Quantum product on the big phase space and the Virasoro conjecture

Quantum cohomology is a family of new ring structures on the space of cohomology classes of a compact symplectic manifold (or a smooth projective variety) V . The quantum products are defined by third order partial derivatives of the generating function of primary Gromov-Witten invariants of V (cf. [RT1]). In a similar way, using the generating function of all descendant Gromov-Witten invariants, we can define products on an infinite dimensional vector space, called the big phase space, which can be thought of as a product of infinite copies of the small phase space H(V ;C). It seems that the products on the big phase space have not gotten enough attention in the literature so far. In this paper we will study some basic structures of such products and apply them to the study of topological recursion relations and the Virasoro conjecture. The Virasoro conjecture predicts that the generating function of the Gromov-Witten invariants is annihilated by infinitely many differential operators, denoted by {Ln | n ≥ −1}, which form a half branch of the Virasoro algebra. This conjecture was proposed by Eguchi, Hori and Xiong [EHX] and also by S. Katz (cf. [CK] [EJX]). It is a natural generalization of a conjecture of Witten (cf. [W2] [Ko] [W2]) and provides a powerful tool in the computation of Gromov-Witten invariants. The genus-0 Virasoro conjecture was proved in [LT] (cf. [DZ2] and [G2] for alternative proofs). The genus-1 Virasoro conjecture for manifolds with semisimple quantum cohomology was proved in [DZ2]. Without assuming semisimplicity, the genus-1 Virasoro conjecture was reduced to the genus-1 L1-constraint on the small phase space in [L1]. It was also proved in [L1] [L2] that the genus-1 Virasoro conjecture holds if the quantum cohomology is not too degenerate (a condition weaker than semisimplicity). The genus-g Virasoro conjecture with g ≥ 1 can be formulated in a way which computes the derivatives of the genus-g generating function along a sequence of vector fields, called the Virasoro vector fields (see Section 5.1). The study of the properties of these vector fields will be important in both proving and applying the Virasoro conjecture in all genera. In this paper we will give a simple recursive description of the Virasoro vector fields (see equation (41) and Theorem 4.7). This recursive description enables us to understand the relations between the Virasoro vector fields and the quantum powers of the Euler vector field defined by equation (27). The action of the Virasoro vector fields on the generating function of genus-g Gromov-Witten invariants is equivalent to the action of a