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New topological recursion relations
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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
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