Confinement, Screening and the Center on S^3 x S^1

Understand

We compute the one-loop effective potential for the Polyakov loop on $S^3 times S^1$ for an asymptotically free gauge theory of arbitrary group $G$ and a generic matter content.

• We apply this result to study the phase structures of $G_2$, SO(N) and $Sp(2N)$ gauge theories which turn out to be in qualitative agreement with the results of lattice calculations.
• On $S^3 \times S^1$, at zero temperature, the Polyakov loop is zero for kinematical reasons.
• For small but non-zero temperature, the Polyakov loop is still zero if the gauge theory has an unbroken center, while it acquires a small vacuum expectation value for gauge theories whose center is trivial or explicitly broken by the presence of dynamical matter fields.