Long-distance interactions of branes: correspondence between supergravity and super Yang-Mills descriptions

TL;DR

The paper addresses subleading corrections to the long-distance brane interaction potentials by extending the known supergravity–SYM correspondence beyond leading order. It proposes that the leading IR part of the L-loop SYM effective action scales as $F^{2L+2}/M^{(7-p)L}$, and for L=2 identifies a universal $F^6$ structure that, when evaluated on carefully chosen SYM backgrounds on dual tori, reproduces the subleading terms in the supergravity potentials for a wide class of 0-brane bound-state configurations, including 1/2, 1/4, and 1/8 BPS states and extremal D=5 and D=4 black holes. The analysis relies on BI-type expansions with a corrected, possibly non-commuting, ${\widehat{\rm STr}} C_6(F)$ to capture cross-terms and commutator contributions; explicit examples demonstrate matching of $V^{(1)}$ and $V^{(2)}$ across various backgrounds, highlighting the subtle role of non-Abelian structure. The work suggests a robust subleading extension of the Matrix theory–supergravity correspondence and outlines directions for direct F^6 computations and string-theory two-loop checks to further validate the framework.

Abstract

We address the issue of correspondence between classical supergravity and quantum super Yang-Mills (or Matrix theory) expressions for the long-distance, low-velocity interaction potentials between 0-branes and various bound states of branes. The leading-order potentials are known to be reproduced by the F^4 terms in the 1-loop SYM effective action. Using self-consistency considerations, we determine a universal combination of F^6 terms in the 2-loop SYM effective action that corresponds to the subleading terms in the supergravity potentials in many cases, including 0-brane scattering off 1/8 supersymmetric 410 and 4440 bound states representing extremal D=5 and D=4 extremal black holes. We give explicit descriptions of the these configurations in terms of 1/4 supersymmetric SYM backgrounds on dual tori. Under a proper choice of the gauge field backgrounds, the 2-loop F^6 SYM action reproduces the full expression for the subleading term in the supergravity potentials, including its subtle v^2 part.