colloque paul gauduchonpalaiseau, 20/05/05 8-dimensional quaternionic geometry 8-dimensional...
TRANSCRIPT
![Page 1: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/1.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
8-DIMENSIONALQUATERNIONIC
GEOMETRY
8-DIMENSIONALQUATERNIONIC
GEOMETRY
Simon SalamonSimon Salamon
Politecnico di TorinoPolitecnico di Torino
![Page 2: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/2.jpg)
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
![Page 3: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/3.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
4-FORMS AND SPINORS
4-FORMS AND SPINORS
![Page 4: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/4.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
4-forms in dimension 84-forms in dimension 8
Possible dimensions include Possible dimensions include
![Page 5: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/5.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
A simple exampleA simple example
![Page 6: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/6.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
A complex variantA complex variant
![Page 7: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/7.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
A complex variantA complex variant
![Page 8: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/8.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
The quaternionic 4-formThe quaternionic 4-form
![Page 9: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/9.jpg)
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
![Page 10: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/10.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
Triality for Sp(2)Sp(1)Triality for Sp(2)Sp(1)
![Page 11: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/11.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
Clifford multiplicationClifford multiplication
X determines X determines
8 = 3 + 5 8 = 3 + 5
![Page 12: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/12.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
TYPES OFQUATERNIONICSTRUCTURES
TYPES OFQUATERNIONICSTRUCTURES
![Page 13: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/13.jpg)
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
![Page 14: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/14.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
Intrinsic torsionIntrinsic torsion
![Page 15: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/15.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
“Q symplectic” manifolds“Q symplectic” manifolds
![Page 16: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/16.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
Quaternionic manifoldsQuaternionic manifolds
“Nijenhuis” = 0“Nijenhuis” = 0
![Page 17: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/17.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
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
![Page 18: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/18.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
DIRAC OPERATORS
DIRAC OPERATORS
![Page 19: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/19.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
Rigidity principleRigidity principle
G acts trivially onG acts trivially on
M Wolf spaceM Wolf space
![Page 20: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/20.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
The tautological sectionThe tautological section
An Sp(2)Sp(1) structure determinesAn Sp(2)Sp(1) structure determines
oror
![Page 21: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/21.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
Proposition [Witt] Proposition [Witt]
The tautological sectionThe tautological section
![Page 22: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/22.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
Killing spinorsKilling spinors
M QK, X an infinitesimal isometryM QK, X an infinitesimal isometry
![Page 23: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/23.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
Killing spinorsKilling spinors
![Page 24: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/24.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
MODEL GEOMETRIES
MODEL GEOMETRIES
![Page 25: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/25.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
M is QK ( )M is QK ( )
M is Einstein ( )M is Einstein ( )
M8 is symmetricM8 is symmetric
Quaternion-Kahler manifoldsQuaternion-Kahler manifolds
![Page 26: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/26.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
Wolf spacesWolf spaces
M8 QK symmetricM8 QK symmetric
![Page 27: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/27.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
1. Projection1. Projection
Links with HK and G2 holonomyLinks with HK and G2 holonomy
![Page 28: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/28.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
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]
![Page 29: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/29.jpg)
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
![Page 30: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/30.jpg)
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
![Page 31: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/31.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
QUATERNIONIC SYMPLECTIC MANIFOLDS
QUATERNIONIC SYMPLECTIC MANIFOLDS
![Page 32: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/32.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
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”
![Page 33: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/33.jpg)
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
![Page 34: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/34.jpg)
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
![Page 35: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/35.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
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:
![Page 36: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/36.jpg)
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
![Page 37: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/37.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
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
![Page 38: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/38.jpg)
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]
![Page 39: Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino](https://reader036.vdocuments.us/reader036/viewer/2022062417/55161d66550346c6758b45a4/html5/thumbnails/39.jpg)
Colloque Paul Gauduchon Palaiseau, 20/05/05
8-DIMENSIONALQUATERNIONIC
GEOMETRY
8-DIMENSIONALQUATERNIONIC
GEOMETRY