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8-DIMENSIONAL QUATERNIO NIC GEOMETRY. Simon Salamon. Politecnico di Torino. Contents. 4-forms and spinors. Types of Q structures. Dirac operators. Model geometries. Q symplectic manifolds. 4-FORMS AND SPINORS. 4 -forms in dimension 8. Possible dimensions include. A simple example. - PowerPoint PPT Presentation
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Colloque Paul Gauduchon Palaiseau, 20/05/05
8-DIMENSIONALQUATERNIONIC
GEOMETRY
8-DIMENSIONALQUATERNIONIC
GEOMETRY
Simon SalamonSimon Salamon
Politecnico di TorinoPolitecnico di Torino
Colloque Paul Gauduchon Palaiseau, 20/05/05
ContentsContents
Dirac operators Dirac operators
Model geometries
Model geometries
4-forms and spinors 4-forms and spinors
Types of Q structures Types of Q structures
Q symplectic manifolds Q symplectic manifolds
Colloque Paul Gauduchon Palaiseau, 20/05/05
4-forms in dimension 84-forms in dimension 8
Possible dimensions include Possible dimensions include
Colloque Paul Gauduchon Palaiseau, 20/05/05
Set of OQS’s Set of OQS’s
Symmetric spacesSymmetric spaces
3-forms
8 = 3 + 5
3-forms
8 = 3 + 5
Colloque Paul Gauduchon Palaiseau, 20/05/05
Clifford multiplicationClifford multiplication
X determines X determines
8 = 3 + 5 8 = 3 + 5
Colloque Paul Gauduchon Palaiseau, 20/05/05
TYPES OFQUATERNIONICSTRUCTURES
TYPES OFQUATERNIONICSTRUCTURES
Colloque Paul Gauduchon Palaiseau, 20/05/05
Reduction of structureReduction of structure
The 4-form determines the metric and Levi-Civita connection
on the
bundle with fibre
The 4-form determines the metric and Levi-Civita connection
on the
bundle with fibre
Colloque Paul Gauduchon Palaiseau, 20/05/05
Quaternionic manifoldsQuaternionic manifolds
“Nijenhuis” = 0“Nijenhuis” = 0
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M8 has an integrable “twistor space”M8 has an integrable “twistor space”
I,J,K can be chosen with I complexI,J,K can be chosen with I complex
Quaternionic manifoldsQuaternionic manifolds
Colloque Paul Gauduchon Palaiseau, 20/05/05
Rigidity principleRigidity principle
G acts trivially onG acts trivially on
M Wolf spaceM Wolf space
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The tautological sectionThe tautological section
An Sp(2)Sp(1) structure determinesAn Sp(2)Sp(1) structure determines
oror
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Proposition [Witt] Proposition [Witt]
The tautological sectionThe tautological section
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Killing spinorsKilling spinors
M QK, X an infinitesimal isometryM QK, X an infinitesimal isometry
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M is QK ( )M is QK ( )
M is Einstein ( )M is Einstein ( )
M8 is symmetricM8 is symmetric
Quaternion-Kahler manifoldsQuaternion-Kahler manifolds
Colloque Paul Gauduchon Palaiseau, 20/05/05
1. Projection1. Projection
Links with HK and G2 holonomyLinks with HK and G2 holonomy
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Complex coadjoint orbitsComplex coadjoint orbits
Any nilpotent orbit N has both QK and
HK metrics
Any nilpotent orbit N has both QK and
HK metrics
The hunt for potentials: [Biquard-Gauduchon,
Swann]
The hunt for potentials: [Biquard-Gauduchon,
Swann]
Colloque Paul Gauduchon Palaiseau, 20/05/05
2. The case SL(3,C)2. The case SL(3,C)
8 = 3 + 5 8 = 3 + 5
Colloque Paul Gauduchon Palaiseau, 20/05/05
2. The case SL(3,C)2. The case SL(3,C)
M8 parametrizes a subset of OQS’s
M8 parametrizes a subset of OQS’s
Colloque Paul Gauduchon Palaiseau, 20/05/05
QUATERNIONIC SYMPLECTIC MANIFOLDS
QUATERNIONIC SYMPLECTIC MANIFOLDS
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Q contact structuresQ contact structures
On hypersurfaces and asymptotic boundaries of QK manifolds with
non-degenerate “Levi form”
On hypersurfaces and asymptotic boundaries of QK manifolds with
non-degenerate “Levi form”
Colloque Paul Gauduchon Palaiseau, 20/05/05
An extra integrability condition is needed for n=1 and allows one to
extend QCS’s on S7 [Duchemin]
An extra integrability condition is needed for n=1 and allows one to
extend QCS’s on S7 [Duchemin]
Without the integrability condition, extension to a Q symplectic metric
is nonetheless possible
Without the integrability condition, extension to a Q symplectic metric
is nonetheless possible
Q contact structuresQ contact structures
Colloque Paul Gauduchon Palaiseau, 20/05/05
3. The case SO(5,C)3. The case SO(5,C)
Fibration based on the reduction Fibration based
on the reduction
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3. The case SO(5,C)3. The case SO(5,C)
Total space is both Kahler and
QK:
Total space is both Kahler and
QK:
Colloque Paul Gauduchon Palaiseau, 20/05/05
3. The case SO(5,C)3. The case SO(5,C)
X6 has a subspace of 3-forms X6 has a subspace of 3-forms
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T2 product examplesT2 product examples
Ingredients:
symplectic with
closed primitive 3-forms
giving closed 4-form
Ingredients:
symplectic with
closed primitive 3-forms
giving closed 4-form
Colloque Paul Gauduchon Palaiseau, 20/05/05
Compact nilmanifold examples have 3 transverse simple closed 3-forms, with reduction
Compact nilmanifold examples have 3 transverse simple closed 3-forms, with reduction
T2 product examplesT2 product examples
Applications to SL/CY geometry [Giovannini,
Matessi]
Applications to SL/CY geometry [Giovannini,
Matessi]
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