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Licence: CC BY: Creative Commons Attribution 4.0 International License
Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Quantum cohomology of twistor spaces and their Lagrangian submanifolds
AU - Evans, Jonathan David
PY - 2014/3/20
Y1 - 2014/3/20
N2 - 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 m0 in the Fukaya–Floer A∞-algebra of a Reznikov Lagrangian and calculate the Lagrangian quantum homology. There is a well-known correspondence between the possible values of m0 for a Lagrangian with nonvanishing Lagrangian quantum homology and eigenvalues for the action of c1 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.
AB - 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 m0 in the Fukaya–Floer A∞-algebra of a Reznikov Lagrangian and calculate the Lagrangian quantum homology. There is a well-known correspondence between the possible values of m0 for a Lagrangian with nonvanishing Lagrangian quantum homology and eigenvalues for the action of c1 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.
UR - http://dx.doi.org/10.4310/jdg/1395321845
U2 - 10.4310/jdg/1395321845
DO - 10.4310/jdg/1395321845
M3 - Journal article
VL - 96
SP - 353
EP - 397
JO - Journal of Differential Geometry
JF - Journal of Differential Geometry
SN - 0022-040X
IS - 3
ER -