Quantum product on the big phase space and the Virasoro conjecture

We first study the quantum product on the big phase space defined by gravitational Gromov-Witten invariants. We then use this product to give an interpretation for various topological recursion relations and also use it to study the Virasoro conjecture proposed by Eguchi-Hori-Xiong and Katz. We will give a recursive formulation to the Virasoro conjecture and study properties of relevent vector fields, which will be useful in proving and applying the Virasoro conjecture for all genera. In the genus-2 case, we will prove that the genus-2 Virasoro conjecture can be reduced to the $L_{1}$ constraint for any manifold. In the case when the quantum cohomology of the underlying manifold is not too degenerate (in particular, is semisimple) we will prove an explicit formula expressing the generating function of genus-2 Gromov-Witten invariants in terms of genus-0 and genus-1 data. This result reduces the genus-2 Virasoro conjecture to a genus-1 problem for such manifolds.