Effective model for pure Yang-Mills theory on ${\mathbb{T}}^2\times {\mathbb{R}}^2$ with Polyakov loops

We investigate the phase diagram and thermodynamics of $SU(N)$ pure Yang-Mills theory on a manifold ${\mathbb{T}}^2\times {\mathbb{R}}^2$ with an effective model that includes two Polyakov loops along two compactified directions. We find that a rich phase structure can appear owing to the spontaneous breaking of two center symmetries for $N=2$ and $3$. Thermodynamic quantities are obtained in the model and compared with recent lattice results. It is shown that two Polyakov loops play significant roles in thermodynamics on ${\mathbb{T}}^2\times {\mathbb{R}}^2$.