A Review of Symmetry Algebras of Quantum Matrix Models in the Large-N Limit

TL;DR

This review introduces two infinite-dimensional Lie algebras associated with quantum matrix models in the large-N limit, unifying their description and exploring their properties and connections to other well-known algebras.

Contribution

It provides a comprehensive, detailed presentation of the symmetry algebras of quantum matrix models, including new properties and their relation to established algebras.

Findings

Identification of two key Lie algebras for open and closed string sectors

These algebras are quotients of a larger algebra

Connections to Cuntz, Witt, and Virasoro algebras

Abstract

This is a review article in which we will introduce, in a unifying fashion and with more intermediate steps in some difficult calculations, two infinite-dimensional Lie algebras of quantum matrix models, one for the open string sector and one for the closed string sector. Physical observables of quantum matrix models in the large-N limit can be expressed as elements of these Lie algebras. We will see that both algebras arise as quotient algebras of a larger Lie algebra. We will also discuss some properties of these Lie algebras not published elsewhere yet, and briefly review their relationship with well-known algebras like the Cuntz algebra, the Witt algebra and the Virasoro algebra. We will also review how Yang--Mills theory, various low energy effective models of string theory, quantum gravity, string-bit models, and quantum spin chain models can be formulated as quantum matrix models. Studying these algebras thus help us understand the common symmetry of these physical systems.