Equivalence of Two Dimensional QCD and the $c=1$ Matrix Model

TL;DR

This paper demonstrates the equivalence between two-dimensional QCD on a circle and the $c=1$ matrix model, showing that the theory can be reduced to free fermions on a circle with specific gauge group considerations.

Contribution

It establishes a connection between 2D QCD with compactified space and the $c=1$ matrix model, including the derivation of the Hamiltonian and reduction to free fermions.

Findings

States consist of interacting winding strings

Hamiltonian expressed in terms of a continuous field

2D QCD reduces to free fermions on a circle

Abstract

We consider two dimensional QCD with the spatial dimension compactified to a circle. We show that the states in the theory consist of interacting strings that wind around the circle and derive the Hamiltonian for this theory in the large $N$ limit, complete with interactions. Mapping the winding states into momentum states, we express this Hamiltonian in terms of a continuous field. For a $U(N)$ gauge group with a background source of Wilson loops, we recover the collective field Hamiltonian found by Das and Jevicki for the $c=1$ matrix model, except the spatial coordinate is on a circle. We then proceed to show that two dimensional QCD with a $U(N)$ gauge group can be reduced to a one- dimensional unitary matrix model and is hence equivalent to a theory of $N$ free nonrelativistic fermions on a circle. A similar result is true for the group $SU(N)$, but the fermions must be modded out by the center of mass coordinate.