Quantum cohomology of twistor spaces and their lagrangian submanifolds

Abstract

We compute the classical and quantum cohomology rings of the twistor spaces of 6-dimensional hyperbolic manifolds and the eigenvalues of quantum multiplication by the first Chern class. Given a half-dimensional totally geodesic submanifold we associate, after Reznikov, a monotone Lagrangian submanifold of the twistor space. In the case of a 3-dimensional totally geodesic submanifold of a hyperbolic 6-manifold, we compute the obstruction term m0 in the Fukaya-Floer A1-algebra of a Reznikov Lagrangian and calculate the Lagrangian quantum homology. There is a well-known correspondence between the possible values of m0 for a Lagrangian with nonvanishing Lagrangian quantum homology and eigenvalues for the action of c1 on quantum cohomology by quantum cup product. Reznikov's Lagrangians account for most of these eigenvalues, but there are four exotic eigenvalues we cannot account for.