Relations Among Universal Equations For Gromov-Witten Invariants

It is well known that relations in the tautological ring of moduli spaces of pointed stable curves give partial differential equations for Gromov-Witten invariants of compact symplectic manifolds. These equations do not depend on the target symplectic manifolds and therefore are called universal equations for Gromov-Witten invariants. In the case that the quantum cohomology of the symplectic manifolds are semisimple, it is expected that higher genus Gromov-Witten invariants are completely determined by such universal equations and genus-0 Gromov-Witten invariants. This has been proved for genus-1 (cf. [DZ]) and genus-2 (cf. [L2]) cases. Universal equations also play very important role in the understanding of the Virasoro conjecture (cf. [EHX]). The genus-0 Virasoro conjecture for all compact symplectic manifolds follows from a universal equation called the genus-0 topological recursion relation (cf. [LT]). For projective varieties, we expect that such universal equations reduce higher genus Virasoro conjecture to an SL(2) symmetry for the generating function of the Gromov-Witten invariants. Again this has been proved for genus-1 ([L1]) and genus-2 ([L2]) cases. In this paper, we will discuss the relation among known universal equations for Gromov-Witten invariants. We hope that the understanding of such relations would be helpful to the study of both Gromov-Witten invariants and the topology of the moduli spaces of pointed curves. Relations among genus-2 universal equations were studied in [L2]. It was proved that the three universal equations in [G2] and [BP] implies certain complicated genus-1 relations (see equations (4) and (6) below). Modulo these genus-1 relations, the three known genus-2 equations can be reduced to only two equations. The main result of this paper is that the genus-1 relations derived in [L2] follow from a known genus-1 relation found in [G1] and the genus-0 and genus-1 topological recursion relations. This completes the discussion of relations among genus-2 equations in [L2]. Part of the work in this paper was done while the author visited Max-Planck Institute for Mathematics at Bonn during the workshop on Frobenius Manifolds, Quantum Cohomology, and Singularity Theory. The author would like to thank the organizers of the workshop for invitation. He would also like to thank E. Getzler for conversations related to this work.