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

TL;DR

This paper introduces a hierarchy of infinite-dimensional Lie superalgebras that describe the symmetries of quantum matrix models in the large-N limit, unifying various physical systems and extending known algebraic structures.

Contribution

It presents a unifying framework of Lie superalgebras for quantum matrix models, generalizing known algebras and linking symmetry structures to physical observables.

Findings

Defined new Lie superalgebras for quantum matrix models

Connected these algebras to physical observables like the Hamiltonian

Showed these algebras generalize known structures like Virasoro and Witt

Abstract

Quantum matrix models in the large-N limit arise in many physical systems like Yang-Mills theory with or without supersymmetry, quantum gravity, string-bit models, various low energy effective models of string theory, M(atrix) theory, quantum spin chain models, and strongly correlated electron systems like the Hubbard model. We introduce, in a unifying fashion, a hierachy of infinite-dimensional Lie superalgebras of quantum matrix models. One of these superalgebras pertains to the open string sector and another one the closed string sector. Physical observables of quantum matrix models like the Hamiltonian can be expressed as elements of these Lie superalgebras. This indicates the Lie superalgebras describe the symmetry of quantum matrix models. We present the structure of these Lie superalgebras like their Cartan subalgebras, root vectors, ideals and subalgebras. They are generalizations of well-known algebras like the Cuntz algebra, the Virasoro algebra, the Toeplitz algebra, the Witt algebra and the Onsager algebra. Just like we learnt a lot about critical phenomena and string theory through their conformal symmetry described by the Virasoro algebra, we may learn a lot about quantum chromodynamics, quantum gravity and condensed matter physics through this symmetry of quantum matrix models described by these Lie superalgebras.