Genus-2 Gromov-Witten invariants for manifolds with semisimple quantum cohomology

In [L2], the author studied universal equations for genus-2 Gromov-Witten invariants given in [Ge1] and [BP] using quantum product on the big phase space. Among other results, the author proved that for manifolds with semisimple quantum cohomology, the generating function for genus-2 Gromov-Witten invariants, denoted by F2, is uniquely determined by known genus-2 universal equations. Moreover, an explicit formula for F2 was given in terms of genus-0 and genus-1 invariants. However, the formula given in [L2] is very complicated to work with. In this paper, we will give a much simpler formula using idempotents of the quantum product on the big phase space, and then use it to prove the genus-2 Virasoro conjecture for manifolds with semisimple quantum cohomology (cf. [EHX] and [CK]). Properties of idempotents on the big phase space were studied in [L4]. Let M be a compact symplectic manifold. In Gromov-Witten theory, the space H(M ;C) is called the small phase space. A product of infinitely many copies of the small phase space is called the big phase space. The generating functions for Gromov-Witten invariants are formal power series on the big phase space. Let N be the dimension of H(M ;C). If the quantum cohomology of M is semisimple, there exist vector fields Ei, i = 1, . . . , N , on the big phase space such that Ei ◦ Ej = δijEi for all i and j, where “◦” stands for the quantum product (see equation (1)). These vector fields are called idempotents. Let ui, i = 1, . . . , N , be the eigenvalues of the quantum multiplication by the Euler vector field (see equation (2)). The first main result of this paper is the following