New M(atrix)-models for Commutative and Noncommutative Gauge Theories

We propose a M(atrix) model for N = 4 SU(k) Super-Yang-Mills theory compactified on T 4. In this model it is possible to make T 4 noncommutative and it is easy to turn on all 6 components of the noncommutativity on T 4. The action of S-duality on the noncommutativity parameters is also manifest. The M(atrix)-model is given by the large N limit of a σ-model on T 2 whose target space is the moduli space of k SU(N) instantons on T 3 × R. We also propose that the SU(k) 2+1D Spin(8) theory (the low-energy description of k M2-branes) on T 3 corresponds to the large N limit of an integral over the latter instanton moduli space. The identification is based on the fact that Euclidean wrapped M2-branes in toroidally compactified M-theory correspond to instantons in the M(atrix)-model. In the new M(atrix) models, operators with nonzero momentum along T 3 (or T 4) correspond to insertions of Wilson lines along a 1-cycle that is determined by the momentum. Momentum is conserved in the large N limit.