Central extensions of gauge transformation groups of higher abelian gerbes

TL;DR

This paper constructs a central extension of the smooth Deligne cohomology group for higher-dimensional manifolds, revealing non-trivial cases and an analogue of the Segal-Witten reciprocity law, extending concepts from loop groups.

Contribution

It generalizes the central extension of loop groups to higher-dimensional manifolds using smooth Deligne cohomology, identifying when these extensions are non-trivial.

Findings

Central extension is trivial for dimensions 3, 7, 11,...

Non-trivial for dimensions 1, 5, 9,...

Established an analogue of the Segal-Witten reciprocity law for non-trivial cases

Abstract

We construct a central extension of the smooth Deligne cohomology group of a compact oriented odd dimensional smooth manifold, generalizing that of the loop group of the circle. While the central extension turns out to be trivial for a manifold of dimension 3, 7, 11,..., it is non-trivial for 1, 5, 9,.... In the case where the central extension is non-trivial, we show an analogue of the Segal-Witten reciprocity law.