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Evaluating tautological classes using only Hurwitz numbers

Author(s): Aaron Bertram; Renzo Cavalieri; Gueorgui Todorov
Journal: Trans. Amer. Math. Soc. 360 (2008), 6103-6111.
MSC (2000): Primary 14N35
Posted: May 22, 2008
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Abstract | References | Similar articles | Additional information

Abstract: Hurwitz numbers count ramified covers of a Riemann surface with prescribed monodromy. As such, they are purely combinatorial objects. Tautological classes, on the other hand, are distinguished classes in the intersection ring of the moduli spaces of Riemann surfaces of a given genus, and are thus ``geometric''. Localization computations in Gromov-Witten theory provide non-obvious relations between the two. This paper makes one such computation, and shows how it leads to a ``master'' relation (Theorem 0.1) that reduces the ratios of certain interesting tautological classes to the pure combinatorics of Hurwitz numbers. As a corollary, we obtain a purely combinatorial proof of a theorem of Bryan and Pandharipande, expressing in generating function form classical computations by Faber/Looijenga (Theorem 0.2).

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Additional Information:

Aaron Bertram
Affiliation: Department of Mathematics, University of Utah, 155 South 1400 East, Rm 233, Salt Lake City, Utah 84112-0090

Renzo Cavalieri
Affiliation: Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, Michigan 48109-1043

Gueorgui Todorov
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112-0090

DOI: 10.1090/S0002-9947-08-04481-4
PII: S 0002-9947(08)04481-4
Received by editor(s): September 27, 2006
Received by editor(s) in revised form: January 20, 2007
Posted: May 22, 2008
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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