Quantum cohomology of twistor spaces and their Lagrangian submanifolds
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arXiv
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| Format: | Preprint |
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2011
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| _version_ | 1866909217684717568 |
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| author | Evans, Jonathan David |
| author_facet | Evans, Jonathan David |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1106_3959 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | Quantum cohomology of twistor spaces and their Lagrangian submanifolds Evans, Jonathan David Symplectic Geometry Differential Geometry 53D12, 53D35, 53D40, 53D45, 32L25 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. |
| title | Quantum cohomology of twistor spaces and their Lagrangian submanifolds |
| topic | Symplectic Geometry Differential Geometry 53D12, 53D35, 53D40, 53D45, 32L25 |
| url | https://arxiv.org/abs/1106.3959 |