Symmetry Algebras of Large-N Matrix Models for Open Strings

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 ${\mathcal{V}}_{\Lambda }$ by a product of more well-known algebras such as $g{l}_{\infty }$ and the Cuntz algebra. ${\mathcal{V}}_{\Lambda }$ 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.

Comments: 44 pages, 8 eps figures, 3 tables, LaTeX2.09; this is the published version