Lessons from crossing symmetry at large N

TL;DR

The paper tackles the problem of constraining the four-point function of the stress-tensor multiplet in ${\cal N}=4$ SYM via crossing symmetry at large $N$ and large 't Hooft coupling. It develops analytic solutions to the superconformal bootstrap equation, revealing a distinguished supergravity term plus an infinite tower of higher-order solutions that are organized as a double expansion in $1/c$ and $1/\Delta_{gap}$, with Mellin-space structure clarifying their form. The key contributions include explicit constructions of the tower $A^{(p,q)}$, demonstration that most extra solutions are suppressed by a gap scale consistent with instanton results (giving $\Delta_{gap}\sim N^{1/4}$), and a demonstration that causality-based positivity constraints in flat space connect to the upper bounds observed numerically. The work shows that the conformal bootstrap in this theory encodes not only the supergravity data but also stringy/UV completion information, and it suggests that $\Delta_{gap}$ may be accessible from bootstrap data alone, linking causality, crossing symmetry, and UV consistency in AdS/CFT.

Abstract

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 ~ 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 Δ_{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/Δ_{gap}. Our solutions are valid to leading order in 1/c and to all orders in 1/Δ_{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.