  We explicitly construct the holographic dual configuration for the four
dimensional $\mathcal{N}=4$ superconformal block containing half-BPS scalar
primary operators by considering its full $AdS_5 \times S^5$ dual geometry. We
extend the embedding space formalism and the related Harmonic analysis to
general $d$-dimensional sphere $S^d$, and obtain precisely the $R$-symmetry
contribution to the half-BPS scalar superconformal blocks, which we refer as
"$R$-symmetry block". We also observe that the $R$-symmetry quadratic Casimir
operator can be mapped to BC$_{2}$ Calogero-Sutherland system Hamiltonian, such
that $R$-symmetry block is in terms identified as its bound state wave
function.
