Higher-Form Symmetries (HFS) of a closed bosonic M2-brane formulated on a compactified target space ℳ_9 × 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 𝒢_1^∇_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. A Wilson surface, identified with the holonomy Hol_∇ one of the Gerbe structures, the flat 𝒢_1^∇_c-gerbe, is naturally introduced as the topological operator characterizing 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 pullback of the Wilson surface.

We study the Kaluza–Klein descent of non-invertible higher-form symmetry defects from eleven-dimensional Supergravity to Type IIA Supergravity. Starting from the eleven-dimensional construction of non-invertible Supergravity defects, we show that the full defect, including its auxiliary topological sector, admits a pushforward along the M-theory circle in the zero-mode Supergravity regime. The seven-dimensional Chern–Simons-like auxiliary theory descends to a six-dimensional BF-type sector. We also show that the compactification of the eleven-dimensional Bianchi sector splits into an invertible H_[3]-sector and a twisted non-invertible F_[4]-sector, controlled by dF_[4]+H_[3]∧ F_[2]=0. The resulting Type IIA defect algebra contains both an invertible Picard subgroup and non-invertible BF-dressed defects, with charged probes identified through the standard M-theory/Type IIA brane dictionary.

The role of torsion in the local and global description of supersymmetric M2-branes with fluxes and parabolic monodromies is determined. The monodromy corresponds to a representation of the fundamental group of the base manifold into the parabolic subgroup of , the group of isotopy classes of area-preserving diffeomorphisms. These are M2-branes with a quantum discrete spectrum with finite multiplicity. The global description of these M2-branes is given by twisted torus bundles with monodromy. Given a representation, they are classified by , or equivalently, by the coinvariants associated with the parabolic monodromy subgroup. Previous constructions are generalized in two different ways. The first one considers parabolic monodromies with . This makes it possible to identify torsion cycles of order greater than one. It is shown that there are well-defined nilmanifolds in three, four, and five dimensions with nontrivial torsion cycles contained in the global description of these M2-branes. The torsion is also manifest in the equivalence classes of M2-brane bundles. The second one is the action of the torsion on the coinvariants of the base manifold, which is analyzed. Together with the flux condition, the torsion defines explicitly the number of coinvariants for a given monodromy.

We discuss the relationship between the Swampland program and QM2-branes, which are supersymmetric M2-branes with discrete spectrum. The worldvolume description of these M2-branes is well-known. The global description is given in terms of twisted torus bundles with monodromy in SL(2,ℤ). In particular, we show that the towers of states that ensure the Swampland Distance Conjecture for type II maximal supergravity in nine dimensions are directly related to QM2-branes with trivial monodromy on a background that is toroidally compactified.

Higher Forms Symmetries (HFS) of a closed bosonic M2-brane theory formulated from the world-volume perspective on a compactified target space ℳ_9 × T^2 are obtained. We show that the cancellation of the 't Hooft anomaly present in the theory is related to a 3-form flux with 1-gerbe structure associated to the world-volume flux quantization condition. A Wilson surface is naturally defined on the topological operator that characterize the holonomy of the M2-brane. The projection of the flux quantization condition onto the spatial part of the worldvolume, leads to a flux quantization on the M2-brane. The topological operators realise discrete symmetries associated with the winding and the flux/monopole condition. The algebra of operators is well defined.

Abstract In this paper, we study the relation of the M2-brane with fluxes and monodromy in SL(2, ℤ), which has a quantum discrete supersymmetric spectrum with finite multiplicity and type IIB gauged supergravities in nine dimensions. SL(2, ℤ) is the group of isotopy classes of the area-preserving diffeomorphisms. The global description of these M2-branes we are considering is formulated on twisted torus bundles, and they are classified in terms of H2(Σ, ℤρ), or equivalently, by their coinvariants for a given monodromy subgroup. We find the ‘gauge’ symmetries between equivalent M2-branes on torus bundles with monodromy that lead to ℝ, SO(2), or SO(1, 1), the symmetry groups of type IIB gauged supergravities in 9d. We obtain an explicit relation between the equivalent classes of M2-brane bundles and the mass parameters that classify the gaugings of type IIB supergravities in 9d. We also find that the symmetries between inequivalent M2-branes on twisted torus bundles for a given monodromy are related to ℤ, ℤ3, ℤ5, ℤ9, or ℤ2n − 7 for n ≥ 5, the U-duality symmetry group, a subgroup of SL(2, ℤ). In distinction, in the case without monodromy, related to type II maximal supergravity at low energies, its U-duality group corresponds to the full SL(2, ℤ).

We consider the Becchi, Rouet, Stora and Tyutin (BRST) invariant effective action of the non-abelian BF topological theory in two dimensions with gauge group Sl(2,R) . By considering different gauge fixing conditions, the zero-curvature field equation gives rise to several well known integrable equations. We prove that each integrable equation together with the associated ghost field evolution equation, obtained from the BF theory, is a BRST invariant system with an infinite sequence of BRST invariant conserved quantities. We construct explicitly the systems and the BRST transformation laws for the Korteweg-de Vries (KdV) sequence (including the KdV, mKdV and CKdV equations) and Harry Dym integrable equation.

An anisotropic model describing gravity-vector gauge coupling at all energy scales is presented. The starting point is the 4+1 dimensional non-projectable Ho.rava-Lifshitz gravity theory subject to a geometrical restriction. Renormalizability arguments require all possible interactions in the potential up to terms with z= 4 spatial derivatives on the geometrical tensor fields: the Riemann and Weyl tensors. The latter being necessary on a 4+1 dimensional formulation. The dimensional reduction to 3+1 dimensions gives rise to a model invariant under foliation-preserving diffeomorphisms (FDiff) and U(1) symmetry groups. The reduced theory on the kinetic conformal point (lambda = 1/3), propagates the same spectrum of the Einstein-Maxwell theory. Moreover, at low energies, on the IR point alpha = 0, beta = 1, its field equations are exactly the Einstein-Maxwell ones in a particular gauge condition. The Minkowski ground state is stable provided several restrictions on the coupling parameters are satisfied, they are explicitly obtained. The quantum propagators of the physical degrees of freedom are obtained and after an analysis of the first and second class constraints the renormalizability by power counting is proved, provided that the aforementioned restrictions on the coupling parameters are satisfied.

Abstract We construct, following [1, 2], a massive M2-brane (supermembrane) as the limit of a genus two M2-brane that becomes a twice punctured Riemann surface with particular boundary conditions on the fields defined on the punctures. The target space is M 9 × LCD, where LCD is a genus one light cone diagram. It contains mass terms and a topological term associated with the non-triviality of the target surface that, at low energies, can be associated with the presence of a cosmological constant. We show that the supergravity background of the M2-brane considered in this formulation requires the presence of M9-branes acting as sources. They correspond to the 11D uplift of the characteristic D8’s of Romans supergravity. To this end, we explicitly show that some of the background singularities of the massive M2-brane can be reproduced by the M9-branes found by [3]. This establishes a relation between the Romans mass and the moduli of the massive M2-brane. When dimensionally reduced, we obtain a worldsheet Hamiltonian of a N=2 type IIA closed superstring in 10D. We denote it massive string. The corresponding massive string inherits a non-vanishing constant term from the topological massive M2-brane that shifts the Hamiltonian. The non-vanishing parameter is related to the non-trivial structure of the massive M2-brane background and it can be related to the Romans mass term. It also contains a modified tension due to the non-trivial dependence on the moduli and on the punctures associated with the target torus.

In this paper, we obtain the explicit expression of the supersymmetric algebra associated with the recently proposed massive supermembrane including all surface terms. We formulate the theory as the limit of a supermembrane on a genus-two compact Riemann surface when one of the handles becomes a string attached to a torus. The formulation reduces to a supermembrane on a punctured torus with a “string spike” (in the sense of [1]), attached to it. In this limit, we identify all surface terms of the algebra and give the explicit expression of the Hamiltonian in agreement with the previous formulation of it. The symmetry under area preserving diffeomorphisms, connected and nonconnected to the identity, is also discussed. Only parabolic Sl(2,Z) discrete symmetries are preserved.

We analyze the dispersion relation for an anisotropic gravity-electromagnetic theory at very high energies. In particular for photons of very high energy. We start by introducing the anisotropic gravity-gauge vector field model. It is invariant under spacelike diffeomorphisms, time parametrization, and U(1) gauge transformations. It includes high-order spacelike derivatives as well as polynomial expressions of the Riemann and field strength tensor fields. It is based on the Hořava-Lifshitz anisotropic proposal. We show its consistency, and the stability of the Minkowski ground state. Finally, we determine the exact zone at which the physical degrees of freedom, i.e. the transverse-traceless tensorial degrees of freedom and the transverse vectorial degrees of freedom propagate according to a linear wave equation. This is so, in spite of the fact that there exists in the zone a non-trivial Newtonian background of the same order. The wave equation contains spatial derivatives up to the sixth order, in the lowest order it exactly matches the relativistic wave equation. We then analyze the dispersion relation at very high energies in the context of recent experimental data. The qualitative predictions of the proposed model, concerning the propagation of highly energetic photons, are different from the ones obtained from the modified dispersion relation of the LIV models.

Abstract We extend the work of Schwarz [1] to show that bound states of type IIB supersymmetric (p, q)-strings on a circle are associated with M2-branes irreducibly wrapped on T 2, or equivalently with nontrivial worldvolume fluxes. Beyond this extension we consider the Hamiltonian of an M2-brane with C ± fluxes formulated on a symplectic torus bundle with monodromy. In particular, we analyze the relevant case when the monodromy is parabolic. We show that the Hamiltonian is defined in terms of the coinvariant module. We also find that the mass operator is invariant under transformations between inequivalent coinvariants. These coinvariants classify the inequivalent classes of twisted torus bundles with nontrivial monodromy for a given flux. We obtain their associated (p, q)-strings via double dimensional reduction, which are invariant under a parabolic subgroup of SL(2, ℚ). This is the origin of the gauge symmetry of the associated gauged supergravity. These bound states could also be related to the parabolic Scherk-Schwarz reductions of type IIB string theory.

We consider the anisotropic gravity-gauge vector coupling in the non-projectable Hořava-Lifshitz theory at the kinetic conformal point, in the low energy regime. We show that the canonical formulation of the theory, evaluated at its constraints, reduces to a canonical formulation solely in terms of the physical degrees of freedom. The corresponding reduced Hamilton defines the ADM energy of the system. We obtain its explicit expression and discuss its relation to the ADM energy of the Einstein-Maxwell theory. We then show that there exists, in this theory, a well–defined wave zone. In it, the physical degrees of freedom ı.e., the transverse–traceless tensorial modes associated to the gravitational sector and the transverse vectorial modes associated to the gauge vector interaction satisfy independent linear wave equations, without any coupling between them. The Newtonian part of the anisotropic theory, very relevant near the sources, does not affect the free propagation of the physical degrees of freedom in the wave zone. It turns out that both excitations, the gravitational and the vectorial one, propagate with the same speed √(β), where β is the coupling parameter of the scalar curvature of the three dimensional leaves of the foliation defining the Hořava–Lifshitz geometry.

We show that in the anisotropic Hořava-Lifshitz gravity there is a well-defined wave zone where the physical degrees of freedom propagate according to a non-relativistic linear evolution equation of high order in spatial derivatives, which reduces to the wave equation at low energy. This is so, provided the coupling parameters satisfy some restrictions which we study in detail. They are imposed to obtain a finite ADM gravitational energy, which depends manifestly on the terms which break the Lorentz symmetry of the formulation. The analysis we perform is beyond the perturbative approach and includes all high order terms of the Hamiltonian potential.

We present the formulation of the bosonic Hamiltonian M2-brane compactified on a twice punctured torus following the procedure proposed in \cite{mpgm14}. In this work we analyse two possible metric choice, different from the one used in \cite{mpgm14}, over the target space and study some of the properties of the corresponding Hamiltonian.

In this work we consider the existence and uniqueness of the ground state of the regularized Hamiltonian of the Supermembrane in dimensions D = 4, 5, 7 and 11, or equivalently the SU(N) Matrix Model. That is, the 0+1 reduction of the 10-dimensional SU(N) Super Yang-Mills Hamiltonian. This ground state problem is associated with the solutions of the inner and outer Dirichlet problems for this operator, and their subsequent smooth patching (glueing) into a single state. We have discussed properties of the inner problem in a previous work, therefore we now investigate the outer Dirichlet problem for the Hamiltonian operator. We establish existence and uniqueness on unbounded valleys defined in terms of the bosonic potential. These are precisely those regions where the bosonic part of the potential is less than a given value V0, which we set to be arbitrary. The problem is well posed, since these valleys are preserved by the action of the SU(N) constraint. We first show that their Lebesgue measure is finite, subject to restrictions on D in terms of N. We then use this analysis to determine a bound on the fermionic potential which yields the coercive property of the energy form. It is from this, that we derive the existence and uniqueness of the solution. As a by-product of our argumentation, we show that the Hamiltonian, restricted to the valleys, has spectrum purely discrete with finite multiplicity. Remarkably, this is in contrast to the case of the unrestricted space, where it is well known that the spectrum comprises a continuous segment. We discuss the relation of our work with the general ground state problem and the question of confinement in models with strong interactions.

We show that in the Hořava–Lifshitz theory at the kinetic-conformal point, in the low energy regime, a wave zone for asymptotically flat fields can be consistently defined. In it, the physical degrees of freedom, the transverse traceless tensorial modes, satisfy a linear wave equation. The Newtonian contributions, among which there are terms which manifestly break the relativistic invariance, are non-trivial but do not obstruct the free propagation (radiation) of the physical degrees of freedom. For an appropriate value of the couplings of the theory, the wave equation becomes the relativistic one in agreement with the propagation of the gravitational radiation in the wave zone of General Relativity. Previously to the wave zone analysis, and in general grounds, we obtain the physical Hamiltonian of the Hořava–Lifshitz theory at the kinetic-conformal point in the constrained submanifold. We determine the canonical physical degrees of freedom in a particular coordinate system. They are well defined functions of the transverse-traceless modes of the metric and coincide with them in the wave zone and also at linearized level.

A spherically symmetric solution of the field equations of the Hořava–Lifshitz gravity–gauge vector interaction theory is obtained and analyzed. It describes a charged throat. The solution exists provided a restriction on the relation between the mass and charge is satisfied. The restriction reduces to the Reissner–Nordström one in the limit in which the coupling constants tend to the relativistic values of General Relativity. We introduce the correct charts to describe the solution across the entire manifold, including the throat connecting an asymptotic Minkowski space-time with a singular 3+1 dimensional manifold. The solution external to the throat on the asymptotically flat side tends to the Reissner–Nordström space-time at the limit when the coupling parameter, associated with the term in the low energy Hamiltonian that manifestly breaks the relativistic symmetry, tends to zero. Also, when the electric charge is taken to be zero the solution becomes the spherically symmetric and static solution of the Hořava–Lifshitz gravity.

A bstract We obtain the Hamiltonian formulation of the 11D Supermembrane theory non-trivially compactified on a twice punctured torus times a 9D Minkowski space-time. It corresponds to a M2-brane formulated in 11D space with ten non-compact dimensions. The critical points like the poles and the zeros of the fields describing the embedding of the Supermembrane in the target space are treated rigorously. The non-trivial compactification generates non-trivial mass terms appearing in the bosonic potential, which dominate the full supersymmetric potential and should render the spectrum of the (regularized) Supermembrane discrete with finite multiplicity. The behaviour of the fields around the punctures generates a cosmological term in the Hamiltonian of the theory. The massive supermembrane can also be seen as a nontrivial uplift of a supermembrane torus bundle with parabolic monodromy in M 9 × T 2 . The moduli of the theory is the one associated with the punctured torus, hence it keeps all the nontriviality of the torus moduli even after the decompactification process to ten noncompact dimensions. The formulation of the theory on a punctured torus bundle is characterized by the (1 , 1) − knots associated with the monodromies.