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Hodge Integrals and Hurwitz Numbers via Virtual Localization

Published online by Cambridge University Press:  04 December 2007

Tom Graber  and
Ravi Vakil
Tom Graber
Affiliation:
Department of Mathematics, Harvard University, Cambridge MA 02138, U.S.A. e-mail: grabber@math.harvard.edu
Ravi Vakil
Affiliation:
Department of Mathematics, Stanford University Bldg. 380, Stanford CA 94305–2125, U.S.A. e-mail: vakil@math.stanford.edu
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Abstract

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We give another proof of Ekedahl, Lando, Shapiro, and Vainshtein's remarkable formula expressing Hurwitz numbers (counting covers of P1 with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. Our proof uses virtual localization on the moduli space of stable maps. We describe how the proof could be simplified by the proper algebro-geometric definition of a ‘relative space’. Such a space has recently been defined by J. Li.

Keywords

Hodge integralsHurwitz numbersvirtual localization
Type
Research Article
Information
Compositio Mathematica , Volume 135 , Issue 1 , January 2003 , pp. 25 - 36
Copyright
© 2003 Kluwer Academic Publishers