Modular transformations of the elliptic hypergeometric functions, Macdonald polynomials, and the shift operator

TL;DR

This paper explores the action of the modular group on elliptic hypergeometric functions related to conformal blocks in CFT, providing explicit formulas involving Macdonald polynomials at roots of unity.

Contribution

It introduces explicit formulas for the modular group action on elliptic hypergeometric functions using Macdonald polynomials, connecting conformal blocks and special functions.

Findings

Derived formulas for modular transformations in terms of Macdonald polynomial values

Established a link between elliptic hypergeometric functions and conformal blocks in CFT

Provided computational tools for analyzing modular actions on these functions

Abstract

We consider the space of elliptic hypergeometric functions of the sl_2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl_2 WZW model of CFT. The modular group acts on this space. We give formulas for the matrices of the action in terms of values at roots of unity of Macdonald polynomials of the sl_2 type.