New topological recursion relations.pdf

a r X i v : 0 8 0 5 .4 8 2 9 v 1 [ m a t h .A G ] 3 0 M a y 2 0 0 8 NEW TOPOLOGICAL RECURSION RELATIONS XIAOBO LIU AND RAHUL PANDHARIPANDE ABSTRACT. Simple boundary expressions for the kth power of the cotangent line class ψ1 on Mg,1 are found for k ≥ 2g. The method is by virtual localization on the moduli space of maps to P1. As a consequence, nontrivial tautological classes in the kernel of the boundary push-forward map ι? : A ?(Mg,2) → A ?(Mg+1) are constructed. The geometry of genus g + 1 curves then provides universal equations in genus g Gromov-Witten theory. As an application, we prove all the Gromov-Witten identities conjectured recently by K. Liu and H. Xu. 0. INTRODUCTION 0.1. Tautological classes. Let M g,n be the moduli space of stable curves of genus g with n marked points. Let A?(M g,n) denote the Chow ring with Q-coefficients. The system of tautological rings is defined in [4] to be the set of smallest Q-subalgebras of the Chow rings, R?(M g,n) ? A ?(M g,n), satisfying the following two properties: (i) The system is closed under push-forward via all maps forgetting markings: π? : R ?(M g,n) → R ?(M g,n?1). (ii) The system is closed under push-forward via all gluing maps: ι? : R ?(M g1,n1∪{?})?Q R ?(Mg2,n2∪{?}) → R ?(M g1+g2,n1+n2), ι? : R ?(M g,n∪{?,?}) → R ?(Mg+1,n), with attachments along the markings ? and ?. Natural algebraic constructions typically yield Chow classes lying in the tautological ring. For ex- ample, the standard ψ, κ, and λ classes in A?(M g,n) all lie in the tautological ring. The tautological rings also possess a rich conjectural structure, see [3] for a detailed discussion. Date: May 30, 2008. 1 2 XIAOBO LIU AND RAHUL PANDHARIPANDE The moduli space M g,n admits a stratification by topological type indexed by decorated graphs. The normalized stratum closures are simply quotients of products of simpler moduli spaces of pointed curves. A descendent stratum class in R?(M g,n) is a push-forward from a stratum S of a monomial i