The star-triangle relation and 3d superconformal indices

Superconformal indices of 3d N=2 supersymmetric field theories are investigated from the Yang-Baxter equation point of view. Solutions of the star-triangle relation, vertex and IRF Yang-Baxter equations are expressed in terms of the q-special functions associated with these 3d indices. For a two-dimensional monopole-spin system on the square lattice a free energy per spin is explicitly determined. Similar to the partition functions, superconformal indices of 3d theories with the chiral symmetry breaking reduce to Dirac delta functions with the support on chemical potentials of the preserved flavor groups.

The first physical application of elliptic hypergeometric integrals consisted in the interpretation of some of them as wave functions or normalizations of wave functions in particular quantum mechanical problems [48]. The most important known application of identity (1.1) was found in [20] in the context of N = 1 supersymmetric field theories within which it has the meaning of the equality of superconformal indices [36,40,41] in Seiberg dual theories [43,44]. Indeed, the integral on the left-hand side of the equality (1.1) is the superconformal index of the 4d N = 1 supersymmetric gauge theory with SU (2) gauge group and N F = 6 flavors, chiral scalar multiplets in the fundamental representation of the flavor group SU (6), while the expression on the right side is the superconformal index for the dual theory without gauge degrees of freedom and the chiral fields in the 15-dimensional totally antisymmetric tensor representation of the same flavor group. In other words, the elliptic beta integral is the manifestation of the s-confinement phenomenon in gauge theories [43]. The superconformal indices techniques is the most convenient tool for searching new Seiberg dualities [50,51,52]. Using properties of the elliptic hypergeometric integrals one can describe uniformly the 't Hooft anomaly matching conditions [53] and the chiral symmetry breaking [54]. A direct consequence of formula (1.1) was used in topological field theories as well [39].
Another application of relation (1.1) has lead to important progress in the study of exactly solvable models of statistical mechanics. Namely, it has been shown to yield new solutions of the star-triangle relations either in functional [8] or operator forms [16]. Actually, the latter form of the star-triangle relation has been found long before as the integral Bailey lemma [47]. Using the results of [8], a correspondence between the quiver gauge theories and integrable lattice models such that the integrability emerges as a manifestation of the Seiberg type dualities has been established in [49].
Degenerations of the 2d spin system of [8] lead to many known models. For instance, the Faddeev-Volkov model [57,7] or its extension [49] can be obtained in this context as follows. One can reduce superconformal indices of 4d theories to the partition functions of 3d N = 2 theories [21]. This reduction leads to the equality of partition functions on the squashed sphere [29] of dual theories expressed in terms of the hyperbolic hypergeometric integral identities.
The star-triangle relation represents a particular form of the Yang-Baxter equations (YBE) standing behind the quantum integrable systems. Another form is the vertex type YBE associated with the integrable spin chains. A powerful techniques for solving such type of equations was developed in [14,15]. The elliptic beta integral (1.1) and related Bailey lemma [47] played a prominent role in building the most complicated known integral operator solutions of the YBE [16]. In particular, this approach has lead to a new rich class of finite-dimensional solutions of the YBE [11].
In this paper, we present a new solution of the star-triangle relation and other forms of YBE in terms of the basic hypergeometric identity presented in [42]. We relate the Yang-Baxter equations to three-dimensional supersymmetric dualities. The new solution corresponds to the generalized superconformal index of certain 3d N = 2 superconformal gauge theory having a distinguished form due to the contribution of monopoles [30,32,35,37]. Detailed presentation of this correspondence is given in the last section.

Notation and definitions
For q, z ∈ C, |q| < 1, we define the infinite q-product (2.1) The (normalized) q-gamma function of Jackson has the form [2] Γ(z; q) : with a similar convention for other generalized gamma functions in (1.1) and other relations below. We need the following q-hypergeometric identity.
Let us define the following generalized q-gamma function as a combination of four q-gamma functions and z m and a m : where a, z ∈ C and n, m ∈ Z. Lemma. One has the following inversion relation: Γ q (a, n; z, m)Γ q (b, −n; z, m) = 1, ab = q. (2.6) Proof. Consider the explicit form of the indicated product of Γ q -functions after the substitution b = q/a: Γ q (a, n; z, m)Γ q ( q a , −n; z, m) Using the relation (a; q) ∞ = (1 − a)(aq; q) ∞ , for n > m > 0 we can rewrite this expression as For other possible values of the integers n and m one gets the same result due to the properties of q-Pochhammer symbols. Now we can rewrite the above q-beta sum-integral in the following compact form.
Introduce the integral-sum operator of the form where Γ q (tx ±1 , ±n; z, m) : = Γ q (tx, n; z, m)Γ q (tx −1 , −n; z, m) and f m (z) is an arbitrary sequence of holomorphic functions. We note that the following permutational symmetries hold true Γ q (tx ±1 , ±n; z, m) = Γ q (tz ±1 , ±m; x, n), Following the original integral generalization [47,48] of the Bailey chains techniques [2], we introduce the notion of Bailey pairs in the present context.
Definition. We say that two sequences of functions α m (z; t) and β m (z; t), m ∈ Z, of complex variables z and t form a Bailey pair with respect to the parameter t if they are related by the integral-sum transform (3.3), β n (x; t) = M (t) x,n;z,m α m (z; t). (3.7) Here we assume that |tx|, |t/x| < 1 and other regions of parameters are reached by the analytical continuation. where s, y ∈ C, l ∈ Z are arbitrary new parameters, form a Bailey pair with respect to the parameter st.
Proof. Let us substitute primed sequences into the relation and use the inversion D(t −1 ; y, l; x, k) = 1/D(t; y, l; x, k). This yields the operator identity M (s) w,k;z,m D(st; y, l; z, m)M (t) z,m;x,j = D(t; y, l; w, k)M (st) w,k;x,j D(s; y, l; x, j) (3.11) known as the star-triangle relation. It is a straightforward consequence of the Rosengren q-beta sum-integral. First we compute the expression on the left-hand side of (3.11) where we used the permutational symmetry of Γ q -function and have denoted a 1 = sw, n 1 = k, a 2 = s w , n 2 = −k, a 3 = q 1/2 y st , n 3 = l, a 4 = q 1/2 sty , n 4 = −l, a 5 = tx, n 5 = j, a 6 = t x , n 6 = −j.
which proves the required identity. We note that the derived solution of the star-triangle relation resembles structurally a different solution obtained in [33]. We stress that the parameters y and l are dummy variables in this construction, i.e. at each step of the walk along the lattice of Bailey pairs one can introduce further new parameters y, l → y , l → . . ..

Coxeter relations and the vertex type Yang-Baxter equation
Consider elementary transposition operators s j , j = 1, . . . , 5, acting on six parameters t = (t 1 , . . . , t 6 ): They generate the permutation group S 6 characterized by the Coxeter relations Define now five operators S j (t), j = 1, . . . , 5, acting on the three-index functions of three complex variables f n1,n2,n3 (z 1 , z 2 , z 3 ): We stress that all these operators depend on the ratios of parameters, S j (t) = S j (t j /t j+1 ). Let us prove that for an appropriate space of test functions the operators S j generate the group S 6 , provided their sequential action is defined via a cocycle condition S j S k := S j (s k (t))S k (t). For this it is necessary to verify the Coxeter relations Indeed, the latter relations are equivalent to algebraic properties of the Bailey lemma entries, in complete analogy with the elliptic hypergeometric case [16]. It is sufficient to establish them for S 1 and S 2 , others will follow by the symmetry. So, we have A substantially more complicated relation is needed for S 1 : for the holomorphic test functions satisfying the reflection symmetry f −m (y −1 ) = f m (y). This fact follows from the residue calculus. For t → 1 two pairs of poles approach the integration contour in M (t) z,m;y,j f j (y) from two sides and pinch it.
To resolve the singularity it is necessary to compute two residues which leads to the expression (f m (z) + f −m (z −1 ))/2, and the reflection symmetry reduces it to one term. We now substitute in the star-triangle relation (3.11) the constraint st = 1.
Using the inversion relation for D-function and D(1; z 1 , n 1 ; z 2 , n 2 ) = 1, the D-terms disappear on both sides and we obtain M (t −1 )M (t) = 1l. Finally, which is precisely the star-triangle relation.
Consider the tensor product of three infinite-dimensional (equal or different) spaces V 1 ⊗ V 2 ⊗ V 3 and associate with each space V j a pair of variables: the spectral parameter u j and the spin variable g j , respectively. Define R-operators R ik (u i , g i |u k , g k ) acting in a non-trivial way in the subspace V i ⊗ V k with the unity operator action in its complement. The vertex type YBE has the form Actually, the R-operators depend on the difference of spectral parameters, where we omitted dependence on the spin variables. Using this notation we can rewrite YBE in the more conventional form It is convenient to single out the permutation operators from the R-operator where the operator P ik interchanges the spaces, Removing these permutation operators from the Yang-Baxter equation (4.7) yields the relation where one sees only two R-operators, R 12 and R 23 . Let us fix the spaces V j as copies of the infinite bilateral sequences of meromorphic functions f j (z), j ∈ Z. Then the triple tensor product of interest takes the form . Define now the composite operators acting in this space R 12 (t), Denoting (4.14) one can identify and check that these operators depend only on the difference of spectral parameters u − v and v − w, respectively.
Proof. Substituting the explicit forms of the R-operators into equality (4.11), we come to the relation which is easily checked using only the cubic Coxeter relations for operators S j in complete analogy with the cases considered in [15,16].

A new two-dimensional solvable lattice model
Let us apply the operator relation (3.11) to a product of the Kronecker and Dirac delta-functions which remove integration over the x-variable and summation over the index j. This yields the functional star-triangle relation of the form where W a (x, j; u, m) = Γ q (e 2πi(a−ξ±x±u) ), e −4πiξ := q, (5.2) and We now define a two-dimensional lattice model associated with this relation. Consider a honeycomb lattice with the spins denoted by labels x, u, w, etc which seat in vertices. Each spin has a discrete internal degree of freedom denoted as m, j, k, l, etc (the monopole number). Neighboring spins (x, j) and (u, m) interact along the edges connecting them with the energy determined by the Boltzmann weight W a (x, j; u, m). The function ρ m (u) describes the self-energy of spins, and ξ is called the crossing parameter. In this picture the "integration-plus-summation" in the star-triangle relation (5.1) means computation of the partition function for an elementary star-shaped cell with contributions coming from all possible values of the continuous spin u sitting in the central vertex and all possible values of the magnetic charge m. The honeycomb lattice can be transformed using the startriangle relation to triangular and square lattices.
Compose now N × M sized two-dimensional square lattice of spins and associate with each horizontal edge the weight W a (x, j; u, m) and with the vertical one the weight W ξ−a (x, j; u, m). Then the partition function of such homogeneous spin system with the internal spin energy ρ m (u) has the form Conjecturally, similar to the models considered in [7,8,49], the value of κ(a) can be found using the reflection method [6]. Namely, one renormalizes the Bolztmann weights  We note that this is the product of the numerator and denominator of the elliptic gamma function. One has the following inversion relation Define the composite function f (x √ pq; p 2 , q) . (5.14) It satisfies the equations Using these relations we can set m(a) = µ(e 4πia ; q, q) = (q 2 e 4πia , qe −4πia ; q, q 2 ) ∞ (qe 4πia , q 2 e −4πia ; q, q 2 ) ∞ (5. 16) and see that this function satisfies the unitarity condition m(−a) = 1 m(a) (5.17) and the key starting equation (5.11). So, − log m(a) provides the explicit expression for the free energy per spin of the discussed two-dimensional "spin" model. For the model with the Boltzmann weights (5.7) the free energy is equal to zero.

Star-star relations and an IRF model Boltzmann weight
We consider the simplest consequence of the Bailey chain of identities for sums of q-hypergeometric integrals described above following the elliptic hypergeometric pattern [47]. For this we use the evident explicit Bailey pair, following from the integration formula (2.9). Namely, let us choose α m (z, t) = 4 j=1 Γ q (a j , n j ; z, m), where a j are arbitrary parameters. Substituting this expression into the integral transformation (3.7), imposing the constraint 4 j=1 n j = 0, and choosing t 2 = q 4 j=1 a −1 j , we derive from the Rosengren identity that We now take definitions of the Bailey lemma entries (3.8) and (3.9) and substitute them into the relation β k (w; st) = M (st) w,k;x,j α j (x; st). This yields the following explicit symmetry transformation law V (a, n; q) = V (ã, n; q) Remind also the constraint t 2 4 j=1 a j = q. Conjecture. Let us take the V -function, whose parameters a j , n j satisfy only the balancing conditions indicated in the definition (6.4) and an additional constraint 4 j=1 n j = 0. Then we conjecture that it satisfies the symmetry transformation (6.3), where ã j = εa j , j = 1, 2, 3, 4 a j = ε −1 t j , j = 5, 6,7,8 ; ε = q a 1 a 2 a 3 a 4 = a 5 a 6 a 7 a 8 q . (6.7) Indeed, using the relation one can verify that a repetition of the transformation (6.3), (6.7) returns back the original V -function, i.e. we deal with a reflection. The map a j →ã j is the key reflection extending the Weyl group S 8 of the root system A 7 to the Weyl group of the exceptional root system E 7 . However, because of the presence of integers n j and the constraint 4 j=1 n j = 0 we do not have the full W (E 7 ) symmetry of the V -function yet. Interestingly, even in this reduced case the Bailey chains techniques yields the symmetry transformation (6.3) only when a pair of integers is forced to take particular values n i + n j = n k + n l = 0, i = j = k = l, which contrasts with the elliptic hypergeometric V -function case [46,48].
Consider a 2d checkerboard lattice [4] where each "black" site has four "white" neighbours and, vice versa, each "white" site has four "black" neighbours. Ascribe to each edge connecting the white and black sites the Boltzmann weight W αi (5.2) with arbitrary parameters α i subject to the constraint 4 j=1 α j = 2ξ. An IRF model is obtained when we integrate out the one-color lattice spins. The Boltzmann weight of the corresponding elementary "cell" containing four vertices determines the energy of this square face. It is given obviously by a special case of the general V -function introduced above when all integer variables n j are paired by the relation n 2i−1 + n 2i = 0. Then, completely similarly to [49], the symmetry transformation (6.3) has now the interpretation as a star-star relation [4]. As shown by Baxter [5] knowledge of the star-star relations automatically leads to the YBE for IRF models.

IRF Yang-Baxter equation with spectral parameter
The Yang-Baxter equation for IRF models (or SOS-type YBE) [12,13] associated with 3d superconformal indices has the following form where we introduced for convenience the shorthand notation for spectral parameters t ij = (t i , t j ). The following statistical weight satisfies this equation It is substantially equal to the V -function (6.4) with particular constraints on the integers n = (±A, ±B, ±C, ±D). We now form the following composite function defined by 6 integrations and 6 discrete summations Finally, we apply the inverse triangle-star relation to the last line product of Γ qfunctions in the square brackets and obtain the left-hand side expression in equation (7.1). The right-hand side expression of this IRF YBE is obtained after performing first the integrations over z 2 , z 4 , z 6 and summations over m 2 , m 4 , m 6 and an application of a similar triangle-star transformation.

The 3d superconformal index and duality
In this section we briefly review some necessary details about superconformal index of three-dimensional supersymmetric theories with four supercharges (N = 2 theories). Here we mainly follow the references [30,32,37].
The superconformal index first was proposed for four-dimensional theories [40,36] and later extended to other dimensions. Three-dimensional index was computed using localization technique by Kim [35] for ABJM theory and it was generalized to N = 2 theories by Imamura and Yokoyama [30] (with a topological symmetry contribution amendment pointed out in [37]). The superconformal index of threedimensional N = 2 superconformal field theory is a twisted partition function defined on S 2 × S 1 [10,35,30]: where F is the fermion number, ∆ is the energy, j 3 is the third component of the angular momentum around S 2 , and F j are the Cartan generators of the global flavor symmetry. The trace is taken over the Hilbert space of the theory. Here, Q is a supersymmetric charge with quantum numbers ∆ = 1 2 and j 3 = − 1 2 and the R-charge is normalized in a such way that Q has R-charge equal to 1. The supercharges Q † = S and Q satisfy the following anti-commutation relation (the full algebra can be found in many papers, for instance, in [19]) where R is the R-charge. Only the BPS states satisfying the bound H = 0 contribute to the index, therefore the index is β-independent.
Using the localization technique [38] the superconformal index can be computed exactly [35,30], and it reduces to the following matrix integral Let us unpack this expression. The summation is over magnetic fluxes on twosphere which appears in the localization procedure as a contribution of monopoles. The dµ G (z) is the Haar measure of the gauge group G. The prefactor |W m | = k i=1 (rankG i )! is the order of the Weyl group of G which is "broken" by the monopoles to the product G 1 × G 2 × · · · × G k . If the theory has the Chern-Simons term it contributes to the index as where tr CS stands for the trace including the Chern-Simons levels, g runs over the maximal torus of the gauge group and m takes values in the Cartan of the gauge group and parametrizes magnetic monopole charges. There is also the one-loop correction to the Chern-Simons term where Φ and ρ∈RΦ represent summations over all chiral multiplets and all weights of the representation R Φ of the gauge group. The term q 0j is the zeropoint contribution to the energy, and 0 is the Casimir energy of the vacuum state on two-sphere with magnetic flux m, where α∈G is the sum over all roots of G and r Φ is the R-charge of the chiral multiplet. The single letter index ind(z, t, x; m) gets contributions from chiral and vector multiplets The single particle index enters the full superconformal index (8.3) via the "plethystic exponential" [9,24] exp ∞ n=1 1 n ind(z n , t n , x n ; m) .
The three-dimensional superconformal index can be written in terms of sums of basic hypergeometric integrals, see e.g. [37,32,25,26]. For instance, let us consider the N = 2 theory with U (N ) gauge group. Then the chiral multiplet Φ with R-charge r Φ in the fundamental representation of the gauge group contributes to the index as 10) and the corresponding vector superfield contributes as Our main object of interest is the so-called generalized superconformal index which includes integer parameters corresponding to global symmetries. In [32] Kapustin and Willett pointed out that it is possible to generalize the superconformal index of 3d N = 2 theory by considering a non-trivial background gauge field coupled to the global symmetries of the theory. Then the superconformal index includes new discrete parameters for global symmetries (one can obtain this expression using the localization technique [22]). For instance, the contribution of the chiral multiplet (8.10) in this case gets the following form where the parameters n j are new discrete variables, and the contribution of gauge fields remains the same. The general expression for such an index has the following form We do not write the contribution of the Chern-Simons term, since we consider theories without this term. We now want to describe the two-dimensional solvable lattice models discussed above in the context of supersymmetric dualities for 3d N = 2 supersymmetric gauge theories. The duality we study is very similar to the initial Seiberg duality for N = 1 four-dimensional supersymmetric quantum chromodynamics. The following two theories are dual to each other [27]: • 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 (gaugeinvariant "mesons") in the 15-dimensional totally antisymmetric tensor representation of the flavor group. More precisely, the first interacting gauge fields theory flows in the infrared limit to the second one. This duality was considered in [56]. The authors calculated the three-dimensional ellipsoid partition functions for dual theories by applying the reduction procedure of [21] to the models considered in [50]. The ordinary superconformal index of the "theory A" with enhanced symmetry was presented in [17] (see also [28] for the N f = 4 case and [25,26] for the similar theory with the broken gauge group). The duality between theories A and B leads to the equality of corresponding superconformal indices expressed by the following q-hypergeometric identity [27] (after denoting with the balancing condition 6 j=1 a j = q, and 6 j=1 n j = 0 . (8.15) This condition is imposed by the effective superpotential W = ηX for the theory A, where X is a monopole operator and η is the four-dimensional instanton factor, which breaks a part of the symmetry (for details, see [1]). Using the relation [18] one can obtain the q-beta sum-integral (2.4) from (8.14). Similarly, the full symmetry transformation (6.3) is a consequence of a duality of two 3d theories with N f = 8. One can guess that there exist proper analogs of all elliptic hypergeometric integral identities described in [48,50,51,52] for sums of qhypergeometric integrals associated with 3d dualities. Actually, the latter dualities are easily found using the reduction of 4d superconformal indices to 3d partition functions [21] which naturally leads to conjectural equalities of corresponding 3d superconformal indices.
By breaking the flavor symmetry to SU (2) × SU (2) × SU (2) in (8.14) we obtain the star-triangle relation (7.3). Then the expression (7.2) corresponds to the generalized superconformal index of a 3d N = 2 theory with the gauge group G = SU (2) and the flavor group F = SU (2) × SU (2) × SU (2) × SU (2). In this picture, the SOS-type YBE (7.1) is nothing else than the equality of superconformal indices of two dual 3d N = 2 supersymmetric quiver gauge theories presented in Fig. 1, where the boxes correspond to SU (2) flavor subgroups and the circles represent SU (2) gauge subgroups.
We note that relation (4.5) describes the chiral symmetry breaking similarly to the 3d partition function case [54]. Indeed, it assumes the following sum-integral evaluation m∈Z [d m z] Γ q (t −1 x ±1 , ±n; z, m)Γ q (ty ±1 , ±j; z, m) = δ(φ y + φ x )δ n+j,0 + δ(φ y − φ x )δ n−j,0 q −j (1 − q j y 2 )(1 − q j y −2 )(1 − t 2 )(1 − t −2 ) , (8.17) where y = e 2πiφy and x = e 2πiφx and δ(φ) is the periodic Dirac delta function with period 1, δ(φ + 1) = δ(φ). On the left-hand side of equality (8.17) we have the 3d superconformal index of a theory with SU (2) gauge group and N f = 4 chiral fields with the naive flavor group SU (2) × SU (2). However, as follows from the the right-hand side expression, the true flavor group is (SU (2) × SU (2)) diag and the superconformal index has, actually, a non-zero support only on the corresponding subset of fugacities. This is precisely the manifestation of chiral symmetry breaking in confining theories similar to the 3d partition functions case [54]. A more detailed and rigorous consideration of this relation between indices and spontaneous breaking of global symmetries is needed, in particular, for the case when one has originally the full naive SU (4) flavor group which is broken to SP (4) group.
is supported in part by the SFB 647 "Raum-Zeit-Materie. Analytische und Geometrische Strukturen", the Research Training Group GK 1504 "Mass, Spectrum, Symmetry" and the International Max Planck Research School for Geometric Analysis, Gravitation and String Theory. After completion of this work we have known that the functional form of the star-triangle relation of Sect. 5 was considered in [34] and a relation of 3d superconformal indices to integrable lattice systems was discussed in [58].