Reflections on Conformal Spectra

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.

13 pages, 5 figures

References in corpus (2)

• Universal Spectrum of 2d Conformal Field Theory in the Large c Limit
• Remarks on the Convergence Properties of the Conformal Block Expansion

Cited by in corpus (10)

• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Crossing Symmetry in Alpha Space
• Bootstrapping Non-Invertible Symmetries
• Casimir recursion relations for general conformal blocks
• ${\mathbb{Z}}_N$ Symmetries, Anomalies, and the Modular Bootstrap
• Modular crossings, OPE coefficients and black holes
• Solutions of modular bootstrap constraints from quantum codes
• Classical Codes and Chiral CFTs at Higher Genus
• Fake Z
• Bounds on CFTs with $W_3$ algebras and AdS${}_3$ higher spin theories