Polynomiality of Hurwitz numbers, Bouchard-Mariño conjecture, and a new proof of the ELSV formula

Abstract

In this paper we give a new proof of the ELSV formula. First, we refine an argument of Okounkov and Pandharipande in order to prove (quasi-)polynomiality of Hurwitz numbers without using the ELSV formula (the only way to do that before used the ELSV formula). Then, using this polynomiality we give a new proof of the Bouchard-Mariño conjecture. After that, using the correspondence between the Givental group action and the topological recursion coming from matrix models, we prove the equivalence of the Bouchard-Mariño conjecture and the ELSV formula (it is a refinement of an argument by Eynard).

26 pages; minor inconsistency corrected, references added

References in corpus (7)

• Instantons and Merons in Matrix Models
• The equivariant Gromov-Witten theory of P^1
• Equivalence of ELSV and Bouchard-Mariño conjectures for $r$-spin Hurwitz numbers
• From matrix models' topological expansion to topological string theories: counting surfaces with algebraic geometry
• Double Hurwitz numbers via the infinite wedge
• Lectures on the ELSV formula
• The asymptotic expansion for the factorial and Lagrange inversion formula

Cited by in corpus (24)

• Quantum curves for Hitchin fibrations and the Eynard-Orantin theory
• Quantum spectral curve for the Gromov-Witten theory of the complex projective line
• Ramifications of Hurwitz theory, KP integrability and quantum curves
• The Eynard-Orantin recursion and equivariant mirror symmetry for the projective line
• Topological recursion for Kadomtsev-Petviashvili tau functions of hypergeometric type
• Loop equations and a proof of Zvonkine's $qr$-ELSV formula
• Explicit closed algebraic formulas for Orlov-Scherbin $n$-point functions
• Chiodo formulas for the r-th roots and topological recursion
• Polynomiality of orbifold Hurwitz numbers, spectral curve, and a new proof of the Johnson-Pandharipande-Tseng formula
• Towards an orbifold generalization of Zvonkine's $r$-ELSV formula
• Quantization of Harer-Zagier formulas
• Cut-and-join equation for monotone Hurwitz numbers revisited
• The Geometry of integrable systems. Tau functions and homology of Spectral curves. Perturbative definition
• Quasi-polynomiality of monotone orbifold Hurwitz numbers and Grothendieck's dessins d'enfants
• Combinatorics of Bousquet-Mélou-Schaeffer numbers in the light of topological recursion
• Quantum curves for the enumeration of ribbon graphs and hypermaps
• Quantizing Weierstrass
• Special cases of the orbifold version of Zvonkine's $r$-ELSV formula
• Bijections for simple and double Hurwitz numbers
• Hurwitz numbers and matrix integrals labeled with chord diagrams
• Log topological recursion through the prism of $x-y$ swap
• Matrix integrals and Hurwitz numbers
• Taking limits in topological recursion
• Topological recursion of Eynard-Orantin and the Harmonic Oscillator