Noncommutative Geometry from D0-branes in a Background B-field

We study D0-branes in type IIA on T 2 with a background B-field turned on. We calculate explicitly how the background B-field modifies the D0-brane action. The effect of the Bfield is to replace ordinary multiplication with a noncommutative product. This enables us to find the matrix model for M-theory on T 2 with a background 3-form potential along the torus and the lightlike circle. This matrix model is exactly the non-local 2+1 dim SYM theory on a dual T 2 proposed by Connes, Douglas and Schwarz. We calculate the radii and the gauge coupling for the SYM on the dual T 2 for all choices of longitudinal momentum and membrane wrapping number on the T .