Unitary matrix model for toroidal compactifications of M theory


A unitary matrix model is proposed as the large-N matrix formulation of M theory on flat space with toroidal topology. The model reproduces the motion of elementary D-particles on the compact space, and admits membrane states with nonzero wrapping around nontrivial 2-tori even at finite N. † poly@calypso.teorfys.uu.se The recently proposed matrix model approach to M-theory consists of the dimensional reduction of 10-dimensional super-Yang-Mills (SYM) to 0+1 dimensions [1]. This model has its origins in D-brane dynamics [2-6] and is known to describe the dynamics of D-particles in the low-energy (nonrelativistic) limit [7-9]. The remarkable conjecture made in [1] is that the full light-cone dynamics of the different physical objects within M-theory are imbedded in the large-N limit of the above matrix model. This conjecture has survived a number of consistency checks [10-15] in a rapidly increasing literature. The above matrix model applies to the case of flat uncompactified spacetime and would need be modified for other topologies. For the case of toroidal compactifications of space, the model should account for the interactions due to virtual strings winding around the compact dimensions, as well as describe membranes wrapped around compact submanifolds. The main proposal for doing that is to enlarge the matrix model to a (K+1)-dimensional SYM filed theory, where K is the number of compact dimensions [1,16]. Although this certainly contains all the relevant degrees of freedom [11-16,17], it most probably contains more than is actually needed at the large-N limit. Furthermore, it takes away from the simplicity of the original matrix model by adding an infinity of degrees of freedom. This is somewhat undesirable. The hope would be that a matrix model with the correct dynamics will contain, in the large-N limit, all the relevant degrees of freedom. The purpose of this note is to point to a possible such model. The proposed model consists of using unitary (rather than hermitian) matrices for each compact dimension. For simplicity, we can assume that all 9 transverse dimensions are compactified with radii Ri. Then the Lagrangian would contain the terms L = trR { R I 2 DU iDUi +R 2 iR 2 j [Ui, Uj ][Ui, Uj ] † + θDθ +Riθ TγiU i [θ, Ui] }

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@inproceedings{Polychronakos1997UnitaryMM, title={Unitary matrix model for toroidal compactifications of M theory}, author={Alexios P. Polychronakos}, year={1997} }