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Quantum cohomology of twistor spaces and their Lagrangian submanifolds

Jonathan David Evans

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 $m_0$ 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.

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 in the Fukaya-Floer -algebra of a Reznikov Lagrangian and calculate the Lagrangian quantum homology. There is a well-known correspondence between the possible values of for a Lagrangian with nonvanishing Lagrangian quantum homology and eigenvalues for the action of 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.

Paper Structure

This paper contains 38 sections, 39 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Acknowledgements
  4. The homogeneous space $SO(2n)/U(n)$
  5. Twistor spaces
  6. Setting
  7. Eells-Salamon twistor correspondence
  8. Classification of $J_-$-holomorphic spheres
  9. Characteristic classes
  10. First Chern class
  11. Pontryagin classes
  12. Some useful formulae
  13. Reznikov Lagrangians
  14. Reznikov's construction
  15. Holomorphic discs
  16. ...and 23 more sections

Key Result

Theorem A

The small quantum cohomology of the twistor space of a hyperbolic 6-manifold $M$ with vanishing Stiefel-Whitney classes is See erratum at the end of the paper. where $\alpha=c_1(Z)$ and $\Lambda=\mathbb{C}[q]$. Moreover, $c_1(\mathcal{H})^2=\alpha^2-4q$, $c_1(\mathcal{H})^3=\alpha^3-4\alpha q$. The twistor space is also uniruled.

Theorems & Definitions (74)

  • Theorem A
  • proof
  • Corollary B
  • Theorem C
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Theorem 2
  • Theorem 3: Eells-Salamon twistor correspondence (ESTC)
  • ...and 64 more