Polyakov-Loops and Fermionic Zero Modes in QCD2 on the Torus

TL;DR

This paper derives the free energy and Polyakov-loop expectation values in 2D QCD using path integral methods, revealing a simplified gauge fixing approach and linking fermionic zero modes to gauge field winding numbers.

Contribution

It provides a straightforward derivation of Polyakov-loop properties in QCD2 and connects fermionic zero modes to gauge field topology in a simplified gauge setting.

Findings

Polyakov-loops become vertex operators in a quantum mechanical model.

The Fadeev-Popov determinant cancels with gauge field integration in the chosen gauge.

Fermionic zero modes are related to winding numbers of the gauge field.

Abstract

A simple derivation of the free energy and expectation values of Polyakov-loops in $QCD_2$ via path integral methods is given. In the chosen gauge (which can be generalized to 4 dimensions) without Gribov-copies the Fadeev-Popov determinant and the integration over the space component of the gauge field cancel exactly and we are left only with an integration over the zero components of the gauge field in the Cartan sub-algebra. This way the Polyakov-loop operators become Vertex-operators in a simple quantum mechanical model. The number of fermionic zero modes is related to the winding-numbers of $A_0$ in this gauge.