Orientation twisted homotopy field theories and twisted unoriented Dijkgraaf-Witten theory

TL;DR

This paper develops a new class of orientation twisted homotopy field theories to extend twisted Dijkgraaf-Witten theories to unoriented settings, introducing novel constructions for unoriented topological quantum field theories.

Contribution

It introduces orientation twisted homotopy field theories and an orientation twisted orbifold procedure for extending twisted Dijkgraaf-Witten theories to unoriented cases.

Findings

Constructed a lift of twisted G-Dijkgraaf-Witten theory to unoriented TQFTs.

Developed a new class of homotopy field theories called orientation twisted.

Introduced an orientation twisted orbifold procedure for unoriented topological field theories.

Abstract

Given a finite $\mathbb{Z}_2$-graded group $\hat{\mathsf{G}}$ with ungraded subgroup $\mathsf{G}$ and a twisted cocycle $\hat{\lambda} \in Z^n(B \hat{\mathsf{G}}; \mathsf{U}(1)_{\pi})$ which restricts to $\lambda \in Z^n(B \mathsf{G}; \mathsf{U}(1))$, we construct a lift of $\lambda$-twisted $\mathsf{G}$-Dijkgraaf--Witten theory to an unoriented topological quantum field theory. Our construction uses a new class of homotopy field theories, which we call orientation twisted. We also introduce an orientation twisted variant of the orbifold procedure, which produces an unoriented topological field theory from an orientation twisted $\mathsf{G}$-equivariant topological field theory.