Extremal couplings, graviton exchange, and gluon scattering in AdS

Extremal cubic couplings in AdS relate bulk fields such that $\Delta_i+\Delta_j=\Delta_k$. Such couplings lead to divergent 3-point Witten diagrams, and do not occur in theories with maximal supersymmetry. We consider the simplest theories where such coupling are non-zero, which is type IIB string theory with $N$ D3 branes probing various configurations of sevenbranes, which are dual to certain 4d $\mathcal{N}=2$ SCFTs. At large $N$, the low energy effective theory is supergravity on $AdS_5\times S^5$ with a singularity, whose fixed point locus is $AdS_5\times S^3$. These theories have infinite towers of graviton modes, as well as gluon modes on the sevenbranes. We compute the nonzero coupling between these modes, which is in general (super)-extremal. We use these couplings to compute the graviton exchange term in the holographic correlator of gluon KK modes $\langle22pp\rangle$, which appears at the same order $1/N^2$ as 1-loop gluon exchange, and receives contributions from a whole tower of graviton modes. We use this graviton exchange term to compute the unmixing of the single trace graviton modes with double traces of gluon modes, which explains the divergent 3-point diagrams. Finally, for $\langle 2222\rangle$ for the simplest 4d $\mathcal{N}=2$ gauge theory, we use supersymmetric localization and the new graviton exchange term to completely fix the correlator at order $1/N^2$.