Relation between Commutative and Non-Commutative Descriptions of D-branes in Large R-R Field Background

TL;DR

This work derives the Seiberg–Witten map to first order in the non-commutativity parameter $\theta$ for D-branes in a large Ramond–Ramond background and extends the map from U($1$) to U($N$) in flat backgrounds. It shows that the commutative description cannot be expressed solely in terms of U($N$) gauge fields, with the SU($N$) sector acquiring genuine nonlocality through covariant derivatives, while the Abelian (U($1$)) sector can be made local. The perturbative analysis yields explicit mappings and covariant field strengths, revealing that the commutative Lagrangian depends on the SU($N$) covariant potential rather than just field strengths, and that a naive DBI promotion fails. The findings illuminate the structure of the DBI action in RR backgrounds and offer insight into M5-brane dynamics, suggesting that non-commutative formulations may be better suited to capture higher-derivative corrections and the nonlocal features of multiple M5-branes.

Abstract

We derive the Seiberg-Witten map to first order in the non-commutativity parameter for D-branes in the presence of a large R-R background field. This result enables a systematic investigation of the commutative formulation of the corresponding Lagrangian. In the SU($N$) sector, the map introduces a non-local operator. In contrast, in the U(1) sector, this non-locality can be removed. This contrast suggests that the essential source of non-local behavior lies in the non-Abelian degrees of freedom. The commutative description obtained here offers further insight into both the Dirac-Born-Infeld structure and its possible extensions to the dynamics of M5-branes.

Figures (1)

• Figure 1: The result's summarization and implication.