Pontrjagin duality on multiplicative Gerbes

TL;DR

This paper develops a framework for multiplicative gerbes over topological groups using Segal-Mitchison cohomology, introduces Pontrjagin duality for these gerbes, and demonstrates their categorical equivalences and examples across various groups.

Contribution

It introduces a novel notion of Pontrjagin duality for multiplicative gerbes and explores their categorical properties and examples.

Findings

Pontrjagin dual multiplicative gerbes have equivalent representation categories.

Their monoidal centers are also equivalent.

Examples include finite, discrete, compact, and non-compact Lie groups.

Abstract

We use Segal-Mitchison's cohomology of topological groups to define a convenient model for topological gerbes. We introduce multiplicative gerbes over topological groups in this setup and we define its representations. For a specific choice of representation, we construct its category of endomorphisms and we show that it induces a new multiplicative gerbe over another topological group. This new induced group is fibrewise Pontrjagin dual to the original one and therefore we called the pair of multiplicative gerbes `Pontrjagin dual'. We show that Pontrjagin dual multipliciative gerbes have equivalent categories of representations and moreover, we show that their monoidal centers are equivalent. Examples of Pontrjagin dual multiplicative gerbes over finite and discrete, as well as compact and non-compact Lie groups are provided.