We revisit 't Hooft anomalies in (1+1)$d$ non-spin quantum field theory, starting from the consistency and locality conditions, and find that consistent U(1) and gravitational anomalies cannot always be canceled by properly quantized (2+1)$d$ classical Chern-Simons actions. On the one hand, we prove that certain exotic anomalies can only be realized by non-reflection-positive or non-compact theories; on the other hand, without insisting on reflection-positivity, the exotic anomalies present a caveat to the inflow paradigm. For the mixed U(1) gravitational anomaly, we propose an inflow mechanism involving a mixed U(1)$\times$SO(2) classical Chern-Simons action with a boundary condition that matches the SO(2) gauge field with the (1+1)$d$ spin connection. Furthermore, we show that this mixed anomaly gives rise to an isotopy anomaly of U(1) topological defect lines. The isotopy anomaly can be canceled by an extrinsic curvature improvement term, but at the cost of creating a periodicity anomaly. We survey the holomorphic $bc$ ghost system which realizes all the exotic consistent anomalies, and end with comments on a subtlety regarding the anomalies of finite subgroups of U(1).

中文翻译：

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关于（1 + 1）$ d $中的异乎寻常的一致异常：鬼故事

我们重新审视 (1+1)$d$ 非自旋量子场论中的 't Hooft 异常，从一致性和局域性条件开始，发现一致的 U(1) 和引力异常并不总是可以通过适当量化 (2 +1)$d$ 经典陈-西蒙斯动作。一方面，我们证明了某些外来异常只能通过非反射正或非紧致理论来实现；另一方面，在不坚持反射积极性的情况下，奇异异常对流入范式提出了警告。对于混合 U(1) 引力异常，我们提出了一种流入机制，涉及混合 U(1)$\times$SO(2) 经典陈-西蒙斯作用，其边界条件与 SO(2) 规范场与(1+1)$d$ 自旋连接。此外，我们表明，这种混合异常会引起 U(1) 拓扑缺陷线的同位素异常。同位素异常可以通过外在曲率改进项来消除，但代价是创建周期性异常。我们调查了实现所有奇异一致异常的全纯 $bc$ 鬼系统，并以对 U(1) 的有限子群异常的微妙之处的评论结束。