## Journal of Differential Geometry

- J. Differential Geom.
- Volume 96, Number 3 (2014), 353-397.

### 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 $\mathbb{m}_0$ 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 $\mathbb{m}_0$ 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.

#### Article information

**Source**

J. Differential Geom., Volume 96, Number 3 (2014), 353-397.

**Dates**

First available in Project Euclid: 20 March 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1395321845

**Digital Object Identifier**

doi:10.4310/jdg/1395321845

**Mathematical Reviews number (MathSciNet)**

MR3189460

**Zentralblatt MATH identifier**

1306.53076

#### Citation

Evans, Jonathan David. Quantum cohomology of twistor spaces and their Lagrangian submanifolds. J. Differential Geom. 96 (2014), no. 3, 353--397. doi:10.4310/jdg/1395321845. https://projecteuclid.org/euclid.jdg/1395321845