D=11 Supermembrane wrapped on calibrated submanifolds

TL;DR

This paper constructs a Hamiltonian for the D=11 Supermembrane with topological conditions, demonstrating a discrete spectrum and linking it to noncommutative Yang-Mills theory and D-brane configurations.

Contribution

It introduces a novel supermembrane model with topological conditions on calibrated submanifolds, establishing spectrum discreteness and connections to noncommutative gauge theories.

Findings

Spectrum of the Hamiltonian is discrete with finite multiplicity.

Configurations correspond to minimal immersions and relate to noncommutative Yang-Mills theory.

Topological conditions allow interpretation as intersections of supermembranes.

Abstract

We construct the Hamiltonian of the D=11 Supermembrane with topological conditions on configuration space. It may be interpreted as a supermembrane theory where all configurations are wrapped in an irreducible way on a calibrated submanifold of a compact sector of the target space. We prove that the spectrum of its Hamiltonian is discrete with finite multiplicity. The construction is explicitly perfomed for a compact sector of the target space being a $2g$ dimensional flat torus and the base manifold of the Supermembrane a genus $g$ compact Riemann surface. The topological conditions on configuration space work in such a way that the $g=2$ case may be interpreted as the intersection of two D=11 Supermembranes over $g=1$ surfaces, with their corresponding topological conditions. The discreteness of the spectrum is preserved by the intersection procedure. Between the configurations satisfying the topological conditions there are minimal configurations which describe minimal immersions from the base manifold to the compact sector of the target space. They allow to map the D=11 Supermembrane with topological conditions to a symplectic noncommutative Yang-Mills theory. We analyze geometrical properties of these configurations in the context of Supermembranes and D-branes theories. We show that this class of configurations also minimizes the Hamiltonian of D-branes theories.