On the spectrum of a matrix model for the D=11 supermembrane compactified on a torus with non-trivial winding

TL;DR

This paper proves that the spectrum of a compactified D=11 supermembrane with non-trivial winding is discrete due to a positive definite potential, contrasting with the continuous spectrum in the non-compact case.

Contribution

It demonstrates that the bosonic potential in the compactified supermembrane model is strictly positive and leads to a discrete spectrum, a novel result for this class of models.

Findings

Bosonic potential is strictly positive definite.

Spectrum of the Hamiltonian is discrete.

Provides an upper bound for eigenvalue distribution.

Abstract

The spectrum of the Hamiltonian of the double compactified D=11 supermembrane with non-trivial central charge or equivalently the non-commutative symplectic super Maxwell theory is analyzed. In distinction to what occurs for the D=11 supermembrane in Minkowski target space where the bosonic potential presents string-like spikes which render the spectrum of the supersymmetric model continuous, we prove that the potential of the bosonic compactified membrane with non-trivial central charge is strictly positive definite and becomes infinity in all directions when the norm of the configuration space goes to infinity. This ensures that the resolvent of the bosonic Hamiltonian is compact. We find an upper bound for the asymptotic distribution of the eigenvalues.