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Evaluating tautological classes using only Hurwitz
"... 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 “g ..."
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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 GromovWitten theory provide nonobvious 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).
0.1 Informal Overview For any fixed integer d consider
, 2008
"... In this paper we describe explicit generating functions for a large class of HurwitzHodge integrals. These are integrals of tautological classes on moduli spaces of admissible covers, a (stackily) smooth compactification of the Hurwitz schemes. Admissible covers and their tautological classes are i ..."
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In this paper we describe explicit generating functions for a large class of HurwitzHodge integrals. These are integrals of tautological classes on moduli spaces of admissible covers, a (stackily) smooth compactification of the Hurwitz schemes. Admissible covers and their tautological classes are interesting mathematical objects on their own, but recently they have proved to be a useful tool for the study of the tautological ring of the moduli space of curves, and the orbifold GromovWitten theory of DM stacks. Our main tool is AtiyahBott localization: its underlying philosophy is to translate an interesting geometric problem into a purely combinatorial
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, 2008
"... We exhibit a set of recursive relations that completely determine all equivariant GromovWitten invariants of [C 3 /Z3]. We interpret such invariants as Z3Hodge integrals, and produce relations among them via AtiyahBott localization on moduli spaces of twisted stable maps to gerbes over P 1. ..."
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We exhibit a set of recursive relations that completely determine all equivariant GromovWitten invariants of [C 3 /Z3]. We interpret such invariants as Z3Hodge integrals, and produce relations among them via AtiyahBott localization on moduli spaces of twisted stable maps to gerbes over P 1.