What can we learn from crossing symmetry at large N?

Abstract: In this talk I will discuss how to construct all solutions consistent with
crossing symmetry in the limit of large central charge c ~ N^2, starting from the
four-point correlator of the stress tensor multiplet in N=4 SYM. Unitarity forces
the introduction of a scale \Delta_{gap} and these solutions organize as a double
expansion in 1/c and 1/\Delta_{gap}. These 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. Comparison with such instanton computations allows to
fix \Delta_{gap}. Using this gap scale one can explain the upper bounds for the
scaling dimension of unprotected operators observed in the numerical superconformal
bootstrap at large central charge. Furthermore, I will present connections between
such upper bounds and positivity constraints arising from causality in flat space
and I will discuss how certain relations derived from causality constraints for
scattering in AdS follow from crossing symmetry.