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Register a new userThe Minimally Immersed 4D Supermembrane
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In this note we summarize some of the properties found in [1], and its relation with [2]. We comment on the construction of the action of the 11D supermembrane with nontrivial central charges minimally immersed on a 7D toroidal manifold is obtained (MIM2).The transverse coordinates to the supermembrane are maps to a 4D Minkowski space-time. The action is invariant under additional symmetries in comparison to the supermembrane on a 11D Minkowski target space. The hamiltonian in the LCG is invariant under conformal transformations on the Riemann surface base manifold. The spectrum of the regularized hamiltonian is discrete with finite multiplicity. Its resolvent is compact. Susy is spontaneously broken, due to the topological central charge condition, to four supersymmetries in 4D, the vacuum belongs to an N=1 supermultiplet. When assuming the target-space to be an isotropic 7-tori, the potential does not contain any flat direction, it is stable on the moduli space of parameters. Moreover due to the discrete symmetries of the hamiltonian, there are only 7 possible minimal holomorphic immersions of the MIM2 on the 7-torus. When these symmetries are identified on the target space, it corresponds to compactify the MIM2 on a orbifold with G2 structure. Once the singularities are resolved it leads to the compactification of the MIM2 on a G2 manifold as shown in [2].
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We show that the supermembrane theory compactified on a torus is invariant under T-duality. There are two different topological sectors of the compactified supermembrane (M2) classified according to a vanishing or nonvanishing second cohomology class. We find the explicit T-duality transformation that acts locally on the supermembrane theory and we show that it is an exact symmetry of the theory. We give a global interpretation of the T-duality in terms of bundles. It has a natural description in terms of the cohomology of the base manifold and the homology of the target torus. We show that in the limit when the torus degenerate into a circle and the M2 mass operator restricts to the string-like configurations, the usual closed string T-duality transformation between the type IIA and type IIB mass operators is recovered. Moreover if we just restrict M2 mass operator to string-like configurations but we perform a generalized T-duality we find the SL(2,Z) non-perturbative multiplet of IIA.
We obtain the Hamiltonian formulation of the 11D Supermembrane theory non-trivially compactified on a twice-punctured torus times a 9D Minkowski spacetime. 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 Supermenbrane in the target space are treated rigorously. The nontrivial 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 generate a cosmological term on 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 9D. 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 to the monodromies.
We analyse the measure of the regularized matrix model of the supersymmetric potential valleys, $Omega$, of the Hamiltonian of non zero modes of supermembrane theory. This is the same as the Hamiltonian of the BFSS matrix model. We find sufficient conditions for this measure to be finite, in terms the spacetime dimension. For $SU(2)$ we show that the measure of $Omega$ is finite for the regularized supermembrane matrix model when the transverse dimensions in the light cone gauge $mathrm{D}geq 5$. This covers the important case of seven and eleven dimensional supermembrane theories, and implies the compact embedding of the Sobolev space $H^{1,2}(Omega)$ onto $L^2(Omega)$. The latter is a main step towards the confirmation of the existence and uniqueness of ground state solutions of the outer Dirichlet problem for the Hamiltonian of the $SU(N)$ regularized $mathrm{D}=11$ supermembrane, and might eventually allow patching with the inner solutions.
In this note we explicitly show how the generalization of the T-duality symmetry of the supermembrane theory compactified in M9xT2 can be reduced to a parabolic subgroup of SL(2,Z) that acts non-linearly on the moduli parameters and on the KK and winding charges of the supermembrane. This is a first step towards a deeper understanding of the dual relation between the parabolic type II gauged supergravity in nine dimensions.
We develop a classification of emph{minimally unbalanced} $3d~mathcal{N}=4$ quiver gauge theories. These gauge theories are important because the isometry group $G$ of their Coulomb branch contains a single factor, which is either a classical or an exceptional Lie group. Concurrently, this provides a classification of hyperkahler cones with isometry group $G$ which are obtainable by Coulomb branch constructions. HyperKahler cones such as Coulomb branches of $3d~mathcal{N}=4$ quivers are indispensable tools for describing Higgs branches of different theories in various dimensions. In particular, they are used to describe Higgs branches of $5d~mathcal{N}=1$ SQCD with gauge group $SU(N_c)$ and $6d~mathcal N = (1,0)$ SQCD with gauge group $Sp(N_c)$ at the respective UV fixed points.
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