Integrable lattice spin models from supersymmetric dualities
Ilmar Gahramanov Shahriyar Jafarzade
math-ph cond-mat.stat-mech hep-th math.MP nlin.SI 81T60, 16T25, 14K25, 33D60, 33E20, 33D90, 39A13







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Integrable lattice spin models from supersymmetric dualities
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Ilmar Gahramanov[ilmar.gahramanov@msgsu.edu.tr]^a,b,c and
Shahriyar Jafarzade[shahriyar.jzade@gmail.com]^a,b 



^a Department of Physics, Mimar Sinan Fine Arts University,
 Bomonti 34380, Istanbul, Turkey 



^b Department of Mathematics, Khazar University, 
 Mehseti St. 41, AZ1096, Baku, Azerbaijan 



^c  Max Planck Institute for Gravitational Physics (Albert Einstein Institute),
 Am Mühlenberg 1, D-14476 Potsdam, Germany 

 










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Dedicated to the memory of Ludvig D. Faddeev



1.5cm Abstract 0.2cm Recently, there has been observed an interesting correspondence between supersymmetric quiver gauge theories with four supercharges and integrable lattice models of statistical mechanics such that the two-dimensional spin lattice is the quiver diagram, the partition function of the lattice model is the partition function of the gauge theory and the Yang-Baxter equation expresses the identity of partition functions for dual pairs.  This correspondence is a powerful tool which enables us to generate new integrable models. The aim of the present paper is to give a short account on a progress in integrable lattice models which has been made due to the relationship with supersymmetric gauge theories.

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MSC classes: 81T60, 16T25, 14K25, 33D60, 33E20, 33D90, 39A13 

Keywords: Yang-Baxter equation, star-triangle relation, integrable models, exactly solvable models, supersymmetry with four supercharge, supersymmetric duality, elliptic hypergeometric function, basic hypergeometric function, hyperbolic hypergeometric function











§ INTRODUCTION



Integrability is a beautiful phenomenon which plays an important role in theoretical and mathematical physics. One of the key structural elements leading to quantum integrability is the Yang-Baxter equation <cit.>

    ℝ_12 (u-v) ℝ_13(u) ℝ_23(v)
    =ℝ_23(v) ℝ_13(u) ℝ_12(u-v)  ,

where the operators ℝ_ik(u) act in the tensor product of some three vector spaces 𝕍⊗𝕍⊗𝕍 and depend on the spectral parameter u. The importance of the Yang-Baxter equation as a condition for integrability was noticed by Ludvig Faddeev who developed (with his Leningrad group) a deep connection between integrability and other areas of mathematical physics.  Nowadays the Yang-Baxter equation has a relation to quantum field theory, knot theory, string theory, statistical physics, conformal field theory etc. 


In this work we consider quantum integrability of two-dimensional square lattice spin models of statistical mechanics with the pair interaction between neighboring spins. The most known example of such models is the two-dimensional Ising model <cit.> which was solved by Onsager <cit.>. Onsager also observed that the Boltzmann weights of the Ising model satisfies the star-triangle relation which is a special form <cit.> of the Yang-Baxter equation for integrable statistical models with spin variables living on sites of the lattice:

    ∑_σ _0𝒮( σ _0)𝒲_η - α( σ _i , σ _0)   𝒲_η - β( σ _j, σ _0) 𝒲_η - γ( σ _k, σ _0) 
    
     =    ℛ(α,β,γ)𝒲_α( σ _j, σ _k)W_β( σ _i, σ _k) 𝒲_γ( σ _j, σ _i)  ,

where 𝒲 and S stand for the Boltzmann weight functions of the model. The star-triangle relation appears as a condition for commuting transfer matrices what makes the model integrable <cit.>.

There have been many developments in the integrable lattice spin models since Onsager's solution. There are by now many solutions[Classification of all solutions <cit.> to the quantum Yang-Baxter equation still remains an open problem.] of the star-triangle equation, i.e. Ising-like lattice models, most notable ones are the Fateev-Zamodchikov model <cit.> (the case N=2 gives the Ising model), Kashiwara-Miwa model <cit.>, chiral Potts model <cit.>, Faddeev-Volkov model <cit.>, Bazhanov-Sergeev model <cit.> etc.

One of the most surprising developments in the field has appeared recently <cit.> coming from a different area of theoretical physics. It was observed a relationship between exact results in supersymmetric quiver gauge theories and exactly solvable two-dimensional lattice models in statistical mechanics <cit.>.  In the  gauge/YBE correspondence, as it is called[The name “gauge/YBE” was first used by Yamazaki in <cit.> probably in an analogue to the gauge/Bethe correspondence <cit.>.], the integrability in statistical models is a direct consequence of supersymmetric duality. Roughly speaking, gauge/YBE correspondence relates the Yang-Baxter equation with the equality of partition functions for supersymmetric dual theories. This relationship has led to the construction of new exactly solvable models of statistical mechanics and we believe that much more are to be found.


In this work we try to present an elementary description of the gauge/YBE correspondence and to list solutions of the star-triangle relation found (or related) by this correspondence. Of course, it is impossible to give all details of this recent subject of research, therefore in some places these notes have a sketchy character. We hope to convince the reader, both with mainly integrability background and supersymmetry alike, that the subject has many interesting applications and new open problems.




The rest of the paper is organized in the following way:




  * In Section 2 we review integrable lattice models and formulate the star-triangle relation (Yang-Baxter equation) for Ising-like lattice models of statistical mechanics. 


  * In Section 3 we give a very brief review of supersymmetric duality, exact results for partition functions and quiver notations.



  * We present a survey of recent progress in integrable statistical models inspired by supersymmetric gauge theory computations in Section 4 and list all recently found solutions to the star-triangle relation. 


  * The paper concludes with comments on the recent status of the correspondence and briefly discusses some open problems in Section 5. 





§ A CRASH COURSE ON INTEGRABLE LATTICE MODELS


The main players in the notes are solvable[The terms “solvable” and “integrable” are the same in the context of this paper and will be used interchangeably.] lattice models and quiver gauge theories with four supercharges in two, three and four dimensions. In this section, we set up basic terminology about the exactly solvable lattice models of statistical mechanics. More details on the subject can be found in the book by Baxter <cit.> and in the review papers <cit.>. The section will mainly follow the exposition in <cit.>.




 §.§ Lattice models in statistical mechanics


An Ising-like model on a two-dimensional square lattice is defined as follows. At each site i there is a “spin” variable σ_i which takes some set of continuous or discrete values (or both as we will see later) in some range. Two adjacent spins i and j interact with an energy ϵ(σ_i, σ_j). The quantities of interest in statistical physics are statistical sums, such as the following partition function

    Z= ∑_{σ} e^-E(σ)/k_BT

where the summation runs over all values of spins;  E(σ), k_B and T   are the energy of the system,  Boltzmann constant and  temperature  respectively.
Let

    𝒲(σ_i, σ_j)  =  e^-ϵ(σ_i, σ_j)/k_B T

be the Boltzmann weight of the edge (i,j), with an interaction energy ϵ(σ_i,σ_j) between spins σ_i and σ_j. Then the partition function can be written in the following way

    Z  =  ∑∏_<i,j>𝒲(σ_i, σ_j)

The “integrable model” means that one can evaluate the partition function (<ref>) in the thermodinamic limit N →∞, where N is a number of sites of the lattice.

There exist  two other types of models in two-dimensional statistical mechanics, the IRF model and the vertex model[As was mentioned above we will not be interested in the solutions of IRF and vertex-type Yang-Baxter equation and talk about them here only for completeness. We would like to point out that for some models in our list discussed in the next sections the IRF and vertex-type solutions are known. We will not discuss those solution and refer to the original papers <cit.>.].

In the “interaction round a face model” (IRF) version of spin models four spins round a face of the lattice interact with each other. This interaction can be determined by the energy of face ε(σ_1, σ_2, σ_3, σ_4) depending on four spins. 




The notable examples of IRF models are the hard hexagon model <cit.>, the cyclic solid-on-solid model <cit.> and the restricted solid-on-solid model <cit.>.

In the vertex model, spin variables are located on the edges of the spin lattice. In this case one associates the local Boltzmann weight with each vertex configuration, namely statistical weights depend on four spins surrounding each site. The most known examples of vertex models are the six-vertex model <cit.>, the eight-vertex model <cit.> and the nineteen-vertex model <cit.>. 






 §.§ Integrability of models: star-triangle relation


The search for the Ising-like integrable models can be reduced to the problem of finding the Boltzmann weights that satisfy the so-called star-triangle relation[Note that in the literature “the star-triangle relation” often is used also for the IRF-type models, see, e.g. <cit.>.]


    ∑_σ𝒮(σ)  𝒲_qr(σ,σ_j)𝒲_pr(σ,σ_k) 𝒲_pq(σ_i,σ) 
                            = ℛ(p,q,r)𝒲_pq(σ_j,σ_k) 𝒲_pr(σ_i,σ_j)𝒲_qr(σ_i,σ_k), 
       ∑_σ𝒮(σ)  𝒲_pq(σ,σ_i)𝒲_pr(σ_k,σ) 𝒲_qr(σ_j,σ) 
                            = ℛ(p,q,r)𝒲_pq(σ_k,σ_j) 𝒲_pr(σ_j,σ_i)𝒲_qr(σ_k,σ_i).

where 


  * Summation is over all spin variables, 𝒲_pq(σ_i,σ_j) and 𝒲_pq(σ_i,σ_j)  are two different kinds of  the Boltzmann weights describe the interaction between two spins;



  * 𝒮(σ) is the rapidity-independent single-spin Boltzmann weight  assigned for each spin σ on the lattice;

  * ℛ(p,q,r) is some factor depending on three rapidity variables and independent of the spins. 

The star-triangle relation is a sufficient condition for the existence of  an infinite set of commuting transfer matrices[Here we do not discuss a transfer matrix method and relation of the star-triangle equation to commutativity of transfer matrices, the interested reader can be find details in many places, for instance, in <cit.>.] and thereby the model can be exactly solved using the transfer matrix method <cit.>. Namely by defining the transfer matrices

    (T_q)_σ, σ̅    = ∏_i=1^L𝒲_pq(σ_i, σ̅_i) 𝒲_pq(σ_i+1, σ̅_i)
    
    (T̅_r)_σ, σ̅    = ∏_i=1 ^L𝒲_pr(σ_i, σ̅_i) 𝒲_pr(σ_i, σ̅_i+1)

with periodic boundary conditions that σ_L+1=σ_1 and σ̅_L+1=σ̅_1 one can prove that T_q T̅_r=T_r T̅_q. If one has such a family of commuting transfer matrices then a partition function (<ref>) can be calculated exactly . 


In all our examples here, the two types of Boltzmann weights 𝒲_pq, 𝒲_pq, depend on rapidity variables only via their difference[The majority of lattice models of statistical mechanics satisfy this property, the most notable exception being the Chiral Potts model <cit.>.] p-q.  Consequently the Boltzmann weights will be written in terms of the spectral variable α=p-q, as 

    𝒲_α(σ_i,σ_j):=𝒲_pq(σ_i,σ_j) ,   and   𝒲_α(σ_i,σ_j):=𝒲_pq(σ_i,σ_j)  .

The two Boltzmann weights are also related by the crossing symmetry 

    𝒲_α(σ_i,σ_j)=𝒲_η-α(σ_i,σ_j)  ,

where η>0 is a real valued, model dependent “crossing parameter”.  Thus all two-spin interactions in the lattice model may be described in terms of the single Boltzmann weight 𝒲_α(σ_i,σ_j). 

The Boltzmann weights considered here are spin reflection symmetric, i.e. unchanged by interchanging the spin variables σ_i and σ_j:

    𝒲_α(σ_i,σ_j)=𝒲_α(σ_j,σ_i)  .


The simple consequence of the star-triangle relation and initial condition gives the unitarity and inversion relations[Note that the inversion relation for the partition function may exist even for a model which is not integrable. ]

    𝒲_α(σ_i,σ_j)𝒲_-α(σ_i,σ_j)    =1


    _σ_0 S(σ_0) 𝒲_η-α(σ_i,σ_0)    𝒲_η+α(σ_0,σ_j)  
       =1/𝒮(σ_i)(δ(x_i+x_j) δ_m_i,-m_j+δ(x_i-x_j) δ_m_i,m_j) .






 












§ A CRASH COURSE ON SUPERSYMMETRIC DUALITIES AND EXACT RESULTS


Obviously, it is impossible to review supersymmetric dualities and exact results in supersymmetric gauge theories in a few pages. Our intention is to give some a short description of the important keywords on the subject.




 §.§ Supersymmetric duality


In this section we very briefly remind some facts about supersymmetric duality. For more details, see e.g. <cit.>.


About two decades ago Seiberg <cit.> and many others found a non-trivial quantum equivalence between different supersymmetric theories, called supersymmetric duality. To be more precise it was shown that two or more different theories may describe the same physics in the far infrared limit, i.e. an observer testing the low energy physics (or physics at long distances) cannot distinguish the dual theories[It is worth mentioning that supersymmetric dual theories are not identical, but they give rise to the same physics at long distances.]. 

The supersymmetric duality was first constructed for four-dimensional 𝒩=1 gauge theory with a matter in the fundamental representation. Later many examples of dualities have been found with complicated matter content, different gauge and flavor groups in different dimensions. Today, supersymmetric duality has become a key tool for studying strongly coupled effects.

The basic example of supersymmetric duality <cit.> is an SU(N_c) electric gauge theory with N_F flavors of quarks which possesses a dual description in terms of N_f magnetic flavors of quarks charged under SU(N_F-N_c) gauge group[In this case the gauge singlets of the dual theory interact with the flavors via the superpotential term.] in the so-called conformal window 3/2 N_c < N_F < 3 N_c. The field content of dual theories is summarized in the table below. These two theories flow to the same infrared fixed point.




    SU(N_c)     SU(N_F)_L     SU(N_F)_R 
 Q     f     f     1 
 
Q'     f̅     1     f̅ 
 


Matter content of the electric theory.






    SU(N_F-N_c)     SU(N_F)_L     SU(N_F)_R 
 q     f     f̅     1 
 
q'     f̅     1     f  
 
M     1     f     f̅ 
 

 Matter content of the magnetic theory.



The main point for us about the supersymmetric dualities is that the partition functions of dual theories are expected to be equal. In the context of gauge/YBE correspondence integrability on the statistical models' side is equivalent to the equality of partition functions of supersymmetric dual theories. It means that one may generate solutions to the Yang-Baxter equation by considering partition functions of suitable duality.










 §.§ Quiver gauge theories


Here we briefly outline quiver notation of gauge theories which is a very useful tool for summarizing the group-theoretical data about a gauge theory in a compact way. Quiver gauge theories have been studied in physics more than forty years, initially, they were used in composite model building in the context of the Standard Model. For more details, see e.g. <cit.>.



Supersymmetric gauge theories considered in the work are specified with gauge group G (in our examples we consider just SU(2) group) and the matter fields transforming as chiral multiplets in a suitable representation. One can encode this information[Note that the superpotential term of the theory is not encoded by the quiver diagram.] in quiver diagrams using nodes for the gauge groups and edges for the matter multiplets.

Consider a theory with the gauge group G as a direct product of simple groups G_i

    G=G_1× G_2 ×…× G_n  .

In the quiver diagram


  * each node g_i corresponds to a vector multiplet in the adjoint representation of a gauge group G_i;


  * each edge corresponds to the matter multiplet in the bifundamental representation. In general, one uses arrows between nodes. The arrow going from g_i to g_j corresponds to a chiral multiplet in the fundamental representation of g_i and the anti-fundamental representation of g_j.


The quiver diagram encoding the SU(N_c) Seiberg duality from the previous section is described in Fig <ref>. 






In this work we deal with supersymmetric (with four supercharges) dualities of quiver gauge theories built from bifundamental matter, namely the matter content of gauge theories are represented as bifundamentals between gauge groups. 










 §.§ Partition functions and corresponding solutions


In this section we collect the expressions for the matrix models associated to the lens index, supersymmetric index and squashed sphere partition functions in four, three and two dimensions. The localization technique enables us to calculate the partition function of supersymmetric gauge theories with four supercharges on different manifolds exactly

    Z =  1/|W|dz_i/2 π i z_i∏_i=1^rankG Z_gauge (z_i; m_i)  ∏_Φ Z_Φ(z_i, t_a,; m_i)  .

In the examples of the next sections we only discuss theories without the Chern-Simons and Fayet-Iliopoulos terms are therefore the classical terms is absent in our expressions.



  * The integral is performed over the Cartan subgroup of the gauge
group. It is parameterized by the diagonal entries of the real scalar
z in the gauge group. 

  * The factor of inverse |W| represents the order of the Weyl group of the gauge group. 








  * The contribution of the vector multiplet is parameterized by the positive roots of the algebra. Actually, the Vandermonde determinant in the measure exactly cancels the one loop determinant of the vector multiplet.  

  * The contribution of the matter multiplet  corresponds to the contribution of the j-th chiral multiplet with R charge Δ_j. Each chiral multiplet is in the corresponding representation of the gauge group G with weight ρ_j(z) and in the corresponding representation of the flavor group F, with weight ϕ_j(a). 




The observation of the correspondence between supersymmetric theories and solvable lattice models is based on the fact that in both fields appears same special functions of hypergeometric type. For instance, the partition functions of supersymmetric theories on different manifolds can be expressed in terms of the following hypergeometric functions (for details see, e.g. <cit.>)




  * S^3 × S^1, S^3/ℤ_r × S^1 : elliptic hypergeometric integral 


  * S^2 × S^1 : basic hypergeometric integral


  * S_b^3, S_b^3/ℤ_r : hyperbolic hypergeometric integral


  * S^2, S^1 × S^1: ordinary hypergeometric integral 










§ INTEGRABILITY FROM DUALITY


In this section, we summarize the present status of the gauge/YBE correspondence and list solutions to the star-triangle relation found (or related) via this correspondence. 

A central phenomenon in the construction of integrable lattice models via the gauge/YBE correspondence is the existence of the corresponding Seiberg duality. Here we consider the special Seiberg duality for supersymmetric theories with four supercharges in different dimensions. All known solutions to the star-triangle relation found via the gauge/YBE correspondence results from the following duality <cit.>




  * Theory A: SU(2) gauge group with N_f=6 flavors, chiral
multiplets in the fundamental representation of the flavor group SU(6) and in the
fundamental representation of the gauge group.



  * Theory B: without gauge degrees of freedom and the chiral fields
(gauge-invariant “mesons”) in the 15-dimensional totally antisymmetric tensor
representation of the flavor group.



Note that we consider only the field content of the dual theories. Of course, this duality has different features in different dimensions, but such details are not crucial for our discussions. For instance, in order to get the right duality, one needs to specify the exact form of the superpotential <cit.>.

It turns out that identity of partition functions of dual theories can be written[One needs to break  the flavor symmetry down to SU(2) × SU(2) × SU(2).] in the form of the star-triangle relation


    _σ_0𝒮( σ _0)𝒲_η - α( σ _i , σ _0)   𝒲_η - β( σ _j, σ _0) 𝒲_η - γ( σ _k, σ _0) 
    
     =    ℛ(α,β,γ)𝒲_α( σ _j, σ _k)W_β( σ _i, σ _k) 𝒲_γ( σ _j, σ _i)  .

In all models we discuss here the scalar factor R can be absorbed into the Boltzmann weights. In Fig <ref> we described the quiver diagram of the above duality which gives the star-triangle relation. In Fig <ref> SU(2) gauge group lives on each node presented by circles and bifundamental matter on each edge, the boxes represents the flavor groups SU(2).




It is not clear in gauge/YBE correspondence why the above duality is special, but it is the duality for which the correspondence takes place. We think that one can find solutions to the star-triangle relation only via this duality, all other supersymmetric dualities may give a solution to the star-star relation. 







In the context of gauge/YBE correspondence the spin lattice
models can be identified with the quiver gauge theory with SU(2) gauge groups on the sites of the lattice. Then the partition function of the corresponding integrable model is equivalent to the supersymmetric partition function of the corresponding supersymmetric quiver gauge
theory. The contribution of chiral and vector multiplets to the supersymmetric partition function correspond to the nearest-neighbor Boltzmann weights and the self-interaction, respectively. In Fig <ref> we describe the correspondence of partition functions pictorially.




We would like to mention that the inversion relation (<ref>) has an interesting counterpart on supersymmetry side of the gauge/YBE correspondence, namely, it is related to the chiral symmetry breaking of the corresponding supersymmetric gauge theory. Such relation can be derived in many different ways, from supersymmetric gauge theory side one can obtain inversion relation by an accurate limit of parameters in the partition functions of dual theories <cit.>.

The Seiberg duality can be realized in the context of the brane language <cit.>, namely one can obtain the duality by exchanging NS5-branes. Such construction gives an opportunity to obtain integrable lattice models directly using brane construction. We will not discuss this direction in the paper and refer the interested reader to the papers <cit.>.

We finish this section by remarking that one can also construct an IRF-type integrable multispin model. For that one needs to take SU(N) gauge theory with SU(N) × SU(N) flavor symmetry. Then the identity of partition functions for dual theories can be written as the star-star relation for the IRF-type model. Unfortunately, it is quite complicated to prove analytically such integral identities <cit.>.

Below we list all solutions to the star-triangle relation found (or related) by gauge/YBE correspondence, but we do not treat them in detail. 



 §.§ S^3/ℤ_r × S^1 partition function and solution



The solution to the star-triangle relation was found[Kels also gives an analytic proof of the integral identity in <cit.>.] by Kels in <cit.> on the bases of the special case of the star-star relation found by Yamazaki[Actually, Yamazaki constructed <cit.> the star-star relation for the dual theories with SU(N) gauge group and SU(N) × SU(N) flavor group.] in <cit.>. 

The idea is that the sum-integral identity for the  four-dimensional lens supersymmetric indices of supersymmetric dual theories can be written as the star-triangle relation


    (p^r,p^r)_∞(q^r,q^r)_∞/2∑_m_0=0^r-1∫_0^1dz   ∏_i=1^6Γ_e((a_i± z),(m_i± m_0);σ,τ)/Γ_e(± 2z,± 2m_0);σ,τ
               =∏_1≤ i<j≤ 6Γ_e((a_i+a_j),(m_i+m_j);σ,τ) ,

 with the  balancing conditions ∑_i=1^6 a_i=σ+τ and ∑_i=1^6 m_i=0. The definition of the lens gamma function is given in Appendix <ref>.
 
 


 
The lattice model has two spin variables on each site, discrete and continuous spin 

    σ_j=(x_j, m_j),   where   0 ≤ x_j < 2π,    m_j=0,1,2,… ,[r/2]  .

The Boltzmann weight and self-interaction term of the model are <cit.>. 

    𝒲_α ( σ _i, σ _j)  = e^-2 α ([m_i-m_j]_±+[m_i+m_j]_±)/r/k( α )   Φ _r, m_i-m_j(x_i-x_j+i α )/Φ _r, m_i-m_j(x_i-x_j-i α )
         ×Φ _r,m_i+m_j(x_i+x_j+i α )/Φ _r,m_i+m_j(x_i+x_j-i α ),


    𝒮(σ_0)     = ε_0/π(p^2r;p^2r)_∞(q^2r;q^2r)_∞e^2η[2m_0]_±/rΦ_r,-2m_0(-2x_0-iη) Φ_r,2m_0(2x_0-iη),

where k( α ) represents the partition function per edge and is defined as follows:
 
    k(α)=exp( ∑_n ≠ 0e^4α n((pq)^rn-(pq)^-rn)/n((pq)^2n-(pq)^-2n)(p^rn-p^-rn)(q^rn-q^-rn)),
  
and   
  
    ε_0= 1/2,    if  m_0=0   or  [r-m_0]_r
    
                1,    otherwise  .

Here we use the notation of <cit.> for the lens elliptic gamma function which is defined in the following way 
  
    Φ _r,m(z)=∏_j,k=0^∞1-e^2izp^-2[m]_r(pq)^2j+1p^2kr+2r/1-e^-2izp^2[m]_r(pq)^2j+1p^2kr1-e^2izq^-2[m]_r(pq)^2j+1q^2kr/1-e^-2izq^-2[m]_r(pq)^2j+1q^2kr+2r,

where [m]_r ∈{0,1,2…, r-1} and [m]_±:= [m]_+[m]_-. 
  

We should mention that the solution in terms of lens gamma functions is the top level known solution to the star-triangle relation at this time which gives almost all known other models in the limiting case. 






 §.§ S^3 × S^1 partition function and solution


The identity for the superconformal indices of the dual theories is the following elliptic beta integral <cit.>

    (q;q)_∞(p;p)_∞/2∮dz/2π i z ∏_i=1^6Γ(a_iz^±;p,q)/Γ(z^± 2;p,q) =∏_1≤ i<j≤6Γ(a_ia_j;p,q) ,

with the balancing condition ∏_i=1^6 a_i = p q. This identity was introduced and proven by Spiridonov in <cit.>. Later Bazhanov and Sergeev interpreted this identity as the star-triangle relation and introduced a new integrable spin lattice model <cit.>. 

In the corresponding integrable lattice model spin variables get continuous values

    0 ≤σ_i< 2π,

and the Boltzmann weights are expressed in terms of elliptic gamma functions

    𝒲_α(σ_i,σ_j)     = 1/k(α)Γ(e^α-η± iσ_i± iσ_j;p,q),


    𝒮(σ_0)  =  (p;p)_ ∞(q;q)_ ∞/4 πθ(e^± 2iσ_0;q),

where

    k(α)=Γ(e^2α(pq)^2;p,q,(pq)^2)/Γ(e^2αpq;p,q,(pq)^2)    and  Γ(z;p,q,t):=∏_i,j,k=0^∞1-z^-1p^i+1q^j+1t^k+1/1-zp^iq^jt^k.


Here we use the notations of <cit.>. In order to keep track of notations of the paper <cit.> by Bazhanov one has to use  the following form of the elliptic gamma  function:
 
    Φ (x) =  Γ (e^-i(x-i η );p^2,q^2);
 
where p=e^iπτ, q=e^iπσ, η=-iπ(τ+σ), Imτ>0 and Imσ>0.Then the Boltzmann weights for the Bazhanov-Sergeev model get the following form:

    𝒲_ α (σ_i,σ_j)=k( α )^-1Φ (σ_i-σ_j+iα)/Φ (σ_i-σ_j-iα)Φ (σ_i+σ_j+i α )/Φ (σ_i+σ_j-i α ),
 
  
    S(σ_0)= e^ η /4/4 πϑ _1(σ_0| τ ) ϑ _1(σ_0|  σ  ),

where 

    k(α)= exp ( ∑_n ≠ 0e^2α n/n(p^n-p^-n)(q^n-q^-n)((pq)^n+(pq)^-n)) .

Here  ϑ _1(x| τ ) is the Jacobi theta function defined in the Appendix <ref>.

This solution of the star-triangle relation can be obtained from the lens supersymmetric index by taking r=1. Note that in <cit.> the authors found a multi-spin generalization[ In terms of supersymmetric gauge theories one needs to consider the dual theories with SU(N) gauge group and SU(N) × SU(N) flavor group.] of this model and constructed the star-star relation for it. 





 §.§ S^2 × S^1  partition function and solution





The identity for S^2 × S^1  partition functions (three-dimensional supersymmetric indices) of daul theories[Note that here we presented the so-called generalized supersymmetric index <cit.>. The ordinary supersymmetric index with enhanced symmetry for the N_f=4 case was considered in <cit.> (see also the cases with broken gauge group in <cit.>).] is the following q-beta hypergeometric sum-integral <cit.>

    ∑_m=-∞^∞∮   ∏_i=1^6 (q^1+(m+n_i)/2/a_iz;q)_∞(q^1+(n_i-m)/2 z/a_i ;q)_∞/(q^(m+n_i)/2a_iz;q)_∞(q^(n_i-m)/2 a_i/z ;q)_∞(1-q^m z^± 2)/q^m z^6mdz/2π i z
                           =2/∏_i=1^6 a_i^n_i∏_1 ≤ i<j  ≤ 6(q^1+(n_i+n_j)/2/a_i a_j;q)_∞/(q^(n_i+n_j)/2a_i a_j;q)_∞,

with the balancing conditions ∏_i=1^6 a_i = q and ∑_i=1^6 n_i =  0. This identity was studied in the context of supersymmetric dualities in <cit.>, integrability in <cit.>, and from point of view of orthogonal polynomials in <cit.>.

In the corresponding integrable model we again have discrete and continuous spin variables

    σ_j=(x_j,m_j)   where    0 ≤ x_j < 1  and     m_j ∈ℤ.

The Boltzmann weights for the model are

    𝒲_α(σ_i,σ_j)     =q^-2i(x_im_i+x_jm_j)/k(α)(q^1+(m_i+m_j)/2q^η-α-i(x_i+x_j);q)_∞/(q^(m_i+m_j)/2q^α-η+i(x_i+x_j);q)_∞(q^1+(m_j-m_i)/2q^η-α+i(x_i-x_j);q)_∞/(q^(m_j-m_i)/2q^α-η-i(x_i-x_j);q)_∞
         ×(q^1-(m_i+m_j)/2q^η-α+i(x_i+x_j);q)_∞/(q^-(m_i+m_j)/2q^α-η-i(x_i+x_j);q)_∞(q^1+(m_i-m_j)/2q^η-α+i(x_j-x_i);q)_∞/(q^(m_i-m_j)/2q^α-η-i(x_j-x_i);q)_∞,



    𝒮(σ_0)= 1/2π q^m(q^± 2x_0+m;q)_∞/(q^± 2x_0+m+1;q)_∞,

where

    k(α)=(q^2e^4π i α,qe^-4π i α;q,q^2)_∞/(qe^4π i α,q^2e^-4π i α;q,q^2)_∞;      
     (z,w;p,q)_∞:=∏_i,j=0^∞(1-zp^iq^j)(1-wp^iq^j).

   
The multi-spin generalization of this solution  was constructed by authors in <cit.>.

In <cit.> Kels uses slightly different notations. In his case the  Boltzmann weight and self interaction terms are expressed as

    𝒲_α(σ_i.σ_j)=e^-2α|m_i-m_j|-2α|m_i+m_j|/k(α)Q(x_i-x_j+iα,m_i-m_j)/Q(x_i-x_j-iα,m_i-m_j)
                                               ×Q(x_i+x_j+iα,m_i+m_j)/Q(x_i+x_j-iα,m_i+m_j),


    𝒮(σ_0)=1/2πQ(2x_0-iη,2m_0)Q(-2x_0-iη,-2m_0),

with

    k(α)=exp{-∑_n≠ 0e^4α n/n((q)^n-(q)^-n)};
           Q(z;n)=(e^2iz (p/q)^-n (pq)^1+|n|;(pq)^2)_∞/(e^-2iz (p/q)^n (pq)^1+|n|;(pq)^2)_∞.








 §.§ S_b^3/ℤ_r partition function and solution



The sum-integral identity for the  three dimensional  lens partition function of supersymmetric dual theories reads as

    ∑_m_0=0^r-1∫ _ℝdx_0/r√(ω_1ω_2)2sinh2π/rω_1(x_0-iω_1m_0)sinh2π/rω_2(x_0+iω_2m_0)                                        
    ×∏_i=1^6ŝ_b,-m_0-m_k(x_0+x_k+iQ/2)/ŝ_b,-m_0+m_k(x_0-x_k-iQ/2)=∏_1≤ i<j≤ 6ŝ_b,-m_j-m_k(x_j+x_k+iQ/2),

with the balancing conditions i∑_i=1^6x_i=Q and ∑_i=1^6m_i=0, where Q=b+1/b.
Here we used the improved double sine function defined as

    ŝ_b,-m(x)=e^iπ/2r([m](r-[m])-(r-1)m^2)φ_r,m(x)  .

The function φ_r,m(z) is generalization of the Faddeev's quantum dilogarithm

    φ_r,m(z) =exp{∫_0^∞ dx (iz/ω_1  ω _2 r x^2-    sinh(2izx-ω_1(r-2[m]x))/2xsinh(ω_1 rx)sinh((ω_1 + ω _2) x) 
         -sinh(2izx+ω_2(r-2[m]x))/2xsinh(ω_2 rx)sinh((ω_1 + ω _2) x) ) }.

The hyperbolic limit of the lens index solution was presented in <cit.> where the authors  also give an interpretation of this solution in terms of supersymmetric gauge theory.


In this model there are continuous and discrete spin variables living on each site of the lattice 

    σ_j=(x_j,m_j)   where   0 ≤ x_j < ∞ and 
        m_j=0,1,2,…,⌊ r/2 ⌋ ,

and the Boltzmann weights are

    𝒲_α ( σ _i, σ _j)     = 1/k(α)φ _m_i+m_j(x_i+x_j+i α )φ _m_i-m_j(x_i-x_j+i α )/φ _m_i+m_j(x_i+x_j-i α )φ _m_i-m_j(x_i-x_j-i α ),
    𝒮( σ _0)     = 4ε_0/ r√(ω _1ω _2)sinh(  2π/ω _1r(x_0-i ω _1m_0) )sinh( 2π/ω _2r(x_0+i ω _2m_0)),

where the normalization constant gets the following form

    k (α)     = exp{∫_0^∞ dx (-α/ω_1  ω _2 r x^2+sinh(4α x)sinh(2rη x)/2xsinh(ω_1 rx)sinh(ω_2 rx)sinh(4η x))       }.



For r=1 the this solution reduces to the Spiridonov's generalization <cit.> of the Faddeev-Volkov model which we will consider in the next section.





 §.§ S_b^3 partition function and solution





The hyperbolic hypergeometric integral identity for the  three-dimensional  squashed sphere partition function[Note that depending on different squashings of three-sphere and on choice of the preserved charges in the supersymmetric localization one can get different partition functions depending on values of squashing parameter b (see <cit.> for details). For the specific choice of the preserved supercharge the partition function on squashed three-sphere with SU_l(2) × U_r(1) isometry gives the b=1 case <cit.> (the case b=1 also corresponds to the round sphere partition function). The integral identity with general values of b  is written for the partition functions of dual theories on three-sphere with U(1) × U(1) isometry (also with SU_l(2) × U_r(1) <cit.>).] of supersymmetric dual theories reads as

    ∫_-∞^∞∏_j=1^6γ^(2)(g_k± i z;ω) /γ^(2)(± 2 i z;ω)  d z
      =2√(ω_1ω_2)∏_1≤ j<k≤ 6γ^(2)(g_j+g_k;ω)  ,

with the balancing condition ∑_j=1^6 g_j = ω_1 + ω_2 := 2η. The definition of the hyperbolic gamma function is given in Appendix <ref>. 

This integral identity for supersymmetric dual theories was computed in <cit.> by applying the reduction procedure of <cit.> to the elliptic beta integral. As a star-triangle relation this identity was considered by Spiridonov in <cit.>. It is a generalization of the Faddeev-Volkov model[Actually from the supersymmetric viewpoint the Faddeev-Volkov model corresponds to above duality with broken gauge group.] <cit.>. In this integrable model spins get continuous values and the Boltzmann weights are

    𝒲_α(σ_i,σ_j)     =1/k(α)γ^(2)(α-η± iσ_i± iσ_j;ω),
    𝒮(σ_0)     =2/√(ω_1ω_2)sinh2πσ_0 /ω_1sinh2πσ_0/ω_2,

where

    k(α)=exp{-π i α^2-π i/24(1-2(b+b^-1)^2)}   (q̃  e^2π i u/b;q̃^2)_∞/(q e^2π iu b;q^2)_∞
                ×   ∏_j,k=0^∞1+e^π i u/(b+b^-1)p̃^j+1q̃^2k/1-e^π i u/(b+b^-1)p̃^j+1q̃^2k,

with b=ω_1 and b^-1=ω_2, q=e^2π i b^2, q̃=e^-2π i /b^2, and p̃=e^-π i/(1+b^2).

This solution can also be  expressed in a different way <cit.>

    𝒲_α(σ_i,σ_j)     =1/k(α)Φ(σ_i-σ_j+iα)/Φ(σ_i-σ_j-iα)Φ(σ_i+σ_j+iα)/Φ(σ_i+σ_j-iα),
    𝒮(σ_0)     = 2 sinh(2πησ_0) sinh(2πη^-1σ_0),

with

    k(α)=exp{1/8∫_PVe^4α w/sinh (η w)sinh (η^-1 w)cosh((η+η^-1)w)dw/w},

where the other version of the hyperbolic gamma function is used

    Φ(z)=exp{1/4∫_PVe^-2izw/sinh (wb)sinh (wb^-1)dw/w} .


One can find the proof of the star-triangle relation in terms of Boltzmann weights (<ref>)-(<ref>) in many places, e.g. see <cit.>. The quasi-classical limit b → 0 of the model was considered in <cit.>. 







 §.§ S^2 partition function and solution




The solution to the star-triangle relation was obtained by Kels in <cit.> (see also <cit.>). The interpretation in terms of supersymmetric sphere partition function and the star-star relation is given in 

The integral identity for sphere partition functions of dual theories reads as

    ∑_m∈ℤ∫_-∞^+∞dz/2π i zΓ(m± 2iz+1)/Γ(m±2iz)∏_i=1^6   Γ(m+n_i/2+a_i+iz)/Γ(1+m+n_i/2-a_i-iz)Γ(m-n_i/2+a_i-iz)/Γ(1+m-n_i/2-a_i+iz)
           = ∏_1≤ i<j≤6Γ(a_i+a_j+n_i+n_j/2)/Γ(1-a_i-a_j-n_i+n_j/2),

with the balancing conditions ∑_i=1^6 a_i=1 and ∑_i=1^6 n_i=0.

In the corresponding models spin variables get discrete and continuous  values 

    σ_j=(x_j,m_j)   where        0 ≤ x_j < 2π and     m_j ∈ Z  .

The Boltzmann weight and self-interaction term for the model are  
 
    𝒲_α(σ_i,σ_j)=Γ(1+α/2)/Γ(1-α/2) Γ(1-α/2±i(x_i+x_j)-(m_i+m_j)/2) Γ(1-α/2±i(x_i-x_j)-(m_i-m_j)/2)/Γ(1+α/2±i(x_i+x_j)+(m_i+m_j)/2) Γ(1+α/2±i(x_i-x_j)+(m_i-m_j)/2),
   

    𝒮(σ_0)=1/2πΓ(m_0± 2ix_0+1)/Γ(m_0± 2ix_0).






 §.§ S^1 × S^1 partition function and solution




Integral identity for the two dimensional supersymmetric indices of dual theory is defined  

    1/2((q;q)^2_∞/θ(y;q)) ∮dz/2π i z∏_i=1^6Δ(a_iz^±1;q,y)/Δ(z^±2;q,y)
         =∏_1≤ i<j≤ 6Δ(a_ia_j;q,y),

with the  balancing condition  ∏_i=1^6 a_i= q/y. Here we used the following ratios of the theta functions 

    Δ(z;q,y):=θ(zy,q)/θ(z,q)

This integral identity was written as the star-triangle relation in <cit.>. In this model spin variables get the cantionuos values 0 ≤σ_i< 2π. The Boltzmann weights and self-interaction term are

    𝒲_α(σ_i,σ_j)=1/k(α)θ( e^-α-η∓ i(σ_i±σ_j);q)/θ(e^α-η± i(σ_i±σ_j);q),


    𝒮(σ_0)=1/4π((q;q)^2_∞/θ(y;q))  θ(e^± 2iσ_0;q)/θ(e^-2η± 2iσ_0;q),

where 

    k(α)=(q^2e^2α,qe^-2α;q,q^2)_∞/(qe^2α,q^2e^-2α;q,q^2)_∞(q^-1e^-2α,e^2α;q,q^2)_∞/(e^-2α,q^-1e^2α;q,q^2)_∞.

and the infinite products in the formula are defined in the second expression of (<ref>).

Actually one can use the standard Jacobi theta function notions for this solution. In that case the integral identity (<ref>) will be the indices  of dual theories  defined for RR sector[Because of spectral duality the index defined for RR sector <cit.> is identical with the index defined for the NS-NS sector <cit.>]. Then the Boltzmann weights have the following form 

    𝒲_α(σ_i,σ_j)=θ_1( e^-α-η∓ i(σ_i±σ_j);q)/θ_1(e^α-η± i(σ_i±σ_j);q),


    𝒮(σ_0)=1/4π(η(q)^3/iθ_1(e^-2η;q)) θ_1(e^± 2iσ_0;q)/θ_1(e^-2η± 2iσ_0;q).

We would like to mention that the multi-spin case of the model was constructed in <cit.>. 
In <cit.> the authors considered the high temperature limit of this model and obtained a new solvable model with the following Boltzmann weights

    𝒲_α(σ_i,σ_j)=sinhπ((-η+α)± i(σ_i±σ_j)+t)/sinhπ((-η+α)± i(σ_i±σ_j)),


    S(σ_0)=1/2-π/sinhπ tsinhπ(± 2 iσ_0) /sinhπ(± 2iσ_0+t),


    R(α,β,γ) =sinhπ(-2α+t)/sinhπ(-2α)sinhπ(-2β+t)/sinhπ(-2β)sinhπ(-2 γ+t)/sinhπ(-2γ)

which is the solution of (<ref>) with continuous spin variables.





§ SUMMARY AND DISCUSSION




In this paper, we review integrable Ising-like square lattice models obtained (or related) from supersymmetric gauge theory computations. 

In all models which we consider in the paper the Boltzmann weights depend only on the differences of the spin variables at the neighbor sites and rapidity variables at the ends of the edge. In principle, it should possible to obtain a model without such symmetries from supersymmetric computations <cit.>.

However, many key questions have not been answered. For example, given a supersymmetric duality with a gauge and matter multiplets in some representation of the gauge and flavor groups, what is the corresponding integrable lattice model? In our opinion, this is one of the most important questions posed by consideration of the subject. Actually, it is absolutely unclear whether gauge/YBE is generic or a feature of a few special duality.




The reader might wonder whether the gauge/YBE could be used for  three-dimensional lattice models. Actually, there are a lot of attempts to extend the idea of integrability to three- <cit.> and higher-dimensional generalization <cit.> of lattice models. The condition of commutativity for the transfer matrices in the three-dimensional case takes the form of the so-called tetrahedron equation by Zamolodchikov <cit.>. It would be interesting to extend the relationship between supersymmetric gauge theory computations and integrable models to higher dimensions and find a solution to the tetrahedron equation. 

We remark that the reader might have expected that one can use integrability methods to study dualities and supersymmetric gauge theories via the correspondence. The closed form expressions for the partition function of the models discussed above may provide insight towards an understanding of supersymmetric quiver gauge theories and dualities for them. This is an important point and much work remains to be done in this direction.

Linear quiver gauge theories can be formulated using brane constructions, hence one can obtain solutions of the Yang-Baxter equation directly from the latter via corresponding topological quantum field theories. We refer to the work by  <cit.> for a discussion on this formulation. 

The Seiberg duality <cit.> for quiver gauge theories corresponds to the cluster mutation <cit.> for cluster algebras. To be more precise, a cluster algebra defined by a quiver and different quivers are related by the so-called cluster mutation (it is also called cluster transformation) and it happens that the mutation action on quivers is exactly the same as Seiberg duality. It would be interesting to find a relation of integrable models to cluster algebras <cit.>.   


















Acknowledgments

The authors would like to thank all participants of “Quantum integrability” seminars at the Physics Department of Mimar Sinan Fine Arts University, especially to Deniz Bozkurt, Doğu Dönmez and Sinan Sevim for discussions on the subject and on some parts of the paper. The authors are also thankful to Zainab Nazari for fruitful discussions. IG would like to express his sincere thanks to Vyacheslav Spiridonov, Andrew Kels, Hjalmar Rosengren and Masahito Yamazaki for valuable discussions on the subject. Sh.J is thankful to ICTP (Trieste, Italy) postgraduate diploma program for supporting and the Max-Planck Institut für Gravitationsphysik (Albert-Einstein-Institut) for warm hospitality. IG would like to thank The Institut des Hautes Etudes Scientifiques (IHES) and  Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna where some parts of the work were done for the hospitality.




§ APPPENDIX







§ NOTATIONS

  
For z∈ℂ, |q|<1, we define the infinite q-product (also called q-Pochhammer symbol)

    (z;q)_∞ :=  ∏_k=0^∞ (1-z q^k).


Denote that we will use the following definition for theta and  gamma functions (Euler, elliptic and hyperbolic) through the paper 

    (a,b;q)_∞:=(a;q)_∞(b;q)_∞  ;   (az^± 1;q)_∞:=(az;q)_∞(az^-1;q)_∞.




§ THETA FUNCTION


The θ(z,q) is the theta function defined by

    θ(z;p)  =  ∏_i=0^∞ (1-z^-1p^i+1)(1-z p^i)
 
It is related to the Jacobi theta functions. The first Jacobi theta function which is used in present paper can be expressed in terms of theta function

    θ_1(τ|z) = - i q^1/8 y^1/2 (q,q)_∞θ(y^-1;q)    with y=e^2 π i z,   q=e^2 π i τ.

As a product form 

    θ_1(τ|z) = -i q^1/8 y^1/2∏_k=1^∞ (1-q^k)(1-y q^k)(1-y^-1q^k-1).

Dedekind eta function is

    η(q) = q^1/24∏_k=1^∞(1-q^k).







§ ELLIPTIC GAMMA FUNCTION


The elliptic gamma function is a meromorphic function of three complex variables with double infinite product <cit.> 

    Γ(u;τ, σ)  =  ∏_i,j=0^∞1-e^2 π i((1+j) τ +(1+i)σ-u) /1-e^2 π i (jτ+iσ+u),

here u, σ, τ∈ℂ and Imτ, Imσ >0. It is convenient to do the following reparametrization

    p=e^2 π i τ,    q=e^2 π i σ,    z=e^2 π i u,

and get the following form 

    Γ(z;p,q):=∏_i,j=0^∞1-z^-1p^i+1q^j+1/1-zp^iq^j,

for |p|,|q|<1 and z∈ℂ^*.


The elliptic gamma function satisfies many interesting properties such as symmetry under exchange of parameters p and q

    Γ(z;p,q) = Γ(z;q,p)  ,

the functional relations

    Γ(qz;p,q) = θ(z;p) Γ(z;p,q), 
    Γ(pz;p,q) = θ(z;q) Γ(z;p,q)  ,
 
and the reflection property

    Γ(z;p,q)  Γ(pq/z;p,q) = 1    .

The elliptic gamma function has zeros at
 
    z∈(p^i+1q^j+1);   (i,j)∈ℤ^≥0

 poles at
 
    z∈(p^-iq^-j);   (i,j)∈ℤ^≥0

and the residue
 
    Res_z=1Γ(z;p,q)=-1/(p,p)_∞(q,q)_∞.










 






Th elliptic Gamma function is an automorphic form of degree 1 associated to a 2-cocycle and it has an SL(3,Z) modular property <cit.> based on the following relations

    Γ(u+τ, τ, τ+σ) Γ(u, τ+σ, σ)  =  Γ(u, τ, σ)   ,


    Γ(z/σ;τ/σ,1/σ) = e^i π Q(z,τ, σ)Γ(z-σ/τ; 1/τ,  σ/τ)  Γ( z; τ,  σ)

Note that the elliptic gamma function is related to the Barnes multiple gamma function of order three <cit.>. Probably this relationship has connection to its modular property. 

Lens elliptic gamma function is defined as 

    Γ_e  (z,m;σ,τ)=e^ϕ_e(z,m;σ,τ)   ∏_i,j=0^∞1-z^-1p^-m(pq)^i+1p^r(j+1)/1-zp^m(pq)^ip^rj
                   ×1-z^-1q^-r+m(pq)^i+1q^r(j+1)/1-zq^r-m(pq)^ip^rj,



    ϕ_e(z,m;σ,τ)=2π i(R_2(z,0;σ,τ)+ R_2(0,m;1/2,-1/2)-R_2(z,m;σ,τ)  )


    R_2(z,m;σ,τ)=R(z+mσ;rσ, σ+τ)+R(z+mσ;rσ, σ+τ)

where z∈ℂ, m∈ℤ and r∈{0,1,2,…}.

Here

    R(z;σ,τ)=B_3,3(z,σ,τ,-1)+B_3,3(z-1,σ,τ,-1)/12

and  the third order Bernoulli polynomial is

    B_3,3(z,ω_1,ω_2,ω_3)=z^3/ω_1ω_2ω_3-   3(ω_1+ω_2+ω_3)z^2/2ω_1ω_2ω_3+(ω_1^2+ω_2^2+ω_3^2)+3(ω_1ω_2+ω_1ω_3+ω_2ω_3)z/2ω_1ω_2ω_3
           -(ω_1+ω_2+ω_3)(ω_1ω_2+ω_1ω_3+ω_2ω_3)/4ω_1ω_2ω_3

where z∈ℂ and ω_1,ω_2,ω_3∈ℂ∖{0}.






§ HYPERBOLIC GAMMA FUNCTION AND ITS EXTENSIONS


The hyperbolic gamma function is defined as

    γ^(2)(u;ω_1, ω_2)=e^-π i B_2,2(u;ω)/2(e^2 π i u/ω_1q̃;q̃)/(e^2 π i u/ω_1;q)  with   q=e^2 π i ω_1/ω_2,   q̃=e^-2 π i ω_2/ω_1  ,

where  B_2,2(u;ω) is the second order Bernoulli polynomial,

    B_2,2(u;ω) =
    u^2/ω_1ω_2 - u/ω_1 -
    u/ω_2 + ω_1/6ω_2 +
    ω_2/6ω_1 + 1/2.


There is a version of the hyperbolic gamma function, called the double sine function which is defined as follows.

    s_b(z)= e^-iπ z^2/2∏_k=0^∞(1+e^2π b ze^2π i b^2(k+1/2))/∏_k=0^∞(1+e^2π z/b e^-2π i /b^2(k+1/2))


The relation between hyperbolic gamma function and double sine functions reads as

    γ^(2)(u;ω_1,ω_2)= e^-π i /2(B_2,2(u;ω_1,ω_2)-u^2-1/4(ω_1+ω_2)^2)+u(ω_1+ω_2))) s^-1_b(iu-i/2(ω_1+ω_2)),

for ω_1=b, ω_2=b^-1 and u=1/2(b+b^-1)-iz.

The reflection identity for a hyperbolic gamma-function is as follows

    γ^(2)(z;ω_1,ω_2)γ^(2)(ω_1+ω_2-z;ω_1,ω_2) = 1,
 
and the asymptotic formulas are

    lim_u →∞
    e^π/2 B_2,2(u;ω_1,ω_2)γ^(2)(u; ω_1,ω_2)
        =  1,   for arg ω_1 < arg  u < arg ω_2 + π,  
    lim_u →∞e^-π/2 B_2,2(u;ω_1,ω_2)γ^(2)(u;ω_1,ω_2)
        =  1,   for arg ω_1 - π < arg  u < arg ω_2.

It has the following useful properties  

    γ^(2)(z+ω_2;ω_1,ω_2)=2sin(π z/ω_1)γ^(2)(z;ω_1,ω_2),
    γ^(2)(z+ω_1;ω_1,ω_2)=2sin(π z/ω_2)γ^(2)(z;ω_1,ω_2).


The hyperbolic gamma function has zeros at
 
    z=ω_1ℤ^≥1+ω_2ℤ^≥1,

 poles at
 
    z=-ω_1ℤ^≤ 0-ω_2ℤ^≤ 0.


There are different notations and modifications of hyperbolic gamma function, relations between some of them can be found in <cit.>. 





utphys





