Unitarity and positivity constraints for CFT at large central charge

TL;DR

The paper analyzes the four-point correlator of the ${\cal N}=4$ SYM stress-tensor multiplet at large central charge using Mellin space to impose crossing, unitarity, and the pole structure from single-trace exchanges. It shows that the sign of these contributions is fixed by unitarity and, when all single-trace twists exceed four, leads to negative anomalous dimensions for twist-four double-trace operators, explained via Mack polynomial positivity and Regge behavior. The work further connects these CFT positivity constraints to bulk EFT causality, showing consistency with forward-limit positivity and the Virasoro–Shapiro amplitude in the flat-space limit, and demonstrates that in the large-twist limit the solutions reproduce EFT-like polynomial towers with correct suppression and signs. It also analyzes the large-n regime to all orders in 1/√λ, revealing bulk-point singularity structure and linking to the bulk S-matrix. Overall, the results establish a coherent CFT framework that mirrors bulk causality and locality constraints in the AdS/CFT setting for ${\cal N}=4$ SYM at large $c$.

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.

Figures (3)

• Figure 1: Contribution to the anomalous dimension $\gamma_{0,0}$ from the symmetric exchange of a scalar primary, in units of $a_{\delta}$ .
• Figure 2: Contribution to the anomalous dimension $\gamma_{0,0}$ and $\gamma_{0,2}$ from the exchange of an intermediate operator of spin zero, in units of the positive OPE coefficient. The behaviour of $\gamma_{0,\ell}$ with $\ell > 0$ is very similar to $\gamma_{0,2}$.
• Figure 3: Contribution to the anomalous dimension $\gamma_{0,0}$, $\gamma_{0,2}$ and $\gamma_{0,4}$ from the exchange of an intermediate operator of spin two, in units of the positive OPE coefficient.