correlation length exponent whose difference is approximately $0.5 \%$ . This is a strong evidence that the orthogonal approximation is very good to give reliable information for the isotropic case in this specific situation. This feature was already encountered for uniaxial anisotropic cases, where the approximation showed its usefulness for three-dimensional uniaxial systems. We hope that these results may stimulate the search for these exponents using Monte Carlo numerical simulations in the particular case of second character isotropic Lifshitz critical points.
The remarkable agreement between the numerical results for the critical indices above mentioned either using the orthogonal approximation or the exact two-loop calculations corroborates previous conjectures that the orthogonal approximation is not only good to describe uniaxial systems pertaining to the anisotropic second character Lifshitz critical behaviors but also arbitrary isotropic higher character Lifshitz points.
In fact the numerical analysis can be carried out for higher-dimensional lattices for higher values of $n$. Extending this argument, the case $L=4n$, $N=1$ and $d=L-1$ yields the same value for the expansion parameter and should not deviate very much when both calculations are compared. Perhaps the study of the most general arbitrary isotropic points via numerical tools might be worthwhile as well, since now we have satisfactory numerical results coming from a purely analytical field theoretical investigation.
Conclusions and perspectives
============================
In this paper we have discussed the field theoretic description of the most general competing system, which has a simple lattice model representation named CECI model. It consists of a modified Ising model presenting the most general type of competing exchange couplings among arbitrary neighbors and includes other models previously reported. We have derived explicitly the critical exponents in the anisotropic as well as in the isotropic situations at least up to two-loop level. The CECI model and its field theory representation generalize the description of the second character Lifshitz universality classes in a nontrivial way. In particular, strong anisotropic scale invariance is exact up to the loop order considered here. The universality class reduction is a general property of both anisotropic and isotropic critical behaviors. It implies that when the interactions beyond first neighbors are turned off, the Ising-like universality classes are recovered. This feature is manifest in all exponents.
The anisotropic exponents were calculated by using the generalized orthogonal approximation. The calculation of loop integrals is consistent, since it is rooted in the physical property of homogeneity. It is required for a satisfactory renormalization group treatment with several independent relevant length scales represented by each correlation length associated to the several competing axes. The fixed point is the same irrespective of the competing axes under consideration. In close analogy to the second character case, this result is expected to hold in arbitrary perturbative orders. The second character Lifshitz exponents are easily recovered as a particular case of the generic anisotropic situation described in the paper. Although it is desirable to have an approach that does not require the use of approximations for the anisotropic behaviors, in the present moment it is not obvious. It is a consequence of the appearance of many competing subspaces simultaneously which makes the exact calculation (if not impossible) very difficult. We hope the techniques developed in the present work shed light on a quest for a solution without the necessity of approximations for the anisotropic cases.
The isotropic behaviors were calculated using two different trends to evaluate the Feynman diagrams. The first of them makes use of the orthogonal approximation. It can be noticed that the isotropic behavior cannot be obtained from any type of anisotropic behavior within the framework of the $\epsilon_{L}$-expansions developed in the present work even though the same kind of approximations are employed in both cases. This generalizes the previous situation taking place for second character Lifshitz points. Next, we attacked the diagrams without making any approximation. The result is that deviations in the calculation of the critical exponents start at two-loop level in comparison to the outcome provenient of the orthogonal approximation. We obtain as a particular case of the generic isotropic behavior the second character isotropic behavior, which extend the results first derived by Hornreich, Luban and Shtrikman [@Ho-Lu-Sh]. In this way, we obtain $\eta_{2}$ up to three-loop order and $\nu_{2}$ at two-loops as well as the remaining exponents via the scaling relations derived previously in [@Leite4], a result which was lacking since the discovery of the second character isotropic Lifshitz point.
The most immediate application of the formalism just presented is the calculation of all universal amplitude ratios of certain quantities close to generic higher character Lifshitz points. It is amazing that a detailed scale theory for these quantities is lacking in the literature, even for the simple second character Lifshitz points. In fact, some results were presented for the susceptibility [@Leite5] and specific heat [@Leite6] amplitude ratios at one-loop order. The latter amplitude ratio proved to be reliable to explain the experimental result associated to the magnetic material $MnP$ [@Be]. Nevertheless, a thorough renormalization group analysis is necessary in order to have a better comprehension of the several scale transformations in each competing subspace and the role played by them in the treatment of generic amplitude ratios. In principle we can extend the formalism presented here for the calculations of amplitude ratios including the most general competing system.
The treatment of finite-size effects for the most general Lifshitz critical behavior can be developed as a direct extension of the approach to the Ising-like critical behavior [@Bre-Zinn; @Nemi]. The applications might include systems which are finite (or semi-infinite) along one (or several) of their dimensions, but of infinite extent in the remaining directions. It would be interesting to see how different competing axes alter the approach to the bulk criticalities, for example in parallel plate geometries. The utilization of different boundary conditions in a layered geometry would be particularly simple and instructive to see how the generalization works for arbitrary competing systems. It could be used to investigate how amplitude ratios change with different boundary conditions with respect to the situation occurring in bulk systems [@Leite7].
Typical examples are systems which are finite in all directions, such as a (hyper) cube of size $L$, and systems which are of infinite size in $d'=d-1$ dimensions but are either of finite thickness $L$ along the remaining direction ($d$-dimensional layered geometry) or of a semi-infinite extension. The presence of geometrical restrictions on the domain of systems also requires the introduction of boundary conditions (periodic, antiperiodic, Dirichlet and Neumann) satisfied by the order parameter on the surfaces. In particular, the validity limits of the $\epsilon_{L}$-expansion for these systems and the approach to bulk criticality in a layered geometry can be studied [@Leite7].
One interesting aspect of the generalized orthogonal approximation is that it can actually address the problem of calculating Feynman integrals originating in field theories in the massless limit with odd (greater than 2) powers of momentum in the propagator as well. This subject goes beyond the realm of critical phenomena in competing systems. It might be useful for treating perturbatively a recent proposal of an effective quantum field theory with cubic kinetic terms [@MP]. In addition, the general framework can be used to treat perturbatively other effective quantum field theories with higher derivative kinetic terms which break Lorentz invariance in the infrared regime as the effect of combined gravitational attraction and repulsion [@AH]. This type of theory resembles very much the second character Lifshitz critical behavior above discussed. One can expect that introducing more and more competing gravitational interactions in the infrared (long distance) limit higher powers will appear in the kinetic terms of the corresponding effective field theory. The perturbative analysis of this system would be quite analogous to the generic higher character discussed in the present work.
To summarize, we have described the generic higher character Lifshitz critical behaviors. New field theoretical tools were exposed resulting in new analytical expressions for all the critical indices in the isotropic as well as in the anisotropic cases at least at $O(\epsilon_{L}^{2})$. Other aspects like crossover phenomena and tricritical behavior for this model remain to be studied. New anisotropic behaviors in the absence of a uniform ordered phase for the CECI model are under current investigation.
Acknowledgments
===============
I thank kind hospitality at the Instituto Tecnológico de Aeronáutica where this work has begun and T. Frederico for discussions. I wish to thank J. X. de Carvalho for the preparation of the figures, V. O. Rivelles for conversations and N. Berkovits for useful discussions on an earlier version of this work. I would like to acknowledge financial support from FAPESP, grant number 00/06572-6.
Feynman integrals for anisotropic behaviors
===========================================
In general the critical exponents and other universal ammounts are independent of the renormalization group scheme employed. The explicit calculations in this section are presented in such a way that the results can be checked using more than one renormalization procedure. We shall now describe the generalized orthogonal approximation (GOA) for the integrals appearing in the anisotropic cases.
In order to accomplish the task of calculating the critical indices at least up to two-loop level, we need a minimal set of Feynman diagrams to work with. The one- two- and three-loop integrals we shall need to determine are given by the following expressions
$$I_2 = \int \frac{d^{d-\sum_{n=2}^{L} m_{n}}q \Pi_{n=2}^{L} d^{m_{n}}k_{(n)}}
{[\bigl(\sum_{n=2}^{L}(k_{(n)} + K_{(n)}^{'})^{2}\bigr)^{n} +
(q + P)^{2}] \left(\sum_{n=2}^{L} (k_{(n)}^{2})^{n} + q^{2} \right)}\;\;\;,$$
$$\begin{aligned}
I_{3} =&& \int \frac{d^{d-\sum_{n=2}^{L} m_{n}}{q_{1}}d^{d-\sum_{n=2}^{L} m_{n}}q_{2} \Pi_{n=2}^{L} d^{m_{n}}k_{1 (n)}\Pi_{n=2}^{L} d^{m_{n}}k_{2 (n)}}
{\left( q_{1}^{2} + \sum_{n=2}^{L}(k_{1 (n)}^{2})^{n} \right)
\left( q_{2}^{2} + \sum_{n=2}^{L}(k_{2 (n)}^{2})^{n}\right)}\nonumber\\
&& \qquad\qquad \times \frac{1}{[(q_{1} + q_{2} + P)^{2} + \bigl(\sum_{n=2}^{L}(k_{1 (n)} + k_{2 (n)} + K_{(n)}^{'})^{2}\bigr)^{n}]}\;\;,\end{aligned}$$
$$\begin{aligned}
I_{4} =&& \int \frac{d^{d-\sum_{n=2}^{L} m_{n}}{q_{1}}d^{d-\sum_{n=2}^{L} m_{n}}q_{2}\Pi_{n=2}^{L} d^{m_{n}}k_{1 (n)}\Pi_{n=2}^{L} d^{m_{n}}k_{2 (n)}}
{\left ( q_{1}^{2} + \sum_{n=2}^{L}(k_{1 (n)}^{2})^{n}\right)
\left( (P - q_{1})^{2} + \sum_{n=2}^{L} \bigl((K'_{(n)} - k_{1 (n)})^{2}\bigr)^{n} \right)}\nonumber\\
&&\qquad \times \frac{1}
{\left( q_{2}^{2} + \sum_{n=2}^{L}(k_{2 (n)}^{2})^{n}\right)[(q_{1} - q_{2} + p_{3})^{2} + \sum_{n=2}^{L}\bigl((k_{1 (n)} - k_{2 (n)} + k_{3 (n)}')^{2}\bigl)^{n}]}\;\;,\end{aligned}$$
$$\begin{aligned}
I_{5} &=&
\int \frac{d^{d-\sum_{n=2}^{L} m_{n}}q_{1} d^{d-\sum_{n=2}^{L} m_{n}}q_{2} d^{d-\sum_{n=2}^{L} m_{n}}q_{3} \Pi_{n=2}^{L} d^{m_{n}}k_{1 (n)}}
{\left( q_{1}^{2} + \sum_{n=2}^{L}(k_{1 (n)}^{2})^{n} \right)
\left( q_{2}^{2} + \sum_{n=2}^{L}(k_{2 (n)}^{2})^{n}\right)
\left( q_{3}^{2} + \sum_{n=2}^{L}(k_{3 (n)}^{2})^{n}\right)}\nonumber\\
&& \qquad\qquad\qquad \times \frac{\Pi_{n=2}^{L} d^{m_{n}}k_{2 (n)} \Pi_{n=2}^{L} d^{m_{n}}k_{3 (n)}}{[ (q_{1} + q_{2} - p)^{2} + \sum_{n=2}^{L} \bigl((k_{1(n)} + k_{2(n)} - k'_{(n)})^{2}\bigr)^{n}]}\nonumber\\
&& \qquad\qquad\qquad \times \frac{1}{[(q_{1} + q_{3} - p)^{2} + \sum_{n=2}^{L}\bigl((k_{1(n)} + k_{3(n)} -
k'_{(n)})^{2}\bigr)^{n}]}.\end{aligned}$$
We stress that in the above expressions $P, p_{3}$ and $p$ are external momenta perpendicular to the various competing axes, whereas $K'_{(n)}, k'_{3(n)}$ and $k'_{(n)}$ are external momenta characterizing the $n$th competing axes, respectively. For arbitrary values of the external momenta, these integrals can be calculated by making use of approximations very similar to those first developed to describe second character Lifshitz points [@Leite2]. As a matter of fact, the generalized dissipative approximation was formerly used to compute the critical exponents out of the anomalous dimension and correlation length exponents corresponding to space directions perpendicular to the competing axes for this model. Indeed, the generalized orthogonal approximation was figured out using a similar analogy. Nevertheless, since the dissipative approximation cannot approach the isotropic cases, we shall not describe it in this paper. Instead, we shall make use of the generalized orthogonal approximation, for it can approach both anisotropic and isotropic behaviors.
Certain useful identities will be derived in order to solve the integrals above. Let us proceed to find out them. Our starting point is the integral derived in [@Leite2], namely $$\int_{-\infty}^{\infty} dx_{1}...dx_{m} exp(- a(x_{1}^{2} + ...+x_{m}^{2})^{n})
= \frac{1}{2n} \Gamma(\frac{m}{2n}) a^{\frac{-m}{2n}} S_{m}.$$ After choosing $r^{2}= x_{1}^{2} + ... + x_{m}^{2}$, one can take $y=r^{n}$ to write last equation in the form $$\int_{0}^{\infty} dy y^{\frac{m}{n} - 1} exp(-ay^{2})
= \frac{1}{2} a^{\frac{-m}{2n}} \Gamma(\frac{m}{2n}).$$
The integral $$\int_{-\infty}^{\infty} exp(-ax^{2k} - b(x + x_{0})^{2k}) dx$$ cannot be solved exactly for all $k$. In fact for generic $k \geq 2$ it has no elementary primitive. Nevertheless, one can select only the homogeneous function of $a$ by using the following approximation. First, make the change of variables $y=x^{k}$. Second perform the approximation $(x+x_{0})^{k} \cong x^{k} + x_{0}^{k}$. Thus, this integral becomes: $$\int_{-\infty}^{\infty} exp(-ax^{2k} - b(x + x_{0})^{2k}) dx =
exp(-by_{0}^{2}) \frac{2}{k}\int_{0}^{\infty} exp(-(a+b)y^{2} - 2byy_{0})
y^{\frac{1}{k} -1}dy$$ Next, perform another change of variables, namely, $y' = y + \frac{by_{0}}{a+b}$. We then obtain: $$\begin{aligned}
&\int_{-\infty}^{\infty} exp(-ax^{2k} - b(x + x_{0})^{2k}) dx =
exp(-\frac{ab}{a+b} y_{0}^{2}) \frac{2}{k} [\int_{0}^{\infty}
exp(-(a+b)y'^{2}) (y' -\frac{by_{0}}{a+b})^{\frac{1}{k}-1}dy' \nonumber\\
& - \int_{0}^{\frac{by_{0}}{a+b}} exp(-(a+b)y'^{2}) (y' -\frac{by_{0}}{a+b})^{\frac{1}{k}-1}dy'].\end{aligned}$$ Since the integrals are convergent, we can make use of the approximation $(y'-\frac{b}{2a})^{\frac{1}{k}-1} = y'^{\frac{1}{k}-1} + ...$. The elipsis stands for the remaining terms that will be subtracted from the last integral, producing a type of error function. The original integral is then approximated by its leading contribution, i.e. $$\int_{-\infty}^{\infty} exp(-ax^{2k} - b(x + x_{0})^{2k}) dx \cong
exp(-\frac{ab}{a+b} x_{0}^{2k}) \frac{1}{k}
\Gamma(\frac{1}{2k}) (a+b)^{-\frac{1}{2k}}.$$ It can be easily shown that the generalization for the $m$-sphere yields $$\begin{aligned}
&\int_{-\infty}^{\infty} exp[-a(x_{1}^{2} + ... + x_{m}^{2})^{k}
- b((x_{1}+x_{01})^{2} + ... + (x_{m}+x_{0m})^{2})]^{k} dx_{1}...dx_{m}
\cong \\
&exp(-\frac{ab}{a+b} x_{0}^{2k}) \frac{1}{2k} S_{m} \Gamma(\frac{m}{2k})
(a+b)^{-\frac{m}{2k}}\nonumber.\end{aligned}$$ We found appropriate to name this approximation the generalized orthogonal approximation, for it is a natural generalization of that discussed in the usual second character case [@Leite2].
Let us now start our calculation of loop integrals given by Eqs.(A1)-(A4). We can calculate the one-loop integral using two Schwinger parameters. Using the formula derived above, the integration over quadratic momenta can be shown to be given by $$\begin{aligned}
&& I_2= \frac{1}{2} S_{(d-\sum_{n=2}^{L}m_{n})}
\Gamma(\frac{(d-\sum_{n=2}^{L}m_{n})}{2}) \int^{\infty}_{0}\int^{\infty}_{0}
d\alpha_{1}d\alpha_{2}\,\exp(- \frac{\alpha_{1}
\alpha_{2}P^{2}}{\alpha_{1} + \alpha_{2}})\nonumber\\
&& \quad\times(\alpha_{1} + \alpha_{2})^{- \bigl(\frac{d-\sum_{n=2}^{L}m_{n}}{2} \bigr)}
\int (\Pi_{n=2}^{L}d^{m_{n}}k_{(n)}) \exp(-\alpha_{1}\sum_{n=2}^{L}(k_{(n)}^{2})^{n} - \alpha_{2} \sum_{n=2}^{L}((k_{(n)} + K_{(n)}^{'})^{2})^{n}).\end{aligned}$$ We can now expand the argument of the last exponentials. Notice that now we have a product of independent integrals corresponding to the momentum components along arbitrary competing subspaces. Those integrals cannot be performed analytically. If we use the homogeneity property of the remaining integrals in the arbitrary competing external momenta scales, we can simplify the calculation by utilizing the generalized orthogonal approximation. In fact, taking $(k_{(n)} + K'_{(n)})^{n} \cong k_{(n)}^{n} + K_{(n)}^{' n}$ and using (A11), we obtain: $$\begin{aligned}
&\int d^{m_{n}}k_{(n)}
exp(-\alpha_{1}k_{(n)}^{2n} -\alpha_{2}(k_{(n)}+ K'_{(n)})^{2n}) = \frac{1}{2n}
S_{m_{n}} \nonumber\\
& \times (\alpha_{1} + \alpha_{2})^{-\frac{m_{n}}{2n}} exp(-\frac{\alpha_{1} \alpha_{2} K_{(n)}^{' 2n}}{\alpha_{1} + \alpha_{2}}) \Gamma(\frac{m_{n}}{2n}) \end{aligned}$$
Therefore, we can write the integral as $$\begin{aligned}
&& I_2= \frac{1}{2} S_{(d-\sum_{n=2}^{L}m_{n})}
\Gamma(\frac{(d-\sum_{n=2}^{L}m_{n})}{2})(\Pi_{n=2}^{L}
\frac{S_{m_{n}} \Gamma(\frac{m_{n}}{2n})}{2n})
\int^{\infty}_{0}\int^{\infty}_{0}d\alpha_{1}d\alpha_{2}\nonumber\\
&&\quad\times \,\exp(- \frac{\alpha_{1}
\alpha_{2}(P^{2} + \sum_{n=2}^{L}((K'_{(n)})^{2})^{n}}{\alpha_{1} + \alpha_{2}})\;
(\alpha_{1} + \alpha_{2})^{- \frac{\bigl(d -\sum_{n=2}^{L}\frac{(n-1)m_{n}}{n}\bigr)}{2}}.\end{aligned}$$ Take $x=\alpha_{1} (P^{2} + \sum_{n=2}^{L}((K'_{(n)})^{2})^{n})$ and $y = \alpha_{2}(P^{2} + \sum_{n=2}^{L}((K'_{(n)})^{2})^{n})$. Then, define $v = \frac{x}{x+y}$. Consequently, the parametric integrals can be performed with this change of variables. Using the identity $$\Gamma (a + b x) = \Gamma(a)\,\Bigl[\,1 + b\, x\, \psi(a) + O(x^{2})\,\Bigr],$$ and expressing everything in terms of the $\epsilon_{L}$ parameter leads to the following expression for $I_{2}$: $$\begin{aligned}
& I_{2} = \frac{1}{2}[S_{(d-\sum_{n=2}^{L}m_{n})}
\Gamma(2 - \sum_{n=2}^{L}\frac{m_{n}}{2n})(\Pi_{n=2}^{L}
\frac{S_{m_{n}} \Gamma(\frac{m_{n}}{2n})}{2n})](1 - \frac{\epsilon_{L}}{2} \psi(2 - \sum_{n=2}^{L}\frac{m_{n}}{2n})) \Gamma(\frac{\epsilon_{L}}{2})\nonumber\\
& \times \; \int_{0}^{1} dv(v(1-v)(P^{2} + \sum_{n=2}^{L}((K'_{(n)})^{2})^{n}))^{\frac{-\epsilon_{L}}{2}}.\end{aligned}$$ This is a homogeneous function (with the same homogeneity degree) in $(P,K'_{(n)})$ . The factor $[S_{(d-\sum_{n=2}^{L}m_{n})}
\Gamma(2 - \sum_{n=2}^{L}\frac{m_{n}}{2n})(\Pi_{n=2}^{L}
\frac{S_{m_{n}} \Gamma(\frac{m_{n}}{2n})}{2n})]$ is going to be absorbed in a redefinition of the coupling constant after performing each loop integral and shall be omited henceforth. We can follow two routes from last equation. The first one is to perform the $v$ integral in terms of products of the Euler $\Gamma$ functions. It will be useful in the calculation of higher-loop integrals since we need the subdiagrams of this type in order to compute the complete integral. Then, we obtain $${I}_{2}(P,K'_{(n)}) = (P^{2} + \sum_{n=2}^{L}((K'_{(n)})^{2})^{n}))^{\frac{-\epsilon_{L}}{2}}
\frac{1}{\epsilon_{L}}\biggl(1 + h_{m_{L}} \epsilon_{L} \biggr)\;\;,$$ This is appropriate to calculate the critical exponents only using normalization conditions. But we would like the solution in a form suitable for minimal subtraction as well. The second possibility convenient for both types of renormalization schemes is to expand the last integral as $$\int_{0}^{1} dv (v(1-v)(P^{2} + \sum_{n=2}^{L}((K'_{(n)})^{2})^{n}))^{\frac{-\epsilon_{L}}{2}} =
1 - \frac{\epsilon_{L}}{2}L(P,K'_{(n)}),$$ where $$L(P,K'_{(n)}) =
\int_{0}^{1} dv \;\;ln[v(1-v)(P^{2} +\sum_{n=2}^{L}((K'_{(n)})^{2})^{n} )].$$ Hence, this integral reads $${I}_{2}(P,K'_{(n)}) =
\frac{1}{\epsilon_{L}}\biggl(1 + (h_{m_{L}} - 1) \epsilon_{L}
-\frac{\epsilon_{L}}{2} L(P,K'_{(n)}) \biggr)\;\;,$$ where $h_{m_{L}}= 1 + \frac{(\psi(1) - \psi(2- \sum_{n=2}^{L}\frac{m_{n}}{2n}))}
{2}$. Notice that whenever $m_{3} = ...=m_{L}=0$, $h_{m_{2}}=[i_{2}]_{m}$, and the usual anisotropic Lifshitz critical behavior is trivially obtained from this more general competing situation. This form is convenient for the renormalization using minimal subtraction. Instead, for normalization conditions we have: $${I}_{2 SP_{1}} = ...={I}_{2 SP_{L}} =
\frac{1}{\epsilon_{L}}\biggl(1 + h_{m_{L}}\,\epsilon_{L} \biggr)\;\;,$$ since $L(SP_{1}=...=SP_{L})=-2$, with $SP_{1}\equiv (P^{2}=1,K'_{(n)}=0)$,..., $SP_{L} \equiv (P=0,(K'_{(L)})^{2}=1)$.
The simplifying condition $(k_{(n)}+K'_{(n)})^{n} = k_{(n)}^{n} + K_{(n)}^{' n}$ for the one-loop integral can be generalized to the higher-loop graphs. It is translated in the statement that [*the loop momenta characterizing a certain competition subspace in a given bubble (subdiagram) do not mix to all loop momenta not belonging to that bubble*]{}. The simplest practical application of this principle can be viewed in the calculation of the “sunset” two-loop integral $I_{3}$ contributing to the two-point function $$\begin{aligned}
I_{3} =&& \int \frac{d^{d-\sum_{n=2}^{L} m_{n}}{q_{1}}d^{d-\sum_{n=2}^{L} m_{n}}q_{2} \Pi_{n=2}^{L} d^{m_{n}}k_{1 (n)}\Pi_{n=2}^{L} d^{m_{n}}k_{2 (n)}}
{\left( q_{1}^{2} + \sum_{n=2}^{L}(k_{1 (n)}^{2})^{n} \right)
\left( q_{2}^{2} + \sum_{n=2}^{L}(k_{2 (n)}^{2})^{n}\right)}\nonumber\\
&& \qquad\qquad \times \frac{1}{[(q_{1} + q_{2} + P)^{2} + \bigl(\sum_{n=2}^{L}(k_{1 (n)} + k_{2 (n)} + K_{(n)}^{'})^{2}\bigr)^{n}]}\;\;,\end{aligned}$$
Defining $K^{''}_{(n)}= k_{1(n)} + K^{'}_{(n)}$ and using the condition $k_{2(n)}.K''_{(n)}=0$, one can solve the integral over $q_{2},k_{2(n)}$ first, picking out only the homogeneous part of each individual integral. The remaining parametric integrals contains the divergence (pole in $\epsilon_{L}$) and can be solved as before. Using Eq.(A17), we obtain: $$\begin{aligned}
I_{3}(P, K') =&& \frac{1}{\epsilon_{L}}(1 +h_{m_{L}}) \nonumber\\
&&\qquad \times \int \frac{d^{d-\sum_{n=2}^{L} m_{n}}q_{1} \Pi_{n=2}^{L}
d^{m_{n}}k_{1 (n)}}{ \left( q_{1}^{2} + \sum_{n=2}^{L}(k_{1 (n)}^{2})^{n} \right)
[(q_{1} + P)^{2} + \sum_{n=2}^{L} \bigl ((k_{1 (n)} + K'_{(n)})^{2}\bigr)^{n}]^{\frac{\epsilon_{L}}{2}}}. \end{aligned}$$ Using Feynman parameters, integrating the loop momenta along with the remaining parametric integrals, and expanding the resulting $\Gamma$ functions in $\epsilon_{L}$ we find: $$I_{3}(P, K') = (P^{2} + \sum_{n=2}^{L}K^{' 2n}_{n})\frac{-1}{8 \epsilon_{L}}
(1+ 2h_{m_{L}} \epsilon_{L} -\frac{3}{4} \epsilon_{L} -
2 \epsilon_{L} L_{3}(P, K'_{(n)})),$$ where $$L_{3}(P, K') = \int_{0}^{1} dx (1-x) ln[(P^{2} + \sum_{n=2}^{L}K^{' 2n}_{n}) x(1-x)].$$ At the symmetry points $SP_{n}$, it can be rewritten as $$I_{3 SP_{1}} =...= I_{3 SP_{L}} = \frac{-1}{8 \epsilon_{L}}
(1+ 2 h_{m_{L}} \epsilon_{L} + \frac{5}{4} \epsilon_{L}).$$ From the above equation we can derive the expressions: $$I_{3 SP_{1}}^{'} (\equiv \frac{\partial I_{3 SP_{1}}}{\partial P^{2}})=...=
I_{3 SP_{L}}^{'} (\equiv \frac{\partial I_{3 SP_{2}}}{\partial K^{' 2L}_{(L)}}) =
\frac{-1}{8 \epsilon_{L}}
(1+2h_{m_{L}} \epsilon_{L} + \frac{1}{4} \epsilon_{L}).$$ To complete our description of the 1$PI$ two-point vertex parts, consider the integral $$\begin{aligned}
I_{5} &=&
\int \frac{d^{d-\sum_{n=2}^{L} m_{n}}q_{1} d^{d-\sum_{n=2}^{L} m_{n}}q_{2} d^{d-\sum_{n=2}^{L} m_{n}}q_{3} \Pi_{n=2}^{L} d^{m_{n}}k_{1 (n)}}
{\left( q_{1}^{2} + \sum_{n=2}^{L}(k_{1 (n)}^{2})^{n} \right)
\left( q_{2}^{2} + \sum_{n=2}^{L}(k_{2 (n)}^{2})^{n}\right)
\left( q_{3}^{2} + \sum_{n=2}^{L}(k_{3 (n)}^{2})^{n}\right)}\nonumber\\
&& \qquad\qquad\qquad \times \frac{\Pi_{n=2}^{L} d^{m_{n}}k_{2 (n)} \Pi_{n=2}^{L} d^{m_{n}}k_{3 (n)}}{[ (q_{1} + q_{2} - p)^{2} + \sum_{n=2}^{L} \bigl((k_{1(n)} + k_{2(n)} - k'_{(n)})^{2}\bigr)^{n}]}\nonumber\\
&& \qquad\qquad\qquad \times \frac{1}{[(q_{1} + q_{3} - p)^{2} + \sum_{n=2}^{L}\bigl((k_{1(n)} + k_{3(n)} -
k'_{(n)})^{2}\bigr)^{n}]},\end{aligned}$$ which is the three-loop diagram contributing to the two-point vertex function. Incidentally, there is a symmetry in the dummy loop momenta $q_{2} \rightarrow q_{3}$ and $k_{2(n)} \rightarrow k_{3(n)}$. Concerning the integrations either over $q_{2}, k_{2(n)}$ or $q_{3}, k_{3(n)}$, we use the condition $(k_{2(n)}+(k_{1} - K'))^{n} =
k_{2(n)}^{n} + (k_{1(n)} - K'_{(n)})^{n}$ when the integration is performed over $k_{2}$ as well as $(k_{3(n)}+(k_{1(n)} - K'_{(n)}))^{n} =
k_{3(n)}^{n} + (k_{1} - K'_{(n)})^{n}$ when the integral over $k_{3}$ is realized. The two internal bubbles, which are represented by the integrals over $(q_{2}, k_{2(n)})$ and $(q_{3}, k_{3(n)})$, respectively, are actually the same, resulting in $I_{2}((q_{1} - P), (k_{1(n)} - K'_{(n)}))$. Next take $P\rightarrow -P$, $K'_{(n)}\rightarrow -K'_{(n)}$. Using a Feynman parameter and proceeding in close analogy to the calculation of $I_{3}$ we find: $$I_{5}(P, K'_{(n)}) = (P^{2} + \sum_{n=2}^{L}K^{' 2n}_{n}) \frac{-1}{6 \epsilon_{L}^{2}}
(1+3h_{m_{L}} \epsilon_{L} - \epsilon_{L} - 3 \epsilon_{L} L_{3}(P, K'_{(n)})),$$ At the symmetry points $SP_{1}$, ..., $SP_{L}$ one obtains: $$I_{5 SP_{1}}^{'} (\equiv \frac{\partial I_{5 SP_{1}}}{\partial P^{2}})=...=
I_{5 SP_{L}}^{'} (\equiv \frac{\partial I_{5 SP_{2}}}{\partial K^{'2L}_{(L)}}) =
\frac{-1}{6 \epsilon_{L}^{2}}
(1+3h_{m_{L}} \epsilon_{L} + \frac{1}{2} \epsilon_{L}).$$ Finally we compute one of the two-loop diagrams contributing to the four-point function, namely $$\begin{aligned}
I_{4} =&& \int \frac{d^{d-\sum_{n=2}^{L} m_{n}}{q_{1}}d^{d-\sum_{n=2}^{L} m_{n}}q_{2}\Pi_{n=2}^{L} d^{m_{n}}k_{1 (n)}\Pi_{n=2}^{L} d^{m_{n}}k_{2 (n)}}
{\left ( q_{1}^{2} + \sum_{n=2}^{L}(k_{1 (n)}^{2})^{n}\right)
\left( (P - q_{1})^{2} + \sum_{n=2}^{L} \bigl((K'_{(n)} - k_{1 (n)})^{2}\bigr)^{n} \right)}\nonumber\\
&&\qquad \times \frac{1}
{\left( q_{2}^{2} + \sum_{n=2}^{L}(k_{2 (n)}^{2})^{n}\right)[(q_{1} - q_{2} + p_{3})^{2} + \sum_{n=2}^{L}\bigl((k_{1 (n)} - k_{2 (n)} + k_{3 (n)}')^{2}\bigl)^{n}]}\;\;.\end{aligned}$$ Recall that $P= p_{1} + p_{2}$, $p_{i}$ ($i=1,...,3$) are external momenta perpendicular to the competing axes. On the other hand, $K'_{(n)}= k'_{1(n)} + k'_{2(n)}$, and $k'_{i(n)}$ ($i=1,...,3$) are the external momenta along arbitrary competition directions. We can integrate first over the bubble $(q_{2}, k_{2(n)})$. It is convenient to choose Schwinger parameters in the calculation. Then, we use two Feynman parameters and solve for the loop momenta to obtain the following parametric form $$\begin{aligned}
I_{4}&=& \frac{1}{2} f_{m}(\epsilon_{L})\,
\frac{\Gamma(\epsilon_L) \Gamma(2 - \sum_{n=2}^{L}\frac{m_{n}}{2n} -\frac{\epsilon_{L}}{2}) S_{(d-\sum_{n=2}^{L} m_{n})}}{\Gamma\biggl(\frac{\epsilon_L}{2}\biggr)}(\Pi_{n=2}^{L} \frac{\Gamma(\frac{m_{n}}{2n}) S_{m{n}}}{2n})\nonumber\\
&&\, \times \int_0^1 dy\, y\,(1-y)^{\frac{1}{2}\epsilon_L-1}
\int_0^1 dz
\biggl[ yz(1-yz)(P^{2} + \sum_{n=2}^{L} K^{'2n}_{(n)})
+y(1-y)(p_{3}^{2} + \sum_{n=2}^{L} k^{'2n}_{3(n)})\nonumber\\
&& -2yz(1-y)(p_3.P + \sum_{n=2}^{L} (-1)^{n} k^{'n}_{3(n)} K^{'n}_{(n)})\biggr]^{-\epsilon_L}.\end{aligned}$$ The integral over $y$ is singular at $y=1$ when $\epsilon_L=0$. Replace the value $y=1$ inside the integral over $z$ [@Am; @Leite2], integrate over $y$ and expand the Gamma functions in $\epsilon_{L}$. This implies that $$I_{4} = \frac{1}{2 \epsilon_{L}^{2}} \Bigl(1 +
2\;h_{m_{L}} \epsilon_{L} -\frac{3}{2} \epsilon_{L} - \epsilon_{L} L(P,K')\Bigr).$$ This form is particularly suitable for the renormalization procedure using minimal subtraction. For the purpose of normalization conditions, the value of this integral at the symmetry points discussed before is given by $$I_{4 SP_{1}} = ... = I_{4 SP_{L}} = \frac{1}{2 \epsilon_{L}^{2}} \Bigl(1 +
2\;h_{m_{L}} \epsilon_{L} +\frac{1}{2} \epsilon_{L}\Bigr).$$ The method proposed here is equivalent to a new regularization procedure to calculate Feynman integrals whose propagators have any combination of even powers of momenta. We can define the measure of the $m_{n}$-dimensional sphere in terms of a half integer measure. In fact, taking $k=p^{2n}$, one has $d^{m_{n}}k \equiv d^{\frac{m_{n}}{2n}}p =
\frac {1}{2n} p^{\frac{m_{n}}{2n}-1}dp d\Omega_{m_{n}}$. Hence, the approximation required to solve the integrals results that the new “measure”$d^{\frac{m_{n}}{2n}}p$ is invariant under translations $p'=p+a$. This is a simple generalization of the same property valid for the usual $m_{2}$-fold Lifshitz behaviors.
Isotropic diagrams in the generalized orthogonal approximation
==============================================================
The computation of the Feynman integrals using the generalized orthogonal approximation is simpler in the isotropic cases, since there is only one subspace to be integrated over. At the Lifshitz point $\delta_{0n} = \tau_{nn'}= 0$ and solely the $2L$th power of momentum appears in the propagator for the case of $L$th character isotropic critical point. The isotropic analogous of the one-loop integral contributing to the four-point vertex part is $$I_2 = \int \frac{d^{m_{n}}k}{\bigl((k + K^{'})^{2}\bigr)^{n} (k^{2})^{n} }\;\;\;.$$ We can use two Schwinger parameters and the orthogonality condition $(k+K')^{n} \cong k^{n} + K^{'n}$, resulting in the expression $$I_{2}(K') = \int d^{m_{n}}k \int_{0}^{\infty} \int_{0}^{\infty}
d \alpha_{1} d \alpha_{2} e^{-(\alpha_{1} + \alpha_{2})(k^{2})^{n}}
e^{-2\alpha_{2} K'^{n} k^{n}} e^{-\alpha_{2}(K'^{2})^{n}}.$$ Turning to polar spherical coordinates, take $r^{2}= k_{1}^{2}+...+ k_{m_{n}}^{2}$. Making the transformation $k^{n}=p$ the volume element becomes $d^{m_{n}}k=\frac{1}{n} p^{\frac{m_{n}}{n} -1}dp d\Omega_{m_{n}} \equiv d^{\frac{m_{n}}{n}}p$. The former integral with a $n$th power of momenta changes to a quadratic integral over $p$. After discarding the infinite terms which change the measure $d^{\frac{m_{n}}{n}}k$ under the translation $y'=y+\frac{b}{2a}$, only the leading contribution is picked out and we have $$\int d^{m_{n}}k e^{-a (k^{2})^{n} -b k^{n}} = \int d^{\frac{m_{n}}{n}}p e^{-a p^{2} -b p} \cong a^{-\frac{m_{n}}{2n}} e^{\frac{b^{2}}{4a}} \frac{1}{2n} \Gamma(\frac{m_{n}}{2n}) S_{m_{n}} .$$ When this result is replaced into the expression of $I_{2}(K')$, we get to $$I_{2}(K') = \frac{S_{m_{n}}}{\epsilon_{n}} [ 1 - \frac{\epsilon_{n}}{2n}
(1+L(K'^{2}))].$$ Henceforth we absorb the factor of $S_{m_{n}}$ in this integral through a redefinition of the coupling constant and shall do so after performing each loop integral for arbitrary vertex parts. Note that this absorption factor is different from that arising in the anisotropic case in the limit $d \rightarrow m_{n} =4n -\epsilon_{n}$. Since the geometric angular factor coming from the anisotropic cases becomes singular in the above isotropic limit the attempt of extrapolating from one case to another is not valid, at least within the framework of the $\epsilon_{L}$-expansion presented in this work. This is a further technical evidence that the isotropic and anisotropic cases have to be tackled differently. Thus, $$I_{2}(K') = \frac{1}{\epsilon_{n}} [ 1 - \frac{\epsilon_{n}}{2n} (1+L(K'^{2}))].$$ The suitable symmetry point $(K'^{2})^{n} = 1$ useful for the purpose of normalization conditions leads to the following simple outcome $$I_{2}(K') = \frac{1}{\epsilon_{n}} [1 +\frac{\epsilon_{n}}{2n}].$$ The next step is the evaluation of the integral $$I_{3} = \int \frac{d^{m_{n}}k_{1} d^{m_{n}}k_{2}}{\bigl((k_{1} + k_{2} + K^{'})^{2}\bigr)^{n} (k_{1}^{2})^{n} (k_{2}^{2})^{n}}\;\;\;,$$ Integrate first over $k_{2}$. Take $K''= k_{1} + K'$ and use the condition $(k_{2}+ K'')^{n} \cong k_{2}^{n} + K^{''n}$ to obtain: $$I_{3} = \frac{1}{\epsilon_{n}}[1 + \frac{\epsilon_{n}}{2n}]
\int \frac{d^{m_{n}}k_{1}}
{[\bigl((k_{1} + K^{'})^{2}\bigr)^{n}]^{\frac{\epsilon_{n}}{2n}} (k_{1}^{2})^{n}}\;.$$ Utilizing a Feynman parameter, we can integrate over $k_{1}$. After the expansion $m_{n}=4n -\epsilon_{n}$ is done inside the argument of the resulting $\Gamma$ functions and using the expression $$\int \frac {d^{\frac{m_{n}}{n}}q}{(q^{2} + 2 k.q + m^{2})^{\alpha}} \cong
\frac{1}{2n} \frac{\Gamma(\frac{m_{n}}{2n}) \Gamma(\alpha - \frac{m_{n}}{2n})
(m^{2} - k^{2})^{\frac{m_{n}}{2n} - \alpha} S_{m_{n}}}{\Gamma(\alpha)},$$ the integral $I_{3}$ can be found to be $$I_{3} = -\frac{(K'^{2})^{n}}{8n \epsilon_{n}}[1 + \epsilon_{n}(\frac{1}{4n}
- \frac{2}{n} L_{3}(K'^{2}))].$$ At the symmetry point, this reduces to $$I_{3} = -\frac{1}{8n \epsilon_{n}}[1 + \frac{9}{4n} \epsilon_{n}],$$ implying that $$\frac{\partial I_{3}}{\partial (K'^{2})^{n}}|_{SP} = I_{3}^{'} =
-\frac{1}{8n \epsilon_{4n}}[1 + \frac{5}{4n} \epsilon_{4n}].$$ The 3-loop integral $I_{5}$ is given by $$I_{5} = \int \frac{d^{m_{n}}k_{1} d^{m_{n}}k_{2}d^{m_{n}}k_{3}}
{\bigl((k_{1} + k_{2} + K^{'})^{2}\bigr)^{n} \bigl((k_{1} + k_{3} + K^{'})^{2}\bigr)^{n} (k_{1}^{2})^{n} (k_{2}^{2})^{n} (k_{3}^{2})^{n} }\;\;\;,$$ where we took for convenience the redefinition $K'\rightarrow -K'$. The integrals over $k_{2}$ and $k_{3}$ are the same. Thus, following analogous steps and employing the same reasoning as in the calculation of $I_{3}$ we get to $$I_{5} = -\frac{(K'^{2})^{n}}{6n \epsilon_{n}^{2}}[1 + \epsilon_{n}(\frac{1}{2n}
- \frac{3}{n}L_{3}(K'^{2}))].$$ At the symmetry point, the following expression follows trivially $$\frac{\partial I_{5}}{\partial (K'^{2})^{n}}|_{SP} = I_{5}^{'} =
-\frac{1}{6n \epsilon_{n}^{2}}[1 + \frac{2}{n} \epsilon_{n}].$$ The two-loop integral $I_{4}$ in the isotropic behavior is $$\begin{aligned}
I_{4}\;\; =&& \int \frac{d^{m_{n}}k_{1}d^{m_{n}}k_{2}}
{(k_{1}^{2})^{n} \bigl((K' - k_{1})^{2}\bigr)^{n}
\left(k_{2}^{n}\right) \bigl((k_{1} - k_{2} + k_{3}')^{2}\bigl)^{n}}\;\;,\end{aligned}$$ where $K'= k_{1}' + k_{2}'$. The integration can be done along the same lines of the computation performed for its anisotropic counterpart. It is straightforward to show that $$I_{4}(K'^{2}) = \frac{1}{2\epsilon_{n}^{2}}
[1 -\frac{\epsilon_{n}}{2n}(1 + 2 L(K'^{2}))].$$ At the symmetry point the integral can be rewritten in the form $$I_{4}(K'^{2}=1) = \frac{1}{2\epsilon_{n}^{2}}
[1 +\frac{3 \epsilon_{n}}{2n}].$$ As was shown above, these results are a natural generalization of those originally developed for the second character Lifshitz points. It can be checked that all integrals reduce to the usual $\phi^{4}$ values for $n=1$ and reproduce the results from [@Leite2] in case $n=2$.
Isotropic integrals in the exact calculation
============================================
An interesting feature of the isotropic case is that it can be calculated exactly. We now proceed to yield the exact solution to the Feynman diagrams without performing approximations. $$I_2 = \int \frac{d^{m_{n}}k}{\bigl((k + K^{'})^{2}\bigr)^{n} (k^{2})^{n} }\;\;\;.$$ Using a Feynman parameter and making the continuation $d=m_{n}=4n - \epsilon_{n}$ we get $$I_{2}(K') = \frac{\Gamma(2n -\frac{\epsilon_{n}}{2}) \Gamma(\frac{\epsilon_{n}}{2}) S_{m_{n}}}
{2 \Gamma(n) \Gamma(n)} [\frac {\Gamma(n) \Gamma(n)}{\Gamma(2n)}
- \frac{\epsilon_{n}}{2}L_{n}(K')],$$ where $L_{n}(K')$ is given by $$L_{n}(K') = \int_{0}^{1} dx x^{n-1} (1-x)^{n-1} ln[x(1-x)K'^{2}].$$ This integral is the analogous of the integral $L(K')$ appearing in the orthogonal approximation. Here it depends explicitly on $n$, and that is the reason we have included a subscript in it to emphasize this dependence. The integration over $x$ together with the $\epsilon_{n}$ expansion of the Gamma functions results in $$I_{2}(K')= \frac{S_{m_{n}}}{\epsilon_{n}}[1
-\frac{\epsilon_{L}}{2}(\psi(2n) - \psi(1) +
\frac{\Gamma(2n)}{\Gamma(n) \Gamma(n)} L_{n}(K'))].$$ As before, we absorb the factor $S_{m_{n}}$ in a redefinition of the coupling constant. We need to do this for each loop integral. Thus, $$I_{2}(K')= \frac{1}{\epsilon_{n}}[1
-\frac{\epsilon_{n}}{2}(\psi(2n) - \psi(1) +
\frac{\Gamma(2n)}{\Gamma(n) \Gamma(n)} L_{n}(K'))].$$
This is a useful result for doing minimal subtraction. Defining the quantity $D(n)=\frac{1}{2} \psi(2n) - \psi(n) +
\frac{1}{2} \psi(1)$, at the symmetry point $K'^{2}=1$ the integral turns out to be $$I_{2 SP}= \frac{1}{\epsilon_{n}}[1 + D(n) \epsilon_{n}].$$
Now, let us calculate the integral $I_{3}.$ As before, take $K''= k_{1} + K'$ and solve for the internal bubble $k_{2}$ using Feynman parameters solving the momentum independent integrals over the Feynman parameters and expanding the Gamma functions in $\epsilon_{n}$, we end up with $$I_{3} = K'^{2n} (-1)^{n} \frac{\Gamma^{2}(2n)}{4 \Gamma(3n) \Gamma(n+1)}
\frac{1}{\epsilon_{n}}[1 + \epsilon_{n}(B_{n} - \frac{L_{3n}(K')}{A_{n}})],$$ where $$A_{n} = \frac{\Gamma(2n) \Gamma(n)}{\Gamma(3n)},$$
$$\begin{aligned}
&& B_{n} = D(n)
- \frac{1}{2} \sum_{p=1}^{2n-1} \frac{1}{p}
+ \sum_{p=1}^{n} \frac{1}{p} \nonumber\\
&& +
\frac{\sum_{p=0}^{2n-1} \frac{(2n-1)! (-1)^{p+1}}{2 p! (2n-1-p)! (n+p)^{2}}}{A_{n}},\end{aligned}$$
$$L_{3n}(K') = \int_{0}^{1} dx x^{2n-1} (1-x)^{n-1}ln[x(1-x)K'^{2}].$$
Again, this integral depends explicitly on $n$ and should be compared with its counterpart arising in the orthogonal approximation. We then learn that for massless propagators with arbitrary power of momenta, the external momentum dependent part of the Feynman integrals generalizes the standard $\phi^{4}$ in the manner prescribed above.
At the symmetry point, the integral $I_{3}$ simplifies to the following expression: $$\begin{aligned}
&& I_{3SP} = (-1)^{n}\frac{\Gamma(2n)^{2}}
{4 \Gamma(n+1) \Gamma(3n) \epsilon_{n}}(1+(D(n) + \frac{3}{4} + \frac{1}{n})\epsilon_{n})\nonumber\\
&& [1 +\frac{3 \epsilon_{n}}{2}
(\sum_{p=3}^{3n-1} \frac{1}{p} - \sum_{p=2}^{2n-1} \frac{1}{p})
+ \frac{\epsilon_{n}}{2} \sum_{p=1}^{n-1} \frac{1}{n-p}].\end{aligned}$$ Notice that while the first term inside the parenthesis contributes for arbitrary values of $n$, the last factor of $O(\epsilon_{n})$ into the brackets are corrections which contribute solely for $n\geq2$. (A similar feature will also take place in the calculation of $I_{4}$ and $I_{5}$.) Therefore, taking the derivative with respect to $K'^{2n}$ at the symmetry point produces the result $$\begin{aligned}
&& I'_{3SP} = (-1)^{n}\frac{\Gamma(2n)^{2}}
{4 \Gamma(n+1) \Gamma(3n) \epsilon_{n}}(1+(D(n) + \frac{3}{4})\epsilon_{n})\nonumber\\
&& [1 +\frac{3 \epsilon_{n}}{2}
(\sum_{p=3}^{3n-1} \frac{1}{p} - \sum_{p=2}^{2n-1} \frac{1}{p})
+ \frac{\epsilon_{n}}{2} \sum_{p=1}^{n-1} \frac{1}{n-p}].\end{aligned}$$
We now calculate the three-loop integral $I_{5}$. Proceeding analogously, we can show that it has the solution $$I_{5} = K'^{2n} (-1)^{n} \frac{\Gamma^{2}(2n)}{3 \Gamma(3n) \Gamma(n+1)}
\frac{1}{\epsilon_{n}^{2}}
[1 + \epsilon_{n}(C_{n} - \frac{3 L_{3n}(K')}{2A_{n}})],$$ where $$\begin{aligned}
&& C_{n} = 2D(n)
- \frac{1}{2} \sum_{p=1}^{2n-1} \frac{1}{p}
+ \frac{3}{2} \sum_{p=1}^{n} \frac{1}{p} \nonumber\\
&& +
\frac{\sum_{p=0}^{2n-1} \frac{(2n-1)! (-1)^{p+1}}{p! (2n-1-p)! (n+p)^{2}}}{A_{n}}.\end{aligned}$$
At the symmetry point this result gets simplified. Taking the derivative with respect to the external momenta we get to: $$\begin{aligned}
&& I'_{5SP} = (-1)^{n}\frac{\Gamma(2n)^{2}}
{3 \Gamma(n+1) \Gamma(3n) \epsilon_{n}^{2}}(1+ D(n) + 1)\epsilon_{L})\nonumber\\
&& [1 + 2 \epsilon_{n}
(\sum_{p=3}^{3n-1} \frac{1}{p} - 2 \sum_{p=2}^{2n-1} \frac{1}{p})
+ \epsilon_{n} \sum_{p=1}^{n-1} \frac{1}{n-p}].\end{aligned}$$ Notice that the $O(\epsilon_{n})$ terms inside the bracket gives a nonvanishing contribution only for $n \geq 2$.
To conclude, let us calculate the integral $I_{4}$. The integral over the bubble $k_{2}$ can be solved directly by taking the effective external momenta as $K''= -k_{1}-k'_{3}$. It has $I_{2}(K'')$ as a subdiagram. Using the information obtained in calculating $I_{2}$ and working out the details we find the intermediate result: $$\begin{aligned}
&& I_{4} = f_{n}(\epsilon_{n})
\frac{\Gamma(2n -\frac{\epsilon_{n}}{2}) \Gamma(\epsilon_{n})}
{2 \Gamma(n)^{2} \Gamma(\frac{\epsilon_{n}}{2})}
\int_{0}^{1} dy y^{2n-1} (1-y)^{\frac{\epsilon_{n}}{2} -1} \nonumber\\
&& \int_{0}^{1} dz [z(1-z)]^{n-1}[yz(1-yz)K^{'2} + y(1-y)k_{3}^{'2}
- 2yz(1-y)K'.k'_{3}]^{-\epsilon_{n}}.\end{aligned}$$ The situation resembles that in the calculation of $I_{4}$ using the orthogonal approximation. Again, set $y=1$ in the integral over $z$ and carry out the integral over $y$ independently. Performing the integral over $y$, we find: $$I_{4} = f_{n}(\epsilon_{n})
\frac{\Gamma(2n -\frac{\epsilon_{n}}{2}) \Gamma(\epsilon_{n}) \Gamma(2n)}
{2 \Gamma(n)^{2} \Gamma(2n + \frac{\epsilon_{n}}{2})}
\int_{0}^{1}dz [z(1-z)]^{n-1} [z(1-z)K^{' 2}]^{-\epsilon_{n}}.$$ Then for the purposes of minimal subtraction, it can be expressed as $$I_{4} = \frac{1}{2 \epsilon_{n}^{2}}[1 +
(D(n) -1 -
\frac{\Gamma(2n) L_{n}}{\Gamma(n)^{2}})\epsilon_{n} -
\epsilon_{n} \sum_{p=1}^{2n-2} \frac{1}{2n-p}].$$ We emphasize that the last $O(\epsilon_{n})$ term in this expression only contributes for $n\geq2$. At the symmetry point, we find $$I_{4SP} = \frac{1}{2 \epsilon_{n}^{2}}[1 +
(D(n) + 1)\epsilon_{n} +
\epsilon_{n}(\sum_{p=1}^{2n-2} \frac{1}{2n-p} - 2 \sum_{p=1}^{n-1}
\frac{1}{n-p})].$$ Once again, the last two terms of $O(\epsilon_{n})$ containing the sums in the above expression correspond to corrections in case $n\geq2$. Since the corrections are absent when starting from the scratch for the $n=1$ case, by neglecting them in the above expressions it can be easily checked that all of these integrals reduce to the values of the ordinary $\lambda \phi^{4}$ in the limit $n \rightarrow 1$. We learn that a general feature of the exact calculation is that higher loop integrals generally receive further contributions to the subleading singularities for $n\geq2$.
In order to make contact with the more concrete case of the usual second character Lifshitz point obtained by Hornreich, Luban and Shtrikman [@Ho-Lu-Sh], we discuss the particular $n=2$ case next.
The n=2 case
------------
Let us now analyse the Feynman integrals involved in the calculation of the critical indices of the ordinary second character Lifshitz critical behavior. This is a mere particular case of the most general isotropic CECI model discussed in the previous subsection. Nevertheless, the discussion of this particular case is useful when comparing with the original previous result obtained by Hornreich, Luban and Shtrikman about three decades ago. Needless to say, both results agree for the exponents $\eta_{L4}$ and $\nu_{L4}$, which in the notation of Section IV correspond to $\eta_{2}$ and $\nu_{2}$. Moreover, our treatment permits to obtain two new results for this behavior: the results for $\eta_{2}$ are extended including corrections up to $O(\epsilon_{2}^{3})$ whereas the remaining exponents are obtained through the complete set of scaling relations derived in [@Leite1] up to $O(\epsilon_{2}^{2})$.
Since the calculation was already indicated in the last subsection, we simply quote the results. For calculating the exponents using minimal subtraction the most appropriate form of the integrals are given by
$$I_{2}(K')= \frac{1}{\epsilon_{L}}
[1 - \frac{11 \epsilon_{L}}{12} - 3 \epsilon_{L} L_{2}(K')].$$
$$I_{3} = K'^{4} \frac{3}{80 \epsilon_{L}}
[1 - \epsilon_{L}(\frac{17}{120} + 20L_{32}(K'))],$$
$$I_{5} = K'^{4} \frac{1}{20 \epsilon_{L}^{2}}
[1 - \epsilon_{L}(\frac{7}{60} + 30L_{32}(K'))],$$
$$I_{4} = \frac{1}{2 \epsilon_{L}^{2}}[1 -
(\frac{23}{12} + 6 L_{12}(K^{' 2}))\epsilon_{L}].$$
For the use of normalization conditions, however, it is convenient expressing these integrals at their symmetry point. Instead of calculating $I_{3}$ and $I_{5}$ at the symmetry point, we need their derivatives with respect to $K^{' 4}$ at the symmetry point. Thus, we find
$$I_{2SP}= \frac{1}{\epsilon_{L}}[1
-\frac{\epsilon_{L}}{12}].$$
$$I_{3SP}^{'} = \frac{3}{80 \epsilon_{L}}
[1 + \frac{131}{120} \epsilon_{L} ],$$
$$I_{5SP}^{'} = \frac{1}{20 \epsilon_{L}^{2}}
[1 + \frac{26}{15}\epsilon_{L}],$$
$$I_{4} = \frac{1}{2 \epsilon_{L}^{2}}[1 - \frac{1}{4} \epsilon_{L}].$$
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FIGURE CAPTIONS
[^1]: e-mail:mleite@df.ufpe.br
[^2]: For an alternative field-theoretic approach for $m$-axial Lifshitz points, see [@Mergulho; @DS].
[^3]: Let us mention briefly another type of anisotropic critical behavior whose theoretical existence is granted from the CECI model structure. In magnetic systems, the absence of the ferromagnetic phase leads to a structure where several modulated phases are mixed among each other. This situation is transliterated in the condition $d=m_{2}+...+m_{L}$. In that case, the normalization conditions can be defined without making reference to the noncompeting subspace. In general the critical dimension will increase and the lowest character modulated phase will play the role of the former ferromagnetic ordered phase. We shall not delve any further into this situation in this paper, but leave the analysis for future work.
[^4]: The maximal deviations also occur for $\alpha_{n}$ and $\gamma_{n}$ for $n=6,7,8$ and decreases for increasing $n$. For $n=8$, the maximal error for $\alpha_{8}$ is about $2.7\%$, whereas the maximal deviation for $\gamma_{8}$ is $1.3\%$.
|
---
author:
- Uffe Haagerup and Sören Möller
bibliography:
- 'WLLN.bib'
title: The law of large numbers for the free multiplicative convolution
---
|
---
address: |
Dept. Physics & Astronomy, Vanderbilt University, Nashville, TN 37235, USA\
E-mail: volker.e.oberacker@vanderbilt.edu
author:
- 'Volker E. Oberacker and A. Sait Umar'
title: 'Mean-field nuclear structure calculations on a Basis-Spline Galerkin lattice'
---
=10000 =10000
Introduction
============
In recent years, the area of nuclear structure physics has shown substantial progress and rapid growth [@NSAC96; @ISOL97]. With detectors such as GAMMASPHERE and EUROGAM, the limits of total angular momentum and deformation in atomic nuclei have been explored, and new neutron rich nuclei have been identified in spontaneous fission studies. Gamma-ray detectors under development such as GRETA [@IYL98] will have improved resolving power and should allow for the identification of weakly populated states never seen before in nuclei. Particularly exciting is the proposed construction of a next-generation ISOL FACILITY in the United States which has been been identified in the 1996 DOE/NSAC Long Range Plan [@NSAC96] as the highest priority for major new construction.
These experimental developments as well as recent advances in computational physics have sparked renewed interest in nuclear structure theory. In contrast to the well-understood behavior near the valley of stability, there are many open questions as we move towards the proton and neutron driplines and towards the limits in mass number (superheavy region). While the proton dripline has been explored experimentally up to Z=83, the neutron dripline represents mostly “terra incognita”. In these exotic regions of the nuclear chart, one expects to see several new phenomena: near the neutron dripline, the neutron-matter distribution will be very diffuse and of large size giving rise to “neutron halos” and “neutrons skins”. One also expects new collective modes associated with this neutron skin, e.g. the “scissors” vibrational mode or the “pygmy” resonance. In proton-rich nuclei, we have recently seen both spherical and deformed proton emitters; this “proton radioactivity” is caused by the tunneling of weakly bound protons through the Coulomb barrier. The investigation of the properties of exotic nuclei is also essential for our understanding of nucleosynthesis in stars and stellar explosions (rp- and r-process). Our primary goal is to carry out high-precision nuclear structure calculations in connection with Radioactive Ion Beam Facilities. Some of the topics of interest are the effective N-N interaction at large isospin, large pairing correlations and their density dependence, neutron halos/skins, and proton radioactivity. Specifically, we are interested in calculating observables such as the total binding energy, charge radii, densities $\rho_{p,n}({\bf r})$, separation energies for neutrons and protons, pairing gaps, and potential energy surfaces.
There are many theoretical approaches to nuclear structure physics. For lack of space, we mention only three of these: in the macroscopic - microscopic method, one combines the liquid drop / droplet model with a microscopic shell correction from a deformed single-particle shell model (Möller and Nix [@MN97], Nazarewicz et al. [@NW94]). For relatively light nuclei, it is possible to diagonalize the nuclear Hamiltonian in a shell model basis. Barrett et al. [@NB97] have recently carried out large-basis no-core shell model calculations for p-shell nuclei. A different approach to the interacting nuclear shell model is the Shell Model Monte Carlo (SMMC) method developed by Dean et al. [@KDL97] which does not involve matrix diagonalization but a path integral over auxiliary fields. This method has been applied to fp-shell and medium-heavy nuclei. Finally, for heavier nuclei one utilizes either nonrelativistic [@DF84; @DN96; @RB97] or relativistic [@ND96; @PV97] mean field theories.
Outline of the theory: HFB formalism in coordinate space
========================================================
As we move away from the valley of stability, surprisingly little is known about the pairing force: For example, what is its density dependence? Large pairing correlations are expected near the drip lines which are no longer a small residual interaction. Neutron-rich nuclei are expected to be highly superfluid due to continuum excitation of neutron “Cooper pairs”. The Hartree-Fock-Bogoliubov (HFB) theory unifies the HF mean field theory and the BCS pairing theory into a single selfconsistent variational theory. The main challenge in the theory of exotic nuclei near the proton or neutron drip line is that the outermost nucleons are weakly bound (which implies a very large spatial extent), and that the weakly-bound states are strongly coupled to the particle continuum. This represents a major problem for mean field theories that are based on the traditional shell model basis expansion method in which one utilizes bound harmonic oscillator basis wavefunctions. As illustrated in Figure \[fig:wsoscill\] a weakly bound state can still be reasonably well represented in the oscillator basis, but this is no longer true for the continuum states. In fact, Nazarewicz et al. [@NW94] have shown that near the driplines the harmonic oscillator basis expansion does not converge even if $N=50$ oscillator quanta are used. This implies that one either has to use a continuum-shell model basis or one has to solve the problem directly on a coordinate space lattice. We have chosen the latter method.
Several years ago, Umar [*et al.*]{} [@CU94] have developed a three-dimensional HF code in Cartesian coordinates using the Basis-Spline discretization technique. The program is based on a density dependent effective N-N interaction (Skyrme force) which also includes the spin-orbit interaction. The code has proven efficient and extremely accurate; it incorporates BCS and Lipkin-Nogami pairing, and various constraints. The configuration space Hartree-Fock approach has had great successes in predicting systematic trends in the global properties of nuclei, in particular the mass, radii, and deformations across large regions of the periodic table.
So far, our attempts to generalize this 3D code to include self-consistent pairing forces (Hartree-Fock-Bogoliubov theory on the lattice) have proven too ambitious. The reason may be the lack of a suitable damping operator in 3D. We have therefore taken a different approach and developed a new Hartree-Fock + BCS pairing code in cylindrical coordinates for axially symmetric nuclei, based on the Galerkin method with B-Spline test functions [@KO96; @K96]. The new code has been written in Fortran 90 and makes extensive use of new data concepts, dynamic memory allocation and pointer variables. Extending this code, we believe that it will be easier to implement HFB in 2D because one can use highly efficient LAPACK routines to diagonalize the lattice Hamiltonian and does not necessarily rely on a damping operator.
We outline now our basic theoretical approach for lattice HFB. As is customary, we start by expanding the nucleon field operator into a complete orthonormal set of s.p. basis states $\phi_i$ $$\hat{\psi}^\dagger ({\bf r},s) = \sum_i \ \hat{c}_i^\dagger \ \phi_i^* ({\bf r},s)$$ which leads to the Hamiltonian in occupation number representation $$\hat{H}= \sum_{i,j} < i|\ t\ |j> \ \hat{c}_i^\dagger \ \hat{c}_j \ +
\frac{1}{4} \sum_{i,j,m,n} <ij|\ \tilde{v}^{(2)} \ |mn>
\ \hat{c}_i^\dagger \ \hat{c}_j^\dagger \ \hat{c}_n \ \hat{c}_m \ .$$ Like in the BCS theory, one performs a canonical transformation to quasiparticle operators $\hat{\beta},\hat{\beta}^\dagger$ $$\left(
\begin{array}{c}
\hat{\beta} \\
\hat{\beta}^\dagger
\end{array}
\right) =
\left(
\begin{array}{cc}
U^\dagger & V^\dagger \\
V^T & U^T
\end{array}
\right)
\left(
\begin{array}{c}
\hat{c} \\
\hat{c}^\dagger
\end{array}
\right) \ .$$ The HFB ground state is defined as the quasiparticle vacuum $$\hat{\beta}_k \ | \Phi_0 > \ = \ 0 \ .$$ The basic building blocks of the theory are the normal density $$\rho_{ij} = < \Phi_0 | \hat{c}_j^\dagger \ \hat{c}_i | \Phi_0 > = (V^*V^T)_{ij}$$ and the pairing tensor $$\kappa_{ij} = < \Phi_0 | \hat{c}_j \ \hat{c}_i | \Phi_0 > = (V^*U^T)_{ij}$$ from which one can form the generalized density matrix $$\mathcal{R} =
\left(
\begin{array}{cc}
\rho & \kappa \\
- \kappa^* & 1 - \rho^*
\end{array}
\right)
\ \Rightarrow \mathcal{R}^\dagger = \mathcal{R}, \ \mathcal{R}^2 = \mathcal{R} \ .$$ Using the definition of the HFB ground state energy $$E' ( \mathcal{R} ) = < \Phi_0 | \hat{H} - \lambda \hat{N} | \Phi_0 >$$ we derive the equations of motion from the variational principle $$\delta \ [ E'( \mathcal{R} ) - {\rm{tr}} \ \Lambda ( \mathcal{R}^2 - \mathcal{R} ) ] \ = \ 0$$ which results in the standard HFB equations $$[ \mathcal{H}, {\mathcal{R}} ] \ = \ 0$$ with the generalized single-particle Hamiltonian $$\mathcal{H} =
\left(
\begin{array}{cc}
h & \Delta \\
- \Delta^* & -h^*
\end{array}
\right)
; \ h = \partial E' / \partial \rho, \ \Delta = \partial E' / \partial \kappa^* \ .$$
Our goal is to transform to a coordinate space representation and solve the resulting differential equations on a lattice. For this purpose, we define two types of quasi-particle wavefunctions $\phi_1,\phi_2$ $$\phi_1^* (E_n, {\bf r} \sigma) \ = \ \sum_i U_{in} \ \phi_i ({\bf r} \sigma) \ ,\ \ \
\phi_2 (E_n, {\bf r} \sigma) \ = \ \sum_i V_{in}^* \ \phi_i ({\bf r} \sigma)$$ in terms of which the particle density matrix for the HFB ground state assumes a very simple mathematical structure [@DN96] $$\begin{aligned}
\rho_0 \ ({\bf r}, \sigma, {\bf r}', \sigma' ) \ = \
< \Phi_0 | \ \hat{\psi}^\dagger ({\bf r}' \sigma') \ \hat{\psi} ({\bf r} \sigma) \ | \Phi_0 > \nonumber \\
= \sum_{i,j} \rho_{ij} \ \phi_i ({\bf r} \sigma) \ \phi_j^* ({\bf r}' \sigma')
= \sum_{E_n > 0}^{\infty}
\phi_2 (E_n, {\bf r} \sigma) \ \phi_2^* (E_n, {\bf r}' \sigma') \ .\end{aligned}$$ In a similar fashion we obtain for the pairing tensor $$\begin{aligned}
\kappa_0 \ ({\bf r}, \sigma, {\bf r}', \sigma' ) \ = \
< \Phi_0 | \ \hat{\psi} ({\bf r}' \sigma') \ \hat{\psi} ({\bf r} \sigma) \ | \Phi_0 > \nonumber \\
= \sum_{i,j} \kappa_{ij} \ \phi_i ({\bf r} \sigma) \ \phi_j ({\bf r}' \sigma')
= \sum_{E_n > 0}^{\infty} \phi_2 (E_n, {\bf r} \sigma) \ \phi_1^* (E_n, {\bf r}' \sigma') \ .\end{aligned}$$
For certain types of effective interactions (e.g. Skyrme forces) the HFB equations in coordinate space are local and have a structure which is reminiscent of the Dirac equation [@DN96] $$\left( \matrix{ ( h-\lambda ) & \tilde h \cr
\tilde h & - ( h-\lambda ) \cr} \right)
\left( \matrix{ \phi_1({\bf r}) \cr
\phi_2({\bf r}) \cr} \right) = E
\left( \matrix{ \phi_1({\bf r}) \cr
\phi_2({\bf r}) \cr} \right) \ ,
\label{eq:hfbeqn}$$ where $h$ is the “particle” Hamiltonian and $\tilde h$ denotes the “pairing” Hamiltonian.
The various terms in $h$ depend on the particle densities $\rho_q(\bf r)$ for protons and neutrons, on the kinetic energy density $\tau_q(\bf r)$, and on the spin-current tensor $J_{ij}(\bf r) $. The pairing Hamiltonian $\tilde h$ has a similar structure, but depends on the pairing densities $\tilde \rho_q(\bf r), \tilde \tau_q(\bf r)$ and $\tilde J_{ij}(\bf r)$ instead. Because of the structural similarity between the Dirac equation and the HFB equation in coordinate space, we encounter here similar computational challenges: for example, the spectrum of quasiparticle energies $E$ is unbounded from above [*and*]{} below. The spectrum is discrete for $|E|<-\lambda$ and continuous for $|E|>-\lambda$. In the case of axially symmetric nuclei, the spinor wavefunctions $\phi_1({\bf r})$ and $\phi_2({\bf r})$ have the structure $$\psi^\Omega (\phi,r,z) = \frac{1}{\sqrt{2 \pi}}
\left( \matrix{ e^{i(\Omega - \frac 12)\phi} \ U(r,z) \cr
e^{i(\Omega + \frac 12)\phi} \ L(r,z) \cr} \right) \ .
\label{eq:spinor}$$
Computational method: Spline-Galerkin lattice representation
============================================================
For nuclei near the p/n driplines, we overcome the convergence problems of the traditional shell-model expansion method by representing the nuclear Hamiltonian on a lattice utilizing a Basis-Spline expansion [@WO95; @K96; @KO96]. B-Splines $B_i^M(x)$ are piecewise-continuous polynomial functions of order $(M-1)$. They represent generalizations of finite elements which are B-splines with $M=2$. A set of fifth-order B-Splines is shown in Figure \[fig:bspline\].
Let us now discuss the Galerkin method with B-Spline test functions. We consider an arbitrary (differential) operator equation $${\cal{O}} \bar{f}(x) - \bar{g}(x) = 0\;.
\label{eq:eq4}$$ Special cases include eigenvalue equations of the HF/HFB type where ${\cal{O}}=h$ and $\bar{g}(x)=E \bar{f}(x)$. We assume that both $\bar{f}(x)$ and $\bar{g}(x)$ are well approximated by Spline functions $$\bar{f}(x) \approx f(x) \equiv \sum_{i=1}^{\cal N} B_i^M(x)a^i\;,\ \ \ \
\bar{g}(x) \approx g(x) \equiv \sum_{i=1}^{\cal N} B_i^M(x)b^i\;.
\label{eq:eq5}$$ Because the functions $f(x)$ and $g(x)$ are approximations to the exact functions $\bar{f}(x)$ and $\bar{g}(x)$, the operator equation will in general only be approximately fulfilled $${\cal{O}} f(x) - g(x) = R(x)\;.
\label{eq:eq6}$$ The quantity $R(x)$ is called the [*residual*]{}; it is a measure of the accuracy of the lattice representation. We multiply the last equation from the left with the spline function $B_k(x)$ and integrate over $x$ $$\int v(x) dx B_k(x) {\cal{O}} f(x) - \int v(x) dx B_k(x) g(x) =
\int v(x) dx B_k(x) R(x)\;\;.
\label{eq:eq7}$$ We have included a volume element weight function $v(x)$ in the integrals to emphasize that the formalism applies to arbitrary curvilinear coordinates. Various schemes exist to minimize the residual function $R(x)$; in the Galerkin method one requires that there be no overlap between the residual and an arbitrary B-spline function $$\int v(x) dx B_k(x) R(x) = 0 \;\;.
\label{eq:eq8}$$ This so called [*Galerkin condition*]{} amounts to a [*global reduction of the residual*]{}. Applying the Galerkin condition and inserting the B-Spline expansions for $f(x)$ and $g(x)$ results in $$\sum_i \left[\int v(x) dx B_k(x) {\cal{O}} B_i(x) \right] a^i -
\sum_i \left[\int v(x) dx B_k(x) B_i(x) \right] b^i = 0\;\;.
\label{eq:eq9}$$ Defining the matrix elements $${\cal{O}}_{k i}= \int v(x) dx B_k(x) {\cal{O}} B_i(x)\;\;,\ \
G_{k i} = \int v(x) dx B_k(x) B_i(x)\;
\label{eq:eq10}$$ transforms the (differential) operator equation into a matrix $\times$ vector equation $$\sum_i {\cal{O}}_{k i} a^i = \sum_i G_{k i} b^i\;
\label{eq:eq11}$$ which can be implemented on modern vector or parallel computers with high efficiency. The matrix $G_{k i}$ is sometimes referred to as the [*Gram*]{} matrix; it represents the nonvanishing overlap integrals between different B-Spline functions (see Fig. \[fig:bspline\]). We eliminate the expansion coefficients $a^i,\ b^i$ in the last equation by introducing the function values at the lattice support points $x_\alpha$ including both interior and boundary points.
The upper $(U)$ and lower $(L)$ components of the spinor wavefunctions defined earlier are represented on the 2-D lattice $(r_\alpha,
z_\beta)$ by a product of Basis Splines $B_i (x)$ evaluated at the lattice support points $$U(r_\alpha, z_\beta) = \sum_{i,j} B_i (r_\alpha) \ B_j (z_\beta) \ U^{ij} \ ,
\ \ \
L(r_\alpha, z_\beta) = \sum_{i,j} B_i (r_\alpha) \ B_j (z_\beta) \ L^{ij}
\ .
\label{eq:eq16}$$ We are also extending our previous B-spline work to include nonlinear grids. Use of a nonlinear lattice should be most useful for loosely bound systems near the proton or neutron drip lines. Non-Cartesian coordinates necessitate the use of fixed endpoint boundary conditions; much effort has been directed toward improving the treatment of these boundaries [@KO96].
Numerical tests and results
===========================
We expect our Spline techniques to be superior to the traditional harmonic oscillator basis expansion method in cases of very strong nuclear deformation. To illustrate this point, we have performed a numerical test using a phenomenological (Woods-Saxon) deformed shell model potential. We calculate the single-particle energy spectrum for neutrons in $^{40}Ca$ for quadrupole deformations ranging from strong oblate ($\beta_2=-1.25$) to extreme prolate ($\beta_2=+2.25$). The results are shown in Fig. \[fig:ca40\]. Apparently, for $\beta_2=0$ we correctly reproduce the spherical shell structure of magic nuclei. As $\beta_2$ approaches large positive values our s.p. potential approaches the structure of two separated potential wells; as expected, we observe pairs of levels with opposite parity that are becoming degenerate in energy. The largest quadrupole deformation corresponds physically to a symmetric fission configuration. Clearly, such configurations cannot be described in a single oscillator basis, which confirms the numerical superiority of the B-Spline lattice technique.
In a second test calculation, we have investigated the properties of a nucleus near the neutron drip line. During the last decade the discovery of a ‘neutron halo’ in several neutron-rich isotopes generated a great deal of interest in the area of weakly bound quantum systems. The halo effect was first observed in $^{11}_{\ 3}$Li, which consists of three protons and six neutrons in a central core and two planetary neutrons which comprise the halo. By adjusting the depth of the Woods-Saxon potential so that the separation energy of the last bound neutron is only $10$ keV, i.e. very close to the continuum, we were able to determine this neutron wavefunction on the lattice which shows a very large spatial extent (see Fig. \[fig:li11\]). We conclude that the B-Spline lattice techniques are well-suited for representing weakly bound states near the drip lines; a similar calculation in the basis expansion method would require a large number of oscillator shells.
We now discuss our numerical results for the selfconsistent Hartree-Fock calculations with Skyrme-M$^*$ interaction and BCS pairing. This is a special case of the HFB equation with a constant pairing matrix element. In Fig. \[fig:gd154\] we display the proton density for a heavy nucleus, $^{154}_{\ 64}$Gd, calculated with our new 2-D (HF+BCS) code. It should be noted that ALL 154 nucleons are treated dynamically (no inert core approximation). The theoretical charge density looks quite similar to the experimental result which is shown on the right hand side.
For several spherical nuclei, we have also compared the selfconsistent s.p. energy levels of our 2-D Spline-Galerkin code with a fully converged 1-D radial calculation. The result is shown in Table 1.
[lcrcrc]{} \
\
& 1D Radial & 2D Spline-Galerkin\
& $\Delta=0.025$fm & $\Delta=0.625$fm\
\
$E_{tot}$ & -127.73 MeV & -127.48 MeV\
$E_{s1/2}(n)$ & -33.31 MeV & -33.29 MeV\
$E_{p3/2}(n)$ & -19.88 MeV & -19.86 MeV\
$E_{p1/2}(n)$ & -13.55 MeV & -13.53 MeV\
$E_{s1/2}(p)$ & -29.74 MeV & -29.72 MeV\
$E_{p3/2}(p)$ & -16.48 MeV & -16.45 MeV\
$E_{p1/2}(p)$ & -10.27 MeV & -10.26 MeV\
Plans and Future Directions
---------------------------
Having validated our new (HF+BCS) code on a 2D lattice with the Spline-Galerkin method, we plan to proceed as follows: We are currently working on the 2D HFB implementation with a pairing delta-force. After that, we will generalize the code utilizing the full SkP force with mean pairing field and pairing spin-orbit term. We will also add appropriate constraints, e.g. $Q_{20},Q_{30},\omega j_x$ for calculating potential energy surfaces and rotational bands. As we compare the observables (e.g. total binding energy, charge radii, densities $\rho_{p,n}({\bf r})$, separation energies for neutrons and protons, pairing gaps) with experimental data from the RIB facilities, it will almost certainly be necessary to develop new effective N-N interactions as we move farther away from the stability line towards the p/n drip lines.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research project was sponsored by the U.S. Department of Energy under contract No. DE-FG02-96ER40975 with Vanderbilt University. Some of the numerical calculations were carried out on CRAY supercomputers at the National Energy Research Scientific Computing Center (NERSC) at Lawrence Berkeley National Laboratory. We also acknowledge travel support from the NATO Collaborative Research Grants Program.
[99]{}
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---
abstract: 'The contextual probabilistic quantization procedure is formulated. This approach to quantization has much broader field of applications, compared with the canonical quantization. The contextual probabilistic quantization procedure is based on the notions of probability context and the Principle of Complementarity of Probabilities. The general definition of probability context is given. The Principle of Complementarity of Probabilities, which combines the ideas of the Bohr complementarity principle and the technique of noncommutative probability, is introduced. The Principle of Complementarity of Probabilities is the criterion of possibility of the contextual quantization.'
author:
- |
Andrei Khrennikov[^1]\
International Center for Mathematical Modelling\
in Physics and Cognitive Sciences,\
University of Växjö, S-35195, Sweden,\
e–mail: [Andrei.Khrennikov@msi.vxu.se]{}\
Sergei Kozyrev[^2]\
Institute of Chemical Physics,\
Russian Academy of Science, Moscow, Russia,\
e–mail: [kozyrev@mi.ras.ru]{}
title: |
The Contextual Quantization and\
the Principle of Complementarity of Probabilities
---
\[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Definition]{} \[theorem\][Remark]{} \[theorem\][Proposition]{}
Introduction
============
The canonical approach to quantization (originated by Heisenberg’s ”matrix mechanics” [@Heisenberg]) is based, in essence, on purely algebraic ideas. But quantum mechanics is a probabilistic theory, see for discussion [@Holevo], [@Ballentine], [@KHR6]. Therefore it would be natural to create a quantization procedure by starting directly with the probabilistic structure of an experiment. Of course the authors are well aware that a probabilistic structure can be added to the Heisenberg algebraic quantization through the representation of noncommutative Heisenberg variables by operators in a Hilbert space and by using Born’s probabilistic interpretation of normalized vectors. P.Dirac developed ideas of W.Heisenberg and created the modern variant of canonical quantization: mapping dynamical variables to quantum observables and Poisson bracket to commutator. Intrinsically Dirac’s canonical quantization is also an algebraic quantization. Probabilistic structure is added via the Hilbert space representation of quantum observables.
In this paper we formulate the purely probabilistic approach to quantization, based on the ideas of the Bohr complementarity principle and the methods of noncommutative probability theory, especially on the approach of contextual probability. For this reason we formulate here the new general definition of probabilistic context as a map of (in general, noncommutative) probability spaces. We find that this new approach to quantization is applicable for considerably broader class of models, compared with the canonical quantization.
Let us remind the algebraic version of canonical approach to quantization. Assume that we have a family of noncommutative involutive algebra ${\cal B}(h)$ over complex numbers with the commutation relations, which depend (say polynomially) on the number parameter $h$, and for $h=0$ the algebra ${\cal B}(0)$ is commutative.
Let us have the commutative Poissonian $*$–algebra ${\cal A}$ over the complex numbers, which is isomorphic to ${\cal B}(h)$ as a linear space, and is isomorphic to ${\cal B}(0)$ as a commutative $*$–algebra.
Therefore the mentioned isomorphism of linear spaces, which relates the algebras ${\cal A}$ and ${\cal B}(h)$, defines the $h$–dependent family of algebras ${\cal B}(h)$ with the elements $x(h)$ depending on $h$ (where $x(0)=x\in {\cal A}$).
Let the Poissonian structure, or the Poissonian bracket, on the algebra ${\cal A}$ (the Lie bracket which is the differentiation of ${\cal A}$), be related to the product in ${\cal B}(h)$ as follows: $$\{x,y\}=\lim_{h\to 0} {i\over h} [x(h),y(h)],\qquad x,y \in {\cal A},
\quad x(h),y(h) \in {\cal B}(h)$$ Due to polynomial dependence of the commutation relation in ${\cal
B}(h)$ on $h$ the RHS (right hand side) of the above is well defined.
Then we say that ${\cal B}(h)$ is the quantization of ${\cal A}$, and the linear map $x\mapsto x(h)$ is the quantization map. Note that the quantization map is non unique in the following sense: the different families $x(h)$ (corresponding to the different isomorphisms of linear spaces) will reproduce the same Poissonian bracket.
This is the canonical approach to quantization, which is sufficient to reproduce the standard models of quantum mechanics. But this approach, in our opinion, does not take into account the probabilistic structure of quantum mechanics and moreover, looks too restrictive.
First, it is based on purely algebraic ideas, when quantum mechanics is a probabilistic theory. In the discussed above standard approach we never used the structure of the state (and any probabilistic arguments).
Second, it is obvious that in this approach one can obtain only the commutation relations for the algebras which become commutative for $h=0$, such as the Heisenberg algebra. It is not possible to reproduce in the canonical approach the relations of the quantum Boltzmann algebra (or the algebra of the free creation and annihilation operators) of the form $$A_iA_j^{\dag}=\delta_{ij}$$ which, for example, describes the statistics of collective excitations in the quantum electrodynamics (and many other interacting theories) [@book]. The relations above has no contradiction with the celebrated spin–statistics theorem, since they are valid for the collective interacting degrees of freedom. This relations are derived by the time averaging procedure (called the stochastic limit), and in [@book] the phenomenon of arising of the quantum Boltzmann relations in the stochastic limit was called the third quantization[^3], see Appendix for details.
It would be worth to include the mentioned above probabilistic arguments into more general frameworks of quantization. In the present paper we investigate the quantization procedure based on the contextual approach to probability. We call this procedure the contextual quantization. This quantization procedure will be relevant to a wide class of noncommutative probability spaces with the statistics different from the Bose or Fermi statistics, for instance for the quantum Boltzmann statistics.
In papers [@KHR1]–[@KHR4] there was developed a contextual probabilistic approach to the statistical theory of measurements over quantum as well as classical physical systems. It was demonstrated that by taking into account dependence of probabilities on complexes of experimental physical conditions, [*physical contexts,*]{} we can derive quantum interference for probabilities of alternatives. Such a contextual derivation is not directly related to special quantum (e.g. superposition) features of physical systems.
In the present paper we are performing the next step in developing of the contextual probabilistic approach — the step from the contextual interpretation of noncommutative probability to the contextual quantization procedure. We show that the contextual probabilistic approach gives us not only interpretation of quantum mechanics, but, after corresponding generalization, allows to formulate a new quantization procedure, which is applicable, in principle, not only to models of standard quantum mechanics, but to a wide class of physical phenomena, which may be described by models of noncommutative probability theory, see discussion of [@nrp], [@nra], [@ncpro].
To formulate the contextual quantization procedure we introduce the following notions.
First, we introduce the general definition of probabilistic context as a deformation of embedding of probability spaces: the probability space ${\bf A}$ has context $f$ in the probability space ${\bf B}$, if $f$ is the map $f: {\bf A}\to{\bf B}$ which is a deformation of embedding of probability spaces, see Section 3 for details. This definition gives a contextual probabilistic formulation of the correspondence principle in quantum mechanics, but our aim is to apply it to more general situation. So the aim of this paper is not just a new reformulation of the conventional quantum formalism, but extension of quantum ideology to new domains.
Second, we propose a probabilistic version of the Bohr complementarity principle, which we call the [*Principle of Complementarity of Probabilities*]{}(the PCP) and show, that this principle is the criterion of possibility of the contextual quantization. The Principle of Complementarity of Probabilities describes possibility of unification of probability contexts of several probability spaces in the frameworks of a new larger probability space.
We discuss the relation of the introduced contextual quantization procedure and the noncommutative replica procedure, introduced in [@nrp], [@nra] and show, that the noncommutative replica procedure is the example of the contextual quantization. This shows that the contextual quantization is applicable in the multiplicity of the fields which is much wider than quantum mechanics. This in some sense reflects the original ideas by Bohr.
Then we discuss applications of the contextual quantization to collective interacting commutation relations of the stochastic limit, see also [@book], [@ncpro] for details.
For the review of quantum probability see [@Holevo]. Quantization of thermodynamics was discussed in [@Maslov]. Preliminary discussion of the mathematical framework for the contextual approach was given in [@KhrVol]. Discussion of the Chameleon point of view on quantum measurements which is similar to contextual approach was given in [@Accardi], [@AR]. Contextual approach to Kolmogorov probability with relations to quantum mechanics was discussed in [@KHR2].
We also underline that our Principle of Complementarity of Probabilities can be coupled to investigations on quantum entropy and information dynamics, see [@Ohya1] – [@Ohya3]. It might be that our principle can be reformulated by using the [*language of quantum entropy and information.*]{}
The structure of the present paper is as follows.
In Section 2 we introduce the Principle of Complementarity of Probabilities.
In Section 3 we introduce the definition of probability context as (deformation of) embeddings of probability spaces.
In Section 4 we introduce the contextual quantization procedure based on the notions of probability context and the Principle of Complementarity of Probabilities.
In Section 5 we discuss the relation of the contextual quantization and the noncommutative replica procedure of [@nrp].
In Section 6 we discuss the stochastic limit approach, in which commutation relations for collective operators, which can not be described by canonical quantization, but can be described by the contextual quantization, were obtained.
The Principle of Complementarity of Probabilities
=================================================
In the present Section we discuss the meaning of the complementarity principle in quantum mechanics from the point of view of the probabilistic interpretation of quantum mechanics and introduce the Principle of Complementarity of Probabilities.
Probabilistic interpretation of quantum mechanics is based on the notion of noncommutative (or quantum) probability space. A [*noncommutative probability space*]{} is a pair ${\bf B}=({\cal
B},\psi)$, where ${\cal B}$, called the algebra of observables, is the involutive algebra with unit over the complex numbers, and $\psi$ is a state (positive normed linear functional) on ${\cal
B}$.
The commutative, or Kolmogorovian, probability space is a particular variant of noncommutative probability space for the case when the algebra is commutative. Such a probability space is called classical.
The state of a quantum system is described by density matrix — positive functional on noncommutative algebra of observables. Non compatible observables correspond to noncommuting operators, which we may consider belonging to different classical (commutative) probability subspaces in the full noncommutative probability space. In particular, in the ordinary quantum mechanics (where ${\cal B}$ is the Heisenberg algebra) the position and momentum observables generate commutative subalgebras of the Heisenberg algebra — algebras of functions of the position and momentum observables, respectively. Measuring the observables from the classical subalgebra we build the (classical) state on the classical subalgebra. Therefore, after the observation of the full set of incompatible observables, we obtain the set of classical states on noncommuting classical subalgebras, or the set of classical probability spaces.
The following formulation presented in Discussion with Einstein on Epistemological Problems in Atomic Physics (see [@Bohr], vol. 2, p. 40), perhaps is Bohr s most refined formulation of what he means by the complementary situations of measurements:
[*Evidence obtained under different experimental conditions \[e.g. those of the position vs. the momentum measurement\] cannot be comprehended within a single picture, but must be regarded as \[mutually exclusive and\] complementary in the sense that only the totality of the \[observable\] phenomena exhausts the possible information about the \[quantum\] objects \[themselves\].*]{}
We propose the following probabilistic version of the complementarity principle. From the beginning we consider very general model, which is essentially wider than the conventional quantum model. Thus our aim is not only a probabilistic reformulation of Bohr’s complementarity principle, but the extension of this principle to more general situation[^4].
Assume we have noncommutative involutive algebra of observables ${\cal B}$, and the set ${\cal B}_i$ of subalgebras in ${\cal B}$ with the states $\psi_i$ defined on the corresponding classical subalgebras ${\cal B}_i$ (thus ${\bf B}_i=({\cal B}_i,\psi_i)$ are probability spaces).
\[Def1\] [*We say that the states $\psi_i$ on the subalgebras ${\cal
B}_i$ of the algebra ${\cal B}$ of observables, corresponding to measurements of physically incompatible observables, satisfy the Principle of Complementarity of Probabilities (or the PCP), if they may be unified into the state $\psi$ on the full algebra of observables ${\cal B}$.* ]{}
In short the PCP may be formulated as follows: probability spaces, corresponding to measurements of physically incompatible observables, may be unifed into a larger probability space.
In particular, the initial probability spaces ${\bf B}_i$ can be classical (commutative) as in the above considerations on classical probability spaces generated by incompatible observables. Note that in this definition we did not claim that the initial probability spaces are necessarily commutative and their unification is noncommutative and our definition is more general. Therefore the PCP is not only a reformulation of the Bohr complementarity principle in the probabilistic language, but it also is an extension the field of applicability of the ideas of complementarity.
One can see, that the PCP is not always trivially satisfied. For example, on the algebra with the relation $$[a,a^*]=-1$$ there is no positive faithful state. Therefore, classical states on the subalgebras, generated by $a+a^*$ and $i(a-a^*)$ can not be unified into the faithful state on the full algebra of observables. Therefore, this algebra satisfies the original Bohr complementarity principle but can not satisfy the Principle of Complementarity of Probabilities.
Probabilistic contexts
======================
In the present Section we discuss the contextual approach to noncommutative probability and introduce the new general definition of probabilistic context.
We remind that a morphism of probability spaces is a $*$–homomorphism of algebras which conserves all the correlation functions, i.e. for the morphism $f:{\bf A}=({\cal A}, \phi)\to
{\bf B}=({\cal B}, \psi)$, one has $\psi(f(a))=\phi(a)$ for any $a\in {\cal A}$. A subspace of probability space is defined by a subalgebra of algebra of observables and the restriction of the state on the algebra of observables on this subalgebra. A morphism is an embedding, if it is an injection as a $*$–homomorphism.
The contextual approach in probability theory, see [@KHR1]–[@KHR4], discusses the definition of probability with respect to the complex of physically relevant conditions, or the context. An attempt to give a formal definition of a context was done in [@KhrVol]. Let us formulate the new general definition of a context in noncommutative probability:
\[Def2\][*The contextual representation (or simply context) of the probability space ${\bf A}=({\cal A}, \phi)$ in the probability space ${\bf B}=({\cal B}, \psi)$ is the injective $*$–homomorphism $f:{\cal A}\to {\cal B}$, which satisfies the correspondence principle for the states $\phi$ and $\psi$.* ]{}
In the following we, if no confusion is possible, will use the term context also for the image of $f$ in ${\bf B}$.
Two contexts are incompatible, if their images can not be considered in the frameworks of commutative (or classical) probability space.
Now we define (in philosofical sense) the correspondence principle, see also [@KHR3] for the discussion. Identifying ${\cal A}$ with its image in ${\cal B}$, we formulate the following:
\[Def3\] [*The states $\phi$ and $\psi$ on the involutive algebra ${\cal A}$ satisfy the correspondence principle, if $\psi$ is a deformation of $\phi$.* ]{}
Of course, the definition above does not make sense, if we do not formulate the definition of deformation of the states. For different algebras ${\cal A}$ and different contexts we may have different definitions of deformation. One of the natural examples is the following.
\[DefD\] [*We say that the state $\psi$ on algebra ${\cal A}$ is the deformation of the state $\phi$, if the state $\psi=\psi_h$ depends on the (real) parameter $h$, $\psi_0=\phi$, and for any $a\in {\cal A}$ we have $\lim_{h\to
0}\psi_h(a)=\phi(a)$.*]{}
For the prototypical example the correspondent states mean simply equal states, and the context in Definition \[Def2\] will be simply an embedding of probability spaces. But this simple case does not cover the case of standard quantum mechanics, although may be useful for some other applications of the contextual quantization.
Discuss the important examples of probability contexts, describing the well known two slit experiment, in which we observe quantum interference of a particle, passing through two slits to a screen. In quantum mechanics it is naturally to define a context by fixing of classical probability subspace (classical subalgebra with the restriction of the state). Note that in the definition of context we noted that we will identify the contextual map with it’s image, if no confusion is possible.
In the two slit experiment we have two important probabilistic contexts:
1\) The context of measurements, in which we perform observations of particles. This context, as a probability space, is generated by projections onto the basis of measurements, and in the considered case is given by probability space in which the operator of coordinate along the screen is diagonal.
2\) The context of dynamics — probability space, in which density matrix of the particle is diagonal.
Since these two contexts are incompatible (i.e. density matrix of the particle is non diagonal in the basis of measurements), we observe quantum interference.
The contextual quantization
===========================
The introduced in Section 2 Principle of Complementarity of Probabilities is a necessary condition for the existence of noncommutative probability space. In the present Section we use the Principle of Complementarity of Probabilities to define the quantization procedure based on probabilistic arguments.
Assume we have the family of probability spaces ${\bf A}_i=({\cal
A}_i, \phi_i)$, and the family $f_{ij}:{\cal A}_i\to {\cal B}_j$ of $*$–homomorphisms of ${\cal A}_i$ onto the subalgebras ${\cal
B}_j$ of $*$–algebra ${\cal B}$, which define the states $\psi_j$ on ${\cal B}_j$: $$\psi_j(f_{ij}(a))=\phi_i(a)$$ and therefore make these subalgebras the probability spaces ${\bf
B}_j=({\cal B}_j, \psi_j)$.
We propose the following definition of the contextual quantization.
\[Def4\]
*Let the images ${\cal B}_j$ of algebras ${\cal A}_i$ generate the $*$–algebra ${\cal B}$, and moreover the states $\psi_j$ on the subalgebras ${\cal B}_j\in {\cal B}$ satisfy the Principle of Complementarity of Probabilities and therefore there exists the state $\psi$ on the whole $*$–algebra ${\cal B}$.*
In this case we will say that the probability space ${\bf
B}=({\cal B},\psi)$ is the contextual quantization of the family of probability spaces ${\bf A}_i$ with respect to the contexts ${\bf B}_j=f_{ij}({\bf A}_i)$.
\[pcpc\][We see that in the definition above ${\bf B}_j$ will be the contexts of ${\bf A}_i$, and in the contextual approach the quantization of the probability space is defined by the family of contexts with the images satisfying the Principle of Complementarity of Probabilities. Hence we will also call this principle the Principle of Complementarity of Probabilistic Contexts.]{}
\[homomorphism\][Note that the context map $f$ in Definition \[Def2\] is a homomorphism, while in the canonical quantization we usually can not use homomorphisms to map classical objects to quantum. This is related to the fact that the context of classical probability space in noncommutative probability space is a classical probability subspace (and classical subalgebra (say of coordinates) in quantum algebra may be related to classical algebra by a homomorphism). The contextual quantization is based on totally different idea, compared to canonical quantization: instead of looking for classical constructions, which will be classical traces of quantum phenomena (as in the canonical quantization), we unify several (not necessarily classical) statistics into more general noncommutative statistics. ]{}
Since in a general situation the morphisms of probability spaces (and moreover their deformations) are highly non unique, the contextual quantization is applicable in the situations which are beyond of the frameworks of the canonical quantization. Moreover, the contextual quantization in principle may be applied to some special classical systems.
Consider the following examples. The first example describes contextual quantization of harmonic oscillator.
[**Example 1.**]{} Consider the classical probability space ${\bf A}=({\cal A},\phi)$ described by the real valued classical random variable $x$ with the Gaussian mean zero state $\phi$, which is the Gibbs state of a classical harmonic oscillator.
Consider the noncommutative probability space ${\bf B}=({\cal B},\psi)$, where ${\cal B}$ is the Heisenberg algebra with the (selfadjoint) generators $q_i$, $p_i$, $i=1,\dots,d$ and the relations $$[p_i,q_j]=-ih\delta_{ij}$$ and the state $\psi$ which is the Gibbs state for the harmonic oscillator with $d$ degrees of freedom: $$\psi(X)=\hbox{ tr }e^{-\beta H}X,\qquad
H={1\over 2}\sum_{i=1}^d \left(p_i^2 + q_i^2\right)$$
Then the set of $2d$ injections \[Ex1\] f\_i:xq\_i,g\_j:xp\_j,i,j=1,…,d of probability spaces ${\bf A}\to {\bf B}$ satisfies the conditions of the Definition \[Def4\] and defines the contextual quantization, which transforms the classical mean zero Gaussian real valued random variable into the quantum probability space describing the quantum harmonic oscillator.
The correspondence principle in this example relates the Gibbs states of classical and quantum harmonic oscillators.
Note that in the described contextual quantization, unlike in the canonical quantization, different degrees of freedom are described by different maps $f_i$, $g_j$ (the initial classical probability space ${\bf A}$ has one degree of freedom). In principle we may distinguish different degrees of freedom from the beginning and start contextual quantization from a family of classical probability spaces.
[**Example 2.**]{}Consider the contextual quantization which generates noncommutative probability space ${\bf B}=({\cal B},\psi)$, where ${\cal B}$ is the quantum Boltzmann algebra, generated by the quantum Boltzmann annihilations $A_i$ and creations $A^{\dag}_i$, $i=1,\dots,d$ with the relations \[qB\] A\_iA\_j\^=\_[ij]{} in the Fock state $\psi(X)=\langle\Omega,X\Omega\rangle$, where the vacuum $\Omega$ is annihilated by all annihilations $A_i$.
Consider the probability space ${\bf A}=({\cal A},\phi)$, where ${\cal A}$ is the quantum Boltzmann algebra with one degree of freedom, i.e. the algebra generated by operators $A$, $A^{\dag}$ with the relation $$AA^{\dag}=1$$ and $\phi$ is the Fock state $\phi(X)=\langle\Omega,X\Omega\rangle$. Note that the algebra ${\cal A}$ is noncommutative.
Consider the set of $d$ embeddings of probability spaces ${\bf
A}\to {\bf B}$ \[Ex2\] AA\_i,A\^A\_i\^,i=1,…,d This set of injections satisfies the conditions of Definition \[Def4\] (where the deformation in the correspondence principle is an identity, i.e. the correspondent states are identically equal) and therefore the probability space ${\bf B}$ (quantum Boltzmann for $d$ degrees of freedom in the Fock state) is the contextual quantization of the probability space ${\bf A}$ (the same for one degree of freedom).
[**Example 3.**]{}In the Example 2 we quantized the noncommutative probability space ${\bf A}$ and obtained the more complicated noncommutative probability space ${\bf B}$. Note that if we consider in the noncommutative probability space ${\bf A}$ the commutative subspace ${\bf A}_0$ with the algebra of observables generated by $X=A+A^{\dag}$ and consider the restriction of the quantization procedure on ${\bf A}_0$: we consider the embeddings ${\bf A}_0\to {\bf B}_0$, \[Ex20\] XX\_i=A\_i+A\_i\^,i=1,…,d where ${\bf
B}_0$ is the probability space with the algebra of observables generated by $X_i$ in the Fock state, then the contextual quantization ${\bf B}_0$ of noncommuting contexts of commutative probability space ${\bf A}_0$ will be a noncommutative probability space.
For further discussion of relations used in Examples 2 and 3 see Appendix.
[**Example 4.**]{}Example 1 can be generalized to quantization in superspace, see [@KHRsup]. We would not like to go into details (since it needs a few new definitions which are not directly relevant to contextual quantization). We just mention that there is considered a quantization of supercommutative superalgebras. Thus initial classical algebras are not commutative, but supercommutative.
The next example describes that one can call a [*cognitive quantization*]{}.
[**Example 5.**]{}The next example has no direct relation to quantum mechanics, but we hope, it will find applications in the future. Assume we have several persons which have different points of view on some complex problem. Each point of view is described by a classical probability space, which describes the distribution of possible opinions. If the problem under consideration is complex enough, it may be possible that different points of view can not be unified within the frameworks of a single point of view, or in our description, within the frameworks of a single commutative probability space. Instead, it may be possible, that different points of view (or corresponding probability spaces) are complementary and may be unified within the frameworks of noncommutative probability space, which gives a relevant complete description of the considered complex problem, while it is not possible to give such a description using classical probability space (or single point of view). The Principle of Complementarity of Probabilities here guarantees, that there is no contradiction between different points of view and these points of view are complementary but not contradictory.
In the next Section we discuss the contextual quantization in relation to the noncommutative replica approach, introduced in [@nrp], [@nra], which was one of the motivations for the definitions of the present paper.
The contextual quantization and replicas
========================================
In the present Section we show that the noncommutative replica procedure for disordered systems is the example of the contextual quantization and the contexts describe the way of averaging of quenched disorder.
In papers [@nrp], [@nra] the noncommutative replica procedure for disordered systems was introduced. The noncommutative replica procedure has the form of the embedding of probability spaces. It was mentioned that this embedding is non unique and moreover the combination of different morphisms can not be considered in a commutative probability space, see [@nrp], [@ncpro].
The noncommutative replica procedure is defined (by S. Kozyrev [@nrp], [@nra]) as follows. Consider the system of random Gaussian $N\times N$ matrices $J_{ij}$ with independent matrix elements with the zero mean and unit dispersion of each of the matrix element. Consider disordered system with the Hamitlonian $H[\sigma,J]$, where $\sigma$ enumerates the states of the system and the disorder $J$ is the mentioned above random matrix.
Introduce the commutative probability space where random variable corresponds to the random matrix $J$ and the correlations are defined as by the correlator \[corr\] J\^k = Z\^[-1]{} [1N\^k]{} E(\_e\^[-H\[,J\]]{} J\^k) where $E$ is the expectation of the matrix elements and $$Z=E\left(\sum_{\sigma}e^{-\beta H[\sigma,J]}\right)$$ This probability space describes annealed disordered system and the averaging over $J$ describes the averaging of the disorder. For $\beta=0$ (\[corr\]) reduces to the Gaussian state $E$ on random matrices.
We consider the family of contexts of the annealed probability space in the larger replica probability space generated by the replicas $J_{ij}^{(a)}$ of the random matrix $J_{ij}$, see [@nrp] for details. These contexts we call the noncommutative replica procedures and they have the form \[Delta\] : J\_[ij]{}\_[a=0]{}\^[p-1]{} c\_a J\_[ij]{}\^[(a)]{}; This embedding of probability spaces describes the way of selfaveraging of the quenched disorder.
The context $\Delta$ maps the matrix element $J_{ij}$ into the linear combination of independent replicas $J_{ij}^{(a)}$, enumerated by the replica index $a$. Here $c_a$ are complex coefficients, which should satisfy the condition $$\sum_{a=0}^{p-1} |c_a|^2 =p$$ Varying coefficients $c_a$ we will obtain different morphisms $\Delta$ of probability spaces.
After the morphism $\Delta$ the probability space is described by the correlation functions (\[corr\]) with $J$ replaced by $\Delta J$ and mathematical expectation $E$ replaced by the analogous expectation for the set of independent random matrices (replicas of $J$).
Discuss the large $N$ limit of the probability space (\[corr\]) for the free case $\beta=0$.
In the large $N$ limit, by the Wigner theorem, see [@Wig]–[@ALV], the system of $p$ random matrices with independent variables will give rise to the quantum Boltzmann algebra with $p$ degrees of freedom with the generators $A_a$, $A_a^{\dag}$, $a=0,\dots,p-1$ and the relations $$A_aA_b^{\dag}=\delta_{ab}$$ The Gaussian state on large random matrices in the $N\to\infty$ limit becomes the Fock, or vacuum, state on the quantum Boltzmann algebra: the Fock state is generated by the expectation $\langle\Omega, X\Omega\rangle$ where $\Omega$ is the vacuum: $A_a\Omega=0$, $\forall a$.
The operators $A_a$ are the limits of the large random matrices $$\lim_{N\to\infty}{1\over N}J_{ij}^{(a)}=Q_a=A_a+A_a^{\dag}$$ where the convergence is understood in the sense of correlators (as in the central limit theorem).
Then in the thermodynamic limit $N\to\infty$ the map (\[Delta\]) will take the form of the following map (for which we use the same notation) of the quantum Boltzmann algebra with one degree of freedom into quantum Boltzmann algebra with $p$ degrees of freedom: $$\Delta: A\mapsto {1\over\sqrt{p}}\sum_{a=0}c_a A_a$$ and correspondingly $$\Delta: Q\mapsto {1\over\sqrt{p}}\sum_{a=0}c_a Q_a$$
We see that the set of contexts $\Delta$ with the different coefficients $c_a$ defines the contextual quantization of the commutative probability space, described in (\[Ex20\]).
Appendix: the Stochastic Limit
==============================
In the present Section we briefly discuss the stochastic limit approach, in which deformations of quantum Boltzmann commutation relations were obtained, see [@book]. In this approach we consider the quantum system with the Hamiltonian in the form $$H=H_0+\lambda H_I$$ where $H_0$ is called the free Hamiltonian, $H_I$ is called the interaction Hamiltonian, and $\lambda\in {\bf R}$ is the coupling constant.
We investigate the dynamics of the system in the new slow time scale of the stochastic limit, taking the van Hove time rescaling [@vanHove] $$t\mapsto t/\lambda^2$$ and considering the limit $\lambda\to 0$. In this limit the free evolutions of the suitable collective operators $$A(t,k)=e^{itH_0}A(k)e^{-itH_0}$$ will become quantum white noises: $$\lim_{\lambda\to 0}{1\over\lambda}A\left({t\over
\lambda^2},k\right)=b(t,k)$$ The convergence is understood in the sense of correlators. The $\lambda\to 0$ limit describes the time averaging over infinitesimal intervals of time and allows to investigate the dynamics on large time scale, where the effects of interaction with the small coupling constant $\lambda$ are important. For the details of the procedure see [@book].
The collective operators describe joint excitations of different degrees of freedom in systems with interaction, and may have the form of polynomials over creations and annihilation of the field, or may look like combinations of the field and particles operators etc.
For example, for nonrelativistic quantum electrodynamics without the dipole approximation the collective operator is \[QED\] A\_j(k)=e\^[ikq]{}a\_j(k) where $a_j(k)$ is the annihilation of the electromagnetic (Bose) field with wave vector $k$ and polarization $j$, $q=(q_1,q_2,q_3)$ is the position operator of quantum particle (say electron), $qk=\sum_{i}q_ik_i$.
The nontrivial fact is that, after the $\lambda\to 0$ limit, depending on the form of the collective operator, the statistics of the noise $b(t,k)$ depends on the form of the collective operator and may be nontrivial.
Consider the following examples.
1\) We may have the following possibility \[Bose\] \[b\_i(t,k),b\_j\^(t’,k’)\]=2\_[ij]{}(t-t’)(k-k’)((k)-\_0) which corresponds to the quantum electrodynamics in the dipole approximation, describing the interaction of the electromagnetic field with two level atom with the level spacing (energy difference of the levels) equal to $\omega_0$. Here $\omega(k)$ is the dispersion of quantum field.
In this case the quantum noise will have the Bose statistics, and different annihilations of the noise will commute $$[b_i(t,k),b_j(t',k')]=0$$
2\) The another possibility is the relation \[qB1\] b\_i(t,k)b\_j\^(t’,k’)=2\_[ij]{}(t-t’)(k-k’)((k)+(p)-(p+k)) which corresponds to the quantum electrodynamics without the dipole approximation with interacting operator (\[QED\]). Here $\omega(k)$ and $\varepsilon(p)$ are dispersion functions of the field and of the particle correspondingly.
In this case the quantum noise will have the quantum Boltzmann statistics [@book], [@QED], and different annihilations of the noise will not commute $$b_i(t,k) b_j(t',k')\ne b_j(t',k') b_i(t,k)$$
The commutation relations of the types (\[Bose\]), (\[qB1\]) are universal in the stochastic limit approach (a lot of systems will have similar relations in the stochastic limit $\lambda\to 0
$).
Appendix: Copenhagen and Växjö complementarities
================================================
In the first vesrion of this paper, see [@KOZKHR], we used the following version of the complementarity principle (Växjö complementarity):
[*To obtain the full information about the state of a quantum system, one has to perform measurements of the set of physically incompatible (noncommuting) observables, say the momentum and the coordinate. Measuring only the momentum or only the coordinate we will not obtain the full information about the state of the quantum system.*]{}
In his Email to one of the authors (A. Yu. Khrennikov) A. Plotnitsky remarked:
“First of all, this formulation does not appear to me to correspond to Bohr’s view to complementarity and indeed implies a conflict bewteen the respective interpretations of quantum mechanics, Bohr’s and your own[^5], or any interpretation consistent with your formulation...”
We agree with A. Plotnitsky and in this paper we use the original Bohr’s formulation. The difference between two formulations, “Copenhagen and Växjö complementarities”, is that N. Bohr considered [*possible information*]{} and we considered [*full information.*]{} To consider “possible information” one need not to use a realists interpretation and to consider “full information” (about something) we need to use a realists interpretation, e.g., the Växjö interpretation.
However, in this paper we do not try to connect our mathematical formulation of an extended principle of complementarity to any fixed interpretation of quantum mechanics. This is just a formal mathematically formalized principle. This principle can be considered as the formalization of either the Bohr’s principle or Växjö principle.
Finally, we remark that problems discussed in this paper are closely related to the problem of understanding of information in quantum theory, see [@CH]– [@CH3], [@Plotnitsky].
**Acknowledgements**
The authors would like to thank L.Accardi and I.V.Volovich, L. Ballentine, S. Albeverio, S. Gudder, W. De Muynck, J. Summhammer, P. Lahti, A. Holevo, B. Hiley for fruitful (and rather critical) discussions.
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[^1]: Supported in part by the EU Human Potential Programme, contact HPRN–CT–2002–00279 (Network on Quantum Probability and Applications) and Profile Math. Modelling in Physics and Cogn.Sci. of Växjö Univ.
[^2]: Supported in part by the EU Human Potential Programme, contact HPRN–CT–2002–00279 (Network on Quantum Probability and Applications) and Profile Math. Modelling in Physics and Cogn.Sci. of Växjö Univ., by INTAS YSF 2002–160 F2, CRDF (grant UM1–2421–KV–02), The Russian Foundation for Basic Research (project 02–01–01084) and by the grant of the President of Russian Federation for the support of scientific schools NSh 1542.2003.1.
[^3]: The terminology third quantization is quite divertive. Some authors use it in a totally different framework.
[^4]: N.Bohr discussed at many occasions the possibility to use the principle of complementarity outside of the quantum domain [@Bohr], see [@Plotnitsky], [@Plotnitsky1], [@KHR5], [@KHR6], [@Heisenberg1]. However, his proposals were presented on merely philosophical level and, as a consequence, we did not see any fruitful applications of the principle of complementarity in other domains. Another reason for the absence of such applications of the principle of complementarity was Bohr’s attitude to present this principle as a kind of [*no go*]{} principle. For N.Bohr the main consequence of the principle of complementarity was that [*in quantum mechanics, we are not dealing with an arbitrary renunciation of a more detailed analysis of atomic phenomena, but with a recognition that such an analysis is [*in principle*]{} excluded*]{} [@Bohr] (Bohr’s emphasis). In the opposite, we shall present the principle of complementarity in a constructive form by concentrating on the possibility of [*unification*]{} of statistical data obtained in incompatible experiments into a single quantum probabilistic model. Here we are coming to the crucial difference between Bohr’s view and our view to complementarity. Bohr’s complementarity was merely [*individual complementarity*]{} and our complementarity is [*probabilistic complementarity*]{}. It is well known that historically N.Bohr came to the principle of complementarity through discussion with W.Heisenberg on his uncertainty principle. The original source of all those considerations was the idea that for a single quantum system the position and momentum could not be simultaneously measured. Our main idea is that statistical data for incompatible observables (e.g. the position and momentum) can not be obtained in a single experiment. But, nevertheless, such data, obtained in different experiments can be unified in a single quantum probabilistic model. Thus, in the opposite to N.Bohr and W.Heisenberg, we present a constructive program of the unification of statistical data and not a [*no go*]{} program. Such a constructive approach gives the possibility to extend essentially the domain of applications of the modified principle of complementarity.
At the same time we see the real bounds of the extension of quantum ideology: models in which statistical data, obtained in experiments with incompatible observables, can not be even in principle unified into a single quantum–like probability model, see the example at the end of this section.
[^5]: A. Plotnitsky mentioned the so called Växjö interpretation of quantum mechanics, see [@VI], [@VI1], cf. [@CH], [@CH1].
|
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abstract: |
For a fully chaotic two-dimensional (2D) microcavity laser, we present a theory that guarantees both the existence of a stable single-mode lasing state and the nonexistence of a stable multimode lasing state, under the assumptions that the cavity size is much larger than the wavelength and the external pumping power is sufficiently large. It is theoretically shown that these universal spectral characteristics arise from the synergistic effect of two different kinds of nonlinearities: deformation of the cavity shape and mode interaction due to a lasing medium. Our theory is based on the linear stability analysis of stationary states for the Maxwell-Bloch equations and accounts for single-mode lasing phenomena observed in real and numerical experiments of fully chaotic 2D microcavity lasers.
[*OCIS codes*]{}: (140.3945) Micorcavities; (140.3410) Laser resonators; (270.3430) Laser theory; (000.1600) Classical and quantum physics.
[doi:10.1364/PRJ.5.000B39](https://doi.org/10.1364/PRJ.5.000B39) © 2017 Optical Society of America. Users may use, reuse, and build upon the article, or use the article for text or data mining, so long as such uses are for non-commercial purposes and appropriate attribution is maintained. All other rights are reserved.
author:
- Takahisa Harayama
- Satoshi Sunada
- Susumu Shinohara
title: 'Universal single-mode lasing in fully chaotic two-dimensional microcavity lasers under continuous-wave operation with large pumping power'
---
Introduction
============
Universality is a key concept in quantum chaos study. We can find a common feature in a quantum system if the corresponding classical system exhibits fully chaotic dynamics [@StoeckmannBook; @Haake; @NakamuraHarayama]. The most representative example is the Bohigas-Giannoni-Schmit conjecture concerning universal spectral fluctuations of quantized fully chaotic systems [@BohigasCasatiBerry; @MullerHeusler]. Universality in quantum chaos has also been observed for electron transport in mesoscopic devices [@Jalabert].
Because of an analogy between the classical-quantum and the ray-wave correspondence, quantum chaos theory can be directly applied to “wave-chaotic” systems, which exhibit chaotic dynamics in the ray-optic limit [@StoneNoeckeletc]. A representative example is a two-dimensional (2D) microcavity laser, whose resonant modes can be viewed as those of quantum billiards [@JanCao; @HarayamaShinohara]. By a quantum-chaos approach, the emission patterns of 2D microcavity lasers with various cavity shapes have been successfully explained and predicted [@StoneNoeckeletc; @JanCao; @HarayamaShinohara]. However, there is another important aspect of 2D microcavity lasers that cannot be elucidated only by quantum chaos theory, that is, nonlinear interactions among resonant modes due to a laser gain medium. From the viewpoint of universality, it is of interest to uncover how this additional nonlinear effect manifests itself depending on the chaoticity or integrability of underlying ray dynamics inside a cavity.
A recent experimental study of semiconductor 2D microcavity lasers has demonstrated that single-mode lasing is achieved with a stadium-shaped (i.e., fully chaotic) cavity, while multimode lasing with an elliptic (i.e., nonchaotic) cavity [@Sunada1; @Sunada2]. This drastic difference was attributed to the difference of spatial modal patterns between the stadium and elliptic cavities. It was numerically shown that for the stadium cavity, an arbitrary low-loss modal pair has a significant spatial overlap, while for the elliptic cavity, there exist low-loss modal pairs whose spatial overlaps are small [@Sunada2]. This means that any lasing mode of the stadium cavity tends to strongly interact with the other modes, while multiple lasing modes can coexist for the elliptic cavity because of small interactions among them.
It is important that all of the various fully chaotic stadium-type cavities of different aspect ratios and sizes studied in the experiments of Ref.[@Sunada2] have shown single-mode lasing while all of various integrable elliptic cavities multimode lasing. Therefore, it was conjectured that single-mode lasing is universal for fully chaotic cavity lasers [@Sunada2]. In order to ascertain this universality, it is important to further examine it experimentally and numerically for various 2D cavities. Such studies are important because they would add evidence of the universality. However, these pieces of evidence cannot directly reveal the insight of the universality. In past studies of universalities not only in quantum chaos but also in second-order phase transitions and critical phenomena, establishment of a theory that verifies a universality was the most difficult, challenging and important task [@MullerHeusler; @Fisher].
For a theoretical verification of the conjecture of universal single-mode lasing, it is necessary to evaluate the stability of a stationary-state solution of a full nonlinear model such as the Maxwell-Bloch equations [@Harayama2003; @Harayama2005; @Tureci2006; @Tureci2007]. Stability analysis for one-dimensional lasers in a low pumping regime just above the lasing threshold has been established by Lamb [@Lamb; @L2; @Sargent]. It also has been applied to 2D microcavity lasers and explained the spontaneous symmetry breaking of a lasing pattern [@HarayamaAsymmetricPRL; @Harayama2005]. However, Lamb’s perturbation theory becomes invalid for a high pumping regime where the universal single-mode lasing can be observed because his perturbation theory expresses the population inversion by a power series expansion of lasing modes.
In this paper, we introduce a different expansion method for the population inversion in the Maxwell-Bloch equations that is applicable to a high pumping regime. Furthermore, we explicitly derive the stability matrix for stationary-state solutions, which describes the interactions among a huge number of lasing modes. The matrix elements turn out to be greatly simplified under the assumptions that the cavity size is much larger than the wavelength and the external pumping power is sufficiently large. Moreover, the eigenvalues of this matrix can be analytically evaluated by applying a theorem in linear algebra. This enables us to theoretically show that for a fully chaotic 2D microcavity laser, at least one single-mode lasing state is stable, while all multimode lasing states are unstable. This result provides a theoretical ground for universal single-mode lasing in fully chaotic 2D microcavity lasers.
Fundamental equations
=====================
For modeling the microcavity, we assume that it is wide in the $xy$-directions and thin in the $z$-direction. This allows us to separate the electromagnetic fields into transverse-magnetic (TM) and transverse-electric (TE) modes. Here we focus only on TM modes, whose electric field vector is expressed as ${\mbox{\boldmath $E$}}$ $=$ $(0,0,E_z)$. We also assume that the atoms in the lasing medium have spherical symmetry and two energy levels. The relaxation due to the interaction with the reservoir can be described phenomenologically with decay constants $\gamma_\perp$ for the microscopic polarization $\rho$ and $\gamma_{\parallel}$ for the population inversion $W$. We also need to include phenomenologically the effect of the external energy injected into the lasing medium by the pumping power $W_{\infty}$.
By applying the slowly varying envelope approximation, one can reduce the Maxwell equation as follows, $$\frac{\partial}{\partial t} \tilde{E} =
\frac{i}{2} \left( \nabla^2_{xy} + 1 \right)
\tilde{E}
-\alpha_L \tilde{E}
+\frac{2 \pi N \kappa \hbar}{\varepsilon} \tilde{\rho} ,
\label{SB-in}$$ where ${\tilde{E}}$ and ${\tilde{\rho}}$ are respectively the slowly varying envelopes of the $z$-component of the electric field and the microscopic polarization, $N$ is the number density of the atoms, $\kappa$ is the coupling strength, $\varepsilon$ is the permittivity, and $\alpha_L$ represents the losses describing absorption inside the cavity. In the above, space and time are made dimensionless by the scale transformation $((n_{in}\omega_s/c)x,$ $(n_{in}\omega_s/c)y)$$\rightarrow$ $(x,y)$, $t\omega_s \rightarrow$ $t$, respectively, where $n_{in}$ denotes the effective refractive index inside the cavity and $\omega_s$ is the oscillation frequency of the fast oscillation part of the electric field. In the same way, we have the equation for the electric field outside the cavity, $$\frac{n^2_{out}}{n_{in}^2} \frac{\partial}{\partial t} \tilde{E} =
\frac{i}{2} \left(\nabla^2_{xy} + \frac{n^2_{out}}{n_{in}^2} \right)
\tilde{E},
\label{SB-out}$$ where $n_{out}$ denotes the refractive index outside the cavity. For the boundary condition at infinity, we adopt the outgoing wave condition.
The optical Bloch equations are also transformed to the following form: $$\frac{\partial}{\partial t} \tilde{\rho} =
- \tilde{\gamma}_\perp \tilde{\rho}
- i\Delta_0\tilde{\rho}
+ \tilde{\kappa} W \tilde{E},
\label{SB-Bloch1}$$ $$\frac{\partial}{\partial t} W =
- \tilde{\gamma}_{\parallel} \left( W - W_\infty \right)
- 2 \tilde{\kappa} \left( \tilde{E} \tilde{\rho}^*
+ \tilde{E}^* \tilde{\rho} \right),
\label{SB-Bloch2}$$ where the dimensionless parameters are defined as follows: $\tilde{\gamma}_\perp \equiv \gamma_\perp / \omega_s$ , $\tilde{\gamma}_{\parallel} \equiv \gamma_{\parallel} / \omega_s$ , $\Delta_0 \equiv [\omega_0-\omega_s]/\omega_s$, and $\tilde{\kappa} \equiv \kappa / \omega_s$ has the dimension of the inverse of the electric field and $\omega_0$ is the transition frequency of the two-level atoms.
A theoretical method to obtain stationary-state solutions of Eq. (\[SB-in\])$\sim$Eq. (\[SB-Bloch2\]) has been developed as “steady-state ab initio laser theory (SALT)” [@Tureci2006; @Tureci2007; @HakanScience; @etc]. However, the existence of a stationary-state solution does not always mean its experimental observability. That is, a stationary-state solution must be stable so that it can be experimentally observed, especially when the experiment is performed with continuous-wave operation, where the long-term dynamical effect is expected to be important. Such a dynamical stability is not explicitly incorporated in the SALT approach.
In the following, we carry out the stability analysis of a stationary-state solution for the Maxwell-Bloch equations. Although the equations for the stability analysis are very complicated in general, they turn out to be greatly simplified thanks to full chaoticity and the short wavelength limit. By applying the analysis to a 2D microcavity laser where the ray dynamics are fully-chaotic, we show that at least one single-mode lasing state is stable and all of the multimode lasing states are unstable when the size of the cavity is much larger than the wavelength and the pumping power is sufficiently large.
Dynamics of almost stationary lasing states
===========================================
We assume that near a stationary state the light field and polarization can be expressed as follows: $${\tilde{E}}=\sum_{i}E_i (t) e^{-i {\Delta}_i t} U_i(x,y) ,$$ $${\tilde{\rho}}=\sum_{i}\rho_i (t) e^{ -i {\Delta}_i t } V_i(x,y) ,$$ where ${\Delta}_i$ represents the lasing oscillation frequency. Note that the lasing mode $i$ depends on the pumping power and can be a fusion of several modes that coalesce by frequency-locking, and separate into individual lasing modes with different frequencies as the pumping power decreases [@Sunada1; @HarayamaAsymmetricPRL]. $U_i$ is supposed to be normalized.
Then, from Eq. (\[SB-in\]), we obtain $$\begin{aligned}
\lefteqn{
\frac{d E_i(t)}{d t}
+\sum_{j\neq i} \frac{d E_j(t)}{d t} e^{ -i \Delta_{ij} t } U_{ij} }
\nonumber \\ & &
= \left\{ i\left(\Delta_i +\frac{1}{2}\right)
-\left( \alpha_L +\tilde{\gamma}_{ii} \right) \right\} E_i(t)
\nonumber \\ & &
+\sum_{j\neq i}
\left[
\left\{ i\left(\Delta_j +\frac{1}{2}\right) - \alpha_L \right\} U_{ij}
-\tilde{\gamma}_{ij}
\right]
E_j(t) e^{ -i \Delta_{ij} t }
\nonumber \\ & &
+\frac{ 2\pi N \kappa \hbar }{\epsilon} \sum_j e^{ -i \Delta_{ij} t }
\rho_j(t) \int_D U_i^* (x,y) V_j (x,y) dxdy , \nonumber \\ & &
\label{E_i}\end{aligned}$$ where $\Delta_{ij}$ denotes the frequency difference between modes ${j}$ and ${i}$, i.e., $\Delta_{ij}\equiv \Delta_j - \Delta_i$ and $U_{ij}$ is defined by the inner product $U_{ij}\equiv\int_D U_i^*
(x,y) U_j (x,y) dxdy$, and it will be shown later that $\tilde{\gamma}_{ii}$ is related to the flux of the light field intensity from inside to outside the cavity through the cavity edge and $\tilde{\gamma}_{ij}$ are defined as follows: $$\tilde{\gamma}_{ij} \equiv -\frac{i}{2}\int_D dxdy
U_i^*(x,y)\nabla^2 U_j(x,y) .$$ In the above, $D$ denotes the area inside the cavity.
From Eq. (\[SB-Bloch1\]), we have $$\rho_j(t)V_j(x,y)=\frac{\tilde{\kappa}W}{{\tilde{\gamma}}_{\perp}-i\Delta_{0j}}
E_j(t) U_j(x,y) .
\label{rho_i}$$
Therefore, when the light field is almost stationary, from Eqs. (\[SB-Bloch2\]) and (\[rho\_i\]), one can express the population inversion $W$ by the light field amplitudes $E_i$ and the spatial patterns $U_i$, i.e., $$\begin{aligned}
\lefteqn{W=W_\infty / } \nonumber \\ & &
\left[
1+
\left\{
\sum_i \sum_j
\frac{
2\tilde{\kappa}^2 E_i E_j^* U_i U_j^* e^{-i\Delta_{ji} t }
}
{
\left\{
{\tilde{\gamma}}_{\perp}+i\Delta_{0j}
\right\}
\left\{ {\tilde{\gamma}}_{\parallel}-i\Delta_{ji} \right\}
}
+ c. c.
\right\}
\right]. \nonumber \\
\label{stationary W}\end{aligned}$$ The conventional approach to treat the nonlinear terms in $W$ is to perturbatively expand the right-hand side of Eq. (\[stationary W\]) for small light field amplitudes [@L2; @Lamb]. This method is only applicable to just above the lasing threshold and cannot correctly describe the case when the external pumping power $W_\infty$ is very large. In the following, we present a different approach applicable to the high-pumping cases. Note that our method can be applied to semiconductor lasers in the same way as the conventional approach [@Sargent].
We introduce the dimensionless quantities $L$ and $C$ related to the total intensity and mode interference, respectively, as follows: $$L(x,y)\equiv
1+\sum_m a_m \left| U_m \right|^2,
\label{I}$$ where $a_m$ denotes the dimensionless light field intensity of the mode $m$ weighted by the Lorentzian gain $g(\Delta_m)\equiv {\tilde{\gamma}}_{\perp} / ({\tilde{\gamma}}_{\perp} ^2 + \Delta_{0m}^2)$, i.e., $a_m \equiv
({4\tilde{\kappa}^2 }/{{\tilde{\gamma}}_{\parallel}}) g(\Delta_m) \left| E_m \right|^2$, and $$C(x,y)\equiv
\sum_{\substack{l,j \\ l\neq j} }
\frac{
2\tilde{\kappa}^2 E_l E_j^* U_l U_j^* e^{-i\Delta_{jl} t }
}
{
\left\{
{\tilde{\gamma}}_{\perp}+i\Delta_{0j}
\right\}
\left\{ {\tilde{\gamma}}_{\parallel}-i\Delta_{jl} \right\}
}
+ c. c.$$ Then the denominator of the term in the right-hand side in Eq. (\[stationary W\]) is expressed as $L+C=L(1+C/L)$. The basic idea of our approach is to expand it in the power series of $C/L$ under the condition $|C|/|L|<1$ almost everywhere in the cavity. From Eqs. (\[E\_i\]), (\[rho\_i\]) and (\[stationary W\]), we obtain $$\begin{gathered}
\frac{d E_i(t)}{d t}
+\sum_{j\neq i} \frac{d E_j(t)}{d t} e^{ -i \Delta_{ij} t } U_{ij}
\\
= \left\{ i\left(\Delta_i +\frac{1}{2}\right)
-\left( \alpha_L +\tilde{\gamma}_{ii} \right) \right\} E_i(t)
\\
+\sum_{j\neq i}
\left[
\left\{ i\left(\Delta_j +\frac{1}{2}\right) - \alpha_L \right\} U_{ij}
-\tilde{\gamma}_{ij}
\right]
E_j(t) e^{ -i \Delta_{ij} t }
\\
+\xi W_\infty
\sum_k
\frac{
e^{ -i\Delta_{ik} t }
E_k
}
{
{\tilde{\gamma}}_{\perp}-i\Delta_{0k}
}
\\
\times \int_D dx dy
\frac{ U_i^* U_k }{ L(x,y) }
\left[
1-\frac{C(x,y)}{L(x,y)}+\left\{\frac{C(x,y)}{L(x,y)}\right\}^2
-\cdots
\right],
\label{C/I expansion}\end{gathered}$$ where $
\xi\equiv 2\pi N \kappa \tilde{\kappa} \hbar / \varepsilon
$. Since we are focusing on the vicinity of the stationary state of the slowly varying envelope, $dE_i(t)/dt$ is very small. Therefore we can assume $E_i(t)\sim e^{\epsilon_i t}$ where $\epsilon_i \ll
|\Delta_{ij}| $ for all $i$ and $j$ ($j\neq i$). When Eq. (\[C/I expansion\]) is integrated over $t$, the second terms on both sides have the coefficients of $1/\left({\epsilon_j -i\Delta_{ij}}\right)$ while the first terms $1/{\epsilon_i}$. Consequently, the contributions of the terms concerning fast oscillations like the second terms are much smaller than those of the first terms. Accordingly, one can ignore the terms oscillating faster than $e^{-i\Delta_{ik} t }$.
By ignoring the terms oscillating faster than $e^{-i\Delta_{ik} t }$, Eq. (\[C/I expansion\]) is reduced to $$\begin{gathered}
\frac{d E_i}{d t} \simeq
\left\{ i\left(\Delta_i +\frac{1}{2}\right)
-\left( \alpha_L +\tilde{\gamma}_{ii} \right) \right\} E_i \\
+
\frac{
\xi W_\infty E_i
}
{
{\tilde{\gamma}}_{\perp}-i\Delta_{0i}
}
\int_D dx dy
\frac {\left|U_i\right|^2} { L(x,y) }
\\
-
\xi W_\infty E_i
\int_D dxdy
\frac{ \left|U_i \right|^2 }
{
\left\{ L(x,y)\right\}^{2}
}
\sum_{\substack{k \\ k\neq i} }
\frac{ 2\tilde{\kappa}^2 \left|E_k\right|^2 \left|U_k\right|^2 }
{
({\tilde{\gamma}}_{\perp}-i\Delta_{0k})( {\tilde{\gamma}}_{\parallel}-i\Delta_{ki} )
}
\\
\times
\left(
\frac{
1
}
{
{\tilde{\gamma}}_{\perp}+i\Delta_{0k}
}
+
\frac{
1
}
{
{\tilde{\gamma}}_{\perp}-i\Delta_{0i}
}
\right)
.\end{gathered}$$ Therefore, we obtain $$\begin{gathered}
\frac{d |E_i|^2}{dt} = \frac{d}{dt} (E_i Ei^*) = E_i^* \frac{dE_i}{dt} + E_i \frac{dE_i^*}{dt} \\
= (-2\alpha_L + \tilde{\gamma}_{ii}+\tilde{\gamma}_{ii}^* ) |E_i|^2
+2\xi W_{\infty} g(\Delta_i) \int_D dxdy \{L(x,y)\}^{-2} \\
\times |E_i|^2 |U_i|^2
\Biggl[ L(x,y) -\sum_{k, k\neq i} \frac{2\tilde{\kappa}^2}{\tilde{\gamma}_{\perp} }
g(\Delta_k) g_{\parallel}(\Delta_i - \Delta_k) \\
\times \Bigl\{ 2\tilde{\gamma}_{\perp}
+ (\Delta_i-\Delta_0)(\Delta_i-\Delta_k)/\tilde{\gamma}_{\perp}
~~~~~~~~~~~~~~~~~~\\
+(\Delta_i-\Delta_k)(\Delta_i+\Delta_k-2\Delta_0)/\tilde{\gamma}_{\parallel}
\Bigr\}
|E_k|^2 |U_k|^2 \Biggr],
\label{raw eq.}
$$ where $g_{\parallel}(\Delta_i - \Delta_k)$ is a Lorentzian defined as $g_{\parallel}(\Delta_i - \Delta_k)
\equiv \tilde{\gamma}_{\parallel} /
\{ \tilde{\gamma}_{\parallel}^2+(\Delta_i - \Delta_k)^2 \}$. If $\Delta_i$ is far from $\Delta_0$, the second term of the second line does not contribute because $g(\Delta_i)$ almost vanishes. Therefore, we assume $|\Delta_i - \Delta_0| \ll \tilde{\gamma}_{\parallel}$. Since the terms concerning $\Delta_k$ contribute in $L(x,y)$ if $\Delta_k$ is as close to $\Delta_0$ as $\Delta_i$ due to the Lorentzian $g(\Delta_k)$, we obtain $$L(x,y) \simeq 1+ \sum_{k, |\Delta_k - \Delta_i| \ll \tilde{\gamma}_{\parallel} }
\frac{4\tilde{\kappa}^2 }{\tilde{\gamma}_{\parallel}} g(\Delta_k)
|E_k|^2 |U_k|^2.$$ Because of the Lorentzian $g_{\parallel}(\Delta_i - \Delta_k)$, only the terms whose $\Delta_k$ values are close to $\Delta_i$ such that $|\Delta_i - \Delta_k|\ll \tilde{\gamma}_{\parallel}$ contribute to the sum over $k$ in Eq. (\[raw eq.\]). Accordingly, we have $g_{\parallel}(\Delta_i - \Delta_k) \simeq 1/{\tilde{\gamma}_{\parallel}}$, and $$\begin{gathered}
2\tilde{\gamma}_{\perp} \gg
(\Delta_i-\Delta_0)(\Delta_i-\Delta_k)/\tilde{\gamma}_{\perp} \\
+(\Delta_i-\Delta_k)(\Delta_i+\Delta_k-2\Delta_0)/\tilde{\gamma}_{\parallel}. \end{gathered}$$ Consequently, we obtain $$\begin{gathered}
\frac{d |E_i|^2}{dt}
= (-2\alpha_L + \tilde{\gamma}_{ii}+\tilde{\gamma}_{ii}^* ) |E_i|^2 \\
+2\xi W_{\infty} g(\Delta_i) \int_D
dxdy \{L(x,y)\}^{-2} |E_i|^2 |U_i|^2 \\
\times
\Biggl\{ 1+
\sum_{k, |\Delta_k - \Delta_i| \ll \tilde{\gamma}_{\parallel} }
\frac{4\tilde{\kappa}^2 }{\tilde{\gamma}_{\parallel}} g(\Delta_k)
|E_k|^2 |U_k|^2 \\
-\sum_{k, k\neq i, |\Delta_k - \Delta_i| \ll \tilde{\gamma}_{\parallel} }
\frac{4\tilde{\kappa}^2 }{\tilde{\gamma}_{\parallel}} g(\Delta_k)
|E_k|^2 |U_k|^2 \Biggr\} \\
= (-2\alpha_L + \tilde{\gamma}_{ii}+\tilde{\gamma}_{ii}^* ) |E_i|^2~~~~~~~~~~~~~~~~~~~~ \\
+2\xi W_{\infty} g(\Delta_i) \int_D
dxdy \{L(x,y)\}^{-2} |E_i|^2 |U_i|^2 \\
\times
\Biggl\{ 1+
\frac{4\tilde{\kappa}^2 }{\tilde{\gamma}_{\parallel}} g(\Delta_i)
|E_i|^2 |U_i|^2
\Biggr\}. ~~~~~~~~~~~~~
$$ Therefore, we finally obtain the equation for the time evolution of the light field intensity $I_i \equiv |E_i|^2$ of the lasing mode $i$, $$\frac{d I_i }{d t} \simeq S_i I_i ,
\label{main1}$$ where $S_i$ denotes the balance of the loss, gain and saturation of the mode $i$, and is defined as $$S_i \equiv -2\left( \alpha_L +\gamma_i \right)
+2\xi W_\infty g(\Delta_i)
\int_D dx dy
\frac{
\left|U_i \right|^2 L_i(x,y)
}
{
\left\{ L(x,y)\right\}^{2}
},
\label{S_i}$$ and $L_i(x,y)$ is related to the dimensionless light field intensity of the mode $i$, i.e., $
L_i(x,y)\equiv 1+a_i \left| U_i \right|^2.
\label{I_i}
$ $\gamma_i$ is derived by applying Green’s theorem to $(\tilde{\gamma}_{ii}+\tilde{\gamma}_{ii}^*)$ and represents the rate of the flux of the light field intensity going outside the cavity through the cavity edge for the lasing mode $i$: $$\gamma_i \equiv -\frac{i}{4} \oint_{\partial D} ds \left(
U_i^*\frac{\partial U_i}{\partial n}-U_i\frac{\partial U_i^*}{\partial n} \right),$$ where $\partial/\partial n$ is a normal derivative on the cavity edge. It is important to note that Eq. (\[main1\]) is derived without any assumption of small intensities, and hence it can be applied to the strongly pumped regimes.
Stability analysis of stationary lasing states
==============================================
Stability matrix {#sect:resonant_modes}
----------------
The light field intensities that make the right-hand side of Eq. (\[main1\]) vanish correspond to stationary-state solutions. The stability of a stationary-state solution is evaluated by the time evolution of the small displacements $\delta I_i$ from the intensities $I_{s,i}$ for the stationary-state subject to the differential equations $d \mbox {\boldmath $\delta I$}/ dt =
\tilde{M} \mbox {\boldmath $\delta I$}$. Here the displacement vector is defined by $\mbox {\boldmath $\delta I$ } \equiv {}^t
(\delta I_1~ \delta I_2~ \cdots~ \delta I_n)$ and the matrix $\tilde{M}$ is given by $
\tilde{M}\equiv P M P^{-1},
$ and $$\begin{gathered}
{M}_{ij} \equiv
\left[ S_j +2 \xi W_\infty b_j^2 \int_D dx dy
\frac{\left| U_j \right|^4}{\left\{ L(x,y)\right\}^{2}} \right] \delta_{ij} \\
-4 \xi W_\infty b_i b_j
\int_D dx dy
\frac{ \left| U_i \right|^2 \left| U_j \right|^2 L_i(x,y) }{ \left\{L(x,y)\right\}^{3}},
\label{M}\end{gathered}$$ where $P\equiv \mbox{diag}(|E_1|,\cdots,|E_N|)$ and $
b_i \equiv (a_i g(\Delta_i))^{1/2}=
2 \tilde{\kappa} g(\Delta_i)|E_i|/ {\tilde{\gamma}}_{\parallel}^{1/2}
$. For simplicity, we introduced the matrix $P$ and assumed its inverse matrix $P^{-1}$ exists. However, $|E_i|^{-1}$ in $P^{-1}$ always cancels out $|E_i|$ in $P$. Therefore, $\tilde{M}$ is always well-defined and the following discussion is valid irrespective of the existence of $P^{-1}$.
Single-mode lasing states
-------------------------
From Eq. (\[main1\]), one can see that the fixed point $(0,\cdots,0,I_{s,j},0,\cdots,0)$ which satisfies $S_j=0$ corresponds to the single-mode lasing state of the mode $j$ whose light field intensity is equal to $I_{s,j}=|E_{s,j}|^2$. Then the matrix $\tilde{M}$ for this fixed point becomes a diagonal matrix, i.e., $$\tilde{M}_{jj}=-2 \xi W_\infty b_j^2 \int_D dx dy
\frac{\left| U_j \right|^4}{L_{s,j}^{2}} <0,
\label{single-diagonal}$$ and for $i\neq j$, $$\tilde{M}_{ii}= 2\xi W_\infty g(\Delta_i)
\int_D dx dy
\left|U_i \right|^2
\frac{ L_{s,i}-L_{s,j}^2}{L_{s,j}^2 L_{s,i}},
\label{single-diagonal-2}$$ where $L_{s,i}$ is related to the light intensity of the single-mode lasing corresponding to the fixed point $(0,\cdots,0,I_{s,i},0,\cdots,0)$, i.e., $L_{s,i}\equiv 1+a_{s,i} \left| U_i \right|^2$ and $a_{s,i}\equiv
({4\tilde{\kappa}^2 }/{{\tilde{\gamma}}_{\parallel}}) g(\Delta_i) I_{s,i} $. From Eq. (\[single-diagonal\]), one can see that the single-mode lasing state of the mode $j$ is stable in the direction ${}^t (0~\cdots~0~\delta I_{j}~0~\cdots~0)$.
It is important to note that $L_{s,i(j)}$ contains the spatial pattern of the mode $i(j)$. As shown in the section \[single\], if the spatial pattern $|U_i|^2$ overlaps with $|U_j|^2$ inside the cavity, $\tilde{M}_{ii}$ can be negative. Then, the single-mode lasing state of the mode $j$ can be stable in the direction ${}^t (0~\cdots~0~\delta I_{i}~0~\cdots~0)$.
Multimode lasing states
-----------------------
From Eq. (\[main1\]), one can see that a multimode lasing state corresponds to the solutions $I_{s,i}$ of the simultaneous equations $S_i=0$ $(i=1,2,\cdots,N)$. The number $N$ of the lasing modes that have nonzero light field intensities is an arbitrary natural number more than 1 because $I_i=0$ always satisfies the stationary-state condition for Eq. (\[main1\]). Then, from Eq. (\[M\]), we have $${M}_{ii} =
2 \xi W_\infty b_i^2 \int_D dx dy
\frac{\left| U_i \right|^4 L_i(x,y)}{\left\{ L(x,y) \right\}^3}
\left\{ \frac{L(x,y)}{L_i(x,y)}-2 \right\},
\label{multi-diagonal}$$ $${M}_{ij} =
-4 \xi W_\infty b_i b_j
\int_D dx dy
\frac{ \left| U_i \right|^2 \left| U_j \right|^2 L_i(x,y) }{ \left\{L(x,y)\right\}^{3}}.
\label{multi-off-diagonal}$$ Consequently, if and only if all of the eigenvalues of the $N\times N$ matrix $
\tilde{M}(\equiv P M P^{-1})
$ are negative, this multimode lasing state is stable.
Fully chaotic 2D microcavity lasers
===================================
Spatial patterns of stationary lasing modes {#Spatial pattern}
-------------------------------------------
For bounded chaotic systems, theories on quantum ergodicity have shown that the probability density of finding a quantum particle in a small area whose state is described by the eigenfunction of a quantized fully chaotic system approaches a uniform measure as the energy of the particle increases and the wavelength becomes shorter [@Shnirelman; @etc].
For open chaotic mapping systems, it has been shown that the long-lived eigenstates tend to be localized on the forward trapped set of the corresponding classical dynamics as the wavelength decreases [@Keating; @etc]. This tendency can be considered as a manifestation of quantum ergodicity in open systems.
A similar tendency has also been numerically observed for chaotic 2D microcavities [@Sunada2], where resonance wave functions of low-loss modes are supported by the forward trapped set of the corresponding ray dynamics with Fresnel’s law [@HaraShinoPRE; @Altmann] in the short wavelength limit. Because of this property, the overlap of wave functions between an arbitrary pair of low-loss modes takes a large value [@Sunada2]. Therefore, we assume that the spatial patterns of the wave functions for low-loss modes are similar to each other and expressed as $
|U_i(x,y)|^2=\left\{ 1+\epsilon_i (x,y) \right\} |U_0(x,y)|^2
$, which implies $$\int_D dxdy \epsilon_i(x,y) |U_0(x,y)|^2=0,
\label{overlap}$$ because of the normalization for all of the spatial patterns of the lasing modes. We assume that the fluctuation $\epsilon_i (x,y)$ is so small almost everywhere and random that $
|\epsilon_i(x,y)|U_0(x,y)|^2 | \ll 1
$ and $
\int_D dxdy \epsilon_i(x,y) \simeq 0
$. We also assume $
\int_D dxdy \epsilon_i(x,y) |U_0(x,y)|^{-2} \simeq 0
$.
Stability of single-mode lasing states {#single}
--------------------------------------
In the case of a fully chaotic 2D microcavity, one can apply the wave function property in (\[overlap\]) to evaluate $\tilde{M}_{ii}$ in Eq. (\[single-diagonal-2\]) whose sign determines the stability of the single-mode lasing state of the mode $j$ in the direction ${}^t (0~\cdots~0~\delta I_{i}~0~\cdots~0)$. Indeed, $\tilde{M}_{ii}$ is reduced for the first order of $\epsilon_i(x,y) |U_0(x,y)|^2$ as follows: $$\begin{gathered}
\tilde{M}_{ii}
= 2\xi W_\infty g(\Delta_i)
\int_S dxdy (1+\epsilon_i)|U_0|^2 \\
\times
\frac{ a_{s,i} (1+\epsilon_i) -a_{s,j}^2(1+\epsilon_j)^2 |U_0|^2 }
{ a_{s,j}^2(1+\epsilon_j)^2 (1+\epsilon_i) a_{s,i} |U_0|^6 }
|U_0|^2 \\
=
\frac{2\xi W_\infty g(\Delta_i)}
{ a_{s,j}^2 a_{s,i} }
\int_S dx dy \Bigl[
\left\{ 1+O(\epsilon_i (x,y) |U_0(x,y)|^2) \right\}\frac{a_{s,i}}{|U_0|^2} \Bigr. \\ \Bigl.
-\left\{ 1+O(\epsilon_j (x,y) |U_0(x,y)|^2) \right\}a_{s,j}^2 \Bigr] \\
\simeq
\frac{2\xi W_\infty g(\Delta_i)}{ a_{s,j}^2 a_{s,i} }
\left[ a_{s,i} \int_S dx dy \frac{1}{|U_0|^2} -A a_{s,j}^2 \right]
,
\label{single-diagonal-final}\end{gathered}$$ where $S$ denotes the support of $|U_0(x,y)|^2$ and $A$ is its area.
If and only if $\tilde{M}_{ii}$ is negative, the single-mode lasing state of the mode $j$ is stable in the direction ${}^t (0~\cdots~0~\delta I_{i}~0~\cdots~0)$. Therefore, from Eq. (\[single-diagonal-final\]), we obtain the stability condition for the single mode lasing of the mode $j$, $$\left( \frac{a_{s,j}}{A} \right)^2 >
\frac{a_{s,i}}{A} \int_S dxdy \frac{1}{ A^2 |U_0|^2} .
\label{single mode stability condition}$$ Accordingly, the single-mode lasing state of the mode $j$ is stable when its intensity is large enough to satisfy Eq. (\[single mode stability condition\]) for all of the other modes $i$ values. Note that the integral in Eq. (\[single mode stability condition\]) is estimated to be approximately unity because $|U_0|^{-2}$ can be approximated to be $A$. Consequently, the mode that has the largest single-mode intensity is stable, and hence there always exists one stable single-mode lasing state at least.
Instability of multimode lasing states
--------------------------------------
Next, we show that all of the multimode lasing states are unstable under the assumptions that all of the spatial patterns $U_i$ of the single-mode lasing states are similar to each other, and the pumping power $W_\infty$ is very high, which implies that the light field intensities are very large.
According to the assumption in Section \[Spatial pattern\] for the spatial patterns of the lasing modes in a fully chaotic microcavity, we obtain $$\begin{aligned}
L_i(x,y)
& \simeq & (1+\epsilon_i) a_i |U_0|^2, \end{aligned}$$ $$\begin{aligned}
L(x,y)
& \simeq &
\sum_{m=1}^N(1+\epsilon_m) a_m |U_0|^2 \end{aligned}$$ for the support $S$ of $|U_0|^2$ and we assumed $a_i\gg 1$. Then, the matrix elements in Eqs. (\[multi-diagonal\]) and (\[multi-off-diagonal\]) can be expressed by using the fluctuations $\epsilon_i(x,y) $. Therefore, from the properties assumed for $\epsilon_i(x,y) $, we obtain for the first order of $\epsilon_i(x,y) |U_0(x,y)|^2$, $${M}_{ii} \simeq
-4 \xi A W_\infty b_i^2
\frac{a_i}{a_{tot}^3}
\left( 1-\frac{a_{tot}}{2a_i} \right),
\label{multi-diagonal2}$$ $${M}_{ij} \simeq
-4 \xi A W_\infty b_i b_j
\frac{a_i}{a_{tot}^3},
\label{multi-off-diagonal2}$$ where $a_{tot}\equiv \sum_{m=1}^N a_m$.
It is important to note that $\tilde{M}(N)$ can be factorized as $
\tilde{M} (N) =
-4\xi A W_{\infty}/( a_{tot}^3 )
P Q B R B P^{-1},
$ where $
Q \equiv \mbox{diag}(a_1,\cdots,a_N),
$ $
B \equiv \mbox{diag}(b_1,\cdots,b_N),
$ and the diagonal elements of $R$ are given by $
R_{ii} \equiv 1-{a_{tot}}/{(2a_i)} ,
$ while the off-diagonal elements $
R_{ij} =1
$ . Then, as is explained in Appendix, all of the eigenvalues of $\tilde{M}(N)$ are equal to those of $M^{\prime}(N)$ defined as
$$\begin{gathered}
M^\prime(N) \equiv
-\frac{
4\xi A W_{\infty}
}
{ a_{tot}^3 }
\mbox{ diag } (\sqrt{a_1},\cdots,\sqrt{a_N}) \\ \times
B R B \mbox{ diag } (\sqrt{a_1},\cdots,\sqrt{a_N}).\end{gathered}$$
All of the eigenvalues of $M^{\prime}(N)$ are real because it is a real symmetric matrix. According to the theorem of linear algebra, the number of the negative eigenvalues of $M^{\prime}(N)$ is equal to that of sign changes of the sequence $\{1,|M^{\prime}(1)|,|M^{\prime}(2)|,\cdots,|M^{\prime}(N)| \}$ where the minor determinants of $M^{\prime}(N)$ are given as explained in Appendix, $$\left|M^\prime(k)\right|
=
\left(
\frac{
2\xi A W_{\infty}
}
{ a_{tot}^2 }
\right)^k
\left( 1-\sum_{i=1}^k \frac{2a_i}{a_{tot}} \right)
\prod_{i=1}^k b_i^2 .
\label{M'-determinant}$$ Since the term $(1-\sum_{i=1}^k {2a_i}/{a_{tot}})$ decreases monotonically for $k$ and equals $-1$ when $k=N$, the above sequence of the minor determinants of $M^{\prime}(N)$ changes the sign once. Accordingly, $\tilde{M}$ has one negative eigenvalue and $(N-1)$ positive eigenvalues, which means the fixed point corresponding to the multimode lasing state is an unstable saddle point.
Summary and discussion
======================
By introducing an expansion method different from the conventional theories [@Lamb; @L2; @Sargent] for the population inversion in the Maxwell-Bloch equations and evaluating the eigenvalues of a stability matrix describing the interactions among a huge number of lasing modes, we theoretically showed that in a fully chaotic 2D microcavity laser, at least one single-mode lasing state is stable, while all multimode lasing states are unstable, when the external pumping power is sufficiently large and the cavity size is much larger than the wavelength to the extent that $ {\tilde{\gamma}}_{\parallel}\gg |\Delta_{ij}| $, where $\Delta_{ij} $ is the difference between the adjacent lasing frequencies. This result provides a theoretical ground for recent experimental observations of universal single-mode lasing in fully chaotic 2D microcavity lasers [@Sunada1; @Sunada2].
It is important to note that the theory presented in this paper should be applied to explain the observation of single-mode lasing in the experiments of continuous-wave pumping cases. Generally, the lifetime of an unstable multimode lasing state can be much longer than a pulse width. Thus, for a pulsed operation, it is likely that the collapse of an unstable multimode lasing state cannot be achieved within a pulse width, even if the size of a fully chaotic 2D microcavity is sufficiently large, and multimode lasing is observed and universal single-mode lasing seems to disappear [@ReddingCao; @ChoiMulti]. In this case, as the pulse width is increased, the number of lasing modes decreases [@Sunada1]. Multimode lasing in a fully chaotic 2D microcavity can also be observed when the condition $ {\tilde{\gamma}}_{\parallel}\gg |\Delta_{ij}| $ is not satisfied [@CerjanOPEX; @Sunada3]. This condition for multimode lasing coincides with that derived for one-dimensional lasers [@FuHaken].
The theory presented in this paper cannot give the threshold pumping power for single-mode lasing, but it is useful for understanding the single-mode lasing mechanism. In addition, it does not take into consideration these phenomena that might affect lasing characteristics such as the thermal effect. However, according to previous studies [@Sunada1; @Sunada2], we can at least say that the threshold for single-mode lasing is achievable in real experiments. It is of interest to further demonstrate the predicted single-mode lasing experimentally for various fully chaotic cavities. It is also important to elucidate experimentally, numerically and theoretically how multimode lasing states in a low pumping regime and/or in a small cavity change into a single-mode lasing state as the pumping power and/or the size of the cavity are increased.
Appendix {#appendix .unnumbered}
========
Let us suppose that $\mbox {\boldmath $x$ } $ is the eigenvector corresponding to the eigenvalue $\lambda$ of the matrix $M \equiv \mbox{diag}(B_1,\cdots,B_N)C\mbox{diag}(A_1,\cdots,A_N)$ where the diagonal element of the matrix C is defined as $C_{ii}\equiv C_i$ and every off-diagonal element is equal to 1, that is, $$M \mbox {\boldmath $x$ } =
\mbox{diag}(B_1,\cdots,B_N)C\mbox{diag}(A_1,\cdots,A_N)\mbox {\boldmath $x$ }
=\lambda \mbox {\boldmath $x$ }.
\label{appendix1}$$ The left-hand side of Eq. (\[appendix1\]) is rewritten as follows: $$\begin{gathered}
\mbox{diag}(B_1,\cdots,B_N)C
\mbox{diag}(A_1^{1/2},\cdots,A_N^{1/2}) \\
\times
\mbox{diag}(A_1^{1/2},\cdots,A_N^{1/2})
\mbox{diag}(B_1^{1/2},\cdots,B_N^{1/2}) \\
\times \mbox{diag}(B_1^{-1/2},\cdots,B_N^{-1/2})
\mbox {\boldmath $x$ }~~~~~~~~~~~~~~~~~~\\
=\mbox{diag}(B_1,\cdots,B_N)C
\mbox{diag}(B_1^{1/2},\cdots,B_N^{1/2})~~~~~~~~~~~~~~~~~~~~~~~ \\
\times \mbox{diag}(A_1^{1/2},\cdots,A_N^{1/2})
\mbox{diag}(B_1^{-1/2},\cdots,B_N^{-1/2}) \\
\times \mbox{diag}(A_1^{1/2},\cdots,A_N^{1/2})
\mbox {\boldmath $x$}.~~~~~~~~~~~~~~~~~~~~~~\end{gathered}$$ On the other hand, the right-hand side of Eq. (\[appendix1\]) is rewritten as follows: $$\begin{gathered}
\lambda
\mbox{diag}(A_1^{-1/2},\cdots,A_N^{-1/2})
\mbox{diag}(A_1^{1/2},\cdots,A_N^{1/2}) \\
\times
\mbox{diag}(B_1^{1/2},\cdots,B_N^{1/2})
\mbox{diag}(B_1^{-1/2},\cdots,B_N^{-1/2})
\mbox {\boldmath $x$ } \\
=
\lambda
\mbox{diag}(A_1^{-1/2},\cdots,A_N^{-1/2})
\mbox{diag}(B_1^{1/2},\cdots,B_N^{1/2}) \\
\times \mbox{diag}(B_1^{-1/2},\cdots,B_N^{-1/2})
\mbox{diag}(A_1^{1/2},\cdots,A_N^{1/2}) \mbox {\boldmath $x$}. \end{gathered}$$ Therefore, we have $$\begin{gathered}
\mbox{diag}(B_1,\cdots,B_N)C
\mbox{diag}(B_1^{1/2},\cdots,B_N^{1/2})~~~~~~~~~~~~~~~~~~~~~~~ \\
\times \mbox{diag}(A_1^{1/2},\cdots,A_N^{1/2})
\mbox {\boldmath $x^{\prime}$ } \\
=
\lambda
\mbox{diag}(A_1^{-1/2},\cdots,A_N^{-1/2})
\mbox{diag}(B_1^{1/2},\cdots,B_N^{1/2})
\mbox {\boldmath $x^{\prime}$},
\label{appendix2}\end{gathered}$$ where $\mbox {\boldmath $x^{\prime}$ } \equiv
\mbox{diag}(B_1^{-1/2},\cdots,B_N^{-1/2})
\mbox{diag}(A_1^{1/2},\cdots,A_N^{1/2}) \mbox {\boldmath $x$}$. Operating $\mbox{diag}(B_1^{-1/2},\cdots,B_N^{-1/2})
\mbox{diag}(A_1^{1/2},\cdots,A_N^{1/2}) $ to both sides of Eq. (\[appendix2\]) yields $$\begin{gathered}
\mbox{diag}(A_1^{1/2},\cdots,A_N^{1/2}) \mbox{diag}(B_1^{1/2},\cdots,B_N^{1/2})C \\
\times \mbox{diag}(B_1^{1/2},\cdots,B_N^{1/2}) \mbox{diag}(A_1^{1/2},\cdots,A_N^{1/2})
\mbox {\boldmath $x^{\prime}$ }
=
\lambda
\mbox {\boldmath $x^{\prime}$}.
\label{appendix3}\end{gathered}$$ From Eq. (\[appendix3\]), one can see that the eigenvalue of the matrix $M^{\prime}$ defined as $$\begin{gathered}
M^{\prime} \equiv
\mbox{diag}(A_1^{1/2},\cdots,A_N^{1/2}) \mbox{diag}(B_1^{1/2},\cdots,B_N^{1/2})C \\
\times \mbox{diag}(B_1^{1/2},\cdots,B_N^{1/2}) \mbox{diag}(A_1^{1/2},\cdots,A_N^{1/2}), \end{gathered}$$ is equal to $\lambda$. Since $M_{ij}^{\prime}=C_{ij}(A_i A_j B_i B_j)^{1/2}$ and $C$ is a real symmetric matrix, $M^{\prime}$ is a real symmetric matrix. The determinant of $M^{\prime}$ is given as follows: $$\begin{gathered}
|M^{\prime}| =
|\mbox{diag}(A_1^{1/2},\cdots,A_N^{1/2})| |\mbox{diag}(B_1^{1/2},\cdots,B_N^{1/2})||C| \\
\times |\mbox{diag}(B_1^{1/2},\cdots,B_N^{1/2})|
|\mbox{diag}(A_1^{1/2},\cdots,A_N^{1/2})|\\
=\left( \prod_{i=1}^N A_i B_i \right) \left( \prod_{i=1}^N (C_i-1) \right)
\left( 1+\sum_{i=1}^N (C_i-1)^{-1} \right).\end{gathered}$$
Funding {#funding .unnumbered}
=======
This work was supported in part by a Waseda University Grant for Special Research Projects (Project number: 2017B-197).
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---
abstract: 'We propose an experimental technique for classifying the topology of band structures realized in optical lattices, based on a generalization of topological charge pumping in quantum Hall systems to cold atoms in optical lattices. Time-of-flight measurement along one spatial direction combined with *in situ* detection along the transverse direction provide a direct measure of the system’s Chern number, as we illustrate by calculations for the Hofstadter lattice. Based on an analogy with Wannier functions techniques of topological band theory, the method is very general and also allows the measurement of other topological invariants, such as the $\mathbb{Z}_2$ topological invariant of time-reversal symmetric insulators.'
author:
- 'Lei Wang$^{1,2}$, Alexey A. Soluyanov$^{1}$ and Matthias Troyer$^{1}$'
bibliography:
- 'hofstadterTOF.bib'
title: Direct measurement of topological invariants in optical lattices
---
Recent advances in experimental techniques have led to realization of synthetic gauge fields [@Lin:2009p15747; @Lin:2011p29660; @Aidelsburger:2011p67955; @JimenezGarcia:2012p56707; @Struck:2012p56790] and spin-orbit coupling [@Wang:2012p61874; @Cheuk:2012p61844] in cold atomic gases. These new developments allow one to study a variety of topological phases of condensed matter physics using cold gases of neutral atoms trapped in optical lattices. Such topological phases occur in systems whose Hilbert space has a non-trivial topological structure, and they are classified according to the value of a corresponding quantum number, the topological invariant.
Particular examples of such phases in condensed matter include the quantum Hall insulators [@Klitzing:1980p64777] or the quantum anomalous Hall insulators [@Haldane:1988p3868], where the topological invariant of the Hilbert space is the so-called Chern number [@Thouless:1982p24208]. Some recent experiments [@Aidelsburger:2011p67955; @JimenezGarcia:2012p56707; @Struck:2012p56790; @Tarruell:2012p52125] point towards the possibility of realizing an optical lattice with non-zero Chern number in the near future.
Once the desired lattice is created the question of experimental verification of the non-trivial topology arises. Unlike condensed matter systems, where a routine measurement of the Hall conductance reveals the Chern number value [@Klitzing:1980p64777], cold atom systems require a special setup [@Brantut:2012p70665] for transport measurements. Edge states probes of condensed matter experiments also become cumbersome in a cold atom environment since the smooth harmonic potential washes out the edge states associated with the quantum Hall state. This problem can potentially be circumvented by enhancing a weak Bragg signal from the topological edge states by means of specifically tuned Raman transitions [@Buchhold:2012p58396; @Goldman:2012p58384]. Other approaches, which do not have an immediate analogy among solid state experiments, might also allow for the measurement of Chern numbers in optical lattices [@Alba:2011p48844; @Zhao:2011p45563; @Atala:2012ts; @Abanin:2012p70250; @Goldman:2012p71551]. However, the quest for a universal method to obtain Chern numbers and other topological invariants directly in a single measurement, avoiding a sophisticated experimental setup or data analysis, remains open.
In this Letter we tackle the problem of detecting topological invariants in optical lattice systems from a very different perspective, drawing an analogy to the theory of electric polarization of crystalline solids [@KingSmith:1993p61262] to suggest a simple and effective method to measure Chern numbers in cold atom systems. We introduce the concept of hybrid time-of-flight (HTOF) images, referring to an *in situ* measurement of the cloud’s density in one direction during free expansion in the other. The HTOF reveals the topology of the optical lattice just as hybrid Wannier functions (HWF) do in band theory [@Coh:2009p26093; @Marzari:2012p65932]. We illustrate our approach by numerical simulations of a square optical lattice that realizes a Hofstadter model [@Hofstadter:1976p4046], and discuss how it works in lattices with a more complicated geometry. Our method does not require the presence of the sharp edge states and is not affected by a soft harmonic trap. It can also be used to detect the $\mathbb{Z}_2$ topological invariant of time-reversal ($\cal{T}$) symmetric topological insulators. The modern theory of electric polarization of crystalline solids [@KingSmith:1993p61262; @Resta:1994p61451] relates the [[electronic polarization]{}]{} $P$ to the geometry of the underlying band structure. For a 1D insulator with a single occupied band $ P$=$\frac{1}{2\pi}\oint_{BZ} {\cal A}(k)dk$, where $k$ is the crystal momentum, ${\cal A}(k)$=$i\langle u_{k}|\partial_k|u_{k}\rangle$ is the Berry connection [@Berry:1984p70668] and the $u$s are the lattice-periodic parts of the Bloch functions. Alternatively, we will use the fact that the polarization can be written [@KingSmith:1993p61262] as the center of mass of the Wannier function constructed for the occupied band [@Kohn:1959p1285], which can be defined as an [[expectation value]{}]{} of position operator projected onto the occupied state [@Kivelson:1982p26324; @Marzari:1997p1458].
In two dimensions (2D) an insulating Hamiltonian can be viewed as a fictitious 1D insulator subject to an external parameter $k_x$. Polarization of this 1D insulator can be defined by means of HWF [@Sgiarovello:2001p68653], which are localized in only one direction retaining Bloch character in the other. The polarization at each $k_{x}$ is given by the center of mass of the corresponding HWF [@Marzari:2012p65932]. The definitions of electronic polarization given above are gauge dependent, meaning that $P$ is defined only modulo a lattice vector. For measurements one has to consider the change in polarization induced by a change in an external parameter [@KingSmith:1993p61262]. In the 2D insulator considered above, $k_x$ plays the role of such a parameter. When $k_x$ is adiabatically changed by $2\pi$, the change in polarization, [*i.e.*]{} the shift of the HWF center, is proportional to the Chern number [@KingSmith:1993p61262]. This is a manifestation of topological charge pumping [@Thouless:1983p23000; @Niu:1990p70657], with $k_{x}$ being the adiabatic pumping parameter.
![Square lattice illustrating the Hofstadter model of Eq. (\[eq:lattice\]). The oval marks $q$=$7$ sites of the unit cell. Small arrows indicate the directions in which the phase of the hopping amplitude is chosen to be positive. Different colors of these arrows correspond to different values of the phase. Large arrows indicate the direction of the free expansion of the atomic cloud in hybrid time of flight (HTOF) images.[]{data-label="fig:lattice"}](lattice.pdf){width="6cm"}
A generalization of these ideas to cold atomic gases is a natural way to measure the Chern number in optical lattices. We replace the HWFs of band theory with hybrid densities $\rho(k_{x},y)$, which are the particle densities resolved along the $y$-direction as a function of $k_{x}$. Note that, while for an extended system the HWF charge center position can not be reconstructed from the single particle density [@Resta:1994p61451], this becomes possible in a finite system where the position operator is well defined [@PhysRevLett.80.1800]. Hence in a system with finite extent $L_y$ in the $y$-direction we can calculate the HWF center as $$\bar{y}(k_{x}) = \frac{\int_{0}^{L_y}\, y \rho(k_{x}, y) \mathrm{d} y} {\int_{0}^{L_y}\, \rho(k_{x}, y) \mathrm{d} y}.
\label{eq:continue}$$
The proportionality between the shift of the HWF center and the Chern number still holds in the open system. The Chern number measures the charge transported from one boundary to the other as $k_x$ is cycled by $2\pi$. An experimental measurement of the shift in the hybrid density will hence directly determine the Chern number.
Experimentally, $\rho(k_{x},y)$ is measured by an HTOF measurement, in which the lattice and trap are switched off along the $x$-direction, while keeping the lattice and harmonic confinement unchanged in the $y$-direction. In the long time limit [@Gerbier:2008p16148] TOF images map out the crystal momentum distribution along $k_{x}$ [^1]. At the same time, the system is still confined in the $y$-direction and a real-space density resolution can be done.
We now show HTOF unambiguously determine the Chern number by performing a numerical simulation of the Hofstadter model [@Hofstadter:1976p4046] on a square lattice. Its Hamiltonian is given by $$H_{\mathrm{lattice}} = -J_x \sum_{m,n} e ^{i2\pi n\Phi}c^{\dagger}_{m+1,n} c_{m,n}- J_y \sum_{m,n}c^{\dagger}_{m,n+1} c_{m,n} + H.c.,
\label{eq:lattice}$$ where $J_\alpha$ is the hopping amplitude in the $\alpha$=$\{x,y\}$ direction and $c_{m,n}$ is the fermionic annihilation operator, with $m$ and $n$ being the column and row indices of the lattice (see Fig \[fig:lattice\]).
An atom hopping clock wise around a plaquette accumulates a phase $\Phi$. We consider $\Phi=p/q$ where $p$ and $q$ are two relatively prime integers. The hopping matrix elements $J_x e ^{i2\pi n\Phi}$ along the $x$-direction depend on the row index $n$ so that each unit cell contains $q$ sites. In the following we fix $q$=$7$ and assume that only the lowest band is occupied. The Chern number $C$ of the lowest band is determined by the Diophantine equation $1$=$qs$+$pC$ [@Thouless:1982p24208; @Kohmoto:1989p19836; @Dana:2000p69285], where $s$ is an integer and $|C|\le q/2$.
We first consider an infinite ribbon of this model with width $L_{y}$=$10$ setting $J_x$=$J_y$=$J$, and $p=1$ which corresponds to a Chern number $C=1$. In the spectrum shown in Fig. \[fig:ribbon\](a) we see that, as expected for $p=1$, two edge states cross the Fermi level. Analogously to the 2D insulator considered above, this setup can be viewed as a finite 1D chain subject to a $k_{x}$-driven pump. From this point of view the hybrid density $\rho(k_{x}, n)$ describes the change in the density of the 1D system as a function of the pumping parameter. Figure \[fig:ribbon\](b) shows that the hybrid density is shifted by one unit cell in the bulk, indicating that a single charge is pumped across the system, as expected for $C=1$.
![Results for a ribbon ($L_{y}=10$) of the Hofstadter lattice model with $p/q=1/7$ and corresponding Chern number $C=1$. (a) Energy spectrum: two edge states (in red) cross the bulk energy gap. (b) Hybrid density $\rho(k_{x},n)$ shifts by one unit cell in a $2\pi$-cycle. (c) The center of mass of the hybrid density Eq. (\[eq:com\]) jumps by one unit cell. []{data-label="fig:ribbon"}](fig3.pdf){width="9cm"}
We calculate the $k_{x}$ dependence of the HWF center by taking a tight-binding limit of Eq. (\[eq:continue\]) $$\bar{n} (k_{x}) = \frac{ \sum_{n} n \rho(k_{x},n)}{ \sum_{n} \rho(k_{x},n)}.
\label{eq:com}$$ As shown in Fig. \[fig:ribbon\](c), $ \bar{n} (k_{x})$ jumps by one unit cell ($q$=$7$ sites), analogous to the HWF shift in Chern insulators [@Coh:2009p26093].
Having established a clear connection between the hybrid density and Chern number, we now turn to a more realistic case by adding a harmonic trapping potential of the form $$H_{\mathrm{trap}}=V_{\mathrm{T}} \sum_{m,n} [(m-L_{x}/2)^{2}+(n-qL_{y}/2)^{2}] c^{\dagger}_{m,n} c_{m,n}.
\label{eq:trap},$$ where $L_{x(y)}$ denotes the number of unit cells in the $x(y)$ direction. The system contains $qL_y$ rows and $L_{x}$ columns. The values of $V_\mathrm{T}$ and the number of atoms $N=300$ are chosen such that the atom cloud has a large insulating region corresponding to $1/q$ filling at the trap center. We now consider the cases $p$=$1$ and $p$=$3$, corresponding to Chern numbers $+1$ and $-2$ respectively.
There is no significant difference in the real space densities of the two states since the harmonic trap smears out the edge states [@Buchhold:2012p58396]. In contrast, the HTOF density profiles allow one to directly read of the Chern numbers. We calculate these HTOF images by first solving the Schrödinger equation $(H_\mathrm{lattice} + H_\mathrm{trap} )\psi_{i}(m,n)= \varepsilon_{i}\psi_{i}(m,n)$ and constructing HWF’s by means of the Fourier transform in the $x$-direction $$\psi_{i}(k_{x}, n) = \sum_{m=0}^{L_{x}-1} e^{i k_{x} m}\psi_{i}(m,n).
\label{eq:fourier}$$ We then use these wavefunctions to construct the hybrid particle density of the HTOF measurement $$\rho(k_{x},n) = \frac{1}{N}\sum_{i=1}^{N}|\psi_{i}(k_{x}, n) |^{2}.
\label{eq:rho}$$ HTOF images obtained in this way are shown in Fig. \[fig:trap\] and clearly exhibit the topological charge pumping effect despite the presence of a trap: the hybrid density shifts by $C$ unit cells along the $y$-direction as $k_{x}$ changes from $-\pi$ to $\pi$. Thus, the hybrid density is an accurate probe of topological properties and allows to directly measure the Chern number. To get deeper understanding we consider the case of vanishing transverse coupling $J_y$=$0$, corresponding to a set of decoupled tight-binding chains with dispersions $\epsilon_n(k_x)=-2J_x \cos(k_{x} - 2\pi n p/q)$. The position of the band minimum shifts by $2\pi p/q$ from one chain to the next, as shown in Fig. \[fig:diophantine\](a) for $p$=$3$. If an infinitesimal coupling $J_y$ is added, the 2D lattice is in the Chern insulating regime. Charge pumping can be inferred by tracing the change in the position of the valence band minima for weakly coupled chains: connecting nearest neighbor points in Fig. \[fig:diophantine\](a) reveals a shift by two unit cells ($|C|$=$2$) in the course of a $2\pi$ change of momentum (dashed red arrow in Fig. \[fig:diophantine\](a)). This illustrates the geometrical interpretation of the Diophantine equation [@Thouless:1982p24208], as also discussed in Ref. [@Huang:2012p70624] in a different context.
![The HTOF images for the Hoftadter lattice ($L_{x}=70$, $L_{y}=10$, $N=300$) in the presence of the harmonic trap: (a) for $p/q=1/7$ with $C=1$ ($V_{T}=0.001$), (b) for $p/q=3/7$ with $C=-2$ ($V_{T}=0.001$). Chern numbers can be determined as the number of unit cells traversed by the hybrid density in the course of a $2\pi$-cycle. Upward (downward) direction of the shift corresponds to positive (negative) Chern number. The broadening of lines correspond to exponential localization of the peaks of the hybrid density.[]{data-label="fig:trap"}](fig2.pdf){width="9cm"}
The above analysis allows for an alternative way of describing the HTOF results presented in Fig. \[fig:trap\](a-b). We introduce sublattice densities $\rho^{a}(k_{x},\tilde{n})$, which correspond to the particle density on the $a$-th site of the $\tilde{n}$-th unit cell. These sublattice densities, shown in Fig. \[fig:diophantine\](b), shift along the $y$-direction as $k_x$ changes, illustrating the motion of charge. The motion of charge can also be tracked in the total sublattice density obtained by summing $\rho^{a}(k_x,\tilde{n})$ over the unit cells: $$\mathcal{N}^{a}(k_{x}) = \sum_{\tilde{n}=0}^{L_{y}-1}\rho^{a}(k_{x}, \tilde{n}).
\label{eq:N}$$ Thus, the topological nature of the state can also be seen in the total sublattice density, which is potentially accessible in a TOF experiment that can distinguish different sublattices [@Folling:2007p55101; @Aidelsburger:2011p67955]. However, such an analysis is specific to the Hofstadter model, while the HTOF measurement is generally applicable.
.[]{data-label="fig:diophantine"}](Diophantine.pdf){width="9cm"}
The square lattice considered so far is particularly simple, since a straightforward partition into 1D chains is possible. HTOF measurement can also work for other lattice geometries such as the honeycomb lattice, which is topologically equivalent to the brick-wall lattice shown in Fig. \[fig:brickwall\](a). Such a lattice was used to create Dirac points in optical lattices [@Tarruell:2012p52125]. This lattice is a potential candidate for realizing the Haldane model [@Haldane:1988p3868], which is a canonical example of a Chern insulator.
A partition of the brick-wall lattice into 1D chains, along which the charge is pumped, is illustrated in Fig. \[fig:brickwall\](a) with solid dark bonds. The chains consist of two sublattices offset from each other in the $x$-direction. Due to this offset the charge pumping is not directly visible in HTOF image unless one separately measures them for each sublattice (for example using the superlattice technique of Ref. [@Folling:2007p55101]). As illustrated in Fig. \[fig:brickwall\](c) the center of mass of the hybrid density along the zig-zag 1D chain is indeed shifted by one unit cell along the 1D chain, revealing the Chern number $C=1$ of the Haldane model.
![(a) The brick-wall lattice split into 1D chains. Thick (black) and light (yellow) bonds indicate intra- and inter-chain hoppings respectively. The shape of an imposed superlattice potential $V_{\rm S}(x)$ is shown schematically (in gray). (b) Energy spectrum of a Haldane model in the ribbon geometry. Edge states are shown in red. (c) The shift of the center of mass of the hybrid density indicates non-trivial Chern number ($C=1$). []{data-label="fig:brickwall"}](brickwall.pdf){width="8cm"}
The HTOF technique can not only determine Chern numbers, but also the $\mathbb{Z}_2$ topological invariant of $\cal{T}$-symmetric insulators [@Hasan:2010p23520]. In band theory this invariant can be obtained by means of HWFs [@Fu:2006p3991; @Soluyanov:2011p33800; @Yu:2011p37026], once they form $\cal{T}$ images of one another. The occupied single particle states of the insulator can always be split into two sets of states (related by $\cal{T}$ symmetry) where each set has a well defined Chern number [@Soluyanov:2012p50967]. For a wide range of models such a splitting can be achieved by projecting the occupied states onto particular spin directions [@Sheng:2006p4513; @Prodan:2009p23865]. If the values of thus obtained spin Chern numbers are odd, the system is in the $\mathbb{Z}_2$-insulating regime. In the context of cold atoms, considering a minimal model with only two occupied bands, a spin-projected HTOF measurement of spin-resolved densities would serve as a direct measurement of the spin Chern numbers, and hence of the $\mathbb{Z}_2$-invariant.
The proposed HTOF technique is not only practical, providing exhaustive information about the topological state of an optical lattice, but is also a conceptually novel idea for using a cold atom lattice as a quantum simulator. Hybrid density measurement as proposed here for cold atom systems are not possible in condensed matter experiments and the HWFs are used only in computer simulations. These experiments can thus access novel probes of topological order and will give rise to further implementations of so far numerical experiments of condensed matter in real experiments on cold atom systems.
#### Acknowledgment
We thank David Vanderbilt, Thomas Uehlinger, Daniel Greif and Gregor Jotzu for helpful discussions. The work was supported by the Swiss National Science Foundation through the NCCR QSIT and the European Research Council. Simulations were run on the Brutus cluster at ETH Zurich.
[^1]: Due to the short coherence length of the band insulator, the Fresnel interference term [@Gerbier:2008p16148] can safely be ignored for typical expansion times. For example, with $^{40}$K atoms in a typical optical lattice at expansion times of $\sim10$ms will suffice.
|
---
abstract: 'We investigate GPU-based parallelization of Iterative-Deepening A\* (IDA\*). We show that straightforward thread-based parallelization techniques which were previously proposed for massively parallel SIMD processors perform poorly due to warp divergence and load imbalance. We propose Block-Parallel IDA\* (BPIDA\*), which assigns the search of a subtree to a block (a group of threads with access to fast shared memory) rather than a thread. On the 15-puzzle, BPIDA\* on a NVIDIA GRID K520 with 1536 CUDA cores achieves a speedup of 4.98 compared to a highly optimized sequential IDA\* implementation on a Xeon E5-2670 core. [^1]'
author:
- Satoru Horie
- |
Alex Fukunaga\
Graduate School of Arts and Sciences\
The University of Tokyo
bibliography:
- 'main.bib'
title: |
Block-Parallel IDA\* for GPUs\
(Extended Manuscript)
---
Introduction
============
Graphical Processing Units (GPUs) are many-core processors which are now widely used to accelerate many types of computation. GPUs are attractive for combinatorial search because of their massive parallelism. On the other hand, on many domains, search algorithms such as A\* tend to be limited by RAM rather than runtime. A standard strategy for addressing limited memory in sequential search is iterative deepening [@korf85]. We present a case study on the GPU-parallelization of Iterative-Deepening A\* [@korf85] for the 15-puzzle using the Manhattan Distance heuristic. We evaluate previous thread-based techniques for parallelizing IDA\* on SIMD machines, and show that these do not scale well due to poor load balance and warp divergence. We then propose Block-Parallel IDA\* (BPIDA\*), which, instead of assigning a subtree to a single thread, assigns a subtree to a group of threads which share fast memory. BPIDA\* achieves a speedup of 4.98 compared to a state-of-the-art 15-puzzle solver on a CPU, and a speedup of 659.5 compared to a single-thread version of the code running on the GPU.
Background and Related Work
===========================
An NVIDIA CUDA architecture GPU consists of a set of [*streaming multiprocessors (SMs)*]{} and a GPU main memory (shared among all SMs). Each SM consists of shared memory, cache, registers, arithmetic units, and a warp scheduler. Within each SM the cores operate in a SIMD manner. However, each SM executes independently, so threads in different SMs can run asynchronously. A [*thread*]{} is the smallest unit of execution. A [*block*]{} is a group of threads which execute on the same SM and share memory. A [*grid*]{} is a group of blocks which execute the same function. Threads in a block are partitioned into [*warps*]{}. A warp executes in a SIMD manner (all threads in the same warp share a program counter). [*Warp divergence*]{}, an instance of [*SIMD divergence*]{}, occurs when threads belonging to the same warp follow different execution paths, e.g., IF-THEN-ELSE branches. [*Shared memory*]{} is shared by a block and is local within a SM, and access to shared memory is much faster than access to the GPU [*global memory*]{} which is shared by all SMs.
Rao et al ([-@rao87]) parallelized each iteration of IDA\* using work-stealing on multiprocessors. Parallel-window IDA\* assigned each iteration of IDA\* to its own processor [@powley89]. Two SIMD parallel IDA\* algorithms are by Powley et al ([-@powley93]) and @mahanti93 ([-@mahanti93]). For each $f$-cost limited iteration of IDA\*, they perform an initial partition of the workload among the processors, and then periodically perform load balancing between IDA\* iterations and within each iteration. Hayakawa et al ([-@hayakawa15]) proposed a GPU-based parallelization of IDA\* for the 3x3x3 Rubik’s cube which searches to a fixed depth $l$ on the CPU, then invokes a GPU kernel for the remaining subproblems. Their domain-specific load balancing scheme relies on tuning $l$ using knowledge of “God’s number” (optimal path length for the most difficult cube instance) and is fragile – perturbing $l$ by 1 results in a 10x slowdown. @zhou15 ([-@zhou15]) proposed a GPU-parallel A\* which partitions OPEN into thousands of priority queues. The amount of global RAM on the GPU (currently $\leq 24GB$) poses a serious limitation for GPU-based parallel A\*. Edelkamp and Sulewski ([-@edelkamp2010perfect]) investigated memory-efficient GPU search. Sulewski et al ([-@sulewski2011exploiting]) proposed a hybrid planner which uses both the GPU and CPU.
Experimental Settings and Baselines {#sec:baseline}
===================================
We used the standard set of 100 15-puzzle instances by Korf ([-@korf85]). These instances are ordered in approximate order of difficulty. All solvers used the Manhattan distance heuristic. Reported runtimes include all overheads such as data transfers between CPU and GPU memories (negligible). All experiments were executed on a non-shared, dedicated AWS EC2 g2.2xlarge instance. The CPU is an Intel Xeon E5-2670. The GPU is an NVIDIA GRID K520, with 4GiB global RAM, 48KiB shared RAM/block, 1536 CUDA cores, warp size 32, and 0.80GHz GPU clock rate.
First, we evaluated 3 baseline IDA\* solvers:\
[**Solver B**]{}: The efficient, Manhattan-Distance heuristic based 15-puzzle IDA\* solver implemented in C++ by @burns12 ([-@burns12]). We used the current version at https://github.com/eaburns/ssearch.\
[**Solver C**]{}: Our own implementation of IDA\* in C (code at http://github.com/socs2017-48/anon48), This is the basis for G1 and all of our GPU-based code.\
[**Solver G1**]{}: A direct port of Solver C to CUDA. The implementation is optimized so that all data structures are in the fast, shared memory (the memory which is local to a SM). [**This baseline configuration uses only 1 GPU block/thread, i.e., only 1 core is used, all other GPU cores are idle**]{}.
The total time to solve all 100 problem instances was 620 seconds for Solver B [@burns12] and 475 seconds for our Solver C. Solver C was consistently 25% faster on every instance. Thus, Solver C is appropriate as a baseline for our GPU-based 15-puzzle solvers.
Next, we compare Solver C (1 CPU thread) to G1 (1 GPU thread). G1 required 62957 seconds to solve all 100 instances, 131 times slower than Solver C. This implies that on the GPU we used with 1536 cores, a perfectly efficient implementation of parallel IDA\* might be able to achieve a speedup of up to 1536/131 = 11.725 compared to Solver C.
Thread-Based Parallel IDA\*
===========================
Most of the previous work on parallel IDA\* parallelizes each iteration of IDA\* using a [*thread-based parallel*]{} scheme [@rao87; @powley93; @mahanti93; @hayakawa15].
We evaluated 3 thread-parallel IDA\* configurations. Since these are relatively straightforward and not novel, we sketch the implementations below. Details are in Appendix. Details are in the extended version [@ExtendedArxivManuscript].
### PSimple (baseline) {#psimple-baseline .unnumbered}
In this baseline configuration, for each $f$-bounded iteration of IDA\*, PSimple performs A\* search from the start state until as many unique states as the \# of threads are in OPEN. Then, each root is assigned to a thread. No load balancing is performed. The subtree sizes under each root state can vary significantly, so some threads may finish their subproblem much faster than other threads. Each $f$-bounded iteration must wait for all threads to complete, so PSimple has very poor load balance. Therefore, [*load balancing*]{} mechanisms which redistribute the work among processors are necessary.
### PStaticLB (static load balancing) {#sec:static-load-balancing .unnumbered}
This configuration adds static load balancing to PSimple. After each $f$-bounded iteration, PStaticLB implements a [*static load balancing*]{} mechanism somewhat similar to that of [@powley93]. In IDA\*, the $i$-th iteration repeats all of the work done in iteration $i-1$. Thus, the \# of states visited under each root state in the iteration $i-1$ can be used to estimate the \# of states which will be visited in the current iteration $i$, and root nodes are redistributed based on these estimates (details in Appendix). (details in [@ExtendedArxivManuscript]).
### PFullLB (thread-parallel with dynamic load balancing) {#sec:dynamic-load-balancing .unnumbered}
This configuration adds dynamic load balancing (DLB) to PStaticLB, which moves work to idle threads from threads with remaining work [*during*]{} an iteration. On a GPU, work can be transferred between two threads [*within*]{} a single block relatively cheaply using the shared memory within a block, while transferring work between two threads in different blocks is expensive because it requires access to the global memory. When dynamic load balancing is triggered, idle threads steal work from threads with remaining work within a block. We experimented with various DLB strategies including variants of policies investigated by [@powley89; @mahanti93], and used a policy we found for triggering DLB based on the policy by @powley89. See Appendix for additional details. See [@ExtendedArxivManuscript] for additional details.
Evaluation of Thread-Parallel IDA\*
-----------------------------------
PSimple on 1536 cores required a total of 3378 seconds to solve all 100 problems, a speedup of only 18.6 compared to G1 (1 core on the GPU).
This is mostly due to extremely poor load balance. We define load balance as $\mathit{maxload}/\mathit{averageload}$, where $\mathit{averageload}$ is the average number of nodes expanded among all threads, and $\mathit{maxload}$ is the number of states expanded by the thread which performed the most work. The load balance for PSimple on the 100 problems was: mean 96.46, min 14, max 680, stddev 113.19. This is extremely unbalanced ($maxload$ is almost 100x $averageload$).
Static load balancing significantly improved load balance (PStaticLB: mean 9.96, min 3, max 56, stddev 8.96), and dynamic load balancing further improved load balance (PFullLB: mean 6.14 min 3 max 19 stddev 3.38). This resulted in speedups of 58.9 and 70.8 compared to G1 (Table \[tab:summary-runtime\]). However, the 70.8 speedup vs G1 achieved by PFullLB is only a parallel efficiency of 70.8/1536 = 4.6%, which is extremely poor. We experimented extensively but could not achieve significantly better results with thread-parallel IDA\*.
Block Parallelization {#sec:BlockParallelization}
=====================
The likely causes for the poor (4.6%) efficiency of PFullLB are: (1) SMs become idle due to poor load balance even after our load balancing efforts, (2) threads stall for warp divergence, and (3) load balancing overhead. All of these can be attributed to the thread-based parallelization scheme in PFullLB and PStaticLB, in which each processor/thread executes an independent subproblem during a single $f$-bound iteration. This scheme, based on parallel IDA\* variants originally designed for SIMD machines [@powley93; @mahanti93], was appropriate for those SIMD architectures where all communications between processors were very expensive – paying the price of SIMD divergence overhead was preferable to incurring communication costs. On the other hand, in NVIDIA GPUs, threads in the same block (which execute on the same SM) can access fast shared memory on the SM with relatively low overhead. We exploit this in a [*block-parallel*]{} approach.
@rocki09 ([-@rocki09]) proposed a GPU-based parallel minimax gametree search algorithm for 8x8 Othello (without any $\alpha\beta$ pruning) which works as follows. Within each block a node $n$ is selected for expansion. If $n$ is a leaf, it is evaluated using a parallel evaluation function (32 threads, 1 thread per 2 positions in the 8x8 board). Otherwise a parallel successor generator function is called (1 thread/position) to generate successors of $n$, which are added to the node queue.
This approach greatly reduced warp divergence, since all threads in the warp are synchronized to perform the fetch-evaluate-expand cycle. Because there is no $\alpha\beta$ pruning, their search trees have uniform depth (i.e., fixed-depth DFS), and also, the \# of possible moves on the othello board (64) conveniently matched a multiple of the CUDA warp size (32).
We now propose a generalization of this approach for IDA\*, which we call [*Block-Parallel IDA\**]{}, shown in Alg. \[alg:Block-Parallel-IDA\*\]. In contrast to the parallel minimax of [@rocki09], BPIDA\* handles variable-depth subtrees (due to the heuristic, IDA\* tree depths are irregular) and does not depend on a fixed number of applicable operators (e.g., 64).
$openList$ is a stack which is shared among all threads in the same block, which supports two key parallel operations: [parallelPop]{} and [atomicPut]{}. [parallelPop]{} extracts $(\mathit{\#threads\_in\_a\_block} / \mathit{\#operators})$ nodes from $openList$. [atomicPut]{} inserts nodes in $t$ into the shared $openList$ concurrently. This is implemented as a linearizable [@herlihy90] operation. The [BPDFS]{} function is similar to a standard, sequential $f$-limited depth-first search, but in each iteration of the repeat-until loop in lines 4-16 (Alg. \[alg:Block-Parallel-IDA\*\]), a warp performs the fetch-evaluate-expand cycle on $(\mathit{\#threads\_in\_a\_block} / \mathit{\#operators})$ nodes. The number of threads per block is set to the warp size (32). This allows the following: (1) When a warp is scheduled for execution, all cores in the SM are active. (2) Since all threads in the block (=warp) share a program counter, explicit synchronizations become unnecessary.
BPIDA\* applies a slightly modified version of the static load balancing used by PStaticLB (Sec. \[sec:static-load-balancing\]). While PStaticLB uses the number of expanded nodes to estimate the work in the next iteration, BPIDA\* uses the number of repetitions executed in lines 4-16. BPIDA\* does not use dynamic load balancing.
$openList \gets root$ $f_{next} = \infty$ $s \gets \Call{parallelPop}{openList}$ $s$ and its parents as a shortest path $a \gets (\mathit{threadID} \enspace \text{mod} \enspace \text{\#actions}) \text{th action}$ $t \gets successor(a, s)$ $f_{new} \gets g(s) + cost(a) + h(t)$ $\Call{atomicPut}{openList, t}$ $f_{next} \gets min(f_{next}, f_{new})$ $f_{next}$ no plan is found\
$roots \gets \Call{CreateRootSet}{start, goals}$ $limit_f \gets \Call{DecideFirstLimit}{roots}$ $limit_f, stat \gets \Call{BPDFS}{root, goals, limit_f}$
Evaluation of BPIDA\*
=====================
[.31]{} {width="\linewidth"}
[.31]{} {width="\linewidth"}
[.31]{} {width="\linewidth"}
### Runtimes {#runtimes .unnumbered}
Figure \[block\_rel\] compares the relative runtime of BPIDA\* vs. PFullLB. BPIDA\* required a total of 95 seconds to solve all 100 problems, a speedup of 9.39 compared to PFullLB. Table \[tab:summary-runtime\] summarizes the total runtimes and speedups for all algorithms in this paper.
[|l|cc|]{} configuration & total runtime & speedup\
& (seconds) & vs. G1\
\
Solver B [@burns12] & 620 & n/a\
Solver C & 475 & n/a\
\
G1 & 62957 & 1\
\
PSimple & 3378& 18.6\
PStaticLB & 1069 & 58.9\
PFullLB & 892& 70.8\
[**BPIDA\***]{} & [**95**]{} & [**659.5**]{}\
### Other metrics {#other-metrics .unnumbered}
There are 3 suspected culprits for the poor performance of thread-based parallel IDA\*: (1) dynamic load overhead, (2) idle SMs (bad load balance), and (3) thread stalls for warp divergence. BPIDA\* doesn’t perform dynamic load balancing, so (1) is irrelevant. For factors (2) and (3), there are related metrics, sm\_efficiency and IPC(instructions per cycle), which can be measured by the CUDA profiler, nvprof. sm\_efficiency is the average % of time at least one warp is active on a SM. High sm\_efficiency shows how busy the SMs are, and high IPC indicates there are few NOPs due to warp divergence. The (mean, min, max, stddev) sm\_efficiency over 100 instances was (65.22, 31.8, 82.7, 7.94) for PFullLB, and (94.29, 32.3, 99.9, 9.76) for BPIDA\*, and for IPC, the results were (0.30, 0.13, 0.39, 0.048) for PFullLB and (0.97, 0.60, 1.06, 0.059) for BPIDA\*. For both metrics, the results of BPIDA\* were better than PFullLB, and close to the ideal values (100% sm\_efficiency and IPC=1.0).
Comparison with Sequential Solver C
-----------------------------------
We now compare BPIDA\* with the CPU-based, sequential Solver C (Sec. \[sec:baseline\]).
Fig. \[one\_rel\] compares the relative runtimes of Solver C (1 CPU core) and BPIDA\* (1536 GPU cores). The y-axis shows Runtime(SolverC)/Runtime(BPIDA\*) for each instance. Comparing the total time to solve all 100 instances, BPIDA\* was 4.98 times faster.
Runtime comparisons between parallel vs. sequential IDA\* can be obfuscated by the fact that they do not necessarily expand the same set of nodes in the final iteration, although they expand the same set of nodes in non-final iterations (the same issue exists with comparisons among parallel IDA\* variants, but from Fig. \[block\_rel\] and Table \[tab:summary-runtime\], it is clear that BPIDA\* significantly outperforms the other parallel algorithms, so above, we simply reported the time to find a single solution, as is standard practice in previous works).
To eliminate differences in search efficiency (node expansion order) from the comparison, the next experiment compares the time required to find [*all optimal-cost solutions*]{} of every problem, i.e., the search does not terminate until all nodes with $f \leq OptimalCost$ have been expanded. This eliminates node ordering effects, allowing comparison of the wall-clock time required to perform the same amount of search. Fig. \[allsol\_rel\] compares the relative runtimes of Solver C (1 CPU core) and BPIDA\* (1536 GPU cores). The y-axis shows Runtime(SolverC)/Runtime(BPIDA\*) to find all optimal solutions for each instance. Comparing the total time to find all optimal solutions for all 100 instances, BPIDA\* was 6.78 times faster.
Conclusions and Future Work
===========================
We proposed Block-Parallel IDA\*, which assigns subtrees to GPU blocks (groups of threads with fast shared memory). Compared to thread-parallel approaches, this greatly reduces warp divergence and improves load balance. BPIDA\* also does not require explicit dynamic load balancing, making it relatively simple to implement. On 1536 cores, BPIDA\* achieves a speedup of 659.5 vs. a 1-thread GPU baseline, i.e., 42% parallel efficiency. Compared to a highly optimized single-CPU IDA\*, BPIDA\* achieves a 6.78x speedup when comparing the time to find all optimal solutions.
The successful parallelization of BPIDA\* on the 15-puzzle with Manhattan distance (MD) heuristic exploits the following factors: (1) compact states, (2) the MD heuristic requires little memory, and (3) standard IDA\* doesn’t perform duplicate state detection. Thus, all work could be performed in the SM local+shared memories, without using global memory. In many domains, data structures representing each state are larger and the IDA\* state stacks will not fit in local memory. Also, some powerful memory-intensive heuristics, e.g,. PDBs [@korf02], will require at least the use of global memory. Finally, standard approaches for reducing duplicate state expansion, e.g., transposition tables [@reinefeld94] requires significant memory. Thus, future work will focus on methods which use GPU global memory effectively so that domains with larger states, memory-intensive heuristics, and memory-intensive duplicate pruning techniques can be used.
[^1]: This is an extended manuscript based on a paper accepted to appear in SoCS2017.
|
---
abstract: 'We prove that a structurally stable diffeomorphism of closed $(2m+1)$-manifold, $m\ge 1$, has no codimension one non-orientable expanding attractors.'
author:
- 'V. Medvedev'
- 'E. Zhuzhoma'
date: Dedicated to Carlos Gutierrez on his 60th birthday
title: 'There are no structurally stable diffeomorphisms of odd-dimensional manifolds with codimension one non-orientable expanding attractors'
---
Introduction
============
Structurally stable diffeomorphisms exist on any closed manifold (say a diffeomorphism $f$ structurally stable if all diffeomorphisms $C^1$-close to $f$ are conjugate to $f$). It is natural to study the question of existence of such diffeomorphisms with some additional conditions. The condition we consider here is the presence of a codimension one non-orientable expanding attractor. Due to well known example of Plykin [@Plykin74], the answer is YES for 2-manifolds. Medvedev and Zhuzhoma [@MedvZh2002] proved that for 3-manifolds the answer is NO. In the paper, we generalize the result of [@MedvZh2002] proving that there are no structurally stable diffeomorphisms with a codimension one non-orientable expanding attractor on closed odd-dimensional manifolds. The proof is shorter than [@MedvZh2002] and includes $d = 3$. As to orientable attractors, the answer is YES for any $d\ge 2$. Namely, starting with a codimension one Anosov diffeomorphism of the $d$-torus $T^d$, $d\ge 2$, a structurally stable diffeomorphism of $T^d$ with an orientable codimension one expanding attractor can be obtained by Smale’s surgery [@Smale67], so-called $DA$-diffeomorphism (see also [@KatokHassenblat95], [@Plykin84], [@Robinson-book99]).
Before the formulation of exact result, we give necessary definitions and notions. Let $ f: M\to M $ be a diffeomorphism of a closed $d$-manifold $M$, $d = \dim M\ge 2$, endowed with some Riemann metric $\rho$ (all definitions in this section can be found in [@KatokHassenblat95] and [@Robinson-book99], unless otherwise indicated). A point $x\in M$ is [*non-wandering*]{} if for any neighborhood $U$ of $x$, $f^n(U)\cap U \neq \emptyset$ for infinitely many integers $n$. Then the non-wandering set $NW(f)$, defined as the set of all non-wandering points, is an $f$-invariant and closed. A closed invariant set $\Lambda \subset M$ is [*hyperbolic*]{} if there is a continuous $f$-invariant splitting of the tangent bundle $T_{\Lambda}M$ into stable and unstable bundles $E^s_{\Lambda}\oplus E^u_{\Lambda}$ with $$\Vert df^n(v)\Vert \le C\lambda ^n\Vert v\Vert ,\quad \Vert df^{-n}(w)\Vert \le C\lambda ^n\Vert w\Vert ,
\quad \forall v\in E^s_{\Lambda}, \forall w\in E^u_{\Lambda}, \forall n\in \mathbb{N},$$ for some fixed $C > 0$ and $\lambda < 1$. For each $x\in \Lambda$, the sets $ W^s(x) = \{y\in M: \lim_{j\to \infty}\rho (f^j(x),f^j(y))\to 0 $, $ W^u(x) = \{y\in M: \lim_{j\to \infty} \rho (f^{-j}(x),f^{-j}(y)) \to 0 $ are smooth, injective immersions of $E_x^s$ and $E_x^u$ that are tangent to $E_x^s$, $E_x^u$ respectively. $W^s(x)$, $W^u(x)$ are called [*stable*]{} and [*unstable manifolds*]{} at $x$.
For a diffeomorphism $ f: M\to M $, Smale [@Smale67] introduced the Axiom A: $NW(f)$ is hyperbolic and the periodic points are dense in $NW(f)$. A diffeomorphism satisfying the Axiom A is called $A$-diffeomorphism. According to Spectral Decomposition Theorem, $NW(f)$ of an $A$-diffeomorphism $f$ is decomposed into finitely many disjoint so-called basic sets $B_1$, $\ldots , B_k$ such that each $B_i$ is closed, $f$-invariant and contains a dense orbit.
A basic set $\Omega$ is called an [*expanding attractor*]{} if there is a closed neighborhood $U$ of $\Omega$ such that $f(U)\subset int~U$, $\cap _{j\ge 0}f^j(U) = \Omega$, and the topological dimension $\dim \Omega$ of $\Omega$ is equal to the dimension $\dim (E^u_{\Omega})$ of the unstable splitting $ E^u_{\Omega} $ (the name is suggested in [@Williams67], [@Williams74]). $\Omega$ is codimension one if $\dim~\Omega = \dim~M - 1$. It is well known that a codimension one expanding attractor consists of the $(d-1)$-dimensional unstable manifolds $W^u(x)$, $x\in \Omega$, and is locally homeomorphic to the product of $(d-1)$-dimensional Euclidean space and a Cantor set. $W^s(x)$ is homeomorphic to $\mathbb{R}$ and can be endowed with some orientation. $W^u(x)$ is homeomorphic to $\mathbb{R}^{d-1}$ and can be endowed with some normal orientation (even if $M$ is non-orientable). Due to hyperbolic structure, any $W^s(x)$ intersects $W^u(x)$ transversally, $x\in \Omega$. Following [@Grines75], say that $\Omega$ is [*orientable*]{} if for every $x\in \Omega$ the index of the intersection $W^s(x)\cap W^u(x)$ does not depend on a point of this intersection (it is either $+1$ or $-1$). The main result is the following theorem.
\[main-theorem\] Let $f: M\to M$ be a structurally stable diffeomorphism of a closed $(2m+1)$-manifold $M$, $m\ge 1$. Then the spectral decomposition of $f$ does not contain codimension one non-orientable expanding attractors.
Our proof does not work for even-dimensional manifolds for which the existence of codimension one non-orientable expanding attractors stay open question (except $d = 2$).
[*Acknowledgment*]{}. The research was partially supported by RFFI grant 02-01-00098. We thank Roman Plykin and Santiago Lopez de Medrano for useful discussions. This work was done while the second author was visiting Rennes 1 University (IRMAR) in March-Mai 2004. He thanks the support CNRS which made this visit possible. He would like to thank Anton Zorich and Vadim Kaimanovich for their hospitality.
Proof of the main theorem
=========================
Later on, $\Omega$ is a codimension one non-orientable expanding attractor of diffeomorphism $f: M\to M$. A point $p\in \Omega$ is called [*boundary*]{} if at least one component of $W^s(p) - p$ does not intersect $\Omega$. Boundary points exist and satisfy to the following conditions [@Grines75], [@Plykin74]:
- There are finitely many boundary points and each is periodic.
- Given a boundary point $p\in \Omega$, there is a unique component of $W^s(p) - p$ denoted by $W^s_{\emptyset}(p)$ which does not intersect $\Omega$.
- Given a point $x\in W^u(p) - p$, there is a unique arc $(x,y)^s\subset W^s(x)$ denoted by $(x,y)^s_{\emptyset}$ such that $(x,y)^s\cap \Omega = \emptyset$ and $y\in \Omega$.
An unstable manifold $W^u(p)$ containing a boundary point is called a [*boundary unstable manifold*]{}. Due to [@Grines75] and [@Plykin84], the accessible boundary of $M - \Omega$ from $M - \Omega$ is a finite union of boundary unstable manifolds that splits into so-called bunches defined as follows. The family $W^u(p_1)$, $\ldots , W^u(p_k)$ is said to be a [*$k$-bunch*]{} if there are points $x_i\in W^u(p_i)$ and arcs $(x_i,y_i)^s_{\emptyset}$, $y_i\in W^u(p_{i+1})$, $1\le i\le k$, where $p_{k+1} = p_1$, $y_k\in W^u(p_1)$, and there are no $(k+1)$-bunches containing the given one.
\[non-orientability\] Let $f: M\to M$ be an $A$-diffeomorphism of a closed $(2m+1)$-manifold $M$, $m\ge 1$. If the spectral decomposition of $f$ contains a codimension one non-orientable expanding attractor, then $M$ is non-orientable.
[*Proof*]{}. The non-orientability of $ \Omega $ implies that $ \Omega $ has at least one 1-bunch, say $W^u(p)$ [@Plykin84]. Therefore, given any point $x\in W^u(p) - p$, there is a unique point $y\in W^u(p) - p$ such that $(x,y)^s = (x,y)^s_{\emptyset}$, and vise versa. Let the map $ \phi : W^u(p) - p \to W^u(p) - p $ be given by $\phi (x) = y$ whenever $(x,y)^s = (x,y)^s_{\emptyset}$. Then $\phi$ is an involution, $\phi ^2 = id$.
Let $r$ be the period of $p$. Since the stable (as well as unstable) manifolds are $f$-invariant, $ f^r\circ \phi |_{W^u(p)} = \phi \circ f^r|_{W^u(p)}$. Since the restriction $f^r|_{W^u(p)}$ is an expantion map with the unique hyperbolic fixed point $p$, $\phi$ can be extended homeomorphically to $W^u(p)$ putting $\phi (p) = p$. By theorem 2.7 and lemma 2.1 [@Plykin84], $\phi$ is conjugate to the antipodal involution, i.e. there exist a homeomorphism $ h: W^u(p)\to \mathbb{R}^{d-1} $ (in the intrinsic topology of $W^u(p)$) and the involution $\theta : \mathbb{R}^{d-1}\to \mathbb{R}^{d-1}$ of the type $\vec v \to -\vec v$ such that $\theta \circ h = h\circ \phi$. This implies that there is the $(d-1)$-dimensional ball $B^{d-1}\subset W^u(p)$ such that $p\in B^{d-1}$, the boundary $\partial B^{d-1}\stackrel{\rm def}{=}S^{d-2}$ is tamely embedded in $W^u(p)$, and $S^{d-2}$ is $\phi$-invariant. Moreover, there is the annulus $ S^{d-2}\times [0,1]\subset W^u(p) $ foliated by $S^{d-2}_t = S^{d-2}\times \{t\}$, $t\in [0,1]$, $ S^{d-2} = S^{d-2}_0 $, such that every $ S^{d-2}_t $ is $\phi$-invariant and bounds the $(d-1)$-dimensional ball $B^{d-1}_t\subset W^u(p)$ containing $p$. Since $\phi ^2 = id$, the set $$B^{d-1}_t\bigcup _{x\in S^{d-2}_t}[x, \phi (x)]\stackrel{\rm def}{=}P_t$$ is homeomorphic to the projective space $ \mathbb{R}P^{d-1} $ for every $t\in [0,1]$. Since $d - 1 = 2m$ is even, $P_t$ is non-orientable. For any $ x\in S^{d-2}_{t_1}$ and $ y\in S^{d-2}_{t_2}$ with $t_1\neq t_2$, $ [x, \phi (x)]^s_{\infty} \cap [y, \phi (y)]^s_{\infty} = \emptyset$. Hence the set $$\bigcup _{x\in S^{d-2}\times [0,1]}[x, \phi (x)] \subset M$$ is homeomorphic to $ \mathbb{R}P^{d-1}\times [0,1] $. Since $ \mathbb{R}P^{d-1}\times [0,1] $ is a non-orientable $d$-manifold, $M$ is non-orientable. $\Box$
[*Proof of theorem \[main-theorem\]*]{}. Assume the converse. Then the spectral decomposition of $f$ contains a codimension one non-orientable expanding attractor, say $\Omega$. According to lemma \[non-orientability\], $M$ is non-orientable. Let $ {\overline }M $ be an orientable manifold such that $ \pi : {\overline }M\to M $ is a (nonbranched) double covering for $M$. Then there exists a diffeomorphism ${\overline }f: {\overline }M\to {\overline }M$ that cover $f$, i.e., $f\circ \pi = \pi \circ {\overline }f$. It is easy to see that ${\overline }f$ is an $A$-diffeomorphism with a codimension one expanding attractor ${\overline }\Omega \subset \pi ^{-1}(\Omega )$. It follows from lemma \[non-orientability\] and orientability of ${\overline }M$ that $ {\overline }\Omega $ is orientable.
Because of $f$ is a structurally stable diffeomorphism, $f$ satisfies to the strong transversality condition [@Mane88] which is a local condition. Since $\pi$ is a local diffeomorphism, ${\overline }f$ satisfies to the strong transversality condition as well. Hence, ${\overline }f$ is structurally stable [@Robinson76].
Take a periodic point $p\in \Omega$ on the boundary unstable manifold $W^u(p)$ that is a 1-bunch. Then the preimage $\pi ^{-1}(W^u(p))$ is a 2-bunch of ${\overline }\Omega$ consisting of unstable manifolds $ W^u(p_1) $, $ W^u(p_2) $, where $\{p_1, p_2\} = \pi ^{-1}(p)$ are boundary periodic points of ${\overline }f$. It was proved in [@GrinesZh2000a], [@GrinesZh2000b] that $W^s_{\emptyset}(p_1)$ and $W^s_{\emptyset}(p_2)$ belong to the unstable manifolds $W^u(\alpha _1)$ and $W^u(\alpha _2)$ respectively of the repelling periodic points $ \alpha _1 $, $\alpha ^{\prime}$ (possibly, $ \alpha _1 = \alpha ^{\prime} $). Moreover, there are repelling periodic points $ \alpha _1 $, $ \ldots , \alpha _{k+1} = \alpha ^{\prime} $ and saddle periodic points $ P_1 = p_1 $, $ P_2 $, $ \ldots , P_{k+1} $, $ P_{k+2} = p_2 $, $ k\ge 0 $, of index $ d - 1 $ such that the following conditions hold:
1. The set $$l = P_1\cup W^s_{\emptyset}(P_1)\cup \alpha _1\cup W^s(P_2)\cup \ldots \cup \alpha _{k+1}\cup W^s_{\emptyset}(P_{k+2})\cup P_{k+2}$$ is homeomorphic to an arc with no self-intersections whose endpoints are $P_1$ and $P_{k+2}$.
2. $ l - (P_1\cup P_{k+2}) \subset {\overline }M - {\overline }\Omega $.
3. The repelling periodic points $\alpha _i$ alternate with saddle periodic points $P_i$ on $l$.
It follows from $f\circ \pi = \pi \circ {\overline }f$ that $\pi$ maps the stable and unstable manifolds of ${\overline }f$ into the stable and unstable manifolds respectively of $f$. Since $ \pi (P_1) = \pi (P_{k+2}) = p $, $$\pi (W^s_{\emptyset}(P_1)) = \pi (W^s_{\emptyset}(P_2)), \quad \pi (\alpha _1) = \pi (\alpha _{k+1}).$$ Hence (if $k\ge 1$), $$\pi (W^s(P_2)) = \pi (W^s(P_{k+1})),\quad \pi (P_2) = \pi (P_{k+1}),
\quad \pi (\alpha _2) = \pi (\alpha _{k}),\quad \ldots .$$ Due to item (3) above, the number of all periodic points on $l$ equals $2k+3$ that is odd. As a consequence, there is either a periodic point $\alpha _i$ with $ \pi (W^s(P_i)) = \pi (W^s(P_{i+1})) $ or a periodic point $P_i$ with $ \pi (W^s_1(P_i)) = \pi (W^s_2(P_{i})) $, where $ \pi (W^s_1(P_i)) $, $\pi (W^s_2(P_{i})) $ are different components of $ W^s(P_i) - P_i $. In both cases, there is a point ($ \alpha _i$ or $P_i$) at which $\pi$ is not a local homeomorphism. This contradiction concludes the proof. $\Box$
[99]{}
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Dept. of Diff. Equat., Inst. of Appl. Math. and Cyber., Nizhny Novgorod State University, Nizhny Novgorod, Russia
[*E-mail address*]{}: medvedev@uic.nnov.ru
Dept. of Appl. Math., Nizhny Novgorod State Technical University, Nizhny Novgorod, Russia
[*E-mail address*]{}: zhuzhoma@mail.ru
[*Current e-mail address*]{}: zhuzhoma@maths.univ-rennes1.fr
|
---
abstract: 'The question raised by \[Bastin and Martin 2003 J. Phys. B: At. Mol. Opt. Phys. [**36**]{}, 4201\] is examined and used to explain in more detail a key point of our calculations. They have sought to rebut criticisms raised by us of certain techniques used in the calculation of the off-resonance case. It is also explained why this result is not a problem for the off-resonance case, but, on the contrary, opens the door to a general situation. Their comment is based on a blatant misunderstanding of our proposal an d as such is simply wrong.'
address: |
$^1$Mathematics Department, Faculty of Science, South Valley University, 82524 Sohag, Egypt.\
$^2$Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City, Cairo, Egypt.
author:
- 'Mahmoud Abdel-Aty$^{1,}$, A.-S. F. Obada$^2$'
title: '**Reply to Comment on “Quantum inversion of cold atoms in a microcavity: spatial dependence”** '
---
[**Submitted to:**]{}
In our paper \[2\] we have used the Hamiltonian (in the mesa mode case)
$$\hat{H}=\frac{p_{z}^{2}}{2M}+\frac{\Delta }{2}\sigma_{z}+\omega (a^{\dagger
}a+\frac{1}{2}\sigma_{z})+\lambda f(z)\{\sigma ^{-}a^{\dagger }+a\sigma
^{+}\}. \label{1}$$
Let us write equation (1) in the following form (in the mesa mode case $%
f(z)=1)$ $$\begin{aligned}
\hat{H} &=&\frac{p_{z}^{2}}{2M}+\hat{V}
\nonumber
\\
\hat{V} &=&\frac{\Delta }{2}\sigma_{z}+\omega (a^{\dagger }a+\frac{1}{2}%
\sigma_{z})+\lambda \{\sigma ^{-}a^{\dagger }+a\sigma ^{+}\}.\end{aligned}$$ It is easy to show that in the $2\times 2$ atomic-photon space the eigenvalues and eigenfunction of the interaction Hamiltonian $\hat{V}$ ($%
\hat{V}|\Phi_{n}^{\pm }\rangle =E_{n}^{\pm }|\Phi_{n}^{\pm }\rangle )$ are given by \[3\], $$E_{n}^{\pm }=\omega (n+\frac{1}{2})\pm \sqrt{\frac{\Delta ^{2}}{4}+\lambda
^{2}(n+1)},$$ $$\begin{aligned}
|\Phi_{n}^{+}\rangle &=&\cos \theta_{n}|n+1,g\rangle +\sin \theta
_{n}|n,e\rangle,
\nonumber
\\
|\Phi_{n}^{-}\rangle &=&-\sin \theta_{n}|n+1,g\rangle +\cos \theta
_{n}|n,e\rangle,\end{aligned}$$ where $$\theta_{n}=\tan ^{-1}\left( \frac{\lambda \sqrt{n+1}}{\sqrt{\frac{\Delta
^{2}}{4}+\lambda ^{2}(n+1)}-\frac{\Delta }{2}}\right),\qquad$$ We write the wave function $|\Psi (z,t)\rangle =\sum\limits_{n}C_{n}^{\pm
}(z,t)|\Phi_{n}^{\pm }\rangle .$ Then using the total Hamiltonian (1) we have $$\begin{aligned}
\hat{H}|\Psi (z,t)\rangle &=&\sum\limits_{n}\left( \frac{p^{2}}{2M}%
+V\right) C_{n}^{\pm }(z,t)|\Phi_{n}^{\pm }\rangle
\nonumber
\\
&=&\sum\limits_{n}\frac{p^{2}}{2M}C_{n}^{\pm }(z,t)|\Phi_{n}^{\pm }\rangle
+\sum\limits_{n}VC_{n}^{\pm }(z,t)|\Phi_{n}^{\pm }\rangle
\nonumber
\\
&=&\sum\limits_{n}\frac{p^{2}}{2M}C_{n}^{\pm }(z,t)|\Phi_{n}^{\pm }\rangle
+\sum\limits_{n}C_{n}^{\pm }(z,t)E_{n}^{\pm }|\Phi_{n}^{\pm }\rangle
\nonumber
\\
&=&\sum\limits_{n}\left( \frac{p^{2}}{2M}C_{n}^{\pm }(z,t)+E_{n}^{\pm
}C_{n}^{\pm }(z,t)\right) |\Phi_{n}^{\pm }\rangle ,\end{aligned}$$ because of the orthonormality of the wavefunctions $|\Phi_{n}^{\pm }\rangle $ then $$i\frac{\partial }{\partial t}C_{n}^{\pm }=\left( -\frac{1}{2M}\frac{\partial
^{2}}{\partial z^{2}}+E_{n}^{\pm }\right) C_{n}^{\pm },$$ with no coupling even in the presence of the detuning (equation (13) in BM comment \[1\]).
It may be worthwhile for the authors to consult some papers that have been published previously (see for example Refs. \[4-6\]) where the detuning has been considered and similar results have been obtained. To be more precisely:
- We have used the interaction picture, so that the term $(n+\frac{1%
}{2})\omega $ does not appear, it can be used as a phase only. Bearing in mind the case of mesa mode is being treated in our paper i.e. $f(z)=1$.
- The most serious point is that Bastin and Martin have overlooked the formulae for $$\cos 2\theta_{n}\qquad and\qquad \sin 2\theta_{n}.$$ From equation (10) raised in Bastin and Martin comment, it is easy to write $$\tan \theta_{n}=\frac{\lambda \sqrt{n+1}}{\sqrt{\frac{\Delta ^{2}}{4}%
+\lambda ^{2}(n+1)}-\frac{\Delta }{2}}=\frac{\sqrt{\frac{\Delta ^{2}}{4}%
+\lambda ^{2}(n+1)}+\frac{\Delta }{2}}{\lambda \sqrt{n+1}}$$ then $$\tan 2\theta_{n}=\frac{\lambda \sqrt{n+1}}{-\frac{\Delta }{2}}$$ Also, it is easy to prove that $$\cos 2\theta_{n}=\frac{-\Delta /2}{\sqrt{\frac{\Delta ^{2}}{4}+\lambda
^{2}(n+1)}}\qquad and\qquad \sin 2\theta_{n}=\frac{\lambda \sqrt{n+1}}{%
\sqrt{\frac{\Delta ^{2}}{4}+\lambda ^{2}(n+1)}}.$$ Once these formulae inserted in equations (18) and (19) of Bastin and Martin comment, we find that, the second terms vanish identically.
Now let us look more carefully at the general case when we take f(z) no longer a constant, i.e. we go beyond the mesa mode case. In this case the orthonormal functions $|\Phi_{n}^{\pm }\rangle $ in the $2\times 2$ system diagonalize the Hamiltonian $V$ and its elements are diagonal in this set of functions with $$\begin{aligned}
V_{n}^{\pm } &=&(n+\frac{1}{2})\omega \pm \sqrt{\frac{\Delta ^{2}}{4}%
+\lambda ^{2}f^{2}(z)(n+1)} \nonumber \\
\tan 2\theta_{n} &=&\frac{\lambda f(z)\sqrt{n+1}}{-\frac{\Delta }{2}}.\end{aligned}$$ The states $|\Phi_{n}^{\pm }\rangle $ are z-dependent through the trigonometric functions, they satisfy $$\begin{aligned}
\frac{\partial }{\partial z}|\Phi_{n}^{\pm }\rangle &=&\pm |\Phi_{n}^{\pm
}\rangle \frac{d\theta }{dz}, \nonumber \\
\frac{\partial ^{2}}{\partial z^{2}}|\Phi_{n}^{\pm }\rangle &=&\pm |\Phi
_{n}^{\pm }\rangle \frac{d^{2}\theta }{dz^{2}}-|\Phi_{n}^{\pm }\rangle
\left( \frac{d\theta }{dz}\right) ^{2}.\end{aligned}$$ Then $|\Psi (z,t)\rangle $ can be expanded in the form $|\Psi (z,t)\rangle
=\sum\limits_{n}C_{n}^{\pm }(z,t)|\Phi_{n}^{\pm }\rangle $ and it satisfies the Schrodinger equation $$i\frac{\partial }{\partial z}|\Psi (z,t)\rangle =H|\Psi (z,t)\rangle .$$
Hence the coefficients $C_{n}^{\pm }(z,t)$ satisfy the coupled equations $$\begin{aligned}
i\frac{\partial C_{n}^{+}}{\partial z} &=&\left( -\frac{1}{2M}\frac{\partial
^{2}}{\partial z^{2}}+V_{n}^{+}-\left( \frac{d\theta }{dz}\right)
^{2}\right) C_{n}^{+}-\left( 2\frac{\partial C_{n}^{-}}{\partial z}\left(
\frac{d\theta }{dz}\right) +C_{n}^{-}\left( \frac{d\theta }{dz}\right)
^{2}\right) ,
\nonumber
\\
i\frac{\partial C_{n}^{-}}{\partial z} &=&-\left( -\frac{1}{2M}\frac{%
\partial ^{2}}{\partial z^{2}}+V_{n}^{-}-\left( \frac{d\theta }{dz}\right)
^{2}\right) C_{n}^{-}+\left( 2\frac{\partial C_{n}^{+}}{\partial z}\left(
\frac{d\theta }{dz}\right) +C_{n}^{+}\left( \frac{d\theta }{dz}\right)
^{2}\right) ,\end{aligned}$$ These equations should replace equations (18) and (19) of the comment of \[1\]. But once $f(z)$ is taken to be constant, then $\frac{d\theta }{dz}$ will vanish and we get equations (7) and the results of \[2,3\].
[9]{} Bastin T and Martin J 2003 J. Phys. B: At. Mol. Opt. Phys. [**36**]{}, 4201.
Abdel-Aty M and Obada A-S F 2002 J. Phys. B: At. Mol. Opt. Phys. **35** 807.
Abdel-Aty M and Obada A-S F 2002 Modern Physics Letters B **16** 117.
Battocletti M and Englert B-G, 1994 J. Phys. II France **4** 1939.
Zhang, Z-M, et al 2000 J.Phys.B: At. Mol. Opt. Phys. **33** 2125.
Zhang, Z-M, He L-S 1998 Opt. Commun. **157** 77.
|
---
abstract: 'Online social media have greatly affected the way in which we communicate with each other. However, little is known about what are the fundamental mechanisms driving dynamical information flow in online social systems. Here, we introduce a generative model for online sharing behavior that is analytically tractable and which can reproduce several characteristics of empirical micro-blogging data on hashtag usage, such as (time-dependent) heavy-tailed distributions of meme popularity. The presented framework constitutes a null model for social spreading phenomena which, in contrast to purely empirical studies or simulation-based models, clearly distinguishes the roles of two distinct factors affecting meme popularity: the memory time of users and the connectivity structure of the social network.'
author:
- 'James P. Gleeson'
- 'Kevin P. O’Sullivan'
- 'Raquel A. Baños'
- 'Yamir Moreno$^{2,}$'
bibliography:
- 'CIC2.bib'
- 'compete\_bib.bib'
date: 30 Mar 2016
title: '[ The effects of network structure, competition and memory time on social spreading phenomena]{}'
---
Introduction
============
Recent advances in communication technologies and the emergence of social media have made it possible to communicate rapidly on a global scale. However, since we receive pieces of information from multiple sources, this has also made the information ecosystem highly competitive: in fact, users’ influence and visibility are highly heterogeneous and topics strive for users’ attention in online social systems. Although several studies have described the dynamics of information flow in popular communication media [@Bakshy11; @Lerman12; @Weng12; @Cheng14; @GleesonPNAS14], the main factors determining the observed patterns have not been identified and there is no theoretical framework that addresses this challenge. Indeed, given the potential for applications—e.g., having more efficient systems to spread information for safety and preparedness in the face of threats—a better understanding of how memes (ideas, hashtags, etc.) emerge and compete in online social networks is critical.
Information often spreads through a social network as a cascade: a person adopts a new behavior or installs a new app, or sends a news item or rumour to their friends (e.g., by tweeting it on Twitter). The avalanche spreads if the friends decide to also adopt the new behavior, and in turn pass on the social influence effect to their own friends, who may further propagate the behavior. Following the usage in the review [@Castellano09], we apply the term “social spreading phenomena” to describe such cascading or “viral” propagation [@Leskovec07]. The latter term is used because the description of information spreading bears some similarity to epidemics of contagious disease; the effects of network structure on disease contagion have been well-studied by physicists [@PastorSatorras01], see [@diseasereview] for a recent review. However, unlike epidemics of a single disease strain, we focus on social spreading phenomena that occur in the presence of competition between a large number of different items of similar type. Examples of the types of items include URLs on Twitter [@Bakshy11; @Lerman12], apps on Facebook [@Onnela10; @GleesonPNAS14], or videos on YouTube [@Miotto]. In each of these examples, users make choices—often influenced by the choices they have seen their friends make—and the accumulation of many choices leads to a distribution of popularity of the items: some items become extremely popular, while other items remain obscure.
To enable a succinct general description, we will call such items by Dawkins’ term [@Dawkins]“memes” because they are all “elements of a culture or system of behavior passed from one individual to another by imitation...”[@memedefinition]. Note that we do not restrict our study only to very popular memes; indeed our interest is in understanding the entire popularity distributions of memes, from the unpopular to the very popular. This definition of a meme has also been used by researchers studying cascades on Facebook [@Adamic14], the spreading of news through blogs [@Leskovec09], and the popularity of hashtags on Twitter [@Weng12; @Weng13], but it can also be applied to analyze popularity distributions of offline items (where copying promotes spreading) such as baby names [@Bentley04], dog breeds [@Herzog04], and even to citations (which are a type of popularity measure) of scientific papers [@Simkin07; @Redner98]. The memes in these examples are all relatively simple units of information that are easily identified in data sets; recent work has also demonstrated that more complex memes (represented by the appearance of common phrases, such as “quantum” or “graphene”, in the scientific literature) can be recognized by their inheritance patterns in the citation network [@Kuhn14].
A notable characteristic of many meme popularity distributions is that they are very fat-tailed: if a power-law distribution is fitted to the data then the power-law exponent $\tau$ is typically between 1.5 and 2, which lies outside the range of exponents produced by models of cumulative-advantage [@Price76; @Newman05; @Perc14] or preferential-attachment [@Barabasi99] type. The statistical physics of avalanches has been studied in the context of condensed-matter systems, where the flip of a single magnetic spin domain can cause its neighboring domains to also flip and so initiate a cascade [@Sethna01]. If the physical parameters of such a system are tuned to place it at a critical point [@Stanleybook] the sizes of avalanches are power-law distributed; the sandpile model of self-organized criticality (SOC) self-tunes so that the system balances at the critical point [@Bakbook]. However, unlike the memoryless particles or magnetic spins that constitute the microscopic entities in condensed-matter avalanches, humans absorb and transmit information on a wide variety of timescales that range from seconds to weeks [@Barabasi05; @Malmgren08]. Models of social interaction must therefore include “memory” effects (non-Markovian aspects) that lead to the emergence of characteristics that are qualitatively different from those seen in condensed-matter avalanches. The non-Markovian aspects of human temporal behavior have attracted considerable recent attention (e.g. [@Karsai11; @Delvenne15; @JoPRX14; @Iribarren09; @Iribarren11]), but we wish to investigate the effects of memory on popularity avalanches caused by users choosing between multiple items that they have seen in the past.
To address this problem, we develop a theoretical framework that models how users choose among multiple sources of incoming information and affect the spreading of memes on a directed social network, like Twitter [@Bakshy11; @Lerman12; @Weng12]. Our probabilistic model, in contrast to other studies [@Weng12; @Cheng14; @GleesonPNAS14; @Bentley11; @Simkin07] that use intensive computational simulations to fit to data, allows us to get analytical insights into the respective roles of the network degree distribution, the memory-time distribution of users, and the competition between memes for the limited resource of user attention. The model is a “null model” in the sense that it is analytically tractable, yet realistic enough to be fitted to empirical data and to reproduce some important characteristics of the data. We show that fitting to time-dependent data requires a non-trivial memory-time distribution, which is not possible with the toy model of Ref. [@GleesonPRL14], where users can remember only one meme. However, the phenomenon of “competition-induced criticality” that was first identified in [@GleesonPRL14] is shown to be robust to the inclusion of memory-times, heterogeneous user activity rates and complex network structures in the more realistic model used here. The current model requires more sophisticated mathematical analysis than that of Ref. [@GleesonPRL14] to deal with the long memory of users, but it enables us to understand how heavy-tailed distributions of meme popularity evolve over a range of timescales, as a few memes “go viral” but the majority become only moderately popular.
We phrase the model in terms of meme propagation on a directed social network (like Twitter) and interpret a “meme” to be any distinct piece of information that is easily copied and transmitted (e.g., a hashtag or URL within a tweet). However, it should be clear that the model and its results can also be extended to the other examples of viral phenomena discussed above. For the adoption of apps on Facebook, for example, the memes are the notifications sent when a user installs an app [@Onnela10]. If a friend is prompted by this notification to also install the app, then the meme propagates on the network and its popularity is measured by the number of installations of the app. We show that the crucial property of the model that poises the system at criticality is the competitive pressure for the limited resource of user attention, and this property is common to a broad range of social spreading phenomena that are characterized by the availability of large time-dependent data sets.
The remainder of the paper is structured as follows. The model is introduced in Sec. \[model\]; in Secs. \[derivation\] and \[analysis\] we derive and analyze a branching-process description of the model dynamics. We confirm the results of this analysis using numerical simulations in Sec. \[resultssynthetic\] and then use the analytical results to fit the model to hashtag popularities extracted from micro-blogging data in Sec. \[resultsdata\], and to explain novel features of the time-dependent data. [ In Sec. \[limitations\] we discuss limitations of the model and possible extensions of it.]{} Note that the Secs. \[derivation\] and \[analysis\] may be omitted on a first reading without affecting the understanding of the model and the main results.
Model
=====
In online communication platforms like Twitter, users follow (receive the broadcasts or “tweets” of) other users. In graph-theoretical terms, these relationships constitute directed links from the followed node (user) to the follower (Fig. 1). The network structure is defined by the joint probability $p_{jk}$ that a randomly-chosen node (user) has in-degree $j$ (i.e., follows $j$ other Twitter users) and out-degree $k$ (i.e., has $k$ followers), but the network is otherwise assumed to be maximally random (a configuration model directed network). The mean degree of the network is $z=\sum_{j,k} k p_{jk} = \sum_{j,k} j p_{jk}$. If we simplify the model by assuming that all users follow $z$ others—as we sometimes do to highlight the role of the out-degree distribution—then $p_{j k}$ can be replaced with $\delta_{j,z} p_k$, where $\delta_{j,z}$ is the Kronecker delta and $p_k$ is the out-degree distribution.
Each user has a “stream” that records all tweets received by the user, time-stamped by their arrival time. We assume that only a fraction $\lambda$ of the tweets received are deemed “interesting” by the user, and only the interesting tweets are considered for possible retweeting by that user. (Here we use the term “retweeting” in a general sense, to include any reuse of a previously-received meme such as a hashtag: note that a meme may be retweeted more than once by a user, unlike the model of Ref. [@Iribarren11]). The activity rate of a user—the average number of tweets that she sends per unit time, i.e., the rate of the Poisson process that describes her tweeting activity—can depend on how well-connected the user is within the social network [@Weng12], and we assume it depends on her in-degree $j$ and out-degree $k$ (her “$(j,k)$-class” for short); this assumption is supported by empirical evidence from Twitter, see Fig. 6 of Ref. [@Hodas13]. The user activity rates $\beta_{jk}$ give the relative activity levels of users in the $(j,k)$ class; the rates are normalized by choosing time units so that $\sum_{j k} \beta_{j k} p_{j k}=1$. If there are $N$ users in the network, this rate implies that an average of $N$ tweets are sent in each model time unit. To simplify the analysis, we will sometimes specialize to the case where all user activity rates are equal: $\beta_{j k}=1$.
When a user decides, at time $t$, to send a tweet, she has two options (see Fig. 1): with probability $\mu$, the user innovates, i.e., invents a new meme, and tweets this new meme to all her followers. The new meme appears in the user’s own stream (it is automatically interesting to the originating user), and in the streams of all her followers (where it may be deemed interesting by each follower, independently, with probability $\lambda$). If not innovating (with probability $1-\mu$), the user instead chooses a meme from her stream to retweet. The meme for retweeting is chosen by looking backwards in time an amount $t_m$ determined by a draw from the memory-time distribution $\Phi(t_m)$, and finding the first interesting meme in her stream that arrived prior to the time $t-t_m$. The retweeted meme then appears in the streams of the user’s followers (time-stamped as time $t$), but because it is a retweet, it does not appear a second time in the stream of the tweeting user. The popularity $n(a)$ of a meme is the total number of times it has been tweeted or retweeted by age $a$, i.e., by a time $a$ after its first appearance (when it was tweeted as an innovation) [@Simkin07]. [ Figure \[fig\_histories\] shows some examples of evolving meme popularities: each panel displays the popularity $n(a)$ of a single meme as a function of its age $a$.]{}
The model as described is a “neutral model” [@Pinto11; @Kimurabook] in the sense that all memes have the same “fitness” [@Bianconi01]: no meme has an inherent advantage in terms of its attractiveness to users. Nevertheless, the competition between memes for the limited resource of user attention causes initial random fluctuations in popularities of memes to be amplified, and leads to the variability across memes seen in Fig. \[fig\_histories\] and to popularity distributions with very heavy tails [@Bentley04]: heavier, for example, than can be generated by models of preferential attachment or cumulative advantage type [@Simkin11; @Simon55; @Price76; @Barabasi99; @Newman05]. This “competition-induced criticality” was studied for a zero-memory ($\Phi(t_m)=\delta(t_m)$) version of this model in Ref. [@GleesonPRL14]. Indeed, the results of Ref. [@GleesonPRL14] can be obtained as a special case of the model described here, by setting $\Phi(t_m)=\delta(t_m)$, $\lambda=1$, $\beta_{j k}\equiv 1$, and $p_{j k}=\delta_{j z} p_k$; numerical simulation results for a closely related model were first reported in Ref. [@Weng12].
A branching process approximation [@Harrisbook; @Iribarren11] for the model enables us to understand how the network structure (via the out-degree distribution $p_k$) and the users’ memory-time distribution ($\Phi(t_m)$) affect the popularity distribution of memes. Defining $q_n (a)$ as the probability that a meme has popularity (total number of (re)tweets) $n$ at age $a$, the branching process provides analytical expressions that determine the probability generating function (PGF) [@Wilfbook; @Newman01] of the popularity distribution, $$H(a;x)=\sum_{n=1}^\infty q_n (a) x^n.\label{defH}$$ The details of the derivation and analysis of the branching-process approximation are given in Sec. \[derivation\] and \[analysis\]. The reader who is mainly interested in the applications of the model may jump straight to Sec. \[resultssynthetic\], while noting that the most important outcome of the analysis is that in the small-innovation limit $\mu \to 0$, the model describes a critical branching process, with power-law distributions of popularity (avalanche size) [@Goh03; @SchwartzCohen; @Adami02; @Zapperi95].
Derivation of branching process approximation {#derivation}
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Derivation of governing equations
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We define $G_{jk}(\tau,\Omega;x)$ as the probability generating function for the size of the “retweet tree”, as observed at time $\Omega$, that grows from the retweeting of a meme that entered, at time $\tau\le\Omega$, the stream of a $(j,k)$-class user, see Fig. \[figS1\]B. To obtain an equation for $G_{jk}$, we consider the stream of a random $(j,k)$-class user (called “user $A$”) with a meme $M$ that entered the stream at time $\tau$ (either by innovation, or because it was received from a followed user and deemed interesting by $A$), see Fig. \[figS1\]A.
The likelihood that meme $M$ is retweeted in the future depends on how quickly other tweets enter the stream of user $A$. In fact, meme $M$ can be considered to “occupy” the stream for a time interval $\ell$ stretching from $\tau$ until the time $\tau+\ell$ when the next interesting meme enters the stream of user $A$. New memes enter the stream as a Poisson process at the constant rate[^1] $$r_{j k} = j \overline{\beta} \lambda + \mu \beta_{j k},\label{rate}$$ so the occupation time $\ell$ of meme $M$—the time it occupies the stream of user $A$—is an exponentially distributed random variable with density $$P_\text{occ}(\ell)= r_{j k} \exp\left(-r_{j k} \ell\right).\label{Pocc}$$ We note in passing that the mean occupation time $$\left< \ell \right>=\int_0^\infty \ell \,P_\text{occ}(\ell) \,d\ell=\frac{1}{j \overline{\beta} \lambda + \mu \beta_{j k}} \label{lmean}$$ is, for small innovation probabilities $\mu$, inversely proportional to $j$, the number of users followed. Thus, a user who follows many others experiences tweets entering his stream at a higher rate than a lower-$j$ user (compare the streams of users $B$ and $C$ in the schematic Fig. 1). Consequently, the high-$j$ user is less likely to see (and so to retweet) a given meme than a low-$j$ user. This aspect of the model clearly reflects empirical data, as seen in Fig. 3 of [@Hodas12] for example.
To determine the size of trees originating from meme $M$, we consider that trees observed at a time $\Omega\ge \tau$ must be created by the retweeting by user $A$, at some time(s) between $\tau$ and $\Omega$, via looking back in her stream to a time $r$, where $r$ lies between $\tau$ and $\min(\tau+\ell,\Omega)$ (i.e., $r$ lies within the time interval where meme $M$ occupies the stream). Let’s consider a time interval of (small) length $dr$, centered at time $r$, and calculate the size of trees that are seeded by a retweet based on a lookback into this interval, from a time $t$, with $t>r$, see Fig. \[figS1\]. In each $dt$ interval centered at time $t$, a tree will be seeded with probability[^2] $$P_\text{seed} = (1-\mu)\beta_{j k} \Phi(t-r) \, dr\, dt,$$ and will grow to a tree with size distribution (at observation time $\Omega$) generated by[^3] $$R_{k}(t,\Omega;x)= x \left[ 1-\lambda+\lambda G(t,\Omega;x)\right]^{k},$$ where $$G(t,\Omega;x) = \sum_{j,k}\frac{j}{z}p_{jk}G_{jk}(t,\Omega;x) \label{Gavg}$$ is the PGF for the sizes of trees originating from the successful insertion at time $t$ of a meme (that is deemed interesting) into the stream of a random follower.
To calculate the total size of the tree seeded by copying from the $dr$-interval, we must add the sizes of trees that are copied into all times $t$ with $t>r$. Since each copying event is independent, the total tree size is generated by $$J(r;x) = \prod_{t=r}^\Omega \left[ 1-P_\text{seed} + P_\text{seed} R_{k}(t,\Omega;x)\right].$$ Taking logarithms of both sides of this equation and expanding to first order in $dt$ gives $$\begin{aligned}
\log J \
& = \sum_{t=r}^\Omega \log\left[1-(1-\mu)\beta_{j k}\Phi(t-r)\,dr \,dt (1- R_{ k}(t,\Omega;x))\right]\nonumber\\
& \approx -(1-\mu)\beta_{j k} \sum_{t=r}^\Omega \Phi(t-r) \,dr\, dt (1-R_{k}(t,\Omega;x)) \nonumber\\
&\rightarrow - (1-\mu)\beta_{j k} \, dr \int_{r}^\Omega \Phi(t-r) (1-R_{k}(t,\Omega;x))\, dt \quad\text{ as }dt\to 0,\end{aligned}$$ so $J(r;x)$ can be written as $$J(r;x) = \exp\left[-(1-\mu)\beta_{j k}\, dr\int_r^\Omega\Phi(t-r) (1-R_{ k}(t,\Omega;x))\, dt\right]. \label{J8}$$ Recall that $J(r;x)$ is the PGF for trees seeded by copying from time $r$. To obtain the total size of all children trees of meme $M$, we must consider trees seeded at all possible times $r$ from $\tau$ to the time $\min(\tau+\ell,\Omega)$ that marks the end of the occupation of user $A$’s stream by meme $M$. Each $dr$ time interval again independently generates trees with sizes distributed according to Eq. (\[J8\]), so the PGF for the total size is found by multiplying together copies of the $J(r;x)$ function for each $dr$ time interval, thus: $$\begin{aligned}
P_\text{size}(\ell) & = \prod_{r=\tau}^{\min(\tau+\ell,\Omega)} J(r;x)\nonumber\\
& =\exp\left[-(1-\mu)\beta_{j k}\sum_{r=\t}^{\min(\tau+\ell,\Omega)} dr\, \int_r^\Omega\Phi(t-r) (1- R_{k}(t,\Omega;x))\, dt\right]\nonumber\\
& \rightarrow \exp\left[-(1-\mu)\beta_{j k}\int_{\t}^{\min(\tau+\ell,\Omega)} dr\, \int_r^\Omega dt\,\Phi(t-r) (1- R_{k}(t,\Omega;x))\right]\quad \text{ as } dr\to 0.\label{P9}\end{aligned}$$ Combining probabilities, by integrating over all possible occupation times $\ell$, gives $$G_{jk}(\tau,\Omega;x) = \int_0^\infty P_\text{occ}(\ell) P_\text{size}(\ell)\, d\ell$$ and combining Eqs. (\[Pocc\]), (\[Gavg\]) and (\[P9\]) yields an integral equation for $G$: $$\begin{aligned}
G(\t,\Omega;x)&=\sum_{j k}\frac{j}{z}p_{j k} \int_0^\infty d\ell\, \left(j \overline{\beta}\lambda + \mu \beta_{j k}\right)\exp\left[-\left(j \overline{\beta}\lambda + \mu \beta_{j k}\right)\ell\right] \times\nonumber\\
&\times\exp\left[-(1-\mu)\beta_{j k}\int_{0}^{\min(\t+\ell,\Omega)} dr\, \int_r^\Omega dt\,\Phi(t-r) (1- x\left[1-\lambda+\lambda G(t,\Omega;x)\right]^k)\right]. \label{E10}\end{aligned}$$ [ Introducing the change of variables $a=\Omega-\tau$, $\tilde r = r-\tau$, $\tilde \tau = \Omega-t$, we rewrite this equation as $$\begin{aligned}
G(\Omega-a,\Omega;x)&=\sum_{j k}\frac{j}{z}p_{j k} \int_0^\infty d\ell\, \left(j \overline{\beta}\lambda + \mu \beta_{j k}\right)\exp\left[-\left(j \overline{\beta}\lambda + \mu \beta_{j k}\right)\ell\right] \times\nonumber\\
&\hspace{-1.0cm}\times\exp\left[-(1-\mu)\beta_{j k}\int_{0}^{\min(\ell,a)} d\tilde{r}\, \int_{0}^{a-\tilde{r}} d\tilde{\t}\,\Phi(a-\tilde{r}-\tilde{\t}) (1- x\left[1-\lambda+\lambda G(\Omega-\tilde{\t},\Omega;x)\right]^k)\right].\label{e19A}\end{aligned}$$ Note that the only appearance of the observation time $\Omega$ in this equation is in the first two arguments of the $G$ function: this reflects the fact that the popularity of memes in this model depends only on their age $a$ (unlike cumulative-advantage models, which exhibit a dependence also on the global time because early-born items have an “early-mover” advantage [@Newman09]). We therefore compress the notation by defining $G$ in terms only of the age $a$ of the memes: $ G(\Omega-\tau;x):=G(\tau,\Omega;x)$, and $G(a;x)$ solves the integral equation $$\begin{aligned}
G(a;x)&=\sum_{j k}\frac{j}{z}p_{j k} \int_0^\infty d\ell\, \left(j \overline{\beta}\lambda + \mu \beta_{j k}\right)\exp\left[-\left(j \overline{\beta}\lambda + \mu \beta_{j k}\right)\ell\right] \times\nonumber\\
&\times\exp\left[-(1-\mu)\beta_{j k}\int_{0}^{\min(\ell,a)} d\tilde{r}\, \int_{0}^{a-\tilde{r}} d\tilde{\t}\,\Phi(a-\tilde{r}-\tilde{\t}) (1- x\left[1-\lambda+\lambda G(\tilde{\t};x)\right]^k)\right],\label{e19}\end{aligned}$$ with initial condition $G(0;x)=1$. ]{}
The popularity of a meme, as observed at time $\Omega$, that is seeded by a single tweet (e.g., by an innovation) at time $\tau$ may be calculated in a similar way to the derivation of Eq. (\[e19\]); the generating function is of the form $$H(\tau,\Omega;x) = \sum_{j,k} \beta_{j k} p_{j k} R_k(\tau,\Omega;x) G_{j k}(\tau,\Omega;x),\label{Hgen}$$ where $\beta_{jk}p_{jk}$ represents the probability that the seed tweet originates from a $(j,k)$-class user, $R_k$ is the PGF for the trees generated from the followers of the user, and $G_{jk}$ is the PGF for the size of the retweet-tree of the meme (see Fig. \[figS1\]C). Introducing the age $a$ of the meme as before and defining $q_n(a)$ as the probability that an age-$a$ meme has popularity $n$, we have the PGF defined in Eq. (\[defH\]), which is given by $$\begin{aligned}
H(a;x)&=\sum_{j k} \beta_{j k} p_{j k} x \left[1-\lambda+\lambda G(a;x)\right]^k \int_0^\infty d\ell\, \left(j \overline{\beta}\lambda + \mu \beta_{j k}\right)\exp\left[-\left(j \overline{\beta}\lambda + \mu \beta_{j k}\right)\ell\right] \times\nonumber\\
&\times\exp\left[-(1-\mu)\beta_{j k}\int_{0}^{\min(\ell,a)} d\tilde{r}\, \int_{0}^{a-\tilde{r}} d\tilde{\t}\,\Phi(a-\tilde{r}-\tilde{\t}) (1- x\left[1-\lambda+\lambda G(\tilde{\t};x)\right]^k)\right];\label{e21}\end{aligned}$$ the initial condition is $H(0;x)=x$ (i.e., all memes have initial popularity 1: $q_n(0)=\delta_{n,1}$).
Distribution of response times {#sec3.2}
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It is worth noting that all agents in the model have constant activity rates, so that the actions of each individual agent constitute a Poisson process. A Poisson process is characterized by an exponential distribution of inter-event times, where each event corresponds to an innovation or a retweeting action. This assumption is contrary to studies such as [@Vazquez07; @Min11; @Iribarren09; @Iribarren11; @JoPRX14; @TemporalNetworks; @Barabasi05; @Malmgren08; @Hoffmann12; @Boguna14; @VanMieghem13; @Kivela14], where heavy-tailed distributions of inter-event times are examined. Despite this, in our model the memory-time distribution $\Phi(t_m)$ directly influences the waiting times (or “response times”) between the receipt of a specific meme, and the retweeting of it. Indeed, if $\Phi(t_m)$ is a heavy-tailed distribution, then a meme received by a given user at time $\tau$ will be retweeted by that user at a time $t$ (with $t\gg\tau$) with probability proportional to $\Phi(t-\tau)$ (the exact relation depends on how long the meme occupies the stream of the user). Therefore, a heavy-tailed memory distribution gives rise to a heavy-tailed waiting-time distribution for individual memes, despite the fact that the activity of each individual user is described by a Poisson process (cf. the heavy-tailed waiting-time distributions found in empirical studies of email correspondence [@Barabasi05; @Malmgren08]). It is clearly important to distinguish between the distributions of inter-event times (for actions of users) and of the waiting times experienced by individual memes: the model assumes each user has exponentially-distributed inter-event times, but it can nevertheless produce heavy-tailed distributions of waiting times for memes to be retweeted.
In particular, if the memory-time distribution $\Phi(t_m)$ is a $\text{Gamma}(k_G,\theta)$ distribution [@Iribarren11] as used in Secs. \[resultssynthetic\] and \[resultsdata\], i.e., $\Phi(t_m)=\frac{1}{\Gamma(k_G)\theta^{k_G}}t_m^{k_G-1}\exp\left(-t_m/\theta\right)$, then $\Phi(t_m)$ is approximately power-law for memory times $t_m$ with $t_m\ll \theta$, with an exponential cutoff at larger times. The corresponding waiting-time distribution shows a similar scaling in this range, like the slow decay noted in empirical response times for Twitter users (e.g., in Fig. 5 of [@Hodas12]). [ In Sec. \[Discussion\] we consider how the model could be extended to incorporate bursty (non-Poisson) user activity.]{}
Analysis
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Criticality of the branching process {#criticality}
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A branching process may be classified by the expected (mean) number $\xi$ of “children” of each “parent”: if this number (called the “branching number”) is less than 1, the process is *subcritical* and if $\xi$ is greater than 1 the process is *supercritical*. *Critical* branching processes, with an average of exactly one child per parent, give rise to power-law distributions of tree-sizes and of durations of growth cascades, and have been used to examine self-organized criticality in sandpile models on networks [@Goh03; @Zapperi95]. Here we demonstrate that the general process derived in Sec. \[derivation\] is a critical branching process in the limit of vanishing innovation $\mu\to0$.
We identify the “parent” in the process as a meme that was accepted into the stream (i.e., deemed interesting) of a $(j,k)$-class user at time $\tau$: see, for example, meme $M$ in the stream of user $A$, as shown in Fig. \[figS1\]. The “children” of this meme are the retweets of it that are accepted into the streams of the followers of $A$ at any time $t>\tau$. The PGF for the number of children of meme $M$ is derived by following the same steps as in Sec. \[derivation\], but replacing $R_k$ by $(1-\lambda+\lambda x)^k$: each power of $x$ then counts a successful insertion of meme $M$ into the stream of one of the $k$ followers of $A$. The resulting PGF, for a meme of age $a$, is (cf. Eq. (\[E10\])) $$\begin{aligned}
K_{jk}(a;x)&= \int_0^\infty d\ell\, P_\text{occ}(\ell) \times\nonumber\\
&\times\exp\left[-(1-\mu)\beta_{j k}\int_{0}^{\min(\ell,a)} d\tilde{r}\, \int_{0}^{a-\tilde{r}} d\tilde{\t}\,\Phi(a-\tilde{r}-\tilde{\t}) (1- \left[1-\lambda+\lambda x\right]^k)\right] \nonumber\\
&= \int_0^\infty d\ell\, P_\text{occ}(\ell)
\exp\left[-(1-\mu)\beta_{j k} (1- \left[1-\lambda+\lambda x\right]^k) \int_{0}^{\min(\ell,a)} C(a-\tilde r) d\tilde{r}\right],
\label{EK}\end{aligned}$$ where $C(t)=\int_0^t \Phi(t_m)dt_m$ is the cumulative distribution function (CDF) for memory times. The expected (mean) number of children for a meme in the $(j,k)$-class stream is determined from the PGF in the usual way [@Wilfbook], by differentiating with respect to $x$ and evaluating at $x=1$, thus: $$\xi_{j k} = \left. \frac{\partial K_{j k}}{\partial x}\right|_{x=1}.$$ In the limit of large ages, $a\to\infty$, we use the fact that $C(\infty)=1$ to obtain $$\begin{aligned}
\xi_{jk}&\sim (1-\mu)\beta_{jk}\lambda k \int_0^\infty \ell\, P_\text{occ}(\ell) \,d\ell\quad \text{ as }a\to \infty \nonumber\\
&= \frac{(1-\mu) \beta_{j k}\lambda k}{j \overline{\beta} \lambda+ \mu \beta_{jk}}.\label{xijk}\end{aligned}$$ Averaging over all $(j,k)$ classes, the effective branching number $\xi$ of the process is the expected number of children of a meme that is accepted into the stream of a random follower: $$\begin{aligned}
\xi & = \sum_{j,k} \frac{j}{z} p_{j k} \xi_{j k} \nonumber\\
& \to \sum_{j,k} \frac{j}{z} p_{j k} \frac{\beta_{jk}\lambda k}{j \overline \beta \lambda} = 1 \quad\text{ as }\mu \to 0 \label{xi}\end{aligned}$$ (recall that $\overline \beta \equiv \sum_{j,k} \frac{k}{z}\beta_{j k} p_{j k}$).
Thus, we have shown that the branching process underlying the model is critical when $\mu=0$. The occupation time of a meme in a users’ stream is due to the competition between neutral-fitness memes for the limited resource of user attention; this competition ensures that the mean number of successful retweets (children) generated during the finite occupation time of the meme is precisely one, and so induces the power-law distributions of cascade sizes that are characteristic of critical branching processes [@Goh03; @Zapperi95].
It is worth noting that the result of Eq. (\[xi\]) can also be derived in a more heuristic fashion, which enables us to discuss possible generalizations of the model in Sec. \[limitations\]. As above, we want to calculate $\xi_{j k}$, the expected number of children of a parent meme $M$ that has been accepted into the stream of a $(j,k)$-class user, called user $A$. We consider a (long) time window of duration $W$ units. During this time window, a total of approximately $(j \overline{\beta} \lambda + \mu \beta_{j k})W$ tweets have been accepted into the stream of user $A$ (see footnote \[footnote2\] and Eq. (\[lmean\])). When user $A$ decides to retweet during the time window, one of these memes is chosen for retweeting. If the times chosen by the user are uniformly distributed over the window then the probability that the chosen meme is meme $M$ is $$P_\text{chosen} = \frac{1}{\text{number of memes in stream}} = \frac{1}{(j \overline{\beta} \lambda + \mu \beta_{j k})W} \label{Pchosen}.$$ Alternatively, this result can be calculated by noting that the average time that a single meme occupies the stream is given by $\left<\ell\right>$ in Eq. (\[lmean\]), so the expected fraction of the total time that meme $M$ occupies the stream of user $A$ over the window of length $W$ is $\left<\ell\right>/W = P_\text{chosen}$.
Recalling that the activity rate of user $A$ is $\beta_{j k}$, the expected number of retweets by this user during the time window is $$N_\text{retweets} = (1-\mu) \beta_{j k} W. \label{Nretweets}$$ Each retweet is broadcast to the $k$ followers of $A$, each of whom finds the retweet interesting with probability $\lambda$, so the expected number of children (memes deemed interesting by followers) per retweet is $\lambda k$. The expected number of children of the parent meme $M$ over the time window is therefore $$\begin{aligned}
\xi_{j k} & = \left(\text{number of retweets by $A$}\right)\times\left(\text{probability meme $M$ is chosen}\right)\times\left(\text{children per retweet}\right) \nonumber\\
& = N_\text{retweets} P_\text{chosen} \lambda k ,\label{xijkB}\end{aligned}$$ which recovers Eq. (\[xijk\]). The expected number $\xi$ of children of a meme that is accepted into the stream of a random follower is then calculated as in Eq. (\[xi\]), giving $\xi\to 1$ in the $\mu \to 0$ limit.
An explicit expression for $q_1(a)$ {#sec:q1}
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The value $q_1(a)$ is the probability that a meme, once created via an innovation event, is not retweeted by the time it reaches age $a$: recall that the popularity $n$ of a meme is set to 1 when it is first tweeted (i.e., at birth); subsequent retweets (if any) increase the value of $n$ above 1. The probability $q_1(a)$ may be calculated explicitly using Eq. (\[e21\]): $$\begin{aligned}
q_1(a) &= \lim_{x\to0}\frac{H(a;x)}{x},\nonumber\\
& = \sum_{j,k} \beta_{jk}p_{jk}\left[1-\lambda+\lambda G(a;0)\right]^k \int_0^\infty d\ell\, P_\text{occ}(\ell)
\exp\left[-(1-\mu)\beta_{j k} \int_{0}^{\min(\ell,a)} C(a-\tilde r) d\tilde{r}\right],\end{aligned}$$ with $G(a;0)$ given, from Eq. (\[e19\]), by $$\begin{aligned}
G(a;0) &= \sum_{j,k}\frac{j}{z} p_{j k }\int_0^\infty d\ell\, P_\text{occ}(\ell)
\exp\left[-(1-\mu)\beta_{j k} \int_{0}^{\min(\ell,a)} C(a-\tilde r) d\tilde{r}\right].\end{aligned}$$ If we consider the large-age limit, $a\to\infty$, than we can approximate the integral of the cumulative distribution function for memory times as $$\int_{0}^{\min(\ell,a)} C(a-\tilde r) d\tilde{r} \approx \ell \,C(a)$$ and the integral over $\ell$ can be calculated to give the large-$a$ approximation $$q_1(a) \sim \sum_{j k}\beta_{j k} p_{j k} \frac{j \overline{\beta} \lambda + \mu \beta_{j k}}{ j \overline{\beta}\lambda + \mu\beta_{jk} + (1-\mu) \beta_{j k}C(a)}\left[1-\lambda+\lambda G(a;0)\right]^k,\label{q1a}$$ with $$G(a;0)\sim \sum_{ j k}\frac{j}{z} p_{j k} \frac{j \overline{\beta} \lambda + \mu \beta_{j k}}{ j \overline{\beta}\lambda + \mu\beta_{jk} + (1-\mu) \beta_{j k}C(a)}.\label{q1b}$$ In the simplified case $p_{jk}=\delta_{j,z}p_k$ and $\beta_{j k}\equiv 1$, Eqs. (\[q1a\]) and (\[q1b\]) reduce to $$q_1(a)\sim \frac{\lambda z+\mu}{\lambda z + \mu+(1-\mu)C(a)}\sum_{k=0}^\infty p_k \left[ 1-\lambda +\lambda \frac{\lambda z+\mu}{\lambda z+\mu+(1-\mu)C(a)}\right]^k. \label{q1}$$
The $a=\infty$ limit of $q_1(a)$ gives the fraction of memes that are *never* retweeted, and so have popularity $n=1$ forever. The value of $q_1(\infty)$ is obtained from Eqs. (\[q1a\]) and (\[q1b\]) by setting $C(a)$ to its $a\to\infty$ limit of 1. The approach of $q_1(a)$ towards the value $q_1(\infty)$ depends, through the CDF $C(a)$, on the tail of the memory-time distribution $\Phi$. If the distribution $\Phi$ is heavy-tailed, there is a non-negligible probability that a meme may be retweeted even if a very long time has elapsed since its birth.
Mean popularity {#sec:mean}
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The age dependence of the mean popularity (i.e., the expected number of tweets/retweets for a meme of age $a$) is given by $$m(a) = \sum_{n=1}^\infty n\, q_n(a) = \left.\frac{\partial H(a;x)}{\partial x}\right|_{x=1}.$$ Differentiating (\[e21\]) and setting $x=1$ yields an integral equation for $m(a)$: $$\begin{aligned}
m(a) &= \sum_{j k}\beta_{j k}p_{j k}\left\{ 1+\lambda\, k\, m_G(a) + (1-\mu)\beta_{j k}\int_0^\infty
d\ell\, \left(j \overline{\beta}\lambda + \mu \beta_{j k}\right)\exp\left[-\left(j \overline{\beta}\lambda + \mu \beta_{j k}\right)\ell\right] \right.\times\nonumber\\
&\left.\times\int_{0}^{\min(\ell,a)} d\tilde{r}\, \int_{0}^{a-\tilde{r}} d\tilde{\t}\,\Phi(a-\tilde{r}-\tilde{\t}) \left[1+\lambda \,k\, m_G(\tilde\tau)\right]\right\},\label{21}
\end{aligned}$$ where $m_G(a)$, defined by $m_G(a) = \left.\frac{\partial G(a;x)}{\partial x}\right|_{x=1}$, is the solution of the integral equation found by differentiating Eq. (\[e19\]): $$\begin{aligned}
m_G(a)&=\sum_{j k}\frac{j}{z} p_{j k}\int_0^\infty
d\ell\, \left(j \overline{\beta}\lambda + \mu \beta_{j k}\right)\exp\left[-\left(j \overline{\beta}\lambda + \mu \beta_{j k}\right)\ell\right] \times\nonumber\\
&\times(1-\mu)\beta_{j k}\int_{0}^{\min(\ell,a)} d\tilde{r}\, \int_{0}^{a-\tilde{r}} d\tilde{\t}\,\Phi(a-\tilde{r}-\tilde{\t}) \left[1+\lambda \,k\, m_G(\tilde\tau)\right].\label{22}\end{aligned}$$ The order of the time integrals may be swapped using the identity $$\int_0^\infty d\ell\int_0^{\min(\ell,a)} d\tilde r = \int_0^a d\tilde r\int_{\tilde r}^\infty d\ell,\label{swap}$$ and the resulting $\ell$ integral can be performed explicitly: $$\int_{\tilde r}^\infty (j \overline \beta \lambda+\mu \beta_{jk})e^{-(j \overline \beta \lambda+\mu \beta_{jk})\ell}d\ell = e^{-(j \overline \beta \lambda+\mu \beta_{jk})\tilde r}.$$ As a result, the expressions (\[21\]) and (\[22\]) can be written as double convolution integrals. Taking Laplace transforms, Eq. (\[21\]) then becomes $$\hat m(s) = \frac{1}{s}+z \overline \beta \lambda \hat{m}_G(s) + (1-\mu) \hat{\Phi}(s) \sum_{j,k} \beta_{j k}^2 p_{j k}\frac{\frac{1}{s}+\lambda k \hat{m}_G(s)}{j \overline \beta \lambda + \mu \beta_{j k}+s}, \label{MH}$$ where hats denote Laplace transforms, e.g., $$\hat \Phi(s) \equiv \int_0^\infty e^{-s t} \Phi(t) dt,$$ and with $\hat{m}_G(s)$ given explicitly from the Laplace transform of Eq. (\[22\]): $$\hat{m}_G(s)=\frac{ (1-\mu)\hat{\Phi}(s) \sum_{j,k} \frac{j}{z}p_{jk}\frac{\beta_{jk}}{j \overline \beta \lambda + \mu \beta_{jk}+s}}{s\left[ 1-(1-\mu)\lambda \hat{\Phi}(s) \sum_{j,k} \frac{j}{z}p_{j k}\frac{k \beta_{j k}}{j \overline \beta \lambda + \mu \beta_{j k}+s}\right]}.$$ If we specialize now to the simplified case where $\beta_{jk}\equiv 1$ for all $(j,k)$ classes, and $p_{jk}=\delta_{j,z} p_k$, we obtain the simpler expression $$\hat{m}_G(s)=\frac{ (1-\mu)\hat{\Phi}(s) \frac{1}{\lambda z +\mu+s}}{s\left[ 1-(1-\mu) \hat{\Phi}(s) \frac{\lambda z}{\lambda z +\mu+s}\right]}. \label{ma0}$$ Substituting for $\hat{m}_G$ into the simplified version of Eq. (\[MH\]) yields $$\hat{m}(s) = \frac{1}{s}+\frac{1-\mu}{s}\frac{(\lambda z +1)\hat\Phi(s)}{\lambda z+ \mu+s-(1-\mu)\lambda z \hat \Phi(s)}. \label{Eq1maintext}$$ Note that, unlike the expression for $q_1$ in Eq. (\[q1\]), the mean popularity depends on the out-degree distribution $p_k$ only through the mean degree $z$, implying that the mean popularity is independent of the finer details of the network structure.
To consider the large-age asymptotics of $m(a)$ from Eq. (\[Eq1maintext\]) we use results from renewal theory [@Iribarren11; @AthreyaNeybook]. If the Malthusian parameter $\alpha$ exists, where $\alpha$ is defined as the solution of the equation $$\frac{(1-\mu)\lambda z \hat\Phi(\alpha)}{\lambda z +\mu +\alpha}=1, \label{Malthusian}$$ then the large-age, small-$\mu$ asymptotic behavior of $m(a)$ can be written as (Theorem IV.4.2 of [@AthreyaNeybook]) $$m(a) \sim \frac{1}{\mu}-\frac{1}{\mu} \, e^{-\frac{\mu(\lambda z+1)}{1+T \lambda z} a} \quad\text{ as } a\to \infty \text{, } \mu\to0.\label{ma2}$$ Here we have used the fact that near criticality (i.e., as $\mu\to 0$) the Malthusian parameter $\alpha$ is determined by Eq. (\[Malthusian\]) to be $\alpha= -\frac{\mu(\lambda z +1)}{1+T \lambda z}+O(\mu^2)$, where $T=\int_0^\infty t_m \Phi(t_m)\,d t_m$ is the mean memory time[^4]. Setting $a=\infty$ in Eq. (\[ma2\]), we obtain the steady-state value of the mean popularity, $m(\infty)=1/\mu$. Although Eq. (\[ma2\]) is a large-$a$ asymptotic result, we may expand the exponential term about $a=0$ provided that the argument of the exponential remains small: this is valid for ages $a$ that obey the constraint $ %\begin{equation}
a\ll \frac{1+T\lambda z}{\mu(\lambda z +1)}.
$ Taking the $\mu\to 0$ limit of Eq. (\[ma2\]) shows that the function $m(a)$ grows linearly with $a$ for ages in this range: $$m(a) \sim \frac{\lambda z +1}{1+ T \lambda z} a.\label{ma3}$$
[ The preceding analysis all assumes that the seed node (i.e., the user who first tweets the meme of interest) is chosen at random from all the network users, with probability weighted by the user activity rate. It is straightforward to repeat the steps of the calculations for the case where the seed node is known to have $k$ followers, and so to investigate the importance of the connectivity of the seed node. Restricting our attention to the simplified case as above, and taking the infinite-age limit, we find that the expected popularity for a meme that is initiated by a seed node of out-degree $k$ is $$% see 13.80
m_k(\infty)= \frac{\lambda z+1}{\lambda z +\mu}\left( 1 + \frac{\lambda(1-\mu)}{\mu(\lambda z+1)}k\right).$$ Note the linear dependence of this expression on the number of followers $k$ of the seed node: memes tweeted by users with a large number of followers are likely to become more popular than memes seeded by less influential nodes. This feature of the model matches well to the observed dependence of the size of information cascades on the connectivity of the initial seed (e.g., Fig. 2 of [@Banos13]). Of course, the earlier results for randomly-chosen seeds are recovered by averaging over all possible seed nodes: $m(\infty)=\sum_k p_k m_k(\infty) = 1/\mu$. ]{}
Infinite-age limit of popularity distribution {#sec:S1.6}
---------------------------------------------
In the infinite-age (steady-state) limit $a \to \infty$, we assume $G(a;x) \to G_\infty(x)$, independent of $a$, and use the fact that $\int_0^\infty \Phi(t) \, dt =1$ in Eq. (\[e19\]) to obtain $$\begin{aligned}
G_\infty(x)&=\sum_{j k}\frac{j}{z}p_{j k} \int_0^\infty d\ell\, \left(j \overline{\beta}\lambda + \mu \beta_{j k}\right)\exp\left[-\left(j \overline{\beta}\lambda + \mu \beta_{j k}\right)\ell\right] \times\nonumber\\
&\times\exp\left[-(1-\mu)\beta_{j k} \ell (1- x\left[1-\lambda+\lambda G_\infty(x)\right]^k)\right].\end{aligned}$$ Calculating the $\ell$ integral then gives the equation satisfied by $G_\infty(x)$: $$G_\infty(x) = \sum_{j k}\frac{j}{z}p_{j k} \frac{ j \overline{\beta}\lambda + \mu \beta_{j k}}{ j \overline{\beta}\lambda + \beta_{j k}-(1-\mu)\beta_{j k} x\left[1-\lambda+\lambda G_\infty(x)\right]^k}.\label{e24}$$ Similarly, the infinite-age limit for $H$ is given in terms of $G_\infty$ by $$H_\infty(x)= \sum_{j k}\beta_{j k} p_{j k} \frac{\left( j \overline{\beta}\lambda + \mu \beta_{j k}\right) x\left[1-\lambda+\lambda G_\infty(x)\right]^k}{ j \overline{\beta}\lambda + \beta_{j k}-(1-\mu)\beta_{j k} x\left[1-\lambda+\lambda G_\infty(x)\right]^k}. \label{e25}$$ Note that these steady-state equations are independent of the memory distribution function $\Phi$. Accordingly, the asymptotic analysis approach used in [@GleesonPRL14] to obtain the large-$n$ behavior of the popularity distribution $q_n(\infty)$ may also be applied here: this is based on writing $x=1-w$ and $G_\infty=1-\phi(w)$ and analyzing the small-$w$, small-$\phi$ asymptotics of Eqs. (\[e24\]) and (\[e25\]). We refer to [@GleesonPRL14] for details, and here summarize the main results for the simplified case $\beta_{j k}\equiv 1$, $p_{jk}=\delta_{j,z} p_k$.
- [**Case 1: $p_k$ has finite second moment**]{}\
The large-$n$ scaling of the popularity distribution is given by a power-law with exponential cutoff: $$q_n(\infty)\sim A \,n^{-\frac{3}{2}} e^{-\frac{n}{\kappa}}\quad\text{ as }n\to \infty,$$ where the prefactor $A$ is[^5] $$%A=(\lambda z+1)\left[ 2\pi \lambda(\lambda\left<k^2\right>+(2-\lambda)z)\right]^{-\frac{1}{2}}
% see 13.58:
A=\frac{z(\lambda z+1)}{\lambda z+\mu}\left[ 2\pi \left(\frac{\left< k^2 \right>(2+\lambda z-\mu)}{\lambda z +\mu}-z\right)\right]^{-\frac{1}{2}}$$ and the cutoff $\kappa$ is $$%\kappa = \frac{2\lambda(\lambda\left<k^2\right>+(2-\lambda)z)}{\mu^2(\lambda z+1)^2}.
% 13.57
\kappa = \frac{2 \lambda^2(1-\mu)^2}{\mu^2(\lambda z+1)^2}\left[\frac{\left< k^2 \right>(2+\lambda z-\mu)}{\lambda z +\mu}-z\right].$$ Note that $\kappa$ is proportional to $1/\mu^2$ for small $\mu$, so in the limit of vanishing innovation probability the exponential cutoff tends to infinity and the power-law part of the popularity distribution extends to all $n$.
- [**Case 2: $p_k\sim D \,k^{-\gamma}$ as $k\to \infty$, with $\gamma$ between 2 and 3**]{}\
Immediately taking the $\mu\to 0$ limit, we find in this case that the popularity distribution has a power-law form with exponent $\gamma/(\gamma-1)$ lying between $3/2$ and $2$ [@Goh03; @SchwartzCohen]: $$q_n(\infty) \sim B \,n^{-\frac{\gamma}{\gamma-1}} \quad\text{ as }n\to \infty \label{steadystate}$$ with prefactor $B$ given by $$B=-(\lambda z+1)\frac{(D \Gamma(1-\gamma))^{-\frac{1}{\gamma-1}}}{\lambda \Gamma\left(\frac{1}{1-\gamma}\right)}\left[(\lambda z)^2\sum_{n=1}^\infty \frac{n^{\gamma-1}}{(\lambda z+1)^{n+1}}\right]^{-\frac{1}{\gamma-1}}, \label{Bprefactor}$$ where $\Gamma$ is the gamma function.
Large-$a$, large-$n$ asymptotics of popularity distribution {#sec:7.1}
-----------------------------------------------------------
[ In Appendix \[AppA\] we consider how the popularity distribution $q_n(a)$ behaves for large, but finite, ages, focussing on the case $\beta_{j k}\equiv 1$, $p_{j k}=\delta_{j,z} p_k$ for simplicity. The result of the asymptotic analysis is an expression for the Laplace transform of the PGF $H(a;x)$ that is valid in the $a\to\infty$ limit, see Eqs. (\[H1\]) and (\[Hasym2\]) for the cases of out-degree distributions $p_k$ that have second moments $\left<k^2\right>$ that are, respectively, infinite or finite. ]{}
Results: numerical simulation {#resultssynthetic}
=============================
\
\
To confirm the accuracy of the branching-process approximation and to explore the interactions of the network structure and the memory-time distribution, we here compare numerical simulations of the model with the theoretical predictions of Sec. \[analysis\]. We generate configuration-model directed networks with prescribed out-degree distribution $p_k$. Each one of $N$ users (nodes) is assigned a random number $k$ (drawn from the distribution $p_k$) of out-links (links to followers). The identities of the $k$ followers are chosen uniformly at random from the set of all users; in the $N\to\infty$ limit, this gives a Poisson in-degree distribution $p_j$ which, for sufficiently large $z$, gives similar results to using the in-degree distribution $p_j=\delta_{j,z}$, i.e., assuming every user follows exactly $z$ others [@GleesonPRL14]. Each user has the same activity rate, so $\beta_{jk}\equiv 1$.
Figure \[fig4\]A shows the fraction of memes that have popularity greater than or equal to $n$, at age $a$. Black symbols are the results of numerical simulations; the colored curves are determined from the large-$a$, large-$n$, $\mu=0$ asymptotics of Eq. (\[H1\]), using the Laplace transform inversion described in Appendix \[sec:inversion\]. The main figure in panel Fig. \[fig4\]A shows results for networks with the scale-free out-degree distribution $p_k\sim D k^{-\gamma}$ for $k\ge 4$ and exponent $\gamma=2.5$ (with $p_k=0$ for $k<4$); the inset shows the results for networks with a Poisson out-degree distribution with mean degree $z=11$ matching that of the scale-free networks. The memory time distribution is $\Phi=\text{Gamma}(k_G,\theta)$ with $k_G=0.1$, $\theta=50$ for the scale-free case and $k_G=0.1$, $\theta=5$ for the Poisson case; the mean memory time for this distribution is $T=k_G \theta$.
Panels B and C of Fig. \[fig4\] show results for various memory time distributions $\Phi$ on networks with the same scale-free out-degree distribution as used in panel A, and panels D and E show the corresponding results for the Poisson network. Panels B and D show the fraction $q_1(a)$ of memes that have not been retweeted by age $a$, along with the large-$a$ asymptotics of Eq. (\[q1\]). The age-dependence of $q_1(a)$ is qualitatively similar in panels B and D: note in both panels that the cases with longer mean memory time $T=5$ (dashed curves) approach their $a\to\infty$ limit more slowly than the $T=1$ cases (solid curves). However, the limiting value of $q_1(a)$ as $a\to\infty$ is different in the two panels, reflecting the effect of the network structure (out-degree distribution). Using Eq. (\[q1\]) (with $C(\infty)=1$) we obtain $q_1(\infty)=0.50$ for the scale-free network with $\lambda=1$, whereas $q_1(\infty)=0.37$ for the Poisson network.
The mean popularity $m(a)$ of age-$a$ memes is shown in panels C and E for the scale-free and Poisson networks, respectively, and for the same memory-time distributions as used in panels B and D. In contrast to the results for $q_1(a)$, we see that the finer details of the network structure have no effect on the $m(a)$ curves: panels C and E are identical, because Eq. (\[Eq1maintext\]) depends on $p_k$ only through the mean degree $z$, which is identical for both networks. The mean memory time $T$ determines the rate of linear growth of $m(a)$ at intermediate ages (see Eq. (\[ma3\])), while at early ages, the gamma memory time distribution $\Phi(t_m)$ (which has significant probability mass at low values of $t_m$) gives a faster-than-linear growth of $m(a)$ that is not present for the exponentially-distributed memory times. The large-age asymptotics are shown in the insets; as discussed in Sec. \[sec:mean\], we find $m(a)\to1/\mu$ as $a\to\infty$. [ As we show in Sec. \[resultsdata\] below, the $m(a)$ curves can be fitted to empirical data on the popularity of Twitter hashtags; note also that the qualitative features identified here (nonlinear early growth; linear intermediate-time growth, saturation at later times) have also been observed in several other measures of information spread on social networks, such as views of YouTube videos [@Szabo10] and the installation of Facebook apps [@GleesonPNAS14]. ]{}
Results: Twitter hashtags data {#resultsdata}
==============================
Data and model inputs
---------------------
To test the ability of the model to fit real-world data, we use a 1-year dataset comprised of the popularities of $1.4\times 10^5$ hashtags related to the 2011 15M protest movement in Spain that were tracked over the 1-year period from March 2011 to March 2012 [@Borge-Holthoefer11; @Gonzalez-Bailon11]. We use all hashtags for which we have at least 200 days of data; each curve in Fig. \[fig6\]A shows the popularity distribution for all hashtags which have the same age (to the nearest day).
The out-degree distribution $p_k$ of the Twitter network is an important input to the model. We determine the empirical distribution by randomly selecting $8.2\times 10^5$ Twitter user ids and recording the number of followers $k$ of each user. The measured mean number of followers is $z=703$, but the distribution $p_k$ is heavy-tailed. The complementary cumulative distribution function (CCDF) of the $k$ values is shown in Fig. \[figSpk\], along with the line $D/(\gamma-1) k^{1-\gamma}$ with $D=240$ and $\gamma=2.13$ that corresponds to an out-degree distribution with tail scaling as $p_k \sim D \,k^{-\gamma}$ as $k\to\infty$ [@Clauset09].
The model parameter $\lambda$ and the memory-time distribution $\Phi(t_m)$ cannot be directly estimated from the data because in cases where users receive multiple copies of the same meme (hashtag) prior to retweeting it, it is impossible to tell which of received memes “caused” the retweet. Therefore, we instead use the analytical results of the model (Eqs. (\[Eq1maintext\]) and (\[H1\])) to find parameter values that fit the model to the statistical characteristics of the data. Guided by the faster-than-linear growth of the mean popularity at early ages $a$ (Fig. \[fig6\]C) and the results of Sec. \[resultssynthetic\], we assume that the memory time distribution $\Phi$ is a Gamma$(k_G,\theta)$ distribution, and fit the distribution parameters $k_G$ and $\theta$, as well as the model parameters $\mu$ and $\lambda$ to give the results in Fig. \[fig6\]C. Note that a delta-function memory-time distribution, as used in the toy model of [@GleesonPRL14], leads to a purely-linear dependence $m(a)\propto a$, and so cannot fit to the early-time growth of the observed mean popularity.
The data does, however, provide an upper bound on the value of the innovation probability $\mu$. Recall that $\mu$ is defined as the probability that a tweeted meme (hashtag) is an innovation, i.e., that the hashtag has never before appeared in the system. Each innovation event thus increases by one the number of distinct hashtags that appear in the dataset, whereas a non-innovative (copying) tweet will instead increase the number of copies of a hashtag that is already present in the dataset. We can therefore calculate an upper bound on the empirical innovation probability from the ratio $$\tilde \mu = \frac{\text{number of distinct hashtags used in the dataset}}{\text{total number of hashtags tweeted by users}}=\frac{322799}{5886837} = 0.055. \label{mubound}$$ Note this upper bound is consistent with the parameter value of $\mu=0.033$ that is fitted in Fig. \[fig6\]. The reason why Eq. (\[mubound\]) gives an upper bound rather than an exact value for $\mu$ is the finite size of the dataset: the data collection started at a specific point in time and so any hashtags that are in fact copied from tweets received prior to the start date will be erroneously counted as “distinct hashtags” in the estimate, thus leading to an overestimate of the true innovation probability.
Results using identical user activity rates
-------------------------------------------
Using the empirical Twitter out-degree distribution $p_k$, we apply the analytical results of Eqs. (\[Eq1maintext\]) and (\[H1\]) (which assume $\beta_{jk}\equiv 1$) to fit the model to the data in Fig. \[fig6\]. Figure \[fig6\]A and \[fig6\]B show that the model-predicted age-dependent popularity distributions match reasonably well to the data, and Fig. \[fig6\]C shows that the age-dependent mean can be fitted very closely by the model. The data collapse seen in Fig. \[fig6\]B is intriguing, and we analyze it further in Sec. \[datacollapse\] below.
Despite these successes, it was not possible to successfully fit the $q_1(a)$ curve (Fig. \[fig6\]D) using the simplified version of the model in which all users have the same activity rates. In Sec. \[hetactivityrates\] below, we therefore investigate the effect of heterogeneous activity rates, and show that an improved fit can be obtained using more realistic rates.
Analysis of the data collapse in Fig. \[fig6\]B {#datacollapse}
-----------------------------------------------
As shown in Fig. \[fig6\]B, the ratio $q_n(a)/m(a)$ is approximately independent of the age $a$, giving a collapse of the popularity distribution data (and of the model predictions) onto a single curve. As in Sec. \[sec:7.1\], the large-$n$ asymptotics of the popularity distribution are found from the small-$w$ expansion (with $w=1-x$) of $h(a;x)=1-H(a;x)$, and for the scale-free out-degree distribution we obtain from Eq. (\[eS56\]) (using the final value theorem for Laplace transforms) the following asymptotic behavior in the $a\to\infty$ limit: $$h(\infty;1-w) \sim (\lambda z+1)C^{-\frac{1}{\gamma-1}} w^\frac{1}{\gamma-1} \quad\text{ as }w\to 0.$$ Understanding the large-$a$ approach to this steady state (i.e., the case where $a$ is large but finite) is a difficult problem in asymptotic analysis, involving the double limits $n\to\infty$ and $a\to\infty$. However, some insight can be obtained by factoring the function $h$ into a product of its infinite-age limit $h(\infty;x)$ and another function $h_1$, with $h_1$ limiting to 1 as $a\to\infty$: $$h(a;x)=h(\infty;x) h_1(a;x). \label{hh1}$$ Taking Laplace transforms gives $$\hat h(s;x)=h(\infty;x) \hat{h}_1(s;x),$$ where $$\hat{h}_1(s;1-w)= \frac{\lambda z(s+\lambda z + \hat \Phi(s))}{s(\lambda z+1)(s+\lambda z)}\frac{(\gamma-1)\lambda D^{\frac{1}{\gamma-1}}\left[\Gamma(1-\gamma)\right]^\frac{1}{\gamma-1} w^\frac{\gamma-2}{\gamma-1}\hat\Phi(s)}{s+\lambda z-\lambda z \hat\Phi(s)+(\gamma-1)\lambda D^{\frac{1}{\gamma-1}}\left[\Gamma(1-\gamma)\right]^\frac{1}{\gamma-1} w^\frac{\gamma-2}{\gamma-1}\hat\Phi(s)}.$$ In particular, note that $\hat{h}_1(s;1-w)$ depends on $w$ only through the factor $w^\frac{\gamma-2}{\gamma-1}$. In the case where $\gamma$ is very close to 2, the exponent $(\gamma-2)/(\gamma-1)$ of the $w$ dependence is close to zero, and the dependence of $h_1$ on $w$ is therefore very weak. It follows that the rate of approach of the corresponding distribution $q_n(a)$ to the steady state $q_n(\infty)$ does not show a strong dependence on $n$, and the CCDFs for various ages appear almost parallel in the log-log plot of Fig. \[fig6\]A (note $\gamma=2.13$ in the Twitter network).
As we saw in Sec. \[sec:mean\] for the large-age asymptotics of the mean popularity, the long-time behavior of the popularity distribution may be obtained by examining the linear (early-age) growth of the inverse transform of Eq. (\[H1\]). The resulting popularity distributions $q_n(a)$ show (for large $n$) a regime of linear-in-age growth, and in the case where $\gamma \approx 2$, the rate of this growth depends only weakly on $n$. Since the mean popularity $m(a)$ is also growing linearly during this age period (see Eq. (\[ma3\])), the division of the CCDFs at various ages by the corresponding mean $m(a)$ leads to the collapse of the data onto the single curve that is seen in Fig. \[fig6\]B.
Heterogeneous activity rates {#hetactivityrates}
----------------------------
Although our analysis methods are quite general, in order to focus on understanding the combined effects of memory and out-degree distribution most of our results thus far are specialized to the case of uniform user activity rates, $\beta_{jk}\equiv 1$. It is interesting, therefore, to examine the impact that more realistic heterogeneous activity rates would have upon the results we have obtained. To this end, we extend here to the case where the activity rate of a user depends on its out-degree $k$ while retaining the assumption $p_{jk}=\delta_{j,z}p_k$, so that $\beta_{jk} = \beta_k$ (normalized so that $\sum_k \beta_k p_k = 1$ and with $\overline \beta = \sum_k \frac{k}{z} \beta_k p_k$).
The mean popularity is given in the general case by Eq. (\[MH\]). Repeating the asymptotic analysis of leading to Eq. (\[ma3\]) for the $\mu\to0$ limit, we again find linear growth of $m(a)$ with age $a$, with a slope that generalizes that found in Eq. (\[ma3\]): $$m(a) \sim \frac{\lambda z \overline \beta+\frac{\overline{\beta^2}}{\overline\beta}}{T \lambda z \overline \beta +1}\, a\quad\text{ as }a\to\infty,\label{ma4}$$ where we have introduced the notation $\overline{\beta^2}\equiv \sum_k\frac{k}{z}\left( \beta_k\right)^2 p_k$.
If we additionally assume that the user activity rates saturate to a constant level $\beta_\infty$ at very large $k$, so $\beta_k \to \beta_\infty$ as $k\to \infty$, then we can repeat the asymptotic approximations of Sec. \[sec:7.1\] to determine a generalized version of Eq. (\[H1\]): $$\begin{aligned}
\hat H(s;x) &\sim\frac{1}{s}\nonumber\\
&\hspace{-0.3cm}- \frac{1}{s} \frac{ \lambda z \overline\beta\left(s+\lambda z \overline\beta+\frac{\overline{\beta^2}}{\overline\beta}\hat\Phi(s)\right)(\gamma-1)(1-x)\hat\Phi(s)}
{(s+\lambda z \overline\beta)\left(s+\lambda z \overline\beta-\lambda z \overline\beta \hat\Phi(s)+\beta_\infty^\frac{1}{\gamma-1}(\gamma-1)\lambda D^\frac{1}{\gamma-1}\left[\Gamma(1-\gamma)\right]^{\frac{1}{\gamma-1}}(1-x)^\frac{\gamma-2}
{\gamma-1}\hat\Phi(s)\right)}.\end{aligned}$$
To demonstrate the effect of heterogeneous activity rates, we consider a model for $\beta_k$ inspired by the data analysis shown in Fig. 6(a) of [@Hodas13], see Appendix \[AppC\] for details. Using this heterogeneous activity rate, Fig. \[figS2\] shows results that correspond closely to the homogeneous-activity example of Fig. \[fig6\]. A comparison of panels D from both figures clearly shows that including heterogeneous activity rates leads to a better fit of the model to the data on the fraction $q_1(a)$ of non-retweeted memes. However, the other results of the model (panels A, B and C of Fig. \[figS2\] compared to same panels in Fig. \[fig6\]) are relatively unaffected by the activity rate, so that the good matches between model and data seen in Fig. \[fig6\] are not compromised by including heterogeneity in activity rates.
Limitations of the model {#limitations}
========================
As we have demonstrated, the analytical tractability of the null model enables it to be fitted to time-dependent data on meme popularity. However, we were required to make a number of assumptions to obtain analytical results and in this section we briefly highlight the most important assumptions and discuss possible extensions to the model.
The network structure is assumed to be that of a directed configuration-model graph defined by the joint probability $p_{j k}$ of a node having in-degree $j$ and out-degree (number of followers) $k$. While this joint probability can encode correlations between the number followed by, and the number of followers of, a node, it does not incorporate edge-based correlations, i.e., the probability that a user with many followers is followed by users who also have high numbers of followers. It may be possible to extend the analysis of the model to deal with at least some types of edge correlation [@Boguna05; @Hurd16], but this would be at the cost of increased complexity of the equations.
A more unrealistic simplification of the configuration model is the fact that it generates networks that are locally tree-like, with few short cycles. In particular, our model does not include bidirectional edges (i.e., reciprocated following relationships, where user $A$ follows user $B$ and $B$ also follows $A$), which are quite common in the Twitter network [@Kwak10], but which violate the independence assumption of a branching process. However, numerical simulations in Ref. [@GleesonPRL14] using a real Twitter network for a zero-memory version of the model (Sec. S4 of [@GleesonPRL14]) gave quite good agreement with branching process theory, despite the presence of a large fraction of reciprocal links in the graph. The conditions under which tree-based theories give good approximations for dynamics on non-tree-like networks remains an active area of research [@Melnik11] and more work is required for further understanding.
An important assumption of the null model is that all memes have equal fitness. This is consistent with random-copying models of human decision-making [@Bentleybook; @Bentley11] where the quality of the product—here, the “interestingness” of the meme—is less important than the social influence of peers’ decisions [@Salganik06]. This neutrality of the model is at the root of the criticality of the dynamical system [@Pinto11]. A related (discrete-time) model for the number of citations gained by scientific papers was analyzed in Ref. [@Simkin07], where the authors also extended their neutral model to include unequal fitnesses of papers. It is likely that our model could be extended in a similar way, to incorporate a fitness parameter for each individual meme. Based on the results of Ref. [@Simkin07], we expect that our main results would be qualitatively unaffected if the distribution of fitness values over the set of all memes is strongly peaked (i.e., if most memes have roughly equal fitness values, with only the high-fitness outliers demonstrating supercritical popularity growth).
Perhaps the most unrealistic aspect of the current model is the assumption that all users have constant activity rates, so their tweeting activity is described by a Poisson process (see the discussion in Sec. \[sec3.2\]). It would be interesting to relax this assumption, for example to allow the activity of users to be described by models such as that of Ref. [@Perra12] or by inhomogeneous Poisson processes: the latter incorporates time-varying activity rates and so could model the 24-hour variability in tweeting levels determined by daily patterns [@Malmgren08]. However, we believe that the near-critical aspect of the model will not be strongly affected by such generalizations. To see this, consider the heuristic derivation of the branching number $\xi$ that was described at the end of Sec. \[criticality\]. Over a sufficiently long time window $W$, the expected number of interesting memes received into the stream of a $(j,k)$-class user is linear in the number $j$ of users followed, and this remains true even for inhomogeneous Poisson (or even non-Poisson) activities, provided that the observation window is long enough (e.g., such that the average rate $\overline \beta$ of incoming tweets should yield approximately similar values when time-averaged over disjoint time windows of length $W$). Similarly, the expected number of retweets by the user during the time window can be written as in Eq. (\[Nretweets\]), but with the Poisson rate $\beta_{j k}$ replaced by its time-averaged value. The calculations of Eq. (\[xijkB\]) then proceed as before, leading to the conclusion that the branching number limits to the critical value of one as $\mu \to 0$, which implies that non-Poisson user activity rates (or burstiness) will not affect the criticality of the model, which is a long-time (i.e., ages of memes limit to infinity) characteristic. Of course, the short-term behaviour of the model (such as the small-$a$ behaviour in panels B–E of Fig. \[fig4\]) would be affected by introducing burstiness; incorporating such realistic features into the model is left as a challenge for further work. As a final comment on this topic, we note that the agreement (in Sec. \[resultsdata\]) of our theoretical results with real data of a spreading process for which users’ activity rates are not constant also provides indirect evidence that the phenomenology discussed is robust to the details of user activity burstiness.
The heuristic calculation of the branching number considered at the end of Sec. \[criticality\] also offers a clue as to how the model can be extended to the spreading of information on undirected social networks (as opposed to the directed networks that we focus on in this paper). Of particular interest is the spreading of app-adoption on Facebook, for which data was analyzed in Ref. [@Onnela10] and a computational model was introduced in Ref. [@GleesonPNAS14]. If the Facebook update messages that inform all friends of user $A$ that she has installed a particular app are considered to be the memes in a version of our model, then the arguments of Sec. \[criticality\] need only slight modifications. The total number of update messages received in the stream of a user with $k$ Facebook friends is linear in $k$ (i.e., the $j$ in the denominator of Eq. (\[Pchosen\]) is replaced by $k$), while the expected number of friends who would be interested in user $A$’s adoption of the app is $\lambda (k-1)$ (since one friend out of $k$ must have adopted before $A$ in order to have spread the message to her). Following very similar steps to calculate the expected number $\xi$ of children of a meme—see the calculations leading to Eq. (\[xijkB\])—we find that $$\begin{aligned}
\xi_\text{undirected} &\to \sum_k\frac{k}{z}p_k \frac{\beta_k \lambda (k-1)}{k \overline{\beta}\lambda} \quad\text{ as }\mu \to 0 \nonumber \\
& = 1-\frac{1}{z}.\end{aligned}$$ Although this branching number is less than one, the mean number $z$ of friends on Facebook is large (e.g., Ref. [@Ugander11] calculated $z\approx 190$) so that $\xi_\text{undirected}$ is in fact very close to unity, implying that the information spread process is close to criticality. Such a near-critical branching process was hypothesized in Ref. [@GleesonPNAS14] to explain to observed fat-tailed distributions of app popularity in Facebook data and the temporal characteristics of the adoption behaviour. The cascade sizes for other forms of “meme” spreading on Facebook have also been observed to have fat-tailed distributions [@Adamic14]. Other undirected networks to which the model should be applicable include YouTube [@Szabo10] and Digg [@Wu07; @Lerman12].
Finally, our focus here has been on the statistical physics of the model, but for completeness we should note the difficulties inherent in applying the model to data sets where memes may not be as simple to recognize and track as hashtags are. In [@Kuhn14] for example, the process of extracting memes (representing popular scientific terms) from data (citation archives of scientific publications) is explained in detail, and considerable such effort will generally be required to identify and track the memes to which this null model might be applied.
A related question is whether the popularity of online memes has any implications in terms of mass social movements in the offline world. This is a complex question that lies beyond the scope of this paper, but we note that Fig. 3 of [@gonzalez13] shows that the usage of hashtags related to the 15M Spanish protest movement was found to be closely correlated with the number of protest-related headlines in newspapers, at least during the main activity of the protests. This indicates that online social spreading phenomena can, at least in some cases, give useful information about real-world social movements and activism.
Discussion {#Discussion}
==========
The extremely wide range of popularities achieved by items on social media poses many challenges for complex systems researchers. These include the identification of the causes [@Coscia14] and structural features [@Goel15] of “viral” propagation, and the prediction of future spreading based on the content or the early-time growth of memes [@Cheng14; @Weng13; @Miotto14; @Thij14], each of which are important in the design of more efficient systems to spread information (e.g. in case of emergency). We argue that null models are fundamentally important in this quest—and complement more data-driven approaches—as they demonstrate, for example, that extreme popularity can arise purely because of random fluctuations in the competition between memes for user attention. While the content of a meme may well be an important factor in its popularity (or predictability [@Miotto14]), definitive statements about the significance of such factors should be referenced to an appropriate null model.
In this paper we have introduced and analyzed a null model of meme spreading that is analytically tractable, yet realistic enough to reproduce several characteristic features of empirical data. The model is sufficiently general to incorporate heterogeneous user activity rates and a joint distribution $p_{jk}$ of the number of users followed $j$ and the number of followers $k$, as well as a memory-time distribution $\Phi$ that gives non-Markovian dynamics. The competition-induced criticality phenomenon identified in a zero-memory model in Ref. [@GleesonPRL14] is found to be robust to the generalizations, giving power-law popularity distributions with characteristic time-dependence similar to data from social spreading phenomena [ (and see Sec. \[limitations\] for a discussion of further possible extensions of the model).]{}
The analytical tractability enables fast fitting of the model to data, as demonstrated in Sec. \[resultsdata\] with hashtag data from Twitter. We find that a simplified version of the model where users all have the same activity rate can be fitted to some, but not all, aspects of the data (see Fig. \[fig6\]). The aim of a null model is not to perfectly reproduce every aspect of a dataset, but rather to help identify which features of the data can be reproduced using relatively simple models, and so to highlight aspects where more detailed modelling (or, perhaps, factors entirely outside the model) are required to match to data. In this respect, the null model highlights the fact that heterogeneity in activity rates is vital to accurately capturing the $q_1(a)$ curve (compare Figs. \[fig6\]D and \[figS2\]D), even though the time-dependence of the bulk of the popularity distribution may be described reasonably well by a model with homogeneous activity rates (Fig. \[fig6\]A-C).
[ As noted in the Introduction, and expanded upon in Sec. \[limitations\], our definition of “memes” is sufficiently general to enable the model to be applied (with minor changes) not just to the spreading of hashtags or URLs on Twitter, but also to the adoption of apps on Facebook, the popularity of videos on YouTube, and to the broad range of imitation-driven spreading dynamics.]{} We anticipate that the analytical results and potential for fast fitting to data will make this null model a useful tool for further work, and we hope it will contribute to the ongoing investigation of the entangled effects of memory, network structure, and competition on social spreading phenomena.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors acknowledge helpful feedback from Davide Cellai, Timothy Duff, Rick Durrett, Freja Elbro, Ali Faqeeh, Peter Fennell, Kristina Lerman, David O’Sullivan, Mason Porter, and Jonathan Ward. This work was supported by Science Foundation Ireland (grant numbers 11/PI/1026, 12/IA/1683, and 09/SRC/E1780, J.P.G. and K.P.O’S.) and by the European Commission through FET-Proactive projects PLEXMATH (FP7-ICT-2011-8; grant number 317614, J.P.G and Y.M.) and MULTIPLEX (FP7-ICT-2011-8; grant 317532, Y.M.). R.A.B. and Y.M. also acknowledge support from MINECO (Grant FIS2011-25167) and Comunidad de Arag[ó]{}n (Spain; grant FENOL). We acknowledge the SFI/HEA Irish Centre for High-End Computing (ICHEC) for the provision of computational facilities. The data used in this paper is available for download from www.ul.ie/gleesonj/twitter15M.
Calculation of the large-$a$, large-$n$ asymptotics of popularity distribution {#AppA}
==============================================================================
In this Appendix we consider how the popularity distribution $q_n(a)$ behaves for large, but finite, ages. To highlight the effect of the out-degree distribution $p_k$ upon the results we here restrict our analysis to the case $\beta_{jk}\equiv 1$, $p_{jk}=\delta_{j,z}p_k$. Taking the $\mu\to 0$ limit, Eq. (\[e19\]) becomes $$\begin{aligned}
G(a;x)&=\sum_{ k}p_{k} \int_0^\infty d\ell\, \lambda z e^{-\lambda z \ell} \times\nonumber\\
&\times\exp\left[-\int_{0}^{\min(\ell,a)} d\tilde{r}\, \int_{0}^{a-\tilde{r}} d\tilde{\t}\,\Phi(a-\tilde{r}-\tilde{\t}) (1- x\left[1-\lambda+\lambda G(\tilde{\t};x)\right]^k)\right].\label{e19b}\end{aligned}$$ Writing $x=1-w$ and $G(a;x)=1-\phi(a;w)$, we observe that the argument of the exponential function vanishes when $w=0$ and $\phi=0$, and so we consider the small-$w$, small-$\phi$ asymptotic behavior by expanding the exponential term to first order in its argument: $$\phi(a;w)\approx \int_0^\infty d\ell\, \lambda z e^{-\lambda z \ell} \int_{0}^{\min(\ell,a)} d\tilde{r}\, \int_{0}^{a-\tilde{r}} d\tilde{\t}\,\Phi(a-\tilde{r}-\tilde{\t}) (1- (1-w)\sum_k p_k \left[1-\lambda\phi(\tilde{\t};w)\right]^k).\label{onetermexpn}$$ We note that retaining only the first-order term in the expansion of the exponential is an approximation. We will estimate the accuracy of this “one-term expansion” by comparing the infinite-age limit determined under the approximation with the corresponding exact values as given in Sec. \[sec:S1.6\].
For the case of a scale-free out-degree distribution with $p_k\sim D \,k^{-\gamma}$ as $k\to \infty$, and $\gamma$ in the range $2<\gamma<3$, the asymptotic form of the summation term in Eq. (\[onetermexpn\]) is given by [@GleesonPRL14] $$1-(1-w)\sum_{k}p_k\left[1-\lambda\phi\right]^k \sim \lambda z \phi - C \phi^{\gamma-1}+w +o(w,\phi) \quad\text{ as }w\to0,\phi\to0,$$ with the constant $C$ given by $C=\lambda^{\gamma-1}D \Gamma(1-\gamma)$. Applying the integral-swapping trick of Eq. (\[swap\]) allows the right hand side of Eq. (\[onetermexpn\]) to be expressed as a double convolution integral. Laplace transforming then yields $$\hat \phi(s;w) = \frac{1}{\lambda z +s}\hat\Phi(s) \mathcal{L}\left[ \lambda z \phi - C \phi^{\gamma-1}+w\right], \label{phia1}$$ where $\mathcal{L}$ denotes the Laplace transform operation applied to the term in square brackets. In the $a\to \infty$ limit, this equation is satisfied by the steady-state solution $$\phi(\infty;w) = C^{-\frac{1}{\gamma-1}} w^{\frac{1}{\gamma-1}},\label{S51}$$ as can be verified using the final value theorem for Laplace transforms. We note that the corresponding expression for $\phi(\infty;w)$ as calculated from the steady state Eq. (\[e24\]) has a additional multiplicative factor of $F(\lambda z,\gamma)$ that is absent in Eq. (\[S51\]), where the function $F(\zeta,\gamma)$ is defined by $$F(\zeta,\gamma) = \left[\zeta^2\sum_{n=1}^\infty \frac{n^{\gamma-1}}{(\zeta+1)^{n+1}}\right]^{-\frac{1}{\gamma-1}}, \label{Fdefn}$$ see Fig. \[figFplot\]. If $\lambda z \gg 1$, then $F(\lambda z,\gamma)\approx 1$ and the one-term expansion gives results that are very close to the exact values (at least in the infinite-age limit $a\to\infty$).
Moreover, even if $\lambda z$ is not large (e.g., $\lambda z=0.32$ for the model fit to Twitter hashtags data in Sec. \[resultsdata\]), the values of $F(\lambda z,\gamma)$ can still be close to unity if $\gamma$ is sufficiently close to 2.
To consider small deviations from the steady state, we define $g(a;w)$ by $$\phi(a;w)=\phi(\infty;w)\left( 1-g(a;w)\right) \label{phia2}$$ with $g(a;w)\to0$ as $a\to\infty$. Assuming that $g$ is sufficiently small to allow the use of the linearizing approximation $$(1-g)^{\gamma-1} \approx 1-(\gamma-1)g,$$ Eq. (\[phia1\]) can be solved for the Laplace transform of $g$: $$\hat g (s;w) = \frac{1}{s}\frac{s+\lambda z - \lambda z \hat\Phi(s)}{s+\lambda z - \lambda z \hat\Phi(s) + (\gamma-1)C^\frac{1}{\gamma-1}w^{\frac{\gamma-2}{\gamma-1}}\hat\Phi(s)}. \label{phia3}$$ The Laplace transform of $\phi$ then follows from Eq. (\[phia2\]) and a similar asymptotic analysis of Eq. (\[e21\]) yields $$\hat H(s;1-w) = \frac{1}{s}-\frac{\lambda z(s+\lambda z +\hat \Phi(s))}{s+\lambda z}\hat\phi(s;w)\label{eS56}$$ Substituting from Eqs. (\[phia2\]) and (\[phia3\]) results in $$\hat H(s;x) = \frac{1}{s}\left[ 1- \frac{\lambda z\left(s+\lambda z+\hat \Phi(s)\right)(\gamma-1)(1-x)\hat \Phi(s)}{(s+\lambda z)\left(s+\lambda z - \lambda z \hat\Phi(s)+(\gamma-1)\lambda D^\frac{1}{\gamma-1}\left[\Gamma(1-\gamma)\right]^\frac{1}{\gamma-1}(1-x)^\frac{\gamma-2}{\gamma-1}
\hat\Phi(s)\right)}\right].\label{H1}$$
A similar analysis can be performed in the case where the out-degree distribution $p_k$ has finite second moment. We again utilize a one-term expansion similar to Eq. (\[onetermexpn\]), but we can also retain a non-vanishing innovation probability $\mu$ in this case. The one-term expansion can be shown to be accurate when $\lambda z \gg 1$; this condition is obeyed in all relevant cases we examine. The resulting large-$a$ asymptotics for the generating function $H(a;x)$ are found by inverting the following Laplace transform: $$\begin{aligned}
&\hat H(s;1-w) = \frac{1}{s}-\phi(\infty;w)\times&\nonumber\\
&\hspace{0cm}\times\frac{(1-\mu)\lambda z(s+\lambda z+\mu +\hat \Phi(s))}{s(s+\lambda z+\mu)}\left[ \frac{2(1-\mu)\frac{w}{\phi(\infty;w)} \hat \Phi(s) - \mu(\lambda z+1)\hat \Phi(s)}{s+\lambda z +\mu - \left(\lambda z(1+\mu)+2\mu\right)\hat \Phi(s) + 2(1-\mu)\frac{w}{\phi(\infty;w)}\hat \Phi(s)}\right], \label{Hasym2}\end{aligned}$$ with $\phi(\infty;w)$ given by $$\phi(\infty;w) = \frac{-\mu(\lambda z +1)+\sqrt{\mu^2(\lambda z+1)^2+2\lambda^2(1-\mu)^2\left(\left<k^2\right>-z\right)w}}{\lambda^2(1-\mu)\left(\left<k^2\right>-z\right)}.
%\frac{1}{\lambda}\sqrt{\frac{2}{\left<k^2\right>-z}}w^{\frac{1}{2}}.$$
Numerical inversion of Laplace transforms and PGFs {#sec:inversion}
==================================================
Many of our results for the popularity distribution $q_n(a)$ are expressed in terms of the corresponding PGF $H(a;x)$. As in [@GleesonPRL14], we use the Fast Fourier Transform method of [@Cavers78; @Newman01; @Marder07; @Abate92] to numerically invert the PGF at a fixed age $a$ to produce, for example, the model distributions in Figs. \[fig6\] and \[figS2\]; see Sec. S2 of [@GleesonPRL14] for further details and links to Octave/Matlab code for implementing the PGF inversion.
The results of the model for the age-dependence of several quantities are expressed in terms of Laplace transforms. To numerically invert the Laplace transforms we use the efficient Talbot algorithm [@Talbot79], in its simplified version described in Sec. 6 of [@Abate06]. The Talbot algorithm is based on a numerical evaluation of the Bromwich (Laplace inversion) integral, using a cleverly-chosen deformation of the contour in the complex-$s$ plane. The Laplace inversion of $\hat H(s;x)$ to obtain $H(a;x)$ at a desired age $a$, for example, can be quickly computed using the $2 M_L-1$ weights $\gamma_k$ and nodes $\delta_k$ defined by [@Abate04] $$\begin{aligned}
\delta_0&= \frac{2M_L}{5}, \quad \delta_k=\frac{2k\pi}{5}(\cot(k\pi/M_L)+i)\quad\text{ for }-M_L+1\le k\le M_L-1,\nonumber\\
\gamma_0&=\frac{1}{2}e^{\delta_0},\nonumber\\
\gamma_k&=[1+i(k\pi/M_L)(1+[\cot(k\pi/M_L)^2])-i\cot(k\pi/M_L)]e^{\delta_k}\quad\text{ for }-M_L+1\le k\le M_L-1,
\end{aligned}$$ (where $i=\sqrt{-1}$) by calculating the sum $$H(a;x) = \frac{1}{5a}\left[ \gamma_0 \hat H\left(\frac{\delta_0}{a};x\right)+\sum_{k=-M_L+1}^{M_L-1} \gamma_k \hat H\left(\frac{\delta_k}{a};x\right)\right].$$ In practice, the precision of the Talbot algorithm is very high, and only relatively small values of $M_L$ are required to obtain accurate results; we used $M_L=25$ in the examples shown.
Model of heterogeneous activity rates {#AppC}
=====================================
In the data analysis of Fig. 6(a) of [@Hodas13], the average activity rate (as measured by the number of tweets by a user in a fixed time period) is found to grow approximately linearly with the number of followers $k$ of that user, for $k$ from 0 to about 100. Then, for $k$ values from about 100 up to the maximum shown in the plot ($k=10^3$), the activity rate grows as a more slowly increasing linear function of $k$. We model these characteristics (which are also seen in other studies, e.g., [@Myers14]), using a piecewise-linear and continuous function of $k$, assuming a saturation of activity at very high $k$, as follows: $$\beta_k \propto \left\{ \begin{array}{cc}
0.35 k & \text{ if }k<100,\\
35+0.044(k-100) & \text{ if }100\le k<10^4.\\
470.6 & \text{ if }k\ge10^4,
\end{array}
\right. \label{betak}$$ where the values are chosen to closely match the linear growth rates in Fig. 6(a) of [@Hodas13], snd the constant of proportionality being set by the condition $\sum_k \beta_k p_k = 1$.
[^1]: User $A$ follows $j$ users, each of which is assumed to tweet at the average rate $\overline{\beta}=\sum_{j k} \frac{k}{z} \beta_{j k}p_{j k}$. Each meme sent by these $j$ users is deemed interesting by $A$ with probability $\lambda$, so the rate at which interesting memes enter the stream of user $A$ is $j \overline \beta \lambda$. Moreover, user $A$ innovates at a rate $\mu \beta_{j k}$, which gives the second term of Eq. (\[rate\]). If either an incoming tweet or an innovation event occurs, a new meme is inserted into the stream of user $A$, and the occupation time of meme $M$ is ended.\[footnote2\]
[^2]: The factor $(1-\mu)\beta_{j k}\, dt$ is the probability that a $(j,k)$-class user becomes active in the $dt$ interval and copies rather than innovates; the factor $\Phi(t-r)\,dr$ is the probability that this user chooses to copy from the $dr$-interval.
[^3]: There are $k$ followers of user $A$, each of whom may deem the tweet “uninteresting” with probability $1-\lambda$, or consider it “interesting”—and accept it into their stream—with probability $\lambda$. The factor of $x$ counts the increase in popularity due to the tweet event.
[^4]: Note that the Malthusian parameter exists for all the memory-time distributions considered in this paper (exponential and gamma distributions). However, if $\Phi$ is a subexponential distribution [@AthreyaNeybook] (such as the lognormal distribution [@Doerr13]), then the large-$a$ asymptotics of the mean popularity are related to the memory time CDF by $$m(a)\sim \frac{1}{\mu}-\frac{(1-\mu)(\lambda z+\mu)}{\mu^2 (\lambda z+1)}\left(1-C(a)\right)$$ instead of Eq. (\[ma2\]).
[^5]: The values of $A$, $\kappa$ and $B$ reported here are not identical to those reported in [@GleesonPRL14]; this is because of an approximation made in the analysis of [@GleesonPRL14] that is not required here (see Eq. (S6) of [@GleesonPRL14]). However, the differences are of order $1/(\lambda z)$, and so are negligible in the case $\lambda z \gg 1$ that is considered in [@GleesonPRL14].
|
hep-ph/0411168\
Updated February 2005
[**Abstract**]{}
This article reviews and updates the Standard Model prediction of the muon $g$$-$$2$. [QED]{}, electroweak and hadronic contributions are presented, and open questions discussed. The theoretical prediction deviates from the present experimental value by 2–3 standard deviations, if $e^+e^-$ annihilation data are used to evaluate the leading hadronic term.
Introduction
============
The evaluation of the Standard Model ([SM]{}) prediction for the anomalous magnetic moment of the muon $\amu \equiv (g_{\mu}-2)/2$ has occupied many physicists for over fifty years. Schwinger’s 1948 calculation [@Sch48] of its leading contribution, equal to the one of the electron, was one of the very first results of [QED]{}, and its agreement with the experimental value of the anomalous magnetic moment of the electron, $a_e$, provided one of the early confirmations of this theory.
While $a_e$ is rather insensitive to strong and weak interactions, hence providing a stringent test of [QED]{} and leading to the most precise determination to date of the fine-structure constant $\alpha$, $\amu$ allows to test the entire [SM]{}, as each of its sectors contribute in a significant way to the total prediction. Compared with $a_e$, $\amu$ is also much better suited to unveil or constrain “new physics” effects. For a lepton $l$, their contribution to $a_l$ is generally proportional to $m_l^2/\Lambda^2$, where $m_l$ is the mass of the lepton and $\Lambda$ is the scale of “new physics”, thus leading to an $(m_{\mu}/m_e)^2 \sim
4\times 10^4$ relative enhancement of the sensitivity of the muon versus the electron anomalous magnetic moment. The anomalous magnetic moment of the $\tau$ would thus offer the best opportunity to detect “new physics”, but the very short lifetime of this lepton makes such a measurement very difficult at the moment.
In a sequence of increasingly more precise measurements [@BNL00; @BNL01; @BNL02; @BNL04], the E821 Collaboration at the Brookhaven Alternating Gradient Synchrotron has reached a fabulous relative precision of 0.5 parts per million (ppm) in the determination of $\amu$, providing a very stringent test of the [SM]{}. Even a tiny statistically significant discrepancy from the [SM]{} prediction could be the harbinger for “new physics” [@CM01].
Several excellent reviews exist on the topic presented here. Among them, I refer the interested reader to refs. [@KM90; @CM99; @HK99; @MT00; @MLR01; @Me01; @Ny03; @Kn03; @DM04]. In this article I will provide an update and a review of the theoretical prediction for $\amu$ in the [SM]{}, analyzing in detail the three contributions into which $\amu^{\mysmall SM}$ is usually split: [QED]{}, electroweak ([EW]{}) and hadronic. They are respectively discussed in secs. \[sec:QED\], \[sec:EW\] and \[sec:HAD\]. A numerical re-evaluation of the two- and three-loop [QED]{} contributions employing recently updated values for the lepton masses is presented in secs. \[sec:QED2\] and \[sec:QED3\]. Comparisons between $\amu^{\mysmall SM}$ results and the current experimental determination $\amu^{\mbox{$\scriptscriptstyle{EXP}$}}$ are given in sec. \[sec:COMP\]. Conclusions are drawn in sec. \[sec:CONC\].
The QED Contribution to $\amu$ {#sec:QED}
===============================
The [QED]{} contribution to the anomalous magnetic moment of the muon is defined as the contribution arising from the subset of [SM]{} diagrams containing only leptons ($e,\mu,\tau$) and photons. As a dimensionless quantity, it can be cast in the following general form [@KM90; @KNO90] \^[QED]{} = A\_1 + A\_2(m\_/m\_e) + A\_2(m\_/m\_) + A\_3(m\_/m\_e,m\_/m\_), \[eq:amuqedgeneral\] where $m_e$, $m_{\mu}$ and $m_{\tau}$ are the masses of the electron, muon and tau respectively. The term $A_1$, arising from diagrams containing only photons and muons, is mass independent (and is therefore the same for the [QED]{} contribution to the anomalous magnetic moment of all three charged leptons). In contrast, the terms $A_2$ and $A_3$ are functions of the indicated mass ratios, and are generated by graphs containing also electrons and taus. The renormalizability of [QED]{} guarantees that the functions $A_i$ ($i=1,2,3$) can be expanded as power series in $\alpha/\pi$ and computed order-by-order A\_i = A\_i\^[(2)]{}( ) + A\_i\^[(4)]{}( )\^[2]{} + A\_i\^[(6)]{}( )\^[3]{} + A\_i\^[(8)]{}( )\^[4]{} + A\_i\^[(10)]{}( )\^[5]{} +.
One-loop Contribution
---------------------
Only one diagram, shown in fig. \[fig:schwinger\], is involved in the evaluation of the lowest-order contribution (second-order in the electric charge); it provides the famous result by Schwinger [@Sch48], $A_1^{(2)}
= 1/2$ ($A_2^{(2)}=A_3^{(2)} = 0$).
![[Lowest-order [QED]{} contribution to $\amu$.]{}[]{data-label="fig:schwinger"}](schwinger.eps){width="8cm"}
Two-loop Contribution {#sec:QED2}
---------------------
At fourth order, seven diagrams contribute to $A_1^{(4)}$, one to $A_2^{(4)}(m_{\mu}/m_e)$ and one to $A_2^{(4)}(m_{\mu}/m_{\tau})$. They are depicted in fig. \[fig:qed2\]. The coefficient $A_1^{(4)}$ has been known for almost fifty years [@So57-58; @Pe57-58]: A\_1\^[(4)]{} = + + (3) - 2 = -0.328 478 965 579 …, \[eq:qedA14\] where $\zeta(s)$ is the Riemann zeta function of argument $s$. The mass-dependent coefficient A\_2\^[(4)]{}(1/x) = \_0\^1 du \_0\^1 dv , where $x=m_l/m_{\mu}$ and $m_l$ is the mass of the virtual lepton in the vacuum polarization subgraph, was also computed in the late 1950s [@SWP57] for $m_l= m_e$ and neglecting terms of $O(m_e/m_{\mu})$. Its exact expression was calculated in 1966 [@El66]. Actually, the full analytic result of [@El66] can be greatly simplified by taking advantage of the properties of the dilogarithm ${\rm Li}_2(z)=-\int_0^z dt \ln(1-t)/t$. As a result of this simplification I obtain A\_2\^[(4)]{}(1/x) &=& - - +x\^2 (4+3x ) +x\^4 +\
&& + (1-5 x\^2) . \[eq:qedA24\] Note that this simple formula, contrary to the one in ref. [@El66], can be directly used both for $0<x<1$ (the case of the electron loop) and for $x\geq 1$ (tau loop). In the latter case, the imaginary parts developed by the dilogarithms ${\rm Li}_2(x)$ and ${\rm Li}_2(x^2)$ are exactly canceled by the corresponding ones arising from the logarithms. For $x=1$ (muon loop), gives $A_2^{(4)}(1) = 119/36 - \pi^2/3$; of course, this contribution is already part of $A_1^{(4)}$ in . Evaluation of with the latest recommended values for the muon-electron mass ratio $m_{\mu}/m_e = 206.768\,2838\,(54)$ [@MT04], and the ratio of $m_{\mu} = 105.658\,3692\,(94)$ MeV [@MT04] and $m_{\tau}=
1776.99\,(29)$ MeV [@PDG04] yields A\_2\^[(4)]{}(m\_/m\_e) & = & 1.0942583111 (84) \[eq:qedA24e\]\
A\_2\^[(4)]{}(m\_/m\_) & = & 0.000078064 (25), \[eq:qedA24tau\] where the standard uncertainties are only caused by the measurement uncertainties of the lepton mass ratios. Eqs. (\[eq:qedA24e\]) and (\[eq:qedA24tau\]) provide the first re-evaluation of these coefficients with the recently updated [CODATA]{} and [PDG]{} mass ratios of refs. [@MT04; @PDG04]. These new values differ visibly from older ones (see refs. [@CM99; @MT00]) based on previous measurements of the mass ratios, but the change induces only a negligible shift in the total [QED]{} prediction. Note that the $\tau$ contribution in provides a $\sim \! 42 \times 10^{-11}$ contribution to $\amu^{\mysmall
QED}$. As there are no two-loop diagrams containing both virtual electrons and taus, $A_3^{(4)}(m_{\mu}/m_e,m_{\mu}/m_{\tau}) = 0$. Adding up eqs. (\[eq:qedA14\]), (\[eq:qedA24e\]) and (\[eq:qedA24tau\]) I get the new two-loop [QED]{} coefficient C\_2 = A\_1\^[(4)]{} + A\_2\^[(4)]{}(m\_/m\_e) + A\_2\^[(4)]{}(m\_/m\_) = 0.765 857 410 (27). \[eq:qedC2\] The uncertainties in $A_2^{(4)}(m_{\mu}/m_e)$ and $A_2^{(4)}(m_{\mu}/m_{\tau})$ have been added in quadrature. The resulting error $\delta C_2 = 2.7 \times 10^{-8}$ leads to a tiny $0.01 \times
10^{-11}$ uncertainty in $\amu^{\mysmall QED}$.
![[The [QED]{} diagrams contributing to the muon $g$$-$$2$ in order $\alpha^2$. The mirror reflections (not shown) of the third and fourth diagrams must be included as well.]{}[]{data-label="fig:qed2"}](qed2.eps){width="14cm"}
Three-loop Contribution {#sec:QED3}
-----------------------
More than one hundred diagrams are involved in the evaluation of the three-loop (sixth-order) [QED]{} contribution. Their analytic computation required approximately three decades, ending in the late 1990s.
The coefficient $A_1^{(6)}$ arises from 72 diagrams. Its calculation in closed analytic form is mainly due to Remiddi and his collaborators [@Remiddi; @LR96], who completed it in 1996 with the evaluation of the last class of diagrams, the non-planar “triple cross” topologies (see, for example, fig. \[fig:qed3\] $A$) [@LR96]. The result reads: A\_1\^[(6)]{} &=& \^2 (3) - (5) + - \^4\
&& + (3) - \^2 2 + \^2 + = 1.181 2414566 …, \[eq:qedA16\] where $a_4=\sum_{n=1}^{\infty} 1/(2^n n^4)= {\rm Li}_4(1/2) = 0.517 \, 479\,
061 \,674 \ldots$.
The calculation of the exact expression for the coefficient $A_2^{(6)}(m/M)$ for arbitrary values of the mass ratio $(m/M)$ was completed in 1993 by Laporta and Remiddi [@La93; @LR93] (earlier works include refs. [@Ki67; @A26early]). For our analysis, $m=m_{\mu}$, and $M=m_e$ or $m_{\tau}$. This coefficient can be further split into two parts: the first one, $A_2^{(6)}(m/M,\mbox{vp})$, receives contributions from 36 diagrams containing electron or $\tau$ vacuum polarization loops (see, for example, fig. \[fig:qed3\] $B$) [@La93], whereas the second one, $A_2^{(6)}(m/M,\mbox{lbl})$, is due to 12 light-by-light scattering diagrams with electron or $\tau$ loops (like the graph of fig. \[fig:qed3\] $C$) [@LR93]. The exact expression for $A_2^{(6)}(m/M)$ in closed analytic form is quite complicated, containing hundreds of polylogarithmic functions up to fifth degree (for the light-by-light diagrams) and complex arguments (for the vacuum polarization contribution). It also includes harmonic polylogarithms [@HarmPol]. As it is very lengthy, it was not listed in the original papers [@La93; @LR93] which provided, however, useful series expansions in the mass ratio $(m/M)$ for the cases of physical relevance. Using the exact expressions in closed analytic form kindly provided to me by the authors, and the latest values for the mass ratios mentioned above, I obtain the following values A\_2\^[(6)]{}(m\_/m\_e,) &=& 1.920 455 130 (33), \[eq:qedA26evac\]\
A\_2\^[(6)]{}(m\_/m\_e,) &=& 20.947 924 89(16), \[eq:qedA26elbl\]\
A\_2\^[(6)]{}(m\_/m\_,)&=& -0.00178233 (48), \[eq:qedA26tauvac\]\
A\_2\^[(6)]{}(m\_/m\_,)&=&0.00214283 (69). \[eq:qedA26taulbl\] The sums of eqs. (\[eq:qedA26evac\])–(\[eq:qedA26elbl\]) and eqs. (\[eq:qedA26tauvac\])–(\[eq:qedA26taulbl\]) are A\_2\^[(6)]{}(m\_/m\_e) &=& 22.868 380 02(20), \[eq:qedA26e\]\
A\_2\^[(6)]{}(m\_/m\_)&=&0.000 360 51 (21); \[eq:qedA26tau\] to determine the uncertainties, the correlation of the addends has been taken into account. Eqs. (\[eq:qedA26evac\])–(\[eq:qedA26tau\]) provide the first re-evaluation of these coefficients with the recently updated [CODATA]{} and [PDG]{} mass ratios of refs. [@MT04; @PDG04]. These new values differ visibly from older ones (see refs. [@CM99; @MT00]) based on previous measurements of the mass ratios, but the change induces only a negligible shift in the total [QED]{} prediction. Note the large contribution from the electron light-by-light diagrams, – its leading term is $(2/3)\pi^2
\ln(m_{\mu}/m_e)$ [@LS77]. More generally, it was shown in [@Ye89] that the $O(\alpha^{2n+1})$ contribution to $\amu^{\mysmall QED}$, from diagrams in which the electron light-by-light subgraph is connected with $2n+1$ photons to the muon, contains a large $\pi^{2n}\ln(m_{\mu}/m_e)$ term with a coefficient of $O(1)$.
The analytic calculation of the three-loop diagrams with both electron and $\tau$ loop insertions in the photon propagator (see fig. \[fig:qed3\] $D$) became available in 1999 [@CS99]. This analytic result yields the numerical value A\_3\^[(6)]{}(m\_/m\_e,m\_/m\_) = 0.000 527 66 (17), \[eq:qedA36\] providing a small $0.7 \times 10^{-11}$ contribution to $\amu^{\mysmall
QED}$. The error, $1.7 \times 10^{-7}$, is caused by the uncertainty of the ratio $m_{\mu}/m_{\tau}$. Combining the three-loop results presented above, I obtain the new sixth-order [QED]{} coefficient C\_3 &=& A\_1\^[(6)]{} + A\_2\^[(6)]{}(m\_/m\_e) + A\_2\^[(6)]{}(m\_/m\_) + A\_3\^[(6)]{}(m\_/m\_e,m\_/m\_)\
&=& 24.050 509 64 (43). \[eq:qedC3\] The error $\delta C_3 = 4.3 \times 10^{-7}$, due to the measurement uncertainties of the lepton masses, has been determined considering the correlation of the addends. It induces a negligible $O(10^{-14})$ uncertainty in $\amu^{\mysmall QED}$.
In parallel to these analytic results, numerical methods were also developed, mainly by Kinoshita and his collaborators, for the evaluation of the full set of three-loop diagrams [@Ki90; @KM90].
![[Examples of [QED]{} diagrams contributing to the muon $g$$-$$2$ in order $\alpha^3$. $A$, a “triple-cross” diagram. $B$ ($C$), sixth-order muon vertex obtained by insertion of an electron or $\tau$ vacuum polarization (light-by-light) subdiagram. $D$, graph with $e$ and $\tau$ loops in the photon propagator.]{}[]{data-label="fig:qed3"}](qed3.eps){width="13cm"}
Four-loop Contribution
----------------------
More than one thousand diagrams enter the evaluation of the four-loop [QED]{} contribution to $\amu$. As only few of them are known analytically [@A8analytic], this eighth-order term has thus far been evaluated only numerically. This formidable task was first accomplished by Kinoshita and his collaborators in the early 1980s [@KL81; @KNO84]. Since then, they made a major effort to continuously improve this result [@KM90; @HK99; @KNO90; @KL83-89; @Ki93; @KN03], also benefiting from fast advances in computing power. The latest analysis appeared in ref. [@KN04]. One should realize that this eighth-order [QED]{} contribution, being about six times larger than the present experimental uncertainty of $\amu$, is crucial for the comparison between the [SM]{} prediction of $\amu$ and its experimental determination.
There are 891 four-loop diagrams contributing to the mass-independent coefficient $A_1^{(8)}$. Its latest published value is $A_1^{(8)} = -1.7502
\, (384)$ [@KN03], where the error is caused by the numerical procedure. This coefficient has undergone a small revision in ref. [@KN03]. In September 2004 Kinoshita reported a new preliminary updated result [@Ki04], A\_1\^[(8)]{} = -1.7093 (42). \[eq:qedA18\] Note the small shift in the central value and the significant reduction of the numerical uncertainty of this new result. I will adopt it for the value of $A_1^{(8)}$. The latest value of the coefficient $A_2^{(8)}(m_{\mu}/m_e)$, arising from 469 diagrams, is [@KN04] A\_2\^[(8)]{}(m\_/m\_e) = 132.6823 (72). \[eq:qedA28e\] This value is significantly higher than the older one, $127.50
\,(41)$ [@HK99] (its precision is impressively higher too) shifting up the value of $\amu^{\mysmall QED}$ by a non-negligible $\sim \! 15 \times
10^{-11}$. This difference is partly accounted for by the correction of a program error described in ref. [@KN03], but is mostly due to the fact that the computation of the older value suffered from insufficient numerical precision. The term $A_2^{(8)}(m_{\mu}/m_{\tau})$ has been roughly estimated to give an $O(10^{-13})$ contribution to $\amu^{\mysmall QED}$ – it can be safely ignored for now [@KN04]. The numerical evaluation of the 102 diagrams containing both electron and $\tau$ loop insertions yields the three-mass coefficient [@KN04] A\_3\^[(8)]{}(m\_/m\_e,m\_/m\_) = 0.037 594 (83), \[eq:qedA28tau\] which provides a small $O(10^{-12})$ contribution to $\amu^{\mysmall QED}$. Adding up the four-loop results described above, we obtain the eighth-order [QED]{} coefficient C\_4 A\_1\^[(8)]{} + A\_2\^[(8)]{}(m\_/m\_e) + A\_3\^[(8)]{}(m\_/m\_e,m\_/m\_) = 131.011 (8). \[eq:qedC4\] Note that this expression does not contain the term $A_2^{(8)}(m_{\mu}/m_{\tau})$, which has been roughly estimated to be of the same order of magnitude of the uncertainty on the r.h.s. of . However, this uncertainty, $0.008$, causes only a tiny $0.02
\times 10^{-11}$ error in $\amu^{\mysmall QED}$.
Five-loop Contribution
----------------------
The evaluation of the five-loop [QED]{} contribution is in progress [@Ki04]. The existing estimates are mainly based on the experience accumulated computing the sixth- and eighth-order terms, and include only specific contributions enhanced by powers of $\ln(m_{\mu}/m_e)$ times powers of $\pi$. The first estimate, $C_5 = 570 \,(140)$, provided by Kinoshita and collaborators in 1990 [@KNO90], considered the contribution of graphs containing an electron light-by-light subdiagram with one-loop vacuum polarization insertions. A few other predictions for $C_5$ exist, and classes of diagrams were computed or estimated with various methods [@Ye89; @MY89; @Ka92; @Br93; @Ka93; @La94; @EKS94; @KS94]. In September 2004 Kinoshita reported a new very preliminary result [@Ki04], C\_5 A\_2\^[(10)]{}(m\_/m\_e) = 677 (40), \[eq:qedC5\] (9080 diagrams contribute to $A_2^{(10)}(m_{\mu}/m_e)$!) corresponding to a $4.6\,(0.3) \times 10^{-11}$ contribution to $\amu^{\mysmall QED}$. This is the value of $C_5$ I will employ. The uncertainty in this new estimate of the tenth-order term ($0.3 \times 10^{-11}$) no longer dominates the error of the total [QED]{} prediction (see next section). Efforts to improve upon the evaluation of $C_5$ are presently being pursued by Kinoshita and Nio.
The Numerical Value of [$\amu^{\mysmall QED}$]{}
-------------------------------------------------
Adding up all the above contributions and using the latest [CODATA]{} recommended value for the fine-structure constant [@MT04], known to 3.3 ppb, \^[-1]{} = 137.035 999 11 (46), \[eq:alphaMT04\] I obtain the following value for the [QED]{} contribution to the muon $g$$-$$2$: \^[QED]{} = 116 584 718.8 (0.3)(0.4) 10\^[-11]{}. \[eq:qed\] The first error is due to the uncertainties of the $O(\alpha^2)$, $O(\alpha^4)$ and $O(\alpha^5)$ terms, and is strongly dominated by the last of them. (The uncertainty of the $O(\alpha^3)$ term is negligible.) The second error is caused by the 3.3 ppb uncertainty of the fine-structure constant $\alpha$. When combined in quadrature, these uncertainties yield $\delta \amu^{\mysmall QED} = 0.5 \times 10^{-11}$. The value of $\amu^{\mysmall QED}$ in is close to that presented by Kinoshita in [@Ki04], $\amu^{\mysmall QED} = 116 \, 584 \, 717.9 \,(0.3)\,(0.9)
\times 10^{-11}$, and has a smaller error. This latter result was in fact derived using the value of $\alpha$ determined from atom interferometry measurements [@alpha], $\alpha^{-1}=137.036\,000\,3\,(10)$ (7.3 ppb), which has a larger uncertainty than the latest [CODATA]{} value employed for .
The Electroweak Contribution {#sec:EW}
============================
The electroweak ([EW]{}) contribution to the anomalous magnetic moment of the muon is suppressed by a factor $(m_{\mu}/\mw)^2$ with respect to the [QED]{} effects. The one-loop part was computed in 1972 by several authors [@ew1loop]. Back then, the experimental uncertainty of $\amu$ was one or two orders of magnitude larger than this one-loop contribution. Today it’s less than one-third as large.
One-loop Contribution
---------------------
The analytic expression for the one-loop [EW]{} contribution to $\amu$, due to the diagrams in fig. \[fig:ew1\], reads \^[EW]{} () = , \[eq:EWoneloop\] where $G_{\mu}=1.16637(1) \times 10^{-5}\gev^{-2}$ is the Fermi coupling constant, $\mz$, $\mw$ and $\mh$ are the masses of the $Z$, $W$ and Higgs bosons, and $\theta_{\mysmall{W}}$ is the weak mixing angle. Closed analytic expressions for $\amu^{\mysmall EW} (\mbox{1 loop})$ taking exactly into account the $m^2_{\mu}/M^2_{\mysmall{B}}$ dependence ($B=Z,W,$ Higgs, or other hypothetical bosons) can be found in refs. [@Studenikin]. Following [@CMV03], I employ for $\sin^2\!\theta_{\mysmall{W}}$ the on-shell definition $\sin^2\!\theta_{\mysmall{W}} =
1-M^2_{\mysmall{W}}/M^2_{\mysmall{Z}}$ [@Si80], where $\mz=91.1875(21)\gev$ and $\mw$ is the theoretical [SM]{} prediction of the $W$ mass. The latter can be easily derived from the simple analytic formulae of ref. [@FOPS], = , \[eq:fops\] (on-shell scheme [II]{} with $\Delta \alpha_h^{(5)}=0.02761 \,(36)$, $\alpha_s(\mz)=0.118 \,(2)$ and $M_{\rm\scriptstyle top}=$ $178.0 \, (4.3)$ GeV [@newTOP]), leading to $\mw =80.383\gev$ for $\mh=150\gev$, compared with the direct experimental value $\mw=80.425 \,(38)\gev$ [@PDG04], which corresponds to a small $\mh$ [@FOS04]. For $\mh=150\gev$, eq. (\[eq:EWoneloop\]) thus gives \^[EW]{} () = 194.8 10\^[-11]{}, \[eq:EWoneloopNumber\] but this value encompasses the predictions derived from a wide range of values of $\mh$ varying from 114.4 GeV, the current lower bound at 95% confidence level [@LEPHIGGS], up to a few hundred GeV.
![[One-loop electroweak contributions to $a_{\mu}$. The diagram with a W and a Goldstone boson ($\phi$) must be counted twice. The diagrams with the Higgs boson loop and with two Goldstone boson couplings to the muon are suppressed by a factor $m^2_{\mu}/M^2_{\mysmall{Z,W,H}}$ and are not drawn.]{}[]{data-label="fig:ew1"}](ew1.eps){width="14cm"}
The contribution of the Higgs diagram alone, part of the $O(m^2_{\mu}/M^2_{\mysmall{Z,W,H}})$ terms of eq. (\[eq:EWoneloop\]), is [@KM90; @Studenikin] \^[EW,H]{} () = , where $R_{\mysmall{H}}=M^2_{\mysmall{H}}/m^2_{\mu}$. Given the current lower bound $M_{\mysmall{H}} > 114.4$ GeV (95% [CL]{}), $\amu^{\mysmall EW,H} (\mbox{1 loop})$ is smaller than $3 \times 10^{-14}$ and can be safely neglected.
Higher-order Contributions
--------------------------
The two-loop [EW]{} contribution to $\amu$ was computed in 1995 by Czarnecki, Krause and Marciano [@CKM95a; @CKM95b]. This remarkable calculation, probably the first (and still one of the very few) complete two-loop electroweak computation, leads to a significant reduction of the one-loop prediction. Naïvely one would expect the two-loop [EW]{} contribution $\amu^{\mysmall EW} (\mbox{2 loop})$ to be of order $(\alpha/\pi) \times \amu^{\mysmall EW} (\mbox{1 loop})$, and thus negligible, but this turns out not to be so. As first noticed in 1992 [@KKSS], $\amu^{\mysmall EW} (\mbox{2 loop})$ is actually quite substantial because of the appearance of terms enhanced by a factor of $\ln(M_{\mysmall{Z,W}}/m_f)$, where $m_f$ is a fermion mass scale much smaller than $\mw$.
The two-loop contributions to $\amu^{\mysmall EW}$ can be divided into fermionic and bosonic parts; the former includes all two-loop [EW]{} corrections containing closed fermion loops, whereas all other contributions are grouped into the latter. The full two-loop calculation involves 1678 diagrams in the linear ’t Hooft-Feynman gauge [@Kaneko95]. As a check, the authors of [@CKM95a; @CKM95b] employed both this gauge and a nonlinear one in which the vertex of the photon, the $W$ and the unphysical charged scalar vanishes. Their result for $\mh=150\gev$ (obtained in the approximation $\mh \gg M_{\mysmall{W,Z}}$ computing the first two terms in the expansion in $M^2_{\mysmall{W,Z}}/\mh^2$) was $\amu^{\mysmall EW}
(\mbox{2 loop})= -42.3(2.0)(1.8)\times 10^{-11}$, leading to a significant reduction of $\amu^{\mysmall EW}$. The first error is meant to roughly reflect low momentum hadronic uncertainties (more below), whereas the second allows for a range of $\mh$ values from 114 GeV to about 250 GeV. Note that the contribution from $\gamma$–$Z$ mixing diagrams is not included in this result: as it is suppressed by ($1-4\sin^2\!\theta_{\mysmall{W}}) \sim 0.1$ for quarks and ($1-4\sin^2\!\theta_{\mysmall{W}})^2$ for leptons, it was neglected in this early calculation. It was later studied in ref. [@CMV03]: together with small contributions proportional to $(1-4\sin^2\!\theta_{\mysmall{W}})(m_t^2/\mw^2)$ induced by the renormalization of $\sin^2\!\theta_{\mysmall{W}}$, it shifts the above value of $\amu^{\mysmall EW}$ down by a tiny $0.4\times 10^{-11}$.
The hadronic uncertainties, above estimated to be $\sim 2\times 10^{-11}$, arise from two types of two-loop diagrams: hadronic photon–$Z$ mixing, and quark triangle loops with the external photon, a virtual photon and a $Z$ attached to them (see fig. \[fig:ew2\]).
![[Hadronic loops in two-loop [EW]{} contributions.]{}[]{data-label="fig:ew2"}](ew2.eps){width="12cm"}
The tiny hadronic $\gamma$–$Z$ mixing terms can be evaluated either in the free quark approximation or via a dispersion relation using data from $e^+e^-$ annihilation into hadrons; the difference was shown to be numerically insignificant [@CMV03]. The contribution from the second type of diagrams (the quark triangle ones), calculated in [@CKM95a] in the free quark approximation, is numerically more important. The question of how to treat properly the contribution of the light quarks was originally addressed in ref. [@PPD95] within a low-energy effective field theory approach and was further investigated in the detailed analyses of refs. [@CMV03; @KPPD02; @CMV03b]. These refinements significantly improved the reliability of the fermionic part of $\amu^{\mysmall EW}(\mbox{two
loop})$ and increased it by $2\times 10^{-11}$ relative to the free quark calculation, leading, for $\mh=150\gev$, to [@CMV03] \^[EW]{} = 154(1)(2)10\^[-11]{}, \[eq:ew\] where the first error corresponds to hadronic loop uncertainties and the second to an allowed Higgs mass range of $114\gev < \mh < 250\gev$, the current top mass uncertainty[^1] and unknown three-loop effects.
The leading-logarithm three-loop contribution to $\amu^{\mysmall EW}$ was first studied via a renormalization group analysis in ref. [@DGi98]. Such an analysis was revisited and refined in [@CMV03], where this contribution was found to be extremely small (indeed, consistent with zero to a level of accuracy of $10^{-12}$). An uncertainty of $0.2\times
10^{-11}$, included in eq. (\[eq:ew\]), has been conservatively assigned to $\amu^{\mysmall EW}$ for uncalculated three-loop nonleading-logarithm terms.
Lastly, I would like to point out that until recently only one evaluation existed of the two-loop bosonic part of $\amu^{\mysmall EW}$, ie, ref. [@CKM95b]. The recent calculation of ref. [@HSW04], performed without the approximation of large Higgs mass previously employed, agrees with the result of [@CKM95b]. Work is also in progress for an independent recalculation based on the numerical methods of refs. [@Topside].
The Hadronic Contribution {#sec:HAD}
=========================
In this section I will analyze the contribution to the muon $g$$-$$2$ arising from [QED]{} diagrams involving hadrons. Hadronic effects in (two-loop) [EW]{} contributions are already included in $\amu^{\mysmall
EW}$ (see the previous section).
Leading-order Hadronic Contribution {#subsec:HLO}
-----------------------------------
The leading hadronic contribution to the muon $g$$-$$2$, $\amu^{\mbox{$\scriptscriptstyle{HLO}$}}$, is due to the hadronic vacuum polarization correction to the internal photon propagator of the one-loop diagram (see diagram in fig. \[fig:hlo\]).
![[Leading hadronic contribution to $\amu$.]{}[]{data-label="fig:hlo"}](hlo.eps){width="7cm"}
The evaluation of this $O(\alpha^2)$ diagram involves long-distance [QCD]{} for which perturbation theory cannot be employed. However, using analyticity and unitarity (the optical theorem), Bouchiat and Michel [@BM61] showed long ago that this contribution can be computed from hadronic $e^+ e^-$ annihilation data via the dispersion integral [@BM61; @LBdRGdR] \^= \^\_[4m\^2\_]{} ds K(s) \^[(0)]{}(s) = \^\_[4m\^2\_]{} K(s) R(s), \[eq:hlo\] where $\sigma^{(0)}(s)$ is the experimental total cross section for $e^+
e^-$ annihilation into any hadronic state, with extraneous [QED]{} radiative corrections subtracted off (more later), and $R(s)$ is the ratio of $\sigma^{(0)}(s)$ and the high-energy limit of the Born cross section for $\mu$-pair production: $R(s) = \sigma^{(0)}(s)/(4\pi \alpha^2\!/3s)$. The kernel $K(s)$ is the well-known function K(s)= \_0\^1 dx (see ref. [@EJ95] for some of its explicit representations and their suitability for numerical evaluations). It decreases monotonically for increasing $s$, and for large $s$ it behaves as $m_\mu^2/3s$ to a good approximation. For this reason the low-energy region of the dispersive integral is enhanced by $\sim 1/s^2$. About 91% of the total contribution to $\amu^{\mbox{$\scriptscriptstyle{HLO}$}}$ is accumulated at center-of-mass energies $\sqrt{s}$ below 1.8 GeV and 73% of $\amu^{\mbox{$\scriptscriptstyle{HLO}$}}$ is covered by the two-pion final state which is dominated by the $\rho(770)$ resonance [@DEHZ03]. Exclusive low-energy $e^+e^-$ cross sections have been mainly measured by experiments running at $e^+e^-$ colliders in Novosibirsk ([OLYA, TOF, ND, CMD, CMD-2, SND]{}) and Orsay ([M3N, DM1, DM2]{}), while at higher energies the total cross section ratio $R(s)$ has been measured inclusively by the experiments $\gamma \gamma 2$, [MARK I, DELCO, DASP, PLUTO, LENA]{}, Crystal Ball, [MD-1, CELLO, JADE, MARK-J, TASSO, CLEO, CUSB, MAC]{}, and [BES]{}. Perturbative [QCD]{} becomes applicable at higher loop momenta, so that at some energy scale one can switch from data to [QCD]{} [@pQCD; @DH98a; @DH98b].
Detailed evaluations of the dispersive integral in eq. (\[eq:hlo\]) have been carried out by several authors [@EJ95; @DEHZ03; @DH98a; @DH98b; @B85; @KNO85; @CLY85; @MD89; @AY95; @BW96; @ADH98; @SN01; @JatSirlin; @dTY01; @CLS01; @DEHZ02; @HMNT02; @J03; @ELZ04; @ELZ03; @HMNT03; @dTY04; @DEHZ04]. A prominent role among all data sets is played by the precise measurements by the [CMD-2]{} detector at the [VEPP-2M]{} collider in Novosibirsk [@CMD2-99; @CMD2-01; @CMD2-03] of the cross section for $e^+e^-\rightarrow \pi^+\pi^-$ at values of $\sqrt s$ between 0.61 and 0.96 GeV (ie, $s \in [0.37,0.93]\,{\rm GeV}^2$). The quoted systematic error of these data is 0.6% [@CMD2-03], dominated by the uncertainties in the radiative corrections (0.4%). In July 2004, also the [KLOE]{} experiment at the [DA$\Phi$NE]{} collider in Frascati presented the final analysis [@KLOE-04] of the 2001 data for the precise measurement of $\sigma(e^+e^-\rightarrow \pi^+\pi^-)$ via the radiative return method [@RadRet] from the $\phi$ resonance. In this case the machine is operating at a fixed center-of-mass energy $W \simeq$ 1.02 GeV, the mass of the $\phi$ meson, and initial-state radiation is used to reduce the invariant mass of the $\pi^+\pi^-$ system. In [@KLOE-04] the cross section $\sigma(e^+e^-\rightarrow \pi^+\pi^-)$ was extracted for the range $s \in [0.35,0.95]\,{\rm GeV}^2$ with a systematic error of 1.3% (0.9% experimental and 0.9% theoretical) and a negligible statistical one. The study of the $e^+e^-\rightarrow \pi^+\pi^-$ process via the initial-state radiation method is also in progress at the [BABAR]{} detector at the [PEP-II]{} collider in [SLAC]{} [@BABAR]. This analysis will be important to further assess the consistency of the $e^+e^-$ data. The [BABAR]{} collaboration has already presented data for the $\pi^+ \pi^-
\pi^0$ final state [@BABAR+-0], and preliminary ones for the process $e^+e^- \rightarrow 2\pi^+ 2\pi^-$ [@BABAR]. On the theoretical side, the properties of analyticity, unitarity and chiral symmetry provide strong constraints for the pion form factor $F_\pi(s)$ in the low-energy region [@GM91; @BERN01; @Le02; @Co03; @VLC; @dTY04]. They can lead to further improvements. Perhaps, also lattice [QCD]{} computations of $\amu^{\mbox{$\scriptscriptstyle{HLO}$}}$, although not yet competitive with the precise results of the dispersive method, may eventually rival that precision [@LATTICE].
The hadronic contribution $\amu^{\mbox{$\scriptscriptstyle{HLO}$}}$ is of order $7000\times 10^{-11}$. Of course, this is a small fraction of the total [SM]{} prediction for $\amu$, but is very large compared with the current experimental uncertainty $\delta
a_{\mu}^{\mbox{$\scriptscriptstyle{EXP}$}} = 60\times 10^{-11}$. Indeed, as $\delta a_{\mu}^{\mbox{$\scriptscriptstyle{EXP}$}}$ is less than one percent of $\amu^{\mbox{$\scriptscriptstyle{HLO}$}}$, precision analyses of this hadronic term as well as full treatment of its higher-order corrections are clearly warranted. Normally, the “bare” cross section $\sigma^{(0)}(s)$ is used in the evaluation of the dispersive integral and the higher-order hadronic corrections (see sec. \[subsec:HHO\]) are addressed separately. But what does “bare” really mean?
The extraction of $\sigma^{(0)}(s)$ from the observed hadronic cross section $\sigma(s)$ requires the subtraction of several radiative corrections ([RC]{}) which, at the level of precision we are aiming at, have a substantial impact on the result. To start with, [RC]{} must be applied to the luminosity determination, which is based on large-angle Bhabha scattering and muon-pair production in low-energy experiments, and small-angle Bhabha scattering at high energies. The first step to derive $\sigma^{(0)}(s)$ consists then in subtracting the initial-state radiative ([ISR]{}) corrections (virtual and real, described by pure [QED]{}) from $\sigma(s)$. The resulting cross section still contains the effects of the photon vacuum polarization corrections ([VP]{}), which can be simply undressed by multiplying it by $\alpha^2/\alpha(s)^2$, where $\alpha(s)$ is the effective running coupling (obviously depending on nonperturbative contributions itself). The problem with data from old experiments is that it’s difficult to find out if (and which of) these corrections have been included (see ref. [@DEHZ02]). The latest analysis from [CMD-2]{} [@CMD2-03] is explicitly corrected for both [ISR]{} and [VP]{} (leptonic as well as hadronic) effects, whereas the preliminary data [@CMD2-99] of the same experiment were only corrected for [ISR]{}. For a thorough analysis of these problems, I refer the reader to [@DEHZ02; @HMNT03; @HGJ02] and references therein.
All hadronic final states should be incorporated in the hadronic contribution to the muon $g$$-$$2$, in particular final states including photons. These final-state radiation ([FSR]{}) effects, although of higher order ($\alpha^3$), are normally included in the leading-order hadronic contribution $\amu^{\mbox{$\scriptscriptstyle{HLO}$}}$. I will stick to this time-honored convention. The precise [CMD-2]{} data for the cross section $e^+e^- \rightarrow \pi^+ \pi^-$ (quoted systematic error of 0.6% dominated by the uncertainties in the [RC]{}) are corrected for [FSR]{} effects using [*scalar*]{} [QED]{}. I find this worrisome. The following is done: their experimental analysis imposes cuts to isolate the two-pion final states. These cuts exclude a large fraction of the $\pi^+ \pi^- \gamma$ states, in particular those where the photon is radiated off at a relatively large angle [@Me01]. The fraction left is then removed using the Monte Carlo simulation based on point-like pions. Finally, the full [FSR]{} contribution is added [*back*]{} using an analytic expression computed in scalar [QED]{} for point-like pions [@Sch89], shifting up the value of $\amu^{\mbox{$\scriptscriptstyle{HLO}$}}$ by $\sim 50 \times
10^{-11}$ [@Me01; @dTY01; @dTY04; @HGJ02; @CGKR03; @DKMPS04]. (This full scalar-[QED]{} [FSR]{} contribution is also added to older $\pi^+ \pi^-$ data.) This procedure is less than perfect, as it introduces a model dependence which could be avoided by a direct measurement of the cross section into hadronic states inclusive of photons. Any calculation that invokes scalar [QED]{} probably falls short of what is needed.
The 2001 final analysis [@CMD2-01] of the precise [CMD-2]{} $\pi^+
\pi^-$ data taken in 1994–95 substantially differed from the preliminary one [@CMD2-99] released two years earlier (based on the same data sample). The difference mostly consisted in the treatment of [RC]{}, resulting in a reduction of the cross section by about 1% below the $\rho$ peak and 5% above. A second significant change occurred during the summer of 2003, when the [CMD-2]{} collaboration discovered an error in the Monte Carlo program for Bhabha scattering that was used to determine the luminosity [@CMD2-03]. As a result, the luminosity was overestimated by 2–3%, depending on energy. (Another problem was found in the [RC]{} for $\mu$-pairs production.) Overall, the pion-pair cross section increased by 2.1–3.8% in the measured energy range [@DEHZ03], a non-negligible shift. The 2004 results of the [KLOE]{} collaboration, obtained via the radiative return method from the $\phi$ resonance, are in fair agreement with the latest energy scan data from [CMD-2]{} [@DEHZ04; @CMD2-03; @KLOE-04]. Here I will only report the evaluations of the dispersive integral in eq. (\[eq:hlo\]) based on the latest [CMD-2]{} reanalysis, as it supersedes all earlier ones. These evaluations are in very good agreement:[^2] \[eq:DEHZ04\] & \^ = & 6934 (53)\_[exp]{} (35)\_[rad]{} 10\^[-11]{},\
\[eq:J03\] & \^ = & 6948 (86) 10\^[-11]{},\
\[eq:ELZ04\] & \^= & 6934 (92) 10\^[-11]{},\
\[eq:HMNT03\] & \^ = & 6924 (59)\_[exp]{} (24)\_[rad]{} 10\^[-11]{},\
\[eq:dTY04\] & \^ = & 6944 (48)\_[exp]{} (10)\_[rad]{} 10\^[-11]{}. The preliminary result in eq. (\[eq:DEHZ04\]) already includes [KLOE]{}’s 2004 data analysis and updates the one of ref. [@DEHZ03], shifting it down by $29 \times 10^{-11}$; two thirds of this shift are due to the inclusion of [KLOE]{}’s data. The preliminary new result in eq. (\[eq:ELZ04\]) updates the value $\amu^{\mbox{$\scriptscriptstyle{HLO}$}}= 6996 \,(85)_{exp} (19)_{rad}
(20)_{proc} \times 10^{-11}$ previously obtained by the same authors [@ELZ03]. Their central value decreased because of an improvement of their integration procedure.
The authors of ref. [@ADH98] pioneered the idea of using vector spectral functions derived from the study of hadronic $\tau$ decays [@ALEPHtau] to improve the evaluation of the dispersive integral in eq. (\[eq:hlo\]). Indeed, assuming isospin invariance to hold, the isovector part of the cross section for $e^+e^-\rightarrow$ hadrons can be calculated via the Conserved Vector Current ([CVC]{}) relations from $\tau$-decay spectra. An updated analysis is presented in [@DEHZ03], where $\tau$ spectral functions are obtained from the results of [ALEPH]{} [@ALEPH02], [CLEO]{} [@CLEO] and [OPAL]{} [@OPAL], and isospin-breaking corrections are applied [@MS88; @CEN01; @CEN02]. In this $\tau$-based evaluation, the $2\pi$ and the two $4\pi$ channels are taken from $\tau$ data up to 1.6 GeV and complemented by $e^+e^-$ data above (the [QCD]{} prediction for $R(s)$ is employed above 5 GeV). Note that $\tau$ decay experiments measure decay rates which are inclusive with respect to radiative photons. Their result is & \^ = & 7110 (50)\_[exp]{} (8)\_[rad]{} (28)\_[SU(2)]{} 10\^[-11]{}, \[eq:DEHZ03tau\] where the quoted uncertainties are experimental, missing radiative corrections to some $e^+e^-$ data, and isospin violation. This value must be compared with their $e^+e^-$-based determination in eq. (\[eq:DEHZ04\]). Also the analysis of [@dTY04] includes information from $\tau$ decay. They obtain & \^ = & 7027 (47)\_[exp]{} (10)\_[rad]{} 10\^[-11]{}, \[eq:dTY04tau\] to be compared with their determination in eq. (\[eq:dTY04\]).
Although the latest [CMD-2]{} $e^+e^-\rightarrow \pi^+\pi^-$ data are consistent with $\tau$ data for the energy region below 850 MeV, there is an unexplained discrepancy for larger energies. This is clearly visible in fig. \[fig:dehz04\], from ref. [@DEHZ04], where the relative comparison of the $\pi^+\pi^-$ spectral functions from $e^+e^-$ and isospin-breaking-corrected $\tau$ data is illustrated. The same figure also shows the $\pi^+\pi^-$ spectral functions derived from [KLOE]{}’s 2004 $e^+e^-$ analysis. They are in fair agreement with those of [CMD-2]{} and confirm the discrepancy with the $\tau$ data.
![[Relative comparison of the $\pi^+\pi^-$ spectral functions from $e^+e^-$ and isospin-breaking-corrected $\tau$ data, expressed as a ratio to the $\tau$ spectral functions. The band shows the uncertainty of the latter. This figure is from ref. [@DEHZ04].]{}[]{data-label="fig:dehz04"}](dehz04.eps){width="12cm"}
Among the possible causes of this discrepancy, one may wonder about inconsistencies in the $e^+e^-$ data, in the $\tau$ data, or in the isospin-breaking corrections applied to the $\tau$ spectral functions. Given the good consistency of the [ALEPH]{} and [CLEO]{} data sets, and the confirmation by [KLOE]{} of the trend exhibited by other $e^+ e^-$ data, further careful investigations of the isospin-violating effects are clearly warranted – see the interesting studies in [@dTY04; @DEHZ04; @Le02; @GJ03; @Da03; @Mo04], in particular the discussion of the possible difference between the masses and the widths of neutral and charged $\rho$-mesons. Until we reach a better understanding of this problem, it is probably safer to discard information from $\tau$ decays for the evaluation of $\amu^{\mbox{$\scriptscriptstyle{HLO}$}}$ [@DEHZ04].
Higher-order Hadronic Contributions {#subsec:HHO}
-----------------------------------
We will now briefly discuss the $O(\alpha^3)$ hadronic contribution to the muon $g$$-$$2$, $\amu^{\mbox{$\scriptscriptstyle{HHO}$}}$, which can be divided into two parts: \^= \^()+ \^(). The first term is the $O(\alpha^3)$ contribution of diagrams containing hadronic vacuum polarization insertions, including, among others, those depicted in figs. \[fig:hho\] $A$ and $B$. The second one is the light-by-light contribution, shown in fig. \[fig:hho\] $C$. Note that the $O(\alpha^3)$ diagram in fig. \[fig:hho\] $D$ has already been included in the leading-order hadronic contribution $\amu^{\mbox{$\scriptscriptstyle{HLO}$}}$ although, unsatisfactorily, using [*scalar*]{} [QED]{} (see discussion in sec. \[subsec:HLO\]). In recent years, $\amu^{\mbox{$\scriptscriptstyle{HHO}$}}(\mbox{vp})$ was evaluated by Krause [@Kr96] and slightly updated in [@ADH98]. Its latest value is [@HMNT03] \^()= -97.9 (0.9)\_[exp]{} (0.3)\_[rad]{} 10\^[-11]{}. \[eq:hhovp\] This result was obtained using the same hadronic $e^+ e^-$ annihilation data described in sec. \[subsec:HLO\]. It changes by about $-3\times 10^{-11}$ if hadronic $\tau$-decay data (again, see sec. \[subsec:HLO\]) are used instead [@DM04].
![[Some of the higher-order hadronic diagrams contributing to $\amu$.]{}[]{data-label="fig:hho"}](hho.eps){width="14cm"}
The hadronic light-by-light contribution changed sign already three times in its troubled life. Contrary to $\amu^{\mbox{$\scriptscriptstyle{HHO}$}}(\mbox{vp})$, it cannot be expressed in terms of experimental observables determined from data and its evaluation therefore relies on purely theoretical considerations. The estimate of the authors of [@Ny03; @KN01; @KNPdR01], who uncovered in 2001 a sign error in earlier evaluations, is & \^()= & +80(40)10\^[-11]{}. \[eq:hholblNy\] Earlier determinations now agree with this result [@HK01; @BPP01]. Further studies include [@BCM01; @RW02]. At the end of 2003 a higher value was reported in [@MV03], & \^()= & +136(25)10\^[-11]{}. \[eq:hholblMV\] It was obtained by including short-distance [QCD]{} constraints previously overlooked. Further independent calculations would provide an important check of this result for $\amu^{\mbox{$\scriptscriptstyle{HHO}$}}(\mbox{lbl})$, a contribution whose uncertainty may become the ultimate limitation of the [SM]{} prediction of the muon $g$$-$$2$.
The Standard Model Prediction vs. Measurement {#sec:COMP}
=============================================
We now have all the ingredients to derive the [SM]{} prediction for $\amu$: \^[SM]{} = \^[QED]{} + \^[EW]{} + \^ + \^() + \^(). \[eq:sm\] For convenience, I collect here the values of each term from eqs. (\[eq:qed\], \[eq:ew\], \[eq:DEHZ04\]–\[eq:dTY04tau\], \[eq:hhovp\]–\[eq:hholblMV\]):
$$\begin{array}{llclr}
\mbox{[this article]} \qquad\qquad &
\amu^{\mbox{$\scriptscriptstyle{QED}$}} &= &
116 \, 584 \, 718.8 \,(0.5) &\times 10^{-11} \\
\mbox{\cite{CMV03}} \qquad\qquad &
\amu^{\mbox{$\scriptscriptstyle{EW}$}} &= &
154(1)(2) &\times 10^{-11} \\
\mbox{\cite{DEHZ04}} \qquad (e^+e^-) &
\amu^{\mbox{$\scriptscriptstyle{HLO}$}} &= &
6934 \, (53)_{exp} (35)_{rad} &\times 10^{-11} \\
\mbox{\cite{J03}} \qquad (e^+e^-) &
\amu^{\mbox{$\scriptscriptstyle{HLO}$}} &= &
6948 \, (86) &\times 10^{-11} \\
\mbox{\cite{ELZ04}} \qquad (e^+e^-) &
\amu^{\mbox{$\scriptscriptstyle{HLO}$}}&= &
6934 \, (92) &\times 10^{-11} \\
\mbox{\cite{HMNT03}} \qquad (e^+e^-) &
\amu^{\mbox{$\scriptscriptstyle{HLO}$}} &= &
6924 \, (59)_{exp} (24)_{rad} &\times 10^{-11} \\
\mbox{\cite{dTY04}} \qquad (e^+e^-) &
\amu^{\mbox{$\scriptscriptstyle{HLO}$}} &= &
6944 \, (48)_{exp} (10)_{rad} &\times 10^{-11} \\
\mbox{\cite{DEHZ03}} \qquad (\tau) &
\amu^{\mbox{$\scriptscriptstyle{HLO}$}} &= &
7110 \, (50)_{exp} (8)_{rad} (28)_{SU(2)} &\times 10^{-11} \\
\mbox{\cite{dTY04}} \qquad (e^+e^-, \tau) &
\amu^{\mbox{$\scriptscriptstyle{HLO}$}} &= &
7027 \, (47)_{exp} (10)_{rad} &\times 10^{-11} \\
\mbox{\cite{HMNT03}} \qquad (e^+e^-) &
\amu^{\mbox{$\scriptscriptstyle{HHO}$}}(\mbox{vp})\,\,&= &
-97.9 \, (0.9)_{exp} (0.3)_{rad} &\times 10^{-11} \\
\mbox{\cite{DM04}} \qquad (\tau) &
\amu^{\mbox{$\scriptscriptstyle{HHO}$}}(\mbox{vp})\,\,&= &
-101 \, (1) &\times 10^{-11} \\
\mbox{\cite{Ny03}} \qquad\qquad &
\amu^{\mbox{$\scriptscriptstyle{HHO}$}}(\mbox{lbl})\,\,&= &
80\,(40) &\times 10^{-11} \\
\mbox{\cite{MV03}} \qquad\qquad &
\amu^{\mbox{$\scriptscriptstyle{HHO}$}}(\mbox{lbl})\,\,&= &
136\,(25) & \times 10^{-11}
\end{array}$$
The values I obtain for $\amu^{\mysmall SM}$ are shown in the first column of table \[tab:EXPvsSM\]. The values employed for $\amu^{\mbox{$\scriptscriptstyle{HLO}$}}$ are indicated by the reference in the last column. I used the latest value available for the hadronic light-by-light contribution $\amu^{\mbox{$\scriptscriptstyle{HHO}$}}(\mbox{lbl})= 136\,(25)\times
10^{-11}$ [@MV03]. Errors were added in quadrature.
The latest measurement of the anomalous magnetic moment of negative muons by the experiment [E821]{} at Brookhaven is [@BNL04] a\_[\^-]{}\^ = 116 592 140 (80)(30) 10\^[-11]{}, \[eq:bnl04\] where the first uncertainty is statistical and the second is systematic. This result is in good agreement with the average of the measurements of the anomalous magnetic moment of positive muons [@BNL00; @BNL01; @BNL02; @oldEXP], as predicted by the [CPT]{} theorem [@Hu03]. The present world average experimental value is [@BNL04] \^ = 116 592 080 (60) 10\^[-11]{} (0.5 ). \[eq:exp\]
The comparison of the [SM]{} results with the present experimental average in eq. (\[eq:exp\]) gives the discrepancies $(\amu^{\mbox{$\scriptscriptstyle{EXP}$}}-
\amu^{\mbox{$\scriptscriptstyle{SM}$}})$ listed in the second column of table \[tab:EXPvsSM\]. The number of standard deviations, shown in the third column, spans a wide range from 0.7 to 2.8. Somewhat higher discrepancies, shown in parentheses in the third column, are obtained if the hadronic light-by-light contribution $\amu^{\mbox{$\scriptscriptstyle{HHO}$}}(\mbox{lbl})= 80\,(40)\times
10^{-11}$ [@Ny03] is used instead of $\amu^{\mbox{$\scriptscriptstyle{HHO}$}}(\mbox{lbl})= 136\,(25)\times
10^{-11}$ [@MV03], with the number of standard deviations spanning the range $[1.3-3.2]$ instead of $[0.7-2.8]$. Note that the entries of the first row in table \[tab:EXPvsSM\] are based on the preliminary result for $\amu^{\mbox{$\scriptscriptstyle{HLO}$}}$ of ref. [@DEHZ04], which already includes the recent data from [KLOE]{} and updates the one of ref. [@DEHZ03], shifting it down by $29 \times 10^{-11}$. As two thirds of this shift are due to the inclusion of the [KLOE]{} data, it is possible that eventually also the $\amu^{\mbox{$\scriptscriptstyle{HLO}$}}$ results of refs. [@J03; @ELZ04; @HMNT03; @dTY04] will undergo some decrease as a consequence of this inclusion, thus increasing the corresponding $\amu^{\mbox{$\scriptscriptstyle{EXP}$}}-
\amu^{\mbox{$\scriptscriptstyle{SM}$}}$ discrepancies.
--------------------------------------------------------------------------------------------------------------------------------------------------------
$\amu^{\mbox{$\scriptscriptstyle{SM}$}} \times 10^{11}$ $(\amu^{\mbox{$\scriptscriptstyle{EXP}$}}- $\sigma$ [HLO]{} Reference
\amu^{\mbox{$\scriptscriptstyle{SM}$}})\times 10^{11}$
--------------------------------------------------------- ---------------------------------------------------------- -------------- --------------------
116591845 (69) 235 (91) 2.6 (3.0) $(e^+e^-)$
116591859 (90) 221 (108) 2.1 (2.5) $(e^+e^-)$
116591845 (95) 235 (113) 2.1 (2.5) $(e^+e^-)$
116591835 (69) 245 (91) 2.7 (3.1) $(e^+e^-)$
116591855 (55) 225 (81) 2.8 (3.2) $(e^+e^-)$
116592018 (63) 62 (87) 0.7 (1.3) $(\tau)$
116591938 (54) 142 (81) 1.8 (2.3) $(e^+e^-,\tau)$
--------------------------------------------------------------------------------------------------------------------------------------------------------
: [ predictions for $\amu$ compared with the current measured world average value. See text for details.]{}[]{data-label="tab:EXPvsSM"}
Conclusions {#sec:CONC}
===========
In the previous sections I presented an update and a review of the contributions to the [SM]{} prediction for the muon $g$$-$$2$. What should we conclude from the wide spectrum of results obtained in sec. \[sec:COMP\]? The discrepancies in table \[tab:EXPvsSM\] between recent [SM]{} predictions and the current world average experimental value range from 0.7 to 3.2 standard deviations, according to the values used for the leading-order and light-by-light hadronic contributions. In particular, the contribution of the hadronic vacuum polarization depends on which of the two data sets, $e^+e^-$ collisions or $\tau$ decays, are employed.
This puzzling discrepancy between the $\pi^+\pi^-$ spectral functions from $e^+e^-$ and isospin-breaking-corrected $\tau$ data could be caused by inconsistencies in the $e^+e^-$ data, in the $\tau$ data, or in the isospin-breaking corrections applied to the latter. Given the fair agreement between the [CMD-2]{} and [KLOE]{} $e^+e^-$ data, and the good consistency of the [ALEPH]{} and [CLEO]{} $\tau$ spectral functions, it is clear that further careful investigations of the isospin violations are highly warranted. Indeed, the question remains whether all possible isospin-breaking effects have been properly taken into account. Until we reach a better understanding of this problem, it is probably safer to discard information from hadronic $\tau$ decays [@DEHZ04]. (Of course, discarding $\tau$ data information still leaves us with the problem of their discrepancy, a troublesome issue on its own, independent of the calculation of the muon $g$$-$$2$.) If $e^+e^-$ annihilation data are used to evaluate the leading hadronic contribution, the [SM]{} prediction of the muon $g$$-$$2$ deviates from the present experimental value by 2–3 standard deviations.
The measurement of the muon $g$$-$$2$ by the [E821]{} experiment at the Brookhaven Alternating Gradient Synchrotron, with an impressive relative precision of 0.5 ppm, is still limited by statistical errors rather than systematic ones. A new experiment, [E969]{}, has been approved (but not yet funded) at Brookhaven in September 2004 [@E969]. Its goal would be to reduce the present experimental uncertainty by a factor of 2.5 to about 0.2 ppm ($\pm 23 \times 10^{-11}$). A letter of intent for an even more precise $g$$-$$2$ experiment was submitted to [J-PARC]{} with the proposal to reach a precision below 0.1 ppm [@JPARC]. While the theoretical predictions for the [QED]{} and [EW]{} contributions appear to be ready to rival these precisions, much effort will be needed in the hadronic sector to test $\amu^{\mysmall SM}$ at an accuracy comparable to the experimental one. Such an effort is certainly well motivated by the excellent opportunity the muon $g$$-$$2$ is providing us to unveil or constrain “new physics” effects.
Acknowledgments {#acknowledgments .unnumbered}
===============
It is a pleasure to thank G. Colangelo, J. Gasser, H. Leutwyler, P. Minkowski, A. Rusetsky and I. Scimemi for many useful discussions of the topic presented here. Several comments and remarks reported in this paper grew out of our ongoing collaboration. I also wish to thank M. Davier, A. Ferroglia, T. Kinoshita, A. Pich, F. Ynduráin and O.V. Zenin for very helpful comments on a number of points. The exact formulae for the evaluation of the three-loop [QED]{} coefficients were kindly provided by S. Laporta. All diagrams were drawn with [Jaxodraw]{} [@Jax].
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[^1]: Indeed, although the result in eq. (\[eq:ew\]) was computed for $m_t=174.3\gev$, I checked that the shift induced in $\amu^{\mysmall EW}$ by the latest experimental value $m_t=178.0
\, (4.3)\gev$ [@newTOP] is well within the quoted error.
[^2]: I have translated the results of ref. [@dTY04] into the notation of the present article.
|
---
abstract: 'The controllability of complex networks has received much attention recently, which tells whether we can steer a system from an initial state to any final state within finite time with admissible external inputs. In order to accomplish the control in practice at the minimum cost, we must study how much control energy is needed to reach the desired final state. At a given control distance between the initial and final states, existing results present the scaling behavior of lower bounds of the minimum energy in terms of the control time analytically. However, to reach an arbitrary final state at a given control distance, the minimum energy is actually dominated by the upper bound, whose analytic expression still remains elusive. Here we theoretically show the scaling behavior of the upper bound of the minimum energy in terms of the time required to achieve control. Apart from validating the analytical results with numerical simulations, our findings are feasible to the scenario with any number of nodes that receive inputs directly and any types of networks. Moreover, more precise analytical results for the lower bound of the minimum energy are derived in the proposed framework. Our results pave the way to implement realistic control over various complex networks with the minimum control cost.'
author:
- 'Gaopeng Duan$^{1,2^*}$, Aming Li$^{3,4^*}$, Tao Meng$^{1}$, Guofeng Zhang$^{2}$ and Long Wang$^{1^\dag}$'
title: Energy cost for controlling complex networks
---
1. Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China
2. Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong
3. Department of Zoology and Oxford Centre for Integrative Systems Biology, University of Oxford, Oxford OX1 3PS, UK
4. Chair of Systems Design, Department of Management, Technology and Economics, ETH Zürich, Weinbergstrasse 56/58, Zürich CH-8092, Switzerland
5. These authors contributed equally to this work
6. Correspondence to: longwang@pku.edu.cn
Introduction
============
An ultimate goal of studying complex systems is to control them on the basis of the underlying topological structures, where nodes indicate units of a system and edges capture who interacts with whom [@Liu2016Rev; @barabasi2016network; @Liu2008Controllability; @Xie2002; @Xie2003]. Indeed, by implementing appropriate external control signals, if we can drive a system from an arbitrary initial state to any final state in finite time, we say that the system is controllable, [*i.e.*]{}, in principle, we are able to steer the system along our expectations. Recently, the problem of finding set of minimal number of nodes that receive external inputs directly to make a network controllable has been investigated[@Liu2011; @Wang2013]. And in the past several years, several important results have elucidated important problems pertaining to node classification[@Jia2013; @Vinayagam03052016], control profiles[@Ruths2014science], target control[@Gao2014], control of edge dynamics[@Nepusz2012], as well as the energy (or cost) required for control[@Yan2012PRL; @energy2014; @Yan2015a; @Chen2016Energy; @Li2017; @Li2017ConEng].
Beyond the basic property, namely controllability of a system, the control energy steering the system from an initial to a final state has received much attention recently. Indeed, the energy tells the cost required to pay in practical control, and thus represents another dimension of difficulty in achieving control. Although theoretically approximate lower bound of control energy and its scaling behavior in terms of the control time have been provided in the literarure for both static and temporal networks, the energy to reach an arbitrary final state in phase space is usually dominated by the upper bound [@Yan2012PRL; @Li2017ConEng]. Analytical forms on the upper bound of control energy are as yet still missing, and the existing results are all based on the myriad numerical calculations. In this article, apart from presenting more precise lower bound of the minimum control energy, we theoretically derive the upper bound for the first time. Furthermore, we show the scaling behavior of both bounds, and numerical validations are also given for both cases.
The minimum energy for controlling complex
===========================================
Here we consider the canonical linear time-invariant dynamics $$\label{sys1}
\dot{{\mathbf{x}}}(t)={\mathbf{A}}{\mathbf{x}}(t)+{\mathbf{B}}{\mathbf{u}}(t),$$ where ${\mathbf{x}}(t)=(x_1(t)~ x_2(t)~ \dots ~x_n(t))^\text{T}$ is the state of the whole network with $x_i(t)$ capturing the state of node $i$; ${\mathbf{u}}(t)=(u_1(t)~ u_2(t)~ \dots ~u_m(t))^\text{T}$ is the control input; ${\mathbf{A}}=(a_{ij})_{n n}$ is the adjacent matrix of the network; ${\mathbf{B}}=(b_{ij})_{n m}$ is the input matrix with size $n\times m$, and the entry at row $i$ and column $j$ is $b_{ij}$, being $1$ if node $i$ receives the external control input signal $u_j(t)$ directly (driver node), being 0 otherwise.
The networked system (\[sys1\]) is said to be controllable, if it can be driven from any initial state ${\mathbf{x}}_0={\mathbf{x}}(t_0)$ toward any target state ${\mathbf{x}}_f={\mathbf{x}}(t_f)$ at a given control time $t_f$, and the corresponding input control energy cost is defined as $E(t_0, t_f)=\int^{t_f}_{t_0}\|{\mathbf{u}}(t)\|^2\text{d}t$ with $\|{\mathbf{u}}(t)\|$ being the Euclidean norm of the vector ${\mathbf{u}}(t)$. To minimize the above energy cost, one can adopt the minimum energy control input ${\mathbf{u}}^*(t)={\mathbf{B}}^\text{T}\text{e}^{{\mathbf{A}}^\text{T}(t_f-t)}{\mathbf{G}}^{-1}\delta$ with ${\mathbf{G}}=\int^{t_f}_{t_0}\text{e}^{{\mathbf{A}}(t-t_0)}{\mathbf{B}}{\mathbf{B}}^\text{T}\text{e}^{{\mathbf{A}}^\text{T}(t-t_0)}\text{d}t$ and $\delta={\mathbf{x}}_{f}-\text{e}^{{\mathbf{A}}t_f}{\mathbf{x}}_0$ [@OptimalBooLewis], which gives the minimum energy cost $E(t_f)=\delta^\text{T}{\mathbf{G}}^{-1}\delta$ from ${\mathbf{x}}_0$ to ${\mathbf{x}}_f$. By assuming $t_0=0$ and ${\mathbf{x}}_0=\mathbf{0}$ for simplicity, we obtain the minimum energy $$\label{E}
E(t_f)={\mathbf{x}}^\text{T}_f{\mathbf{G}}^{-1}{\mathbf{x}}_f,$$ and note that here the matrix ${\mathbf{G}}$ is positive definite when system (\[sys1\]) is controllable [@Kalman63]. Note that when we refer to control energy later, we mean the minimum control energy. Clearly, for the normalized control distance $\|{\mathbf{x}}_f\|=1$ we have $$\label{EB}
\frac{1}{\lambda_{\max}({\mathbf{G}})}\leq E(t_f)\leq\frac{1}{\lambda_{\min}({\mathbf{G}})}.$$ In what follows, for ease of presenting our framework, we consider undirected networks, where ${\mathbf{A}}$ corresponds to the real symmetric matrix. Subsequently, we have ${\mathbf{A}}={\mathbf{P}}{\mathbf{\Xi}}{\mathbf{P}}^\text{T}$ with ${\mathbf{P}}{\mathbf{P}}^\text{T}={\mathbf{P}}^\text{T}{\mathbf{P}}=\mathbf{I}$, where ${\mathbf{\Xi}}=\text{diag}(\lambda_1, \lambda_2, \dots, \lambda_n)$, and $\lambda_i, (i=1, 2, \dots, n)$ is the eigenvalue of ${\mathbf{A}}$ with the ascending order $\lambda_1\leq\lambda_2\leq\dots\leq\lambda_n$. By letting ${\mathbf{Q}}={\mathbf{P}}^\text{T}{\mathbf{B}}{\mathbf{B}}^\text{T}{\mathbf{P}}=(q_{ij})_{n n}$ and ${\mathbf{F}}=(f_{ij})_{n n}$ with $f_{ij}=\frac{1}{\lambda_i+\lambda_j}\left[\text{e}^{(\lambda_i+\lambda_j)t_f}-1\right]$, we have $\int^{t_f}_0\text{e}^{{\mathbf{\Xi}}t}{\mathbf{P}}^\text{T}{\mathbf{B}}{\mathbf{B}}^\text{T}{\mathbf{P}}\text{e}^{{\mathbf{\Xi}}t}\text{d}t=(q_{ij} f_{ij})_{n n}$. Note that the limit of $f_{ij}$ is $t_f$ as $\lambda_i+\lambda_j\rightarrow0$, which keeps the above expression of $f_{ij}$ alive when $\lambda_i+\lambda_j=0$. Furthermore, we can calculate ${\mathbf{G}}$ by $$\label{G}
{\mathbf{G}}={\mathbf{P}}\int^{t_f}_0\text{e}^{{\mathbf{\Xi}}t}{\mathbf{P}}^\text{T}{\mathbf{B}}{\mathbf{B}}^\text{T}{\mathbf{P}}\text{e}^{{\mathbf{\Xi}}t}\text{d}t{\mathbf{P}}^\text{T}={\mathbf{P}}{\mathbf{M}}{\mathbf{P}}^\text{T},$$ where ${\mathbf{M}}=(m_{ij})_{n n}$ with $m_{ij}=q_{ij} f_{ij}$. Based on similarity between matrices ${\mathbf{G}}$ and ${\mathbf{M}}$, we know that they have the same eigenvalues. Therefore, by calculating the eigenvalues of ${\mathbf{M}}$ we can find the lower and upper bounds of the minimum energy $E(t_f)$ given in Eq. (\[EB\]).
Results
=======
As discussed in the previous section, driver nodes are nodes who receive external control inputs directly. In this section, for different numbers of driver nodes, we derive the analytical bounds of the control energy separately. For simplicity, here we assume that each single input only injects on a single driver node, and each node only receives an input at most.
$n$ driver nodes {#nd=n}
----------------
In the case of $n$ driver nodes, [*i.e.*]{} all nodes receive external inputs directly, we have $m=n$, and ${\mathbf{B}}={\mathbf{Q}}={\mathbf{I}}$, which leads to a diagonal matrix ${\mathbf{M}}$ with $m_{ii}=f_{ii}$. According to the magnitude of the control time $t_f$, the corresponding bounds are given as follows.
When $t_f$ is small, we have $\text{e}^{2\lambda_it_f}\approx1+2\lambda_it_f$, and all eigenvalues of ${\mathbf{M}}$ can be approximated by $t_f$. Then both the upper and lower bounds of the minimum energy are $t_f^{-1}$ (see Fig. \[fig1\]).
When $t_f$ is large and ${\mathbf{A}}$ is indefinite (ID), [*i.e.*]{} $\lambda_{i-1}<0, \lambda_i=\dots=\lambda_{i+j}=0,$ $0<\lambda_{i+j+1}$, the $p$th eigenvalue of ${\mathbf{M}}$ is given by: (i) $\frac{1}{2|\lambda_p|}$ for $p=1, 2, \dots, i-1$; (ii) $t_f$ for $p=i, i+1, \dots, i+j$; and (iii) $\frac{\text{e}^{2\lambda_pt_f}-1}{2\lambda_p}$ for $p=i+j+1, \dots, n$. Therefore, we have $\lambda_{\max}({\mathbf{M}})=\frac{\text{e}^{2\lambda_nt_f}-1}{2\lambda_n}$ and $\lambda_{\min}({\mathbf{M}})\approx\frac{1}{2|\lambda_1|}$ with large $t_f$, which tells that the upper bound $\overline{E}\approx2|\lambda_1|$ and the lower bound $\underline{E}=\frac{2\lambda_n}{\text{e}^{2\lambda_nt_f}-1}\sim \text{e}^{-2\lambda_nt_f}\rightarrow0$.
Similarly, for large $t_f$, when ${\mathbf{A}}$ is negative definite (ND, $\lambda_i<0$), $m_{ii}=\frac{\text{e}^{2\lambda_it_f}-1}{2\lambda_i}\approx\frac{-1}{2\lambda_i}$ holds. Therefore, all eigenvalues of ${\mathbf{M}}$ are approximately $\frac{1}{2|\lambda_i|}, i=1, 2, \dots, n$, respectively. Then we can obtain the upper bound of energy cost $\overline{E}\approx2|\lambda_1|$ and the lower bound of energy cost $\underline{E}\approx2|\lambda_n|$. When ${\mathbf{A}}$ is negative semi-definite (NSD, $\lambda_{i-1}<0, \lambda_i=\dots=\lambda_n=0$), all eigenvalues of ${\mathbf{M}}$ approximate $ \frac{1}{|2\lambda_1|}, \frac{1}{|2\lambda_{2}|}, \dots, \frac{1}{|2\lambda_{i-1}|},$ $t_f, t_f, \dots, t_f$, respectively. Therefore, $\lambda_{\max}({\mathbf{M}})=t_f$ and $\lambda_{\min}({\mathbf{M}})\approx\frac{1}{2|\lambda_1|}$ with large $t_f$. Then $\overline{E}\approx2|\lambda_1|$ and $\underline{E}=\frac{1}{t_f}$. When ${\mathbf{A}}$ is positive semi-definite (PSD, $\lambda_1=\dots=\lambda_{i-1}=0, 0<\lambda_i$), all eigenvalues of ${\mathbf{M}}$ are $t_f, t_f, \dots, t_f, \frac{\text{e}^{2\lambda_it_f}-1}{2\lambda_i}, \frac{\text{e}^{2\lambda_{i+1}t_f}-1}{2\lambda_{i+1}}, \dots, $ $\frac{\text{e}^{2\lambda_nt_f}-1}{2\lambda_n} $. Thus $\lambda_{\max}({\mathbf{M}})=\frac{\text{e}^{2\lambda_nt_f}-1}{2\lambda_n}\sim \text{e}^{2\lambda_nt_f}$ and $\lambda_{\min}({\mathbf{M}})=t_f$ for large $t_f$. Accordingly, the upper bound of energy is $\overline{E}=t^{-1}_f$ and the lower bound is $\underline{E}=\frac{2\lambda_n}{\text{e}^{2\lambda_nt_f}-1}\sim \text{e}^{-2\lambda_nt_f}$. When ${\mathbf{A}}$ is positive definite (PD, $0<\lambda_i$), all eigenvalues of ${\mathbf{M}}$ are $\frac{\text{e}^{2\lambda_1t_f}-1}{2\lambda_1}, \frac{\text{e}^{2\lambda_{2}t_f}-1}{2\lambda_{2}}, \dots, \frac{\text{e}^{2\lambda_nt_f}-1}{2\lambda_n}$. Obviously, $\lambda_{\max}({\mathbf{M}})=\frac{\text{e}^{2\lambda_nt_f}-1}{2\lambda_n}$ and $\lambda_{\min}({\mathbf{M}})=\frac{\text{e}^{2\lambda_1t_f}-1}{2\lambda_1}$. Consequently, $\overline{E}=\frac{2\lambda_1}{\text{e}^{2\lambda_1t_f}-1}\sim \text{e}^{-2\lambda_1t_f}$ and $\underline{E}=\frac{2\lambda_n}{\text{e}^{2\lambda_nt_f}-1}\sim \text{e}^{-2\lambda_nt_f}$.
All the above analytical scaling laws are confirmed by numerical simulations presented in Fig. \[fig1\].
One driver node {#nd=1}
---------------
In the case of one driver node, the scaling behavior of the lower bound $\underline{E}$ is given in [@Yan2012PRL], in which the maximum eigenvalue of ${\mathbf{G}}$ is approximated by the trace of ${\mathbf{G}}$. In order to analytically obtain both the upper and lower bounds of the control energy $E$ shown in (\[EB\]), we adopt the approach presented in [@lam2011estimates] to approximate the maximum and minimum eigenvalues of ${\mathbf{M}}$ by $$\label{upM}
\lambda_{\max}({\mathbf{M}})\approx f(\overline{\alpha}, \overline{\beta})$$ and $$\label{lowM}
\lambda_{\min}({\mathbf{M}})\approx \frac{1}{f(\underline{\alpha}, \underline{\beta})}$$ where $f(\alpha, \beta)= \sqrt{\frac{\alpha}{n}+\sqrt{\frac{n-1}{n}(\beta-\frac{{\alpha}^2}{n})}}$, $
\overline{\alpha}=\text{trace}({\mathbf{M}}^2),
\overline{\beta}=\text{trace}({\mathbf{M}}^4),
\underline{\alpha}=\text{trace}(({\mathbf{M}}^{-1})^2),
$ and $
\underline{\beta}=\text{trace}(({\mathbf{M}}^{-1})^4).
$ From Fig. \[fig2’\] we can see that it is feasible to employ (\[upM\]) and (\[lowM\]) to approximate respectively the maximum and the minimum eigenvalues of the real symmetric matrix with high accuracy. Specially, for positive definite matrix ${\mathbf{G}}$, the accuracy is more pronounced, as shown in Fig. S1 in SI.
In the literature, it is common to use the trace of ${\mathbf{G}}$ to estimate the maximum eigenvalue of ${\mathbf{G}}$ [@Yan2012PRL; @Li2017ConEng]. For the lower bound of $E$, we make a comparison of the precision between the existing result and the result obtained in this paper. From Fig. \[fig2\], we find that the lower bounds derived in this paper are more exact.
By (\[EB\]) with (\[upM\]) and (\[lowM\]), we have $$\label{upE}
\overline{E}\approx f(\underline{\alpha}, \underline{\beta}),$$ and $$\label{lowE}
\underline{E}\approx \frac{1}{f(\overline{\alpha}, \overline{\beta})}.$$ With only one driver node, we denote the node $h$ as the sole driver node with $b_{h1}=1$ and $b_{i1}=0 (i\neq h).$ Since $m_{ij}=q_{ij} f_{ij}$ and $q_{ij}=p_{hi}p_{hj}$, we obtain $
m_{ij}=\frac{p_{hi}p_{hj}}{\lambda_i+\lambda_j}(\text{e}^{(\lambda_i+\lambda_j)t_f}-1).
$ Furthermore, we have $
{\mathbf{M}}^2(i,i)=\sum^n_{k=1}\frac{p^2_{hk}p^2_{hi}}{(\lambda_k+\lambda_i)^2}(\text{e}^{(\lambda_k+\lambda_i)t_f}-1)^2
$ and $
{\mathbf{M}}^4(i,i)=\sum^n_{l=1}\left[\sum^n_{k=1}\frac{p^2_{hk}p_{hi}p_{hl}}{(\lambda_k+\lambda_i)(\lambda_k+\lambda_l)} \right.$ $\left.(\text{e}^{(\lambda_k+\lambda_i)t_f}-1)(\text{e}^{(\lambda_k+\lambda_l)t_f}-1)\right]^2.
$ Note that $\text{trace}(\L^2)=\|\L\|_F$ for arbitrary square matrix $\L$. Then, we get the values of $\overline{\alpha}$ and $\overline{\beta}$ as $$\label{overalpha}
\overline{\alpha}=\text{trace}({\mathbf{M}}^2)=\sum^n_{i=1}\sum^n_{k=1}\frac{p^2_{hk}p^2_{hi}}{(\lambda_k+\lambda_i)^2}(\text{e}^{(\lambda_k+\lambda_i)t_f}-1)^2,$$ and $$\label{overbeta}
\overline{\beta}=\text{trace}({\mathbf{M}}^4)=\sum^n_{i=1}\sum^n_{l=1}\left[\sum^n_{k=1} \frac{p^2_{hk}p_{hi}p_{hl}}{(\lambda_k+\lambda_i)(\lambda_k+\lambda_l)}(\text{e}^{(\lambda_k+\lambda_i)t_f}-1) (\text{e}^{(\lambda_k+\lambda_l)t_f}-1)\right]^2.$$
Based on Eqs. (\[overalpha\]) and (\[overbeta\]), we have discussed and calculated the parameters $\overline{\alpha}$ and $\overline{\beta}$ in different cases (see Supplementary Information Sec. S3). Accordingly, the upper and lower bounds of energy cost are given in Tables S1 and S2 in SI, and numerical validations of our analytical results are shown in Fig. \[fig4\].
$d$ driver nodes {#nd=d}
----------------
In the case of $d$ driver nodes, we label them $m_1, m_2, \dots, m_d$. Hence ${\mathbf{B}}=[e_{m_1}, e_{m_2}, \dots, e_{m_d}]\in R^{n\times d}$, where $e_i=(0\,\dots\, 0\,\, 1\,\, 0\,\, \dots\,\, 0)^\text{T}\in R^n$ with all elements as $0$, except $i$th element as $1$. Let ${\mathbf{P}}_1={\mathbf{B}}^\text{T}{\mathbf{P}}$, where ${\mathbf{P}}_1$ is a $d\times n$ matrix constituted by the rows $m_1$, $m_2$, $\dots$, $m_d$ of ${\mathbf{P}}$. Thus ${\mathbf{Q}}={\mathbf{P}}^\text{T}_1{\mathbf{P}}_1$ with $q_{ij}=\sum^d_{k=1}p_{m_k i}p_{m_k j}$. By comparing the form of $m_{ij}=q_{ij}f_{ij}$ between the cases of one driver node and $d$ driver nodes, we find that only the form of $q_{ij}$ is different. Therefore, in subsequent analysis and calculation, we can refer to the Sec. \[nd=1\] to derive $\overline{\alpha}$ and $\overline{\beta}$ (see Sec. S4 in SI for details). We summarize the lower bound of energy under $d$ driver nodes for different scenarios in Table S3 and the corresponding numerical validations are presented in Fig. \[fig5\]. In addition, the upper bound of energy is presented in Table S4.
Discussion
==========
In this paper, we have investigated the scaling behavior of the bounds of minimum control energy for controlling complex networks in terms of the time given to achieve control. The bounds of minimum energy is determined by the maximum and the minimum eigenvalues of ${\mathbf{G}}$. The maximum eigenvalue is usually approximated by the trace of ${\mathbf{G}}$, while the approximation of the minimum eigenvalue has not yet been discussed in the existing literature. Here, we employ an effective method which not only provides more precise analytical expression than the trace for the approximation of the maximum eigenvalue, but also tells the analytical form of the minimum eigenvalues. All the derived theoretical laws are confirmed by numerical simulations.
Our framework also applies to weighted directed networks. When system (\[sys1\]) is controllable, the matrix ${\mathbf{G}}$ is positive definite. When ${\mathbf{A}}$ is asymmetrical for directed networks, we can still obtain the specific form of ${\mathbf{G}}$. Based on ${\mathbf{G}}$, the lower bound of energy cost can be calculated by Eq. (\[lowE\]) with the traces of ${\mathbf{G}}^2$ and ${\mathbf{G}}^4$. For the upper bound of energy cost, we can apply the method to get the scaling behavior of energy by solving the inverse of ${\mathbf{G}}$ (see Sec. S3 in SI).
Although natural systems are believed to operate with nonlinear dynamics, the type of nonlinearity and empirical parameterization are usually hard to detect, especially for large systems. Besides, the generality of results cannot be guaranteed for some specific nonlinear systems. In contrast, the linear dynamics we analyzed here allows us to derive the theoretical insights, which is suitable for analyzing various complex networks. Even that we only consider static complex networks, our framework can also be employed to derive bounds of energy cost for controlling temporal networks by virtue of the effective matrix given in [@Li2017]. Specifically, utilizing estimations of the maximum and the minimum eigenvalues and some approximation techniques introduced in this paper, the of energy for controlling temporal networks can be obtained.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported by the National Natural Science Foundation of China (NSFC) under grants no. 61751301 and no. 61533001. A.L. acknowledgements the Human Frontier Science Program Postdoctoral Fellowship (Grant: LT000696/2018-C), the generous support from Foster Lab at Oxford, and the Chair of Systems Design at ETH Zürich. ETH Zürich, Weinbergstrasse 56/58, Zürich CH-8092, Switzerland. G.Z. acknowledgements the financial support from the Hong Kong Research Grant council (RGC) grants (No. 15206915, No. 15208418).
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$1$ $d$ $n$
-- ------------- ----------------------------------- ----------------------------------- ----------------------------------
$ t_f^{-1}$ $\sim t_f^{-1}$ $t^{-1}_f$
ND $C_1$ $C_2$ $2|\lambda_n|$
NSD $ t_f^{-1}$ $\sim t_f^{-1}$ $t^{-1}_f$
Not ND $\sim \text{e}^{-2\lambda_n t_f}$ $\sim \text{e}^{-2\lambda_n t_f}$ $\sim \text{e}^{-2\lambda_nt_f}$
: The lower bound of control energy $\underline{E}$. No matter how many driver nodes there are, for small $t_f$, $\underline{E}\sim t^{-1}_f$. For large $t_f$, when ${\mathbf{A}}$ is ND (negative definite), $\underline{E}$ approaches to a constant irrespective of $t_f$, ($C_1$ for one driver node, $C_2$ for $d$ driver nodes and $2|\lambda_n|$ for $n$ driver nodes), where $C_1$ and $C_2$ are given as Eq. (\[lowE\]) with Eqs. (S6) (S7) in Sec. S3 and with Eqs. (S45) (S46) in Sec. S4 of SI, respectively. When ${\mathbf{A}}$ is NSD (negative semi-definite) with large $t_f$, $\underline{E}\approx t^{-1}_f$ under $1$ and $n$ driver nodes; while it approaches $t^{-1}_f$(detailed forms are given as Eq. (\[lowE\]) with Eqs. (S47) and (S48) in SI). In addition, when ${\mathbf{A}}$ is not ND (including the cases of indefinite, positive semi-definite, and positive definite), $\underline{E}\sim \text{e}^{-2\lambda_nt_f}$ holds for large $t_f$.[]{data-label="tb5"}
\[table1\]
$1$ $d$ $n$
-- -------------------------------- ---------------------------------- ---------------------------------- ----------------------------------
$\sim t_f^{-(N_0-N_{\min})/2}$ $\sim t_f^{-(N'_0-N'_{\min})/2}$ $t^{-1}_f$
PD $\sim \text{e}^{-2\lambda_1t_f}$ $\sim \text{e}^{-2\lambda_1t_f}$ $\sim \text{e}^{-2\lambda_1t_f}$
PSD $\sim t^{-1}_f$ $\sim t^{-1}_f$ $t^{-1}_f$
Not PD $C_3$ $C_4$ $2|\lambda_1|$
: The upper bound of control energy $\overline{E}$. For small $t_f$, both $N_0-N_{\min}$ and $N'_0-N'_{\min}$ are much larger than $1$, where the detailed meanings of $N_0, N_{\min}, N'_0$ and $N'_{\min}$ are given in Secs. S3 and S4 of SI. For large $t_f$, when ${\mathbf{A}}$ is PD (positive definite), $\overline{E}\sim \text{e}^{-2\lambda_1t_f}$ for arbitrary number of driver nodes; when ${\mathbf{A}}$ is PSD (positive semi-definite), $\overline{E}\sim t^{-1}_f$; when ${\mathbf{A}}$ is not PD (including the cases of indefinite, negative semi-definite, and negative definite), $\overline{E}$ approaches to a constant irrespective of the magnitude of $t_f$ for large $t_f$ ($C_3$ for one driver node, $C_4$ for $d$ driver nodes, and $2|\lambda_1|$ for $n$ driver nodes), where $C_3$ has different forms for different ${\mathbf{A}}$ (detailed forms are presented in Table S2 of Sec. S3 of SI). []{data-label="tb6"}
![The lower and upper bounds of control energy for $n$ driver nodes. By controlling all nodes directly, here we show the numerical and analytical results for lower ($\underline{E}$) and upper ($\overline{E}$) bounds of control energy for different types of ${\mathbf{A}}$. To adjust the maximum (minimum) eigenvalue of ${\mathbf{A}}$ intuitively, we set the link weight $a_{ij}$ uniformly from $[0, 1]$ in (a) to (d) and from $[-1, 0]$ in (e) and (f); each self-loop (diagonal element) is set as $a+s_i$ with $s_i=-\sum^n_{j=1}a_{ij}$. In (a), we set $a=-5$, which guarantees ${\mathbf{A}}$ is ND with eigenvalues in $[-14.0266, -5]$. Similarly, in (b), $a=0$ and ${\mathbf{A}}$ is NSD with eigenvalues in $[-8.5243, 0]$. In (c) and (d), we have $a=5$, and ${\mathbf{A}}$ is ID with eigenvalues in $[-4.0266, 5]$. In (e), we set $a=0$, and hence ${\mathbf{A}}$ is PSD with all eigenvalues in $[0, 8.3062]$. In (f), $a=5$ and ${\mathbf{A}}$ is PD with all eigenvalues in $[5, 13.7144]$. In each panel, triangles (blue and purple) represent results obtained by numerical calculations and full lines indicate analytical derivations under our framework (see Sec. \[nd=n\] and Table \[table1\]). For small $t_f$, from each panel with horizontal axis $\text{ln}(t_f)$, we see that all slopes are $-1$, which confirm our analytical results that both $\overline{E}$ and $\underline{E}$ approximate $\frac{1}{t_f}$ for different types of ${\mathbf{A}}$. For large $t_f$, subgraphs with horizontal axis $t_f$ or ln($t_f$) show the analytical scaling behaviors of the bounds of energy precisely. Here we adopt the BA scale-free network with $n=50$, and network is constructed based on the preferential attachment with average degree 5.8 [@Albert1999Diameter]. []{data-label="fig1"}](fig1.pdf){width="100.00000%"}
![Veracity of eigenvalues estimation based on Eqs. (\[upM\]) and (\[lowM\]) for an arbitrary symmetric positive definite matrix. Here, we randomly generate $25$ matrices with minimum eigenvalue being $i\cdot 4$, $i=1, 2, \dots, 25$, where $i$ is the index of the matrix. The horizontal and vertical coordinates represent the true eigenvalues and estimated eigenvalues by Eqs. (\[upM\]) and (\[lowM\]), from which it is clear the generated pattern almost overlaps with $y=x$. The inset presents ratio errors of differences between approximated eigenvalues by Eqs. (\[upM\]), (\[lowM\]) and the true eigenvalues, which indicates the accuracy of estimation is reliable, especially the estimation of minimum eigenvalues by (\[lowM\]). []{data-label="fig2’"}](fig2.pdf){width="45.00000%"}
![The lower bound of energy comparisons between the methods shown in [@Yan2012PRL] and this paper. Here we randomly generate BA scale-free networks with ${\mathbf{A}}$ being ND (other parameters are the same as those in Fig. \[fig1\]) and $a_{ij}$ is selected from $[1, 3]$ uniformly with $a=-2$. For approximating the maximum eigenvalue of ${\mathbf{M}}$, here we use the method shown in (\[upM\]), while in [@Yan2012PRL], it is inferred by the corresponding trace. Since the existing results only consider the scenario for one driver node, we follow this setting. The network size is chosen as 10, 20, 40, 60, 80, 100 accordingly. For all cases, we can see that the method we employed generates much more precise $\underline{E}$ compared to the existed tools.[]{data-label="fig2"}](fig3.pdf){width="100.00000%"}
![The lower and upper bounds of energy for one driver node. The scaling behavior of the lower and upper bounds of energy cost is given for one driver node, and the summation of analytical results are presented in Tables \[tb5\] and \[tb6\]. In (a)-(c), with small $t_f$, $\underline{E}\sim t^{-1}_f$ for all ${\mathbf{A}}$. In (d)-(f) for upper bound, the slope of triangular trajectory is much less than $-1$. Parameters are selected the same as those given in Fig. \[fig1\]. The interval of the uniform distribution is $[0, 1]$ in (a)-(c), $[1, 3]$ in (d), $[-1,0]$ in (e), and $[-5, -2]$ in (f). In (a), $a=-5$, by which ${\mathbf{A}}$ is ND with eigenvalues in $[-14.0266, -5]$. Similarly, in (b) and (e), $a=0$ such that ${\mathbf{A}}$ is NSD and PSD, respectively. In (c) and (d), $a=5$ such that ${\mathbf{A}}$ is ID. In (f), $a=3$, such that the minimum eigenvalue of ${\mathbf{A}}$ is $3$. []{data-label="fig4"}](fig4.pdf){width="100.00000%"}
{width="100.00000%"}
|
---
abstract: 'We show that the image of the Abel–Jacobi map admits functorially a model over the field of definition, with the property that the Abel–Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also recover a result of Deligne on complete intersections of Hodge level one.'
address:
- 'Colorado State University, Department of Mathematics, Fort Collins, CO 80523, USA'
- 'University of Colorado, Department of Mathematics, Boulder, CO 80309, USA '
- 'Universität Bielefeld, Fakultät für Mathematik, Postfach 100131, D-33501, Germany'
author:
- 'Jeffrey D. Achter'
- 'Sebastian Casalaina-Martin'
- Charles Vial
bibliography:
- 'DCG.bib'
title: Distinguished models of intermediate Jacobians
---
[^1]
Let $X$ be a smooth projective variety defined over the complex numbers. Given a nonnegative integer $n$, denote $\operatorname{CH}^{n+1}(X)$ the Chow group of codimension-$(n+1)$ cycle classes on $X$, and denote $\operatorname{CH}^{n+1}(X)_{\mathrm{hom}}$ the kernel of the cycle class map $\operatorname{CH}^{n+1}(X) \to H^{2({n+1})}(X,\mathbb Z({n+1})).$ In the seminal paper [@griffiths], Griffiths defined a complex torus, the *intermediate Jacobian*, $J^{2n+1}(X)$ together with an *Abel–Jacobi map* $$AJ : \operatorname{CH}^{n+1}(X)_{\mathrm{hom}} \to J^{2n+1}(X).$$ While $J^{2n+1}(X)$ and the Abel–Jacobi map are transcendental in nature, the image of the Abel–Jacobi map restricted to $\operatorname{A}^{{n+1}}(X)$, the sub-group of $\operatorname{CH}^{n+1}(X)$ consisting of algebraically trivial cycle classes, is a complex sub-torus $J^{2n+1}_a(X)$ of $J^{2n+1}(X)$ that is naturally endowed via the Hodge bilinear relations with a polarization, and hence is a complex abelian variety. The first cohomology group of $J^{2n+1}_a(X)$ is naturally identified via the polarization with $\operatorname{N}^nH^{2n+1}(X,\rat(n))$; i.e., the $n$-th Tate twist of the $n$-th step in the geometric coniveau filtration (see ).
If now $X$ is a smooth projective variety defined over a sub-field $K \subseteq \cx$, it is natural to ask whether the complex abelian variety $J^{2n+1}_a(X_\cx)$ admits a model over $K$. In this paper, we prove that $J^{2n+1}_a(X_\cx)$ admits a unique model over $K$ such that $$AJ : \operatorname{A}^{n+1}(X_\cx) \to J^{2n+1}_a(X_\cx)$$ is $\operatorname{Aut}(\cx/K)$-equivariant, thereby generalizing the well-known cases of the Albanese map $\operatorname{A}^{\dim X}(X_\cx) \to \operatorname{Alb}_{X_\cx}$ and of the Picard map $\operatorname{A}^{1}(X_\cx) \to \operatorname{Pic}_{X_{\mathbb C}}^0$, as well as the case of $AJ : \operatorname{A}^{2}(X_\cx) \to J^{3}_a(X_\cx)$ which was treated in our previous work [@ACMVdcg].
\[T:main\] Suppose $X$ is a smooth projective variety over a field $K\subseteq \cx$, and let $n$ be a nonnegative integer. Then $J^{2n+1}_a(X_\cx)$, the complex abelian variety that is the image of the Abel–Jacobi map $AJ :
\operatorname{A}^{n+1}(X_\cx) \to J^{2n+1}(X_\cx)$, admits a distinguished model $J$ over $K$ such that the Abel–Jacobi map is $\operatorname{Aut}(\cx/K)$-equivariant. Moreover, there is an algebraic correspondence $\Gamma \in \operatorname{CH}^{\dim (J)+n}(J\times_K
X)$ inducing, for every prime number $\ell$, a split inclusion of $\operatorname{Gal}(K)$-representations $$\label{E:Tmain-1}
\Gamma_* : H^1(J_{{{\overline{K}}}},\rat_\ell) \hookrightarrow H^{2n+1}(X_{{{\overline{K}}}},\rat_\ell(n))$$ with image $\operatorname{N}^nH^{2n+1}(X_{{{\overline{K}}}},\rat_\ell(n))$.
By Chow’s rigidity theorem (see [@conradtrace Thm. 3.19]), an abelian variety $A/\mathbb C$ descends to at most one model defined over ${{\overline{K}}}$. On the other hand, an abelian variety $A/{{\overline{K}}}$ may descend to more than one model defined over $K$. Nevertheless, since $AJ:\operatorname{A}^{n+1}(X_{\mathbb C})\to
J_a^{2n+1}(X_{\mathbb C})$ is surjective, the abelian variety $J_a^{2n+1}(X_{\mathbb C})$ admits at most one structure of a scheme over $K$ such that $AJ$ is $\operatorname{Aut}(\cx/K)$-equivariant. This is the sense in which $J_a^{2n+1}(X_{\mathbb C})$ admits a *distinguished model* over $K$.
Our proof of Theorem \[T:main\] uses a different strategy than we took in [@ACMVdcg], and as a result improves on the results of that paper in three ways:
1. In [@ACMVdcg Thm. B], only the case $n=1$ of Theorem \[T:main\] was treated. An essential step in the proof in [@ACMVdcg Thm. B] was a result of Murre [@murre83 Thm. C], relying on the theorem of Merkurjev and Suslin, asserting that $J^3_a(X_\cx)$ is an *algebraic representative*, meaning that it is universal among regular homomorphisms from $\operatorname{A}^2(X_\cx)$ (as defined in §\[S:Tors-Gen\]). In general, little is known about when higher intermediate Jacobians are algebraic representatives, or even when algebraic representatives exist. In this paper we completely avoid the use of Murre’s result, or indeed the existence of an algebraic representative. Instead, we use a new approach to show that for each $n$ there is a model of $J^{2n+1}_a(X_\cx)$ over $K$ which makes the Abel–Jacobi map $\operatorname{Aut}(\mathbb C/K)$-equivariant.
2. The results of [@ACMVdcg Thm. A] concerning descent for $J^{2n+1}_a(X_\cx)$ for $n>1$ only show that the *isogeny class* of $J^{2n+1}_a(X_\cx)$ descends to $K$, and this is under the further restrictive assumption that the Abel–Jacobi map be surjective (or under some other constraint on the motive of $X$; see [@ACMVdcg Thm. 2.1]). In contrast, the present Theorem \[T:main\] provides a *distinguished model* of $J^{2n+1}_a(X_\cx)$ over $K$, without any additional hypothesis. Moreover, we show the assignment in Theorem \[T:main\] is functorial (Proposition \[P:functoriality\]). The new technical input begins with Proposition \[P:niveauK\], which shows that $J^{2n+1}_a(X_\cx)$ is dominated, via the induced action of a correspondence defined over $K$, by the Jacobian of a pointed, geometrically integral, smooth projective curve $C$ defined over $K$, strengthening [@ACMVdcg Prop. 1.3]. The key point is that this strengthening, together with the fact that Bloch’s map [@bloch79] factors through the Abel–Jacobi map on torsion, makes it possible to show directly that $J^{2n+1}_a(X_\cx)$ admits a unique model over $K$ making the Abel–Jacobi map $AJ : \operatorname{A}^{n+1}(X_\cx) \to
J^{2n+1}_a(X_\cx)$ Galois equivariant on torsion. In short, avoiding the use of algebraic representatives, and the motivic techniques employed in [@ACMVdcg], we obtain a stronger result. We then make a careful analysis of Galois equivariance for regular homomorphisms, strengthening some statements in [@ACMVdcg], to conclude that the Abel–Jacobi map is Galois equivariant on all points – and not merely on torsion points (Proposition \[P:Tors-Gen\]); this is crucial to the proof of Theorem \[Ta:MazQ1\] below.
3. Finally, while a splitting in [@ACMVdcg Thm. A] analogous to was established by some explicit computations involving correspondences, here we utilize André’s powerful theory of *motivated cycles* [@AndreIHES] in order to establish the more general splitting . This also provides a proof that the coniveau filtration splits (Corollary \[C:CNFSplit\]), as well as a short motivic proof that the isogeny class of $J^{2n+1}_a(X_\cx)$ descends, without any of the restrictive hypotheses in [@ACMVdcg].
The structure of the proof of Theorem \[T:main\] is broken into three parts. First we give a proof of Theorem \[T:main\], up to the statement of the splitting of the inclusion, and where we focus only on the $\operatorname{Aut}(\cx/K)$-equivariance of the Abel–Jacobi map on torsion (Theorem \[T:JacDesc\]). The proof of Theorem \[T:JacDesc\] relies on showing that $\operatorname{N}^nH^{2n+1}(X_{{{\overline{K}}}},\rat_\ell(n))$ is spanned via the action of a correspondence over $K$ by the first cohomology group of a pointed, geometrically integral curve; this is proved in Proposition \[P:niveauK\]. Next, in §\[S:Tors-Gen\], we show that if the Abel–Jacobi map is $\operatorname{Aut}(\mathbb C/K)$-equivariant on torsion, then it is fully equivariant. This is a consequence of more general results we establish for surjective regular homomorphisms. Finally, the splitting of is then proved in Theorem \[T:mot\]. Note that when $n=1$ the result of [@ACMVdcg Thm. A] is more precise in that the splitting of is shown to be induced by an algebraic correspondence over $K$.
As a first application of Theorem \[T:main\], we recover a result of Deligne [@deligneniveau] regarding intermediate Jacobians of complete intersections of Hodge level $1$ (§ \[S:Deligne\]).
Another application is to the following question due to Barry Mazur. Given an effective polarizable weight-$1$ $\mathbb Q$-Hodge structure $V$, there is a complex abelian variety $A$ (determined up to isogeny) so that $H^1(A,\mathbb
Q)$ gives a weight-$1$ $\mathbb Q$-Hodge structure isomorphic to $V$. On the other hand, let $K$ be a field, and let $\ell$ be a prime number (not equal to the characteristic of the field). It is not known (even for $K=\mathbb Q$) whether given an effective polarizable weight-$1$ $\operatorname{Gal}(K)$-representation $V_\ell$ over $ \mathbb Q_\ell$, there is an abelian variety $A/K$ such that $H^1(A_{{{\overline{K}}}},\mathbb Q_\ell)$ is isomorphic to $V_\ell$. A *phantom abelian variety for $V_\ell$* is an abelian variety $A/K$ together with an isomorphism of $\operatorname{Gal}(K)$-representations $$\xymatrix{
H^1(A_{{{\overline{K}}}},\mathbb Q_\ell)\ar[r]^<>(0.5){\cong} & V_\ell.
}$$ Such an abelian variety, if it exists, is determined up to isogeny; this is called the *phantom isogeny class for $V_\ell$*. Mazur asks the following question [@mazurprobICCM p.38] : *Let $X$ be a smooth projective variety over a field $K\subseteq \cx$, and let $n$ be a nonnegative integer. If $H^{2n+1}(X_{\mathbb C},\mathbb Q)$ has Hodge coniveau $n$ (i.e., $H^{2n+1}(X_{\mathbb C},\mathbb
C)=H^{n,n+1}(X) \oplus H^{n+1,n}(X)$), does there exist a phantom abelian variety for $H^{2n+1}(X_{{{\overline{K}}}},\mathbb Q_\ell(n))$?*
Theorem \[T:main\] answers this affirmatively under the stronger, but according to the generalized Hodge conjecture equivalent, assumption that the Abel–Jacobi map $AJ :
\operatorname{A}^{n+1}(X_\cx) \to J^{2n+1}(X_\cx)$ is surjective. This assumption is known to hold in many cases (e.g., uniruled threefolds). Theorem \[T:main\] in fact shows a stronger statement, namely that the *distinguished* model over $K$ of the image of the Abel–Jacobi map $AJ :
\operatorname{A}^{n+1}(X_\cx) \to J^{2n+1}(X_\cx)$ provides a phantom abelian variety for the $\operatorname{Gal}(K)$-representation $\operatorname{N}^nH^{2n+1}(X_{{{\overline{K}}}},\rat_\ell(n))$. Moreover, the arguments via motivated cycles of Section \[S:Andre\] give a second proof of the existence of a phantom abelian variety, although not the descent of the image of the Abel–Jacobi map. In summary, these results strengthen our answer to Mazur’s question, given in [@ACMVdcg].
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As another application, we provide an answer to a second question of Mazur, which was not addressed in [@ACMVdcg]. Over the complex numbers the image of the Abel–Jacobi map is dominated by Albaneses of resolutions of singularities of products of irreducible components of Hilbert schemes. Since Hilbert schemes are functorial, and in particular defined over $K$, and since the image of the Abel–Jacobi map descends to $K$, one might expect the phantom abelian variety to be linked to the Albanese of a Hilbert scheme. Motivated by concrete examples where this holds (e.g., the intermediate Jacobian of a smooth cubic threefold $X$ is the Albanese variety of the Fano variety of lines on $X$ [@CG]), Mazur asks the following question [@mazurprobICCM Que. 1]: *Can this phantom abelian variety be constructed as – or at least in terms of – the Albanese variety of some Hilbert scheme geometrically attached to $X$?* We provide an affirmative answer for $\operatorname{N}^nH^{2n+1}(X_{{{\overline{K}}}},\rat_\ell(n))$:
\[Ta:MazQ1\] Suppose $X$ is a smooth projective variety over a field $K\subseteq \cx$. Then the phantom abelian variety $J/K$ for $\operatorname{N}^nH^{2n+1}(X_{{{\overline{K}}}},\rat_\ell(n))$ given in Theorem \[T:main\] is dominated by the Albanese variety of (a finite product of resolutions of singularities of some finite number of) components of a Hilbert scheme parameterizing codimension-$(n+1)$ subschemes of $X$ over $K$.
The proof of the theorem, given in §\[S:Hilb\] (Theorem \[T:MazQ1\]), uses in an essential way the $\operatorname{Aut}(\cx/K)$-equivariance of the Abel–Jacobi map as stated in Theorem \[T:main\]. For the sake of generality, the proof is framed in the language of Galois-equivariant regular homomorphisms, as described in [@ACMVdcg §4]. As a consequence, some related results are obtained for algebraic representatives of smooth projective varieties over perfect fields of arbitrary characteristic.
For concreteness, we mention the following consequence of Theorems \[T:main\] and \[Ta:MazQ1\], providing a complete answer to Mazur’s questions for uniruled threefolds (see §\[S:Hilb\]):
\[Ca:MazQ1uni\] Suppose $X$ is a smooth projective threefold over a field $K\subseteq \mathbb
C$ and assume that $X_\cx$ is uniruled. Then the intermediate Jacobian $J^3(X_{\mathbb C})$ descends to an abelian variety $J/K$, which is a phantom abelian variety for $H^{3}(X_{{{\overline{K}}}},\rat_\ell(1))$, and is dominated by the Albanese variety of (a product of resolutions of singularities of a finite number of) components of a Hilbert scheme parameterizing dimension-$1$ subschemes of $X$ over $K$.
.2cm
**Acknowledgments.** We would like to thank Ofer Gabber for comments that were instrumental in arriving at Theorem \[T:main\]. We also thank the referee for helpful suggestions.
.2 cm
**Conventions.** We use the same conventions as in [@ACMVdcg]. A *variety* over a field is a geometrically reduced separated scheme of finite type over that field. A *curve* (resp. *surface*) is a variety of pure dimension $1$ (resp. $2$). Given a variety $X$, $\operatorname{CH}^i(X)$ denotes the Chow group of codimension $i$ cycles modulo rational equivalence, and $\operatorname{A}^i(X)\subseteq \operatorname{CH}^i(X)$ denotes the subgroup of cycles algebraically equivalent to $0$. If $X$ is a smooth projective variety over the complex numbers, then we denote by $J^{2n+1}(X) =
\operatorname{F}^{n+1}H^{2n+1}(X,\cx)\backslash H^{2n+1}(X,\cx)/H^{2n+1}(X,\integ)$ the complex torus that is the $(2n+1)$-th intermediate Jacobian of $X$, and we denote $J_a^{2n+1}(X)$ the image of the Abel–Jacobi map $\operatorname{A}^{n+1}(X) \to
J^{2n+1}(X)$. A choice of polarization on $X$ naturally endows the complex torus $J_a^{2n+1}(X)$ with the structure of a polarized complex abelian variety, and $H^1(J_a^{2n+1}(X),\rat) \iso
\operatorname{N}^nH^{2n+1}(X,\rat)(n)$. If $C/K$ is a smooth projective geometrically irreducible curve over a field, we will sometimes write $J(C)$ for $\operatorname{Pic}^\circ_{C/K}$. Given a field $K$, we denote by ${{\overline{K}}}$ a separable closure. Finally, given an abelian group $A$, we denote by $A[N]$ the kernel of the multiplication-by-$N$ map; and if $A$ is an abelian group scheme over a field $K$, we write $A[N]$ for $A({{\overline{K}}})[N]$.
A result on cohomology
======================
The main point of this section is to prove Proposition \[P:niveauK\], strengthening [@ACMVdcg Prop. 1.3]. Recall that if $X$ is a smooth projective variety over a field $K$, then the geometric coniveau filtration $\operatorname{N}^\nu H^{i}(X_{{{\overline{K}}}},\rat_\ell)$ is defined by:
$$\label{E:coniveau}
\operatorname{N}^\nu H^{i}(X_{{{\overline{K}}}},\rat_\ell) := \sum_{\substack{ Z\subseteq
X\\\text{closed, codim }\ge \nu}} \operatorname{ker}\left(
H^{i}(X_{{{\overline{K}}}},\rat_\ell) \rightarrow
H^{i}(X_{{{\overline{K}}}}
\backslash Z_{{{\overline{K}}}}, \rat_\ell)\right).$$
If $K=\mathbb C$, the geometric coniveau filtration $\operatorname{N}^\nu
H^{i}(X,\rat)$ is defined similarly. We direct the reader to [@ACMVdcg § 1.2] for a review of some of the properties we use here. Sometimes, we will abuse notation slightly and denote the $r$-th Tate twist of step $\nu$ in the geometric coniveau filtration by $\operatorname{N}^\nu H^{i}(X_{{{\overline{K}}}},\rat_\ell(r)):=(\operatorname{N}^\nu H^{i}(X_{{{\overline{K}}}},\rat_\ell))\otimes_{\mathbb Q_\ell} \mathbb Q_\ell(r)$, and similarly for Betti cohomology.
\[P:niveauK\] Suppose $X$ is a smooth projective variety over a field $K\subseteq \cx$, and let $n$ be a nonnegative integer. Then there exist a geometrically integral smooth projective curve $C$ over $K$, admitting a $K$-point, and a correspondence $\gamma \in
\operatorname{CH}^{n+1}(C\times_K
X)_{\mathbb{Q}}$ such that for all primes $\ell$, the induced morphism of $\operatorname{Gal}(K)$-representations $$\gamma_*: H^1(C_{\overline K},{\mathbb{Q}}_\ell) \rightarrow
H^{2n+1}(X_{\overline K},{\mathbb{Q}}_\ell(n))$$ has image $\operatorname{N}^nH^{2n+1}(X_{\overline K},{\mathbb{Q}}_\ell(n))$. Likewise, the morphism of Hodge structures $$\gamma_*: H^1(C_{\mathbb{C}},{\mathbb{Q}}) \rightarrow
H^{2n+1}(X_{\mathbb{C}},{\mathbb{Q}}(n))$$ has image $\operatorname{N}^nH^{2n+1}(X_{\mathbb{C}},{\mathbb{Q}}(n))$; in particular, the image of $\gamma_*
: J(C_\cx) \to J^{2n+1}(X_\cx)$ is $J^{2n+1}_a(X_\cx)$.
\[R:P:niveau\] The result [@ACMVdcg Prop. 1.3] differs from Proposition \[P:niveauK\] only in the sense that is it not shown there that $C$ can be taken to admit a $K$-rational point or to be geometrically integral.
There are three main ingredients in the proof of Proposition \[P:niveauK\]: the Bertini theorems, the Lefschetz type result in Lemma \[L:ladicLef\] below describing cohomology in degree $1$, and Proposition \[P:curveCoh\] regarding the cohomology of curves. While we expect Proposition \[P:curveCoh\] is well-known, for lack of a reference we include a proof in Appendix \[A:Appendix\].
\[L:ladicLef\] Suppose $X$ is a smooth projective variety over a field $K$ with separable closure $\overline K$. There exist a smooth curve $C\hookrightarrow X$ over $K$, which is a (general) linear section for an appropriate projective embedding of $X$, and a correspondence $\gamma \in
\operatorname{CH}^{1}(C\times_K
X)_{\mathbb{Q}}$ such that for all $\ell\not = \operatorname{char}(K)$, the induced morphism of $\operatorname{Gal}(K)$-representations $$\xymatrix{
\gamma_*: H^1(C_{\overline K},{\mathbb{Q}}_\ell) \ar@{->>}[r]&
H^{1}(X_{\overline K},{\mathbb{Q}}_\ell)
}$$ is surjective. Moreover, if $X$ is geometrically integral (resp. admits a $K$-point), then $C$ can be taken to be geometrically integral (resp. to admit a $K$-point).
By Bertini [@poonen], let $\iota : C \hookrightarrow X$ be a one-dimensional smooth general linear section of an appropriate projective embedding of $X$. Note that by the irreducible Bertini theorems [@charlespoonen], if $X$ is geometrically integral (resp. admits a $K$-point), then $C$ can also be taken to be geometrically integral (resp. to admit a $K$-point); (see e.g., [@ACMV Thm. B.1] for the version we use here).
The hard Lefschetz theorem [@deligneweil2 Thm. 4.1.1] states that intersecting with $C$ yields an isomorphism $$\iota_*\iota^*:H^1(X_{{{\overline{K}}}},\mathbb Q_\ell) \hookrightarrow H^1(C_{{{\overline{K}}}},\mathbb
Q_\ell) \twoheadrightarrow H^{2\dim X-1}(X_{{{\overline{K}}}},\mathbb Q_\ell(\dim X-1)).$$ The Lefschetz Standard Conjecture is known for $\ell$-adic cohomology and for Betti cohomology in degree $\leq 1$ (see [@kleiman Thm. 2A9(5)]), meaning in our case that the map $\left(\iota_*\iota^*\right)^{-1}$ is induced by a correspondence, say $\Lambda \in \operatorname{CH}^1(X\times_K
X)_\rat$. Therefore, the composition $$\label{E:LCoNi}
\xymatrix@C=2em{
H^1(C_{{{\overline{K}}}},\mathbb Q_\ell)
\ar@{->>}[r]_<>(0.5){\iota_*}^<>(0.5){(\Gamma_{\iota})_*} &
H^{2\dim X-1}(X_{{{\overline{K}}}},\mathbb Q_\ell(\dim X-1)) \ar@{->}[r]^<>(0.5){
\Lambda_*}_<>(0.5)\cong & H^1(X_{{{\overline{K}}}},\mathbb Q_\ell)
}$$ is surjective and is induced by the correspondence $\gamma := \Lambda \circ
\Gamma_\iota$, where $\Gamma_\iota$ denotes the graph of $\iota$ .
Up to working component-wise, we can and do assume that $X$ is irreducible, say of dimension $d_X$. Since $K\subseteq \mathbb C$, we have from the characterization of coniveau (see e.g., [@ACMVdcg (1.2)]) that there exist a smooth projective variety $Y$ (possibly disconnected) of pure dimension $d_Y=d_X-n$ over $K$, and a $K$-morphism $f:Y\to X$ such that $$\operatorname{N}^nH^{2n+1}(X_{{{\overline{K}}}},{\mathbb{Q}}_\ell(n)) =
\mathrm{Im}\left(
f_*:H^1({Y}_{{{\overline{K}}}},{\mathbb{Q}}_\ell)
\rightarrow
H^{2n+1}(X_{{{\overline{K}}}},{\mathbb{Q}}_\ell(n))\right).$$ Using Lemma \[L:ladicLef\] applied to $Y$, there exist a smooth projective curve $C$ over $K$ (possibly disconnected) and a correspondence $\Gamma\in
\operatorname{CH}^1(C\times_KY)_{\mathbb Q}$ such that the composition $$\xymatrix@C=2em{
H^1(C_{{{\overline{K}}}},\mathbb Q_\ell) \ar@{->>}[r]^<>(0.5){\Gamma_*} & H^1(Y_{{{\overline{K}}}},\mathbb Q_\ell)
\ar[r]^<>(0.5){f_*} & H^{2n+1}(X_{{{\overline{K}}}},\mathbb Q_\ell(n))\\
}$$ has image $\operatorname{N}^nH^{2n+1}(X_{{{\overline{K}}}},{\mathbb{Q}}_\ell(n))$.
As recalled in Proposition \[P:curveCoh\], there is a morphism $\beta:C\to \operatorname{Pic}^\circ_{C/K}$ inducing an isomorphism $\beta^*=(\Gamma_\beta^t)_*:H^1(\operatorname{Pic}^\circ_{C_{{{\overline{K}}}}/{{\overline{K}}}},\mathbb Q_\ell)\to
H^1(C_{{{\overline{K}}}},\mathbb Q_\ell)$. Observe that $\operatorname{Pic}^\circ_{C/K}$ is geometrically integral and admits a $K$-point. Lemma \[L:ladicLef\] yields a smooth geometrically integral curve $D/K$ endowed with a $K$-point, and a surjection $
H^1(D_{{{\overline{K}}}},\mathbb Q_\ell)\twoheadrightarrow
H^1(\operatorname{Pic}^\circ_{C_{{{\overline{K}}}}/{{\overline{K}}}},\mathbb Q_\ell)
$ induced by a correspondence $\widetilde \Gamma$ over $K$. The composition $$\xymatrix@C=1.4em{
H^1(D_{{{\overline{K}}}},\mathbb Q_\ell) \ar@{->>}[r]^<>(0.5){\widetilde
\Gamma_*}_<>(0.5){\operatorname{Lef}} & H^1(\operatorname{Pic}^\circ_{C_{{{\overline{K}}}}/{{\overline{K}}}},\mathbb Q_\ell)
\ar[r]^<>(0.5){(\Gamma_\beta^t)_*}_<>(0.5){\cong}&H^1(C_{{{\overline{K}}}},\mathbb Q_\ell)
\ar@{->>}[r]^<>(0.5){\Gamma_*}_<>(0.5){\operatorname{Lef}} & H^1(Y_{{{\overline{K}}}},\mathbb Q_\ell)
\ar@{->>}[r]^<>(0.5){f_*}_<>(0.5){\operatorname{Def}} &
\operatorname{N}^nH^{2n+1}(X_{{{\overline{K}}}},\mathbb
Q_\ell(n)),
}$$ induced by the associated composition of correspondences $\gamma$, provides the desired surjection $$\xymatrix{\gamma_*:H^1(D_{{{\overline{K}}}},\mathbb Q_\ell) \ar@{->>}[r]&
\operatorname{N}^n H^{2n+1}(X_{{{\overline{K}}}},\mathbb Q_\ell(n))}.$$
Finally, the compatibility of the comparison isomorphisms in cohomology with Gysin maps and the action of correspondences (see e.g., [@ACMVdcg §1.2]), or simply rehashing the argument above after pull-back to $\mathbb C$, establishes that the image of the induced morphism of Hodge structures $
\gamma_*:
H^1(D_{\mathbb C},{\mathbb{Q}})
\rightarrow
H^{2n+1}(X_{\mathbb C},{\mathbb{Q}}(n))
$ is $\operatorname{N}^nH^{2n+1}(X_{\mathbb C},{\mathbb{Q}}(n)) =
H^1(J^{2n+1}_a(X_{\mathbb C},\rat))$. Using the equivalence of categories between polarizable effective weight one Hodge structures and complex abelian varieties, we see that this morphism of Hodge structures is induced by a surjection of abelian varieties $\gamma_*:
J(D_\cx) \twoheadrightarrow J^{2n+1}_a(X_\cx)$.
Proof of Theorem \[T:main\]: Part I, descent of the image of the Abel–Jacobi map
================================================================================
In this section we establish the following theorem, proving the first part of Theorem \[T:main\].
\[T:JacDesc\] Suppose $X$ is a smooth projective variety over a field $K\subseteq \mathbb C$, and let $n$ be a nonnegative integer. Then the image of the Abel–Jacobi map $J^{2n+1}_a(X_\mathbb C)$ admits a distinguished model $J$ over $K$ such that the induced map $AJ[N]:\operatorname{A}^{n+1}(X_{\mathbb C})[N]\to
J_a^{2n+1}(X_{\mathbb C})[N]$ on $N$-torsion is $\operatorname{Aut}(\cx/K)$-equivariant for all positive integers $N$.
Moreover, there is a correspondence $\Gamma\in \operatorname{CH}^{\dim
(J)+n}(J\times_K X)$ such that for each prime number $\ell$, we have that $\Gamma$ induces an inclusion of $\operatorname{Gal}(K)$-representations $$\label{E:cycles4}
\xymatrix{
H^1(J_{{{\overline{K}}}},\mathbb Q_\ell) \ar@{^(->}[r]^<>(0.5){\Gamma_*}&
H^{2n+1}(X_{{{\overline{K}}}},\mathbb Q_\ell(n)),
}$$ with image $\operatorname{N} ^n H^{2n+1}(X_{{{\overline{K}}}},\mathbb Q_\ell(n))$.
We will prove the theorem in several steps contained in the following subsections.
\[R:Distinguished\] As explained below the statement of Theorem \[T:main\], an abelian variety over $\cx$ may admit several models over $K$, if it admits any. However, it admits at most one model over $K$ such that the induced map $AJ[N] :\operatorname{A}^{n+1}(X_{\mathbb C})[N]\to
J_a^{2n+1}(X_{\mathbb C})[N]$ on $N$-torsion is $\operatorname{Aut}(\cx/K)$-equivariant for all positive integers $N$. Indeed, by Chow’s rigidity theorem (see [@conradtrace Thm. 3.19]), an abelian variety $A/\mathbb C$ descends to at most one model defined over ${{\overline{K}}}$; moreover, there is at most one model of $A$ defined over $K$ that induces a given action of $\operatorname{Gal}(K)$ on the ${{\overline{K}}}$-points of $A$. Therefore, since $AJ[N]:\operatorname{A}^{n+1}(X_{\mathbb C})[N]\to
J_a^{2n+1}(X_{\mathbb C})[N]$ is surjective for all $N$ not divisible by a finite number of fixed primes (this is a general fact about regular homomorphisms; see §\[S:Tors-Gen\] and Lemma \[L:Tor-to-Tor\](b)), and since torsion points of order not divisible by a finite number of fixed primes are dense, the abelian variety $J_a^{2n+1}(X_{\mathbb C})$ admits at most one structure of a scheme over $K$ such that $AJ[N]$ is $\operatorname{Aut}(\cx/K)$-equivariant for all positive integers $N$ not divisible by the finite number of given primes. This is the sense in which $J_a^{2n+1}(X_{\mathbb C})$ admits a distinguished model over $K$.
Chow rigidity and $L/K$-trace: descending from $\mathbb C$ to ${{\overline{K}}}$ {#S:rigidity}
--------------------------------------------------------------------------------
The first step in the proof consists in using Chow rigidity and $\mathbb C/{{\overline{K}}}$-trace to descend the image of the Abel–Jacobi map from $\mathbb C$ to ${{\overline{K}}}$. We follow the treatment in [@conradtrace], and refer the reader to [@ACMVdcg § 3.3] where we review the theory in the setting we use here.
For the convenience of the reader, we briefly recall a few points. We focus on the case where $L/K$ is an extension of algebraically closed fields of characteristic $0$. First, we reiterate that by Chow’s rigidity theorem (see [@conradtrace Thm. 3.19]), an abelian variety $B/L$ descends to at most one model, up to isomorphism, defined over $K$. Given an abelian variety $B$ defined over $L$, while $B$ need not descend to $K$, there is [@conradtrace Thm. 6.2, Thm. 6.4, Thm. 6.12, p.72, p.76, Thm. 3.19] an abelian variety ${\underline{\underline{B}}}$ defined over $K$ equipped with an injective homomorphism of abelian varieties $$\xymatrix{
{\underline{\underline{B}}}_L \ar@{^(->}[r]^\tau& B
}$$ (together called the $L/K$-trace) with the property that for any abelian variety $A/K$, base change gives an identification $$\begin{aligned}
\operatorname{Hom}_{\mathsf {Ab}/K}(A,{\underline{\underline{B}}}) &= \operatorname{Hom}_{\mathsf
{Ab}/L}(A_L,B) \\
f &\mapsto \tau \circ f_L.\end{aligned}$$ It follows that if there is an abelian variety $A/K$ and a surjective homomorphism $A_L\to
B$, then $\tau$ is surjective and hence an isomorphism; in other words, $B$ descends to $K$ (as ${\underline{\underline{B}}}$).
In the notation of Theorem \[T:JacDesc\], we wish to show that $J^{2n+1}_a(X_{\mathbb
C})$ descends to an abelian variety over ${{\overline{K}}}$. We have shown in Proposition \[P:niveauK\] that there exist a smooth projective geometrically integral curve $C/K$, admitting a $K$-point, and a correspondence $\gamma \in
\operatorname{CH}^{n+1}(C\times_KX)_{\mathbb{Q}}$ which induces a surjection $\gamma_*:J(C_{\mathbb C}) \twoheadrightarrow J^{2n+1}_a(X_{\mathbb C})$. Thus from the theory of the ($\mathbb C/{{\overline{K}}}$)-trace, and the fact that $J(C_{\mathbb C})= J(C_{{{\overline{K}}}})_{\mathbb C}$ is defined over ${{\overline{K}}}$, $J^{2n+1}_a(X_{\mathbb C})$ descends to ${{\overline{K}}}$ as its ($\mathbb C/{{\overline{K}}}$)-trace $ {\underline{\underline{J}}}^{2n+1}_a(X_{\mathbb C})$, and there is a surjective homomorphism of abelian varieties over ${{\overline{K}}}$ $$\xymatrix{
J(C_{{{\overline{K}}}}) \ar@{->>}[r]^<>(0.5){{\underline{\underline{\gamma_*}}}}& {\underline{\underline{J}}}^{2n+1}_a(X_{\mathbb C}).
}$$ Moreover, the Abel–Jacobi map on torsion $AJ[N] :
\operatorname{A}^{n+1}(X_{\cx})[N] \to J_a^{2n+1}(X_\cx)[N]$ is $\operatorname{Aut}(\cx/{{\overline{K}}})$-equivariant for all positive integers $N$. Indeed, $\operatorname{Aut}(\cx/{{\overline{K}}})$ acts trivially on $J_a^{2n+1}(X_\cx)[N]={\underline{\underline{J}}}^{2n+1}_a(X_{\mathbb C})[N]$ and it also acts trivially on $\operatorname{A}^{n+1}(X_{\cx})[N]$ by Lecomte’s rigidity theorem [@lecomte86] (see e.g., [@ACMVdcg Thm. 3.8(b)]).
Descending from ${{\overline{K}}}$ to $K$ {#S:algdescent}
-----------------------------------------
In the notation of Theorem \[T:JacDesc\], we have found a smooth projective geometrically integral curve $C/K$, admitting a $K$-point, and a correspondence $\gamma \in \operatorname{CH}^{n+1}(C\times_KX)_{\mathbb{Q}}$ inducing a surjective homomorphism of abelian varieties over ${{\overline{K}}}$ $$\label{E:Pdef}
\xymatrix{
0\ar[r]& P \ar[r]& J(C_{{{\overline{K}}}}) \ar@{->>}[r]^<>(0.5){{\underline{\underline{\gamma_*}}}}&
{\underline{\underline{J}}}^{2n+1}_a(X_{\mathbb C}) \ar[r] & 0
}$$
where $P$ is defined to be the kernel. We will show that $ {\underline{\underline{J}}}^{2n+1}_a(X_{\mathbb C})$ descends to an abelian variety $J$ over $K$ by showing that $P$ descends to $K$, using the following elementary criterion:
\[L:torsdescends\] Let $A/K$ be an abelian variety over a perfect field $K$, let $\Omega/K$ be an algebraically closed extension field, and let ${{\overline{A}}} = A_{\Omega}$. Suppose that ${{\overline{B}}}
\subset {{\overline{A}}}$ is a closed sub-group scheme. Then ${{\overline{B}}}_{{\rm
red}}$ descends to a sub-group scheme over $K$ if and only if, for each natural number $N$, we have ${{\overline{B}}}[N](\Omega)$ is stable under $\operatorname{Aut}(\Omega/K)$.
It is well-known that, since the fixed field of $\Omega$ under $\operatorname{Aut}(\Omega/K)$ is $K$ itself, a subvariety $W$ of ${{\overline{A}}}$ descends to $K$ if and only if $W(\Omega)$ is stable under $\operatorname{Aut}(\Omega/K)$ (e.g., [@milneAG Prop. 6.8]). In fact, to show $W$ descends to $K$ it suffices to verify that there is a Zariski-dense subset $S \subset W(\Omega)$ which is stable under $\operatorname{Aut}(\Omega/K)$. (Indeed, if $\sigma \in \operatorname{Aut}(\Omega/K)$, then $W^\sigma$ contains the Zariski closure of $S^\sigma$, which is $W$ itself.) Now use the fact that, over an algebraically closed field, torsion points are Zariski dense in any abelian variety or étale group scheme.
We wish to show that the abelian variety ${\underline{\underline{J}}}^{2n+1}_a(X_{\mathbb C})$ over ${{\overline{K}}}$, obtained in Step 1 of the proof, descends to an abelian variety over $K$. In the notation of Step 1, let $P$ be the kernel of ${\underline{\underline{\gamma_*}}}$, as in . We use the criterion of Lemma \[L:torsdescends\] to show that $P$ descends to $K$. To this end, let $N$ be a natural number. We have a commutative diagram of abelian groups:
$$\label{E:DescBigD}
\vcenter{\xymatrix@R=.6em@C=1em{
&P[N] \ar@{^(->}[dd]\\
P_\cx[N] \ar@{=}[ur] \ar@{^(->}[dd] &\\
& J(C_{{{\overline{K}}}})[N] \ar[rr]^{\simeq} \ar@{->}'[d]^{{\underline{\underline{\gamma}}}_{*,N}}[dd]&&H^1_{\text{\'et}}(C_{{{\overline{K}}}},\mmu_N) \ar[dd]^{\gamma_{*,N}} \\
J(C_\cx)[N] \ar[dd]_{\gamma_{*,N}} \ar@{=}[ur]\ar[rr]_(0.6){\simeq} &&
H^1_{\text{an}}(C_\cx,\mmu_N)\ar@{=}[ur] \ar[dd]_{\gamma_{*,N}}\\
& {\underline{\underline{J_a}}}^{2n+1}(X_\cx)[N]&&H^{2n+1}_{\text{\'et}}(X_{{{\overline{K}}}},\mmu_N^{\otimes(n+1)}) \\
J_a^{2n+1}(X_\cx)[N] \ar@{=}[ur] \ar@{^(->}[r] & J^{2n+1}(X_\cx)[N]
\ar@{=}[r ] & H^{2n+1}_{\text{an}}(X_\cx,\mmu_N^{\otimes(n+1)})\ar@{=}[ur]
}}$$
Here, the identification $J^{2n+1}(X_\cx)[N]= H^{2n+1}_{\text{an}}(X_\cx,\mmu_N^{\otimes(n+1)})$ is given by the definition of the intermediate Jacobian $J^{2n+1}(X_\cx)$, since $H^{2n+1}_{\text{an}}(X_\cx,\mmu_N^{\otimes(n+1)})=H^{2n+1}_{\text{an}}(X_\cx,\mathbb Z/n\mathbb Z)=H^{2n+1}_{\text{an}}(X_\cx,\frac{1}{n}\mathbb Z/\mathbb Z)$. The key point then is that, by commutativity, the composition of arrows along the back of the diagram $$\label{E:back}
\xymatrix{
J(C_{{{\overline{K}}}})[N] \ar[r]^<>(0.5)\simeq & H_{\text{\'et}}^1(C_{{{\overline{K}}}},\mmu_{N}) \ar[r]^<>(0.5){\gamma_{*,N}} &
H_{\text{\'et}}^{2n+1}(X_{{{\overline{K}}}},\mmu_{N}^{\otimes {(n+1)}})
}$$ has the same kernel as the arrow ${\underline{\underline{\gamma}}}_{*,N}$, namely $P[N]$. Moreover, each arrow of the composition is $\operatorname{Gal}(K)$-equivariant. Therefore, $P[N] = \operatorname{ker}{\underline{\underline{\gamma}}}_{*,N}$ is $\operatorname{Gal}(K)$-stable for each $N$, and $P$ descends to $K$. Therefore the abelian variety ${\underline{\underline{J}}}^{2n+1}_a(X_{\mathbb C})$ over ${{\overline{K}}}$ admits a model $J$ over $K$ such that the ${{\overline{K}}}$-homomorphism ${\underline{\underline{\gamma_*}}} : J(C_{{{\overline{K}}}}) \to {\underline{\underline{J}}}^{2n+1}_a(X_{\mathbb C})$ descends to a $K$-homomorphism $f : J(C) \to J$.
The Abel–Jacobi map is Galois-equivariant on torsion
----------------------------------------------------
In the notation of Theorem \[T:JacDesc\], we have so far established that $J^{2n+1}_a(X_{\mathbb C})$ descends to an abelian variety $J$ over $K$. We now wish to show that with respect to this given structure as a $K$-scheme, the Abel–Jacobi map on torsion $$\label{E:AJtorsion}
\xymatrix{AJ :
\operatorname{A}^{n+1}(X_{\cx})[N] \ar[r]& J^{2n+1}_a(X_\cx)[N] = J[N]}$$ is $\operatorname{Aut}(\cx/K)$-equivariant. In Step 1, we already showed that $AJ$ is $\operatorname{Aut}(\cx/{{\overline{K}}})$-equivariant when restricted to torsion. Therefore, in order to conclude, it only remains to prove that the map ${\underline{\underline{AJ}}} : \operatorname{A}^{n+1}(X_{{{\overline{K}}}})[N] \to
J[N]$ is $\operatorname{Gal}({{\overline{K}}}/K)$-equivariant.
.2 cm For future reference, we have the following elementary lemma.
\[L:ez-emma\] Let $G$ be a group and let $A,B,C$ be $G$-modules. Let $\phi : A \to B$ and $ \psi : B \to C$ be homomorphisms of abelian groups. We have:
If $\phi$ is surjective and if $\phi$ and $\psi\circ \phi $ are $G$-equivariant, then $\psi$ is $G$-equivariant.
If $\psi $ is injective and if $\psi$ and $\psi\circ \phi $ are $G$-equivariant, then $\phi$ is $G$-equivariant.
Fix $J/K$ to be the model of $J^{2n+1}_a(X_{\mathbb C})$ from Step 2. We wish to show that for any positive integer $N$, the restriction of the Abel–Jacobi map to $N$-torsion is $\operatorname{Aut}(\cx/K)$-equivariant. As mentioned above, it only remains to prove that the map ${\underline{\underline{AJ}}} :
\operatorname{A}^{n+1}(X_{{{\overline{K}}}})[N] \to {\underline{\underline{J}}}^{2n+1}_a(X_\cx)[N]$ is $\operatorname{Gal}({{\overline{K}}}/K)$-equivariant.
For this, observe that the Bloch map $\lambda^{n+1} :
\operatorname{A}^{n+1}(X_{{{\overline{K}}}})[N] \longrightarrow
H_{\text{\'et}}^{2n+1}(X_{{{\overline{K}}}}, \mmu_N^{\otimes (n+1)})$ is Galois-equivariant, since it is constructed via natural maps of sheaves, all of which have natural Galois actions. Moreover, on torsion the Bloch map factors through the Abel–Jacobi map. Indeed, over $\cx$, we have [@bloch79 Prop. 3.7] $$\xymatrix@C=3em {
\lambda^{n+1} : \operatorname{A}^{n+1}(X_{\cx})[N] \ar[r]^<>(0.5){{AJ}[N]} &
J_{\cx}[N] \ar@{^(->}[r]& H_{\text{\'et}}^{2n+1}(X_{\cx}, \mmu_N^{\otimes
(n+1)}).
}$$ Using rigidity for torsion cycles [@lecomte86], rigidity for torsion on abelian varieties, and proper base change, we obtain the analogous statement over ${{\overline{K}}}$: $$\xymatrix@C=3em {
\lambda^{n+1} : \operatorname{A}^{n+1}(X_{{{\overline{K}}}})[N] \ar[r]^<>(0.5){{\underline{\underline{AJ}}}[N]} &
J_{{{\overline{K}}}}[N] \ar@{^(->}[r]& H_{\text{\'et}}^{2n+1}(X_{{{\overline{K}}}}, \mmu_N^{\otimes
(n+1)}).
}$$ As described in , the inclusion $J_{{{\overline{K}}}}[N]
\hookrightarrow H_{\text{\'et}}^{2n+1}(X_{{{\overline{K}}}}, \mmu_N^{\otimes (n+1)})$ is also Galois-equivariant. By Lemma \[L:ez-emma\](b), we find that $
{\underline{\underline{AJ}}}[N]$ is Galois-equivariant.
The Galois representation
-------------------------
We now conclude the proof of Theorem \[T:JacDesc\] by constructing the correspondence $\Gamma\in \operatorname{CH}^{\dim (J)+n}(J\times_K X)$ inducing the desired morphism of Galois representations.
Let $J/K$ be the model of $J^{2n+1}_a(X_{\mathbb C})$ from Step 2. (which was shown to be distinguished in Step 3; see Remark \[R:Distinguished\]). We will now construct a correspondence $\Gamma\in \operatorname{CH}^{\dim
(J)+n}(J\times_K X)$ such that for each prime number $\ell$, the correspondence $\Gamma$ induces an inclusion of $\operatorname{Gal}(K)$-representations $$\xymatrix{
H^1(J_{{{\overline{K}}}},\mathbb Q_\ell) \ar@{^(->}[r]^<>(0.5){\Gamma_*}&
H^{2n+1}(X_{{{\overline{K}}}},\mathbb Q_\ell(n)),
}$$ with image $\operatorname{N} ^n H^{2n+1}(X_{{{\overline{K}}}},\mathbb Q_\ell(n))$.
Let $C$ and $\gamma \in \operatorname{CH}^{n+1}(C\times_K
X)_{\mathbb{Q}}$ be the smooth, geometrically integral, pointed projective curve and the correspondence provided by Proposition \[P:niveauK\]. As we have seen (in Steps 1 and 2 of the proof of Theorem \[T:JacDesc\]), $\gamma$ induces a surjective homomorphism of complex abelian varieties $J(C_\cx) \to J_a^{2n+1}(X_\cx)$ that descends to a homomorphism $f : J(C) \to J$ of abelian varieties defined over $K$. Consider then the composite morphism $$\label{E:cycles4-5'}
\xymatrix{
H^1(J_{{{\overline{K}}}},\mathbb Q_\ell) \ar@{^(->}[r]^<>(0.5){f^*}& H^1(J(C)_{{{\overline{K}}}},\mathbb Q_\ell) \ar[r]^<>(0.5){\operatorname{alb}^*}_<>(0.5)\simeq
&H^1(C_{{{\overline{K}}}},\mathbb Q_\ell) \ar[r]^<>(0.5){\gamma_*}&H^{2n+1}(X_{{{\overline{K}}}},\mathbb Q_\ell(n)),
}$$ where $\operatorname{alb} : C \to J(C)$ denotes the Albanese morphism induced by the $K$-point of $C$. This morphism is clearly injective and induced by a correspondence on $J\times_K X$, and we claim that its image is $\operatorname{N} ^n H^{2n+1}(X_{{{\overline{K}}}},\mathbb
Q_\ell(n))$. Indeed, the complexification of together with the comparison isomorphisms yields a diagram $$\label{E:cycles4-5''}
\xymatrix{
H^1(J_{a}^{2n+1}(X_\cx),\mathbb Q) \ar@{^(->}[r]^<>(0.5){(f_\cx)^*}&
H^1(J(C)_{\cx},\mathbb Q)
\ar[r]^<>(0.5){(\operatorname{alb}_\cx)^*}_<>(0.5)\simeq &H^1(C_{\cx},\mathbb Q)
\ar[r]^<>(0.5){(\gamma_\cx)_*}&H^{2n+1}(X_{\cx},\mathbb Q(n)),
}$$ where $(\operatorname{alb}_\cx)^*\circ (f_\cx)^*$ is easily seen to be the dual (via the natural choice of polarizations) of $(\gamma_\cx)_*$. Since the Hodge structure $H^1(C_{\cx},\mathbb Q)$ is polarized by the cup-product, we conclude by [@ACMVdcg Lemma 2.3] that the image of is equal to the image of $(\gamma_\cx)_*$, that is, to $\operatorname{N} ^n
H^{2n+1}(X_{\cx},\mathbb
Q(n))$. Invoking the comparison isomorphism settles the claim.
This completes the proof of Theorem \[T:JacDesc\].
Proof of Theorem \[T:main\]: Part II, regular homomorphisms and torsion points {#S:Tors-Gen}
==============================================================================
In order to upgrade Theorem \[T:JacDesc\] to a statement about equivariance for arbitrary cycle classes, we reconsider and extend the theory of *regular homomorphisms*. Given a smooth projective complex variety $X$, a fundamental result of Griffiths [@griffiths] (and also [@griffiths68 p. 826]) is that the Abel–Jacobi map ${AJ}:\operatorname{A}^{n+1}(X)
\longrightarrow J_a^{2n+1}(X)$ is a [regular homomorphism]{}. This means that for every pair $(T,Z)$ with $T$ a pointed smooth integral complex variety, and $Z\in \operatorname{CH}^i(T\times X)$, the composition $$\begin{CD}
T(\cx)@> w_Z >> \operatorname{A}^i(X)@>\phi >>J_a^{2n+1}(X)
\end{CD}$$ is induced by a morphism of complex varieties $\psi_Z:T\to J_a^{2n+1}(X)$, where, if $t_0\in T(\cx)$ is the base point of $T$, $w_Z:T(\cx)\to \operatorname{A}^i(X)$ is given by $t\mapsto Z_t-Z_{t_0}$; here $Z_t$ is the refined Gysin fiber. Likewise, one defines regular homomorphisms for smooth projective varieties defined over an arbitrary algebraically closed field. We direct the reader to [@ACMVdcg §3] for a review of the material we use here on regular homomorphisms and *algebraic representatives*, and to [@ACMVdcg §4] for the notion of a *Galois-equivariant regular homomorphism*. In this section we provide some results regarding equivariance of regular homomorphisms; the main results are Propositions \[P:Tors-Gen’\] and \[P:Tors-Gen\].
Preliminaries
-------------
We will utilize the following facts:
\[P:algcycles\] Let $X/K$ be a scheme of finite type over a perfect field $K$. If $\alpha \in \operatorname{CH}^i(X_{{{\overline{K}}}})$ is algebraically trivial, then there exist an abelian variety $A/K$, a cycle $Z \in \operatorname{CH}^i( A\cross_K
X)$, and a ${{\overline{K}}}$-point $t\in
A({{\overline{K}}})$ such that $\alpha = Z_{t}-Z_{0}$.
We have shown in [@ACMV Thm. 2] that there exist an abelian variety $ A'/K$, a cycle $ Z'$ on $ A'\cross_K
X$, and a pair of ${{\overline{K}}} $-points $t_1,t_0 \in A'({{\overline{K}}})$ such that $\alpha = Z'_{t_1} -
Z'_{t_0}$. Let $p_{13},p_{23}: A'\cross_{K} A' \cross_{K} X
\to A' \cross_{K} X$ be the obvious projections. Let ${Z}$ be defined as the cycle $
{Z}:= p_{13}^*Z'-p_{23}^*Z'
$ on $ A'\times_{K} A' \times_{K} X$. For points $t_1,t_0\in \underline A'({{{\overline{K}}}})$, we have $
{Z}_{(t_1,t_0)} = Z'_{t_1} - Z'_{t_0}$. Thus setting $ A= A'\times_K A'$, we are done.
\[L:Tor-to-Tor\] Let $X$ be a scheme of finite type over an algebraically closed field $k$, and let $A/k$ be an abelian variety.
Let $Z\in \operatorname{CH}^i(A\times_k X)$. The map $w_Z : A(k) \to \operatorname{A}^i(X)$ is a homomorphism on torsion; more precisely, for each natural number $N$, $w_Z$ restricted to $A(k)[N]$ gives a homomorphism $w_Z[N]:A(k)[N]\to \operatorname{A}^i(X)[N]$.
Let $\phi:\operatorname{A}^i(X)\twoheadrightarrow A(k)$ be a surjective regular homomorphism. There exists a natural number $r$ such that for any natural number $N$ coprime to $r$, $\phi$ is surjective on $N$-torsion; i.e., $\phi[N]:\operatorname{A}^i(X)[N]\twoheadrightarrow A(k)[N]$ is surjective.
\(a) Since $w_Z$ factors as $A(k) \stackrel{\tau}{\longrightarrow} \operatorname{A}_0(A) \stackrel{Z_*}{\longrightarrow}
\operatorname{A}^i(X)$, where $\tau(a):=[a]-[0]$, and $Z_*$ is the group homomorphism induced by the action of the correspondence $Z$, it suffices to observe that $\tau$ is a homomorphism on torsion. In fact, $\tau$ is an isomorphism on torsion [@beauvillefourier Prop. 11, Lem. p.259] (which is based on [@bloch76 Thm. (0.1)] and [@roitman80]).
\(b) By [@murre83 Cor. 1.6.3] (see also [@ACMVdcg Lem. 4.9]) there exists a $Z\in \operatorname{CH}^i(A\times_k X)$ so that the composition $\psi_Z:A(k) \stackrel{w_Z}{\longrightarrow} \operatorname{A}^i(X) \stackrel{\phi}{\longrightarrow}A(k)$ is induced by $r\cdot \operatorname{Id}_A$ for some integer $r$. Let $N$ be any natural number coprime to $r$. Then $\psi_Z[N]$ is surjective, and therefore it follows from (a) that $\phi[N]$ is surjective.
Note that the proof of Lemma \[L:Tor-to-Tor\](b) actually shows that for all $N$, we have a surjection $\operatorname{A}^i(X)[rN] \to A(k)[N]$. In particular, a surjective regular homomorphism $\phi:\operatorname{A}^i(X)\twoheadrightarrow A(k)$ (*e.g.* the Abel–Jacobi map) induces a surjective homomorphism $\operatorname{A}^i(X)_{tors}\twoheadrightarrow A(k)_{tors}$ on torsion.
Algebraically closed base change and equivariance of regular homomorphisms
--------------------------------------------------------------------------
In this section we will utilize traces for algebraically closed field extensions in arbitrary characteristic. The main results of this paper focus on the characteristic $0$ case, which we reviewed in §\[S:rigidity\]. The properties of the trace that we utilize here in positive characteristic are reviewed in [@ACMVdcg §3.3.1]; the main difference is that we must potentially keep track of some purely inseparable isogenies.
\[L:T-G-AC\] Let $\Omega/k$ be an extension of algebraically closed fields, and let $X$ be a smooth projective variety over $k$. Let $A$ be an abelian variety over $\Omega$ and let $\phi: \operatorname{A}^i(X_{\Omega})\to A(\Omega)$ be a surjective regular homomorphism. Setting $\tau : {\underline{\underline{A}}}_\Omega \to A$ to be the $\Omega/k$-trace of $A$, we have that $\tau$ is a purely inseparable isogeny, which is an isomorphism in characteristic $0$. Moreover, there is a regular homomorphism $({\underline{\underline{\phi}}})_{\Omega}:\operatorname{A}^i(X_\Omega)\to {\underline{\underline{A}}}_\Omega
(\Omega)$ making the following diagram commute $$\label{E:comL}
\vcenter{ \xymatrix@R=.5cm{
\operatorname{A}^i(X_\Omega) \ar[r]^<>(0.5){({\underline{\underline{\phi}}})_{\Omega}} \ar@{=}[d] & {\underline{\underline{A}}}_\Omega(\Omega) \ar[d]_{\simeq}^{\tau(\Omega)}\\
\operatorname{A}^i(X_\Omega) \ar[r]^<>(0.5){\phi}&
A(\Omega). }}$$
Let us start by recalling some of the set-up from [@ACMVdcg Thm. 3.7]. First, consider the regular homomorphism ${\underline{\underline{\phi}}} : \operatorname{A}^i(X) \to {\underline{\underline{A}}}(k)$ constructed in Step 2 of the proof [@ACMVdcg Thm. 3.7]. It fits into a commutative diagram [@ACMVdcg (3.9)] $$\label{E:com0}
\vcenter{\xymatrix@R=.5cm{
\operatorname{A}^i(X) \ar[r]^<>(0.5){{\underline{\underline{\phi}}}} \ar[dd] _{\text{ base change}}&
{\underline{\underline{A}}}(k) \ar[d]^{\text{ base change}}\\
& {\underline{\underline{A}}}_\Omega(\Omega) \ar[d]^{\tau(\Omega)}\\
\operatorname{A}^i(X_\Omega) \ar[r]^<>(0.5){\phi}&
A(\Omega).
}}$$ Since we are assuming that $\phi: \operatorname{A}^i(X_{\Omega})\to A(\Omega)$ is surjective, Step 3 of the proof of [@ACMVdcg Thm. 3.7] yields that ${\underline{\underline{\phi}}} : \operatorname{A}^i(X) \to {\underline{\underline{A}}}(k)$ is surjective, and that $\tau : {\underline{\underline{A}}}_\Omega \rightarrow A$ is a purely inseparable isogeny, and thus an isomorphism in characteristic $0$. In particular, $\tau(\Omega) : {\underline{\underline{A}}}_\Omega(\Omega) \rightarrow A(\Omega)$ is an isomorphism.
Now consider the regular homomorphism ${\underline{\underline{\phi}}}_\Omega : \operatorname{A}^i(X_{\Omega})\to {\underline{\underline{A}}}_\Omega(\Omega)$ constructed in Step 1 of the proof of [@ACMVdcg Thm. 3.7], which by *loc. cit.* is surjective. We can therefore fill in diagram to obtain: $$\label{E:com}
\vcenter{ \xymatrix@R=.5cm{
\operatorname{A}^i(X) \ar[r]^<>(0.5){{\underline{\underline{\phi}}}} \ar[d]_{\text{ base change}}&
{\underline{\underline{A}}}(k) \ar[d]^{\text{ base change}}\\
\operatorname{A}^i(X_\Omega) \ar[r]^<>(0.5){({\underline{\underline{\phi}}})_{\Omega}} \ar@{=}[d] & {\underline{\underline{A}}}_\Omega(\Omega) \ar[d]_{\simeq}^{\tau(\Omega)}\\
\operatorname{A}^i(X_\Omega) \ar[r]^<>(0.5){\phi}&
A(\Omega). }}$$ We claim that is commutative. To start, the commutativity of the top square is established in Step 1 of the proof of [@ACMVdcg Thm. 3.7], and we have already confirmed the commutativity of the outer rectangle, above. For the bottom square we argue as follows.
By rigidity for torsion cycles on $X$ ([@jannsen15; @lecomte86]; see also [@ACMVdcg Thm. 3.8(b)]) and for torsion points on ${\underline{\underline{A}}}$, the vertical arrows in diagram are isomorphisms on torsion. A little more naively (i.e., without using [@jannsen15]), one can simply fix a prime number $\ell$ not equal to $\operatorname{char}k$, and consider torsion to be $\ell$-power torsion, and the rest of the argument goes through without change. The top square and outer rectangle are commutative, and thus is commutative on torsion. Now let $\alpha\in
\operatorname{A}^i(X_\Omega)$. By Weil [@weil54 Lem. 9] (e.g., Proposition \[P:algcycles\]) there exist an abelian variety $B/\Omega$, a cycle class $Z\in \operatorname{CH}^i(B\times_\Omega X_\Omega)$, and an $\Omega$-point $t\in B(\Omega)$ such that $\alpha=Z_t-Z_0$. Then consider the following diagram (not *a priori* commutative): $$\label{E:com1}
\vcenter{\xymatrix@R=.5cm{
B(\Omega) \ar[r]^{w_Z} \ar@{=}[d]& \operatorname{A}^i(X_\Omega) \ar[r]^<>(0.5){({\underline{\underline{\phi}}})_{\Omega}} \ar@{=}[d] & {\underline{\underline{A}}}_\Omega(\Omega) \ar[d]_{\simeq}^{\tau(\Omega)}\\
B(\Omega) \ar[r]^{w_Z}& \operatorname{A}^i(X_\Omega) \ar[r]^<>(0.5){\phi}&
A(\Omega). }}$$ The left-hand square is obviously commutative. We have shown that the right-hand square is commutative on torsion. The horizontal arrows on the left send torsion points to torsion cycle classes (Lemma \[L:Tor-to-Tor\](a)). Therefore the whole diagram is commutative on torsion. Since torsion points are Zariski dense in abelian varieties, the diagram is commutative if we replace $\operatorname{A}^i(X_\Omega)$ with $\operatorname{Im}(w_Z)$. Since $\alpha\in
\operatorname{Im}(w_Z)$, we see that $(\tau(\Omega)\circ ({\underline{\underline{\phi}}})_\Omega)(\alpha)=\phi(\alpha)$. Thus, since $\alpha$ was arbitrary, the lemma is proved.
\[P:Tors-Gen’\] Let $\Omega/k$ be an extension of algebraically closed fields of characteristic $0$, and let $X$ be a smooth projective variety over $k$. Let $A$ be an abelian variety over $\Omega$ and let $\phi: \operatorname{A}^i(X_{\Omega})\to A(\Omega)$ be a surjective regular homomorphism. Then $A$ admits a model over $k$, the $\Omega/k$-trace of $A$, such that $\phi$ is $\operatorname{Aut}(\Omega/k)$-equivariant.
This follows directly from Lemma \[L:T-G-AC\]. Indeed, by the construction of $({\underline{\underline{\phi}}})_\Omega$ in Step 1 of [@ACMVdcg Thm. 3.7], $({\underline{\underline{\phi}}})_\Omega$ is $\operatorname{Aut}(\Omega/k)$-equivariant. Then, since $\tau: {\underline{\underline{A}}}_\Omega \to A$ is an isomorphism, we are done.
More generally, if $\operatorname{char} k\ne 0$, then in the notation of Proposition \[P:Tors-Gen’\], the abelian variety $A$ admits a purely inseparable isogeny to an abelian variety over $\Omega$ that descends to $k$, namely the $\Omega/k$-trace. Moreover, under this purely inseparable isogeny, the $\Omega$-points of both abelian varieties are identified, and under the induced action of $\operatorname{Aut}(\Omega/k)$ on $A(\Omega)$, we have that $\phi$ is $\operatorname{Aut}(\Omega/k)$-equivariant.
Galois-equivariant regular homomorphisms and torsion points
-----------------------------------------------------------
The main point of this subsection is to prove Proposition \[P:Tors-Gen\]. This allows us to utilize results of [@ACMVdcg] on regular homomorphisms in the setting of torsion points. We start with the following lemma.
\[L:key\] Let $A$ be an abelian variety over a perfect field $K$ and let $\phi: \operatorname{A}^i(X_{{{\overline{K}}}})\to A({{\overline{K}}})$ be a regular homomorphism. Assume that there is a prime $\ell \not
=\operatorname{char}(K)$ such that for all positive integers $n$ we have that the map $\phi[\ell^n]: \operatorname{A}^i(X_{{{\overline{K}}}})[\ell^n]\to A[\ell^n]$ is $\operatorname{Gal}(K)$-equivariant. Let $B/K$ be an abelian variety and let $Z \in \operatorname{CH}^i(B\times_KX)$ be a cycle class. Then the induced morphism $\psi_{Z_{{{\overline{K}}}}}: B_{{{\overline{K}}}} \to A_{{{\overline{K}}}}$ is defined over $K$.
Since (geometric) $\ell$-primary torsion points are Zariski dense in the graph of $\psi_{Z_{{{\overline{K}}}}}$ inside $B\times_K A$, it suffices to show that the induced morphism $B({{\overline{K}}}) \to A({{\overline{K}}})$ is Galois-equivariant on $\ell$-primary torsion. Since the map $w_Z : B({{\overline{K}}}) \to
\operatorname{A}^i(X_{{{\overline{K}}}})$ is Galois-equivariant and since $\phi: \operatorname{A}^i(X_{{{\overline{K}}}})\to A({{\overline{K}}})$ is Galois-equivariant on $\ell^n$-torsion for all positive integers $n$, it is even enough to show that the map $w_Z : B({{\overline{K}}}) \to \operatorname{A}^i(X_{{{\overline{K}}}})$ sends torsion points of $B({{\overline{K}}})$ to torsion cycles in $\operatorname{A}^i(X_{{{\overline{K}}}})$. This is Lemma \[L:Tor-to-Tor\](a).
We can now prove:
\[P:Tors-Gen\] Let $A$ be an abelian variety over a perfect field $K$ and let $\phi: \operatorname{A}^i(X_{{{\overline{K}}}})\to A({{\overline{K}}})$ be a regular homomorphism. Assume that there is a prime $\ell \not
=\operatorname{char}(K)$ such that for all positive integers $n$ the map $\phi[\ell^n]: \operatorname{A}^i(X_{{{\overline{K}}}})[\ell^n]\to A[\ell^n]$ is $\operatorname{Gal}(K)$-equivariant. Then $\phi$ is $\operatorname{Gal}(K)$-equivariant.
Let $\alpha \in \operatorname{A}^i(X_{{{\overline{K}}}})$, and let $\sigma\in
\operatorname{Gal}(K)$. From Proposition \[P:algcycles\], we have an abelian variety $B/K$, a cycle $Z \in \operatorname{CH}^i( B\cross_K
X)$, and a ${{\overline{K}}}$-point $t\in
B({{\overline{K}}})$ such that $\alpha = Z_{t}-Z_{0}$. Now consider the following diagram (not *a priori* commutative): $$\begin{CD}
B({{\overline{K}}})@>w_{Z_{{{\overline{K}}}}} >> \operatorname{A}^i(X_{{{\overline{K}}}}) @>\phi>> A({{\overline{K}}})\\
@V\sigma_B^*VV @V\sigma_X^*VV @V\sigma_A^*VV\\
B({{\overline{K}}})@>w_{Z_{{{\overline{K}}}}} >> \operatorname{A}^i(X_{{{\overline{K}}}}) @>\phi>> A({{\overline{K}}}).\\
\end{CD}$$ Since $Z$ is defined over $K$, and the base point $0$ is defined over $K$, the left-hand square is commutative (e.g., [@ACMVdcg Rem. 4.3]). It follows from Lemma \[L:key\] that the outer rectangle is also commutative. Therefore, from Lemma \[L:ez-emma\](a), the right-hand square in the diagram is commutative on the image of $w_{Z_{{{\overline{K}}}}}$. In particular, $\phi(\sigma_X^*\alpha)=\sigma_B^*\phi(\alpha)$.
Proof of Theorem \[T:main\]: Part III, the coniveau filtration is split {#S:Andre}
=======================================================================
We now complete the proof of Theorem \[T:main\] by showing that the coniveau filtration is split (Corollary \[C:CNFSplit\]). For this purpose, we use Yves André’s theory of motivated cycles [@AndreIHES]. Along the way, we show in Theorem \[T:mot\] that the existence of a phantom isogeny class for $\operatorname{N}^nH^{2n+1}(X_{{{\overline{K}}}},\rat_\ell(n))$ for all primes $\ell$ follows directly from André’s theory. Note that we already proved this in Theorem \[T:JacDesc\] in a more precise form, namely by showing that there exists a *distinguished* phantom abelian variety within the isogeny class.
.2 cm For clarity, we briefly review the setup of André’s theory of motivated cycles, and fix some notation. Given a smooth projective variety $X$ over a field $K$ and a prime $\ell \neq
\operatorname{char}(K)$, let us denote $\operatorname{B}^j(X)_\rat$ the image of the cycle class map $\operatorname{CH}^j(X)_\rat \to H^{2j}(X_{{{\overline{K}}}},
\rat_\ell(j))$. A *motivated cycle* on $X$ with rational coefficients is an element of the graded algebra $\bigoplus_r
H^{2r}(X_{{{\overline{K}}}}, \rat_\ell(r))$ of the form $\mathrm{pr}_* (\alpha \cup *
\beta)$, where $\alpha$ and $\beta$ are elements of $\operatorname{B}^*(X\times_K Y)_\rat$ with $Y$ an arbitrary smooth projective variety over $K$, $\mathrm{pr} : X\times_K Y \to X$ is the natural projection, and $*$ is the Lefschetz involution on $\bigoplus_r H^{2r}((X\times_K Y)_{{{\overline{K}}}}, \rat_\ell(r))$ relative to any polarization on $X\times_K Y$. The set of motivated cycles on $X$, denoted $\operatorname{B}^\bullet_{\mathrm{mot}}(X)_\rat$, forms a graded $\rat$-sub-algebra of $\bigoplus_r H^{2r}(X_{{{\overline{K}}}},
\rat_\ell(r))$, with $\operatorname{B}^r_{\mathrm{mot}}(X)_\rat\subseteq
H^{2r}(X_{{{\overline{K}}}},
\rat_\ell(r))$; *cf.* [@AndreIHES Prop. 2.1]. Taking $Y=\operatorname{Spec}K$ above, we have an inclusion $\operatorname{B}^r(X)_{\mathbb Q}\subseteq
\operatorname{B}^r_{\mathrm{mot}}(X)_\rat$. Moreover there is a notion of motivated correspondences between smooth projective varieties, and there is a composition law with the expected properties.
\[P:mot\] Let $Y$ and $X$ be smooth projective varieties over a field $K\subseteq \mathbb
C$. Consider a motivated cycle $\gamma \in
\operatorname{B}_{\operatorname{mot}}^{d_Y+r}(Y\times_K X)_\rat$ and its action $$\xymatrix{\gamma_* : H^j(Y_{{{\overline{K}}}}, \rat_\ell) \ar[r]& H^{j+2r}(X_{{{\overline{K}}}},
\rat_\ell(r))}.$$ Then $\operatorname{Im}(\gamma_*)$ (resp. $\operatorname{ker}(\gamma_*)$) is a direct summand of the $\operatorname{Gal}(K)$-representation $H^{j+2r}(X_{{{\overline{K}}}}, \rat_\ell(r))$ (resp. $H^j(Y_{{{\overline{K}}}}, \rat_\ell)$).
We are going to show that if $\gamma \in
\operatorname{B}^{d_Y+r}_{\mathrm{mot}}(Y\times_K X)_\rat$ is a motivated correspondence, then there exists an idempotent motivated correspondence $p \in
\operatorname{B}^{d_Y}_{\mathrm{mot}}(Y\times_K Y)_\rat$ such that $p_*H^j(Y_{{{\overline{K}}}}, \rat_\ell) = \operatorname{ker}(\gamma_*)$. Assuming the existence of such a $p$, this would establish that $\operatorname{ker}(\gamma_*)$ is a direct summand of $H^j(Y_{{{\overline{K}}}},
\rat_\ell)$ as a $\mathbb Q_\ell$-vector space. But then by [@AndreIHES Scolie 2.5], motivated cycles on a smooth projective variety $Y$ over $K$ are exactly the $\operatorname{Gal}(K)$-invariant motivated cycles on $Y_{{{\overline{K}}}}$; therefore $\operatorname{ker}(\gamma_*)$ is indeed a direct summand of $H^j(Y_{{{\overline{K}}}}, \rat_\ell)$ as a $\operatorname{Gal}(K)$-representation, completing the proof. The statement about the image of $\gamma_*$ follows by duality.
The existence of $p$ follows formally from [@AndreIHES Thm. 0.4]: the $\otimes$-category of pure motives $\mathscr{M}$ over a field $K$ of characteristic zero obtained by using motivated correspondences rather than algebraic correspondences is a graded, abelian semi-simple, polarized, and Tannakian category over $\rat$. Indeed, using the notations from [@AndreIHES §4] and viewing $\gamma$ as a morphism from the motive $\mathfrak{h}(Y)$ to the motive $\mathfrak{h}(X)(r)$, we see by semi-simplicity that there exists an idempotent motivated correspondence $p \in \operatorname{B}^{d_Y}_{\mathrm{mot}}(Y\times_K
Y)_\rat$ such that $\operatorname{ker}(\gamma) = p\mathfrak{h}(Y)$. Now the Tannakian category $\mathscr{M}$ is neutralized by the fiber functor to the category of $\rat_\ell$-vector spaces given by the $\ell$-adic realization functor. Since by definition a fiber functor is exact, $p_*H^j(Y_{{{\overline{K}}}},
\rat_\ell)
= \operatorname{ker}(\gamma_*)$ as $\rat_\ell$-vector spaces.
\[T:mot\] Suppose $X$ is a smooth projective variety over a field $K\subseteq \mathbb C$, and let $n$ be a nonnegative integer. The $\operatorname{Gal}(K)$-representation $\operatorname{N}^nH^{2n+1}(X_{{{\overline{K}}}},\rat_\ell(n))$ admits a phantom; more precisely there exist an abelian variety $J'$ over $K$ and a correspondence $\Gamma' \in
\operatorname{CH}^{\dim{J'}+n}(J'\times_K X)$ such that the morphism of $\operatorname{Gal}(K)$-representations $$\label{E:splitinclusion}
\xymatrix{ \Gamma'_* : H^1(J'_{{{\overline{K}}}},\rat_\ell) \ar@{^(->}[r]&
H^{2n+1}(X_{{{\overline{K}}}},\rat_\ell(n))}$$ is split injective with image $\operatorname{N}^nH^{2n+1}(X_{{{\overline{K}}}},\rat_\ell(n))$.
Let $C$ and $\gamma \in \operatorname{CH}^{n+1}(C\times_K
X)_{\mathbb{Q}}$ be the pointed curve and the correspondence provided by Proposition \[P:niveauK\]. By Proposition \[P:mot\] and its proof, there is an idempotent motivated correspondence $q \in \operatorname{B}^{1}_{\mathrm{mot}}(C\times_K C)_\rat$ such that $q_*H^1(C_{{{\overline{K}}}}, \rat_\ell) \stackrel{\gamma_*}{\longrightarrow}
H^{2n+1}(X_{{{\overline{K}}}},\rat_\ell(n))$ is a monomorphism of $\operatorname{Gal}(K)$-representations with image $
\operatorname{N}^nH^{2n+1}(X_{{{\overline{K}}}},\rat_\ell(n))$, which is itself a direct summand of $H^{2n+1}(X_{{{\overline{K}}}},\rat_\ell(n))$.
Now we claim that for smooth projective varieties defined over a field of characteristic zero, we have $\operatorname{B}^{1}_{\mathrm{mot}}(-)_\rat = \operatorname{B}^{1}(-)_\rat$. Over an algebraically closed field of characteristic zero this is a consequence of the Lefschetz $(1,1)$-theorem. Over a field $K$ of characteristic zero, the claim follows from the following two facts: (1) if $Y$ is a smooth projective variety over $K$, then $\operatorname{B}^{r}(Y)_\rat $ consists of the $\operatorname{Gal}(K)$-invariant classes in $\operatorname{B}^{r}(Y_{{{\overline{K}}}})_\rat$ by a standard norm argument, and similarly (2) $\operatorname{B}_{\mathrm{mot}}^{r}(Y)_\rat $ consists of the $\operatorname{Gal}(K)$-invariant classes in $\operatorname{B}_{\mathrm{mot}}^{r}(Y_{{{\overline{K}}}})_\rat$ by [@AndreIHES Scolie 2.5].
Therefore the motivated idempotent $q$ is in fact an idempotent correspondence in $\operatorname{B}^{1}(C\times_K C)_\rat$, and thus defines, up to isogeny, an idempotent endomorphism of $\operatorname{Pic}^\circ(C)$. Its image $J'$, which is only defined up to isogeny, is the sought-after abelian variety such that $q_*H^1(C_{{{\overline{K}}}}, \rat_\ell)
\cong H^1(J'_{{{\overline{K}}}}, \rat_\ell)$. Composing the transpose of the graph of the morphism $C \hookrightarrow \operatorname{Pic}^\circ(C) \twoheadrightarrow J'$ with the algebraic correspondence $\gamma$ yields the desired correspondence $\Gamma'
\in \operatorname{CH}^{\dim J'+n}(J'\times_K X)$.
The main difference with [@ACMVdcg Thm. 2.1] is that we do not know if the splitting in Theorem \[T:mot\] is induced by an algebraic correspondence over $K$. In that respect [@ACMVdcg Thm. 2.1] is more precise.
A nice consequence of Proposition \[P:mot\] is the following:
\[C:CNFSplit\] Let $X$ be a smooth projective variety over a field $K\subseteq \mathbb C$. The geometric coniveau filtration on the $\operatorname{Gal}(K)$-representation $H^n(X_{{{\overline{K}}}},\rat_\ell)$ is split.
Let $r$ be a nonnegative integer. Using the coniveau hypothesis, resolution of singularities, mixed Hodge theory, and comparison isomorphisms, there exist a smooth projective variety $Y$ of dimension $\dim X-r$ over $K$ and a morphism $f : Y \to X$ such that the induced morphism of $\operatorname{Gal}(K)$-representations $$f_*: H^{n-2r}(Y_{\overline K},{\mathbb{Q}}_\ell(-r)) \rightarrow
H^{n}(X_{\overline K},{\mathbb{Q}}_\ell)$$ has image $\operatorname{N}^rH^{n}(X_{\overline K},{\mathbb{Q}}_\ell)$; see e.g. [@illusiemiscellany Sec. 4.4(d)]. The splitting of the coniveau filtration follows from Proposition \[P:mot\] and the Krull–Schmidt theorem.
Everything except for the splitting of the inclusion in Theorem \[T:main\] is shown by combining Theorem \[T:JacDesc\] with Propositions \[P:Tors-Gen’\] and \[P:Tors-Gen\]. The splitting follows from Corollary \[C:CNFSplit\].
A functoriality statement
=========================
Recall that if $X$ and $Y$ are smooth projective varieties over a field $K\subseteq \cx$, and we are given a correspondence $Z\in \operatorname{CH}^{m-n+\dim X}(X\times_K Y)$, then $Z$ induces functorially a homomorphism of complex abelian varieties $$\psi_{Z_\cx} : J^{2n+1}_a(X_\cx) \to J^{2m+1}_a(Y_\cx)$$ that is compatible with the Abel–Jacobi map.
\[P:functoriality\] Denote $J$ and $J'$ the distinguished models of $ J^{2n+1}_a(X_\cx) $ and $
J^{2m+1}_a(Y_\cx)$ over $K$. Then the homomorphism $\psi_{Z_\cx}$ descends to a $K$-homomorphism of abelian varieties $\psi_Z : J \to J'$.
In particular, given a morphism $f:X\to Y$ defined over $K$, the graph of $f$ and its transpose induce homomorphisms $$f_* : J^{2n+1}_{X/K}\to J^{2(n-\dim X+\dim Y)+1}_{Y/K} \quad \text{and}\quad f^* : J^{2n+1}_{Y/K}\to J^{2n+1}_{X/K}.$$ This makes our descent functorial for morphisms of smooth projective varieties over $K$.
By Theorem \[T:main\], the Abel–Jacobi map $AJ:\operatorname{A}^{n+1}(X_\cx) \to J_a^{2n+1}(X_\cx)$ is $\operatorname{Aut}(\cx/K)$-equivariant. Applying Lemma \[L:ez-emma\](a) to the commutative square $$\xymatrix@C=3.5em {\operatorname{A}^{n+1}(X_\cx) \ar@{->>}[r]^{AJ}
\ar[d]_{(Z_{\cx})_*} & J_a^{2n+1}(X_\cx) \ar[d]^{\psi_{Z_\cx}}\\
\operatorname{A}^{m+1}(Y_\cx) \ar@{->>}[r]^{AJ} & J_a^{2m+1}(Y_\cx)
}$$ shows that $\psi_{Z_\cx}$ is $\operatorname{Aut}(\cx/K)$-equivariant. From the theory of the $\mathbb C/{{\overline{K}}}$-trace, $\psi_{Z_\cx}$ descends to a morphism ${\underline{\underline{\psi}}}_{Z_\cx}: {\underline{\underline{J}}}^{2n+1}_a(X_{\mathbb
C})\to {\underline{\underline{J}}}_a^{2m+1}(Y_\cx)$ over ${{\overline{K}}}$. Then the $\operatorname{Aut}(\cx/K)$-equivariance of $\psi_{Z_\cx}$ on $\mathbb C$-points implies ${\underline{\underline{\psi}}}_{Z_\cx}$ is $\operatorname{Gal}({{\overline{K}}}/K)$-equivariant on ${{\overline{K}}}$-points, and so descends from ${{\overline{K}}}$ to $K$. Alternately, $\psi_{Z_\cx}$ descends to $K$ simply by $\cx/K$-descent.
Proposition \[P:functoriality\] could have been proved earlier by using Theorem \[T:JacDesc\], together with the fact (see Lemma \[L:Tor-to-Tor\](b)) that $AJ[N]:\operatorname{A}^{n+1}(X_{\mathbb C})[N]\to
J_a^{2n+1}(X_{\mathbb C})[N]$ is surjective for all $N$ not divisible by a finite number of fixed primes and the fact that torsion points on an abelian variety of order not divisible by a finite number of fixed primes are dense.
Deligne’s theorem on complete intersections of Hodge level $1$ {#S:Deligne}
==============================================================
We recapture Deligne’s result [@deligneniveau] on intermediate Jacobians of complete intersections of Hodge level $1$ (Deligne’s primary motivation was to establish the Weil conjectures for those varieties; of course Deligne established the Weil conjectures in full generality a few years later):
Let $X$ be a smooth complete intersection of odd dimension $2n+1$ over a field $K\subseteq \cx$. Assume that $X$ has Hodge level $1$, that is, assume that $h^{p,q}(X_\cx) = 0$ for all $|p-q| >1$. Then the intermediate Jacobian $J^{2n+1}(X_\cx)$ is a complex abelian variety that is defined over $K$.
First note that the assumption that $X$ has Hodge level 1 implies that the cup product on $H^{2n+1}(X_\cx,\mathbb Z)$ endows the complex torus $J^{2n+1}(X_\cx)$ with a Riemannian form so that $J^{2n+1}(X_\cx)$ is naturally a principally polarized complex abelian variety. Deligne’s proof that this complex abelian variety is defined over $K$ uses the irreducibility of the monodromy action of the fundamental group of the universal deformation of $X$ on $H^{2n+1}(X_\cx,\rat)$ and on $H^{2n+1}(X_\cx,\mathbb Z / \ell)$ for all primes $\ell$. Here, we give an alternate proof based on our Theorem \[T:main\].
Denote $V_m(a_1,\ldots,a_k)$ a smooth complete intersection of dimension $n$ of multi-degree $(a_1,\ldots,a_k)$ inside $\mathbb P^{m+k}$. A complete intersection $X$ of Hodge level $1$ of odd dimension is of one of the following types: $V_{2n+1}(2), V_{2n+1}(2,2), V_{2n+1}(2,2,2), V_3(3), V_3(2,3), V_5(3),
V_3(4)$; see for instance [@rapoport Table 1]. In the cases where $X$ is one of the above and $X$ has dimension $3$, then $X$ is Fano and as such is rationally connected, and therefore $\operatorname{CH}_0(X_\cx) = \mathbb Z$. In all of the other listed cases, it is known [@otw Cor. 1] that $\operatorname{CH}_0(X_\cx)_\rat,\ldots,
\operatorname{CH}_{n-1}(X_\cx)_\rat$ are spanned by linear sections. By [@esnaultlevine Thm. 3.2], which is based on a decomposition of the diagonal argument [@BlSr83], it follows that if $X$ is a complete intersection of Hodge level 1, then the Abel–Jacobi map $\operatorname{A}^n(X_\cx) \to J^{2n+1}(X_\cx)$ is surjective, i.e., that $J^{2n+1}(V_{\mathbb C}) = J^{2n+1}_a(V_{\mathbb C})$. Theorem \[T:JacDesc\] implies that the complex abelian variety $J^{2n+1}(X_{\mathbb C})$ has a distinguished model over $K$.
Albaneses of Hilbert schemes {#S:Hilb}
============================
Over the complex numbers the image of the Abel–Jacobi map is dominated by Albaneses of resolutions of singularities of products of irreducible components of Hilbert schemes. Since Hilbert schemes are functorial, and in particular defined over the field of definition, and since the image of the Abel–Jacobi map descends to the field of definition, one might expect this abelian variety to be dominated by Albaneses of resolutions of singularities of products of irreducible components of Hilbert schemes defined over the field of definition. In this section, we show this is the case, thereby proving Theorem \[Ta:MazQ1\]. Our approach utilizes the theory of Galois equivariant regular homomorphisms, and consequently, we obtain some related results over perfect fields in arbitrary characteristic.
Regular homomorphisms and difference maps {#S:DiffMaps}
-----------------------------------------
In this section we give an equivalent theory of regular homomorphisms and algebraic representatives that does not rely on pointed varieties.
Let $X/k$ be a smooth projective variety over the algebraically closed field $k$, let $T/k$ be a smooth integral variety and let $Z$ be a codimension-$i$ cycle on $T \times_{k}X$. Let $p_{13},p_{23}: T\cross_{k} T \cross_{k} X
\to T\cross_{k} X$ be the obvious projections. Let $\widetilde {
Z}$ be defined as the cycle $$\widetilde {Z}:= p_{13}^*Z-p_{23}^*Z$$ on $T\times_{k} T\times_{k} X$. For points $t_1,t_0\in T({k})$, we have $
{{\widetilde{Z}}}_{(t_1,t_0)} = Z_{t_1} - Z_{t_0}$. We therefore have a map $$\label{E:defdiffk}
\xymatrix@R=.1cm{
(T\cross_{k} T)({k}) \ar[r]^<>(0.5){y_{Z}} & \operatorname{A}^i(X) \\
(t_1,t_0) \ar@{|->}[r]& Z_{t_1} - Z_{t_0}.
}$$
\[L:DiffReg\] Let $X/k$ be a smooth projective variety over an algebraically closed field ${k}$, and let $A/k$ be an abelian variety. A homomorphism of groups $
\phi:
\operatorname{A}^i(X)\to A(k)
$ is regular if and only if for every pair $(T, Z)$ with $T$ a smooth integral variety over ${k}$ and $Z\in \operatorname{CH}^i( T\times_{k} X)$, the composition $$\begin{CD}
(T\times_{k}T)({k})@> y_{Z} >> \operatorname{A}^i(X)@>\phi >>A({k})
\end{CD}$$ is induced by a morphism of varieties $\xi_{ Z}:T\times_{k} T\to A$.
If $\phi:\operatorname{A}^i(X) \to A(k)$ is a regular homomorphism to an abelian variety, then $\phi\circ y_{ Z}$ is induced by a morphism of varieties $T\cross_{k} T \to A$; indeed after choosing any diagonal base point $(t_0,t_0)\in (T\times_{k} T)(k)$, the maps $\phi\circ y_{ Z}$ and $\phi\circ w_{\widetilde Z,(t_0,t_0)}$ agree. Conversely, suppose $\phi\circ y_{ Z}$ is induced by a morphism $\xi_{Z}$ of varieties, and let $t_0 \in T({k})$ be any base point. Let $\iota$ be the inclusion $\iota: T \to T\times {\left\{t_0\right\}} \subset T\times T$. Then $w_{Z,t_0} = y_{Z}|_{\iota(T)}$, and $\phi\circ w_Z$ is induced by the morphism $\xi_{ Z} \circ \iota$.
Now suppose that $X$ is a smooth projective variety over $K$, that $T$ is a smooth integral quasi-projective variety over $K$, and that $Z$ is a codimension-$i$ cycle on $T\times_K X$. The cycle ${{\widetilde{Z}}} = p_{13}^*Z -p_{23}^*Z$ on $T\times_K T \times_K X$ is again defined over $K$.
\[L:DiffDesc\] Let $K$ be a perfect field, suppose $X$, $Z$ and $T$ are as above, and let $A/K$ be an abelian variety. If $\phi:
\operatorname{A}^i(X_{{{\overline{K}}}}) \to A({{\overline{K}}})$ is a $\operatorname{Gal}(K)$-equivariant regular homomorphism, then the induced morphism $\xi_{Z_{{{\overline{K}}}}}:
(T\times_K T)_{{{\overline{K}}}} \to A_{{{\overline{K}}}}$ is also $ \operatorname{Gal}(K)$-equivariant on ${{\overline{K}}}$-points, and thus $\xi_{Z_{{{\overline{K}}}}}$ descends to a morphism $\xi_Z: T \times_K T \to A$ of varieties over $K$.
For each $\sigma \in \operatorname{Gal}(K)$ there is a commutative diagram (see [@ACMVdcg Rem. 4.3]) $$\xymatrix@C=1.5cm@R=.75cm{
(T\times_K T)({{{\overline{K}}}}) \ar@{->}[r]^{y_{ Z_{{{\overline{K}}}}}}
\ar@{->}[d]^{\sigma_{T\times
T}^*} & \operatorname{A}^i(X_{{{\overline{K}}}}) \ar@{->}[r]^{\phi}
\ar@{->}[d]^{\sigma_X^{*}} &
A({{{\overline{K}}}}) \ar@{->}[d]^{\sigma_{A}^*}\\
(T\times_K T)({{{\overline{K}}}}) \ar@{->}[r]^{y_{Z_{{{\overline{K}}}}}}
&\operatorname{A}^i(X_{{{\overline{K}}}})
\ar@{->}[r]^{\phi}& A({{{\overline{K}}}}). \\
}$$ Now $\phi$ is $\operatorname{Gal}(K)$-equivariant by hypothesis, and $y_{Z_{{{\overline{K}}}}}$ is $\operatorname{Gal}(K)$-equivariant since ${{\widetilde{Z}}}$, $T$ and $X$ are defined over $K$. Consequently, $\xi_{Z_{{{\overline{K}}}}}$ is $\operatorname{Gal}(K)$-equivariant, as claimed.
Albaneses of Hilbert schemes and the Abel–Jacobi map
----------------------------------------------------
We are now in a position to prove the following theorem, which will allow us to prove Theorem \[Ta:MazQ1\].
\[T:MazQ1\] Suppose $X$ is a smooth projective variety over a perfect field $K$, and let $n$ be a nonnegative integer. Let $A/K$ be an abelian variety defined over $K$, and let $$\xymatrix{
\phi:\operatorname{A}^{n+1}(X_{{{\overline{K}}}})\ar@{->>}[r] &A_{{{\overline{K}}}}({{\overline{K}}})
}$$ be a surjective Galois-equivariant regular homomorphism. Then there are a finite number of irreducible components of the Hilbert scheme $\operatorname{Hilb}^{n+1}_{X/K}$ parameterizing codimension $n+1$ subschemes of $X/K$, so that by taking a finite product $H$ of these components, and then denoting by $\operatorname{Alb}_{\widetilde H/K}$ the Albanese variety of a smooth alteration $\widetilde H$ of $H$, there is a surjective morphism $$\label{E:albanesetheorem}
\xymatrix{
\operatorname{Alb}_{\widetilde H/K} \ar@{->>}[r]& A
}$$ of abelian varieties over $K$.
Let $Z$ be the cycle on $A\times_ K X$ from [@ACMVdcg Lem. 4.9(d)] so that the composition $$\begin{CD}
A({{\overline{K}}})@>w_{{{\overline{Z}}}}>> \operatorname{A}^{n+1}(X_{{{\overline{K}}}})@>{\phi}>>A({{\overline{K}}})
\end{CD}$$ is induced by the $K$-morphism $
r\cdot \operatorname{Id}:A\to A
$ for some positive integer $r$.
Now using Bertini’s theorem, let $C$ be a smooth projective curve that is a linear section of $A$ passing through the origin (so it has a $K$-point), and such that the inclusion $C\hookrightarrow A$ induces a surjective morphism $J_{C/K}\twoheadrightarrow A$. Denote again by $Z$ the refined Gysin restriction of the cycle $Z$ to $C$. We have a commutative diagram $$\label{E:HilbThm1}
\xymatrix@R=1em{
C({{\overline{K}}}) \ar[d] \ar@{^(->}[r]& A({{\overline{K}}})\ar[r]^<>(0.5){w_{{{\overline{Z}}}}}
\ar@{->>}@/_2pc/[rr]_{r\cdot \operatorname{Id}}& \operatorname{A}^{n+1}(X_{{{\overline{K}}}}) \ar@{->>}[r]^{{\phi}} & A({{\overline{K}}})\\
J_{C/K}({{\overline{K}}}) \ar@{->>}[ru]&
}$$
Discarding extra components, we may assume that $Z$ is flat over $C$. Write $Z=\sum_{j=1}^mV^{(j)}-\sum_{j=m+1}^{m'}V^{(j)}$, where $V^{(1)},\dots,V^{(m')}$ are (not necessarily distinct) integral components of the support of $Z$, which by assumption are flat over $C$. Let $\operatorname{Hilb}^{(j)}_{X/K}$ be the component of the Hilbert scheme, with universal subscheme $U^{(j)}\subseteq
\operatorname{Hilb}^{(j)}_{X/K}\times_K X$ such that $V^{(j)}$ is obtained by pull-back via a morphism $f^{(j)}:C\to \operatorname{Hilb}^{(j)}_{X/K}$. Let $H=\prod_{j=1}^{m'}\operatorname{Hilb}^{(j)}_{X/K}$, and let $U_H:=\sum_{j=1}^m
\operatorname{pr}_j^*U^{(j)}-\sum_{j=m+1}^{m'}\operatorname{pr}_j^*U^{(j)}$, where $\operatorname{pr}_j : H \to \operatorname{Hilb}^{(j)}_{X/K}$ is the natural projection. There is an induced morphism $f:C\to H$ and we have $Z=f^*U_H$; the pull-back is defined since all the cycles are flat over the base.
Now let $\nu:\widetilde H\to H$ be a smooth alteration of $H$ and let $\widetilde U=\nu^*U_H$. Let $\mu:\widetilde C\to C$ be an alteration such that there is a commutative diagram $$\xymatrix{
\widetilde C \ar[r]^{\tilde f} \ar[d]_\mu& \ar[d]_\nu \widetilde H\\
C \ar[r]^f& H.
}$$ Let $\widetilde Z=\mu^*Z$. We obtain maps $$\xymatrix{
(\widetilde C\times_{ K} \widetilde C)({{\overline{K}}}) \ar@{^(->}[r]& (\widetilde
H\times_{K}\widetilde H)({{\overline{K}}}) \ar[r]^{y_{\widetilde U_{{{\overline{K}}}}}} &
\operatorname{A}^{n+1}(X_{{{\overline{K}}}}) \ar@{->>}[r]^\phi& A({{\overline{K}}}).
}$$ By Lemma \[L:DiffDesc\], these descend to $K$-morphisms $$\xymatrix{
\widetilde C\times_{ K} \widetilde C\ar@{^(->}[r]& \widetilde
H\times_{K}\widetilde H \ar[r]^<>(0.5){{\xi}_{\widetilde U}}
& A.
}$$ Recall that if $W/K$ is any variety, then there exist an abelian variety $\operatorname{Alb}_{W/K}$ and a torsor $\operatorname{Alb}^1_{W/K}$ under $\operatorname{Alb}_{W/K}$, equipped with a morphism $W \to \operatorname{Alb}^1_{W/K}$ which is universal for morphisms from $W$ to abelian torsors. Taking Albanese torsors we obtain a commutative diagram $$\xymatrix@R=1em{
\widetilde C\times_{ K} \widetilde C\ar@{^(->}[r] \ar[d] \ar@/_5pc/[ddd]&
\widetilde H\times_{K}\widetilde H \ar[r]^<>(0.5){{\xi}_{\widetilde
U}} \ar[d]& A \ar@{=}[ddd]\\
\operatorname{Alb}^1_{\widetilde C/K}\times_{ K} \operatorname{Alb}^1_{\widetilde
C/K} \ar@{->}[r] \ar@{->>}[d] & \operatorname{Alb}^1_{\widetilde
H/K}\times_{K}\operatorname{Alb}^1_{\widetilde H/K} \ar[ru] & \\
J_{ C/K}\times_{ K} J_{C/K}\ar@{->>}[rrd]& &\\
C\times_{ K} C \ar[u] \ar@{->}[rr]_<>(0.5){{\xi}_{Z}}& & A.\\
}$$ The surjectivity of the map $J_{ C/K}\times_{ K} J_{C/K}\to A$ follows from . A diagram chase then shows that the map $\operatorname{Alb}^1_{\widetilde H/K}\times_{ K} \operatorname{Alb}^1_{\widetilde
H/K}\to A$ is surjective. In general, if $T$ is a torsor under an abelian variety $B/K$, and if $T \twoheadrightarrow A'$ is a surjection to an abelian variety, then there is a surjection $B \to A'$ over $K$. (Indeed, the surjection $T \twoheadrightarrow A'$ induces an inclusion $\operatorname{Pic}^0_{A'/K} \hookrightarrow \operatorname{Pic}^0
_{T/K}$; but $\operatorname{Pic}^0_{A'/K}$ is isogenous to $A'$, while $\operatorname{Pic}^0_{T/K}$ is isogenous to $B$.) Applying this to the surjection $\operatorname{Alb}^1_{\widetilde H/K}\times_{ K} \operatorname{Alb}^1_{\widetilde
H/K}\twoheadrightarrow A$, we obtain the surjection $\operatorname{Alb}_{\widetilde H/K}\times_{ K} \operatorname{Alb}_{\widetilde
H/K}\to A$. Theorem \[T:MazQ1\] now follows, where the ${{\widetilde{H}}}$ in is the product ${{\widetilde{H}}} \times_K {{\widetilde{H}}}$ considered here.
We now use Theorem \[T:MazQ1\] to prove Theorem \[Ta:MazQ1\].
Recall the fundamental result of Griffiths [@griffiths68 p. 826] asserting that the Abel–Jacobi map ${AJ}:\operatorname{A}^{n+1}(X_{\cx})
\longrightarrow J_a^{2n+1}(X_\cx)$ is a surjective regular homomorphism. By Theorem \[T:main\] and its proof, $J_a^{2n+1}(X_\cx)$ descends uniquely to an abelian variety $J/K$ such that the surjective regular homomorphism ${\underline{\underline{AJ}}}:\operatorname{A}^{n+1}(X_{{{\overline{K}}}}) \longrightarrow J_{{{\overline{K}}}}$ defined in the proof of Lemma \[L:T-G-AC\] is Galois-equivariant. Now employ Theorem \[T:MazQ1\].
A uniruled threefold has Chow group of zero-cycles supported on a surface. A decomposition of the diagonal argument [@BlSr83] shows that the threefold has geometric coniveau $1$ in degree $3$.
Theorem \[T:MazQ1\] also gives the following result for algebraic representatives.
\[C:ChowAlgRep\] Let $X$ be a smooth projective variety over a perfect field $K$, let $\Omega/{{\overline{K}}}$ be an algebraically closed field extension, with either $\Omega={{\overline{K}}}$ or $\operatorname{char}(K)=0$, and let $n$ be a nonnegative integer. Assume there is an algebraic representative $\phi^{n+1}_{\Omega}:\operatorname{A}^{n+1}(X_{\Omega})\to
\operatorname{Ab}^{n+1}(X_{\Omega})(\Omega)$ (e.g., $n=0$, $1$, or $\dim X-1$).
Then the abelian variety $\operatorname{Ab}^{n+1}(X_{ \Omega})$ descends to an abelian variety $\underline{\operatorname{Ab}}^{n+1}(X_{{{\overline{K}}}})$ over $K$, and there are a finite number of irreducible components of the Hilbert scheme $\operatorname{Hilb}^{n+1}_{X/K}$ parameterizing codimension $n+1$ subschemes of $X/K$, so that by taking a finite product $H$ of these components, and then denoting by $\operatorname{Alb}_{\widetilde H/K}$ the Albanese of a smooth alteration $\widetilde H$ of $H$, there is a surjective morphism $
\operatorname{Alb}_{\widetilde H/K}\twoheadrightarrow
\underline{\operatorname{Ab}}^{n+1}(X_{{{\overline{K}}}})
$ of abelian varieties over $K$.
The fact that $\operatorname{Ab}^{n+1}(X_{ \Omega})$ descends to ${{\overline{K}}}$ to give $\operatorname{Ab}^{n+1}(X_{{{\overline{K}}}})$ is [@ACMVdcg Thm. 3.7]. It is then shown in [@ACMVdcg Thm. 4.4] that $\operatorname{Ab}^{n+1}(X_ {{{\overline{K}}}})$ descends to an abelian variety over $K$ and that the map $\phi^{n+1}_{{{\overline{K}}}}:\operatorname{A}^{n+1}(X_{{{\overline{K}}} })\to
\operatorname{Ab}^{n+1}(X_{{{\overline{K}}}})({{\overline{K}}})$ is $\operatorname{Gal}(K)$-equivariant. Therefore, we may employ Theorem \[T:MazQ1\] to conclude.
Cohomology of Jacobians of curves via Abel maps {#A:Appendix}
===============================================
Let $C$ be a smooth projective curve over a field $K$ with separable closure ${{\overline{K}}}$. For any $n$ invertible in $K$, the Kummer sequence of étale sheaves on $C$: $$\xymatrix{
1 \ar[r] & \mmu_n \ar[r] & \gp_m \ar[r]^{[n]} & \gp_m \ar[r] & 1
}$$ gives an isomorphism $$H^1({{\overline{C}}},\mmu_n) \iso\operatorname{Pic}_{C/K}[n] = \operatorname{Pic}^\circ_{C/K}[n],$$ where we have written ${{\overline{C}}}$ for $C_{{{\overline{K}}}}$. After taking the inverse limit over all powers of a fixed prime $n=\ell$, we obtain isomorphisms of $\operatorname{Gal}(K)$-representations $$H^1({{\overline{C}}}, \integ_\ell(1)) \iso T_\ell \operatorname{Pic}^\circ_{C/K} \iso (T_\ell \widehat{\operatorname{Pic}^\circ_{C/K}})^\vee(1) \iso H^1(\widehat{\operatorname{Pic}^\circ_{C/K,{{\overline{K}}}}},\integ_\ell(1)).$$ After twisting by $-1$, the canonical (principal) polarization on the Jacobian gives an isomorphism $$\label{E:H1isom}
H^1({{\overline{C}}},\integ_\ell) \iso H^1(\operatorname{Pic}^\circ_{{{\overline{C}}}/{{\overline{K}}}},
\integ_\ell).$$ In this appendix we show that, up to tensoring with $\rat_\ell$, the isomorphism is induced by a $K$-morphism $C \ra \operatorname{Pic}^\circ_{C/K}$.
\[P:curveCoh\] Let $C$ be a smooth projective curve over a field $K$. Then there exists a morphism $\beta:C \to \operatorname{Pic}^\circ_{C/K}$ over $K$ which induces an isomorphism $$\xymatrix{
\beta^*:H^1(\operatorname{Pic}^\circ_{{{\overline{C}}}/{{\overline{K}}}},\integ_\ell) \ar[r]^<>(0.5)\sim &
H^1({{\overline{C}}},\integ_\ell)
}$$ of $\operatorname{Gal}(K)$-representations for all but finitely many $\ell$. For all $\ell$ invertible in $K$, we have that the pull-back $\beta^*:H^1(\operatorname{Pic}^\circ_{{{\overline{C}}}/{{\overline{K}}}},\rat_\ell) \to H^1({{\overline{C}}},\rat_\ell)$ is an isomorphism.
The case of an integral curve over an algebraically closed field is standard (e.g., [@milneAV Prop. 9.1,p.113]). The case where $C$ is geometrically irreducible and $C(K)$ is nonempty is certainly well-known; even if $C$ admits no $K$-points, the result follows almost immediately from the case $K={{\overline{K}}}$:
\[L:appspecialcase\] If $C/K$ is geometrically irreducible, then Proposition \[P:curveCoh\] holds for $C$.
Let $d$ be a positive integer such that $C$ admits a line bundle $ L$ of degree $d$ over $K$. Let $\beta$ denote the composition $$\xymatrix@C=3.5em{
\beta:C \ar[r]^<>(0.5)a & \operatorname{Pic}^1_{C/K} \ar[r]^{[d] = (-)^{\tensor
d}}_{\text{isogeny}} & \operatorname{Pic}^d_{C/K} \ar[r]^{(-)\tensor
{ L}^\vee}_{\sim} & \operatorname{Pic}^\circ_{C/K},
}$$ where $\operatorname{Pic}^e_{C/K}$ denotes the torsor under $\operatorname{Pic}^\circ_{C/K}$ consisting of degree $e$ line bundles on $C/K$, and $a$ is the Abel map (e.g., [@kleimanPIC Def. 9.4.6, Rem. 9.3.9]).
We claim that if $\ell \nmid d$, then $\beta^*: H^1(\operatorname{Pic}^\circ_{{{\overline{C}}}/{{\overline{K}}}},\integ_\ell) \to H^1({{\overline{C}}},\integ_\ell)$ is an isomorphism. After passage to ${{\overline{K}}}$, we may find a line bundle ${M}$ such that ${ M}^{\tensor d} \iso L$. We have a commutative diagram $$\xymatrix{
{{\overline{C}}} \ar[r]^<>(0.5)a \ar[dr]_{a_{ M}}& \operatorname{Pic}^1_{{{\overline{C}}}/{{\overline{K}}}} \ar[r]^{[d]}
\ar[d]^{(-)\tensor { M}^\vee}&
\operatorname{Pic}^d_{{{\overline{C}}}/{{\overline{K}}}}\ar[d]^{(-)\tensor { L}^\vee} \\
& \operatorname{Pic}^\circ_{{{\overline{C}}}/{{\overline{K}}}} \ar[r]^{[d]} & \operatorname{Pic}^\circ_{{{\overline{C}}}/{{\overline{K}}}}
}$$ Since the diagonal arrow is the usual Abel–Jacobi embedding of ${{\overline{C}}}$ in its Jacobian, where the assertion about pull back of cohomology is well known (e.g., [@milneAV Prop. 9.1, p.113]), and the lower horizontal arrow is an isogeny of degree $d^{2g(C)}$, the commutativity of the diagram implies that $\beta$ has the asserted properties.
Components of the Picard scheme
-------------------------------
Now suppose that $C$ is irreducible but ${{\overline{C}}}$ is reducible. Continue to let $\operatorname{Pic}^\circ_{{{\overline{C}}} / {{\overline{K}}}}$ denote the connected component of identity of the Picard scheme, and for each $d$ let $\operatorname{Pic}^d_{{{\overline{C}}}/{{\overline{K}}}}$ be the space of line bundles of total degree $d$. (This has the unfortunate notational side effect that $\operatorname{Pic}^\circ_{{{\overline{C}}}/{{\overline{K}}}}$ does not coincide with $\operatorname{Pic}^0_{{{\overline{C}}}/{{\overline{K}}}}$, but we will never have cause to study the space of line bundles of total degree zero.) Then $\operatorname{Pic}^d_{{{\overline{C}}}/{{\overline{K}}}}$ is no longer a torsor under $\operatorname{Pic}^\circ_{{{\overline{C}}}/{{\overline{K}}}}$, and we need to work slightly harder to identify suitable geometrically irreducible, $K$-rational components of the Picard scheme of $C$.
Let $\Pi_0({{\overline{C}}})$ be the set of irreducible components of ${{\overline{C}}}$. (Since ${{\overline{K}}}$ is separably closed, each such component is geometrically irreducible.) Fix a component ${{\overline{D}}}\in \Pi_0(C_{{{\overline{K}}}})$, and let $H\subset \operatorname{Gal}(K)$ be its stabilizer. Since $C$ is irreducible, we have $${{\overline{C}}} = \bigsqcup_{[\sigma] \in \operatorname{Gal}(K)/H} {{\overline{D}}}^\sigma,$$ where viewing $\sigma$ as an automorphism of ${{\overline{C}}}$, we set ${{\overline{D}}}^\sigma
=\sigma({{\overline{D}}})$. Let $e = \#\Pi_0({{\overline{C}}})$. Inside the $de$-th symmetric power $S^{de}(C)_{{{\overline{K}}}} = S^{de}({{\overline{C}}})$ we identify the irreducible component $$S^{\Delta_d}({{\overline{C}}}) := \prod_{[\sigma] \in \operatorname{Gal}(K)/H} S^d({{\overline{D}}}^\sigma).$$ Since this element of $\Pi_0(S^{de}({{\overline{C}}}))$ is fixed by $\operatorname{Gal}(K)$, it descends to $K$ as a geometrically irreducible variety.
Similarly, inside the Picard scheme $\operatorname{Pic}_{{{\overline{C}}}/{{\overline{K}}}}$ we single out $$\operatorname{Pic}^{\Delta_d}_{{{\overline{C}}}/{{\overline{K}}}} = \prod_{[\sigma] \in \operatorname{Gal}(K)/H}
\operatorname{Pic}^d_{{{\overline{D}}}^\sigma/{{\overline{K}}}}.$$ It is visibly irreducible and, since it is stable under $\operatorname{Gal}(K)$, it descends to $K$. Note that $\operatorname{Pic}^{\Delta_d}_{{{\overline{C}}}/{{\overline{K}}}}$ is a $\operatorname{Pic}^{\circ}_{{{\overline{C}}}/{{\overline{K}}}}$-torsor.
The $(de)$-th Abel map $S^{de}(C) \to \operatorname{Pic}^{de}_{C/K}$ then restricts to a morphism $$\xymatrix{
S^{\Delta_d}(C) \ar[r]^{a_{\Delta_d}} & \operatorname{Pic}^{\Delta_d}_{C/K}
}$$ of geometrically irreducible varieties over $K$.
One (still) has the canonical Abel map $$\xymatrix{
C \ar[r]^<>(0.5)a & \operatorname{Pic}^1_{C/K}.
}$$ Over ${{\overline{K}}}$, the image of the Abel map $a_{{{\overline{K}}}}$ lands in $$\operatorname{Pic}^\one_{{{\overline{C}}}/{{\overline{K}}}} = \bigsqcup_{[\sigma] \in \operatorname{Gal}(K)/H} \left(
\operatorname{Pic}^1_{{{\overline{D}}}^\sigma} \times \prod_{[\tau]\not = [\sigma]} \operatorname{Pic}^\circ_{{{\overline{D}}}^\tau} \right).$$ Although $\operatorname{Pic}^\one_{{{\overline{C}}}/{{\overline{K}}}}$ has $e$ components, $\operatorname{Gal}(K)$ acts transitively on them, and we have an irreducible variety $\operatorname{Pic}^\one_{C/K}$ over $K$.
In conclusion, the canonical Abel map induces a morphism $$\xymatrix{
C \ar[r]^<>(0.5)a & \operatorname{Pic}^\one_{C/K}
}$$ of irreducible varieties over $K$.
We need two more $K$-rational morphisms:
Let $C/K$ be a smooth projective integral curve. Let $s$ be the map $$\xymatrix@R=.5em{
\operatorname{Pic}^\one_{{{\overline{C}}}/{{\overline{K}}}} \ar[r]^s & \operatorname{Pic}^{\Delta_1}_{{{\overline{C}}}/{{\overline{K}}}} \\
L \ar@{|->}[r] & \bigotimes_{[\sigma] \in \operatorname{Gal}(K)/H}
\sigma^*{ L}.
}$$ Let $t$ be the map $$\xymatrix{
{{\overline{C}}} \ar[r]^<>(0.5)t & S^{\Delta_1}({{\overline{C}}})
}$$ such that, if $P \in {{\overline{D}}}^\tau(K) \subset C({{\overline{K}}})$, then the components of $t(P)$ are given by $$t(P)_\sigma = \sigma\tau\inv(P) \in {{\overline{D}}}^\sigma$$ Then $s$ and $t$ descend to morphisms over $K$.
Each is $\operatorname{Gal}(K)$-equivariant on ${{\overline{K}}}$-points.
Isomorphisms on cohomology
--------------------------
\[L:sona\] Let $C/K$ be a smooth projective irreducible curve. Then the composition $$\xymatrix{
C \ar[r]^<>(0.5)a & \operatorname{Pic}^\one_{C/K} \ar[r]^s & \operatorname{Pic}^{\Delta_1}_{C/K}
}$$ induces an isomorphism of $\operatorname{Gal}(K)$-representations $$H^1(\operatorname{Pic}^{\Delta_1}_{{{\overline{C}}}/{{\overline{K}}}}\integ_\ell) \to
H^1({{\overline{C}}},\integ_\ell).$$
It suffices to analyze $s \circ a$ after base change to ${{\overline{K}}}$. Choose a base point $P_\sigma \in {{\overline{D}}}^\sigma$ for each irreducible component of ${{\overline{C}}}$. We have a commutative diagram $$\xymatrix{
{{\overline{C}}} \ar[r]^a \ar[d]^t & \operatorname{Pic}^{\one}_{{{\overline{C}}}/{{\overline{K}}}} \ar[r]^s &
\operatorname{Pic}^{\Delta_1}_{{{\overline{C}}}/{{\overline{K}}}} \ar[d]^{\prod_{[\sigma]} (-)\tensor
{\mathcal O}(-P_\sigma)} \\
S^{\Delta_1}({{\overline{C}}}) \ar[rru]^{a_{\Delta_1}}
\ar[rr]_{\prod_{[\sigma]}a_{P_\sigma}}
& &\operatorname{Pic}^\circ_{{{\overline{C}}}/{{\overline{K}}}},
}$$ where the bottom arrow is the product of Abel maps associated to the points $P_\sigma$. Since the right-most vertical arrow is an isomorphism of schemes, it suffices to verify that $t$ and $\prod a_{P_\sigma}$ induce isomorphisms on first cohomology groups. On one hand, since cohomology takes coproducts to products, we have $H^1({{\overline{C}}},\integ_\ell) \iso
\prod_\sigma H^1({{\overline{D}}}^\sigma,\integ_\ell)$. On the other hand, since each ${{\overline{D}}}^\sigma$ is connected, the Künneth formula implies that $H^1(S^{\Delta_1}({{\overline{C}}}), \integ_\ell) =H^1(\prod_\sigma {{\overline{D}}}^\sigma,
\integ_\ell) \iso \oplus_\sigma
\operatorname{pr}_\sigma^* H^1({{\overline{D}}}^{\sigma}, \integ_\ell)$. Since the composition $
\xymatrix{
{{\overline{D}}}^\tau \ar[r]^<>(0.5)t & \prod_\sigma {{\overline{D}}}^\sigma
\ar[r]^<>(0.5){\operatorname{pr}_\tau} & {{\overline{D}}}^\tau
}
$ is the identity, $$\xymatrix{
H^1(S^{\Delta_1}({{\overline{C}}}), \integ_\ell) \ar[r]^<>(0.5){t^*} &
H^1({{\overline{C}}}, \integ_\ell)
}$$ is an isomorphism as well.
Finally, since each Abel–Jacobi map $a_{P_\sigma}$ induces an isomorphism $H^1(\operatorname{Pic}^\circ_{{{\overline{D}}}^\sigma/{{\overline{K}}}},\integ_\ell) \iso H^1({{\overline{D}}}^\sigma,
\integ_\ell)$, their product yields an isomorphism $(\prod_{[\sigma]}a_{P_\sigma})^*:H^1(\operatorname{Pic}^\circ_{{{\overline{C}}}/{{\overline{K}}}},\mathbb Z_\ell)
\to H^1(S^{\Delta_1}({{\overline{C}}}), \integ_\ell)$.
It is now straight-forward to provide a proof of the main result of this appendix.
Since both the Picard functor and cohomology take coproducts to products, we may and do assume that $C$ is irreducible. Choose $d$ such that $\operatorname{Pic}^{\Delta_d}_{C/K}$ admits a $K$-point $L$. Let $\beta$ be the composition $$\xymatrix{
C \ar[r]^<>(0.5)a & \operatorname{Pic}^\one_{C/K} \ar[r]^s & \operatorname{Pic}^{\Delta_1}_{C/K}
\ar[r]^{[d]}_{\operatorname{isog.}} & \operatorname{Pic}^{\Delta_d}_{C/K} \ar[r]^{(-)\tensor
L^\vee}_\cong &
\operatorname{Pic}^\circ_{C/K}.
}$$ By Lemma \[L:sona\], $\beta^*: H^1(\operatorname{Pic}^\circ_{{{\overline{C}}}/{{\overline{K}}}},\integ_\ell)
\to H^1({{\overline{C}}},\integ_\ell)$ is an isomorphism as long as $\ell
\nmid d$.
[^1]: The first author was partially supported by grants from the NSA (H98230-14-1-0161, H98230-15-1-0247 and H98230-16-1-0046). The second author was partially supported by a Simons Foundation Collaboration Grant for Mathematicians (317572) and NSA grant H98230-16-1-0053. The third author was supported by EPSRC Early Career Fellowship EP/K005545/1.
|
---
abstract: 'Surfaces bombarded with low energy ion beams often display development of self assembled patterns and quasi-periodic structures. Kinetic Monte Carlo simulations have been performed to describe ion sputtered Tantalum surfaces. A weak nonlinearity in the relaxation process has been introduced and the results show that the Positive Schwoebel barrier, produced by the nonlinear Hamiltonian, is necessary in describing ion bombarded Tantalum surfaces. Furthermore, their scaling exponents suggest presence of a class other than KPZ.'
author:
- Shalik Ram Joshi
- Trilochan Bagarti
- 'Shikha Varma[^1]'
title: Kinetic Monte Carlo simulations of self organized nanostructures on Ta Surface Fabricated by Low Energy Ion Sputtering
---
Introduction
============
Non-equilibrium surfaces produced via bombardment with energetic ion beams, often exhibit well ordered patterns having several potential applications [@Qia; @Eli; @Mar]. The surface morphology, here, develops as a consequence of competition between a variety of processes like roughening dynamics, relaxation processes, generation of defects, material transport etc. [@Mak; @Fro]. Occurrence of a wide array of surface morphologies complicates the prediction of dominant mechanism controlling their evolution. Non-metallic surfaces generally display hills and depressions for normal incident ion beams [@Fac], while showing quasi-periodic ripple morphology under off-normal conditions [@Cha; @Car]. For metallic surfaces, ripples develop even at normal incidence [@Rus]. Symmetry breaking anisotropy in surface diffusion can cause these ripples to rotate with substrate temperature [@Rus; @Val]. For low energy ion beam induced patterning, although the erosive processes are dominant, enhanced surface diffusion due to the defect mobility also becomes important especially under the low ion-flux conditions [@Mak; @Cha]. The surface morphology in this scenario is governed by the non-equilibrium biased diffusion current.
Continuum model approach has proved to be a very successful technique in describing the surface evolution but involvement of several complex phenomena make it difficult to relate with the experimental results [@Cha1]. An alternative approach is the Kinetic Monte Carlo(KMC) method where the kinetic behavior of the surface is simulated at microscopic level for a discrete surface. During diffusion or sputtering, the surface gets modified in the units of whole atoms, only at the specific site [@Bar]. Various models have been proposed to understand the effect of sputtering related erosion on surface evolution. For instance, in model by Cuerno et.al [@Cue], the local surface morphology has been used to determine the sputtering yield of the surface, while in binary collision approximation, the atom gets removed from the surface with a probability proportional to the energy deposited, in its near surface region, by incoming ions [@Bro; @Har; @Ste]. Surface diffusion has been formulated via relaxation of the surface to its minimum energy through a series of atomic jumps with probability that depends upon the energy of the initial and the final state [@Sie]. Thermally activated hopping, where an atom hops over an energetic barrier with a barrier height that depends on the local configuration, has also been considered for studying surface diffusion [@Bro; @Ste; @Yew].
In the present article, KMC results are presented in 1+1 dimension for ion beam modified metallic surfaces. Experimentally, the presence of ripple morphology on ion irradiated Tantalum surfaces has been observed [@Ram; @Sub]. We have developed a model based on earlier work by Cuerno [[*et. al.*]{}]{}[@Cue] which was not able to describe the surface morphology of ion sputtered metallic Tantalum surface. The Schwoebel effect is found to be important for Tantalum surface, and has been incorporated in our model, by including a weak non-linearity in Hamiltonian for relaxation of diffusing atoms on the sputtered surface. Simulation results, presented here, shows that the presence of Schwoebel effect produces the surface morphology and scaling exponents that are consistent with our experimental observations. The scaling exponents indicates that the morphology of ion sputtered Tantalum surfaces may belong to universality class other than Edward-Wilkinson(EW) and Kardar-Parisi-Zhang(KPZ). The paper is organized as follows. In Section \[model\] we describe the KMC model. The experimental details are described in Section \[exp\] and in Section \[result\] we discuss the results.
The Model {#model}
=========
In our model we assume that ion irradiation causes the surface to evolve by two dominant processes, namely erosion and surface diffusion. The erosion process consists of the removal of atoms from the surface due to the impinging ions. Ions bombarded on the surface penetrate deep into the substrate releasing energy to the neighboring atoms along the trajectory. If the energy gained by a surface atom is sufficiently large, it gets eroded from the surface. Sputtering yield, $Y(\phi)$, defined as the number of atoms eroded for every incoming ion at an angle $\phi$ to the surface normal. It depends on the energy of the ion and local morphology of the interface and, is a measure of the efficiency of the sputtering process [@Ken]. We assume $Y(\phi) = a + b \phi^2 + c \phi^4$, where $a, b$ and $c$ are constant such that $Y(\pi/2)=0$ and for some critical $\phi_0$, $Y(\phi)$ has a maxima. The erosion process brings in an effective negative surface tension that causes the surface to become rough [@Bra].
Surface diffusion, on the other hand, consists of the random migration of surface atoms on the surface such that the surface energy is minimized. Its strength depends on the temperature of the substrate and the binding energy of the atoms. The negative surface tension, during erosion, leads the system towards instability and as a result the system responds to restore stability by surface diffusion [@Cue].
We consider a one dimensional lattice with periodic boundary conditions. The surface at any instant of time *t* is described by the height $h_{i}(t)$ at each lattice site $i=1,\ldots,L $. We consider initially (at time $t=0$) a flat surface $h_i(0) = \mbox{const}$. The erosion takes place with probability $p$ while the diffusive process occurs with probability $(1-p)$ at a randomly chosen lattice site. The surface is evolved by following dynamical rules.
\(i) [*Erosion:*]{} The particle on the surface at site $i$ gets eroded with a probability $Y(\phi_i)p_e$ where $\phi_i = \tan^{-1}((h_{i+1}-h_{i-1})/2)$ and $1/7 \leq p_e \leq 1$ is the ratio of the number of occupied neighbors to the total number of neighboring sites [@Ken]. The ratio $p_{e}$ accounts for the unstable erosion mechanism due to the finite penetration depth of the bombarding ions into the eroded substrate [@Cue; @Bra; @Cue1].\
(ii) [*Surface diffusion:*]{} The surface diffusion process is taken into account by nearest neighbor hopping. The hopping rate from an initial state $i$ to a final state $f$ is given by $w_{i,f} = [1+\exp(\beta \Delta H_{i\rightarrow f})]^{-1}$ where $\Delta H_{i\rightarrow f} = H_{f} - H_{i}$ is the difference in the energy of the states and $\beta^{-1} = K_{B} T$ where $T$ is the surface temperature and K$_{B}$ is the Boltzmann constant. The surface Hamiltonian is given by $$H = \frac{J}{2} \sum_{\langle i,j \rangle} |h_i-h_j|^2,
\label{hamiltonian1}$$ where ${\langle i,j \rangle}$ denotes sum over nearest neighbor sites and $J$ is the coupling strength. Similar relaxation behaviour has been considered by Cuerno [[*et. al.*]{}]{}[@Cue] to describe the surfaces evolving from initial ripple structure into rough surfaces of KPZ class. Morphology of several other non-metallic surfaces, post ion irradiation, have also been describe sufficiently well by this model [@Ell; @Yan].
The morphology of metallic surfaces, however, cannot be described by the above model alone. In these cases, a diffusing atom is repelled from the lower step and preferably diffuses in the uphill direction. This Schwoebel effect is controlled by the potential barrier, Schwoebel barrier, and has been considered to be crucial for MBE grown surfaces [@Sie; @Fer; @Sie1]. Growth models based on MBE studies, have shown that positive Schwoebel effect can be incorporated in the relaxation process Hamiltonian by the inclusion of a quartic term [@Sie]. In the present study, we model the relaxation behaviour on ion sputtered Tantalum metal surfaces by modifying [Eq. (\[hamiltonian1\])]{} to include the Schwoebel effect in the Hamiltonian: $$H = \frac{J}{2} \sum_{\langle i,j \rangle} |h_i-h_j|^2 + \epsilon |h_i-h_j|^4.
\label{hamiltonian2}$$ Here $0<\epsilon<1$ is a non-linearity parameter which controls the intensity of Schwoebel effect. The additional quartic term in [Eq. (\[hamiltonian2\])]{}, as the results presented here shows, can be crucial for relaxation after ion irradiation of metallic surfaces as it is responsible for an uphill current which results in the formation of sharp peaked ’groove’ structures.
The algorithm for the Monte Carlo simulation is following. A site $1 \leq i \leq N$ is chosen at random and is subjected to follow erosion process with probability $p$ or the diffusive process with probability $1-p$. If erosion process is chosen, the angle $\phi_i$ is computed and a particle is eroded with probability $Y(\phi_i)p_e$. On the other hand if diffusive process is chosen, $w_{i,f}$ is computed using the surface Hamiltonian [Eq. (\[hamiltonian2\])]{} and the new configuration is updated. Time $\it{t}$ is incremented by one unit.
Experimental Details {#exp}
====================
High purity (99.99$\%$) Tantalum foils were bombarded by 3keV Ar ions under UHV conditions. The angle of incidence for ion beam was 15$^\circ$ w.r.t surface normal and its flux was 10$^{13}$ions/cm$^2$. Scanning Probe Microscopy (SPM) studies have been conducted on the surfaces by using Bruker (Nanoscope V) system in tapping mode.
Results and discussion {#result}
======================
[Fig. \[fig:afmimg\]]{} displays an SPM image from a Tantalum surface bombarded by Ar$^+$ ions at fluence of 3.6$\times10^{16}$ions/cm$^{2}$. A quasi periodic ripple pattern with a wavelength of $\sim$80nm is observed. 1-dimension height profile from the experiment (section marked in [Fig. \[fig:afmimg\]]{}) and from KMC simulations are presented in [Fig. \[fig:morph\]]{}.
For KMC simulations, first we study the model that considers erosion and includes relaxation mechanisms via only quadratic term in the Hamiltonian i.e for $\epsilon = 0.0$ in [Eq. (\[hamiltonian2\])]{}. A range of parameters were chosen for simulations and the results are presented in [Fig. \[fig:morph\]]{}, for $p=0.1$, $J_{c}/K_{B}T=0.25$. The height profile shows a periodic structure, usually similar to the morphologies observed for non metallic surfaces [@Jav] where the Schwoebel effects are not essential. Experimental height profile has several sharp peaks and grooves while the simulated profile has only smooth morphology.
Next we examine the model where we consider relaxation of the surface by including both quadratic and quartic terms in the Hamiltonian [Eq. (\[hamiltonian2\])]{}. The non-linear parameter $\epsilon$ is varied between 0.001 and 1.0. The surface morphology with $\epsilon =0.01$ is presented in [Fig. \[fig:morph\]]{} for $p=0.1$, $J_{c}/K_{B}T=0.25$. The simulated height profile here, with $\epsilon=0.01$, displays good agreement with experiments where formation of groove like structures are clearly observed. These features are characteristic signature of positive Schwoebel barrier that force the atom to move in uphill direction by breaking the translational invariance symmetry. A high diffusion rate, as observed here ($p=0.1$), is expected for metallic surfaces [@Val]. The steady state height profile for KMC simulations are also shown in [Fig. \[fig:morph\]]{} (inset).
The scaling behaviours and related exponents have also been explored here to investigate the nature of the growing surface. The exponents are useful as they depend on the growth condition and not on the microscopic details of the system. The correlation length $\xi$, which represents the typical wavelength of fluctuations on the growing surface, also characterizes the phenomenon of kinetic roughening. The width of the surface grows as $W(t) \sim \xi(t)^{\alpha}$ for roughness exponent ${\alpha}$. The scale invariant surfaces lead to scaling laws for correlation functions. The equal time height-height correlation (HHC) function can be written as: $$G(\mathbf{r},t)=L^{-1}\sum_{\mathbf{r'}}\langle[h(\mathbf{r+r'},t)-h(\mathbf{r'},t)]^{2}\rangle.
\label{hhc.eqn}$$ Here $\mathbf{r}$ is the Translational length along lateral direction of the 1-d lattice and $\langle\cdot\rangle$ denotes the ensemble average. This HHC function has the following scaling form: $$G(\mathbf{r},t)= r^{2\alpha}g(r/\xi(t)).
\label{hhcscale.eqn}$$ with $g(x)\sim\mbox{constant}$ for $ x \ll 1$ and $g(x)\sim x^{-2\alpha}$ for $x \gg 1$
[Fig. \[fig:hhc\]]{} presents the 1-dimensional height-height correlation function for experiment (using [Fig. \[fig:afmimg\]]{}) as well as KMC simulations. By utilizing the phenomenological scaling function of form $H(r)\sim [1-\exp(-(r/\xi)^{2\alpha})]$, values of $\xi$ and $\alpha$ have been obtained and are listed in Table 1. Although in absence of any Schwoebel effect ($\epsilon = 0.0$), the simulation results are very different from experimental HHC function, after inclusion of Schwoebel effect ($\epsilon = 0.01$) the results are in agreement. These results demonstrate that a small nonlinearity parameter with $\epsilon =0.01$ is essential for achieving experimentally consistent HHC functional form, $\xi$ and $\alpha$. This indicates that Schwoebel effect is necessary for understanding correct growth behaviour on Tantalum surface. Value of $\alpha$ obtained here for KMC simulation, in absence of Schwoebel effect ($\epsilon =0.0$), is similar to that observed in literature for MBE models with linear Hamiltonian [@Wol; @Das].
$\alpha$ $\xi$
----------------- --------------- ---------------
Experiment 1.22$\pm$0.26 5.66$\pm$0.36
$\epsilon=0.0$ 1.63$\pm$0.35 2.63$\pm$0.50
$\epsilon=0.01$ 1.20$\pm$0.07 7.14$\pm$0.05
: Roughness exponent $\alpha$ and correlation length $\xi$.[]{data-label="tabl"}
Obtaining $\alpha$ from G(r,t) can be difficult when the correlation length reaches the system size, specially in the steady state regime. In order to neglect finite size effects $\alpha$ can be computed in $0\le r \le L/2$ regime, where $\it{L}$ is the system size. Structure Factor mentioned below does not encounter this problem. The Structure Factor can be defined as [@Sie]: $$S(\mathbf{k},t)=\langle\hat{h}(\mathbf{k},t)\hat{h}(-\mathbf{k},t)\rangle.$$ Here, $\hat{h}(\mathbf{r},t)=L^{-d/2}\sum_{\mathbf{r}}[h(\mathbf{r,t})-\overline{h}]e^{\mathbf{ikr}}$, is the associated correlation function and $\overline{h}$ is the spatial average of h(r,t). This function has the following scaling form: $$S(\mathbf{k},t)={k}^{-\gamma} s(\mathbf{k}^{zt}).$$ with $\gamma = 2\alpha +d$. The scaling function [*s*]{} approaches a constant for large argument but behaves differently in the short time limit $x \ll 1$ where it has form $$s(x) \sim \left\{\begin{array}{c l}
x \mbox{~if~} \gamma\le z \\
x^{\gamma/z} \mbox{~if~} \gamma \ge z
\end{array}\right.$$
In [Fig. \[fig:struct\]]{}, the steady state structure factor $S(\mathbf {k})=S(\mathbf{k},t \rightarrow \infty)$ is shown for experiment as well as from KMC simulations The result clearly demonstrate that the non-linear Hamiltonian, with $\epsilon= 0.01$, agrees quite well with the experimental results. For $\epsilon =0.0$, we observe that $S(\mathbf{k})$ qualitatively differs from the experimental results. The value of exponent $\gamma$ obtained by linear Hamiltonian is 4.00$\pm$0.12 while for non-linear Hamiltonian, the value is 3.27$\pm$0.07. For the ion beam modified Tantalum surfaces, $\gamma =3.04\pm0.06$ has been observed here. This value of $\gamma$ is not expected from the universality classes of EW or KPZ. Thus the results indicate that positive Schwoebel effect is necessary for understanding the morphological growth of ion sputtered Tantalum surfaces which does not belong to the well known EW or the KPZ class.
Conclusion
==========
In Conclusion, we have presented a KMC model to describe the morphology of ion sputtered Tantalum surfaces. We have shown that a positive Schwoebel effect is needed to explain the characteristics of self organized nanostructures observed in the experiment. The Schwoebel effect has been used earlier for MBE growth models. Here, we have shown that, it can also be used for ion sputtered metallic surfaces such as Tantalum. The scaling exponents computed from the simulation and experimental data agrees quite well and indicates the presence of universality class that differs from that of non-metallic surfaces.
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![Morphology of Tantalum surface: SPM image (500nm$\times$500nm) after ion beam irradiation at a fluence of 36$\times10^{15}$ions/cm$^{2}$. []{data-label="fig:afmimg"}](Ta.eps){width="70.00000%"}
[^1]: shikha@iopb.res.in
|
---
author:
- Debraj Chakrabarti
title: 'Sets of Approximation and Interpolation in ${{\mathbb{C}}}$ for manifold-valued maps'
---
Introduction {#intro}
============
Notation
--------
This article is devoted to the study of approximation and interpolation of maps from a compact $K\subset{{\mathbb{C}}}$ into a complex manifold ${{\mathcal{M}}}$, and in particular to giving examples of sets $K$ for which certain types of approximation and interpolation are possible. For brevity, we introduce three properties $A_1$, $A_2$ and $A_3$ that compact subsets of the plane may possess. We first define property $A_2$.
For a compact $K\subset {{\mathbb{C}}}$, let ${{K}^{\circ}}$ be the interior of $K$. We will let ${\mathcal{A}\left({K},{{{\mathcal{M}}}}\right)}$ denote the continuous maps from $K$ into the complex manifold ${{\mathcal{M}}}$ which are holomorphic on ${{K}^{\circ}}$. Let ${\mathcal{O}\left({K},{{{\mathcal{M}}}}\right)}\subset{\mathcal{A}\left({K},{{{\mathcal{M}}}}\right)}$ denote the subspace of those maps $f$ which extend to a holomorphic map from some neighborhood $U_f$ of $K$ to ${{\mathcal{M}}}$. We endow ${{\mathcal{M}}}$ with an arbitrary metric, which makes ${\mathcal{A}\left({K},{{{\mathcal{M}}}}\right)}$ and ${\mathcal{O}\left({K},{{{\mathcal{M}}}}\right)}$ into metric spaces with the uniform metric on maps. We will say that a compact $K\subset{{\mathbb{C}}}$ has the property $A_2$, if, [*for every complex manifold ${{\mathcal{M}}}$, every finite set $\mathcal{P}\subset{{\mathcal{M}}}$, and every $\epsilon>0$, we can approximate any $f\in{\mathcal{A}\left({K},{{{\mathcal{M}}}}\right)}$ by a map $f_\epsilon\in{\mathcal{O}\left({K},{{{\mathcal{M}}}}\right)}$ such that ${{\ensuremath{\operatorname{dist}}}}_{{\mathcal{M}}}(f,f_\epsilon)<\epsilon$, and for each $p\in\mathcal{P}$, we have $f_\epsilon(p)=f(p)$.*]{} That is, $f_\epsilon$ is an uniform approximation to $f$ which interpolates the values of $f$ on $\mathcal{P}$.
We now define the property $A_1$, which will be the special case of $A_2$ with ${{\mathcal{M}}}={{\mathbb{C}}}$. More precisely, we will say that $K\subset{{\mathbb{C}}}$ has the property $A_2$, if, [*for any finite $\mathcal{P}\subset
K$, any function in $\mathcal{A}(K){:=}{\mathcal{A}\left({K},{{{\mathbb{C}}}}\right)}$ can be uniformly approximated by functions in $\mathcal{O}(K){:=}{\mathcal{O}\left({K},{{{\mathbb{C}}}}\right)}$ which interpolate the values of $f$ on $\mathcal{P}$.*]{} Obviously for a set $K$, $A_2\Rightarrow
A_1$.
It is easy to see that a set $K$ has property $A_1$ iff $\mathcal{O}(K)$ is dense in $\mathcal{A}(K)$. Note the trivial fact that there is a constant $C>0$ such that for $g\in\mathcal{C}(K)$, if $L_\mathcal{P}(g)$ is the Lagrange polynomial which interpolates the values of $g$ on $\mathcal{P}$, we have ${\left\|{L_\mathcal{P}(g)}\right\|}_K<C{\left\|{g}\right\|}_K$. If $\tilde{f}\in\mathcal{O}(K)$ be such that ${\left|{f-\tilde{f}}\right|}<\frac{\epsilon}{C+1}$, then ${\left|{f-f_\epsilon}\right|}<\epsilon$, and $f(p)=f_\epsilon(p)$ for $p\in\mathcal{P}$, where $f_\epsilon= \tilde{f} +
L_\mathcal{P}(f-\tilde{f})$.
The compact sets $K\subset{{\mathbb{C}}}$ such that $\mathcal{O}(K)$ is dense in $\mathcal{A}(K)$ can be characterized by Vituškin’s theorem ( [@vitushkin]). A sufficient condition is that ${{\mathbb{C}}}\setminus K$ has finitely many connected components. Also, such approximation can be localized, in the sense that $\mathcal{O}(K)$ is dense in $\mathcal{A}(K)$ iff every point $z\in K$ has a neighborhood $U_z$ in ${{\mathbb{C}}}$ such that $\mathcal{O}(K\cap \overline{U_z})$ is dense in $\mathcal{A}(K\cap \overline{U_z})$.
We now define property $A_3$. Let $\phi:{{\mathcal{M}}}\rightarrow{{\mathbb{C}}}$ be a holomorphic submersion such that $\phi(K)\supset {{\mathcal{M}}}$. Let ${\mathcal{A}_\phi\left({K},{{{\mathcal{M}}}}\right)}$ (resp. ${\mathcal{O}_\phi\left({K},{{{\mathcal{M}}}}\right)}$ ) be the subspace of ${\mathcal{A}\left({K},{{{\mathcal{M}}}}\right)}$ (resp. ${\mathcal{O}\left({K},{{{\mathcal{M}}}}\right)}$) consisting of sections of $\phi$ over $K$, i.e. maps $s:K\rightarrow{{\mathcal{M}}}$ such that $\phi\circ s=
{{\mathbb{I}}}_K$. We will say that a compact $K\subset {{\mathbb{C}}}$ has property $A_3$ if [*for every complex manifold ${{\mathcal{M}}}$ and every holomorphic submersion $\phi$ such that $\phi({{\mathcal{M}}})\supset K$, and every finite set $\mathcal{P}\subset K$, every section $\sigma\in{\mathcal{A}_\phi\left({K},{{{\mathcal{M}}}}\right)}$ can be uniformly approximated by sections in ${\mathcal{O}_\phi\left({K},{{{\mathcal{M}}}}\right)}$ which interpolate the values of $\sigma$ on $\mathcal{P}$*]{}.
We also note the following elementary fact:
\[aone\] For a compact $K\subset {{\mathbb{C}}}$, $A_3\Rightarrow A_2 $.
Assume that $K$ satisfies $A_3$, and let $\mathcal{P}$ and ${{\mathcal{M}}}$ have the same meaning as above, and let $f\in{\mathcal{A}\left({K},{{{\mathcal{M}}}}\right)}$. Consider the complex manifold $\mathcal{N}={{\mathcal{M}}}\times {{\mathbb{C}}}$ and let $\phi$ and $\pi$ be the projections onto ${{\mathbb{C}}}$ and ${{\mathcal{M}}}$ respectively. If $F:K\rightarrow
\mathcal{N}$ is defined by $F(z)=(f(z),z)$, then clearly $F\in{\mathcal{A}_\phi\left({K},{\mathcal{N}}\right)}$ and since $K$ has the property $A_3$, we can approximate $F$ by maps $G\in {\mathcal{O}_\phi\left({K},{\mathcal{N}}\right)}$ such that $G(p)=F(p)=(f(p),p)$ for each $p\in\mathcal{P}$. Then $g=\pi\circ G$ is in ${\mathcal{O}\left({K},{{{\mathcal{M}}}}\right)}$ and an approximation to $f$ with $g(p)=f(p)$ for each $p\in\mathcal{P}$, which shows that $K$ has property $A_2$.
(We may remark here that while holomorphic submersions $\phi:{{\mathcal{M}}}\rightarrow{{\mathbb{C}}}$ always exist if ${{\mathcal{M}}}$ is Stein ([@forstneric:noncritical]), in general such ${{\mathcal{M}}}$ may be highly nontrivial, see e.g. [@demailly:nonsteinbundle].)
It is natural to ask the question: [*which are the compact sets in the plane which satisfy property $A_2$ or property $A_3$?*]{} Given that approximation is local in nature, one can conjecture that using patching arguments in the manifold ${{\mathcal{M}}}$ we may be able to show that every set with property $A_1$ has property $A_2$, or even $A_3$. We do not know if this program can be carried out. In this article, we confine ourselves to to the much more modest goal of giving examples of sets $K$ for which properties $A_2$ and $A_3$ hold (see Theorems \[onedim\],\[atwo\] and \[athree\] below.)
Known results
-------------
Some results have been obtained recently regarding the approximation and interpolation of manifold-valued maps.
The following was proved in [@michiganpaper]. Define a [*Jordan domain*]{} to be an open set $\Omega\Subset{{\mathbb{C}}}$, such that the boundary $\partial\Omega$ has finitely many connected components, each of which is homeomorphic to a circle. (We explicitly allow $\Omega$ to be multiply connected.)
\[thesis\] Let $K=\overline{\Omega}$, where $\Omega\Subset{{\mathbb{C}}}$ is a Jordan domain. For any complex manifold ${{\mathcal{M}}}$, the subspace ${\mathcal{O}\left({K},{{{\mathcal{M}}}}\right)}$ is dense in ${\mathcal{A}\left({K},{{{\mathcal{M}}}}\right)}$.
This is purely a statement about approximation, no interpolation being involved. The following result is contained in what was proved by Drinovec-Drnovšek and Forstnerič in [@bddforstneric:holocurves], Theorem 5.1:
\[forst\] Let $\Omega\Subset{{\mathbb{C}}}$ have $\mathcal{C}^2$ boundary, ${{\mathcal{M}}}$ be a complex manifold, and let $f:\overline{\Omega}\rightarrow{{\mathcal{M}}}$ be a map of class $\mathcal{C}^r$ ($r\geq 2$) which is holomorphic in $\Omega$. Given finitely many points $z_1,\ldots, z_l\in\Omega$, and an integer $k\in\mathbb{}{N}$, there is a sequence of maps $f_\nu\in{\mathcal{O}\left({K},{{{\mathcal{M}}}}\right)}$ such that $f_\nu$ agrees with $f$ to order $k$ at $z_j$ for $j=1,\ldots,l$ and $\nu\in\mathbb{N}$, and the sequence $f_\nu$ converges to $f$ in $\mathcal{C}^r(\overline{\Omega})$ as $\nu\rightarrow\infty$.
This can in fact be proved when $\Omega\Subset S$, where $S$ is a non-compact Riemann surface, and ${{\mathcal{M}}}$ is a complex space. Observe that there is a stronger assumption on the smoothness of the map than in property $A_2$, and also a stronger conclusion regarding approximation and interpolation. However, the set of points of interpolation is restricted to the interior of $\overline{\Omega}$. The authors subsequently proved the following:
\[forst2\] If $\Omega\Subset{{\mathbb{C}}}$ is a domain with $\mathcal{C}^2$ boundary, ${{\mathcal{M}}}$ a complex manifold, and $\phi:{{\mathcal{M}}}\rightarrow{{\mathbb{C}}}$ a holomorphic submersion such that $\phi({{\mathcal{M}}})\supset\overline{\Omega}$, then ${\mathcal{O}_\phi\left({\overline{\Omega}},{{{\mathcal{M}}}}\right)}$ is dense in ${\mathcal{A}_\phi\left({\overline{\Omega}},{{{\mathcal{M}}}}\right)}$.
This is a (very) special case of [@bddforstneric:approximation], Theorem 5.1 regarding the approximation of sections of submersions over strongly pseudoconvex sets in Stein manifolds, which may be thought of as a far-reaching generalization of the Henkin-Ramírez-Kerzman approximation theorem for functions on such domains ([@henkinleiterer:book], Theorem 2.9.2. )
Sets with property $A_2$ or $A_3$
---------------------------------
We recall some definitions from elementary point-set topology. Let $X$ be a topological space. For an integer $n\geq 0$, we say that ${\ensuremath{\operatorname{dim}}}(X)\leq n$ if the following holds : given any open cover $\mathcal{A}$ of $X$, there is a refinement $\mathcal{B}$ of $\mathcal{A}$ such that any point of $X$ is contained in at most $n+1$ of the sets in the cover $\mathcal{B}$. If ${\ensuremath{\operatorname{dim}}}(X)\leq n$ holds, but ${\ensuremath{\operatorname{dim}}}(X)\leq n-1$ does not, we say that the dimension ${\ensuremath{\operatorname{dim}}}(X)$ of $X$ is $n$. The dimension is clearly a topological invariant of $X$. Also, if ${\ensuremath{\operatorname{dim}}}(X)=n$, then each connected component of $X$ (or more generally, any closed subset) has dimension less than or equal to $n$.
We say that a topological space is [*locally contractible*]{} if there is a basis of the topology consisting of contractible open sets. We can now state the following:
\[onedim\] Let $K\subset {{\mathbb{C}}}$ be a connected compact set, such that ${\ensuremath{\operatorname{dim}}}(K)=1$, and suppose that $K$ is locally contractible. Then $K$ has property $A_3$.
Examples of sets that satisfy the hypotheses are
1. arcs in ${{\mathbb{C}}}$, where By an [*arc*]{} $\alpha$ in a topological space $X$ we mean a continuous [*injective*]{} map $\alpha:[0,1]\rightarrow X$ from the unit interval. The injectivity avoids pathologies like space filling curves. By standard abuse of language, we will refer to the image $\alpha([0,1])$ as the arc $\alpha$.
2. planar realizations of connected [*graphs*]{} i.e. finite simplicial complexes having only $0$-simplices (“vertices") and $1$-simplices (“edges".) We can think of these as unions of arcs in the plane, any two of which meet at at most one endpoint of each.
Let $K\subset {{\mathbb{C}}}$, then ${\ensuremath{\operatorname{dim}}}(K)\leq 1$ iff ${{K}^{\circ}}=\emptyset$ ([@ems:gtopology1], p.133,Theorem 20.) However, it is possible for a set $K$ with ${{K}^{\circ}}=\emptyset$, to be [*not*]{} locally contractible, and yet to satisfy $A_1$ (recall that $A_1$ is equivalent to $\mathcal{O}(K)$ being dense in $\mathcal{A}(K)$.) An example is the “bouquet of circles" $K=\bigcup_{n=1}^{\infty} C_n$, where $C_n$ is the circle in the plane with center at $\frac{1}{n}+i0$, and radius $\frac{1}{n}$. It can be shown that this $K$ has property $A_1$, yet Theorem \[onedim\] does not apply to it. (We don’t know if $K$ has property $A_3$ or $A_2$.)
For completeness, we record the following easy fact:
\[zerodim\] Let $K$ be a compact subset of ${{\mathbb{C}}}$ with ${\ensuremath{\operatorname{dim}}}(K)=0$. Then $K$ has property $A_3$.
We now go on to give examples of two-dimensional sets which have properties $A_2$ or $A_3$. Let $k\geq 0$ be an integer, $\infty$ or $\omega$. We introduce a class of sets in the plane denoted by $\mathfrak{C}_k$. We will let $\mathfrak{C}_k$ denote the class of compact sets $K\subset{{\mathbb{C}}}$ such that there is an integer $N$ and domains $\Omega_j$ , where $j=1,\ldots, N$ with the following properties:
- For $k\not=0$, each $\Omega_j$ is a domain with $\mathcal{C}^k$ boundary, and for $k=0$, each $\Omega_j$ is a Jordan domain.
- The $\Omega_j$’s are pairwise disjoint : $\Omega_i\cap \Omega_j=\emptyset$ if $i\not=j$.
- $K=\cup_{i=1}^N \overline{\Omega_j}$. We will refer to each $\Omega_j$ as a [*summand*]{} of $K$.
- whenever $i\not = j$, the set $$P_{ij}{:=}\overline{\Omega_i}\cap \overline{\Omega_j}=\partial\Omega_i\cap \partial\Omega_j$$ is finite.
- If $k\not= 0$, at each point of $P_{ij}$, the boundaries $\partial\Omega_i$ and $\partial\Omega_j$ are tangent to each other.
We can now provide a supply of sets which have property $A_2$:
\[atwo\] Each compact set $K$ of class $\mathfrak{C}_0$ has property $A_2$.
This can be thought of as a generalization of Proposition \[thesis\]. The second result gives examples of sets with property $A_3$:
\[athree\] Each compact set $K$ of class $\mathfrak{C}_2$ has property $A_3$.
This again can be thought of as a generalization of Proposition \[forst\].
In [§\[goodpairsection\]]{} below, we prove a general result (Theorem \[goodpairapprox\] ) regarding approximation of sections of submersions over a set $K$ which admits a decomposition $K=K_1\cup K_2$ into a “good pair" $(K_1,K_2)$ of compact sets. This is a direct generalization of the results in [@michiganpaper], $\S$4.1 for maps into manifolds. Proofs of all new statements are given in detail, although they are similar to the proofs in the case of maps. This is the tool for patching maps which is used to prove Theorems \[onedim\] and \[athree\].
In [§\[athreesection\]]{}, we give proofs of Lemma \[zerodim\], Theorem \[onedim\], and Theorem \[athree\]. For the last two, the crucial ingredient is three successive applications of Theorem \[goodpairapprox\] to produce a section which is holomorphic in a neighborhood of the given compact $K$. For Theorem \[athree\] we also require a result from [@michiganpaper] (Theorem \[arcs\]). In [§\[atwosection\]]{} we deduce Theorem \[atwo\] from Theorem \[athree\]. This requires a result from [@macgregor:interpolation] regarding conformal mappings continuous to the boundary.
It should be noted that the results proved here have analogs for the approximation and interpolation of $\mathcal{A}^k$ maps, i.e., maps which are $\mathcal{C}^k$ and holomorphic in the interior. The proofs are exactly the same. For notational simplicity we stick to the case of maps which are only continuous to the boundary. Also, this article is largely self contained, with the exception of the proofs of Theorem \[arcs\] and Proposition \[univalent\].
Approximation on Good Pairs {#goodpairsection}
===========================
Some definitions
----------------
We will call a set $K\subset {{\mathbb{C}}}$ [*nicely contractible*]{} if there is a homotopy $c:[0,1]\times K\rightarrow K$ with the following properties:
1. for each $t$, the map $z\mapsto c(t,z)$ is in ${\mathcal{A}\left({K},{{{\mathbb{C}}}}\right)}$,
2. $z\mapsto c(1,z)$ is the identity map on $K$.
3. there is a $z_0\in K$ such that $c(0,z)\equiv z_0$.
4. for $t\not=0$, $z\mapsto c(t,z)$ maps ${{K}^{\circ}}$ into ${{K}^{\circ}}$.
Of course, convex sets are nicely contractible, as are strongly star-shaped sets (these are sets $K$ such that the maps $c(t,z)$ may be taken as dilations with stretch factor $t$ and center $z_0\in
K$.) Another important class of nicely contractible sets are arcs. The property of nicely contractible sets which is used in the proof of Lemma \[cartan\] (and therefore Theorem \[goodpairapprox\] is the following: if $K$ is nicely contractible and ${{\mathcal{M}}}$ is a connected complex manifold, then ${\mathcal{A}\left({K},{{{\mathcal{M}}}}\right)}$ is in fact contractible. We can take the contraction to be the map $\phi_t$ from ${\mathcal{A}\left({K},{{{\mathcal{M}}}}\right)}$ to itself given by $\phi_t(f)(z)= f(c(t,z)).$
Let $K_1$ and $K_2$ be compact subsets of ${{\mathbb{C}}}$. We will say that $(K_1,K_2)$ is a [*good pair*]{} if the following hold:
1. $\overline{K_1\setminus K_2}\cap \overline{K_2\setminus
K_1}=\emptyset$.
2. $K_{1,2}{:=}K_1\cap K_2$ has finitely many connected components, each of which is nicely contractible.
Let ${{\mathcal{M}}}$ be a complex manifold of complex-dimension $n$, and $\phi:{{\mathcal{M}}}\rightarrow{{\mathbb{C}}}$ be a submersion. We will say that an open set $U\subset {{\mathcal{M}}}$ is [*$\phi$-adapted*]{}, if $U$ is biholomorphic to an open set in ${{\mathbb{C}}}^n$, and there are holomorphic coordinates $(z_1,\cdots,z_n)$ on $U$ such that with respect to these coordinates (and the standard coordinate on ${{\mathbb{C}}}$), the map $\phi$ takes the form $(z_1,\cdots,z_n)\mapsto z_n$. (This is of course the same as saying $z_n=\phi|_U$.)
The main result
---------------
We introduce the following notation and definitions:
1. Let $K=K_1\cup K_2$, where $(K_1,K_2)$ is a good pair.
2. Let ${{\mathcal{M}}}$ be a complex manifold and $\phi:{{\mathcal{M}}}\rightarrow{{\mathbb{C}}}$ be a holomorphic submersion such that $\phi({{\mathcal{M}}})\supset K$.
3. Let $B\subset{{\mathbb{C}}}$ be compact and such that $B\cap
K_1=\emptyset$, and each function $g\in\mathcal{A}(K_2)$ can be approximated uniformly on $K_2$ by functions in $\mathcal{A}(K_2\cup
B)$ ;i.e., if $g\in \mathcal{A}(K_2)$ and $\epsilon>0$, then there is a $g_\eta\in \mathcal{A}(K_2\cup B)$ such that ${\left|{g-g_\eta}\right|}<\eta$ on $K_2$.
4. Let $\mathcal{P}$ be a finite subset of $K$.
We now state the following result regarding the approximation of sections of $\phi$ over $K$:
\[goodpairapprox\] With $K_1, K_2,{{\mathcal{M}}},\phi$, $B$, and $\mathcal{P}$ as above, let $s\in{\mathcal{A}_\phi\left({K},{{{\mathcal{M}}}}\right)}$ be a section of $\phi$ such that each $s(K_j)$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$ for $j=1,2$. Then, given $\eta>0$, there is an $s_\eta\in {\mathcal{A}_\phi\left({K},{{{\mathcal{M}}}}\right)}$ such that ${{\ensuremath{\operatorname{dist}}}}(s,s_\eta)<\eta$ on $K$, $s_\eta $ extends as a holomorphic section of $\phi$ to a neighborhood $B_\eta$ of $K_2\cap B$, and for each $p\in\mathcal{P}$, we have $s_\eta(p)= s(p)$. Moreover, $s_\eta(K_2\cup B_\eta)$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$.
The proof of \[goodpairapprox\] will be reduced to the solution of a non-linear patching problem in Euclidean space by the use of coordinate in neighborhoods of $s(K_j)$. We now describe the ingredients required in the proof . The most important is the following version (resembling that A. Douady in [@douady:modules], pp.47-48)of H. Cartan’s lemma on holomorphic matrices.
\[cartan\] Let $\mathfrak{G}$ be a complex connected Lie Group, and let $(K_1, K_2)$ be a good pair, and let $g\in \mathcal{A}(K_{1,2},\mathfrak{G})$. Then, for $j=1,2$ there are $g_j\in \mathcal{A}(K_j,\mathfrak{G})$ such that $g=g_2\cdot
g_1$ on $K_{1,2}$.
For a proof, see [@michiganpaper], Lemma 4.4, where it is assumed that $\mathfrak{G}=GL_n({{\mathbb{C}}})$, and each component of the intersection is star shaped, but the proof is valid in general.
If $\mathcal{P}\subset {{\mathbb{C}}}$ is a finite set, we will let $\mathcal{A}^\mathcal{P}(K,{{{\mathbb{C}}}^n})$ denote the closed subspace of the Banach space ${\mathcal{A}\left({K},{{{\mathbb{C}}}^n}\right)}$ consisting of those functions which vanish at each $p\in K\cap\mathcal{P}$. We will require the following version of the solution of the additive Cousin problem continuous to the boundary (the case with $\mathcal{P}=\emptyset$ is in fact used in the proof of Lemma \[cartan\] ):
\[additive\] Let $K_1,K_2$ be a compact subsets of the plane such that $\overline{K_1\setminus K_2}\cap\overline{K_2\setminus
K_1}=\emptyset$, and let $\mathcal{P}$ be a finite subset of the plane. There exist bounded linear maps $T_j:
\mathcal{A}^\mathcal{P}(K_{1,2},{{\mathbb{C}}})\rightarrow
\mathcal{A}^\mathcal{P}(K_j,{{\mathbb{C}}})$ such that for any function $f$ in $\mathcal{A}^\mathcal{P}(K_{1,2},{{\mathbb{C}}})$ we have on $K_{1,2}$, $$\label{additive_eq}
T_1f + T_2f = f,$$
We reduce the problem to a ${\overline{\partial}}$ equation in the standard way. Let $\chi$ be a smooth cutoff which is 1 near $\overline{K_1\setminus K_2}$ and 0 near $\overline{K_2\setminus K_1}$. Let $\lambda{:=}f.{\frac{\partial{\chi}}{\partial\overline{z}}}$, so that $\lambda\in \mathcal{A}(K_{1,2},{{\mathbb{C}}})$. Let $$\Lambda_f(z) =\frac{1}{2\pi i}\int_{K_{1,2}}\frac{\lambda(\zeta)}{\zeta-z}
d\overline{\zeta}\wedge d\zeta$$ Let $q_f$ be the Lagrange interpolation polynomial with the property that $q_f(p)=\Lambda_f(p)$ for $p\in \mathcal{P}$. Observe that both $\Lambda_f$ and $q_f$ are linear in $f$ and continuous in the sup norm when restricted to compact sets.
We can now define (assuming $(1-\chi).f=0$ where $\chi=1$ even if $f$ is not defined): $$(T_1 f)(z) = (1-\chi(z)).f(z) +
\Lambda_f(z) -q_f(z)$$
and (assuming $\chi.f=0$ where $\chi=0$ even if $f$ is not defined): $$(T_2 f)(z) = \chi(z).f(z)
- \Lambda_f(z) +q_f(z),$$ Since $T_jf$ is clearly holomorphic (resp. continuous) where $f$ is, the result follows.
We will use the following standard result regarding Banach spaces, which can be proved by iteration (see [@lang:realanalysis] pp. 397-98):
\[surjective\] In a metric space $X$, let $B_X(p,r)$ denote the open ball in $X$ of radius $r$ centered at $p$. Let $\mathcal{E}$ and $\mathcal{F}$ be Banach Spaces and let $\Phi: B_{\mathcal{E}}(p,r)\rightarrow\mathcal{F}$ be a $\mathcal{C}^1$ map. Suppose there is a constant $C>0$ such that:
- for each $h\in B_{\mathcal{E}}(p,r)$, the linear operator $\Phi'(h):\mathcal{E}\rightarrow\mathcal{F}$ is surjective and the equation $\Phi'(h)u=g$ can be solved for $u$ in $\mathcal{E}$ for all $g$ in $\mathcal{F}$ with the estimate ${\left\|{u}\right\|}_{\mathcal{E}}\leq C {\left\|{g}\right\|}_{\mathcal{F}}$.
- for any $h_1$ and $h_2$ in $B_{\mathcal{E}}(p,r)$ we have ${\left\|{
\Phi'(h_1)-\Phi'(h_2)}\right\|}\leq \frac{1}{2C}.$
Then, $$\Phi(B_{\mathcal{E}}(p,r)) \supset B_\mathcal{F}\left( \Phi(p),\frac{r}{2C}\right).$$
We will use the three lemmas above to give a proof of the following result (the main component of which is due to Rosay, [@rosay:preprint], and comments in [@rosay:korean]) regarding the solution of a non-linear Cousin problem (see Lemma 4.5 of [@michiganpaper].) It is simply a translation of Theorem \[goodpairapprox\] to coordinates.
\[rosay\] Let $\omega$ be an open subset of ${{\mathbb{C}}}^n$ and let $\mathfrak{F}:\omega\rightarrow{{\mathbb{C}}}^n$ be a holomorphic immersion. Assume that $\mathfrak{F}$ preserves the last coordinate, i.e., $\mathfrak{F}$ is of the form $\mathfrak{F}(z_1,\cdots,z_n)=
\left(F(z_1,\cdots,z_n),z_n\right)$ for some map $F:\omega\rightarrow{{\mathbb{C}}}^{n-1}$.
Let $(K_1,K_2)$ denote a good pair of compact subsets of ${{\mathbb{C}}}$, let $\mathcal{P}$ be a finite subset of $K=K_1\cup K_2$, and suppose for each $p\in K_2\setminus K_1$ we are given a point $q(p)\in{{\mathbb{C}}}^{n-1}$. Let $u_1\in {\mathcal{A}\left({K_1},{{{\mathbb{C}}}^n}\right)}$ be such that
- $u_1(K_{1,2})\subset\omega$, and
- $u_1$ is of the form $u_1(z)=
\left(t_1(z),z\right)$ where $t_1:K_1\rightarrow{{\mathbb{C}}}^{n-1}$
Given any $\epsilon>0$, there exists $\delta>0$ such that if $u_2\in{\mathcal{A}\left({K_2},{{{\mathbb{C}}}^n}\right)}$ be such that
- ${\left\|{u_2-\mathfrak{F}\circ u_1}\right\|}<\delta$ on $K_2\cap K_2$,
- for $p\in \mathcal{P}\cap \left(K_{1,2}\right)$, we have $u_2(p)=\mathfrak{F}(u_1(p))$,
- for $p\in K_2\setminus K_1$ we have $u_2(p)=(q(p),p)\in{{\mathbb{C}}}^n$, and
- $u_2(z) =(t_2(z),z)$, where $t_2:K_2\rightarrow{{\mathbb{C}}}^{n-1}$
then for $j=1,2$ there exist $v_j\in
\mathcal{A}^\mathcal{P}(K_j,{{\mathbb{C}}}^n)$ such that
- ${\left\|{v_j}\right\|}<\epsilon$,
- $u_2+v_2=\mathfrak{F}(u_1+v_1)$, and
- the last coordinate function of each of $v_1$ and $v_2$ is 0.
We now give a proof of Lemma \[rosay\].
In order to apply Lemma \[surjective\] we choose the Banach spaces $\mathcal{E}$, $\mathcal{F}$ and the map $\Phi$ as follows:
- For $j=1,2$, let $\mathcal{B}_j$ be the closed subspace of the Banach space $\mathcal{A}^\mathcal{P}(K_j,{{\mathbb{C}}}^n)$ consisting of those maps whose last coordinate function is 0, i.e., $\mathcal{B}_j=\mathcal{A}^\mathcal{P}(K_j,{{\mathbb{C}}}^n)\oplus 0$. We now let $\mathcal{E}$ the Banach space $\mathcal{B}_1\oplus\mathcal{B}_2$, which we endow with the norm ${\left\|{\cdot}\right\|}_{\mathcal{E}} {:=}\max\left({\left\|{\cdot}\right\|}_{\mathcal{A}(K_1,{{\mathbb{C}}}^n)},{\left\|{\cdot}\right\|}_{\mathcal{A}(K_2,{{\mathbb{C}}}^n)}\right)$.
- Let $\mathcal{F}$ be the Banach space $\mathcal{A}^\mathcal{P}(K_{1,2},{{\mathbb{C}}}^{n-1})$.
- Let $\pi:{{\mathbb{C}}}^{n}\rightarrow{{\mathbb{C}}}^{n-1}$ denote the projection on the first $n-1$ coordinates, and let the open subset $\mathcal{U}$ of $\mathcal{E}$ be given by $\{(w_1,w_2): (u_1+w_1)(K_{1,2})\subset
\omega\}$. (Observe that $\{0\}\times\mathcal{B}(K_2,{{\mathbb{C}}}^n)\subset
\mathcal{U}$.) Let $P=(P_1,\ldots,P_n)$ be an $n$-tuple of polynomials such that $P_n(z)\equiv z$, for $p\in K_{1,2}\cap
\mathcal{P}$, we have $P(p)=\mathfrak{F}(u_1(p))$, and for $p\in
K_2\setminus K_1$ we have $(P_1(p),\ldots, P_{n-1}(p))=q(p)$.
Let the map $\Phi:\mathcal{U}\rightarrow\mathcal{F}$ be given by $$\Phi(w_1,w_2){:=}\pi\circ\left[(P+w_2)|_{K_{1,2}}-
\mathfrak{F}\circ ((u_1+w_1)|_{K_{1,2}})\right].$$ (Observe that, since $\mathfrak{F}$ preserves the last coordinate, the last coordinate function of $(P+w_2)|_{K_{1,2}}- \mathfrak{F}\circ
((u_1+w_1)|_{K_{1,2}})$ is actually 0. So the precomposition with $\pi$ simply drops a coordinate which is identically $0$.)
A computation shows that $\Phi'(w_1,w_2)$ is the bounded linear map from $\mathcal{E}$ to $\mathcal{F}$ given by $$(v_1,v_2)\mapsto \pi\circ \left[v_2|_{K_{1,2}}-
\mathfrak{F}'((u_1+w_1)|_{K_{1,2}})(v_1|_{K_{1,2}})\right].$$ Observe that $w_2$ plays no role whatsoever in this expression, and therefore $\Phi'(w_1,w_2)\in BL(\mathcal{E},\mathcal{F})$ is in fact a smooth function of $w_1$ alone, and we will henceforth denote it by $\Phi'(w_1,*)$.
Let $\mathfrak{G}\subset GL_n({{\mathbb{C}}})$ be the complex Lie subgroup of matrices of the form $$\left(\begin{array}{cc}A & b\\{\bf 0}&1\end{array}\right),$$ where $A\in GL_{n-1}({{\mathbb{C}}})$, $b$ is an $1\times (n-1)$ column vector, and ${\bf 0}$ is the zero row vector of size $(n-1)\times 1$. It is easy to see that $\mathfrak{G}$ is connected.
We construct a right inverse to $\Phi'(u_1,*)$ . Let $\gamma=\mathfrak{F}'\circ\left(u_1|_{K_{1,2}}\right)$. Since $\mathfrak{F}$ preserves the last coordinate, $\gamma\in
\mathcal{A}(K_{1,2},\mathfrak{G})$, and thanks to Lemma \[cartan\] above, we may write $\gamma= \gamma_2\cdot \gamma_1$, where $\gamma_j\in \mathcal{A}(K_j,\mathfrak{G})$. (We henceforth suppress the restriction signs.) Let $j:\mathcal{F}\rightarrow{\mathcal{A}\left({K_{1,2}},{{{\mathbb{C}}}^n}\right)}$ be the continuous inclusion induced by the map ${{\mathbb{C}}}^{n-1}\rightarrow{{\mathbb{C}}}^{n}$ given by $(z_1,\cdots, z_{n-1})\mapsto (z_1,\cdots,z_{n-1},0)$. For $g\in \mathcal{F}$, let $$S(g)=(-\gamma_1^{-1} T_1(\gamma_2^{-1}j(g)),\gamma_2
T_2(\gamma_2^{-1}j(g))),$$ where the $T_j$ are as in equation \[additive\_eq\] above (and extended to ${{\mathbb{C}}}^n$ componentwise). Observe that $-\gamma_1^{-1} T_1(\gamma_2^{-1}j(g))$ and $\gamma_2 T_2(\gamma_2^{-1}j(g))$ vanish at each $p\in\mathcal{P}$ and they belong to $\mathcal{B}_1$ and $\mathcal{B}_2$ respectively. It is easy to see that $S:\mathcal{F}\rightarrow\mathcal{E}$ is a bounded linear operator, and a computation shows that $\Phi'(u_1,*)\circ S={{\mathbb{I}}}_\mathcal{F}$. Choose $\theta>0$ so small so that if $w_1\in
B_\mathcal{F}(u_1,\theta),$
1. ${\left\|{\Phi'(w_1,*)-
\Phi'(u_1,*)}\right\|}_{{\ensuremath{\operatorname{op}}}}< \frac{1}{8{\left\|{S}\right\|}}$ (possible by continuity), and
2. the equation $\Phi'(w_1,*)u=g$ can be solved with the estimate ${\left\|{u}\right\|}\leq 2 {\left\|{S}\right\|} {\left\|{g}\right\|}$, (possible from the fact that small perturbations of a surjective linear operator is still surjective.)
Consequently, if $\epsilon<\theta$ and $\tilde{u}_2\in\mathcal{B}(K_2,{{\mathbb{C}}}^n)$, for the ball $B_{\mathcal{E}}((0,\tilde{u}_2),\epsilon)$ the hypothesis of Lemma \[surjective\] are verified with $C=2{\left\|{S}\right\|}$. We have therefore, $$\begin{aligned}
\Phi\left(B_\mathcal{E}((0,\tilde{u}_2),\epsilon)\right)&\supset&
B_\mathcal{F}\left(\Phi(0,\tilde{u}_2),\frac{\epsilon}{2C}\right)\\
& =& B_\mathcal{F}\left(P+\tilde{u}_2-
\mathfrak{F}({u_1}),\frac{\epsilon}{2C}\right).\end{aligned}$$ So, if ${\left\|{P+\tilde{u}_2- F(u_1)}\right\|}< \frac{\epsilon}{4C}$, we have $0\in\Phi\left(B_\mathcal{E}((0,\tilde{u}_2),\epsilon)\right)$. This is exactly the conclusion required, since any $u_2$ such that $u_2(p)=P(p)$ for each $p\in K_2\cap \mathcal{P}$ can be written as $u_2= P+\tilde{u}_2$ for some $\tilde{u}_2\in\mathcal{B}(K_2,{{\mathbb{C}}}^n)$.
We will now prove Theorem \[goodpairapprox\].
We omit the restriction signs on maps for notational clarity. For $j=1,2$ let $V_j$ be a $\phi$-adapted neighborhood of $s(K_j)$, and let $\mathfrak{F}_j:V_j\rightarrow{{\mathbb{C}}}^n$ be a coordinate system such that $\mathfrak{F}_j(z)=\left(F_j(z),\phi(z)\right)$. Let $\mathfrak{F}=\mathfrak{F}_2\circ\mathfrak{F}_1^{-1}$ be the associated transition function.Then $\mathfrak{F}$ is a biholomorphism from the open set $\omega=\mathfrak{F}_1(V_1\cap
V_2)$ onto the open set $\mathfrak{F}_2(V_2\cap V_1)$, and $\mathfrak{F}$ preserves the last coordinate, i.e., $\mathfrak{F}$ is of the form $\mathfrak{F}(z_1,\cdots,z_n)=
\left(F(z_1,\cdots,z_n),z_n\right)$ for some map $F:\omega\rightarrow{{\mathbb{C}}}^{n-1}$. Any section of $\phi$ over $K_j$ is represented in the coordinate system $\mathfrak{F}_j$ by a map of the form $t_j:K_j\rightarrow{{\mathbb{C}}}^n$, where $t_j(z)=
(\tilde{t}_j(z),z)$, with $\tilde{t}_j$ a map from $K_j$ to ${{\mathbb{C}}}^{n-1}$. Also, for $j=1,2$, given maps $t_j:K_j\rightarrow{{\mathbb{C}}}^n$ of the form $t_j=(\tilde{t}_j,z)$, they glue together to form a section over $K_1\cup K_2$ (i.e. there is a section $\lambda$ of $\phi$ over $K_1\cup K_2$ such that $t_j =\mathfrak{F}_j\circ
\lambda$) iff $t_2 = \mathfrak{F}\circ t_1$.
Since $B\cap
K_1=\emptyset$ by hypothesis, the pair of compact sets $(K_1,
K_2\cup B)$ is good. Let $\epsilon>0$, and let $u_1=\mathfrak{F}_1\circ s$. Observe that $u_1(K_{1,2})\subset
\omega$, and $u_1$ is of the form $u_1(z)= (t_1(z),z)$, where $t_1:K_1\rightarrow{{\mathbb{C}}}^{n-1}$. Then Lemma \[rosay\] gives a $\delta>0$ corresponding to $u_1$, the good pair $(K_1, K_2\cup B)$, $\omega$ and $\mathfrak{F}$.
Let $w_2 = \mathfrak{F}_2\circ (s|_{K_2})$, then $w_2\in{\mathcal{A}\left({K_2},{{{\mathbb{C}}}^n}\right)}$, and is of the form $w_2(z)=
(\tilde{w}_2(z),z)$. Thanks to the hypothesis regarding uniform approximation of functions in ${\mathcal{A}\left({K_2},{{{\mathbb{C}}}}\right)}$ by functions in ${\mathcal{A}\left({K_2\cup B},{{{\mathbb{C}}}}\right)}$, we can find a $u_2\in{\mathcal{A}\left({K_2\cup B},{{{\mathbb{C}}}^n}\right)}$ of the same form as $w_2$ such that ${\left\|{u_2 -
\mathfrak{F}(u_1)}\right\|}<\delta$ on $K_{1,2}$ (Since $\mathfrak{F}(u_1)=
w_2$ on $K_{1,2}$.) Further we may assume that for $p\in\mathcal{P}\cap K_2$, we have $u_2(p)=w_2(p)$ and that the last coordinate function of $u_2$ is $z$.
Then, by Proposition \[rosay\], there is a $v_1\in\mathcal{A}^\mathcal{P}(K_1,{{\mathbb{C}}}^n)$ and a $v_2\in\mathcal{A}^\mathcal{P}(K_2\cup B,{{\mathbb{C}}}^n)$, such that ${\left\|{v_j}\right\|}<\epsilon$ and $u_2 + v_2 = \mathfrak{F}(u_1 + v_1)$, and the last coordinate functions of the $v_j$ are 0’s. Hence the maps $u_1+v_1$ and $u_2+v_2$ glue together to form a section of $\phi$ (which we call $\tilde{s}_\epsilon$ given by $$\tilde{s}_{\epsilon}{:=}\left\{ \begin{array}{ccc}
\mathfrak{F}_1^{-1}(u_1+v_1)& {\ensuremath{\operatorname{ on }}} & K_1\\
\mathfrak{F}_2^{-1}(u_2+v_2) & {\ensuremath{\operatorname{ on }}} & K_2 {\ensuremath{\operatorname{ and ~near
}}} K_2\cap B
\end{array} \right.$$ Clearly, $\tilde{s}_\epsilon$ is in ${\mathcal{A}_\phi\left({K_1\cup K_2},{{{\mathcal{M}}}}\right)}$, and extends to a holomorphic map near $K_2\cap B$. Moreover, ${\ensuremath{\operatorname{dist}}}(\tilde{s}_\epsilon, s)= O(\epsilon)$. By construction, we have $s_\epsilon(p)=s(p)$. Therefore, given $\eta>0$, we can find $s_\eta$ with required properties.
Sets with property $A_3$ {#athreesection}
========================
In this section we prove Lemma \[zerodim\], and Theorems \[onedim\] and \[athree\]. In each we show that a certain set $K$ has property $A_3$. We will let ${{\mathcal{M}}}$ be a complex manifold, $\phi:{{\mathcal{M}}}\rightarrow{{\mathbb{C}}}$ a holomorphic submersion such that $\phi({{\mathcal{M}}})\supset K$, $\mathcal{P}$ a finite subset of $K$, and $s$ a section of $\phi$, $s\in{\mathcal{A}_\phi\left({K},{{{\mathcal{M}}}}\right)}$. We let $\epsilon>0$, and want to show that there is an $s_\epsilon\in{\mathcal{O}_\phi\left({K},{{{\mathcal{M}}}}\right)}$ such that ${{\ensuremath{\operatorname{dist}}}}(s,s_\epsilon)<\epsilon$ and $s_\epsilon(p)=s(p)$ for each $p\in\mathcal{P}$.
Proof of Lemma \[zerodim\]
--------------------------
If $K$ is not finite, it can be written as a disjoint union of finitely many singletons and a closed subset $C$ homeomorphic to the Cantor middle-third set. (See e.g., [@ems:gtopology1], pp. 108-109.) In particular, ${{\mathbb{C}}}\setminus K$ is connected, and $K$ has property $A_1$.
Observe that $s\in{\mathcal{A}_\phi\left({K},{{{\mathcal{M}}}}\right)}$ is simply a continuous section of $\phi$ over $K$. We can cover $K$ by finitely many sets $\{U_j\}$ such that each $s(\overline{U_j})$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$. We can choose a refinement $\{V_j\}$ of this cover, such that the $V_j$’s are pairwise disjoint. Observe that $K_j{:=}K\cap \overline{V_j}$ has property $A_1$. Now, with respect to the $\phi$-adapted coordinate system around $s(K_j)$, the map $s$ has a representation on $K_j$ of the form $s(z)= (\tilde{s}(z),z)$, where $\tilde{s}$ is continuous and takes values in ${{\mathbb{C}}}^{n-1}$. Approximating $\tilde{s}$ by a holomorphic map in a neighborhood of $K_j$ our result follows.
Proof of Theorem \[onedim\]
---------------------------
We first introduce some combinatorial preliminaries. Let $vw$ be an edge in a graph $\Gamma$. We can construct a new graph $\Gamma'$ with one more vertex by “splitting the edge $vw$." More formally, if $V$ is the vertex set of $\Gamma$, and $E$ its edge set, the vertex set of $\Gamma'$ is $V\cup\{u\}$ (where $u\not\in V$), and the edge set is $\left(E\setminus\{ vw\}\right)\cup \{vu,uw\}$.
Recall that a graph is [*$n$-colorable*]{}, if there is an assignment of $n$ colors to its vertices so that no two adjacent vertices have the same color. By a classical result of König (see [@harary:book], Theorem 12.1), the condition that a graph is $2$-colorable is that it does not have a circuit (closed non-self intersecting path) of odd length. We now have the following:
\[bicolorable\] Given any finite graph $G$, there are edges $e_1,\ldots, e_k$ such that the graph $G'$ obtained after successively splitting these edges is 2-colorable.
Let $V$ be the vector space over the field $\mathbb{Z}/2\mathbb{Z}$ with basis the set of edges of $G$. Given a circuit $C$ in $G$, traversing the edges $x_1,x_2,\ldots, x_m$, associate with it an element $c$ of $V$ given by $c= x_1+x_2+\cdots+x_m$. Let $Z$ be the subspace of $V$ spanned by all $c$ for each circuit $C$. Pick a basis $c_1,\ldots, c_k$. Since any circuit can be written as a linear combination of the $c_j$’s it is sufficient to split any edge of those $c_j$ which have an odd number of summands. Moreover, from the fact that the $c_j$ are linearly independent over $\mathbb{Z}/2\mathbb{Z}$ it follows that each $c_j$ has an edge not contained in any $c_k$, $k\not=j$. The result follows.
Now we can prove Theorem \[onedim\]. Note that as in the proof of Lemma \[onedim\] above, ${{K}^{\circ}}=\emptyset$, so $s$ is simply a continuous section of $\phi$ over $K$.
For $z\in K$, there is a neighborhood $U_z$ of $z$ in $K$ such that $s(\overline{U_z})$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$. Select a finite subcover $\mathcal{F}$ of $\{U_z\}_{z\in K}$, and let $\mathcal{G}$ be a refinement of $\mathcal{F}$ which has only double intersections (this is possible since ${\ensuremath{\operatorname{dim}}}(K)=1$, and $K$ is connected.) We can write $\mathcal{G}=\{V_1,\ldots,V_M\}$.
We associate to $\mathcal{G}$ a graph $N_\mathcal{G}$ by taking the nerve: the vertices $V_i$ and $V_j$ are joined by an edge in $N_\mathcal{G}$ iff $V_i\cap V_j\not=\emptyset$. Let $V$ and $W$ be open sets in $\mathcal{G}$ such that $V\cap W\not=\emptyset$. We define a new open cover $S_{VW}(\mathcal{G})$ of $K$ in the following way. Replace the sets $V$ and $W$ by $V'$, $U$ and $W'$, where $V'\subset V$, $W'\subset W$, and $U$ is a neighborhood of $\overline{V\cap W}$ such that $V'\cup U\cup W'= V\cup W$, and $V\cap U\not=\emptyset$, $U\cap W\not=\emptyset$ but $V\cap W=
\emptyset$. If $U$ is chosen small enough, then $\left(\mathcal{G}\setminus\{V,W\}\right)\cup \{V',U,W'\}$ is again an open cover of $K$ with only double intersections. This is our $\mathcal{G}'=S_{VW}(\mathcal{G})$. At the level of nerves $N_{\mathcal{G}'}$ is obtained by splitting the edge $VW$ in the graph $N_{\mathcal{G}}$, as in Lemma \[bicolorable\]. Thanks to Lemma \[bicolorable\] we can obtain after finitely many rounds of edge-splitting a new cover $\mathcal{H}$, with only double intersections such that the corresponding nerve $N_\mathcal{H}$ is $2$-colorable. It also follows that for each $U\in \mathcal{H}$, the set $s(\overline{U})$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$. Further, by shrinking the $U$’s we can assume that whenever an intersection $U\cap U'$ of two sets in $\mathcal{H}$ is non-empty, it is contractible. Let us call the colors used in coloring $N_\mathcal{H}$ red and blue. We can now define $$K_1=\bigcup_{{\stackrel{U\in\mathcal{H}}{ U\mbox{~{\scriptsize{red}}}}}}
\overline{U},\mbox{~~and~~~}
K_2=\bigcup_{\stackrel{U\in\mathcal{H}}{U\mbox{~{\scriptsize{blue}}}}}
\overline{U}.$$ It is easy to verify that $(K_1,K_2)$ is a good pair, and for $j=1,2$ the set $s(K_j)$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$.
We also define sets $K_1'\subset K_1$ and $K_2'\subset K_2$ in the following way. For $U,U'\in\mathcal{H}$, let $\mathcal{W}_{UU'}$ be a simply connected neighborhood of $U\cap U'$ in ${{\mathbb{C}}}$ with $\mathcal{C}^2$ boundary. Further we can assume that $\overline{\mathcal{W}_{UU'}\cap K}$ is contractible and $s(\overline{\mathcal{W}_{UU'}\cap K})$ is contained in a $\phi$-adapted open set of ${{\mathcal{M}}}$. We set $$\mathcal{W}= \bigcup_{\stackrel{U,U'\in\mathcal{H}}{U\cap
U'\not=\emptyset}}\mathcal{W}_{UU'},$$ so that $\mathcal{W}$ is a neighborhood in ${{\mathbb{C}}}$ of the points in $K$ which are contained in two sets of the cover $\mathcal{H}$.
Let $V\in\mathcal{H}$. We set $$V^s= V\setminus \overline{W}$$
So that $V^s\subset V$, and points of $V^s$ do not belong to any other set of $\mathcal{H}$ apart from $V$. We set: $$K_1'=\bigcup_{{\stackrel{U\in\mathcal{H}}{ U\mbox{~{\scriptsize{red}}}}}}
\overline{U^s},\mbox{~~and~~~}
K_2'=\bigcup_{\stackrel{U\in\mathcal{H}}{U\mbox{~{\scriptsize{blue}}}}}
\overline{U^s}.$$
We now apply Theorem \[goodpairapprox\] to the good pair $(K_1,
K_2)$, and obtain an $s_1\in {\mathcal{O}_\phi\left({K\cup\mathcal{B}_1},{{{\mathcal{M}}}}\right)}$, where $\mathcal{B}_1$ is a neighborhood of $K_1'$ in ${{\mathbb{C}}}$, such that ${{\ensuremath{\operatorname{dist}}}}(s,s_1)<\frac{\epsilon}{3}$, and $s(p)=s_1(p)$ for $p\in\mathcal{P}$. We can further assume that $\mathcal{B}_1$ is a disjoint union of simply connected neighborhoods of the sets $V^s$ for $V$ red, $\partial\mathcal{B}_1$ is smooth, and $\partial\mathcal{B}_1$ and $\partial\mathcal{W}$ meet transversely at each point of intersection.
Observe now that $(K_1\cup \mathcal{B}_1, K_2)$ is a good pair, and we can apply Theorem \[goodpairapprox\] to it. We obtain a $s_2\in {\mathcal{A}_\phi\left({K\cup \mathcal{B}_1\cup \mathcal{B}_1},{{{\mathcal{M}}}}\right)}$ with ${{\ensuremath{\operatorname{dist}}}}(s_2,s_1)<\frac{\epsilon}{3}$, and $s_2(p)= s_1(p)$ for $p\in\mathcal{P}$. As before, $\mathcal{B}_2$ is a disjoint union of simply connected neighborhoods of the sets $V^s$ for $V$ blue, $\partial\mathcal{B}_2$ is smooth, and $\partial\mathcal{B}_2$ and $\partial\mathcal{W}$ meet transversely at each point of intersection.
We set $\mathcal{B}=\mathcal{B}_1\cup \mathcal{B}_2$ so that $\partial\mathcal{B}=\partial\mathcal{B}_1\cup
\partial\mathcal{B}_2$ meets $\partial\mathcal{W}$ transversely, so that $\partial\mathcal{B}\cap\partial\mathcal{W}$ is a finite set. Let $\mathcal{U}$ be a simply connected neighborhood of $K$ in ${{\mathbb{C}}}$ such that all points of $\partial\mathcal{B}\cap\partial\mathcal{W}$ lie outside $\mathcal{U}$ and $\partial\mathcal{U}$ meets both $\partial\mathcal{B}$ and $\partial\mathcal{W}$ transversely. Note that $K_1\cup K_2\subset\mathcal{W}\setminus\mathcal{B}$. Let $$L_1 = \partial\mathcal{U}\cap\partial\mathcal{B},$$ and $$L_2 = \partial\mathcal{U}\cap \partial\mathcal{W}.$$ It is easy to verify that $(L_1,L_2)$ is a good pair, and each of $s(L_j)$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$. Setting $L=L_1\cup L_2$, we apply Theorem \[goodpairapprox\] to $(L_1,L_2)$ to obtain a map $s_3\in{\mathcal{A}_\phi\left({L\cup \mathcal{C}},{{{\mathcal{M}}}}\right)}$, where $\mathcal{C}$ is a neighborhood of $K_1\cap K_2$ in ${{\mathbb{C}}}$, ${{\ensuremath{\operatorname{dist}}}}(s_3,s_2)<\frac{\epsilon}{3}$, and $s_3(p)=s_2(p)$ for all $p\in\mathcal{P}$. Clearly, $s_3\in {\mathcal{O}_\phi\left({K},{{{\mathcal{M}}}}\right)}$, and we are done.
Proof of Theorem \[athree\]
---------------------------
In [§\[arcsection\]]{} we record a result that will be used in the proof. In [§\[smoothathree\]]{} we use Theorem \[arcs\] and Theorem \[goodpairapprox\] to give a proof of Theorem \[athree\] in the special case when $N=1$, i.e. $K$ is the closure of a smoothly bounded domain. As in the proof of Theorem \[onedim\] this uses a “Triple bumping." Finally, in [§\[athreeproofsection\]]{} we complete the proof by reducing the general case to the case considered in [§\[smoothathree\]]{}.
### Arcs in Complex Manifolds {#arcsection}
If $X$ is a differentiable manifold, and $\alpha$ is an arc (continuous injective map from $[0,1]$) which is at least $\mathcal{C}^1$, we will say $\alpha$ is [*embedded*]{} if for each $t\in[0,1]$, we have $\alpha'(t)\not=0$. A proof of the following result, required in the proof of Theorem \[athree\], can be found in [@michiganpaper], Theorem 2:
\[arcs\]
Let ${{\mathcal{M}}}$ be a complex manifold, and $\alpha$ be an arc in ${{\mathcal{M}}}$. Assume that
1. there is a complex-valued submersion $\phi$ defined in a neighborhood of $\alpha$ in ${{\mathcal{M}}}$ such that $\phi\circ\alpha$ is a $\mathcal{C}^1$ embedded arc in ${{\mathbb{C}}}$.
2. there is a finite subset $P\subset [0,1]$ such that $\alpha$ is $\mathcal{C}^3$ on $[0,1]\setminus P$.
Then $\alpha$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$.
### Case of a smooth domain {#smoothathree}
In this section we prove the following special case of Theorem \[athree\] (Compare with Proposition \[forst\]):
\[smoothathreeprop\] Let $\Omega\Subset{{\mathbb{C}}}$ be a domain with $\mathcal{C}^2$ boundary. Then $\overline{\Omega}$ has property $A_3$.
The first step in the proof is the following application of Sard’s Theorem:
\[sard1\] There is a unit vector $\mathbf{v}$ in the plane such that:
- every straight line in the plane parallel to $\mathbf{v}$ meets $\partial\Omega$ in only finitely many points.
- the number of straight lines parallel to $\mathbf{v}$ which are tangent to $\partial\Omega$ is finite.
In fact these hold for almost all unit vectors $\mathbf{v}$ in the unit circle $S^1$ (with respect to the standard measure on $S^1$.)
Fix an arbitrary orientation on $\partial\Omega$, and define the Gauss map $G:\partial\Omega\rightarrow S^1$ by mapping the point $z\in\partial\Omega$ to the unit tangent vector $G(z)$ to $\partial\Omega$ at the point $z$. This is a $\mathcal{C}^1$ map, and the set of its critical values is of measure 0 by Sard’s Theorem. If $\mathbf{v}$ is any regular value of $G$, such that $-\mathbf{v}$ is also a regular value, it follows that the sets $G^{-1}(\mathbf{v})$ and $G^{-1}(-\mathbf{v})$ are discrete in $\partial\Omega$ (since $G$ is a diffeomorphism near each of them.) Since $\partial\Omega$ is compact, it follows that the sets $G^{-1}(\pm\mathbf{v})$ are finite. From this, the conclusions follow immediately.
After a rotation if required, we will assume that $\mathbf{v}$ is vertical, i.e., $\mathbf{v}=\pm i$. Now we introduce some notation. for $c\in{{\mathbb{R}}}$, denote by $L(c)$ the vertical straight line $\Re(z)=c$ in ${{\mathbb{C}}}$, and for $a<b$, denote by $M[a,b]$ the vertical strip $\{z\in{{\mathbb{C}}}\colon a\leq\Re(z)\leq b\}$. Also set $l(c)=L(c)\cap\overline{\Omega}$ and $m[a,b]=
M[a,b]\cap\overline{\Omega}$. We will assume without loss of generality that $\Omega\subset M[0,1]$.
Thanks to Lemma \[sard1\] and the choice $\mathbf{v}=\pm i$, it follows that for each $c$, the set $l(c)$ has only finitely many components, each of which is either a point, or a line segment. Also, $l(c)\cap\partial\Omega$ is finite. We now make the following observation.
\[delta\] There is a $\delta>0$ such that for each $c\in[0,1]$, the set $s(m[c-\delta,c+\delta])$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$.
First we show that for each $c\in[0,1]$, the set $s(l(c))$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$. Since $s$ is injective (it has a left-inverse $\phi$), it is sufficient to show that each component of $l(c)$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$. For a component of $l(c)$ that reduces to a point, this is trivial. Therefore, consider a component which is a (vertical) line segment. We can think of this component as an arc in the plane, and parameterize it as $\lambda(t) = z_0+ iat$, where $z_0$ is the lower end point of the segment, and $a$ is a positive real number. Then $s\circ\lambda$ is a continuous arc in ${{\mathcal{M}}}$, which is real analytic except at finitely many points, and $\phi$ is a submersion from ${{\mathcal{M}}}$ to ${{\mathbb{C}}}$ such that $\phi\circ\left(s\circ\lambda\right)=\lambda$. Thanks to Proposition \[arcs\] above, $s(\lambda)$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$. Therefore, $s(l(c))$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$.
It follows that there is a $\delta_c$ such that $s(m[c-\delta_c,c+\delta_c])$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$. The uniform choice of $\delta$ follows by compactness.
We will also require the following simple fact (a proof may be found in [@michiganpaper] , Observation 4.8 )
\[strip\] Let $u$ and $v$ be real valued $\mathcal{C}^1$ functions defined on a neighborhood of $0$ in ${{\mathbb{R}}}$ such that for each $x$, we have $u(x)<0<v(x)$. Then there is an $\eta>0$ such that for $0<\theta\leq\eta$, the vertical strip $$S {:=}\{ (x,y)\in {{\mathbb{R}}}^2\colon x\in [-\theta,\theta],
{u}(x)\leq y \leq {v}(x) \}$$ is star shaped with respect to the origin.
The proof of Proposition \[smoothathree\] will parallel that of Proposition \[onedim\] in that both require a “Triple bumping", i.e. three successive applications of Theorem \[goodpairapprox\]. However, the good pairs are obtained by different methods.
Let $\mathcal{E}\subset[0,1]$ be the set of $c$ such that the line $L(c)$ is tangent to some component of $\partial\Omega$. $\mathcal{E}$ is finite by Lemma \[sard1\]. If $c\not\in\mathcal{E}$, $L(c)$ meets $\partial\Omega$ transversely at each point of intersection, so that (1) each component of $l(c)$ is a line segment, and (2) (by Lemma \[strip\] ) there is a $\theta_c$ such that for $\epsilon\leq\theta_c$, each component of $m([c-\epsilon,c+\epsilon])$ is strongly star shaped.
By compactness, we can find $c_1<c_2<\ldots<c_M$, and $\eta_j>0$, $j=1,\ldots,M$, such that if $m_j= m[c_j-\eta_j,c_j+\eta_j]$, we have $\overline{\Omega}=\bigcup_{j=1}^M m_j$. We can assume that the $\eta_j<\frac{\delta}{100}$, where $\delta$ is as in Lemma \[delta\]. We see that for each $j$, $s(m_j)$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$. We will further impose the following conditions on the $m_j$’s
1. Each point of $\overline{\Omega}$ is in at most two of the $m_j$’s (this can be done by shrinking the $\eta_j$’s.) Therefore $m_j\cap m_k=\emptyset$ if ${\left|{j-k}\right|}>1$.
2. Each $c\in\mathcal{E}$ occurs in the list $\{c_j\}_{j=1}^M$.
3. Each component of $m_j\cap m_{j+1}$ is strongly star shaped. Observe that by the previous step, $m_j\cap m_{j+1}$ does not contain any $l(c)$ for $c\in\mathcal{E}$. Therefore, this can be achieved by shrinking the $\eta_j$’s.
Now let $$K_1=\bigcup_{j\mbox{~~{\scriptsize{odd}}}}m_j ,\mbox{~~and~~}
K_2=\bigcup_{j\mbox{~~{\scriptsize{even}}}}m_j.$$ It is easy to see that $(K_1,K_2)$ is a good pair. Since each $s(m_j)$ has a $\phi$-adapted neighborhood, and $K_1,K_2$ are disjoint union of $m_j$’s, it follows that each of $s(K_1)$ and $s(K_2)$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$.
Let $I_1\subset \partial\Omega\cap K_1$ and $I_2\subset
\partial\Omega\cap K_2$ be such that (1) $ I_1\cap K_2= I_2\cap K_2
=\emptyset,$ and (2) Each connected component $\partial\Omega\setminus (I_1\cup I_2)$ is contained in a vertical strip of width $\frac{\delta}{2}$, where $\delta$ is as in Lemma \[delta\]. (This is possible, since $\eta_j<\frac{\delta}{100}$.)
Let $B$ be a neighborhood of $I_1$ such that $B\cap K_2=\emptyset$. We apply Theorem \[goodpairapprox\] to the good pair $(K_1,K_2)$ to obtain an $s_1\in {\mathcal{A}_\phi\left({\overline{\Omega}\cup B_1},{{{\mathcal{M}}}}\right)}$, where $B_1$ is a neighborhood of $I_1$ contained in $B$, such that ${{\ensuremath{\operatorname{dist}}}}(s,s_1)<\frac{\epsilon}{3}$, and $s(p)=s_1(p)$ for $p\in
\mathcal{P}$. Observe that for $B_1$ small enough, $s_1(K_1\cup
B_1)$ is contained in a $\phi$-adapted open set of ${{\mathcal{M}}}$, and $(K_1\cup B_1, K_2)$ is again a good pair. We now apply Theorem \[goodpairapprox\] again to this good pair to obtain an $s_2\in{\mathcal{A}_\phi\left({\overline{\Omega}\cup B_1\cup B_2},{{{\mathcal{M}}}}\right)}$ where $B_2$ is a neighborhood of $I_2$, such that ${{\ensuremath{\operatorname{dist}}}}(s_1,s_2)<\frac{\epsilon}{3}$, and $s_1(p)= s_2(p)$ for $p\in\mathcal{P}$.
Let $I_3=\partial\Omega\setminus (I_1\cup I_2).$ By construction, each connected component of $I_3$ is contained in a vertical strip of width $\frac{\delta}{2}$. Let $\Omega'$ be a domain with $\mathcal{C}^2$ boundary such that $\overline{\Omega}\cup B_1\cup
B_2\supset \Omega'\supset \Omega\cup I_1\cup I_2$ (i.e. $\Omega'$ is obtained by smoothly bumping $\Omega$ along $I_1$ and $I_2$.) We can assume that $\Omega'$ and $\Omega$ are so close that for $\delta$ as in Lemma \[delta\], and any $c$, the set $s(M[c-\delta,c+\delta]\cap\overline{\Omega'})$ is contained in a $\phi$-adapted open set in ${{\mathcal{M}}}$. We can now repeat the constructions that gave us $K_1$ and $K_2$ to obtain a good pair $(L_1,L_2)$ such that (1) $L_1\cup L_2=\overline{\Omega'}$, (2) each of $s(L_1)$ and $s(L_2)$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$, and (3) $I_3\subset L_1\setminus L_2$. We can now apply Theorem \[goodpairapprox\] to the good pair $(L_1,L_2)$ to obtain an $s_3\in{\mathcal{A}_\phi\left({\overline{\Omega'}\cup C},{{{\mathcal{M}}}}\right)}$, where $C$ is a neighborhood of $I_3$ in ${{\mathbb{C}}}$, such that ${{\ensuremath{\operatorname{dist}}}}(s_3,s_2)<\frac{\epsilon}{3}$ and $s_3(p)=s_2(p)$, for each $p\in\mathcal{P}$. Then $s_3\in{\mathcal{O}_\phi\left({\overline{\Omega}},{{{\mathcal{M}}}}\right)}$, and we are done.
### End of proof of Theorem \[athree\] {#athreeproofsection}
Let $K$ be of class $\mathfrak{C}_2$. Recall that $K=\cup_{i=1}^N\overline{\Omega_i}$, and for $i\not =j$ , we have $\partial\Omega_i$ and $\partial\Omega_j$ meet at a set of finitely many points $P_{ij}$. Set $P=\cup_{i\not=j}P_{ij}$. We can refer to the points in $P$ as [*nonsmooth points*]{} of $\partial K$.
The proof in this section is very similar to that in [§\[smoothathree\]]{}. The first step is to establish the following version of Lemma \[sard1\] for this case:
\[sard2\] Let $K$ be of class $\mathfrak{C}_2$. Then there is a unit vector $\bf{v}$ in the plane with the following properties:
1. every straight line in the plane parallel to $\mathbf{v}$ meets $\partial K$ in only finitely many points.
2. The number of straight lines parallel to $v$ which are tangent to $\partial K$ at smooth points is finite.
3. Let $p\in P_{ij}$ be a non-smooth point of $\partial K$. Then, the straight line through $p$ parallel to $\mathbf{v}$ is transverse to both $\partial\Omega_i$ and $\partial\Omega_j$ at $p$.
Applying lemma \[sard1\] separately to each $\partial\Omega_j$, we conclude that properties (1) and (2) hold for almost all unit vectors $\mathbf{v}$. Let $p$ be a non-smooth point of $\partial K$, so that for some $i,j$, we have $p\in P_{ij}=\partial\Omega_i\cap
\partial\Omega_j$. Let $\mathbf(t)(p)$ be a common unit tangent vector to $\partial\Omega_i$ and $\partial\Omega_j$ at the point $p$. We can choose $\mathbf{v}\not= \pm \mathbf{t}(p)$ for all $p\in
P_{ij}$ for all $i$ and $j$.
As in [§\[smoothathree\]]{} we can assume that $\mathbf{v}=\pm i$. Let $L(c)$ and $M[a,b]$ have the same meaning as in the last section, and set $l'(c)= L(c)\cap K$, $m'[a,b]= M$. We can assume that $K\subset M[0,1]$.
We let $\mathcal{E}$ be the finite set of points $c\in[0,1]$ such that either (1) $L(c)$ is tangent to $\partial\Omega_i$ for some $i$, or (2) $L(c)$ passes through a nonsmooth point of $\partial K$. Arguing as in the previous section, we can find $c_1<c_2<\ldots c_M$ and $\eta_j>0$, $j=1,\ldots, M$, such that if $m_j'=m'[c_j-\eta_j,c_j+\eta_j]$, we have $K=\cap_{j=1}^M m_j'$ and each $s(m_j'$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$. We can further impose the following conditions, (the first three are just as in the last section, the last is a new condition):
1. Each point in $K$ is contained in at most two of the $m_j'$’s, i.e., $m_j'\cap m_k'=\emptyset$ if ${\left|{j-k}\right|}>1$.
2. Each $c\in\mathcal{E}$ occurs among the $\{c_j\}_{j=1}^M$.
3. Each component of $m_j'\cap m_{j+1}'$ is strongly star shaped (Note that, thanks to the last step, in this case we have ensured that $m_j'\cap m_{j+1}'$ does not contain any nonsmooth points.)
4. Each nonsmooth point is contained in an $m_j'$ with an even $j$. This can be ensured by introducing additional $c_j's$. Observe that if a nonsmooth point $p\in m_j'$, then $p\not\in m_{j-1}'$ and $p\not \in m_{j+1}'$.
As in the proof of Proposition \[smoothathreeprop\] $$K_1=\bigcup_{j\mbox{~~{\scriptsize{odd}}}}m_j' ,\mbox{~~and~~}
K_2=\bigcup_{j\mbox{~~{\scriptsize{even}}}}m_j'.$$ It is easy to see that $(K_1,K_2)$ is a good pair. Since each $s(m_j')$ has a $\phi$-adapted neighborhood, and $K_1,K_2$ are disjoint union of $m_j'$’s, it follows that each of $s(K_1)$ and $s(K_2)$ has a $\phi$-adapted neighborhood in ${{\mathcal{M}}}$. Moreover, the set $P$ of nonsmooth points of $\partial K$ is contained in $K_2\setminus K_1$.
Let $B$ be a neighborhood of the nonsmooth points $P$ such that $B\cap K_1=\emptyset$. Thanks to Theorem \[goodpairapprox\] we can find an $s'\in{\mathcal{A}_\phi\left({K\cup B'},{{{\mathcal{M}}}}\right)}$ (where $B'$ is a neighborhood of $P$ contained in $B$), such that ${{\ensuremath{\operatorname{dist}}}}(s,s')<\frac{\epsilon}{2}$, and $s'(p)=s(p)$ for each $p\in\mathcal{P}$.
Now we can find an open set $\Omega'$ with $\mathcal{C}^2$ boundary such that $\overline{\Omega'}\subset K\cup B'$ but $\Omega'\supset
\Omega\cup P$. Then $s'\in{\mathcal{A}\left({\overline{\Omega'}},{{{\mathcal{M}}}}\right)}$, and thanks to Proposition \[smoothathreeprop\], we can find $s_\epsilon\in
{\mathcal{O}_\phi\left({\overline{\Omega'}},{{{\mathcal{M}}}}\right)}$ such that ${{\ensuremath{\operatorname{dist}}}}(s_\epsilon,s')<\frac{\epsilon}{2}$ and $s_\epsilon(p)=s'(p)$ for $p\in \mathcal{P}$. Since $s_\epsilon\in {\mathcal{O}_\phi\left({K},{{{\mathcal{M}}}}\right)}$, the proof is complete.
Proof of Theorem \[atwo\] {#atwosection}
=========================
$\mathcal{A}$-equivalence
-------------------------
Let $K$ and $K'$ be compact subsets of ${{\mathbb{C}}}$. We will say that $K$ and $K'$ are [*$\mathcal{A}$-equivalent*]{} if there is a homeomorphism $\chi:K\rightarrow K'$ such that $\chi|_{{{K}^{\circ}}}$ is a conformal map of ${{K}^{\circ}}$ onto ${{K'}^{\circ}}$. We will call $\chi$ an [*$\mathcal{A}$-equivalence* ]{} from $K$ to $K'$. A well-known example of $\mathcal{A}$-equivalence is the following ([@tsuji:bible], Theorems IX.35 and IX.2):
\[circular\] Let $\Omega$ be a [*Jordan Domain*]{}, Then there is a domain $\omega$ in the plane bounded by [*circles*]{} such that $\overline{\Omega}$ and $\overline{\omega}$ are $\mathcal{A}$-equivalent.
The significance of this notion in the current investigation is explained by the following observation:
\[aeq\] Suppose that two compact sets $K$ and $L$ in ${{\mathbb{C}}}$ are $\mathcal{A}$-equivalent. If $K$ has property $A_2$ and $L$ has property $A_1$ then $L$ has property $A_2$.
Let ${{\mathcal{M}}}$ be a complex manifold, $f\in{\mathcal{A}\left({L},{{{\mathcal{M}}}}\right)}$, and $\mathcal{P}$ be a finite subset of $K$. We want to approximate $f$ by maps $f_n$ in ${\mathcal{O}\left({L},{{{\mathcal{M}}}}\right)}$ such that $f_n(p)=f(p)$ for $p\in\mathcal{P}$. Let $\chi:K\rightarrow L$ be an $\mathcal{A}$-equivalence,let $\mathcal{Q}=\chi^{-1}(\mathcal{P})$, and let $g=f\circ\chi$. Then $g\in{\mathcal{A}\left({K},{{{\mathcal{M}}}}\right)}$, and consequently there is a sequence $g_n\in{\mathcal{O}\left({K},{{{\mathcal{M}}}}\right)}$ such that $g_n\rightarrow g$ uniformly, and $g_n(q)=g(q)$ for $q\in\mathcal{Q}$. Let $\zeta=\chi^{-1}$, so that $\zeta\in{\mathcal{A}\left({L},{{{\mathbb{C}}}}\right)}$. Since $L$ has property $A_1$, we can find $\zeta_n\in{\mathcal{O}\left({L},{{{\mathbb{C}}}}\right)}$ such that $\zeta_n\rightarrow\zeta$ on $L$, with $\zeta_n(p)=\zeta(p)$ for $p\in\mathcal{P}$. Then $f_n{:=}g_n\circ\zeta_n\in{\mathcal{O}\left({L},{{{\mathcal{M}}}}\right)}$, $f_n\rightarrow f$ uniformly, and $f_n(p)=f(p)$ for $p\in\mathcal{P}$.
Proof of Theorem \[atwo\] {#proof-of-theorematwo}
-------------------------
Thanks to Lemma \[aeq\] and Theorem \[athree\], it is sufficient to prove the following result:
\[touching\] For each $K$ in $\mathfrak{C}_0$ there is an $L$ in $\mathfrak{C}_\omega$ such that $K$ and $L$ are $\mathcal{A}$-equivalent.
We will require two results, the first from Combinatorics. A vertex $v$ of a graph $G$ is said to be a [*cutpoint*]{} of $G$ if the graph $G^{\{v\}}$ obtained by removing from $G$ the vertex $v$ along with all edges incident at $v$ has at least one more connected component than $G$ has. (So for example, if $G$ is connected, $G^{\{v\}}$ is [*not*]{} connected.) We will need the following elementary fact.
\[graph\]([@harary:book], Theorem 3.4, p. 29) Let $G$ be a graph with more than one vertex. Then there are at least two vertices of $G$ which are [*not*]{} cutpoints.
The second result is the following boundary interpolation theorem for conformal maps, due to MacGregor and Tepper ([@macgregor:interpolation], Theorem 1). $\Delta\subset{{\mathbb{C}}}$ is the open unit disc, and $\{z_1,z_2,\ldots,z_n\}\subset
\partial\Delta$ and $\{w_1,w_2,\ldots,w_n\}\subset \partial\Delta$ are given finite subsets of the unit circle.
\[univalent\] There is a function $f$ which is analytic and univalent in the union of $\Delta$ and a neighborhood of $\{z_1,z_2,\ldots,z_n\}$ and continuous on $\overline{\Delta}$ such that $f(z_k)=w_k$ for $k=1,\ldots,n$. Furthermore, ${\left|{f(z)}\right|}=1$ if ${\left|{z}\right|}=1$ and $z$ is sufficiently near any of the points $z_k$, and also $f(\Delta)\subset\Delta.$
For a compact connected set $K$ in the plane, by the outer boundary we mean the boundary of the unbounded component of ${{\mathbb{C}}}\setminus K$ (this is also a component of $\partial K$.) We use Proposition \[univalent\] to prove the following lemma.
\[interpolation\] Let $\Omega\Subset{{\mathbb{C}}}$ be a Jordan domain, and let $\gamma$ be its outer boundary. Suppose we are given a finite set of points $\{z_1,\ldots, z_n\}$ on $\gamma$ and the same number of points $\{w_1,\ldots,w_n\}$ on the unit circle $\partial\Delta$. Then there is a continuous map $f:\overline{\Omega}\rightarrow\overline{\Delta}$ such that
1. $f|_\Omega$ is conformal,
2. $f(z_k)= w_k$, for $k=1,\ldots, n$,
3. let $W=f(\Omega)$. Then $\partial W$ is $\mathcal{C}^\omega$, and at each $w_k$, $\partial W$ is tangent to $\partial \Delta$.
By Lemma \[circular\] there is a domain $D$ bounded by circles and an $\mathcal{A}$-equivalence $f_0:\overline{\Omega}\rightarrow\overline{D}$. After applying an inversion of the plane if required, we can assume further than the outer boundary $\gamma$ of $\Omega$ is mapped onto the outer boundary of $D$, which we may assume is the unit circle $\partial\Delta$. Set $z_k'= f_0(z_k)$ for $k=1,\ldots,n$.
Let $D'$ be a simply connected open set in ${{\mathbb{C}}}$ with $\mathcal{C}^\infty$ boundary such that $\Delta\subset D'$, $\partial D'\cap \partial \Delta =\{z_1',\ldots, z_n'\}$ where at each $z_k'$, the boundaries $\partial D'$ and $\partial \Delta$ are tangent to each other. Let $f_1:D'\rightarrow \Delta$ be a conformal map of $D'$ onto $\Delta$. Since $\partial D'$ has $\mathcal{C}^\infty$ boundary, $f_0$ extends to a diffeomorphism of the closures. Set $z_k''=f_1(z_k')$, and $D''=f_1(D')$. Observe that $f_1\circ f_0$ maps $\Omega$ to a subdomain $\Omega''$ of $\Delta$ and the boundary $\partial\Omega''$ is tangent to $\partial\Delta$ at each point of intersection $z_k''$.
We now apply Proposition \[univalent\] to obtain a continuous $f_2:\overline{\Delta}\rightarrow\overline{\Delta}$ such that $f_2(z_k'')=w_k$, $f_2$ is conformal on the union of $\Delta$ with a neighborhood of $\{z_1'',\ldots,z_n''\}$, and $f_2$ maps a piece of $\partial\Delta$ near each $z_k''$ onto a piece of $\partial\Delta$ near $w_k$. It follows immediately that if $W=f_2(\Omega'')$, $\partial W$ meets $\partial\Delta$ tangentially at each $w_k$. We set $f{:=}f_2\circ f_1\circ f_0$. The properties claimed are easily verified.
We now prove Theorem \[touching\].
It is clear that we only need to consider the case in which $K$ is connected. We use induction on $N$, the number of summands of $K$.
When $N=1$, the result is reduced to Lemma \[circular\]. Now suppose that the result has been proved for some $N\geq 1$, and let $K= \cup_{i=1}^{N+1} \overline{\Omega_j}$. Let $G$ be a graph whose vertices $v_i$ correspond to the sets $\overline{\Omega_i}$, and there is an edge connecting $v_i$ and $v_j$ iff $\overline{\Omega_i}\cap \overline{\Omega_j}\not=\emptyset$. Since we have assumed that $K$ is connected, it follows that $G$ is a connected graph. Thanks to Lemma \[graph\] above, we can assume (after a renumbering of the vertices of $G$) that $v_{N+1}$ is [*not*]{} a cutpoint of $G$. Let $K'=\cup_{i=1}^N \overline{\Omega_j}$, and let $P\subset K'$ be the finite set $K'\cap\overline{\Omega_{N+1}}=K'\cap\partial\Omega_{N+1}$. Then $K'$ is connected, therefore is contained in exactly one connected component $U$ of ${{\mathbb{C}}}\setminus\Omega_{N+1}$. Let $\gamma=\partial
U$. Clearly $\gamma$ is a connected component of $\partial\Omega_{N+1}$. It follows that the set $P\subset\gamma$. Moreover, as $\overline{\Omega_{N+1}}$ is connected, it follows that $\Omega_{N+1}$ is contained in exactly one component of ${{\mathbb{C}}}\setminus K'$.
We claim that [*we can assume that $\gamma$ is the outer boundary of $\Omega_{N+1}$*]{}. To show this it is sufficient to show that for some $\mathcal{A}$-equivalence $\Phi:K\rightarrow
\hat{K}\subset{{\mathbb{C}}}$, $\Phi(\gamma)$ is the outer boundary of $\Phi(\Omega_{N+1})$.
If $\gamma$ not already the outer boundary of $\Omega_{N+1}$ let $z_0\in U\setminus K'$, where $U\Subset{{\mathbb{C}}}$ is the component of ${{\mathbb{C}}}\setminus \Omega_{N+1}$ which contains $K'$ (then $\gamma=\partial U$). Let $\rho>0$ be small enough so that $B_{{\mathbb{C}}}(z_0,\rho)\Subset U\setminus K'$, and define the inversion $\Phi:{{\mathbb{C}}}\setminus\{z_0\}\rightarrow{{\mathbb{C}}}$ by $$\Phi(z) = \frac{\rho^2}{z-z_0}.$$ Then $\Phi(K)$ is contained in the ball $B_{{\mathbb{C}}}(0,\rho)$, and since $z_0$ is mapped to the point at infinity, it follows that $U\setminus K'$ is mapped to the unbounded component of ${{\mathbb{C}}}\setminus K$. Since $\gamma=\partial U$, we see that $\gamma\subset \partial \left( U\setminus K'\right)$, so that $\Phi(\gamma)$ is the outer boundary of $\Phi(\Omega_{N+1})$.
Now, by induction hypothesis, there is an $L'\in\mathfrak{C}_\omega$, and an $\mathcal{A}$-equivalence $\chi':K'\rightarrow L'$. Using Lemma \[univalent\] we will extend $\chi'$ to an $\mathcal{A}$-equivalence $\chi$ defined on $K=K'\cup
\overline{\Omega_{N+1}}$.
Let us write $P=\{\zeta_1,\ldots, \zeta_n\}$, and let $\zeta_k' =
\chi'(\zeta_k)$. Then the $\zeta_k'$’s lie at the boundary of a single connected component $U$ of ${{\mathbb{C}}}\setminus L'$. Let $U'$ be a simply connected domain, $U\subset U'$ such that $\partial U'$ is $\mathcal{C}^\omega$ and passes through each $\zeta_k'\in\partial
U$, and further at each $\zeta_k'$, $\partial U'$ is tangent to $\partial U$, i.e. to $\partial L$. Let $\theta$ be a conformal map of $U'$ onto the disc $\Delta$. Then $\theta$ extends to a holomorphic map of a neighborhood of $\overline{U'}$. We set $w_k=\theta(\zeta_k')$.
Thanks to Lemma \[interpolation\] above, there is a map $\lambda\in{\mathcal{A}\left({\overline{\Omega_{N+1}}},{{{\mathbb{C}}}}\right)}$ such that $\lambda|_{\Omega_{N+1}}$ is conformal, $\lambda(\Omega_{N+1})\subset \Delta$ $f(\zeta_k)=w_k$, $\partial(\lambda(\Omega_{N+1}))$ is real analytic and tangent to $\partial\Delta$ at each point $w_k$. We can now define $$\chi{:=}\left\{\begin{array}{ccc} \chi'&\mbox{on}& K'\\
\theta^{-1}\circ\lambda &\mbox{on} &
\overline{\Omega_{N+1}}.\end{array}\right.$$ Let $\omega_{N+1}=\chi(\Omega_{N+1})$, and $L = L'\cup
\overline{\omega_{N+1}}$. Then $L$ is in $\mathfrak{C}_\omega$, and $\chi$ is an $\mathcal{A}$-equivalence between $K$ and $L$.
A Problem
=========
We conclude this article by stating an open problem. An solution will lead to a clearer picture of sets with properties $A_2$ and $A_3$.
[*Is it possible to prove an analog of Theorem \[goodpairapprox\] for [*three* ]{} sets $K_1$, $K_2$, $K_3$?*]{} That is, given $s: K\rightarrow{{\mathcal{M}}}$, (where $K=K_1\cup
K_2\cup K_3$), such that $s(K_j)$ lies in a subset of ${{\mathcal{M}}}$ homeomorphic to an open set in ${{\mathbb{C}}}^n$, obtain an approximation to $s$, after assuming reasonable hypotheses. In particular, we should have $K_1\cap K_2\cap K_3\not=\emptyset$.
Such a result will be necessary if we want to avoid the use of results like Theorem \[arcs\] two prove approximation results for two-dimensional sets. Observe that the use of Theorem \[arcs\] resulted in assumptions regarding the smoothness of the sets on which we want to do approximation.
Acknowledgements
================
It is a great pleasure to thank André Boivin and Rasul Shafikov for all their help in the preparation of this article.
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---
author:
- |
V. Mallet$^1$, D. E. Keyes$^2$, F. E. Fendell$^3$\
[$^1$ École des Ponts, Marne-la-Vallée, France]{}\
[$^2$ Columbia University, City of New York, USA]{}\
[$^3$ Northrop Grumman Space Technology, Redondo Beach, CA, USA]{}
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date: August 2007
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with Level Set Methods
---
*Level set methods are versatile and extensible techniques for general front tracking problems, including the practically important problem of predicting the advance of a firefront across expanses of surface vegetation. Given a rule, empirical or otherwise, to specify the rate of advance of an infinitesimal segment of firefront arc normal to itself (i.e., given the firespread rate as a function of known local parameters relating to topography, vegetation, and meteorology), level set methods harness the well developed mathematical machinery of hyperbolic conservation laws on Eulerian grids to evolve the position of the front in time. Topological challenges associated with the swallowing of islands and the merger of fronts are tractable.*
The principal goals of this paper are to: collect key results from the two largely distinct scientific literatures of level sets and firespread; demonstrate the practical value of level set methods to wildland fire modeling through numerical experiments; probe and address current limitations; and propose future directions in the simulation of, and the development of decision-aiding tools to assess countermeasure options for, wildland fires. In addition, we introduce a freely available two-dimensional level set code used to produce the numerical results of this paper and designed to be extensible to more complicated configurations.\
Keywords: Wildland firespread, level set methods, Multivac software
Introduction
============
Wildland fire modeling has received attention for decades, due to the sometimes disastrous consequences of large fires, and the tremendous costs of often ineffectual, possibly even counterproductive firefighting [@pyne04tending]. For the practically important scenario of wind-aided firespread, one seeks a computationally efficient model, useful not only offline (for pre-crisis planning, e.g., placement of access roads, firebreaks, and reservoirs, and scoping of fuel-reduction burning, and post-crisis review, e.g., personnel training, litigation), but also during a crisis (i.e., real-time guidance for evacuation and firefighting). For computational efficiency, such that the benefits of ensemble forecasting [@palmer05representing] are readily accessible from a model, advantage should be taken of the inherent scale separation of: (1) the kilometer-and-larger, landscape-dominated scales of the local atmospheric dynamics; and (2) the one-meter-and-smaller scales of the local combustion dynamics. Even with advanced techniques and access to exceptional contemporary computing facilities, numerical simulations (of turbulent flows) that proceed from fundamental principles are challenged to resolve accurately in real time phenomena with spatial scales spanning much more than two orders of magnitude \[H.R. Baum, private communication\]. Thus, the feasibility of a direct numerical simulation encompassing the multivaried processes of wildland fire propagation [@coen03wildfire] may be decades off [@jenkins01forest]. Moreover, at least many attempts (albeit usually problematic) at parameterization of subgridscale phenomena in terms of gridscale variables have been undertaken by meteorologists for cumulus convection, turbulent transport, and radiative transfer. However, meteorologists have extremely limited experience with the parameterization of combustion dynamics for weather-dependent wildland firespread; even if such parameterization be possible, it remains unknown. Furthermore, data collection in wildland fires is so piecemeal, irregular, and of uncertain accuracy that, for many years to come, the data better suit reinitialization of a simplistic model than assimilation into an ongoing calculation with a highly detailed model.
Accordingly, in this study, attention is focused on a minimalist treatment of the firefront, idealized as an interface between expanses of burned and unburned vegetation. This treatment is consistent with the typically limited, only gross characterization available for the vegetation at issue, since the vagaries of ignition events are difficult to anticipate, and maintaining an updated inventory for the huge area of wildlands in (say) the USA is daunting. This simplistic interfacial approach to the fire dynamics, easily executed in minutes on a laptop given the requisite meteorological and other input fields, reserves computational resources for the difficult, more critical, and mostly yet-to-be-undertaken landscape-scale weather forecasting targeted for real-time wildfire applications.
The upshot is that simple persistence models are adopted for the wind field (and thermodynamic variables) in the study undertaken here. Also, attention is limited to a one-way interaction between the meteorology and the firespread, though future extension to two-way interaction by use of an iterative procedure may be envisioned. Simplistic modeling still may provide the key macroscopic fire behavior sufficiently accurately for practical purposes (including estimates of smoke and pollutant generation), even for circumstances for which the simplification is not formally justifiable. In fact, observational data of wind-aided firefront progression in wildland are today typically sparse, so that not much more than the output of a simplistic model can be meaningfully validated and tuned. Moreover, the use of relevant mathematical methods to perform model selection, to carry out efficient parameter estimation, and to account for the uncertainty in predictions is facilitated by focusing on less detailed models with fewer parameters. In this paper, we mainly address the first step, which is to achieve proper forward simulations.
One of the most widely used models was devised by Rothermel [@rothermel72mathematical] to predict the rate of firespread, with focus on the head of a wind-aided fire. Because predictions of the Rothermel treatment have been found to be at odds with some observations, efforts to improve this spatially one-dimensional semi-empirical treatment, and to supplement the data upon which it is based, have been undertaken, especially in recent years [@carlton03impact]. Extension from a focus exclusively on the head of the fire seeks to evolve the configuration of the entire fire perimeter, possibly of multiple fire perimeters. In this study, and typically, the firefront, even a moderate fraction of an hour after a localized ignition in fire-prone vegetation, is taken to be a closed curve projected on a plane (the ground may not be flat). Such simulations of firespread have been performed [@finney98farsite] with the so-called marker technique, which discretizes a front into a set of marker particles, and advances the front through updates of the particle positions. Parenthetically, as a problematic step, the updating by Finney takes each marker on the front to evolve identically to an idealization of how a front evolves from a single isolated ignition site in an unbounded expanse of vegetation, in the presence of a wind. In any case, even though applied projects have supported software development [@finney98farsite], still from a computational point of view, only a few, largely equivalent methodological developments have been undertaken [e.g. @andre06forest]. In this paper, we apply [*level set methods*]{} [@osher88fronts; @sethian99level] to calculate firefront evolution.
In Section \[sec:fire\_model\], we introduce wildland firespread models, especially a semi-empirical, equilibrium-type model proposed in [@fendell01forest] for wind-aided firespread across surface-layer, chaparral-type, burning-prone vegetation. (In commonly adopted equilibrium-type models, the firespread rate depends on only the parametric values holding locally and instantaneously, so the firespread rate is taken to adjust indefinitely rapidly to any temporal and spatial change.) Section \[sec:level\_set\] provides a brief introduction to level set methods. Section \[sec:code\] describes the Multivac level set package that has been applied in this paper to the firespread problem. A quick description of its performance is presented in Section \[sec:complexity\]. Finally, results of firespread simulations with different idealized environmental conditions are reported in Section \[sec:applying\].
Front Propagation Functions for Wildland Fires {#sec:fire_model}
==============================================
Even if theory and/or measurement furnished complete, perfect knowledge of the topography, vegetation, and meteorology at a site at a given time (e.g., furnished the locally pertinent values of all parameters in functional forms capable of representing these three types of input), still one currently possesses very incomplete, imperfect knowledge of the “rules” that would yield the physically observed rate of firespread from the input. Achieving knowledge of firespread “rules” sufficiently accurate for practical purposes may well lag emplacing means for observing and collecting exhaustive input data.
As already noted, a fire-growth simulation such as FARSITE [@finney98farsite] seems unlikely to reach its potential as long as it seeks to describe the rate of firespread at all orientations to the direction of the sustained low-level ambient wind from spread-rate modeling focused on the direction of the wind [e.g. @rothermel72mathematical]. On the other hand, posing a different rule for the spread rate at every possible orientation to the wind defeats the goal of simplicity.
Wind-aided wildland fire spread
-------------------------------
Fendell and Wolff [@fendell01forest] addressed this dilemma in developing a model dedicated to wind-aided wildland fires that spread rapidly over level terrain with dry, moderately sparse fuel, taken here to be uniformly distributed to permit concentration on wind effects. Parenthetically, for consistency with modeling in which the firefront is idealized as an interface moving according to a semi-empirical rule, only a minimal amount of information about the surface-layer fuel is required, mainly the mass loading consumed with firefront passage (“available”-fuel loading).
The Fendell and Wolff model focuses on front velocities at the rear of the front (where propagation is against the wind), at the head of the front (where propagation is with the wind), and on the flanks (where propagation is across the wind direction) – see Figure \[fig:front\]. The firespread velocities primarily depend on the wind velocity $U$. At the rear, the front advances relatively slowly against the oncoming wind, since hot combustion products tend to be blown over an already burned area. The velocity at the rear is denoted $\varepsilon(U)$. At the head, the velocity $h(U)$ is relatively large, since hot combustion products tend to be blown over a yet-to-burn area, in which discrete fuel elements are heated toward ignition by convective-conductive transfer. Both analytic modeling and laboratory experiments have shown that $h(U)$ is roughly proportional to $\sqrt{U}$ [@wolff91wind]. At the flanks, the (spread-aiding) wind component along the normal to the front is zero, but observationally the front advances faster than in the absence of wind. As a speculation, a more meticulous treatment would find that, at the nominal flank, the configuration is convoluted, and firespread is alternately with and against the wind. Of course, were the wind direction constant, limiting attention to the head would seem adequate, but, in fact, change in wind direction may (rapidly) result in an interchange of the locations of the flank and head – an interchange sometimes associated with tragic consequence for firefighters.
![Fendell and Wolff model introduces velocities at the rear (against the wind), at the head (in the wind direction) and at the flanks. [@fendell01forest][]{data-label="fig:front"}](figures/model.eps){width="80.00000%"}
The velocities (the terminology henceforth adopted, for brevity, in place of firespread rates) proposed in [@fendell01forest] are $$\varepsilon(U) = \varepsilon_0 \exp(-\varepsilon_1 U), \quad f(U) =
\varepsilon_0 + c_1 U \exp(-c_2 U), \quad h(U) = \varepsilon_0 + a \sqrt{U},
\label{eq:velocities}$$ where $\varepsilon_0$, $\varepsilon_1$, $c_1$, $c_2$ and $a$ are parameters (with readily inferred dimensionality) depending on the mass loading of fuel and other parameters characterizing the fuel bed, but independent of $U$.
The velocity is then provided at any point on the front through a “trigonometric interpolation”: $$\begin{array}{l}
F(U, \theta) = f(U \sin^m \theta) + h(U \cos^n \theta) \quad \textrm{if
$|\theta| \leq \frac{\pi}{2}$},\\
F(U, \theta) = f(U \sin^m \theta) + \varepsilon(U \cos^2 \theta) \quad
\textrm{if $|\theta| > \frac{\pi}{2}$},
\end{array}
\label{eq:ext_velocities}$$ where $\theta$ is the angle between the wind direction and the normal to the front. We set $m = 2$. In this paper, parameter $n$ is set to $\frac{3}{2}$ and is significant since it determines the overall shape of the front from the flanks to the head.
To summarize, the velocity is, for all $U\in{\mathbb{R}}_+$ and $ \theta \in ]-\pi,
\pi[$, $$\begin{array}{rl}
F(U, \theta) = & \varepsilon_0 \sin^2 \theta + c_0 U \sin^2 \theta
\hspace{0.5mm} \exp \left(-c_1 U \sin^2 \theta \right) \\ & + \left\{
\begin{array}{llll}
\displaystyle \varepsilon_0 \cos^2 \theta + a \sqrt{U}
\cos^n \theta \quad & \textrm{if $|\theta| \leq
\frac{\pi}{2}$}\\ \displaystyle \varepsilon_0 \cos^2 \theta
\hspace{0.5mm} \exp \left(-\varepsilon_1 U \cos^2 \theta \right) \quad
& \textrm{if $|\theta| > \frac{\pi}{2}$}
\end{array}
\right..
\end{array}
\label{eq:fire_model}$$
Simplified model
----------------
Based on the numerical experiments carried out with the level set code Multivac (Section \[sec:code\]), the model (\[eq:fire\_model\]) proposed in [@fendell01forest] has been modified. First, the parameter $n$ has been set to $\frac{3}{2}$ instead of $1$. Second, the model has been simplified without losing its main features, primarily the overall shape of the firefront. The new model reads $$\begin{array}{ll}
F(U, \theta) = \varepsilon_0 + c_1 \sqrt{U} \cos^n \theta & \quad
\textrm{if $|\theta| \leq \frac{\pi}{2}$},\\
F(U, \theta) = \varepsilon_0 (\alpha + (1 - \alpha)|\sin \theta|) & \quad
\textrm{if $|\theta| > \frac{\pi}{2}$},
\end{array}
\label{eq:simplified_model}$$ where $\alpha \in [0, 1]$ is the ratio between the velocity at the rear ($\alpha \varepsilon_0$) and the velocity at the flanks ($\varepsilon_0$). Velocities at the rear and at the flanks no longer depend on the wind, since their dependence on the wind speed is hard to model accurately and has little impact on the overall front location. The velocity at the head is the same as in the “full” model (\[eq:fire\_model\]).
The simplified model is easier to tune, either via direct trials or with systematic methods for parameter estimation (which may require derivatives of the model with respect to its parameters). All results in this paper are for the simplified model. However, results for the “full” model would appear roughly the same.
Level Set and Fast Marching Methods {#sec:level_set}
===================================
First introduced in @osher88fronts, level set methods are Eulerian schemes for tracking fronts propagating according to a given speed function. In this section, we explain basic features of the level set methods used for firespread modeling.
Mathematical basis and technique
--------------------------------
### Definitions
Assume the front evolves from the initial time $t = 0$ to the final time $t =
T_f$. For all $t \in \left[0, T_f\right]$, the front at time $t$ is the set of points (in ${\mathbb{R}}^N$) $\Gamma(t)$. We define $\Gamma_0=\Gamma(0)$ as the initial front.
For all $t \in \left[0, T_f\right]$, each point $X \in \Gamma(t)$ with a well-defined normal moves in the direction normal to the front with a given speed $F(X, \Gamma,t)$. Notice that $F$ may depend on the position, on the time and on local properties of the front itself (certainly the normal direction, not always defined, and possibly the local curvature or other properties).
The problem is to approximate $\Gamma: \left[0, T_f\right] \rightarrow
{\mathbb{R}}^N$, given $\Gamma_0$ and $F$.
### Strategy
The main idea is to evolve a function $\varphi: {\mathbb{R}}^N \times \left[0,
T_f\right] \rightarrow {\mathbb{R}}$ such that $${\forall \hspace*{0.7mm}}t \in \left[0, T_f\right] \qquad \Gamma(t) = \left\{ x \in
{\mathbb{R}}^N \Big{/} \varphi(x, t) = 0 \right\}.
\label{eq:level_set_def}$$
$\varphi$ is called the level set function. At any time, the zero level set of $\varphi$ is the front itself. [*A priori*]{}, $\varphi$ could be any function satisfying equation (\[eq:level\_set\_def\]). However, some assumptions (e.g., smoothness) and practical issues (e.g., initialization of $\varphi$) make it convenient to define $\varphi$ as the signed distance to the front.
Then, if $d$ is the Euclidean distance on ${\mathbb{R}}^N$, we define, for any given curve $\Upsilon$, the distance $d_{\Upsilon}$ to $\Upsilon$: $${\forall \hspace*{0.7mm}}x \in {\mathbb{R}}^N \qquad d_{\Upsilon}(x) = \min \left\{ d(x, P) \Big{/} P
\in \Upsilon \right\}.$$
Hence the signed distance $\varphi$ for all $x\in{\mathbb{R}}^N$ and $t\in\left[0,
T_f\right]$: $$\varphi(x, t) = \left\{
\begin{array}{ll}
d_{\Gamma(t)}(x) & \textrm{if $x$ lies outside the front
$\Gamma(t)$}\\
- d_{\Gamma(t)}(x) & \textrm{if $x$ lies inside the front
$\Gamma(t)$}\\
\end{array}
\right..
\label{eq:phi}$$
It can be shown that $\varphi$ obeys the equation $${\forall \hspace*{0.7mm}}x \in {\mathbb{R}}^N\quad {\forall \hspace*{0.7mm}}t \in [0, T_f] \qquad
\varphi_t(x, t) + F(x,\varphi(\cdot, t), t) {\| \nabla_x \varphi(x,
t) \|}_2 = 0,
\label{eq:level_set}$$ where the velocity $F$ is now defined everywhere in ${\mathbb{R}}^N$ and depends on the front through its dependence upon $\varphi$. Details may be found in @sethian99level.
Recall that $\varphi(\cdot, 0)$ is known as well as $\Gamma_0$; $\varphi(0)$ is the signed distance to $\Gamma_0$: $${\forall \hspace*{0.7mm}}x\in{\mathbb{R}}^N \quad \varphi(x, 0) = \left\{
\begin{array}{ll}
d_{\Gamma_0}(x) & \textrm{if $x$ lies outside the front
$\Gamma_0$}\\
- d_{\Gamma_0}(x) & \textrm{if $x$ lies inside the front
$\Gamma_0$}\\
\end{array}
\right..
\label{eq:initial_cond}$$
Equations (\[eq:level\_set\]) and (\[eq:initial\_cond\]) define the initial-value problem that is to be solved. Zero level sets of $\varphi$ yield the front points.
This nonstationary problem involves the Hamilton-Jacobi equation (\[eq:level\_set\]). There may be multiple solutions to this equation. P.-L. Lions and M. G. Crandall defined the so-called “viscosity solution” of Hamilton-Jacobi equations [@lions82generalized; @crandall83viscosity], which turns out to be the unique physical solution for which we search. Under given assumptions (mainly on the speed function $F$), existence and uniqueness of the viscosity solution of the problem (\[eq:level\_set\])–(\[eq:initial\_cond\]) can be proved.
Advantages and disadvantages of level set methods
-------------------------------------------------
Several methods may be relevant to simulate the propagation of firefronts. One may want to use marker techniques, in which the front is discretized by a set of points. At each time step, each point is advanced according to the speed function. This Lagrangian methodology leads to low-cost computations, but requires care in the handling of topological changes.
Volume-of-fluid methods represent the front by the amount of each grid-cell that is inside the front. In each cell, the front is approximated by a straight line (horizontal or vertical, in most methods). Such methods can deal with topological changes, but the front representation can be inaccurate. In wildland firespread, the direction normal to the front is crucial because of the wind-direction-dependent speed function (see Section \[sec:fire\_model\]).
Level set methods automatically deal with topological changes that occur in wildland firespread, such as fronts merging and front convergence (in connection with unburnt “islands”). The level set description enables a fair estimate of the normal to the front, making it well suited to the fire propagation problem.\
However, level set methods have disavantages. First, they embed the front in a higher-dimensional space. Helpfully, the narrow band level set method [@adalsteinsson95fast] is an efficient algorithm which almost decreases the problem dimension by one. Moreover, when it can be used, the fast marching method [@sethian96fast] provides a highly efficient algorithm.
The main reservation may be the lack of proof of convergence of numerical schemes for certain problems. For a given class of speed functions, the problem (\[eq:level\_set\])–(\[eq:initial\_cond\]) may routinely be solved numerically [@crandall84two]. However, no proof of convergence in mesh parameter or time step is yet available for some situations.
Quick review of numerical approximations
----------------------------------------
Numerical approximation to solutions of Hamilton-Jacobi equations is closely related to numerical approximation to hyperbolic conservation laws[^1]. The point is to introduce a numerical Hamiltonian to approximate the Hamiltonian $H = F \cdot{\| \nabla_x \varphi \|}_2$.
Crandall and Lions have proven that, for given Hamiltonians and initial conditions, a consistent, monotonic and locally Lipschitzian numerical Hamiltonian yields a solution that converges to the viscosity solution. Formal results may be found in @crandall84two and @souganidis85approximation.
In one dimension, $\varphi_t + H(\nabla_x \varphi)=0$ may lead to the following approximation: $$\varphi^{n+1}_{j} = \varphi^n_{j} - \Delta t \hspace{1mm} g
\left(\frac{\varphi_{j+1} - \varphi_{j}}{\Delta x}, \frac{\varphi_{j} -
\varphi_{j-1}}{\Delta x} \right).
\label{eq:approx}$$
For instance, if the Hamiltonian is not convex, the Lax-Friedrichs scheme may be used; then, the numerical Hamiltonian is $${\forall \hspace*{0.7mm}}a, b \in {\mathbb{R}}\qquad g(a, b) = H\left(\frac{a+b}{2}\right)
- \vartheta \frac{b-a}{2},
\label{eq:Lax-Friedrichs}$$ where the monotonicity is satisfied on $[-R, R]$ if $\displaystyle\vartheta =
\max_{-R \leq a \leq R} |H'(a)|$.
Several schemes have been developed, from simple and efficient schemes as that of Engquist-Osher to high-order essentially nonoscillatory schemes [@osher91high].
Overview of complexity issues
-----------------------------
Let the mesh (in ${\mathbb{R}}^N$) be orthogonal with $M$ points along each direction. Assume that the front is described by $\mathcal{O}(M^{N-1})$ points. The narrow band level set method makes it sufficient to update the level set function only in a narrow band (of width $k$) around the front. For each time step, the complexity of the algorithm is therefore $\mathcal{O}(kM^{N-1})$.
For an explicit temporal discretization the number of iterations is related to the Courant-Friedrichs-Lewy condition. Along $x$, the Courant number must be less than $1$: $$\frac{\max \left|H'\right|\Delta t}{\Delta x} \leq 1.
\label{eq:CFL}$$
Usually, controlling the accuracy of approximation is subordinate to space discretization, which means that the time step is adjusted so that the Courant number is taken close to $1$.\
Calculations may sometimes be sped up by reformulating the level set problem as a stationary problem. This leads to the so-called fast marching method [@sethian96fast]. Nevertheless, restrictions on the Hamiltonian prevent the use of this technique for some applications. The work of Sethian and Vladimirsky has overcome some limitations [@sethian01ordered], but restrictive conditions still remain (e.g., convexity of the Hamiltonian).
Code {#sec:code}
====
Introduction to the Multivac level set package
----------------------------------------------
Multivac is a level set package freely available (under the GNU GPL license) at <http://vivienmallet.net/fronts/>. It is designed to be both efficient and extensible, so that it may be used for a large range of applications. To achieve these goals, Multivac is built as a fully object-oriented library in C++.
Multivac was designed independently of the firespread application described herein, but easily enabled firespread simulations, and is presently distributed with firespread-motivated functions. It has also been used in modeling the growth of Si-based nanofilms [@phan03modeling] and image segmentation.
The latest stable version available at the time of submission is Multivac 1.10.
Structure
---------
The modularity of Multivac comes from its object-oriented framework, in which the main components of a simulation have been split into an equal number of objects. A simulation is defined by the following objects:
- the *mesh*;
- the *level set function*;
- the *velocity*, which provides the propagation rate of the front according to its position, its normal, its curvature, and the time;
- the *initial front*;
- the *initializer*, which manages first initializations and initializations required by level set methods (e.g., the narrow band reconstruction);
- the *numerical scheme*, which advances the front in time;
- the *output management*.
For each item, a set of classes[^2] with a common interface is available. For instance, several speed (i.e., propagation rate) functions are available through several classes, e.g. [`CConstantSpeed`]{} or [`CFireModel`]{}. All speed functions have the same interface, which allows users to define their own speed function on the same basis. The user principally provides speed rates as a function of the position, the time, the normal to the front and the curvature (these values are computed by Multivac itself).
Calling sequence
----------------
The whole is managed by an object of the class [`CSimulator`]{}. This object simply calls the *initializer* to perform the first initializations. Then it manages the loop in time (or iterations, in the case of the fast marching method) into which the *numerical scheme* is called to advance the front. The *initializer* is called again to reinitialize the signed distance function for the new step, and the object dedicated to post-processing requirements is called to save any needed data.
In each step, objects communicate with one another through methods (i.e., functions) of their interface. For example, the *velocity* object provides speed rates to the *numerical scheme*.
Overview of available classes
-----------------------------
Multivac package (version 1.10) includes several classes which are listed in Table \[tab:classes\].
Category Available classes
-------------------- ----------------------------------------------
Mesh Orthogonal mesh
Level set function Defined on an orthogonal mesh
Velocity Constant speed
Piecewise constant speed
Fire model
Simplified fire model
Image intensity
Image gradient
Initial front Circle
Two or three circles
One or two circles with an island inside
Front defined by any set of points
Initializer Basic initialization (no velocity extension)
Extends the velocity with the closest
neighbor on the front
Numerical scheme Engquist-Osher, first order
(narrow band) Lax-Friedrichs, first order
Engquist-Osher, ENO, second order
Chan-Vese algorithm [@chan01active]
Numerical scheme Engquist-Osher, first order
(fast marching)
: Basic classes available in Multivac 1.10.[]{data-label="tab:classes"}
Other strengths, limitations and future work
--------------------------------------------
Multivac takes advantage of C++ exceptions to track errors, and several debugging levels are defined, from a safe mode, in which all is checked, to a fast mode, in which performance is the primary concern.
There are currently two main limitations. First, Multivac deals only with uniform orthogonal meshes. However, extensions of level set methods to unstructured meshes exist (e.g., @barth98numerical) and they could be implemented within the Multivac framework. Adaptively refined meshes are also accommodated with additional mathematical complexity, though the implementation effort would be substantial. Second, Multivac deals only with two-dimensional problems.
Work is planned to allow inverse modeling (parameter estimation based on data assimilation) within the framework of Multivac. The main idea is to replace the class [`CSimulator`]{} with a class dedicated to inverse modeling. Preliminary results show the framework extendibility, but this capability is not yet available in distributed versions. Future versions should include this feature, based on an innovative method for integrating sensitivities along with the front itself.
Complexity and Convergence Studies {#sec:complexity}
==================================
Convergence studies
-------------------
In this section, we report convergence studies that are necessary to validate the code. As in @adalsteinsson98fast, tests are carried out for a circle that expands in time with a unitary velocity. Details of the simulation are summarized in Table \[tab:test\_case\].
Data Value Comment
----------------------- ------------------------------------------------- ---------------
Domain $\Omega = [0, 3] \times [0, 3]$
Initial front Circle
Circle center $\left(x_c, y_c\right) = \left(1.5, 1.5\right)$ Domain center
Initial circle radius $r_{initial} = 0.5$
Final circle radius $r_{final} = 0.9$
Velocity $F = 1.0$ Constant
Duration $T_f = 0.4$
Time step $\Delta t = 10^{-4}$
: Simulation test-case.[]{data-label="tab:test_case"}
We introduce three norms. The first is $$\label{eq:es}
e_{spatial}^1 = |r_{simulated} - r_{final}|,$$ where $r_{simulated}$ is the simulated radius, estimated as follows: $$r_{simulated} = \frac{1}{\mathrm{card}(\Gamma_d)} \sum_{(x, y)\in\Gamma_d}
\textrm{d} \left( (x, y), \left(x_c, y_c\right) \right),$$ where $\Gamma_d$ is the discretized front as returned by the simulation (at time $T_f$) and $\textrm{d}$ is the Euclidian distance.
Additionally, if $T_{true}(x, y)$ is the time at which the front is supposed to reach the point $(x, y)$: $$\label{eq:et}
e_{time}^2 = \sqrt{ \frac{1}{\mathrm{card}(\Gamma_d)} \sum_{(x,
y)\in\Gamma_d} \left( T_f - T_{true}(x, y) \right)^2 }.$$
The last norm is an infinity norm: $$\label{eq:ef}
e_{time}^\infty = \max_{(x, y)\in\Gamma_d} \left| T_f - T_{true}(x,
y) \right|.$$
Table \[tab:eo\] shows results for the first-order Engquist-Osher scheme with the narrow band method. The width of the band is 12 cells and the front lies within a band whose width is 6 cells.
--------------------------------------------------------------------------------------------------------------------------------------------------------------
$\Delta x = \Delta y$ $N_x = N_y$ $e_{spatial}^1$ $\left(\times 10^3\right)$ $e_{time}^2$ $\left(\times $e_{time}^\infty$ $\left(\times 10^3\right)$
10^3\right)$
----------------------- ------------- -------------------------------------------- ---------------------------- ----------------------------------------------
0.01 301 1.634 1.753 2.377
0.005 601 0.855 0.901 1.191
0.0025 1,201 0.460 0.474 0.600
0.00125 2,401 0.244 0.247 0.299
--------------------------------------------------------------------------------------------------------------------------------------------------------------
: Errors versus spatial discretization.[]{data-label="tab:eo"}
The first-order Lax-Friedrichs scheme and the second ENO Engquist-Osher scheme were also checked successfully. As for the second-order scheme, the full-matrix method, that is, without the narrow-band restriction, was used because the front reconstruction destroys the second-order accuracy.
Complexity issues
-----------------
Multivac was compiled under Linux with GNU/g++ 3.3, and the reference simulation (see Table \[tab:test\_case\]) was launched on a Pentium 4 running at 2.6 Ghz. The width of the narrow band was 12 cells and the width of the inner band, in which the front lies, was 6 cells. If $N_x=N_y=1001$ (one million cells), the $4,000$ iterations were achieved in 14 s.
The complexity of the narrow band level set method is close to $\mathcal{O}(N)$, where $N=N_x=N_y$. Table \[tab:timings\] shows that linear complexity of the method is not observed. Instead, the complexity seems to be $\mathcal{O}(N^2)$. This is the complexity of the suboptimal algorithm currently used to rebuild the front. Moreover, the number of front reconstructions increases with the mesh refinement since the width of the narrow band does not change.
$\Delta x = \Delta y$ $N_x = N_y$ Timings (s)
----------------------- ------------- ------------- -- --
0.03 101 0.4
0.015 201 0.9
0.01 301 1.6
0.0075 401 2.6
0.006 501 4.0
0.005 601 5.6
0.004285714 701 7.4
0.00375 801 9.5
0.003333333 901 11.9
0.003 1001 14.1
: Timings versus spatial discretization.[]{data-label="tab:timings"}
Applying Level Set Methods to Firespread Applications {#sec:applying}
=====================================================
Method and numerical scheme
---------------------------
The speed function (\[eq:fire\_model\]) introduced in the level set equation (\[eq:level\_set\]) provides an Hamiltonian with nontrivial dependencies. Because of these dependencies (particularly the non-convexity of the Hamiltonian), neither the fast marching method nor its extension to anisotropic problems can be applied. The narrow-band level set method is more relevant.
A highly accurate numerical scheme is not required for the investigations reported here. The discrepancies between the numerical simulation and the exact solution should be considered in the context of other approximations: the model itself is simplistic; input parameters such as wind speed or fuel density are typically not accurately estimated; the location of the initial front introduces further uncertainties. A first-order scheme suffices for our purposes.
Since the Hamiltonian involved is not convex with respect to spatial derivatives of the level set function, the first-order Lax-Friedrichs scheme (refer to equation (\[eq:Lax-Friedrichs\])) is well suited. To minimize introduction of diffusivity, a local Lax-Friedrichs scheme may be used as well.
As previously advocated, the timestep $\Delta t$ is chosen according to the Courant-Friedrichs-Lewy condition (\[eq:CFL\]): $$\Delta t = \frac{\alpha\Delta x}{\max \left|H'\right|},$$ where $\alpha \leq 1$; $\alpha$ is not kept constant in the tests that we undertake. Nevertheless, the Courant-Friedrichs-Lewy condition is estimated at every iteration with an ([*a priori*]{}) approximation to $\max
\left|H'\right|$ along $x$ and $y$, which leads to: $$\Delta t \leq \frac{\Delta x}{a (m + 1) \sqrt{U}}.$$
The main characteristics of the simulation, including model parameters (refer to equation (\[eq:fire\_model\])), are gathered in Table \[tab:param\].
Results
-------
The simulation described by Table \[tab:param\] is shown in Figure \[fig:basic\_case\]. The figure shows snapshots of the front, initially circular, at subsequent times, under a constant-magnitude wind blowing from left to right. Since thoroughly burnt areas cannot be burnt again (on the time scale of the simulation), the area enclosed by the front increases with time. The rear, the flanks and the head of the front are clearly identifiable.
![Basic simulation described by Table \[tab:param\].[]{data-label="fig:basic_case"}](figures/basic_case.eps){width="80.00000%"}
The reference simulation is slightly modified to show the ability to deal with multiple fronts – Figure \[fig:merge\_and\_island\]. It demonstrates the capability to deal with the merging of fronts (two main fronts), and to deal with the so-called islands, i.e. an unburnt area surrounded by a burnt area.
![Two main fronts merge, and an island – the unburnt area within the biggest front – is burnt.[]{data-label="fig:merge_and_island"}](figures/merge_and_island.eps){width="80.00000%"}
In Figures \[fig:fuel\_slow\] and \[fig:fuel\_fast\], we use the same parameters as in Table \[tab:param\] but $\Delta t = 2.5\cdot 10^{-5}$, and $a$ depends on $x$, $a$ being equal to $0.5$ if $x<1.7$, and $a=0.25$ (Figure \[fig:fuel\_slow\]) or $a=1.0$ (Figure \[fig:fuel\_fast\]) if $x>1.8$, and $a$ being linearly interpolated for intermediate values of $x$. Since $a$ takes into account the available fuel loading, these two simulations roughly show the influence of the inhomogeneous available fuel loading, should it increase (Figure \[fig:fuel\_slow\]) or decrease (Figure \[fig:fuel\_fast\]). The inherent decrease of the radius of curvature at the head for a constant-direction wind suggests that some vacillation of wind direction contributes when the head broadens under otherwise uniform conditions.
![The front slows down at the head for $a = 0.25$ if $x>1.8$. The final time is changed to $T_f = 1.5$.[]{data-label="fig:fuel_slow"}](figures/fuel_slow.eps){width="80.00000%"}
![The front advances faster at the head for $a = 1.0$ if $x>1.8$.[]{data-label="fig:fuel_fast"}](figures/fuel_fast.eps){width="80.00000%"}
Figure \[fig:rotating\] shows the impact of a rotating wind direction. If north is toward the top of the figure, then the wind is oriented first west-to-east and tends later to south-to-north.
![Same as the reference simulation, but with a changing wind direction.[]{data-label="fig:rotating"}](figures/rotating.eps){width="80.00000%"}
The next two Figures \[fig:counterflow\_up\] and \[fig:counterflow\_middle\] show the behavior of two fronts subject to a simple-counterflow wind, i.e., a wind defined as: $$\overrightarrow{U}(x, y) = \left(
\begin{array}{c}
-u x\\
u y
\end{array}
\right)$$ where $u$ is set to $100$. A counterflow exemplifies wind conditions well suited for setting a backfire, to preburn the vegetation in the path of a wind-aided fire.
![Evolution of the merged front from initially two mirror-image fronts, one to each side of the stagnation line for a converging $x$-component wind, but both to one side of the stagnation line for a diverging $y$-component wind.[]{data-label="fig:counterflow_up"}](figures/counterflow_up.eps){width="80.00000%"}
![Evolution of the merged front from initially two mirror-image fronts, here symmetrically sited relative to a simple counterflow wind.[]{data-label="fig:counterflow_middle"}](figures/counterflow_middle.eps){width="80.00000%"}
The last Figure \[fig:topography\] shows a front that propagates over an idealized hill. Where the slope is positive (between $x=1.6$ and $x=1.7$), the firefront typically advances faster. Downhill the front typically slows down [@luke78bushfires pp. 94–97]. The speed function is therefore modified to take into account the slope $s$: $$\label{eq:topography}
F_{\mathrm{topography}} = F \times e^{2s},$$ where $s$ is in radians.
![Taking into account topography: the front propagates over an idealized hill.[]{data-label="fig:topography"}](figures/topography.eps){width="80.00000%"}
Conclusion and Future Prospects {#sec:conclusion}
===============================
A semi-empirical, equilibrium-type firespread rate has been used to model a wind-aided firefront propagation across wildland surface vegetation. In this formulation, the rate depends primarily on the wind speed, and the angle between the wind direction and the normal to the firefront (idealized as a one-dimensional interface). In scenarios arising in practice, the front may consist of several closed curves (possibly nested) that can merge as they propagate.
Level set methods appear capable of treating the model formulated to simulate wildland fire evolution. They treat readily the topological changes that may occur to the firefront, and they are known to converge to the physical solution of front tracking problem.
They were applied via the Multivac package. This open-source library is designed to handle a wide range of applications without loss of computing performance. It includes several algorithms and numerical schemes, primarily for the narrow-band level set method, which is more computationally efficient than the full level set method.
A possible direction for future work is to focus on parameter estimation within the context of the simple model illustrated herein. A cost is introduced to measure the distance between the simulated front and ground, aerial, and/or satellite observations. The discrepancy between the simulated and observed positions of the front may be based either on the front arrival times (at monitored locations), or on distances between the simulated front and the monitored locations (at arrival times). For gradient-based optimization methods, the main challenge is to compute the derivative of the cost function with respect to the parameters. An adjoint code being difficult to construct, alternative methods should be sought.
This work could help guide fire-control tactics. The objective function would then penalize front advance into societal assets, and penalize the cost of the firefighting activity. The parameters would be the model variables modifiable by firefighting countermeasures. The links between this optimization problem and shape optimization should be investigated.
Acknowledgments {#acknowledgments .unnumbered}
===============
The support of the National Science Foundation under grant CCF-03-52334 and the U.S. Department of Agriculture Forest Service under grant SFES 03-CA-11272169-33, administered by the Riverside Forest Fire Laboratory, a research facility of the Pacific Southwest Research Station, is gratefully acknowledged. The authors are particularly indebted to Dr. Francis M. Fujioka of Riverside for enhancing the relevance of our research through his technical advice, and for his support for the training of summer students in the computational technology of firespread and fire imaging.
[^1]: Notice that, from equation (\[eq:level\_set\]), $\varphi_x$ satisfies a hyperbolic conservation law in the one-dimensional case.
[^2]: A class is a user-defined type, in the manner of structures in C. Classes encapsulate data (called attributes) and functions (called methods).
|
---
abstract: 'We prove that the space of coinvariants of functions on an affine variety by a Lie algebra of vector fields whose flow generates finitely many leaves is finite-dimensional. Cases of the theorem include Poisson (or more generally Jacobi) varieties with finitely many symplectic leaves under Hamiltonian flow, complete intersections in Calabi-Yau varieties with isolated singularities under the flow of incompressible vector fields, quotients of Calabi-Yau varieties by finite volume-preserving groups under the incompressible vector fields, and arbitrary varieties with isolated singularities under the flow of all vector fields. We compute this quotient explicitly in many of these cases. The proofs involve constructing a natural ${\mathcal{D}}$-module representing the invariants under the flow of the vector fields, which we prove is holonomic if it has finitely many leaves (and whose holonomicity we study in more detail). We give many counterexamples to naive generalizations of our results. These examples have been a source of motivation for us.'
address:
- 'MIT Dept of Math, Room 2-176, 77 Massachusetts Ave, Cambridge, MA 02139, USA'
- 'U. Texas at Austin, Math Dept, RLM 8.100, 2515 Speedway Stop C1200, Austin, TX 78712, USA'
author:
- Pavel Etingof
- Travis Schedler
bibliography:
- 'master.bib'
date: 2012
title: Coinvariants of Lie algebras of vector fields on algebraic varieties
---
Introduction
============
Vector fields on affine schemes
-------------------------------
Let ${\mathbf{k}}$ be an algebraically closed field of characteristic zero, and let $X = {\operatorname{\mathsf{Spec}}}{\mathcal{O}}_X$ be an affine scheme of finite type over ${\mathbf{k}}$ (we will generalize this to nonaffine schemes in §\[ss:nonaffine\] below). Our examples will be varieties, so the reader interested only in these (rather than the general theory, which profits from restriction to nonreduced subschemes) can freely make this assumption.
We also remark that our results can be generalized to the analytic setting using the theory of analytic ${\mathcal{D}}$-modules, except that in these cases, the coinvariants need no longer be finite-dimensional, since analytic varieties can have infinite-dimensional cohomology in general (e.g., a surface with infinitely many punctures). But we will not discuss this here.
When we say $x \in X$, we mean a closed point, which is the same as a point of the reduced subvariety $X_{{\text{red}}}$. Note that (since ${\mathbf{k}}$ has characteristic zero) it is well-known that all vector fields on $X$ (which by definition means derivations of ${\mathcal{O}}_X$) are parallel to $X_{\text{red}}$ (dating to at least [@Sei-dirfgt Theorem 1]). Hence, there is a restriction map ${\operatorname{Vect}}(X) \to {\operatorname{Vect}}(X_{{\text{red}}})$, although this is not an isomorphism unless $X=X_{\text{red}}$. In particular, for all global vector fields $\xi \in {\operatorname{Vect}}(X)$ and all $x \in X$, $\xi|_x \in
T_x X_{\text{red}}$. Let ${\mathfrak{v}}\subseteq {\operatorname{Vect}}(X)$ be a Lie subalgebra of the Lie algebra of vector fields (which is allowed to be all vector fields). We are interested in the coinvariant space, $$({\mathcal{O}}_X)_{{\mathfrak{v}}} := {\mathcal{O}}_X / {\mathfrak{v}}({\mathcal{O}}_X).$$ (We remark that one could more generally consider an arbitrary set of vector fields, but the coinvariants coincide with those of the Lie algebra generated by that set. One could also consider more generally differential operators of order $\leq 1$: see Remark \[r:diffop-leq1\].)
Our main results show that, under nice geometric conditions, this coinvariant space is finite-dimensional, and in fact that the corresponding ${\mathcal{D}}$-module generated by ${\mathfrak{v}}$ is holonomic. This specializes to the finite-dimensionality theorems [@BEG Theorem 4] and [@ESdm Theorem 3.1] in the case of Poisson varieties. It also generalizes a standard result about coinvariants under the action of a reductive algebraic group (see Remarks \[r:red\] and \[r:nonfinite\] below).
Our first main result can be stated as follows.
\[t:main1\] Suppose that, for all $i \geq 0$, the locus of $x \in X$ where the evaluation ${\mathfrak{v}}|_x$ has dimension $\leq i$ has dimension at most $i$. Then the coinvariant space $({\mathcal{O}}_X)_{{\mathfrak{v}}}$ is finite-dimensional.
In particular, the hypothesis implies that, on an open dense subvariety of $X_{\text{red}}$, ${\mathfrak{v}}$ generates the tangent bundle; as we will explain below, the hypothesis is equivalent to the statement that $X_{\text{red}}$ is stratified by locally closed subvarieties with this property.
Goals and outline of the paper
------------------------------
First, in §\[s:gen\], we reformulate Theorem \[t:main1\] geometrically and prove it, along with more general finite-dimensionality and holonomicity theorems. The main tool involves the definition of a right ${\mathcal{D}}$-module, $M(X,{\mathfrak{v}})$, generalizing [@ESdm], such that ${\operatorname{Hom}}(M(X,{\mathfrak{v}}), N) \cong N^{\mathfrak{v}}$ for all ${\mathcal{D}}$-modules $N$, i.e., the ${\mathcal{D}}$-module which represents invariants under the flow of ${\mathfrak{v}}$. Then the theorem above is proved by studying when this ${\mathcal{D}}$-module is holonomic.
The next goal, in §\[s:Csla\], is to study examples related to Cartan’s classification of simple infinite-dimensional transitive Lie algebras of vector fields on a formal polydisc which are complete with respect to the jet filtration. Namely, according to Cartan’s classification [@Car-gtcis; @GQS-ic], there are four such Lie algebras, as follows. For $\xi \in {\operatorname{Vect}}(X)$, let $L_\xi$ denote the Lie derivative by $\xi$. Let $\hat {\mathbf{A}}^n$ be the formal neighborhood of the origin in ${\mathbf{A}}^n$, which is a formal polydisc of dimension n. Then, Cartan’s classification consists of:
1. The Lie algebra ${\operatorname{Vect}}(\hat {\mathbf{A}}^n)$ of all vector fields on $\hat {\mathbf{A}}^n$;
2. The Lie algebra $H(\hat {\mathbf{A}}^{2n},\omega)$ of all Hamiltonian vector fields on $\hat {\mathbf{A}}^{2n}$, i.e., preserving the standard symplectic form $\omega = \sum_i dx_i \wedge dy_i$; explicitly, $\xi$ such that $L_\xi \omega = 0$;
3. The Lie algebra $H(\hat {\mathbf{A}}^{2n+1},\alpha)$ of all contact vector fields on an odd-dimensional formal polydisc, with respect to the standard contact structure $\alpha = dt + \sum_i x_i dy_i$, i.e., those vector fields satisfying $L_\xi \alpha \in {\mathcal{O}}_X \cdot
\alpha$;
4. The Lie algebra $H(\hat {\mathbf{A}}^n,{\mathsf{vol}})$ of all volume-preserving vector fields on $\hat {\mathbf{A}}^n$ equipped with the standard volume form ${\mathsf{vol}}= dx_1 \wedge \cdots \wedge dx_n$, i.e., vector fields $\xi$ such that $L_\xi {\mathsf{vol}}= 0$.
In §\[s:Csla\], we define generalizations of each of these examples to the global (but still affine), singular, degenerate situation. For example, (a) becomes vector fields on arbitrary schemes of finite type. For (b)–(d), we define generalizations of the structure on the variety, which in case (b) yields Poisson varieties. Then, there are essentially two different choices of the Lie algebra of vector fields. In case (b), these are Hamiltonian vector fields or Poisson vector fields. We recall that Hamiltonian vector fields are of the form $\{f,-\}$ for $f \in {\mathcal{O}}_X$, and Poisson vector fields are all vector fields which preserve the Poisson bracket, i.e., such that $\xi\{f,g\} = \{\xi(f),g\}+\{f,\xi(g)\}$; this includes all Hamiltonian vector fields.
In each of the cases (a)–(d), we study the leaves under the flow of ${\mathfrak{v}}$ and the condition for the associated ${\mathcal{D}}$-module to be holonomic (and hence for $({\mathcal{O}}_X)_{\mathfrak{v}}$ to be finite-dimensional).
In §\[s:ex-dloc\] we discuss the globalization of these examples to the nonaffine setting, which turns out to be straightforward for Hamiltonian vector fields and all vector fields, but quite nontrivial for Poisson vector fields (and hence their generalizations). We do not need this material for the remainder of the paper.
In the remainder of the paper we study in detail three specific examples for which the ${\mathcal{D}}$-module has an interesting and nontrivial structure which reflects the geometry. In these examples, we explicitly compute the ${\mathcal{D}}$-module and the coinvariants $({\mathcal{O}}_X)_{\mathfrak{v}}$.
In §\[s:cy-cplte-int-is\], we consider the case of divergence-free vector fields on complete intersections in Calabi-Yau varieties. Holonomicity turns out to be equivalent to having isolated singularities, and we restrict to this case. Then, the structure of the ${\mathcal{D}}$-module and the coinvariant functions $({\mathcal{O}}_X)_{{\mathfrak{v}}}$ is governed by the Milnor number and link of the isolated singularities. This is because there is a close relationship between these coinvariants and the de Rham cohomology of the formal (or analytic) neighborhood of the singularity, which was studied in, e.g., [@Gre-GMZ; @HLY-plhdRc; @Yau-vniis] (which were a source of motivation for us).
In §\[s:fqcyvar\], we consider quotients of Calabi-Yau varieties by finite groups of volume-preserving automorphisms. In this case, it turns out that the ${\mathcal{D}}$-module associated to volume-preserving vector fields is governed by the most singular points, where the stabilizer is larger than that of any point in some neighborhood. More generally, rather than working on the quotient $X/G$ where $X$ is Calabi-Yau and $G$ is a group of volume-preserving automorphisms, we study the Lie algebra of $G$-invariant volume-preserving vector fields on $X$ itself. Finally, in §\[s:sym\], we consider symmetric powers $(S^n X,
{\mathfrak{v}})$ of smooth varieties $(X,{\mathfrak{v}})$ on which ${\mathfrak{v}}$ generates the tangent space everywhere (which we call *transitive*). This includes the symplectic, locally conformally symplectic, contact, and Calabi-Yau cases. In these situations, we explicitly compute the ${\mathcal{D}}$-module and the coinvariant functions. Dually, the main result says that the invariant functionals on ${\mathcal{O}}_{S^n X}$ form a polynomial algebra whose generators are the functionals on diagonal embeddings $X^i \to S^n X$ obtained by pulling back to $X^i$ and taking a products of invariant functionals on each factor of $X$. For the ${\mathcal{D}}$-module, this expresses $M(S^n X, {\mathfrak{v}})$ as a direct sum of external tensor products of copies of $M(X,{\mathfrak{v}})$ along each diagonal embedding.
Acknowledgements
----------------
The first author’s work was partially supported by the NSF grant DMS-1000113. The second author is a five-year fellow of the American Institute of Mathematics, and was partially supported by the ARRA-funded NSF grant DMS-0900233.
General theory {#s:gen}
==============
Let $\Omega^\bullet_X := \wedge_{{\mathcal{O}}_X}^\bullet \Omega^1_X$ be the algebraic de Rham complex, where $\Omega^1_X$ is the sheaf of Kähler differentials on $X$. We will frequently use the de Rham complex modulo torsion, $\tilde \Omega^\bullet_X := \Omega^\bullet_X /
\text{torsion}$.
By *polyvector fields* of degree $m$ on $X$, we mean skew-symmetric multiderivations $\wedge^m_{{\mathbf{k}}} {\mathcal{O}}_X \to {\mathcal{O}}_X$. Let $T^m_X$ be the sheaf of such multiderivations. Equivalently, $T^m_X =
{\operatorname{Hom}}_{{\mathcal{O}}_X}(\Omega^m_X, {\mathcal{O}}_X)$, where $\xi \in T^m_X$ is identified with the homomorphism sending $df_1 \wedge \cdots \wedge df_m$ to $\xi(f_1 \wedge \cdots \wedge f_m)$. This also coincides with ${\operatorname{Hom}}_{{\mathcal{O}}_X}(\tilde \Omega^m_X, {\mathcal{O}}_X)$.
When $X$ is smooth, then $\tilde \Omega^\bullet_X = \Omega^\bullet_X$, and its hypercohomology (which, for $X$ affine, is the same as the cohomology of its complex of global sections) is called the algebraic de Rham cohomology of $X$. Over ${\mathbf{k}}={\mathbf{C}}$, this cohomology coincides with the topological cohomology of $X$ under the complex topology, by a well-known theorem of Grothendieck. For arbitrary $X$, we will denote the cohomology of the space of global sections, $\Gamma(\tilde
\Omega^\bullet_X)$, by $H_{DR}^\bullet(X)$, and the hypercohomology of the complex of sheaves $\tilde \Omega^\bullet_X$ by ${\mathbf{H}}_{DR}^\bullet(X)$ (very often we will use these when $X$ is smooth and affine, where they both coincide with topological cohomology).
We caution that, when $X$ is smooth, $\Omega_X$ (without a superscript) will denote the canonical right ${\mathcal{D}}_X$-module of volume forms, which as a ${\mathcal{O}}_X$-module coincides with $\Omega_X^{\dim X}$ under the above definition, when $X$ has pure dimension.
By a *local system* on a variety, we mean an ${\mathcal{O}}$-coherent right ${\mathcal{D}}$-module on the variety. Moreover, from now on, when we say ${\mathcal{D}}$-module, we always will mean a right ${\mathcal{D}}$-module.
Reformulation of Theorem \[t:main1\] in terms of leaves
-------------------------------------------------------
Recall that $(X, {\mathfrak{v}})$ is a pair of an affine scheme $X$ of finite type and a Lie algebra ${\mathfrak{v}}\subseteq {\operatorname{Vect}}(X)$ of vector fields on $X$. We will give a more geometric formulation of Theorem \[t:main1\] in terms of *leaves* of $X$ under ${\mathfrak{v}}$, followed by a strengthened version in these terms.
An *invariant subscheme* is a locally closed subscheme $Z
\subseteq X$ preserved by ${\mathfrak{v}}$; set-theoretically, this says that, at every point $z \in Z$, the evaluation ${\mathfrak{v}}|_z$ lies in the tangent space $T_z Z_{\text{red}}$. A *leaf* is a connected invariant (reduced) subvariety $Z$ such that, at every point $z$, in fact ${\mathfrak{v}}|_z = T_z Z$. A *degenerate invariant subscheme* is an invariant subscheme $Z$ such that, at every point $z \in Z$, ${\mathfrak{v}}|_z \subsetneq T_z Z_{\text{red}}$.
When an invariant subscheme is reduced, we call it an invariant subvariety. An invariant subscheme $Z \subseteq X$ is degenerate if and only if the invariant subvariety $Z_{\text{red}}$ is degenerate. Note that the closure of any degenerate invariant subscheme is also such. Also, leaves are necessarily smooth. Although the same is clearly not true of degenerate invariant subschemes, we can restrict our attention to those with smooth reduction by first stratifying $X_{\text{red}}$ by its (set-theoretic) singular loci, in view of the classical result:
[@Sei-dirfgt Corollary to Theorem 12]\[t:sing-pres\] The set-theoretic singular locus of $X_{\text{red}}$ is preserved by all vector fields on $X$.
We give a proof of a more general assertion in the proof of Proposition \[p:decomp\] below.
\[r:sch-sing\] Note that, for the set-theoretic singular locus to be preserved by all vector fields, we need to use that the characteristic of ${\mathbf{k}}$ is zero; otherwise the singular locus is not preserved by all vector fields: e.g., in characteristic $p > 0$, one has the derivation $\partial_x$ of ${\mathbf{k}}[x,y]/(y^2-x^p)$, which does not vanish at the singular point at the origin.
On the other hand, in arbitrary characteristic, the scheme-theoretic singular locus of a variety of pure dimension $k \geq 0$ is preserved, where we define this by the Jacobian ideal: for a variety cut out by equations $f_i$ in affine space, this is the ideal generated by determinants of $(k \times k)$-minors of the Jacobian matrix $(\frac{\partial f_i}{\partial x_j})$ (this is preserved by [@Har-dcr], where it is shown that it coincides with the smallest nonzero Fitting ideal of the module of Kähler differentials). In the above example it would be defined by the ideal $(y)$ when $p > 2$. This is evidently preserved by all vector fields, which are all multiples of $\partial_x$. Note, however, that we will not make use of the scheme-theoretic singular locus in this paper (except in §\[s:cy-cplte-int-is\], where we will explicitly define it), nor will we consider the case of positive characteristic.
Say that $(X, {\mathfrak{v}})$ has *finitely many leaves* if $X_{\text{red}}$ is a (disjoint) union of finitely many leaves.
For example, when $X$ is a Poisson variety and ${\mathfrak{v}}$ is the Lie algebra of Hamiltonian vector fields, then this condition says that $X$ has finitely many symplectic leaves.
We caution that, when $(X, {\mathfrak{v}})$ does not have finitely many leaves, it does *not* follow that there are infinitely many algebraic leaves, or any at all:
\[ex:noleaves\] Consider the two-dimensional torus $X = ({\mathbf{A}}^1\setminus \{0\})^2$, and let ${\mathfrak{v}}= \langle \xi \rangle$ for some global vector field $\xi$ which is not algebraically integrable, e.g., $x\partial_x - c
y\partial_y$ where $c$ is irrational. The analytic leaves of this are the level sets of $x^cy$, which are not algebraic. There are in fact *no* algebraic leaves at all.
However, it is always true that, in the formal neighborhood $\hat
X_x$ of every point $x \in X$, there exists a formal leaf of $X$ through $x$: this is the orbit of the formal group obtained by integrating ${\mathfrak{v}}$. In the above example, this says that the level sets of $x^cy$ do make sense in the formal neighborhood of every point $(x,y) \in X$.
The condition of having finitely many leaves is well-behaved:
\[p:decomp\] Let $X_i := \{x \in X \mid \dim {\mathfrak{v}}|_x = i\} \subseteq
X_{\text{red}}$. Then $X_i$ is an invariant locally closed subvariety. If $X$ has finitely many leaves, then the connected components of the $X_i$ are all leaves.
We prove this below. We first note the consequence:
\[c:finleaves\] There can be at most one decomposition of $X_{\text{red}}$ into finitely many leaves. The following are equivalent:
- $X$ has finitely many leaves;
- $X$ contains no degenerate invariant subvariety;
- For all $i$, the dimension of $X_i$ is at most $i$.
For the first statement, suppose that $X = \sqcup_i Z_i = \sqcup_i
Z'_i$ are two decompositions into leaves. Then each nonempty pairwise intersection $Z_i \cap Z_j'$ is evidently a leaf. Now, for each $i$, $Z_i = \sqcup_j (Z_i \cap Z_j')$ is a decomposition of $Z_i$ as a disjoint union of locally closed subvarieties of the same dimension as $Z_i$. Since $Z_i$ is connected, this implies that this decomposition is trivial, i.e., $Z_j' = Z_i$ for some $j$.
For the equivalence, first we show that (i) implies (ii). Indeed, if $X$ were a union of finitely many leaves and also $X$ contained a degenerate invariant subvariety $Z$, we could assume $Z$ is irreducible. Then there would be some $X_i$ such that $X_i \cap Z$ is open and dense in $Z$. But then the rank of ${\mathfrak{v}}$ along $X_i
\cap Z$ would be less than the dimension of $Z$, and hence less than the dimension of $X_i$, a contradiction. To show (ii) implies (iii), note that, if $\dim X_i > i$, then any open subset of $X_i$ of pure maximal dimension is degenerate. To show (iii) implies (i), note that the decomposition of Proposition \[p:decomp\] must be into leaves if $\dim X_i \leq i$ for all $i$ (in fact, in this case, each $X_i$ is a (possibly empty) finite union of leaves of dimension $i$).
First, to see that the $X_i$ are locally closed, it suffices to show that $Y_j := \bigsqcup_{i \leq j} X_i$ is closed for all $j$. This statement would be clear if ${\mathfrak{v}}$ were finite-dimensional; for general ${\mathfrak{v}}$ we can write ${\mathfrak{v}}$ as a union of its finite-dimensional subspaces, and $Y_j({\mathfrak{v}})$ is the intersection of $Y_j({\mathfrak{v}}')$ over all finite-dimensional subspaces ${\mathfrak{v}}' \subseteq
{\mathfrak{v}}$.
Next, we claim that, for all $i \leq k$, the subvariety $X_{i,k} \subseteq X$ of points $x \in X_i$ at which $\dim T_x X = k$ is preserved by all vector fields from ${\mathfrak{v}}$.
Let $S := {\operatorname{\mathsf{Spec}}}{\mathbf{k}}[\![t]\!]$ and $X_S := {\operatorname{\mathsf{Spec}}}{\mathcal{O}}_X[\![t]\!]$. For every $\xi \in {\mathfrak{v}}$, consider the automorphism $e^{t \xi}$ of ${\mathcal{O}}_{X_S}$. For any point $x \in X_{i,k}$, consider the corresponding $S$-point $x_S \in X_S$, i.e., ${\mathcal{O}}_S$-linear homomorphism ${\mathcal{O}}_{X_S} \to {\mathcal{O}}_S$. Let $\mathfrak{m} =
\mathfrak{m}_{x_S}$ be its kernel, i.e., $\mathfrak{m}_x[\![t]\!]$. Then, let $\tilde x_S = e^{t \xi} x_S$, another $S$-point of $X_S$, and let $\tilde {\mathfrak{m}} = \mathfrak{m}_{x_S}$ be the kernel of its associated homomorphism ${\mathcal{O}}_{X_S} \to {\mathcal{O}}_S$.
Let the cotangent space to $X_S$ at $x_S$ be defined as $T^*_{x_S} X_S =
\mathfrak{m} / \mathfrak{m}^2$, and similarly $T^*_{x_S'} X_S =
\tilde {\mathfrak{m}}/\tilde {\mathfrak{m}}^2$. Since $T^*_{x_S} X_S$ is a free ${\mathcal{O}}_S$-module of rank $k$, the same holds for $T^*_{\tilde x_S} X_S$.
Moreover, we can view ${\mathfrak{v}}[\![t]\!]$ as a space of vector fields on $X_S$ over $S$, i.e., as a subspace of ${\mathcal{O}}_S$-derivations ${\mathcal{O}}_{X_S}
\to {\mathcal{O}}_S$. Since $e^{t\xi}$ is an automorphism preserving ${\mathfrak{v}}[\![t]\!]$, it follows as for $x_S \in X_S$ that the image of ${\mathfrak{v}}[\![t]\!] \to \ {\operatorname{Hom}}_{{\mathcal{O}}_S}(T^*_{\tilde x_S} X_S, {\mathcal{O}}_S)$ is a free ${\mathcal{O}}_S$-module of rank $i$. We conclude that $\tilde x_S \in
(X_{i,k})_S = {\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X_{i,k}}[\![t]\!] \subseteq X_S$.
We conclude from the preceding paragraphs that ${\mathfrak{v}}$ is parallel to $X_{i,k}$, as desired.
This can also be used to prove Theorem \[t:sing-pres\]: setting $i =
k = \dim X + 1$, we conclude that the intersection of the (set-theoretic) singular locus with the union of irreducible components of $X$ of top dimension is preserved by all vector fields; one can then induct on dimension. Alternatively, one can apply the above argument, replacing $X_{i,k}$ by the set-theoretic singular locus of $X$.
For the final statement of the proposition, note that, if $X$ has a degenerate invariant subvariety $Z \subseteq X$, then it cannot be a union of finitely many leaves, since one of them would have to be open in $Z$, which is impossible.
In the case that ${\mathbf{k}}={\mathbf{C}}$, we could prove the proposition by embedding $X$ into ${\mathbf{C}}^k$ and locally analytically integrating the flow of vector fields of ${\mathfrak{v}}$ (which individually noncanonically lift to ${\mathbf{C}}^k$), which must preserve the singular locus and the rank of ${\mathfrak{v}}$.
In view of Corollary \[c:finleaves\], Theorem \[t:main1\] above can be restated as:
\[t:main1-alt\] If $(X,{\mathfrak{v}})$ has finitely many leaves, then $({\mathcal{O}}_X)_{{\mathfrak{v}}}$ is finite-dimensional.
In the aforementioned Poisson variety case, the theorem is a special case of [@ESdm Theorem 1.1]. Note that the converse to the theorem does not hold: see Remark \[r:hol-fd\].
\[r:red\] Suppose that $X$ is irreducible and that ${\mathfrak{v}}$ acts locally finitely and semisimply on ${\mathcal{O}}_X$, e.g., if ${\mathfrak{v}}$ is the Lie algebra of a reductive algebraic group acting on $X$. If, moreover, ${\mathfrak{v}}$ acts with finitely many leaves, then Theorem \[t:main1-alt\] is elementary. In fact, it is enough to assume that ${\mathfrak{v}}$ has a dense leaf. Then, $\dim ({\mathcal{O}}_X)_{{\mathfrak{v}}}= 1$. This is because, by local finiteness and semisimplicity, the canonical map $({\mathcal{O}}_X)^{{\mathfrak{v}}} \to ({\mathcal{O}}_X)_{{\mathfrak{v}}}$ is an isomorphism, and the former has dimension one.
\[r:nonfinite\] One can obtain examples where $\dim ({\mathcal{O}}_X)_{{\mathfrak{v}}} > 1$ when ${\mathfrak{v}}$ is semisimple and transitive, but does not act locally finitely. For example, let $X \subseteq
{\mathbf{A}}^2$ be any nonempty open affine subvariety such that $0 \notin
X$. Let $\mathfrak{sl}_2$ act on $X$ by the restriction of its action on ${\mathbf{A}}^2$. This is the Lie algebra of linear Hamiltonian vector fields with respect to the usual symplectic structure on ${\mathbf{A}}^2$. Since $X$ is affine symplectic, if $H(X)$ denotes the Hamiltonian vector fields, $({\mathcal{O}}_X)_{H(X)} \cong H^{\dim X}(X) = H_{DR}^{2}(X)$, by the usual isomorphism $[f] \mapsto f \cdot
{\mathsf{vol}}_X$.[^1] On the other hand, $\mathfrak{sl}_2 \subseteq H(X)$, so $\dim ({\mathcal{O}}_X)_{\mathfrak{sl}_2}
\geq \dim H^2(X)$ (in fact this is an equality since $\mathfrak{sl}_2 \cdot {\mathcal{O}}_X = H(X) \cdot {\mathcal{O}}_X$ inside ${\mathcal{D}}_X$, as $\mathfrak{sl}_2$ is transitive and volume-preserving, cf. Proposition \[p:inc-os\] below). There are many examples of such varieties $X$ which have $\dim H^2(X) > 1$. For example, if $X$ is the complement of $n+1$ lines through the origin, then $\dim H^2(X) = n$: the Betti numbers of $X$ are $1, n+1$, and $n$, since the Euler characteristic is zero, each deleted line creates an independent class in first cohomology, and there can be no cohomology in degrees higher than two as $X$ is a two-dimensional affine variety. This produces an example as desired for $n \geq 2$.
The ${\mathcal{D}}$-module defined by ${\mathfrak{v}}$ {#ss:ddv}
------------------------------------------------------
The proof of the theorems above is based on a stronger result concerning the ${\mathcal{D}}$-module whose solutions are invariants under the flow of ${\mathfrak{v}}$. This construction generalizes $M(X)$ from [@ESdm] in the case $X$ is Poisson and ${\mathfrak{v}}$ is the Lie algebra of Hamiltonian vector fields. Namely, we prove that this ${\mathcal{D}}$-module is holonomic when $X$ has finitely many leaves. We will explain a partial converse in §\[ss:inc\], and discuss holonomicity in more detail in §\[ss:hol-bicond\] below.
Let ${\mathcal{D}}_X$ be the canonical right ${\mathcal{D}}$-module on $X$, which is equipped with a left action by ${\operatorname{Vect}}(X)$. Explicitly, under Kashiwara’s equivalence, if $X {\hookrightarrow}V$ is an embedding into a smooth affine variety $V$, this corresponds to $I_X \cdot {\mathcal{D}}_V \setminus {\mathcal{D}}_V$ together with the left action by derivations which preserve $I_X$.
Define $$M(X,{\mathfrak{v}}) := {\mathfrak{v}}\cdot {\mathcal{D}}_X \setminus {\mathcal{D}}_X,$$ where ${\mathfrak{v}}\cdot {\mathcal{D}}_X$ is the right submodule generated by the action of ${\mathfrak{v}}$ on ${\mathcal{D}}_X$. (We will also use the same definition when $X$ is replaced by its completion $\hat X_x$ at points $x \in X$, even though $\hat X_x$ does not have finite type.)
Explicitly, if $i: X \to V$ is an embedding into a smooth affine variety $V$, let $\widetilde {\mathfrak{v}}\subseteq {\operatorname{Vect}}(V)$ be the subspace of vector fields which are parallel to $X$ and restrict on $X$ to elements of ${\mathfrak{v}}$. Then, the image of $M(X,{\mathfrak{v}})$ under Kashiwara’s equivalence is $$M(X,{\mathfrak{v}},i) = (I_X + \widetilde {\mathfrak{v}}) {\mathcal{D}}_V \setminus {\mathcal{D}}_V.$$
Let $\pi: X \to {\operatorname{\mathsf{Spec}}}{\mathbf{k}}$ be the projection to a point, and $\pi_0$ the functor of underived direct image from ${\mathcal{D}}$-modules on $X$ to those on ${\mathbf{k}}$, i.e., ${\mathbf{k}}$-vector spaces.
\[p:pushfwd\] $\pi_0 M(X,{\mathfrak{v}}) = ({\mathcal{O}}_X)_{{\mathfrak{v}}}$.
Fix an affine embedding $X {\hookrightarrow}V$. Then, the underived direct image is $$\pi_0 M(X,{\mathfrak{v}}) = (I_X + \widetilde {\mathfrak{v}}) {\mathcal{D}}_V \setminus {\mathcal{D}}_V \otimes_{{\mathcal{D}}_V} {\mathcal{O}}_V =
({\mathcal{O}}_X)_{{\mathfrak{v}}}. \qedhere$$
If $Z \subseteq X$ is an invariant closed subscheme, we will repeatedly use the following relationship between $M(X,{\mathfrak{v}})$ and $M(Z, {\mathfrak{v}}|_Z)$:
\[p:inv-sub-qt\] If $i: Z \to X$ is the tautological embedding of an invariant closed subscheme, then there is a canonical surjection $M(X,{\mathfrak{v}}) {\twoheadrightarrow}i_*
M(Z, {\mathfrak{v}}|_Z)$.
This follows because $i_* M(Z, {\mathfrak{v}}|_Z) = (({\mathfrak{v}}+I_Z)\cdot
{\mathcal{D}}_X)\setminus {\mathcal{D}}_X$, where $I_Z$ is the ideal of $Z$.
\[r:vs\] As pointed out in the previous subsection, one could more generally allow ${\mathfrak{v}}$ to be an arbitrary subset of ${\operatorname{Vect}}(X)$. However, it is easy to see that the ${\mathcal{D}}$-module is the same as for the Lie algebra generated by this subset. So, no generality is lost by assuming that ${\mathfrak{v}}$ be a Lie algebra.
\[ntn:la\] By a Lie algebroid in ${\operatorname{Vect}}(X)$, we mean a Lie subalgebra which is also a coherent subsheaf.
\[r:diffop-leq1\] One could more generally (although equivalently in a sense we will explain) allow ${\mathfrak{v}}\subseteq
{\mathcal{D}}_X^{\leq 1}$ to be a space of differential operators of order $\leq 1$. One then sets, as before, $M(X,{\mathfrak{v}}) = {\mathfrak{v}}\cdot {\mathcal{D}}_X
\setminus {\mathcal{D}}_X$. In this case, one obtains the same ${\mathcal{D}}$-module not merely by passing to the Lie algebra generated by ${\mathfrak{v}}$, but in fact one can also replace ${\mathfrak{v}}$ by ${\mathfrak{v}}\cdot {\mathcal{O}}_X$. Let $\sigma: {\mathcal{D}}_X^{\leq 1} \to {\operatorname{Vect}}(X)$ denote the principal symbol. Then, we conclude that $\sigma({\mathfrak{v}}) \subseteq {\operatorname{Vect}}(X)$ is actually a Lie algebroid (cf. Notation \[ntn:la\]).
This is actually equivalent to using only vector fields, in the following sense: Given any pair $(X, {\mathfrak{v}})$ with ${\mathfrak{v}}\subseteq
{\mathcal{D}}_X^{\leq 1}$, one can consider the pair $({\mathbf{A}}^1 \times X, \hat
{\mathfrak{v}})$ where, for $x$ the coordinate on ${\mathbf{A}}^1$, $\hat {\mathfrak{v}}$ contains the vector field $\partial_x$ together with, for every differential operator $\theta \in {\mathfrak{v}}$, $\sigma(\theta) -
(\theta-\sigma(\theta)) x \partial_x$. Since $(x \partial_x + 1)
= \partial_x \cdot x \in (\partial_x \cdot {\mathcal{D}}_{{\mathbf{A}}^1})$, one easily sees that $M({\mathbf{A}}^1 \times X, \hat {\mathfrak{v}}) \cong \Omega_{{\mathbf{A}}^1}
\boxtimes M(X,{\mathfrak{v}})$. So, in this sense, one can reduce the study of pairs $(X, {\mathfrak{v}})$ to the study of affine schemes of finite type with Lie algebras of vector fields. In particular, our general results extend easily to the setting of differential operators of order $\leq 1$.
Similarly, one can reduce the study of pairs $(X, {\mathfrak{v}})$ to the case where $X$ is affine space. Indeed, if $X {\hookrightarrow}{\mathbf{A}}^n$ is any embedding, and $I_X$ is the ideal of $X$, we can consider the Lie algebroid $$I_X \cdot {\mathcal{D}}_{{\mathbf{A}}^n}^{\leq 1} + {\mathfrak{v}}\subseteq {\mathcal{D}}_{{\mathbf{A}}^n}^{\leq 1}.$$ This makes sense by lifting elements of ${\mathfrak{v}}$ to vector fields on ${\mathbf{A}}^n$, and the result is independent of the choice. We can then apply the previous remark to reduce everything to Lie algebras of vector fields on affine space. (This is not really helpful, though: in our examples, ${\mathfrak{v}}$ is naturally associated with $X$ (e.g., Hamiltonian vector fields on $X$), so it is not natural to replace $X$ with an affine space.)
Holonomicity and proof of Theorems \[t:main1\] and \[t:main1-alt\]
------------------------------------------------------------------
Recall that a nonzero ${\mathcal{D}}$-module on $X$ is *holonomic* if it is finitely generated and its singular support is a Lagrangian subvariety of $T^*X$ (i.e., its dimension equals that of $X$). We always call the zero module holonomic. (Derived) pushforwards of holonomic ${\mathcal{D}}$-modules are well-known to have holonomic cohomology. Since a holonomic ${\mathcal{D}}$-module on a point is finite-dimensional, this implies that, if $M$ is holonomic and $\pi: X
\to {\text{pt}}$ is the pushforward to a point, then $\pi_* M$ (by which we mean the cohomology of the complex of vector spaces), and in particular $\pi_0 M$, is finite-dimensional. Therefore, if we can show that $M(X, {\mathfrak{v}})$ is holonomic, this implies that $({\mathcal{O}}_X)_{{\mathfrak{v}}}
= \pi_0 M(X, {\mathfrak{v}})$ is finite-dimensional, along with the full pushforward $\pi_* M(X,{\mathfrak{v}})$. This reduces Theorem \[t:main1-alt\] and equivalently Theorem \[t:main1\] to the statement:
\[t:main2\] If $(X,{\mathfrak{v}})$ has finitely many leaves, then $M(X,{\mathfrak{v}})$ is holonomic. In this case, the composition factors are intermediate extensions of local systems along the leaves.
The converse does not hold: see, e.g., Example \[ex:hol-nf\].
The equations ${\operatorname{\mathsf{gr}}}{\mathfrak{v}}$ are satisfied by the singular support of $M(X,{\mathfrak{v}})$. These equations say, at every point $x \in X$, that the restriction of the singular support of $M(X,{\mathfrak{v}})$ to $x$ lies in $({\mathfrak{v}}|_x)^\perp$. Thus, if $Z \subseteq X$ is a leaf, then the restriction of the singular support of $M(X,{\mathfrak{v}})$ to $Z$ lies in the conormal bundle to $Z$, which is Lagrangian. If $X$ is a finite union of leaves, it follows that the singular support of $M(X,{\mathfrak{v}})$ is contained in the union of the conormal bundles to the leaves, which is Lagrangian. The last statement immediately follows from this description of the singular support.
We will be interested in the condition on ${\mathfrak{v}}$ for $M(X,{\mathfrak{v}})$ to be holonomic, which turns out to be subtle.
Call $(X,{\mathfrak{v}})$, or ${\mathfrak{v}}$, holonomic if $M(X,{\mathfrak{v}})$ is.
We will often use the following immediate consequence:
\[p:hol-fd\] If ${\mathfrak{v}}$ is holonomic, then ${\mathcal{O}}_{{\mathfrak{v}}}$ is finite-dimensional.
\[r:hol-fd\] The converse to Proposition \[p:hol-fd\] does not hold in general (although we will have a couple of cases where it does: the Lie algebras of all vector fields (Proposition \[p:allvfds\]) and of Hamiltonian vector fields preserving a top polyvector field (Corollary \[c:vtop-hol\])). A simple example where this converse does not hold is $(X,{\mathfrak{v}})=({\mathbf{A}}^2, \langle \partial_x \rangle)$ (where $x$ is one of the coordinates on ${\mathbf{A}}^2$), where $M(X,{\mathfrak{v}}) = \Omega_{{\mathbf{A}}^1}
\boxtimes {\mathcal{D}}_{{\mathbf{A}}^1}$ is not holonomic, but ${\mathcal{O}}_{{\mathfrak{v}}} = 0$. This example also has infinitely many leaves, namely all lines parallel to the $x$-axis.
Incompressibility and a weak converse {#ss:inc}
-------------------------------------
We say that a vector field $\xi$ preserves a differential form $\omega$ if the Lie derivative $L_\xi$ annihilates $\omega$.
Say that ${\mathfrak{v}}$ flows *incompressibly* along an irreducible invariant subvariety $Z$ if there exists a smooth point $z \in Z$ and a volume form on the formal neighborhood of $Z$ at $z$ which is preserved by ${\mathfrak{v}}$.
There is an alternative definition using divergence functions which does not require formal localization, which we discuss in §\[s:div\]; see also Proposition \[p:inc-subvar\].(iii). When $X$ is irreducible and ${\mathfrak{v}}$ flows incompressibly on $X$, we omit the $X$ and merely say that ${\mathfrak{v}}$ flows incompressibly. Note that this is equivalent to flowing generically incompressibly.
In §\[ss:inc-subvar-pf\] we will prove
\[p:inc-subvar\] Let $X$ be an irreducible affine variety. The following conditions are equivalent:
- ${\mathfrak{v}}$ flows incompressibly;
- $M(X,{\mathfrak{v}})$ is fully supported;
- For all $\xi_i \in {\mathfrak{v}}$ and $f_i \in {\mathcal{O}}_X$ such that $\sum_i f_i \xi_i = 0$, one has $\sum_i \xi_i(f_i) = 0$.
Moreover, the equivalence (ii) $\Leftrightarrow$ (iii) holds when $X$ is an arbitrary affine scheme of finite type, if one generalizes (ii) to the condition: (ii’) The annihilator of $M(X,{\mathfrak{v}})$ in ${\mathcal{O}}_X$ is zero.
\[r:inc-subvar\] We can alternatively state (ii’) and (iii) as follows, in terms of global sections of ${\mathfrak{v}}\cdot {\mathcal{D}}_Z \subseteq {\mathcal{D}}_Z$ (cf. §\[ss:inc-subvar-pf\] below): (ii’) says that $({\mathfrak{v}}\cdot {\mathcal{D}}_Z) \cap {\mathcal{O}}_Z =
0$, and (iii) says that $({\mathfrak{v}}\cdot {\mathcal{O}}_Z) \cap {\mathcal{O}}_Z = 0$.
Motivated by this proposition, we will generalize the notion of incompressibility to the case of nonreduced subschemes in §\[ss:supp\] below, to be defined by conditions (ii’) or (iii) above.
\[ex:poiss-inc\] In the case that $X$ is a Poisson variety, ${\mathfrak{v}}$ is the Lie algebra of Hamiltonian vector fields, and $Z \subseteq X$ is a symplectic leaf (i.e., a leaf of ${\mathfrak{v}}$), then ${\mathfrak{v}}$ flows incompressibly on $Z$, since it preserves the symplectic volume along $Z$.
Say that ${\mathfrak{v}}$ has *finitely many incompressible leaves* if it has no degenerate invariant subvariety on which ${\mathfrak{v}}$ flows incompressibly.
As before, if ${\mathfrak{v}}$ does not have finitely many incompressible leaves, one does *not* necessarily have infinitely many incompressible leaves, or any at all (see Example \[ex:noleaves\], which does not have finitely many incompressible leaves, but has no algebraic leaves).
In §\[ss:hol-finc-pf\] below we will prove
\[t:hol-finc\]
- For every incompressible leaf $Z \subseteq X$, letting $i:
\bar Z {\hookrightarrow}X$ be the tautological embedding of its closure, the canonical quotient $M(X,{\mathfrak{v}}) {\twoheadrightarrow}i_* M(\bar Z, {\mathfrak{v}}|_{\bar Z})$ is an extension of a nonzero local system on $Z$ to $\bar Z$.
- If $(X,{\mathfrak{v}})$ is holonomic, then it has finitely many incompressible leaves.
Note that the converse to (i) does not hold: see Example \[ex:a3-inf-hol-qt\]. We will give a correct converse statement in §\[ss:hol-bicond\] below. Also, the converse to (ii) does not hold, as we will demonstrate in Example \[ex:fin-infl\].
We conclude from the Theorems \[t:main2\] and \[t:hol-finc\] that $$\label{e:finleaf-hol-inc}
\text{finitely many leaves} \,\, \Rightarrow \,\, \text{holonomic}
\,\, \Rightarrow \,\, \text{finitely many incompressible leaves},$$ but neither converse direction holds, as mentioned (see Examples \[ex:hol-nf\] and \[ex:fin-infl\], respectively). However, we will see below that the second implication is generically a biconditional for irreducible varieties $X$, i.e., $X$ generically has finitely many incompressible leaves if and only if $X$ is generically holonomic.
\[ex:poiss-finsym\] When $X$ is Poisson and ${\mathfrak{v}}$ the Lie algebra of Hamiltonian vector fields, then Theorem \[t:hol-finc\] and Example \[ex:poiss-inc\] imply that ${\mathfrak{v}}$ is holonomic *if and only if* $X$ has finitely many symplectic leaves. More precisely, if $Z \subseteq X$ is any invariant subvariety, then in the formal neighborhood of a generic point $z \in Z$, we can integrate the Hamiltonian flow and write $\hat Z_z = V \times V'$ for formal polydiscs $V$ and $V'$, where the Hamiltonian flow is along the $V$ direction, and transitive along fibers of $(V \times V') {\twoheadrightarrow}V'$. Then Hamiltonian flow preserves the volume form $\omega_V \otimes \omega_{V'}$, where $\omega_V$ is the canonical symplectic volume, and $\omega_{V'}$ is an arbitrary volume form on $V'$. Therefore, all $Z$ are incompressible. (In particular, all leaves are incompressible, preserving the canonical symplectic volume.) Then shows that $H(X)$ is holonomic if and only if there are finitely many leaves.
\[ex:fin-infl\] We demonstrate that $({\mathcal{O}}_X)_{{\mathfrak{v}}}$ need not be finite-dimensional if we only assume that $X$ has finitely many incompressible leaves. Therefore, ${\mathfrak{v}}$ is not holonomic (although non-holonomicity also follows directly in this example). Let $X={\mathbf{A}}^2 \times ({\mathbf{A}}\setminus \{0\}) \subseteq {\mathbf{A}}^3$, with ${\mathbf{A}}^2 = {\operatorname{\mathsf{Spec}}}{\mathbf{k}}[x,y]$ and ${\mathbf{A}}\setminus \{0\} = {\operatorname{\mathsf{Spec}}}{\mathbf{k}}[z,z^{-1}]$. Let ${\mathfrak{v}}= \langle
y^2 \partial_x, y \partial_y + z \partial_z, \partial_z\rangle$. Then this has an incompressible open leaf, $\{y \neq 0\}$, preserving the volume form $\frac{1}{y^2} dx \wedge dy \wedge
dz$. The complement consists of the leaves $\{x=c, y=0\}$ for all $c
\in {\mathbf{k}}$, which are not incompressible since the restriction of ${\mathfrak{v}}$ to each such leaf (or to their union, $\{y=0\}$) includes both $\partial_z$ and $z \partial_z$.
We claim that the coinvariants $({\mathcal{O}}_X)_{{\mathfrak{v}}}$ are infinite-dimensional, and isomorphic to ${\mathbf{k}}[x] \cdot yz^{-1}$ via the quotient map ${\mathcal{O}}_X {\twoheadrightarrow}({\mathcal{O}}_X)_{{\mathfrak{v}}}$. Indeed, $
y^2 \partial_x ({\mathcal{O}}_X) = y^2 {\mathcal{O}}_X$, $(y \partial_y + z \partial_z) {\mathcal{O}}_X = {\mathbf{k}}[x] \cdot \langle y^i z^j
\mid i+j \neq 0 \rangle$, and $\partial_z ({\mathcal{O}}_X) = {\mathbf{k}}[x,y] \cdot
\langle z^i \mid i \neq -1\rangle$. The sum of these vector subspaces is the space spanned by all monomials in $x, y, z$, and $z^{-1}$ except for $x^i yz^{-1}$ for all $i \geq 0$.
\[ex:hol-nf\] It is easy to give an example where ${\mathfrak{v}}$ is holonomic but has infinitely many leaves: for $Y$ any positive-dimensional variety, consider $X={\mathbf{A}}^1 \times Y$, ${\mathfrak{v}}:= \langle \partial_x,
x \partial_x \rangle$, where $x$ is the coordinate on ${\mathbf{A}}^1$. Then the leaves of $(X,{\mathfrak{v}})$ are of the form ${\mathbf{A}}^1 \times \{y\}$ for $y
\in Y$, but $M(X,{\mathfrak{v}})=0$, which is holonomic.
\[ex:a3-inf-hol-sub\] For a less trivial example, which is a generically nonzero holonomic ${\mathcal{D}}$-module without finitely many leaves, let $X={\mathbf{A}}^3$ with coordinates $x, y$, and $z$, and let ${\mathfrak{v}}$ be the Lie algebra of all incompressible vector fields (with respect to the standard volume) which along the plane $x=0$ are parallel to the $y$-axis. Then we claim that the singular support of $M(X,{\mathfrak{v}})$ is the union of the zero section of $T^* X$ and the conormal bundle of the plane $x=0$, which is Lagrangian, even though there are not finitely many leaves. Actually, from the computation below, we see that $M(X,{\mathfrak{v}})$ is isomorphic to $j_!\Omega_{{\mathbf{A}}^1 \setminus \{0\}}
\boxtimes \Omega_{{\mathbf{A}}^2}$, where $j: {\mathbf{A}}^1 \setminus \{0\} {\hookrightarrow}{\mathbf{A}}^1$ is the inclusion (which is an affine open embedding, so $j_!$ is an exact functor on holonomic ${\mathcal{D}}$-modules). This is an extension of $\Omega_{{\mathbf{A}}^3}$ by $i_*\Omega_{{\mathbf{A}}^2}$, where $i:
{\mathbf{A}}^2 = \{0\} \times {\mathbf{A}}^2 {\hookrightarrow}{\mathbf{A}}^3$ is the closed embedding, i.e., there is an exact sequence $$0 \to i_* \Omega_{{\mathbf{A}}^2} {\hookrightarrow}M(X,{\mathfrak{v}}) {\twoheadrightarrow}\Omega_{{\mathbf{A}}^3} \to 0.$$ Thus, there is a single composition factor on the open leaf and a single composition factor on the degenerate (but not incompressible) invariant subvariety $\{x=0\}$.
To see this, note first that $\partial_y \in {\mathfrak{v}}$. We claim that $1+x\partial_x$ and $\partial_z$ are in ${\mathfrak{v}}\cdot {\mathcal{D}}_X$: $$\begin{gathered}
\partial_y \cdot y - (y \partial_y - x \partial_x) = 1 + x \partial_x; \\
(1 + x \partial_x) \cdot \partial_z - (x \partial_z)
\cdot \partial_x = \partial_z.\end{gathered}$$ Thus, $\langle 1+x\partial_x, \partial_y, \partial_z \rangle \subseteq
{\mathfrak{v}}\cdot {\mathcal{D}}_X$. Conversely, we claim that ${\mathfrak{v}}\subseteq \langle
1+x\partial_x, \partial_y, \partial_z \rangle \cdot {\mathcal{D}}_X$. Indeed, given an incompressible vector field of the form $\xi = xf\partial_x +
g \partial_y + xh \partial_z \in {\mathfrak{v}}$ for $f,g,h \in {\mathcal{O}}_X$, we can write $$\xi = (1 + x \partial_x) \cdot f + \partial_y \cdot g + x \partial_z \cdot h,$$ where the RHS is a vector field (and not merely a differential operator of order $\leq 1$) because $\xi$ is incompressible. Explicitly, the condition for this RHS to be a vector field, and the condition for $\xi$ to be incompressible, are both that $\partial_x(xf) + \partial_y(g) + \partial_z(xh) = 0$.
We conclude that $\langle 1+x\partial_x, \partial_y, \partial_z
\rangle \cdot {\mathcal{D}}_X = {\mathfrak{v}}\cdot {\mathcal{D}}_X$. Therefore, $M(X,{\mathfrak{v}}) \cong
j_! \Omega_{{\mathbf{A}}^1 \setminus \{0\}} \boxtimes \Omega_{{\mathbf{A}}^2}$, as claimed.
\[ex:a3-inf-hol-qt\] We can slightly modify Example \[ex:a3-inf-hol-sub\], so that (again for $X:={\mathbf{A}}^3$ and $i: \{0\} \times {\mathbf{A}}^2 {\hookrightarrow}{\mathbf{A}}^3$), $i_*
\Omega_{{\mathbf{A}}^2}$ appears as a quotient of $M(X,{\mathfrak{v}})$ rather than as a submodule. More precisely, we will have $M(X,{\mathfrak{v}}) \cong j_*
\Omega_{{\mathbf{A}}^1 \setminus \{0\}} \boxtimes \Omega_{{\mathbf{A}}^2}$. To do so, let ${\mathfrak{v}}$ be the Lie algebra of all incompressible vector fields preserving the volume form $\frac{1}{x^2} dx \wedge dy \wedge dz$ (cf. Example \[ex:fin-infl\]), which again along the plane $x=0$ are parallel to the $y$-axis. Note also that, in this example, the subvariety $\{0\} \times {\mathbf{A}}^2$ is still not incompressible (since $\partial_y$ and $y \partial_y$ are both in ${\mathfrak{v}}|_{0 \times {\mathbf{A}}^2}$, and these cannot both preserve the same volume form), even though this subvariety now supports a quotient $i_* \Omega_{{\mathbf{A}}^2}$ of $M(X)$.
To see this, we claim that ${\mathfrak{v}}\cdot {\mathcal{D}}_Z = \langle 1 -
x \partial_x, \partial_y, \partial_z \rangle \cdot {\mathcal{D}}_Z$. For the containment $\supseteq$, we show that $1 - x \partial_x$ and $\partial_z$ are in ${\mathfrak{v}}\cdot {\mathcal{D}}_Z$. This follows from $$\begin{gathered}
\partial_y \cdot y - (y \partial_y + x \partial_x) = 1 - x \partial_x; \\
(1 - x \partial_x) \cdot \partial_z + (x \partial_z)
\cdot \partial_x = \partial_z.\end{gathered}$$ Then, as in Example \[ex:a3-inf-hol-sub\], if $\xi = xf \partial_x + g \partial_y + xh \partial_z$ preserves the volume form $\frac{1}{x^2} dx \wedge dy \wedge dz$, then $$\xi = -(1 - x \partial_x) \cdot f + \partial_y \cdot g + x \partial_z \cdot h.$$ Therefore, we also have the opposite containment, ${\mathfrak{v}}\cdot {\mathcal{D}}_Z
\subseteq \langle 1 - x \partial_x, \partial_y, \partial_z \rangle
\cdot {\mathcal{D}}_Z$. As a consequence, $M(X,{\mathfrak{v}}) \cong j_* \Omega_{{\mathbf{A}}^1
\setminus \{0\}} \boxtimes \Omega_{{\mathbf{A}}^2}$. We therefore have a canonical exact sequence $$0 \to \Omega_{{\mathbf{A}}^3} {\hookrightarrow}M(X,{\mathfrak{v}}) {\twoheadrightarrow}i_* \Omega_{{\mathbf{A}}^2} \to 0.$$
The transitive case {#ss:transitive}
-------------------
In this section we consider the simplest, but important, example of ${\mathfrak{v}}$ and the ${\mathcal{D}}$-module $M(X,{\mathfrak{v}})$, namely when ${\mathfrak{v}}$ has maximal rank everywhere:
A pair $(X, {\mathfrak{v}})$ is called *transitive at $x$* if ${\mathfrak{v}}|_x =
T_x X$. We call the pair $(X, {\mathfrak{v}})$ transitive if it is so at all $x \in X$.
In other words, the transitive case is the one where every connected component of $X$ is a leaf. Note that, in particular, $X$ must be a smooth variety. Also, we remark that $X$ is generically transitive if and only if it is not degenerate.
\[p:tr\] If $(X,{\mathfrak{v}})$ is transitive and connected, then $M(X,{\mathfrak{v}})$ is a rank-one local system if ${\mathfrak{v}}$ flows incompressibly, and $M(X,{\mathfrak{v}})=0$ otherwise.
By taking associated graded of $M(X,{\mathfrak{v}})$, in the transitive connected case, one obtains either ${\mathcal{O}}_X$ (where $X \subseteq T^*
X$ is the zero section) or zero. So $M(X, {\mathfrak{v}})$ is either a one-dimensional local system on $X$, or zero. In the incompressible case, in a formal neighborhood of some $x \in X$, a volume form is preserved, so there is a surjection $M(\hat X_x, {\mathfrak{v}}|_{\hat X_x})
{\twoheadrightarrow}\Omega_{\hat X_x}$, and hence in this case $M(X, {\mathfrak{v}})$ is a one-dimensional local system. Conversely, if $M(X, {\mathfrak{v}})$ is a one-dimensional local system, then in a formal neighborhood of any point $x \in X$, it is a trivial local system, and hence it preserves a volume form there.
\[ex:cy\] In the case when $X$ is connected and Calabi-Yau and ${\mathfrak{v}}$ preserves the global volume form (which includes the case where $X$ is symplectic and ${\mathfrak{v}}$ is the Lie algebra of Hamiltonian vector fields), then we conclude that $M(X,{\mathfrak{v}}) \cong \Omega_X$. Thus, for $\pi: X \to {\text{pt}}$ the projection to a point, $({\mathcal{O}}_X)_{\mathfrak{v}}= \pi_0
\Omega_X = H_{DR}^{\dim X}(X)$, the top de Rham cohomology. Taking the derived pushforward, we conclude that $\pi_* M(X,{\mathfrak{v}}) = \pi_*
\Omega_X = H_{DR}^{\dim X - *}(X)$. In the Poisson case, where $({\mathcal{O}}_X)_{{\mathfrak{v}}}$ is the zeroth Poisson homology, in [@ESdm Remark 2.27] this motivated the term *Poisson-de Rham homology*, $HP^{DR}_*(X) = \pi_* M(X,{\mathfrak{v}})$, for the derived pushforward. More generally, if ${\mathfrak{v}}$ preserves a *multivalued* volume form, then $M(X,{\mathfrak{v}})$ is a nontrivial rank-one local system and $\pi_*
M(X,{\mathfrak{v}}) = H_{DR}^{\dim X - *}(X, M(X,{\mathfrak{v}}))$ is the cohomology of $X$ with coefficients in this local system (identifying $M(X,{\mathfrak{v}})$ with its corresponding local system under the de Rham functor). See the next example for more details on how to define such ${\mathfrak{v}}$.
\[ex:mvvol\] The rank-one local system need not be trivial when ${\mathfrak{v}}$ does not preserve a global volume form. For example, let $X =
({\mathbf{A}}^1\setminus\{0\}) \times {\mathbf{A}}^1 = {\operatorname{\mathsf{Spec}}}{\mathbf{k}}[x,x^{-1},y]$. Then we can let ${\mathfrak{v}}$ be the Lie algebra of vector fields preserving the multivalued volume form $d(x^{r}) \wedge dy$ for $r \in {\mathbf{k}}$. It is easy to check that this makes sense and that the resulting Lie algebra ${\mathfrak{v}}$ is transitive. Then, $M(X,{\mathfrak{v}})$ is the rank-one local system whose homomorphisms to $\Omega_X$ correspond to this volume form, which is nontrivial (but with regular singularities) when $r$ is not an integer. For ${\mathbf{k}}={\mathbf{C}}$, the local system $M(X,{\mathfrak{v}})$ thus has monodromy $e^{-2 \pi i r}$.
More generally, if $X$ is an arbitrary smooth variety of pure dimension at least two, and $\nabla$ is a flat connection on $\Omega_X$, we can think of the flat sections of $\nabla$ as giving multivalued volume forms, and define a corresponding Lie algebra ${\mathfrak{v}}$ so that ${\operatorname{Hom}}_{{\mathcal{D}}_X}(M(X,{\mathfrak{v}}), \Omega_X)$ returns these forms on formal neighborhoods. Precisely, we can let ${\mathfrak{v}}$ be the Lie algebra of vector fields preserving formal flat sections of $\nabla$. We need to check that ${\mathfrak{v}}$ is transitive, which is where we use the hypothesis that $X$ has pure dimension at least two: see §\[ss:ham-df\] and in particular Proposition \[p:div-enough-vfds\] (alternatively, we could simply impose the condition that ${\mathfrak{v}}$ be transitive, which is immediate to check in the example of the previous paragraph). Then $M(X,{\mathfrak{v}}) \cong
(\Omega_X, \nabla)^* \otimes_{{\mathcal{O}}_X} \Omega_X$, via the map sending the canonical generator $1 \in M(X,{\mathfrak{v}})$ to the identity element of ${\operatorname{End}}_{{\mathcal{O}}_X}(\Omega_X)$. Conversely, if $(X,{\mathfrak{v}})$ is transitive and $M(X,{\mathfrak{v}})$ is nonzero (hence a rank-one local system), then ${\operatorname{Hom}}_{{\mathcal{O}}_X}(M(X,{\mathfrak{v}}), \Omega_X)$ canonically has the structure of a local system on $\Omega_X$ with formal flat sections given by ${\operatorname{Hom}}_{{\mathcal{D}}_X}(M(X,{\mathfrak{v}}),\Omega_X)$, and one has a canonical isomorphism $$M(X,{\mathfrak{v}}) \cong {\operatorname{Hom}}_{{\mathcal{O}}_X}(M(X,{\mathfrak{v}}), \Omega_X)^*
\otimes_{{\mathcal{O}}_X} \Omega_X.$$
On the other hand, if $X$ is one-dimensional and ${\mathfrak{v}}$ is transitive, then $M(X,{\mathfrak{v}})$ cannot be a nontrivial local system, since there are no vector fields defined in any Zariski open set preserving a nontrivial local system. More precisely, assuming $X$ is a connected smooth curve, in order to be incompressible, ${\mathfrak{v}}$ must be a one-dimensional vector space. Then, if $\xi \in {\mathfrak{v}}$ is nonzero, then the inverse $\xi^{-1}$ defines the volume form preserved by ${\mathfrak{v}}$.
\[c:inc\] If $(X,{\mathfrak{v}})$ is a variety, then $M(X,{\mathfrak{v}})$ is fully supported on $X$ if and only if ${\mathfrak{v}}$ flows incompressibly on every irreducible component of $X$. In this case, the dimension of the singular support of $M(X,{\mathfrak{v}})$ on each irreducible component $Y \subseteq X$ is generically $\dim Y + (\dim Y - r)$, where $r$ is the generic rank of ${\mathfrak{v}}$ on $Y$.
\[c:inc2\] If $(X,{\mathfrak{v}})$ is an irreducible variety, then ${\mathfrak{v}}$ is generically holonomic if and only if it is either generically transitive or not incompressible.
It suffices to assume $X$ is irreducible, since the statements can be checked generically on each irreducible component. For generic $x \in X$, in the formal neighborhood $\hat X_x$, we can integrate the flow of ${\mathfrak{v}}$ and write $\hat X_x \cong (V \times V')$, where $V$ and $V'$ are two formal polydiscs about zero, mapping $x \in
\hat X_x$ to $(0,0) \in (V \times V')$, and such that ${\mathfrak{v}}$ generates the tangent space in the $V$ direction everywhere, i.e., ${\mathfrak{v}}|_{(v,v')} = T_{v} V \times \{0\}$ at all $(v,v') \in (V \times
V')$.
Since $\hat {\mathcal{O}}_{X,x} \cdot {\mathfrak{v}}= T_V \boxtimes {\mathcal{O}}_{V'}$, inside ${\mathfrak{v}}\cdot \hat {\mathcal{O}}_{X,x}$ we have, for every $\xi \in T_V$, an element of the form $\xi + D(\xi)$, for some $D(\xi) \in \hat
{\mathcal{O}}_{X,x}$. Namely, this is true because, when $\xi \in {\mathfrak{v}}$ and $f
\in \hat {\mathcal{O}}_{X,x}$, $\xi \cdot f = f \cdot \xi + \xi(f) \in {\mathfrak{v}}\cdot \hat {\mathcal{O}}_{X,x}$, and $T_V$ is contained in the span of such $f
\cdot \xi$.
Now assume that ${\mathfrak{v}}$ preserves a volume form $\omega$ on $\hat
X_x$. Recall that this means that, for all $\xi \in {\mathfrak{v}}$, one has $L_\xi \omega = 0$. Since the right ${\mathcal{D}}$-module action of vector fields $\xi \in {\operatorname{Vect}}(X)$ on $\Omega_X$ is by $\omega \cdot \xi :=
-L_{\xi} \omega$, we conclude that $D(\xi) = L_\xi \omega / \omega$. Write $\omega = f \cdot \omega_V \wedge \omega_{V'}$ where $\omega_V$ and $\omega_{V'}$ are volume forms on $V$ and $V'$ and $f
\in \hat {\mathcal{O}}_{X,x}$ is a unit. Then we conclude that $M(\hat X_x,
{\mathfrak{v}}|_{\hat X_x}) \cong \Omega_V \boxtimes {\mathcal{D}}_{V'}$, the quotient of ${\mathcal{D}}_{X,x}$ by the right ideal generated by $\omega_V$-preserving vector fields on $V$.
Conversely, assume that $M(X,{\mathfrak{v}})$ is fully supported. Since $x$ was generic, $M(\hat X_x, {\mathfrak{v}}|_{\hat X_x})$ is also fully supported. Thus, for every $\xi \in T_V$, there is a unique $D(\xi)$ such that $\xi + D(\xi) \in {\mathfrak{v}}\cdot \hat {\mathcal{D}}_{X,x}$ (and in fact this is in ${\mathfrak{v}}\cdot \hat {\mathcal{O}}_{X,x}$).
Let $\partial_1, \ldots, \partial_n$ be the constant vector fields on $V \times V'$. We conclude that ${\mathfrak{v}}\cdot \hat {\mathcal{D}}_{X,x} = \{
\xi + D(\xi): \xi \in T_V \boxtimes {\mathcal{O}}_{V'}\} \cdot {\operatorname{\mathsf{Sym}}}\langle \partial_1, \ldots, \partial_n \rangle$. Since $M(\hat X_x,
{\mathfrak{v}}|_{\hat X_x})$ is fully supported, this implies that ${\operatorname{\mathsf{gr}}}({\mathfrak{v}}\cdot \hat {\mathcal{D}}_{X,x}) = T_V \cdot {\operatorname{\mathsf{Sym}}}_{\hat {\mathcal{O}}_{X,x}} T_{\hat
X_x}$, and hence that $M(\hat X_x, {\mathfrak{v}}|_{\hat X_x}) \cong
\Omega_V \boxtimes {\mathcal{D}}_{V'}$. Then, ${\mathfrak{v}}$ also preserves a formal volume form, since ${\operatorname{Hom}}(M(\hat X_x, {\mathfrak{v}}|_{\hat X_x}), \Omega_{V
\times V'}) \neq 0$. (Explicitly, for the unique (up to scaling) volume form $\omega_V$ on $V$ preserved by ${\mathfrak{v}}|_V$, these are of the form $\omega_V \boxtimes \omega_{V'}$ for arbitrary volume forms $\omega_{V'}$ on $V'$.)
For the final statement, the proof shows that, in the incompressible (irreducible) case, the dimension of the singular support is generically $\dim V + 2 \dim V'$, which is the same as the claimed formula when we note that $\dim V = r$ and $\dim V + \dim V' = \dim
Y$.
Proof of Proposition \[p:inc-subvar\] {#ss:inc-subvar-pf}
-------------------------------------
By Corollary \[c:inc\], conditions (i) and (ii) are equivalent, when $X$ is an irreducible affine variety. Now let $X$ be an arbitrary affine scheme of finite type. We prove that (ii’) and (iii) are equivalent.
In view of Remark \[r:inc-subvar\], the implication (ii’) $\Rightarrow$ (iii) is immediate. To make Remark \[r:inc-subvar\] precise, we should define ${\mathfrak{v}}\cdot {\mathcal{O}}_X$ as a subspace of global sections of ${\mathcal{D}}_X$. One way to do this is to take an embedding $i: X
\to V$ into a smooth affine variety $V$ as in §\[ss:ddv\]; in the notation there, the global sections of $i_*({\mathfrak{v}}\cdot {\mathcal{D}}_X)$ then identify as $$\label{e:gs-vd}
\Gamma(V, i_*({\mathfrak{v}}\cdot {\mathcal{D}}_X))
= I_X {\mathcal{D}}_V \setminus \bigl( (\widetilde {\mathfrak{v}}+I_X) \cdot {\mathcal{D}}_V\bigr).$$ Then, by ${\mathfrak{v}}\cdot {\mathcal{O}}_X$ we mean the subspace $$\label{e:vox}
{\mathfrak{v}}\cdot {\mathcal{O}}_X =
(I_X {\mathcal{D}}_V \cap \widetilde {\mathfrak{v}}\cdot {\mathcal{O}}_V) \setminus
\widetilde {\mathfrak{v}}\cdot {\mathcal{O}}_V.$$ Finally, by ${\mathcal{O}}_X$ itself, we mean the subspace $$\label{e:ox}
{\mathcal{O}}_X = (I_X {\mathcal{D}}_V \cap {\mathcal{O}}_V) \setminus {\mathcal{O}}_V.$$ Then, it follows that (ii’) is equivalent to $({\mathfrak{v}}\cdot {\mathcal{D}}_X) \cap
{\mathcal{O}}_X = 0$ and that (iii) is equivalent to $({\mathfrak{v}}\cdot {\mathcal{O}}_X)
\cap {\mathcal{O}}_X = 0$, as desired. In other words, it is equivalent to ask that $(\widetilde {\mathfrak{v}}\cdot {\mathcal{D}}_V) \cap {\mathcal{O}}_V \subseteq I_X$ and $(\widetilde {\mathfrak{v}}\cdot {\mathcal{O}}_V) \cap {\mathcal{O}}_V \subseteq I_X$.
We now prove that (iii) implies (ii’). Assume that $V = {\mathbf{A}}^n = {\operatorname{\mathsf{Spec}}}{\mathbf{k}}[x_1, \ldots, x_n]$. Note that $$(\widetilde {\mathfrak{v}}+ I_X) \cdot {\mathcal{D}}_V = (\widetilde {\mathfrak{v}}+ I_X) \cdot
{\mathcal{O}}_V \cdot {\operatorname{\mathsf{Sym}}}\langle\partial_1, \ldots, \partial_n \rangle.$$ Thus, the fact that $\bigl((\widetilde {\mathfrak{v}}+ I_X) \cdot {\mathcal{O}}_V\bigr) \cap {\mathcal{O}}_V = I_X$, i.e., (iii), implies also that $\bigl(
(\widetilde {\mathfrak{v}}+ I_X) \cdot {\mathcal{D}}_V \bigr) \cap {\mathcal{O}}_V = I_X$, i.e., (ii’).
\[r:inc-subvar-alt\] For irreducible affine varieties, we can also show that (i) and (iii) are equivalent directly without using Remark \[r:inc-subvar\], and hence by Corollary \[c:inc\], that (ii) and (iii) are equivalent. Suppose (i). By Corollary \[c:inc\], ${\mathfrak{v}}$ flows incompressibly on $Z$. Let $z \in Z$ be a smooth point and $\omega \in \Omega_{\hat Z_z}$ be a formal volume preserved by ${\mathfrak{v}}$. Then, if $f_i \in {\mathcal{O}}_Z$ and $\xi_i \in {\mathfrak{v}}|_Z$ satisfy $\sum_i f_i \xi_i = 0$, we have $0 = L_{f_i \xi_i} \omega = \sum_i
\xi_i(f_i)$, which proves (iii).
Conversely, suppose that (iii) is satisfied. Let $z \in Z$ be a smooth point where the rank of ${\mathfrak{v}}|_Z$ is maximal. Then, in a neighborhood $U \subseteq Z$ of $z$, ${\mathcal{O}}_U \cdot {\mathfrak{v}}$ is a free submodule of $T_{U}$, and hence has a basis $\xi_1, \ldots, \xi_j$. In the language of §\[s:div\], one can define a divergence function $D:
{\mathcal{O}}_U \cdot {\mathfrak{v}}\to T_U, D(\sum_i f_i \xi_i) = \sum_i \xi_i(f_i)$. Therefore, by Proposition \[p:reinc\], ${\mathfrak{v}}$ flows incompressibly on $U$, and hence on $Z$.
Proof of Theorem \[t:hol-finc\] {#ss:hol-finc-pf}
-------------------------------
Part (i) is an immediate consequence of Proposition \[p:inv-sub-qt\] and Corollary \[c:inc\].
For part (ii), suppose that $X$ does not have finitely many incompressible leaves. Then, there is a degenerate invariant subvariety $i: Z {\hookrightarrow}X$ such that ${\mathfrak{v}}$ flows incompressibly on $Z$. By Proposition \[p:inv-sub-qt\] and Corollary \[c:inc\], there is a nonholonomic quotient of $M(X,{\mathfrak{v}})$ supported on the closure of $Z$. So $M(X,{\mathfrak{v}})$ is not holonomic.
Support and saturation {#ss:supp}
----------------------
To proceed, note that in some cases, $M(X, {\mathfrak{v}})$ is actually supported on a proper subvariety, e.g., in Example \[ex:hol-nf\], where it is zero; more generally, by Proposition \[p:inc-subvar\], this happens if and only if ${\mathfrak{v}}$ does not flow incompressibly. In this case, it makes sense to replace $X$ with the support of $M(X,{\mathfrak{v}})$, and define an equivalent system there. More precisely, we define a scheme-theoretic support of $M(X,{\mathfrak{v}})$:
\[d:supp\] The support of $(X,{\mathfrak{v}})$ is the closed subscheme $X_{\mathfrak{v}}\subseteq X$ defined by the ideal $({\mathfrak{v}}\cdot {\mathcal{D}}_X) \cap {\mathcal{O}}_X$ of ${\mathcal{O}}_X$.
To make sense of this definition, we work in the space of global sections of ${\mathfrak{v}}\cdot {\mathcal{D}}_X$, using and . Note that here it is *essential* that we allow $X_{\mathfrak{v}}$ to be nonreduced (this was our motivation for working in the nonreduced context).
We immediately conclude
Let $i: X_{\mathfrak{v}}\to X$ be the natural closed embedding. Then, there is a canonical isomorphism $M(X, {\mathfrak{v}}) \cong i_* M(X_{\mathfrak{v}},
{\mathfrak{v}}|_{X_{\mathfrak{v}}})$.
The above remarks say that, when $X$ is a variety, $X=X_{\mathfrak{v}}$ if and only if ${\mathfrak{v}}$ flows incompressibly. Moreover, ${\mathfrak{v}}$ flows incompressibly on an invariant subvariety $Z \subseteq X$ if and only if $Z = Z_{{\mathfrak{v}}|_Z}$. With this in mind, we extend the definition of incompressibility to subschemes:
We say that ${\mathfrak{v}}$ flows incompressibly on an invariant subscheme $Z$ if $Z = Z_{{\mathfrak{v}}|_Z}$.
With this definition, as promised, the conditions (i), (ii’), and (iii) of Proposition \[p:inc-subvar\] are equivalent for arbitrary affine schemes of finite type.
\[p:inc-supp\] Let $Z \subseteq X$ be an irreducible closed subvariety. Then there exists a quotient of $M(X,{\mathfrak{v}})$ whose support is $Z$ if and only if $Z$ is invariant and ${\mathfrak{v}}$ flows incompressibly on some infinitesimal thickening of $Z$. In this case, this quotient factors through the quotient $M(X,{\mathfrak{v}}) {\twoheadrightarrow}i_* M(Z',
{\mathfrak{v}}|_{Z'})$, for some infinitesimal thickening $Z'$, with inclusion $i: Z' {\hookrightarrow}X$.
Here, an *infinitesimal thickening* of a subvariety $Z \subseteq
X$ is a subscheme $Z' \subseteq X$ such that $Z'_{\text{red}}= Z$. Note that it can happen that ${\mathfrak{v}}$ flows incompressibly on $Z'$ but not on $Z$, as in Example \[ex:a3-inf-hol-qt\]. We caution that, on the other hand, $M(X,{\mathfrak{v}})$ could have a *submodule* supported on $Z$ even if ${\mathfrak{v}}$ does not flow incompressibly on any infinitesimal thickening of $Z$: see Example \[ex:a3-inf-hol-sub\].
$M(X,{\mathfrak{v}}) = {\mathfrak{v}}\cdot {\mathcal{D}}_X \setminus {\mathcal{D}}_X$ admits a quotient supported on $Z$ if and only if, for some $N \geq 1$, $({\mathfrak{v}}+ I_Z^N)
\cdot {\mathcal{D}}_X$ is not the unit ideal. This is equivalent to saying that $M(Z',{\mathfrak{v}}|_{Z'}) \neq 0$ for some infinitesimal thickening $Z'$ of $Z$. This can only happen if $Z$ is invariant. By definition, such a restriction is fully supported if and only if ${\mathfrak{v}}$ flows incompressibly on $Z'$. For the final statement, note that the quotient morphism must factor through a map $M(X,{\mathfrak{v}})
{\twoheadrightarrow}({\mathfrak{v}}+ I_Z^N) {\mathcal{D}}_X \setminus {\mathcal{D}}_X$, and the latter is $M(Z', {\mathfrak{v}}|_{Z'})$, where we define $Z'$ by $I_{Z'} =
I_Z^N$.
Next, even if $X=X_{\mathfrak{v}}$, there can be many choices of ${\mathfrak{v}}$ that give rise to the same ${\mathcal{D}}$-module. This motivates
The *saturation* ${\mathfrak{v}}^{s}$ of ${\mathfrak{v}}$ is ${\operatorname{Vect}}(X) \cap ({\mathfrak{v}}\cdot
{\mathcal{D}}_X)$. Precisely, in the language of §\[ss:inc-subvar-pf\] for an embedding $i: X {\hookrightarrow}V$, $${\mathfrak{v}}^s =
\Bigl( {\operatorname{Vect}}(V) \cap \bigl( (\widetilde {\mathfrak{v}}+ I_X) \cdot {\mathcal{D}}_V \bigr)\Bigr)|_X.$$
It is easy to check that the definition of the saturation does not depend on the choice of embedding. We next define a smaller, but more computable, saturation:
\[d:os\] The *${\mathcal{O}}$-saturation* ${\mathfrak{v}}^{os}$ of ${\mathfrak{v}}$ is ${\operatorname{Vect}}(X) \cap
({\mathfrak{v}}\cdot {\mathcal{O}}_X)$, precisely, $${\mathfrak{v}}^{os} := \{\sum_i f_i \xi_i \mid f_i \in {\mathcal{O}}_X, \xi_i \in {\mathfrak{v}},
\text{ s.t. } \sum_i \xi_i(f_i) = 0\}.$$
Equivalently, for any embedding $X \subseteq V$ as above, $${\mathfrak{v}}^{os} = \Bigl( {\operatorname{Vect}}(V) \cap \bigl( ({\mathfrak{v}}_V + I_X) \cdot {\mathcal{O}}_V
\bigr)\Bigr)|_X.$$ Note that, by definition, ${\mathfrak{v}}^{os} \subseteq {\mathcal{O}}_X \cdot {\mathfrak{v}}$; however, the same does *not* necessarily hold for ${\mathfrak{v}}^s$, as in Examples \[ex:a3-inf-hol-sub\] and \[ex:a3-inf-hol-qt\]. In particular, in those examples, ${\mathfrak{v}}^s$ has rank two on the locus $x=0$, whereas ${\mathfrak{v}}^{os}$ has rank one.
However, generically on incompressible affine varieties, ${\mathfrak{v}}^{os}={\mathfrak{v}}^s$. More precisely:
If $(X,{\mathfrak{v}})$ is incompressible, then call a vector field $\xi\in {\mathcal{O}}_X \cdot {\mathfrak{v}}$ *incompressible* if, writing $\xi = \sum_i f_i \xi_i$ for $f_i \in {\mathcal{O}}_X$ and $\xi_i \in {\mathfrak{v}}$, one has $\sum_i \xi_i(f_i) = 0$.
Note that we used incompressibility for the definition to make sense; otherwise there could be multiple expressions $\sum_i f_i \xi_i$ for $\xi$ which yield different values $\sum_i \xi_i(f_i)$.
When $X$ is a variety, $\xi \in {\mathcal{O}}_X \cdot {\mathfrak{v}}$ is incompressible if and only if, for every irreducible component of $X$, at a smooth point with a formal volume preserved by ${\mathfrak{v}}$, then $\xi$ also preserves that volume.
\[p:inc-os\] If ${\mathfrak{v}}$ flows incompressibly, then ${\mathfrak{v}}^{os}$ is the subspace of ${\mathcal{O}}_X \cdot {\mathfrak{v}}$ of incompressible vector fields. If $X$ is additionally a variety, then for some open dense subset $U \subseteq X$, $({\mathfrak{v}}|_U)^s = ({\mathfrak{v}}|_U)^{os}$ is the subspace of ${\mathcal{O}}_U \cdot {\mathfrak{v}}$ of incompressible vector fields.
For the first statement, if $X$ is incompressible and $f_i \in
{\mathcal{O}}_X, \xi_i \in {\mathfrak{v}}$ are such that $\sum_i \xi_i(f_i) = 0$, then it follows that $\sum_i f_i \cdot \xi_i = \sum_i \xi_i \cdot f_i$.
For the second statement, first note that, by Corollary \[c:inc\], since ${\mathfrak{v}}$ is incompressible and $X$ is a variety, on each irreducible component, ${\mathfrak{v}}^s$ must generically have the same rank as ${\mathfrak{v}}$. Now let $U \subseteq X$ be the locus of smooth points $x
\in X$ such that, if $Y \subseteq X$ is the irreducible component containing $x$, the dimension ${\mathfrak{v}}|_x$ is maximal along $Y$. Then ${\mathcal{O}}_U \cdot {\mathfrak{v}}|_U$ is locally free. It follows that this also equals ${\mathcal{O}}_U \cdot ({\mathfrak{v}}|_U)^s$. Since $M(U,({\mathfrak{v}}|_U)^s) = M(U,
{\mathfrak{v}}|_U)$ is fully supported, $({\mathfrak{v}}|_U)^s$ is incompressible. By the first part, we therefore have $({\mathfrak{v}}|_U)^s \subseteq
({\mathfrak{v}}|_U)^{os}$; the opposite inclusion is true by definition. Finally, note that, by definition, $U$ is open and dense.
\[ex:red-gen-sat\] When $X = X_{\mathfrak{v}}$ is reduced and irreducible, in the formal neighborhood of a generic point of $x \in X$, one has $\hat X_x
\cong (V \times V')$ for formal polydiscs $V$ and $V'$, and ${\mathfrak{v}}^{s}={\mathfrak{v}}^{os} = {\mathcal{O}}_{V'} \cdot H(V)$ where $V$ is equipped with its standard volume form (this also gives an alternative proof of part of Proposition \[p:inc-os\]). So, up to isomorphism, this only depends on the dimension of $X$ and the generic rank of ${\mathfrak{v}}$.
\[r:supp-sat\] There is a close relationship between the saturation and the support ideal. In the language of Remark \[r:diffop-leq1\], if we generalize ${\mathfrak{v}}$ to the setting of differential operators of order $\leq 1$, then the natural saturation becomes $({\mathfrak{v}}\cdot {\mathcal{D}}_X)
\cap {\mathcal{D}}_X^{\leq 1}$. In the case ${\mathfrak{v}}\subseteq {\operatorname{Vect}}(X)$, this saturation contains both ${\mathfrak{v}}^s$ and the ideal of $X_{\mathfrak{v}}$; by a computation similar to that of § \[ss:inc-subvar-pf\], in fact, this saturation is ${\mathfrak{v}}^s \cdot
{\mathcal{O}}_X$.
By Remark \[r:supp-sat\], one obtains an alternative formula for the support ideal, call it $I_{X_{\mathfrak{v}}}$, of $X$: this is $I_{X_{\mathfrak{v}}}
= ({\mathfrak{v}}^s \cdot {\mathcal{O}}_X) \cap {\mathcal{O}}_X$. This can be viewed as a generalization of the equivalence of Proposition \[p:inc-subvar\], (ii’) $\Leftrightarrow$ (iii), in the case that ${\mathfrak{v}}= {\mathfrak{v}}^s$ is saturated.
Holonomicity criteria {#ss:hol-bicond}
---------------------
\[t:hol-bicond\] The following conditions are equivalent:
- $(X,{\mathfrak{v}})$ is holonomic;
- For every (degenerate closed) invariant subscheme $Z'
\subseteq X$ on which ${\mathfrak{v}}$ flows incompressibly, for $i:Z := Z'_{\text{red}}\to
Z'$ the inclusion, $i^! M(Z',{\mathfrak{v}}|_{Z'})$ is generically a local system;
- $X$ has only finitely many invariant closed subvarieties $Z$ on which ${\mathfrak{v}}$ flows incompressibly in some infinitesimal thickening $i: Z {\hookrightarrow}Z' \subseteq X$, and for all of them, in formal neighborhoods of generic $z \in Z$ there is a canonical isomorphism $$i^! M(Z', {\mathfrak{v}}_{Z'}) \cong \Omega_{\hat Z_z} \otimes ((i_*
\Omega_{\hat Z_z})^{{\mathfrak{v}}|_{Z'}})^*.$$
In this case, $M(X,{\mathfrak{v}})$ admits a filtration $$0 \subseteq M_{\geq \dim X}(X, {\mathfrak{v}}) \subseteq M_{\geq \dim X-1}(X, {\mathfrak{v}})
\subseteq \cdots \subseteq M_{\geq 0}(X,{\mathfrak{v}}) = M(X,{\mathfrak{v}}),$$ whose subquotients $M_{\geq j}(X,{\mathfrak{v}}) / M_{\geq (j+1)}(X,{\mathfrak{v}})$ are direct sums of indecomposable extensions of local systems on open subvarieties of the dimension $j$ varieties appearing in (iii) by local systems on subvarieties of their boundaries.
Here $(i_* \Omega_{\hat Z_z})^{{\mathfrak{v}}|_{Z'}}$ is the (finite-dimensional) vector space of distributions along $Z$ preserved by the flow of ${\mathfrak{v}}|_{Z'}$. For example, in the case that there exists a product decomposition $\widehat {Z'_z} \cong \hat Z_z \times
S$ for some zero-dimensional scheme $S$, for which the inclusion of $\hat Z_z$ is the obvious one to $\hat Z_z \times \{0\}$, then $i_*
\Omega_{\hat Z_z} \cong (\Omega_{\hat Z_z} \otimes_{\mathbf{k}}{\mathcal{O}}_S^*)$, where $\Omega_{\hat Z_z}$ is the space of formal volume forms on $\hat
Z_z$ and ${\mathcal{O}}_S^*$ is the (finite-dimensional) space of algebraic distributions on $S$.
We remark that the theorem also gives another proof of Proposition \[p:tr\] (which we don’t use in the proof of the theorem), since a connected transitive variety $(X,{\mathfrak{v}})$ is a single leaf and therefore ${\mathfrak{v}}$ is holonomic.
Using part (iii) of the Theorem, we immediately conclude
When $(X,{\mathfrak{v}})$ is holonomic, an invariant subscheme $Z' \subseteq
X$ is incompressible if and only if, for generic $z \in Z :=
Z'_{\text{red}}$, with $i: Z {\hookrightarrow}Z'$ the inclusion, $(i_*\Omega_{\hat
Z_z})^{{\mathfrak{v}}|_{Z'}} \neq 0$.
Note that, when $Z'$ is a variety, the corollary is tantamount to the definition of incompressibility, and does not require holonomicity.
In particular, we can weaken the holonomicity criterion of Theorem \[t:main2\], adding in the word “incompressible”: $$\text{no incompressible degenerate invariant closed subschemes}
\Rightarrow \text{holonomic}.$$ For a counterexample to the converse implication, recall Example \[ex:a3-inf-hol-qt\].
Since holonomic ${\mathcal{D}}$-modules are always of finite length and their composition factors are intermediate extensions of local systems, and since in our case it is clear that any local systems must be on invariant subvarieties, it is immediate that (i) $\Rightarrow$ (iii) $\Rightarrow$ (ii); we only need to explain the formula in (iii). First, note that, by Kashiwara’s equivalence (i.e., via the restriction functor of ${\mathcal{D}}$-modules from $Z'$ to $Z$), the categories of ${\mathcal{D}}$-modules on $Z'$ and on $Z$ are canonically equivalent. Then, the multiplicity space $((i_* \Omega_{\hat
Z_z})^{{\mathfrak{v}}|_{Z'}})^*$ is explained by the canonical isomorphism $$(i_* \Omega_{\hat Z_z})^{{\mathfrak{v}}|_{Z'}} \cong {\operatorname{Hom}}_{\hat {\mathcal{D}}_{X,z}}(M(\widehat{Z'_z},
{\mathfrak{v}}|_{\widehat{Z'_z}}), i_* \Omega_{\hat Z_z}),$$ looking at the image of the canonical generator of $M(\widehat{Z'_z},
{\mathfrak{v}}|_{\widehat{Z'_z}})$, and viewing ${\mathcal{D}}$-modules on $Z'$ as ${\mathcal{D}}$-modules on the ambient space $X$.
So, we prove that (ii) implies (i). Suppose (ii) holds. We prove holonomicity by induction on the dimension of $X$. There is an open dense subset $Y \subseteq X$ such that $M(X,{\mathfrak{v}})|_Y$ is a local system (viewed as a ${\mathcal{D}}$-module on $Y_{\text{red}}$). Take $Y$ to be maximal for this property, i.e., the set-theoretic locus where $M(X,{\mathfrak{v}})$ is a local system in some neighborhood.
Let $j: Y {\hookrightarrow}X$ be the open embedding. Then by adjunction, since $j^* M(X,{\mathfrak{v}}) = M(X,{\mathfrak{v}})|_Y$ is holonomic, we obtain a canonical map $H^0 j_! M(X, {\mathfrak{v}})|_Y \to M(X,{\mathfrak{v}})$. The cokernel is supported on the closed invariant subvariety $Z := X \setminus Y$, which has strictly smaller dimension than that of $X$. By Proposition \[p:inc-supp\], the quotient factors through $M(Z',{\mathfrak{v}}|_{Z'})$ for some infinitesimal thickening $Z'$ of $Z$. Then, by induction, $M(Z',
{\mathfrak{v}}|_{Z'})$ is holonomic. This implies the result.
The final statement follows from the inductive construction of the previous paragraph, if we note that the image of $H^0j_! M(X,{\mathfrak{v}})|_Y$ is an extension of the local system $M(X,{\mathfrak{v}})|_Y$ by local systems on boundary subvarieties, none of which split off the extension.
Note that, by Example \[ex:a3-inf-hol-sub\], in general the extensions appearing (iii) can contain composition factors supported on invariant subvarieties which do not themselves appear in (iii).
Global generalization {#ss:nonaffine}
---------------------
Now, suppose that $X$ is not necessarily affine. Since $X$ does not in general admit (enough) global vector fields, we need to generalize ${\mathfrak{v}}$ to a *presheaf* of vector fields, i.e., a sub-presheaf of ${\mathbf{k}}$-vector spaces of the tangent sheaf. As we will see, even for affine $X$, this is more natural and more flexible: for example, even in the case of Hamiltonian vector fields, we will see that Zariski-locally Hamiltonian vector fields need not coincide with Hamiltonian vector fields, so that the natural presheaf ${\mathfrak{v}}$ is not even a sheaf, let alone constant; see Remark \[r:zarlh\] below.
Nonetheless, all of the main examples and results of this paper are already interesting for affine varieties and do not require this material, so the reader interested only in the affine case can feel free to skip this subsection.
Let $X$ be a not necessarily affine variety and ${\mathfrak{v}}$ a presheaf of Lie algebras of vector fields on $X$. For any open affine subset $U \subseteq X$, we can define the ${\mathcal{D}}$-module on $U$, $M(U,{\mathfrak{v}}(U))$, as above. Recall that this is defined as a certain quotient of ${\mathcal{D}}_U$. Therefore, to show that the $M(U,{\mathfrak{v}}(U))$ glue together to a ${\mathcal{D}}$-module on $X$, it suffices to check that the restriction to $U \cap U'$ of the submodules of ${\mathcal{D}}_U$ and ${\mathcal{D}}_{U'}$ whose quotients are $M(U,{\mathfrak{v}}(U))$ and $M(U',{\mathfrak{v}}(U'))$, respectively, are the same. This does *not* hold in general, but it does hold if one has the following condition:
Say that $(X,{\mathfrak{v}})$ is (Zariski) *${\mathcal{D}}$-localizable* if, for every open affine subset $U \subseteq X$ and every open affine $U' \subseteq U$, $$\label{e:d-loc}
{\mathfrak{v}}(U') {\mathcal{D}}_{U'} = {\mathfrak{v}}(U)|_{U'} {\mathcal{D}}_{U'}.$$
If $X$ is already affine, the definition is still meaningful (and this is the case we will primarily be interested in here). In this case we can restrict to $U=X$ in .
\[ex:const-local\] If $X$ is irreducible and ${\mathfrak{v}}$ is a constant sheaf, then it is immediate that ${\mathfrak{v}}$ is ${\mathcal{D}}$-localizable. More generally, for reducible $X$ and ${\mathfrak{v}}\subseteq {\operatorname{Vect}}(X)$, we can consider the associated presheaf ${\mathfrak{v}}(U) := {\mathfrak{v}}|_U$, and this is ${\mathcal{D}}$-localizable. If the irreducible components of $X$ are $X_j$, then the sheafification of this ${\mathfrak{v}}$ is ${\mathfrak{v}}(U) = \bigoplus_{j
\mid X_j \cap U \neq \emptyset} {\mathfrak{v}}(X_j \cap U)$.
We will use below the following basic
\[l:vd-sheaf\] Let $X$ be an affine scheme of finite type and ${\mathfrak{v}}\subseteq {\operatorname{Vect}}(X)$ an arbitrary subset of vector fields. Then for every affine open $U \subseteq X$, one has the equality of sheaves on $U$, $$({\mathfrak{v}}\cdot {\mathcal{D}}_X)|_U = {\mathfrak{v}}|_U \cdot {\mathcal{D}}_U.$$ In particular, as a sheaf, the sections of ${\mathfrak{v}}\cdot {\mathcal{D}}_X$ on $U$ coincide with the global sections of ${\mathfrak{v}}|_U \cdot {\mathcal{D}}_U$.
Similarly, for every $x \in X$, we have $({\mathfrak{v}}\cdot {\mathcal{D}}_X)|_{\hat X_x} = {\mathfrak{v}}|_{\hat X_x} \cdot \hat {\mathcal{D}}_{X,x}$.
We use . In these terms, for $X {\hookrightarrow}V$ an embedding into a smooth affine variety $V$, let $U' \subseteq V$ be an affine open subset such that $U' \cap X = U$. Then $({\mathfrak{v}}\cdot
{\mathcal{D}}_X)|_U$ identifies with the ${\mathcal{D}}$-module restriction of to $U'$, which is then ${\mathfrak{v}}|_U \cdot {\mathcal{D}}_U$. We conclude the statements of the first paragraph. The second paragraph is similar.
Given a presheaf $\mathcal{C}$, let ${\operatorname{Sh}}(\mathcal{C})$ be its sheafification.
\[p:local\] Suppose that $(X, {\mathfrak{v}})$ is ${\mathcal{D}}$-localizable. Then the following hold:
1. The $M(U,{\mathfrak{v}}(U))$ glue together to a ${\mathcal{D}}$-module $M(X,{\mathfrak{v}})$ on $X$.
2. For every open affine $U$ and every open affine $U' \subseteq U$, $M(X,{\mathfrak{v}})|_{U'} = M(U',{\mathfrak{v}}(U'))$.
3. $(X,{\operatorname{Sh}}({\mathfrak{v}}))$ is also ${\mathcal{D}}$-localizable, and $M(X,{\operatorname{Sh}}({\mathfrak{v}})) = M(X,{\mathfrak{v}})$.
For (i), note that applied to $U' := U \cap V$ implies that $M(U, {\mathfrak{v}}(U))$ and $M(V, {\mathfrak{v}}(V))$ glue. Then, (ii) is an immediate consequence of .
It remains to prove (iii). Suppose that $U$ is an affine open, $U'
\subseteq U$ is affine open, and $\xi \in {\operatorname{Sh}}({\mathfrak{v}})(U')$. Let $u
\in U'$. By definition, there exists a neighborhood $U'' \subseteq
U'$ of $u$ such that $\xi|_{U''} \in {\mathfrak{v}}(U'')$. By , $\xi|_{U''} \in {\mathfrak{v}}(U)|_{U''} \cdot {\mathcal{D}}_{U''}$. Thus, by Lemma \[l:vd-sheaf\], $\xi$ is a section of the ${\mathcal{D}}$-module ${\mathfrak{v}}(U)|_{U'} \cdot {\mathcal{D}}_{U'} = ({\mathfrak{v}}(U) \cdot
{\mathcal{D}}_U)|_{U'}$ on $U'$. This proves the first statement. This also proves the second statement, since $U' \subseteq U$ and $\xi \in
{\operatorname{Sh}}({\mathfrak{v}})(U')$ were arbitrary.
\[r:vsloc\] As in Remark \[r:vs\], we could have allowed ${\mathfrak{v}}$ to be an arbitrary presheaf of vector fields (rather than a sheaf of Lie algebras of vector fields). However, it is easy to see that it is then ${\mathcal{D}}$-localizable if and only if the presheaf of Lie algebras generated by it is, and that the resulting ${\mathcal{D}}$-module is the same. So, no generality is lost by requiring that ${\mathfrak{v}}$ be a presheaf of Lie algebras.
Using the above, in the ${\mathcal{D}}$-localizable setting, the results of this section extend to nonaffine schemes of finite type. We omit further details (but we will discuss ${\mathcal{D}}$-localizability more in §\[s:ex-dloc\] below).
Generalizations of Cartan’s simple Lie algebras {#s:Csla}
===============================================
In this section we state and prove general results on Lie algebras of vector fields on affine varieties which generalize the simple Lie algebras of vector fields on affine space as classified by Cartan. Namely, we will consider the Lie algebras of all vector fields; of Hamiltonian vector fields on Poisson varieties; of Hamiltonian vector fields on Jacobi varieties (this generalizes both the previous example and the setting of contact vector fields on contact varieties); and of Hamiltonian vector fields on varieties equipped with a top polyvector field, or more generally equipped with a divergence function. The last example, which seems to not have been studied before, generalizes the volume-preserving or divergence-free vector fields on ${\mathbf{A}}^n$ or on Calabi-Yau varieties. We also consider invariants of these Lie algebras under the actions of finite groups (we will continue this study in §$\!$§\[s:fqcyvar\] and \[s:sym\]).
Namely, in this section we compute the leaves under the flow of these vector fields and determine when they are holonomic, and hence their coinvariants are finite-dimensional.
We will state all examples in the affine setting; in §\[s:ex-dloc\] below we will explain how to generalize them to the nonaffine setting (which will at least work for the cases of all vector fields and Hamiltonian vector fields).
The case of all vector fields {#ss:allvfds}
-----------------------------
Consider the case where ${\mathfrak{v}}$ is the Lie algebra of all vector fields. In this case we have a basic result:
\[p:supp-allvfds\] The support, $Z = X_{{\operatorname{Vect}}(X)}$, of ${\operatorname{Vect}}(X)$ is the locus where all vector fields vanish, i.e., the scheme of the ideal $({\operatorname{Vect}}(X)({\mathcal{O}}_X))$. Moreover, $$M(X,{\operatorname{Vect}}(X)) = {\mathcal{D}}_Z :=
{\operatorname{Vect}}(X)({\mathcal{O}}_X) \cdot {\mathcal{D}}_X \setminus {\mathcal{D}}_X,$$ and $({\mathcal{O}}_X)_{{\operatorname{Vect}}(X)} = {\mathcal{O}}_Z$.
The support is evidently incompressible, and is the union of zero-dimensional leaves at every point. Therefore, ${\operatorname{Vect}}(X)$ is holonomic if and only if this vanishing locus is finite.
Given $\xi \in {\operatorname{Vect}}(X)$, the submodule ${\mathfrak{v}}\cdot {\mathcal{D}}_X$ contains $[\xi, f]=\xi(f)$ for all $f \in {\mathcal{O}}_X$. These generate the ideal $({\operatorname{Vect}}(X)({\mathcal{O}}_X))$ over ${\mathcal{O}}_X$, which defines the vanishing scheme of ${\operatorname{Vect}}(X)$. Conversely, notice that the principal symbol of any product of vector fields lies in the submodule $({\operatorname{Vect}}(X)({\mathcal{O}}_X))
\cdot {\mathcal{D}}_X$. Thus, ${\mathfrak{v}}\cdot {\mathcal{D}}_X = ({\operatorname{Vect}}(X)({\mathcal{O}}_X)) \cdot {\mathcal{D}}_X$. The last statement follows immediately.
This motivates the
A point $x \in X$ is *exceptional* if all vector fields on $X$ vanish at $x$.
Clearly, all exceptional points are singular, but not conversely: for example, if $X = Y \times Z$ where $Z$ is smooth and of purely positive dimension, then $X$ will have no exceptional points, regardless of how singular $Y$ is.
\[p:allvfds\] The following are equivalent:
1. The quotient $({\mathcal{O}}_X)_{{\operatorname{Vect}}(X)}$ is finite-dimensional;
2. $X$ has finitely many exceptional points;
3. ${\operatorname{Vect}}(X)$ (i.e., $M(X,{\operatorname{Vect}}(X))$) is holonomic.
First, (ii) and (iii) are equivalent by Proposition \[p:supp-allvfds\], since ${\mathcal{D}}_Z$ is holonomic if and only if $Z$ has dimension zero, i.e., set-theoretically $Z$ is finite. By the proposition, with $Z$ the support of ${\operatorname{Vect}}(X)$, then $Z_{\text{red}}$ is the locus of exceptional points of $X$ and $M(X,{\operatorname{Vect}}(X)) = {\mathcal{D}}_Z$, so the equivalence follows. Similarly, these are equivalent to (i), since $({\mathcal{O}}_X)_{{\operatorname{Vect}}(X)} = {\mathcal{O}}_Z$.
Note that the implication (i) $\Rightarrow$ (iii) above, a converse to Proposition \[p:hol-fd\], is special to the case ${\mathfrak{v}}=
{\operatorname{Vect}}(X)$. See, e.g., Remarks \[r:hol-fd\] and \[r:hp0-van\].
If $X$ has a finite exceptional locus $Z \subseteq X$ (i.e., ${\mathfrak{v}}$ is holonomic), then $$M(X, {\operatorname{Vect}}(X)) \cong \bigoplus_{z \in Z} \delta_z \otimes (\hat
{\mathcal{O}}_{X,z})_{Vect(\hat {\mathcal{O}}_{X, z})}.$$
This follows immediately by formally localizing at each exceptional point.
Under the same assumptions as in the previous corollary, if $\pi: X
\to {\text{pt}}$ is the projection to a point, $$\pi_* M(X,{\operatorname{Vect}}(X)) = \pi_0 M(X,{\operatorname{Vect}}(X)) \cong \bigoplus_{z \in Z} (\hat
{\mathcal{O}}_{Z,z})_{Vect(\hat {\mathcal{O}}_{Z, z})}.$$
This follows since $\pi_* \delta_x = \pi_0 \delta_x = {\mathbf{k}}$ for any point $x \in X$.
Suppose that $X$ has finitely many exceptional points. Then, the dual space $(({\mathcal{O}}_X)_{{\operatorname{Vect}}(X)})^* = ({\mathcal{O}}_X^*)^{{\operatorname{Vect}}(X)}$, of functionals invariant under all vector fields, includes the evaluation functionals at every exceptional point. These are linearly independent. However, they need not span all invariant functionals. In other words, the multiplicity spaces $(\hat
{\mathcal{O}}_{X,x})_{Vect(\hat {\mathcal{O}}_{X, x})}$ in the corollaries need not be one-dimensional.
For example, if one takes a curve $X \subset {\mathbf{A}}^2$ of the form $P(x,y) + Q(x,y) = 0$ in the plane with $P(x,y)$ and $Q(x,y)$ homogeneous of degrees $n$ and $n+1$, then we claim that, if $n \geq
5$ and $P$ and $Q$ are generic, all vector fields on $X$ vanish to degree at least two at the singularity at the origin. Therefore, the coinvariants $({\mathcal{O}}_X)_{{\operatorname{Vect}}(X)}$ have dimension at least three, even though $0$ is the only singularity of $X$.
Indeed, up to scaling, any vector field which sends $P$ to a constant multiple of $P$ up to higher degree terms is of the form $a
{\operatorname{Eu}}+ v$, where $a \in {\mathbf{k}}$ and $v$ vanishes up to degree at least two at the origin. Suppose that such a vector field preserves the ideal $(P+Q)$, i.e., that it sends $P + Q$ to a multiple of $P+Q$. We claim that $a=0$. Otherwise, we can assume up to scaling that $a=1$. Then $({\operatorname{Eu}}+v)(P+Q) = f(P+Q)$ for some polynomial $f$. By comparing the parts of degree $n$, we conclude that $f(0)=n$. Writing $f=n+bx+cy+g$, where $g$ vanishes to degree at least two at the origin, we conclude that $Q = (-v + (bx+cy){\operatorname{Eu}})P$. So there exists a quadratic vector field $w = -v + (bx+cy){\operatorname{Eu}}$ which takes $P$ to $Q$. The space of all quadratic vector fields is six-dimensional, whereas the space of all possible $Q$ is of dimension $n+2$. So for $n \geq 5$, we obtain a contradiction, since $P$ and $Q$ are assumed to be generic.
Here is an explicit example for the smallest case, $n=5$, of such a $P$ and $Q$: Let $P=x^5+y^5$ and $Q=x^3y^3$. Then it is clear that the equation $Q = -v(P) + (bx+cy)P$ cannot be satisfied for any quadratic vector field $v$ and any $b, c \in {\mathbf{k}}$.
\[ex:inf-exc\] One example of a variety with infinitely many exceptional points, and hence infinite-dimensional $({\mathcal{O}}_X)_{{\operatorname{Vect}}(X)}$ and non-holonomic ${\operatorname{Vect}}(X)$, is a nontrivial family of affine cones of elliptic curves: one can take $X = {\operatorname{\mathsf{Spec}}}{\mathbf{k}}[x,y,z,t] / (x^3 + y^3 +
z^3 + txyz)$, which is a family over ${\mathbf{A}}^1={\operatorname{\mathsf{Spec}}}{\mathbf{k}}[t]$ whose fibers are affine cones of elliptic curves in ${\mathbf{P}}^2$. Then, we claim that all singular points $x=y=z=0$ are exceptional.
The proof is as follows: Take any vector field on $X$ and lift it to a vector field $\xi$ on ${\mathbf{A}}^4$ parallel to $X$. Then $\xi(x^3 + y^3 + z^3 + txyz) = f (x^3+y^3+z^3+txyz)$ for some $f \in {\mathcal{O}}_{{\mathbf{A}}^4}$. Replacing $\xi$ by $\xi - (1/3)f \cdot
(x \partial_x + y \partial_y + z\partial_z)$, we can assume that $f = 0$. Restricting to $t=t_0$, we obtain $$\label{e:xieqn}
\xi|_{t=t_0} (x^3 + y^3 + z^3 + t_0 xyz) = - \xi(t)|_{t=t_0}\cdot xyz.$$ Suppose that $\xi$ did not vanish at $(0,0,0,t_0)$. We can assume $\xi$ is homogeneous with respect to the grading $|x|=|y|=|z|=1$ and $|t|=0$. Then $\xi|_{t=t_0}$ is either constant or linear. By , $\xi|_{t=t_0}$ annihilates $x^3+y^3+z^3$, but no constant or linear vector field can do that, which is a contradiction.
The Poisson case {#ss:poisson}
----------------
Suppose that $X$ is an affine Poisson scheme of finite type, i.e., ${\mathcal{O}}_X$ is a Poisson algebra. Let $\pi$ be the Poisson bivector field on $X$. Then, we can let ${\mathfrak{v}}$ be the Lie algebra of Hamiltonian vector fields, $H(X)=H_\pi(X)$. In particular, these vector fields are $\xi_f := \pi(df)$ for $f \in {\mathcal{O}}_X$. In this case, $({\mathcal{O}}_X)_{\mathfrak{v}}=
{\mathsf{HP}}_0({\mathcal{O}}_X)$, the zeroth Poisson homology of ${\mathcal{O}}_X$. As pointed out in Example \[ex:poiss-finsym\], $H(X)$ is holonomic if and only if $X$ has finitely many symplectic leaves.
There are several natural larger Lie algebras to consider than $H(X)$. Note that $H(X)$ is the space of vector fields obtained by contracting $\pi$ with exact one-forms. So, one can consider instead $LH(X)=LH_\pi(X) = \pi(\tilde \Omega^1_X)$, the space of vector fields obtained by contracting $\pi$ with *closed* one-forms modulo torsion (note that contracting $\pi$ with torsion yields zero, since ${\mathcal{O}}_X$ is torsion-free). Here we will denote the resulting vector field by $\eta_\alpha := \pi(\alpha)$. Thus, when $X$ is generically symplectic, $LH(U)/H(U) \cong H^1_{DR}(U)$ for all open affine $U
\subseteq X$. (Recall from the beginning of §\[s:gen\] that, over ${\mathbf{k}}={\mathbf{C}}$, if $U$ is smooth, this coincides with the first topological cohomology of $U$).
Here, $LH$ stands for “locally Hamiltonian;” in a smooth affine open subset, in the case that ${\mathbf{k}}={\mathbf{C}}$, these are the vector fields which are locally Hamiltonian in the analytic topology. In general, in a smooth open subset, these are the vector fields which, restricted to a formal neighborhood of a point, are Hamiltonian. However, as explained in the next example, in formal neighborhoods of singular points not all locally Hamiltonian vector fields are Hamiltonian:
\[ex:fn-sing-poiss\] In the formal neighborhood of singular points, locally Hamiltonian vector fields need not be Hamiltonian, since the first de Rham cohomology modulo torsion need not vanish in such a neighborhood, and as mentioned above, when $X$ is generically symplectic, then $LH(X)/H(X) \cong H^1(\tilde \Omega_X^\bullet)$.
Here is an example where this cohomology does not vanish. Suppose $Z \subseteq {\mathbf{A}}^n$ is a complete intersection with an isolated singularity at $z \in Z$. By below, in this case $\tilde
\Omega_{Z,z}^\bullet$ is acyclic except in degree $k=\dim Z$, where $\dim H^k(\tilde \Omega_{Z,z}^\bullet) = \mu_z - \tau_z$, where $\mu_z$ and $\tau_z$ are the Milnor and Tjurina numbers of $z$ (see §\[s:cy-cplte-int-is\] below; we will not use the general definition here). In the case when $Z \subseteq {\mathbf{A}}^2$ is a reduced curve cut out by $Q \in {\mathbf{k}}[x,y]$ with an isolated singularity at the origin, then all one-forms modulo torsion are closed, but they are not all exact in general. Explicitly, $H^1(\tilde
\Omega_{Z,0}^\bullet) \cong (Q, \partial_x Q, \partial_y Q)_0 /
(\partial_x Q, \partial_y Q)_0$, where $(-)_0 \subseteq \hat
{\mathcal{O}}_{{\mathbf{A}}^2,0}$ is the ideal in the completed local ring at the origin.
Specifically, take $Q = x^3 + x^2 y + y^4$, where $$\begin{gathered}
(Q,\partial_x Q, \partial_y Q) = (3x^2+2xy,x^2+4y^3,x^3+x^2y+y^4)=
(3x^2+2xy,x^2+4y^3,y^4) \\ \neq (3x^2+2xy,x^2+4y^3)=(\partial_x
Q,\partial_y Q).\end{gathered}$$ One therefore obtains a nonexact (closed) one-form. Such a form is $\alpha := x \cdot dy$: one can compute that $$\alpha \wedge dQ = (-3x^3 -2x^2 y) \cdot dx \wedge dy \equiv 2y^4
\cdot dx \wedge dy \pmod { d {\mathbf{k}}[\![x,y]\!] \wedge dQ + (Q) dx
\wedge dy + (x,y)^5 dx \wedge dy },$$ and this is not equivalent to zero modulo $d {\mathbf{k}}[\![x,y]\!] \wedge dQ
+ (Q) dx \wedge dy + (x,y)^5 dx \wedge dy$.
Then, consider the Poisson variety $X = Z \times {\mathbf{A}}^1$ with the Poisson structure $(\partial_x \wedge \partial_y)(dQ)
\wedge \partial_t$, with $t$ the coordinate on ${\mathbf{A}}^1$. This is generically symplectic, so provides an example where $LH(X) \neq
H(X)$. Specifically, the vector field $\eta_\alpha =
(-3x^3-2x^2y) \partial_t$ is locally Hamiltonian on $X$, but in the formal neighborhood of the origin it is not Hamiltonian. By the above computation, this spans $LH(\hat X_0) / H(\hat X_0)$.
Note that the fact that $LH(X)$ and $H(X)$ are Lie algebras follow from the fact that $[LH(X), LH(X)] \subseteq H(X)$, since $\{\eta_\alpha, \eta_\beta\} = \xi_{i_{\eta_\alpha} \beta}$ for closed one-forms $\alpha$ and $\beta$.
Next, one can consider $P(X)=P_\pi(X)$, the space of all Poisson vector fields, i.e., those $\xi$ such that $L_\xi(\pi)=0$. Clearly, we have $H(X) \subseteq LH(X) \subseteq P(X)$. If $X$ is symplectic (which for us in particular means $X$ is smooth), then it is well-known that $LH(X) = P(X)$, but this may not be true in general (even if $X$ has finitely many symplectic leaves: see Example \[ex:poisson-not-ham\]). However, there is a certain generalization of this equality to the mildly singular case, as explained in the next remark.
\[r:n-cod2-poiss\] In the case that $X$ is normal and generically symplectic, then the following conditions are equivalent:
- $X$ is symplectic on its smooth locus;
- On each irreducible component, $X$ is symplectic outside of a codimension two subset.
This is because the degeneracy locus of a Poisson structure is given by a single equation $\pi^{\wedge \lceil\dim Y/2\rceil} = 0$, so on the smooth locus this consists of divisors (if it is generically nondegenerate).
If we assume that either of these conditions is satisfied, then letting $X^\circ \subseteq X$ be the smooth locus (which is not affine unless $X=X^\circ$) we claim that $P(X) = P(X^\circ) = LH(X^\circ)$, where here by $P(X^\circ)$ we mean global Poisson vector fields on the nonaffine $X^\circ$, and by $LH(X^\circ)$ we mean the collection of vector fields $\eta_\alpha$ for $\alpha \in \Gamma(X^\circ,
\Omega_{X^{\circ}})$ a closed one-form regular on $X^\circ$.
Indeed, in this case, all vector fields which are regular on $X^\circ$ extend to all of $X$. Thus $P(X)=P(X^\circ)$. Moreover, if $\xi \in
P(X)$ is a global Poisson vector field, then dividing by the Poisson bivector, we obtain a closed one-form regular on $X^\circ$, and conversely.
The leaves of $X$ under both $H(X)$ and $LH(X)$ are the symplectic leaves. For $H(X)$, this is the definition of symplectic leaves; for $LH(X)$, this is true because, since all one-forms (and in particular all closed one forms) are spanned over ${\mathcal{O}}_X$ by exact one-forms, the evaluations at each point of the contraction of $\pi$ with either span the same subspace of the tangent space. That is, $H(X)|_x = LH(X)|_x$ for all $x \in X$, as subspaces of $T_x X$. In fact, $H(X)$ and $LH(X)$ define the same ${\mathcal{D}}$-module, since they have the same ${\mathcal{O}}$-saturation, as defined in §\[ss:supp\]:
\[p:poiss-hlh\] The ${\mathcal{O}}$-saturations are equal: $H(X)^{os} = LH(X)^{os}$. Hence, $M(X,H(X)) \cong
M(X,LH(X))$.
Given any closed one-form $\alpha := \sum_i f_i dg_i \in T_X^*$, for $f_i, g_i \in {\mathcal{O}}_X$, we claim that $\eta_\alpha = \sum_i \xi_{g_i}
\cdot f_i$. This follows because $\sum_i [\xi_{g_i}, f_i] = \sum_i
\xi_{g_i}(f_i) = \pi(d\alpha) = 0$. Hence $LH(X) \cdot {\mathcal{O}}_X
\subseteq H(X) \cdot {\mathcal{O}}_X$. For the opposite inclusion, note that $H(X) \subseteq LH(X)$.
In the case that $X$ has finitely many symplectic leaves, then $P(X)$ also has these as its leaves, since in this case every Poisson vector field must be parallel to the symplectic leaves. On the other hand, it can happen that $P(X)$ has finitely many leaves but not $LH(X)$:
If $\pi = x \partial_x \wedge \partial_y$ on ${\mathbf{A}}^2$, then there are infinitely many symplectic leaves: the $y$-axis is a degenerate invariant subvariety with respect to $LH(X)$. On the other hand, the vector field $\partial_y$ is Poisson, so the $y$-axis is a leaf with respect to $P(X)$.
For $LH(X)$, the same argument as for $H(X)$ shows that, in the notation of Proposition \[p:decomp\], all of the $X_i$ are incompressible, and hence $LH(X)$ is holonomic if and only if it has finitely many leaves (the symplectic leaves); or one can use Proposition \[p:poiss-hlh\]. So, again, Theorem \[t:main2\] is the same as Theorem \[t:main1-alt\].
On the other hand, it can happen that $P(X)$ is holonomic even though it does not have finitely many leaves:
If $X$ is a variety equipped with the zero Poisson structure, then $P(X)$ is the Lie algebra of all vector fields, and as explained in §\[ss:allvfds\], this is holonomic if and only if there are finitely many exceptional points. This can happen without having finitely many leaves, e.g., if one takes a product $X= {\mathbf{A}}^1 \times
Y$ where $Y$ has infinitely many exceptional points (cf. Example \[ex:inf-exc\]). Moreover, this is an example where the $X_i$ are not incompressible (if $x$ is the coordinate on ${\mathbf{A}}^1$, $P(X)$ contains both $\partial_x$ and $x \partial_x$, so cannot be incompressible on any of the $X_i = {\mathbf{A}}^1 \times Y_{i-1}$).
\[ex:poiss-cplte-int\] If $Y$ is an $n$-dimensional Calabi-Yau variety (e.g., $Y = {\mathbf{A}}^n$) and $X = Z(f_1, \ldots, f_{n-2}) \subseteq Y$ is a surface which is a complete intersection $f_1=\cdots=f_{n-2}=0$, then there is a standard *Jacobian* Poisson structure on $X$, given by $i_{\Xi}df_1 \wedge \cdots \wedge df_{n-2}$, where $\Xi =
{\mathsf{vol}}_Y^{-1}$ is the inverse to the volume form on $Y$, which we then contract with the exact $n-2$-form $df_1 \wedge \cdots \wedge
df_{n-2}$. It is then standard that the result is a Poisson bivector field. Then $H(X)$ is holonomic if and only if $X$ has only isolated singularities. Already in the case $Y = {\mathbf{A}}^3$ and $X =
Z(f)$ for $f$ a (quasi)homogeneous surface with an isolated singularity at zero, this is quite interesting; ${\mathsf{HP}}_0({\mathcal{O}}_X)=({\mathcal{O}}_X)_{H(X)}$ was computed in [@AL] (although, as we will explain in §\[s:cy-cplte-int-is\], it follows from older results of [@Gre-GMZ]); we plan to compute $M(X,H(X))$ in [@ES-ciiss]. See Example \[ex:cy-cplte-int\] and § \[s:cy-cplte-int-is\].
\[ex:poiss-prod\] If $X$ and $Y$ are Poisson schemes of finite type, then for any of the three Lie algebras defined above, the coinvariants are multiplicative in the sense that $({\mathcal{O}}_{X \times Y})_{H(X \times Y)}
= ({\mathcal{O}}_X)_{H(X)} \otimes ({\mathcal{O}}_Y)_{H(Y)}$ and similarly for $LH$ and $P$. Similarly, the leaves of $X \times Y$ are the products of leaves from $X$ and of leaves from $Y$. These facts follow from the following formula, which also holds for $LH$ and $P$ replacing $H$: $$\label{e:prod-vfds}
H(X) \oplus H(Y) \subseteq
H(X \times Y)
\subseteq
({\mathcal{O}}_X \boxtimes H(Y)) \oplus (H(X) \boxtimes {\mathcal{O}}_Y).$$ The first inclusion holds because, for $f \in {\mathcal{O}}_X$ and $g \in
{\mathcal{O}}_Y$, $\xi_{(f \otimes 1)+ (1 \otimes g)}=\xi_f+\xi_g$. The second follows because, for $f \in {\mathcal{O}}_X$ and $g \in {\mathcal{O}}_Y$, $\xi_{f \otimes g}(h) = f \xi_g + g \xi_f$. To extend to the case of $LH(X \times Y)$, it remains only to consider also the action of Hamiltonian vector fields of closed one-forms modulo torsion generating $H^1_{DR}(X \times Y) =
H^1_{DR}(X) \oplus H^1_{DR}(Y)$ (assuming for simplicity that $X$ and $Y$ are connected). So it suffices to consider Hamiltonian vector fields of closed one-forms modulo torsion on $X$ and $Y$ separately. One concludes that holds for $LH$ replacing $H$. Finally, for $P(X \times Y)$, one also has with $P$ replacing $H$, since $\pi_{X \times Y} =
\pi_X \oplus \pi_Y$ and ${\operatorname{Vect}}(X \times Y) = ({\operatorname{Vect}}(X) \boxtimes
{\mathcal{O}}_Y) \oplus ({\mathcal{O}}_X \boxtimes {\operatorname{Vect}}(Y))$.
\[ex:poisson-not-ham\] Here we give an example of a variety $X$ with finitely many symplectic leaves for which $LH(X) \subsetneq P(X)$. Namely, suppose $X$ is a homogeneous cubic hypersurface, $Q=0$, in ${\mathbf{A}}^3$ with an isolated singularity at the origin, i.e., the cone over a smooth curve of genus one. Then $X$ is equipped with the Poisson bivector given by contracting the top polyvector field $\partial_x
\wedge \partial_y \wedge \partial_z$ on ${\mathbf{A}}^3$ with $dQ$, where $x,
y$, and $z$ are the coordinate functions on ${\mathbf{A}}^3$. This has two symplectic leaves: the origin and its complement.
We claim that the Euler vector field is Poisson but not locally Hamiltonian. This is because the Poisson bracket preserves total degree, so the Euler vector field is Poisson, but it cannot be Hamiltonian since the Poisson bivector vanishes to degree two at the origin, i.e., $\pi(df \wedge dg) \subseteq \mathfrak{m}_0^2$ for all $f,g \in {\mathcal{O}}_{X}$, with $\mathfrak{m}_0$ the maximal ideal of functions vanishing at the origin. Hence all Hamiltonian vector fields vanish to degree two at the origin as well.
For example, $X$ could be the hypersurface $x^3+y^3+z^3 =0$, which is the cone over the Fermat curve. Then $\{x,y\}=3z^2$, $\{y,z\} =
3x^2$, and $\{z,x\} = 3y^2$, and it is clear that the Euler vector field is Poisson but not (locally) Hamiltonian.
\[r:hp0-van\] We note that, unlike for all vector fields, the converse to Proposition \[p:hol-fd\] does not hold in the Poisson case. Indeed, one can consider ${\mathbf{A}}^3$ with the Poisson structure $\partial_x \wedge \partial_y$, which has infinitely many leaves (hence is not holonomic) but vanishing ${\mathsf{HP}}_0$.
Finally, if $X$ is an affine Poisson scheme of finite type with finitely many symplectic leaves, and $f: X \to Y$ is a finite map, then the argument of [@ESdm] showed that the Lie algebra of Hamiltonian vector fields of Hamiltonian functions from $f^*{\mathcal{O}}_Y$ has finitely many leaves. We recover the result from *op. cit.* that ${\mathcal{O}}_X / \{{\mathcal{O}}_X, {\mathcal{O}}_Y\}$ is finite-dimensional. This includes the case, for example, where $X = V$ is a symplectic vector space, and $Y
= V/G$ for $G < {\mathsf{Sp}}(V)$ a finite subgroup (or even any finite subgroup $G < {\mathsf{GL}}(V)$). If $G < {\mathsf{Sp}}(V)$ then we obtain the $G$-invariant Hamiltonian vector fields, $H(X)^G$. Note that, in this case, if $q: X \to X/G$ is the projection, then $q_* M(X,H(X)^G)^G \cong M(X/G,H(X/G))$.
Jacobi schemes {#ss:jacobi}
--------------
A Jacobi structure [@Lic-jac] is a generalization of a Poisson structure, which includes both symplectic and contact manifolds (see the examples below), and can be thought of as a degenerate or singular version of both. By definition, it is a Lie bracket on ${\mathcal{O}}_X$ which need not satisfy the Leibniz rule, but instead satisfies that $\{f, -\}$ is a differential operator of order $\leq 1$ for all $f \in {\mathcal{O}}_X$. Equivalently, the Lie bracket is given by a pair of a bivector field $\pi$ and a vector field $u$ via the formula $$\{f,g\} = \pi(df \wedge dg) + u(fdg - gdf).$$ Here, by a degree $k$ polyvector field, we mean a skew-symmetric multiderivation of ${\mathcal{O}}_X$ of degree $k$, i.e., a linear map $\wedge^k {\mathcal{O}}_X \to {\mathcal{O}}_X$ which is a derivation in each component.
The Jacobi identity is then equivalent to the identities $$[u, \pi] = 0, \quad [\pi, \pi] = 2u \wedge \pi,$$ where $[-,-]$ is the Schouten-Nijenhuis bracket on polyvector fields.
To any affine Jacobi scheme $X$ of finite type, one naturally associates the Lie algebra of Hamiltonian vector fields $\xi_f$ for $f \in {\mathcal{O}}_X$, given by the principal symbol of the differential operator $\{f, -\}$, i.e., $$\xi_f = \pi(df) + f u, \quad \text{i.e.,} \quad \xi_f(g) = \{f,g\} + g
u(f).$$ It is well-known and easy to verify that one has the identity $$[\xi_f, \xi_g] = \xi_{\{f,g\}},$$ so this indeed forms a Lie algebra. Call it $H(X) := H_{\pi,u}(X)$.
We can also define a version $P(X) := P_{\pi,u}(X)$ of vector fields *preserving* the Jacobi structure, i.e., vector fields $\xi$ such that $\xi(\{f,g\}) = \{\xi(f),g\} + \{f, \xi(g)\} = 0$ for all $f,g
\in {\mathcal{O}}_X$. However, unlike before, it is no longer true that $H(X)
\subseteq P(X)$. In particular, to have $\xi_f \in P(X)$, we require that $[u,\xi_f]=\xi_{u(f)} = 0$. So to have $H(X) \subseteq P(X)$, we would need to have $u=0$, i.e., the structure has to be Poisson.
It seems that we cannot define an analogue of $LH(X)$ in this setting since there is no way to obtain Hamiltonian vector fields from closed one-forms. In a neighborhood of a smooth point, one could consider vector fields that restrict in a formal neighborhood of the point to a Hamiltonian vector field, but in general this will not coincide with the definition of $LH(X)$ in the Poisson case, in neighborhoods of singular points where the first de Rham cohomology does not vanish in the formal neighborhood; see Example \[ex:fn-sing-poiss\].
\[r:jac-prod\] Unlike the Poisson case, given Jacobi varieties $X$ and $Y$, there is no natural way to define a Jacobi structure on the product $X
\times Y$: if one set $\pi_{X \times Y} = \pi_X \oplus \pi_Y$ and $u_{X \times Y} = u_X \oplus u_Y$, then the identity $[\pi,\pi] = 2u
\wedge \pi$ would no longer be satisfied: $\pi_X \wedge u_Y$ and $\pi_Y \wedge u_X$ would appear on the RHS but not the LHS. However, one can still equip $X \times Y$ with the Lie algebra of vector fields ${\mathfrak{v}}_X \oplus {\mathfrak{v}}_Y$; in this general situation (i.e., for any ${\mathfrak{v}}_X$ and ${\mathfrak{v}}_Y$), one always has $({\mathcal{O}}_{X
\times Y})_{{\mathfrak{v}}_X \oplus {\mathfrak{v}}_Y} \cong ({\mathcal{O}}_X)_{{\mathfrak{v}}_X} \otimes
({\mathcal{O}}_Y)_{{\mathfrak{v}}_Y}$.
\[ex:j-tr\] The analogue of symplectic varieties in this setting is a smooth Jacobi variety for which $H(X)$ has full rank everywhere, i.e., it has only one leaf (assuming $X$ is connected). This is called a *transitive* Jacobi variety.
As pointed out in, e.g., [@MS-gm] (this is in the smooth context, but the result is proved using a formal neighborhood and works in general), there are two types of connected transitive varieties. One is called *locally conformally symplectic*, and is the situation where $\pi$ is nondegenerate (recall we assumed $X$ was smooth). Therefore, $X$ is even-dimensional. In this case, $u$ is equivalent to the data of a closed one-form $\phi$ satisfying $d
\omega = \phi \wedge \omega$, where $\omega$ is the inverse of $\pi$, and $\phi = u(\omega)$. Then, in the formal neighborhood of any point $x \in X$, we can write $\phi = df$ for some function $f$, and then $H(X)$ preserves the formal volume form $(e^{-f}\omega)^{\wedge \dim X}$ (cf. Example \[ex:st-lcs\] below). This need not be a global volume form, so $M(X,H(X))$ is a rank-one local system which need not be trivial.
The other type of transitive Jacobi variety is an odd-dimensional contact variety. In this case, the Jacobi structure is equivalent to the structure of a *contact one-form* $\alpha$, i.e., a one-form such that ${\mathsf{vol}}_X := \alpha \wedge (d\alpha)^{\wedge (\dim X-1)/2}$ is a nonvanishing volume form. This determines $u$ and $\pi$ uniquely by the formulas $$u(d\alpha)=0, u(\alpha) = 1, \quad \pi(\alpha, \beta) = 0, \quad \pi(\beta \wedge d\alpha) = -\beta + u(\beta)\alpha, \forall \beta \in T_X^*.$$ By the next example, in this case ${\mathfrak{v}}$ does not flow incompressibly, so by Proposition \[p:tr\], $M(X,H(X)) = 0$. On the other hand, we will see that $P(X)$ does flow incompressibly and transitively, preserving the volume form ${\mathsf{vol}}_X$, so $M(X,P(X)) = \Omega_X$ and $\pi_*M(X,P(X)) \cong H_{DR}^{\dim X - *}(X)$. In particular, $({\mathcal{O}}_X)_{P(X)} = H_{DR}^{\dim X}(X)$.
\[ex:st-cont\] The standard example of a contact variety is ${\mathbf{A}}^{2d+1}$ with the standard contact structure, $\alpha = dt + \sum_i x_i dy_i$. Also, note that an arbitrary contact variety restricts to one isomorphic to this in the formal neighborhood of any point. We claim that no volume form is preserved by $H({\mathbf{A}}^{2d+1})$, and hence the flow of $H(X)$ on an arbitrary contact variety is not incompressible. Indeed, let ${\operatorname{Eu}}$ be the weighted Euler vector field on ${\mathbf{A}}^{2d+1}$ assigning weights $|x_i|=1=|y_i|$ and $|t|=2$, i.e., ${\operatorname{Eu}}= 2t
\frac{\partial}{\partial t} + \sum_i x_i \frac{\partial}{\partial
x_i} + y_i \frac{\partial}{\partial y_i}$. Then, we have $\pi =
-\sum_i \frac{\partial}{\partial x_i} \wedge
(\frac{\partial}{\partial y_i} - x_i \frac{\partial}{\partial t})$ and $u = \frac{\partial}{\partial t}$. In this case, $\xi_1 =
\frac{\partial}{\partial t}$, $\xi_{x_i} = -\frac{\partial}{\partial
y_i} + x_i \frac{\partial}{\partial t}$, $\xi_{y_i} =
\frac{\partial}{\partial x_i}$, and $\xi_{t} = - \sum_i x_i \frac{\partial}{\partial x_i}$. In particular, the Lie algebra $H(X)$ does not preserve any volume form (if it did, for this form to be preserved by $\xi_1, \xi_{x_i}$, and $\xi_{y_i}$, it would have to preserve the constant vector fields, and hence the form would have to be the standard volume form, i.e., the one determined by the contact structure; however this form is not preserved by $\xi_t$.)
Finally, note that, in the above case, $P({\mathbf{A}}^{2d+1})$, the Lie algebra of all vector fields that commute with both $\pi$ and $u$, is the subspace of Hamiltonian vector fields $\xi_f$ where $f$ is independent of $t$. So $P({\mathbf{A}}^{2d+1}) \subsetneq H({\mathbf{A}}^{2d+1})$. This still flows transitively, since it includes the constant vector fields as above. As a result, for arbitrary odd-dimensional contact varieties, $P(X) \subsetneq H(X)$. In fact, $P(X)$ does flow incompressibly, since it preserves the standard volume form (it is clear that it preserves the inverse top polyvector field, $\pm \pi^{\wedge (\dim X - 1)/2} \wedge u$).
\[ex:st-lcs\] By the Darboux theorem, every locally conformally symplectic variety $X$ of dimension $2d$ has the form, in a formal neighborhood of a point $x \in X$, $\omega = e^f \omega_0$ and $\phi = df$, where $\omega_0$ is the standard symplectic form on $\hat {\mathbf{A}}^{2d} \cong
\hat X_x$. In this case $\pi = e^{-f} \pi_0$ where $\pi_0$ is the standard Poisson bivector on ${\mathbf{A}}^{2d}$, and $u=\pi(df)$ is $e^{-f}$ times the Hamiltonian vector field of $f$ under the standard symplectic structure. Thus, $H(\hat X_x)$ is identical with the Lie algebra of Hamiltonian vector fields preserving the standard symplectic form $\omega_0$ (in this formal neighborhood), so it flows incompressibly. However, as noted above, $H(\hat X_x) \not \subseteq P(\hat X_x)$. In fact, in this case, as in the case of odd-dimensional contact varieties, $P(\hat X_x) \subsetneq H(\hat X_x)$. Indeed, $P(\hat
X_x)$ consists of $\xi_g$ such that $u(g)=0$, i.e., $\{f,g\}=0$.
We see as a consequence of the above that, in general, the leaves of $H(X)$ consist of odd-dimensional contact varieties and locally conformally symplectic varieties. The former are not incompressible (without passing to an infinitesimal neighborhood), whereas the latter are. As a consequence, we conclude from Corollary \[c:inc\] that
Let $X$ be a Jacobi variety. Then $X = X_{H(X)}$ if and only if the generic rank of $H(X)$ is even on each irreducible component.
(Recall from Definition \[d:supp\] that $X_{H(X)}$ is the support of $M(X,H(X))$ on $X$.)
Here is an example of a Jacobi variety where there is an odd-dimensional leaf having an infinitesimal neighborhood which is incompressible. Let $X = {\mathbf{A}}^2$ with $\pi = -x \partial_x
\wedge \partial_t$ and $u = \partial_t$. Then $H(X)$ has rank two except along $x=0$, where it has rank one. Moreover, the distribution $\phi := \partial_x(\delta_{x=0}) \boxtimes dt$ is preserved by $H(X)$: for $\xi_{x^it^j}$ with $i\geq 2$ this clearly annihilates $\phi$; then $\xi_{xt^j} = jx^2t^{j-1} \partial_x$ and $\xi_{t^j} = jxt^{j-1} \partial_x + t^j \partial_t$ also do (recall that the action of differential operators on distributions is a right action; the action of vector fields is given by $(\psi \cdot
\xi)(f) = \psi(\xi(f))$ for $\psi$ a distribution and $f$ a function). The final vector field, $\xi_{t^j}$, can alternatively be rewritten in $H(X) \cdot {\mathcal{O}}_X$ as $$\xi_{t^j} = j(x\partial_x - 1) t^{j-1} + \partial_t \cdot t^j,$$ and note that $x\partial_x - 1$ and $\partial_t$ annihilate $\phi$, which implied that $\xi_{t^j}$ does.
Let $X_{\text{even}}$ be the closure of the locus where the rank of ${\mathfrak{v}}$ is even. Then, is the set-theoretic support, $(X_{H(X)})_{\text{red}}$, of $(X,H(X))$ equal to $X_{\text{even}}$? If the answer is negative, is there an example where $H(X)$ has everywhere odd rank, but $M(X,H(X)) \neq 0$?
Varieties with a top polyvector field {#ss:vtop}
-------------------------------------
Motivated by the idea that a Poisson structure is a singular and/or degenerate generalization of a symplectic structure, we define a similar analogue of Calabi-Yau structures, and their associated Lie algebras of incompressible vector fields. These are also motivated by the relationship between incompressibility and holonomicity.
In the Poisson case, one replaces a nondegenerate two-form by a possibly degenerate two-bivector, which in the nondegenerate case is inverse to the symplectic form. Thus, by analogy, we replace a volume form by a top polyvector field, which is allowed to vanish on some locus. On the nondegenerate, smooth locus, one recovers a symplectic form by taking the inverse of the polyvector field.
Specifically, let $X$ be an affine variety of dimension $n$ equipped with a global top polyvector field, i.e., a multiderivation $\Xi:
\wedge^{n} {\mathcal{O}}_X \to {\mathcal{O}}_X$. Then, as in the Poisson case, there are three natural Lie algebras to consider: the Lie algebra $H_\Xi(X)$ of vector fields obtained by contracting $\Xi$ with exact $(n-1)$-forms; the Lie algebra $LH_\Xi(X)$ of vector fields obtained by contracting $\Xi$ with closed $(n-1)$-forms; and the Lie algebra $P_\Xi(X)$ of all incompressible vector fields, i.e., vector fields $\xi$ such that $L_\xi(\Xi) = 0$ (vector fields *preserving* $\Xi$). Note that, in this case, when $X$ is irreducible and $\Xi$ is nonzero, it is immediate that all three flow incompressibly on $X$.
\[ex:cy2\] As in Example \[ex:cy\], if $X$ is affine Calabi-Yau and $\Xi$ is the inverse of the volume form, then all three Lie algebras coincide and equal the Lie algebra of volume-preserving vector fields. Then $M(X,H(X)) = \Omega_X$ and $\pi_* M(X,H(X)) \cong H_{DR}^{\dim X - *}(X)$.
As for generically symplectic varieties with their associated (locally) Hamiltonian vector fields, for arbitrary irreducible $(X,\Xi)$ with $\Xi \neq 0$, one has $LH_\Xi(U) / H_\Xi(U) \cong
H^{\dim X - 1}_{DR}(U)$ for all open affine $U \subseteq X$. Moreover, when $U$ is additionally smooth, $LH_\Xi(X)$ coincides with those vector fields which, in formal neighborhoods of all $x
\in U$, are Hamiltonian.
As in the Poisson case (see Example \[ex:fn-sing-poiss\]), in the formal neighborhood of a singular point $x \in X$, not all locally Hamiltonian vector fields need be Hamiltonian, since $H^{\dim X -
1}_{DR}(\hat X_x)$ need not vanish. Indeed, as in Example \[ex:fn-sing-poiss\], when $X = {\mathbf{A}}^1 \times Z$ where $Z$ is a complete intersection with an isolated singularity at $z \in Z$, then $\dim H^{\dim X - 1}_{DR}(\hat X_{(t,z)}) = \mu_z-\tau_z$, which need not be zero (already for the case of a hypersurface in ${\mathbf{A}}^n$). Then, equipped with the polyvector field $\Xi_{{\mathbf{A}}^1}
\boxtimes \Xi_Z$ where $\Xi_Z$ is as in Example \[ex:cy-cplte-int\] (which in the case $Z = \{Q=0\} \subseteq
{\mathbf{A}}^n$ is $\Xi_{{\mathbf{A}}^n}(dQ)$), one concludes that $LH(\hat X_{(t,z)})
/ H(\hat X_{(t,z)}) \cong H^{\dim X -1}(\hat X_{(t,z)}) \neq 0$.
As in the Poisson case, these are Lie algebras since $[LH(X), LH(X)]
\subseteq H(X)$, as we explain. Given a $(n-2)$-form (modulo torsion) $\alpha \in \tilde \Omega^{n-2}_X$, let $\xi_{\alpha} := \Xi(d\alpha)$ be its associated Hamiltonian vector field. Similarly, given a closed $(n-1)$-form modulo torsion, $\gamma \in \tilde \Omega^{n-1}_X$, let $\eta_\gamma := \Xi(\gamma)$ be its associated locally Hamiltonian vector field. Then the fact that $[LH(X), LH(X)] \subseteq H(X)$ follows from the formula, where $\alpha$ and $\beta$ are closed $(n-1)$-forms modulo torsion, $$\label{e:toppol-lie}
[\eta_\alpha, \eta_\beta] = \xi_{i_{\eta_\alpha}(\beta)},$$ which can be verified in a formal neighborhood of a smooth point of $X$ where $\Xi$ is nonvanishing, and hence which holds globally.
As in the Poisson case (Proposition \[p:poiss-hlh\]), $H(X)$ and $LH(X)$ define the same ${\mathcal{D}}$-modules on $X$:
\[p:vtop-hlh\] The ${\mathcal{O}}$-saturations are equal: $H(X)^{os} = LH(X)^{os}$. Thus, $M(X,H(X))
\cong M(X,LH(X))$.
Given a closed $n-1$ form $\alpha = \sum_i f_i d \beta_i$, we see that $\eta_\alpha = \sum_i \eta_\beta \cdot f_i$, since $\sum_i
\eta_\beta(f_i) = \Xi(d\alpha) = 0$. Thus, $LH(X) \cdot {\mathcal{O}}_X
\subseteq H(X) \cdot {\mathcal{O}}_X$, and the proposition follows since $H(X)
\subseteq LH(X)$.
Next, we compute the leaves of $H_\Xi(X)$ and of $LH_\Xi(X)$. All non-open leaves turn out to be points. We will use a general
Given a Lie algebra of vector fields ${\mathfrak{v}}$ on $X$, the *degenerate locus* of ${\mathfrak{v}}$ is the locus of $x \in X$ such that ${\mathfrak{v}}|_x \neq T_x X_{\text{red}}$.
Note that the degenerate locus includes the singular locus of $X_{\text{red}}$ (which equals $X$ in this subsection, although the preceding definition makes sense more generally).
If $X$ is irreducible, then we claim that the degenerate locus is the same as the locus of $x$ such that $\dim {\mathfrak{v}}|_x < \dim X$, i.e., such that ${\mathfrak{v}}$ does not have maximal rank. Thus, in terms of Proposition \[p:decomp\], the degenerate locus is the union of $X_i$ for $i < \dim X$. To prove the claim, we only have to show that, along the singular locus, the rank of ${\mathfrak{v}}$ is strictly less than $\dim X$. This is true at generic singular points, where the singular locus is smooth, since ${\mathfrak{v}}$ must be parallel to the singular locus. Then, the result follows for the entire singular locus, by replacing $X$ by its singular locus and inducting on the dimension of $X$.
Now return to our assumption that $(X,\Xi)$ is a variety with a top polyvector field $\Xi$. For ${\mathfrak{v}}= H_\Xi(X), LH_\Xi(X)$, or $P_\Xi(X)$, it is clear that the degenerate locus is the union of the singular locus with the vanishing locus of $\Xi$. We will also call this *the degenerate locus of $\Xi$*.
\[t:inc-leaves\] Let $(X,\Xi)$ be a variety equipped with a top polyvector field. If ${\mathfrak{v}}:= H_\Xi(X)$ or $LH_\Xi(X)$, then every degenerate point is a (zero-dimensional) leaf. That is, ${\mathfrak{v}}|_x \neq T_x X$ implies ${\mathfrak{v}}|_x = 0$.
We remark that the theorem is in stark contrast to the previous subsections, where in general there can exist leaves of positive dimension less than the dimension of $X$. For surfaces, where $\Xi$ is the same as a Poisson structure, the theorem reduces to the statement that all symplectic leaves have dimension zero or two.
It suffices to show that $\Xi$ vanishes on the singular locus of $X$. Let $Z$ be an irreducible component of the singular locus. Then $\dim Z < \dim X$, and ${\mathfrak{v}}$ is parallel to $Z$. Hence, $(\wedge^{\dim Z} {\mathfrak{v}})|_Z = 0$ (this holds at smooth points of $Z$, hence generically on $Z$, and hence on all of $Z$).
\[c:inc-degloc\] For $(X,{\mathfrak{v}})$ as in the theorem, assuming also that $X$ is purely of positive dimension, the following are equal:
1. The degenerate locus of ${\mathfrak{v}}$;
2. The set-theoretic support of the ideal generated by ${\mathfrak{v}}({\mathcal{O}}_X)$;
3. The set of points $x$ such that $(\hat {\mathcal{O}}_{X,x})_{{\mathfrak{v}}} \neq 0$.
It is easy to see that (ii) and (iii) coincide with the vanishing locus of ${\mathfrak{v}}$ since $X$ is positive-dimensional. The theorem implies that this coincides with (i).
\[c:vtop-leaves\] For $(X,{\mathfrak{v}})$ as in the theorem, $X$ is the union of finitely many open leaves and the degenerate (set-theoretic) locus of $\Xi$. There are finitely many leaves if and only if the degenerate locus is finite.
The connected components of the open locus where ${\mathfrak{v}}|_x = T_x X$ are the open leaves (of which there are finitely many), and the vanishing locus of ${\mathfrak{v}}|_x$ is the union of all points which are leaves. By the theorem, the union of these is all of $X$.
\[c:vtop-hol\] Let ${\mathfrak{v}}:= H_\Xi(X)$ or $LH_\Xi(X)$. Then, the following are equivalent:
- $({\mathcal{O}}_X)_{{\mathfrak{v}}}$ is finite-dimensional;
- The degenerate locus of $\Xi$ is finite;
- ${\mathfrak{v}}$ is holonomic.
By the corollary, $X$ has finitely many leaves if and only if it has finitely many zero-dimensional leaves. Since zero-dimensional leaves are automatically incompressible, this shows that (ii) and (iii) are equivalent. Moreover, since zero-dimensional leaves always support linearly independent evaluation functionals in $(({\mathcal{O}}_X)_{{\mathfrak{v}}})^*$, (i) implies (ii) and (iii). The implication (iii) $\Rightarrow$ (i) is immediate.
Note that, in contrast to $H_\Xi(X)$ and $LH_\Xi(X)$, $P_\Xi(X)$ can be holonomic even without having finitely many leaves (e.g., in the case when $\Xi = 0$, this happens if and only if $X$ has finitely many exceptional points).
One example of a variety with a top polyvector field is an even-dimensional (affine) Poisson variety, with $\Xi = \pi^{\wedge \dim
X/2}$, for $\pi$ the Poisson bivector field. Note that $P_\Xi(X)
\supseteq P_{\pi}(X)$. We claim that this is a proper containment if and only if $\dim X > 2$. For $\dim X = 2$ it is clear these are equal. Otherwise, since $\Xi \neq 0$ if and only if $\pi$ is generically symplectic, passing to a formal neighborhood of a point, the statement reduces to the case $X={\mathbf{A}}^{2n}$ with $n > 1$ and the usual symplectic structure, where it is well-known and easy to check.
As noted in example \[ex:cy\], if $X$ is a symplectic variety, then in particular it is Calabi-Yau and $M(X,H_\pi(X)) =
M(X,H_\Xi(X))=\Omega_X$, whether we use the Poisson bivector $\pi$ or the top polyvector field $\Xi = \wedge^{\dim X/2} \pi$ (cf. Example \[ex:cy2\]). However, for general Poisson varieties, again setting $\Xi = \wedge^{\dim X/2}$, this does not hold. For example, if the Poisson bivector field $\pi$ has generic rank two and $\dim X \geq 4$, then the top exterior power, $\Xi = \pi^{\wedge
(\dim X / 2)}$, is zero, so $H_\Xi = LH_\Xi = 0$, and $P_\Xi =
{\operatorname{Vect}}(X)$, but this is clearly not true of $H_\pi, LH_\pi$, and $P_\pi$, and the coinvariants will differ in general.
\[ex:cy-cplte-int\] Generalizing Example \[ex:poiss-cplte-int\], we can let $(Y,\Xi_Y)$ be any $n$-dimensional variety with a top polyvector field, and let $X=Z(f_1,\ldots,f_{k}) \subseteq Y$ be a complete intersection. Then we can set $\Xi_X = i_{\Xi_Y}(df_1 \wedge \cdots \wedge df_{k})$, which is a top polyvector field on $X$. Then, by Corollary \[c:vtop-hol\], $H(X)$ is holonomic if and only if $X$ has only isolated singularities, and the degenerate locus of $Y$ meets $X$ at only finitely many points. In this case, we explicitly compute $({\mathcal{O}}_X)_{H(X)}$ in §\[s:cy-cplte-int-is\].
\[r:inc-prod\] Unlike Example \[ex:poiss-prod\], a product formula does not hold for the above Lie algebras of vector fields on $X \times Y$, when $X$ and $Y$ are equipped with top polyvector fields $\Xi_X$ and $\Xi_Y$ and $X \times Y$ is equipped with the tensor product $\Xi_X \boxtimes \Xi_Y$. First of all, for the Lie algebras $P$, note that, in general, $$P(X \times Y) \not \subseteq (P(X) \boxtimes {\mathcal{O}}_{Y}) \oplus ({\mathcal{O}}_{X}
\boxtimes P(Y)).$$ For example, when $X$ and $Y$ admit vector fields ${\operatorname{Eu}}_X, {\operatorname{Eu}}_Y$ such that $L_{{\operatorname{Eu}}_X}(\Xi_X)=\Xi_X$ and $L_{{\operatorname{Eu}}_Y}(\Xi_Y)=\Xi_Y$, then ${\operatorname{Eu}}_X - {\operatorname{Eu}}_Y$ is in the LHS but not the RHS above. (This holds, for example, when $X$ and $Y$ are conical with top polyvector fields $\Xi_X$ and $\Xi_Y$ which are homogeneous of nonzero weight under the scaling action, replacing the standard Euler vector fields by suitable nonzero multiples).
Using this, one can see that a product formula does not hold for coinvariants: suppose $({\mathcal{O}}_X)_{P(X)} \ncong
({\mathcal{O}}_X)_{{\operatorname{Vect}}(X)}$. Suppose that $\xi \in {\operatorname{Vect}}(X)$ is a vector field such that $\xi({\mathcal{O}}_X) \not \subseteq P(X)({\mathcal{O}}_X)$ and $L_\xi(\Xi_X) =
\Xi_X$. Then $P(X \times X)({\mathcal{O}}_{X \times X})$ contains $(\xi
\boxtimes 1 - 1 \boxtimes \xi)({\mathcal{O}}_X \boxtimes 1) = \xi({\mathcal{O}}_X)$, but this is not contained in $(P(X)({\mathcal{O}}_X) \boxtimes {\mathcal{O}}_X) + ({\mathcal{O}}_X
\boxtimes P(X)({\mathcal{O}}_X))$. Since also $P(X \times X)$ contains horizontal and vertical vector fields, $P(X) \boxtimes 1$ and $1
\boxtimes P(X)$, we conclude that $({\mathcal{O}}_{X \times X})_{P(X \times X)}$ is quotient of $({\mathcal{O}}_X)_{P(X)}^{\boxtimes 2}$ by a nontrivial vector subspace. For an explicit example, we could let $X$ be the cuspidal curve $x^2=y^3$ in the plane ${\mathbf{A}}^2$. Then, $P(X) = \langle 2x \partial_y +
3y^2 \partial_x \rangle$ and hence $({\mathcal{O}}_X)_{P(X)}$ surjects (in fact isomorphically by a special case of Corollary \[c:is-qh\]; cf. Remark \[r:is-qh\]) to $({\mathcal{O}}_X)/(2x,3y^2)$, which is two-dimensional; on the other hand, since ${\operatorname{Vect}}(X)$ contains the Euler vector field $3x\partial_x +2 y \partial_y$, $({\mathcal{O}}_X)_{{\operatorname{Vect}}(X)}
= ({\mathcal{O}}_X)/(x,y)$ is one dimensional. In particular, in this case, $({\mathcal{O}}_{X^2})_{P(X^2)}$ is two-dimensional, whereas $({\mathcal{O}}_X)_{P(X)}^{\otimes 2}$ is four-dimensional.
For the Lie algebras of Hamiltonian and locally Hamiltonian vector fields, let $(Y,\Xi_Y)$ be any (affine) variety with $({\mathcal{O}}_Y)_{H(Y)} =
0$ (by Corollary \[c:inc-degloc\] and Example \[ex:cy2\], this is equivalent to $Y$ being Calabi-Yau with $H^{\dim Y}(Y) = 0$), and $(X,\Xi_X)$ be a positive-dimensional (affine) variety. Then, we claim that $({\mathcal{O}}_{X \times Y})_{H(X \times Y)} \cong ({\mathcal{O}}_X)/(H(X) \cdot
{\mathcal{O}}_X) \boxtimes {\mathcal{O}}_{Y}$, where now $(H(X) \cdot {\mathcal{O}}_X)$ is the *ideal* generated by $H(X) \cdot {\mathcal{O}}_X$. That is, we claim that the vector space $H(X \times Y) \cdot {\mathcal{O}}_{X \times Y}$ is $(H(X)
\cdot {\mathcal{O}}_X) \boxtimes {\mathcal{O}}_Y$.
To see this, note that the ideal $(H(X) \cdot {\mathcal{O}}_X)$ is identified with the image of the contraction of $\Xi_X$ with top differential forms on $X$. Now, on the product variety $X \times Y$, top differential forms are spanned by exterior products of top differential forms on $X$ with top differential forms on $Y$. The same is true for top polyvector fields: a derivation of ${\mathcal{O}}_X \otimes
{\mathcal{O}}_Y$ is uniquely determined by its restriction to ${\mathcal{O}}_X \otimes 1$ and $1 \otimes {\mathcal{O}}_Y$, by the formula $D(f \otimes g) = D(f) \otimes g
+ f \otimes D(g)$. Thus, skew-symmetric multiderivations of degree $\dim X + \dim Y$ are of the form $\Xi_X \boxtimes \Xi_Y$ for $\Xi_X$ and $\Xi_Y$ top polyvector fields on $X$ and $Y$, respectively.
Therefore, the contraction of top polyvector fields on $X \times Y$ with top differential forms lies in the ideal $(H(X) \cdot {\mathcal{O}}_X)
\otimes {\mathcal{O}}_Y$ (in fact, they are equal, in view of the assumption that $({\mathcal{O}}_Y)_{H(Y)} = 0$, or by the next argument). Thus we get the inclusion of $H(X \times Y) \cdot {\mathcal{O}}_{X \times Y}$ in $(H(X) \cdot
{\mathcal{O}}_X) \otimes {\mathcal{O}}_Y$.
Conversely, for any element $f \in (H(X) \cdot {\mathcal{O}}_X) \subseteq
{\mathcal{O}}_X$, suppose that $f = \Xi_X(\alpha)$ for some top differential form $\alpha$. For any $g \in {\mathcal{O}}_Y$, write $g = \Xi_Y(d\beta)$ for some $(\dim Y-1)$-form $\beta$. Then, $(f \otimes g) = (\Xi_X \wedge
\Xi_Y)(\alpha \wedge d\beta)$. Therefore, $(f\otimes g) \in H(X
\times Y) \cdot {\mathcal{O}}_{X \times Y}$. This gives the opposite inclusion.
Note that the ideal $(H(X) \cdot {\mathcal{O}}_X)$ is supported at the zero-dimensional leaves of $X$, which by Theorem \[t:inc-leaves\] is the degenerate locus of $\Xi_X$. More generally, for arbitrary $X$ and $Y$, the leaves of $H(X \times Y)$ and $LH(X \times Y)$ consist of the open leaves obtained as products of open leaves in $X$ with open leaves in $Y$, and zero-dimensional leaves at every point of the degenerate locus.
Finally, as in the Poisson case, one can also consider, for every map $f: X \to Y$, the smaller Lie algebra of vector fields obtained by contracting $\Xi$ with exact (or closed) $(n-1)$-forms pulled back from $Y$. The leaves of the resulting Lie algebra consist of open leaves, which are the restriction of the open leaves in $X$ to the noncritical locus, together with zero-dimensional leaves at the critical points of $f$ together with the degenerate locus of $\Xi$. This example includes, for every subgroup $G < {\mathsf{SL}}(n)$ (or even ${\mathsf{GL}}(n)$), the map $X = {\mathbf{A}}^n \to {\mathbf{A}}^n / G = Y$. The coinvariants of ${\mathcal{O}}_{{\mathbf{A}}^n}$ under the resulting Lie algebra is finite-dimensional if and only if the critical locus of $f$ is finite, i.e., no nontrivial element of $G$ has one as an eigenvalue; equivalently, this says that $G$ acts freely on the $2n-1$-sphere of unit vectors in ${\mathbf{C}}^n$. More generally, we can take a quotient of an arbitrary pair $(X, \Xi)$ by a finite group of automorphisms preserving $\Xi$, and the coinvariants of the resulting Lie algebra are finite-dimensional if and only if the degenerate locus of $X$ is finite and all elements of the group have only isolated fixed points.
One can alternatively consider, for a finite group quotient $X {\twoheadrightarrow}X/G$, the Lie algebras $H(X)^G$, $LH(X)^G$, and $P(X)^G$. We can do this slightly more generally, where $G$ only preserves $\Xi$ up to scaling (then $G$ still acts on $H(X), LH(X)$, and $P(X)$).
\[p:hg-vtop-lf\] Suppose $\dim X \geq 2$ and let $G$ be a finite group of automorphisms of $X$ which acts on $\Xi$ by rescaling. Let ${\mathfrak{v}}$ be $H(X)$ or $LH(X)$. Then the leaves of ${\mathfrak{v}}^G$ consist of the points of the degenerate locus of $X$, together with the connected components of the subvarieties of the open leaf whose stabilizers are fixed subgroups of $G$.
If the degenerate locus of $X$ is finite, then ${\mathfrak{v}}^G$ has finitely many leaves, and the same result holds for ${\mathfrak{v}}=P(X)$.
Call a subgroup $K < G$ *parabolic* if it occurs as the stabilizer of a point in $X$, i.e., it is the stabilizer of one of the leaves of ${\mathfrak{v}}^G$.
It is clear that ${\mathfrak{v}}^G$ must flow parallel to the given subvarieties. Therefore, since $H(X) \subseteq LH(X)$, we only have to show that $H(X)^G$ flows transitively along each of the given subvarieties. Also, the last statement is immediate from this, the fact that $P(X)$ preserves the given subvarieties (since the degenerate locus is finite, it cannot flow along it), and $H(X)
\subseteq P(X)$.
Since $H(X)$ is ${\mathcal{D}}$-localizable, the same is true for $H(X)^G$, so we can remove the vanishing locus of $\Xi$ and assume that $X$ is Calabi-Yau.
Let $K < G$ be parabolic and let $Z$ be a connected component of $\{x \in X \mid {\operatorname{Stab}}_G(x) = K\}$, as mentioned in the proposition. We have to show that, for $z \in Z$, $H(X)^G$ spans $T_z Z$.
Fix $z \in Z$ and $w \in T_z Z$. We will find $\xi \in H(X)^G$ such that $\xi|_z = w$. Since $X$ is Calabi-Yau, there exists $\xi \in
H(X)$ such that $\xi|_z = w$. Let $\phi \in
\tilde \Omega_X^{\dim X-2}$ be such that $\xi=\xi_\phi$. Let $f \in
{\mathcal{O}}_X$ be such that $f(z)=1$ and $f(y)=0$ for all $y \in G \cdot z
\setminus\{z\}$, and moreover such that $df|_{G \cdot z} = 0$.
Now, consider $\eta := |K|^{-1} \sum_{g \in G}g^* \xi_{f \phi} \in
H(X)^G$. Then $(\xi_\psi)|_z = w$, as desired.
Using Theorem \[t:main1-alt\], we immediately conclude:
In the situation of Proposition \[p:hg-vtop-lf\], the coinvariants $({\mathcal{O}}(X))_{H(X)^G}$ are finite-dimensional.
Note that, when $X$ is normal and $G$ acts by automorphisms on $(X,\Xi)$ (preserving $\Xi$) with critical locus of codimension at least two, then $P(X)^G = P(X/G)$. This is because, by Hartogs’ theorem, vector fields on $X/G$ are the same as $G$-invariant vector fields on $X$, and such vector fields preserve $\Xi_X$ if and only if they preserve $\Xi_{X/G}$. In particular, we conclude in this case that $({\mathcal{O}}_{X/G})_{P(X/G)} = ({\mathcal{O}}(X))_{P(X)^G}^G$, and that this, as well as $({\mathcal{O}}_X)_{P(X)^G}$ itself, are finite-dimensional if and only if the degenerate locus of $X$ is finite. Moreover, $M(X/G,P(X/G)) \cong q_* M(X,P(X)^G)^G$, where $q: X \to X/G$ is the projection. We caution, however, that $H(X/G)$ and $LH(X/G)$ are in general much smaller than $P(X/G)$ (even for $X$ Calabi-Yau), owing to the fact that $G$-invariant $k$-forms on $X$ do not in general descend to $k$-forms on $X/G$ when $k > 1$. In fact, by Theorem \[t:inc-leaves\], $({\mathcal{O}}_{X/G})_{H(X/G)}$ and $({\mathcal{O}}_{X/G})_{LH(X/G)}$, as well as $({\mathcal{O}}_X)_{H(X/G)}$ and $({\mathcal{O}}_X)_{LH(X/G)}$, are finite-dimensional if and only if the critical locus of $G$ is *finite* and $X$ has a finite degenerate locus.
Divergence functions and incompressibility {#s:div}
------------------------------------------
The preceding example can be generalized to the setting of degenerate versions of *multivalued* volume forms (i.e., Calabi-Yau structures) rather than of ordinary volume forms. We formulate this in terms of *divergence functions*, which also yield an alternative definition of incompressibility (Proposition \[p:reinc\]).
We assume throughout that $X$ is irreducible and reduced. Recall the definitions of polyvector fields $T_X^\bullet$ and differential forms $\Omega_X^\bullet$ and $\tilde \Omega_X^\bullet$ from §\[s:gen\].
\[d:df\] Let $N \subseteq T_X$ be an ${\mathcal{O}}_X$-submodule. A *divergence function* $D$ on $N$ is a morphism of sheaves of vector spaces $D: N \to {\mathcal{O}}_X$ satisfying $D(f \xi) = f D(\xi) + \xi(f)$ for all $\xi \in N$ and $f \in {\mathcal{O}}_X$. When $N = T_X$, we call this a divergence function on $X$.
As we will explain, divergence functions should be viewed as a degenerate, *multivalued* version of Calabi-Yau structures: they simultaneously generalize flat sections of flat connections on the canonical bundle (which includes volume forms), discussed in Example \[ex:mvvol\], and top polyvector fields on possibly singular schemes of finite type, discussed in §\[ss:vtop\].
For the latter, given $(X, \Xi)$, we let $N \subseteq T_X$ be the submodule of $\xi \in T_X$ such that $L_{\xi}(\Xi)$ is a multiple of $\Xi$. This is a submodule in view of the identity $L_{f \xi}(\Xi) =
f L_{\xi}(\Xi) - \xi(f) \cdot \Xi$, which can be checked in local formal coordinates where $\Xi$ is nondegenerate (where we can take $\Xi$ to be the inverse to the standard volume form on the formal neighborhood of the origin in affine space). Next, define $D$ by the formula $D(\xi)\cdot \Xi = -L_{\xi}(\Xi)$. Note that, on the nondegenerate locus of $\Xi$, call it $X^\circ \subseteq X$, we have $N|_{X^\circ} = T_{X^\circ}$, since $X^\circ$ is symplectic.
Next, we explain how divergence functions generalize multivalued volume forms:
\[p:div-conncan\] If $X$ is normal and of pure dimension $n$, then the following are in natural bijection:
1. Divergence functions $D$ on $N \subseteq T_X$;
2. Connections $N \times \tilde \Omega^{n}_X \to \tilde \Omega_{X}^n$ on $\tilde \Omega^{n}_X$ along $N$.
3. Connections $N \times T^{n}_X \to T^{n}_X$ on $T^{n}_X$ along $N$.
The equivalence between (i) and (ii) is given by the correspondences, for $\xi \in N$ and $\omega \in \tilde \Omega_X^{n}$, $$\begin{gathered}
D \mapsto \nabla^D, \quad \nabla^D_\xi(\omega) = L_\xi(\omega) - D(\xi) \cdot \omega; \\
\nabla \mapsto D_\nabla, \quad D_\nabla(\xi) = L_\xi - \nabla_\xi \in {\operatorname{End}}_{{\mathcal{O}}_X}(\tilde \Omega_X^{n}) \cong {\mathcal{O}}_X.\end{gathered}$$ The equivalence between (i) and (iii) is given by the formulas, for $\xi \in N$ and $\Xi \in T_X^{n}$, $$\begin{gathered}
D \mapsto \nabla^D, \quad \nabla^D_\xi(\Xi) = L_\xi(\Xi) + D(\xi) \cdot \Xi; \\
\nabla \mapsto D_\nabla, \quad D_\nabla(\xi) = \nabla_\xi - L_\xi \in {\operatorname{End}}_{{\mathcal{O}}_X}(T_X^{n}) \cong {\mathcal{O}}_X.\end{gathered}$$ Finally, the constructions $D \mapsto \nabla^D$ are valid even when $X$ is not normal.
We will need the elementary
\[l:norm-tfc2\] Suppose that $X$ is normal and that $F$ is a torsion-free coherent sheaf on $X$ which is a line bundle outside of codimension two. Then ${\operatorname{End}}(F) = {\mathcal{O}}_X$.
For any $a \in {\operatorname{End}}(F)$, on some open subset $U \subseteq X$ where $F$ is a line bundle and $X
\setminus U$ has codimension at least two, $a|_U \in {\operatorname{End}}({\mathcal{O}}_U) =
\Gamma({\mathcal{O}}_U)$. By normality, this is the restriction of a function $f_a \in {\mathcal{O}}_X$. Since ${\mathcal{O}}_X \subseteq {\operatorname{End}}(F)$, we conclude that $f_a-a \in {\operatorname{End}}(F)$ has zero restriction to $U$, and hence is zero since $F$ is torsion-free.
Suppose that $D$ is a divergence function. Then $\nabla^D_\xi(f
\cdot \omega) = f\nabla^D_\xi(\omega) + \xi(f) \cdot
\omega$. Similarly, $\nabla^D_{f\xi}(\omega) = f \nabla^D(\omega) +
\xi(f)-\xi(f) = f \nabla^D(\omega)$. We deduce that $\nabla^D_\xi$ is a connection. Similarly, if $\nabla$ is a connection on $\tilde
\Omega_X$, then first of all $L_\xi(f\omega) - \nabla_{\xi}(f\omega)
= f \bigl( L_\xi(\omega) - \nabla_\xi(\omega) \bigr)$, so $D_\nabla(\xi)$ is indeed a well-defined ${\mathcal{O}}_X$-module endomorphism of $\tilde \Omega_X$. By Lemma \[l:norm-tfc2\], this is the same as an element of ${\mathcal{O}}_X$. Then, $D_\nabla(f\xi) = f
D_\nabla(\xi) + \xi(f)$, so $D_\nabla$ is a divergence function. One immediately checks that $D_{\nabla^D} = D$ and $\nabla^{D_\nabla} = \nabla$.
The proof of the equivalence between (i) and (iii) is similar, so we omit the details. For the final statement, note that well-definition of $\nabla^D$ did not require normality.
In fact, in Proposition \[p:div-conncan\], we can replace $T_X^n$ and $\tilde \Omega_X^n$ by any torsion-free coherent sheaves which coincide with $T_X^n$ and $\tilde \Omega_X^n$, respectively, outside of codimension two; the proof then goes through unchanged.
For not necessarily normal $X$, but still of pure dimension $n$, Proposition \[p:div-conncan\] generalizes to give an equivalence between divergence functions of the form $D: N
\to {\operatorname{End}}_{{\mathcal{O}}_X}(\tilde \Omega_X^{n}) \supseteq {\mathcal{O}}_X$ and connections $N
\times \tilde \Omega^{n}_X \to \tilde \Omega^{n}_X$. Similarly, we obtain an equivalence between divergence functions valued in ${\operatorname{End}}_{{\mathcal{O}}_X}(T_X^{n}) \supseteq {\mathcal{O}}_X$ and connections on $T_X^{n}$ along $N$.
Divergence functions yield the following alternative formulation of the incompressibility condition. Let ${\mathcal{O}}_X \cdot {\mathfrak{v}}$ denote the ${\mathcal{O}}_X$-linear span of ${\mathfrak{v}}$ and similarly for ${\mathcal{O}}_{X'}$ where $X'$ is an open subvariety of $X$ (we will also use this notation for formal neighborhoods, etc.).
\[p:reinc\] Let $X$ be an arbitrary affine variety and ${\mathfrak{v}}$ a Lie algebra of vector fields ${\mathfrak{v}}$ on $X$. Then, the flow of ${\mathfrak{v}}$ along $X$ is incompressible if and only if there exists an open dense subset $X^\circ \subseteq X$ and a divergence function on ${\mathcal{O}}_{X^\circ}
\cdot {\mathfrak{v}}|_{X^\circ}$ annihilating ${\mathfrak{v}}|_{X^\circ}$. In this case, in the formal neighborhood of every point of $X^\circ$, there exists a volume form preserved by ${\mathfrak{v}}$.
We can restate this in terms of connections using:
\[p:div-lie\] In terms of Proposition \[p:div-conncan\], when $X$ is normal and of pure dimension, a divergence function $D$ on $N$ annihilates ${\mathfrak{v}}\subseteq T_X$ if and only if $\nabla^D_\xi = L_\xi$ for all $\xi \in {\mathfrak{v}}$.
The proof is immediate from the definition of $\nabla^D$ and $D_\nabla$.
### Flat divergence functions
In terms of Proposition \[p:div-conncan\], we can describe what it means for a divergence function to be flat. As before, assume that $X$ is a variety of pure dimension $n$. Assume that $N \subseteq T_X$ is a Lie subalgebroid.
Consider the extension of $D$ to an operator $\tilde D:
\wedge_{{\mathcal{O}}_X}^\bullet N \to \wedge_{{\mathcal{O}}_X}^{\bullet-1} N$ given by $$\begin{gathered}
\label{e:divfn-cplx}
\xi_1 \wedge \cdots \wedge \xi_k
\mapsto
\sum_{i=1}^k (-1)^{i-1} D(\xi_i)
\xi_1 \wedge \cdots \wedge \hat \xi_i \wedge
\cdots \wedge \xi_k
\\ + \sum_{i,j} (-1)^{i+j-1} [\xi_i, \xi_j] \wedge \xi_1 \wedge \cdots \wedge \hat \xi_i \wedge \cdots \wedge \hat \xi_j \wedge \cdots \wedge \xi_k. \end{gathered}$$ Note that, since we take the exterior algebra over ${\mathcal{O}}_X$, one must check that the formula is well-defined, i.e., that one obtains the same result if we multiply $\xi_i$ by $f$ as if we multiply $\xi_j$ by $f$, for all $i < j$ and all $f \in {\mathcal{O}}_X$. This is easy to check.
Call a divergence function $D$ flat if the associated operator has square zero: $\tilde D^2=0$.
\[ex:neqtx\] Suppose that $N=T_X$ and $X$ is smooth. Then we can replace with $$\label{e:divfn-cplx-smth}
\Omega_X^{\dim X -
\bullet} \otimes_{{\mathcal{O}}_X} T_X^n,$$ equipped with the derivation $d_D = d \otimes{\operatorname{Id}}+ {\operatorname{Id}}\otimes \bar \nabla^D$, where $\bar \nabla^D: T_X^n \to \Omega^1_X \otimes_{{\mathcal{O}}_X} T_X^n$ is the usual ${\mathbf{k}}$-linear operator associated to the connection $\nabla^D$. This is isomorphic to by contracting $\Omega^\bullet$ with $T_X^n$. Thus, $d_D^2=0$ if and only if $D$ is flat.
\[ex:nlocfree\] More generally than Example , suppose that $X$ is smooth and $N$ is locally free of rank $n-k$ and the vanishing locus of a collection of (linearly independent) one-forms $df_1, \ldots, df_k$. Then, we can consider the $k$-form $\alpha = df_1 \wedge \cdots \wedge df_k$, and replace by $$\label{e:divfn-cplx-locfree}
(\Omega_X^{\dim X - k -
\bullet} \wedge \alpha) \otimes_{{\mathcal{O}}_X} T_X^n.$$ This is equipped with the derivation $d_D$ defined as before, and with this derivation, the contraction map produces an isomorphism of with . Thus, it remains true that $d_D^2=0$ if and only if $D$ is flat. Moreover, by Frobenius’s theorem, in a formal neighborhood of a smooth point $x \in
X$, such $f_1, \ldots, f_k$ always exist since $N$ is integrable.
\[p:flat-lie\] Let $D: N \to {\mathcal{O}}_X$ be a divergence function with $N \subseteq T_X$ a Lie subalgebroid, and let ${\mathfrak{v}}:= \{\xi \in N
\mid D(\xi)=0\}$. Suppose moreover that $N = {\mathcal{O}}_X \cdot
{\mathfrak{v}}$. Then $D$ is generically flat if and only if ${\mathfrak{v}}$ is a Lie algebra.
By generically flat, we mean that, restricted to an open dense subset of $X$, $D$ is flat. Note that the condition $N = {\mathcal{O}}_X \cdot {\mathfrak{v}}$ is automatic if we replace $X$ with a formal neighborhood $\hat X_x$ for generic $x \in X$ and define ${\mathfrak{v}}\subseteq T_{\hat X_x}$ as above, since $N$ is integrable, so we can write $\hat X_x \cong V
\times V'$ for formal polydiscs $V, V'$ such that $N$ identifies with the subsheaf of $T_{\hat X_x}$ in the $V$ direction.
First, if $D$ is generically flat, then on some open dense subset of $X$, given any $\xi, \eta \in {\mathfrak{v}}$, we have $\tilde D^2(\xi \wedge
\eta) = 0$ (since $D(\xi)=0=D(\eta)$), which implies that $[\xi,
\eta]\in {\mathfrak{v}}$ as well.
Consider now the reverse implication. It suffices to restrict to a formal neighborhood of a smooth point $x \in X$ (on each connected component of $X$). Then, as noted in Example \[ex:nlocfree\], we can assume $N$ is the vanishing locus of $k$ nonvanishing one-forms $df_1, \ldots, df_k$. Set $\alpha = df_1 \wedge \cdots \wedge df_k$ and replace by . By Proposition \[p:div-lie\], ${\mathfrak{v}}$ consists of those $\xi \in N$ such that $\nabla^D_\xi = L_\xi$ on $\Omega^n_{\hat X_x}$, or equivalently on $(\Omega^{n-k}_{\hat X_x} \wedge \alpha)$.
Assume that ${\mathfrak{v}}$ is a Lie algebra. Then, for $\xi, \eta \in
{\mathfrak{v}}$, $$[\nabla^D_\xi, \nabla^D_\eta] = [L_\xi, L_\eta] = L_{[\xi,\eta]} = \nabla^D_{[\xi, \eta]}.$$ Note that this also implies that $[\nabla^D_\xi, \nabla^D_\eta] =
\nabla^D_{[\xi, \eta]}$ for all $\xi, \eta \in \hat {\mathcal{O}}_{X,x} \cdot
N$, since this equality remains true when replacing $\xi$ by $f \cdot
\xi$ for $f \in\hat {\mathcal{O}}_{X,x}$, and it is biadditive in $\xi$ and $\eta$. Since $N = {\mathcal{O}}_X \cdot {\mathfrak{v}}$, and hence $N|_{\hat X_x} = \hat
{\mathcal{O}}_{X,x} \cdot {\mathfrak{v}}|_{\hat X_x}$, the equality holds for all $\xi,
\eta \in \hat {\mathcal{O}}_{X,x}$. Now, the identity $[\nabla^D_\xi,
\nabla^D_\eta] = \nabla^D_{[\xi, \eta]}$ on $\Omega^n_{\hat X_x}$ implies in the standard way that the derivation $d_D$ on $(\Omega_{\hat X_x}^{\dim X - k - \bullet} \wedge \alpha)
\otimes_{{\mathcal{O}}_X} T_{\hat X_x}^n$ has square zero. Namely, one can verify that $d_D^2$ is given by contraction with the two-form $\alpha$ given by $$\alpha(\xi \wedge \eta) = [\nabla^D_\xi, \nabla^D_\eta] -
\nabla^D_{[\xi, \eta]} \in {\operatorname{End}}(T_{\hat X_x}^n) = \hat
{\mathcal{O}}_{X,x}. \qedhere$$
### Hamiltonian vector fields on varieties with flat divergence functions {#ss:ham-df}
Now we define, analogously to §\[ss:vtop\], Hamiltonian and incompressible vector fields preserving flat divergence functions (i.e., preserving the formal volumes associated to them).
Let $X$ be a variety of pure dimension $n$ and $N \subseteq T_X$ an ${\mathcal{O}}_X$-submodule, and $D: N \to {\mathcal{O}}_X$ be a flat divergence function. Then first we have the Lie algebra $P(X,D) \subseteq N$ of all incompressible vector fields in $N$. Note that the ${\mathcal{O}}_X$-linear span of $P(X,D)$ need not be all of $N$.
Next, given any element $\tau \in \wedge_{{\mathcal{O}}_X}^2 N$, consider the image $\theta_\tau := \tilde D(\tau) \in N$. By construction, $D(\theta_\tau) = 0$. We call $\theta_\tau$ the Hamiltonian vector field of $\tau$. Since $[\theta_\tau, \theta_{\tau'}] =
\theta_{L_{\theta_\tau}(\tau')}$, these form a Lie subalgebra of $P(X,D)$, $$H(X,D) := \langle \theta_\tau \rangle
\subseteq P(X,D).$$
If $X$ is Calabi-Yau and $D$ is the associated divergence function, we again recover $H(X,D)=P(X,D)=H(X)$, the Lie algebra of volume-preserving vector fields.
As long as $N$ has rank at least two, then $H(X,D)$ has enough vector fields, in the sense that ${\mathcal{O}}_X \cdot H(X,D) = N$; more precisely:
\[p:div-enough-vfds\] Suppose that the image of $N$ at the tangent fiber $T_x X$ has dimension at least two. Then $H(X,D)|_x = N|_x$, i.e., $H(X,D)
\subseteq N$ spans the same tangent space at $x$ as $N$. In particular, if $N = T_X$ and $X$ has pure dimension at least two, then $H(X,D)$ is transitive.
As a consequence, the same result holds for $P(X,D) \supseteq H(X,D)$.
Let $x \in X$ be a point, and $\xi, \eta \in N$ two vector fields linearly independent at $x$. Let $f \in {\mathcal{O}}_X$ be a function such that $\xi(df)(x)=1$ and $\eta(df)(x)=0$. Then $\bigl(\tilde D(f \xi
\wedge \eta) - f \tilde D(\xi \wedge \eta)\bigr)|_x = \eta|_x$.
On the other hand, if $N$ has rank one, then $P(X,D)$ can be zero, e.g., when $X$ is a smooth curve and $D$ is a divergence function preserving a multivalued volume form which is not single valued (cf. Example \[ex:mvvol\]).
Consider the case of Example \[ex:nlocfree\], i.e., where $N$ is locally free of rank $n-k$ and the zero locus of (linearly independent) exact one-forms $df_1, \ldots, df_k$. Set $\alpha :=
df_1 \wedge \cdots \wedge df_k$ and replace by . Given any element $\beta \in (\Omega^{n-k-2}_X\wedge \alpha)
\otimes T_X^n$, we can define the Hamiltonian vector field $$\xi_{\beta} = {\operatorname{ctr}}(\nabla^D(\beta)),$$ where ${\operatorname{ctr}}$ is the operator $${\operatorname{ctr}}: \tilde \Omega^\bullet_X \otimes_{{\mathcal{O}}_X} T_X^n \to
T_X^{n-\bullet}, {\operatorname{ctr}}(\omega \otimes \tau) = i_\tau(\omega).$$ These vector fields coincide with $H(X,D)$ as defined above, since is isomorphic to via the contraction operation.
Next, call an element of $(\Omega^{n-k-\bullet}_X \wedge \alpha) \otimes_{{\mathcal{O}}_X} T_X^n$ $\nabla^D$-closed if it is in the kernel of $\nabla^D$. Then, if $\gamma \in
(\Omega^{n-k-1}_X\wedge \alpha)
\otimes T_X^n$ is $\nabla^D$-closed, we can define the locally Hamiltonian vector field $$\eta_\gamma := {\operatorname{ctr}}(\gamma).$$ These vector fields coincide with $P(X,D)$ as defined above, since via the contraction isomorphism of complexes and , the vector fields $\eta_\gamma$ are precisely those elements of $N$ with zero divergence.
Suppose $(X, \Xi)$ is a variety of pure dimension $n$ equipped with a generically nonvanishing top polyvector field $\Xi$ as in § \[ss:vtop\], and define $N$ and $D$ as at the beginning of § \[s:div\]. Then we see immediately that $P(X) = P(X,D)$, consisting of the vector fields $\xi$ such that $L_\xi \Xi = 0$. On the other hand, $H(X)$ need not equal $H(X,D)$ in general. Indeed, although it is true that, given $\alpha \in \Omega^{n-2}_X$, then $\theta_{i_\Xi(\alpha)} = \xi_{\alpha}$, we have $\wedge^2 N \neq
i_{\Xi}(d\Omega^{n-2}_X)$ in general, and sometimes $H(X) \neq
H(X,D)$.
### Proof of Proposition \[p:reinc\] {#ss:reinc-pf}
We can assume that $X=X^\circ$ is smooth and that ${\mathfrak{v}}$ has constant (i.e., maximal) rank. Therefore $\Omega_X = \tilde \Omega_X$, and we omit the tilde from now on. We show that ${\mathfrak{v}}$ flows incompressibly on $X$ if and only if there exists a connection $\nabla$ on $\Omega_X$ along $N:= {\mathcal{O}}_X \cdot {\mathfrak{v}}$ such that $\nabla_\xi=L_\xi$ for all $\xi
\in {\mathfrak{v}}$.
First, suppose that ${\mathfrak{v}}$ flows incompressibly on $X$. Let $x \in X$ be a point and $\omega \in \Omega_{\hat X_x}$ a formal volume form preserved by ${\mathfrak{v}}$. Let $\nabla$ be the unique flat connection whose flat sections are multiples of $\omega$. Then $\nabla_\xi \omega = 0 =
L_\xi \omega$ for all $\xi \in {\mathfrak{v}}$. Therefore, the restriction of $\nabla$ to $N$ is as desired.
Conversely, suppose that $\nabla$ is a connection on $\Omega_X$ along $N$ such that $\nabla_\xi = L_\xi$ for all $\xi \in {\mathfrak{v}}$. Since ${\mathfrak{v}}$ is a Lie algebra, Proposition \[p:flat-lie\] implies that $\nabla$ is generically flat. Thus, at a generic point $x \in X$, $\hat N_x$ is free over $\hat {\mathcal{O}}_{X,x}$, and we can write $T_{\hat X_x} = \hat N_x \oplus L$ for some complementary free $\hat {\mathcal{O}}_{X,x}$-submodule $L$. Then the connection $\nabla$ can be extended to a flat connection on $T_{\hat X_x}$. Let $\omega
\in\Omega_{\hat X_x}$ be a nonzero flat formal section of $\nabla$. Then $\nabla_\xi(\omega)=0$ for all $\xi \in T_X$. Hence $L_\xi(\omega)=0$ for all $\xi \in {\mathfrak{v}}$. Therefore, $\omega$ is preserved by ${\mathfrak{v}}$.
Smooth curves {#ss:curves}
-------------
Let $X$ be a smooth connected curve. In this section we explicitly compute $M(X,{\mathfrak{v}})$. We may assume that ${\mathfrak{v}}$ is nonzero. Let $Z
\subseteq X$ be the vanishing locus of ${\mathfrak{v}}$, which is zero-dimensional. Let $X^\circ:= (X \setminus Z) \subseteq X$ be the complement.
If ${\mathfrak{v}}$ is one-dimensional, then $M(X,{\mathfrak{v}})|_{X^\circ} = \Omega_{X^\circ}$. Otherwise, $M(X,{\mathfrak{v}})|_{X^\circ} = 0$.
By our assumptions, ${\mathfrak{v}}|_{X^\circ}$ is transitive. Moreover, if ${\mathfrak{v}}$ is one-dimensional, then any nonzero element $\xi \in {\mathfrak{v}}$ is a top polyvector field on $X$ vanishing on $Z$, so $\xi^{-1}$ defines a nondegenerate volume form on $X^\circ$ preserved by ${\mathfrak{v}}$. Therefore we conclude that $M(X^\circ,{\mathfrak{v}})\cong\Omega_{X^\circ}$ by Proposition \[p:tr\]. On the other hand, if ${\mathfrak{v}}$ is at least two-dimensional, then if $\xi_1, \xi_2 \in {\mathfrak{v}}$ are linearly independent, then on some open subset $U \subseteq X^\circ$, $\xi_1^{-1}$ and $\xi_2^{-1}$ both define nondegenerate volume forms which are not scalar multiples of each other. Then there can be no volume form on $U$ preserved by both, even restricted to $\hat U_x$ for every $x \in U$.
If $\dim {\mathfrak{v}}\geq 2$, then $M(X,{\mathfrak{v}}) \cong
\bigoplus_{z \in Z} \delta_z \otimes (\hat
{\mathcal{O}}_{X,z})_{{\mathfrak{v}}}$. Moreover, $\dim (\hat {\mathcal{O}}_{X,z})_{{\mathfrak{v}}}$ is the minimum order of vanishing of vector fields of ${\mathfrak{v}}$ at $z$.
Note in particular that each $\dim (\hat {\mathcal{O}}_{X,z})_{{\mathfrak{v}}}$ is positive.
By the lemma, we immediately conclude that $M(X,{\mathfrak{v}})$ is a direct sum of copies of delta-function ${\mathcal{D}}$-modules at points of $Z$, which is finite. Then, the result follows from the fact that $${\operatorname{Hom}}(M(X,{\mathfrak{v}}), \delta_z) = {\operatorname{Hom}}({\mathcal{D}}_X, \delta_z)^{{\mathfrak{v}}} =
((\hat {\mathcal{O}}_{X,z})^*)^{\mathfrak{v}}. \qedhere$$
Now, assume that ${\mathfrak{v}}= \langle \xi \rangle$, so that $M(X,{\mathfrak{v}}) =
\xi \cdot {\mathcal{D}}_X \setminus {\mathcal{D}}_X$ for $\xi \in {\operatorname{Vect}}(X)$. Then, by the lemma and the argument of the proposition, we have an exact sequence $$0 \to j_! \Omega_{X^\circ} = \Omega_X {\hookrightarrow}M(X,{\mathfrak{v}}) {\twoheadrightarrow}\bigoplus_{z \in Z}
\delta_z \otimes (\hat
{\mathcal{O}}_{X,z})_{{\mathfrak{v}}} \to 0.$$ It turns out that this sequence is maximally nonsplit. Namely, at each $z \in Z$, ${\operatorname{Ext}}(\delta_z, \Omega_X) = {\mathbf{k}}$, since $X$ is a smooth curve.
When ${\mathfrak{v}}$ is one-dimensional, then $M(X,{\mathfrak{v}}) = N \oplus
\bigoplus_{z \in Z} \delta_z \otimes (\hat {\mathcal{O}}_{X,z})_{{\mathfrak{v}}}/{\mathbf{k}}$, where $N$ is an indecomposable ${\mathcal{D}}$-module fitting into an exact sequence $$0 \to j_! \Omega_{X^\circ} = \Omega_X {\hookrightarrow}N
{\twoheadrightarrow}\bigoplus_{z \in Z} \delta_z \to 0.$$ As before, $\dim (\hat {\mathcal{O}}_{X,z})_{{\mathfrak{v}}}$ is the minimum order of vanishing of vector fields of ${\mathfrak{v}}$ at $z$.
By formally localizing at $z \in Z$, it is enough to assume that ${\mathfrak{v}}= \langle x^k \partial_x \rangle$ for ${\mathbf{A}}^1 = {\operatorname{\mathsf{Spec}}}{\mathbf{k}}[x]$ and $k \geq 1$. In this case, it suffices to prove that $${\operatorname{Hom}}({\mathcal{D}}_{{\mathbf{A}}^1} / x^k \partial_x \cdot {\mathcal{D}}_{{\mathbf{A}}^1}, \Omega_{{\mathbf{A}}^1}) = 0.$$ But, no volume form on ${\mathbf{A}}^1$ is annihilated by $L_{x^k \partial_x}$ (even in a formal neighborhood of zero): the rational volume form annihilated by $L_{x^k \partial_x}$ is $x^{-k} dx$. The last statement follows as in the previous proof.
Finite maps {#ss:fmaps}
-----------
Let $f: X \to Y$ be a finite surjective map of affine varieties. In this section we explain how to construct more examples using finite maps, which generalizes the aforementioned Lie algebras of Hamiltonian vector fields of Hamiltonians pulled back from $Y$. We will not need the material of this section for the remainder of the paper.
Let ${\operatorname{Vect}}_X(Y) \subseteq {\operatorname{Vect}}(Y)$ be the subspace of vector fields $\xi$ on $Y$ such that there exists a vector field $f^* \xi$ on $X$ such that $f_* (f^* \xi|_x) = \xi|_{f(x)}$ for all $x \in X$.
Algebraically, ${\operatorname{Vect}}_X(Y)$ consists of the derivations of ${\mathcal{O}}_Y$ which extend to derivations of ${\mathcal{O}}_X$.
Since $f$ is finite and $X$ and $Y$ are reduced, it is generically a covering map. Therefore, when $f^* \xi$ exists, it is unique.
\[ex:normcrit2\] If $X$ is a normal variety and the critical locus of $f$ has codimension at least two, then by Hartogs’ theorem, vector fields on $X$ outside the singular and critical locus extend to all of $X$. Therefore, ${\operatorname{Vect}}_X(Y) = {\operatorname{Vect}}(Y)$, since $f$ is a covering map when restricted to this latter locus.
Suppose that $X$ and $Y$ are varieties and ${\mathfrak{v}}_Y \subseteq {\operatorname{Vect}}_X(Y)$. Let ${\mathfrak{v}}_X := f^* {\mathfrak{v}}_Y$.
- $(X,{\mathfrak{v}}_X)$ has finitely many leaves if and only if $(Y,{\mathfrak{v}}_Y)$ does.
- $(X,{\mathfrak{v}}_X)$ has finitely many incompressible leaves of and only if $(Y,{\mathfrak{v}}_Y)$ does.
- $(X,{\mathfrak{v}}_X)$ has finitely many zero-dimensional leaves if and only if $(Y,{\mathfrak{v}}_Y)$ does.
Restricted to any invariant subvariety $Z \subseteq X$, $f$ is still finite and therefore generically a covering map. This reduced the statement to the case where $f$ is a covering map of smooth varieties. Then, the statements (i)–(ii) follow from the basic facts that (i) $X$ is generically transitive if and only if $Y$ is; (ii) $X$ is incompressible if and only if $Y$ is. Statement (iii) follows from the fact that $f$ restricts to a finite map from the vanishing locus of ${\mathfrak{v}}_X$ onto the vanishing locus of ${\mathfrak{v}}_Y$.
In the situation of Example \[ex:normcrit2\], $X$ has finitely many leaves under the flow of all vector fields if and only if the same is true of $Y$, and $X$ has finitely many exceptional points if and only if $Y$ does. Thus, $({\mathcal{O}}_X)_{{\operatorname{Vect}}(X)}$ is finite-dimensional if and only if $({\mathcal{O}}_Y)_{{\operatorname{Vect}}(Y)}$ is.
If $f: X \to Y$ is a finite Poisson map of varieties with finitely many symplectic leaves and $X$ is normal, one recovers the observation at the end of §\[ss:poisson\] in the setting of Poisson maps (note that the critical locus of $f$ is automatically of codimension at least two, since $f$ is nondegenerate over the open leaves of $Y$). Thus, one recovers [@ESdm Theorem 3.1] in this setting, i.e., that $f^*H(Y)$ is holonomic (similarly one obtains that $f^*LH(Y)$ and $f^*P(Y)$ are holonomic). Here, we only used the conditions that $X$ is normal and $Y$ has finitely many symplectic leaves to assure that $H(Y) \subseteq {\operatorname{Vect}}_Y(X)$; to drop these assumptions, one can observe that $H(Y) \subseteq
{\operatorname{Vect}}_Y(X)$, since $f^* \xi_h = \xi_{f^* h}$ (which also allows one to drop the condition that $Y$ is Poisson altogether, using Hamiltonian vector fields on $X$ of the form $\xi_{f^* h}$); similarly we can conclude in this setting that $LH(Y) \subseteq {\operatorname{Vect}}_Y(X)$.
Suppose $f: X \to Y$ is a finite map of varieties equipped with top polyvector fields $\Xi_X$ and $\Xi_Y$ such that $f_*(\Xi_X|_x) =
\Xi_Y|_{f(x)}$ for all $x \in X$ (an “incompressible” finite map). If $X$ is normal, $\Xi_Y$ has a finite degenerate locus, and the dimension of $X$ is at least two, one concludes that $f^* H(Y)$ is holonomic (as well as $f^* LH(Y)$ and $f^* P(Y)$), and hence that $({\mathcal{O}}_X)_{f^* H(Y)}$ is finite-dimensional; this recovers an observation at the end of §\[ss:vtop\] in a special case. As in the previous remark, we can drop the assumptions that $X$ is normal and $\Xi_Y$ has a finite degenerate locus, since those were only used to show that $H(Y) \subseteq {\operatorname{Vect}}_Y(X)$, but this is automatic since we can pull back closed $(n-1)$-forms from $Y$ to $X$ (this also applies to $LH(Y)$, but not necessarily to $P(Y)$).
In the case $Y=X/G$ where $G$ is a finite group acting on $(X,
\Xi_X)$, one similarly recovers the observation from the end of § \[ss:vtop\], that $({\mathcal{O}}_{X/G})_{P(X/G)} = ({\mathcal{O}}_X)_{f^*P(X/G)}^G =
({\mathcal{O}}_X)_{P(X)^G}^G$, as well as $({\mathcal{O}}_X)_{P(X)^G}$, are finite-dimensional if and only if $\Xi_X$ has a finite degenerate locus, i.e., if and only if $({\mathcal{O}}_X)_{P(X)}$ is finite-dimensional.
Globalization and Poisson vector fields {#s:ex-dloc}
=======================================
Hamiltonian vector fields are ${\mathcal{D}}$-localizable
---------------------------------------------------------
In order to prove that our main examples are ${\mathcal{D}}$-localizable (for all vector fields and Hamiltonian vector fields), we prove the following more general result, which roughly states that a Lie algebra of vector fields generated by a coherent sheaf $E$ of “potentials” is ${\mathcal{D}}$-localizable (in the Poisson case with ${\mathfrak{v}}=H(X)$, or in the case ${\mathfrak{v}}={\operatorname{Vect}}(X)$, $E={\mathcal{O}}_X$, as we will explain):
\[t:h-loc\] Let $E$ be a coherent sheaf on an affine variety $X$ equipped with a map $v: E \to T_X$ of ${\mathbf{k}}$-linear sheaves, such that, for all $e
\in E$, the bilinear map $$\pi_e(f,g) := v(f \cdot e)(g) - f \cdot v(e)(g)$$ defines a skew-symmetric biderivation ${\mathcal{O}}_X^{\otimes 2} \to {\mathcal{O}}_X$. Then (the Lie algebra generated by) $v(E)$ is ${\mathcal{D}}$-localizable.
The condition of the theorem can alternatively be stated as: $v: E \to
T_X$ is a differential operator of order $\leq 1$ whose principal symbol $\sigma(v): E \to T_X \otimes T_X$ is skew-symmetric.
Let $X \subseteq {\mathbf{A}}^n$ be an embedding into affine space, and let $x_1, \ldots, x_n$ be the coordinate functions on ${\mathbf{A}}^n$. Let $U \subseteq X$ be an open affine subset. We need to show that, for every $g
\in {\mathcal{O}}_{U}$ and $e \in E(X)$, then $v(g \cdot e) \in v(E(X)) \cdot
{\mathcal{D}}_U$. Let $V \subseteq {\mathbf{A}}^n$ be an open affine subset such that $V \cap X = U$. We claim that, in ${\mathcal{D}}_U$, for all $f \in {\mathcal{O}}_V$, $$\label{e:loc}
v(f \cdot e) = v(e) \cdot f + \sum_{i=1}^n \bigl(v(x_i \cdot e) - v(e) \cdot x_i \bigr) \cdot \frac{\partial f}{\partial x_i},$$ which immediately implies the statement. To prove , we first rewrite it (putting vector fields on the left-hand side and functions on the right hand side) as $$(v(f \cdot e) - f \cdot v(e)) - \sum_{i=1}^n \frac{\partial
f}{\partial x_i} \bigl(v(x_i \cdot e) - x_i v(e) \bigr) = v(e)(f) +
\sum_{i=1}^n \bigl( -\frac{\partial f}{\partial x_i} v(e)(x_i) +
(v(x_i \cdot e) - x_i v(e))(\frac{\partial f}{\partial x_i}) \bigr).$$ So the statement is equivalent to showing that both sides of the above desired equality are zero. For the LHS, this follows from the fact that, for fixed $e \in E$, the map $f \mapsto v(f \cdot e) - f \cdot
v(e)$ is a derivation of $f$; in more detail, this implies that this is obtained from a linear map $\Omega^1 \to T_X, df \mapsto v(f \cdot
e) - f \cdot v(e)$, and then we write $df = \sum_i \frac{\partial
f}{\partial x_i} dx_i$. For the RHS, the fact that $v(e) \in T_X$ is a derivation implies that $v(e)(f) + \sum_{i=1}^n -\frac{\partial
f}{\partial x_i} v(e)(x_i) = 0$, just as before. It remains to show that $$\sum_{i=1}^n \bigl( v(x_i \cdot e) - x_i \cdot v(e)
\bigr)\bigl(\frac{\partial f}{\partial x_i}\bigr) = 0.$$ Using the definition of $\pi_e$, we can rewrite the LHS of this expression as $$\sum_i \pi_e\bigl(x_i, \frac{\partial f}{\partial x_i}\bigr).$$ Now, viewing $\pi_e$ as a bivector field (i.e., a skew-symmetric biderivation), this can be rewritten as $$\sum_i \pi_e \bigl(dx_i \wedge d(\frac{\partial f}{\partial x_i})\bigr) =
\pi_e d(df) = 0. \qedhere$$
\[c:all-local\] $(X,{\operatorname{Vect}}(X))$ is ${\mathcal{D}}$-localizable. More generally, if $E
\subseteq {\operatorname{Vect}}(X)$ is a coherent subsheaf, then (the Lie algebra generated by) $E$ is ${\mathcal{D}}$-localizable.
Take $v = {\operatorname{Id}}$ in the theorem.
Let $X$ be either Poisson, Jacobi, or equipped with a top polyvector field. Then the presheaf ${\mathcal{H}}(X)$ of Hamiltonian vector fields is ${\mathcal{D}}$-localizable. Moreover, in the Poisson and top polyvector field cases, the presheaf ${\mathcal{LH}}(X)$ of locally Hamiltonian vector fields is also ${\mathcal{D}}$-localizable, and defines the same ${\mathcal{D}}$-module.
Similarly, when $X$ is equipped with a coherent subsheaf $N
\subseteq {\mathcal{T}}_X$ and a divergence function $D: N \to {\mathcal{O}}_X$, then the presheaf $\mathcal{H}(X,D)$ is ${\mathcal{D}}$-localizable, setting $E :=
\wedge^2_{{\mathcal{O}}_X} N$.
In the Poisson and Jacobi cases, we can take $E = {\mathcal{O}}_X$ and $v(f) =
\xi_f$. Then it is easy to check that $\pi_e$ is a skew-symmetric biderivation for all $e \in E$, so the theorem implies that ${\mathcal{H}}_X$ is ${\mathcal{D}}$-localizable. In the case of a top polyvector field $\Xi$, we take $E = \tilde \Omega_X^{n-2}$ and again let $v(\alpha) = \xi_\alpha =
\Xi(d\alpha)$.
For the second statement, it suffices to recall from Propositions \[p:poiss-hlh\] and \[p:vtop-hlh\] that, in the Poisson and top polyvector field cases, $H(X) \cdot {\mathcal{O}}_X = LH(X) \cdot {\mathcal{O}}_X$ for all affine $X$.
The final statement follows in the same manner.
On the other hand, $P(X)$ need not be ${\mathcal{D}}$-localizable: see § \[ss:pvfd-loc\] for a detailed discussion.
\[r:zarlh\] We note that, in general, ${\mathcal{H}}_X$ is *not* a sheaf, and neither is ${\mathcal{LH}}_X$.
For an example where ${\mathcal{H}}_X$ and ${\mathcal{LH}}_X$ are not sheaves, let $X$ be the complement in ${\mathbf{A}}^3$ of the plane $x+y=0$, equipped with the Poisson structure given by the potential $f(x,y,z) =
\frac{xy}{x+y}$, i.e., $$\{x,y\}=0, \{y,z\}= \frac{y^2}{(x+y)^2}, \{z,x\} = \frac{x^2}{(x+y)^2}.$$ Consider the vector field $\xi := (x+y)^{-2} \partial_z$. This is regular, and on the open set where $x \neq 0$, it is the Hamiltonian vector field of $x^{-1}$, and on the open set where $y \neq 0$, it is the Hamiltonian vector field of $-y^{-1}$. But it is not globally Hamiltonian, since if $\xi = \xi_f$ for some $f \in {\mathcal{O}}_X$, then $\frac{1}{x^2+y^2} = \{f,z\}$, but the RHS must live in $\frac{(x^2,y^2)}{x^2+y^2}$, where $(x^2,y^2)$ is the ideal generated by $x^2$ and $y^2$. This is impossible.
The same argument shows that $\xi$ is not given by a global one-form: otherwise, $\frac{1}{x^2+y^2} = f_1 \{x, z\} + f_2 \{y, z\}$ for some $f_1, f_2 \in {\mathcal{O}}_X$, and one concludes as before that this is impossible.
On the other hand, in the case that $X$ is generically symplectic, it follows that ${\mathcal{H}}_X$ and ${\mathcal{LH}}_X$ are sheaves, since in this case any vector field which is Hamiltonian in some neighborhood must be given by a unique Hamiltonian function up to locally constant functions, and this is then defined and Hamiltonian on the regular locus of that function (and similarly in the locally Hamiltonian case).
Note similarly that, in the case of a variety with a top polyvector field $\Xi$, ${\mathcal{H}}_X$ and ${\mathcal{LH}}_X$ are sheaves, since if $\Xi$ is nonzero, then on its nonvanishing locus a Hamiltonian vector field is once again given by a unique Hamiltonian.
\[r:f-loc\] In the examples above, the presheaves also are equipped naturally with spaces of sections on formal neighborhoods $\hat X_x$ of every point $x \in X$; the presheaf condition requires only that these contain the restrictions of sections on open subsets containing $x$. Thus it makes sense to define the notion of *formal ${\mathcal{D}}$-localizability*, i.e., that, for every open affine $U$ and $x \in U$, $$\label{e:f-d-loc}
{\mathfrak{v}}(\hat X_x) {\mathcal{D}}_{\hat X_x} = {\mathfrak{v}}(U)|_{\hat X_x} {\mathcal{D}}_{\hat X_x}.$$ Formal localizability implies usual localizability: indeed, if ${\mathfrak{v}}$ is formally localizable, and $\xi \in {\mathfrak{v}}(U')$ for some $U' \subseteq
U$, then at every $x \in U'$, it follows that $\xi|_{\hat
X_x} \in {\mathfrak{v}}(U) \cdot {\mathcal{D}}_{\hat X_x}$, and hence $\xi \in {\mathfrak{v}}(U)
\cdot {\mathcal{D}}_U$, by Lemma \[l:vd-sheaf\].
Theorem \[t:h-loc\] extends to show that, under the assumptions there, ${\mathfrak{v}}$ is formally ${\mathcal{D}}$-localizable, by formally localizing the embedding $X \to {\mathbf{A}}^n$ to $\hat X_x \to \hat {\mathbf{A}}^n_0$. Then, the same proof applies. We conclude as before that the presheaves of (locally) Hamiltonian vector fields are formally ${\mathcal{D}}$-localizable, as well as ${\operatorname{Vect}}(X)$ and all coherent subsheaves thereof.
${\mathcal{D}}$-localizability of Poisson vector fields {#ss:pvfd-loc}
-------------------------------------------------------
An interesting question raised in the previous subsection is whether $P(X)$ is ${\mathcal{D}}$-localizable. This turns out to have an interesting answer, which we discuss here. The material of this subsection will not be needed for the rest of the paper, and our motivation is partly to illustrate the nontriviality of ${\mathcal{D}}$-localizability. We will first give the statements and examples, and postpone the proofs of the propositions to the end of the subsection, for the purpose of emphasizing the statements and counterexamples to their generalization.
\[p:ploc\] Let $X$ be an irreducible affine Poisson variety on which $P(X)$ flows incompressibly. If $P(X)$ is ${\mathcal{D}}$-localizable, then the generic rank of $P(X)$ must equal that of $P(U)$ for every open subset $U \subseteq X$.
Conversely, suppose that $X$ is a smooth affine Poisson variety on which the rank of $P(U)$ equals that of $H(U)$ everywhere, for all affine open $U \subseteq X$, and that this rank is constant on $X$. Then, for all affine open $U \subseteq X$, one has an equality of ${\mathcal{O}}$-saturations $P(U)^{os} = H(U)^{os}$. Hence, $P(X)$ is ${\mathcal{D}}$-localizable.
The assertion of the second paragraph follows from the more general
\[l:inc-rank-sat\] Suppose ${\mathfrak{v}}\subseteq \mathfrak{w}$ is an inclusion of Lie algebras of vector fields on a smooth affine variety $X$. Suppose that the rank of ${\mathfrak{v}}$ is constant and equals that of $\mathfrak{w}$ everywhere, and moreover that $\mathfrak{w}$ flows incompressibly. Then ${\mathfrak{v}}^{os} = \mathfrak{w}^{os}$. In particular, $M(X,{\mathfrak{v}}) = M(X, \mathfrak{w})$.
Since the ranks of ${\mathfrak{v}}$ and $\mathfrak{w}$ are constant and equal, we conclude that, for every $x \in X$, there exists an open subset $U \subseteq X$ containing $x$ such that ${\mathcal{O}}_U \cdot {\mathfrak{v}}|_U = {\mathcal{O}}_U \cdot
\mathfrak{w}$, and hence in fact ${\mathcal{O}}_X \cdot {\mathfrak{v}}= {\mathcal{O}}_X \cdot
\mathfrak{w}$. Now, if $\mathfrak{w}$ flows incompressibly, and hence also ${\mathfrak{v}}$, then ${\mathfrak{v}}^{os}=\mathfrak{w}^{os}=$ the subspace of ${\mathcal{O}}_X \cdot {\mathfrak{v}}$ of incompressible vector fields, by Proposition \[p:inc-os\].
Lemma \[l:inc-rank-sat\] generalizes to affine schemes of finite type, if we replace the rank condition by the condition that ${\mathcal{O}}_X \cdot {\mathfrak{v}}=
{\mathcal{O}}_X \cdot \mathfrak{w}$.
We also give a localizability result that does not require $X$ to be smooth, in the situation of Remark \[r:n-cod2-poiss\], where $P(X) = LH(X^\circ)$ for $X^\circ$ the smooth locus of $X$.
\[ntn:gdr\] Given any not necessarily affine scheme $Y$, we will let $H^\bullet_{DR}(Y) := H^\bullet(\Gamma(\tilde \Omega_Y))$ denote the cohomology of the complex of global sections of de Rham differential forms modulo torsion.
For all $x \in X$, let ${\mathcal{O}}_{X,x}$ be the *uncompleted* local ring of $X$ at $x$.
\[p:ploc-is\] Suppose $X$ is Poisson, normal, and symplectic on its smooth locus. Let $S$ be its singular locus.
1. For every $s \in S$, let $E_s \subseteq S$ be the union of all irreducible components of $S$ containing $s$. Suppose that, for all $s \in S$, the natural map $$\label{e:is-l-surj}
H^1_{DR}(X \setminus E_s) \oplus H^1_{DR}({\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s}) \to
H^1_{DR}( {\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s} \setminus E_s)$$ is surjective. Then, $X$ is Poisson localizable. Moreover, for all $s \in S$, $$\label{e:imp-ploc}
P({\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s}) =P(X)|_{{\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s}} \cdot {\mathcal{O}}_{X,s} + LH({\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s}).$$
2. Now suppose that $S$ is finite and ${\mathbf{k}}={\mathbf{C}}$. Then the hypothesis of (i) is satisfied if, for all $s \in S$ and all affine Zariski open neighborhoods $U$ of $s$, the natural map on *topological* cohomology, $$\label{e:is-l-top-surj}
H^1_{{\operatorname{top}}}(X \setminus \{s\}) \oplus H^1_{{\operatorname{top}}}(U) \to H^1_{{\operatorname{top}}}(U \setminus \{s\})$$ is surjective. In particular, in this case, $X$ is Poisson localizable, and holds.
When $X$ has a contracting ${\mathbf{G}}_m$ action (where this is the multiplicative group), i.e., ${\mathcal{O}}_X$ is nonnegatively graded with ${\mathbf{k}}$ in degree zero, then $H^\bullet(X)={\mathbf{k}}$, and in particular $H^2(X)=0$. Therefore, if $X$ has an isolated singularity at the fixed point for the action, using the Mayer-Vietoris sequence below, is surjective. Thus, if $X$ is also normal and generically symplectic, the conditions of the proposition are satisfied, so $P(X)$ is ${\mathcal{D}}$-localizable. Also, in this case, $P(U) = P(X)|_U + LH(U)$ for all open sets (and for those $U$ which don’t contain the singularity we have $P(U)=H(U)$, since then $U$ is symplectic).
For such an example where $P(U)/LH(U)$ is nonzero, let $X$ be the locus $x^3+y^3+z^3=0$ (or a more general elliptic singularity); then $P(U)/LH(U)$ is generated by the Euler vector field in $P(X)$ for all open affine $U$.
Here is a simple example of a non-normal $X$ which is not ${\mathcal{D}}$-localizable. Suppose $X = {\operatorname{\mathsf{Spec}}}{\mathbf{k}}[x^2,x^3,y,xy]$ and $\{x,y\}=y$. This is generically symplectic but not normal. Then we claim that every global Poisson vector field vanishes at $y=0$. Indeed, $\xi = f \partial_x + g \partial_y$ is Poisson if and only if $\frac{\partial f}{\partial x} + \frac{\partial g}{\partial
y} = \frac{g}{y}$. Writing $g=yh$, we obtain $\frac{\partial
f}{\partial x} + y\frac{\partial h}{\partial y} = 0$. So $y \mid
\frac{\partial f}{\partial x}$. Since $\xi$ is a vector field on $X$, $f$ vanishes at the origin, and hence $y \mid f$. This proves the claim. On the other hand, in the complement $U$ of any hyperplane through the origin, $\partial_x$ is a Poisson vector field; but this can only be in $P(X) \cdot {\mathcal{D}}_U$ when the hyperplane was $y=0$. Thus $P(X)$ is not ${\mathcal{D}}$-localizable.
In the case $X$ is smooth, if it has finitely many symplectic leaves, it is in fact symplectic. However, there are many cases where $X$ is smooth and generically symplectic, and $P(X)$ has finitely many leaves even though $X$ has infinitely many symplectic leaves; e.g., $\pi = x \partial_x \wedge \partial_y$ on ${\mathbf{A}}^2$, as mentioned in §\[ss:poisson\].
We give an example where $X$ is smooth but $P(X)$ is not ${\mathcal{D}}$-localizable:
Let ${\mathfrak{g}}$ be the Lie algebra ${\mathfrak{g}}:=
\mathfrak{sl}_2$ and let $X = {\mathfrak{g}}^*$, equipped with the induced Poisson bracket on ${\mathcal{O}}_X = {\operatorname{\mathsf{Sym}}}{\mathfrak{g}}$. Then, all global Poisson vector fields are Hamiltonian, since $H^1({\mathfrak{g}}, {\operatorname{\mathsf{Sym}}}{\mathfrak{g}}) = 0$ (this implies that all derivations ${\mathfrak{g}}\to {\operatorname{\mathsf{Sym}}}{\mathfrak{g}}$ are inner, and hence all derivations of ${\operatorname{\mathsf{Sym}}}{\mathfrak{g}}$, i.e., vector fields on ${\mathfrak{g}}^*$, are Hamiltonian). It is clear that the Poisson bivector has rank two, except at the origin, where the rank is zero; hence this is the rank of $P(X)$. However, we claim that the rank of the space of generic Poisson vector fields is three. Indeed, write ${\mathfrak{g}}= \langle e, h, f \rangle$ with the standard bracket $[e,f]=h,
[h,e]=2e, [h,f]=-2f$. So $e, h, f \in {\mathcal{O}}_X$ are linear coordinates. Let $C = 2ef + \frac{1}{2} h^2 \in {\mathcal{O}}_X$ be the Casimir function, so $\{C,g\}=0$ for all $g \in {\mathcal{O}}_X$. Then, if we localize where $e\neq 0$, we can consider the coordinate system $(e,h,C)$ and take the directional derivative in the $C$ direction, which in the original coordinates $(e,h,f)$ is $\xi :=
\frac{1}{2e}\partial_f$. Since the Poisson bivector field is tangent to the planes where $C$ is constant, this vector field is Poisson, which is also immediate from explicit computation (it is enough to check that $\{\xi(x), y\} + \{x, \xi(y)\} = \xi\{x,y\}$ for $x,y \in {\mathcal{O}}_X$, which clearly reduces to the case $x=f, y=h$, where $\{\xi(f), h\} = \frac{1}{e} = \xi(2f) = \xi\{f,h\}$.)
By incompressibility and Corollary \[c:inc\], the generic rank of $P(X)$ equals $2 \dim X$ minus the dimension of the singular support of $M(U',P(X)|_{U'})$ for small enough open subsets $U'$ (viewing $P(X)|_{U'}$ as a vector space). Thus ${\mathcal{D}}$-localizability implies that this must also equal the generic rank of $P(U)$ for every open subset $U \subseteq X$.
The second statement follows from Lemma \[l:inc-rank-sat\], provided we can show that $P(X)$ flows incompressibly. By assumption, $P(X)$ flows parallel to the symplectic leaves. But, to be Poisson, such vector fields must preserve the symplectic form along the leaves, and hence they are incompressible along the leaves. Thus, as for $H(X)$ (see Example \[ex:poiss-finsym\]), one concludes that $P(X)$ flows incompressibly on $X$.
\(i) Suppose that $\xi \in P({\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s})$. As explained in Remark \[r:n-cod2-poiss\], this means that $\xi = \eta_\alpha$ where $\alpha$ is a closed one-form on ${\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s} \setminus E_s$. By the hypothesis , we can write $$\label{e:ploc-is-pf1}
\alpha = \alpha_{{\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s}} + \alpha_{X \setminus E_s} + df,$$ where $\alpha_{{\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s}}$ is a closed one-form modulo torsion on ${\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s}$, $\alpha_{X \setminus E_s}$ is a closed one-form modulo torsion on $X \setminus E_s$, and $f \in
\Gamma({\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s} \setminus E_s)$. By normality, $f \in {\mathcal{O}}_{X,s}$. Thus $\xi_f \in H({\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s})$. Note that $\eta_{\alpha_{{\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s}}} \in LH({\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s})$, by definition. As in Remark \[r:n-cod2-poiss\], we obtain that $\eta_{X \setminus E_s} \in P(X)$. Therefore, applying the operation $\beta \mapsto \eta_\beta$ to both sides of , we obtain , since $\xi$ was arbitrary.
As a consequence, we deduce that $P({\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s}) \subseteq P(X)|_{{\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s}} \cdot {\mathcal{O}}_{X,s}$. Now, $s \in X$ was an arbitrary singular point. At smooth points $x \in X$, we have $P({\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,x}) = H({\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,x}) \subseteq
H(X)|_{{\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,x}} \cdot {\mathcal{O}}_{X,x}$.
Now, for arbitrary open affine $U \subseteq X$, $P(X)|_U \cdot {\mathcal{D}}_U$ is a sheaf on $U$, by Lemma \[l:vd-sheaf\]. By the above, $P(U)|_{{\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,x}} \subseteq P(X)|_{{\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,x}} \cdot
{\mathcal{D}}_{X,x}$, where the latter is the Zariski localization of ${\mathcal{D}}_X$ at $x$. By Lemma \[l:vd-sheaf\], $P(X)|_{{\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,x}} \cdot
{\mathcal{D}}_{X,x} = (P(X)|_U \cdot {\mathcal{D}}_U)|_{{\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,x}}$. We conclude that, for all $x \in U$, $$P(U)|_{{\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,x}} \subseteq (P(X)|_U \cdot {\mathcal{D}}_U)|_{{\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,x}},$$ and since $P(X)|_U \cdot {\mathcal{D}}_U$ is a sheaf, this implies that $P(U)
\subseteq P(X)|_U \cdot {\mathcal{D}}_U$. As $U$ was arbitrary, we conclude Poisson localizability.
\(ii) In order to prove , it suffices to prove the statement when ${\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s}$ is replaced by sufficiently small Zariski open neighborhoods $U$ of $s$. This is because every closed one-form modulo torsion in ${\operatorname{\mathsf{Spec}}}{\mathcal{O}}_{X,s} \setminus E_s$ is actually regular on $U \setminus E_s$ for some Zariski open neighborhood $U$ of $s$, and we are free to shrink it.
Now, assuming that $S$ is finite, $E_s = \{s\}$ for all $s \in S$. By the preceding paragraph, it suffices to show that implies that the map $$\label{e:is-l-usurj}
H^1_{DR}(X \setminus \{s\}) \oplus H^1_{DR}(U) \to
H^1_{DR}(U \setminus \{s\})$$ is surjective. To see this, we first note that, for $Y$ smooth but not necessarily affine, we have an isomorphism by Grothendieck’s theorem, $$\mathbf{H}^\bullet_{DR}(Y) \cong H^\bullet_{{\operatorname{top}}}(Y),$$ where $\mathbf{H}^\bullet_{DR}(Y)$ denotes the *hypercohomology* of the complex of sheaves $\Omega_Y^\bullet = \tilde \Omega_Y^\bullet$.
Next, there is a natural map $H^1_{DR}(U) \to H^1_{{\operatorname{top}}}(U)$, obtained by integrating along cycles; one can slightly perturb a closed path in $U$ to miss the isolated singularities of $U$, and integrating against a one-form on $U$ which is closed mod torsion (hence closed when restricted to the smooth locus of $U$) produces a well-defined answer, which depends only on the homology class in $U$ of the closed path.
Then, the restriction map $H^1_{DR}(U) ={\mathbf{H}}^1_{DR}(U) \to {\mathbf{H}}^1_{DR}(U \setminus \{s\}) = H^1_{{\operatorname{top}}}(U \setminus \{s\})$ factors through the map $H^1_{DR}(U) \to H^1_{{\operatorname{top}}}(U)$, which is surjective by the main result of [@BH-drcas].
Then, implies that we have a surjection $$\label{e:is-l-hdr-surj}
{\mathbf{H}}^1_{DR}(X \setminus \{s\}) \oplus H^1_{DR}(U) \to
{\mathbf{H}}^1_{DR}(U \setminus \{s\}),$$ where here we note that $H^1_{DR}(U) = {\mathbf{H}}^1_{DR}(U)$ since $U$ is affine.
Since $X = (X \setminus \{s\}) \cup U$, we have an exact Mayer-Vietoris sequence on hypercohomology of the triple $(X, X \setminus \{s\}), U)$, which in part takes the form $$\label{e:mv-hyper}
{\mathbf{H}}^1_{DR}(X \setminus \{s\}) \oplus H^1_{DR}(U) \to
{\mathbf{H}}^1_{DR}(U \setminus \{s\}) \to H^2_{DR}(X) \to {\mathbf{H}}^2_{DR}(X \setminus \{s\})
\oplus H^2_{DR}(U).$$ By , the first map is surjective, and hence the last map is injective.
We also have a Mayer-Vietoris sequence for ordinary $H^\bullet_{DR}$, associated to the exact sequence of complexes of global sections, $$0 \to \Omega^\bullet_X {\hookrightarrow}\Gamma(\Omega^{\bullet}_{X \setminus
\{s\}}) \oplus \Omega^\bullet_U {\twoheadrightarrow}\Gamma(\Omega^{\bullet}_{U
\setminus \{s\}}).$$ This has the form $$\label{e:mv-ord}
H^1_{DR}(X \setminus \{s\}) \oplus H^1_{DR}(U) \to
H^1_{DR}(U \setminus \{s\}) \to H^2_{DR}(X) \to H^2_{DR}(X \setminus \{s\})
\oplus H^2_{DR}(U).$$ Now, the final map in factors through the final map in (since $X$ is affine). Therefore the last map in must also be injective. We conclude that the first map of , which is the same as , is surjective. This completes the proof.
Formal ${\mathcal{D}}$-localizability of Poisson vector fields
--------------------------------------------------------------
It turns out that formal ${\mathcal{D}}$-localizability of Poisson vector fields is a stronger condition, which implies (in the incompressible case) that $X$ is generically symplectic.
\[p:p-floc\] If $X$ is irreducible affine Poisson and $P(X)$ flows incompressibly, then if $P(X)$ is formally ${\mathcal{D}}$-localizable, then $X$ must be generically symplectic.
Note that this in particular implies that the condition of Proposition \[p:ploc\] is satisfied: $P(U)$ has generic rank equal to $\dim X$ for all $U$.
Suppose that $X$ is not generically symplectic. Then, in the neighborhood of some sufficiently generic smooth point, $\hat X_x
\cong V \times V'$ as a formal Poisson scheme, where $V$ is a symplectic formal polydisc and $V'$ is a positive-dimensional formal polydisc with the zero Poisson bracket. So $P(\hat X_x) = P(V)
\otimes {\mathcal{O}}_{V'} \oplus {\operatorname{Vect}}(V')$. This is evidently not incompressible since $V'$ is positive-dimensional. Thus $M(\hat
X_x, P(\hat X_x)) = 0$. However, if we assume $P(X)$ flows incompressibly, then $M(X, P(X))|_{\hat X_x} \neq 0$ for sufficiently generic $x$ (with $P(X)$ here the constant sheaf). Thus $P(X)$ is not formally localizable.
We can also give a positive result parallel to Proposition \[p:ploc-is\]:
\[p:fploc-is\] Suppose $X$ is affine Poisson, normal, and symplectic on its smooth locus. Let $S$ be its singular locus.
1. For every $s \in S$, let $E_s \subseteq S$ be the union of all irreducible components of $S$ containing $s$. Suppose that, for all $s \in S$, the natural map $$\label{e:is-fl-surj}
H^1_{DR}(X \setminus E_s) \oplus H^1_{DR}({\operatorname{\mathsf{Spec}}}\hat {\mathcal{O}}_{X,s}) \to
H^1_{DR}( {\operatorname{\mathsf{Spec}}}\hat {\mathcal{O}}_{X,s} \setminus E_s)$$ is surjective. Then, $X$ is formally Poisson localizable. Moreover, for all $s \in S$, $$\label{e:imp-fploc}
P({\operatorname{\mathsf{Spec}}}\hat {\mathcal{O}}_{X,s}) =P(X)|_{{\operatorname{\mathsf{Spec}}}\hat {\mathcal{O}}_{X,s}} \cdot \hat {\mathcal{O}}_{X,s} +
LH({\operatorname{\mathsf{Spec}}}\hat {\mathcal{O}}_{X,s}).$$
2. Suppose that $S$ is finite and ${\mathbf{k}}={\mathbf{C}}$. Then the hypothesis of (i) is satisfied if, for sufficiently small neighborhoods $U$ of $s$ in the complex topology, $H^1_{{\operatorname{top}}}(X \setminus \{s\}) \to H^1_{{\operatorname{top}}}(U \setminus \{s\})$ is surjective. In particular, in this case, $X$ is formally Poisson localizable, and holds.
\[r:is-l-top-surj\] The condition of (ii) is equivalent to asking that $H^1_{{\operatorname{top}}}(X
\setminus \{s\}) \to H^1_{{\operatorname{top}}}(U \setminus \{s\})$ be surjective for any fixed contractible neighborhood $U$ of $s$ (whose existence was proved in [@Gil-eavlc]). Thus, the condition of (ii) is the same as that of , except replacing Zariski open subsets by analytic neighborhoods, and using holomorphic functions rather than algebraic functions.
The proof of part (i) of the proposition is the same as in Proposition \[p:ploc-is\], except replacing $U$ by $\hat X_x$. We omit the details. Note that, when $x \notin S$, one has $P(\hat X_x) = H(\hat
X_x)$, since then $\hat X_x$ is symplectic.
For part (ii), we use holomorphic functions and analytic neighborhoods and results about them contained in §\[ss:comp\] below. As in Proposition \[p:ploc-is\], for every analytic neighborhood $U$ of $s$, the assumption of (ii) together with Grothendieck’s theorem implies that the map on hypercohomology, $${\mathbf{H}}^1_{DR}(X
\setminus \{s\})\to {\mathbf{H}}^{1,{\operatorname{an}}}_{DR}(U \setminus \{s\}),$$ is surjective. Using the Mayer-Vietoris sequence for the exact sequence of complexes of sheaves ( below for $Y=X$, $Z = \{s\}$, and $V = U$), we conclude that the map $$H^2_{DR}(X) \to
{\mathbf{H}}^2_{DR}(X \setminus \{s\}) \oplus H^{2,{\operatorname{an}}}(U)$$ is injective. This map factors through the map from ordinary cohomology to hypercohomology, so we conclude that the map $$H^2_{DR}(X) \to
H^2_{DR}(X \setminus \{s\}) \oplus H^{2,{\operatorname{an}}}(U)$$ is also injective. Using the Mayer-Vietoris sequence for ordinary cohomology (using the global sections of , which is an exact sequence of complexes since $X$ is affine), we conclude that $$\label{e:h1x}
H^1_{DR}(X \setminus \{s\}) \oplus H^{1,{\operatorname{an}}}_{DR}(U) \to H^{1,{\operatorname{an}}}_{DR}(U
\setminus \{s\})$$ is surjective. Then, by Theorem \[t:form-an-comp\] below, we conclude that $$H^1_{DR}(X \setminus \{s\}) \oplus H^1_{DR}(\hat X_s) \to H^1_{DR}(\hat X_s \setminus \{s\})$$ is also surjective, as desired.
In fact, we did not need the full strength of Theorem \[t:form-an-comp\] below, but only the fact that the maps $H^1_{DR}(U) \to H^1_{DR}(\hat X_s)$ and $H^1_{DR}(U \setminus
\{s\}) \to H^1_{DR}(\hat X_s \setminus \{s\})$ are surjective. At least the first fact can be proved in an elementary way by lifting closed formal differential forms to closed analytic differential forms, and does not require resolution of singularities as used in the proof of Theorem \[t:form-an-comp\].
Here is an example of a surface with an isolated singularity, which is normal and symplectic away from the singularity, which is ${\mathcal{D}}$-localizable (in fact satisfying ) but not formally ${\mathcal{D}}$-localizable (so in particular not satisfying ). This example was pointed out to us by J. McKernan.
Let $E \subseteq {\mathbf{P}}^2$ be a smooth cubic curve. Then, under the intersection pairing on ${\mathbf{P}}^2$, $E \cdot E = 9$. Now, blow up ${\mathbf{P}}^2$ at twelve generic points of $E$. Let $Y$ be the resulting projective surface, and let $E' \subseteq Y$ be the proper transform of $E$. Then $E' \cdot E' = 9 - 12 = -3$, so we can blow down $E'$ to obtain a new surface, call it $Z$, where the image of $E'$ is a singular point, call it $s$, whose formal neighborhood $\hat Z_s$ is isomorphic to the cone over an elliptic curve.
Note that $H^1_{{\operatorname{top}}}(Z \setminus \{s\}) \cong H^1_{{\operatorname{top}}}(Y
\setminus E') \cong H^1_{{\operatorname{top}}}({\mathbf{P}}^2 \setminus E) = 0$, since $E
\subseteq {\mathbf{P}}^2$ has a nontrivial normal bundle.
Next, embed $Z$ into projective space ${\mathbf{P}}^N$ of some dimension $N >
2$. Let $C \subseteq Z$ be the intersection of $Z$ with a generic hyperplane, and let $X := Z \setminus C$ be the resulting affine surface. Since ${\mathcal{O}}(C)$ is (very) ample, $C$ has a nontrivial normal bundle. Hence, the restriction map induces isomorphisms $H^1_{{\operatorname{top}}}(Z) {{\;\stackrel{_\sim}{\to}\;}}H_{{\operatorname{top}}}^1(X)$ and $H^1_{{\operatorname{top}}}(Z \setminus
\{s\}) {{\;\stackrel{_\sim}{\to}\;}}H^1_{{\operatorname{top}}}(X\setminus \{s\})$. In particular, these are zero as well.
Thus, ${\mathbf{H}}^1_{DR}(X \setminus \{s\}) = 0$. We claim that $H^1_{DR}(X \setminus \{s\}) = 0$ as well. More generally, this follows from the following statement:
\[l:h-bh-inj\] Let $V$ be a scheme or complex analytic space. Then the map $H^1_{DR}(V) \to {\mathbf{H}}^1_{DR}(V)$ is injective.
We remark that, in the case $V$ is a smooth variety (as with $V = X \setminus
\{s\}$ above), by Grothendieck’s theorem we can replace ${\mathbf{H}}^1_{DR}(V)$ by the topological first cohomology of $V$, and then the statement follows because, if an algebraic or analytic one-form is the differential of a smooth ($C^\infty$) function, then the function must actually be algebraic (or analytic).
Consider the spectral sequence $H^i(R^j \Gamma (\Omega_V)) \Rightarrow {\mathbf{H}}_{DR}^{i+j}(V)$. In total degrees $\leq 2$, the second page has the form $$H^0_{DR}(V) \to H^1_{DR}(V) \oplus H^0(R^1 \Gamma(\Omega_V)) \to
H^2_{DR}(V) \oplus H^1(R^1 \Gamma(\Omega_V)) \oplus
H^0(R^2 \Gamma(\Omega_V)).$$ The first map above is zero, and the restriction of the second map above to $H^1_{DR}(V)$ is zero. Therefore the summand of $H^1_{DR}(V)$ maps injectively to a summand of the third page of the spectral sequence. The same argument shows that, at every page, $H^1_{DR}(V)$ maps injectively to the next page, so the map $H^1_{DR}(V) \to {\mathbf{H}}^1_{DR}(V)$ is injective.
Now, since $X \setminus \{s\}$ is symplectic, all global Poisson vector fields are locally Hamiltonian given by a global closed one-form. By the above, $H^1_{DR}(X \setminus \{s\}) = 0$, so that locally Hamiltonian vector fields are Hamiltonian. Since global functions on $X \setminus \{s\}$ coincide with those on $X$, all global Poisson vector fields on $X$ are Hamiltonian.
On the other hand, not all Poisson vector fields on $\hat X_s$ are Hamiltonian, since $\hat X_s$ is isomorphic to the formal neighborhood of the vertex in the cone over an elliptic curve, and there one has the Euler vector field which is not Hamiltonian. Hence, $P(X)$ is not formally ${\mathcal{D}}$-localizable. (In fact, $P(X)$ is not étale-locally ${\mathcal{D}}$-localizable either, since the Euler vector field exists in an étale neighborhood of $x$, or equivalently in the Henselization of the local ring at $x$.)
On the other hand, we claim that $P(X)$ is ${\mathcal{D}}$-localizable, and in fact that holds. Let $U \subseteq X$ be any affine open subset containing $s$. Since $Z$ is rational (as $Y$, and hence $Z$, is birational to ${\mathbf{P}}^2$), so is $U$. Now, we claim that the map $H^1_{DR}(U)
\to H^1_{DR}(U
\setminus \{s\})$ is surjective. Consider the sequence for the pair $(U, \{x\})$: this yields the exact sequence $$H^1_{DR}(U) \to H^1_{DR}(U \setminus \{s\}) \oplus H^1_{DR}(\hat U_s)
\to H^1_{DR}(\hat U_s \setminus \{s\}).$$ It suffices to show that the map $H^1_{DR}(U \setminus \{s\}) \to
H^1_{DR}(\hat U_s \setminus \{s\})$ is zero. By Lemma \[l:h-bh-inj\] above, this is equivalent to showing that the map $H^1_{DR}(U \setminus \{s\}) \to {\mathbf{H}}^1_{DR}(\hat U_s \setminus \{s\})$ is zero. This map factors through the hypercohomology of any punctured neighborhood of $s$ contained in $U \setminus \{s\}$, which by Grothendieck’s theorem is the same as the topological cohomology of that punctured neighborhood. Such punctured neighborhoods, for sufficiently small contractible $U$, are homotopic to nontrivial $S^1$-bundles over an elliptic curve, and their fundamental group is isomorphic to that of the elliptic curve. If the map $H^1_{DR}(U
\setminus \{s\}) \to {\mathbf{H}}^1_{DR}(\hat U_s \setminus \{s\})$ were nonzero, then a nontrivial period of the elliptic curve would be computable by integrals of closed algebraic one-forms on $U \setminus \{s\}$. However, as remarked, $U$ is rational. Thus this would imply that a nontrivial period of the elliptic curve were computable by integrals of rational closed one-forms along contours in ${\mathbf{C}}^2$. This is well-known to be impossible, since these periods are given by transcendental hypergeometric functions with infinite monodromy. Thus, $P(X)$ is ${\mathcal{D}}$-localizable. (Note that this paragraph also gives another proof that $P(X)$ is not formally ${\mathcal{D}}$-localizable, and in fact that $P(U)$ is not formally ${\mathcal{D}}$-localizable for every open affine neighborhood $U$ of $s$: these periods are computable in a formal neighborhood of $s$, but by the above, they are not computable using global closed one-forms. Passing from closed one-forms to Poisson vector fields via the symplectic form on $U \setminus \{s\}$, this yields that $P(U)$ is not formally ${\mathcal{D}}$-localizable.)
We give an example where $X$ is smooth and ${\mathcal{D}}$-localizable but not formally ${\mathcal{D}}$-localizable. By Propositions \[p:ploc\] and \[p:p-floc\], one way this happens is if the rank of $P(U)$ equals that of $H(U)$ and is constant but less than the dimension of $X$ (which in particular is not generically symplectic).
Let $X = ({\mathbf{A}}^\times)^3 = {\operatorname{\mathsf{Spec}}}{\mathbf{k}}[x^{\pm 1}, y^{\pm 1}, z^{\pm
1}]$ with the Poisson bracket $\{x,y\}=xyz$ and $\{x,z\}=\{y,z\}=0$. The Poisson bivector, call it $\pi$, is $\pi=xyz \partial_x \wedge \partial_y$. Then $H(U)$ has rank two everywhere, for every open affine subset $U \subseteq X$. We claim that any rational Poisson vector field on $X$ annihilates $z$. Therefore, the rank of every vector field in $P(U)$ is also everywhere two, for every open subset $U \subseteq X$, as desired.
To prove that every rational Poisson vector field annihilates $z$, it is enough to assume that ${\mathbf{k}}={\mathbf{C}}$. Let $\xi$ be a rational Poisson vector field and let $c \in {\mathbf{C}}$ be such that it does not have a pole at $z=c$. Then the irregular locus of $\xi$ in $\{z=c\}$ is an algebraic curve in ${\mathbf{A}}^2$. One can show that such a curve must avoid a real two-torus $T=\{|x|=r, |y|=s\}$, and then $\int_{T \times \{z\}} \pi^{-1}$ is a nonzero constant multiple of $\frac{1}{z}$, for $z$ in some topological neighborhood of $c$ (which is independent of $r$ and $s$). Since $\xi$ preserves $\pi$, one concludes that it must be parallel to the level sets of $z$, i.e., it annihilates $z$.
We can also give an elementary algebraic proof that, in the above example, every rational Poisson vector field annihilates $z$. Any rational Poisson vector field must send $z$ to a rational function of $z$, since these are all the rational Casimirs. Moreover, any such vector field is still Poisson after multiplying by an arbitrary rational function of $z$. Hence, if such a vector field exists which does not annihilate $z$, then there must be one of the form $\partial_z + f \partial_x + g \partial_y$ for some rational functions $f,g$ on $X$.
On the other hand, we can explicitly write one such non-rational vector field, $\xi := \partial_z + \frac{x \log x}{z} \partial_x$. This vector field is best understood by writing the Poisson bracket in coordinates $(u,v,z)=(\log x, \log y, z)$, as $\{u, v\} = z$, and the vector field as $\xi=\partial_z + \frac{u}{z} \partial_{u}$.
Thus, given a rational vector field $\partial_z + f \partial_x +
g \partial_y$, taking the difference, we would obtain a non-rational Poisson vector field of the form $\eta = -\frac{x \log x}{z} \partial_x +
f \partial_x + g \partial_y$. But no such vector field can be Poisson, since a vector field parallel to the symplectic leaf is Poisson if and only if its symplectic divergence, $-L_{\eta}\pi / \pi$, vanishes. But this symplectic divergence is $$x\partial_x(-(\log x)/z) + x \partial_x(f/x) + y\partial_y(g/y)
= -\frac{1}{z} + x \partial_x(f/x) + y\partial_y(g/y).$$ Dividing by $x$ and taking the residue at $x=0$, we would obtain $$-\frac{1}{z} + y \partial_y(g/y)|_{x=0} = 0.$$ Now, dividing by $y$ and taking the residue at $y=0$, we would obtain $$=- \frac{1}{z} = 0,$$ which is a contradiction.
Analytic-to-formal comparison for de Rham cohomology {#ss:comp}
----------------------------------------------------
### Preliminaries on analytic forms and Mayer-Vietoris sequences
We will need to use holomorphic differential forms, on an algebraic or (complex) analytic variety $Y$ which need not be affine.
\[d:an\] Let $Y$ be an algebraic or analytic variety over ${\mathbf{k}}=
{\mathbf{C}}$. Let $\Omega_Y^{\bullet,{\operatorname{an}}}$ denote the complex of sheaves of holomorphic Kähler differential forms, and $\tilde \Omega_Y^{\bullet,{\operatorname{an}}}$ its quotient modulo torsion. Let ${\mathbf{H}}_{DR}^{\bullet,{\operatorname{an}}}(Y)$ denote the hypercohomology of this complex, and $H_{DR}^{\bullet,{\operatorname{an}}}(Y)$ denote the cohomology of the complex of global sections $\Gamma(\Omega_Y^{\bullet,{\operatorname{an}}})$.
Grothendieck’s theorem also extends to the holomorphic setting, where we obtain that ${\mathbf{H}}_{DR}^{\bullet,{\operatorname{an}}}(U) \cong H_{{\operatorname{top}}}^\bullet(U)$ if $U$ is smooth.
For $Z \subseteq Y$ a subvariety and $V$ an analytic neighborhood of $Z$, we will make use of the Mayer-Vietoris sequence associated to the exact sequence of complexes, $$\label{e:mv}
0 \to \tilde \Omega^\bullet_Y {\hookrightarrow}\tilde \Omega^\bullet_{Y \setminus Z}
\oplus \tilde \Omega^{\bullet,{\operatorname{an}}}_V {\twoheadrightarrow}\tilde \Omega^{\bullet,{\operatorname{an}}}_{V \setminus Z} \to 0.$$ Similarly, we will need the corresponding sequence when $V$ is replaced by a formal neighborhood of $Z$: $$\label{e:mv-formal}
0 \to \tilde \Omega^\bullet_Y {\hookrightarrow}\tilde \Omega^\bullet_{Y \setminus Z}
\oplus \tilde \Omega^{\bullet}_{\hat Y_Z} {\twoheadrightarrow}\tilde \Omega^{\bullet}_{\hat Y_Z
\setminus Z} \to 0.$$ Note that there is a natural map by restriction from the sequence to . This forms the commutative diagram with exact rows, $$\label{e:mvseqs}
\xymatrix{
\cdots \ar[r] & {\mathbf{H}}_{DR}^{i-1}(Y) \ar[r] \ar@{=}[d] &
{\mathbf{H}}_{DR}^{i-1}(Y \setminus Z)
\oplus {\mathbf{H}}_{DR}^{i-1,{\operatorname{an}}}(V) \ar[r] \ar[d] & {\mathbf{H}}^{i-1,{\operatorname{an}}}_{DR}(V \setminus Z) \ar[r] \ar[d]
& {\mathbf{H}}_{DR}^i(Y) \ar[r] \ar@{=}[d] & \cdots \\
\cdots \ar[r] & {\mathbf{H}}_{DR}^{i-1}(Y) \ar[r] & {\mathbf{H}}_{DR}^{i-1}(Y \setminus Z)
\oplus {\mathbf{H}}_{DR}^{i-1}(\hat Y_Z) \ar[r] & {\mathbf{H}}^{i-1}_{DR}(\hat Y_Z \setminus Z) \ar[r]
& {\mathbf{H}}_{DR}^i(Y) \ar[r] & \cdots \\
}$$ Finally, note that, when $Y$ is affine, we can also consider the same diagram for ordinary rather than hypercohomology, since the sequences and remain exact on the level of global sections.
### Comparison isomorphisms for smooth varieties
Now consider the case that $Y$ is smooth. Then, we will need the result that a small enough tubular neighborhood $V$ of $Z$ retracts onto $Z$. By Grothendieck’s theorem, this implies $$\label{e:tubnbhd}
{\mathbf{H}}_{DR}^{\bullet,{\operatorname{an}}}(V) \cong H^\bullet_{{\operatorname{top}}}(Z),$$ where as before ${\mathbf{H}}$ denotes hypercohomology (which is necessary since we do not require $Y$ to be affine).
Hartshorne’s theorem [@Har-adrc; @Har-drcav] gives an algebraic analogue of the above statement: $$\label{e:hartshorne}
{\mathbf{H}}_{DR}^\bullet(\hat Y_Z) \cong H^\bullet_{{\operatorname{top}}}(Z).$$ Moreover, the isomorphism composed with the restriction ${\mathbf{H}}_{DR}^{\bullet,{\operatorname{an}}}(V) \to {\mathbf{H}}_{DR}^\bullet(\hat Y_Z)$ is the natural isomorphism . Put together, we deduce that the restriction map is an isomorphism, $$\label{e:comp-smooth1}
{\mathbf{H}}_{DR}^{\bullet,{\operatorname{an}}}(V) {{\;\stackrel{_\sim}{\to}\;}}{\mathbf{H}}_{DR}(\hat Y_Z).$$ Therefore, the five-lemma implies that the vertical arrows in are all isomorphisms. In particular, this yields also $$\label{e:comp-smooth2}
{\mathbf{H}}_{DR}^{\bullet,{\operatorname{an}}}(V \setminus Z) {{\;\stackrel{_\sim}{\to}\;}}{\mathbf{H}}_{DR}^{\bullet}(\hat Y_Z \setminus Z).$$ Note that, when $Y$ is affine, we can also replace hypercohomology with ordinary cohomology (in the second isomorphism), by using for ordinary cohomology.
### Comparison theorem for isolated singularities
\[t:form-an-comp\] Suppose that $X$ is a complex algebraic variety with an isolated singularity at $x \in X$. Then, for sufficiently small contractible neighborhoods $U$ of $x$, there are canonical isomorphisms $$\label{e:an-to-form}
{\mathbf{H}}^{\bullet,{\operatorname{an}}}_{DR}(U) {{\;\stackrel{_\sim}{\to}\;}}{\mathbf{H}}^{\bullet}_{DR}(\hat X_x),
\quad {\mathbf{H}}^{\bullet,{\operatorname{an}}}_{DR}(U \setminus \{x\}) {{\;\stackrel{_\sim}{\to}\;}}{\mathbf{H}}^\bullet_{DR}(\hat X_x \setminus \{x\}).$$ If in addition $U$ is Stein, then we have canonical isomorphisms on cohomology of global sections, $$\label{e:an-to-form-stein}
H^{\bullet,{\operatorname{an}}}_{DR}(U) {{\;\stackrel{_\sim}{\to}\;}}H^{\bullet}_{DR}(\hat X_x),
\quad
H^{\bullet,{\operatorname{an}}}_{DR}(U \setminus \{x\}) {{\;\stackrel{_\sim}{\to}\;}}H^{\bullet}_{DR}(\hat X_x \setminus \{x\}).$$
The theorem also extends to the case where $X$ is an analytic variety with an isolated singularity at $x$, with the same proof as below, since Hironaka’s theorem on resolution of singularities also applies to analytic varieties. (This is a strict generalization of the theorem, since every algebraic variety is also analytic, and the objects above are the same.)
Let $Y \to X$ be a resolution of singularities, and let $Z \subseteq
Y$ be the fiber over $x$. Let $V$ be a tubular neighborhood of $Z$ which retracts to $Z$ and $U$ its image under the resolution, which therefore retracts to $x$. Then the resolution maps restrict to isomorphisms $Y \setminus Z
{{\;\stackrel{_\sim}{\to}\;}}X \setminus \{x\}$, $V \setminus Z {{\;\stackrel{_\sim}{\to}\;}}U \setminus \{x\}$, and $\hat Y_Z \setminus Z {{\;\stackrel{_\sim}{\to}\;}}\hat X_x \setminus \{x\}$. By , we conclude the second isomorphism in .
Now, the above was for specific neighborhoods $U$, namely those obtainable from tubular neighborhoods $V$ of $Z \subseteq Y$. For any smaller contractible neighborhood $U' \subseteq U$ of $x$, the restriction map $H_{{\operatorname{top}}}^\bullet(U \setminus \{x\}) \to
H_{{\operatorname{top}}}^\bullet(U' \setminus \{x\})$ is an isomorphism by Grothendieck’s theorem, and hence the second isomorphism of holds for sufficiently small contractible neighborhoods of $x$.
Consider now for the pair $(X,\{x\})$, with $U$ such that the second isomorphism of holds. The five-lemma then implies that the vertical arrows are all isomorphisms, which implies the first isomorphism of .
Next, the first isomorphism of follows immediately, since $U$ is Stein, so hypercohomology of $U$ and $\hat X_x$ coincides with the cohomology of global sections. Finally, since $X$ is affine, we can consider for the pair $(X,\{x\})$ using ordinary rather than hypercohomology. The five-lemma now implies that the vertical arrows are once again isomorphisms, yielding the second isomorphism of .
Complete intersections with isolated singularities {#s:cy-cplte-int-is}
==================================================
In this section, we explicitly compute $({\mathcal{O}}_X)_{\mathfrak{v}}$, $M(X,{\mathfrak{v}})$, and $\pi_* M(X,{\mathfrak{v}})$, in the case that $X \subseteq Y$ is a locally complete intersection of positive dimension, $Y$ is affine Calabi-Yau, and $X$ has only isolated singularities, equipping $X$ with a top polyvector field as in Example \[ex:cy-cplte-int\]. For $M(X,{\mathfrak{v}})$ itself, the assumption that $Y$ (and hence $X$) is affine is not necessary, using §\[s:ex-dloc\].
We set ${\mathfrak{v}}= H(X)$ (one could equivalently use $LH(X)$, in view of Proposition \[p:vtop-hlh\].) Note that, in the case $X$ is two-dimensional, then $X$ is a Poisson variety and $H(X)$ is the Lie algebra of Hamiltonian vector fields.
Complete intersections: Greuel’s formulas
-----------------------------------------
Here we recall from [@Gre-GMZ] an explicit formula for the de Rham cohomology of an analytic neighborhood of $x$. This is also closely related to the results of [@HLY-plhdRc; @Yau-vniis].
Embed $\hat X_x \subseteq {\mathbf{A}}^n$ cut out by equations $f_1, \ldots,
f_k$ such that $(f_1, \ldots, f_i)$ has only isolated singularities for all $i$. Then define the ideals $$\label{e:jxi-defn}
J_{X,x,i} = (f_1, \ldots, f_{i-1}, \frac{\partial(f_1, \ldots, f_i)}
{\partial(x_{j_1}, \ldots, x_{j_i})}, 1 \leq j_1 \leq \cdots \leq j_i \leq n)
\subseteq {\mathcal{O}}_{\hat {\mathbf{A}}^n,x}.$$ Here $\frac{\partial(f_1, \ldots, f_i)}{\partial(x_{j_1}, \ldots,
x_{j_i})}$ is the determinant of the matrix of partial derivatives $\partial_{x_{j_p}}(f_q), 1 \leq p,q \leq i$. Then, the Milnor number, $\mu_x$, of the singularity of $X$ at $x$ is given by $$\mu_x = \sum_{i=1}^k (-1)^{k-i}{\operatorname{codim}}_{\hat {\mathcal{O}}_{{\mathbf{A}}^n,x}} J_{X,x,i}.$$
\[d:tj-no\] Let $X$ and $x$ be as above. Define the singularity ring, $\mathcal{C}_{X,x}$, of $X$ at $x$ to be $$\mathcal{C}_{X,x} := \hat {\mathcal{O}}_{{\mathbf{A}}^n,x} / (J_{X,x,k},f_k),$$ and define the Tjurina number, $\tau_x$, to be the dimension of $\mathcal{C}_{X,x}$.
Note that the ring $\mathcal{C}_{X,x}$ does not depend on the embedding $\hat X_x \subseteq \hat {\mathbf{A}}^n_x$ and is also definable intrinsically as the quotient of $\hat {\mathcal{O}}_{X,x}$ by the $m$-th Fitting ideal of $\Omega^1_{\hat X_x}$; cf. [@Har-dcr] and Remark \[r:sch-sing\].
[@Gre-GMZ Proposition 5.7.(iii)] \[t:greuel\] If $x$ is an isolated singularity which is locally a complete intersection in the analytic topology, then $$\label{e:gre-fla}
H^\bullet(\tilde \Omega^{\bullet,{\operatorname{an}}}_{X,x})
\cong {\mathbf{k}}^{\mu_x - \tau_x}[-\dim X].$$
Here, $V[-\dim X]$ is the graded vector space concentrated in degree $\dim X$ with underlying vector space $V$.
General structure
-----------------
Since $H(X)$ has finitely many leaves, $M(X,H(X))$ is holonomic. Let $i: Z {\hookrightarrow}X$ be the (finite) singular locus of $X$.
Note that $i_* H^0 i^* M(X,{\mathfrak{v}})$ is the maximal quotient of $M(X,{\mathfrak{v}})$ supported on $Z$. Let $N$ be its kernel. Let $X^\circ :=
X \setminus Z$ and let ${\operatorname{IC}}(X) = j_{!*} \Omega_{X^\circ}$ be the intersection cohomology ${\mathcal{D}}$-module of $X$, i.e., the intermediate extension of $\Omega_{X^\circ}$. Since $j^! M(X,{\mathfrak{v}}) \cong
\Omega_{X^\circ}$, this is a composition factor of $M(X,{\mathfrak{v}})$, and all other composition factors are delta function ${\mathcal{D}}$-modules of points in $Z$. Since $N$ has no quotient supported on $Z$, it must be an indecomposable extension of the form $$\label{e:ciis-ext}
0 \to K {\hookrightarrow}N {\twoheadrightarrow}{\operatorname{IC}}(X) \to 0,$$ where $K$ is supported at $Z$. Then, the structure of $M(X,{\mathfrak{v}})$ reduces to computing $i_*H^0 i^* M(X,{\mathfrak{v}})$, the extension , and how these two are extended. The first question has a nice general answer:
\[t:is-str\] For every $z \in Z$, with $i_z: \{z\} \to X$ the embedding, there is a canonical exact sequence $$\label{e:is-str}
0 \to H_{DR}^{\dim X}(\hat X_z) {\hookrightarrow}H^0 i_z^* M(X,{\mathfrak{v}}) {\twoheadrightarrow}\mathcal{C}_{X,z} \to 0.$$
By Theorem \[t:form-an-comp\], there is a canonical isomorphism $H^{\dim X}(\Omega_{X,z}^{\bullet,{\operatorname{an}}}) {{\;\stackrel{_\sim}{\to}\;}}H_{DR}^{\dim X}(\hat
X_z)$. By Theorem \[t:greuel\], the former has dimension $\mu_z-\tau_z$. On the other hand, $\dim \mathcal{C}_{X,z} =
\tau_z$. We conclude
$i_* H^0 i^* M(X,{\mathfrak{v}}) \cong \bigoplus_{z \in Z} \delta_z^{\mu_z}$.
The following basic result will be useful in the theorem and later on. For an arbitrary scheme $X$ and point $x \in X$, let $(\hat
{\mathcal{O}}_{X,x})^*$ be the continuous dual of $\hat {\mathcal{O}}_{X,x}$ with respect to the adic topology.
\[l:maxquot-coinv\] Let $(X,{\mathfrak{v}})$ and $x \in X$ be arbitrary. Then ${\operatorname{Hom}}(M(X,{\mathfrak{v}}), \delta_x)
\cong ((\hat {\mathcal{O}}_{X,x})^*)^{\mathfrak{v}}$.
Note that ${\operatorname{Hom}}({\mathcal{D}}_X, \delta_x) \cong (\hat {\mathcal{O}}_{X,x})^*$, since the latter are exactly the delta function distributions at $x$. By definition of $M(X,{\mathfrak{v}})$, each $\phi \in {\operatorname{Hom}}(M(X,{\mathfrak{v}}), \delta_x)$ is uniquely determined by $\phi(1)$, which can be any element of $\delta_x$ which is invariant under ${\mathfrak{v}}$.
The theorem can therefore be restated as
\[t:is-coinv\] For all $z \in Z$, there is a canonical exact sequence $$\label{e:is-coinv}
0 \to H_{DR}^{\dim X}(\hat X_z) \to (\hat
{\mathcal{O}}_{X,z})_{\mathfrak{v}}\to \mathcal{C}_{X,z} \to 0.$$ In particular, $\dim (\hat {\mathcal{O}}_{X,z})_{{\mathfrak{v}}} = \mu_z$.
In the case that $Y={\mathbf{A}}^3$ and $X$ is a quasihomogeneous hypersurface, the consequence that $\dim ({\mathcal{O}}_{X,z})_{\mathfrak{v}}= \mu_z=\tau_z$ was discovered in [@AL] without using the earlier results of [@Gre-GMZ].
Let $n := \dim Y$, $m := \dim X$, and $k := n-m$. Let $I_X := (f_1,
\ldots, f_k)$ be the ideal defining $X$. Consider the map $$\Phi: \tilde \Omega^\bullet_X \to \Omega^{\bullet+k}_{Y} / I_X \cdot \Omega^{\bullet+k}_{Y},
\quad \alpha \mapsto \alpha\wedge df_1 \wedge \cdots \wedge df_k,$$ which induces also a map taking the completion at $z$, which we also denote by $\Phi$. Note that, in this formula, we have to lift $\alpha$ to a form on $Y$, but the map is independent of the choice of lift. Furthermore, $\Phi$ is injective, since $X\setminus Z$ is locally transversely cut out by $f_1, \ldots, f_k$. Let $\widetilde{H(X)} \subseteq H(Y)$ be the Lie algebra of vector fields obtained from the $(n-2)$-forms $\Phi(\tilde \Omega^{m-2}_X)$. Then we have an identification $$\label{e:is1}
(\hat {\mathcal{O}}_{X,z})_{{\mathfrak{v}}} {{\;\stackrel{_\sim}{\to}\;}}\Omega^n_{\hat Y_z} / (\widetilde{H(X)}(\hat {\mathcal{O}}_{Y,z}) + I_X) \cdot {\mathsf{vol}}_{\hat Y_z},$$ obtained by multiplying by ${\mathsf{vol}}_{\hat Y_z}$. In turn, $\widetilde{H(X)}(\hat {\mathcal{O}}_{Y,z}) \cdot {\mathsf{vol}}_{\hat Y_z}$ identifies with $d \Phi(\Omega_{\hat Y_z}^{m-1})$. Therefore, $$\label{e:is2}
(\hat {\mathcal{O}}_{X,z})_{{\mathfrak{v}}} {{\;\stackrel{_\sim}{\to}\;}}\Omega^n_{\hat Y_z} / (d \Phi(\Omega^{m-1}_{\hat X_z}) + I_X \Omega^n_{\hat Y_z}).$$ We now compute the RHS. Recall that $\Phi$ is an injection of complexes. The image of $H^m(\tilde \Omega^\bullet_{\hat X_z})$ is a subspace of . Moreover, the quotient of $\Omega^n_{\hat Y_z}$ by this image is $$\label{e:is4}
\mathcal{C}_{X,z} = \Omega^n_{\hat Y_z} / (I_X \Omega^n_{\hat Y_z} + \Phi(\Omega^m_{\hat X_z})).$$ We obtain the desired canonical exact sequence .
We can be more specific about the meaning of $K$ in and use this to describe the derived pushforward $\pi_* M(X,{\mathfrak{v}})$, where $\pi: X \to {\text{pt}}$ is the projection to a point. Let $\pi_i := H^i \pi_*$. If we apply $\pi_*$ to , we obtain isomorphisms $\pi_i N \cong
\pi_i {\operatorname{IC}}(X)$ for $i > 1$, and an exact sequence $$0 \to \pi_1 N {\hookrightarrow}{\operatorname{IH}}^{\dim X - 1}(X) \to \pi_0 K \to \pi_0 N {\twoheadrightarrow}{\operatorname{IH}}^{\dim X}(X) \to 0.$$ Here ${\operatorname{IH}}^*(X)$ denotes the intersection cohomology of $X$, ${\operatorname{IH}}^*(X) := \pi_{\dim X - *} {\operatorname{IC}}(X)$.
Similarly, from the exact sequence $0 \to N {\hookrightarrow}M(X,{\mathfrak{v}}) {\twoheadrightarrow}i_*
H^0 i^* M(X,{\mathfrak{v}}) \to 0$, we obtain isomorphisms $\pi_i(N) \cong \pi_i
M(X,{\mathfrak{v}})$, $i \geq 1$, and a split exact sequence $$0 \to \pi_0 N {\hookrightarrow}({\mathcal{O}}_X)_{{\mathfrak{v}}} {\twoheadrightarrow}H^0 i^* M(X,{\mathfrak{v}}) \to 0.$$ Put together, we obtain
For $i \geq 2$, $\pi_i M(X,{\mathfrak{v}}) \cong {\operatorname{IH}}^{\dim X - i}(X)$. For some decomposition $K = K' \oplus K''$, one has a split exact sequence $$0 \to \pi_1 M(X,{\mathfrak{v}}) {\hookrightarrow}{\operatorname{IH}}^{\dim X - 1}(X) {\twoheadrightarrow}\pi_0 K' \to 0,$$ and an isomorphism $$({\mathcal{O}}_X)_{{\mathfrak{v}}} \cong {\operatorname{IH}}^{\dim X}(X) \oplus \bigoplus_{z \in Z} (\hat {\mathcal{O}}_{X,z})_{\mathfrak{v}}\oplus \pi_0 K''.$$
We plan to show in [@ES-ciiss] that $N = H^0j_! \Omega_{X\setminus\{0\}}$, so one obtains an exact sequence $$0 \to K {\hookrightarrow}N {\twoheadrightarrow}IC(X) \to 0.$$ Moreover, we plan to show that $K = K' = \bigl({\operatorname{Ext}}({\operatorname{IC}}(X), \delta_0)^* \otimes \delta_0\bigr)$. Finally, will then conclude that $\pi_\bullet M(X,{\mathfrak{v}}) \cong H^{\dim X - \bullet}_{{\operatorname{top}}}(X) \oplus
{\mathbf{k}}^{\mu_z}$.
The quasihomogeneous case
-------------------------
Now suppose that $X \subseteq {\mathbf{A}}^n$ where ${\mathbf{A}}^n = {\operatorname{\mathsf{Spec}}}{\mathbf{k}}[x_1,\ldots,x_n]$, each of the $x_i$ is assigned a weight $m_i \geq
1$, and $X$ is cut out by $k:=n-\dim X$ weighted-homogeneous polynomials in the $x_i$. In this case, ${\mathsf{HP}}_0({\mathcal{O}}_X)$ is a nonnegatively graded vector space by weight. Moreover, $M(X,H(X))$ is a weakly ${\mathbf{G}}_m$-equivariant ${\mathcal{D}}$-module which decomposes into weight submodules. Hence, $H^0 i^* M(X,H(X))$ is weight-graded. Then, the proofs of the preceding results generalize to this context (considering also [@Gre-GMZ] and references therein). Moreover, by [@Fer-chdcas] (cf. [@Gre-GMZ Korollar 5.8]), in this case $H_{DR}^\bullet(X)=0$ and implies that $\mu_z=\tau_z$, which is the dimension of the singularity ring (see Definition \[d:tj-no\]). By using the weight-graded versions of the arguments of [@Gre-GMZ] one deduces, for $X_{\text{sing}}$ the scheme-theoretic singular locus of $X$, defined by the ideal $(J_{X,0,k},f_k)$,
\[t:is-qh\] The graded vector space $H^0 i^* M(X,H(X))$ has Poincaré polynomial $$\label{e:qh-iim} P(H^0 i^* M(X,H(X));t) =
P({\mathcal{O}}_{X_{\text{sing}}};t) = P({\mathcal{O}}_{{\mathbf{A}}^n}/(J_{X,0,k},f_k);t)=
\sum_{i=1}^k (-1)^{k-i} P({\mathcal{O}}_{{\mathbf{A}}^n}/J_{X,0,i};t).$$
Since ${\mathcal{O}}_X$ is nonnegatively graded and $X$ is connected, $H(X)$ is spanned by homogeneous vector fields, and $({\mathcal{O}}_X)_{H(X)}$ is finite-dimensional, we conclude that $(\hat {\mathcal{O}}_X)_{H(X)} \cong
({\mathcal{O}}_X)_{H(X)}$. Therefore, Lemma \[l:maxquot-coinv\] implies
\[c:is-qh\] $P(({\mathcal{O}}_X)_{H(X)};t) = P({\mathcal{O}}_{{\mathbf{A}}^n}/(J_{X,0,k},f_k);t)$.
In particular, in this case, ${\operatorname{IH}}^{\dim X}(X) = 0$ and $K''=0$ (i.e., $K=K'$).
\[r:is-qh\] In the case that $k=1$, i.e., $X$ is a quasihomogeneous hypersurface $Z(f)$, the ideal of the singular locus of $X$ is also known as the Jacobi ideal $J_X = (\partial_i f) = (\partial_i f, f)$. For the last equality, let $m_i$ be the weight of $x_i$ for all $i$ as above, and set $m := \sum_i m_i$. Then $f = \frac{1}{m} \sum_i m_i x_i \partial_i f$.
In this case, one can prove the theorem in an elementary way. Namely, we need to show that $$H(X)({\mathcal{O}}_X) = J_X / (f).$$ Equivalently, we have to show that $$\label{e:qhs-cond}
\Omega^{n-1}_X \wedge df + I_X \cdot \Omega^n_{{\mathbf{A}}^n}
= d\Omega^{n-2}_X \wedge df + I_X \cdot \Omega^n_{{\mathbf{A}}^n}.$$ For this, let ${\operatorname{Eu}}:= \sum_i m_i x_i \partial_i$ be the Euler vector field on ${\mathbf{A}}^n$. Set ${\operatorname{Eu}}^\vee := i_{{\operatorname{Eu}}}({\mathsf{vol}}_{{\mathbf{A}}^n}) \in
\Omega^{n-1}_{{\mathbf{A}}^n}$. Then, for all $g \in {\mathcal{O}}_{{\mathbf{A}}^n}$, we have the identities $${\operatorname{Eu}}^\vee \wedge dg = {\operatorname{Eu}}(g) \cdot {\mathsf{vol}}_{{\mathbf{A}}^n}, \quad d(g{\operatorname{Eu}}^\vee) = ({\operatorname{Eu}}(g)+ m \cdot g) \cdot {\mathsf{vol}}_{{\mathbf{A}}^n}.$$ Therefore, we conclude that, for all quasihomogeneous $\alpha \in
\Omega^{n-1}_X$, letting $|\cdot|$ denote the weighted degree function, $$\bar \alpha := \alpha -(|\alpha|+m)^{-1} (d\alpha / {\mathsf{vol}}_{{\mathbf{A}}^n}) {\operatorname{Eu}}^\vee \in d \Omega^{n-2}_{{\mathbf{A}}^n}.$$ Moreover, $$\alpha \wedge df \equiv \bar \alpha \wedge df
\pmod{I_X \cdot \Omega_{{\mathbf{A}}^n}^n}.$$ We conclude , and hence the theorem in this case.
Finite quotients of Calabi-Yau varieties {#s:fqcyvar}
========================================
Let $X$ be an affine connected Calabi-Yau variety and $\Xi$ the top polyvector field inverse to the volume form; for instance, we could have $X={\mathbf{A}}^n$ with the inverse to the standard volume form. In this case, $H(X)=LH(X)=P(X)$. Let $G$ be a finite group acting by automorphisms on $X$ preserving $\Xi$. In this section we will compute the ${\mathcal{D}}$-module $M(X,H(X)^G)$. Everything generalizes without change to the case where $X$ is not affine, using §\[s:ex-dloc\].
Note that, if $X$ is one-dimensional, then $G$ must be trivial. Therefore, we will assume until the end of the section that $\dim X
\geq 2$.
As noticed at the end of §\[ss:vtop\], using the induced top polyvector field on $X/G$, $H(X)^G=P(X/G)$. So we also deduce $M(X/G,P(X/G))=q_*M(X,H(X)^G)^G$ where $q: X \to X/G$ is the projection, and hence also its underived pushforward to a point, $({\mathcal{O}}_{X/G})_{P(X/G)}$. We note that, by Proposition \[p:hg-vtop-lf\], since $\dim X \geq 2$, $H(X)^G$ has finitely many leaves and hence is holonomic, so $P(X/G)$ is as well; however, in general, $H(X/G)$ and $LH(X/G)$ are not holonomic (by Corollary \[c:inc-degloc\], they are holonomic if and only if $X/G$ has only finitely many singular points, i.e., only finitely many points of $X$ have nontrivial stabilizers in $G$).
Recall from §\[ss:vtop\] that we call a subgroup $K < G$ *parabolic* if there exists a point $x \in X$ such that ${\operatorname{Stab}}_G(x)=K$. Let ${\operatorname{Par}}(G)$ be the set of parabolic subgroups of $G$. For $K \in {\operatorname{Par}}(G)$, the connected components of $X^K$ are called *parabolic subvarieties* of $X$. By Proposition \[p:hg-vtop-lf\], these are exactly the closures of the leaves of ${\mathfrak{v}}$, which are the connected components of $(X^K)^\circ = \{x \in X
\mid {\operatorname{Stab}}_G(x)=K\}$.
Let ${\operatorname{Parpt}}(X,G)$ be the collection of points which are parabolic subvarieties; call them *parabolic points*.
Equivalently, the parabolic points $x \in X$ are those such that, for some open neighborhood $U$ containing $x$, ${\operatorname{Stab}}_G(x)$ is strictly larger than the stabilizer of any point in $U \setminus \{x\}$.
There is a canonical isomorphism $$M(X,{\mathfrak{v}}) \cong \Omega_X \oplus \bigoplus_{x \in {\operatorname{Parpt}}(X,G)}
\delta_x \otimes (\hat {\mathcal{O}}_{X,x})_{{\mathfrak{v}}},$$ and each $(\hat {\mathcal{O}}_{X,x})_{{\mathfrak{v}}}$ is finite-dimensional.
Let $X^\circ \subseteq X$ be the inclusion of the open locus where $G$ acts freely. Then, $M(X, H(X)^G)|_{X^\circ} \cong
\Omega_{X^\circ}$. Since $G$ preserves volume, it follows that $X
\setminus X^\circ$ has codimension at least two, since this is true in the local setting $X = {\mathbf{A}}^n$ and $G < {\mathsf{SL}}_n$.
We claim that there are no nontrivial extensions between local systems supported on $X \setminus X^\circ$ and $\Omega_X$. Indeed, this reduces in a formal neighborhood of an arbitrary point to the statement that ${\operatorname{Ext}}^1({\mathcal{O}}_{{\mathbf{A}}^m}, \delta) = 0 = {\operatorname{Ext}}^1(\delta,
{\mathcal{O}}_{{\mathbf{A}}^m})$ when $\delta$ is the delta-function ${\mathcal{D}}$-module of a proper subspace of ${\mathbf{A}}^m$ of codimension $k \geq 2$. Then, by the Künneth theorem, ${\operatorname{Ext}}^i({\mathcal{O}}_{{\mathbf{A}}^m}, \delta)$ and ${\operatorname{Ext}}^i(\delta,
{\mathcal{O}}_{{\mathbf{A}}^m})$ vanish for $i \neq k$, since the statement is true when $m=1$. Therefore, the extension $H^0 j_! \Omega_{X^\circ}$ of $\Omega_X$ must be trivial, i.e., the canonical map $H^0 j_!
\Omega_{X^\circ} \to \Omega_X$ is an isomorphism.
By adjunction, we have a map $H^0 j_!
\Omega_{X^\circ} =\Omega_X \to M(X, H(V)^G)$, and the cokernel of this map is supported on a union of proper parabolic subvarieties of $V$. Suppose that $U \subseteq V^K$ is a maximal such subvariety for $K \in {\operatorname{Par}}(G)$. We claim that $U$ is zero-dimensional, i.e., a finite union of points. By formally localizing in the neighborhood of a generic point of $U$, it suffices to assume that $K=G$. This reduces the claim to the statement:
\[l:nopropqt\] Suppose that $U$ and $W$ are positive-dimensional vector spaces and $G < {\mathsf{GL}}(W)$ is finite. Then $M(U \times W, H(U \times W)^G)$ admits no quotients supported on proper subvarieties of $U \times W$.
Let $X := U \times W$ and ${\mathfrak{v}}:= H(U \times W)^G$. Suppose there were a quotient of $M(X, {\mathfrak{v}})$ supported on $U \times W^K$ for some parabolic subgroup $K < G$. By formally localizing in a neighborhood of a generic point of $U \times W^K$, we can reduce to the case that $K=G$; let us assume this. So we have to show that there is no quotient supported at $U \times \{0\}$.
Since ${\mathfrak{v}}$ includes constant vector fields in the $U$ direction, the defining quotient ${\mathcal{D}}_{X} {\twoheadrightarrow}M(X, {\mathfrak{v}})$ factors through ${\mathcal{D}}_{X} {\twoheadrightarrow}\Omega_U \boxtimes {\mathcal{D}}_W$. Moreover, given a vector field $\xi \in {\mathfrak{v}}$, write $\xi=\xi_1 + \xi_2$ where $\xi_1
\in {\operatorname{Vect}}(U) \otimes {\mathcal{O}}_W$ and $\xi_2 \in {\mathcal{O}}_U \otimes
{\operatorname{Vect}}(W)^G$. Let $D: {\operatorname{Vect}}(X) \to {\mathcal{O}}_X$ be the standard divergence function, i.e., $D(\xi)=L_{\xi} \omega / \omega$, where $\omega$ is the standard volume form on $X$. Then, since ${\mathfrak{v}}$ includes constant vector fields in the $U$ direction, $\xi_1 + D(\xi_1) \in
{\mathfrak{v}}\cdot {\mathcal{D}}_X$. Thus, $\xi_2 - D(\xi_1) = \xi_2 + D(\xi_2) \in
{\mathfrak{v}}\cdot {\mathcal{D}}_X$ as well. Conversely, the constant vector fields in the $U$ direction together with elements $\xi_2 + D(\xi_2)$ span ${\mathfrak{v}}\cdot {\mathcal{D}}_X$. We conclude that $M(X,{\mathfrak{v}}) = {\mathfrak{v}}\cdot {\mathcal{D}}_X \setminus {\mathcal{D}}_X$ is of the form $$M(X,{\mathfrak{v}}) = \Omega_U \boxtimes N, \quad N = \langle \xi +
D(\xi) \mid \xi \in {\operatorname{Vect}}(W)^G \rangle \cdot {\mathcal{D}}_W \setminus {\mathcal{D}}_W.$$ Therefore, the lemma reduces to showing that $N$ admits no quotient supported at $0 \in W$. First of all, let ${\operatorname{Eu}}_W \in {\operatorname{Vect}}(W)$ be the Euler vector field on $U$. Then ${\operatorname{Eu}}_W + D({\operatorname{Eu}}_W) = ({\operatorname{Eu}}_W + \dim(W))
\in {\mathfrak{v}}\cdot {\mathcal{D}}_X$. On the other hand, since $\dim(W) > 0$, ${\operatorname{Eu}}_W + \dim(W)$ acts by an automorphism on every quotient supported at zero (note that sections of the delta function ${\mathcal{D}}$-module are in nonpositive polynomial degree, and homogeneous sections in degree $m
\leq 0$ are annihilated by ${\operatorname{Eu}}+ m$ (since we are using right ${\mathcal{D}}$-modules)). Thus, $N$ admits no such quotient.
We conclude that the cokernel of the inclusion $\Omega_X {\hookrightarrow}M(X,
{\mathfrak{v}})$ is supported at finitely many points, i.e., it is a direct sum of delta-function ${\mathcal{D}}$-modules at these points. Since we assumed that $\dim X \geq 2$, ${\operatorname{Ext}}(\Omega_X, \delta)=0$ when $\delta$ is such a delta-function ${\mathcal{D}}$-module (this follows because it is true in the case $X={\mathbf{A}}^n$ and the point is the origin). Therefore, $M(X,{\mathfrak{v}})$ is semisimple.
It remains only to compute the multiplicity of $\delta_x$. Note that this must be finite-dimensional since $M(X,{\mathfrak{v}})$ is holonomic. The result thus follows from Lemma \[l:maxquot-coinv\].
The results of this section can be generalized to the case where $G$ acts on $X$ preserving $\Xi$ only up to scaling. Then, $X/G$ is no longer equipped with a top polyvector field, but it is equipped with a divergence function from $X$. Indeed, since $G$ preserves the flat connection which annihilates the volume form on $X$, and this equips $X/G$ with a flat connection on its (possibly nontrivial) canonical bundle. So in this case one still has $q_* M(X,H(X)^G)^G \cong
M(X/G, P(X/G))$, where $q: X \to X/G$ is the quotient map, and $P(X/G)$ is interpreted as in §\[s:div\].
In this case, the above results go through without change if $X$ has no parabolic subvarieties of codimension one. However, when there are parabolic subvarieties of codimension one, then $M(X, H(X)^G)$ is no longer semisimple: although Lemma \[l:nopropqt\] still implies that it has no quotients supported on proper subvarieties of $X$, we can have nontrivial extensions on the bottom by submodules supported on proper subvarieties. Let $j^\circ: X^\circ \to X$ be the inclusion of the locus where $G$ acts freely, and $j': X' \to X$ the inclusion of the possibly larger locus which is the complement of codimension-one parabolic subspaces (an affine subvariety). Then, $H^0 j^\circ_! \Omega_{X^\circ} = j'_! \Omega_{X'}$, which is not equal to $\Omega_X$ when $X' \neq X$. We then obtain by the argument of the proof an exact sequence $$j'_! \Omega_{X'} \to M(X, H(X)^G) \to \bigoplus_{x \in {\operatorname{Parpt}}(X,G)}
\delta_x \otimes (\hat {\mathcal{O}}_{X,x})_{{\mathfrak{v}}} \to 0.$$ Moreover, the computation of Lemma \[l:nopropqt\] shows that the first map in the sequence above is injective in codimension one, i.e., restricted to the formal neighborhood of a generic point of any component of $X \setminus X'$, it is injective. So, for $X \neq X'$, $M(X, H(X)^G)$ is not semisimple.
Symmetric powers of varieties {#s:sym}
=============================
Given $(X, {\mathfrak{v}})$, note that ${\mathfrak{v}}$ also acts naturally on the symmetric powers $S^n X := X^n / S_n$. Then, the diagonal embedding of $X$ into $S^n X$ is invariant under the flow of ${\mathfrak{v}}$, and more generally, arbitrary diagonal embeddings are invariant.
In this section, we compute the coinvariants $({\mathcal{O}}_{S^n X})_{{\mathfrak{v}}}$ as well as the ${\mathcal{D}}$-module $M(S^n X, {\mathfrak{v}})$ for all $n \geq 1$ in the transitive (affine) cases of §\[s:Csla\] (the “global” versions of the simple Lie algebras of vector fields). In the symplectic case this specializes to the main result of [@hp0weyl]. Our main result says that, in the Calabi-Yau and symplectic cases, this is a direct sum of the pushforwards under $X^n {\twoheadrightarrow}S^n X$ of the canonical ${\mathcal{D}}$-modules $\Omega_\Delta$ as $\Delta$ ranges over the diagonal subvarieties $\Delta \subseteq X^n$ up to the action of $S_n$. In other words, these are the intersection cohomology ${\mathcal{D}}$-modules of the diagonal subvarieties of $S^n X$. In the locally conformally symplectic case, and in a more general transitive setting that includes all of these cases, we prove the same result, except replacing $\Omega_\Delta$ by the diagonal embedding of $M(X,{\mathfrak{v}})$. Moreover, when $X$ is a contact variety and ${\mathfrak{v}}=H(X)$, or $X$ is smooth and ${\mathfrak{v}}={\operatorname{Vect}}(X)$, we show that $M(S^n X, {\mathfrak{v}}) = 0$, and extend these cases to a more general transitive setting where ${\mathfrak{v}}$ does not flow incompressibly.
More generally, we will prove general structure theorems on $M(S^n X,
{\mathfrak{v}})$ in the case that ${\mathfrak{v}}$ is transitive and satisfies a certain condition we call *quasi-locality*, which essentially says that its restriction to the $m$-th infinitesimal neighborhood of every finite set is equal to the sum of its restrictions to the $m$-th infinitesimal neighborhood of each point in the set. For convenience, we will also generally assume that $X$ is connected; it is easy to remove this assumption.
Relation to Lie algebras for $S^n X$
------------------------------------
The study of $S^n X$ under ${\mathfrak{v}}$ is closely related to the study of $S^n X$ under its own associated Lie algebras of vector fields. Note that ${\mathcal{O}}_{S^n X} = {\operatorname{\mathsf{Sym}}}^n {\mathcal{O}}_X$ is spanned by elements $f^{\otimes n}$ for $f \in {\mathcal{O}}_X$. Let ${\text{symm}}: {\mathcal{O}}_X^{\otimes n} \to {\operatorname{\mathsf{Sym}}}^n {\mathcal{O}}_X$ be the symmetrization map, $${\text{symm}}(f_1 \otimes \cdots \otimes f_n) = \frac{1}{n!} \sum_{\sigma \in
S_n} f_{\sigma(1)} \otimes \cdots \otimes f_{\sigma(n)}.$$ Note that, if $X$ is Poisson with bivector field $\pi$, then so is $S^n X$, using the unique Poisson bracket on ${\operatorname{\mathsf{Sym}}}^n {\mathcal{O}}_X$ obtained from the Leibniz rule; in other words, one can consider the bivector field $\sum_{i=1}^n \pi^{i}$ on $X^n = {\operatorname{\mathsf{Spec}}}{\mathcal{O}}_X^{\otimes n}$, where $\pi^i = {\operatorname{Id}}^{\otimes (i-1)} \otimes \pi \otimes {\operatorname{Id}}^{\otimes (n-i)}
\in (\wedge_{{\mathcal{O}}_X} T_X)^{\otimes n}$ denotes $\pi$ acting on the $i$-th component. This then restricts to symmetric functions ${\mathcal{O}}_{S^n X} = {\operatorname{\mathsf{Sym}}}^n {\mathcal{O}}_X$.
If $X$ is even-dimensional and equipped with a top polyvector field $\Xi$, then $S^n X$ is equipped with the top polyvector field $\wedge^n \Xi$.
As discussed in Remark \[r:jac-prod\], when $X$ is Jacobi, there is no natural Jacobi structure induced on $X^n$ and hence neither on $S^n
X$.
We then have the following elementary proposition (the first part was essentially used in [@hp0weyl]):
\[p:lie-snx\]
1. If $X$ is Poisson, then $M(S^n X, H(X)) \cong M(S^n X, H(S^n X))$;
2. For $X$ even-dimensional and equipped with a top polyvector field, $P(X) \subseteq P(S^n X)$;
3. For $X$ equipped with a divergence function $D$ on a coherent subsheaf $N \subseteq T_X$, one has $P(X,D) \subseteq P(S^n X,D)$, where $S^n$ is equipped with a divergence function on ${\mathcal{O}}_{S^n X} \cdot N$, using the natural embedding of vector spaces $N \subseteq T_X
{\hookrightarrow}T_{S^n X}$ (via extending derivations from ${\mathcal{O}}_X$ to ${\mathcal{O}}_{S^n X} = {\operatorname{\mathsf{Sym}}}^n_{\mathbf{k}}{\mathcal{O}}_X$);
4. For general $X$, ${\operatorname{Vect}}(X) \subseteq {\operatorname{Vect}}(S^n X)$.
\(i) Given $f \in {\mathcal{O}}_X$, it is evident that (up to normalization) $n\cdot \xi_{{\text{symm}}(f \otimes 1^{\otimes (n-1)})}$ identifies with $\xi_f
\in H(X)$. Hence $H(X) \subseteq H(S^n X)$ (this is also a special case of part (ii)). Next, $H(S^n X)$ is spanned by the vector fields $\xi_{f^{\otimes n}} = n \cdot {\text{symm}}(\xi_f \otimes f^{\otimes (n-1)})$ for $f \in {\mathcal{O}}_X$. Note the identities $\xi_f(f)=0$ and $\xi_{f^i} = i
f^{i-1} \xi_f$. Thus, for all $i \geq 1$, $$\begin{gathered}
{\text{symm}}(\xi_{f^i} \otimes 1^{\otimes (n-1)}) \cdot {\text{symm}}(f^{\otimes
(n-i-1)} \otimes 1^{\otimes (i+1)})
\\ = \frac{i}{n}{\text{symm}}(\xi_{f^i} \otimes f^{\otimes (n-i-1)} \otimes 1^{\otimes i}) \\+
\frac{n-i}{n}{\text{symm}}(\frac{i}{i+1} \cdot \xi_{f^{i+1}} \otimes
f^{\otimes (n-i-2)} \otimes 1^{\otimes (i+1)}).\end{gathered}$$ The LHS is in $H(X) \cdot {\mathcal{D}}_X$, and the RHS terms, taken over all $i \geq 1$, generate ${\text{symm}}(\xi_f \otimes f^{\otimes (n-1)})$, as desired.
\(ii) It is evident that, if a vector field preserves a top polyvector field $\Xi$ on $X$, then it also preserves $\wedge^n \Xi$ on $S^n X$.
\(iii) Similarly, if a vector field $\xi$ preserves a divergence function $D$, i.e., $D(\xi) = 0$, then also it preserves the induced divergence function on $S^n X$, i.e., the induced divergence function on $S^n X$ by definition also kills $\xi$, viewed as a vector field on $S^n X$.
\(iv) Similarly, given a vector field $\xi \in {\operatorname{Vect}}(X)$, we can take the sum $\sum_{i=1}^n \xi^i \in {\operatorname{Vect}}(X^n)$ which descends to ${\operatorname{Vect}}(S^n X)$.
Note that the isomorphism of (i) does not extend, in general, to the cases of top polyvector fields. For instance, when $X$ is symplectic, then by part (i), viewed as a Poisson variety, $H(S^n
X)$ and $H(X)$ determine the same ${\mathcal{D}}$-module, which is holonomic since $S^n X$ has finitely many symplectic leaves (the images of the diagonal embeddings). However, since the singular locus of $S^n X$ is infinite for $n \geq 2$, by Corollary \[c:vtop-leaves\], $H(S^n
X,\wedge^n {\mathsf{vol}}_X^{-1})$ does not have finitely many leaves, and by Corollary \[c:vtop-hol\] the associated ${\mathcal{D}}$-module is not holonomic.
Diagonal embeddings
-------------------
Let $\Delta_i: X \to X^i$ be the standard diagonal embeddings for all $i \geq 1$. Let ${\operatorname{pr}}_n: X^n \to S^n X$ be the projection. Recall that a partition $\lambda$ of $n$, which we denote by $\lambda \vdash n$, is a tuple $(\lambda_1, \ldots, \lambda_k)$ with $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k \geq 1$ and $\lambda_1 + \cdots + \lambda_k = n$. In this case the length, $|\lambda|$, of $\lambda$ is defined by $|\lambda|:=k$. Given a partition $\lambda \vdash n$, define the product of diagonal embeddings $$\Delta_\lambda:= \Delta_{\lambda_1} \times \cdots \times
\Delta_{\lambda_{|\lambda|}}: X^{|\lambda|} \to X^n.$$ Now, composing with ${\operatorname{pr}}_n$, we obtain a map $X^{|\lambda|} \to S^n
X$. On the complement of diagonals in $X^{|\lambda|}$, this is a covering onto its image whose covering group is the subgroup $S_\lambda <
S_{|\lambda|}$ preserving the partition $\lambda$. Explicitly, $S_{\lambda} = S_{r_1} \times \cdots \times S_{r_k}$ where, for all $j$, $$\lambda_{r_1+\cdots+r_j} > \lambda_{r_1+\cdots+r_j+1} = \lambda_{r_1 +
\cdots + r_j + 2} = \cdots = \lambda_{r_1 + \cdots + r_j + r_{j+1}}.$$
A morphism of graded algebras
-----------------------------
Consider the canonical morphism of graded algebras $$\label{e:sym-tr-invts}
\Phi: {\operatorname{\mathsf{Sym}}}(t \cdot (({\mathcal{O}}_X)^*)^{{\mathfrak{v}}}[t]) \to \bigoplus_{n \geq 0}
(({\mathcal{O}}_{S^n X})^*)^{{\mathfrak{v}}},$$ given by the formula $$\Phi(t^{r_1} \phi_1 \otimes \cdots \otimes t^{r_k} \phi_k)(f_1 \otimes
\cdots \otimes f_{r_1+\cdots+r_k}) = \prod_{i=1}^k
\phi_i(f_{r_1+\cdots+r_{i-1}+1} \cdots f_{r_1+\cdots+r_i}).$$ Let us explain the graded algebra structures in . First, the grading is by degree in $t$ on the left-hand side and by degree in $n$ on the right-hand side. The algebra structure on the left-hand side is as in a symmetric algebra. The algebra structure on the right-hand side is obtained from the natural inclusions $${\mathcal{O}}_{S^{n+m}X} {\hookrightarrow}{\mathcal{O}}_{S^n X} \otimes {\mathcal{O}}_{S^m X}.$$ In other words, the above maps are the symmetrization maps, $$(f_1 \otimes \cdots \otimes f_{m+n}) \mapsto \frac{m!n!}{(m+n)!}
\sum_{\underset{|I|=n}{I \subseteq \{1,\ldots, m+n\}}} f_I \otimes f_{I^c},$$ where $f_I := \bigotimes_{i \in I} f_i$, and $I^c$ is the complement of $I$.
This induces a coproduct on $\bigoplus_{n \geq 0} {\mathcal{O}}_{S^n X}$ and hence an algebra structure on $\bigoplus_{n \geq 0} {\mathcal{O}}_{S^n
X}^*$. The ${\mathfrak{v}}$-invariants form a subalgebra.
Moreover, replacing $({\mathcal{O}}_{S^n X})_{{\mathfrak{v}}}$ by the derived pushforward $\pi_\bullet M(S^n X, {\mathfrak{v}})$ for $\pi: S^n X \to {\text{pt}}$ the projection to a point, we obtain a bigraded algebra $\bigoplus_{n \geq 0} \pi_\bullet M(S^n
X, {\mathfrak{v}})^*$, in de Rham and homological degrees. Then becomes $$\label{e:sym-can-invts}
\Phi: {\operatorname{\mathsf{Sym}}}(t \cdot \pi_\bullet M(X,{\mathfrak{v}})^*[t]) \to \bigoplus_{n \geq 0}
\pi_\bullet M(S^n X, {\mathfrak{v}})^*.$$ Here, $\bullet$ is the homological degree, and the symmetric algebra is supersymmetric where the parity is given by the homological degree (note that this *differs* from the de Rham parity in the case that $\dim X$ is odd). By Proposition \[p:tr\] and Example \[ex:cy2\], in the case that $X$ is symplectic or Calabi-Yau, can be restated as $$\label{e:sym-can-invts2}
{\operatorname{\mathsf{Sym}}}(t \cdot H^{\dim X - \bullet}(X)^*[t]) \to \bigoplus_{n \geq 0}
\pi_\bullet M(S^n X, {\mathfrak{v}})^*.$$
Quotients of $M(S^nX)$ supported on diagonals
---------------------------------------------
For arbitrary $(X, {\mathfrak{v}})$, since each $\Delta_\lambda$ is a closed embedding, one has a natural epimorphism $$M(X^n, {\mathfrak{v}}) {\twoheadrightarrow}(\Delta_\lambda)_* M(X^{|\lambda|}, {\mathfrak{v}}),$$ Next, note that $M(X^n, {\mathfrak{v}})$ is an $S_n$-equivariant ${\mathcal{D}}$-module, and one has $({\operatorname{pr}}_n)_* M(X^n, {\mathfrak{v}})^{S_n} \cong
M(S^n X, {\mathfrak{v}})$. The morphism above descends to a natural map $$M(S^n X, {\mathfrak{v}}) {\twoheadrightarrow}({\operatorname{pr}}_n)_* (\Delta_\lambda)_* M(X^{|\lambda|},
{\mathfrak{v}})^{S_\lambda}.$$ Summing over $\lambda$, we obtain a natural map $$\label{e:gen-sn}
M(S^n X, {\mathfrak{v}}) \to \bigoplus_{\lambda \vdash n} ({\operatorname{pr}}_n)_*
\bigl( (\Delta_\lambda)_* M(X^{|\lambda|},{\mathfrak{v}}) \bigr)^{S_\lambda}$$ In the case that $X$ is symplectic or Calabi-Yau, by Proposition \[p:tr\] and Example \[ex:cy2\], can be restated as $$\label{e:gen-sn2}
M(S^n X, {\mathfrak{v}}) \to \bigoplus_{\lambda \vdash n} ({\operatorname{pr}}_n)_*
\bigl( (\Delta_\lambda)_* \Omega_X^{\boxtimes |\lambda|} \bigr)^{S_\lambda}.$$
Main result
-----------
\[t:sym\]
1. If $X$ has pure dimension at least two and is locally conformally symplectic or Calabi-Yau, then with ${\mathfrak{v}}= H(X)$, and are isomorphisms.
2. If $(X,{\mathfrak{v}})$ is an (odd-dimensional) contact variety with ${\mathfrak{v}}=H(X)$, or $(X,{\mathfrak{v}})$ is connected, smooth, and positive-dimensional with ${\mathfrak{v}}={\operatorname{Vect}}(X)$, then $M(S^n X, {\mathfrak{v}}) =
0$.
For the case where $X$ is a Calabi-Yau curve, ${\mathfrak{v}}$ is one-dimensional, and $M(S^n X, {\mathfrak{v}})$ is not holonomic for $n > 1$.
In the symplectic and Calabi-Yau cases, one can alternatively consider $H(S^n X)$, $LH(S^n X)$, and $P(S^n X)$, where now $S^n X$ is viewed as either a Poisson variety (when $X$ is symplectic) or as a variety equipped with a top polyvector field (when $X$ is even-dimensional Calabi-Yau) or more generally one can consider $H(S^n X,D)$ and $P(S^n X,D)$ when $X$ is odd-dimensional and equipped with a divergence function on $T_{S^n X} =
T_{X^n}^{S_n}$ obtained from the Calabi-Yau divergence function on $X^n$. It is easy to see that the image of the map in is invariant under all of these, since on each leaf, i.e., the complement in a diagonal ${\operatorname{pr}}_n \circ
\Delta_\lambda(X^{|\lambda|})$ of smaller diagonals, the image of the corresponding functionals on the left-hand side are supported on this diagonal and invariant under all vector fields that preserve the given structure. Moreover, in the symplectic case, if we instead use $H(S^n X)$ (or $LH(S^n X)$), we will obtain the same result in view of Proposition \[p:lie-snx\].(i) (as already noticed in [@hp0weyl]). This recovers the main result of [@hp0weyl] (where this observation was also used in the proof).
In the Calabi-Yau case, one can replace ${\mathfrak{v}}$ on the RHS of by $P(S^n X)$, since here one also has $P(X)
\subseteq P(S^n X)$, so the isomorphism factors through the same expression with $P(S^n X)$-invariants.
However, in the Calabi-Yau case, one cannot replace the RHS with $H(S^n X)$ or $LH(S^n X)$-invariants, since $H(X)$ is not contained in these in general. In fact, for $n \geq 2$, these invariants are infinite-dimensional: already when $X = {\mathbf{A}}^2$ equipped with the standard volume form, $S^2 {\mathbf{A}}^2 \cong ({\mathbf{A}}^2 / ({\mathbf{Z}}/2)) \times
{\mathbf{A}}^2$, so the coinvariants $({\mathcal{O}}_{S^2 {\mathbf{A}}^2})_{H_{\wedge^2 \Xi}(S^2
{\mathbf{A}}^2)} = ({\mathcal{O}}_{S^2 {\mathbf{A}}^2})_{LH_{\wedge^2 \Xi}(S^2 {\mathbf{A}}^2)}$ are infinite-dimensional by Remark \[r:inc-prod\].
Theorem \[t:sym\] may generalize in some form to the case where $X$ is not necessarily transitive, but has a finite degenerate locus. As a first step, in [@ESsym], the authors prove that, when $X
\subseteq {\mathbf{A}}^3$ is a quasihomogeneous isolated surface singularity and ${\mathfrak{v}}=H(X)$, then abstractly one still has an isomorphism $$\label{e:abst-iso-symqh}
{\operatorname{\mathsf{Sym}}}(t \cdot (({\mathcal{O}}_X)^*)^{{\mathfrak{v}}}[t]) \cong \bigoplus_{n \geq 0}
(({\mathcal{O}}_{S^n X})^*)^{{\mathfrak{v}}},$$ but only as algebras graded by symmetric power degree, not by the weight degree in ${\mathcal{O}}_X$. (To correct this, one can assign $t$ weight degree $-d$, where the hypersurface cutting out $X$ has weight $d$ (note that here ${\mathcal{O}}_X$ has nonnegative weight and $({\mathcal{O}}_X)^*$ has nonpositive weight). Then one does obtain an isomorphism of graded algebras.)
Does the abstract algebra isomorphism , graded only by symmetric power degree, extend to the case where $X \subseteq {\mathbf{A}}^n$ is an arbitrary quasihomogeneous complete intersection with an isolated singularity, equipped with its top polyvector field from Example \[ex:cy-cplte-int\]? Can it be corrected to an abstract bigraded isomorphism by assigning $t$ the appropriate weight?
Does the abstract algebra isomorphism extend to the case of arbitrary (not necessarily quasihomogeneous) complete intersections with isolated singularities? What about if the complete intersection condition is dropped?
Finally, we remark that, even as nonequivariant ${\mathcal{D}}$-modules, the two sides of are *not* in general isomorphic, because $M(S^n X, {\mathfrak{v}})$ is not in general semisimple. For the case where $X$ is a quasihomogeneous surface with an isolated singularity, the authors hope to compute $M(S^n X, {\mathfrak{v}})$ in [@ES-ciiss]. There, only in the case where $X$ has genus zero, i.e., the hypersurface is a du Val singularity, does it hold that $M(X,{\mathfrak{v}})$ is semisimple; in all other cases, the two sides of are not isomorphic as (nonequivariant) ${\mathcal{D}}$-modules for $n \geq 2$.
In the case of the du Val singularities, the two sides of are only abstractly isomorphic as nonequivariant ${\mathcal{D}}$-modules, by [@hp0weyl §1.3]. One can introduce a correction analogous to the above one to the RHS which makes the two sides isomorphic as ${\mathbf{G}}_m$-equivariant ${\mathcal{D}}$-modules, but we do not know of any natural isomorphism between the two.
Smooth and contact varieties {#ss:sym-sc}
----------------------------
By Theorem \[t:sym\], in the case that $(X,{\mathfrak{v}})$ is either $(X,
{\operatorname{Vect}}(X))$ for smooth $X$, or $(X, H(X))$ for $X$ an odd-dimensional contact variety, then $M(S^n X, {\mathfrak{v}}) = 0$ for all $n \geq 0$. However, it turns out that $M(X^n, {\mathfrak{v}})$ itself is *nonzero* when $n > \dim X$. Moreover, this can be explicitly computed as an $S_n$-equivariant ${\mathcal{D}}$-module.
We first construct some canonical quotients $M(X^n, {\operatorname{Vect}}(X)) {\twoheadrightarrow}(\Delta_n)_* \Omega_X$. Let $d := \dim X$. We can identify global sections of $(\Delta_n)_* \Omega_X$ with ${\mathcal{O}}_{\Delta_n(X)}$-linear polydifferential operators $\hat {\mathcal{O}}_{X^n, \Delta_n(X)} \to
\Omega_{\Delta_n(X)}$. Then, we consider the operator $$\phi_{n,d}: (f_1 \otimes \cdots \otimes f_n) \mapsto
f_{d+2} \cdots f_n \sum_{\sigma \in S_{d+1}}
\frac{1}{(d+1)!} {\operatorname{sign}}(\sigma)
f_{\sigma(1)} df_{\sigma(2)} \wedge \cdots \wedge df_{\sigma(d+1)}.$$ We can see that $\phi_{n,d}$ is ${\mathcal{O}}_{\Delta_n(X)}$-linear (to ensure this, we had to skew-symmetrize over $S_{d+1}$ rather than $S_d$). Moreover, ${\mathbf{k}}[S_n] \cdot \phi_{n,d}$ is actually preserved by ${\mathbf{k}}[S_{n+1}]$, and as a representation of $S_{n+1}$, it is $${\operatorname{Ind}}_{S_d \times S_{n-d-1}} ({\operatorname{sign}}\boxtimes \, {\mathbf{k}}).$$ Thus, ${\mathbf{k}}[S_n] \cdot \phi_{n,d}$ has dimension ${n-1 \choose d}$. Let $L_n$ be the $S_n$-equivariant local system supported on $\Delta_n(X)$ of rank ${n-1 \choose d}$ corresponding to this quotient (as a nonequivariant local system, it is $((\Delta_n)_* \Omega_X)^{\oplus
{n-1 \choose d}}$). More generally, given a decomposition $\{1,\ldots,n\} = P_1 \sqcup \cdots \sqcup P_m$ into cells, let $L_{P_1} \boxtimes \cdots \boxtimes L_{P_m}$ denote the corresponding tensor product of local systems $L_{|P_i|}$ in the components $P_i$ (i.e., these are all obtained by permutation of components from the local system $L_{|P_1|} \boxtimes \cdots \boxtimes L_{|P_m|}$). This is equivariant with respect to the subgroup of $S_n$ preserving the decomposition, which is isomorphic to $S_{|P_1|} \times \cdots \times S_{|P_m|}$. Note that it is nonzero if and only if $|P_i| > d$ for all $i$.
Suppose that $(X,{\mathfrak{v}})$ is either $(X,
{\operatorname{Vect}}(X))$ for smooth $X$, or $(X, H(X))$ for $X$ an odd-dimensional contact variety. Then, we have an isomorphism as $S_n$-equivariant local systems, $$M(X^n, {\mathfrak{v}})
= \bigoplus_{m \geq 1, P_1 \sqcup \cdots \sqcup P_m = \{1,\ldots,n\}}
L_{P_1} \boxtimes \cdots \boxtimes L_{P_m}.$$
Note in the theorem that, even though the individual summands on the RHS are not $S_n$-equivariant, the direct sum is canonically $S_n$-equivariant.
Quasi-locality and a generalization of Theorem \[t:sym\] {#ss:ql}
--------------------------------------------------------
\[d:ql\] Say that $(X, {\mathfrak{v}})$ is *quasi-local* if, for every $n$-tuple of distinct points $x_1, \ldots, x_n \in X$, and every choice of positive integers $m_1, \ldots, m_{n-1} \geq 1$, the subspace of ${\mathfrak{v}}$ of vector fields vanishing to orders $m_i$ at $x_i$ for all $1 \leq i \leq n-1$ topologically span ${\mathfrak{v}}|_{\hat X_{x_n}}$.
Equivalently, as stated in the beginning of the section, the evaluation of ${\mathfrak{v}}$ at every subscheme supported at a finite subset $S \subseteq X$ is the direct sum of its evaluations at each connected component of $S$ (i.e., at each subscheme of $S$ supported on a point of $S_{\text{red}}$).
If $(X, {\mathfrak{v}})$ is quasi-local, then the leaves of $(S^n X, {\mathfrak{v}})$ are the images of the products of leaves of $X$ under ${\operatorname{pr}}_n$. In particular, if $(X,{\mathfrak{v}})$ has finitely many leaves, so does $(S^n X,
{\mathfrak{v}})$, and the latter is holonomic.
At each point ${\operatorname{pr}}_n \circ \Delta_\lambda(x_1, \ldots, x_{|\lambda|})$, the pushforward $({\operatorname{pr}}_n)_*$ induces an isomorphism of vector spaces, $$\label{e:ql-mtr}
(T_{\Delta_\lambda(x_1, \ldots, x_{|\lambda|})} X^n)^{S_\lambda}
{{\;\stackrel{_\sim}{\to}\;}}{\mathfrak{v}}|_{{\operatorname{pr}}_n \circ \Delta_\lambda(x_1, \ldots, x_{|\lambda|})}.$$ Therefore, along each diagonal, the flow of ${\mathfrak{v}}$ is transitive along the images of the products of leaves of $X$.
If $X$ is Jacobi or equipped with a top polyvector field, then $(X,H(X))$ is quasi-local. Similarly, $(X, {\operatorname{Vect}}(X))$ is quasi-local.
We first consider the Jacobi case. Given points $x_1,
\ldots, x_n \in X$, and any orders $m_1, \ldots, m_{n-1} \geq 1$, we can consider functions which vanish up to order $m_i$ at $x_i$ for $1 \leq i \leq n-1$. Since the $x_i$ are distinct, these functions topologically span $\hat {\mathcal{O}}_{X,x_n}$. Therefore, the Hamiltonian vector fields of such functions topologically span all Hamiltonian vector fields in the formal neighborhood $\hat X_{x_n}$.
Next consider the Calabi-Yau case. This is similar: we replace functions which vanish up to order $m_i$ at $x_i$ for $1 \leq i \leq
n-1$ by $(\dim X - 2)$-forms with this vanishing property. Again, these topologically span $\Omega_{\hat X_{x_n}}$, and we conclude the result.
For the case of all vector fields, this is immediate.
\[t:sym-ql\] Suppose that $(X,{\mathfrak{v}})$ is transitive and quasi-local and that $X$ has pure dimension at least $2$.
- If ${\mathfrak{v}}$ flows incompressibly, then is an isomorphism if and only if:
- For all $n$, and any (or every) $x \in X$, the space of ${\mathfrak{v}}$-invariant polydifferential operators ${\operatorname{\mathsf{Sym}}}^n \hat {\mathcal{O}}_{X,x}
\to \hat {\mathcal{O}}_{X,x}$ is spanned by the multiplication operator.
- If ${\mathfrak{v}}$ does not flow incompressibly, then $M(S^n X, {\mathfrak{v}}) = 0$ for all $n \geq 1$ if and only if, for all $n \geq 1$, there are no ${\mathfrak{v}}$-invariant polydifferential operators ${\operatorname{\mathsf{Sym}}}^{n} \hat {\mathcal{O}}_{X,x} \to \Omega_{\hat X_x}$.
We will prove this theorem as a consequence of more general results in the not-necessarily quasi-local case below. First we explain why this theorem implies Theorem \[t:sym\]:
- Let $(X,H(X))$ be locally conformally symplectic or Calabi-Yau of pure dimension at least two. Then (\*) of Theorem \[t:sym-ql\] is satisfied.
- In the case where $(X,{\mathfrak{v}})$ is either an odd-dimensional contact variety with ${\mathfrak{v}}= H(X)$, or smooth with ${\mathfrak{v}}={\operatorname{Vect}}(X)$, then for all $x \in X$, all ${\mathfrak{v}}$-invariant polydifferential operators $\hat {\mathcal{O}}_{X,x}^{\otimes n} \to \Omega_{\hat X,x}$ are spanned over ${\mathbf{k}}[S_n]$ by the operator $$(f_1 \otimes \cdots \otimes f_n) \mapsto f_1 \cdots f_{n-\dim X}
df_{n-\dim X+1} \wedge \cdots \wedge df_n.$$ In particular there are no symmetric such operators.
\(i) This relies on the Darboux theorem, following [@hp0weyl Lemma 2.1.8]. In a formal neighborhood $\hat X_x$, we can reduce to the case of the standard symplectic or Calabi-Yau structure, since in the locally conformally symplectic case, $H(\hat
X_x)$ equals $H(\hat X_x, \omega_0)$, where $\omega_0$ is a standard symplectic structure, as explained in Example \[ex:st-lcs\].
Now, given a polydifferential operator $\phi: {\operatorname{\mathsf{Sym}}}^n \hat {\mathcal{O}}_{X,x}
\to \hat {\mathcal{O}}_{X,x}$, view it as a polynomial function $\bar \phi:
\hat {\mathcal{O}}_{X,x} \to \hat {\mathcal{O}}_{X,x}$ on the pro-vector space $\hat
{\mathcal{O}}_{X,x}$. Then $\bar \phi$ is uniquely determined by its restriction to functions with nonvanishing first derivative, since the complement has codimension at least two. Let $f \in \hat
{\mathcal{O}}_{X,x}$ be such a function. Let $G_{X,x}$ be the formal group obtained by integrating $H(X)$, which acts on $\hat {\mathcal{O}}_{X,x}$. By the Darboux theorem, there is a coordinate change by $G_{X,x}$ that takes $f$ to a coordinate function $x_1$ of $X$. Now, if a polydifferential operator is invariant under $H(X)$, it must take $x_1$ to a function invariant under the formal subgroup of $G_{X,x}$ preserving $x_1$, i.e., to a polynomial in $x_1$. Now, to be invariant under automorphisms in $G_{X,x}$ sending $x_1$ to $\lambda
x_1$, $\phi$ must have the form $x_1 \mapsto c \cdot x_1^n$ for some $c \in {\mathbf{k}}$. It remains to note that, if $f,g \in \hat {\mathcal{O}}_{X,x}$ are two functions with nonvanishing first derivative, again by the Darboux theorem there is an element of $G_{X,x}$ sending $f$ to $g$, so the constant $c$ must be independent of the choice of $f$. Therefore, $\bar \phi(g)=c g^n$ for all $g$. We can easily see that this is ${\mathfrak{v}}$-invariant.
\(ii) Restricting to $\hat X_x$, suppose first that ${\mathfrak{v}}$ is arbitrary such that, in some coordinate system, it contains the constant vector fields and an Euler vector ${\operatorname{Eu}}=
\sum_i m_i \partial_i$ for $m_i > 0$. Let $m := \sum_i m_i$. Let ${\mathsf{vol}}$ be the standard volume form in this coordinate system. The polydifferential operators $\hat {\mathcal{O}}_{X,x}^{\otimes n} \to
\Omega_{\hat X_x}$ invariant under the aforementioned vector fields are spanned by $$(F_1 \otimes \cdots \otimes F_n) \cdot {\mathsf{vol}}, \quad |F_1|+\cdots+|F_n| = -m,$$ where each $F_i$ is a constant-coefficient monomial in the $\partial_i$, and here $|\cdot|$ denotes the weighted degree with respect to ${\operatorname{Eu}}$. This is a finite-dimensional vector space.
Now, in the case where ${\mathfrak{v}}= {\operatorname{Vect}}(X)$, in order to be invariant under all possible Euler vector fields, the operator must be a linear combination of terms such that $F_1 \cdots F_n$ is linear in each coordinate. Moreover, to be invariant under volume-preserving linear changes of basis, i.e., under ${\mathsf{SL}}(T_x X)$, we conclude that the operator is spanned by images under $S_n$ of $({\mathsf{vol}}^{-1}
\otimes 1^{\otimes (n-\dim X)}) \cdot {\mathsf{vol}}$, as desired.
In the case ${\mathfrak{v}}= H(X)$ and $X$ is odd-dimensional contact variety, then we can take ${\operatorname{Eu}}$ as in Example \[ex:st-cont\], so that $F_1 \cdots F_n$ must have total degree $-(\dim X + 1)$ (since $|x_i|=|y_i|=1$ and $|t|=2$, and the partial derivatives have negative this degree). Also, the polydifferential operator must be preserved by all linear changes of basis preserving $\partial_t$. In particular, since it is preserved by ${\mathsf{GL}}(\langle \partial_{x_i}, \partial_{y_i} \rangle)$, the operator must be in the ${\mathbf{k}}[S_n]$-span of $$\bigl(
\wedge^{\dim X - 1} \langle \partial_{x_i}, \partial_{y_i} \rangle \otimes
\partial_t \otimes 1^{\otimes (n-\dim X)} \bigr) \cdot {\mathsf{vol}}.$$ Since it is preserved by transformations $x_i \mapsto x_i + \lambda t$ for $\lambda \in {\mathbf{k}}$, we conclude in fact that it is in the ${\mathbf{k}}[S_n]$-span of $({\mathsf{vol}}^{-1} \otimes 1^{\otimes (n-\dim X)}) \cdot {\mathsf{vol}}$, as desired.
General decomposition statement
-------------------------------
First, we generalize Theorem \[t:sym-ql\] by replacing (\*) by a general decomposition statement about $M(S^n X, {\mathfrak{v}})$. Then (\*) becomes a multiplicity-one condition.
Given a smooth affine variety $X$ and an integer $m \geq 1$, let ${\operatorname{PDiff}}({\mathcal{O}}_X,\Omega_X, m+1)$ be the space of polydifferential operators ${\mathcal{O}}_X^{\otimes m} \to \Omega_X$ of degree $m$, i.e., linear maps which are differential operators in each component.
Note that there there is a natural action of $S_{m+1}$ on ${\operatorname{PDiff}}({\mathcal{O}}_X, \Omega_X, m+1)$ given by viewing these operators as distributions along on the diagonal in $X^{m+1}$, i.e., as sections of the ${\mathcal{D}}_{X^{m+1}}$-module $(\Delta_{m+1})_* \Omega_X$, which has its natural $S_{m+1}$-action. The $S_m$ action is just by permutation of components, and the extension to $S_{m+1}$ is explicitly given by the integration by parts rule. For example, when $X = {\mathbf{A}}^1$ with the standard volume, this action restricted to the span of partial derivatives $\partial_1, \ldots, \partial_m$ is the reflection representation of $S_{m+1}$ (viewed as a type $A_m$ Weyl group); explicitly this can be viewed as the usual permutation action on $\partial_1, \ldots, \partial_{m+1}$ where we set $\partial_{m+1} =
-\sum_{i=1}^m \partial_i$.
For all $m \geq 1$, let $L_m$ be the maximal quotient of $M(S^m X,
{\mathfrak{v}})$ supported on the diagonal, i.e., $L_m = ({\operatorname{pr}}_m \circ \Delta_m)_*
H^0 ({\operatorname{pr}}_m \circ \Delta_m)^* M(S^m X, {\mathfrak{v}})$ (which at least makes sense when $M(S^m
X, {\mathfrak{v}})$ is holonomic, as in the quasi-local transitive case).
\[t:sym-ql2\] Suppose that $(X,{\mathfrak{v}})$ is quasi-local and transitive and has pure dimension at least two. Then, there is a canonical isomorphism $$\label{e:msn-dec1}
M(S^n X, {\mathfrak{v}}) {{\;\stackrel{_\sim}{\to}\;}}\bigoplus_{\lambda \vdash n} ({\operatorname{pr}}_n)_* (\Delta_\lambda)_*
L_\lambda^{S_\lambda}, \quad L_\lambda := L_{\lambda_1} \boxtimes \cdots \boxtimes
L_{\lambda_{|\lambda|}}.$$ Moreover, the rank of $L_m$ is equal to the dimension of $({\operatorname{PDiff}}(\hat
{\mathcal{O}}_{X,x}, \Omega_{\hat X_x},m)^{{\mathfrak{v}}})^{S_m}$.
The canonical isomorphism is given by the direct sum of the morphisms $$\label{e:msn-dec1-fact}
M(S^n X, {\mathfrak{v}})
{\twoheadrightarrow}({\operatorname{pr}}_n)_* (\Delta_\lambda)_*
L_\lambda^{S_\lambda},$$ obtained by adjunction from the canonical quotients $H^0 ({\operatorname{pr}}_n \circ
\Delta_\lambda)^* M(S^n X, {\mathfrak{v}}) {\twoheadrightarrow}L_\lambda$.
The theorem implies that composition factors from distinct leaves do not appear in nontrivial extensions:
\[c:d-ie\] In the situation of the theorem, $M(S^n X, {\mathfrak{v}})$ is a direct sum of intermediate extensions of local systems on the leaves (locally closed diagonals).
For each diagonal $X_\lambda := {\operatorname{pr}}_n \circ
\Delta_\lambda(X^{|\lambda|})$, let $j_\lambda: X_\lambda^\circ
{\hookrightarrow}X_\lambda$ be the open embedding of the complement of smaller diagonals, i.e., such that $X_\lambda^\circ$ is a leaf of $S^n X$. Let $\tilde j_\lambda: \tilde X_\lambda^\circ {\hookrightarrow}\Delta_\lambda(X) \subseteq X^n$ be the preimage of $X_\lambda^\circ$. Then, for each factor in , $$j_\lambda^* ({\operatorname{pr}}_n)_* (\Delta_\lambda)_*
L_\lambda^{S_\lambda} \cong ({\operatorname{pr}}_n)_* \tilde j_\lambda^* (\Delta_\lambda)_*
L_\lambda^{S_\lambda}.$$ Since ${\operatorname{pr}}_n$ is a covering of $\tilde X_\lambda^\circ$ onto its image (with covering group $S_\lambda$), the above is a local system on $X_\lambda^\circ$. It now suffices to prove that $$({\operatorname{pr}}_n)_* (\Delta_\lambda)_*
L_\lambda^{S_\lambda} \cong j_{!*}j_\lambda^* ({\operatorname{pr}}_n)_* (\Delta_\lambda)_*
L_\lambda^{S_\lambda}.$$ This follows because, since ${\operatorname{pr}}_n$ is finite, the singular support of $({\operatorname{pr}}_n)_* (\Delta_\lambda)_* L_\lambda^{S_\lambda}$ is the closure of the conormal bundle of the leaf ${\operatorname{pr}}_n \circ
\Delta_\lambda((X^{|\lambda|})^\circ)$, where $(X^{|\lambda|})^\circ$ is the complement in $X^{|\lambda|}$ of the images of all diagonal embeddings of $X^r$ for all $r < |\lambda|$.
We can make a similar statement about $M(X^n, {\mathfrak{v}})$ itself: Let $\tilde L_m = (\Delta_m)_* H^0 \Delta_m^* M(X^m, {\mathfrak{v}})$ be the maximal quotient of $M(X^m, {\mathfrak{v}})$ supported on the diagonal. This is $S_m$-equivariant, and $(\tilde L_m)^{S_m} = L_m$.
\[t:sym-ql3\] Let $(X,{\mathfrak{v}})$ be as in Theorem \[t:sym-ql2\]. Then, there is a canonical isomorphism $$\label{e:msn-dec2}
M(X^n, {\mathfrak{v}}) {{\;\stackrel{_\sim}{\to}\;}}\bigoplus_{\lambda \vdash n} S_n
(\widetilde{L_\lambda}), \quad
\widetilde{L_\lambda} := \widetilde{L_{\lambda_1}} \boxtimes \cdots
\boxtimes \widetilde{L_{\lambda_{|\lambda|}}}.$$ Here, $S_n (\widetilde{L_\lambda})$ is the $S_n$-equivariant local system on the $S_n$-orbit of $\Delta_\lambda(X)$ whose restriction to $\Delta_\lambda(X)$ is the $N_{S_n}(S_\lambda)/S_\lambda$-equivariant local system $\tilde L_m$.
Moreover, the rank of $\tilde L_m$ is the dimension of ${\operatorname{PDiff}}(\hat
{\mathcal{O}}_{X,x},\Omega_{\hat X_x},m)^{{\mathfrak{v}}}$.
As in Corollary \[c:d-ie\], it follows from this that the entire pushforward $({\operatorname{pr}}_n)_* M(X^n, {\mathfrak{v}})$ on $S^n X$ is a direct sum of intermediate extensions of $S_\lambda$-equivariant local systems on the diagonals corresponding to partitions $\lambda \vdash n$.
Proof of Theorems \[t:sym-ql2\] and \[t:sym-ql3\]
-------------------------------------------------
We will work with $M(X^n, {\mathfrak{v}})$. Since this is $S_n$-equivariant and $M(S^n X, {\mathfrak{v}}) = ({\operatorname{pr}}_n)_* M(X^n, {\mathfrak{v}})^{S_n}$, this will also compute the latter.
By transitivity and quasi-locality, the closures of the leaves of $M(X^n, {\mathfrak{v}})$ are the diagonals $\Delta_\lambda(X^{|\lambda|})$ together with the diagonals obtained from these by the action of $S_n$. Hence, $M(X^n, {\mathfrak{v}})$ is holonomic and its composition factors are intermediate extensions of local systems on these leaves. Similarly to , one has canonical surjections $$\label{e:msn-dec2-fact}
M(X^n, {\mathfrak{v}})
{\twoheadrightarrow}(\Delta_\lambda)_*
L_\lambda,$$ and similarly for the orbits of these under $S_n$ (there is one of these for each coset in $S_n / (N_{S_n}(S_\lambda))$, and each is a local system on the image of $\Delta_\lambda(X^{|\lambda|})$ under the element of $S_n$ which is equivariant under the corresponding conjugate of the subgroup $N_{S_n}(S_\lambda) < S_n$). It suffices to prove the following:
1. The quotient is the maximal quotient supported on $\Delta_\lambda(X^{|\lambda|})$, i.e., it is $(\Delta_\lambda)_* H^0 (\Delta_\lambda)^* M(X^n, {\mathfrak{v}})$;
2. For distinct $\lambda$ or distinct orbits for a fixed $\lambda$, that the above factors have no nontrivial extensions (i.e., the ${\operatorname{Ext}}$ group of the two is zero).
For (i), by restricting to a formal neighborhood of a generic point $y = \Delta_\lambda(x)$ of $\Delta_\lambda(X^{|\lambda|})$, it suffices to find an isomorphism $${\operatorname{Hom}}_{\hat {\mathcal{D}}_{X^n,y}}(M(X^n,{\mathfrak{v}})|_{\widehat{X^n}_y},
(\Delta_\lambda)_* \Omega_{\widehat{X^{|\lambda|}}_x}) \cong
\bigotimes_{i=1}^{|\lambda|} {\operatorname{PDiff}}(\hat {\mathcal{O}}_{X,x},\Omega_{\hat X_x}, \lambda_i)^{{\mathfrak{v}}}.$$ By quasi-locality, it suffices to restrict to the case $|\lambda|=1$ (for all $n$). For this, note that there is a canonical isomorphism $$(\Delta_n)_* \Omega_{\widehat{X,x}} \cong{\operatorname{PDiff}}(\hat {\mathcal{O}}_{X,x},n).$$ Moreover, for any ${\mathcal{D}}_{X^n}$-module $N$, we have a canonical isomorphism ${\operatorname{Hom}}(M(X^n, {\mathfrak{v}}), N) \cong N^{\mathfrak{v}}$, by considering the image of the canonical generator of $M(X^n, {\mathfrak{v}})$. Putting these together, we deduce part (i).
For (ii), note that the factors $(\Delta_\lambda)_* L_\lambda$, as well as their images under the action of $S_n$, are local systems on smooth closed subvarieties of $X^n$. Moreover, the intersection of two of these subvarieties has codimension a multiple of $\dim X$ in each, which in particular is codimension at least $2$. Thus, the result follows from the following basic lemma:
\[l:extsubvars\] [@hp0weyl Lemma 2.1.1] Suppose that $Z$ is a smooth variety, and $Z_1, Z_2 \subseteq Z$ as well as $Z_1 \cap Z_2$ are smooth closed subvarieties, all of pure dimension. Let $\mathcal{L}_1, \mathcal{L}_2$ be local systems on $Z_1$ and $Z_2$, respectively, and let $i_1: Z_1 \to Z$ and $i_2: Z_2 \to Z$ be the inclusions. Then, $$\label{e:extsubvars}
{\operatorname{Ext}}^j((i_1)_* \mathcal{L}_1, (i_2)_* \mathcal{L}_2) = 0, \text{ for } j <
(\dim Z_1 - \dim Z_1 \cap Z_2) + (\dim Z_2 - \dim Z_1 \cap Z_2).$$
Proof of Theorem \[t:sym-ql\]
-----------------------------
\(i) If ${\mathfrak{v}}$ flows incompressibly, then we have an isomorphism of modules over the Lie algebra ${\mathfrak{v}}$, $\hat {\mathcal{O}}_{X,x}
{{\;\stackrel{_\sim}{\to}\;}}\Omega_{\hat X_x}$, obtained from the formal volume at $x$ preserved by ${\mathfrak{v}}$. Therefore, in the theorem, we can replace the polydifferential operators described by $({\operatorname{PDiff}}(\hat {\mathcal{O}}_{X,x},
\Omega_{\hat X_x},n+1)^{\mathfrak{v}})^{S_n}$. Then, the result is almost immediate from Theorem \[t:sym-ql2\], except Theorem \[t:sym-ql\] deals with $S_n$-invariant polydifferential operators, whereas the multiplicity spaces of Theorem \[t:sym-ql2\] are more symmetric: they are $({\operatorname{PDiff}}(\hat {\mathcal{O}}_{X,x}, \Omega_{\hat X_x},n+1)^{\mathfrak{v}})^{S_{n+1}}$.
Thus, it suffices to show that, if such ${\mathfrak{v}}$-invariant $S_{n+1}$-invariant polydifferential operators of degree $n$ are spanned by the multiplication operator *for all n*, then the same is true requiring only $S_n$-invariance.
For this, note that, given a ${\mathfrak{v}}$-invariant polydifferential operator $\phi$ on $\hat {\mathcal{O}}_{X,x}$ of degree $n$, then the space of ${\mathfrak{v}}$-invariant polydifferential operators of degree $n+1$ includes the space ${\operatorname{Ind}}_{S_n \times S_1}^{S_{n+1}} \langle \phi \boxtimes
{\mathbf{k}}\rangle$ spanned over $S_{n+1}$ by the operator $f \boxtimes g
\mapsto \phi(f) \cdot g$ for all $f \in \hat {\mathcal{O}}_{X,x}^{\otimes n}$ and $g \in \hat {\mathcal{O}}_{X,x}$.
To proceed, we will need the following technical combinatorial result, which we prove below:
\[l:tech-sn\] Suppose that $\phi$ generates a $S_{n+1}$-representation $V$, and that $\phi$ is $S_n$-invariant but not $S_{n+1}$-invariant. Then the $S_{n+1}$-representation ${\operatorname{Ind}}_{S_n
\times S_1}^{S_{n+1}} (V|_{S_n} \boxtimes {\mathbf{k}})$ extends to a unique $S_{n+2}$-representation, up to isomorphism, and this has a nonzero $S_{n+2}$-invariant vector.
Let us use the lemma to finish the proof of the first statement. We conclude from the lemma that there exists a $S_{n+2}$-invariant, ${\mathfrak{v}}$-invariant polydifferential operator $\phi$ on $\hat {\mathcal{O}}_{X,x}$ of degree $n+1$. We claim that this is not the multiplication operator (up to scaling). Indeed, we could have assumed that $\phi$ were homogeneous of positive order (since ${\mathfrak{v}}$ preserves the grading by order of differential operators), so the latter $S_{n+2}$-operator can be assumed to have positive order. This contradicts our hypothesis. Hence, (\*) of Theorem \[t:sym-ql\] is indeed satisfied.
\(ii) If ${\mathfrak{v}}$ does not flow incompressibly, $M(X,
{\mathfrak{v}}) = 0$, by Proposition \[p:tr\]. Next, suppose that there existed an $S_n$-invariant, ${\mathfrak{v}}$-invariant polydifferential operator $\phi: \hat
{\mathcal{O}}_{X,x}^{\otimes n} \to \Omega_{\hat X_x}$ but not a $S_{n+1}$-invariant one. Again, we can form the polydifferential operator $(\phi \boxtimes 1)$, sending $f_1 \otimes \cdots \otimes
f_{n+1}$ to $\phi(f_1 \otimes \cdots \otimes f_n) f_{n+1}$. So as before, we would obtain that, as $S_{n+1}$-representations, ${\operatorname{PDiff}}(\hat {\mathcal{O}}_{X,x}, \Omega_{\hat X_x}, n+2)^{{\mathfrak{v}}} \supseteq
{\operatorname{Ind}}_{S_n \times S_1}^{S_{n+1}} ({\mathbf{k}}^n \boxtimes {\mathbf{k}})$. By the same argument as above, this would contain an $S_{n+2}$-invariant operator. Thus, $M(S^{n+2} X, {\mathfrak{v}}) \neq 0$. So, if $M(S^n X, {\mathfrak{v}}) = 0$ for all $n \geq 1$, then there are no $S_n$-invariant, ${\mathfrak{v}}$-invariant polydifferential operators $\phi: \hat
{\mathcal{O}}_{X,x}^{\otimes n} \to \Omega_{\hat X_x}$, for all $n$. The converse is clear from Theorem \[t:sym-ql2\].
Under the assumption, $V$ must include a summand isomorphic to the reflection representation ${\mathbf{k}}^n$ ($V$ is either this or the direct sum of ${\mathbf{k}}^n$ with a trivial representation). As a representation of $S_n$, ${\mathbf{k}}^n$ is the standard representation.
Thus, we can assume that $V = {\mathbf{k}}^n$. As an $S_n$-representation, $V \cong {\mathbf{k}}^{n-1} \oplus {\mathbf{k}}$. Then, for $n \geq 3$, one computes the decomposition into irreducible $S_{n+1}$-representations: $${\operatorname{Ind}}_{S_n \times S_1}^{S_{n+1}} V|_{S_n} \boxtimes {\mathbf{k}}\cong \rho_{(1,1)[n+1]} \oplus \rho_{(2)[n+1]} \oplus {\mathbf{k}}^n \oplus
{\mathbf{k}}^n \oplus {\mathbf{k}},$$ where, given a partition $\lambda \vdash (n+1)$, the representation $\rho_\lambda$ is the irreducible representation with Young diagram $\lambda$. Moreover, given $\lambda' \vdash m$, we let $\lambda'[n+1]$ denotes the diagram obtained from $\lambda'$ by adding a new row on top with $n+1-m$ boxes (which makes sense if $n+1-m \geq \lambda'_1$.) Now, if the $S_{n+1}$-structure above extends to a $S_{n+2}$-structure, then the decomposition into irreducible $S_{n+2}$-representations (up to isomorphism) must be $$\rho_{(1,1)[n+2]} \oplus \rho_{(2)[n+2]} \oplus {\mathbf{k}}.$$ We conclude that ${\operatorname{Ind}}_{S_n \times S_1}^{S_{n+1}} \langle \phi
\boxtimes {\mathbf{k}}\rangle$ must contain a $S_{n+2}$-fixed vector.
In the case that $n=2$, the second decomposition (as $S_{n+2}=S_4$-representations) above is still valid, so we still obtain the $S_{n+2}$-fixed vector.
[^1]: Dually, in the complex case ${\mathbf{k}}={\mathbf{C}}$, the second homology of $X$ as a topological space produces the functionals on ${\mathcal{O}}_X$ invariant under $H(X)$ (and hence also those invariant under $\mathfrak{sl}_2$) by $C \in H_2(X) \mapsto \Phi_C$, $\Phi_C(f) = \int_C f {\mathsf{vol}}_X$.
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[**Addition formulas for the $\boldsymbol{_pF_p}$ and $\boldsymbol{_{p+1}F_p}$ generalized hypergeometric functions with arbitrary parameters and their Kummer- and Euler-type transformations** ]{}\
Krishna Choudhary\
Gladstone Institutes, 1650 Owens St, San Francisco, CA 94158\
krishna.choudhary@gladstone.ucsf.edu, kchoudhary@ucdavis.edu
Abstract {#abstract .unnumbered}
========
We obtain addition formulas for $_pF_p$ and $_{p+1}F_p$ generalized hypergeometric functions with general parameters. These are utilized in conjunction with integral representations of these functions to derive Kummer- and Euler-type transformations that express $_pF_p\left(x\right)$ and $_{p+1}F_p\left(x\right)$ in the form of sums of $_pF_p\left(-x\right)$ and $_{p+1}F_p\left(-x\right)$ functions, respectively.
Introduction
============
A generalized hypergeometric function with $p$ numerator parameters and $q$ denominator parameters is defined as $$\begin{aligned}
_pF_q\left(
\begin{smallmatrix}
a_1, & a_2, & \ldots, & a_p\\
b_1, & b_2, & \ldots, & b_q \\
\end{smallmatrix};x
\right) & = & \sum_{i=0}^\infty \frac{\left(a_1\right)_i \left(a_2\right)_i \ldots \left(a_p\right)_i}{\left(b_1\right)_i \left(b_2\right)_i \ldots \left(b_q\right)_i}\frac{x^i}{i!},\end{aligned}$$ where the parameters and the argument can take complex values except that the denominator parameters cannot be negative integers, and $\left(a\right)_i$ is the Pochhammer symbol for the ascending factorial $\prod_{j=0}^{i-1}\left(a+j\right)$ with $\left(a\right)_0 = 1$. For example, confluent hypergeometric function with $p=q=1$ and Gaussian hypergeometric function with $p=q+1=2$ are two of its special cases that are well-known.
The generalized hypergeometric functions are of significant interest for their applications in diverse areas, notably including mathematical physics and mathematical statistics [@slater1966generalized; @seaborn2013hypergeometric]. Insights into their theory as well as their applications are facilitated by various transformation formulas and identities, many of which have been collected in standard references [@slater1966generalized; @bateman1953higher; @abramowitz1970handbook; @beals2016special]. As mathematical models in physics or statistics grow in complexity and involve systems that are described in terms of multiple parameters, their numerical or analytical solutions may have to contend with cases where $p,q\geq2$. In the last two decades, a number of transformation and summation formulas have been derived for such cases. Paris utilized the addition theorem for $_1F_1\left(x\right)$ in conjunction with an integral representation of hypergeometric functions to derive a Kummer-type transformation formula that connects $_2F_2\left(x\right)$ with general parameters to $_2F_2\left(-x\right)$ [@paris2005kummer]. A number of formulas for functions with special relationships between the numerator and denominator parameters or special values of the argument have also been derived [@miller2011euler; @kim2012two; @rakha2014extension; @srivastava2019extensions]. For example, Miller and Paris derived Kummer- and Euler-type transformation formulas respectively for $_pF_p\left(x\right)$ and $_{p+1}F_p\left(x\right)$ with integral differences between the numerator and denominator parameters [@miller2013transformation]. However, to the best of my knowledge, such formulas are not available for $_pF_p\left(x\right)$ and $_{p+1}F_p\left(x\right)$ with general parameters. If addition formulas were available for the $_pF_p$ and $_{p+1}F_p$ functions, one could follow the method of Paris [@paris2005kummer] and derive the said transformation formulas. However, to the best of my knowledge, addition formulas, which are important in their own right [@koelinkEOM], are also not available for the $_pF_p$ and $_{p+1}F_p$ functions. Here, I fill this gap by stating and proving addition formulas for these functions, which I use in conjunction with their integral representations to derive transformation formulas that connect $_pF_p\left(x\right)$ and $_{p+1}F_p\left(x\right)$ with $_pF_p\left(-x\right)$ and $_{p+1}F_p\left(-x\right)$, respectively for general parameter values.
Theorem 1 (addition formula for $\boldsymbol{_pF_p}$).
======================================================
For $p \geq 1$ and $b_i \neq 0, -1, -2, ...$ $\forall$ $1 \leq i \leq p$, $$\begin{aligned}
_{p}F_{p}\left(
\begin{smallmatrix}
a_1 , & a_2, & ..., & a_p \\
b_1 , & b_2, & ..., & b_p \\
\end{smallmatrix};x+y
\right) & = & e^x \sum_{j_1=0}^\infty \sum_{j_2=0}^\infty ... \sum_{j_p=0}^\infty \prod_{q=1}^p \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q}} \frac{\left(-x\right)^{j_q}}{j_q!} \cdot \nonumber \\
& & \hspace{35pt} {_{p}}F_{p}\left(
\begin{smallmatrix}
a_1 + u_0, & a_2 + u_1, & a_3 + u_2, & ..., & a_p + u_{p-1}\\
b_1 + u_1, & b_2 + u_2, & b_3 + u_3, & ..., & b_p + u_p \\
\end{smallmatrix};y
\right), \label{eq: addition_theorem_pFp}\end{aligned}$$ where $u_0 = 0$, $u_1 = j_1$, $u_2 = j_1+j_2$, ..., $u_p = \sum_{q=1}^p j_q$.
### Proof {#proof .unnumbered}
We can use the method of induction to prove this theorem. First, note that for $p=1$, Eq. \[eq: addition\_theorem\_pFp\] becomes $$\begin{aligned}
_{1}F_{1}\left(
\begin{smallmatrix}
a_1 \\
b_1 \\
\end{smallmatrix};x+y
\right) & = & e^x \sum_{j_1=0}^\infty \frac{\left(b_1 - a_1\right)_{j_1}}{\left(b_1\right)_{j_1}} \frac{\left(-x\right)^{j_1}}{j_1!} \cdot {_{1}}F_{1}\left(
\begin{smallmatrix}
a_1 \\
b_1 + j_1 \\
\end{smallmatrix};y
\right), \end{aligned}$$ which is correct (see Eq. 2.3.5 in Slater [@slater1960confluent]). Next, let us assume that Eq. \[eq: addition\_theorem\_pFp\] is valid for some $p=k$. Then, using the integral representation in Eq. 4.8.3.12 of Slater [@slater1966generalized], for $p=k+1$ and $\text{Re}\left(b_{k+1}\right) > \text{Re}\left(a_{k+1}\right)>0$, $$\begin{aligned}
& _{k+1}F_{k+1}\left(
\begin{smallmatrix}
a_1, & ..., & a_{k+1} \\
b_1, & ..., & b_{k+1} \\
\end{smallmatrix};x+y
\right) \nonumber \\
& = \frac{\Gamma\left(b_{k+1}\right)}{\Gamma\left(a_{k+1}\right) \Gamma\left(b_{k+1} - a_{k+1}\right)} \int_0^1 t^{a_{k+1} - 1} \left(1-t\right)^{b_{k+1} - a_{k+1} - 1} {_{k}}F_{k}\left(
\begin{smallmatrix}
a_1, & ..., & a_{k} \\
b_1, & ..., & b_{k} \\
\end{smallmatrix};xt + yt
\right) dt. \label{eq: pFp_add_thrm_proof_start}\end{aligned}$$ Here, using Eq. \[eq: addition\_theorem\_pFp\] for $p=k$ given $b_i \neq 0, -1, -2, ...$ $\forall$ $1 \leq i \leq k$ yields $$\begin{aligned}
& \frac{\Gamma\left(b_{k+1}\right)}{\Gamma\left(a_{k+1}\right) \Gamma\left(b_{k+1} - a_{k+1}\right)} \int_0^1 t^{a_{k+1} - 1} \left(1-t\right)^{b_{k+1} - a_{k+1} - 1} \cdot \nonumber \\
& \hspace{35pt} e^{xt} \sum_{j_1=0}^\infty \sum_{j_2=0}^\infty ... \sum_{j_k=0}^\infty \prod_{q=1}^k \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q}} \frac{\left(-xt\right)^{j_q}}{j_q!} {_{k}}F_{k}\left(
\begin{smallmatrix}
a_1 + u_0, & ..., & a_k + u_{k-1}\\
b_1 + u_1, & ..., & b_k + u_k \\
\end{smallmatrix};yt
\right) dt, \nonumber\end{aligned}$$ where $u_0 = 0$, $u_1 = j_1$, $u_2 = j_1+j_2$, ..., $u_k = \sum_{q=1}^k j_q$. Upon rearrangement of terms and reversal of the order of summation and integration, this expression can be written as $$\begin{aligned}
& e^x \frac{\Gamma\left(b_{k+1}\right)}{\Gamma\left(a_{k+1}\right) \Gamma\left(b_{k+1} - a_{k+1}\right)} \sum_{j_1=0}^\infty \sum_{j_2=0}^\infty ... \sum_{j_k=0}^\infty \prod_{q=1}^k \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q}} \frac{\left(-x\right)^{j_q}}{j_q!} \cdot \nonumber \\
& \hspace{35pt} \int_0^1 t^{a_{k+1} + u_k - 1} \left(1-t\right)^{b_{k+1} - a_{k+1} - 1} e^{-x\left(1-t\right)} {_{k}}F_{k}\left(
\begin{smallmatrix}
a_1 + u_0, & ..., & a_k + u_{k-1}\\
b_1 + u_1, & ..., & b_k + u_k \\
\end{smallmatrix};yt
\right) dt. \nonumber\end{aligned}$$ Next, using the power series expansion for $e^{-x\left(1-t\right)}$ in the above expression, we write it as $$\begin{aligned}
& e^x \frac{\Gamma\left(b_{k+1}\right)}{\Gamma\left(a_{k+1}\right) \Gamma\left(b_{k+1} - a_{k+1}\right)} \sum_{j_1=0}^\infty \sum_{j_2=0}^\infty ... \sum_{j_k=0}^\infty \prod_{q=1}^k \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q}} \frac{\left(-x\right)^{j_q}}{j_q!} \cdot \nonumber \\
& \int_0^1 t^{a_{k+1} + u_k - 1} \left(1-t\right)^{b_{k+1} - a_{k+1} - 1} \sum_{j_{k+1}=0}^\infty \frac{\left(-x\right)^{j_{k+1}} \left(1-t\right)^{j_{k+1}}}{j_{k+1}!} {_{k}}F_{k}\left(
\begin{smallmatrix}
a_1 + u_0, & ..., & a_k + u_{k-1}\\
b_1 + u_1, & ..., & b_k + u_k \\
\end{smallmatrix};yt
\right) dt, \nonumber\end{aligned}$$ which upon reversal of the order of integration and summation yields $$\begin{aligned}
& e^x \frac{\Gamma\left(b_{k+1}\right)}{\Gamma\left(a_{k+1}\right) \Gamma\left(b_{k+1} - a_{k+1}\right)} \sum_{j_1=0}^\infty \sum_{j_2=0}^\infty ... \sum_{j_k=0}^\infty \prod_{q=1}^k \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q}} \frac{\left(-x\right)^{j_q}}{j_q!} \cdot \nonumber \\
& \sum_{j_{k+1}=0}^\infty \frac{\left(-x\right)^{j_{k+1}} }{j_{k+1}!} \int_0^1 t^{a_{k+1} + u_k - 1} \left(1-t\right)^{b_{k+1} - a_{k+1} +j_{k+1} - 1} {_{k}}F_{k}\left(
\begin{smallmatrix}
a_1 + u_0, & ..., & a_k + u_{k-1}\\
b_1 + u_1, & ..., & b_k + u_k \\
\end{smallmatrix};yt
\right) dt. \nonumber\end{aligned}$$ Once again, by using the integral representation in Eq. 4.8.3.12 of Slater [@slater1966generalized] in the above expression, we get $$\begin{aligned}
& e^x \frac{\Gamma\left(b_{k+1}\right)}{\Gamma\left(a_{k+1}\right) \Gamma\left(b_{k+1} - a_{k+1}\right)} \sum_{j_1=0}^\infty \sum_{j_2=0}^\infty ... \sum_{j_k=0}^\infty \prod_{q=1}^k \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q}} \frac{\left(-x\right)^{j_q}}{j_q!} \cdot \nonumber \\
& \sum_{j_{k+1}=0}^\infty \frac{\left(-x\right)^{j_{k+1}} }{j_{k+1}!} \frac{\Gamma\left(b_{k+1} - a_{k+1} + j_{k+1}\right) \Gamma\left(a_{k+1} + u_k\right)}{\Gamma\left(b_{k+1} + u_{k+1}\right)} \cdot {_{k+1}}F_{k+1}\left(
\begin{smallmatrix}
a_1 + u_0, & ..., & a_{k+1} + u_k\\
b_1 + u_1, & ..., & b_{k+1} + u_{k+1}, \\
\end{smallmatrix};y
\right), \nonumber\end{aligned}$$ where $u_{k+1} = \sum_{q=1}^{k+1}j_{q}$. This expression can be more compactly written as $$\begin{aligned}
& e^x \sum_{j_1=0}^\infty \sum_{j_2=0}^\infty ... \sum_{j_{k+1}=0}^\infty \prod_{q=1}^{k+1} \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q}} \frac{\left(-x\right)^{j_q}}{j_q!} \cdot {_{k+1}}F_{k+1}\left(
\begin{smallmatrix}
a_1 + u_0, & ..., & a_{k+1} + u_{k}\\
b_1 + u_1, & ..., & b_{k+1} + u_{k+1} \\
\end{smallmatrix};y
\right) \nonumber\end{aligned}$$ to get the right hand side of Eq. \[eq: addition\_theorem\_pFp\] for $p=k+1$. While we have shown that the left hand side in Eq. \[eq: pFp\_add\_thrm\_proof\_start\] is equal to the above expression given that $\text{Re}\left(b_{k+1}\right) > \text{Re}\left(a_{k+1}\right)>0$, they are equal for general values of $b_{k+1}$ and $a_{k+1}$ as well due to analytic continuation since they are both analytic functions of these parameters, given $b_{k+1} \neq 0, -1, -2, ...$. This implies that if the theorem holds for $p=k$, it also holds for $p=k+1$. Since we know that it holds for $p=1$, by induction, it holds for all $p \geq 1$ given $b_{i} \neq 0, -1, -2, ...$ $\forall$ $1 \leq i \leq p$, thereby completing the proof.
Theorem 2 (addition formula for $\boldsymbol{_{p+1}F_p}$).
==========================================================
For $p \geq 1$ and given $b_i \neq 0, -1, -2, ...$ $\forall$ $1 \leq i \leq p$, $\left|x+y\right| < 1$, $\left|y\right| < \left|x\right|$ and $\text{Re}\left(x\right) < 1/2$, $$\begin{aligned}
& _{p+1}F_{p}\left(
\begin{smallmatrix}
a_0, & a_1 , & ..., & a_p \\
& b_1 , & ..., & b_p \\
\end{smallmatrix};x+y
\right) \nonumber \\
& = \left(\frac{1}{1-x}\right)^{a_0} \sum_{j_1 = 0}^\infty \sum_{j_2 = 0}^\infty ... \sum_{j_p = 0}^\infty \left(a_0\right)_{u_p} \prod_{q=1}^p \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q} j_q !} \left(\frac{x}{x-1}\right)^{j_q} \cdot \nonumber \\
& \hspace{25pt} {_{p+1}}F_{p}\left(
\begin{smallmatrix}
a_0+u_p, & a_1 + u_0, & a_2 + u_1, & a_3 + u_2, & ..., & a_p + u_{p-1}\\
& b_1 + u_1, & b_2 + u_2, & b_3 + u_3, & ..., & b_p + u_p\\
\end{smallmatrix};\frac{-y}{x-1}
\right), \label{eq: add_theorem_p+1Fp}\end{aligned}$$ where $u_0 = 0$, $u_1 = j_1$, $u_2 = j_1+j_2$, ..., $u_p = \sum_{q=1}^p j_q$.
### Proof {#proof-1 .unnumbered}
The addition formula for $_{p+1}F_{p}$ can be derived using the addition formula for $_{p}F_{p}$ by appealing to a relationship between the two derived by application of Mellin transform (see Eq. 4.8.3.3 in Slater [@slater1966generalized]), $$\begin{aligned}
_{p+1}F_{p}\left(
\begin{smallmatrix}
a_0, & a_1 , & a_2 , & ..., & a_p \\
& b_1 , & b_2 , & ..., & b_p \\
\end{smallmatrix};x+y
\right) & = & \frac{1}{\Gamma\left(a_0\right)} \int_0^\infty t^{a_0 -1} e^{-t} {_p}F_{p}\left(
\begin{smallmatrix}
a_1 , & a_2, & ..., & a_p \\
b_1 , & b_2, & ..., & b_p \\
\end{smallmatrix};xt+yt
\right) dt \nonumber\end{aligned}$$ given $\left|x+y\right| < 1$. Now, we work with the right hand side of the above equation and show that it equals the right hand side of Eq. \[eq: add\_theorem\_p+1Fp\]. Given $b_i \neq 0, -1, -2, ...$ $\forall$ $1 \leq i \leq p$, we use Eq. \[eq: addition\_theorem\_pFp\] to rewrite the right hand side of the above equation as $$\begin{aligned}
& \frac{1}{\Gamma\left(a_0\right)} \int_0^\infty t^{a_0 -1} e^{-t} e^{xt} \sum_{j_1=0}^\infty \sum_{j_2=0}^\infty ... \sum_{j_p=0}^\infty \prod_{q=1}^p \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q}} \frac{\left(-xt\right)^{j_q}}{j_q!} \cdot \nonumber \\
& \hspace{55pt} {_{p}}F_{p}\left(
\begin{smallmatrix}
a_1 + u_0, & a_2 + u_1, & ..., & a_p + u_{p-1}\\
b_1 + u_1, & b_2 + u_2, & ..., & b_p + u_p \\
\end{smallmatrix};yt
\right) dt, \nonumber \end{aligned}$$ where $u_0 = 0$, $u_1 = j_1$, $u_2 = j_1+j_2$, ..., $u_p = \sum_{q=1}^p j_q$. Upon changing the order of integration and summation and using the series expansion for $_pF_p$, it yields $$\begin{aligned}
& \sum_{j_1=0}^\infty \sum_{j_2=0}^\infty ... \sum_{j_p=0}^\infty \prod_{q=1}^p \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q}} \frac{\left(-x\right)^{j_q}}{j_q!} \frac{1}{\Gamma\left(a_0\right)} \cdot \nonumber \\
& \hspace{35pt} \int_0^\infty t^{a_0 + u_p -1} e^{-t\left(1-x\right)} \sum_{i=0}^\infty \prod_{r=1}^{p} \frac{\left(a_r + u_{r-1}\right)_i}{\left(b_r + u_{r}\right)_i} \frac{y^i t^i}{i!} dt. \nonumber \end{aligned}$$ Once again, rearranging the terms and changing the order of integration and summation, we get $$\begin{aligned}
& \sum_{j_1=0}^\infty \sum_{j_2=0}^\infty ... \sum_{j_p=0}^\infty \prod_{q=1}^p \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q}} \frac{\left(-x\right)^{j_q}}{j_q!} \frac{1}{\Gamma\left(a_0\right)} \cdot \nonumber \\
& \hspace{35pt} \sum_{i=0}^\infty \prod_{r=1}^{p} \frac{\left(a_r + u_{r-1}\right)_i}{\left(b_r + u_{r}\right)_i} \frac{y^i}{i!} \int_0^\infty t^{a_0 + u_p + i -1} e^{-t\left(1-x\right)} dt. \nonumber\end{aligned}$$ Next, we substitute $v=t\left(1-x\right)$ in the integral to get $$\begin{aligned}
& \sum_{j_1=0}^\infty \sum_{j_2=0}^\infty ... \sum_{j_p=0}^\infty \prod_{q=1}^p \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q}} \frac{\left(-x\right)^{j_q}}{j_q!} \frac{1}{\Gamma\left(a_0\right)} \cdot \nonumber \\
& \hspace{35pt} \sum_{i=0}^\infty \prod_{r=1}^{p} \frac{\left(a_r + u_{r-1}\right)_i}{\left(b_r + u_{r}\right)_i} \frac{y^i}{i!} \left(\frac{1}{1-x}\right)^{a_0+u_p+i} \int_0^\infty v^{a_0 + u_p + i -1} e^{-v} dv. \nonumber \end{aligned}$$ The integral in the above expression can be written as a gamma function yielding $$\begin{aligned}
& \sum_{j_1=0}^\infty \sum_{j_2=0}^\infty ... \sum_{j_p=0}^\infty \prod_{q=1}^p \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q}} \frac{\left(-x\right)^{j_q}}{j_q!} \frac{1}{\Gamma\left(a_0\right)} \cdot \nonumber \\
& \hspace{35pt} \sum_{i=0}^\infty \prod_{r=1}^{p} \frac{\left(a_r + u_{r-1}\right)_i}{\left(b_r + u_{r}\right)_i} \frac{y^i}{i!} \left(\frac{1}{1-x}\right)^{a_0+u_p+i} \cdot \Gamma\left(a_0 + u_p + i\right), \nonumber\end{aligned}$$ which upon rearranging the terms and using $\left(a_0\right)_{u_p + i} = \left(a_0\right)_{u_p} \left(a_0 + u_p\right)_{i}$ can be written as $$\begin{aligned}
& \left(\frac{1}{1-x}\right)^{a_0} \sum_{j_1=0}^\infty \sum_{j_2=0}^\infty ... \sum_{j_p=0}^\infty \left(a_0\right)_{u_p} \prod_{q=1}^p \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q}} \left(\frac{x}{x-1}\right)^{j_q} \frac{1}{j_q!} \cdot \nonumber \\
& \hspace{35pt} \sum_{i=0}^\infty \left(a_0+u_p\right)_{i} \prod_{r=1}^{p} \frac{\left(a_r + u_{r-1}\right)_i}{\left(b_r + u_{r}\right)_i} \frac{1}{i!} \left(\frac{-y}{x-1}\right)^{i}. \nonumber\end{aligned}$$ Finally, since $\left|y\right| < \left|x-1\right|$ given $\left|y\right| < \left|x\right|$ and $\text{Re}\left(x\right) < 1/2$, it can be expressed more compactly as $$\begin{aligned}
& \left(\frac{1}{1-x}\right)^{a_0} \sum_{j_1 = 0}^\infty \sum_{j_2 = 0}^\infty ... \sum_{j_p = 0}^\infty \left(a_0\right)_{u_p} \prod_{q=1}^p \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q} j_q !} \left(\frac{x}{x-1}\right)^{j_q} \cdot \nonumber \\
& \hspace{25pt} {_{p+1}}F_{p}\left(
\begin{smallmatrix}
a_0+u_p, & a_1 + u_0, & a_2 + u_1, & a_3 + u_2, & ..., & a_p + u_{p-1}\\
& b_1 + u_1, & b_2 + u_2, & b_3 + u_3, & ..., & b_p + u_p\\
\end{smallmatrix};\frac{-y}{x-1}
\right), \nonumber\end{aligned}$$ which is the same as the right hand side of Eq. \[eq: add\_theorem\_p+1Fp\], thereby proving the theorem.
Theorem 3 (Kummer-type transformation for $\boldsymbol{_pF_p}$ with general parameters).
========================================================================================
For $p \geq 1$ and $b_i \neq 0, -1, -2, ...$ $\forall$ $1 \leq i \leq p+1$, $$\begin{aligned}
& _{p+1}F_{p+1}\left(
\begin{smallmatrix}
a_1 , & a_2 , & a_3 , & ..., & a_p , & a_{p+1}\\
b_1 , & b_2 , & b_3 , & ..., & b_p , & b_{p+1} \\
\end{smallmatrix};x
\right) \nonumber \\
& = e^x \sum_{j_1=0}^\infty \sum_{j_2=0}^\infty ... \sum_{j_p=0}^\infty \prod_{q=1}^p \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q}} \frac{\left(-x\right)^{j_q}}{j_q!} \cdot \nonumber \\
& \hspace{55pt} {_{p+1}}F_{p+1}\left(
\begin{smallmatrix}
a_1 + u_0, & a_2 + u_1, & a_3 + u_2, & ..., & a_p + u_{p-1}, & b_{p+1} - a_{p+1}\\
b_1 + u_1, & b_2 + u_2, & b_3 + u_3, & ..., & b_p + u_p, & b_{p+1} \\
\end{smallmatrix};-x
\right), \label{eq: kummer_transform_pFp}\end{aligned}$$ where $u_0 = 0$, $u_1 = j_1$, $u_2 = j_1+j_2$, ..., $u_p = \sum_{q=1}^p j_q$.
### Proof {#proof-2 .unnumbered}
To prove the theorem, we utilize the approach followed by Paris [@paris2005kummer] for a Kummer-type transformation of $_2F_2\left(x\right)$. Hence, we utilize the integral representation for $_{p+1}F_{p+1}\left(x\right)$ function, which is available from Slater [@slater1966generalized] (see Eq. 4.8.3.12) and the addition formula for $_pF_p\left(x\right)$, which we obtained above (theorem 1). The integral representation for $_{p+1}F_{p+1}\left(x\right)$ is $$\begin{aligned}
_{p+1}F_{p+1}\left(
\begin{smallmatrix}
a_1, & ..., & a_{p+1}\\
b_1, & ..., & b_{p+1}\\
\end{smallmatrix};x
\right) & = & \frac{\Gamma\left(b_{p+1}\right)}{\Gamma\left(a_{p+1}\right) \Gamma\left(b_{p+1}-a_{p+1}\right)} \nonumber \\
& & \hspace{-40pt} \int_0^1 t^{a_{p+1}-1} \left(1-t\right)^{b_{p+1}-a_{p+1}-1} {_pF_p}\left(
\begin{smallmatrix}
a_1, & ..., & a_{p}\\
b_1, & ..., & b_{p}\\
\end{smallmatrix};xt
\right) dt \label{eq: integral_representation_p+1Fp+1_original} \\
\implies {_{p+1}F_{p+1}}\left(
\begin{smallmatrix}
a_1, & ..., & a_{p+1}\\
b_1, & ..., & b_{p+1}\\
\end{smallmatrix};x
\right) & = & \frac{\Gamma\left(b_{p+1}\right)}{\Gamma\left(a_{p+1}\right) \Gamma\left(b_{p+1}-a_{p+1}\right)} \nonumber \\
& & \hspace{-40pt} \int_0^1 t^{b_{p+1}-a_{p+1}-1} \left(1-t\right)^{a_{p+1}-1} {_pF_p}\left(
\begin{smallmatrix}
a_1, & ..., & a_{p}\\
b_1, & ..., & b_{p}\\
\end{smallmatrix};x-xt
\right) dt \label{eq: integral_representation_p+1Fp+1_altered}\end{aligned}$$ if $\text{Re}\left(b_{p+1}\right) > \text{Re}\left(a_{p+1}\right) > 0$. Using the addition formula in Eq. \[eq: addition\_theorem\_pFp\], to replace $_pF_p\left(x-xt\right)$ on the right hand side in Eq. \[eq: integral\_representation\_p+1Fp+1\_altered\], given $b_i \neq 0, -1, -2, ...$ $\forall$ $1 \leq i \leq p$, we rewrite it after switching the order of integral and summation as $$\begin{aligned}
e^x \sum_{j_1=0}^\infty \sum_{j_2=0}^\infty ... \sum_{j_p=0}^\infty \prod_{q=1}^p \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q}} \frac{\left(-x\right)^{j_q}}{j_q!} \cdot \frac{\Gamma\left(b_{p+1}\right)}{\Gamma\left(a_{p+1}\right) \Gamma\left(b_{p+1}-a_{p+1}\right)} \nonumber \\
\int_0^1 t^{b_{p+1}-a_{p+1}-1} \left(1-t\right)^{a_{p+1}-1} {_{p}}F_{p}\left(
\begin{smallmatrix}
a_1 + u_0, & a_2 + u_1, & a_3 + u_2, & ..., & a_p + u_{p-1}\\
b_1 + u_1, & b_2 + u_2, & b_3 + u_3, & ..., & b_p + u_p \\
\end{smallmatrix};-xt
\right)dt,\end{aligned}$$ where $u_0 = 0$, $u_1 = j_1$, $u_2 = j_1+j_2$, ..., $u_p = \sum_{q=1}^p j_q$. Now, using Eq. \[eq: integral\_representation\_p+1Fp+1\_original\], the above expression can be rewritten as $$\begin{aligned}
& e^x \sum_{j_1=0}^\infty \sum_{j_2=0}^\infty ... \sum_{j_p=0}^\infty \prod_{q=1}^p \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q}} \frac{\left(-x\right)^{j_q}}{j_q!} \cdot \nonumber \\
& \hspace{30pt}{_{p+1}}F_{p+1}\left(
\begin{smallmatrix}
a_1 + u_0, & a_2 + u_1, & a_3 + u_2, & ..., & a_p + u_{p-1}, & b_{p+1} - a_{p+1}\\
b_1 + u_1, & b_2 + u_2, & b_3 + u_3, & ..., & b_p + u_p, & b_{p+1} \\
\end{smallmatrix};-x
\right),\end{aligned}$$ which upon substitution in Eq. \[eq: integral\_representation\_p+1Fp+1\_altered\] yields the Kummer-type transformation in Eq. \[eq: kummer\_transform\_pFp\]. While we derived the result requiring that $\text{Re}\left(b_{p+1}\right) > \text{Re}\left(a_{p+1}\right) > 0$, the result holds for general parameters by appealing to analytic continuation, since both sides in Eq. \[eq: kummer\_transform\_pFp\] are analytic functions of $a_{p+1}$ and $b_{p+1}$ (given $b_{p+1} \neq 0, -1, -2, ...$). Note that for $p=1$, Eq. \[eq: kummer\_transform\_pFp\] is the same as the Kummer transformation derived by Paris (see their Eq. 3) for $_2F_2\left(x\right)$ with general parameters [@paris2005kummer].
Theorem 4 (Euler-type transformation for $\boldsymbol{_{p+1}F_p}$ with general parameters).
===========================================================================================
For $p \geq 1$ and given $b_i \neq 0, -1, -2, ...$ $\forall$ $1 \leq i \leq p+1$, $\left|x\right| < 1$ and $\text{Re}\left(x\right) < 1/2$, $$\begin{aligned}
& _{p+2}F_{p+1}\left(
\begin{smallmatrix}
a_0, & a_1 , & a_2 , & a_3 , & ..., & a_p , & a_{p+1}\\
& b_1 , & b_2 , & b_3 , & ..., & b_p , & b_{p+1} \\
\end{smallmatrix};x
\right) \nonumber \\
& = \left(\frac{1}{1-x}\right)^{a_0} \sum_{j_1 = 0}^\infty \sum_{j_2 = 0}^\infty ... \sum_{j_p = 0}^\infty \left(a_0\right)_{u_p} \prod_{q=1}^p \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q} j_q !} \left(\frac{x}{x-1}\right)^{j_q} \cdot \nonumber \\
& \hspace{25pt} {_{p+2}}F_{p+1}\left(
\begin{smallmatrix}
a_0+u_p, & a_1 + u_0, & a_2 + u_1, & a_3 + u_2, & ..., & a_p + u_{p-1}, & b_{p+1} - a_{p+1}\\
& b_1 + u_1, & b_2 + u_2, & b_3 + u_3, & ..., & b_p + u_p, & b_{p+1} \\
\end{smallmatrix};\frac{x}{x-1}
\right), \label{eq: euler_transform_p+1Fp}\end{aligned}$$ where $u_0 = 0$, $u_1 = j_1$, $u_2 = j_1+j_2$, ..., $u_p = \sum_{q=1}^p j_q$.
### Proof {#proof-3 .unnumbered}
To prove this theorem, we follow the same approach as that for theorem 3. Hence, we utilize the integral representation for $_{p+2}F_{p+1}\left(x\right)$ function, which is available from Slater [@slater1966generalized] (see Eq. 4.8.3.12) and the addition formula for $_{p+1}F_p\left(x\right)$, which we obtained previously (theorem 2). The integral representation for $_{p+2}F_{p+1}\left(x\right)$ is $$\begin{aligned}
_{p+2}F_{p+1}\left(
\begin{smallmatrix}
a_0, & a_1, & ..., & a_{p+1}\\
& b_1, & ..., & b_{p+1}\\
\end{smallmatrix};x
\right) & = & \frac{\Gamma\left(b_{p+1}\right)}{\Gamma\left(a_{p+1}\right) \Gamma\left(b_{p+1}-a_{p+1}\right)} \nonumber \\
& & \hspace{-90pt} \int_0^1 t^{a_{p+1}-1} \left(1-t\right)^{b_{p+1}-a_{p+1}-1} \cdot {_{p+1}F_p}\left(
\begin{smallmatrix}
a_0, & a_1, & ..., & a_{p}\\
& b_1, & ..., & b_{p}\\
\end{smallmatrix};xt
\right) dt \label{eq: integral_representation_p+2Fp+1_original} \\
\implies {_{p+2}F_{p+1}}\left(
\begin{smallmatrix}
a_0, & a_1, & ..., & a_{p+1}\\
& b_1, & ..., & b_{p+1}\\
\end{smallmatrix};x
\right) & = & \frac{\Gamma\left(b_{p+1}\right)}{\Gamma\left(a_{p+1}\right) \Gamma\left(b_{p+1}-a_{p+1}\right)} \nonumber \\
& & \hspace{-90pt} \int_0^1 t^{b_{p+1}-a_{p+1}-1} \left(1-t\right)^{a_{p+1}-1} \cdot {_{p+1}F_p}\left(
\begin{smallmatrix}
a_0, & a_1, & ..., & a_{p}\\
& b_1, & ..., & b_{p}\\
\end{smallmatrix};x-xt
\right) dt \label{eq: integral_representation_p+2Fp+1_altered}\end{aligned}$$ if $\text{Re}\left(b_{p+1}\right) > \text{Re}\left(a_{p+1}\right) > 0$ and $\left|x\right| < 1$. Using the addition formula in Eq. \[eq: add\_theorem\_p+1Fp\] on the right hand side in Eq. \[eq: integral\_representation\_p+2Fp+1\_altered\], given $b_i \neq 0, -1, -2, ...$ $\forall$ $1 \leq i \leq p$, we rewrite it after switching the order of integral and summation as $$\begin{aligned}
& \left(\frac{1}{1-x}\right)^{a_0} \sum_{j_1 = 0}^\infty \sum_{j_2 = 0}^\infty ... \sum_{j_p = 0}^\infty \left(a_0\right)_{u_p} \prod_{q=1}^p \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q} j_q !} \left(\frac{x}{x-1}\right)^{j_q} \frac{\Gamma\left(b_{p+1}\right)}{\Gamma\left(a_{p+1}\right) \Gamma\left(b_{p+1}-a_{p+1}\right)} \nonumber \\
& \int_0^1 t^{b_{p+1}-a_{p+1}-1} \left(1-t\right)^{a_{p+1}-1} {_{p+1}}F_{p}\left(
\begin{smallmatrix}
a_0+u_p, & a_1 + u_0, & a_2 + u_1, & a_3 + u_2, & ..., & a_p + u_{p-1}\\
& b_1 + u_1, & b_2 + u_2, & b_3 + u_3, & ..., & b_p + u_p\\
\end{smallmatrix};\frac{xt}{x-1}
\right)dt.\end{aligned}$$ Now, using the relationship in Eq. \[eq: integral\_representation\_p+2Fp+1\_original\], the above expression can be rewritten as $$\begin{aligned}
\left(\frac{1}{1-x}\right)^{a_0} \sum_{j_1 = 0}^\infty \sum_{j_2 = 0}^\infty ... \sum_{j_p = 0}^\infty \left(a_0\right)_{u_p} \prod_{q=1}^p \frac{\left(b_q - a_q\right)_{j_q} \left(a_q\right)_{u_{q-1}}}{\left(b_q\right)_{u_q} j_q !} \left(\frac{x}{x-1}\right)^{j_q} \cdot \nonumber \\
\hspace{25pt} {_{p+2}}F_{p+1}\left(
\begin{smallmatrix}
a_0+u_p, & a_1 + u_0, & a_2 + u_1, & a_3 + u_2, & ..., & a_p + u_{p-1}, & b_{p+1} - a_{p+1}\\
& b_1 + u_1, & b_2 + u_2, & b_3 + u_3, & ..., & b_p + u_p, & b_{p+1} \\
\end{smallmatrix};\frac{x}{x-1}
\right),\end{aligned}$$ which upon substitution in Eq. \[eq: integral\_representation\_p+2Fp+1\_altered\] yields the Euler-type transformation in Eq. \[eq: euler\_transform\_p+1Fp\]. Once again, we note by appealing to analytic continuation that the result holds for general parameters even if the requirement $\text{Re}\left(b_{p+1}\right) > \text{Re}\left(a_{p+1}\right) > 0$ is relaxed because both sides in Eq. \[eq: euler\_transform\_p+1Fp\] are analytic functions of $a_{p+1}$ and $b_{p+1}$ (given $b_{p+1} \neq 0, -1, -2, ...$).
Summary
=======
In summary, we have obtained addition formulas for $_pF_p$ and $_{p+1}F_p$ generalized hypergeometric functions, which we utilized to derive Kummer- and Euler-type transformations of these functions with general parameters. The theorems stated herein could be used to derive transformation formulas for other special cases of interest.
Declaration of interest {#declaration-of-interest .unnumbered}
=======================
The author declares that he has no competing interests.
|
---
abstract: 'We study the many-body effects on coherent atom-molecule oscillations by means of an effective quantum field theory that describes Feshbach-resonant interactions in Bose gases in terms of an atom-molecule hamiltonian. We determine numerically the many-body corrections to the oscillation frequency for various densities of the atomic condensate. We also derive an analytic expression that approximately describes both the density and magnetic-field dependence of this frequency near the resonance. We find excellent agreement with experiment.'
author:
- 'R.A. Duine'
- 'H.T.C. Stoof'
date: 'February 18, 2003'
title: 'Many-body aspects of coherent atom-molecule oscillations'
---
[*Introduction*]{} — One of the most remarkable applications of Feshbach resonances in doubly spin-polarized alkali gases [@stwalley; @eite; @inouye] is the observation of coherent atom-molecule oscillations [@elisabeth2]. In this last experiment Donley [*et al.*]{} used the Feshbach resonance at $B_0 \simeq 154.9$ Gauss in the $|
f=2;m_f=-2\rangle$ state of $^{85}$Rb to perform a Ramsey-type experiment, consisting of two short pulses in the magnetic field towards resonance separated by a longer evolution time. As a function of this evolution time an oscillation in the number of condensate atoms was observed. Over the investigated range of magnetic field during the evolution time, the frequency of this oscillation agreed exactly with the molecular binding energy found from a two-body coupled-channels calculation [@servaas], indicating coherence between atoms and molecules.
Very recently, Claussen [*et al.*]{} have performed a similar series of measurements over a larger range of magnetic fields [@claussen2003]. It was found that close to resonance the frequency of the oscillation deviates from the two-body molecular binding energy, which indicates that many-body effects play an important role in this regime. It is the main purpose of this Letter to present the theory that explains this deviation quantitatively.
The first mean-field theory for Feshbach-resonant interactions in Bose-Einstein condensed gases is due to Drummond [*et al.*]{} [@drummond1998] and Timmermans [*et al.*]{} [@timmermans] and introduces the physical picture of an interacting atomic condensate coupled to a noninteracting molecular condensate. Although this theory contains the correct resonant scattering amplitude for the atoms, it does not contain the correct molecular binding energy. By studying the fluctuations around this mean-field theory within the Hartree-Fock-Bogoliubov approximation, it is possible to also incorporate the correct binding energy [@servaas; @mackie2002; @kohler2002]. This comes about because the latter approach explicitly contains a so-called anomalous density, or pairing field, for the atoms which, after elimination, leads to a shift in the coupling constants and the molecular binding energy. Unfortunately, however, the elimination does not lead to a proper renormalization of all the coupling constants. In particular the interaction between condensate atoms and non-condensate atoms, which is ultimately responsible for the many-body corrections to the molecular binding energy that are of interest to us here, is not correctly described within the Hartree-Fock-Bogoliubov approximation.
Relying on the anomalous density for the description of the molecular properties also makes the theory inapplicable above the critical temperature for Bose-Einstein condensation, since the anomalous density is proportional to the atomic condensate density. To overcome all these problems, it is convenient to formulate an effective quantum field theory that incorporates the exact two-body physics not at the mean-field level but at the quantum level and is hence applicable both above and below the critical temperature. We have recently derived such an effective quantum field theory by starting from the microscopic hamiltonian for an atomic gas with a Feshbach resonance and explicitly summing all the ladder diagrams [@rembert4]. Here, we apply the mean-field theory for the Bose-Einstein condensed phase of the gas that results from this effective quantum field theory, to the study of the recent experiments by Claussen [*et al.*]{} [@claussen2003].
[*Atom-molecule coherence*]{} — The mean-field equations consist of coupled equations for the macroscopic wave function $\psi_{\rm a} ({\bf x},t)$, describing the atomic Bose-Einstein condensate, and the macroscopic wave function $\psi_{\rm m} ({\bf
x}, t)$ that describes the condensate of bare molecules, i.e., with the appropriate bound-state wave function in the closed channel potential of the Feshbach problem. The mean-field equations for the coupled atom-molecule system are ultimately given by $$\begin{aligned}
\label{eq:mfe}
i \hbar \frac{\partial \psi_{\rm a} ({\bf x},t)}{\partial t}
&=& \left[-\frac{\hbar^2 {\bf \nabla}^2}{2m}
+T^{\rm 2B}_{\rm bg} |\psi_{\rm a} ({\bf x},t)|^2
\right]
\psi_{\rm a} ({\bf x},t) \nonumber \\ &&+ 2 g \psi_{\rm a}^* ({\bf x},t)
\psi_{\rm m}({\bf x},t)~, \nonumber \\
i \hbar \frac{\partial\psi_{\rm m} ({\bf x},t)}{\partial t}
&=&\left[-\frac{\hbar^2 {\bf \nabla}^2}{4m}+ \delta (B(t))
\right] \psi_{\rm m} ({\bf x},t) +
g \psi_{\rm a}^2 ({\bf x},t) \nonumber \\ && \hspace*{-0.7in}
- g^2 \frac{m^{3/2}}{2 \pi \hbar^3} i \sqrt{i \hbar \frac{\partial}{\partial t}
+ \frac{\hbar^2 {\bf \nabla}^2}{4m} -2 \hbar \Sigma^{\rm HF}}
\psi_{\rm m} ({\bf x},t)~,\end{aligned}$$ where $T^{\rm 2B}_{\rm bg}=4 \pi a_{\rm bg} \hbar^2/m$ is the off-resonant two-body T(ransition) matrix with $a_{\rm bg}$ the off-resonant background scattering length and $m$ the mass of one atom. The atom-molecule coupling $g$ is found from experiment by adiabatically eliminating the molecular wave function and using the fact that the resulting magnetic-field dependent scattering length of the atoms must be equal to $$\label{eq:ascatofb}
a(B) = a_{\rm bg} \left( 1-\frac{\Delta B}{B-B_0} \right),$$ with $\Delta B$ and $B_0$ the experimental width and position of the resonance, respectively. This procedure results in $g=\hbar \sqrt{2 \pi
a_{\rm bg} \Delta B \Delta \mu /m}$, where we have made use of the fact that the detuning of the bare molecular state is given by $\delta(B) = \Delta \mu (B-B_0)$ with $\Delta \mu \simeq -2.2 \mu_{\rm
B}$ for $^{85}$Rb [@servaas] and $\mu_{\rm B}$ the Bohr magneton.
The mean-field equation for the molecular condensate contains a fractional derivative, corresponding to the retarded self energy of the molecules. In momentum and frequency space, this self energy reads $\hbar \Sigma^{(+)}_{\rm m} ({\bf k},\omega)=-
(g^2 m^{3/2}/2 \pi \hbar^3) i
\sqrt{\hbar \omega - \hbar^2{\bf k}^2/4m -2 \hbar \Sigma^{\rm HF}}$ [^1]. The square-root behavior is a result of the Wigner threshold law for the decay of a molecule with total energy $\hbar \omega$ and center-of-mass momentum $\hbar {\bf k}$ into the two-atom continuum. Due to the mean-field interaction of the noncondensed atoms with the condensate, the decaying molecule has to overcome a mean-field barrier given by $2 \hbar \Sigma^{\rm
HF}$. Here, $\hbar \Sigma^{\rm HF}$ denotes the Hartree-Fock self energy for the noncondensed atoms. Neglecting the momentum dependence of this self energy, it is given by $$\begin{aligned}
\label{eq:sigmahf}
&&\hbar \Sigma^{\rm HF} =
2 n_{\rm a} \Bigl( T^{\rm 2B}_{bg} \nonumber \\
&&+\frac{2 g^2}{\hbar \Sigma^{\rm HF}+\mu
-\delta (B) - g^2 \frac{m^{3/2}}{2 \pi \hbar^3} \sqrt{\hbar
\Sigma^{\rm HF}-\mu}} \Bigr)~,\end{aligned}$$ where $n_{\rm a}=|\psi_{\rm a}|^2$ is the density of the atomic condensate, $\mu$ its chemical potential, and we have used that a collision between a condensed atom and a noncondensed atom has a mean-field shift of $\mu+\hbar \Sigma^{\rm HF}$. Far from resonance the energy-dependence of the interactions can be safely ignored and the Hartree-Fock self energy becomes equal to $8 \pi a(B) \hbar^2 n_{\rm
a}/m$, as expected.
The molecular binding energy is given by the pole of the molecular propagator at zero momentum, which from Eq. (\[eq:mfe\]) is seen to be given by $$\label{eq:gmkw}
G_{\rm m}^{(+)} ({\bf 0},\omega)=
\frac{\hbar}{\hbar \omega+i0 -\delta (B)
+ \frac{g^2 m^{3/2}}{2 \pi \hbar^3} i
\sqrt{\hbar \omega - 2\hbar \Sigma^{\rm HF}}}~.$$ In the limit of vanishing condensate density $n_{\rm a}$ and for negative detuning, it has a pole at $$\label{eq:bse}
\epsilon_{\rm m} (B)= \delta (B) + \frac{g^4 m^3}{8 \pi^2 \hbar^6}
\left[\sqrt{1-\frac{16 \pi^2 \hbar^6}{g^4 m^3} \delta (B)} -1\right].$$ Close to the resonance it thus follows that , which is the correct molecular binding energy in vacuum [@servaas]. Note that in the absence of the molecular self energy the binding energy would be equal to the detuning, which is incorrect close to resonance. Moreover, the residue of the pole is given by $$\begin{aligned}
\label{eq:factorz}
Z(B)&=&\left[1-\frac{\partial \Sigma^{(+)}_{\rm m}({\bf 0},\omega)}
{\partial \omega}\right]^{-1}
\nonumber \\ &=& \left[1 + \frac{g^2 m^{3/2}}{4 \pi \hbar^3
\sqrt{|\epsilon_{\rm m}(B)|}}\right]^{-1}\end{aligned}$$ and always smaller than one. Physically, the latter can be understood from the fact that the dressed molecular bound state near the Feshbach resonance is given by $$\begin{aligned}
\label{eq:wavefctmol}
| \chi_{\rm m}; {\rm dressed} \rangle&=&
\sqrt{Z(B)}| \chi_{\rm m} ; {\rm bare} \rangle
\nonumber \\
&&+ \int \frac{d {\bf k}}{(2 \pi)^3} C({\bf k})
| {\bf k}, -{\bf k};{\rm open} \rangle ~,\end{aligned}$$ where the coefficient $C({\bf k})$ denotes the amplitude of the dressed molecular state to be in the open channel of the Feshbach problem and the two atoms having momenta ${\bf k}$ and $-{\bf k}$, respectively. They are normalized as $\int d {\bf k} |C({\bf k})|^2/(2 \pi)^3 = 1-Z(B)$. The dressed molecular state therefore only contains with an amplitude $\sqrt{Z(B)}$ the bare molecular state $|\chi_{\rm m};{\rm bare} \rangle$. Close to resonance we have that $Z(B) \ll 1$, whereas it approaches one far off resonance. With respect to this remark it is important to note that the result of the Hartree-Fock-Bogoliubov theory for the density of the molecular condensate should be multiplied by a factor $1/Z(B) \gg 1$ to obtain the density of real dressed molecules, since in this theory always the density of bare molecules is calculated [@rembert5; @braaten2003].
![\[fig:fig1\] Dispersion relation for the collective modes of the atom-molecule system for an atomic condensate density of $n_{\rm a}=2 \times 10^{12}$ cm$^{-3}$, at a magnetic field of $B=156$ G. The momentum is indicated in units of $1/\xi$, where $\xi = 1/\sqrt{16 \pi a n_{\rm a}}$ is the coherence length. The upper solid line shows the gapless dispersion relation for phonon-like excitations. The lower solid line indicates the dispersion for atom-molecule oscillations. The dashed line shows the effective Bogoliubov dispersion. ](fig1.eps)
[*Collective modes*]{} — To study the many-body effects on the frequency of the coherent atom-molecule oscillations it is important to realize that these oscillations are in fact a collective mode where the atomic condensate density oscillates out-of-phase with the molecular condensate density. It is thus worthwhile to study the collective modes of the mean-field equations in Eq. (\[eq:mfe\]) and look for solutions of the form $$\begin{aligned}
\label{eq:linearization}
\psi_{\rm m} ({\bf x},t)\!&=&\!\left[ \psi_{\rm m}
+u'_{\bf k} e^{-i \omega t + i {\bf k} \cdot {\bf x}}
+{v'_{\bf k}}^* e^{+i \omega t -i {\bf k} \cdot {\bf x}}\right]e^{-i 2 \mu
t/\hbar}~,
\nonumber \\
\psi_{\rm a} ({\bf x},t)\!&=&\!\left[ \psi_{\rm a} +u_{\bf k}
e^{-i \omega t + i {\bf k} \cdot {\bf x}}
+ v_{\bf k}^* e^{+i \omega t -i {\bf k} \cdot {\bf x}}\right]e^{-i \mu
t/\hbar}~.\end{aligned}$$ After substitution into the mean-field equations in Eq. (\[eq:mfe\]) the eigenmodes are found by diagonalizing the resulting $4 \times 4$ matrix. This yields a dispersion relation with two branches, one corresponding to the gapless Bogoliubov modes and one that corresponds to the atom-molecule oscillations. The zero-momentum part of the latter corresponds to the experimentally observed frequency of the coherent atom-molecule oscillations. Note that for this calculation the evaluation of the molecular self energy can be performed exactly because every term in the right-hand side of Eq. (\[eq:linearization\]) is an eigenfunction of the operator under the square root.
In Fig. \[fig:fig1\] we show the dispersion relations found by means of the above procedure, for a fixed atomic condensate density of $n_{\rm a}=2
\times 10^{12}$ cm$^{-3}$ at a fixed magnetic field of $B=156$ G. Physically, the upper branch corresponds to the phonon-like excitations in the atomic condensate and the lower branch to coherent atom-molecule oscillations. The dashed line denotes the Bogoliubov dispersion for the scattering length $a(B)$. At low momenta the phonon branch corresponds with the Bogoliubov dispersion, as expected. At higher momenta the dispersion starts to deviate from the Bogoliubov result, due to the energy and momentum dependence of the resonant interactions that reduce the scattering amplitude. The dispersion of the lower branch obeys $\hbar\omega_{\bf k} \simeq -\hbar\omega_{\rm J} + \hbar^2{\bf k}^2/4m$, where $\hbar\omega_{\rm J}$ is the Josephson frequency that is observed by Claussen [*et al*]{}. in their Ramsey experiment. This dispersion is negative due to the fact that we are dealing with a metastable situation. For negative detuning the true ground state contains almost all atoms in the form of molecules.
![\[fig:fig2\] Frequency of coherent atom-molecule oscillations. The solid lines show the frequency as a function of the magnetic field for three different densities of the atomic condensate. The dashed line shows the two-body result for the molecular binding energy. The experimental points are taken from Ref. [@claussen2003]. ](fig2.eps)
In Fig. \[fig:fig2\] the results are shown for the frequency of the atom-molecule oscillations as a function of the magnetic field, for three different densities. Clearly, for increasing density the frequency starts to deviate from the two-body result. In addition, we show in Fig. \[fig:fig3\] the frequency of the atom-molecule oscillations relative to the two-body binding energy, as a function of the atomic condensate density. The calculation is performed in this case for several values of the magnetic field. Both the magnetic field and the atomic density dependence of the frequency can be understood as follows. We first observe that for the range of magnetic fields and atomic densities that are explored experimentally, the difference between the energy of the dressed molecular state and the threshold of the two-particle continuum is almost independent of density and equal to $\epsilon_{\rm m}(B)$. It is interesting to note that only this difference is independent of density, whereas both quantities individually show a substantial mean-field shift of about $2 \hbar \Sigma^{\rm HF}$ [@rembert4]. As a result of the cancellation, however, also the wave-function renormalization factor $Z$ is almost density independent and equal to $Z(B)$. Expressing our mean-field equations in Eq. (\[eq:mfe\]) in terms of the condensate wave function for the dressed molecules by replacing $\psi_{\rm m}({\bf x},t)$ by $\sqrt{Z(B)} \psi_{\rm m}({\bf
x},t)$, we see that the coupling between the atomic condensate and the dressed molecular condensate is reduced by a factor of $\sqrt{Z(B)} \ll 1$ [@rembert5]. We thus expect the frequency to obey $$\label{eq:josephson}
\hbar\omega_{\rm J} \simeq
\sqrt{16 Z(B) g^2 n_{\rm a} + (\epsilon_{\rm m} (B))^2}~.$$ This analytic expression indeed turns out to give a first approximation to the deviation of the frequency from the two-body result.
![\[fig:fig3\] Frequency of coherent atom-molecule oscillations as a function of the atomic condensate density, relative to the two-body binding energy. The solid lines show the result for different magnetic fields. ](fig3.eps)
In order to confront our results with the experimental data we have to realize that the experiments are performed in a magnetic trap. Taking only the ground-states into account for both the atomic and the molecular condensates, this implies effectively that the atom-molecule coupling $g$ is reduced by an overlap integral. Hence we define the effective homogeneous condensate density by means of $n_{\rm a} = N_{\rm a} \left[
\int d {\bf x} \phi_{\rm a}^2 ({\bf x} ) \phi_{\rm m} (\bf x) \right]^2$, where $N_{\rm a}$ denotes the number of condensed atoms and $\phi_{\rm a}
({\bf x} )$ and $\phi_m (\bf x)$ denote the atomic and molecular ground state wave function, respectively. For the experiments of Claussen [*et al.*]{} this results in an effective density of $n_{\rm a}\simeq 2 \times
10^{12}$ cm$^{-3}$ [@claussen2003]. Fig. \[fig:fig2\] clearly shows an excellent agreement with the experimentally observed frequency for this density.
It is important to note that there are two hidden assumptions in the above comparison. First, we have used that the dressed molecules are trapped in the same external potential as the atoms. This is not obvious because the bare molecular state involved in the Feshbach resonance is high-field seeking and therefore not trapped. However, Eq. (\[eq:factorz\]) shows that near resonance almost all the amplitude of the dressed molecule is in the low-field seeking open channel and its magnetic moment is therefore almost equal to twice the atomic magnetic moment. Second, we have determined the frequency of the coherent atom-molecule oscillations in equilibrium. In contrast, the observed oscillations in the number of condensate atoms is clearly a nonequilibrium phenomenon. This is, however, expected not to play an important role because the Ramsey-pulse sequence is performed on such a fast time scale that the response of the condensate wave function can be neglected.
[*Conclusions*]{} — With the linear-response calculation presented in this Letter we have obtained excellent agreement with the experimental results on the frequency of coherent atom-molecule oscillations. The next step is a more detailed understanding of other quantities that are of interest for the two-pulse experiments [@elisabeth2; @claussen2003]. This includes a quantitative study of the number of condensed atoms as a function of time that goes beyond the linear approximation discussed here. Experimentally an overall decay of the number of condensed atoms is observed and also that the oscillations have a finite damping rate. Both effects increase as one approaches the resonance. We expect that an important contribution to these effects is due to the so-called rogue-dissociation process [@mackie2002], which has its physical origin in the decay of a molecule into two noncondensed atoms that in the center-of-mass system have opposite momenta ${\bf k}$ and $-{\bf k}$, respectively. In equilibrium this process is always forbidden due to energy conservation, since the molecular state lies below the two-atom continuum threshold. However, in the nonequilibrium setting of the experiments of interest, it may occur due to the fact that the detuning is strongly time-dependent. In the mean-field equations in Eq. (\[eq:mfe\]) it is the fractional derivative term corresponding to the self energy of the molecules that automatically incorporates the rogue-dissociation effect. A full numerical simulation of these equation is challenging due to the fact that we are essentially dealing with a term that is nonlocal in time. Nevertheless, work in this direction is in progress and will be reported in a future publication.
[00]{} W.C. Stwalley, Phys. Rev. Lett. [**37**]{}, 1628 (1976). E. Tiesinga, B.J. Verhaar, and H.T.C. Stoof, Phys. Rev. A [**47**]{}, 4114 (1993). S. Inouye, M.R. Andrews, J. Stenger, H.-J. Miesner, D.M. Stamper-Kurn, W. Ketterle, Nature [**392**]{}, 151 (1998). E.A. Donley, N.R. Claussen, S.T. Thompson, and C.E. Wieman, Nature [**417**]{}, 529 (2002). S.J.J.M.F. Kokkelmans and M.J. Holland, Phys. Rev. Lett. [**89**]{}, 180401 (2002). N.R. Claussen, S.J.J.M.F. Kokkelmans, S.T. Thompson, E.A. Donley, and C.E. Wieman (cond-mat/0302195). P.D. Drummond, K.V. Kheruntsyan, and H. He, Phys. Rev. Lett. [**81**]{}, 3055 (1998). E. Timmermans, P. Tommasini, H. Hussein, A. Kerman, Phys. Rep. [**315**]{}, 199 (1999). M. Mackie, K.-A. Suominen and J. Javanainen, Phys. Rev. Lett. [**89**]{}, 180403 (2002). T. Köhler, T. Gasenzer and K. Burnett, Phys. Rev. A [**67**]{}, 013601 (2002). R.A. Duine and H.T.C. Stoof (cond-mat/0210544). R.A. Duine and H.T.C. Stoof (cond-mat/0211514). See also E. Braaten, H.-W. Hammer and M. Kusunoki (cond-mat/0301489).
[^1]: The full expression for the self energy of the molecules is given by $\hbar\Sigma^{(+)}_{\rm m} ({\bf k},\omega)= - (g^2
m^{3/2}/2 \pi \hbar^3) i
\sqrt{z} (1+i a_{\rm bg}\sqrt{m z}/\hbar)^{-1}$, with $z = \hbar \omega - \hbar^2{\bf k}^2/4m -2 \hbar \Sigma^{\rm
HF}$. Close to resonance only the square-root in the numerator is important. We will use this approximation throughout the paper.
|
---
abstract: 'We consider a single server retrial queue with the server subject to interruptions and classical retrial policy for the access from the orbit to the server. We analyze the equilibrium distribution of the system and obtain the generating functions of the limiting distribution.'
author:
- |
T<span style="font-variant:small-caps;">ewfik</span> K<span style="font-variant:small-caps;">ernane</span>\
*Department of Mathematics*\
*Faculty of Sciences*\
*King Khalid University*\
*Abha, Saudi Arabia*\
tkernane@gmail.com
title: '**A Single Server Retrial Queue with Different Types of Server Interruptions**'
---
Introduction
============
Queueing systems with retrials of the attempts are characterized by the fact that an arrival customer who finds the server occupied is obliged to join a group of blocked customers, called orbit, and reapply after random intervals of time to obtain the service. These systems are useful in the stochastic modeling of much of situations in practice. We can find them in aviation, where a plane which finds the runway occupied remakes its attempt of landing later and we say in this case that it is in orbit. In Telephone systems where a telephone subscriber who obtains a busy signal repeats the call until the required connection is made. In data processing, we find them in protocol of access CSMA/CD. They appear in the modeling of the systems of maintenance and the problems of repair among others. For details on these models, see the book of Falin and Templeton [@Fal] or the recent book by Artalejo and Gomez-Corral [@Art2].
We study in this article single server retrial queues with various types of interruptions of the server. From a practical point of view, it is more realistic to consider queues with repetitions of calls and the server exposed to random interruptions. Queueing models with interruptions of service proved to be a useful abstraction in the situations where a server is shared by multiple queues, or when the server is subject to breakdowns. Such systems were studied in the literature by many authors. Fiems et *al*. [@Fie] considered an M/G/1 queue with various types of interruptions of the server and our work is a generalization to the case of retrial queues. White and Christie [@Whi] were the first to study queues with interruptions of service by considering a queueing system with exponentially distributed interruptions. Times of interruptions and services generally distributed are considered by Avi-Itzhak and Naor [@Avi] and Thiruvengadam [@Thi]. Other generalisations were considered in the literature ( phase-type [@Fed], approximate analysis [@Van], Markov-modulated environment [@Tak] and [@Mas], processor sharing [@Nun]). Gaver [@Gav] considers the case where the service is repeated or repeated and begin again after the interruption.
We consider in this paper a single server retrial queue with server interruptions and the classical retrial policy where each customer in orbit conduct its own attempts to get served independently of other customers present in the orbit. We can then assume that the probability of a retrial during the time interval $(t,t+dt)$, given that $j$ customers were in orbit at time $t,$ is $j\theta dt+\circ \left( dt\right) $. Kulkarni and Choi [Kul1]{} studied a single server linear retrial queue with server subject to breakdowns and repairs and they obtained the generating functions of the limiting distribution and performance characteristics. Artalejo [@Art1] obtained sufficient conditions for ergodicity of multiserver retrial queues with breakdowns and a recursive algorithm to compute the steady-state probabilities for the M/G/1 linear retrial queue with breakdowns. The detailed analysis for reliability of retrial queues with linear retrial policy was given by Wang, Cao and Li [@Wan1].
The remainder of paper is organized as follows. In the following section, we describe the model and give the necessary and sufficient conditions so that the system is stable. In section 3, we analyze the equilibrium distribution of the system in study.
Model Description
=================
Consider a single server queueing system in which customers arrive in accordance with a Poisson process with arrival rate $\lambda $. If at the instant of arrival the customer finds the server free, it takes its service and leaves the system. Otherwise, if the server is busy or in interruption, the arriving customer joins an unlimited queue called orbit and makes retrials for getting served after random time intervals. We consider the classical policy where each customer in orbit conducts his own attempts to obtain service independently from the other customers present in the orbit. We can then assume that the probability of a retrial during the time interval $(t,t+dt)$, given that $j$ customers were in orbit at time $t,$ is $j\theta dt+\circ \left( dt\right) $. Service times constitute a series of independent and identically distributed (i.i.d.) random variables with common distribution function $B(t)$, density function $b(t)$, and corresponding Laplace–Stieltjes transform (LST) $\beta (s)$ and finite first two moments $\beta _{k}=\left( -1\right) ^{k}\beta ^{(k)}(0),$ $k=1,2$. Interruptions of the service may occur according to a Poisson process with rate $\nu $ if the server is busy and this type of interruption can be disruptive with probability $p_{d}$ (or rate $\nu _{d}=p_{d}\nu $) or non-disruptive with probability $p_{n}=1-p_{d}$ (or rate $\nu _{n}=p_{n}\nu $). In the case of a disruptive interruption the customer being served repeats his service at the end of the interruption, in the other type the customer continues his stopped service. If the server is idle, another type of interruptions may occur according to a Poisson process with rate $\nu
_{i} $. We call this type *idle interruption*. The lengths of the consecutive disruptive (non-disruptive, idle time) interruptions constitute a series of i.i.d. positive random variables with distribution function $B_{d}(t)$ ($B_{n}(t)$, $B_{i}(t)$), density function $b_{d}(t)$ ($b_{n}(t)$, $b_{i}(t)$), corresponding Laplace–Stieltjes transform (LST) $\beta _{d}(s)$ ($\beta _{n}(s)$, $\beta _{i}(s)$) and finite first two moments $\beta
_{k}^{d},$ ($\beta _{k}^{n},$ $\beta _{k}^{i}$) $k=1,2$.
Denote by $N(t)$ the number of customers in orbit at time $t$. Let $C(t)$ be the state of the server at time $t$ : $C(t)=F$ if the server is free (and functions normally), $C(t)=S$ if the server is busy (and functions normally), $C(t)=D$ if the server is on a disruptive interruption, $C(t)=N$ if the server is on a non-disruptive interruption, $C(t)=I$ if the server is taking an idle interruption. We introduce the random variables $\mathcal{\xi
}(t),$ $\mathcal{\xi }_{D}(t),$ $\mathcal{\xi }_{N}(t)$ and $\mathcal{\xi }_{I}(t)$ defined as follows. If $C(t)=S$ then $\mathcal{\xi }(t)$ represents the elapsed service time at time $t$; if $C(t)=D,$ then $\mathcal{\xi }_{D}(t)$ represents the elapsed disruptive interruption time at $t$; if $C(t)=N,$ then $\mathcal{\xi }_{N}(t)$ represents the elapsed non-disruptive interruption time at $t$; and if $C(t)=I$ then $\mathcal{\xi }_{I}(t)$ is the elapsed idle interruption time at $t.$
Stability Analysis
==================
We first study the condition for the system to be stable. The following theorem provides the necessary and sufficient stability condition.
The system with classical retrial policy and interruptions is stable if and only if the following condition is fulfilled$$\frac{\lambda \left( 1-\beta (\nu _{d})\right) }{\nu _{d}\beta (\nu _{d})}\left( 1+\nu _{d}\beta _{1}^{d}+\nu _{n}\beta _{1}^{n}\right) <1.
\label{stab}$$
Let $\{s_{n};$ $n\in
\mathbb{N}
\}$ be the sequence of epochs of service completion time. We consider the process $Y_{n}=\left( N(s_{n}+),C(s_{n}+)\right) $ embedded immediately after time $s_{n}.$ It is readily to see that $\{Y_{n};$ $n\in
\mathbb{N}
\}$ is an irreducible aperiodic Markov chain. To determine the stability of the system it remains to prove that $\{Y_{n};$ $n\in
\mathbb{N}
\}$ is ergodic under the suitable stability condition. Let us first consider the generalized service time $\widetilde{S}$ of a customer which includes, in addition to the original service time $S$ of the customer, possible interruption times during the service period of the customer.* *Fiems et *al*. [@Fie] showed that the generalized service time has the Laplace transform$$\widetilde{\beta }(s)=\frac{\left[ s+\nu -\nu _{n}\beta _{n}(s)\right] \beta
\left( s+\nu -\nu _{n}\beta _{n}(s)\right) }{\left[ s+\nu -\nu _{n}\beta
_{n}(s)\right] -\nu _{d}\beta _{d}(s)\left[ 1-\beta \left( s+\nu -\nu
_{n}\beta _{n}(s)\right) \right] },$$hence its expected value is given by$$E\widetilde{S}=-\widetilde{\beta }^{\prime }(0)=\frac{\left( 1-\beta (\nu
_{d})\right) }{\nu _{d}\beta (\nu _{d})}\left( 1+\nu _{d}\beta _{1}^{d}+\nu
_{n}\beta _{1}^{n}\right) .$$For the sufficiency, we shall use Foster’s criterion, which states that a Markov chain $\{Y_{n};$ $n\in
\mathbb{N}
\}$ is ergodic if there exists a nonnegative function $f(k),$ $k\in
\mathbb{N}
,$ and $\delta >0$ such that for all $k\neq 0$ the mean drift$$\chi _{k}=E\left[ f(Y_{n+1})-f(Y_{n})\mid Y_{n}=k\right] ,$$satisfies $\chi _{k}\leq -\delta $ and $E\left[ f(Y_{n+1})\mid Y_{n}=0\right]
<\infty .$ If we choose $f(k)=k$ we obtain $$E\left[ f(Y_{n+1})\mid Y_{n}=0\right] =\lambda E\widetilde{S}=\frac{\lambda
\left( 1-\beta (\nu _{d})\right) }{\nu _{d}\beta (\nu _{d})}\left( 1+\nu
_{d}\beta _{1}^{d}+\nu _{n}\beta _{1}^{n}\right) <\infty ,$$and we can easily check that$$\chi _{k}=\lambda E\widetilde{S}-1=\left[ \lambda \left( 1-\beta (\nu
_{d})\right) /\nu _{d}\beta (\nu _{d})\right] \left( 1+\nu _{d}\beta
_{1}^{d}+\nu _{n}\beta _{1}^{n}\right) -1.$$If we set$$\delta =1-\frac{\lambda \left( 1-\beta (\nu _{d})\right) }{\nu _{d}\beta
(\nu _{d})}\left( 1+\nu _{d}\beta _{1}^{d}+\nu _{n}\beta _{1}^{n}\right)$$then the condition (\[stab\]) is sufficient for ergodicity.To prove that the condition (\[stab\]) is necessary, we use theorem 1 of Sennot et *al.* [@Sen] which states that if the Markov chain $\{Y_{n};$ $n\in
\mathbb{N}
\}$ satisfies Kaplan’s condition, namely $\chi _{k}<\infty $ for all $k\geq
0 $ and there is an $k_{0}$ such that $\chi _{k}\geq 0$ for $k\geq k_{0}$, then $\{Y_{n};$ $n\in
\mathbb{N}
\}$ is not ergodic. Indeed, if$$\frac{\lambda \left( 1-\beta (\nu _{d})\right) }{\nu _{d}\beta (\nu _{d})}\left( 1+\nu _{d}\beta _{1}^{d}+\nu _{n}\beta _{1}^{n}\right) \geq 1$$then for $f(k)=k$, there is a $k_{0}$ such that $p_{ij}=0$ for $j<i-k_{0}$ and $i>0$, where $P=(p_{ij})$ is the one-step transition matrix associated to $\{Y_{n};$ $n\in
\mathbb{N}
\}.$The stability of the system follows from Burke’s theorem (see Cooper [Coo]{} p187) since the input flow is a Poisson process.
Steady-state analysis
=====================
We investigate in this section the steady-state distribution of the system. Define the functions $\mu (x),$ $\mu _{D}(x),$ $\mu _{N}(y)$ and $\mu
_{I}(x) $ as the conditional completion rates for service, disruptive interruption, non-disruptive interruption and idle interruption, respectively, i.e., $\mu (x)=b(x)/\left( 1-B(x)\right) ,$ $\mu
_{D}(x)=b_{d}(x)/\left( 1-B_{d}(x)\right) ,$ $\mu _{N}(x)=b_{n}(x)/\left(
1-B_{n}(x)\right) $ and $\mu _{I}(x)=b_{i}(x)/\left( 1-B_{i}(x)\right) .$
We now introduce the following set of probabilities for $j\geq 0$:$$\begin{aligned}
p_{F,j}(t) &=&P\left\{ N(t)=j,\text{ }C(t)=F\right\} , \\
p_{B,j}(t,x)dx &=&P\left\{ N(t)=j,\text{ }C(t)=S,\text{ }x\leq \mathcal{\xi }(t)<x+dx\right\} , \\
p_{D,j}(t,x)dx &=&P\left\{ N(t)=j,\text{ }C(t)=D,\text{ }x\leq \mathcal{\xi }_{D}(t)<x+dx\right\} , \\
p_{N,j}(t,x,y)dy &=&P\left\{ N(t)=j,\text{ }C(t)=N,\text{ }\mathcal{\xi }(t)=x,\text{ }y\leq \mathcal{\xi }_{N}(t)<y+dy\right\} , \\
p_{I,j}(t,x)dx &=&P\left\{ N(t)=j,\text{ }C(t)=I,\text{ }x\leq \mathcal{\xi }_{I}(t)<x+dx\right\} .\end{aligned}$$where $t\geq 0$ and $x,y\geq 0.$
The usual arguments lead to the differential difference equations by letting $t\rightarrow +\infty $$$\begin{aligned}
\left( \lambda +j\theta +\nu _{i}\right) p_{F,j} &=&\int\limits_{0}^{+\infty
}\mu (x)p_{B,j}(x)dx+\int\limits_{0}^{+\infty }\mu _{I}(x)p_{I,j}(x)dx,
\label{pfj} \\
\left( \frac{\partial }{\partial x}+\lambda +\nu +\mu (x)\right) p_{B,j}(x)
&=&\left( 1-\delta _{0j}\right) \lambda
p_{B,j-1}(x)+\int\limits_{0}^{+\infty }\mu _{N}(y)p_{N,j}(x,y)dy,
\label{pbj} \\
\left( \frac{\partial }{\partial x}+\lambda +\mu _{D}(x)\right) p_{D,j}(x)
&=&\left( 1-\delta _{0j}\right) \lambda p_{D,j-1}(x), \label{pdj} \\
\left( \frac{\partial }{\partial y}+\lambda +\mu _{N}(y)\right) p_{N,j}(x,y)
&=&\left( 1-\delta _{0j}\right) \lambda p_{N,j-1}(x,y), \label{pnj} \\
\left( \frac{\partial }{\partial x}+\lambda +\mu _{I}(x)\right) p_{I,j}(x)
&=&\left( 1-\delta _{0j}\right) \lambda p_{I,j-1}(x). \label{pij}\end{aligned}$$With boundary conditions$$\begin{aligned}
p_{B,j}(0) &=&\left( j+1\right) \theta p_{F,j+1}+\lambda
p_{F,j}+\int\limits_{0}^{+\infty }\mu _{D}(x)p_{D,j}(x)dx, \label{b0j} \\
p_{D,j}(0) &=&\nu _{d}\int\limits_{0}^{+\infty }p_{B,j}(x)dx, \label{d0j} \\
p_{N,j}(x,0) &=&\nu _{n}p_{B,j}(x), \label{n0j} \\
p_{I,j}(0) &=&\nu _{i}p_{F,j}. \label{i0j}\end{aligned}$$The normalising equation is$$\sum\limits_{j=0}^{+\infty }p_{F,j}+\sum\limits_{j=0}^{+\infty
}\int\limits_{0}^{+\infty }p_{B,j}(x)dx+\sum\limits_{j=0}^{+\infty
}\int\limits_{0}^{+\infty }p_{D,j}(x)dx+\sum\limits_{j=0}^{+\infty
}\int\limits_{0}^{+\infty }\int\limits_{0}^{+\infty }p_{N,j}(x,y)dxdy$$$$+\sum\limits_{j=0}^{+\infty }\int\limits_{0}^{+\infty }p_{I,j}(x)dx=1$$Define the generating functions$$\begin{aligned}
P_{F}(z) &=&\dsum\limits_{j=0}^{\infty }p_{F,j}~z^{j}, \\
P_{B}(x,z) &=&\dsum\limits_{j=0}^{\infty }p_{B,j}(x)~z^{j}, \\
P_{D}(x,z) &=&\dsum\limits_{j=0}^{\infty }p_{D,j}(x)~z^{j}, \\
P_{N}(x,y,z) &=&\dsum\limits_{j=0}^{\infty }p_{N,j}(x,y)~z^{j}, \\
P_{I}(x,z) &=&\dsum\limits_{j=0}^{\infty }p_{I,j}(x)~z^{j},\end{aligned}$$for $\left\vert z\right\vert \leq 1.$We introduce $h(z)=\left[ \nu -\nu _{n}\beta _{n}\left( \lambda -\lambda
z\right) +\lambda -\lambda z\right] $ and$\chi \left( z\right) =h(z)-\nu _{d}\beta _{d}\left( \lambda -\lambda
z\right) \left( 1-\beta \left( h(z)\right) \right) $ to simplify notation.
We have the following theorem
In steady state, the joint distribution of the server state and queue length is given by$$\begin{aligned}
P_{F}(z) &=&P_{F}(1)\exp \left\{ \int\limits_{1}^{z}\Psi (u)du\right\} , \\
P_{B}(x,z) &=&P_{B}(0,z)\left( 1-B(x)\right) \exp \left\{ -h(z)x\right\} , \\
P_{D}(x,z) &=&P_{B}(0,z)\frac{\nu _{d}\left( 1-\beta (h(z))\right) }{h(z)}\left( 1-B_{d}(x)\right) \exp \left\{ -\left( \lambda -\lambda z\right)
x\right\} , \\
P_{N}(x,y,z) &=&P_{B}(0,z)\nu _{n}\left( 1-B_{n}(y)\right) \left(
1-B(x)\right) \exp \left\{ -h(z)x\right\} \exp \left[ -\left( \lambda
-\lambda z\right) y\right] , \\
P_{I}(x,z) &=&\nu _{i}P_{F}(z)\left( 1-B_{i}(x)\right) \exp \left[ -\left(
\lambda -\lambda z\right) x\right] ,\end{aligned}$$where$$\begin{aligned}
P_{F}(1) &=&\frac{\nu _{d}^{2}\beta \left( \nu _{d}\right) \left( 1-\rho
\right) }{\left( 1+\nu _{i}\beta _{1}^{i}\right) \left[ \nu _{d}^{2}\beta
\left( \nu _{d}\right) +\lambda \nu _{n}\beta _{1}^{n}\left( 1-\beta (\nu
_{d})\right) ^{2}-\lambda \nu _{n}\beta _{1}^{n}\nu _{d}\left( 1-\beta (\nu
_{d})\right) \right] }, \\
\Psi (z) &=&\frac{\lambda h(z)\beta \left( h(z)\right) -\left[ \lambda +\nu
_{i}\left( 1-\beta _{i}\left( \lambda -\lambda z\right) \right) \right] \chi
\left( z\right) }{\theta \left( z\chi \left( z\right) -h(z)\beta \left(
h(z)\right) \right) }, \\
P_{B}(0,z) &=&\frac{h(z)\left( \lambda +\theta \Psi (z)\right) }{\chi \left(
z\right) }P_{F}(z).\end{aligned}$$
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abstract: |
Given a graph $ G $ with $ n $ vertices and a set $ S $ of $ n $ points in the plane, a point-set embedding of $ G $ on $ S $ is a planar drawing such that each vertex of $ G $ is mapped to a distinct point of $ S $. A straight-line point-set embedding is a point-set embedding with no edge bends or curves. The point-set embeddability problem is NP-complete, even when $ G $ is $ 2 $-connected and $ 2 $-outerplanar. It has been solved polynomially only for a few classes of planar graphs. Suppose that $ S $ is the set of vertices of a simple polygon. A straight-line polygon embedding of a graph is a straight-line point-set embedding of the graph onto the vertices of the polygon with no crossing between edges of graph and the edges of polygon. In this paper, we present $ O(n) $-time algorithms for polygon embedding of path and cycle graphs in simple convex polygon and same time algorithms for polygon embedding of path and cycle graphs in a large type of simple polygons where $n$ is the number of vertices of the polygon.\
**keywords:** graph drawing, point-set embedding, Halin graph, fan graph.
author:
- 'Hamid Hoorfar[^1]'
- Alireza Bagheri
bibliography:
- 'bibfile.bib'
title: 'Geometric Embedding of Path and Cycle Graphs in Pseudo-convex Polygons'
---
Introduction
============
Geometric embedding of graphs has wide applications in circuit schematics, algorithm animation, and software engineering[@di1994algorithms]. This problem has been studied extensively for planar graphs in various fields. Planar graph drawing is the problem of drawing a graph in the plane such that its edges intersect only at common vertices. It has been proved that any planar graph can be drawn without crossing, even when its edges are drawn as a straight-line segment joining its vertices. Schnyder[@schnyder1990embedding] showed that each plane graph with $n\ge 3$ vertices has a straight line embedding on the $\left(n-1\right)\times\left(n-1\right)$ grid. He proved that the embedding is computable in time $ O(n) $. In the point-set embeddability problem given a planar graph $ G $ and a set $ P $ of points in the plane, deciding whether there is a planar straight-line embedding of $ G $ such that the vertices are embedded onto the points $ P $ is NP-complete, even when $ G $ is $ 2 $-connected and $ 2 $-outerplanar[@cabello2006planar]. NP-hardness for $ 3 $-connected graphs was shown by Durocher and Mondal[@durocher2012hardness]. For outer-planar graphs was presented an $ O(n log^{3} n) $ time and $ O(n) $ space algorithm to compute a straight-line embedding of $ G $ in $ P $[@bose2002embedding; @castaneda1996straight; @pach1991embedding].There is an $\Omega \left(n\mathrm{log}n\right)$ lower bound for the point-set embeddability problem[@bose1997optimal]. Several papers studied point-set embeddability for special graph classes. Nishat et al.[@nishat2012point], Moosa and Rahman[@moosa2011improved] and Hosseinzadegan and Bagheri[@Hoseinzadegan] presented algorithms to test point-set embeddability of triangulated planar graphs of treewidth $ 3 $ and wheel graphs. Optimal embedding of a planar graph $ G $ means an embedding of $ G $ such that the total edge length of embedding is the minimum. Optimal embedding application is in the design of distributed computing networks, a special case of the problem is the Euclidean Travelling Salesman Problem (TSP), where graph $ G $ is a cycle of $ n $ nodes[@Hoseinzadegan]. There is an algorithm for embedding rooted trees and degree-constrained trees in arbitrary point set, computable in $ O(n \log n) $-time[@ikebe1994rooted]. A wheel graph $ W_{n} $ is a graph with $ n $ vertices, formed by connecting a single vertex to all vertices of an $ (n-1) $-cycle. The single central vertex will be referred to as *hub* and others vertices as *rim* vertices[@yang2009beyond]. Chambers et al[@chambers2010drawing] describe an algorithm for drawing a planar graph with a prescribed outer face shape. The input consists of an embedded planar graph $ G $, a partition of the outer face of the embedding into a set $ S $ of $ k $ chord-free paths, and a $ k $-sided polygon $ P $; the output of their algorithm is a drawing of $ G $ within $ P $ with each path in $ S $ drawn along an edge of $ P $. In our problem, we are not allowed to draw any edge of $ G $ along an edge of $ P $, even with intersection and we should not use any point, except vertices of $ P $, for graph embedding. In similar problem by[@mchedlidze2013drawing]. given a planar graph $ G $ with a fixed planar embedding and a simple cycle $ C $ in $ G $ whose vertices are in convex position, they studied the question whether this drawing can be extended to a planar straight-line drawing of $ G $. they characterize when this is possible in terms of simple necessary and sufficient conditions. For this purpose, they described a linear-time testing algorithm. Hoseinzadegan and Bagheri[@Hoseinzadegan] consider optimal embedding of wheel graphs and a sub-class of 3-trees, that are not outer-planar. they presented optimal $ O(n \log n) $-time algorithms for embedding and $ O(n^{2}) $-time algorithms for optimal embedding of wheel graphs. Wheel graphs are a sub-class of Halin graphs. A Halin graph $ H $,also known as a roofless polyhedron, is obtained by a planar drawing of a tree having four or more vertices, having no nodes of degree $ 2 $ in the plane, and then connecting all leaves of the tree with a cycle $ C $ which passes around the tree’s boundary in such a way that the resulting graph is planar. Halin graphs are edge $ 3 $-connected and Hamiltonian[@cornuejols1983halin].
![[Some examples of Halin graphs]{}[]{data-label="fig:figure02"}](./figure02){width="0.7\linewidth"}
A gear graph is obtained from the wheel $ W_{n} $ by adding a vertex between every pair of adjacent vertices of the $ n $-cycle[@gallian2014dynamic]. In 2010, Bagheri and Razzazi[@bagheri2010planar] showed that the problem of deciding whether there is a planar straight-line point-set embedding of an $ n $-node tree $ T $ on a set $ P $ of $ n $ points in the plane that includes a partial embedding $ E $ of $ T $ on $ P $ is NP-complete. In another paper[@sepehri2010point], Sepehri and Bagheri presented an algorithm for embedding a tree with $ n $ vertices on a set of $ n $ points inside a simple polygon with $ m $ vertices so that number of bends is minimum. They showed that time complexity of the presented algorithm is $ O(n^{2}m+n^{4}) $. A geometric graph H is a graph $ G(H) $ together with an injective mapping of its vertices into the plane. An edge of the graph is drawn as a straight-line segment joining its vertices. We use $ V(H) $ for the set of points where the vertices of $ G(H) $ are mapped to, and we do not make a distinction between the edges of $ G(H) $ and $ H $. A planar geometric graph is a geometric graph such that its edges intersect only at common vertices. In this case, we say that H is a geometric planar embedding of $ G(H) $. In the next section, we present some definitions and preliminaries that that will be used throughout the paper.
Preliminaries
=============
We use basic notions of graph drawing that is used in [@di2009point]. Let $ G = ( V , E ) $ be a simple graph with $ n $ vertices in the node set $ V $ and $ E $ be the edge set of $ G $. A $ planar graph $ is a graph with at least a drawing without edge crossing except at the nodes, where the edges are incident. A *plane graph* is a planar graph with a fixed embedding on the plane. let $ S $ be a set of $ n $ points in the plane. A point-set embedding of $ G $ onto $ S $ , denoted as $\varGamma ( G , S ) $, is a planar drawing of $ G $ such that each vertex is mapped to a distinct point of $ S $ . $ \varGamma ( G , S ) $ is called a geometric (straight-line) point-set embedding if each edge is drawn as a straight-line segment. Let $ D ( S ) $ be a straight-line drawing whose vertices are points of a subset of $ S $. An optimal point-set embedding of a planar graph $ G $ is a point-set embedding of $ G $, where the total edge length (or the area) of the embedding is minimized. A path graph $ P_{n} $ is a tree with $ n-2 $ nodes of degree $ 2 $ and two nodes of degree one. A cycle graph $ C_{n} $, known as an $ n $-cycle is a graph with $ n $ nodes containing a single cycle through all nodes. In cycle graph $ C_{n} $, every nodes are of degree $ 2 $. A complete graph $ K_{n} $ is a graph with $ n $ nodes and $ \frac{n\left(n-1\right)}{2} $ edges, in which each pair of graph vertices is connected by an edge. A complete graph that is embedded in an other graph is called clique. A wheel graph $ W_{n} $, is a planar graph with at least $ 3 $ nodes, such that a certain node is connected to all other nodes of a $ (n-1) $-cycle. Wheel graph is extended to Halin graph. A Halin graph (so called roofless polyhedron) is a planar graph formed of a tree $ T $, without any $ 2 $-degree nodes, and a cycle $ C $ connecting leaves of $ T $ in the cyclic order determined by a plane embedding of $ T $. Therefore, A Halin Graph $ H_{n,m} $ has $ n+m $ nodes, $ n $ of them belongs to tree $ T $ and $ m $ of them belongs to cycle $ C $. If $ T $ be a star, Halin graph becomes a wheel graph. A fan graph $ F_{n,m} $ is defined as a path graph $ P_{m} $ and empty graph $ E_{n} $ with $ n $ nodes such that every node belongs $ E_{n} $ is connected to every node belong $ P_{m} $ with additional edges. The convex hull of a point set $ S $ is a minimum area convex polygon $ CH(S) $ that is contain all points of $ P $. Convex hull $ CH(S) $ is a partition of set $ S $ into two subsets are named *inner points* and *outer points*. The points that is placed on boundary of $ CH(S) $ belong to outer points set and other points that is placed inside $ CH(S) $(without boundary) belong to inner points set, denoted as $ S_{out} $, $ S_{in} $, respectively. It is obvious to get the following result, $\left|S\right|\ge\left|{S}_{out}\right|\ge 3$. Convex hull of a point set with $ n $ points can be compute in $O\left(n\mathrm{log}n\right)$ time [@de2000computational]. We introduce new version of graph drawing, named as *polygon embedding* . Let $ G=(V,E) $ be a simple graph with $ n $ vertices in the node set $ V $ and $ E $ be the edge set of $ G $. Furthermore, Let $ P=(\varSigma,\varPi) $ be a simple polygon with $m(\ge n)$ vertices in the vertex set $ \varSigma $ and $ \varPi $ be the edge set of $ P $. A *polygon embedding* of $ G $ on $ P $ is drawing of $ G $ such that each vertex of $ G $ mapped to a distinct vertex of $ P $ without edge crossing between edges in $ E$ and edges in $\Pi$ except at the nodes, where the edges are incident and $ E\cap \Pi =\phi $. For each edge $ e $ of $ G $, must be $ e\subseteq P $. See figure \[fig:figure04\]. A polygon embedding of $ G $ into $ P $ is named *planar* such that drawing be without edge crossing between edges of graph $ G $ and called a *geometric (straight-line) polygon embedding* if each edge is drawn as a straight-line segment.
![[]{data-label="fig:figure04"}](./figure04){width="0.5\linewidth"}
Polygon embedding is a special case of point set embedding with some additional constraints which make problem more complex than before. For example, connecting between some pair of points in $ S $ (vertices of polygon) is not allowed. In many cases, polygon embedding is possible if number of vertices of $ P $ be greater than number of vertices of $ G $. Therefore, always mapping must be one-to-one but sometimes not onto. There is an $ \varOmega(n\log n) $ lower bound for the point-set embeddability problem. For path and cycle graphs was presented an $ \varTheta(n\log n) $ time and $ O(n) $ space algorithm to compute a straight-line embedding in point set $ S $. We will review optimum-time algorithms to embedding these two kinds of graphs in a given point set in the next section. After that, we will present algorithms for straight-line polygon embedding of path and cycle graphs in polygon $ P $ and compute upper bound on maximum edges of graph that can be embeddable in $ P $. A convex polygon is a polygon with all its interior angles less than $ \pi $. A concave polygon is a polygon that is not convex. A simple polygon is concave if at least one of its internal angles is greater than $ \pi $. An orthogonal polygon is one whose edges are all aligned with a pair of orthogonal coordinate axes, which we take to be horizontal and vertical without loss of generality. Thus, the edges alternate between horizontal and vertical, and always meet orthogonally, with internal angles of either $ \frac{\pi }{2} $ or $ \frac{3\pi }{2} $ [@o1987art]. An orthogonal polygon is defined *orthoconvex* if the intersection of the polygon with a horizontal or a vertical line is a single line segment [@nandy2010recognition]. Orthoconvex with $ n\geqslant 6 $ vertices is a concave polygon. Every orthogonal polygon with $ n $ vertices has $ \frac{n-4}{2} $ angles of $ \frac{3\pi }{2} $ and $ \frac{n+4}{2} $ angles of $ \frac{\pi }{2} $, exactly. Let $ p $ and $ q $ are two points in polygon $ P $. $ p $ and $ q $ are said *visible* to each other, if the line segment that joins them does not intersect any edge of $ P $. So, $ p $ and $ q $ are said to be *invisible* to each other, if they are not visible. A *visibility graph* is a graph for polygon $ P $, as denoted $ G_{v}^{P}(V,E) $, that each node in $ V $ represents a point location in $ P $, and there is an edge between two nodes $ v_{i} $ and $ v_{j} $ if they are visible from each other. A visibility graph $ G_{v}^{P}(V,E) $ is named *polygon visibility graph*, if $ V$ be vertex set of $ P $. Note that in the whole of paper, size of graph means number of its edges.
Preliminary Algorithms and Results {#s:s03}
==================================
In the following, we review a exact $ O(n \log n) $-algorithm for embedding of path graph $ P_{n} $ onto point set $ S $ with $ n $ points which is always possible. First, order point set $ S $ according to $ X $-axes in list $ L=\left[{p}_{0},{p}_{1},{p}_{2,\dots },{p}_{n}\right] $, ascending. Then, connect points according to this ordering successive. There is a similar $ O(n \log n) $-algorithm for embedding of cycle graph $ C_{n} $ onto point set $ S $, as following:
1. Order point set $ S $ according to $ X $-axes in list $ L=\left[{p}_{0},{p}_{1},{p}_{2,\dots },{p}_{n-1}\right] $, ascending.
2. Make list $ L_{down}=\left[{d}_{0},{d}_{1},{d}_{2,\dots },{d}_{\frac{n}{2}-1}\right] $ such that $ {d}_{i}=\underset{y}{\mathrm{min}}\left\{{p}_{2i},{p}_{2i-1}|i< \frac{n}{2},i\in N\right\} $.
3. Make list $ L_{up}=\left[{u}_{0},{u}_{1},{u}_{2,\dots },{u}_{\frac{n}{2}-1}\right] $ such that $ {u}_{i}=\underset{y}{\mathrm{max}}\left\{{p}_{2i},{p}_{2i-1}|i< \frac{n}{2},i\in N\right\} $.
4. Connect points in each list according to their ordering successive.
5. Connect $ d_{0} $ to $ u_{0} $ and $ d_{\frac{n}{2}-1} $ to $ u_{\frac{n}{2}-1} $.
![[(a)Embedding of path graph and (b)cycle graph in the point set.]{}[]{data-label="fig:figure05"}](./figure05){width="0.6\linewidth"}
See figure \[fig:figure05\]. These graph embeddings are not unique and optimum. Time complexity of both algorithms is equal to time complexity of sorting algorithm that puts elements of list in a certain order. Therefore, time complexity is $ \varTheta(n \log n) $ and it is tight. Maximum path(cycle)that can be draw in point set of size $ n $ is $ n $, but where we want to embed a path (cycle) graph in polygon with $ n $ vertices, maximum size is not $ n $. We show that maximum size of path graph that can be embedded planar in polygon with $ n $ vertices is not greater than $ n-3 $ and for cycle is not greater than $ \lfloor\frac{n}{2}\rfloor $, where polygon is convex. Therefore, we have the following obviously results. Given a polygon $ P $ with $ n $ vertices, maximum size of path graph that can be embedded planar in $ P $ is lesser than or equal to $ n-3 $. Edges of path graph that can be embedded planar in $ P $ must be chords of $ P $ without crossing except at their end points. Maximum number of chords in a polygon with $ n $ vertices without crossing is $ n-3 $. Therefore, maximum size of path graph that can be embedded planar in $ P $ can not be greater than $ n-3 $. Also, maximum size of every graphs that can be embedded planar in polygon $ P $ with $ n $ vertices is $ n-3 $. By the way, given a polygon $ P $ with $ n $ vertices, maximum size of cycle graph that can be embedded planar in $ P $ is lesser than or equal to $ \lfloor\frac{n}{2}\rfloor $. Let cycle graph $ C $ is embedded in $ P $ and $ \{c_{i}|0\leq i \leq m\} $ is set of its nodes, ordered counter clockwise. Each node $ c_{i} $ of $ C $ is degree two and must be connected to two different other nodes [(as named $ c_{i-1} $ and $ c_{i+1} $)]{} which are vertices of $ P $. Let nodes $ c_{i-1}$, $c_{i} $ and $ c_{i+1} $ of cycle are mapped to vertices $ p_{j}$, $ p_{k} $ and $ p_{l} $ in $ P $ [(ordered counter clockwise)]{}. Therefore, ($ p_{j}$,$ p_{k} $) and ($ p_{k} $,$ p_{l} $) can not be edges of $ P $ because edges of cycle are not allowed to be edges of $P$. Cycle $ C $ is planar, so, any other nodes of $ C $ can not mapped to vertices of $ P $ between $ p_{j}$, $ p_{k} $ [(as ordered, $ {p}_{j+1 },\dots,{p}_{k-1} $)]{} and between $ p_{k} $, $ p_{l} $ [(as ordered, $ {p}_{k+1 },\dots,{p}_{l-1} $]{}), see figure \[fig:figure06\]. Suppose node $ {c}_{f} $ is mapped to $ {p}_{g} $ and [$ j<g<k $]{}, so, edge ($ c_{f-1}$,$ c_{f} $) and ($ c_{f} $,$ c_{f+1} $) of $ C $ must cross edge ($ c_{i-1} $,$ c_{i} $) and It is contradiction. In the other words, if nodes $ c_{i-1}$, $c_{i} $ and $ c_{i+1} $ of cycle are mapped to vertices $ p_{j}$, $ p_{k} $ and $ p_{l} $ in $ P $, consecutively, then vertices $ {p}_{j+1 },\dots,{p}_{k-1} $ and $ {p}_{k+1 },\dots,{p}_{l-1} $ are out of reach to be in cycle $ C $. ($ p_{j}$,$ p_{k} $) and ($ p_{k} $,$ p_{l} $) are not adjacent in $ P $. Therefore, at least one vertex between $ p_{j}$ and $ p_{k} $ and one vertex between $ p_{k} $ and $ p_{l} $ are out of reach to be in planar cycle $ C $. It happens for every $ c_{i}$, [$0\leq i \leq m $]{}. Hence, for every two successive nodes in $ S $ at least one vertex of $ P $ must be blocked and never can be mapped to any node of $ C $. Maximum size of $ C $ is lesser than or equal to $ \lfloor\frac{n}{2}\rfloor $.
![[Embedding of cycle graph in the polygon.]{}[]{data-label="fig:figure06"}](./figure06){width="0.3\linewidth"}
Also, maximum clique that can be embedding in polygon $ P $ with $ n $ vertices such that nodes of clique are mapped to vertices of $ P $ is $ K_{\lfloor \frac{n}{2}\rfloor}$, where $ P $ is convex. There is a linear-time reduction from sorting problem to point-set embeddability problem that be used to show the second problem is at least as difficult as the first. For time complexity, $ \varOmega(n\log n) $ is lower bound for the point-set embeddability problem. There are several algorithms for embedding trees in arbitrary point set, computable in $ O(n \log n)$-time[@ikebe1994rooted]. All of them using a sorting algorithm on point set and ordering it corresponding to a axes line or angular. In the polygon embeddability problem, given point set is placed on vertices set. The vertices of a polygon have a kind of ordering by itself. We study the following lemmas.
\[le:le03\] There is a linear-time algorithm for straight-line planar embedding of a path graph $ P_{m} $ into convex polygon $ Q_{m+3} $ such that each node of $ P_{m} $ mapped to a vertex of $ C $.
\[pr:pr03\] Let $ \{q_{0},q_{1},\dots,q_{m+2}\} $ be the set of vertices of $ Q $ ordered counter clockwise. Traverse vertices of $ Q $ in following order:
1. Traverse from $ {q}_{m+2} $ to $ {q}_{m} $.
2. Traverse from $ {q}_{m-i} $ to $ {q}_{i} $ [(for $ 0\leq i<\frac{m}{2} $)]{} if current location index is greater than $ \frac{m}{2} $.
3. Traverse from $ {q}_{i} $ to $ {q}_{m-i-1} $ [(for $ 0\leq i< \frac{m}{2}-1 $)]{} if current location index is lesser than $ \frac{m}{2} $.
Because of convexity, vertices of convex polygon are visible from each others. Therefore, according to algorithm, path $ P $ has exactly $ m $ edges which are all planar chords of $ Q $. Time complexity is $ \theta \left(m\right) $, see figure \[fig:figure07\](a).
![[Embedding of maximum (a)path and (b)cycle in convex polygon.]{}[]{data-label="fig:figure07"}](./figure07){width="0.5\linewidth"}
There is an $ \varOmega(n) $ lower bound on time complexity for the polygon embeddability problem(in comparison with lower bound for point set embeddability that was $ \varOmega(n \log n) $ is better).
\[le:le04\] There is a linear-time algorithm for straight-line planar embedding of a cycle graph $ C_{\lfloor \frac{m}{2}\rfloor} $ into convex polygon $ P_{m} $ such that each node of $ C_{m} $ mapped to a vertex of $ P $.
There are similarities between this proof and previous one. Let $ \{p_{0},p_{1},\dots,p_{m-1}\} $ be vertices set of $ P_{m} $ ordered counter clockwise. Tour vertices of $ P_{m} $ in following order:
1. Traverse from $ {p}_{2i} $ to $ {p}_{2i+2} $, [for $ 0\le i <\lfloor\frac{m}{2}\rfloor-1 $]{}.
2. Turn back from $ {p}_{2\lfloor\frac{m}{2}\rfloor-2} $ to $ {p}_{0}$.
Because of convexity, vertices of convex polygon are visible from each others. Therefore, according to algorithm, $ C $ has exactly $ \lfloor\frac{m}{2}\rfloor $ edges which are all planar chords of $ P $. Time complexity is $ \theta \left(m\right) $, see figure \[fig:figure07\](b).
Our algorithms for embedding path and cycle use segments from the polygon chords to avoid intersections between embedded edges.
Embedding a cycle graph with maximum size in pseudo-convex polygon
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In this section, we first definite *pseudo-convex* polygon and provide a linear-time algorithm for straight-line planar embedding of maximum cycle $ C_{max} $ in a pseudo-convex polygon $ P $ with $ n $ vertices. If polygon $ P $ is not convex, then there are some pairs of vertices that is not visible from each others. Therefore, let $ (v_{i},v_{j}) $ be an invisible pair of vertices, no edge of graph can be mapped to it. In the other words, finding maximum cycle in $ P $ is reduced to finding maximum cycle of polygon visibility graph of $ P $. Visibility graph of a polygon computes in time of $ O(n^{2}) $. In addition, finding maximum cycle in a graph is known as NP-hard.
A reflex vertex in polygon $ P $ that is adjacent with another reflex is named *u-turn vertex* and an edge in $ P $ that its both endpoints are u-turn vertices is named *u-turn edge*, see figure \[fig:figure08\](a).
Let $ P $ be a polygon and $ V=\left\{{p}_{0},{p}_{1},{p}_{2},\dots ,{p}_{n-1}\right\} $ be vertex set of $ P $ ordered counter clockwise, if there is no successive reflex vertices in $ V $, $ P $ is named *pseudo-convex polygon*. In the other word, $ P $ is pseudo-convex polygon if it has no u-turn vertex (edge), see figure \[fig:figure08\](b).
Let $ P $ be a polygon, $ V=\left\{{p}_{0},{p}_{1},{p}_{2},\dots ,{p}_{n-1}\right\} $ be vertex set of $ P $ ordered counter clockwise and $ S_{i} $ be the set of vertices that are visible from $ {p}_{i} $. $ {p}_{i} $ is named *isolated vertex*, if it has one of the following conditions:
- Cardinality of $ S_{i} $ be lesser than five (note that $ {p}_{i}\in S_{i} $).
- Cardinality of $ S_{i} $ be equal to five and both members of $ S_{i}-\{p_{i-1},p_{i},p_{i+1}\} $ be adjacent.
![[(a)A polygon with u-turn vertices and edges: bold disks are u-turned vertices and fat segment are u-turned edges. (b)A pseudo-convex polygon: bold disks are reflex vertices. There are no successive reflex vertices. (c)Vertices $v_{i}$ and $v_{k}$ can not be a part of an embedded cycle graph but other vertices can be at least in one cycle those are drawn by the cycles.]{}[]{data-label="fig:figure08"}](./figure08){width="0.7\linewidth"}
See figure \[fig:figure08\](c), $ v_{i} $ and $ v_{k} $ are isolated vertices. Convex and orthoconvex polygon are two examples for pseudo-convex polygon. If a polygon has a u-turn vertex, actually, it has two successive u-turn vertices or more. In the following, we provide a linear-time algorithm for straight-line embedding of maximum cycle graph $ C $ in a pseudo-convex polygon $ P $. A vertex $ v_{i} $ of $ P $ can be in the cycle, if it is visible from at least two different vertices which are not adjacent to $ v_{i} $. Because the degree of every node $ c_{j} $ in the cycle $ C $ is two, that means, it must be connect to two others nodes $ c_{j-1} $ and $ c_{j+1} $. Therefore, if $ c_{j} $ is mapped to $ v_{i} $, $ P $ must have two chords from $ v_{i} $ to other vertices which the edges must be mapped to. Otherwise straight-line embedding is not possible by this mapping. Hence, a mapped vertex $ v_{i} $ in every embedded cycle must be visible from at least two other non-adjacent vertices $ v_{s} $ and $ v_{t} $ of $ P $. In addition, if $ v_{s} $ and $ v_{t} $ are adjacent, being part of a cycle is not still possible. There is no isolated vertex of $ P $ in cycle graph $ C $. See figure \[fig:figure08\](c).\
In the following, we want to study straight-line planar embedding of a cycle graph with maximum size in pseudo-convex polygon. Let $ C_{m} $ be a cycle graph with maximum number of edges that is embedded in pseudo-convex polygon $ P_{n} $ with $ n $ vertices. Let $ \{p_{0},p_{1},\dots,p_{n-1}\} $ be vertex set of $ P_{n} $ ordered counter clockwise and $ \{c_{0},c_{1},\dots,c_{n-1}\} $ be node set of $ C_{m} $ ordered counter clockwise. If node $ c_{i} $ is mapped to vertex $ p_{j} $, then $ p_{j-1} $ and $ p_{j+1} $ can not be nodes in $ C $. If node $ c_{i} $ is mapped to vertex $ p_{j} $ and vertex $ p_{j+1} $ is a reflex vertices, then it is better that $ c_{i} $ be mapped to vertex $ p_{j+1} $ instead of $ p_{j}$, and size of $ C $ will not change and still remain maximum. We must probe this explained claim. Consequently, we prove the following lemmas.
Let $ p_{j-1} $, $ p_{j} $ and $ p_{j+1} $ be three successive non-isolated vertices in pseudo-convex polygon $ P $. There is at least one node in maximum embedded cycle that is mapped to one of these three successive vertices.
Suppose that there is a maximum embedded cycle $ C $ such that any node of $ C $ is not mapped to these three successive vertices $ p_{j-1} $, $ p_{j} $ and $ p_{j+1} $. Let $ p_{s} $ be the first previous vertex (first before $ p_{j-1} $) such that a node of $ C $ is mapped to it and $ p_{t} $ be the first next vertex (first after $ p_{j+1} $) such that a node of $ C $ is mapped to it. There must exist a straight-line planar polygon embedded path, as denoted $ \varPi $, between $ p_{s} $ and $ p_{t} $ that is contained at least one of $ p_{j-1} $, $ p_{j} $ or $ p_{j+1} $ without edge crossing with cycle edges except at the nodes $ p_{s} $ and $ p_{t} $. The size of $ \varPi $ is greater than one, if we remove edge $( p_{s} , p_{t}) $ from $ C $ and replace path $ \varPi $ instead of it, we should have a new cycle that its size is greater than size of $ C $. It means $ C $ was not maximum cycle and it is contradiction.
At least one and at most two vertices of every three successive non-isolated vertices in pseudo-convex polygon $ P $ are on a cycle graph with maximum number of edges that is embedded in $ P $.
\[le:le05\] There is a straight-line planar embedded cycle graph with maximum size such that all reflex vertices of $ P_{n} $ are on it, where $ P_{n} $ is a pseudo-convex polygon.
Let $ C_{m} $ be a cycle graph with maximum number of edges that is embedded in pseudo-convex polygon $ P_{n} $ with $ n $ vertices. Let $ \{p_{0},p_{1},\dots,p_{n-1}\} $ be vertex set of $ P_{n} $ ordered counter clockwise and $ \{c_{0},c_{1},\dots,c_{n-1}\} $ be node set of $ C_{m} $ ordered counter clockwise. Cycle graph $ C_{m} $ has $ m $ nodes that is mapped to $ m $ vertices of $ P_{n} $. Suppose that $ p_{i} $ is a reflex vertex and any node of $ C_{m} $ is not mapped to it. So, if $ p_{i} $ is a reflex vertex, then $ p_{i-1} $ and $ p_{i+1} $ must be convex vertices, we find it from being pseudo-convexity. In the first case, let there is a node, as denoted $ c_{j} $, that is mapped to $ p_{i-1} $. So, $ c_{j+1} $ can not mapped to $ p_{i+1} $, because $ p_{i-1} $ and $ p_{i+1} $ is not visible from each other. Let $ c_{j+1} $ is mapped to $ p_{t} $ and Let $ c_{j-1} $ is mapped to $ p_{s} $. Now, replace edges $ \left({p}_{s},{p}_{i-1}\right) $ and $ \left({p}_{i-1},{p}_{t}\right) $ in $ C_{m} $ with $ \left({p}_{s},{p}_{i}\right) $ and $ \left({p}_{i},{p}_{t}\right) $. The cycle graph $ C_{m} $ will contain reflex vertex $ p_{i} $ with the size as same as being before replacement. See figure \[fig:figure09\](a). For every reflex vertices $ p_{v} $ in this situation do same operation till it remain no reflex vertex like that. In the second case, let there is a node, as denoted $ c_{j} $, that is mapped to $ p_{i+1} $. Let $ c_{j+1} $ and $ c_{j-1} $ are mapped to $ p_{s} $ and $ p_{t} $. Now, replace edges $ \left({p}_{s},{p}_{i+1}\right) $ and $ \left({p}_{i+1},{p}_{t}\right) $ in $ C_{m} $ with $ \left({p}_{s},{p}_{i}\right) $ and $ \left({p}_{i},{p}_{t}\right) $. The cycle graph $ C_{m} $ will contain reflex vertex $ p_{i} $ with the size as same as being before replacement. See figure \[fig:figure09\](b). For every reflex vertices $ p_{w} $ in second explained situation do same operation till it remain no reflex vertex like that. Finally, $ C_{m} $ will be contained all reflex vertices on $ P_{n} $, where $ P_{n} $ is a pseudo-convex polygon.
![[The illumination of the proof]{}[]{data-label="fig:figure09"}](./figure09){width="0.8\linewidth"}
Let $ C_{m} $ has the property that is explained in lemma \[le:le05\]. We provide a linear-time algorithm to find $ C_{m} $. Hence, We have the following theorem.
\[th:th01\] There is a linear-time algorithm for finding straight-line planar embedding of a cycle graph with maximum size in pseudo-convex polygon $ P_{n} $ with $ n $ vertices such that each node of cycle mapped to a vertex on $ P $.
Let $ \{p_{0},p_{1},\dots,p_{n-1}\} $ be vertices set of $ P_{n} $ ordered counter clockwise. As lemma \[le:le05\], there is a straight-line planar embedded cycle graph with maximum size such that all reflex vertices of $ P_{n} $ are on it. We provide an algorithm to find such a cycle, as denoted $ C_{m} $, if $ m $ be number of its edges. We will find $ m $ after tracing algorithm. Therefore, all reflex vertices of $ P_{n} $ is selected to being in $ C_{m} $.
Traverse vertices of $ P $ in following order:
1. Traverse from $ {q}_{m+2} $ to $ {q}_{m} $.
2. Traverse from $ {q}_{m-i} $ to $ {q}_{i} $ [(for $ 0\leq i<\frac{m}{2} $)]{} if current location index is greater than $ \frac{m}{2} $.
3. Traverse from $ {q}_{i} $ to $ {q}_{m-i-1} $ [(for $ 0\leq i< \frac{m}{2}-1 $)]{} if current location index is less than $ \frac{m}{2} $.
Because of convexity, the vertices of convex polygon are visible from each others. Therefore, according to the algorithm, path $ P $ has exactly $ m $ edges which are all planar chords of $ Q $. Time complexity is $ \theta \left(m\right) $, see figure \[fig:figure09\].
Conclusion
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In this paper, we present a linear-time algorithms for embedding the maximum-size path and cycle in a convex polygon with $ n $ vertices. After that, we use a similar algorithm for pseudo-convex polygons which have no u-turn vertex and time complexity is remained linear. But, time complexity of the problem for simple polygon remains open. If it is NP-hard, finding a fix parameter algorithm is interesting for this problem.
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[^1]: Department of Computer Engineering and Information Technology, Amirkabir University of Technology (Tehran Polytechnic), [{hoorfar,ar.bagheri}@aut.ac.ir]{}
|
---
abstract: 'We present the results of a high-resolution imaging survey for brown dwarf binaries in the Pleiades open cluster. The observations were carried out with the Advance Camera for Surveys [@ACS_INSTR_HANDBOOK] onboard the Hubble Space Telescope. Our sample consists of 15 bona-fide brown dwarfs. We confirm 2 binaries and detect their orbital motion, but we did not resolve any new binary candidates in the separation range between 5.4 AU and 1700 AU and masses in the range 0.035–0.065 M$_{\sun}$. Together with the results of our previous study [@2003ApJ...594..525M], we can derive a visual binary frequency of 13.3$^{+13.7}_{-4.3}$% for separations greater than 7 AU masses between 0.055–0.065 M$_{\sun}$ and mass ratios between 0.45–0.9$<q<$1.0. The other observed properties of Pleiades brown dwarf binaries (distributions of separation and mass ratio) appear to be similar to their older counterparts in the field.'
author:
- 'H. Bouy'
- 'E. Moraux'
- 'J. Bouvier'
- 'W. Brandner'
- 'E. L. Martín'
- 'F. Allard'
- 'I. Baraffe'
- 'M. Fernández'
title: A Hubble Space Telescope ACS Search for Brown Dwarf Binaries in the Pleiades Open Cluster
---
Introduction
============
Young open clusters offer the advantage that both the age and distance are precisely known so that brown dwarfs candidates are more easily identified from their positions in colour-magnitude diagrams, relative to the expected position of the cluster’s sub-stellar isochrone. Over the last few years, a large number of authors have published results of large surveys looking for substellar members of the Pleiades . Using theoretical models [@2000ApJ...542..464C], the magnitude of an object can be readily converted to a mass (given the age and distance of the cluster) and the resulting IMF estimated. While detailed studies of the IMF of the Pleiades’ very low mass stars and brown dwarfs have already been performed [see e.g @2002MNRAS.335L..79D; @2002MNRAS.335..853J; @1999MNRAS.303..835H], the contribution of multiple systems to the IMF has rarely been taken into account. In this study, we obtained high angular resolution images of a sample of brown dwarfs in the Pleiades cluster in order to investigate the occurrence of multiple systems among sub-stellar objects, and its implications on:
1. the formation and evolution processes of brown dwarfs
2. the properties of these multiple systems in comparison with those of the field and in star forming regions.
3. the contribution of sub-stellar objects to the IMF
The Pleiades is one of the best studied open clusters. Its age [105-140 Myr @martin_cancun2006] and distance are well known and its IMF has been well studied over the stellar mass range. All the targets come from the same star forming region: they formed under similar initial conditions and are now following identical evolutionary paths, which is not the case of field brown dwarfs for which in general we know neither the age nor the distance precisely. Moreover, the Pleiades cluster offers two important advantages for our study in comparison with other clusters, star forming regions or associations. First of all, there exists a relatively large sample of confirmed brown dwarfs, which is of prime importance in making a good statistical study. Secondly because the cluster is not so far away as to exclude a search for close visual binaries. These considerations make this cluster the ideal place for a complementary study to the field ultracool dwarfs presented by @2005ApJ...621.1023S [@2003AJ....126.1526B; @2003ApJ...586..512B; @2003ApJ...587..407C; @2003AJ....125.3302G].
In a first attempt to investigate brown dwarf binaries, @1998ApJ...509L.113M [@2000ApJ...543..299M] surveyed 34 very low mass Pleiades members with HST and adaptive optics at CFHT. They found only one binary at a resolution of 02 or larger (27 AU, but it failed the lithium test and was therefore not confirmed as a Pleiades member. More recently, @2003ApJ...594..525M used the HST/WFPC2 and found only four binary candidates at a resolution of $\sim$0060 or larger (8.1 AU at 135 pc) among a total sample of 25 objects. In this paper, we present the result of our complementary. higher resolution ACS observations. In section \[obs\_strat\], we present the new sample, the observations and the data analysis. In section \[mutliple\_syst\], we present the results on the resolved multiple systems. In section \[conf\_photom\], we discuss the confirmed and unresolved photometric binary candidates. In sections \[bin\_frequ\] and \[discussion\], we calculate and discuss the binary frequency.
Observational strategy and techniques \[obs\_strat\]
====================================================
In order to refine the previous studies of Pleiades brown dwarfs binaries [@2000ApJ...543..299M; @2003ApJ...594..525M], we used the higher angular resolution provided by HST/ACS-HRC (program SNAP-9831, P.I. Bouy). Using PSF fitting, the observations we obtained with HST/ACS allow us to resolve multiple systems with separations as low as $\sim$0040 ($\sim$5.4 AU at the distance of the Pleiades). This is more than 5 times better than the NICMOS study of @2000ApJ...543..299M and 1.5 times as good as the WFPC2/PC study of @2003ApJ...594..525M. Moreover, the sensitivity of HST/ACS in the chosen filter is $\sim$5 times greater than the WFPC2/PC [see @WFPC..Instrument..Handbook]. This allows us to investigate systems with close companions and with low flux ratios between the companion and the primary.\
Sample
------
The initial sample consists of 32 brown dwarfs (spectral types later than M7) in the magnitude range I=18.0 mag to I=22.9 mag, identified from deep, wide-field surveys of the Pleiades cluster . Six objects (the binaries CFHT-PL-12, IPMBD 25 and IPMBD 29, and the unresolved objects CFHT-PL-15, CFHT-Pl-21 and CFHT-Pl-24) had already been observed with WFPC2 by @2003ApJ...594..525M, and two more (CFHT-Pl-11 and CFHT-Pl-13) with NICMOS by @2000ApJ...543..299M. All targets have been identified as brown dwarfs using near-infrared and optical photometry analysis and/or spectroscopy. The sample covers a mass range from 0.025 to 0.080 M$_{\sun}$ (see Table \[pleiades\_acs\_sample\]). The membership of our targets has been already confirmed by proper motion measurements or spectroscopy .
[llllllllll]{} & 03 47 39.0 & +24 36 22.1 & 17.91 &\
[**$^{\star}$**]{} & 03 53 55.1 & +23 23 36.4 & 17.87 & 1.04\
[****]{} & 03 52 06.72 & +24 16 00.76 & 17.82 & 0.90\
[****]{} & 03 55 12.5 & +23 17 38.0 & 18.62 &\
[****]{} & 03 44 35.3 & +25 13 44.0 & 18.47 & 1.11\
[****]{} & 03 43 00.2 & +24 43 52.1 & 18.47 & 0.96\
[****]{} & 03 51 25.6 & +23 45 21.2 & 18.88 & 1.07\
[****]{} & 03 52 18.64 & +24 04 28.41 & 19.32 & 1.11\
[****]{} & 03 43 40.29 & +24 30 11.34 & 19.38 & 1.12\
[****]{} & 03 54 05.37 & +23 33 59.47 & 19.69 & 1.21\
[****]{} & 03 44 31.29 & +25 35 14.42 & 21.88 & 1.14\
[****]{} & 03 55 23.07 & +24 49 05.01 & 17.81 & 0.90\
[****]{} & 03 51 33.48 & +24 10 14.16 & 20.30 & 1.10\
[****]{} & 03 44 48.66 & +25 39 17.52 & 20.85 & 1.20\
[****]{} & 03 54 14.03 & +23 17 51.39 & 21.01 & 1.23\
[****]{} & 03 41 40.92 & +25 54 23.0 & 17.82 & 0.96\
[**$^{\star}$**]{} & 03 45 31.3 & +24 52 48.0 & 18.35 &\
& 03 51 44.97 & +23 26 39.47 & 18.66 & 1.03\
& 03 44 27.27 & +25 44 41.28 & 22.47 & 1.23\
& 03 55 04.4 & +26 15 49.3 & 18.94 & 1.14\
& 03 53 32.39 & +26 07 01.2 & 18.94 & 1.14\
& 03 51 29.43 & +24 00 36.79 & 22.32 & 1.35\
& 03 51 26.69 & +23 30 10.65 & 19.44 & 1.08\
& 03 56 16.37 & +23 54 51.44 & 19.56 & 1.10\
& 03 55 27.66 & +25 49 40.72 & 19.80 & 1.17\
& 03 52 44.3 & +24 24 50.04 & 20.58 & 1.16\
& 03 49 45.29 & +26 50 49.88 & 21.03 & 1.27\
& 03 51 15.6 & +23 47 05.38 & 22.1 & 1.18\
& 03 51 47.65 & +24 39 59.51 & 21.05 & 1.26\
& 03 46 36.24 & +25 33 36.21 & 22.59 & 1.24\
& 03 48 12.13 & +25 54 28.4 & 18.46 & 1.12\
$^{\star}$ & 03 46 26.1 & +24 05 10.0 & 17.82 &\
Observations
------------
Observations were carried out during cycle 12 between July 2003 and August 2004 as part of the HST Snapshot SNAP-9831 program. Each object was observed in the F814W filter, which provides the best compromise between the efficiency, the sensitivity to our cold objects, and the S/N ratio. Only one band was obtained in order to maximize exposure times, minimize the visit times and thus optimize schedulability.
Diffraction limited imaging with ACS-HRC at 814 nm gives us a spatial resolution of 0085. With its 0027 pixel scale, the ACS-HRC thus provides the required critical sampling of the PSF, which was not the case of the WFPC2/PC camera. Using PSF fitting, we are thus able to resolve even closer companions than in the case of WFPC2. Integration times were 400 s, spread over 4 exposures in CR-SPLIT mode [@ACS_INSTR_HANDBOOK]. Figure \[limit\_detection\_all\] shows that we are sensitive to companions 5.9 mag fainter than their primary (3-$\sigma$ detection limit), corresponding to a lower limit on the mass ratio between 0.4 and 0.7 at separations greater than 0250, depending on the brightness of the primary. Considering the total field of view of the ACS camera (26$\times$29) we were sentitive to companions up to separation as high as $\sim$1700 AU.
{height="0.7\textheight"}
Seventeen objects among the 33 submitted have been observed, but in 2 cases a problem with the guidance sensor resulted in moved exposures, as shown in Figure \[moved\_acs\_targets\]. The corresponding images are useless. We thus obtained images for 15 targets, 2 of which were already known binaries.
{width="80.00000%"}
Data Analysis
-------------
### Search for the multiple systems
In order to look for multiple systems, we used the same method as described in @2005AJ....129..511B. Briefly, it consists in a quantitative analysis of the relative intensity of the residuals after PSF subtraction. Any multiple system is expected to show higher residuals than an unresolved one. The technique and its limitations are fully described in the above mentioned article. Figure \[ri\] shows the result of this analysis. Two systems appear to have clearly higher residuals, indicating that they are very likely to be multiple. These two objects had already been resolved in a previous HST program [see @2003ApJ...594..525M]. Some objects at lower SNR also show slightly higher residuals (at about $\sim$1-$\sigma$), but a careful visual inspection of the images and of the PSF subtraction does not show any convincing evidence of multiplicity. As a sanity check, all images have been inspected visually.
{width="80.00000%"}
### PSF fitting
The ACS-HRC data have been processed with the same PSF fitting program described in @2003AJ....126.1526B, adapted to ACS-HRC. Briefly: the program performs a dual-PSF fit of the binary, fitting both component at the same time. The relative astrometry and photometry are obtained when the residuals reach their minimum value. The method and its limitations are fully described in @2004PhDT.........5B [@2003AJ....126.1526B].
Results for the individual objects \[mutliple\_syst\]
=====================================================
We confirm 2 binaries previously discovered in @2003ApJ...594..525M study, and report no new binary in the angular separation 0045–026 and apparent brightness range 18$<$I$_{C}<$22.8.
Considering the relatively high proper motion of the Pleiades cluster , and the small relative motion of their respective components (see Tables \[astrometry\_cfhtpl12\] and \[astrometry\_IPMBD\_29\]), we conclude that CFHT-PL-12AB and IPMBD-29AB are common proper motion pairs. Tables \[astrometry\_cfhtpl12\] and \[astrometry\_IPMBD\_29\] show the astrometric measurements of the two objects. For both binaries the separation measured in 2003 is smaller than that measured in 2000. This is an effect of the eccentricity of the orbits and a selection bias due to the resolution limit of the WFPC2 survey.
Cl\* Melotte 22 CFHT-Pl 12
--------------------------
Cl\* Melotte 22 CFHT-Pl 12 is a binary with a separation of 0062$\pm$0002 and a position angle (P.A) of 266.7$\pm$1.7 (14th November 2000), corresponding to a physical separation of 8.4$\pm$0.3 AU at 135 pc. Correcting for a statistical factor of 1.26 as explained in @1992ApJ...396..178F, it leads to a semi-major axis of 10.5$\pm$0.3 AU. Its proper motion and the presence of Li absorption in its spectrum indicate that it is substellar and belongs to the Pleiades cluster . Table \[pleiades\_bin\] gives a summary of its astrometric and photometric properties. Using the NextGen models for the primary and the DUSTY models for the fainter (and therefore cooler) secondary and assuming an age of 120 Myr, we can estimate the masses of each component to be M$_{A}$=0.066$\pm$0.001 M$_{\sun}$ and M$_{B}$=0.052$\pm$0.002 M$_{\sun}$, corresponding to a mass ratio of $q=0.79$ (see Figure. \[cmd\]). According to Kepler’s third laws [@1609QB41.K32.......], the corresponding period is $\sim$99$\pm$5 years. The small relative motion of 15 in 3 years corresponds to an orbital period of $\sim$70 years, which is of the same order than the orbital period derived from the theoretical masses and the semi-major axis, but a more precise comparison between dynamical masses and theoretical masses requires more astrometric monitoring.
[lccccc]{} 14/11/2000 & WFPC2 & 62$\pm$3 & 266.7$\pm$4.5 & 0.98$\pm$0.15 & F814W\
07/11/2003 & ACS & 50$\pm$3 & 251.4$\pm$0.75 & 0.43$\pm$0.15 & F814W\
{width="80.00000%"}
Cl\* Melotte 22 IPMBD 29
------------------------
Cl\* Melotte 22 IPMBD 29 was confirmed as a Pleiades member via proper motion measurements by @1999MNRAS.303..835H. It was observed twice: the first time with WFPC2 (18th September 2000), and the second time with ACS (13th December 2003). Table \[astrometry\_IPMBD\_29\] gives a summary of the astrometric and photometric properties measured at both epochs. Unfortunately a satellite crossed the field of our ACS image exactly on the target (see Figure \[IPMBD29\_satellite\]). The flux of the satellite track is relatively low. Measuring the number of counts in an area of 11 pixels around the source and in another area centered on the satellite track away from the source, we can estimate that the flux of the satellite track corresponds to less than 5% of that of the source. The elongation and the duplicity are nevertheless real, since it appears clearly on the 3 individual exposures of the CR-SPLIT that have not been affected by the satellite track. It is moreover confirmed by the previous detection in the WFPC2 image 3 years earlier, with consistent relative astrometry of the two components. The difference of magnitude is different at the two epochs. They agree within 3-$\sigma$, but the WFPC2 value should be considered with more caution than the ACS value. The ACS image is indeed much better sampled (the pixel-scale of ACS is twice as good as that of WFPC2), and the separation is below the sampling limit of WFPC2, while it is above that of ACS. We therefore consider that the ACS value is more reliable than the WFPC2 one. Uncertainties on the relative photometry at such short separations should always be considered with caution, since we are much below the diffraction limit of HST at this wavelength. The difference between the measurements obtained with two different instruments on-board HST illustrate the limitations of the PSF fitting.
Cl\* Melotte 22 IPMBD 29 is a binary with a separation of 0050$\pm$0003 and P.A of 85.6$\pm$0.75 corresponding to a physical separation of 6.75$\pm$0.4 AU at 135 pc. Correcting for a statistical factor of 1.26 as explained in @1992ApJ...396..178F, it leads to a semi-major axis of 8.5$\pm$0.5 AU. Using the NextGen models for the primary and the DUSTY models for the fainter secondary and assuming an age of 120 Myr, we can estimate the masses of each component to be M$_{A}$=0.056$\pm$0.002 M$_{\sun}$ and M$_{B}$=0.047$\pm$0.002 M$_{\sun}$, corresponding to a mass ratio of $q=0.83$ (see Figure. \[cmd\]). According to Kepler’s third laws, the corresponding period is $\sim$77$\pm$9 years. The small relative motion of 5/yr corresponds to an orbital period of $\sim$75 years, consistent with the period derived from the Kepler’s laws.
[lccccc]{} 18/07/2000 & WFPC2 & 58$\pm$3 & 103$\pm$4.5 & 1.25$\pm$0.15 & F814W\
13/12/2003 & ACS & 50$\pm$3 & 85.6$\pm$0.75 & 0.22$\pm$0.30 & F814W\
{width="80.00000%"}
Confirmed photometric binary candidates \[conf\_photom\]
========================================================
From its position in the H-R diagram, suspected CFHT-Pl-12 to be a brown dwarf binary. Similarly, from their photometric analysis, @2003MNRAS.342.1241P suspected this object to be multiple. Using our WFPC2 and ACS images, we resolve CFHT-Pl-12 and calculate a mass ratio consistent with the one they derive from the photometry.
It is interesting to note that the two resolved binaries IPMBD-25 and IPMBD-29, which have $I_{C}$ and $K$ photometric measurements available, fall just on the binary sequence of the $K$ vs. ($I_{C}-K$) colour-magnitude diagram (CMD) defined by @2003MNRAS.342.1241P, as shown in Figure \[pinfield\], although they were not included in their study. From this diagram we can predict a mass ratio of 0.6–0.9 for IPMBD-25, very similar to that of CFHT-Pl-12 since the two objects are very close in the diagram, and consistent with the mass ratio we derive from the relative photometry of the two components. Similarly, the CMD predict a mass ratio of 0.7–1.0 for IPMBD-29, in good agreement with the one we derive from the relative photometry of the two components.
{width="90.00000%"}
Unresolved photometric binary candidates
========================================
From their positions in the H-R diagram, suspected CFHT-Pl-16 to be a brown dwarf binary. It is not resolved in our ACS images. From their photometric study, @2003MNRAS.342.1241P also classify this object as binary, and derive a mass ratio of about 0.75–1. According to the DUSTY models, this mass ratio corresponds to a difference of magnitude between 0.0$\le\Delta$mag$\le$6 mag in the $I$ band, thus just at/above the limit of sensitivity of our study. This indicates that, if multiple, this system should have a separation less than 5.4–34 AU depending on the flux ratio (see Figure \[limit\_detection\_all\] and Table \[binary\_candidates\]).
Due to its peculiar proper motion, suggested that CFHT-Pl-15 might be a multiple system. @2000ApJ...543..299M found evidence for high residuals after PSF subtraction on their NICMOS image, and suspected the presence of a companion at a separation less than 022. Using ACS, we do not resolve any companion at separation larger than 0040. If multiple, this object should have a separation smaller than 5.4 AU and/or a difference in magnitude larger than 5.9 mag in the F814W band.
From their photometric analysis, @2003MNRAS.342.1241P suspected CFHT-Pl-25, CFHT-Pl-23 and CFHT-Pl-21 to be binaries. Using our ACS images, we do not find any evidence of companions around these three objects. @2003MNRAS.342.1241P also predict mass ratios of $q\sim$1 for CFHT-Pl-23, $q<$0.75–1 for CFHT-Pl-25, and 0.5$<q<$0.7 for CFHT-Pl-21, corresponding to differences of magnitude of respectively 0 mag, $>$0–3 mag, and 3.3–8.8 mag. Together with our ACS study, this constrains the separations of CFHT-Pl-23 to be smaller than 5.4 AU and that of CFHT-Pl-25 to be smaller than $\sim$5.4–13 AU while that of CFHT-Pl-21 should be less than 13 AU (see Figure \[limit\_detection\_all\]). Spectroscopic studies would be currently the only way to test the possibility that these objects are binaries. Table \[binary\_candidates\] summarizes this analysis.
[lcccc]{} CFHT-Pl-16 & 0.75–1.0 & 18.7 & 0.0–6.0 & $<$5.4–34.0\
CFHT-Pl-21 & 0.5–0.7 & 19.0 & 3.5–8.8 & $<$13.0–34.0\
CFHT-Pl-23 & $\sim$1 & 19.3 & $\sim$0.0 & $<$5.4\
CFHT-Pl-25 & $<$0.75–1.0 & 19.7 & $>$0.0–3.5 & $<$5.4–13.0\
[lccccccccccc]{} CFHT-Pl 12 & 18.34$\pm$0.11 & 19.32$\pm$0.11 & & 17.57$\pm$0.11 & 18.48$\pm$0.11 & 0.062$\pm$0.002 & 10.5$\pm$0.3 & 266.7$\pm$1.7 & 0.066 & 0.79 & 99\
IPMBD 25 & 17.93$\pm$0.09 & 19.38$\pm$0.09 & & 17.22$\pm$0.09 & 18.74$\pm$0.09 & 0.094$\pm$0.003 & 16.0$\pm$0.5 & 340.5$\pm$2.1 & 0.063 & 0.62 & 200\
IPMBD 29 & 18.70$\pm$0.15 & 19.95$\pm$0.15 & & 17.81$\pm$0.11 & 19.06$\pm$0.11 & 0.058$\pm$0.004 & 8.6$\pm$0.5 & 103.0$\pm$4.5 & 0.056 & 0.83 & 77\
Analysis: Binary frequency \[bin\_frequ\]
=========================================
Our sample of bona-fide brown dwarfs Pleiades members include 15 objects. Two of them were peviously known binaries, and should therefore be excluded from the statistics. This gives an observed visual binary frequency of $<$7.7% for separations greater than 5.4 AU and primary masses between 0.030–0.065 M$_{\sun}$. The binary frequency is defined here as the number of binaries divided by the total number of objects in the sample. Upper limit uncertainty is derived as explained in @2003ApJ...586..512B.
@2003ApJ...594..525M noticed that the primaries of the only two binaries resolved with WFPC2 are brighter than $I$=18.5 mag, suggesting breaking the statistical analysis in two bins of magnitudes. In the first bin, between 17.7$<I<$ 18.5 mag corresponding to 0.055$<M<$0.065 M$_{\sun}$, they reported a binary frequency of 22$^{+19}_{-8}$%, with 2 binaries among a sample of 9 objects. In the same magnitude bin, and over the same separation range ($>$7–12 AU, we have 6 new objects and 0 new binary. The combination of the two results gives a total of 2 binaries over 15 objects, leading to a refined binary frequency of 13.3$^{+13.7}_{-4.3}$%. In the second magnitude bin, between 18.5$<I<$21.0 corresponding to 0.035$<M<$0.055 M$_{\sun}$, @2003ApJ...594..525M reported 0 binary among a total of 6 objects. In the same magnitude bin and over the same separation range ($>$7–12 AU, we report 5 new objects and 0 new binary. The combination of the two results gives a total of 0 binary over 11 objects, leading to a refined limit on the visual binary frequency of $f_{vis} <$9.1%.
In the new separation range that we were able to investigate with ACS, between 5.4–7.0 AU (for the brightest objects only, 17.7$<I<$18.5 mag or 0.055$<M<$0.065 M$_{\sun}$, see Fig. \[limit\_detection\_all\]), we report 0 binaries among a total of 6 objects, leading to a limit on the visual binary frequency of $f_{vis} <$16.7%, consistent with that reported in the separation range between 7–12 AU for the same range of masses.
To summarize, we obtain the following binary frequencies: in the separation range $>$5.4–7.0 AU and in the range of mass between 0.055$<M<$0.065 M$_{\sun}$, we report a visual binary frequency of $f_{vis} = \frac{0}{6}<$16.7%. In the separation range $>$7-12 AU and in the mass range 0.055$<M<$0.065 M$_{\sun}$, we report a visual binary frequency of $f_{vis} = \frac{2}{15}=$13.3$^{+13.7}_{-4.3}$%. In the separation range $>$7-12 AU and in the mass range 0.035$<M<$0.055 M$_{\sun}$, we report a visual binary frequency of $f_{vis} = \frac{0}{11}<$9.1%. Table \[pleiades\_study\] gives an overview of these results.
The three binaries observed in the WFPC2 study all have separations less than 12 AU. The mass ratios are all larger than 0.62. PPL 15, the spectroscopic binary brown dwarf discovered by @1999AJ....118.2460B, has a semi-major axis of 0.03 AU and a mass ratio of 0.87. Although this sample is too small for allowing any meaningful statistical study, it is interesting to note that these results are consistent with that obtained in the field for slightly more massive objects, for which a cut-off in the separation range at 20$\sim$30 AU and a possible lack of small mass ratios[^1] are observed [$q\le$0.5 @2005ApJ...621.1023S; @2003AJ....126.1526B; @2003ApJ...587..407C; @2003AJ....125.3302G].
[lcccccc]{} @2000ApJ...543..299M & 34 & 0 & $>$24 & $>$0.090 & 0.6 & $<$3%\
@2003ApJ...594..525M & 13 & 2 & $>$7–12 & 0.040–0.065 & 0.45–0.9 & 15$^{+15}_{-5}$%\
@2003ApJ...594..525M & 9 & 2 & $>$7–12 & 0.055–0.065 & 0.45–0.9 & 22$^{+19}_{-8}$%\
[**ACS+@2003ApJ...594..525M**]{} & 15 & 2 & $>$7–12 & 0.055–0.065 & 0.45–0.9 & 13.3$^{+13.7}_{-4.3}$%\
[**this ACS study**]{} & 6 & 0 & $>$5.4–7.0 & 0.055–0.065 & 0.9 & $<$16.7%\
@2003ApJ...594..525M & 6 & 0 & $>$7–12 & 0.035–0.055 & 0.45–0.9 & $<$16.7%\
this ACS study & 5 & 0 & $>$7–12 & 0.035–0.055 & 0.45–0.9 & $<$20.0%\
[**ACS+@2003ApJ...594..525M**]{} & 11 & 0 & $>$7–12 & 0.035–0.055 & 0.45–0.9 & $<$9.1%\
Discussion \[discussion\]
=========================
Properties of multiplicity and the mass
---------------------------------------
Both the present ACS study and @2003ApJ...594..525M WFPC2 study suggest that there might be an important change in the properties of multiplicity within the brown dwarf regime. Although statistically inconclusive because of the small number statistics and the relatively large uncertainties, the binary fractions in the two ranges of mass 0.035–0.055 M$_{\sun}$ ($f_{vis}<$9.1%) and 0.055–0.065 M$_{\sun}$ ($f_{vis}=$13.3$^{+13.7}_{-4.3}$%) seems to be notably different. This could mean that the brown dwarf binaries at lower masses are tighter, as already suggested by @2003ApJ...587..407C, and therefore were not resolved by any of the ACS or WFPC2 studies. The small separations reported for the 3 field binary T-dwarfs currently known are consistent with this result.
Properties of multiplicity and the environment \[binarity\_environment\]
------------------------------------------------------------------------
Figure \[bin\_freq\_vs\_spt\_Pleiades+field\] shows that the observed binary frequency among the Pleiades brown dwarfs (13.3$^{+13.7}_{-4.3}$% for separation greater than 7–12 AU is similar to the values reported in the field: 1) for slightly more massive objects [see @2005ApJ...621.1023S; @2003AJ....126.1526B; @2003ApJ...587..407C; @2003AJ....125.3302G 10$\sim$15% of late-M, L-dwarfs]; 2) for field brown dwarfs, as reported by @2003ApJ...586..512B [9$^{+15}_{-4}$% for T5 to T8 field brown dwarfs].
This indicates that the statistical properties, and therefore the formation and evolution processes, of field and Pleiades binary brown dwarfs are probably similar. This would imply that the evolution processes of very low mass binaries do not depend much on the age after 120 Myrs, as expected. The formation, the evolution and, possibly, the disruption of binaries responsible for the low rate of binaries and the cut-off in the separation range would thus have to occur during the early stages of the cluster, when its density and the probability of gravitational encounters are higher. N-body simulations performed by @1995MNRAS.277.1491K [@1995MNRAS.277.1522K] have shown that in dense stellar clusters, such as the Pleiades during its early stages, the binary fraction could drop from 100% to $\sim$50% in less than 1 Myr. More recent hydrodynamical simulations undertaken by @2005MmSAI..76..223D led to similar conclusions, with a typical decay-time for multiple systems of $\sim$10 Myr, consistent with the preliminary conclusion we draw here.
In their numerical simulations of the dynamical interactions in stellar clusters, show that the different properties cited above (binary fraction and distribution of separation) can be nicely reproduced when considering a small-N cluster model (N$<$10) where stars and brown dwarfs form from progenitor clumps. Choosing specific clump and stellar mass spectra, they were able to generate a cluster with an IMF consistent with that observed. Using Monte-Carlo simulations they could then study the small-N cluster decay dynamics and compute the properties of brown dwarfs and brown dwarf binaries. Their study shows that a simple gravitational point-mass dynamics, with weighting factors for the pairing probabilities as a function of the mass evaluated in the first of a two step process, gives results consistent with the observations over the entire range of mass. In particular, they obtain a binary fraction for brown dwarfs of 8–18%, consistent with the binary fraction we report here (13.3$^{+13.7}_{-4.3}$%). They also model a distribution of separation in remarkable agreement with that reported for the field brown dwarfs and for the three Pleiades binaries of our study, with a peak around 4 AU and most ($\sim$85%) objects with separations less than 20 AU. On the other hand, they produce a flat distribution of mass ratio in the range 0.2$<q<$1.0, which is apparently not observed in the field and in the Pleiades. @2005ApJ...621.1023S [@2003AJ....126.1526B; @2003ApJ...586..512B; @2003ApJ...587..407C; @2003AJ....125.3302G] showed that their observations in the field, although statistically incomplete, suggest that there is a preference for equal mass systems. showed also that the mass ratio distribution of spectroscopic binaries among field and Pleiades F–G dwarfs is not flat but bimodal. Finally, in a similar recent study performed on the decay of accreting[^2] triple systems, [@2005ApJ...623..940U] shows that they are also able to reproduce nicely both the distribution of separation observed for field brown dwarfs, with a cut-off around 20 AU.
Photometric binary frequency \[photom\_freq\]
---------------------------------------------
Our work allows the measurement of the binary frequency among brown dwarfs in the Pleiades Open Cluster for separations greater than 7 AU masses between 0.055–0.065 M$_{\sun}$, and mass ratios in the range 0.45–0.9$<q<$1, with $f_{vb}=$13.3$^{+13.7}_{-4.3}$% (visual binaries). We will compare this result to that obtained for slightly more massive objects by @2003MNRAS.342.1241P via the study of binary sequences in colour-magnitude diagrams.
The results of @2003MNRAS.342.1241P do not agree with the observations we report here. From their study of $IK$, $JK$ and $JHK$ colour-magnitude diagrams, they measure a binary frequency of 50$^{+11}_{-10}$% for brown dwarfs in the Pleiades in the mass range 0.05–0.07 M$_{\sun}$ with mass ratio between 0.5$<q<$1.0, thus comparable to the ranges covered by our study. This result is much higher than any of the two values reported in our WFPC2 and ACS studies. If correct, these results together would imply that most ($\sim$85%) of the Pleiades brown dwarf binaries in the range 0.055–0.065 M$_{\sun}$ and 0.5$<q<$1.0 have separations less than 7 AU. From their simulations, @2005MNRAS.tmpL..67M have recently shown that the spectroscopic binary fraction might be as high as 17–30% for separations less than 2.6 AU This value, together with the one we report for separations greater than 7 AU adds up to 30–43% for objects with separations less than 2.6 AU or greater than 7 AU (with a gap between the two). Over the whole separation range, it probably adds up to a binary fraction close to that reported by @2003MNRAS.342.1241P. On the other hand, a recent spectroscopic surveys among Cha I brown dwarfs [@2005astro.ph..9134J no binary candidate out of a sample of 10 objects] show that the spectroscopic binary fraction seems to be relatively low at young ages.
If confirmed by spectroscopic surveys, it would contrast with the results obtained for late type G–K dwarfs in the Pleiades and for early-M dwarfs in the field. found indeed that only $\sim$30% of the G–K Pleiades binaries have separations smaller than 5 AU. Similarly, @2004ASPC..318..166D [@2003sf2a.confE.248M] found that only $\sim$30% of the early-M field binaries have separations smaller than 5 AU. These two values are much smaller than the above mentioned 85%. Assuming that the properties of brown dwarf binaries in that range of masses are similar to that of field or Pleiades late type stars is of course a strong assumption, although we showed in Section \[binarity\_environment\] that the current results tend to confirm it.
The discrepancy between the photometric binary frequency and our visual binary frequency cannot be due to the companions we missed because of their small mass ratios, since the study of @2003MNRAS.342.1241P is sensitive to a similar range of mass ratio as our study. Moreover found that $\sim$60% of the F–G Pleiades spectroscopic binaries have a mass ratio larger than 0.5, and @2004ASPC..318..166D [@2003sf2a.confE.248M] report that $\sim$75% of the field early M-dwarfs have a mass ratio larger than 0.5. If once again we make the assumption that field and Pleiades late type binaries have similar properties to Pleiades brown dwarfs binaries, we should have missed between 25–40% of the multiple systems “only”, leading to a corrected binary fraction of 15–19%, still far from the 50% reported by @2003MNRAS.342.1241P.
In addition to the spectroscopic binaries we miss, we suspect that the large discrepancy between the observations we report and the photometric binary frequency of @2003MNRAS.342.1241P could be due to a combination of the following effects:
- underestimations of the photometric uncertainties, and of possible intrinsic photometric variability due, for example, to weather effects or magnetically driven surface features. Weather effects are known to be producing variability in the luminosity, up to 0.05 mag in I as observed by , and magnetically driven surface features modulation of up to 0.1 mag in J [for young Cha-1 brown dwarfs, @2003ApJ...594..971J].
- spread in the age of the objects. According to the DUSTY evolutionnary models, a spread in the age between 80 and 125 Myr translates into differences of magnitude of up to 0.1 mag in I.
- contamination by field objects. Only 14 of 39 brown dwarfs of their sample have been confirmed as cluster members by proper motion and/or Li detection, while all the objects of our sample have been confirmed by one or both tests. The remaining 25 objects (64% of the sample) have been classified as brown dwarfs on the only basis of their photometric properties. From their photometric (I vs I-Z) and proper motion surveys, estimated that the contamination by foreground M-dwarfs in their sample of Pleiades brown dwarfs can be as high as 30%. From a three colour photometric study (I,Z, and K), they estimate the remaining contamination to be of the order of 10%. A similar non-negligible level of contamination could be expected in @2003MNRAS.342.1241P sample and explain some of the red objects identified as binaries. Since the contaminating objects would be foreground (i.e closer) M-dwarfs, most of them would indeed appear close to the Pleiades binary sequence. The binary is an example of such contaminating objects [@2000ApJ...543..299M].
- effect of rotation: brown dwarfs are known to be fast rotators , and a correlation between the rotation and the luminosity, by up to 0.1 mag, could affect the colours of some objects, as measured by @1982Msngr..28...15V. Deformation of the objects due to their fast rotation can produce variable light curves. A rapidly rotating brown dwarf seen pole-on may be reddened enough to perhaps be identified as a binary by the photometric technique.
- contamination by non-physical pairs in unresolved blends
The binary frequency we report here for brown dwarfs in the Pleiades is consistent with that observed for similar objects, similar separation and mass ratio ranges than in the field, as shown in Figure \[bin\_freq\_vs\_spt\_Pleiades+field\]. It is comparable to that of slightly more massive field late-M/early-L dwarfs, and close to the frequency observed for field T-dwarfs, which have masses comparable to the brown dwarfs of our Pleiades sample.
Deep spectroscopic surveys on unbiased samples should provide answers to these questions and determine how many small mass ratio/small separation binaries we missed.
{width="\textwidth"}
Separations and mass ratios
---------------------------
In his statistical analysis of the photometric binary properties in the Pleiades, shows that the distribution of mass ratios for late type stars should be similar to that in the field. The distribution is expected to be bimodal, with a major peak at $q$=0.4 and a minor one at $\sim$1. In a more recent observational study of unbiased samples of spectroscopic binaries of F to K dwarfs in the field and in the Pleiades cluster, refine the results of in the range of periods shorter than 10 yrs. They report a mass ratio distribution with a primary peak at $q$=1, decreasing towards smaller mass ratios, with a broad secondary peak around $q$=0.4. They observe no difference between the distributions of mass ratio of F–G and K stars, and find that these are identical in the field and in the Pleiades.
If confirmed, the lack of multiple systems with small mass ratios would then imply a major difference between the distributions of mass ratios (and therefore the formation and evolution processes) of late type stars and brown dwarfs. The current studies are inconclusive regarding that question since the observed lack might well be due to a combination of the following reasons:
- the bias toward bright magnitudes in favor of binaries with large mass ratios [@1924Oepik]
- the current limit of sensitivity: $q>$0.4 for separation larger than 30 AU and only $q>$0.7 for separations larger than 10 AU (see Figure \[limit\_detection\_all\])
Deep spectroscopic surveys on unbiased samples should allow to answer these questions, and see how many binaries of small mass ratios and small separations we missed.
Conclusions
===========
Our new high angular resolution survey for brown dwarf binaries leads to a visual binary fraction in the Pleiades of 13.3$^{+13.7}_{-4.3}$% for separations larger than 7 AU mass ratio between 0.45–0.9, and masses between 0.055–0.65 M$_{\sun}$. The preliminary results show that there might be a difference in the properties of multiplicity within the brown dwarf regime itself, with smaller separations at smaller masses. The binary frequency we report here is a lower limit of the overall binary frequency. It is much lower than the value reported by @2003MNRAS.342.1241P for photometric binaries over a slightly higher range of masses in the Pleiades, but a similar range of mass ratio. As suggested by the recent results of @2005MNRAS.tmpL..67M, the difference could well be due to the spectroscopic binaries missed in our survey. While several surveys looking for visual binaries have already been successfully performed, spectroscopic surveys are only starting to provide results. @2005MNRAS.tmpL..67M results, as well as the present study, show that there is strong need for such systematic surveys looking for close companions, in the Pleiades but also in the field or in star forming regions. The large difference between the results of the two above mentioned independent and complementary studies, and the remaining uncertainties on the overall binary frequency must remind us that any value of the multiplicity fraction must be very carefully used, and [**always considered within its limits (separation range, mass ratio range, mass range)**]{} before a meaningful comparison with other binary frequencies or theoretical predictions can be done.
We are grateful to the STSci team and in particular to our program coordinator Tricia Royle for their kind and efficient support. We also thank our anonymous referee for his comments and corrections. This research is based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, and was funded by HST Grants SNAP-9831.
Facilities:
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[^1]: this latter result might be due to observational biases
[^2]: simulations were purely dynamical, neglecting accretion, but considering small-N clusters rather than triple systems
|
---
abstract: |
The Australian Government uses the means-test as a way of managing the pension budget. Changes in Age Pension policy impose difficulties in retirement modelling due to policy risk, but any major changes tend to be ‘grandfathered’ meaning that current retirees are exempt from the new changes. In 2015, two important changes were made in regards to allocated pension accounts – the income means-test is now based on deemed income rather than account withdrawals, and the income-test deduction no longer applies. We examine the implications of the new changes in regards to optimal decisions for consumption, investment, and housing. We account for regulatory minimum withdrawal rules that are imposed by regulations on allocated pension accounts, as well as the 2017 asset-test rebalancing. The new policy changes are modelled in a utility maximizing lifecycle model and solved as an optimal stochastic control problem. We find that the new rules decrease the benefits from planning the consumption in relation to the means-test, while the housing allocation increases slightly in order to receive additional Age Pension. The difference in optimal drawdown between the old and new policy are only noticeable early in retirement until regulatory minimum withdrawal rates are enforced. However, the amount of extra Age Pension received for many households is now significantly different due to the new deeming income rules, which benefit slightly wealthier households who previously would receive no Age Pension due to the income-test and minimum withdrawals.\
*Keywords:* Dynamic programming, Stochastic control, Optimal policy, Retirement, Means-tested age pension, Defined contribution pension\
*JEL classification:* D14 (Household Saving; Personal Finance), D91 (Intertemporal Household Choice; Life Cycle Models and Saving), G11 (Portfolio Choice; Investment Decisions), C61 (Optimization Techniques; Programming Models; Dynamic Analysis)
author:
- 'Johan G. Andréasson [^1] , Pavel V. Shevchenko [^2]'
bibliography:
- 'bibliography.bib'
title:
-
- 'The 2015-2017 policy changes to the means-tests of Australian Age Pension: implication to decisions in retirement'
---
Introduction
============
Australia relies on a defined-contribution pension system that is based on the superannuation guarantee, private savings, and a government provided Age Pension. The superannuation guarantee mandates that employers contribute a set percentage of the employee’s gross earnings to a superannuation fund, which accumulates and is invested until retirement. The current contribution rate is set to 9.5%, where contributions in addition to this often comes with tax benefits. Private savings comprise of these contributions, but also include savings outside the superannuation fund such as investment accounts, dwelling, and other assets. Finally, the Age Pension is a government managed safety net which provides the retiree with a means-tested Age Pension. This means-test determines whether the retiree qualifies for full, partial, or no Age Pension once the entitlement age is reached. In the means-test, income and assets are evaluated individually, and a certain taper rate reduces the maximum payments once income or assets surpass set thresholds (which are subject to family status and homeownership). Income from different sources are also treated differently; financial assets are expected to generate income based on a progressive deeming rate, while income streams such as labor and non account-based annuity payments are assessed based on their nominal value.
Since the Australian retirement system is relatively young, the long-term effects of this new pension system are not yet known. Changes in this system are expected to occur frequently due to fiscal reasons, and once the effects policy changes have on a retiree’s personal wealth (and the economy in general) becomes evident. Variables directly related to the means-test such as entitlement age, means-test thresholds, taper rates, and pension payments can all be adjusted to meet budget needs by the government. On a larger scale, regulatory changes may include whether the family home is included in the means-tested assets, the elimination of minimum withdrawal[^3] rules, changes in mandatory savings rates, or additional taxes on superannuation savings. From a mathematical modelling perspective, this poses difficulties in terms of future model validity, as regulatory risk and policy changes can quickly make a model obsolete if it is not modified to account for the new rules.
The motivation for this paper was the recent changes for allocated pension accounts, where assets now generate a deemed income and which no longer have an income-test deduction. Account-based pensions (such as allocated pension accounts) are accounts that have been purchased with superannuation and generate an income stream throughout retirement. Prior to 2015, these types of accounts allowed for an income-test deduction that was determined upon account opening, and withdrawals were considered to be income in the means-test. The income-test deduction allowed the retiree to withdraw slightly more every year without missing out on Age Pension. However, in 2015 the rules changed. Existing accounts were ‘grandfathered’ and will continue to be assessed under the old rules, while the new rules will be applied to any new accounts. The argument for the changes were simplicity (people with the same level of assets should be treated the same no matter how they are invested), to increase incentive to maximize total disposable income rather than maximizing Age Pension payments, and to level how capital growth and interest paying investments were assessed [@Fahcsia2016]. From a fiscal point of view, the recommendations to introduce the new rules were based on estimated unchanged costs[^4] [@Henry2009], however the 2015-2016 budget stated expected savings of \$57m for 2015-2016, and \$129m and \$136m for subsequent years [@TheCommonwealthofAustralia2015]. The Age Pension post in the 2015-2016 budget includes all changes to the Age Pension in a combined viewpoint, so a specific impact of the deeming rule changes is not known. We adapt the model previously developed in [@Andreasson2016] to examine the impact of this policy change on an individual retiree. The model used is an extension with stochastic factors (mortality, risky investments and sequential family status), to what was originally presented in [@Ding2013; @Ding2014], which is an expected utility model for the retirement behavior in the decumulation phase of Australian retirees subject to consumption, housing, investment, bequest and government provided means-tested Age Pension.
Problems with decisions that span over multiple time periods are typically modelled with lifecycle models and solved with backward recursion (@Cocco2005, [-@Cocco2005]; @Cocco2012, [-@Cocco2012]; @Blake2014, [-@Blake2014] to name a few). While there is a plethora of research on the subject internationally, there is still rather limited research modelling the Australian Age Pension, and even less that enforces the minimum withdrawal rules. The original model in [@Ding2013] does not constrain drawdown with minimum withdrawal, which would limit the author from finding a closed form solution. Similarly, other authors that focus on means-tested pension also do not enforce minimum withdrawal rates, such as [@Hulley2013] who use CRRA utility to understand consumption and investment behavior, or [@Iskhakov2015] who investigate how annuity purchases changes in relation to Age Pension. It should be noted that their assumptions do not include Allocated Pension accounts, thus minimum withdrawal rates may not apply. There is surprisingly limited research conducted on implications of the regulatory minimum withdrawal rates, even though a large number of retirees are using such accounts (or similar phased withdrawal products). The exception is [@Bateman2007], who compare the welfare of retirees when the current minimum withdrawal rates were introduced 2007 against the previous rules and alternative drawdown strategies. The authors use a rather simple CRRA model to examine the effect of different risk aversion and investment strategies but find that the minimum withdrawal rules increase the welfare for retirees although slightly less than optimal drawdown does. In [@Andreasson2016] the minimum withdrawal rules are included in part of the model outcome, but is by no means exhaustive and only provides a brief introduction to the effects.
The minimum withdrawal rules are designed to exhaust the retiree’s account around year 100, however after year 85 (subject to investment returns) the withdrawn dollar amount starts decreasing quickly. In a recent report from [@PFL2016] it is identified that only 5% of retirees exhaust their accounts completely, though this number is expected to increase as life expectancy increases and the population ages. They find that retirees tend to follow the minimum withdrawal rules as guidelines for their own withdrawal, as few withdraw more than the minimum amount. This is further confirmed in [@Shevchenko2016]. Even so, [@RiceWarner2015] argues that the minimum withdrawal rates should be cut by 25-50% to prevent retirees from exhausting their superannuation prematurely due to increased longevity. The current rates are simply too high for many retirees, thus is not sustainable for people living longer than the average life expectancy, and are significantly higher than what is optimal in [@Andreasson2016].
The contribution of this paper is to improve the understanding of the effect the new policy rules, and with minimum withdrawal enforced, has on a typical retiree’s optimal decisions. Both policy rules are implemented in a utility maximization model with stochastic mortality and risky investments, as well as sequential family status, which explains the behavior of Australian retirees well. We then examine the differences in optimal decisions between an allocated pension account opened prior to 2015 with the one opened post 2015, as well as compare with the planned 2017 asset-test adjustments and the previous results where minimum withdrawal is not enforced. The paper is structured as follows: In Section 2 we summarize the model and present the Age Pension function, as well as explain the parameterization. Section 3 contains a discussion of the results. Finally, in Section 4 we present our concluding remarks.
Model
=====
The model utilized is from [@Andreasson2016], where the Age Pension function has been updated to account for the policy changes in 2015. For a complete description of the model, its calibration to the data and numerical solution, and a discussion of the construction and assumptions, please see the reference.
The objective of the retiree is to maximize expected utility generated from consumption, housing, and bequest. The retiree starts off with a total wealth $\mathsf{W}$, and at the time of retirement $t=t_0$ is given the option to allocate wealth into housing $H$ (and if he already is a homeowner, the option to adjust current allocation by up- or downsizing). The remaining (liquid) wealth $W_{t_0} = \mathsf{W} - H$ is placed in an allocated pension account, which is a type of account that does not have a tax on investment earnings and is subject to the regulatory minimum withdrawal rates. A retiree can either start as a couple or single household, where this information is contained in a family state variable
$$G_t \in \mathcal{G} = \{\Delta, 0, 1, 2\},$$
where $\Delta$ corresponds to the agent already deceased at time $t$, $0$ corresponds to the agent died during $(t-1,t]$, $1$ and $2$ correspond to the agent being alive at time $t$ in a single or couple households respectively. Evolution in time of the family state variable $G_t$ is subject to survival probabilities. In the case of a couple household, there is a risk each time period that one of the spouses passes away, in which case it is treated as a single household model for the remaining years.
At the start of each year $t=t_0, t_0+1...,T-1$ the retiree will receive a means-tested Age Pension $P_t$, and decide what amount of saved liquid wealth $W_t$ will be used for consumption (defined as proportion drawdown $\alpha_t$ of liquid wealth), and the proportion $\delta_t$ of remaining liquid wealth that will be invested in risky assets. The change in wealth after the decision to next period is then defined as $$\label{eq:transition}
W_{t+1} = \left[ W_t - \alpha_t W_t \right] \left[\delta_t e^{Z_{t+1}} + (1-\delta_t) e^{r_t} \right],$$ where $Z_{t+1}$ is the stochastic return on risky assets modelled as independent and identically distributed random variables from Normal distribution $\mathcal{N}(\mu-\widetilde{r},\sigma)$ with mean $\mu - \widetilde{r}$, variance $\sigma^2$ and inflation[^5] rate $\widetilde{r}$. Any wealth not allocated to risky assets is assumed to generate a deterministic real risk-free return $r_t$ (risk-free interest rate adjusted for inflation). Each period the agent receives utility based on the current state of family status $G_t$: $$\label{RewardFunction}
R_{t}(W_t,G_t,\alpha_t,H) = \left\{ \begin{array}{ll}
U_C(C_t,G_t,t) + U_H(H,G_t), & \mbox{if $G_t = 1,2$},\\
U_B(W_t), & \mbox{if $G_t = 0$},\\
0, & \mbox{if $G_t = \Delta.$}\end{array} \right.$$ That is if the agent is alive he receives reward (utility) based on consumption $U_C$ and housing $U_H$, if he died during the year the reward comes from the bequest $U_B$, and if he is dead there is no reward. Note that the reward received when the agent is alive depends on whether the family state is a couple or single household due to differing utility parameters and Age Pension thresholds.
Finally, $t=T$ is the maximum age of the agent beyond which survival is deemed impossible, and the terminal reward function is given as
$$\label{TerminalRewardFunction}
\widetilde{R}(W_T,G_T) = \left\{ \begin{array}{ll}
U_B(W_T), & \mbox{if $G_T \geq 0,$}\\
0, & \mbox{if $G_T = \Delta.$}\end{array} \right.$$
The retiree has to find the decisions that maximize expected utility with respect to the decisions for consumption, investment, and housing. This is defined as a stochastic control problem, where decisions (controls) at time $t$ depend on stochastic variable realization at time $t$ but where future realizations are unknown. The problem can be defined as
$$\label{eq:FinalValueFunction}
\underset{H}{\max} \left[ \underset{\boldsymbol{\alpha}, \boldsymbol{\delta}}{\sup} \: \mathbb{E}^{\boldsymbol{\alpha}, \boldsymbol{\delta}}_{t_0} \left[\beta_{t_0,T} \widetilde{R}(W_T,G_T) + \sum_{t={t_0}}^{T-1} \beta_{t_0,t} R_{t}(W_t,G_t,\alpha_t,H)\right] \right],$$
where $\mathbb{E}^{\alpha, \delta}_{t_0}[\cdot]$ is the expectation conditional on information at time $t=t_0$ if we use control $\boldsymbol{\alpha} = (\alpha_{t_0}, \alpha_{t_0+1}, ..., \alpha_{T-1})$ and $\boldsymbol{\delta} = (\delta_{t_0}, \delta_{t_0+1}, ..., \delta_{T-1})$ for $t=t_0, t_0+1, ..., T-1$. The subjective discount rate $\beta_{t,t'}$ is a proxy for personal impatience between time $t$ and $t'$. This problem can be solved numerically with dynamic programming by using backward induction of the Bellman equation. The state variables are discretized on a grid, and the Gaussian Quadrature method is used for integration between periods; for details, see [@Andreasson2016].
Utility functions
-----------------
Utility in the model is measured with time-separable additive functions based on the commonly used HARA utility function, subject to different utility parameters for singles and couples, as follows.
- **Consumption preferences**. It is assumed that utility comes from consumption exceeding the consumption floor, weighted with a time-dependent health status proxy[^6]. The utility function for consumption is defined as
$$\label{eq:consumption}
U_C (C_t,G_t,t) = \frac{1}{\psi^{t-t_0} \gamma_d} \left(\frac{C_t - \overline{c}_d}{\zeta_d} \right)^{\gamma_d}, \quad d = \left\{ \begin{array}{ll}
\mathrm{C}, & \mbox{if $G_t = 2 \quad \text{(couple),}$}\\
\mathrm{S}, & \mbox{if $G_t = 1 \quad \text{(single),}$}\end{array} \right.$$
where $\gamma_d \in (-\infty,0)$ is the risk aversion and $\overline{c}_d$ is the consumption floor parameters. The scaling factor $\zeta_d$ normalizes the utility a couple receives in relation to a single household. The utility parameters $\gamma_d$, $\overline{c}_d$ and $\zeta_d$ are subject to family state $G_t$, hence will have different values for couple and single households. Also, $\psi \in [1,\infty)$ is the utility parameter for the health status proxy, which controls the declining consumption between current time $t$ and time of retirement $t_0$.
- **Bequest preferences**. Utility is also received from *luxury* bequest, hence the home is not included in the bequest [@Ding2014]. The utility function for bequest is then defined as
$$U_B(W_t) = \left(\frac{\theta}{1-\theta}\right)^{1-\gamma_\mathrm{S}} \frac{\left(\frac{\theta}{1-\theta} a+W_t\right)^{\gamma_\mathrm{S}}}{\gamma_\mathrm{S}},$$
where $W_t$ is the liquid assets available for bequest, and $\gamma_\mathrm{S}$ the risk aversion parameters for single households[^7]. The parameter $\theta \in \left[0,1\right)$ is the degree of altruism which controls the preference of bequest over consumption, and $a \in \mathbb{R}^+$ is the threshold for luxury bequest up to where the retiree leaves no bequest[^8].
- **Housing preferences**. The utility from owning a home comes in the form of preferences over renting but is approximated by the home value. The housing utility is defined as $$U_H (H,G_t) = \frac{1}{\gamma_\mathrm{H}} \left(\frac{\lambda_d H}{\zeta_d} \right)^{\gamma_\mathrm{H}},$$ where $\gamma_\mathrm{H}$ is the risk aversion parameter for housing (allowed to be different from risk aversion for consumption and bequest), $\zeta_d$ is the same scaling factor as in equation (\[eq:consumption\]), $H > 0$ is the market value of the family home at time of purchase $t_0$ and $\lambda_d \in (0,1]$ is the preference of housing defined as a proportion of the market value.
Age Pension
-----------
The Age Pension policy changes over time, and all income streams of allocated pension accounts opened after the 1st January 2015 are assumed to generate deemed income. Accounts opened prior to this are ‘grandfathered’ hence will continue to be assessed under the old rules [@Fahcsia2016], where instead drawdown is considered income.
The Age Pension rules state that the entitlement age is 65 for both males and females, with the means-test thresholds and taper rates for July 2016 presented in Table \[table:PensionRates\] and discussed in detail later in this section. The new rules have introduced a ‘Work bonus’ deduction for the income-test, but as the model assumes the retiree is no longer in the workforce this has been left out.
------------------------------------------------------------------------------------------------------------------------
Couple
-------------------- ----------------------------------------------------------------------------- ---------- ----------
$P^d_\mathrm{max}$ Full Age Pension per annum \$22,721 \$34,252
**[Income-Test]{}\
$L^{d}_\mathrm{I}$ & Threshold & \$4,264 & \$7,592\
$\varpi^d_{\mathrm{I}}$ & Rate of Reduction & \$0.5 & \$0.5\
& **[Asset-Test]{}\
$L^{d,h=1}_\mathrm{I}$ & Threshold: Homeowners & \$209,000 & \$296,500\
$L^{d,h=0}_\mathrm{I}$ & Threshold: Non-homeowners & \$360,500 & \$448,000\
$\varpi^d_{\mathrm{A}}$ & Rate of Reduction & \$0.039 & \$0.039\
& **[Deeming Income]{}\
$\kappa^d$ & Deeming Threshold & \$49,200 & \$81,600\
$\varsigma_-$ & Deeming Rate below $\kappa^d$& 1.75% & 1.75%\
$\varsigma_+$ & Deeming Rate above $\kappa^d$& 3.25% & 3.25%\
******
------------------------------------------------------------------------------------------------------------------------
: Age Pension rates published by Centrelink as at September 2016.[]{data-label="table:PensionRates"}
### Deemed income
Deemed income refers to the assumed returns from financial assets, without reference to the actual returns on the assets held. The deemed income only applies to financial assets and account based income streams and is calculated as a progressive rate of assets. The income-test can therefore depend on both labor income (if any), deemed income from financial investments not held in the allocated pension account, drawdown from allocated pension accounts if opened prior to 2015, or deemed income on such accounts if opened after January 1^st^ 2015.
The deeming rates are subject to change in relation to interest rates and stock market performance[^9]. Two different deeming rates may apply based on the value of the account; a lower rate $\varsigma_{-}$ for assets under the deeming threshold $\kappa_d$ and a higher rate $\varsigma_{+}$ for assets exceeding the threshold, as shown in Table \[table:PensionRates\].
### Age Pension function
The Age Pension received is modelled with respect to the current liquid assets, where the account value is used for the asset-test. Since the model assumption states that no labor income is possible, all income for the income-test comes from either deemed income (new rules) or generated from withdrawals of liquid assets (old rules). The Age Pension function can thus be defined as
$$P_t := f(W_t) = \max \left[0, \min \left[P^{d}_\mathrm{max}, \min \left[P_{\mathrm{A}}, P_{\mathrm{I}}\right] \right] \right],$$
where $P^d_{\mathrm{max}}$ is the full Age Pension, $P_\mathrm{A}$ is the asset-test and $P_\mathrm{I}$ is the income-test functions. The $P_\mathrm{A}$ function is the same for rules prior and post 2015, and is defined as
$$P_{\mathrm{A}}:= P^d_\mathrm{max} - (W_t-L^{d,h}_{\mathrm{A}})\varpi^d_{\mathrm{A}},$$
where $L_\mathrm{A}^{d,h}$ is the threshold for the asset-test and $\varpi^d_\mathrm{A}$ the taper rate for assets exceeding the thresholds. Superscript $d$ is a categorical index indicating couple or single household status as defined in equation (\[eq:consumption\]). The variables are subject to whether it is a single or couple household, and the threshold for the asset-test is also subject to whether the household is a homeowner or not ($h=\{0,1\}$). Although the $P_{\mathrm{A}}$ function is the same for both the old and new policies, the $P_{\mathrm{I}}$ function is different. For the new policy rules, it can be written as
$$P_{\mathrm{I}}:= P^d_\mathrm{max} - (P_{\mathrm{D}}(W_t) - L^{d}_{\mathrm{I}})\varpi^d_{\mathrm{I}},$$
$$P_{\mathrm{D}}(W_t) = \varsigma_- \min \left[ W_t, \kappa^d \right] + \varsigma_+ \max \left[0, W_t - \kappa^d \right],$$
where $L_\mathrm{i}^d$ is the threshold for the income-test and $\varpi^d_\mathrm{I}$ the taper rate for income exceeding the threshold. $P_{\mathrm{D}}(W_t)$ calculates the deemed income, where $\kappa_d$ is the deeming threshold, and $\varsigma_{-}$ and $\varsigma_{+}$ are the deeming rates that apply to assets below and above the deeming threshold respectively.
Under the previous policy, the $P_I$ function is defined as
$$P_{\mathrm{I}}:= P^d_\mathrm{max} - (\alpha_t W_t - M(t) - L^{d}_{\mathrm{I}})\varpi^d_{\mathrm{I}},$$
$$M(t) = \frac{W_{t_0}}{e_{t_0}}(1+\widetilde{r})^{t_0-t},$$
where the function $M(t)$ represents the income-test deduction that was available for accounts opened prior to 2015, $e_{t_0}$ is the lifetime expected at age $t_0$ and $\widetilde{r}$ the inflation. As the model is defined in real terms, the future income-test deductions must discount inflation. Current values of the function parameters are given in Table \[table:PensionRates\].
Parameters
----------
The model parameters are taken from [@Andreasson2016], where calibration was performed on empirical data from [@Statistics2011]. However, the consumption floor $\overline{c}_d$ and the threshold for luxury bequest $a$ must be adjusted as they represent monetary values. Since the previous model was defined in real terms, we need to set a new base year for the comparison. We therefore adjust these parameters based on the Age Pension adjustments from 2010 to 2016. Currently, the Age Pension payments are adjusted to the higher of the Consumer Price Index (CPI) and Male Average Weekly Total Earnings (MTAWE). The increase in full Age Pension payments from 2010 to 2016 equals approximately 4.5% increase per year. We assume that the utility parameters representing monetary values have increased in the same manner. All utility model parameter values are shown in Table \[table:parameters\].
$\gamma_d$ $\gamma_\mathrm{H}$ $\theta$ $a$ $\overline{c}_d$ $\psi$ $\lambda$ $\zeta_d$
------------------- ------------ --------------------- ---------- ---------- ------------------ -------- ----------- ----------- --
Single household -1.98 -1.87 0.96 \$27,200 \$13,284 1.18 0.044 1.0
Couples household -1.78 -1.87 0.96 \$27,200 \$20,607 1.18 0.044 1.3
: Model parameter values adjusted for 2016[]{data-label="table:parameters"}
On the 1^st^ of January 2017 the thresholds of the asset-test will be ‘rebalanced’, hence will change significantly [@AustralianGovernmentDepartmentofVeteransAffairs2016]. The thresholds for the asset-test-will be increased and the taper rate $\varpi^d_{\mathrm{A}}$ will double. This effectively means that retirees will receive full Age Pension for a higher level of wealth, but once the asset-test binds, the partial Age Pension will decrease twice as fast, causing them to receive no Age Pension at a lower level of wealth than before. At the time of writing of this paper there are no proposed adjustments to the full Age Pension or income-test threshold for January 2017, hence these Age Pension parameters do not have to be adjusted other than updating the asset-test thresholds and taper rate according to the changes. The parameters for the Age Pension in 2016 are shown in Table \[table:PensionRates\], and the 2017 Age Pension parameters for the updated asset-test are shown in Table \[table:PensionRates2017\]. In addition to this, we set the following.
- A retiree is eligible for Age Pension at age $t=65$ and lives no longer than $T=100$.
- Real risky returns follow $Z_t \sim \mathcal{N}(0.056, 0.133)$, and the real risk-free rate is set to $r_t = 0.005$. These parameters were estimated from S&P/ASX 200 Total Return and the deposit rate [@Andreasson2016].
- The lower threshold for housing is set to \$30,000. A retiree with wealth below this level can therefore not be a homeowner, hence $H\in \{0,[30000, \mathrm{W}]\}$.
- A unisex survival probability is used to avoid separating the sexes. The survival probabilities for a couple are assumed to be mutually exclusive, based on the oldest partner in the couple. The actual mortality probabilities are taken from Life Tables published in [@ABSMortality2014].
- The subjective discount rate $\beta$ is set in relation to the real interest rate so that $\beta_{t,t'} = e^{-\sum_{i=t}^{t'}r_i}$.
--------------------------------------------------------------------------------------
Couple
-- ----------------------------------------------------------------------------- -- --
**[Asset-Test]{}\
$L^{d,h=1}_\mathrm{I}$ & Threshold: Homeowners & \$250,000 & \$375,000\
$L^{d,h=0}_\mathrm{I}$ & Threshold: Non-homeowners & \$450,000 & \$575,000\
$\varpi^d_{\mathrm{A}}$ & Rate of Reduction & \$0.078 & \$0.078\
**
--------------------------------------------------------------------------------------
: Planned 2017 Age Pension rates published by Centrelink as at September 2016 (*https://www.humanservices.gov.au/customer/services/centrelink/age-pension*).[]{data-label="table:PensionRates2017"}
Minimum withdrawal rates for allocated pension accounts are shown in Table \[table:minwithdrawal\] [@ATO2016]. The rates impose a lower bound on optimal consumption, hence withdrawals from liquid wealth must be larger or equal to these rates.
--------------- ---------- ------- ------- ------- ------- ------- ----------
Age $\le$ 64 65-74 75-79 80-84 85-89 90-94 95 $\le$
Min. drawdown 4% 5% 6% 7% 9% 11% 14%
--------------- ---------- ------- ------- ------- ------- ------- ----------
: Minimum regulatory withdrawal rates for allocated pension accounts for the year 2016 and onwards (*https://www.ato.gov.au/rates/key-superannuation-rates-and-thresholds/*).[]{data-label="table:minwithdrawal"}
Results
=======
In the income-test, the policy change to replace asset drawdown with deemed income leads to some interesting implications for the retirees in all three decision variables (housing, consumption, and risky asset allocation). The main difference is that assets are now included twice in the means-test, as the income-test is now based on assets only rather than the actual drawdown of assets. Optimal decisions are then becoming more sensitive to changes in liquid assets, although the retiree has now less control to optimize utility in relation to the Age Pension.
Below we present and compare results for optimal decisions under the old rules (‘Pre 2015’), new deeming rules for income-test (‘Post 2015’), and new deeming rules with the new asset test (‘Post 2015, new asset test’) that starts in 2017.
Optimal Consumption
-------------------
The optimal consumption consists of the drawdown from liquid wealth and the Age Pension received, and exemplifies a behavior consistent with traditional utility models (Figure \[fig:OptDD\_C\_W\]). The curve is generally a smooth, concave, and monotone function of wealth hence becomes flatter as wealth increases due to decreasing marginal utility. The curve becomes flatter as the retiree ages, which is the desired effect from the model’s health proxy as to reflect the lower consumption resulting from decreasing health. However, this general behavior starts to deviate as the retiree ages due to the minimum withdrawal rates. For a retiree aged 65 with an account of \$500,000, the optimal consumption for a couple is roughly 11.1% which is more than the minimum withdrawal rate of 5% (Table \[table:minwithdrawal\]). As the retiree ages his consumption tends to decrease, but around age 85 the minimum withdrawal rates crosses over the optimal consumption hence the drawdown curve becomes proportional to wealth. This deviation occurs at an even earlier age for wealthier retirees.
{width=".9\linewidth"} \[fig:OptDD\_C\_W\]
{width=".9\linewidth"} \[fig:OptDD\_S\_W\]
Age Pension only contributes to the consumption, rather than being a means of planning for optimizing utility or amount of Age Pension received. This is in contrast to the results in [@Andreasson2016] with the policy rules prior to 2015, which showed that drawdown was highly sensitive to the means-test and could be utilized in financial planning (right column in Figure \[fig:OptDD\_C\_W\]). There is a marginal effect when the retiree goes from no Age Pension to receiving partial Age Pension, especially for the 2017 asset-test adjustment, shown as a tiny dent where the consumption and drawdown curve intersect (the threshold between no pension and partial pension due to asset-test). This implies that a retiree should consume slightly more when his wealth is close to this threshold in order to receive partial Age Pension, but the additional utility would be so small that it is negligible in planning. Another level where such an optimizing decision would be expected is when the income-test binds over the asset-test (the threshold between partial pension due to income-test and asset-test). It occurs around \$508,066 for single households without a family home (\$248,352 for homeowners) and \$574,242 for couple households (\$314,527 for homeowners), and can be seen as a slight change in the drawdown curve due to different taper rates for the partial Age Pension. No apparent effect is identified in the consumption however, hence this would provide no additional utility. Note that as the retiree ages, and the minimum withdrawal rate is higher than the unconstrained optimal consumption, Age Pension simply adds to the consumption rather than being included in desired consumption. This is in line with [@Bateman2007], which finds that welfare decreases slightly when minimum withdrawal rules are enforced over unconstrained optimal withdrawals, especially for higher levels of risk aversion.
An interesting outcome is when the consumption paths over a lifetime are compared with the new and old rules (Figure \[fig:LifetimeComparison\]). Since the optimal drawdown rules are very similar before and after the change, and minimum withdrawal rates quickly binds, the consumption in turn follows the same pattern. However, since the income-test is now based on deemed income, more Age Pension is received in relation to wealth and drawdown assuming the deeming rates stay constant. This is especially true at older ages. Figure \[fig:AgePensionFunction\] clearly shows the difference in Age Pension payments, where the new rules lead to more partial Age Pension (but less for wealthier households in the 2017 asset-test adjustment). As the withdrawal rate increases the difference in partial Age Pension increases as well. One of the reasons for changing the policy was for the government generate savings, but the deeming rules will not have the desired outcome on allocated pension accounts unless the deeming rates increase. Only when the minimum withdrawals are removed (or at least decreased), which in turn could lead to lower withdrawals for given wealth levels, could current rates lead to Age Pension payments being less under the new policy[^10]. The effect can clearly be seen in Figure \[fig:LifetimeComparison\]. Under the old rules, the relatively high drawdown (income) for the retiree would most often lead to no Age Pension due to the income-test, while under the new rules the retiree would receive a significant amount of Age Pension over his lifespan. Wealth paths throughout retirement, however, are almost identical - the difference between the new and old policy is solely in consumption from additional Age Pension.
![Comparison of consumption, Age Pension and wealth over a retiree’s lifetime with the three different policy scenarios and with unconstrained (no minimum withdrawal) optimal consumption. The retiree starts with \$1m liquid wealth which grows with the expected return each year, and drawdown follows the optimal drawdown paths under each policy.[]{data-label="fig:LifetimeComparison"}](Figure_3)
![Comparison of the Age Pension function with the three policy scenarios. The retiree is a single household aged 65-74 and consumption is assumed to be the minimum withdrawal rate of 5%.[]{data-label="fig:AgePensionFunction"}](Figure_4)
Optimal risky asset allocation
------------------------------
The exposure to risky assets in the portfolio is highly dependent on wealth and age, and even more so compared to the old rules. This is expected since the means-test is now based on wealth in both the asset and the income-test, which means investment returns will have a larger impact on expected utility. The risky allocation displays similar characteristics as the older rules and can be explained with the expected marginal utility conditional on wealth. When marginal utility increases with wealth, the risky allocation will always suggest 100% risky assets. This is the case for the black bottom area (Figure \[fig:OptR\_C\_Post\_W\]-\[fig:OptR\_S\_Pre\_W\]), where the upper bound to the left indicates the maximum marginal utility from consumption, and the upper bound to the right is the maximum marginal utility from bequest. If utility from consumption is considered individually, then lower levels of wealth will have higher marginal utility. If marginal utility from bequest is instead isolated, the same effect will occur albeit at a higher level than for consumption ($\sim\$$450,000). It is, therefore, optimal up to these levels to allocate 100% to risky assets, as the reward is larger than the risk.
The marginal utility is also affected by the means-test, as a result of the ‘buffer’ effect. This buffer occurs when the decreasing wealth that stems from an investment loss is partially offset via increased Age Pension and can be seen as the comparatively darker area around the upper white line (indicating where partial pension becomes no pension) in Figure \[fig:OptR\_C\_Post\_W\] or \[fig:OptR\_S\_Post\_W\]. The buffer effect is, therefore, strongest for a retiree who has no Age Pension but is close to receiving partial Age Pension. An investment loss, in this instance, would be offset by partial Age Pension, whereas an investment profit would not cause the retiree to miss out on Age Pension that he would otherwise receive. The taper rate is steeper for the asset-test than the income-test (especially for 2017), hence marginal utility is lower when the asset-test is binding. For very low levels of wealth, the buffer effect is the opposite; investment losses can never lead to more than full Age Pension, and investment profits will decrease the amount of partial Age Pension received, which will result in lower marginal utility.
Another interesting effect occurs as the minimum withdrawal rates cross above unconstrained optimal drawdown. When the retiree is forced to withdraw more from his account than is optimal to consume, the marginal utility drops significantly. This occurs approximately at age 75 for both single and couple households, though slightly later for less wealthy households. The marginal utility received from consumption is essentially zero after this age, thus the utility consists of an increasingly larger proportion bequest as the retiree ages (and mortality risk increases). This switch occurs where the bottom black area starts to increase towards the right bound, as it moves from utility from consumption to utility from bequest (this is more apparent for couple households in Figure \[fig:OptR\_C\_Post\_W\]). These characteristics are very similar to the surface generated by the old rules when minimum withdrawals are enforced. In fact, once the minimum withdrawal rates exceed the optimal drawdown, they become nearly identical. The difference is therefore only for the initial years of retirement, ages 65-80, because of how the income-test is constructed. In regards to the 2017 asset test changes, the buffer feature are slightly stronger, but the characteristics are similar to the 2015 rules, hence are not shown in a graph. With the old rules, the income-test is binding most of the time, whereas with the new rules it binds for only roughly one-third of the partial Age Pension — and even less than that for homeowners.
{width=".9\linewidth"} \[fig:OptR\_C\_Post\_W\]
{width=".9\linewidth"} \[fig:OptR\_C\_Pre\_W\]
{width=".9\linewidth"} \[fig:OptR\_S\_Post\_W\]
{width=".9\linewidth"} \[fig:OptR\_S\_Pre\_W\]
Optimal housing allocation
--------------------------
The decision variable for the allocation of assets into a family home is expected to change slightly due to the increased focus on assets in the means-test. The decision made at the time of retirement shows that under the new policy rules it is optimal to invest slightly less than under the old rules, up to a total wealth level of approximately \$735,000 for single households and \$1,155,000 for couple households (see Figure \[fig:Housing\]). This would leave approximately \$144,000 and \$247,000 respectively as liquid wealth. Households with total wealth above this level, meanwhile, are recommended to invest slightly more. These allocation decisions leave liquid wealth just below the thresholds for receiving full Age Pension, and the difference in the housing curves can be explained by the income-test changes. For a given wealth, the new rules provide the retiree with more partial pension than with the old rules. Early in retirement, the optimal consumption is high which causes the income-test to bind under the old rules. The new rules alternatively have a deemed income for the income-test that is much lower than before, which ultimately results in more partial Age Pension. The effect decreases 10-15 years into retirement, but when the minimum withdrawal rates exceed unconstrained optimal drawdown at older age ranges, the same occurs again. This can be seen by comparing plots in column 1 or 2 with column 3 in Figure \[fig:OptDD\_C\_W\] or Figure \[fig:OptDD\_S\_W\]. Since a certain level of liquid wealth under the new rules will lead to higher expected utility, it is optimal to allocate slightly more in housing compared with the old rules (as long as the liquid wealth is not very low) to benefit from receiving partial Age Pension. The effects of the 2017 asset test changes are very similar and cannot be distinguished visually from the 2015 policy, hence have been left out in Figure \[fig:Housing\].
![Optimal housing allocation given by total wealth $\mathsf{W}$ for single and couple households under the pre 2015 and post 2015 policy.[]{data-label="fig:Housing"}](Figure_9)
Conclusions
===========
In this paper, we adapt the stochastic retirement utility model from [@Andreasson2016] to implement the Age Pension policy changes from 2015, which affect all Allocated Pension accounts opened after January 1^st^ 2015. These changes affect the treatment of income for the Age Pension income-test, and lead to different optimal decisions for consumption, investments, and housing. We also evaluate the new policy rules with the current Age Pension asset-test, as well as the planned asset-test adjustments for 2017.
We find that optimal consumption only applies early in retirement, as minimum withdrawal rates exceed unconstrained optimal drawdown rates between ages 75-85, depending on wealth level. Only before this point, it is possible to plan withdrawals in order maximize utility, but these possibilities are almost nonexistent under the new policy rules compared with previous. Optimal drawdown equals minimum withdrawal after age 85 (as it becomes a binding lower constraint for withdrawal), thus the new and old policy rules are identical after this age. That said, since the income-test tends to bind for the old rules while the asset-test dominates for the new rules, the retiree will now tend to receive more partial pension under the optimal withdrawal rules. Even with the steeper taper rate that will be introduced in January 2017, the retiree will receive a more generous Age Pension compared with the old policy.
Since income (which was considered drawdown from the allocated pension account) is now replaced by deemed income, the assets are means-tested twice, which means risky asset allocation becomes more sensitive. The changes in optimal risky asset allocation over time and wealth are similar to the old rules, but the changes are slightly more aggressive and depend on marginal utility from consumption and bequest, as well as the level of buffering against investment losses the Age Pension provides. This effect dies off as the minimum withdrawal rates bind, and the bequest motive becomes more important.
Providing that the retiree’s remaining liquid wealth is close to (or higher) than the threshold between full and partial Age Pension at the time of retirement, it is optimal to invest slightly more in housing than before. This will allow the retiree to receive more partial Age Pension, and to increase his expected utility in the long term. If the retiree instead has lower total wealth than the threshold, he is alternatively recommended to invest marginally less than before.
One surprising finding is that a retiree with an income stream where minimum withdrawal rules are enforced will receive more Age Pension over the course of their lifetime with the new policy rules. Due to the minimum withdrawal requirement, the drawdown tends to be higher than what is optimal for most ages, which under the old rules would result in no or low partial Age Pension. The new rules combined with the current historically low deeming rates will generate significant Age Pension payments from the same drawdown and wealth levels. This, in turn, affects both the decision for allocation in housing as well as risky investments. The government’s goal of reducing incentives for maximizing Age Pension payments and focusing on maximizing total disposable income is however met - the new policy is not as sensitive to optimal withdrawal decisions in order to maximize Age Pension payments as the old policy was.
Acknowledgment {#acknowledgment .unnumbered}
==============
This research was supported by the CSIRO-Monash Superannuation Research Cluster, a collaboration among CSIRO, Monash University, Griffith University, the University of Western Australia, the University of Warwick, and stakeholders of the retirement system in the interest of better outcomes for all. Pavel Shevchenko acknowledges the support of Australian Research Council’s Discovery Projects funding scheme (project number DP160103489).
[^1]: CSIRO, Australia; School of Mathematical and Physical Sciences, University of Technology, Sydney, Broadway, PO Box 123, NSW 2007, Australia; email: johan.andreasson@uts.edu.au
[^2]: Applied Finance & Actuarial Studies, Macquarie University; email: pavel.shevchenko@mq.edu.au
[^3]: Certain account types for retirement savings have a minimum withdrawal rate once the owner is retired.
[^4]: The recommendations to introduce deeming was made in [@Henry2009] where the fiscal sustainability is evaluated with the general equilibrium model ‘KPMG Econtech MM900’ [@KPMG2010]. The model shows the estimation over a 10-year window hence we do not know the short term or year-to-year estimates. In addition to this, the model includes additional suggested tax and budget related changes, hence the effect of introducing deeming rates cannot be isolated.
[^5]: By defining the model in real terms (adjusted for inflation), time-dependent variables do not have to include inflation which otherwise would be an additional stochastic variable.
[^6]: Note that the purpose is not to model health among the retirees, but rather to explain decreasing consumption with age.
[^7]: The risk aversion is considered to be the same as consumption risk aversion for singles since a couple is expected to become a single household before bequeathing assets.
[^8]: Because the marginal utility is constant for the bequest utility with zero wealth, in a model with perfect certainty and CRRA utility the optimal solution will suggest consumption up to level $a$ before it is optimal to save wealth for bequest [@Lockwood2014].
[^9]: The current rates are at a historical low. In 2008 the deeming rates $\varsigma_{-} /\varsigma_{+}$ were as high as 4%/6%, but in March 2013 they were set to 2.5%/4% due to decreasing interest rates, then in November 2013 to 2%/3.5% and to current levels of 1.75%/3.25% in March 2015.
[^10]: It should be noted that the findings are for the account-based pension only, as other products which do not enforce the minimum withdrawal rates could incur additional savings for the government under the new rules.
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---
author:
- 'Ken’ichi [Takano]{}[^1]'
title: ' Dimer-Monomer Ground State for Extended Spin-1/2 Diamond Chain '
---
Spin systems with frustration in low dimensions are interesting, since they produce various types of quantum spin liquids. The diamond chain depicted in Fig. \[diamond\_chain\] is a frustrated spin system which has several different spin-liquid ground states along with the change of a parameter. The full ground-state phase diagrams are found by both rigorous and numerical analyses for the $S$=1/2 and 1 diamond chains.[@tks; @ht2017] In particular, the dimer-monomer (DM) state is commonly a ground state for diamond chains of any spin magnitudes $S$; we know the exact form,[@tks] which is shown with the lattice in Fig. \[diamond\_chain\]. The DM state is characteristic in that it includes effective free spins, monomers, in strong exchange interactions.
When a distortion is introduced to the Hamiltonian representing the $S$=1/2 diamond chain, detailed numerical analysis is performed.[@ottk; @otk] Under distortion, the exact DM state is the ground state but is continuously changed to a gapless spin-liquid state without free spins. Experimentally, azurite Cu$_3$(CO$_3$)$_2$(OH)$_2$ is found to be a substance in such a spin-liquid state for a distorted diamond chain;[@kfcm; @kfcm_sup] in particular, a 1/3 magnetic plateau is observed. Cu$_3$Cl$_6$(H$_2$O)$_2\cdot$2H$_8$C$_4$SO$_2$ a distorted diamond chain, Theoretical analyses including arguments on values of exchange parameters under distortion are given.[@hl] Recently, substances for extended versions of diamond chains found.[@fkmh; @ftfk; @mfks] They include next-nearest-neighbor exchange interactions as well as distortions. The exchange interaction between monomers theoretically and experimentally considered.[@jokv; @ots; @fkmm] search for substances realizing extended diamond chains are going to develop. In view of the present situation, more information about extended diamond chains is expected to be brought.
In this Letter, we investigate the extended $S$=1/2 diamond chain the DM state is the ground state. The method we use is based on representing the Hamiltonian in a complete square form. This method developed from the method projection operators.[@t1994jpa; @t1994jps] In particular, Ref. is the first paper to treat exactly the diamond chain in a special case, where the method of complete square form is applied at the phase boundary between the DM phase and the tetramer-dimer (TD) phase.
interactions we consider is shown in Fig. \[unit\_extend\]. They all the interactions in a simple choice of unit cell. Then the Hamiltonian for this extended $S=1/2$ diamond chain is written as
$$\begin{aligned}
\label{HamOrg}
\Ham = \sum_l
&\Big( J_{\perp} \vtone_{l} \!\cdot \vttwo_{l}
\nonumber\\[-2.5 mm]
& + J_{-} \vtone_{l} \!\cdot \vS_{l+1}
+ J'_{-} \vttwo_{l} \!\cdot \vS_{l+1}
\nonumber\\
& + J_{+} \vtone_{l+1} \!\cdot \vS_{l+1}
+ J'_{+} \vttwo_{l+1} \!\cdot \vS_{l+1}
\nonumber\\
& + J_{a} \vtone_{l} \!\cdot \vtone_{l+1}
+ J'_{a} \vttwo_{l} \!\cdot \vttwo_{l+1}
\nonumber\\
& + J_{b} \vttwo_{l} \!\cdot \vtone_{l+1}
+ J'_{b} \vtone_{l} \!\cdot \vttwo_{l+1}
\nonumber\\
& + J_{\mathrm{m}} \vS_{l} \!\cdot \vS_{l+1} \Big) , \end{aligned}$$
where $\vtau^{(1)}_{l}$, $\vtau^{(2)}_{l}$, and $\vS_{l}$ are spin operators with magnitude 1/2 in the $l$th unit cell. The sum, with respect to $l$, is taken over $N$ unit cells with large $N$ limit, and . For this Hamiltonian, $\vec{T}_l^2$ with $\vec{T}_l \equiv \vtone_{l}+\vttwo_{l}$ is not generally conserved.
The DM state is written as $$\begin{aligned}
\DM = \bigotimes_l {\left\vert {a_l} \right\rangle} \otimes {\left\vert {0_l} \right\rangle} ,
\label{DMstate}\end{aligned}$$ where ${\left\vert {a_l} \right\rangle}$ is any state of $\vS_{l}$ and ${\left\vert {0_l} \right\rangle}$ is the singlet state of $\vtone_{l}$ and $\vttwo_{l}$. There are infinite degenerate DM states owing to the arbitrariness of ${\left\vert {a_l} \right\rangle}$’s, and $\DM$ represents one of them.
To analyze $\Ham$, we construct another Hamiltonian $\Hsq$ in a complete square form. $\Hsq$ is the linear combination of the squares of the spin summations of some spins except for a constant term. It includes the terms of any spin satisfying the rule: the spins in a group form a partial eigenstate with total spin magnitude 0 or 1/2 for the DM state, and any two of spins in a group are connected by in Fig. \[unit\_extend\]. All the possible grouping are represented in Fig. \[sq\_grouping\]. Then the Hamiltonian is $$\begin{aligned}
\label{HamSq}
\Hsq &= \sum_l
\bigg\{ \frac{1}{2} A \, \left(\vtone_{l}+\vttwo_{l}\right)^2
\nonumber\\
&+ \frac{1}{2} B_{-} \left[ \left(\vtone_{l}+\vttwo_{l} + \vS_{l+1} \right)^2 - \frac{3}{4} \right]
\nonumber\\
&+ \frac{1}{2} B_{+} \left[ \left(\vtone_{l+1}+\vttwo_{l+1} + \vS_{l+1} \right)^2 - \frac{3}{4} \right]
\nonumber\\
&+ \frac{1}{2} C'_{-} \left[ \left(\vtone_{l}+\vttwo_{l} + \vtone_{l+1} \right)^2 - \frac{3}{4} \right]
\nonumber\\
&+ \frac{1}{2} C_{-} \left[ \left(\vtone_{l}+\vttwo_{l} + \vttwo_{l+1} \right)^2 - \frac{3}{4} \right]
\nonumber\\
&+ \frac{1}{2} C_{+} \left[ \left(\vtone_{l+1}+\vttwo_{l+1} + \vtone_{l} \right)^2 - \frac{3}{4} \right]
\nonumber\\
&+ \frac{1}{2} C'_{+} \left[ \left(\vtone_{l+1}+\vttwo_{l+1} + \vttwo_{l} \right)^2 - \frac{3}{4} \right]
\nonumber\\
&+ \frac{1}{2} D \left(\vtone_{l}+\vttwo_{l} + \vtone_{l+1}+\vttwo_{l+1} \right)^2
\nonumber\\
&+ \frac{1}{2} E \left[ \left(\vtone_{l}+\vttwo_{l} + \vtone_{l+1}+\vttwo_{l+1} + \vS_{l+1} \right)^2 - \frac{3}{4} \right] \bigg\}
\nonumber\\
&+ U_{0} \end{aligned}$$ with $$\begin{aligned}
U_{0} &= - \frac{3}{4} ( A + B_{-} + B_{+} + C'_{-} + C_{-}
\nonumber\\
&\qquad\qquad + C_{+} + C'_{+} + 2D + 2E ) N ,
\label{GeneCoef}\end{aligned}$$ where $A, B_{-}, B_{+}, C'_{-}, C_{-}, C_{+}, C'_{+}, D$, and $E$, are constant coefficients. Each term in $\Hsq$ vanishes except for $U_0$, if it operates on an eigenstate of the lowest spin magnitude, 0 or 1/2, for the total spin within the term. Hence the DM state (\[DMstate\]) is the ground state of $\Hsq$, if all the coefficients, $A, B_{-}, B_{+}, C'_{-}, C_{-}, C_{+}, C'_{+}, D$, and $E$, are nonnegative. Then the ground-state energy is $U_{0}$, since all the terms other than $U_{0}$ the minimum value, zero.
For $\Hsq$, there is no ground state the DM state in the following cases: (a) $A$ is positive; (b) one of $B_{-}$ and $B_{+}$, and one of $C'_{-}, C_{-}, C_{+}$, and $C'_{+}$ are positive; and (c) two of $C'_{-}, C_{-}, C_{+}$, and $C'_{+}$ are positive. In case (a), $A>0$ enforces the two spins, $\vtone_{l}$ and $\vttwo_{l}$, to form a singlet dimer in the ground state. Then, all $\vS_{l}$s become free, since there is no interaction between $\vS_{l}$ and $\vS_{l+1}$ for all $l$. As for cases (b) and (c), we consider only $C'_{-}$ is positive, for example. Then, there are two ground states: one is the DM state, and the other is the state in which $\vttwo_{l}$ and $\vtone_{l+1}$ form a dimer for each $l$. Another positive coefficient, $B_{-}, B_{+}, C_{-}, C_{+}$, or $C'_{+}$, excludes the latter. only $B_{-}$ and $B_{+}$ are positive, In fact, we can locally replace the dimer of $\vtone_{l}$ and $\vttwo_{l}$ and the monomers of $\vS_{l}$ and $\vS_{l+1}$ in the DM state with two dimers of $\vtone_{l}$ and $\vS_{l}$ and of $\vttwo_{l}$ and $\vS_{l+1}$ without increasing the energy.
The DM state is the ground state of $\Ham$, if $\Ham = \Hsq$ and all the coefficients in $\Hsq$ are nonnegative. $\Ham = \Hsq$, by expanding $\Hsq$ and comparing coefficients, we have
\[CoefJ\] $$\begin{aligned}
A + B_{+} + B_{-} &+ C_{+} + C'_{+} \nonumber\\
+ \, C'_{-} &+ C_{-} + 2D + 2E = J_{\perp} ,
\label{CoefJperp}\\
B_{-} + E &= J_{-} = J'_{-} ,
\label{CoefJminus}\\
B_{+} + E &= J_{+} = J'_{+} ,
\label{CoefJplus}\\
C'_{-} + C_{+} &+ D + E = J_{a} ,
\label{CoefJa}\\
C_{-} + C'_{+} &+ D + E = J'_{a} ,
\label{CoefJad}\\
C'_{-} + C'_{+} &+ D + E = J_{b} ,
\label{CoefJb}\\
C_{-} + C_{+} &+ D + E = J'_{b} ,
\label{CoefJbd}\\
0 &= \Jm .
\label{CoefJm}\end{aligned}$$
These equations mean that the DM state is an eigenstate of $\Ham$ owing to $\Ham=\Hsq$. In (\[CoefJ\]), we directly restrictions exchange parameters:
\[JJdJm\] $$\begin{aligned}
J'_{+} &= J_{+} ,
\label{JpJdp}\\
J'_{-} &= J_{-} ,
\label{JmJdm}\\
\Jm &= 0 .
\label{Jmzero}\end{aligned}$$
the consistency of (\[CoefJa\]) to (\[CoefJbd\]) provides another restriction $$\begin{aligned}
J_{a} + J'_{a} = J_{b} + J'_{b} .
\label{JadaJbdb}\end{aligned}$$ Then (\[CoefJa\]) to (\[CoefJbd\]) are not independent. Hence, we exclude (\[CoefJbd\]) hereafter. We also know that the nonnegative coefficients require nonnegative values for the exchange parameters, owing to (\[CoefJ\]). By using (\[CoefJperp\]), the energy of the DM state (\[GeneCoef\]) is written as $$\begin{aligned}
U_{0} = - \frac{3}{4} J_{\perp} N .
\label{GeneJ}\end{aligned}$$
We now have 6 linear simultaneous equations, (\[CoefJperp\]) to (\[CoefJb\]), for 9 variables, $A, B_{-}, B_{+}, C'_{-}, C_{-}, C_{+}, C'_{+}, D$, and $E$. We take $C_{+}$, $C_{-}$, and $E$ as arbitrary numbers. Then the solution is written as
\[CoefEq\] $$\begin{aligned}
A &= J_{\perp} - J_{+} - J_{-} - J_{a} - J'_{a} + 2E ,
\label{CoefEq_a}\\
B_{+} &= J_{+} - E ,
\label{CoefEq_bp}\\
B_{-} &= J_{-} - E ,
\label{CoefEq_bm}\\
C'_{+} &= - J_{a} + J_{b} + C_{+} ,
\label{CoefEq_cdp}\\
C'_{-} &= - J'_{a} + J_{b} + C_{-} ,
\label{CoefEq_cdm}\\
D &= J_{a} + J'_{a} - J_{b} - E - C_{+} - C_{-} .
\label{CoefEq_d}\end{aligned}$$
To (\[CoefEq\]), we apply the other condition that all coefficients are nonnegative. inequalities,
\[Jcond\] $$\begin{aligned}
J_{\perp} - J_{+} - J_{-} &- J_{a} - J'_{a} \ge -2E ,
\label{Jcond_a}\\
J_{+} &\ge E ,
\label{Jcond_b}\\
J_{-} &\ge E ,
\label{Jcond_c}\\
J_{a} - J_{b} &\le C_{+} ,
\label{Jcond_d}\\
J'_{a} - J_{b} &\le C_{-} ,
\label{Jcond_e}\\
J_{a} + J'_{a} - J_{b} &\ge E + C_{+} + C_{-} ,
\label{Jcond_f}\end{aligned}$$
where $C_{+}$, $C_{-}$, and $E$ are arbitrary nonnegative numbers.
In conclusion, the Hamiltonian (\[HamOrg\]) has the DM ground state (\[DMstate\]), if (i) the exchange parameters are all nonnegative, (ii) (\[JJdJm\]) and (\[JadaJbdb\]) are satisfied, and (iii) there exist nonnegative numbers, $C_{+}$, $C_{-}$, and $E$, satisfying (\[Jcond\]). Then the ground state energy is (\[GeneJ\]). Further, there is no ground state except for the DM state, if (a) $A$ is positive; (b) one of $B_{-}$ and $B_{+}$, and one of $C'_{-}, C_{-}, C_{+}$, and $C'_{+}$ are positive; or (c) two of $C'_{-}, C_{-}, C_{+}$, and $C'_{+}$ are positive. The positivities are known by (\[CoefEq\]).
We examine, as a simple example, the case that (\[JJdJm\]) is satisfied and $J_{a} = J'_{a} = J_{b} = J'_{b} = 0$. (\[JadaJbdb\]) is satisfied. (\[Jcond\_f\]) requires $C_{+} = C_{-} = E = 0$. Then (\[Jcond\_b\]) to (\[Jcond\_e\]) are satisfied. (\[Jcond\_a\]) reduces to $$\begin{aligned}
J_{\perp} \ge J_{+} + J_{-} .
\label{Jspecial}\end{aligned}$$ (\[CoefEq\_cdp\]) and (\[CoefEq\_cdm\]) deduce $C'_{+} = C'_{-} = 0$ so that the conditions (b) and (c) are not satisfied. As to the condition (a), (\[CoefEq\_a\]) mentions that only the DM state is the ground state if $J_{\perp} > J_{+} + J_{-}$. In the symmetric case of $J_{+} = J_{-}$, (\[Jspecial\]) reproduces the range of the DM phase for the original spin-1/2 diamond chain in Ref. . This type of distorted diamond chain is examined by the coupled cluster method.[@jlty]
We also examine another example to see that the condition for the DM ground state has a nontrivial solution. Our example is the case that (\[JJdJm\]) is satisfied and $J_{\perp} = 2.09$, $J_{+} = 0.9$, $J_{-} = 1.1$, $J_{a} = 0.05$, $J'_{a} = 0.05$, $J_{b} = 0.01$, and $J'_{b} = 0.09$ in arbitrary energy unit. First, (\[JadaJbdb\]) is satisfied. Since (\[Jcond\_a\]) reduces to $E \ge 0.005$, we take $E = 0.005$ for trial. Then (\[Jcond\_b\]) and (\[Jcond\_c\]) are satisfied. Since (\[Jcond\_d\]) and (\[Jcond\_e\]) reduce to $C_{+} \ge 0.04$ and $C_{-} \ge 0.04$, respectively, we try to take $C_{+} = C_{-} = 0.04$. Then (\[Jcond\_f\]) becomes $0.09 \ge 0.085$, which is a consistent inequality. Therefore the DM state is the ground state. , since $C_{+}$ and $C_{-}$ are positive, only the DM state is the ground state.
To summarize, we obtained a sufficient condition that the DM state is the ground state for the extended diamond chain (\[HamOrg\]); the method deriving the condition is based on representing the Hamiltonian in a complete square form. The DM state is the ground state for a wide range of nonsymmetric Hamiltonians with next-nearest-neighbor exchange interactions and distortions. They do not generally have the conservation law of $(\vtone_{l}+\vttwo_{l})^2$ and the space-reflection symmetries with respect to the site of $\vS_l$ and to the line of $\vtone_{l}$ and $\vttwo_{l}$. Experimentally, substances satisfying (\[JJdJm\]) and (\[JadaJbdb\]) are not found in . However, the recent discovery of diamond-chain substances, it that substances satisfying the present condition will be found in near future.
I would like to thank Kazuo Hida for discussions. This work is supported by JSPS KAKENHI Grant Number JP26400411.
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[^1]: E-mail: takano@toyota-ti.ac.jp
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---
abstract: 'We study replication identities satisfied by conformal characters of a 2D CFT, providing a natural framework for a physics interpretation of the famous Hauptmodul property of Monstrous Moonshine, and illustrate the underlying ideas in simple cases.'
author:
- 'P. Bantay'
bibliography:
- 'C:/Users/mester/Documents/bibfiles/conformalcharacters.bib'
- 'C:/Users/mester/Documents/bibfiles/modularfunctions.bib'
- 'C:/Users/mester/Documents/bibfiles/CFT.bib'
- 'C:/Users/mester/Documents/bibfiles/my.bib'
- 'C:/Users/mester/Documents/bibfiles/math.bib'
- 'C:/Users/mester/Documents/bibfiles/strings.bib'
- 'C:/Users/mester/Documents/bibfiles/VOA.bib'
- 'C:/Users/mester/Documents/bibfiles/orbifold\_permutation.bib'
- 'C:/Users/mester/Documents/bibfiles/phys.bib'
- 'C:/Users/mester/Documents/bibfiles/Moonshine.bib'
title: Character relations and replication identities in 2d Conformal Field Theory
---
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Introduction
============
The remarkable interaction between mathematics and physics around the turn of the century has been to a large part spurred by the new mathematical structures underlying String Theory [@GSW; @Polch], leading to such interesting new mathematical concepts as Vertex Operator Algebras [@Borcherds1; @FLM1] and Modular Tensor Categories [@Turaev; @Bakalov-Kirillov]. These developments in turn were strongly influenced by Monstrous Moonshine [@Thompson1; @Thompson2; @Convay-Norton], the amazing connection between the representation theory of the Monster $\mathbb{M}$, the largest sporadic finite simple group, with the classical theory of modular forms. Actually, VOA theory grew out from the need to provide a conceptual explanation of Moonshine.
It has been recognized pretty early [@Dixon-Ginsparg-Harvey] that, to a large extent, the Moonshine conjectures find a natural physics explanation by interpreting the relevant quantities as describing string propagation in a suitable (rather exotic) background, the Moonshine orbifold, obtained as the result of orbifolding [@Dixon_orbifolds1] the Moonshine module by the Monster. From this point of view, many strange-looking properties [@generalizedMoonshine] of the Thompson-McKay series involved in Moonshine follow from general physical principles, with one notable exception: the so-called Hauptmodul property, which states (roughly speaking) that Thompson-McKay series generate the field of meromorphic functions of suitable genus zero Riemann surfaces, does not find any obvious interpretation from a physics perspective [@25years; @Gannon2010].
There has been several attempts to remedy this situation and find a physics explanation of the Hauptmodul property, see e.g. [@Duncan2010; @Tuite1995; @Tuite2010], but none proposed to this date seems completely satisfactory. The aim of the present paper is to present a new approach to the problem, based on the notion of character relations and replication identities, which generalizes to arbitrary 2D Conformal Field Theories [@BPZ; @DiFrancesco-Mathieu-Senechal], and which provides an equivalent formulation of the Hauptmodul property in the special case of the Moonshine orbifold. Roughly speaking, this approach relates the Hauptmodul property to symmetries of second quantized string propagation [@elliptic_genera] on the Moonshine orbifold. While the precise nature of these symmetries is still unclear (because identifying them would require a thorough analysis of the higher symmetric products of the Moonshine orbifold, a pretty challenging task in view of the intricate computations involved), the above identification could prove to be a first step in a better understanding of the problem. That the above approach can be made to work is demonstrated in the comparatively much simpler case of the Ising model, where the analysis can be explicitly performed (at least for low degrees), and the resulting replication identities related precisely to actual symmetries of symmetric products.
Conformal characters, the modular representation and character relations
========================================================================
Among the important characteristics of a 2D CFT [@BPZ; @DiFrancesco-Mathieu-Senechal], a prominent role is played by the conformal characters of the ’primaries’, the trace functions of irreducible modules in the language of ($C_{2}$-cofinite rational) Vertex Operator Algebras. As a consequence of conformal symmetry, the chiral symmetry algebra contains the Virasoro algebra, whose zero mode $L_{0}$ plays the role of (chiral) Hamiltonian. The commutation rules of the Virasoro generators imply that, in each irreducible module separately, the eigenvalues of $L_{0}$ are integrally spaced, hence the spectrum of $L_{0}$ can be characterized by specifying the lowest eigenvalue, called the conformal weight of the primary, and the generating function of the eigenvalue multiplicities. For a primary $p$ of conformal weight $\cw p$, the conformal character reads $$\chi_{p}\!\left(q\right)=q^{{\scriptscriptstyle \nicefrac{-c}{24}}}\sum_{n=0}^{\infty}d_{n}q^{n+\cw p}\label{eq:chardef}$$ where $d_{n}$ denotes the multiplicity of $n\!+\!\cw p$ as an eigenvalue of $L_{0}$ and $c$ the central charge of the model. One can show that the above (fractional) power series is absolutely convergent in the disk $\bigl|q\bigr|<1$, hence defines an analytic function there.
Besides characterizing the spectrum of $L_{0}$ in the irreducible modules, the conformal characters also provide the basic building blocks of the torus partition function. In the simplest case of diagonal theories, the torus partition function reads $$Z\!\left(\tau,\overline{\tau}\right)=\sum_{p}\bigl|\chi_{p}\!\left(\ex{\tau}\right)\bigr|^{2}\label{eq:partfun}$$ where $\tau$ denotes the modular parameter of the torus, and the sum runs over all primaries; more generally, the torus partition function is a sesquilinear combination of the conformal characters. Combining this observation with the invariance [@modinv] of the torus partition function under modular transformations i.e. transformations of the modular parameter $\tau$ that do not change the conformal equivalence class, one arrives at the conclusion that the modular group $\FC$ is represented on the linear span of the characters, i.e. for any $\smat abcd\!\in\!\FC$ there exists a unitary representation matrix $M\!=\!\rho\smat abcd$ such that $$\chi_{p}\!\left(\frac{a\tau\!+\!b}{c\tau\!+\!d}\right)=\sum_{s}M{}_{ps}\chi_{s}\!\left(\tau\right)\label{eq:transrule}$$
Two remarks are in order here: first, the modular representation is actually a *matrix* representation, meaning that each individual modular matrix element has an invariant meaning. This is particularly clear when considering Verlinde’s celebrated formula [@Verlinde1988] expressing the fusion rules of the theory in terms of modular matrix elements, or its various generalizations [@Moore-Seiberg; @Bantay2003c]. From a technical point of view, this means that the linear space $\msp$ affording the modular representation comes equipped with a distinguished basis $\mb\!=\!\left\{ \mbv_{p}\right\} $ labeled by the primaries, and a different choice of basis would correspond to a different theory.
The second observation is that the transformation rule Eq.(\[eq:transrule\]) does not always determine the modular representation matrices. The reason for this is that the conformal characters, as functions of the modular parameter $\tau$, are not necessarily linearly independent, i.e. there may exist nontrivial relations of the form $$\sum_{p}R_{p}\chi_{p}\!\left(\tau\right)=0\label{eq:charrel}$$ with coefficients $R_{p}$ independent of $\tau$. The existence of such nontrivial character relations is actually pretty common, e.g. the characters of charge conjugate primaries are automatically equal $$\chi_{\overline{p}}\!\left(\tau\right)=\chi_{p}\!\left(\tau\right)\label{eq:ccrel}$$ As a consequence of the character relations, the linear span $\csp$ of the characters is usually only a subspace of $\msp$, and the individual modular matrix elements cannot be determined from Eq.(\[eq:transrule\]), only suitable linear combinations of them. Actually, the example of charge conjugation is a good indication for the origin of such character relations: they are the reflections of (possibly hidden) global symmetries of the theory[^2].
To illustrate this last point, let us consider the orbifold line of $c\!=\!1$ theories [@Ginsparg1988]. It is well known that, at compactification radii for which $N\!=\!2r_{\mathtt{orb}}^{2}$ is an integer, these theories have exactly $N\!+\!7$ primary fields with conformal characters $$\begin{aligned}
\FI u\!\left(\tau\right) & =\frac{1}{2\eta\!\left(\tau\right)}\theta_{3}\!\left(2N\tau\right)\pm\sqrt{\frac{\eta}{2\theta_{2}}}\!\left(\tau\right)=\frac{1}{2\eta\!\left(\tau\right)}\left\{ \FJ 00{2N\tau}\pm\theta_{4}\!\left(2\tau\right)\right\} \nonumber \\
\chi_{k}\!\left(\tau\right) & =\frac{1}{\eta\!\left(\tau\right)}\FJ{\frac{k}{2N}}0{2N\tau}\textrm{ ~for }k\!=\!1,\ldots,N-1\nonumber \\
\FI{\phi}\!\left(\tau\right) & =\frac{1}{2\eta\!\left(\tau\right)}\theta_{2}\!\left(2N\tau\right)=\frac{1}{2\eta\!\left(\tau\right)}\FJ{\frac{1}{2}}0{2N\tau}\label{eq:atrels}\\
\FI{\sigma}\!\left(\tau\right) & =\frac{1}{2}\left\{ \sqrt{\frac{\eta}{\theta_{4}}}\!\left(\tau\right)+\sqrt{\frac{\eta}{\theta_{3}}}\!\left(\tau\right)\right\} =\frac{1}{2\eta\!\left(\tau\right)}\left\{ \theta_{2}\!\left(\frac{\tau}{2}\right)+\exi[-]{24}\theta_{2}\!\left(\!\frac{\tau\!+\!1}{2}\!\right)\right\} \nonumber \\
\FI{\tau}\!\left(\tau\right) & =\frac{1}{2}\left\{ \sqrt{\frac{\eta}{\theta_{4}}}\!\left(\tau\right)-\sqrt{\frac{\eta}{\theta_{3}}}\!\left(\tau\right)\right\} =\frac{1}{2\eta\!\left(\tau\right)}\left\{ \theta_{2}\!\left(\frac{\tau}{2}\right)-\exi[-]{24}\theta_{2}\!\left(\!\frac{\tau\!+\!1}{2}\!\right)\right\} \nonumber \end{aligned}$$ where $$\FJ ab{\tau}=\sum_{n\in\mathbb{Z}}\mathtt{e}^{\mathtt{i}\pi\tau\left(n-a\right)^{2}}\mathtt{e}^{-2\pi\mathtt{i}bn}\label{eq:thetadef}$$ and $$\eta\!\left(\tau\right)=q^{\frac{1}{24}}\prod_{n=1}^{\infty}\left(1-q^{n}\right)\label{eq:etadef}$$ denotes Dedekind’s eta function (with $q\!=\!\ex{\tau}$), while $$\begin{aligned}
{2}
\theta_{2} & =\FJ{\frac{1}{2}}0{\tau}~ & =2q^{\nicefrac{1}{8}}\prod_{n=1}^{\infty}\left(1-q^{n}\right)\left(1+q^{n}\right)^{2}\\
\theta_{3} & =\FJ 00{\tau}~ & =~\prod_{n=1}^{\infty}\left(1-q^{n}\right)\left(1+q^{n-\nicefrac{1}{2}}\right)^{2}\\
\theta_{4} & =\FJ 0{\frac{1}{2}}{\tau}~ & =~\prod_{n=1}^{\infty}\left(1-q^{n}\right)\left(1-q^{n-\nicefrac{1}{2}}\right)^{2}\end{aligned}$$ are the classical theta functions of Jacobi.
Let’s restrict our attention to the models with even $N$. Since charge conjugation is trivial in this case, the obvious character relations $$\begin{aligned}
\phi_{{\scriptscriptstyle -}} & =\phi_{{\scriptscriptstyle +}}\nonumber \\
\sigma_{{\scriptscriptstyle -}} & =\sigma_{{\scriptscriptstyle +}}\label{eq:ATcrel}\\
\tau_{{\scriptscriptstyle -}} & =\tau_{{\scriptscriptstyle +}}\nonumber \end{aligned}$$ must have a different origin: they are a manifestation of the dihedral $\mathbb{D}_{4}$ symmetry underlying these models [@Ginsparg1988], which follows from the fact that the orbifold line may be obtained as the conformal limit of Ashkin-Teller models, i.e. two Ising spins coupled locally via their energy density. Clearly, the transformations that flip each Ising spin separately, together with the one that exchanges the two, form a $\mathbb{D}_{4}$ symmetry group, explaining the above character relations. In case $N\!=\!4$ (corresponding to the 4-state Potts model) this symmetry is extended to a full $\mathbb{S}_{4}$, resulting in the extra character relations $$\begin{aligned}
\phi_{{\scriptscriptstyle \pm}} & =u_{{\scriptscriptstyle -}}\nonumber \\
\chi_{1} & =\sigma_{{\scriptscriptstyle \pm}}\label{eq:at4crel}\\
\chi_{3} & =\tau_{{\scriptscriptstyle \pm}}\nonumber \end{aligned}$$
Another interesting case is that of $N\!=\!16$, when the generic character relations Eq.(\[eq:ATcrel\]) get supplemented by $$\begin{aligned}
\chi_{8}-\phi_{{\scriptscriptstyle \pm}} & =u_{{\scriptscriptstyle -}}\nonumber \\
\chi_{2}+\chi_{14} & =\sigma_{{\scriptscriptstyle \pm}}\label{eq:at16crel}\\
\chi_{6}+\chi_{10} & =\tau_{{\scriptscriptstyle \pm}}\nonumber \end{aligned}$$ More generally, such extra character relations occur whenever $N$ is the square of an even integer, $N\!=\!\left(2n\right)^{2}$, when one has $$\begin{aligned}
\sum_{k=1}^{n-1}(-1)^{k-1}\chi_{4nk}-(-1)^{n}\phi_{{\scriptscriptstyle \pm}} & =u_{{\scriptscriptstyle -}}\nonumber \\
\sum_{k=0}^{\left[\frac{n-1}{2}\right]}\left\{ \chi_{n(8k+1)}+\chi_{n(8k+7)}\right\} & =\sigma_{{\scriptscriptstyle \pm}}\label{eq:atcrel}\\
\sum_{k=0}^{\left[\frac{n-1}{2}\right]}\left\{ \chi_{n(8k+3)}+\chi_{n(8k+5)}\right\} & =\tau_{{\scriptscriptstyle \pm}}\nonumber \end{aligned}$$ as a consequence of the general identity $$\sum_{k=0}^{N-1}\ex{\frac{kb}{N}}\FJ{a+\frac{k}{N}}0{\tau}=\FJ{-Na}{\frac{b}{N}}{\frac{\tau}{N^{2}}}\label{eq:thetaid}$$ valid for integer $b$ and $N$, as well as the theta relations[^3] $$\begin{aligned}
\theta_{4}\!\left(2\tau\right) & =\sqrt{\theta_{3}\!\left(\tau\right)\theta_{4}\!\left(\tau\right)}\nonumber \\
\theta_{2}\!\left(\frac{\tau}{2}\right) & =\sqrt{2\theta_{2}\!\left(\tau\right)\theta_{3}\!\left(\tau\right)}\label{eq:thetarel}\\
\theta_{2}\!\left(\!\frac{\tau\!+\!1}{2}\!\right) & =\mathtt{e}^{\frac{\mathtt{i}\pi}{16}}\sqrt{2\theta_{2}\!\left(\tau\right)\theta_{4}\!\left(\tau\right)}\nonumber \end{aligned}$$ The origin of these extra relations Eq.(\[eq:atcrel\]) may be traced back to the fact that the corresponding models may be constructed as dihedral orbifolds of the compactified boson at radius $r\!=\!\nicefrac{1}{\sqrt{2}}$ [@Ginsparg1988].
From a technical point of view, nontrivial character relations indicate that the modular representation $\rho$ is reducible. Indeed, as a consequence of the $\tau$ independence of the coefficients $R_{p}$ in Eq.(\[eq:charrel\]), the linear span $\csp$ of the characters (considered as a subspace of $\msp$) is invariant under $\rho$. In particular, this means that in order to fully characterize the modular properties of the characters, it is not enough to specify the matrix representation $\rho$, but one should amend this by a description of the invariant subspace $\csp$ (e.g. by specifying a basis of it). Formally, one could think that this last step can be avoided by directly reducing the modular representation to the invariant subspace $\csp$: after all, this subspace is the linear span of the conformal characters, thus it contains all the physically relevant information. But this is far from being true. For example, application of Verlinde’s formula [@Verlinde1988; @Moore-Seiberg], one of the cornerstones of the whole theory, necessitates the consideration of the full modular representation, with all individual matrix elements. Similarly, computation of Frobenius-Schur indicators [@Bantay1997a], or the application of the trace identities of [@Bantay2003c] require the knowledge of each matrix element separately.
Symmetric products and replication identities
=============================================
Consider a system made up of $n$ identical subsystems, each described by the same CFT $\mathcal{C}$. The whole system will be still conformally invariant, described by the $n$-fold tensor power of $\mathcal{C}$, and any permutation of the identical subsystems will leave the whole system invariant. Consequently, for any permutation group $\Omega\!<\!\sn n$ of degree $n$, one could consider the permutation orbifold[^4] $\mathcal{C}\wr\Omega$ obtained by orbifolding the tensor power by the twist group $\Omega$ [@Klemm-Schmidt; @Borisov-Halpern-Schweigert; @Bantay1998a]. Because of the universal nature of the action of $\Omega$, all relevant quantities (like correlation and partition functions, fusion rules, modular matrix elements, etc.) of $\mathcal{C}\!\wr\!\Omega$ may be expressed in terms of the relevant quantities of $\mathcal{C}$, namely as polynomial expressions of these quantities evaluated on suitable $n$-sheeted covering surfaces of the world sheet, see [@Bantay2001; @Bantay2002] for details. In particular, the conformal characters of the permutation orbifold are completely determined by those of $\mathcal{C}$ and the twist group $\Omega$ [@Bantay1998a]. We note that all relevant relations can be subsumed under a general group theoretic construct, the orbifold transform, described in detail in [@Bantay2008a].
A particularly interesting case is when the twist group $\Omega$ is maximal, i.e. when $\Omega$ is the full symmetric group $\sn n$ of degree $n$: the resulting permutation orbifold $\mathcal{C}\!\wr\!\sn n$ is called the $n$-th symmetric product of $\mathcal{C}$, and plays an important role in the description of second quantized strings [@elliptic_genera; @Dijkgraaf_disctors; @Bantay2003a]. The analysis of symmetric products is greatly simplified by the exponential identity [@Bantay2008a], a general combinatorial identity satisfied by the orbifold transform, which provides closed expressions for the characteristic quantities of symmetric products.
According to the general theory [@Bantay1998a], the conformal characters (evaluated at some specific modulus $\tau$) of the $n$-fold symmetric product $\mathcal{C}\wr\sn n$ may be expressed as polynomial expressions of the conformal characters of $\mathcal{C}$ evaluated on the different $n$-sheeted (unbranched) coverings of a torus with modulus $\tau$. But all theses coverings have genus $1$, hence each connected component is itself a torus of modulus $$\frac{a\tau+b}{d}\label{eq:covertau}$$ for suitable non-negative integers $a,b,d$ characterizing the relevant covering. The precise form of the polynomial expressions is irrelevant at this point, the only thing to note is that all possible coverings occur in the process. This means that a character relation of the symmetric product $\mathcal{C}\wr\sn n$ is nothing but a polynomial relation between quantities of the form $$\chi_{p}\!\left(\!\frac{a\tau+b}{d}\!\right)$$ We shall call such relations replication identities, because in the specific case of the Moonshine orbifold they yield precisely the replication formulas satisfied by the generalized Thompson-McKay series. It should be emphasized that the above notion of replication identities is pretty general, far from being confined to derivatives of the Moonshine module or to rational conformal models.
As explained in the previous section, character relations for a given CFT are usually reflections of outer symmetries relating different irreducible modules of the chiral algebra. Consequently, one may view replication identities as an indication to the existence of suitable symmetries of the higher symmetric products. To illustrate the above ideas, let us consider the Ising model, i.e. the Virasoro minimal model of central charge $c\!=\!\nicefrac{1}{2}$. In this case there are three primary fields, $\boldsymbol{0},~\boldsymbol{\epsilon}$ and $\boldsymbol{\sigma}$, of respective conformal weights $0,\nicefrac{1}{2}$ and $\nicefrac{1}{16}$, with conformal characters $$\begin{aligned}
\chi_{\boldsymbol{{\scriptscriptstyle 0}}} & =\frac{1}{2}\!\left(\sqrt{\frac{\theta_{3}}{\eta}}+\sqrt{\frac{\theta_{4}}{\eta}}~\right)\nonumber \\
\chi_{\boldsymbol{\epsilon}} & =\frac{1}{2}\!\left(\sqrt{\frac{\theta_{3}}{\eta}}-\sqrt{\frac{\theta_{4}}{\eta}}~\right)\label{eq:isingchars}\\
\chi_{\boldsymbol{\sigma}} & =\sqrt{\frac{\theta_{2}}{2\eta}}\nonumber \end{aligned}$$ Note that $$\begin{aligned}
{2}
\sqrt{\frac{\theta_{3}}{\eta}}= & ~{\displaystyle q^{\textrm{-}1/48}\prod_{n=0}^{\infty}}\left(1+q^{n+\frac{1}{2}}\right) & ~=\frac{\eta\!\left(\tau\right)^{2}}{\eta\!\left(\frac{\tau}{2}\right)\eta\!\left(2\tau\right)}\nonumber \\
\sqrt{\frac{\theta_{4}}{\eta}}= & ~q^{\textrm{-}1/48}\prod_{n=0}^{\infty}\left(1-q^{n+\frac{1}{2}}\right) & =\frac{\eta\!\left(\frac{\tau}{2}\right)}{\eta\!\left(\tau\right)}~~~~~~~~\label{eq:thetaprod}\\
\sqrt{\frac{\theta_{2}}{2\eta}}= & ~{\displaystyle q^{1/24}~\prod_{n=1}^{\infty}}\left(1+q^{n}\right) & =\frac{\eta\!\left(2\tau\right)}{\eta\!\left(\tau\right)}~~~~~~~~\nonumber \end{aligned}$$
The modular representation, characterized by the matrix $$S=\frac{1}{2}\left(\begin{array}{rrr}
1 & 1 & \sqrt{2}\\
1 & 1 & -\sqrt{2}\\
\sqrt{2} & -\sqrt{2} & 0
\end{array}\right)\label{eq:isingS}$$ is irreducible, hence has no non-trivial invariant subspace; consequently, the conformal characters of Ising are linearly independent.
As a consequence of the identities $$\begin{aligned}
\sqrt{\frac{\theta_{4}\!\left(2\tau\right)}{\eta\!\left(2\tau\right)}} & =\frac{\sqrt{\theta_{3}\!\left(\tau\right)\theta_{4}\!\left(\tau\right)}}{\eta\!\left(\tau\right)}\nonumber \\
\sqrt{\frac{\theta_{2}\!\left(\frac{\tau}{2}\right)}{\eta\!\left(\frac{\tau}{2}\right)}} & =\frac{\sqrt{\theta_{2}\!\left(\tau\right)\theta_{3}\!\left(\tau\right)}}{\eta\!\left(\tau\right)}\label{eq:thetarel2}\\
\exi[-]{24}\sqrt{\frac{\theta_{2}\!\left(\frac{\tau+1}{2}\right)}{\eta\!\left(\frac{\tau+1}{2}\right)}} & =\frac{\sqrt{\theta_{2}\!\left(\tau\right)\theta_{4}\!\left(\tau\right)}}{\eta\!\left(\tau\right)}\nonumber \end{aligned}$$ that follow easily from Eqs.(\[eq:thetarel\]), one gets that $$\begin{aligned}
\chi_{\boldsymbol{{\scriptscriptstyle 0}}}\!\left(2\tau\right)-\chi_{\boldsymbol{\epsilon}}\!\left(2\tau\right) & =\chi_{\boldsymbol{{\scriptscriptstyle 0}}}\!\left(\tau\right)^{2}-\chi_{\boldsymbol{\epsilon}}\!\left(\tau\right)^{2}\nonumber \\
\chi_{\boldsymbol{\sigma}}\!\left(\frac{\tau}{2}\right) & =\frac{\chi_{\boldsymbol{{\scriptscriptstyle 0}}}\!\left(\tau\right)\!+\!\chi_{\boldsymbol{\epsilon}}\!\left(\tau\right)}{2}\chi_{\boldsymbol{\sigma}}\!\left(\tau\right)\label{eq:Is2rels}\\
\chi_{\boldsymbol{\sigma}}\!\left(\!\frac{\tau\!+\!1}{2}\!\right) & =\exi{24}\frac{\chi_{\boldsymbol{{\scriptscriptstyle 0}}}\!\left(\tau\right)\!-\!\chi_{\boldsymbol{\epsilon}}\!\left(\tau\right)}{2}\chi_{\boldsymbol{\sigma}}\!\left(\tau\right)\nonumber \end{aligned}$$ These are prime examples of replication identities, involving values of characters on different covering surfaces. Consequently, they should be related to character relations, and ultimately to symmetries of symmetric products of the Ising model. Let’s see how this comes about!
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According to the general theory [@Bantay1998a], the 2-fold symmetric product of Ising has central charge $c\!=\!1$ (twice the central charge of the Ising model) and a total of $\frac{3\left(3+7\right)}{2}\!=\!15$ primary fields, whose conformal characters read $$\begin{aligned}
\chipq pq\!\left(\tau\right) & =\chi_{p}\!\left(\tau\right)\chi_{q}\!\left(\tau\right)~~~\enspace~\quad~\quad~~~\textrm{for }~p\neq q\nonumber \\
\tw u{\pm}p\!\left(\tau\right) & =\frac{1}{2}\left\{ \chi_{p}\!\left(\tau\right)^{2}\pm\chi_{p}\!\left(2\tau\right)\!\right\} \label{eq:Is2chars}\\
\tw t{\pm}p\!\left(\tau\right) & =\frac{1}{2}\left\{ \chi_{p}\!\left(\frac{\tau}{2}\right)\pm\mathtt{e}^{-\mathtt{i}\pi\left(\cw p-\nicefrac{1}{48}\right)}\chi_{p}\!\left(\!\frac{\tau\!+\!1}{2}\!\right)\!\right\} \nonumber \end{aligned}$$ for $p,q\!\in\!\left\{ \boldsymbol{0},\boldsymbol{\epsilon},\boldsymbol{\sigma}\right\} $, with $\cw p$ denoting the conformal weight of the primary $p$. By inspecting the $q$-expansions of these characters, one arrives at the character relations $$\begin{aligned}
\tw u-{\boldsymbol{{\scriptscriptstyle 0}}}\!\left(\tau\right) & =\tw u-{\boldsymbol{\epsilon}}\!\left(\tau\right)\nonumber \\
\chipq{\boldsymbol{0}}{\boldsymbol{\sigma}}\!\left(\tau\right) & =\tw t+{\boldsymbol{{\scriptscriptstyle \sigma}}}\!\left(\tau\right)\label{eq:Is2charrels}\\
\chipq{\boldsymbol{\boldsymbol{\epsilon}}}{\boldsymbol{\sigma}}\!\left(\tau\right) & =\tw t-{\boldsymbol{{\scriptscriptstyle \sigma}}}\!\left(\tau\right)\nonumber \end{aligned}$$ which reduce to $$\begin{aligned}
\chi_{\boldsymbol{{\scriptscriptstyle 0}}}\!\left(\tau\right)^{2}-\chi_{\boldsymbol{{\scriptscriptstyle 0}}}\!\left(2\tau\right) & =\chi_{\boldsymbol{\epsilon}}\!\left(\tau\right)^{2}-\chi_{\boldsymbol{\epsilon}}\!\left(2\tau\right)\nonumber \\
\chi_{\boldsymbol{\sigma}}\!\left(\frac{\tau}{2}\right)+\exi[-]{24}\chi_{\boldsymbol{\sigma}}\!\left(\!\frac{\tau\!+\!1}{2}\!\right) & =\chi_{\boldsymbol{{\scriptscriptstyle 0}}}\!\left(\tau\right)\chi_{\boldsymbol{\sigma}}\!\left(\tau\right)\label{eq:Is2relsbis}\\
\chi_{\boldsymbol{\sigma}}\!\left(\frac{\tau}{2}\right)-\exi[-]{24}\chi_{\boldsymbol{\sigma}}\!\left(\!\frac{\tau\!+\!1}{2}\!\right) & =\chi_{\boldsymbol{\epsilon}}\!\left(\tau\right)\chi_{\boldsymbol{\sigma}}\!\left(\tau\right)\nonumber \end{aligned}$$ upon taking into account the expressions Eqs.(\[eq:Is2chars\]). Clearly, these are equivalent to the replication identities Eqs.(\[eq:Is2rels\]), and we see that, indeed, the latter are nothing but character relations for the second symmetric product. What remains to do is to find out which symmetries are responsible for this.
Since the full moduli space of $c\!=\!1$ conformal models is known, it is a simple matter to identify the second symmetric product of the Ising model: it lies on the orbifold line at radius $r_{\mathtt{orb}}\!=\!2$, i.e. at $N\!=\!8$. Furthermore, it is an easy exercise to identify the respective primary fields, in particular one gets
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But, as explained in the previous section, this model exhibits a dihedral $\mathbb{D}_{4}$ symmetry as a consequence of its Ashkin-Teller origin, leading to the character relations Eqs.(\[eq:ATcrel\]), which, taking into account the field identifications Eq.(\[eq:at8fieldids\]), yield precisely Eqs.(\[eq:Is2charrels\]). In this case the analysis of the relevant symmetries is relatively easy thanks to the identification of the symmetric product as an Ashkin-Teller model, but the underlying idea should be clear.
The third symmetric product of Ising has central charge $c\!=\!\nicefrac{3}{2}$, and can be identified with an isolated $N\!=\!1$ superconformal model [@Cappelli2001], which has a total of $49$ primaries. This superconformal model has $9$ independent character relations, but it turns out that all of these follow from the replication identities (\[eq:Is2rels\]). New replication identities could come from the character relations of the 4-fold symmetric product: unfortunately, this latter model of central charge $c\!=\!2$ has $171$ different primary fields, with $59$ independent relations between their characters, whose connection to the symmetries of the model is far from being easy to determine.
outlook and conclusion
======================
Trying to find a physics interpretation of the Hauptmodul property of Monstrous Moonshine, we considered the question of character relations and replication identities in Conformal Field Theory. Character relations play an important role in understanding the structure of specific models, and should be viewed as one of the basic ingredients (besides the modular representation) to fully specify their modular properties, while replication identities are nothing but character relations for symmetric products. Since character relations can be traced back ultimately to suitable symmetries of the model under study, replication identities should correspond to symmetries of its symmetric products.
The Hauptmodul property of Monstrous Moonshine is a consequence of the replication identities obeyed by the (generalized) Thompson-McKay series. Based on this, we suggest that it is actually a manifestation of the inherent symmetries of second quantized string propagation on the Moonshine orbifold, the string background obtained by orbifolding the Moonshine module by the Monster, and whose primary characters are linear combinations of the Thompson-McKay series. Let us stress that this approach does not give us an alternate proof of the Hauptmodul property, just a possible physics interpretation for it. However, if correct, it could have interesting consequences even from a purely mathematical perspective, e.g. providing suitably generalized versions of the replication identities for higher genus analogues of the Thompson-McKay series.
While the arguments leading to the above could seem straightforward, the actual implementation, i.e. the identification of the relevant symmetries might be far from simple. The proliferation of character relations in higher symmetric products makes the analysis pretty difficult even for the Ising model, and one should expect worse in more complicated cases. But there are various arguments suggesting that, notwithstanding all computational difficulties involved, the identification of the relevant symmetries might be nevertheless carried out.
The first observation is that, for any two permutation groups $\Omega_{1}$ and $\Omega_{2}$ such that $\Omega_{1}$ is a subgroup of $\Omega_{2}$, the character relations of the $\Omega_{1}$ permutation orbifold are inherited by the $\Omega_{2}$ permutation orbifold. This is actually the reason why it is sufficient to look only at symmetric products when considering replication identities. Combining this with the obvious embeddings of wreath products into symmetric groups and the transitivity property of permutation orbifolds [@Bantay1998a; @Bantay2002], one can see that many of the replication identities of a given degree are trivial consequences of lower degree ones, and in particular of character relations, which are nothing but replication identities of degree one. As a result, it is enough to understand the ’primitive’ identities that do not follow from identities coming from lower degrees, and these are clearly much less abundant, hopefully forming a set that can be dealt with.
The second point is that one does not even need the precise identification of all of the symmetries responsible for the primitive identities, it is enough to identify only a generating set, which can turn out to be pretty small. Since all replication identities for Moonshine are known, this should simplify the job to a large extent. Of course, even in case of a few generators the actual identification of the relevant symmetries could require some ingenuity, but one could expect that special properties of the Monster and the Moonshine module should allow the use of ad hoc techniques to solve this problem: after all, such considerations allow the determination of the character table of the Monster (with cca. $10^{54}$ elements), while a brute force computation for a group with only a few million of elements is already a time and resource consuming task.
Even if the above program can be completed and all relevant symmetries responsible for the replication identities of the Moonshine orbifold identified, there would still remain the question of what is so special about this particular model. After all, while non-trivial replication identities are not uncommon for rational models, they are usually not restrictive enough to force the chiral characters to be actually Hauptmoduls; this seems to be connected with a particularly high degree of symmetry inherent to symmetric products of the Moonshine orbifold. It would be interesting to find out other models that show similar features, and whether this could be linked with other approaches [@Duncan2010; @Tuite1995] to the Hauptmodul property of Moonshine. We believe that further elaboration of these issues could lead to a better understanding of the whole subject.
[^1]: Work supported by grant OTKA 79005.
[^2]: Indeed, character relations, as linear relations between suitable (chiral) correlators, may be considered as Ward identities related to some global symmetry. Of course, the precise nature of the relevant symmetry might be pretty hard to pin down.
[^3]: An interesting consequence of Eq.(\[eq:atcrel\]) is that in this case the characters of the orbifold can be expressed as linear combinations of the characters of the original theory, i.e. the compactified boson at radius $r\!=\!\sqrt{2}n$.
[^4]: The origin of the wreath product notation for permutation orbifolds is explained in [@Bantay1998a].
|
---
abstract: 'We investigate the production of the newly found pentaquark exotic baryon $\Xi_5$ in the $\bar{K}N\to K\Xi_5$ and the $\bar{K}N\to K^{*}\Xi_5$ reactions at the tree level. We consider both positive- and negative-parities of the $\Xi_5$. The reactions are dominated by the $s$- and the $u$-channel processes, and the resulting cross sections are observed to depend very much on the parity of $\Xi_5$ and on the type of form factor. We have seen that the cross sections for the positive-parity $\Xi_5$ are generally about a hundred times larger than those of the negative-parity one. This large difference in the cross sections will be useful for further study of the pentaquark baryons.'
author:
- 'Seung-il Nam'
- Atsushi Hosaka
- 'Hyun-Chul Kim'
title: |
Production of the Pentaquark Exotic Baryon $\Xi_5$ in $\bar{K}N$ Scattering:\
$\bar{K}N\to K\Xi_5$ and $\bar{K}N\to K^{*}\Xi_5$
---
introduction
============
The experimental observation of the $\Theta^+$ performed by the LEPS collaboration at SPring-8 [@Nakano:2003qx], which is motivated by Diakonov [*et al.*]{} [@Diakonov:1997mm], has paved the way for intensive studies on the exotic five-quark baryon states, also known as [*pentaquarks*]{}, experimentally [@experiment] as well as theoretically [@Praszalowicz:2003tc; @Jaffe:2003sg; @Hosaka:2003jv; @Kondo:2004rn; @Stancu:2003if; @Glozman:2003sy; @Huang:2003we; @Sugiyama:2003zk; @Zhu:2003ba; @Sasaki:2003gi; @Csikor:2003ng; @Liu:2003rh; @Hyodo:2003th; @Oh:2003kw; @nam1; @nam2; @nam3; @nam4; @Thomas:2003ak; @Hanhart:2003xp; @Liu:2003zi; @Zhao:2003gs; @Yu:2003eq; @Kim:2003ay; @Huang:2003bu; @Liu:2003ab; @Li:2003cb]. As a consequence of the finding of the $\Theta^{+}$, the existence of other pentaquark baryons, such as the $N_5$, $\Sigma_5$, and $\Xi_5$, which have also been predicted theoretically, is anticipated.
The NA49 [@Alt:2003vb] collaboration reported a signal for the pentaquark baryon $\Xi_5$, which was also predicted theoretically. The $\Xi_5$ was found to have a mass of $1862\,{\rm MeV}$, a strangeness $S = -2$, and an isospin $I = 3/2$. It is characterized by its narrow decay width of $\sim
18\,{\rm MeV}$, like that of the $\Theta^+$. However, we have thus far no concrete experimental evidence for its quantum numbers such as [*spin*]{} and [*parity*]{}.
As for the parity of the $\Theta^+$, a consensus has not been reached. For example, the chiral soliton model [@Diakonov:1997mm; @Praszalowicz:2003tc], the diquark model [@Jaffe:2003sg], the chiral potential model [@Hosaka:2003jv], and constituent quark models with spin-flavor interactions [@Stancu:2003if; @Glozman:2003sy; @Huang:2003we] prefer a positive-parity for the $\Theta^+$ whereas the QCD sum rule approach [@Sugiyama:2003zk; @Zhu:2003ba] [^1] and the quenched lattice QCD [@Sasaki:2003gi; @Csikor:2003ng] have supported a negative-parity. In the meanwhile, various reactions for $\Theta^{+}$ production [@Liu:2003rh; @Hyodo:2003th; @Oh:2003kw; @nam1; @nam2; @nam3; @nam4; @Thomas:2003ak; @Hanhart:2003xp; @Liu:2003zi; @Zhao:2003gs; @Yu:2003eq; @Kim:2003ay; @Huang:2003bu; @Liu:2003ab; @Li:2003cb] have been investigated, where the determination of the parity of $\Theta^+$ has been emphasized. In many cases, the total cross-sections of the positive-parity $\Theta^+$ production is typically about ten times larger than those of the negative-parity one. Liu [*et al.*]{} [@Liu:2003dq] evaluated the $\gamma N\to K\Xi_5$ reactions, assuming the positive-parity $\Xi_5$ and its spin $J=1/2$. However, since the parity of the $\Xi_5$ is not known yet, it is worthwhile studying the dynamics of $\Xi_5$ production with two different parities taken into account.
However, we note that negative results for the pentaquark baryons have emerged recently. Especially, the CLAS collaboration at Jefferson laboratory could not see any obvious evidence for the $\Theta^+$ pentaquark, which was expected to have a peak at about $1530\sim1540$ MeV in the reaction $\gamma p \to \bar K^0 \Theta^+$ [@DeVita:2005CLAS]. Moreover, the existence of an $S=-2$ penataquark, such as the $\Xi_5$ or the charmed pentaquark ($\Theta^+_c$), has not been completely confirmed yet.
In the present work, nonetheless, for the unclear status of the pentaquark, we want to investigate the $\Xi_5$ production from the $\bar{K}N\to K\Xi_5$ and the $\bar{K}N\to
K^{*}\Xi_5$ reactions. Due to the exotic strangeness quantum number the $\Xi_5$ has, the reaction process at the tree level becomes considerably simplified. This is a very specific feature of the process containing the exotic strangeness quantum number. We will follow the same framework as in Refs. [@Oh:2003kw; @nam1; @nam2; @nam3; @nam4; @Zhao:2003gs; @Yu:2003eq]. We assume that the spin of the $\Xi_5$ is $1/2$ [@Liu:2003dq]. Then, we estimate the total and the differential cross-sections for the production of $\Xi_5$ with positive and negative parities.
This paper is organized as follows: In Section II, we define the effective Lagrangians and construct the invariant amplitudes. In Section III, we present the numerical results for the total and the differential cross-sections for both positive- and negative-parity $\Xi_5$. Finally, in Section IV, we briefly summarize our discussions.
Effective Lagrangians and amplitudes
====================================
We study the reactions $\bar{K}N\to K\Xi_5$ and $\bar{K}N\to K^{*}\Xi_5$ by using an effective Lagrangian at the tree level of Born diagrams. The reactions are schematically presented in Fig. \[nmset00\], where we define the four momenta of each particle for the reactions by $p_{1, \cdots 4}$. There is no $t$-channel contribution because strangeness-two ($S=2$) mesons do not exist. As discussed in Ref. [@Oh:2003fs], we do not include the ${B}_{\bar{10}}M_{8}B_{10}$ coupling because it is forbidden in exact SU(3) flavor symmetry. Hence, the interaction Lagrangians can be written as $$\begin{aligned}
\mathcal{L}_{KN\Sigma}&=&ig_{KN\Sigma}\bar{\Sigma}
\gamma_{5}KN\,+\,{\rm (h.c.)},
\nonumber \\
\mathcal{L}_{K\Sigma\Xi_5}&=&ig_{K\Sigma\Xi_5}\bar{\Xi}_5
\Gamma_{5}K\Sigma\,+\,{\rm (h.c.)},
\nonumber \\
\mathcal{L}_{K^{*}N\Sigma}&=&g_{K^{*} N\Sigma}\bar{\Sigma}
\gamma_{\mu}K^{*\mu}N\,+\,{\rm (h.c.)},
\nonumber \\
\mathcal{L}_{K^{*}\Sigma\Xi_5}&=&g_{K^{*}\Sigma\Xi_5}\bar{\Xi}_5
\gamma_{\mu}\hat{\Gamma}_{5}K^{*\mu}\Sigma\,+\,{\rm (h.c.)},
\label{lagrangians} \end{aligned}$$ where $\Sigma$, $\Xi_5$, $N$, $K$, and $K^{*}$ denote the corresponding fields for the octet $\Sigma$, the antidecuplet $\Xi_5$, the nucleon, the pseudo-scalar $K$, and the vector $K^{*}$, respectively. The isospin operators are dropped because we treat the isospin states of the fields explicitly. We define $\Gamma_{5} =
\gamma_{5}$ for the positive-parity $\Xi_5$ whereas $\Gamma_{5} = {\bf 1}_{4\times4}$ for the negative-parity one. $\hat{\Gamma}_{5}$ is also defined by $\Gamma_{5}\gamma_{5}$ for the vector meson $K^{*}$.
![Born diagrams, $s$– (left) and $u$–channels (right) for $\Xi_5$ productions[]{data-label="nmset00"}](paper6f0.eps){width="10.0cm"}
The values of the coupling constants $g_{KN\Sigma}$ and $g_{K^* N
\Sigma}$ are taken from the new Nijmegen potential [@stokes] as $g_{KN\Sigma} = 3.54$ and $g_{K^{*}N\Sigma} = -2.99$ whereas we assume ${\rm SU(3)}$ flavor symmetry for $g_{K \Sigma
\Xi_5}$ so that we obtain the relation $g_{K \Sigma \Xi_5} = g_{KN\Theta}$ [@Oh:2003fs]. Employing the decay width $\Gamma_{\Theta \to KN} = 15$ MeV and $M_\Theta = 1540$ MeV, we obtain $g_{KN\Theta} = g_{K \Sigma \Xi_5} = 3.77\, (0.53)$ for the positive (negative) parity. The remaining one, $g_{K^*\Sigma\Xi_5}$, is not known, which we will discuss in the next section.
The invariant scattering amplitude for $\bar{K}N\to K\Xi_5$ can be written as $$i\mathcal{M}_{x,K} =
ig_{K\Sigma \Xi_5}g_{KN\Sigma}F^{2}_{x}(q^2)
\bar{u}(p_4)\Gamma_{5}\frac{
{q}_{x}+M_{\Sigma}}
{q^{2}_{x}-M^{2}_{\Sigma}}\gamma_{5}u(p_2)\, ,
\label{amplitudes1}$$ where $x$ labels either the $s$-channel or the $u$-channel, and the corresponding momenta are $q_{s}=p_{1}+p_{2}$ and $q_{u}=p_{2}-p_{3}$. For $\bar{K}N\to K^{*}\Xi_5$, we have $$\begin{aligned}
i\mathcal{M}_{s,K^{*}}&=&
g_{K^{*}\Sigma \Xi_5}g_{KN \Sigma}
F^{2}_{s}(q^2)\bar{u}(p_4)\rlap{/}{\epsilon}\hat{\Gamma}_{5}
\frac{\rlap{/}{q}_{s}+M_{\Sigma}}{q^{2}_{s}-M^{2}_{\Sigma}}\gamma_{5}u(p_2),\nonumber\\
i\mathcal{M}_{u,K^{*}}&=&
g_{K^{*}N\Sigma}g_{K\Sigma\Xi_5}F^{2}_{u}(q^2)\bar{u}(p_4){\Gamma}_{5}
\frac{\rlap{/}{q}_{u}+M_{\Sigma}}{q^{2}_{u}-M^{2}_{\Sigma}}\rlap{/}{\epsilon}u(p_2).
\label{amplitudes2} \end{aligned}$$ As indicated in Eq. (\[amplitudes1\]), the coupling constants are commonly factored out for the $s$- and the $u$-channels in the $K$ production. Therefore, there is no ambiguity due to the sign of the coupling constants. On the contrary, there is such an ambiguity due to the unknown sign of $g_{K^*\Sigma\Xi_5}$ in the case of $K^*$ production,
Since the baryon has an extended structure, we need to introduce a form factor. We employ the form factor [@nam3] $$F_{1}(x) =
\frac{\Lambda^{2}_{1}}{\sqrt{\Lambda^{4}_{1}+(x-M^{2}_{\Sigma})^{2}}}
\label{ff1}$$ in such a way that the singularities appearing in the pole diagrams can be avoided. Here, $\Lambda_{1}$ and $M_{\Sigma}$ stand for the cutoff parameter and the $\Sigma$ mass, respectively. We set the cutoff parameter $\Lambda_{1} = 0.85\,{\rm GeV}$ as in Ref. [@nam3]. This value was used to reproduce the cross sections of $\gamma p
\to K^{+}\Lambda$. In order to verify the dependence of the form factor, we consider also the three-dimensional form factor $$\begin{aligned}
F_{2}(\vec{q}^{2}) = \frac{\Lambda^{2}_{2}}{\Lambda^{2}_{2}+|\vec{q}^{2}|},
\label{ff2}\end{aligned}$$ where $\vec{q}$ denotes the three momentum of the external meson. As for the cutoff parameter, we set $\Lambda_{2} = 0.5\,{\rm GeV}$, which was deduced from the $\pi N \to K\Lambda$ reaction [@Liu:2003rh].
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![Energy dependence of the squared form factors $F^{2}_1(s)$, $F^{2}_1(u)$ and the $F^{2}_2(\vec q^2)$ (a), and angular dependence of the squared form factor $F^{2}_{1}$ for the $s$- and the $u$-channels at three CM energies. The types of curves are explained by the labels in the figure.[]{data-label="nmset0"}](paper6f5.eps "fig:"){width="7.5cm"} ![Energy dependence of the squared form factors $F^{2}_1(s)$, $F^{2}_1(u)$ and the $F^{2}_2(\vec q^2)$ (a), and angular dependence of the squared form factor $F^{2}_{1}$ for the $s$- and the $u$-channels at three CM energies. The types of curves are explained by the labels in the figure.[]{data-label="nmset0"}](paper6f7.eps "fig:"){width="7.5cm"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
In the left panel (a) of Fig. \[nmset0\], we show the dependence of the two form factors $F_1$ and $F_2$ on the CM energy while in the left panel (b) the angular dependence of the $F_1$ form factor is drawn. The $F_2$ form factor does not have any angular dependence. Obviously, they show very different behaviors. For instance, the $F_2$ decreases much faster than $F_1$ as the center-of-mass (CM) energy grows. The form factor $F_{1}$ in the $u$-channel shows a strong enhancement in a backward direction as the CM energy increases. As we will see, this feature has a great effect on the angular dependence of the differential cross-sections.
Numerical results
=================
$\bar{K}N\to K\Xi_5$
--------------------
In this subsection, we discuss the results for the reaction $\bar{K}N\to K\Xi_5$. Due to isospin symmetry, we can verify that the two possible reactions $\bar{K}^{0}p\to
K^{0}\Xi^{+}_5$ and ${K}^{-}n\to K^{+}\Xi^{--}_5$ are exactly the same in the isospin limit. In Fig. \[nmset1\], we present the total and the differential cross-sections in the left and the right panels, respectively. The average values of the total cross-sections are $\sigma\sim 2.6\,\mu b$ with the $F_{1}$ form factor and $\sigma\sim 1.5\,\mu b$ with the $F_{2}$ in the energy range $E_{CM}^{\rm th} = 2.35 \, {\rm GeV}
\le
E_{CM}\le 3.35\,{\rm GeV}$ (from the threshold to the point of 1 GeV larger). Though the average total cross-sections for the different form factors are similar in order of magnitude, the energy and the angular dependences are very different from each other. They are largely dictated by the form factor, as shown in Fig. \[nmset0\]. The angular distributions are drawn in the right panel (b) of Fig. \[nmset1\], where $\theta$ represents the scattering angle between the incident and the final kaons in the CM system. We show the results at $E_{CM} = 2.4$, $2.6$, and $2.8$ GeV. As shown there, when $F_1$ is used, the backward production is strongly enhanced while the cross sections are almost flat apart from a tiny increase in the backward region, when $F_2$ is employed. Note that the angular dependence of the latter is the same as that of the bare cross section without the form factor.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![ Cross sections for production of the positive parity $\Xi_5$ in the reaction $\bar{K}^{0}p\to K^{0}\Xi^{+}_5$. (a) The left panel shows the total cross-sections as functions of the center-of-mass energy $E_{\rm CM}$. (b) The right panel shows the differential cross-sections as functions of the scattering angle $\theta$ for incident energies $E_{\rm CM} = 2.4$, $2.6$, and $2.8$ GeV. In both cases, results using the form factors $F_1$ and $F_2$ are shown as indicated by the labels in the figures. []{data-label="nmset1"}](paper6f1.eps "fig:"){width="7.5cm"} ![ Cross sections for production of the positive parity $\Xi_5$ in the reaction $\bar{K}^{0}p\to K^{0}\Xi^{+}_5$. (a) The left panel shows the total cross-sections as functions of the center-of-mass energy $E_{\rm CM}$. (b) The right panel shows the differential cross-sections as functions of the scattering angle $\theta$ for incident energies $E_{\rm CM} = 2.4$, $2.6$, and $2.8$ GeV. In both cases, results using the form factors $F_1$ and $F_2$ are shown as indicated by the labels in the figures. []{data-label="nmset1"}](paper6f2.eps "fig:"){width="7.5cm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![ Cross sections for production of the negative parity $\Xi_5$ in the reaction $\bar{K}^{0}p\to K^{0}\Xi^{+}_5$. Notations are the same as in Fig. \[nmset1\].[]{data-label="nmset2"}](paper6f3.eps "fig:"){width="7.5cm"} ![ Cross sections for production of the negative parity $\Xi_5$ in the reaction $\bar{K}^{0}p\to K^{0}\Xi^{+}_5$. Notations are the same as in Fig. \[nmset1\].[]{data-label="nmset2"}](paper6f4.eps "fig:"){width="7.5cm"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
In Fig. \[nmset2\], we plot the total and differential cross-sections for the negative-parity $\Xi_5$. The energy dependence of the total cross section looks similar to that for the positive-parity one. We find that $\sigma\sim 26\,nb$ for $F_{1}$ and $\sim 12\,nb$ for $F_{2}$ in average for the CM energy region $E_{CM}^{\rm th}\,\le\,E_{CM}\,\le\,3.35\,{\rm GeV}$. We see that the total cross-sections are almost a hundred times smaller than that for the positive-parity $\Xi_5$. The difference between the results of the two parities is even more pronounced than in the previously investigated reactions, such as $\gamma N$, $K N$, and $N N$ scattering [@Liu:2003rh; @Hyodo:2003th; @Oh:2003kw; @nam1; @nam2; @nam3; @nam4; @Liu:2003zi; @Zhao:2003gs; @Yu:2003eq; @Kim:2003ay; @Huang:2003bu; @Liu:2003ab; @Li:2003cb], where typically the difference was about an order of ten. In the present reaction, the interference between the $s$- and the $u$-channels becomes important, in addition to the kinematical effect in the p-wave coupling for the positive-parity (but not in the s-wave for the negative-parity), which is proportional to $\vec
\sigma \cdot \vec q$ and enhance the amplitude at high momentum transfers. In the case of the positive parity, the two terms which are kinematically enhanced are interfered constructively, while for the negative-parity $\Xi_5$, the relatively small amplitudes without the enhancement factor is done destructively. These two effects are simultaneously responsible for the large difference in the cross sections.
In the right panel (b) of Fig. \[nmset2\], the angular distributions for the production of the negative-parity $\Xi_5$ are plotted. Here, the angular dependence changes significantly as compared with the positive-parity case. When the form factor $F_2$ is used, forward scattering significantly increases because the bare amplitude shows an enhancement in the forward direction. When using $F_1$, however, due to its strong enhancement in the backward direction, the cross sections get quite larger in the backward direction, except for those in the vicinity of the threshold, [*i.e.*]{}, $E_{CM} \le 2.45$ GeV.
$\bar{K}N\to K^{*}\Xi_5$
------------------------
In this subsection, we discuss the $K^*$ production. As explained in the previous section, the appearance of the coupling constant $g_{K^* \Sigma \Xi_5}$ raises the problem of the relative sign in the amplitude. First, we briefly discuss possible relations to determine the magnitude of the $g_{K^* \Sigma \Xi_5}$ coupling. If we use the SU(3) relation this coupling may be set equal to $g_{K^* N \Theta}$. There are several discussions on the $g_{K^* N \Theta}$ coupling. For example, a small value of the $g_{K^* N \Theta}$ was chosen according to the relation $g_{K^{*}N\Theta}/g_{KN\Theta} = 1/2$ as inferred from a phenomenological study of the hyperon coupling constants [@Janssen:2001wk], while in the quark model, the decay of the pentaquark states predicts a positive parity $\Theta^+$ by using the relation $g_{K^{*}N\Theta}/g_{KN\Theta} =\sqrt{3}$ [@Close:2004tp]. In the meanwhile, we find $g_{K^{*}N\Theta}/g_{KN\Theta} =1/\sqrt{3}$ for the negative parity. Since we are not able to determine the sign of the coupling constant in this study, we will present the results for four different cases: $g_{K^* \Sigma \Xi_5} = \pm \sqrt{3} g_{KN\Theta} = \pm 6.53$ and $g_{K^* \Sigma \Xi_5} = \pm 1/2 g_{KN\Theta} = \pm 1.89$ for positive parity, and $g_{K^* \Sigma \Xi_5} = \pm \sqrt{3} g_{KN\Theta} = \pm 0.91$ and $g_{K^* \Sigma \Xi_5} = \pm 1/2 g_{KN\Theta} = \pm 0.27$ for negative parity.
Figures \[nmset3\] and \[nmset4\] show the total and the differential cross-sections for the positive parity $\Xi_5$ with the $F_1$ and the $F_2$ form factors, respectively. We present the results with the four different coupling constants for the total cross-sections while for the differential cross-sections, we present those with the two positive coupling constants at three different energies, $E_{CM} = 2.8$, $3.0$, and $3.2$ GeV. The results for the negative coupling constants are qualitatively similar to each other. When the $F_1$ form factor is used, the results do not depend on the choice of $g_{K^* \Sigma \Xi_5}$ because the $u$-channel is the dominant component. In this case, similar discussions can be made as in the previous case of the $K$ production reaction. However, when $F_2$ is used, the results are very sensitive to the sign of $g_{K^* \Sigma \Xi_5}$, which determines whether the $s$- and the $u$-channels interfere constructively or not.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![Cross sections for production of the positive parity $\Xi_5$ in the reaction $\bar{K}^{0}p\to K^{*0}\Xi^{+}_5$ with the $F_1$ form factor employed. The total cross-sections in (a) are calculated for four different $g_{K^* \Sigma \Xi_5}$ coupling constants as indicated by the labels. The angular distributions in (b) are calculated for three different CM energies and two different $g_{K^* \Sigma \Xi_5}$ coupling constants as indicated by the labels.[]{data-label="nmset3"}](paper6f8.eps "fig:"){width="7.5cm"} ![Cross sections for production of the positive parity $\Xi_5$ in the reaction $\bar{K}^{0}p\to K^{*0}\Xi^{+}_5$ with the $F_1$ form factor employed. The total cross-sections in (a) are calculated for four different $g_{K^* \Sigma \Xi_5}$ coupling constants as indicated by the labels. The angular distributions in (b) are calculated for three different CM energies and two different $g_{K^* \Sigma \Xi_5}$ coupling constants as indicated by the labels.[]{data-label="nmset3"}](paper6f10.eps "fig:"){width="7.5cm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![Cross sections for production of the positive parity $\Xi_5$ in the reaction $\bar{K}^{0}p\to K^{*0}\Xi^{+}_5$ with the $F_2$ form factor employed. For notations, see the caption of Fig. \[nmset3\].[]{data-label="nmset4"}](paper6f9.eps "fig:"){width="7.5cm"} ![Cross sections for production of the positive parity $\Xi_5$ in the reaction $\bar{K}^{0}p\to K^{*0}\Xi^{+}_5$ with the $F_2$ form factor employed. For notations, see the caption of Fig. \[nmset3\].[]{data-label="nmset4"}](paper6f11.eps "fig:"){width="7.5cm"}
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Fig. \[nmset5\] and Fig. \[nmset6\] show the results for the negative-parity $\Xi_5$ with the $F_1$ and the $F_2$ form factors used, respectively. Similar discussions apply for this case as for the positive parity case, but the values of cross sections are reduced by about a factor of a hundred.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![ Cross sections for production of the negative parity $\Xi_5$ in the reaction $\bar{K}^{0}p\to K^{*0}\Xi^{+}_5$ with the $F_1$ form factor employed. The total cross-sections in (a) are calculated for four different $g_{K^* \Sigma \Xi_5}$ coupling constants as indicated by the labels. The angular distributions in (b) are calculated for three different CM energies and two different $g_{K^* \Sigma \Xi_5}$ coupling constants as indicated by the labels.[]{data-label="nmset5"}](paper6f12.eps "fig:"){width="7.5cm"} ![ Cross sections for production of the negative parity $\Xi_5$ in the reaction $\bar{K}^{0}p\to K^{*0}\Xi^{+}_5$ with the $F_1$ form factor employed. The total cross-sections in (a) are calculated for four different $g_{K^* \Sigma \Xi_5}$ coupling constants as indicated by the labels. The angular distributions in (b) are calculated for three different CM energies and two different $g_{K^* \Sigma \Xi_5}$ coupling constants as indicated by the labels.[]{data-label="nmset5"}](paper6f14.eps "fig:"){width="7.5cm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![ Cross sections for production of the negative parity $\Xi_5$ in the reaction $\bar{K}^{0}p\to K^{*0}\Xi^{+}_5$ with the $F_2$ form factor employed. For notations, see the caption of Fig. \[nmset5\]. []{data-label="nmset6"}](paper6f13.eps "fig:"){width="7.5cm"} ![ Cross sections for production of the negative parity $\Xi_5$ in the reaction $\bar{K}^{0}p\to K^{*0}\Xi^{+}_5$ with the $F_2$ form factor employed. For notations, see the caption of Fig. \[nmset5\]. []{data-label="nmset6"}](paper6f15.eps "fig:"){width="7.5cm"}
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Summary and Discussion
======================
We have studied the production of the pentaquark exotic baryon $\Xi_5\,({\rm mass} = 1862\,{\rm MeV},\,I = 3/2\,S = -2,\,{\rm spin} =
1/2\,({\rm assumed}))$ in the reactions $\bar{K}N\to K\Xi_5$ and $\bar{K}N\to K^{*}\Xi_5$. We have employed two different phenomenological form factor Eqs. (\[ff1\]) and (\[ff2\]), with appropriate parameters for the coupling strengths and the cutoff parameters. In the present reactions, since two units of strangeness are transferred, only $s$- and $u$- channel diagrams are allowed at the tree level. On one hand, this fact simplifies the reaction mechanism and, hence, the computation. Furthermore, there is no ambiguity in the relative signs of coupling constants for the case of $K$ production. On the other hand, the cross sections strongly depend on the choice of form factors. In fact, Fig. \[nmset1\]$\sim$\[nmset6\] show that we have found a rather different energy and angular dependence when using different form factors. At this moment, it is difficult theoretically to say which is better. Nevertheless, it would be useful to summarize the present result for the total cross-sections in Table \[table1\]. There, we see once again that the total cross-sections are, generally, much larger for positive-parity $\Xi_5$ than for positive-parity one by about factor of a hundred because there is a cancellation due to destructive interference.
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Reaction $F_{1}$ $F_{2}$ Reaction $F_{1}$ $F_{2}$
--------------------------------------- ------------- ------------- ------------------------------------------- ------------- ----------------------
$\sigma_{\bar{K}N\to K\Xi_5}(P = +1)$ 2.6 $\mu b$ 1.5 $\mu b$ $\sigma_{\bar{K}N\to K^{*}\Xi_5}(P = 1.6 $\mu b$ $\lesssim$ 2 $\mu b$
+1)$
$\sigma_{\bar{K}N\to K\Xi_5}(P = -1)$ 26 $nb$ 12 $nb$ $\sigma_{\bar{K}N\to K^{*}\Xi_5}(P = -1)$ 14 $nb$ $\lesssim$ 20 $nb$
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: Summary for the average total cross-sections in the CM energy region $2.35\,{\rm GeV}\,\le\,E_{\rm CM}\,\le\,3.35\,{\rm GeV}$ for $\bar{K}N\to K\Xi_5$ and $2.75\,{\rm
GeV}\,\le\,E_{\rm CM}\,\le\,3.75\,{\rm GeV}$ for $\bar{K}N\to K^{*}\Xi_5$. For $K^*$ production with the $F_2$ form factor used, only the upper values are quoted because the interference suppresses them.[]{data-label="table1"}
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are supported by a Korea Research Foundation grant (KRF-2003-041-C20067). The work of S.i.N. was supported by a scholarship endowed by the Ministry of Education, Science, Sports and Culture of Japan. He is also supported by a grant for Scientific Research (Priority Area No. 17070002) from the Ministry of Education, Culture, Science and Technology, Japan.
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[^1]: Recently, Ref. [@Kondo:2004rn] pointed out that the exclusion of the non-interacting $KN$ state from the two-point correlation function may reverse the parity of $\Theta^+$.
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