Relations among universal equations for Gromov-Witten invariants

Abstract

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.KeywordsVector FieldModulus SpaceCovariant DerivativeQuantum CohomologyString EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.