Matching of Wilson loop eigenvalue densities in 1+1, 2+1 and 3+1 dimensions

TL;DR

This paper demonstrates that eigenvalue densities of Wilson loops in SU(N) gauge theories across 1+1, 2+1, and 3+1 dimensions are nearly identical under certain conditions, with implications for phase transitions and large-N behavior.

Contribution

It shows that eigenvalue densities match across dimensions to second order in strong-coupling and perturbation theory, extending understanding of Wilson loop properties in different spacetime dimensions.

Findings

Eigenvalue densities match in different dimensions when loop traces are equal.

Matching persists to second order in strong-coupling and perturbation theory.

Numerical evidence confirms matching and identifies small deviations away from ideal conditions.

Abstract

We investigate the matching of eigenvalue densities of Wilson loops in SU(N) lattice gauge theory: the eigenvalue densities in 1+1, 2+1 and 3+1 dimensions are nearly identical when the traces of the loops are equal. We show that the matching is present to at least second order in the strong-coupling expansion, and also to second order in perturbation theory. We find that in the continuum limit there is matching at all values of the trace for bare Wilson loops. We confirm numerically that there is matching in these limits and find there are small violations away from them. We discuss the implications for the bulk transitions and for non-analytic gap formation at N = infinity in 2+1 and 3+1 dimensions.