A note on T 5 / Z 2 compactification of the M - theory matrix model

We study the T 5/Z2 orbifold compactification of the M-theory matrix model. This model was originally studied by Dasgupta, Mukhi, and Witten. It was found that one had to add 16 5-branes to the system to make the compactification consistent. We demonstrate how this is mimicked in the matrix model. E-mail: [email protected] E-mail: [email protected] A recent proposal by Banks, Fischler, Shenker and Susskind (BFSS) [1] that M-theory in an infinite momentum frame is described by a certain matrix model has excited much attention and activity. This proposal has passed a number of non-trivial tests. There are indications, however, that there are certain elements still lacking in this formulation. By putting the matrix model to various tests one hopes to understand its strengths and weaknesses leading ultimately to, hopefully, the correct formulation of M-theory. The purpose of this note is to put the matrix model to another test which, in our opinion, the matrix model passes successfully. We consider compactification of the matrix theory on a five dimensional torus modded out by a Z2 action which is supposed to be the analogue of a compactification considered by Dasgupta, Mukhi and Witten [2, 3]. This compactification is rather involved due to constraints coming from gravitational anomaly cancellation and cancellation of total charge on the compact manifold. It is of some interest then to see how these constraints arise in the matrix model compactification. Compactification on a d−dimensional torus, according to the rules and explicit constructions of references [4, 5, 6, 7], results in supersymmetric Yang-Mills theory in d+ 1 dimensions (SYMd+1) with 16 super-charges. For example, compactification on T 3 results in N = 4 supersymmetric Yang-Mills theory in d = 4 [5, 6]. Toroidal compactifications always result in non-chiral gauge theories. As in ordinary string theory compactifications, one can get chiral theories by considering orbifolds of tori. The BFSS model is N = 1 supersymmetric Yang-Mills in 10 dimensions with gauge group SU(N) dimensionally reduced to d = 1. The resulting theory is matrix quantum mechanics of 9 bosonic and 16 fermionic matrices transforming in the adjoint of the SU(N) gauge group. The limit N → ∞ is to be taken. This theory has only 16 supercharges which is the appropriate number of manifest supersymmetries in the infinite momentum frame for a theory with 32 supercharges. The Lagrangian for the theory is given by: