The analytic bootstrap equations of non-diagonal two-dimensional CFT

TL;DR

This paper extends the conformal bootstrap approach to non-diagonal 2D CFTs with nonzero conformal spins, deriving equations for structure constants and validating them through numerical crossing symmetry checks.

Contribution

It generalizes the analytic bootstrap method to non-diagonal 2D CFTs, revealing the parametrization of spectra and structure constants for these theories.

Findings

Derived equations for structure constants in non-diagonal CFTs

Numerically verified crossing symmetry in a non-rational limit of D-series minimal models

Connected results to cluster connectivities in the critical Potts model

Abstract

Under the assumption that degenerate fields exist, diagonal CFTs such as Liouville theory can be solved analytically using the conformal bootstrap method. Here we generalize this approach to non-diagonal CFTs, i.e. CFTs whose primary fields have nonzero conformal spins. Assuming generic values of the central charge, we find that the non-diagonal sector of the spectrum must be parametrized by two integer numbers. We then derive and solve the equations that determine how three- and four-point structure constants depend on these numbers. In order to test these results, we numerically check crossing symmetry of a class of four-point functions in a non-rational limit of D-series minimal models. The simplest four-point functions in this class were previously argued to describe connectivities of clusters in the critical Potts model.