Quantum Mechanics of the Vacuum State in Two-Dimensional QCD with Adjoint Fermions

TL;DR

This paper investigates the quantum properties of the vacuum in two-dimensional QCD with adjoint fermions, focusing on symmetry, topology, and anomalies, providing analytical insights especially for small circle limits.

Contribution

It offers a detailed analysis of symmetry, topology, and anomalies in 2D QCD with adjoint fermions, including an analytical vacuum description on small circles.

Findings

Residual $Z_N$ gauge symmetry is characterized.

Vacuum structure is analytically described in the small circle limit.

Discrepancies with other approaches to fermion condensates are discussed.

Abstract

A study of two-dimensional QCD on a spatial circle with Majorana fermions in the adjoint representation of the gauge groups SU(2) and SU(3) has been performed. The main emphasis is put on the symmetry properties related to the homotopically non-trivial gauge transformations and the discrete axial symmetry of this model. Within a gauge fixed canonical framework, the delicate interplay of topology on the one hand and Jacobians and boundary conditions arising in the course of resolving Gauss's law on the other hand is exhibited. As a result, a consistent description of the residual $Z_N$ gauge symmetry (for SU(N)) and the ``axial anomaly" emerges. For illustrative purposes, the vacuum of the model is determined analytically in the limit of a small circle. There, the Born-Oppenheimer approximation is justified and reduces the vacuum problem to simple quantum mechanics. The issue of fermion condensates is addressed and residual discrepancies with other approaches are pointed out.