Elliptic beta integrals and solvable models of statistical mechanics

TL;DR

This paper explores the elliptic beta integral's role in solvable statistical mechanics models, connecting elliptic hypergeometric integrals with dualities and star-triangle relations, and introduces new solutions and higher-dimensional analogues.

Contribution

It introduces a new Faddeev-Volkov type solution of the star-triangle relation and discusses higher-dimensional analogues, linking elliptic hypergeometric integrals to dualities in statistical models.

Findings

Elliptic beta integral interpreted as star-triangle relation.

New Faddeev-Volkov type solution of STR introduced.

Connections established between dualities and elliptic hypergeometric integrals.

Abstract

The univariate elliptic beta integral was discovered by the author in 2000. Recently Bazhanov and Sergeev have interpreted it as a star-triangle relation (STR). This important observation is discussed in more detail in connection to author's previous work on the elliptic modular double and supersymmetric dualities. We describe also a new Faddeev-Volkov type solution of STR, connections with the star-star relation, and higher-dimensional analogues of such relations. In this picture, Seiberg dualities are described by symmetries of the elliptic hypergeometric integrals (interpreted as superconformal indices) which, in turn, represent STR and Kramers-Wannier type duality transformations for elementary partition functions in solvable models of statistical mechanics.