# Presheaves of symmetric tensor categories and nets of $C^*$-algebras

October 16, 2012

Motivated by algebraic quantum field theory, we study presheaves of symmetric
tensor categories defined on 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, provided
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.

Keywords:

Superselection Theory, QFT on curved spacetimes