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March 2014 Quantum cohomology of twistor spaces and their Lagrangian submanifolds
Jonathan David Evans
J. Differential Geom. 96(3): 353-397 (March 2014). DOI: 10.4310/jdg/1395321845

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.

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Jonathan David Evans. "Quantum cohomology of twistor spaces and their Lagrangian submanifolds." J. Differential Geom. 96 (3) 353 - 397, March 2014. https://doi.org/10.4310/jdg/1395321845

Information

Published: March 2014
First available in Project Euclid: 20 March 2014

zbMATH: 1306.53076
MathSciNet: MR3189460
Digital Object Identifier: 10.4310/jdg/1395321845

Rights: Copyright © 2014 Lehigh University

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Vol.96 • No. 3 • March 2014
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