Quantization of the Modular Functor and Equivariant Elliptic cohomology

TL;DR

This paper develops a categorical quantization framework for loop group representations and constructs a sheaf over an elliptic curve, linking it to equivariant elliptic cohomology of G-spaces.

Contribution

It introduces a novel categorical quantization approach for loop group representations and connects it to equivariant elliptic cohomology via a holomorphic sheaf construction.

Findings

Constructs a holomorphic sheaf over a universal elliptic curve valued in dominant K-theory.

Shows each stalk of the sheaf is a cohomological functor of the G-space M.

Provides a model for equivariant elliptic cohomology consistent with existing theories.

Abstract

Given a simple, simply connected compact Lie group G, let M be a G-space. We describe the quantization of the category of positive energy representations of the loop group of G at a given level and parametrized over the loop space LM. This procedure is described in terms of dominant K-theory of the loop group evaluated on the phase space given by the tangent bundle of basic gauge fields about a circle (parametrized over LM and with gauge symmetries given by the loop group LG). As such, our construction gives rise to a (categorical) BV-BRST type quantization for families of rational 2d CFTs with gauge symmetries 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 and others.