Abstract
We analyze 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 cohomology of the Hilbert scheme on the boundary.
Highlights
The symmetric product orbifold conformal field theory on a two-dimensional complex surface M plays an important role in the anti-de Sitter/conformal field theory correspondence in three bulk dimensions [1]
We reproduce the convolution product (2.3) with the additional requirement of degree conservation from the product of the chiral ring operators in the symmetric orbifold conformal field theory. This is in full accord with an alternative route, which is to mathematically abstract [31, 32] the cohomology coded in orbifold conformal field theory [34, 35], to prove that it is equivalent to the Hilbert scheme cohomology [30]
We have established that calculations in the cohomology ring of the symmetric orbifold conformal field theory on the complex two-plane C2 reduce to calculations in the symmetric group Sn
Summary
The symmetric product orbifold conformal field theory on a two-dimensional complex surface M plays an important role in the anti-de Sitter/conformal field theory correspondence in three bulk dimensions [1]. We believe that the efficient mathematical description of the operator ring of the boundary theory provides a scheme for the bulk analysis, and for a proof of a topological subsector of an AdS/CFT correspondence. We point out that the ring is a canonical quotient ring of the chiral ring of the symmetric orbifold conformal field theory on (quasi-projective) compact complex surfaces M. The ring is described by a (symmetric orbifold) Frobenius algebra that can be constructed on the basis of the cohomology ring of M combined with permutation group combinatorics This gives a compact description of the chiral ring of the topologically twisted symmetric orbifold conformal field theory Symn(M ) of the manifold M. Appendix A contains a technical bridge to the physics literature
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