arXiv 1809.050782018-09-13hep-th

自由费米子Anomaly与钉群的表示

The Free Fermion Anomaly and Representations of the Pin Groups

Yakov Landau

摘要：

我们考虑了环上玻色子与自由费米子之间的对偶性，并证明了它们具有对应于O（2）的双重覆盖下的态变换的反常。O（2）有两个（通常不是等距的）双重覆盖，称为Pin $_+(2)$和Pin $_-(2)$。通常可以通过检查反射是否与$\pm 1$成直角来在组级别区分这些。我们证明了在复Hilbert空间的不可约表示中，时间反转和电荷共轭的交换给出了区分Pin $_+(2)$和Pin $_-(2)$的另一种方法.虽然我们只证明了单个费米子的对偶性，但这种异常也存在于任何数量的自由费米子中。对于偶数个费米子，我们证明了O（2n）的两个双覆盖是同构的。对于奇数个费米子，我们证明了Pin $_+$和Pin $_-$的不可约表示之间的区别仍然可以通过时间反转和电荷共轭的交换来检测，即Pin $_\pm(2n)$将具有$TCT = \pm C$。

英文摘要：

We consider a duality between a boson on a ring and a free fermion and show that they have an anomaly which corresponds to the states transforming under double covers of O(2). There are two (in general not isometric) double covers of O(2), known as Pin$_+(2)$ and Pin$_-(2)$. These can in general be distinguished at the group level by checking whether reflections square to $\pm 1$. We show that in irreducible representations in complex Hilbert spaces the commutation of time reversal and charge conjugation gives another method for distinguishing Pin$_+(2)$ and Pin$_-(2)$. While we only demonstrate the duality for a single fermion, the anomaly is also present in any number of free fermions. For an even number of fermions we show that the two double covers of O(2n) are isomorphic. For an odd number of fermions we show that the distinction between irreducible representations of Pin$_+$ and Pin$_-$ can still be detected by the commutation of time reversal and charge conjugation namely Pin$_\pm(2n)$ will have $TCT = \pm C$.

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英文摘要：

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