M2-branes, Higher Form Symmetries and 1-Gerbes
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 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. The resulting topological operators realize discrete symmetries associated with the winding and the flux/monopole sectors, and their operator algebra is well-defined: the monopole operator acts non trivially on a vortex-dressed operator, while the winding operator acts on the transgression of the Wilson surface.