Gerbes and Lie Groups

TL;DR

This paper reviews the theory of bundle gerbes, especially their connections to Lie groups, loop groups, and applications in two-dimensional field theories, highlighting their algebraic structures and geometric significance.

Contribution

It provides a comprehensive overview of bundle gerbes on Lie groups, including their constructions, algebraic structures, and applications in physics, which is a valuable synthesis of geometric and algebraic aspects.

Findings

Canonical bundle gerbes exist on compact Lie groups.

Connections between gerbes and loop groups are established.

Applications to Wess-Zumino terms in field theories are discussed.

Abstract

We present a review of bundle gerbes, emphasizing their relations to Lie groups. Indeed, compact Lie groups do not only carry the structure of a Riemannian manifold, but also canonical families of bundle gerbes. We recall the construction of these bundle gerbes and their relation to loop groups. We discuss several algebraic structures for bundle gerbes with connection such as Jandl structures, gerbe modules and gerbe bimodules, and indicate their applications to Wess-Zumino terms in two-dimensional field theories.