Semisimple Frobenius structures at higher genus

In the context of equivariant Gromov-Witten theory of tori actions with isolated fixed points, we compute genus g > 1 Gromov-Witten potentials and their generalizations with gravitational descendents. Both formulas, with and without descendents, are stated in a form applicable to any semisimple Frobenius structure and therefore can be considered as definitions in the axiomatic context of Frobenius manifolds. In (nonequivariant) Gromov-Witten theory, they become conjectures expressing higher genus GW-invariants in terms of genus 0 GW-invariants of symplectic manifolds with generically semisimple quantum cup-product.