Symmetry Algebras of Large-N Matrix Models for Open Strings

TL;DR

This paper uncovers a Lie algebra structure in gauge invariant observables of large-N matrix models, connecting them to non-commutative geometry, quantum spin chains, and exactly solvable models.

Contribution

It introduces a new Lie algebra extension related to matrix models and establishes isomorphisms with quantum spin chains, revealing new solvable models.

Findings

Identifies a Lie algebra extension involving ${\

Establishes an isomorphism between certain matrix models and quantum spin chains.

Expresses the QCD Hamiltonian explicitly within the Lie algebra framework.

Abstract

We have discovered that the gauge invariant observables of matrix models invariant under U($N$) form a Lie algebra, in the planar large-N limit. These models include Quantum Chromodynamics and the M(atrix)-Theory of strings. We study here the gauge invariant states corresponding to open strings (`mesons'). We find that the algebra is an extension of a remarkable new Lie algebra ${\cal V}_{\Lambda}$ by a product of more well-known algebras such as $gl_{\infty}$ and the Cuntz algebra. ${\cal V}_{\Lambda}$ appears to be a generalization of the Lie algebra of vector fields on the circle to non-commutative geometry. We also use a representation of our Lie algebra to establish an isomorphism between certain matrix models (those that preserve `gluon number') and open quantum spin chains. Using known results on quantum spin chains, we are able to identify some exactly solvable matrix models. Finally, the Hamiltonian of a dimensionally reduced QCD model is expressed explicitly as an element of our Lie algebra.