A higher Kac-Moody extension for two-dimensional gauge groups

TL;DR

This paper constructs a higher central extension of the smooth double loop group associated with a finite-dimensional Lie group, revealing a non-trivial degree 3 cohomology class and extending Kac-Moody theory to higher dimensions.

Contribution

It introduces a novel higher central extension for double loop groups using higher category theory, generalizing Kac-Moody extensions to two-dimensional gauge groups.

Findings

The constructed cohomology class is non-trivial in general.

The higher extension relates to the Kac-Moody extension of the single loop group.

Application to the noncommutative 2-torus group extension.

Abstract

Let $\Gamma$ be a finite dimensional Lie group and consider the smooth double loop group, i.e. the Fr\'echet Lie group of smooth maps from the 2-torus to $\Gamma$. For a finite dimensional Hilbert space V, let H denote the Hilbert space of vector valued $L^2$-functions on the 2-torus. The purpose of this paper is to construct a higher central extension of the smooth double loop group from the representation of the smooth double loop group on H induced by a smooth action of $\Gamma$ on V. This higher central extension comes from an action of the smooth double loop group on a 2-category and yields a group cohomology class of degree 3 on the smooth double loop group. We show by a concrete computation that this group cohomology class is non-trivial in general. We relate our higher central extension to the Kac-Moody extension of the smooth single loop group as a higher dimensional analogue of the latter. More generally, given a group G acting on a bipolarised Hilbert space, we apply higher category theory to construct a group cohomology class of degree 3 on G. As a second motivating example, we use these ideas to introduce a higher central extension of the group of invertible smooth functions on the noncommutative 2-torus.