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.
In this paper, we study the existence of higher form symmetries of degree p (HFS(p)), t'Hooft anomaly cancellation, and the presence of gerbe structures on the bosonic M2-brane. The presence of these gerbe structures induces two effects. Firstly, it leads to the breaking of global symmetries into discrete gauged symmetries. Secondly, it is related to the emergence of flux quantization conditions over the bosonic M2-brane. This final result is a necessary condition for the consistency of the gauged theory in toroidal backgrounds. It is noteworthy that the spectral analysis of the supersymmetric M2-brane compactified on toroidal backgrounds also requires the imposition of a flux condition for the discreteness of the mass spectrum of the toroidally compactified supermembrane, [14], [15]. Symmetries are of fundamental importance in physics. They underlie the unification of fundamental interactions, constrain the dynamics of quantum field theories, and have been used for the classification of phases of matter. Conventionally, a symmetry is defined as a transformation acting on the fields of a physical system under which, the action or the equations of motion remain invariant. Such transformations are characterized by groups, finite-or infinitedimensional. For the case of continuous symmetries Noether's theorem guarantee the existence of conserved currents and charges generated by local operators. In a more general framework, these symmetries are often referred to as 0-form symmetries, reflecting the fact the symmetry operators act on point-like objects. Despite their success, Noether symmetries do not characterize the full symmetry structure of quantum field theories. A limitation of Noether symmetries is that they are unable to indicate the presence of spontaneous symmetry breaking or phase transitions; for instance, transitions from deconfinement to confined phases in quantum field theory (QFT). These limitations motivated the development of a more general notion of symmetry, denoted as Generalized Symmetries in their seminal paper, [21]. Generalized symmetries also known as Higher Form Symmetries, (HFS) represent a vast concept that generalizes the notion of symmetry [21]. The operator representing the HFS(p) are non-local and acts on extended charged objects (of dimension p, larger than zero), as for example, Wilson lines, Wilson surfaces, branes etc. The symmetry operators are defined on submanifolds of codimension p + 1. They extend the notion of Noether symmetry and Ward identities to extended objects instead of considering point-like sources. Furthermore, the breaking of HFS to their discrete subgroup may allow the characterizations of the phases such as confinement, Higgs or topological phases in the context of High Energy physics but also in condensed matter physics, see for example [28]. Symmetry operators are topological and gauge-invariant operators. They are classified as invertible when they satisfy group-like fusion rules. In the context of unification theories, global symmetries are not expected to appear in any consistent quantum gravity theory [2]. The gauging of these HFS is realized through the introduction of gauge Background fields which appear in the action of the theory as a topological field term but can be dynamical in manifolds of higher dimensions. In our paper we do not consider dynamical contributions from the backgrounds fields. In the invertible cases, there is an obstruction to the gauging of the HFS, that implies the existence of anomalies in the path integral, whose cancellation involves the inclusion of new terms constructed with these background fields, like the anomaly inflow terms [44], [13].
To illustrate this point, consider four-dimensional Maxwell theory with its two one-form symmetries. It is a consistent theory. However, when gauging these global symmetries a mixed ’t Hooft anomaly appears, which can be canceled by coupling the theory to appropriate background fields, namely higher-form gauge connections (in particular, U(1) 2-form connections, i.e. G ∇ 1 -gerbe connections) [21]. In the context of Supergravity, the ABJ anomaly emerges due to the non-invariance of the topological operators under the global HFS symmetry, as a consequence of the Chern-Simons term. The generators of the HFS are non-invertible [24]. The original symmetry groups in the presence of sources generally break into their discrete subgroups.
Recently, there has been a growing interest in exploring the potential applications of HFS to various gauge and gravitational theories. See for example in the context of Supergravity [5], string theory [6], [7], F-theory [8] and M-theory [9] among others. The potential applications of the study of the HFS have yet to be fully realized, with the possibility of further applications emerging. The present paper focuses on M-theory, adopting the M2-brane as the fundamental object of study. We search for topological effects related to HFS without introducing Sup
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