Five points for the Polyakov Bootstrap

TL;DR

This paper extends the Polyakov bootstrap approach from 4-point to 5-point functions in 1D conformal field theories, providing new tools for extracting data from multi-point crossing equations and improving bootstrap methods.

Contribution

It generalizes Polyakov bootstrap functionals to 5-point functions, deriving sum-rules and constructing crossing symmetric blocks for enhanced analysis in 1D CFTs.

Findings

Derived sum-rules for 5-point correlators.

Constructed crossing symmetric Polyakov blocks.

Demonstrated advantages in truncated 5-point bootstrap.

Abstract

Higher-point correlation functions encode the data of infinitely many 4-point correlators in conformal field theory (CFT). In this paper, we develop new tools to efficiently extract this data from multi-point crossing equations. Concretely, we generalize the functionals constituting the so-called Polyakov bootstrap of 4-point correlators to the case of 5-point functions in one-dimensional CFTs. We first construct the crossing symmetric Polyakov blocks, and then derive sum-rules by requiring consistency with the operator product expansion (OPE). This procedure leads to two classes of functionals controlling OPE coefficients of double- and triple-twist families. After extensively checking the validity of the associated sum-rules, we apply our functionals to the truncated 5-point bootstrap where we find several advantages with respect to more standard derivative functionals.