Quantum cohomology of twistor spaces and their Lagrangian submanifolds

Details Quantum cohomology of twistor spaces and their Lagrangian submanifolds Quantum cohomology of twistor spaces and their Lagrangian submanifolds

2011-06-20

arxiv.org/abs/1106.3959v2

Mathematics

Symplectic Geometry

43 pages; some canonical references added

Scientific paper

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 $m_0$ in the Fukaya-Floer $A_{\infty}$-algebra of a Reznikov Lagrangian and calculate the Lagrangian quantum homology. There is a well-known correspondence between the possible values of $m_0$ for a Lagrangian with nonvanishing Lagrangian quantum homology and eigenvalues for the action of $c_1$ 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.

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