The jet-space of a Frobenius manifold and higher-genus Gromov-Witten invariants

Abstract

Let X be a projective manifold, and let H be the associated small phase space; this is a Frobenius manifold with canonical choice of fundamental solution for the Dubrovin connection. The large phase space of X may be identified with the jet-space of curves in H. In this paper, we formulate the differential equations satisfied by the higher-genus potentials F_g of X, such as topological recursion relations and the Virasoro constraints, in an intrinsic fashion on the jet space of H, that is, in such a way that the equations do not depend on the choice of fundamental solution. This effort is rewarded by a closer relationship between the resulting theory and the geometry of moduli spaces of stable curves. A consequence of our analysis is the proof of a conjecture of Eguchi and Xiong: F_g is a function on the jet space of jets of order 3g-2.