Gromov-Witten invariants count isolated stable maps from Riemann surfaces to a fixed symplectic manifold $(M,\omega)$ subject to some incidence conditions. If we instead replace the target manifold with a locally Hamiltonian fibration, we get parametrized (or family) GW-theory (see Seidel, McDuff-Buse, Le-Ono,...)
It seems obvious that certain properties (e.g., the divisor axiom or WDVV...) extend to this setting. My question is:
- Is the expectation that the Virasoro conjecture extends to this setting?
- Are there any problems with extending at least the genus-zero proof of Liu-Tian to this setting?
- Is there any known case where there is a proof of such a thing for higher genus?
