The "Parity" Anomaly On An Unorientable Manifold

TL;DR

The paper extends the conventional parity anomaly in 2+1 dimensions to unorientable manifolds, showing a mod-4 constraint on the number of charge-1 Dirac fermions required for gauge-invariant quantization with time-reversal or reflection symmetry. It develops a precise eta-invariant framework (via the Dai–Freed construction) to determine when the fermion path integral depends on gauge data and identifies stable-triviality criteria for the associated real vector bundles, yielding concrete conditions such as $y\equiv 0\pmod{4}$ for U(1) with charges $n_i$. The authors then apply these results to two broad contexts: (i) gapped, ${\sf T}$-invariant boundaries of 3+1D topological superconductors, including detailed constructions at $\nu=16$ and nonperturbative gapping via gauge dynamics; and (ii) the consistency of M2-brane path integrals in M-theory on unorientable worldvolumes, introducing a refined flux quantization and a measure ${\mathcal Z}_U=|{\rm Pf}(\slashed{D})|\exp(-i\pi\eta_U/2)\exp(i\int_U G)$. The work thus links refined anomaly physics, cobordism-invariant bulk terms, and explicit boundary and membrane constructions, with implications for condensed matter, string theory, and M-theory consistency on unorientable backgrounds.

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 $ν=16$ 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.

Figures (4)

• Figure 1: Two manifolds $Y$ and $Y'$ glued along a component of their common boundary to make a manifold $Y^*$ that itself may have a boundary.
• Figure 2: Two manifolds $X$ and $X'$ with common boundary $W$ are glued together along their common boundary to build a compact manifold $X^*$ without boundary.
• Figure 3: Schematic depiction of a four-manifold $U=S^3\times S^1$.
• Figure 4: Like peas in a pod, the disjoint union of a four-manifold $X$ and a four-sphere $S^4$ is cobordant to their connected sum $X'$. The connected sum is defined by cutting an open ball out of each and gluing them together along their boundaries. The normal bundles are trivialized in the cutting and gluing region before this is done.