Representations of affine Lie algebras, parabolic differential equations, and Lame functions

Abstract

We consider correlation functions for the Wess-Zumino-Witten model on the torus with the insertion of a Cartan element; mathematically this means that we consider the function of the form $F=\Tr (Φ_1 (z_1)\ldots Φ_n (z_n)q^{-\d}e^{h})$ where $Φ_i$ are intertwiners between Verma modules and evaluation modules over an affine Lie algebra $\ghat$, $\d$ is the grading operator in a Verma module and $h$ is in the Cartan subalgebra of $\g$. We derive a system of differential equations satisfied by such a function. In particular, the calculation of $q\frac{\d} {\d q} F$ yields a parabolic second order PDE closely related to the heat equation on the compact Lie group corresponding to $\g$. We consider in detail the case $n=1$, $\g = \sltwo$. In this case we get the following differential equation ($q=e^{πıτ}$): $ \left( -2\piı(K+2)\frac{\d}{\dτ} +\frac{\d^2}{\d x^2}\right) F = (m(m+1)\wp(x+\fracτ{2}) +c)F$, which for $K=-2$ (critical level) becomes Lamé equation. For the case $m\in\Z$ we derive integral formulas for $F$ and find their asymptotics as $K\to -2$, thus recovering classical Lamé functions.