Extremal couplings and gluon scattering in M-theory

TL;DR

This work extends the study of extremal and super-extremal bulk couplings to M-theory backgrounds with fixed-point AdS$_{d+1}\times S^3$ for $d=3,6$, deriving explicit cubic couplings $\beta_{pqk_r}$ between gluon and graviton KK modes and using them to compute the graviton-exchange contribution $M_R$ to gluon four-point functions $\langle 22pp\rangle$. A central methodological advance is the introduction of a reduced correlator in Mellin space, allowing the superconformal Ward identities for theories with eight supercharges in $3\le d\le6$ to be solved efficiently and to organize exchange diagrams in terms of reduced Mellin amplitudes $\mathcal{M}^{ABCD}(s,t)$. The paper confirms that the computed $M_R$ matches the expected flat-space limit across the two holographic setups: AdS$_4\times S^7/\mathbb{Z}_{N_f}$ with $c_T,c_J$ fixed by bulk couplings and AdS$_7\times S^4/\mathbb{Z}_2$ with its own large-$N$ data, and it additionally analyzes operator mixing in the 6d case, yielding explicit mixing eigenvalues and unmixed contributions. Together with prior work, this work completes the extremal-coupling analysis in all relevant dimensions except $d=5$, and it provides a Mellin-space toolkit likely to benefit future analytic bootstrap and holographic studies in theories with eight supercharges.

Abstract

We consider M-theory on the backgrounds AdS$_4\times S^7/\mathbb{Z}_{N_f}$ and AdS$_7\times S^4/\mathbb{Z}_2$, which have fixed point locii AdS$_{d+1}\times S^3$ for $d=3,6$. These theories are holographically dual to certain CFTs in $d=3,6$ with eight supercharges. We compute the bulk cubic couplings between graviton KK modes and gluon KK modes living on the fixed points of these theories, which are generically extremal. We use these couplings to compute the graviton exchange term that appears in the strong coupling expansion of holographic correlators of gluon KK modes $\langle 22pp\rangle$ in these theories, and check that it matches the expected flat space limit. We express the answer in terms of a new reduced correlator solution to the superconformal Ward identities, which we derive for all CFTs with eight supercharges in $3\leq d\leq6$.