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

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.