On the Spectral properties of Multi-branes, M2 and M5 branes

In this note we summarize some of the properties found in several papers. We characterize spectral properties of the quantum mechanical hamiltonian of theories with fermionic degrees of freedom beyond semiclassical approximation. We obtain a general class of bosonic polynomial potentials for which the Schr\"oedinger operator has a discrete spectrum. This class includes all the scalar potentials in membrane, 5-brane, p-branes, multiple M2 branes, BLG and ABJM theories. We also give a sufficient condition for discreteness of the spectrum for supersymmmetric and non supersymmetric theories with a fermionic contribution. We characterize then the spectral properties of different theories: the BMN matrix model, the supermembrane with central charges and a bound state of $N$ D2 with $m$ D0. We show that, while the first two models have a purely discrete spectrum with finite multiplicity, the latter has a continuous spectrum starting from a constant given in terms of the monopole charge.Comment: 10pg, Latex, Contributions to the Conference XVI European Workshop on String Theory 2010, Madrid June 14-18, 201

Arguments towards the construction of a matrix model groundstate

We discuss the existence and uniqueness of wavefunctions for inhomogenoeus boundary value problems associated to x^2y^2-type matrix model on a bounded domain of R^2. Both properties involve a combination of the Cauchy-Kovalewski Theorem and a explicit calculations.Comment: 3 pages, Latex Proceedings for the XIX Simposio Chileno de Fisica, SOCHIFI 2014 Conference, 26-28 November 2014, held at Concepcion U., Chil

On the groundstate of octonionic matrix models in a ball

In this work we examine the existence and uniqueness of the groundstate of a SU(N)x G2 octonionic matrix model on a bounded domain of R^N. The existence and uniqueness argument of the groundstate wavefunction follows from the Lax-Milgram theorem. Uniqueness is shown by means of an explicit argument which is drafted in some detail.Comment: Latex, 6 page

The supermembrane with central charges:(2+1)-D NCSYM, confinement and phase transition

The spectrum of the bosonic sector of the D=11 supermembrane with central charges is shown to be discrete and with finite multiplicities, hence containing a mass gap. The result extends to the exact theory our previous proof of the similar property for the SU(N) regularised model and strongly suggest discreteness of the spectrum for the complete Hamiltonian of the supermembrane with central charges. This theory is a quantum equivalent to a symplectic non-commutative super-Yang-Mills in 2+1 dimensions, where the space-like sector is a Riemann surface of positive genus. In this context, it is argued how the theory in 4D exhibits confinement in the N=1 supermembrane with central charges phase and how the theory enters in the quark-gluon plasma phase through the spontaneous breaking of the centre. This phase is interpreted in terms of the compactified supermembrane without central charges.Comment: 33 pages, Latex. In this new version, several changes have been made and various typos were correcte

The ground state of the D=11 supermembrane and matrix models on compact regions

We establish a general framework for the analysis of boundary value problems of matrix models at zero energy on compact regions. We derive existence and uniqueness of ground state wavefunctions for the mass operator of the $D=11$ regularized supermembrane theory, that is the $\mathcal{N}=16$ supersymmetric $SU(N)$ matrix model, on balls of finite radius. Our results rely on the structure of the associated Dirichlet form and a factorization in terms of the supersymmetric charges. They also rely on the polynomial structure of the potential and various other supersymmetric properties of the system.Comment: Latex, 26 pages. We have added some comments at the introduction in order to make it easier for the reader. Results of the paper unchange

Massless ground state for a compact SU(2) matrix model in 4D

We show the existence and uniqueness of a massless supersymmetric ground state wavefunction of a SU(2) matrix model in a bounded smooth domain with Dirichlet boundary conditions. This is a gauge system and we provide a new framework to analyze the quantum spectral properties of this class of supersymmetric matrix models subject to constraints which can be generalized for arbitrary number of colors.Comment: 12 pages, Latex. Somme clarifications. Minor changes. Version to appear at NP