Reflections on Conformal Spectra

TL;DR

This paper explores approximate reflection symmetries in conformal field theory spectra using modular invariance and crossing symmetry, leading to universal bounds on spectra and OPE coefficients in various dimensions.

Contribution

It introduces new approximate reflection symmetries in conformal spectra and derives universal bounds on spectra and operator coefficients using these symmetries.

Findings

Identifies approximate reflection symmetries in 2D and higher dimensions.

Derives universal upper bounds on spectra and OPE coefficients.

Discusses symmetry implications for the Cardy formula and light spectrum conditions.

Abstract

We use modular invariance and crossing symmetry of conformal field theory to reveal approximate reflection symmetries in the spectral decompositions of the partition function in two dimensions in the limit of large central charge and of the four-point function in any dimension in the limit of large scaling dimensions $\Delta_0$ of external operators. We use these symmetries to motivate universal upper bounds on the spectrum and the operator product expansion coefficients, which we then derive by independent techniques. Some of the bounds for four-point functions are valid for finite $\Delta_0$ as well as for large $\Delta_0$. We discuss a similar symmetry in a large spacetime dimension limit. Finally, we comment on the analogue of the Cardy formula and sparse light spectrum condition for the four-point function.