Introduction to the theory of elliptic hypergeometric integrals

TL;DR

This paper reviews the properties and generalizations of elliptic hypergeometric integrals, highlighting their mathematical structure and applications in supersymmetric quantum field theories.

Contribution

It provides a comprehensive overview of elliptic hypergeometric integrals, including their key properties, generalizations, and connections to physics and algebraic structures.

Findings

Elliptic hypergeometric integrals generalize classical beta and hypergeometric functions.

They are connected to superconformal indices in quantum field theories.

The paper outlines the elliptic Fourier transformation and Bailey lemma techniques.

Abstract

We give a brief account of the key properties of elliptic hypergeometric integrals -- a relatively recently discovered top class of transcendental special functions of hypergeometric type. In particular, we describe an elliptic generalization of Euler's and Selberg's beta integrals, elliptic analogue of the Euler-Gauss hypergeometric function and some multivariable elliptic hypergeometric functions on root systems. The elliptic Fourier transformation and corresponding integral Bailey lemma technique is outlined together with a connection to the star-triangle relation and Coxeter relations for a permutation group. We review also the interpretation of elliptic hypergeometric integrals as superconformal indices of four dimensional supersymmetric quantum field theories and corresponding applications to Seiberg type dualities.