String Theory, Matrix Model, and Noncommutative Geometry

Compacti cation of Matrix Model on a Noncommutative torus is obtained from strings ending on D-branes with background B eld. The BPS spectrum of the system and a novel SL(2; Z) symmetry are discussed. Noncommutativity of space-time coordinates emerged in string theory recently in the context of coincident Dbranes [1]; in fact the embedding coordinates of D-branes turned out to be noncommutative. These noncommutative coordinates in the case of 0-branes are elevated to the dynamical variables of Matrix Theory, which is conjectured to describe the strong coupling limit of string theory, or M-theory, in the in nite momentum frame [2]. Another kind of noncommutativity in spce coordinates has been recently observed in Matrix Theory which is super cially di erent from the above kind. It comes from the application of the non-commutative geometry (NCG) techniques pioneered by A. Connes to the Matrix Theory compacti cations [3]. As a formulation of M-theory, Matrix Theory must describe string theory when compacti ed on a circle; further compacti cations being neccessary to accomodate low energy physics. A class of toroidal compacti cations have been known , which relies on a certain commutative subalgebra of matrices [4,5]. The subalgebra being an equivalent description of the manifold of torus on which compacti cation is performed. It was observed by Connes, Douglas and Schwarz (CDS) that a nonabelian generalization of this algebraic description of the manifold of compacti cation, in the spirit of NCG, it is possible to arrive at a di erent compacti cation of Matrix-model, with the subsequent novel physical result of appearance of a constant background of the 3-form eld in the 11 dimensional supergravity limit. It was immediately observed by Douglas and Hull [6] that a consequent deformed SYM theory and, therefore indirectly, the noncommutative torus (NCT) compacti cation is a natural consequence of certain D-brane con gurations in string theory. The subject has been pursued in recent works [7,8,9,10,11]. Thus there is a close connection between constant background Kalb-Ramond eld B and the nonabelian torus compacti cation of the Matrix Theory. But, it is not obvious how a background B eld can make the coordinates noncommutative and how this noncommutativity di ers from that of the coincident D-branes. We will show explicitly how the CDS noncommutativity arises from D-branes in the presence of B eld backgraound and compare it with the noncommutativity due to coincident D-branes [12,13,21]. This noncommutativity persists in higher tori. The dynamical variables of Marix Theory are N N matrices which are function of time, with N going to in nity and with the supersymmetric action,