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Transactions of the American Mathematical Society

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Wall-crossing and recursion formulae for tropical Jucys covers


Authors: Marvin Anas Hahn and Danilo Lewański
Journal: Trans. Amer. Math. Soc. 373 (2020), 4685-4711
MSC (2010): Primary 14N10, 14N35, 14T05
DOI: https://doi.org/10.1090/tran/8006
Published electronically: March 27, 2020
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Abstract: Hurwitz numbers enumerate branched genus $ g$ covers of the Riemann sphere with fixed ramification data or equivalently certain factorisations of permutations. Double Hurwitz numbers are an important class of Hurwitz numbers, obtained by considering ramification data with a specific structure. They exhibit many fascinating properties, such as a beautiful piecewise polynomial structure, which has been well-studied in the last 15 years. In particular, using methods from tropical geometry, it was possible to derive wall-crossing formulae for double Hurwitz numbers in arbitrary genus. Further, double Hurwitz numbers satisfy an explicit recursive formula. In recent years several related enumerations have appeared in the literature. In this work, we focus on two of those invariants, so-called monotone and strictly monotone double Hurwitz numbers. Monotone double Hurwitz numbers originate from random matrix theory, as they appear as the coefficients in the asymptotic expansion of the famous Harish-Chandra-Itzykson-Zuber integral. Strictly monotone double Hurwitz numbers are known to be equivalent to an enumeration of Grothendieck dessins d'enfants. These new invariants share many structural properties with double Hurwitz numbers, such as piecewise polynomiality. In this work, we enlarge upon this study and derive new explicit wall-crossing and recursive formulae for monotone and strictly monotone double Hurwitz numbers. The key ingredient is a new interpretation of monotone and strictly monotone double Hurwitz numbers in terms of tropical covers, which was recently derived by the authors. An interesting observation is the fact that monotone and strictly monotone double Hurwitz numbers satisfy wall-crossing formulae, which are almost identical to the classical double Hurwitz numbers.


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Additional Information

Marvin Anas Hahn
Affiliation: Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 6-8, 60325 Frankfurt am Main, Germany
Email: hahn@math.uni-frankfurt.de

Danilo Lewański
Affiliation: Inst. Hautes Études Scientifiques (IHES), 35 Route de Chartres, 91440 Bures-sur-Yvette, Paris, France. Inst. de Physique Théorique (IPhT), Commissariat à l’énergie atomique, Orme des Merisiers Bat 774, 91191 Gif-sur-Yvette, Paris, France.
Email: danilo.lewanski@ipht.fr

DOI: https://doi.org/10.1090/tran/8006
Keywords: Hurwitz numbers, Jucys correspondence, flows
Received by editor(s): May 7, 2019
Received by editor(s) in revised form: July 9, 2019, and September 6, 2019
Published electronically: March 27, 2020
Additional Notes: The first author gratefully acknowledges financial support as part of the LOEWE research unit “Uniformized structures in Arithmetic and Geometry”
The second author was supported by the Max Planck Gesellschaft.
Article copyright: © Copyright 2020 by the authors