Generating functions for weighted Hurwitz numbers

Abstract

Double Hurwitz numbers enumerating weighted $n$-sheeted branched coverings of the Riemann sphere or, equivalently, weighted paths in the Cayley graph of $S_n$ generated by transpositions are determined by an associated weight generating function. A uniquely determined $1$-parameter family of 2D Toda $\tau$-functions of hypergeometric type is shown to consist of generating functions for such weighted Hurwitz numbers. Four classical cases are detailed, in which the weighting is uniform: Okounkov's double Hurwitz numbers, for which the ramification is simple at all but two specified branch points; the case of Belyi curves, with three branch points, two with specified profiles; the general case, with a specified number of branch points, two with fixed profiles, the rest constrained only by the genus; and the signed enumeration case, with sign determined by the parity of the number of branch points. Using the exponentiated quantum dilogarithm function as weight generator, three new types of weighted enumerations are introduced. These determine {\em quantum} Hurwitz numbers depending on a deformation parameter $q$. By suitable interpretation of $q$, the statistical mechanics of quantum weighted branched covers may be related to that of Bosonic gases. The standard double Hurwitz numbers are recovered in the classical limit.

Final version, appearing in J. Math. Phys. 58, 083503 (2017)

References in corpus (10)

• Virasoro constraints and topological recursion for Grothendieck's dessin counting
• Hypergeometric τ-functions, Hurwitz numbers and enumeration of paths
• The matrix model for dessins d'enfants
• The matrix model for hypergeometric Hurwitz numbers
• Weighted Hurwitz numbers and topological recursion: an overview
• Reflection factorizations of Singer cycles
• Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers
• Multispecies Weighted Hurwitz Numbers
• Calabi-Yau Threefolds Fibred by Kummer Surfaces Associated to Products of Elliptic Curves
• Semiclassical asymptotics of quantum weighted Hurwitz numbers

Cited by in corpus (21)

• Toda hierarchies and their applications
• Weighted Hurwitz numbers and topological recursion: an overview
• Weighted Hurwitz numbers and topological recursion
• Fermionic approach to weighted Hurwitz numbers and topological recursion
• Non-orientable branched coverings, $b$-Hurwitz numbers, and positivity for multiparametric Jack expansions
• On W-operators and superintegrability for dessins d'enfant
• $b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral
• Quantum Hurwitz numbers and Macdonald polynomials
• Hurwitz numbers from Feynman diagrams
• Rationally weighted Hurwitz numbers, Meijer $G$-functions and matrix integrals
• Elements of spin Hurwitz theory: closed algebraic formulas, blobbed topological recursion, and a proof of the Giacchetto-Kramer-Lewanski conjecture
• Grothendieck's Dessins d'Enfants in a Web of Dualities. III
• Generating weighted Hurwitz numbers
• Spectral curves for hypergeometric Hurwitz numbers
• Matrix model generating function for quantum weighted Hurwitz numbers
• Asymptotics of quantum weighted Hurwitz numbers
• Whittaker vectors at finite energy scale, topological recursion and Hurwitz numbers
• Weighted Hurwitz numbers, $\tau$-functions and matrix integrals
• Hurwitz numbers for reflection groups III: Uniform formulas
• 2D Toda $\tau$ Functions, Weighted Hurwitz Numbers and the Cayley Graph: Determinant Representation and Recursion Formula
• $\mathfrak{b}$-Hurwitz numbers from refined topological recursion