Non-Abelian gauge field dynamics on matrix D-branes in curved space and two-dimensional $\sigma$-models

TL;DR

This paper surveys the formulation of a $\sigma$-model for open strings in curved backgrounds with matrix-valued D-branes and gauge fields, deriving equations of motion and addressing matrix ordering and T-duality conflicts.

Contribution

It provides a detailed derivation of D-brane and gauge field equations from conformal invariance, resolving matrix ordering issues and analyzing T-duality conflicts in the context of curved space.

Findings

Derived equations of motion for D-branes and gauge fields from conformal invariance.

Resolved the matrix ordering problem in the $\sigma$-model.

Identified conflicts between classical T-duality rules and renormalization group analysis.

Abstract

A survey is given of the formulation of a $\sigma$-model describing an open string moving in general target space background fields and coupling to both a matrix-valued D-brane position and a matrix-valued gauge field on the D-brane. The equations of motion for the D-brane and the gauge field are derived from the conformal invariance condition on the string world sheet in lowest order of $\ap$. The ordering problem of the involved matrices is solved. In addition to our previous work we discuss a conflict between the classical T-duality rules and renormalization. The calculation of the RG $\beta$-functions does not yield the mass term obtained by formal application of these rules in the case of target space separated D-brane copies.