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 ${\mathbb{T}}^2=S_{\beta }^1\times S_L^1$, with $S_{\beta }^1$ being a thermal circle with period $\beta =1/T$ ($T$ is the temperature) while $S_L^1$ 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 nonperturbative 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.

• Received 17 October 2016

DOI:https://doi.org/10.1103/PhysRevD.95.034031