The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT

TL;DR

This paper develops new techniques for analyzing the 3d Ising conformal field theory using the lightcone bootstrap, deriving analytic approximations for operator spectra that match numerical results and provide insights into crossing symmetry constraints.

Contribution

It introduces novel methods for summing SL(2,R) conformal blocks and solves the lightcone bootstrap analytically to all orders in large spin, advancing understanding of the 3d Ising CFT spectrum.

Findings

Analytic approximations match numerical data for about 100 operators.

New techniques enable summing infinite conformal block series.

Constraints on initial data derived from crossing symmetry.

Abstract

We compute numerically the dimensions and OPE coefficients of several operators in the 3d Ising CFT, and then try to reverse-engineer the solution to crossing symmetry analytically. Our key tool is a set of new techniques for computing infinite sums of SL(2,R) conformal blocks. Using these techniques, we solve the lightcone bootstrap to all orders in an asymptotic expansion in large spin, and suggest a strategy for going beyond the large spin limit. We carry out the first steps of this strategy for the 3d Ising CFT, deriving analytic approximations for the dimensions and OPE coefficients of several infinite families of operators in terms of the initial data $\{\Delta_\sigma,\Delta_\epsilon,f_{\sigma\sigma\epsilon},f_{\epsilon\epsilon\epsilon},c_T\}$. The analytic results agree with numerics to high precision for about 100 low-twist operators (correctly accounting for O(1) mixing effects between large-spin families). Plugging these results back into the crossing equations, we obtain approximate analytic constraints on the initial data.