A Monodromy Graph Approach to the Piecewise Polynomiality of Simple, Monotone and Grothendieck Dessins d’enfants Double Hurwitz Numbers View Full Text


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Article Info

DATE

2019-03-09

AUTHORS

Marvin Anas Hahn

ABSTRACT

Hurwitz numbers count genus g, degree d covers of the complex projective line with fixed branched locus and fixed ramification data. An equivalent description is given by factorisations in the symmetric group. Simple double Hurwitz numbers are a class of Hurwitz-type counts of specific interest. In recent years a related counting problem in the context of random matrix theory was introduced as so-called monotone Hurwitz numbers. These can be viewed as a desymmetrised version of the Hurwitz-problem. A combinatorial interpolation between simple and monotone double Hurwitz numbers was introduced as mixed double Hurwitz numbers and it was proved that these objects are piecewise polynomial in a certain sense. Moreover, the notion of strictly monotone Hurwitz numbers has risen in interest as it is equivalent to a certain Grothendieck dessins d’enfant count. In this paper, we introduce a combinatorial interpolation between simple, monotone and strictly monotone double Hurwitz numbers as triply interpolated Hurwitz numbers. Our aim is twofold: using a connection between triply interpolated Hurwitz numbers and tropical covers in terms of so-called monodromy graphs, we give algorithms to compute the polynomials for triply interpolated Hurwitz numbers in all genera using Erhart theory. We further use this approach to study the wall-crossing behaviour of triply interpolated Hurwitz numbers in genus 0 in terms of related Hurwitz-type counts. All those results specialise to the extremal cases of simple, monotone and Grothendieck dessins d’enfants Hurwitz numbers. More... »

PAGES

1-38

References to SciGraph publications

  • 1891-03. Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkten in MATHEMATISCHE ANNALEN
  • 2001-11. Hurwitz numbers and intersections on moduli spaces of curves in INVENTIONES MATHEMATICAE
  • 2010-09. Tropical Hurwitz numbers in JOURNAL OF ALGEBRAIC COMBINATORICS
  • 2006-03-02. Hermitian matrix model free energy: Feynman graph technique for all genera in JOURNAL OF HIGH ENERGY PHYSICS
  • 2015-08. Virasoro Constraints and Topological Recursion for Grothendieck’s Dessin Counting in LETTERS IN MATHEMATICAL PHYSICS
  • 2016-05. Ramifications of Hurwitz theory, KP integrability and quantum curves in JOURNAL OF HIGH ENERGY PHYSICS
  • Journal

    TITLE

    Graphs and Combinatorics

    ISSUE

    N/A

    VOLUME

    N/A

    Author Affiliations

    Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00373-019-02030-5

    DOI

    http://dx.doi.org/10.1007/s00373-019-02030-5

    DIMENSIONS

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