Chamber behavior of double Hurwitz numbers in genus 0

S. Shadrin; M. Shapiro; A. Vainshtein

doi:10.1016/j.aim.2007.06.016

Chamber behavior of double Hurwitz numbers in genus 0

S. Shadrin, M. Shapiro, A. Vainshtein

Research output: Contribution to journal › Article › peer-review

Abstract

We study double Hurwitz numbers in genus zero counting the number of covers C P¹ → C P¹ 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.

Original language	English
Pages (from-to)	79-96
Number of pages	18
Journal	Advances in Mathematics
Volume	217
Issue number	1
DOIs	https://doi.org/10.1016/j.aim.2007.06.016
State	Published - 15 Jan 2008

Bibliographical note

Funding Information:
The authors are grateful to Max-Planck-Institut für Matematik, Bonn, and to Institut Mittag-Leffler, Djursholm, Sweden, for hospitality in Summer 2004, and Fall 2006. Our sincere thanks go to M. Kazaryan, R. Kulkarni and D. Zvonkine for useful discussions. S.S. was supported by the grants RFBR-05-01-01012-a, RFBR-05-01-02806-CNRS-a, NSh-1972.2003.1, MK-5396.2006.1, NWO-RFBR-047.011.2004.026 (RFBR-05-02-89000-NWO-a), by the Göran Gustafsson foundation, and by Pierre Deligne’s fund based on his 2004 Balzan prize in mathematics. M.S. was supported by the grants DMS-0401178 and PHY-0555346. M.S and A.V. were supported by the grant BSF-2002375.

Keywords

Chambers
Double Hurwitz numbers
Piecewise polynomiality
Wall crossing

ASJC Scopus subject areas

General Mathematics

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10.1016/j.aim.2007.06.016

Cite this

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title = "Chamber behavior of double Hurwitz numbers in genus 0",

abstract = "We study double Hurwitz numbers in genus zero counting the number of covers C P1 → C P1 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.",

keywords = "Chambers, Double Hurwitz numbers, Piecewise polynomiality, Wall crossing",

author = "S. Shadrin and M. Shapiro and A. Vainshtein",

note = "Funding Information: The authors are grateful to Max-Planck-Institut f{\"u}r Matematik, Bonn, and to Institut Mittag-Leffler, Djursholm, Sweden, for hospitality in Summer 2004, and Fall 2006. Our sincere thanks go to M. Kazaryan, R. Kulkarni and D. Zvonkine for useful discussions. S.S. was supported by the grants RFBR-05-01-01012-a, RFBR-05-01-02806-CNRS-a, NSh-1972.2003.1, MK-5396.2006.1, NWO-RFBR-047.011.2004.026 (RFBR-05-02-89000-NWO-a), by the G{\"o}ran Gustafsson foundation, and by Pierre Deligne{\textquoteright}s fund based on his 2004 Balzan prize in mathematics. M.S. was supported by the grants DMS-0401178 and PHY-0555346. M.S and A.V. were supported by the grant BSF-2002375.",

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AU - Shapiro, M.

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N2 - We study double Hurwitz numbers in genus zero counting the number of covers C P1 → C P1 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.

AB - We study double Hurwitz numbers in genus zero counting the number of covers C P1 → C P1 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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Chamber behavior of double Hurwitz numbers in genus 0

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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