String Field Theory of Two Dimensional QCD on a Cylinder: A Realization of W-infinity Current Algebra

TL;DR

This paper formulates 2D QCD on a cylinder using gauge-invariant operators, revealing a $W_infty$ algebra structure in meson and gluon sectors, and finds a glueball spectrum matching $c=1$ string theory states.

Contribution

It introduces a novel gauge-invariant operator framework for 2D QCD on a cylinder, uncovering $W_infty$ algebra structures and solving for the gluon spectrum in the small circle limit.

Findings

Gluon and meson sectors satisfy $W_infty$ current algebra.

Gluon spectrum matches $c=1$ string theory discrete states.

Effective gluon theory is solvable in the small circle limit.

Abstract

We consider 2-dimensional QCD on a cylinder, where space is a circle of length $L$. We formulate the theory in terms of gauge-invariant gluon operators and multiple-winding meson (open string) operators. The meson bilocal operators satisfy a $W_\infty$ current algebra. The gluon sector (closed strings) contains purely quantum mechanical degrees of freedom. The description of this sector in terms of non-relativistic fermions leads to a $W_\infty$ algebra. The spectrum of excitations of the full theory is therefore organized according to two different algebras: a wedge subalgebra of $W_\infty$ current algebra in the meson sector and a wedge subalgebra of $W_\infty$ algebra in the glueball sector. In the large $N$ limit the theory becomes semiclassical and an effective description for the gluon degrees of freedom can be obtained. We have solved the effective theory of the gluons in the small $L$ limit. We get a glueball spectrum which coincides with the `discrete states' of the (Euclidean) $c=1$ string theory. We remark on the implications of these results for (a) QCD at finite temperature and (b) string theory.