Bound States of D-Branes and the Non-Abelian Born-Infeld Action

TL;DR

This paper investigates the correct non-abelian generalisation of the Born–Infeld action for D-branes by computing the two-loop open-string beta functions in both bosonic and supersymmetric settings. It finds that fermionic contributions in the open superstring cancel the bosonic two-loop divergences, making Tseytlin’s non-abelian BI action consistent with $\beta=0$ at leading orders, while the bosonic case shows deviations from Tseytlin’s proposal. Through T-duality, it derives a bound-state description of interacting D-branes, where non-abelian coordinates acquire matrix-valued nature and massive string exchanges lead to a mass term in a massive Yang–Mills theory. In special self-dual cases the non-abelian BI action reduces to Yang–Mills, with instantons as BPS states, highlighting the interplay between non-abelian BI dynamics and familiar gauge theory structures. The results provide a coherent framework for non-abelian D-brane dynamics and point to higher-loop and D0-brane analyses as important future directions.

Abstract

We attempt to settle the issue as to what is the correct non-abelian generalisation of the Born-Infeld action, via a consideration of the two-loop $β$--function for the non-abelian background gauge field in open string theory. An analysis of the bosonic theory alone shows the recent proposal of Tseytlin's to be somewhat lacking. For the superstring, however, this proposal would seem to be correct, and not just within the approximation used in \cite{tseytlin}. Since it is this latter case that is relevant to the description of D-branes we, in effect, obtain an independent verification of Tseytlin's result. Some issues involved in the concept of non-abelian T--duality are discussed; and it is shown how the interaction between separated and parallel branes, in the form of massive string states, emerges.