Hodge integrals and Hurwitz numbers via virtual localization
Tom Graber, Ravi Vakil
TL;DR
This paper proves a formula linking Hurwitz numbers and Hodge integrals using virtual localization, providing a new geometric proof and potential simplifications through a proper algebro-geometric framework.
Contribution
It offers a rigorous proof of a known formula connecting Hurwitz numbers and Hodge integrals via virtual localization techniques.
Findings
Proof of the Hurwitz-Hodge formula using virtual localization.
Description of how to simplify the proof with a proper relative space.
Insights into the geometric structure of moduli spaces.
Abstract
Ekedahl, Lando, Shapiro, and Vainshtein announced a remarkable formula expressing Hurwitz numbers (counting covers of the projective line with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. We give a proof of this formula using virtual localization on the moduli space of stable maps, and describe how the proof could be simplified by the proper algebro-geometric definition of a "relative space".
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology · Advanced Algebra and Geometry