Differential Geometry of Relative Gerbes

TL;DR

This thesis develops the theory of relative gerbes for smooth manifold maps, classifies their equivalence classes via relative cohomology, and applies this to pre-quantization in symplectic geometry.

Contribution

It introduces the concept of relative gerbes, classifies them using relative cohomology, and connects this to pre-quantization of Lie group-valued moment maps.

Findings

Classifies relative gerbes by relative integral cohomology in degree three.

Establishes an equivalence between pre-quantization of Lie group-valued moment maps and infinite-dimensional Hamiltonian loop group spaces.

Develops the differential geometry framework for relative gerbes.

Abstract

This thesis introduces the notion of "relative gerbes" for smooth maps of manifolds, and discusses their differential geometry. The equivalence classes of relative gerbes are classified by the relative integral cohomology in degree three. Furthermore, by using the concept of relative gerbes, the pre-quantization of Lie group-valued moment maps is developed, and its equivalence with the pre-quantization of infinite-dimensional Hamiltonian loop group spaces is established.