Bootstrapping Conformal Field Theories with the Extremal Functional Method

TL;DR

The paper introduces the Extremal Functional Method (EFM), a numerical approach to determine operator spectra and OPE coefficients of boundary CFTs, demonstrated on the 2D Ising model without requiring full algebraic structures.

Contribution

The paper develops the EFM, a novel numerical technique to extract detailed CFT data at the boundary of the allowed parameter space.

Findings

Successfully rederived the 2D Ising model spectrum and OPE coefficients.

Demonstrated EFM's ability to work with minimal input data.

Serves as a benchmark for future applications to less understood CFTs.

Abstract

The existence of a positive linear functional acting on the space of (differences between) conformal blocks has been shown to rule out regions in the parameter space of conformal field theories (CFTs). We argue that at the boundary of the allowed region the extremal functional contains, in principle, enough information to determine the dimensions and OPE coefficients of an infinite number of operators appearing in the correlator under analysis. Based on this idea we develop the Extremal Functional Method (EFM), a numerical procedure for deriving the spectrum and OPE coefficients of CFTs lying on the boundary (of solution space). We test the EFM by using it to rederive the low lying spectrum and OPE coefficients of the 2d Ising model based solely on the dimension of a single scalar quasi-primary -- no Virasoro algebra required. Our work serves as a benchmark for applications to more interesting, less known CFTs in the near future.