The M(atrix) model of M-theory
Washington Taylor
TL;DR
The M(atrix) theory notes present a concrete, nonperturbative framework for M-theory based on a supersymmetric $N\times N$ matrix quantum mechanics, arising either from regularizing the 11D supermembrane or from a DLCQ construction of M-theory. The core idea is that matrix degrees of freedom encode all M-theory objects (supergravitons, membranes, and potentially 5-branes) and that gravitational interactions emerge from Yang–Mills dynamics at long range; the BFSS conjecture posits this matrix model becomes a complete description in the $N\to\infty$ limit. The notes carefully derive the matrix model from the light-front quantization of the supermembrane, discuss the construction of M-theory objects from matrices, and review the status of reproducing 11D supergravity interactions, including extensions to curved and compact backgrounds and covariant quantization attempts. Finally, they discuss the unresolved issues, such as membrane instability and the extent to which matrix theory captures full Lorentz invariance, while highlighting the deep connections between D-brane gauge dynamics and gravity in M-theory.
Abstract
These lecture notes give a pedagogical and (mostly) self-contained review of some basic aspects of the Matrix model of M-theory. The derivations of the model as a regularized supermembrane theory and as the discrete light-cone quantization of M-theory are presented. The construction of M-theory objects from matrices is described, and gravitational interactions between these objects are derived using Yang-Mills perturbation theory. Generalizations of the model to compact and curved space-times are discussed, and the current status of the theory is reviewed.