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Chamber behavior of double Hurwitz numbers in genus 0


Shadrin, S; Shapiro, M; Vainshtein, A (2008). Chamber behavior of double Hurwitz numbers in genus 0. Advances in Mathematics, 217(1):79-96.

Abstract

We study double Hurwitz numbers in genus zero counting the number of covers View the MathML source with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points. We describe the partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of two such polynomials for two neighboring chambers. Besides, we provide an explicit formula for the polynomial in a certain chamber called totally negative, which enables us to calculate double Hurwitz numbers in any given chamber as the polynomial for the totally negative chamber plus the sum of the differences between the neighboring polynomials along a path connecting the totally negative chamber with the given one.

Abstract

We study double Hurwitz numbers in genus zero counting the number of covers View the MathML source with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points. We describe the partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of two such polynomials for two neighboring chambers. Besides, we provide an explicit formula for the polynomial in a certain chamber called totally negative, which enables us to calculate double Hurwitz numbers in any given chamber as the polynomial for the totally negative chamber plus the sum of the differences between the neighboring polynomials along a path connecting the totally negative chamber with the given one.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Double Hurwitz numbers; Piecewise polynomiality; Chambers; Wall crossing
Language:English
Date:2008
Deposited On:09 Nov 2009 02:42
Last Modified:06 Dec 2017 20:39
Publisher:Elsevier
ISSN:0001-8708
Publisher DOI:https://doi.org/10.1016/j.aim.2007.06.016
Related URLs:http://arxiv.org/abs/math/0611442

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