We study $D$-dimensional gauge theory with an extra dimension of a circle atfinite temperature. We mainly focus on the expectation value of the gauge fieldfor the direction of the extra dimension, which is the order parameter of thegauge symmetry breaking. We evaluate the effective potential in the one-loopapproximation at finite temperature. We show that the vacuum configuration ofthe theory at finite temperature is determined by a $(D-1)$-dimensional gaugetheory defined by removing the Euclidean time coordinate as well as all of thefermions from the original $D$-dimensional gauge theory on the circle. It ispointed out that gauge symmetry nonrestoration and/or inverse symmetry breakingcan occur at high temperature in a class of gauge theories on circles and thatphase transitions (if they occur) are, in general, expected to be first order.
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机译：我们研究了在无限温度下具有额外维度的$ D $维规范理论。我们主要关注规范区域对额外尺寸方向的期望值，这是规范对称性破坏的顺序参数。我们在有限温度下评估一回路逼近中的有效电位。我们证明了在有限温度下该理论的真空构型由一个（D-1）$维规范理论确定，该理论通过从欧几里德时间坐标以及原始的$ D $维规范理论中删除所有推论而定义。圈。指出，在圆上的一类规范理论中，规范对称性不还原和/或逆对称破坏可能在高温下发生，并且一般认为相变（如果发生）是一阶的。
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