Cubic and higher-order supergravity couplings for AdS vacua using Exceptional Field Theory

We show how to use Exceptional Field Theory to efficiently compute $n$-point couplings of all Kaluza-Klein modes for vacua that can be uplifted from maximal gauged supergravities to 10/11 dimensions via a consistent truncation. Via the AdS/CFT correspondence, these couplings encode the $n$-point functions of holographic conformal fields theories. Our methods show that these $n$-point couplings are controlled by the $n$-point invariant of scalar harmonics of the maximally symmetric point of the truncation, allowing us to show that infinitely-many $n$-point couplings vanish for any vacua of the truncation, even though they may be allowed by the remnant symmetry group of the vacua. This gives new results even for the maximally supersymmetric AdS$_5 \times S^5$, AdS$_4 \times S^7$ and AdS$_7 \times S^4$ vacua of string and M-theory, where we prove old conjectures about the vanishing of $n$-point extremal and near-extremal couplings. Focusing in particular on cubic couplings for vacua of 5-dimensional gauged supergravity, we derive explicit universal formulae encoding these couplings for any vacuum within a consistent truncation. We use this to compute known and new couplings involving spin-0, spin-1, spin-2 for the AdS$_5 \times S^5$ vacuum of IIB string theory.