The Hurwitz enumeration problem of branched covers and Hodge integrals

Stefano Monni; Jun S. Song; Yun S. Song

doi:10.1016/j.geomphys.2003.11.003

The Hurwitz enumeration problem of branched covers and Hodge integrals

Stefano Monni, Jun S. Song, Yun S. Song

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

Abstract

We use algebraic methods to compute the simple Hurwitz numbers for arbitrary source and target Riemann surfaces. For an elliptic curve target, we reproduce the results previously obtained by string theorists. Motivated by the Gromov-Witten potentials, we find a general generating function for the simple Hurwitz numbers in terms of the representation theory of the symmetric group S_n. We also find a generating function for Hodge integrals on the moduli space M̄_g,2 of Riemann surfaces with two marked points, similar to that found by Faber and Pandharipande for the case of one marked point.

Original language	English (US)
Pages (from-to)	223-256
Number of pages	34
Journal	Journal of Geometry and Physics
Volume	50
Issue number	1-4
DOIs	https://doi.org/10.1016/j.geomphys.2003.11.003
State	Published - Apr 2004
Externally published	Yes

Keywords

Descendant Gromov-Witten invariants
Generating functions
Hodge integrals
Hurwitz numbers

ASJC Scopus subject areas

Mathematical Physics
General Physics and Astronomy
Geometry and Topology

Online availability

10.1016/j.geomphys.2003.11.003

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title = "The Hurwitz enumeration problem of branched covers and Hodge integrals",

abstract = "We use algebraic methods to compute the simple Hurwitz numbers for arbitrary source and target Riemann surfaces. For an elliptic curve target, we reproduce the results previously obtained by string theorists. Motivated by the Gromov-Witten potentials, we find a general generating function for the simple Hurwitz numbers in terms of the representation theory of the symmetric group Sn. We also find a generating function for Hodge integrals on the moduli space {\=M}g,2 of Riemann surfaces with two marked points, similar to that found by Faber and Pandharipande for the case of one marked point.",

keywords = "Descendant Gromov-Witten invariants, Generating functions, Hodge integrals, Hurwitz numbers",

author = "Stefano Monni and Song, {Jun S.} and Song, {Yun S.}",

note = "Funding Information: We gratefully acknowledge Ravi Vakil for numerous valuable discussions and suggestions. JSS thanks Prof. Gang Tian and Y.-H. He for discussions. JSS is supported in part by an NSF Graduate Fellowship and the US Department of Energy under cooperative research agreement #DE-FC02-94ER40818. YSS is supported in part by an NSF Graduate Fellowship and the US Department of Energy under contract #DE-AC03-76SF00515. ",

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AU - Song, Yun S.

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PY - 2004/4

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N2 - We use algebraic methods to compute the simple Hurwitz numbers for arbitrary source and target Riemann surfaces. For an elliptic curve target, we reproduce the results previously obtained by string theorists. Motivated by the Gromov-Witten potentials, we find a general generating function for the simple Hurwitz numbers in terms of the representation theory of the symmetric group Sn. We also find a generating function for Hodge integrals on the moduli space M̄g,2 of Riemann surfaces with two marked points, similar to that found by Faber and Pandharipande for the case of one marked point.

AB - We use algebraic methods to compute the simple Hurwitz numbers for arbitrary source and target Riemann surfaces. For an elliptic curve target, we reproduce the results previously obtained by string theorists. Motivated by the Gromov-Witten potentials, we find a general generating function for the simple Hurwitz numbers in terms of the representation theory of the symmetric group Sn. We also find a generating function for Hodge integrals on the moduli space M̄g,2 of Riemann surfaces with two marked points, similar to that found by Faber and Pandharipande for the case of one marked point.

KW - Descendant Gromov-Witten invariants

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KW - Hodge integrals

KW - Hurwitz numbers

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The Hurwitz enumeration problem of branched covers and Hodge integrals

Abstract

Keywords

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