M2-branes, Higher Form Symmetries and 1-Gerbes

TL;DR

The paper investigates higher-form symmetries of the bosonic M2-brane on $\mathcal{M}_9 \times T^2$, uncovering a mixed 't Hooft anomaly when gauging these symmetries in the presence of background fields. Anomaly cancellation is achieved by a 4D anomaly inflow term of BF type, involving flat $U(1)$ connections and a torsion $\,\mathcal{G}_1^{\nabla_c}\,$-gerbe on the worldvolume, which together with nontrivial winding breaks the continuous $U(1)$ HFS to discrete subgroups and imposes a worldvolume flux condition. The work introduces Wilson-surface topological operators as holonomies of the gerbe structures (Hol$_{\nabla}$) and develops a detailed operator algebra in which monopole and winding sectors act nontrivially on dressed operators and Wilson surfaces. The findings link discrete winding and flux/monopole sectors to the gerbe data and suggest implications for spectral discreteness in toroidally compactified or supersymmetric M2-brane theories, while outlining future directions including the interaction with Wess-Zumino terms and ABJ-type anomalies.

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.