QCD(1+1) with massless quarks and gauge covariant Sugawara construction

TL;DR

This paper provides a non-perturbative Hamiltonian analysis of massless QCD in 1+1 dimensions, revealing its structure as a gauge theory of Kac-Moody currents and establishing its equivalence to a gauged Wess-Zumino-Witten model.

Contribution

It introduces a gauge-covariant Sugawara construction and demonstrates the full non-perturbative structure of massless QCD(1+1) with multiple flavors and gauge groups.

Findings

Explicit non-perturbative Hamiltonian formulation

Gauge-covariant Sugawara construction for energy-momentum tensor

Equivalence to gauged Wess-Zumino-Witten model

Abstract

We use the Hamiltonian framework to study massless QCD$_{1+1}$, i.e.\ Yang-Mills gauge theories with massless Dirac fermions on a cylinder (= (1+1) dimensional spacetime $S^1\times \R$) and make explicite the full, non-perturbative structure of these quantum field theory models. We consider $N_F$ fermion flavors and gauge group either $\U(N_C)$, $\SU(N_C)$ or another Lie subgroup of $\U(N_C)$. In this approach, anomalies are traced back to kinematical requirements such as positivity of the Hamiltonian, gauge invariance, and the condition that all observables are represented by well-defined operators on a Hilbert space. We also give equal time commutators of the energy momentum tensor and find a gauge-covariant form of the (affine-) Sugawara construction. This allows us to represent massless QCD$_{1+1}$ as a gauge theory of Kac-Moody currents and prove its equivalence to a gauged Wess-Zumino-Witten model with a dynamical Yang-Mills field.