M2-branes, Higher Form Symmetries and 1-Gerbes

TL;DR

This paper explores higher-form symmetries of M2-branes on compact spaces, revealing anomalies, symmetry breaking, and topological operators related to gerbes and flux conditions.

Contribution

It introduces a novel analysis of higher-form symmetries, gerbe structures, and topological operators in M2-brane theories, highlighting anomaly cancellation and symmetry breaking mechanisms.

Findings

Identification of a mixed 't Hooft anomaly obstructing symmetry gauging.

Demonstration of symmetry breaking from continuous to discrete due to winding and flux.

Construction of Wilson surface operators as topological invariants of gerbe structures.

Abstract

Higher-Form Symmetries (HFS) of a closed bosonic M2-brane formulated on a compactified target space $\mathcal{M}_9 \times T^2$ are investigated. We show that there is an obstruction to the gauging of these global symmetries in the presence of background fields, a mixed 't~Hooft anomaly. Its cancellation is obtained by the inflow term constructed in terms of gauge fields which are flat connections on a $U(1)$-principal bundle and a torsion $\mathcal{G}_1^{\nabla_c}$-gerbe on the M2-brane worldvolume. The effect of these gauge structures together with non trivial \textit{winding} embedding maps ensures the breaking of the continuous HFS $U(1)$ symmetry to a discrete subgroup and a worldvolume flux condition on the M2-brane. A Wilson surface, identified with the holonomy Hol$_\nabla$ one of the Gerbe structures, the flat $\mathcal{G}_1^{\nabla_c}$-gerbe, is naturally introduced as the topological operator characterizing the M2-brane. The resulting topological operators realize discrete symmetries associated with the \textit{winding} and the flux/\textit{monopole} sectors, and their operator algebra is well-defined: the \textit{monopole} operator acts non trivially on a \textit{vortex-dressed} operator, while the winding operator acts on the pullback of the Wilson surface.