# Orientation Twisted Homotopy Field Theories and Twisted Unoriented Dijkgraaf–Witten Theory

Communications in Mathematical Physics, Jun 2019

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.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs00220-019-03478-5.pdf

Matthew B. Young. Orientation Twisted Homotopy Field Theories and Twisted Unoriented Dijkgraaf–Witten Theory, Communications in Mathematical Physics, 2019, 1-47, DOI: 10.1007/s00220-019-03478-5