On Exotic Consistent Anomalies in (1+1)$d$: A Ghost Story

TL;DR

This work analyzes 't Hooft anomalies in (1+1)d non-spin QFT from consistency and locality, showing that some anomalies resist cancellation by standard (2+1)d Chern–Simons inflow. It develops a mixed U(1)–gravity inflow via a relativistic bulk CS term with boundary spin-connection matching, and reveals isotopy and periodicity anomalies for U(1) defect lines. The holomorphic bc ghost system is shown to realize all discussed exotic anomalies, providing a concrete, minimal model. The paper also clarifies subtleties in embedding finite subgroups of U(1) and discusses limitations and opportunities for future higher-dimensional generalizations.

Abstract

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).

Figures (2)

• Figure 1: The square torus $\tau = i, \, \bar{\tau} = -1$ is symmetric under 90 degree rotations (modular $S$) and reflections. The partition function on the square torus transforms with a phase $\theta_S$ under the former, and must be positive in a reflection-positive theory due to the later.
• Figure 2: Deforming a symmetry defect line from the curve ${\mathcal{C}}$ across the domain ${\mathcal{D}}$ to the new curve ${\mathcal{C}}'$.