Three-point correlators of stress tensors in maximally-supersymmetric conformal theories in d=3 and d=6

TL;DR

This work computes stress-tensor two- and three-point functions in free maximal superconformal theories in $d=3$ and $d=6$ and contrasts them with their AdS$_4$ and AdS$_7$ supergravity duals. It derives general free-field results for the stress-tensor correlators of a conformal $k$-form in $d=2k+2$, and specializes to the $(2,0)$ tensor multiplet in $d=6$ and the $N=8$ multiplet in $d=3$, obtaining explicit constants $C_T$, $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$. The key finding is that the ratio $\langle TTT\rangle/\langle TT\rangle$ matches between the free theory and the large-$N$ AdS description for both $d=3$ and $d=6$, differing only by overall normalization factors $4\sqrt{2}/(3\pi) N^{3/2}$ and $4N^{3}$, respectively. This suggests a universality of short-multiplet correlators and supports modeling the interacting $d=6$ CFT via free multiplets with internal indices, akin to the $d=4$ SYM non-renormalization story, while hinting at deeper structure in the M-brane CFTs.

Abstract

We consider free superconformal theories of n=8 scalar multiplet in d=3 and (2,0) tensor multiplet in d=6 and compute 2-point and 3-point correlators of their stress tensors. The results for the 2-point and 3-point correlators for a single d=3 and d=6 multiplet differ from the "strong-coupling" AdS_4 and AdS_7 supergravity predictions by the factors $4\sqrt2 \over 3π}N^{3/2}$ and $4N^3$ respectively. These are the same factors as found earlier in hep-th/9703040 in the comparison of the brane free field theory and the 11-d supergravity predictions for the absorption cross-sections of longitudinally polarized gravitons by multiple M2 and M5 branes. While the correspondence of the results for the cross-sections and 2-point functions was expected on the basis of unitarity, the fact that the same coefficients appear in the ratio of the free-theory and supergravity 3-point functions is non-trivial. Thus, like in the d=4 SYM case, in both d=3 and d=6 theories the ratio of the 3-point and 2-point correlators <TTT>/<TT> is exactly the same in the free field theory and in the interacting CFT as described (to leading order in large $N$) by the 11-d supergravity on AdS_{d+1} x S^{10-d}.