Implications of the AdS/CFT Correspondence on Spacetime and Worldsheet OPE Coefficients

We explore the connection between the operator product expansion (OPE) in the boundary and worldsheet conformal field theories in the context of AdS$_{d+1}$/CFT$_d$ correspondence. Considering single trace scalar operators in the boundary theory and using the saddle point analysis of the worldsheet OPE [1], we derive an explicit relation between OPE coefficients in the boundary and worldsheet theories for the contribution of single trace spin $\ell$ operators to the OPE. We also consider external vector operators and obtain the relation between OPE coefficients for the exchange of single trace scalar operators in the OPE. We revisit the relationship between the bulk cubic couplings in the Supergravity approximation and the OPE coefficients in the dual boundary theory. Our results match with the known examples from the case of AdS$_3$/CFT$_2$. For the operators whose two and three point correlators enjoy a non renormalization theorem, this gives a set of three way relations between the bulk cubic couplings in supergravity and the OPE coefficients in the boundary and worldsheet theories