Five points for the Polyakov bootstrap

A bstract 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.