Integral representation of solutions of the elliptic Knizhnik--Zamolodchikov--Bernard equations

TL;DR

This paper provides an integral representation for solutions of elliptic KZB equations, connecting conformal blocks, integrable systems, and theta function identities across simple Lie algebras.

Contribution

It introduces a general integral formula for elliptic KZB solutions applicable to all simple Lie algebras, linking conformal field theory and integrable systems.

Findings

Formulas for conformal blocks on a torus at positive integer levels

Asymptotic analysis yields eigenfunctions of Euler-Calogero-Moser systems

Derivation of new integral identities involving classical theta functions

Abstract

We give an integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations for arbitrary simple Lie algebras. If the level is a positive integer, we obtain formulas for conformal blocks of the WZW model on a torus. The asymptotics of our solutions at critical level gives eigenfunctions of Euler-Calogero-Moser integrable $N$-body systems. As a by-product, we obtain some remarkable integral identities involving classical theta functions.