The Matrix model and the non-commutative geometry of the supermembrane

TL;DR

This paper explores the connection between the Matrix model and non-commutative geometry of the supermembrane, proposing a topological expansion of M-theory represented by Moyal Yang-Mills theory, with implications for finite N and cellular structures.

Contribution

It introduces a novel perspective linking the Matrix model with non-commutative geometry of supermembranes through topological expansion and finite quantum mechanics representations.

Findings

Matrix model corresponds to non-commutative Yang-Mills theory of membranes.

Finite N case reveals cellular structure via finite quantum mechanics.

Topological charge and instanton sector are analyzed with Bogomol'nyi bounds.

Abstract

This is a short note on the relation of the Matrix model with the non-commutative geometry of the 11-dimensional supermembrane. We put forward the idea that M-theory is described by the 't Hooft topological expansion of the Matrix model in the large N-limit where all topologies of membranes appear. This expansion can faithfully be represented by the Moyal Yang-Mills theory of membranes. We discuss this conjecture in the case of finite N, where the non-commutative geometry of the membrane is given be the finite quantum mechanics. The use of the finite dimensional representations of the Heisenberg group reveals the cellular structure of a toroidal supermembrane on which the Matrix model appears as a non-commutatutive Yang-Mills theory. The Moyal star product on the space of functions in the case of rational values of the Planck constant \hbar represents exactly this cellular structure. We also discuss the integrability of the instanton sector as well as the topological charge and the corresponding Bogomol'nyi bound.