Elliptic Gromov-Witten Invariants And Virasoro Conjecture

The Virasoro conjecture predicts that the generating function of Gromov-Witten invariants is annihilated by infinitely many differential operators which form a half branch of the Virasoro algebra. This conjecture was proposed by Eguchi, Hori and Xiong [EHX2] and also by S. Katz [Ka] (see also [EJX]). It provides a powerful tool in the computation of Gromov-Witten invariants. In [LT], the author and Tian proved the genus-0 part of the Virasoro conjecture. The main purpose of this paper is to study the genus-1 part of this conjecture. The system of Gromov-Witten invariants relevant to this paper are the so called descendant Gromov-Witten invariants. These invariants arised in the theory of topological sigma model coupled to gravity [W2]. Mathematical definition for such invariants was given in [RT2] for semipositive symplectic manifolds. Using virtual moduli cycles ([LiT1], [LiT2], and also [BF]), these invariants can also be defined for all compact symplectic manifolds and, in purely algebraic geometric setting, for smooth projective varieties. In this paper, we consider descendant Gromov-Witten invariants for a smooth projective variety V . For simplicity, we assume that H(V,C) = 0. Fix a basis {γ1, . . . , γN} of H(V,C) with γ1 equal to the identity of the cohomology ring of V and γα ∈ H αα(V,C) for every α. For any non-negative integer g and A ∈ H2(V, Z), let 〈τn1,α1 . . . τnk,αk〉g,A be the genus g degree A descendant Gromov-Witten invariants associated with cohomology classes γα1, . . . , γαk and non-negative integers n1, . . . , nk (See Section 1.1 for the definition of Gromov-Witten invariants). Summing up the Gromov-Witten invariants over all degrees, we obtain a quantity which is called the k-point correlators in the theory of topological sigma model: