Abstract
Hurwitz numbers count ramified genus $g$, degree $d$ coverings of the projective line with fixed branch locus and fixed ramification data. Double Hurwitz numbers count such covers, where we fix two special profiles over $0$ and $\infty$ and only simple ramification else. These objects feature interesting structural behaviour and connections to geometry. In this paper, we introduce the notion of pruned double Hurwitz numbers, generalizing the notion of pruned simple Hurwitz numbers in Do and Norbury. We show that pruned double Hurwitz numbers, similar to usual double Hurwitz numbers, satisfy a cut-and-join recursion and are piecewise polynomial with respect to the entries of the two special ramification profiles. Furthermore, double Hurwitz numbers can be computed from pruned double Hurwitz numbers. To sum up, it can be said that pruned double Hurwitz numbers count a relevant subset of covers, leading to considerably smaller numbers and computations, but still featuring the important properties we can observe for double Hurwitz numbers.
Highlights
Hurwitz numbers are important enumerative objects connecting numerous areas of mathematics, such as algebraic geometry, algebraic topology, operator theory, representation theory of the symmetric group and combinatorics
We introduce the notion of pruned double Hurwitz numbers, generalizing the notion of pruned simple Hurwitz numbers in [6]
There are various equivalent definitions of Hurwitz numbers and several different settings, among which the most well-studied one is the case of simple Hurwitz numbers, which we denote by Hg(μ)
Summary
Hurwitz numbers are important enumerative objects connecting numerous areas of mathematics, such as algebraic geometry, algebraic topology, operator theory, representation theory of the symmetric group and combinatorics. While there are still a lot of open questions, much is known about these objects as well and they admit many results, which are similar to those about simple Hurwitz numbers Among those is a cut-and-join recursion for double Hurwitz numbers and a definition in terms of factorizations in the symmetric group. They admit a cut-and-join recursion similar to the one for simple Hurwitz numbers Using these results and the ELSV formula, another proof for Witten’s Conjecture was given in [6]. It was proved, that pruned simple Hurwitz numbers admit an Eynard-Orantin topological recursion. The computations agree with the predictions made by the formulas of Theorem 15 and Theorem 24
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
