Polyakov blocks for the 1D CFT mixed correlator bootstrap

TL;DR

This paper develops crossing-symmetric Polyakov block expansions for 1D CFT correlators, deriving sum rules and bounds that improve understanding of mixed correlator bootstrap constraints.

Contribution

It introduces a new basis of Polyakov blocks for 1D CFTs, enabling efficient computation and diagonalization of mixed correlator sum rules.

Findings

Derived crossing-symmetric expansions for 1D CFT correlators

Established sum rules for mixed correlator systems

Obtained optimal bounds saturated by generalized free field models

Abstract

We introduce manifestly crossing-symmetric expansions for arbitrary systems of 1D CFT correlators. These expansions are given in terms of certain Polyakov blocks which we define and show how to compute efficiently. Equality of OPE and Polyakov block expansions leads to sets of sum rules that any mixed correlator system must satisfy. The sum rules are diagonalized by correlators in tensor product theories of generalized free fields. We show that it is possible to do a change of a basis that diagonalizes instead mixed correlator systems involving elementary and composite operators in a single field theory. As an example, we find the first non-trivial examples of optimal bounds, saturated by the mixed correlator system $\phi,\phi^2$ in the theory of a single generalized free field.