1 Introduction

The gauge/YBE correspondence [1, 2] connecting supersymmetric gauge theories and integrable lattice models of statistical mechanics provides a powerful tool for studying spin models. It turns out that most of integrable edge-interacting (Ising-like) models in statistical mechanics [3,4,5] (and some IRF models [6,7,8]) can be obtained by this correspondence. We will start with a very short account of this topic, the interested reader can find an exhaustive review on the subject in [9, 10], where necessary information about the correspondence presented. Similar identities appear in integrable models of statistical mechanics. In this work, we present some new hypergeometric integral identities of hyperbolic and trigonometric types.

One of the striking features of recent developments in non-perturbative supersymmetric gauge theories is their deep relationship with interesting mathematical structures, see, e.g. [11,12,13]. At present, they provide the main source of many new identities for hypergeometric functions [14,15,16,17,18].

In a recent work [19], the authors constructed a new solution to the star-triangle equation. This was achieved by using a certain three-dimensional supersymmetric dual theories on the lens space \(S_b^3/\mathbb {Z}_r\). The sufficient condition for the integrability of the lattice spin models is the star-star relation [20]. In this paper we present the corresponding star-star relation for the model studied in [19] and also for models discussed in [21,22,23] (the corresponding gauge theories live on the squashed sphere \(S_b^3\) and \(S^2 \times S^1\)). In the context of the gauge/YBE correspondence, this relation can be obtained from the equality of three-dimensional \(\mathcal N=2\) supersymmetric partition functions for a certain dual SQED theories. Our first two solutions to the star-star relation are given in terms of hyperbolic hypergeometric integrals (they are written in terms of hyperbolic gamma functions) and the last solution is a trigonometric type written in terms of basic hypergeometric integrals.

The paper is organized as follows. In Sect. 2 we briefly recall the star-star relation for the IRF models. Then we present solutions to the star-star equation.

2 Star-star relation

We deal here with the integrable interaction-round-a-face (IRF) lattice spin models [24, 25]. In the IRF models, spin variables are located on the sites of the square lattice and interact via face. The sufficient condition for the integrability is the Yang–Baxter equation. In [26] it was shown that the Yang–Baxter equation for the face models can be reduced to study of the so-called star-star relation. The star-star relation contains essentially all the information needed to solve the lattice spin model. Here we mainly follow the work of BaxterFootnote 1 [26], therefore we refer the reader to the original paper for details.

We consider an IRF model with the face Boltzmann weight

$$\begin{aligned} R\left( \begin{array}{ll} \sigma _4 &{} \sigma _3 \\ \sigma _1 &{} \sigma _2 \end{array}\right)&=\sum _{m_i}\int dx_i W(\sigma _1,\sigma _i)W(\sigma _2,\sigma _i)\nonumber \\&\quad \times W(\sigma _3,\sigma _i)W(\sigma _4,\sigma _i), \end{aligned}$$
(2.1)

where the \(\sigma _i=(x_i; m_i)\) stands for the continuous valued spin \(x_i\) and discrete valued spin \(m_i\) and \(W(\sigma _i, \sigma _j)\) denotes the interaction between spins \(\sigma _i\) and \(\sigma _j\). The Boltzmann weight \(W(\sigma _i, \sigma _j)\) solves the star-triangle relation for a certain integrable Ising-type lattice model

$$\begin{aligned}&\sum _{m_i} \int dx_i W_{\alpha _1}(\sigma _1,\sigma _i)W_{\alpha _2}(\sigma _2,\sigma _i)W_{\alpha _3}(\sigma _3,\sigma _i) \nonumber \\&\quad =\mathcal {R}(\alpha _1,\alpha _2,\alpha _3) W_{\eta -\alpha _1}(\sigma _1,\sigma _2)\nonumber \\&\qquad \times W_{\eta -\alpha _2}(\sigma _1,\sigma _3)W_{\eta -\alpha _3}(\sigma _2,\sigma _3), \end{aligned}$$
(2.2)

where \(\alpha _1+\alpha _2+\alpha _3=\eta \). Here \(\alpha _i\) stands for the rapidity parameter (spectral parameter) and \(\mathcal R\) is a spin-independent function.

Fig. 1
figure 1

The star-star relation

Full size image

Now let us define the following Boltzmann weight

$$\begin{aligned} {R\left( \begin{array}{lll} &{} \sigma _1 &{}\\ \sigma _2 &{} &{} \sigma _3 \\ &{}\sigma _4 &{} \end{array}\right) }=W(\sigma _1,\sigma _3)W(\sigma _2,\sigma _4) R\left( \begin{array}{ll} \sigma _4 &{} \sigma _3 \\ \sigma _1 &{} \sigma _2 \end{array}\right) . \end{aligned}$$
(2.3)

The system is integrable if the Boltzmann weight satisfies the following star-star relation

$$\begin{aligned} R_{(left)}\left( \begin{array}{lll} &{} \sigma _1 &{}\\ \sigma _2 &{} &{} \sigma _3 \\ &{}\sigma _4 &{} \end{array}\right) =R_{(right)}\left( \begin{array}{lll} &{} \sigma _1 &{}\\ \sigma _2 &{} &{} \sigma _3 \\ &{}\sigma _4 &{} \end{array}\right) . \end{aligned}$$
(2.4)

The identity (2.4) is illustrated in Fig. 1, where we skipped the rapidity lines for convenience, for the full picture and explicit expressions the reader is referred to [20].

One can also see (2.4) as the following way with the definition (2.3) to be convinced with the pictorial representation of the star-star relation in Fig. 1

$$\begin{aligned} R\left( \begin{array}{ll} \sigma _4 &{} \sigma _3 \\ \sigma _1 &{} \sigma _2 \end{array}\right) =\frac{W(\sigma _1,\sigma _2)W(\sigma _1,\sigma _4)}{W(\sigma _3,\sigma _2)W(\sigma _3,\sigma _4)} R\left( \begin{array}{ll} \sigma _4 &{} \sigma _3 \\ \sigma _1 &{} \sigma _2 \end{array}\right) . \end{aligned}$$
(2.5)

where \(R\left( \begin{array}{ll} \sigma _4 &{} \sigma _3 \\ \sigma _1 &{} \sigma _2 \end{array}\right) \) functions differ with the spectral parameters [20] omitted in this study.

By using the star-star relation one obtains the following IRF Yang–Baxter equation (it is depicted in Fig. 2)

$$\begin{aligned}&\sum _{m_i\in \mathbb {Z}}\int d x_i ~~ {R\left( \begin{array}{ccc} &{} \sigma _5 &{} \\ \sigma _6 &{} &{} \sigma _i \\ &{} \sigma _1 &{} \end{array}\right) } {R\left( \begin{array}{cc} \sigma _i &{} \sigma _3 \\ \sigma _1 &{} \sigma _2 \end{array}\right) } {R\left( \begin{array}{ll} \sigma _5 &{} \sigma _4 \\ \sigma _i &{} \sigma _3 \end{array}\right) } \nonumber \\&\quad = \sum _{m_j\in \mathbb {Z}} \int dx_j ~~ {R\left( \begin{array}{cc} \sigma _6 &{} \sigma _j \\ \sigma _1 &{} \sigma _2 \end{array}\right) } {R\left( \begin{array}{cc} \sigma _5 &{} \sigma _4 \\ \sigma _6 &{} \sigma _j \end{array}\right) } {R\left( \begin{array}{ccc} &{} \sigma _4 &{} \\ \sigma _j &{} &{} \sigma _3 \\ &{} \sigma _2 &{} \end{array}\right) }\,, \end{aligned}$$
(2.6)

where the summation and integration stand for the discrete and continuous spin variables, respectively. Note that there are several solutions to the IRF Yang–Baxter equation obtained via gauge/YBE correspondence [2, 3, 5, 7, 8].

Fig. 2
figure 2

Yang–Baxter relation via the star-star relation. Spin variables live on vertices and interact via edges

Full size image

3 Solutions to the star-star equation

By using the gauge/YBE correspondence one can systematically derive solution of the Yang–Baxter equation from calculations of supersymmetric gauge theory. In the context of this correspondence the Yang–Baxter equation expresses the identity of partition functions for supersymmetric dual pairs. Therefore the main step is to choose appropriate supersymmetric duality. Here we consider the following three-dimensional \(\mathcal N=2\) dual theories [19, 23, 41]

  • theory A has U(1) gauge group, six chiral multiplets with \(SU(3) \times SU(3) \times U(1)\) global symmetry group

  • theory B consists of nine free “mesons” with the same global symmetry group as theory A.

The supersymmetric localization techniqueFootnote 2 [29] enables us to calculate the partition function on different manifolds. The results of Coulomb branch localization on \(S_b^3\) , \(S_b^3/{\mathbb Z}_r\) and \(S^2 \times S^1\) are known (see, e.g. [30,31,32]) and we will use these results in order to construct the star-triangle relation and corresponding star-star relation in the next sections.

3.1 Notations

Let us introduce some definitions and notations of special functions which we use in the paper. The q-Pochhammer symbol is defined as follows

$$\begin{aligned} (z;q)_{\infty }=\prod _{i=0}^{\infty }(1-zq^i). \end{aligned}$$
(3.1)

We use the shorthand notation

$$\begin{aligned} (z,x;q)_\infty =(z;q)_\infty (x;q)_\infty . \end{aligned}$$
(3.2)

We also use hyperbolic gamma function which can be defined as

$$\begin{aligned} \gamma ^{(2)}(z;\omega _{1},\omega _{2})=e^{\frac{\pi i}{2}B_{2,2}(z;\omega _{1},\omega _{2})}\frac{(e^{-2\pi i\frac{z}{\omega _{2}}}\tilde{q};\tilde{q})_\infty }{(e^{-2\pi i\frac{z}{\omega _{1}}};q)_\infty } , \end{aligned}$$
(3.3)

with the parameters \(\tilde{q}=e^{2\pi i \omega _{1}/\omega _{2}}\) and \(q=e^{-2\pi i \omega _{2}/\omega _{1}}\) and the \(B_{2,2} (z; \omega _1,\omega _2)\) stands for the Bernoulli polynomial

$$\begin{aligned} B_{2,2}(z;\omega _{1},\omega _{2})=\frac{z^2-z(\omega _{1}+\omega _{2})}{\omega _{1}\omega _{2}}+\frac{\omega _{1}^2+3\omega _{1}\omega _{2}+\omega _{2}^2}{6\omega _{1}\omega _{2}}. \end{aligned}$$
(3.4)

The hyperbolic gamma function also has an integral representationFootnote 3

$$\begin{aligned} \gamma ^{(2)}(z;\omega _{1},\omega _{2})&=\exp \left( -\int _{0}^{\infty }\frac{dx}{x}\left[ \frac{\sinh {x(2z-\omega _{1}-\omega _{2})}}{2\sinh {(x\omega _{1})}\sinh {(x\omega _{2})}}\right. \right. \nonumber \\&\qquad \left. \left. \quad -\frac{2z-\omega _{1}-\omega _{2}}{2x\omega _{1}\omega _{2}}\right] \right) , \end{aligned}$$
(3.5)

where \(Re(\omega _{1}),Re(\omega _{2})>0\) and \(Re(\omega _{1}+\omega _{2})>Re(z)>0\).

Additionally, we will use the following reflection property for hyperbolic gamma function

$$\begin{aligned} \gamma ^{(2)}(\omega _1+\omega _2 - z;\omega _1,\omega _2)\gamma ^{(2)}(z;\omega _1,\omega _2) = 1. \end{aligned}$$
(3.6)

3.2 Solution via \(S_b^3\) supersymmetric partition function

The equivalence of the partition functions for dual theories on \(S_b^3\) gives the following hyperbolic hypergeometric integral identity [1, 35, 36]

$$\begin{aligned}&\int _{-i \infty }^{i \infty } \prod _{i=1}^3 \gamma ^{(2)}(a_i - x;\omega _1,\omega _2) \gamma ^{(2)}(b_i + x;\omega _1,\omega _2) \frac{dx}{i \sqrt{\omega _1\omega _2}} \nonumber \\&\quad = \prod _{i,j=1}^3 \gamma ^{(2)}(a_i+b_j;\omega _1,\omega _2), \end{aligned}$$
(3.7)

with the balancing condition \(\sum _{i=1}^3 (a_i+b_i) = \omega _1+\omega _2\). In [21, 37] it was shown that this integral gives the star-triangle relation for the Faddeev–Volkov model which has the following Boltzmann weight

$$\begin{aligned} W_{\alpha }(x_i,x_j)=&\gamma ^{(2)}(-\alpha +x_i-x_j;\omega _{1},\omega _{2}) \nonumber \\&\gamma ^{(2)} (-\alpha -x_i+x_j;\omega _1,\omega _{2}) , \end{aligned}$$
(3.8)

where we introduced new variables \( a_i =-\alpha _i+x_{i} \) and \( b_i=-\alpha _i-x_{i}\). Here \(\alpha \) is a spectral parameter, \(x_i\) is a spin variable and \(\omega _1, \omega _2\) are temperature-like parameters.

The corresponding star-star relation for this model (for details, see Appendix A) has the following form

$$\begin{aligned}&\int _{-i \infty }^{i \infty } \prod _{i=1}^4 \gamma ^{(2)}(a_i - x;\omega _1,\omega _2) \gamma ^{(2)}(b_i + x;\omega _1,\omega _2) \frac{dx}{i \sqrt{\omega _1\omega _2}} \nonumber \\&\quad = \frac{\prod _{i,j=1}^2 \gamma ^{(2)}(a_i+b_j;\omega _1,\omega _2)}{\prod _{i,j=3}^4 \gamma ^{(2)}(\tilde{a_i}+\tilde{b_j};\omega _1,\omega _2)}\nonumber \\&\qquad \times \int _{-i \infty }^{i \infty } \prod _{i=1}^4 \gamma ^{(2)}(\tilde{a_i} - z;\omega _1,\omega _2) \gamma ^{(2)} (\tilde{b_i} + z;\omega _1,\omega _2) \frac{dz}{i\sqrt{\omega _1\omega _2}}, \end{aligned}$$
(3.9)

with a new balancing condition \(\sum _{i=1}^4 (a_i+b_i) =2( \omega _1+\omega _2)\) and we used the following notations

$$\begin{aligned} \begin{aligned} \tilde{a}_i&= a_i+s,&\tilde{b}_i&= b_i+s,&\text {if} \;\;\; i=1,2, \\ \tilde{a}_i&= a_i-s,&\tilde{b}_i&=b_i-s,&\ \text {if} \;\;\; i=3,4, \end{aligned} \end{aligned}$$
(3.10)

with

$$\begin{aligned} \begin{aligned} s&=\frac{1}{2}(\omega _1+\omega _2-a_1-a_2-b_1-b_2)\\&=\frac{1}{2}(-\omega _1-\omega _2+a_3+a_4+b_3+b_4). \end{aligned} \end{aligned}$$
(3.11)

This integral identity was obtained in [38]. The physical interpretation of this identity discussed in [39].

3.3 Solution via \(S_b^3/{\mathbb Z}_r\) supersymmetric partition function

We again start with the equivalence of the partition functions for dual theories. This time we consider the dual theories on \(S_b^3/{\mathbb Z}_r\) and obtain the following hyperbolic hypergeometric integral identityFootnote 4 [19]

$$\begin{aligned}&\sum _{y=0}^{[ r/2 ]}\epsilon (y) e^{-\pi iy}\int _{-\infty }^{\infty } \prod _{i=1}^3 \nonumber \\&\gamma ^{(2)}(-i(a_i-x)-i\omega _1(u_i- y);-i\omega _1r,-i\omega ) \nonumber \\&\gamma ^{(2)}(-i(a_i-x)-i\omega _2(r-(u_i- y));-i\omega _2r,-i\omega ) \nonumber \\&\gamma ^{(2)}(-i(b_i+x)-i\omega _1(v_i+ y);-i\omega _1r,-i\omega ) \nonumber \\&\gamma ^{(2)}(-i(b_i+x)-i\omega _2(r-(v_i+y));\nonumber \\&\quad -i\omega _2r,-i\omega ) \frac{dx}{r\sqrt{-\omega _1\omega _2}}= e^{\frac{-\pi i}{2}\sum _{i=1}^3(u_i-v_i)}\nonumber \\&\prod _{i,j=1}^3 \gamma ^{(2)}(-i(a_i+ b_j)-i\omega _1(u_i+ v_j);\nonumber \\&-i\omega _1r,-i\omega )\gamma ^{(2)}(-i(a_i+ b_j)-i\omega _2\nonumber \\&\qquad \times (r-(u_i+ v_j));-i\omega _2r,-i\omega )) , \end{aligned}$$
(3.12)

with the balancing conditions \(\sum _{i=1}^3a_i+b_i=\omega _1+\omega _2\) and \(\sum _{i=1}^3u_i+v_i=0\). The \(\epsilon (y)\) function is defined as \(\epsilon (0)=\epsilon (\lfloor \frac{r}{2}\rfloor )=1\) and \(\epsilon (y)=2\) otherwise. The main difference from the expression (3.7) is that here the summation is over the holonomies \(y=\frac{r}{2\pi } \int A_\mu d x^{\mu }\), where the integration is over a non-trivial cycle on \(S_b^3/{\mathbb Z}_r\) and \(A_\mu \) is the gauge field, see, e.g. [31].

By introducing new variables \(a_i =-\alpha _i+x_{i} \) and \( b_i=-\alpha _i-x_{i} \) with the condition \(u_i=-v_i\) for \(i=\overline{1,3}\), one can rewrite the integral identity (3.12) as the star-triangle equation with the following Boltzmann weight

$$\begin{aligned}&W_{\alpha }(x_i,x_j,u_i,u_j)\nonumber \\&\quad =e^{-\pi i(u_i+u_j)}\gamma ^{(2)}(-i(-\alpha +x_i-x_j)\nonumber \\&\qquad -i\omega _1(u_i-u_j);-i\omega _{1}r,-i\omega )\nonumber \\&\qquad \times \gamma ^{(2)}(-i(-\alpha +x_i-x_j)\nonumber \\&\qquad -i\omega _2(r-(u_i-u_j));-i\omega _2r,-i\omega )\nonumber \\&\qquad \times \gamma ^{(2)}(-i(-\alpha -x_i+x_j)\nonumber \\&\qquad -i\omega _1(u_j-u_i);-i\omega _{1}r,-i\omega )\nonumber \\&\qquad \times \gamma ^{(2)}(-i(-\alpha -x_i+x_j)\nonumber \\&\qquad -i\omega _2(r-(u_j-u_i));-i\omega _2r,-i\omega ). \end{aligned}$$
(3.13)

The model with the Boltzmann weight (3.13) is an exactly solvable lattice spin model with discrete and continuous spin variables living on sites, where \(x_i\) represents continuous spin and \(u_i\) represents discrete spin variable. The \(r=1\) case corresponds to the Faddeev–Volkov model from the previous sectionFootnote 5.

Using the similar technique presented in Appendix A one can construct the star-star relation for this model

$$\begin{aligned}&\sum _{y=0}^{[ r/2 ]}\epsilon (y) \int _{-\infty }^{\infty } \prod _{i=1}^4 \gamma ^{(2)}(-i(a_i-x)-i\omega _1(u_i- y);-i\omega _1r,-i\omega ) \nonumber \\&\qquad \times \gamma ^{(2)}(-i(a_i-x)-i\omega _2(r-(u_i- y));-i\omega _2r,-i\omega ) \nonumber \\&\qquad \times \gamma ^{(2)}(-i(b_i+x)-i\omega _1(v_i+ y);-i\omega _1r,-i\omega ) \nonumber \\&\qquad \times \gamma ^{(2)}(-i(b_i+x)-i\omega _2(r-(v_i+y));-i\omega _2r,-i\omega )\frac{dx}{r\sqrt{-\omega _1\omega _2}} \nonumber \\&\quad =\frac{e^{\frac{\pi i}{2}\sum _{i=1}^2(u_i-v_i)}}{e^{\frac{\pi i}{2}\sum _{i=3}^4(\tilde{u_i}-\tilde{v_i})}}\nonumber \\&\qquad \times \frac{\prod _{i,j=1}^2 \gamma ^{(2)}(-i(a_i+ b_j)-i\omega _1(u_i+ v_j);-i\omega _1r,-i\omega ) }{\prod _{i,j=3}^4 \gamma ^{(2)}(-i(\tilde{a_i}+\tilde{b_j})-i\omega _1(\tilde{u_i}+\tilde{v_j});-i\omega _1r,-i\omega ) } \nonumber \\&\qquad \times \frac{\prod _{i,j=1}^2 \gamma ^{(2)}(-i(a_i+ b_j)-i\omega _2(r-(u_i+ v_j));-i\omega _2r,-i\omega )}{\prod _{i,j=3}^4 \gamma ^{(2)}(-i(\tilde{a_i}+\tilde{b_j})-i\omega _2(r-(\tilde{u_i}+\tilde{v_j}));-i\omega _2r,-i\omega ) } \nonumber \\&\qquad \times \sum _{m=0}^{[ r/2 ]}\epsilon (m) \int _{-\infty }^{\infty } \prod _{i=1}^4 \gamma ^{(2)}(-i(\tilde{a_i}-z)-i\omega _1(\tilde{u_i}- m);-i\omega _1r,-i\omega ) \nonumber \\&\qquad \times \gamma ^{(2)}(-i(\tilde{a_i}-z)-i\omega _2(r-(\tilde{u_i}- m));-i\omega _2r,-i\omega ) \nonumber \\&\qquad \times \gamma ^{(2)}(-i(\tilde{b_i}+z)-i\omega _1(\tilde{v_i}+ m);-i\omega _1r,-i\omega ) \nonumber \\&\qquad \times \gamma ^{(2)}(-i(\tilde{b_i}+z)-i\omega _2(r-(\tilde{v_i}+m));-i\omega _2r,-i\omega )\nonumber \\&\qquad \times \frac{dz}{r\sqrt{-\omega _1\omega _2}} , \end{aligned}$$
(3.14)

where the balancing conditions are \(\sum _{i=1}^4a_i+b_i=2(\omega _1+\omega _2)\) and \(\sum _{i=1}^4u_i+v_i=0\), and we used the following notations

$$\begin{aligned} \begin{aligned}&\tilde{a}_i = a_i+s, \tilde{b}_i = b_i+s, \tilde{u}_i =u_i+p, \\&\tilde{v}_i = v_i+p, \text {if} \; i=1,2,\\&\tilde{a}_i = a_i-s, \tilde{b}_i = b_i-s, \tilde{u}_i = u_i-p, \\&\tilde{v}_i =v_i-p, \text {if} \; i=3,4 , \end{aligned} \end{aligned}$$
(3.15)

where

$$\begin{aligned} s&=\frac{1}{2}(\omega _1+\omega _2-a_1-a_2-b_1-b_2)\nonumber \\&=\frac{1}{2}(-\omega _1-\omega _2+a_3+a_4+b_3+b_4),\\ p&=-\frac{1}{2}(u_1+u_2+v_1+v_2)=\frac{1}{2}(u_3+u_4+v_3+v_4).\nonumber \end{aligned}$$
(3.16)

3.4 Solution via \(S^2 \times S^1\) supersymmetric partition function

In this section we present a new trigonometric solution to the star-triangle equation and to the star-star equation. In

[22, 23, 41] authors considered the following basic hypergeometric sum/integral identityFootnote 6 which represents the equivalence of the superconformal indices for dual theories (partition function on \(S^2 \times S^1\)) discussed in Sect. 3

$$\begin{aligned}&\sum _{y=-\infty }^\infty \oint \prod _{i=1}^3\nonumber \\&\qquad \times \frac{(q^{1+(y+u_i)/2}(a_ix)^{-1},q^{1+(v_i-y)/2}xb_i^{-1};q)_\infty }{(q^{(y+u_i)/2}a_ix,q^{(v_i-y)/2}b_ix^{-1};q)_\infty }\frac{1}{x^{3y}}\frac{dx}{2\pi i x} \nonumber \\&\quad =\frac{1}{\prod _{i=1}^3a_i^{u_i}b_i^{v_i}}\prod _{i,j=1}^3\frac{(q^{1+(u_i+v_j)/2}(a_ib_j)^{-1};q)_\infty }{(q^{(u_i+v_j)/2}a_ib_j;q)_\infty }, \end{aligned}$$
(3.17)

where the balancing conditions are \(\prod _{i=1}^3a_ib_i=q\) and \(\sum _{i=1}^3u_i+v_i=0\).

This identity can be written as the star-triangle relation by introducing the new fugacities \(a_i=\alpha _i^{-1}x_{i} \) and \(b_i=\alpha _i^{-1}x_{i}^{-1}\) and using the condition \(u_i=-v_i\). The resulting Boltzmann weight then has the following form

$$\begin{aligned}&W_{\alpha }(x_i,x_j,u_i,u_j)=\frac{(q^{1+(u_i-u_j)/2}(\alpha ^{-1}x_ix_j^{-1})^{-1};q)_\infty }{(q^{(u_i-u_j)/2}\alpha ^{-1} x_ix_j^{-1};q)_\infty }\nonumber \\&\quad \times \frac{(q^{1+(u_j-u_i)/2}(\alpha ^{-1}x_i^{-1}x_j)^{-1};q)_\infty }{(q^{(u_j-u_i)/2}\alpha ^{-1}x_i^{-1}x_j;q)_\infty }, \end{aligned}$$
(3.18)

where \(\alpha \) stands for the spectral parameter. The corresponding statistical mechanics model is a square lattice model with edge interaction and discrete and continuous spin variables. This model is a special case (with broken gauge symmetry) of the integrable lattice spin model considered in [3] and it is a trigonometric analogue of the Faddeev–Volkov model.

One we can construct the star-star relation for this model

$$\begin{aligned}&\sum _{y=-\infty }^\infty \oint \prod _{i=1}^4\frac{(q^{1+(y+u_i)/2}(a_ix)^{-1},q^{1+(v_i-y)/2}x(b_i)^{-1};q)_\infty }{(q^{(y+u_i)/2}a_ix,q^{(v_i-y)/2}b_jx^{-1};q)_\infty }\frac{1}{x^{5y}}\frac{dx}{2\pi i x} =\frac{1}{\frac{\prod _{i=1}^2a_i^{u_i}b_i^{v_i}}{\prod _{i=3}^4(\tilde{a_i})^{\tilde{u_i}}(\tilde{b_i})^{\tilde{v_i}}}} \frac{\prod _{i,j=1}^2\frac{(q^{1+(u_i+v_j)/2}(a_ib_j)^{-1};q)_\infty }{(q^{(u_i+v_j)/2}a_ib_j;q)_\infty }}{\prod _{i,j=3}^4\frac{(q^{1+(\tilde{u_i}+\tilde{v_j})/2}(\tilde{a_i}\tilde{b_j})^{-1};q)_\infty }{(q^{(\tilde{u_i}+\tilde{v_j})/2}\tilde{a_i}\tilde{b_j};q)_\infty }} \nonumber \\&\qquad \times \sum _{m=-\infty }^\infty \oint \prod _{i=1}^4\frac{(q^{1+(m+ \tilde{u_i})/2}(\tilde{a_i}z)^{-1},q^{1+(\tilde{v_i}-m)/2}z\tilde{b_i}^{-1};q)_\infty }{(q^{(m+\tilde{u_i})/2}\tilde{a_i}z,q^{(\tilde{v_i}-m)/2}\tilde{b_i}z^{-1};q)_\infty }\frac{1}{z^{5m}}\frac{dz}{2\pi i z} , \end{aligned}$$
(3.19)

with the new balancing conditions \(\prod _{i=1}^4a_ib_i=q^2\) and \(\sum _{i=1}^4u_i+v_i=0\). In the latter expression we used the following notations

$$\begin{aligned} \begin{aligned}&\tilde{a}_i = a_is, \tilde{b}_i = b_is, \tilde{u}_i =u_i+p, \\&\tilde{v}_i = v_i+p, \text {if} \; i=1,2,\\&\tilde{a}_i = a_is^{-1}, \tilde{b}_i = b_is^{-1}, \tilde{u}_i = u_i-p, \\&\tilde{v}_i =v_i-p, \text {if} \; i=3,4, \end{aligned} \end{aligned}$$
(3.20)

where

$$\begin{aligned} \begin{aligned} s&=\sqrt{\frac{q}{a_1a_2b_1b_2}}=\sqrt{\frac{a_3a_4b_3b_4}{q}}, \\ p&=-\frac{1}{2}(u_1+u_2+v_1+v_2)=\frac{1}{2}(u_3+u_4+v_3+v_4). \end{aligned} \end{aligned}$$
(3.21)

4 Conclusions

In this work, we constructed hyperbolic and trigonometric solutions to the star-star equation. We obtained new solutions from the equality of three-dimensional \(\mathcal N=2\) supersymmetric partition functions for certain dual SQED theories via the gauge/YBE correspondence.

There are several ways of constructing solutions to the star-star equation. One can use the Bailey pair construction starting from the star-triangle relation for the models discussed here. It is possible to obtain the solution by breaking the gauge symmetry from SU(2) group to the U(1) for the supersymmetric dual theories with SU(2) gauge group and SU(6) flavor group considered in [42].

There are many interesting limits of the solutions considered here, for instance it would be interesting to construct solution to the star-star equation in terms of Euler gamma functions [43].

The gauge/YBE correspondence has revealed various interesting connections among integrable models and supersymmetric gauge theories. There are underlying mathematical structures such as quantum algebras related to the solutions of the Yang–Baxter equations obtained via gauge/YBE correspondence. It would be interesting to pursue this direction for these solutions to the star-star relation.