Kels' 2015 star-triangle relations for lattice models with continuous and discrete spin components, built on the lens elliptic gamma function and containing the Bazhanov-Sergeev master solution as the r=1r=1 case, raised four questions about their interpretation and extension. Later literature has settled two of them: the multicomponent star-star relations were proven as AnA_n/BCnBC_n lens elliptic hypergeometric sum/integral transformations (Kels-Yamazaki 2018), and the elliptic relation was identified with Seiberg duality of 4d N=1\mathcal{N}=1 lens superconformal indices, with its hyperbolic reduction corresponding to 3d lens partition functions; the 2d sphere partition functions correspond instead to the further gamma-function limit (41) taken after the r→∞r\to\infty relation (35), and the duality interpretation of the two-nome relation (35) itself remains open. What survives is the question Kels himself emphasized: whether the elliptic relation arises in the strong-coupling limit of some as yet unknown star-triangle relation, which would amount to a new summation/integration identity in more general special functions and a new supersymmetric duality, together with the precise relation between the two-nome r→∞r\to\infty identity and the pentagon identities for 3d N=2\mathcal{N}=2 superconformal indices. The surviving core of Kels' question remains open.

Background

An Ising-type lattice model assigns Boltzmann weights to edges, where each spin variable takes values in a continuous and/or discrete set. The model is exactly solvable if the weights satisfy the star-triangle relation (STR), a form of the Yang-Baxter equation: it equates the partition function of a star (three edges meeting at a central spin, with that spin summed or integrated out) with that of a triangle, and implies an infinite family of commuting transfer matrices. For spins parametrizing a circle with elliptic-function interactions, the natural building block is the elliptic gamma function G(z;p,q)G(z;p,q), a double infinite product depending on two elliptic nomes ∣p∣,∣q∣<1|p|,|q|<1. Bazhanov and Sergeev found the 'master solution' of the STR: weights built from the elliptic gamma function whose STR is equivalent to Spiridonov's elliptic beta integral (the exact evaluation of an integral whose integrand is built from the elliptic gamma function), and which contains essentially all previously known single-spin solutions as degenerations. Kels (2015) generalized the master solution by replacing the elliptic gamma function with the lens elliptic gamma function G(z,m;p,q,r)G(z,m;p,q,r), which depends on an additional positive integer rr and an integer variable mm defined modulo rr; the resulting lattice models have spins with a continuous angular component and a discrete Zr\mathbb{Z}_r component. The paper derives three star-triangle relations: the elliptic relation (25) for r>1r>1; a limit (35) as r→∞r\to\infty in which the discrete spins become unbounded integers while both nomes survive and the weights reduce to ordinary qq-Pochhammer products ∏k≥0(1−zqk)\prod_{k\geq 0}(1-zq^{k}); and a gamma-function relation (41) obtained by a further limit. The gauge/YBE correspondence of Yamazaki connects these identities to supersymmetric gauge theories: a Seiberg duality (an infrared equivalence between two distinct supersymmetric gauge theories) implies equality of their superconformal indices (certain protected partition functions on S1×S3S^1\times S^3), and for special quivers this equality of elliptic hypergeometric integrals is exactly an STR; the lens elliptic gamma function is the building block of the lens index on S1×S3/ZrS^1\times S^3/\mathbb{Z}_r. Analogous 3d N=2\mathcal{N}=2 indices on lens spaces Sb3/ZrS^3_b/\mathbb{Z}_r involve hyperbolic hypergeometric integrals, and their dualities yield pentagon identities, integral identities associated with the 2-3 Pachner move of a triangulation, such as the tetrahedron-index identity of Gahramanov and Rosengren. A key mechanism in this subject is the strong-coupling limit: Bazhanov, Mangazeev and Sergeev showed that the hyperbolic beta integral (the analog of the elliptic beta integral built from the hyperbolic gamma function, reached when one elliptic nome tends to zero) develops delta-function peaks at integer values of its variable, so that the strong-coupling limit of a continuous-spin STR with hyperbolic-gamma weights produces a model with discrete integer spins. Kels raised the converse question for his elliptic relation: does (25) itself arise as the strong-coupling limit of a still more general, as yet unknown STR, which would then also have implications for supersymmetric gauge theories and provide a new summation/integration identity, perhaps in terms of more general special functions?

Problem Statement

Determine the strong-coupling origin and the complete limit/duality interpretation of the lens elliptic star-triangle relation of Kels (2015). The task has two required parts, (A) and (B); part (B) in turn comprises two required sub-questions, (B1) and (B2). All three sub-questions must be addressed: an answer that addresses only some of them is a partial resolution of the task. (A) Does the elliptic star-triangle relation (25), the r>1r>1 relation whose weights are built from the lens elliptic gamma function, arise in the strong-coupling limit of some as yet unknown star-triangle relation? A constructive answer specifies the spin variables and Boltzmann weights of such a parent relation (presumably in terms of a new, more general special function), proves its star-triangle identity, and shows how the strong-coupling limit recovers the weights of (25) with their discrete Zr\mathbb{Z}_r spin components; following Kels, such a parent relation would be expected to have implications for supersymmetric gauge theories via the gauge/YBE correspondence and to provide an interesting new summation/integration identity, perhaps in terms of more general special functions. A proof that no such parent star-triangle relation exists also resolves (A). (B1) Establish the precise relation between the r→∞r\to\infty star-triangle relation (35), which retains both nomes p,qp,q and whose discrete spins become unbounded integers, and the pentagon identities for 3d N=2\mathcal{N}=2 superconformal indices (the Gahramanov-Rosengren tetrahedron index). (B2) Determine whether (35) admits a direct interpretation as a duality identity for the r→∞r\to\infty reduction of the 4d lens superconformal index on S1×S3/ZrS^1\times S^3/\mathbb{Z}_r. A proof that no such relation between (35) and the pentagon identities holds, or that no such duality interpretation exists, resolves the corresponding sub-question.

References

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