D-branes and Matrix Theory in Curved Space

TL;DR

This work argues for defining Matrix theory in curved backgrounds via gauged nonlinear sigma-models, motivated by the D-brane gauge theory–gravity link and the BFSS conjecture that 11D physics emerges from large $N$ gauge theory. The authors outline a D-geometry program that promotes matrix-valued coordinates to encode curved spaces, showing how off-diagonal modes reproduce leading one-loop gravitational interactions (e.g., the $v^4/d^7(x,y)$ term) and identifying Ricci-flat backgrounds ($R_{\mu\nu}=0$) as natural constraints under isotropic mass conditions. They further propose a large-$N$ renormalization-group approach to probe the flow of background geometry and discuss the necessity of higher-derivative or nonlocal corrections to capture full locality, especially in non-compact settings. While the exact, fully local, background-independent formulation remains elusive, these steps provide a concrete, testable framework linking gauge theory in the BFSS/Beckers-DLCQ spirit to curved-space supergravity, with implications for compactifications and M-theory in nontrivial backgrounds.

Abstract

We discuss the relation between supersymmetric gauge theory of branes and supergravity; as it was discovered in D-brane physics, and as it appears in Matrix theory, with emphasis on motion in curved backgrounds. We argue that gauged sigma model Lagrangians can be used as definitions of Matrix theory in curved space. Lecture given at Strings '97; June 20, 1997.