Presheaves of symmetric tensor categories and nets of C*-algebras

TL;DR

This paper explores presheaves of symmetric tensor categories in the context of algebraic quantum field theory, linking superselection sectors to holonomy representations, characteristic classes, and gerbes of C*-algebras, revealing complex gauge structures.

Contribution

It introduces a framework connecting presheaves of tensor categories with holonomy, characteristic classes, and gerbes, extending the understanding of superselection sectors in quantum field theory.

Findings

Section categories are Tannaka duals of locally constant group bundles.

Superselection sectors define holonomy representations and gerbes.

Existence and uniqueness of gauge groups are generally not guaranteed.

Abstract

Motivated by algebraic quantum field theory, we study presheaves of symmetric tensor categories defined over the base of a space, intended as a spacetime. Any section of a presheaf (that is, any "superselection sector", in the applications that we have in mind) defines a holonomy representation whose triviality is measured by Cheeger-Chern-Simons characteristic classes, and a non-abelian unitary cocycle defining a Lie group gerbe. We show that, given an embedding in a presheaf of full subcategories of the one of Hilbert spaces, the section category of a presheaf is a Tannaka-type dual of a locally constant group bundle (the "gauge group"), which may not exist and in general is not unique. This leads to the notion of gerbe of C*-algebras, defined on the given base.