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.
Categories: hep-th hep-lat hep-ph
