Anomalies and Bounds on Charged Operators

TL;DR

This work reveals a sharp, anomaly-dependent constraint on the spectrum of charged operators in 2d bosonic CFTs: a universal bound on the lightest $\\mathbb Z_2$-odd operator exists if and only if the $\\mathbb Z_2$ symmetry is anomalous. Using a modular bootstrap framework that incorporates defect lines, the authors derive analytic and numerical bounds, with saturations by several WZW and Ising-like models, and show how the bound disappears in the non-anomalous case. They extend the discussion to $U(1)$ symmetries, showing that anomalous $U(1)$s admit bounds while non-anomalous ones do not, and connect the results to AdS$_3$/CFT$_2$ insights like the weak gravity conjecture. Analytic bounds are provided for small central charge, and large-$c$ asymptotics are obtained for the relevant sectors, clarifying how anomaly controls the infrared content of gapless and gapped phases. The work also outlines broader implications for higher dimensions, non-invertible defects, and potential generalizations to fermionic theories and more general symmetry structures.

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.

Figures (19)

• Figure 1: Upper bound on the lightest $\mathbb{Z}_2$ odd operator in a 2d CFT with an anomalous $\mathbb{Z}_2$ symmetry, as a function of the central charge $c$ for $c \ge 1$. The region below the curve is allowed. The $\widehat{\mathfrak{su}(2)}_1$ WZW model with $\Delta_\text{gap}^- = {1\over2}$ saturates the bound at $c=1$.
• Figure 2: The black line depicts the topological defect line $\cal L$ for the global symmetry $\mathbb{Z}_2$. The $\mathbb{Z}_2$ action on the Hilbert space can be realized by wrapping the line around the compact circle on the cylinder.
• Figure 3: The topological lines obey the group multiplication law under fusion.
• Figure 4: As we sweep the $\mathbb{Z}_2$ line past a local operator $\phi$, the correlation function might change by a sign.
• Figure 5: The defect Hilbert space ${\cal H}_{\cal L}$ of a $\mathbb{Z}_2$ line quantized on a circle $S^1$. A state in the defect Hilbert space is mapped to an operator living at the end of the $\mathbb{Z}_2$ line via the operator-state correspondence.