BCJ relations in ${AdS}_5 \times S^3$ and the double-trace spectrum of super gluons

We revisit the four-point function of super gluons in $AdS_5 \times S^3$ in the spirit of the large $p$ formalism and show how the integrand of a generalised Mellin transform satisfies various non-trivial properties such as $U(1)$ decoupling identity, BCJ relations and colour-kinematic duality, in a way that directly mirrors the analogous relations in flat space. We unmix the spectrum of double-trace operators at large $N$ and find all anomalous dimensions at leading order. The anomalous dimensions follow a very simple pattern, resembling those of other theories with hidden conformal symmetries.