Quantization of the modular functor and equivariant elliptic cohomology

Abstract

Given a simple, simply connected compact Lie group G, let M be a G-space. We study the quantization of the category of parametrized positive energy representations of the loop group of G at a given level. This procedure is described in terms of dominant K- theory (Ki) of the loop group evaluated on the space of basic classical gauge fields about a circle, for families of two dimensional conformal field theories parametrized over M. More concretely, we construct a holomorphic sheaf over a universal elliptic curve with values in dominant K-theory of the loop space LM, and show that each stalk of this sheaf is a cohomological functor of M. We also interpret this theory as a model of equivariant elliptic cohomology of M as constructed by Grojnowski (G).