Hurwitz numbers, matrix models and enumerative geometry

  title={Hurwitz numbers, matrix models and enumerative geometry},
  author={Vincent Bouchard and Marcos Mari{\~n}o},
  journal={arXiv: Algebraic Geometry},
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… 

Figures from this paper

A new spin on Hurwitz theory and ELSV via theta characteristics
We study spin Hurwitz numbers, which count ramified covers of the Riemann sphere with a sign coming from a theta characteristic. These numbers are known to be related to Gromov-Witten theory of
Classical Hurwitz numbers and related combinatorics
In 1891 Hurwitz [30] studied the number Hg,d of genus g ≥ 0 and degree d ≥ 1 coverings of the Riemann sphere with 2g + 2d− 2 fixed branch points and in particular found a closed formula for Hg,d for
Chiodo formulas for the r-th roots and topological recursion
We analyze Chiodo’s formulas for the Chern classes related to the r-th roots of the suitably twisted integer powers of the canonical class on the moduli space of curves. The intersection numbers of
A matrix model for simple Hurwitz numbers, and topological recursion
Double Hurwitz numbers: polynomiality, topological recursion and intersection theory
Double Hurwitz numbers enumerate branched covers of $\mathbb{CP}^1$ with prescribed ramification over two points and simple ramification elsewhere. In contrast to the single case, their underlying
Towards the topological recursion for double Hurwitz numbers
Single Hurwitz numbers enumerate branched covers of the Riemann sphere with specified genus, prescribed ramification over infinity, and simple branching elsewhere. They exhibit a remarkably rich
Geometrical interpretation of the topological recursion, and integrable string theories
Symplectic invariants introduced in math-ph/0702045 can be computed for an arbitrary spectral curve. For some examples of spectral curves, those invariants can solve loop equations of matrix
Tevelev degrees and Hurwitz moduli spaces
We interpret the degrees which arise in Tevelev’s study of scattering amplitudes in terms of moduli spaces of Hurwitz covers. Via excess intersection theory, the boundary geometry of the Hurwitz
TOPICAL REVIEW: Topological recursion in enumerative geometry and random matrices
We review the method of symplectic invariants recently introduced to solve matrix models' loop equations in the so-called topological expansion, and further extended beyond the context of matrix
Orbifold Hurwitz numbers and Eynard-Orantin invariants
We prove that a generalisation of simple Hurwitz numbers due to Johnson, Pandharipande and Tseng satisfy the topological recursion of Eynard and Orantin. This generalises the Bouchard-Marino


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
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