M(atrix) Theory : a Pedagogical Introduction

TL;DR

The paper provides a thorough, pedagogical overview of the BFSS matrix model as a nonperturbative formulation of M-theory in the infinite momentum frame. It connects D0-brane quantum mechanics to 11D supergravity via strong coupling of IIA string theory, showing how 11D KK modes, membranes, and the full graviton Fock space arise from a $U(N)$ matrix quantum mechanics in the $N\to\infty$ limit. Key results include the correct one-loop graviton scattering, emergence of membranes in the large-$N$ limit, and a consistent interpretation of D0-brane coordinates as noncommuting matrices encoding brane interactions. The work highlights the BFSS conjecture as a viable framework linking M-theory, D-branes, and matrix dynamics, with significant implications for nonperturbative quantum gravity and the microscopic structure of M-theory. The findings underscore the deep correspondence between matrix models, eleven-dimensional physics, and brane dynamics, suggesting a concrete nonperturbative description of M-theory in the IMF with practical checks against supergravity predictions.

Abstract

I attempt to give a pedagogical introduction to the matrix model of M-theory as developed by Banks, Fischler, Shenker and Susskind (BFSS). In the first lecture, I introduce and review the relevant aspects of D-branes with the emergence of the matrix model action. The second lecture deals with the appearance of eleven-dimensional supergravity and M-theory in strongly coupled type IIA superstring theory. The third lecture combines the material of the two previous ones to arrive at the BFSS conjecture and explains the evidence presented by these authors. The emphasis is not on most recent developments but on a hopefully pedagogical presentation.

Figures (2)

• Figure 1: A D$p$-brane is a hypersurface with $p$ space-like and one time-like dimensions on which open strings with Dirichlet boundary conditions end.
• Figure 2: A Wilson line breaking ${\rm U}(N)$ to ${\rm U}(1)^N$ yields $N$ D-branes at positions $\theta_j \hat{R}$.