Gerbes in Geometry, Field Theory, and Quantisation

TL;DR

This survey explores bundle gerbes and their applications in geometry, field theory, and quantisation, emphasizing their classification, surface holonomy, and role in geometric quantisation of various symplectic structures.

Contribution

It provides a comprehensive overview of bundle gerbes with connection, their classification via differential cohomology, and their applications in field theory and geometric quantisation.

Findings

Classification of bundle gerbes with connection via differential cohomology

Surface holonomy of bundle gerbes leads to a smooth bordism-type field theory

Application of bundle gerbes in geometric quantisation of symplectic and shifted symplectic forms

Abstract

This is a mostly self-contained survey article about bundle gerbes and some of their recent applications in geometry, field theory, and quantisation. We cover the definition of bundle gerbes with connection and their morphisms, and explain the classification of bundle gerbes with connection in terms of differential cohomology. We then survey how the surface holonomy of bundle gerbes combines with their transgression line bundles to yield a smooth bordism-type field theory. Finally, we exhibit the use of bundle gerbes in geometric quantisation of 2-plectic as well as 1- and 2-shifted symplectic forms. This generalises earlier applications of gerbes to the prequantisation of quasi-symplectic groupoids.