Non-integral central extensions of loop groups

TL;DR

This paper investigates non-integral level central extensions of loop groups, demonstrating the existence of associated Lie groupoids for real levels and analyzing their bundle structures using categorified principal bundles.

Contribution

It extends the understanding of central extensions of loop groups to arbitrary real levels by constructing relevant Lie groupoids and examining their bundle properties.

Findings

Existence of Lie groupoids for each real level k

Construction of categorified principal bundles for these groupoids

Analysis of the bundle structure of the constructed groupoids

Abstract

It is well-known that the central extensions of the loop group of a compact, simple and 1-connected Lie group are parametrised by their level $k \in Z$. This article concerns the question how much can be said for arbitrary $k \in R$ and we show that for each $k$ there exists a Lie groupoid which has the level $k$ central extension as its quotient if $k \in Z$. By considering categorified principal bundles we show, moreover, that the corresponding Lie groupoid has the expected bundle structure.