Journal of Differential Geometry

Quantum cohomology of twistor spaces and their Lagrangian submanifolds

Jonathan David Evans

Full-text: Open access

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


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