On exotic consistent anomalies in (1+1)$d$: A ghost story

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