The basic bundle gerbe on unitary groups

TL;DR

This paper extends the construction of the basic bundle gerbe from SU(n) to various unitary groups, providing explicit connections, curvings, and calculating the Dixmier-Douady class using advanced operator calculus.

Contribution

It generalizes the basic bundle gerbe construction to a broader class of unitary groups with explicit geometric data and cohomological invariants.

Findings

Constructed explicit connections and curvings for the extended gerbes.

Calculated the real Dixmier-Douady class for these gerbes.

Demonstrated the use of holomorphic functional calculus in the construction.

Abstract

We consider the construction of the basic bundle gerbe on SU(n) introduced by Meinrenken and show that it extends to a range of groups with unitary actions on a Hilbert space including U(n), diagonal tori and the Banach Lie group of unitary operators differing from the identity by an element of a Schatten ideal. In all these cases we give an explicit connection and curving on the basic bundle gerbe and calculate the real Dixmier-Douady class. Extensive use is made of the holomorphic functional calculus for operators on a Hilbert space.