A conformal block Farey tail

TL;DR

This paper explores how crossing symmetry constrains 2D CFT correlation functions, showing they can be constructed via PSL(2,Z) averaging, with implications for holography and semi-classical saddle points.

Contribution

It introduces a method to construct crossing-symmetric functions from conformal blocks using PSL(2,Z) averaging and relates this to holographic interpretations in AdS3.

Findings

Correlation functions in certain 2D CFTs equal PSL(2,Z) averages of vacuum blocks

Unique determination of 3-point coefficients from central charge in Ising models

Holographic interpretation of PSL(2,Z) sum as semi-classical saddle points

Abstract

We investigate the constraints of crossing symmetry on CFT correlation functions. Four point conformal blocks are naturally viewed as functions on the upper-half plane, on which crossing symmetry acts by PSL(2,Z) modular transformations. This allows us to construct a unique, crossing symmetric function out of a given conformal block by averaging over PSL(2,Z). In some two dimensional CFTs the correlation functions are precisely equal to the modular average of the contributions of a finite number of light states. For example, in the two dimensional Ising and tri-critical Ising model CFTs, the correlation functions of identical operators are equal to the PSL(2,Z) average of the Virasoro vacuum block; this determines the 3 point function coefficients uniquely in terms of the central charge. The sum over PSL(2,Z) in CFT2 has a natural AdS3 interpretation as a sum over semi-classical saddle points, which describe particles propagating along rational tangles in the bulk. We demonstrate this explicitly for the correlation function of certain heavy operators, where we compute holographically the semi-classical conformal block with a heavy internal operator.