A Lie Algebra for Closed Strings, Spin Chains and Gauge Theories

TL;DR

This paper introduces a new Lie algebra called the heterix algebra, which describes the symmetries of closed strings, spin chains, and gauge theories in the large-N limit, enabling solvable matrix models and linking them to quantum spin systems.

Contribution

The paper discovers the heterix algebra as a novel Lie algebra structure acting on matrix-invariant states, and introduces the cyclix algebra as its quotient, connecting gauge theories, string models, and spin chains.

Findings

The heterix algebra extends the Virasoro algebra with general linear algebra.

The cyclix algebra models the dynamical symmetries of certain gauge theories and spin chains.

Some matrix models become solvable in the planar limit using this algebraic framework.

Abstract

We consider quantum dynamical systems whose degrees of freedom are described by $N \times N$ matrices, in the planar limit $N \to \infty$. Examples are gauge theoires and the M(atrix)-theory of strings. States invariant under U(N) are `closed strings', modelled by traces of products of matrices. We have discovered that the U(N)-invariant opertors acting on both open and closed string states form a remarkable new Lie algebra which we will call the heterix algebra. (The simplest special case, with one degree of freedom, is an extension of the Virasoro algebra by the infinite-dimensional general linear algebra.) Furthermore, these operators acting on closed string states only form a quotient algebra of the heterix algebra. We will call this quotient algebra the cyclix algebra. We express the Hamiltonian of some gauge field theories (like those with adjoint matter fields and dimensionally reduced pure QCD models) as elements of this Lie algebra. Finally, we apply this cyclix algebra to establish an isomorphism between certain planar matrix models and quantum spin chain systems. Thus we obtain some matrix models solvable in the planar limit; e.g., matrix models associated with the Ising model, the XYZ model, models satisfying the Dolan-Grady condition and the chiral Potts model. Thus our cyclix Lie algebra described the dynamical symmetries of quantum spin chain systems, large-N gauge field theories, and the M(atrix)-theory of strings.