# Anomalies and Bounds on Charged Operators

## TL;DR

This paper explores how 't Hooft anomalies influence bounds on charged operators in 2D conformal field theories with $Z_2$ and $U(1)$ symmetries, revealing universal bounds and differences based on anomaly presence.

## Contribution

It provides the first universal bootstrap bounds on (1+1)D CFTs with $Z_2$ symmetry considering anomalies, highlighting how anomalies affect operator bounds.

## Key findings

- Universal upper bound on the lightest $Z_2$ odd operator if symmetry is anomalous.
- No bound on the lightest $Z_2$ odd operator if symmetry is non-anomalous.
- Constraints on the spectrum of defect states in non-anomalous cases.

## Abstract

We study the implications of 't Hooft anomaly (i.e. obstruction to gauging) on conformal field theory, focusing on the case when the global symmetry is $\mathbb{Z_2}$. Using the modular bootstrap, universal bounds on (1+1)-dimensional bosonic conformal field theories with an internal $\mathbb{Z_2}$ global symmetry are derived. The bootstrap bounds depend dramatically on the 't Hooft anomaly. In particular, there is a universal upper bound on the lightest $\mathbb{Z_2}$ odd operator if the symmetry is anomalous, but there is no bound if the symmetry is non-anomalous. In the non-anomalous case, we find that the lightest $\mathbb{Z_2}$ odd state and the defect ground state cannot both be arbitrarily heavy. We also consider theories with a $U(1)$ global symmetry, and comment that there is no bound on the lightest $U(1)$ charged operator if the symmetry is non-anomalous.