Roles of Polyakov loops in Yang-Mills theory on 2 × ℝ2

We present an effective model of SU(N) pure Yang-Mills theory on 2 × ℝ2, where two directions are compactified with periodic boundary conditions. Our model includes two Polyakov loops serving as the order parameters of two center symmetries. Based on the model, for N = 2 and N = 3 we show that a rich phase diagram in terms of the center symmetries on 2 × ℝ2 is obtained. Besides, we demonstrate roles of the Polyakov loops by comparing with the recent lattice results focusing on thermodynamic quantities on 2 × ℝ2. We expect that analysis on 2 × ℝ2 provides us with a new clue toward further understanding of pure YM theory with the Polyakov loop at finite temperature