Quantum cohomology of twistor spaces and their Lagrangian submanifolds

TL;DR

This paper computes the quantum cohomology rings of twistor spaces of hyperbolic 6-manifolds, analyzes associated Lagrangian submanifolds, and explores eigenvalues of quantum multiplication, revealing both known and exotic eigenvalues.

Contribution

It provides explicit calculations of quantum cohomology and Lagrangian quantum homology for twistor spaces, including the identification of exotic eigenvalues.

Findings

Computed quantum cohomology rings of twistor spaces.

Analyzed Lagrangian submanifolds and their quantum properties.

Identified four exotic eigenvalues not explained by Reznikov's Lagrangians.

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.