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

TL;DR

This paper explores how a background B-field in type IIA string theory on a torus induces noncommutative geometry in the D0-brane matrix model, connecting it to non-local SYM theories on dual tori.

Contribution

It explicitly derives the noncommutative matrix model for M-theory on a torus with background fields, linking string theory, noncommutative geometry, and SYM theories.

Findings

Modified D0-brane action with noncommutative product

Derived the matrix model for M-theory with background 3-form

Calculated radii and gauge coupling for dual SYM theory

Abstract

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 B-field 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 $T^2$.