Gravitational quantum cohomology

Abstract

We discuss how the theory of quantum cohomology may be generalized to “gravitational quantum cohomology ” by studying topological σ-models coupled to twodimensional gravity. We first consider σ-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 = CP N the Lax operator is given by L = Z1 +Z2 + · · ·+ZN +etZ −1 1 Z−1 2 · · · Z −1 N and agrees with the potential of the affine Toda theory of the AN type. We also obtain Lax