Linde problem in Yang-Mills theory compactified on $\mathbb{R}^2 \times \mathbb{T}^2$

TL;DR

This paper analyzes the limitations of perturbative expansions in $SU(3)$ Yang-Mills theory on a compactified space, revealing breakdowns at low orders due to non-perturbative effects related to the geometry of the space.

Contribution

It extends the understanding of the Linde problem to Yang-Mills theory on $ ext{R}^2 imes ext{T}^2$, showing perturbation theory's breakdown at order $g^2$ due to a non-perturbative scale.

Findings

Perturbative expansion breaks down at order $g^2$ in the studied setup.

Non-perturbative scale $ ext{~} g imes ext{sqrt}(TM)$ influences the breakdown.

Results likely extend to other genus-one compact surfaces.

Abstract

We investigate the perturbative expansion in $SU(3)$ Yang-Mills theory compactified on $\mathbb{R}^2\times \mathbb{T}^2$ where the compact space is a torus $\mathbb{T}^2= S^1_{\beta}\times S^1_{L}$, with $S^1_{\beta}$ being a thermal circle with period $\beta=1/T$ ($T$ is the temperature) while $S^1_L$ is a circle with finite length $L=1/M$, where $M$ is an energy scale. A Linde-type analysis indicates that perturbative calculations for the pressure in this theory break down already at order $\mathcal{O}(g^2)$ due to the presence of a non-perturbative scale $\sim g \sqrt{TM}$. We conjecture that a similar result should hold if the torus is replaced by any other compact surface of genus one.