Light-Cone Bootstrap of Higher Point Functions and Wilson Loop Duality

We initiate an exploration of the conformal bootstrap for n>4 point correlation functions. Here we bootstrap correlation functions of the lightest scalar gauge invariant operators in planar non-Abelian conformal gauge theories as their locations approach the cusps of a null polygon. For that we consider consistency of the OPE in the so-called snowflake channel with respect to cyclicity transformations which leave the null configuration invariant. For general non-Abelian gauge theories this allows us to strongly constrain the OPE structure constants of up to three large spin J_{j} operators (and large polarization quantum number l_{j}) to all loop orders. In N=4 we fix them completely through the duality to null polygonal Wilson loops and the recent origin limit of the hexagon explored by Basso, Dixon, and Papathanasiou.