On Certain Vanishing Identities For Gromov-Witten Invariants

Let V be a compact symplectic manifold and {γα | α = 1, · · ·N} be a basis for H(V ;C). We choose γ1 to be the identity of cohomology ring of V . Let γ α = ηγβ with (η) representing the inverse matrix of the Poincáre intersection pairing. As a convention, repeated Greek letter indices are summed up over their entire ranges from 1 to N . Recall that the big phase space for Gromov-Witten invariants is ∏∞ n=0H (V ;C) with standard basis {τn(γα) | α = 1, . . . , N, n ≥ 0}. Let t α n be the coordinates on the big phase space with respect to the standard basis. The genus-g generating function Fg is a formal power series of t = (tn) with coefficients given by genus-g Gromov-Witten invariants. Derivatives of Fg with respect to t α1 n1 , . . . , tk nk are denoted by 〈〈 τn1(α1) · · · τnk(αk) 〉〉g. The following conjecture was proposed by K. Liu and H. Xu in [LkX]: