A $T_2 \times R^2$ roadmap to Confinement in SU(2) Yang-Mills theory

TL;DR

This paper investigates the confinement mechanism in SU(2) Yang-Mills theory on a twisted torus, confirming semiclassical predictions and exploring the transition from dilute vortex gas to non-dilute regimes through lattice simulations.

Contribution

It provides the first detailed lattice study of SU(2) Yang-Mills on a twisted torus, confirming semiclassical vortex gas behavior and identifying the transition scale to non-dilute regimes.

Findings

Confirmed semiclassical vortex gas behavior at small torus sizes.

Determined the scale signaling transition to non-dilute vortex regimes.

Observed the string tension approaching infinite volume values.

Abstract

We study the behaviour of \SU{2} Yang-Mills fields on a $T_2\times R^2$ geometry where the two-torus is equipped with twisted boundary conditions. We monitor the evolution of the dynamics of the system as a function of the torus size $l_s$. For small sizes the behaviour of the system is well understood in terms of semiclassical predictions. In our case, the long distance structure is that of a two-dimensional gas of vortex-like fractional instantons with size and density growing with $l_s$. Our lattice Monte Carlo simulations confirm the semiclassical predictions and allow the determination of the relevant scale signalling the transition to the non-dilute situation. At low densities the string tension takes the standard value of a 2D center-vortex gas, growing with the density and approaching the value measured at infinite volume. Our work includes preliminary studies of the extension to \SU{N} and to the region of large sizes in which boundary conditions are irrelevant, and all physical scales are determined by the $\Lambda$ parameter.