Unitarity and positivity constraints for CFT at large central charge

TL;DR

This paper explores unitarity and positivity constraints on four-point stress tensor correlators in large central charge ${ m N}=4$ SYM, revealing how these constraints influence operator dimensions and relate to causality in effective field theories.

Contribution

It establishes positivity conditions for Mellin amplitudes in large $c$ CFTs and connects these to causality constraints in effective field theories in AdS.

Findings

Positivity of Mack polynomials constrains Mellin amplitudes.

Solutions at large $g^2N$ match local bulk interactions.

Signs of anomalous dimensions align with causality constraints.

Abstract

We consider the four-point correlator of the stress tensor multiplet in ${\cal N}=4$ SYM in the limit of large central charge $c \sim N^2$. For finite values of $g^2N$ single-trace intermediate operators arise at order $1/c$ and this leads to specific poles in the Mellin representation of the correlator. The sign of the residue at these poles is fixed by unitarity. We consider solutions consistent with crossing symmetry and this pole structure. We show that in a certain regime all solutions result in a negative contribution to the anomalous dimension of twist four operators. The reason behind this is a positivity property of Mack polynomials that leads to a positivity condition for the Mellin amplitude. This positivity condition can also be proven by assuming the correct Regge behaviour for the Mellin amplitude. For large $g^2N$ we recover a tower of solutions in one to one correspondence with local interactions in a effective field theory in the $AdS$ bulk, with the appropriate suppression factors, and with definite overall signs. These signs agree with the signs that would follow from causality constraints on the effective field theory. The positivity constraints arising from CFT for the Mellin amplitude take a very similar form to the causality constraint for the forward limit of the S-matrix.