String field theory of two-dimensional QCD on a cylinder: a realization of W_∞ current algebra

Abstract

We consider two-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_∞ current algebra. The gluon sector (closed strings) contains purely quantum-mechanical degrees of freedom. The description of this sector in terms of nonrelativistic fermions leads to a W8 algebra. The spectrum of excitations of the full theory is therefore organized according to two different algebras: a wedge subalgebra of W_∞ current algebra in the meson sector and a wedge subalgebra of W_∞ 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.