Topological recursion and a quantum curve for monotone Hurwitz numbers

Abstract

Classical Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. Monotone Hurwitz numbers restrict the enumeration by imposing a further monotonicity condition on such factorisations. In this paper, we prove that monotone Hurwitz numbers arise from the topological recursion of Eynard and Orantin applied to a particular spectral curve. We furthermore derive a quantum curve for monotone Hurwitz numbers. These results extend the collection of enumerative problems known to be governed by the paradigm of topological recursion and quantum curves, as well as the list of analogues between monotone Hurwitz numbers and their classical counterparts.

arXiv comment23 pages

References in corpus (15)

• Hurwitz numbers and intersections on moduli spaces of curves
• A matrix model for simple Hurwitz numbers, and topological recursion
• The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
• Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models
• Spectral curves and the Schroedinger equations for the Eynard-Orantin recursion
• The spectral curve and the Schroedinger equation of double Hurwitz numbers and higher spin structures
• Orbifold Hurwitz numbers and Eynard-Orantin invariants
• Quantum spectral curve for the Gromov-Witten theory of the complex projective line
• Gromov-Witten invariants of $\bp^1$ and Eynard-Orantin invariants
• All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds
• Intersection numbers on Deligne-Mumford moduli spaces and quantum Airy curve
• Invariants of spectral curves and intersection theory of moduli spaces of complex curves
• Quantum Mirror Curves for ${\mathbb{C}}^3$ and the Resolved Confiold
• Quantum curves for the enumeration of ribbon graphs and hypermaps
• Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture

Cited by in corpus (6)

• Reconstructing WKB from topological recursion
• Quantum curves and conformal field theory
• Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers
• Monotone orbifold Hurwitz numbers
• Integration on complex Grassmannians, deformed monotone Hurwitz numbers, and interlacing phenomena
• On the convergence of monotone Hurwitz generating functions