Constructing the Virasoro groups using differential cohomology

TL;DR

This paper introduces a new geometric method to construct Virasoro groups as central extensions of diffeomorphism groups of the circle, using differential cohomology and lifts of characteristic classes.

Contribution

It provides a novel geometric construction of Virasoro groups via differential cohomology, answering a question posed by Freed and Hopkins.

Findings

Constructed Virasoro groups using differential lifts of the first Pontryagin class.

Established a geometric framework for central extensions of diffeomorphism groups.

Addressed and answered a question in the mathematical physics community.

Abstract

The Virasoro groups are a family of central extensions of $\mathrm{Diff}^+(S^1)$, the group of orientation-preserving diffeomorphisms of $S^1$, by the circle group $\mathbb T$. We give a novel, geometric construction of these central extensions using "off-diagonal" differential lifts of the first Pontryagin class, thus affirmatively answering a question of Freed-Hopkins.