Hurwitz numbers, matrix models and enumerative geometry

Understand

We propose a new, conjectural recursion solution for Hurwitz numbers at all genera.

• This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric Calabi-Yau manifolds, which we briefly review to provide some background for our conjecture.
• We show in particular how this B-model solution, combined with mirror symmetry for the one-leg, framed topological vertex, leads to a recursion relation for Hodge integrals with three Hodge class insertions.
• Our conjecture in Hurwitz theory follows from this recursion for the framed vertex in the limit of infinite framing.