Gravitational quantum cohomology

We discuss how the theory of quantum cohomology may be generalized to ''gravitational quantum cohomology'' by studying topological a models coupled to two-dimensional gravity. We first consider sigma models defined on a general Fano manifold M (manifold with a positive first Chern class) and derive new recursion relations for its two-point functions. We then derive bi-Hamiltonian structures of the theories and show that they are completely integrable at least at the level of genus 0. We next consider the subspace of the phase space where only a marginal perturbation (with a parameter t) is turned on and construct Lax operators (superpotentials) L whose residue integrals reproduce correlation functions. In the case of M = CPN the Lax operator is given by L = Z(1) + Z(2) + ... + Z(N) + e(t)Z(1)(-1)Z(2)(-1)...(-1)(N) and agrees with the potential of the affine Toda theory of the AN type. We also obtain Lax operators for various Fano manifolds; Grassmannians, rational surfaces, etc. In these examples the number of variables of the Lax operators is the same as the dimension of the original manifold. Our result shows that Fano manifolds exhibit a new type of mirror phenomenon where mirror partner is a noncompact Calabi-Yau manifold of the type of an algebraic torus C*(N) equipped with a specific superpotential.