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

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.

77 pages, 21 eps figures, 1 table, LaTeX2.09; an invited review article

References in corpus (6)

• Matrix String Theory
• A Lie Algebra for Closed Strings, Spin Chains and Gauge Theories
• Symmetries of Large N Matrix Models for Closed Strings
• Symmetry Algebras of Large-N Matrix Models for Open Strings
• A Model of Interacting Partons for Hadronic Structure Functions
• Integrability of Supersymmetric Quantum Matrix Models in the Large-N Limit

Cited by in corpus (6)

• Yangian Symmetries of Matrix Models and Spin Chains: The Dilatation Operator of $\mathcal{N}$$=4$ SYM
• The Dilatation Operator of $\mathcal{N}$ $=$ 4 SYM and Classical Limits of Spin Chains and Matrix Models
• Extraction of parton distributions from lattice QCD
• SU(2|2) for Theories with Sixteen Supercharges at Weak and Strong Coupling
• Large N limit of SO(N) scalar gauge theory
• Unitary Irreducible Representations of a Lie Algebra for Matrix Chain Models