Generating functions of Gromov-Witten invariants of compact
symplectic manifolds behave very much like tau-functions of Integrable
systems. It was conjectured by Eguchi-Hori-Xiong and S. Katz that
Gromov-Witten invariants of smooth projective varieties should
satisfy the Virasoro constraints, which also exist for many integrable
systems (e.g, Gelfand-Dickey hierarchies). It was conjectured by Witten
that the generating functions on moduli spaces of spin curves are
tau-functions of Gelfand-Dickey hierarchy. In a joint work with Kimura, we
showed that it is possible to use Virasoro constraints of a point and
the sphere to derive universal equtions for Gromov-Witten invariants of
all compact symplectic manifolds. Such equations can also be used to
compute certain intersection numbers on moduli spaces of spin curves which
coincide with predictions of Witten's conjecture.