High Temperature Symmetry Nonrestoration and Inverse Symmetry Breaking on Extra Dimensions

We study $D$-dimensional gauge theory with an extra dimension of a circle at finite temperature. We mainly focus on the expectation value of the gauge field for the direction of the extra dimension, which is the order parameter of the gauge symmetry breaking. We evaluate the effective potential in the one-loop approximation at finite temperature. We show that the vacuum configuration of the theory at finite temperature is determined by a $(D-1)$-dimensional gauge theory defined by removing the Euclidean time coordinate as well as all of the fermions from the original $D$-dimensional gauge theory on the circle. It is pointed out that gauge symmetry nonrestoration and/or inverse symmetry breaking can occur at high temperature in a class of gauge theories on circles and that phase transitions (if they occur) are, in general, expected to be first order.