Symmetric orbifold OPE from string theory

A bstract 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 AdS 3 × X in the near-boundary limit and k = 1 hybrid strings in AdS 3 × S 3 × $$ {\mathbbm{T}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> . 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.