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.
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