Hurwitz numbers, matrix models and enumerative geometry

Abstract

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.

21 pages, 5 figures, small corrections, references added

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Cited by in corpus (19)

• Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory
• The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
• Quantum curves for Hitchin fibrations and the Eynard-Orantin theory
• The spectral curve of the Eynard-Orantin recursion via the Laplace transform
• A generalized topological recursion for arbitrary ramification
• Equivalence of ELSV and Bouchard-Mariño conjectures for $r$-spin Hurwitz numbers
• Monotone Hurwitz numbers and the HCIZ integral II
• Geometrical interpretation of the topological recursion, and integrable string theories
• Super Quantum Airy Structures
• Connection between cut-and-join and Casimir operators
• Topological Recursion in The Ramond Sector
• Multispecies Weighted Hurwitz Numbers
• Quantizing Weierstrass
• Mirror symmetry for extended affine Weyl groups
• Matrix Field Theory
• $\mathcal{N}=1$ Super Topological Recursion
• Les Houches lecture notes on moduli spaces of Riemann surfaces
• A new kind of anomaly: on W-constraints for GKM
• Quenched free energy from spacetime D-branes