On Superconformal Four-Point Mellin Amplitudes in Dimension $d>2$

We present a universal treatment for imposing superconformal constraints on Mellin amplitudes for $\mathrm{S}\mathrm{C}\mathrm{F}{\mathrm{T}}_{\mathrm{d}}$ with $3\le d\le 6$. This leads to a new technique to compute holographic correlators, which is similar but complementary to the ones introduced in [1,2]. We apply this technique to theories in various spacetime dimensions. In addition to reproducing known results, we obtain a simple expression for next-next-to-extremal four-point functions in $Ad{S}_7\times S^4$. We also use this machinery on $Ad{S}_4\times S^7$ and compute the first holographic one-half BPS four-point function. We extract the anomalous dimension of the R-symmetry singlet double-trace operator with the lowest conformal dimension and find agreement with the 3d $\mathcal{N}=8$ numerical bootstrap bound at large central charge.