Hurwitz numbers, matrix models and enumerative geometry

@article{Bouchard2007HurwitzNM,
  title={Hurwitz numbers, matrix models and enumerative geometry},
  author={Vincent Bouchard and Marcos Mari{\~n}o},
  journal={arXiv: Algebraic Geometry},
  year={2007}
}
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… 
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153 Citations

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References

SHOWING 1-10 OF 59 REFERENCES
Gromov-Witten theory, Hurwitz numbers, and Matrix models, I
The main goal of the paper is to present a new approach via Hurwitz numbers to Kontsevich's combinatorial/matrix model for the intersection theory of the moduli space of curves. A secondary goal is…
Hodge Integrals and Hurwitz Numbers via Virtual Localization
We give another proof of Ekedahl, Lando, Shapiro, and Vainshtein's remarkable formula expressing Hurwitz numbers (counting covers of P1 with specified simple branch points, and specified branching…
Mariño-Vafa formula and Hodge integral identities
Based on string duality Marino and Vafa [10] conjectured a closed formula on certain Hodge integrals in terms of representations of symmetric groups. This formula was first explicitly written down by…
The Gromov–Witten Potential of A Point, Hurwitz Numbers, and Hodge Integrals
Hurwitz numbers, which count certain covers of the projective line (or, equivalently, factorizations of permutations into transpositions), have been extensively studied for over a century. The…
Gromov-Witten theory, Hurwitz theory, and completed cycles
We establish an explicit equivalence between the stationary sector of the Gromov-Witten theory of a target curve X and the enumeration of Hurwitz coverings of X in the basis of completed cycles. The…
Recursions, formulas, and graph-theoretic interpretations of ramified coverings of the sphere by surfaces of genus 0 and 1
We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count…
Recursion Between Mumford Volumes of Moduli Spaces
We propose a new proof, as well as a generalization of Mirzakhani’s recursion for volumes of moduli spaces. We interpret those recursion relations in terms of expectation values in Kontsevich’s…
Knot Invariants and Topological Strings
Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy
We present in this paper a differential version of Mirzakhani's recursion relation for the Weil-Petersson volumes of the moduli spaces of bordered Riemann surfaces. We discover that the…
Phase transitions, double-scaling limit, and topological strings
Topological strings on Calabi–Yau manifolds are known to undergo phase transitions at small distances. We study this issue in the case of perturbative topological strings on local Calabi–Yau…
...
...