Presheaves of symmetric tensor categories and nets of $C^{\ast }$-algebras

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 $C\ast$-algebras, defined on the given base.