Parallel transport of higher flat gerbes as an extended homotopy quantum field theory

Abstract

We prove that the parallel transport of a flat n − 1-gerbe on any given target space gives rise to an n-dimensional extended homotopy quantum field theory. In case the target space is the classifying space of a finite group, we provide explicit formulae for this homotopy quantum field theory in terms of transgression. Moreover, we use the geometric theory of orbifolds to give a dimension-independent version of twisted and equivariant Dijkgraaf-Witten models. Finally, we introduce twisted equivariant Dijkgraaf-Witten theories giving us in the 3-2-1-dimensional case a new class of equivariant modular tensor categories which can be understood as twisted versions of the equivariant modular categories constructed by Maier, Nikolaus and Schweigert.

’t Hooft Anomalies of Discrete Gauge Theories and Non-abelian Group Cohomology

“…The groupoid of symmetries acts only projectively on this state space. Using a result of [MW18] we show that the 2-cocycle twisting this projective representation is the transgression of θ to the groupoid of G-bundles. With this construction we provide an explicit demonstration of the anomaly inflow mechanism at the level of both partition functions and state spaces, which renders the composite bulk-boundary field theory free from anomalies.…”

“…We provide a brief summary of the construction of an invertible extended field theory E ω depending on an n-cocycle ω ∈ Z n (BD; U (1)) which is given in [MW18], to which we refer for more details and all proofs. We denote the corresponding unextended field theory by L ω .…”

“…In [MW18] it is shown that these coherence morphisms are closely related to the transgression of ω. They encode interesting physical properties as we will see in Section 4.1.…”

“…A more systematic way of constructing Z ω is given by applying the orbifold construction of [SW19] to the classical field theory L ω . In [MW18] the extended orbifold construction of [SW18a] is used to construct Dijkgraaf-Witten theories as extended field theories. Three-dimensional extended Dijkgraaf-Witten theories are also constructed in [Mor15].…”

“…In the case of trivial 2-cocycle they have been studied as extended field theories in [MNS12]. Extended G-equivariant Dijkgraaf-Witten theories with non-trivial 2-cocycles are constructed in [MW18].…”