Lessons from crossing symmetry at large N

We consider the four-point correlator of the stress tensor multiplet in N = 4 SYM. We construct all solutions consistent with crossing symmetry in the limit of large central charge c similar to N-2 and large g(2)N. While we find an infinite tower of solutions, we argue most of them are suppressed by an extra scale Delta(gap) and are consistent with the upper bounds for the scaling dimension of unprotected operators observed in the numerical superconformal bootstrap at large central charge. These solutions organize as a double expansion in 1/c and 1/Delta(gap). Our solutions are valid to leading order in 1/c and to all orders in 1/Delta(gap) and reproduce, in particular, instanton corrections previously found. Furthermore, we find a connection between such upper bounds and positivity constraints arising from causality in flat space. Finally, we show that certain relations derived from causality constraints for scattering in AdS follow from crossing symmetry.