Noncommutative Open String and D-brane

TL;DR

This work demonstrates that quantizing open strings ending on D-branes in a background $B$-field induces noncommutativity on the string end-points, which in turn renders the D-brane worldvolume a noncommutative space. By introducing a gauge-invariant field strength ${\\cal F}=B-F$ and a time-averaged symplectic form, the authors derive a consistent quantization in which endpoint coordinates satisfy $[x_0^i,x_0^j]= i \, 2\\pi \, \\alpha' (M^{-1}{\\cal F})^{ij}$ with $M_{ij}=\\eta_{ij}-{\\cal F}_i{}^k{\\cal F}_{kj}$. They show that for D2-branes this reproduces the familiar boundary noncommutativity, and they establish agreement with Matrix theory results via T-duality radii, generalizing to $Dp$-branes through $[X^k,X^l]=\pm 2\\pi i\\alpha' (M^{-1}{\\cal F})^{kl}$. The findings provide a string-theoretic derivation of noncommutative gauge theories on D-branes, linking open-string quantization to Matrix model and duality descriptions, with broad implications for brane dynamics and noncommutative geometry.

Abstract

In this paper we consider the quantization of open strings ending on D-branes with a background B field. We find that spacetime coordinates of the open string end-points become noncommutative, and correspondingly the D-brane worldvolume also becomes noncommutative. This provides a string theory derivation and generalization of the noncommutativity obtained previously in the Matrix model compactification. For Dp-branes with p>=2 our results are new and agree with that of Matrix theory for the case of A=0 (where $A$ is the worldvolume gauge field) if the T-duality radii are used.