Gamma function solutions to the star-triangle equation

In the paper, we clarify some relations between solutions to the star-triangle equation by giving the gauge/YBE formulation of them. We consider two solutions to the star-triangle relation in terms of Euler’s gamma function. We derive these solutions from the reduction of certain basic and hyperbolic hypergeometric integral identities. These identities can be interpreted as equality of the supersymmetric partition functions of a specific three-dimensional N = 2 supersymmetric dual theories.


Introduction and conclusion
There is a remarkable connection [1][2][3], called gauge/YBE correspondence, between supersymmetric gauge theories and integrable lattice models of statistical mechanics. This correspondence is used to be quite useful in the construction of the new integrable lattice spin models [3][4][5][6][7][8][9][10][11]. Curiously, almost all known solutions to the star-triangle relation make an appereance in the context of this correspondence and the subject is under active investigation, see, e.g. [12][13][14]. The purpose of this paper by using the gauge/YBE correspondence is to interpret some integrable lattice spin models with Boltzmann weights in terms of Euler's gamma function from the supersymmetric gauge theory point of view. Therefore we will not discuss details of the gauge/YBE correspondence here, more details can be found in the original works mentioned above and in the review papers [15,16].
In this paper we will re-derive two solutions to the star-triangle relation presented in [5,20] and in [13,21]. The Boltzmann weights of both models are given in terms of Euler's gamma function. It turns out that the solutions presented in [5,20] and [13,21] can be obtained from higher level hypergeometric solutions. From supersymmetric gauge theory side, we explicitly work out reduction procedure of three-dimensional N = 2 supersymmetric sphere partition function and superconformal index to the two-dimensional N = (2, 2) sphere partition function for the supersymmetric dual theories. From mathematical point of view, we make several reductions of the basic and hyperbolic hypergeometric integral identities to the ordinary hypergeometric integral identities.
The first model has the following vertical and horizontal Boltzmann weights [13,21] the following self-interaction term and the spin-independent coefficient and R(α, β, γ) = 4ω 1 πΓ(2α) Γ(2β) Γ(2γ) . (1.4) We use the notation that multiple parameters or ± signs in special functions indicate a product of functions. In the context of the gauge/YBE correspondence, this solution can be obtained from the equality of two-dimensional vortex partition functions for a certain dual theories.
The second integrable lattice spin model which we discuss here has the following nearestneighbouring Boltzmann weight [5,20] , (1.5) 1 In discrete spin case, one needs to replace integration by summation.
following self interaction term Here we choose such normalization for the Boltzmann weights that the spin-independent parameter is equal to one. The Boltzmann weights (1.5-1.6) solve the star-triangle relation of the following form where the crossing parameter η = α + β + γ. One can obtain the relation (1.7) from (1.1) by restricting the Boltzmann weights, see, e.g. [22,23]. Further we will see that there are several ways to obtain the solution (1.5-1.6).
The main goal of the paper is to show connections between different solutions to the startriangle relation explicitly.
It would be interesting to consider also the reduction of RP 2 × S 1 function [24][25][26][27], which gives also an integral in terms of gamma functions.
It is also possible to generalize the reductions discussed here to other dualities, see, e.g. [28] and obtain new or known integral identities in terms of the gamma function, expressing the equivalence of dual partition functions. There are several solutions to the Yang-Baxter equation in terms of gamma functions [29][30][31][32][33], it would be interesting to consider other models in this context. The diagram in Fig. 1 demonstrates the plan of the paper pictorially. The rest of the paper is organized as follows: • A brief review of necessary information on supersymmetric partition functions and N = 2 supersymmetric duality are contained in Sections 2 and 3.
• The relations between different solutions to the star-triangle relation and the main results are presented in Sec. 4.
• In Sec. 4.1 we present the reduction of S 3 b partition function to the two-dimensional vortex partition function and the solution (1.3)-(1.4) to the star-triangle relation. In [20], Kels presented another limit b → 0 for the squashed sphere partition function, which gives the sphere partition function.
• The reduction of S 2 ×S 1 partition function (three-dimensional superconformal index) to S 2 partition function (S 1 shrinks to zero size) and the solution (1.5)-(1.6) to the star-triangle equation are discussed in Sec. 4.2. We make the same limiting procedure discussed in the paper [5] using slightly different notations, see also [15].
• In Sec. 4.3 we consider the r → ∞ limit of the S 3 /Z r partition function. This reduction was discussed in [34], here we worked out the limit in detail on the level of special functions and obtained the solution (1.5)-(1.6).

Supersymmetric partition functions
Let us briefly summarize the basic ingredients which one needs to know about threedimensional theories with four supercharges (N = 2 theories). The vector multiplets consisting of a gauge field, a complex Dirac fermion, a real scalar field and an auxiliary scalar field belong to the adjoint representation of gauge group G, whereas chiral multiplets consisting of a complex scalar field, a complex Dirac fermion and a complex auxiliary belong to a suitable representation of gauge goup G and flavor group F . The supersymmetry algebra contains the SO(2) R-symmetry which rotates supercharges. In the context of gauge/YBE correspondence, the R-charge plays a role of spectarl parameter.
The partition function of three-dimensional N = 2 supersymmetric gauge theory on a certain compact manifold can be computed exactly due to the supersymmetric localization technique [35]. Using the supersymmetric localization, one ends with the following matrix Here Z vector and Z chiral stand for the contribution of the vector and chiral multiplets, respectively. The integral is performed over the Cartan subgroup of the gauge group. |W | represents the order of the Weyl group of gauge group. In our examples, we will consider theories without the Chern-Simons and Fayet-Iliopoulos term 3 , therefore we skip contributions of these terms to (2.1).

Three-dimensional N = 2 superconformal index
Three-dimensional N = 2 superconformal index 4 was studied in [40][41][42]. The one-loop contribution of chiral multiplets to the index is given by where j labels chiral multiplets, ρ j , φ j , are the weights of the representation of the gauge and flavor groups, respectively and ∆ j is the Weyl weight of j'th chiral multiplet. Here we use the usual q-Pochhammer symbol 3) The one-loop contribution of the vector multiplet combined with the Vandermonde determinant, is given also in terms of q-Pochammer symbols for the case of superconformal index the summation in (2.1) is over the GNO quantized fluxes [43] (monopole charges) m i = 1 2π S 2 F i .

Squashed three-sphere partition function
N = 2 supersymmetric partition function on the squashed three-sphere S 3 b was studied in [34,38,44]. The one-loop contribution of chiral multiplets to the partition function can be expressed as where j labels chiral multiplets, ρ j , φ j , are the weights of the representation of the gauge and flavor groups, respectively and ∆ j is the Weyl weight of j'th chiral multiplet. Here is the hyperbolic gamma function 5 defined as follows [45] Actually, it seems that in some cases Fayet-Iliopoulos term plays an important role the integrability, see [39]. 4 In the path-integral formulation superconformal index is a partition function of a theory defined on the background S 2 × S 1 . 5 We should mention that when the squasing parameter is real, the infinite product representation (2.5) is not valid, and one needs to use the integral representation, see [1].
where B 2,2 (u; ω 1 , ω 2 ) is the second order Bernoulli polynomial, The one-loop contribution of the vector multiplet combined with the Vandermonde determinant, is given by where the product is over the positive roots α of the gauge group G.
Note that the squashed partition function has a form of (2.1) without summation.

Orbifold partition function
Supersymmetric partition function on S 3 b /Z r , called orbifold partition function, was studied in [7,34,38,44]. The one-loop contribution of chiral multiplets to the partition function is given by 6 where j labels chiral multiplets, ρ j , φ j , are the weights of the representation of the gauge and flavor groups, respectively and ∆ j is the Weyl weight of j'th chiral multiplet. Here Q = b + 1 b with the squashing parameter b 2 = ω 2 /ω 1 . The function Γ h is a version of the improved double sine function [7], which can be written as a product of hyperbolic gamma functions Γ h (z, m; ω 1 , ω 2 ) = e φ h (m) γ h (z, m; ω 1 , ω 2 ) , 6 In supersymmetry literature, usually one uses the other special function, namely the contribution of the chiral multiplet has a form where the functionŝ b,−m is the improved double sine function [38] and Bϕ is a particular combination of multiple Bernoulli polynomials given by Bϕ(z, m, ω1, ω2) := B2,2(iz + ω1 m + η, rω1, 2η) + B2,2(iz + ω2(r − m ) + η, rω2, 2η) . (2.9) The one-loop contribution of the vector multiplet is given by and The one-loop contribution of the vector multiplet combined with the Vandermonde determinant, is given by where the product is over the positive roots α of the gauge group G.
In the orbifold partition function case the summation in the formula (2.1) is over holonomies where the integration countour is over non-trivial cycle on S 3 b /Z r and A µ is the gauge field.

Two-dimensional sphere partition function
The N = (2, 2) supersymmetric partition function on S 2 was obtained in [46,47]. The one-loop contribution of chiral multiplets to the partition function is given by where j labels chiral multiplets, ρ j , φ j , are the weights of the representation of the gauge and flavor groups, respectively and ∆ j is the Weyl weight of j'th chiral multiplet. Here Γ(z) function is the usual Euler's gamma function.
The one-loop contribution of the vector multiplet for theory with non-abelian gauge group combined with the Vandermonde determinant, is given by where the product is over the positive roots α of the gauge group G.
In the two-dimensional sphere partition function case, the integration contour in formula (2.1) is defined along the real lines and summation is taken over the magnetic fluxes.

Supersymmetric duality
For our purposes we consider the following three-dimensional N = 2 supersymmetric duality [48]: The first theory is the SQCD with SU (2) gauge group and with SU (6) flavor group, chiral multiplets are in the fundamental representation of the gauge group and the flavor group, a vector multiplet is in the adjoint representation of the gauge group.
The second theory, i.e. the dual one, has no gauge degrees of freedom, fifteen chiral multiplets of the theory are in the totally antisymmetric tensor representation of the flavor group.
One of the tools for checking supersymmetric dualities is to compute the partition function, which is expected to be the same for dual theories.
In superconformal index case, the equality of the dual indices can be expressed in terms of the basic hypergeometric sum and the integral identity [48] ∞ with the conditions 6 j=1 g i = q; and 6 j=1 n j = 0. The integral identity for the squashed sphere partition functions can be expressed in terms of the following hyperbolic hypergeometric functions [45] where the parameters g j 's obey the balancing condition 6 j=1 g j = ω 1 + ω 2 . The integral identity for the duality at the level of orbifold partition functions has the following form [7] 1 2r with the balancing condition j g j = ω 1 + ω 2 and (0) = 1 and (m) = 2 for m > 0.
All these integral identities can be written in the form of the star-triangle relation [1,4,5,7].   4) to the star-triangle equation from the Spiridonov's generalization of Faddeev-Volkov model, we will consider the reduction of the three-dimensional supersymmetric squashed sphere partition function to the identity for the two-dimensional supersymmetric vortex partition function. This reduction procedure was studied in [49]. Now let us reduce the identity (3.2) to the hyprgeometric level. Using the balancing condition for the identity (3.2), we rewrite one of the fugacities (in this case g 6 ) as g 6 = 2η − 5 j=1 g j and use the reflection identity for the hyperbolic gamma function to bring terms containing g 6 to the denominator on both sides of the identity (3.2). Then we apply the following reduction formula of the hyperbolic gamma function to the integral identity (3.2) and obtain 7 5 k=1 Γ g k ±iz Now we define again a new fugacity g 6 via the following equation (a new balancing condition) Then the integral (4.4) can be rewritten as Actually, here we have used the asymptotic behaviour of γ (2) (z, ω1, ω2) function, lim z→∞ e πi 2 B 2,2 (z;ω 1 ,ω 2 ) γ (2) (z; ω1, ω2) = 1, for arg ω1 < arg z < arg ω2 + π, (4.3) lim z→∞ e − πi 2 B 2,2 (z;ω 1 ,ω 2 ) γ (2) (z; ω1, ω2) = 1, for arg ω1 − π < arg z < arg ω2.
and canceled a factor of (ω2 2πω1) 3 , which appears on both sides of the equation (4.4).

Reduction of S 3 b /Z r partition function
In this section we consider the Euler Gamma function limit by taking b = 1 and r → ∞ from the solution corresponding to S 3 b /Z r partition function. In order to do it, we use the limit from S 3 /Z r to S 2 partition function.
For ω 1 = ω 2 = ω performing the asymptotic formula for the γ (2) (z) function, one finds the following formula . (4.21) By normalizing the fugacities as z → z 2iω and t i → g i 2ω , we obtain the following integral identity . It is not difficult to show that this identity is equivalent to the integral identity (4.16) (see Appendix A), and therefore the identity (4.22) gives the same solution to the star-triangle equation, i.e. one can obtain the solution (1.5)-(1.6).
From the special function point of view, an alternative method leading to the reduction of S 3 /Z r partition function to the S 2 partition function is described in [34].

A Appendix
Here we present the equality of integal identities (4.16) and (4.22). Noting the definition of the (m) function,