The conformal block Farey tail

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 Vira-soro 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 AdS 3 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.