Theta、时间延迟和温度
Theta, Time Reversal, and Temperature
摘要：
$SU(N)$规范理论在$θ=0$和$θ=π$处是时间反演不变的。我们证明了在$θ=π$处存在一个离散的t Hooft反常，它涉及到时间反演和中心对称性。这种反常导致了对理论真空的限制。由此可见，在$θ=π$，真空不可能是平凡的非简并带隙态。(By相反，在$θ=0$的真空是带隙的、非简并的和平凡的。）由于异常，该理论承认非平凡域墙支持低维理论。根据$θ=π$的真空性质，可能有几种相图。假设类空环的面积定律，我们得到一个不等式，它涉及到CP和中心对称性恢复时的温度。我们还分析了$SU(2)$规范理论的替代方案。$θ=π$的基本对称性是8个元素的二面体群。如果允许去限制循环，则可以有两个$O(2)$对称不动点。也可能是围绕$θ=π$的四维理论是无间隙的，例如库仑相位可以匹配潜在的异常。
英文摘要：
$SU(N)$ gauge theory is time reversal invariant at $θ=0$ and $θ=π$. We show that at $θ=π$ there is a discrete 't Hooft anomaly involving time reversal and the center symmetry. This anomaly leads to constraints on the vacua of the theory. It follows that at $θ=π$ the vacuum cannot be a trivial non-degenerate gapped state. (By contrast, the vacuum at $θ=0$ is gapped, non-degenerate, and trivial.) Due to the anomaly, the theory admits nontrivial domain walls supporting lower-dimensional theories. Depending on the nature of the vacuum at $θ=π$, several phase diagrams are possible. Assuming area law for space-like loops, one arrives at an inequality involving the temperatures at which CP and the center symmetry are restored. We also analyze alternative scenarios for $SU(2)$ gauge theory. The underlying symmetry at $θ=π$ is the dihedral group of 8 elements. If deconfined loops are allowed, one can have two $O(2)$-symmetric fixed points. It may also be that the four-dimensional theory around $θ=π$ is gapless, e.g. a Coulomb phase could match the underlying anomalies.
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