New topological recursion relations

Abstract

Simple boundary expressions for the k t h k^{th} power of the cotangent line class ψ 1 \psi _1 on M ¯ g , 1 \overline {M}_{g,1} are found for k ≥ 2 g k\geq 2g . The method is by virtual localization on the moduli space of maps to P 1 \mathbb {P}^1 . As a consequence, nontrivial tautological classes in the kernel of the boundary push-forward map \[ ι ∗ : A ∗ ( M ¯ g , 2 ) → A ∗ ( M ¯ g + 1 ) \iota _*:A^*( \overline {M}_{g,2}) \rightarrow A^*(\overline {M}_{g+1}) \] are constructed. The geometry of genus g + 1 g+1 curves, then provides universal equations in genus g g Gromov-Witten theory. As an application, we prove all the Gromov-Witten identities conjectured recently by K. Liu and H. Xu.