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

TL;DR

This paper investigates exotic anomalies in (1+1)d quantum field theories, revealing their limitations, proposing new inflow mechanisms, and analyzing their implications for topological defect lines and ghost systems.

Contribution

It demonstrates that certain exotic anomalies cannot be canceled by classical Chern-Simons actions and introduces a novel inflow mechanism involving mixed U(1) and SO(2) Chern-Simons terms.

Findings

Exotic anomalies are realizable only in non-reflection-positive theories.

A new inflow mechanism for mixed U(1) gravitational anomalies is proposed.

The isotopy anomaly of U(1) defect lines can be canceled by extrinsic curvature terms.

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).