Yang-Mills theories on the space-time $S_1 \times R$ cylinder: equal-time quantization in light-cone gauge and Wilson loops

TL;DR

This paper studies the quantization of pure Yang-Mills theories on a cylindrical space-time using light-cone gauge, highlighting the role of zero modes and topological variables, and analyzing Wilson loops in abelian and non-abelian cases.

Contribution

It introduces a method for equal-time quantization of Yang-Mills theories on a cylinder in light-cone gauge, emphasizing zero modes and topological effects, with exact results for abelian Wilson loops.

Findings

Exact area law for abelian Wilson loops

Zero modes encode topological variables

Difficulties arise in non-abelian Wilson loop calculations

Abstract

Pure Yang-Mills theories on the $S_1\times R$ cylinder are quantized in light-cone gauge $A_-=0$ by means of ${\bf equal-time}$ commutation relations. Positive and negative frequency components are excluded from the ``physical" Hilbert space by imposing Gauss' law in a weak sense. Zero modes, related to the winding on the cylinder, provide non trivial topological variables of the theory. A Wilson loop with light-like sides is studied: in the abelian case it can be exactly computed obtaining the expected area result, whereas difficulties are pointed out in non abelian cases.