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

This is a short note on the relation of the Matrix model with the non-commu tative geometry of the 11-dimensional supermembrane. We put forward the ide a that M-theory is described by the 't Hooft topological expansion of the M atrix 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 th e non-commutative geometry of the membrane is given be the finite quantum m echanics. The use of the finite dimensional representations of the Heisenbe rg group reveals the cellular structure of a toroidal supermembrane on whic h the Matrix model appears as a non-commutatutive Yang-Mills theory. The Mo yal star product on the space of functions in the case of rational values o f the Planck constant (h) over bar represents exactly this cellular structu re. We also discuss the integrability of the instanton sector as well as th e topological charge and the corresponding Bogomol'nyi bound. (C) 1999 Publ ished by Elsevier Science B.V. All rights reserved.