Evaluating tautological classes using only Hurwitz numbers

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). Introduction Clever applications of the Atiyah-Bott localization theorem in the context of Gromov-Witten theory have generated volumes of enumerative data as well as many insights into the structure of the tautological rings of the moduli spaces of curves ([FP00], [FP05], [GJV01], [GJV06]). The general idea is to exploit torus actions on a target manifold (often just P) to obtain torus actions on the moduli spaces of stable maps to the target. An analysis of the fixed loci for the torus action then produces intersection numbers and relations among tautological classes. Recent work ([GV01], [GV03a],[GV03b], [Ion05]) has shown that the same idea, when applied to moduli of “relative” stable maps reveals even more information. In this paper, we follow ideas of the second author in [Cav06a], applying localization in the context of the ultimate relative stable map spaces to P, the admissible cover spaces of [ACV01], to tie together enumerative data of apparently quite different natures by means of one succinct formula. Theorem 0.1. For each fixed integer d ≥ 1, CY(u) = (−1)d! dd H(d)(u)e , (1) 1 where H(d)(u) is the generating function for degree d, one-part simple Hurwitz numbers (properly defined in section 1.1). D(u) is the generating function for λgλg−1 Hodge integrals on the moduli spaces Agdd (a.k.a. the evaluation of [A g dd] in R (Mg))(discussed in the next paragraph and in section 1.2). CY(u) is the generating function for the fully ramified Calabi-Yau cap invariants of degree d maps in [Cav05b], briefly described in section 1.3. This new formula combines with other known formulas to explicitly reduce the enumerative invariants involved in the local Gromov-Witten theory of curves ([BP04]) to the “pure combinatorics” of single Hurwitz numbers. One class of invariants appearing in Theorem 0.1 is worth singling out. The tautological ring R(Mg) of the moduli stack of curves of genus g is the subring of the Chow ring (with rational coefficients) generated by Mumford’s “kappa” classes (see [Mum83]). It is conjectured in [Fab99] that this ring satisfies Poincaré duality with a (one-dimensional) socle in degree g − 2. We focus on a particular family of degree g− 2 tautological classes, which arise naturally from Hurwitz theory. Consider the (2g − 1)-dimensional “Hurwitz spaces:”