The 3d Stress-Tensor Bootstrap

TL;DR

This paper applies the conformal bootstrap to 3d CFTs with stress tensors, deriving bounds on the central charge and stress-tensor 3-point functions through numerical semidefinite programming, and explores implications for theories like the 3d Ising model.

Contribution

It develops a comprehensive bootstrap framework for stress-tensor correlators in 3d CFTs, including new bounds and constraints on operator spectra and 3-point functions.

Findings

Reproduces conformal collider bounds numerically.

Establishes bounds on the central charge as a function of stress-tensor 3-point coefficients.

Provides bounds on spectral gaps and stress-tensor 3-point functions for theories with large gaps.

Abstract

We study the conformal bootstrap for 4-point functions of stress tensors in parity-preserving 3d CFTs. To set up the bootstrap equations, we analyze the constraints of conformal symmetry, permutation symmetry, and conservation on the stress-tensor 4-point function and identify a non-redundant set of crossing equations. Studying these equations numerically using semidefinite optimization, we compute bounds on the central charge as a function of the independent coefficient in the stress-tensor 3-point function. With no additional assumptions, these bounds numerically reproduce the conformal collider bounds and give a general lower bound on the central charge. We also study the effect of gaps in the scalar, spin-2, and spin-4 spectra on the central charge bound. We find general upper bounds on these gaps as well as tighter restrictions on the stress-tensor 3-point function coefficients for theories with moderate gaps. When the gap for the leading scalar or spin-2 operator is sufficiently large to exclude large N theories, we also obtain upper bounds on the central charge, thus finding compact allowed regions. Finally, assuming the known low-lying spectrum and central charge of the critical 3d Ising model, we determine its stress-tensor 3-point function and derive a bound on its leading parity-odd scalar.