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Elliptic beta integrals and solvable models of statistical mechanics

V. P. Spiridonov

TL;DR

The paper develops a unified framework linking elliptic hypergeometric integrals to solvable statistical-mechanical models via star-triangle and star-star relations. It centers on the elliptic beta integral and the $V$-function, whose symmetries realize Yang-Baxter-type dualities and encode Seiberg dualities as integral identities. A hyperbolic degeneration yields a Faddeev–Volkov-type STR, while partition-function analysis reveals how lattice models emerge from restricted elliptic integrals and how normalization sets the free energy per edge. The work suggests deep connections between 4D supersymmetric dualities, 2D integrable systems, and higher-rank generalizations, pointing to rich future directions in elliptic Selberg-type theories and their physical realizations.

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.

Elliptic beta integrals and solvable models of statistical mechanics

TL;DR

The paper develops a unified framework linking elliptic hypergeometric integrals to solvable statistical-mechanical models via star-triangle and star-star relations. It centers on the elliptic beta integral and the -function, whose symmetries realize Yang-Baxter-type dualities and encode Seiberg dualities as integral identities. A hyperbolic degeneration yields a Faddeev–Volkov-type STR, while partition-function analysis reveals how lattice models emerge from restricted elliptic integrals and how normalization sets the free energy per edge. The work suggests deep connections between 4D supersymmetric dualities, 2D integrable systems, and higher-rank generalizations, pointing to rich future directions in elliptic Selberg-type theories and their physical realizations.

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.

Paper Structure

This paper contains 7 sections, 157 equations, 4 figures.

Table of Contents

  1. The simplest elliptic hypergeometric integrals
  2. The elliptic beta integral STR solution and star-star relation
  3. A hyperbolic beta integral STR solution
  4. Partition functions
  5. Conclusion
  6. The modified $q$-gamma function
  7. General multiple gamma functions

Figures (4)

  • Figure 1: The star-triangle relation.
  • Figure 2: A honeycomb-triangluar lattice transformation induced by the star-triangle relation.
  • Figure 3: A triangluar-rectangular lattice transformation induced by the star-triangle relation.
  • Figure 4: A star-star relation for the square lattice. Additional Boltzmann weights $p$ and $m$ are indicated by edges connecting corresponding vertices on the right-hand side.