The Weyl map and bundle gerbes

TL;DR

This paper introduces a new class of bundle gerbes called the Weyl bundle gerbe, explores their properties under the Weyl map, and compares their holonomy to existing gerbes, revealing they are not D-stably isomorphic.

Contribution

It defines the Weyl bundle gerbe on T x SU(n)/T and demonstrates its relation to the basic bundle gerbe on SU(n), providing new insights into their equivariant structures and holonomy.

Findings

The Weyl bundle gerbe is constructed on T x SU(n)/T.

Pullback of the basic bundle gerbe is stably isomorphic to the Weyl bundle gerbe.

The holonomy analysis shows these gerbes are not D-stably isomorphic.

Abstract

We introduce the notion of a general cup product bundle gerbe and use it to define the Weyl bundle gerbe on T x SU(n)/T. The Weyl map from T x SU(n)/T to SU(n) is then used to show that the pullback of the basic bundle gerbe on SU(n) defined by the second two authors is stably isomorphic to the Weyl bundle gerbe as SU(n)-equivariant bundle gerbes. Both bundle gerbes come equipped with connections and curvings and by considering the holonomy of these we show that these bundle gerbes are not D-stably isomorphic.