TASI Lectures on Matrix Theory

TL;DR

This paper presents Matrix Theory (BFSS) as a nonperturbative DLCQ formulation of M Theory in backgrounds with at least six noncompact dimensions, highlighting how gravity enforces a Hamiltonian constraint and how Lorentz-invariant dynamics emerge in the large-$N$ limit. It develops the light-cone framework, holographic underpinnings, and the spectrum of gravitons, membranes, and fivebranes, connecting to 11D supergravity, IIA/IIB string theories, and the AdS/CFT correspondence. The talk surveys compactifications on circles and tori, revealing a rich landscape of dualities, including the emergence of 1+1D SYM descriptions, ADHM instanton moduli spaces, and little string theories with Hagedorn spectra that are not conventional quantum field theories. It identifies both successes—precise matches to known S-matrix elements and dualities—and outstanding challenges, notably isolating the $1/N$-order dynamics and extending the formalism to Calabi–Yau compactifications, while suggesting that DLCQ M Theory provides a powerful, though incomplete, nonperturbative framework for quantum gravity.

Abstract

This is a summary of key issues in Matrix Theory and its compactifications. It is emphasized that Matrix Theory is a valid Discrete Light Cone Quantization of M Theory with at least 6 noncompact asymptotically flat dimensions and 16 or 32 Supersymmetry Charges. The background dependence of the quantum mechanics of M Theory, and the necessity of working in light cone frame in asymptotically flat spacetimes are explained in terms of the asymptotic density of states of the theory, which follows from the Bekenstein-Hawking entropy formula. In four noncompact dimensions one is led to expect a Hagedorn spectrum in light cone energy. This suggests the possible relevance of ``little string theories'' (LSTs) to the quantum description of four dimensional compactifications, because one can argue that their exact high energy spectrum has the Hagedorn form. Some space is therefore devoted to a discussion of the properties of LSTs, which were first discovered as the proper formulation of Matrix Theory on the five torus.