  We analyse topological orbifold conformal field theories on the symmetric
product of a complex surface M. By exploiting the mathematics literature we
show that a canonical quotient of the operator ring has structure constants
given by Hurwitz numbers. This proves a conjecture in the physics literature on
extremal correlators. Moreover, it allows to leverage results on the
combinatorics of the symmetric group to compute more structure constants
explicitly. We recall that the full orbifold chiral ring is given by a
symmetric orbifold Frobenius algebra. This construction enables the computation
of topological genus zero and genus one correlators, and to prove the vanishing
of higher genus contributions. The efficient description of all topological
correlators sets the stage for a proof of a topological AdS/CFT correspondence.
Indeed, we propose a concrete mathematical incarnation of the proof, relating
Gromow-Witten theory in the bulk to the quantum cohomology of the Hilbert
scheme on the boundary.
