On The spectrum of a Noncommutative Formulation of the D=11 Supermembrane with Winding

TL;DR

This paper develops a regularized SU(N) model for the double compactified D=11 supermembrane with nontrivial winding, showing its convergence to a noncommutative geometric description and analyzing the spectrum, which is discrete in certain sectors.

Contribution

It introduces a novel SU(N) regularization for the supermembrane with winding, connecting it to noncommutative geometry and analyzing spectral properties.

Findings

The SU(N) regularized model converges to the noncommutative supermembrane description as N approaches infinity.

The spectrum contains string-like spikes with zero energy in general, but is discrete in sectors with non-zero winding.

The model explicitly incorporates nontrivial winding conditions via a nontrivial line bundle.

Abstract

A regularized model of the double compactified D=11 supermembrane with nontrivial winding in terms of SU(N) valued maps is obtained. The condition of nontrivial winding is described in terms of a nontrivial line bundle introduced in the formulation of the compactified supermembrane. The multivalued geometrical objects of the model related to the nontrivial wrapping are described in terms of a SU(N) geometrical object which in the $ N\to \infty$ limit, converges to the symplectic connection related to the area preserving diffeomorphisms of the recently obtained non-commutative description of the compactified D=11 supermembrane.(I. Martin, J.Ovalle, A. Restuccia. 2000,2001) The SU(N) regularized canonical lagrangian is explicitly obtained. In the $ N\to \infty$ limit it converges to the lagrangian in (I.Martin, J.Ovalle, A.Restuccia. 2000,2001) subject to the nontrivial winding condition. The spectrum of the hamiltonian of the double compactified D=11 supermembrane is discussed. Generically, it contains local string like spikes with zero energy. However the sector of the theory corresponding to a principle bundle characterized by the winding number $n \neq 0$, described by the SU(N) model we propose, is shown to have no local string-like spikes and hence the spectrum of this sector should be discrete.