Relations between Non-Commutative and Commutative Spacetime

TL;DR

This paper reviews how non-commutative spacetime arises in string theory, especially through D-brane configurations and boundary conditions, and discusses the Seiberg-Witten map relating non-commutative and commutative gauge theories.

Contribution

It provides a detailed review of the relationship between non-commutative and commutative spacetimes in string theory, highlighting the role of D-branes, boundary conditions, and the Seiberg-Witten map.

Findings

D-brane configurations relate non-commutative and commutative descriptions.

Point splitting regularization leads to non-commutative D-branes.

The Seiberg-Witten map connects non-commutative gauge fields to ordinary gauge fields.

Abstract

Spacetime non-commutativity appears in string theory. In this paper, the non-commutativity in string theory is reviewed. At first we review that a Dp-brane is equivalent to a configuration of infinitely many D($p-2$)-branes. If we consider the worldvolume as that of the Dp-brane, coordinates of the Dp-brane is commutative. On the other hand if we deal with the worldvolume as that of the D($p-2$)-branes, since coordinates of many D-branes are promoted to matrices the worldvolume theory is non-commutative one. Next we see that using a point splitting reguralization gives a non-commutative D-brane, and a non-commutative gauge field can be rewritten in terms of an ordinary gauge field. The transformation is called the Seiberg-Witten map. And we introduce second class constraints as boundary conditions of an open string. Since Neumann and Dirichlet boundary conditions are mixed in the constraints when the open string is coupled to a NS B field, the end points of the open string is non-commutative.