Generally Covariant Actions for Multiple D-branes

TL;DR

This paper presents a formalism for constructing generally covariant actions for multiple D-branes, utilizing covariant matrix fields and a matrix delta function to describe brane positions in curved space.

Contribution

It introduces a covariant framework for multiple D-branes with manifest general covariance, including a covariant distribution function and scalar single-trace actions.

Findings

Actions are expressed as integrals over curved space with covariant matrix fields.

Diagonal matrices localize to individual brane positions, reproducing single-brane actions.

The formalism simplifies transformation laws of matrix coordinates under coordinate changes.

Abstract

We develop a formalism that allows us to write actions for multiple D-branes with manifest general covariance. While the matrix coordinates of the D-branes have a complicated transformation law under coordinate transformations, we find that these may be promoted to (redundant) matrix fields on the transverse space with a simple covariant transformation law. Using these fields, we define a covariant distribution function (a matrix generalization of the delta function which describes the location of a single brane). The final actions take the form of an integral over the curved space of a scalar single-trace action built from the covariant matrix fields, tensors involving the metric, and the covariant distribution function. For diagonal matrices, the integral localizes to the positions of the individual branes, giving N copies of the single-brane action.