Superconformal Block from Holographic Geometry

Abstract

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 $Ad{S}_5\times S^5$ 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.

21 pages+appendices; v2: clarifications and references added