The "Parity" Anomaly On An Unorientable Manifold

TL;DR

This paper fully characterizes the parity anomaly for fermions on unorientable 2+1 dimensional manifolds, with applications to topological superconductors and M-theory, expanding understanding beyond orientable cases.

Contribution

It provides a comprehensive description of the parity anomaly on unorientable manifolds, extending previous studies limited to orientable spaces, and explores implications for topological phases and M-theory.

Findings

Describes the parity anomaly on unorientable manifolds in 2+1 dimensions.

Shows how the anomaly influences the construction of gapped boundary states in topological superconductors.

Ensures the consistency of M-theory membrane path integrals on unorientable worldvolumes.

Abstract

The "parity" anomaly -- more accurately described as an anomaly in time-reversal or reflection symmetry -- arises in certain theories of fermions coupled to gauge fields and/or gravity in a spacetime of odd dimension. This anomaly has traditionally been studied on orientable manifolds only, but recent developments involving topological superconductors have made it clear that one can get more information by asking what happens on an unorientable manifold. In this paper, we give a full description of the "parity" anomaly for fermions coupled to gauge fields and gravity in $2+1$ dimensions on a possibly unorientable spacetime. We consider an application to topological superconductors and another application to M-theory. The application to topological superconductors involves using knowledge of the "parity" anomaly as an ingredient in constructing gapped boundary states of these systems and in particular in gapping the boundary of a $\nu=16$ system in a topologically trivial fashion. The application to M-theory involves showing the consistency of the path integral of an M-theory membrane on a possibly unorientable worldvolume. In the past, this has been done only in the orientable case.