Comments on spinning OPE blocks in AdS$_{3}$/CFT$_{2}$

Abstract: We extend the work of \cite{Ferrara:1971vh},\cite{Ferrara:1973vz}, to obtain an integral expression of OPE blocks for spinning primaries in CFT${2}$. We observe, when the OPE blocks are made out of conserved spinning primaries, the integral becomes a product of two copies of weighted AdS${2}$ fields, smeared along geodesics. In this way, conserved current OPE blocks in CFT${2}$ have a different representation in terms of AdS${2}$ geodesic operators, in stead of viewing them as AdS${3}$ geodesic operators. We also show, how this representation can be related to AdS${3}$ massless higher spin fields through HKLL bulk field reconstruction. Using this picture, we consistently obtain the closed form expression of four point spinning conformal block as a product of two AdS$_{2}$ Geodesic Witten diagrams.