  We describe generating functions for arbitrary-genus Gromov-Witten invariants
of the projective space with any number of marked points explicitly. The
structural portion of this description gives rise to uniform energy bounds and
vanishing results for these invariants. They suggest deep conjectures relating
Gromov-Witten invariants of symplectic manifolds to the energy of
pseudo-holomorphic maps and the expected dimension of their moduli space.
