Snowmass White Paper: The Analytic Conformal Bootstrap

TL;DR

This Snowmass White Paper surveys the analytic conformal bootstrap, a nonperturbative framework for constraining and solving CFTs using symmetry, causality, and unitarity. It highlights advances such as the lightcone bootstrap, the Lorentzian inversion formula, dispersive sum rules, and Polyakov blocks, and explains how these yield universal large-spin structure, bounds on OPE data, and connections to AdS/CFT and gravity. The paper also discusses extensions to nonconformal fixed points, higher-point and defect observables, finite-temperature and curved manifolds, and explorations beyond AdS/CFT to flat-space holography and the swampland program. By linking bootstrap constraints to gravitational EFTs, energy conditions, and mathematical structures (e.g., sphere packing), the work argues that analytic bootstrap provides rigorous guidance toward UV completions and deeper insights into quantum gravity and strongly coupled systems.

Abstract

The analytic conformal bootstrap is an array of techniques to characterize, constrain, and solve strongly interacting quantum field theories using symmetries, causality, unitarity, and other general principles. In the last decade, bolstered by the development of new Lorentzian methods, it has been used to solve conformal field theories at large spin; to place bounds on energy distributions, event shapes, operator product coefficients, and other observables; and to understand aspects of quantum gravity in anti-de Sitter space. We review these advances and highlight several promising areas for future exploration. Targets include developing new methods to close the gap between numerical and analytic bounds, extending the bootstrap beyond conformal fixed points, applications to quantum gravity and cosmology, and building on ties to condensed matter theory and mathematics.