The Algebras of Large N Matrix Mechanics

TL;DR

This paper develops a Hamiltonian framework for large N matrix models, revealing new algebraic structures like symmetric Cuntz algebras and uncovering large N phenomena such as nonlocality and hidden conserved quantities.

Contribution

It introduces a comprehensive algebraic framework for large N matrix mechanics, including novel free algebras and conserved quantities, advancing understanding of large N limits in matrix models.

Findings

Discovery of symmetric Cuntz algebras and their variants.

Identification of new conserved quantities at large N.

Observation of nonlocality in trace class operators.

Abstract

Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: The Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N.