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 ${\mathcal{G}}_1^{{\mathrm{\nabla }}_c}$-gerbe on the M2-brane worldvolume. The effect of these gauge structures together with non trivial $\mathit{\text{winding}}$ 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${}_{\mathrm{\nabla }}$ one of the Gerbe structures, the flat ${\mathcal{G}}_1^{{\mathrm{\nabla }}_c}$-gerbe, is naturally introduced as the topological operator characterizing the M2-brane. The resulting topological operators realize discrete symmetries associated with the $\mathit{\text{winding}}$ and the flux/$\mathit{\text{monopole}}$ sectors, and their operator algebra is well-defined: the $\mathit{\text{monopole}}$ operator acts non trivially on a $\mathit{\text{vortex-dressed}}$ operator, while the winding operator acts on the pullback of the Wilson surface.