J un 1 99 9 A Lie Algebra for Closed Strings , Spin Chains and Gauge Theories

We consider quantum dynamical systems whose degrees of freedom are described by N × N matrices, in the planar limit N → ∞. Examples are gauge theories 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 operators 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 describes the dynamical symmetries of quantum spin chain systems, large-N gauge field theories, and the M(atrix)-theory of strings. The string is emerging as a verstile concept in physics, second only to the notion of a particle in its usefulness. The instantaneous configuration of a string can be thought of roughly as a curve in space, with an energy proportional to its length. The curve may be closed, or open with some extra degrees of freedom stuck at its endpoints. The string is being intensely studied as the fundamental object in the quantum theory of gravity and perhaps even the unified theory of all forces of nature. The modern notion of a relativistic string theory originated in attempts to understand hadron dynamics, in the sixties and early seventies. However , it is now established that Quantum Chromodynamics (QCD) is the fundamental theory of strong interactions: the hadrons are not elementary particles but instead are bound states of quarks and gluons. In spite of this, attempts to understand hadron dynamics in terms of strings have not altogether ceased. Many features of hadron dynamics which originally prompted people to construct the string theory still do not have …