Generalized Symmetries of QED$_4$ on Non-Spin Manifolds

Abstract

Generalized symmetries are sensitive to the tangential structure of spacetime. We study this interplay for massless QED on oriented non-spin four-manifolds, where the photon becomes a spin$^c$ connection and the path integral sums over spin$^c$ structures. The shifted flux quantization extends the magnetic 1-form symmetry into a $w_2$-twisted 1-form symmetry: its $2π$ rotation acts as a gravitational sign, fixed by the normal bundle of the surface the operator wraps. The obstruction to treating it as an ordinary magnetic 1-form symmetry is an order-two mixed magnetic-gravitational anomaly, which is classified by reduced oriented bordism and detected on the Wu manifold. In a fixed spin$^c$ sector, this anomaly is removable by a counterterm, so it is a property of the sum over spin$^c$ structures. We discuss its consequences for gapped, Coulomb, and Higgs phases. The chiral anomaly is governed by the index of the spin$^c$ Dirac operator, so rational chiral defects carry gravitational as well as electromagnetic responses. A spin$^c$ Hall topological field theory cancels the electromagnetic part but leaves a framing anomaly. At the level of anomaly inflow, this is canceled for every rational angle by stacking a neutral, generally noninvertible Witten-Reshetikhin-Turaev layer. The same framing subtlety affects the chiral defects of QED on curved spin manifolds. The complex projective plane illustrates each of these features explicitly.