轴向$U(1)$不可逆对称性对't Hooft线算符的作用：一项格点规范理论研究
Action of the axial $U(1)$ non-invertible symmetry on the 't~Hooft line operator: A lattice gauge theory study
AI总结：
本文在格点规范理论中研究轴向$U(1)$不可逆对称性对't Hooft线算符的作用，发现对称算符扫过't Hooft圈算符时不产生效应，与连续场论结论不等价，并通过半空间gauging验证了该结果。
摘要：
我们采用改进的Villain型格点形式，研究轴向$U(1)$不可逆对称性的对称算符如何作用于$U(1)$规范理论中的't Hooft线算符。我们用一个紧致标量玻色子，即“QED轴子”，来建模轴向反常。为了保证规范不变性，由对偶$U(1)$规范势的线积分定义的简单't Hooft线算符必须由标量场和$U(1)$规范场进行“ dressing ”。在包含带 dressing 因子的't Hooft算符的反常Ward–Takahashi恒等式的基础上，以及在格点上对对称算符的精确定义的基础上，仔细考虑后表明，当对称算符扫过't Hooft圈算符时，它不产生任何效应。这一结果似乎与连续场论中得出的现象不等价。在附录中，我们证明，当在适当的格点框架中表述时，磁$\mathbb{Z}_N$ 1-form对称性的半空间 gauging 会得出与上述相同的结论。对于轴子弦算符，也得到了类似的结果。
英文摘要：
We study how the symmetry operator of the axial $U(1)$ non-invertible symmetry acts on the 't~Hooft line operator in the $U(1)$ gauge theory by employing the modified Villain-type lattice formulation. We model the axial anomaly by a compact scalar boson, the ``QED axion''. For the gauge invariance, the simple 't~Hooft line operator, which is defined by a line integral of the dual $U(1)$ gauge potential, must be ``dressed'' by the scalar and $U(1)$ gauge fields. A careful consideration on the basis of the anomalous Ward--Takahashi identity containing the 't~Hooft operator with the dressing factor and a precise definition of the symmetry operator on the lattice shows that the symmetry operator leaves no effect when it sweeps out a 't~Hooft loop operator. This result appears inequivalent with the phenomenon concluded in the continuum theory. In an appendix, we demonstrate that the half-space gauging of the magnetic $\mathbb{Z}_N$ 1-form symmetry, when formulated in an appropriate lattice framework, leads to the same conclusion as above. A similar result is obtained for the axion string operator.
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