Parallel Transport of Higher Flat Gerbes as an Extended Homotopy Quantum Field Theory

TL;DR

This paper demonstrates that parallel transport of flat higher gerbes yields extended homotopy quantum field theories, providing explicit formulas and new twisted Dijkgraaf-Witten models with applications to equivariant modular tensor categories.

Contribution

It establishes a link between flat higher gerbes and extended homotopy quantum field theories, with explicit formulas and new models for twisted Dijkgraaf-Witten theories.

Findings

Explicit formulas for homotopy quantum field theories from flat gerbes.

Dimension-independent twisted and equivariant Dijkgraaf-Witten models.

Introduction of twisted equivariant Dijkgraaf-Witten theories leading to new modular tensor categories.

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.