The Topological Symmetric Orbifold

TL;DR

This paper studies topological orbifold conformal field theories on symmetric products of complex surfaces, establishing connections with Hurwitz numbers and Frobenius algebras to compute correlators and explore topological AdS/CFT correspondence.

Contribution

It proves a conjecture on extremal correlators, links operator ring structure constants to Hurwitz numbers, and develops methods to compute topological correlators explicitly.

Findings

Proved a conjecture on extremal correlators.

Connected operator ring structure constants to Hurwitz numbers.

Computed topological genus zero and one correlators, showing higher genus vanishing.

Abstract

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.