Presheaves of superselection structures in curved spacetimes

TL;DR

This paper develops a categorical framework for superselection structures in curved spacetimes, linking them to group bundles and gerbes, and explores their role as gauge symmetries in quantum field theory.

Contribution

It introduces presheaves of symmetric tensor categories to describe superselection sectors and connects them to Tannaka duality and gauge group candidates in curved spacetimes.

Findings

Superselection structures form categories of sections of presheaves of symmetric tensor categories.

Embedding functors relate superselection structures to locally constant group bundles acting on C*-algebras.

Examples of gerbes of C*-algebras constructed from fundamental group representations are provided.

Abstract

We show that superselection structures on curved spacetimes, that are expected to describe quantum charges affected by the underlying geometry, are categories of sections of presheaves of symmetric tensor categories. When an embedding functor is given, the superselection structure is a Tannaka-type dual of a locally constant group bundle, which hence becomes a natural candidate for the role of gauge group. Indeed, we show that any locally constant group bundle (with suitable structure group) acts on a net of C*-algebras fulfilling normal commutation relations on an arbitrary spacetime. We also give examples of gerbes of C*-algebras, defined by Wightman fields and constructed using projective representations of the fundamental group of the spacetime, that we propose as solutions for the problem that existence and uniqueness of the embedding functor are not guaranteed.