Symmetric orbifold OPE from string theory

We discuss how to obtain the symmetric orbifold fusion rule/OPE from the dual string theory. We consider two explicit examples: $k_b=3$ bosonic strings in $\mathrm{A}\mathrm{d}{\mathrm{S}}_3\times X$ in the near-boundary limit and $k=1$ hybrid strings in $\mathrm{A}\mathrm{d}{\mathrm{S}}_3\times {\mathrm{S}}^3\times {\mathbb{T}}^4$. The main advantage of these two examples is that they have explicit expressions for the vertex operators in the $x$-basis. We show that the OPE of such vertex operators explicitly captures the longest cycle contribution in the symmetric orbifold fusion rule/OPE. We then argue how one can obtain the shorter-cycle contributions using the screening operators existing in the theories. We also discuss how our general result reduces to the earlier results in the literature.