1. Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino

2. Colloque Paul GauduchonPalaiseau, 20/05/05 Contents Dirac operators Model geometries 4-forms and spinors Types of Q structures Q symplectic manifolds

3. Colloque Paul GauduchonPalaiseau, 20/05/05 4-FORMS AND SPINORS 4-FORMS AND SPINORS

4. Colloque Paul GauduchonPalaiseau, 20/05/05 4-forms in dimension 8 Possible dimensions include

5. Colloque Paul GauduchonPalaiseau, 20/05/05 A simple example

6. Colloque Paul GauduchonPalaiseau, 20/05/05 A complex variant

7. Colloque Paul GauduchonPalaiseau, 20/05/05 A complex variant

8. Colloque Paul GauduchonPalaiseau, 20/05/05 The quaternionic 4-form

9. Colloque Paul GauduchonPalaiseau, 20/05/05 Set of OQSs Symmetric spaces 3-forms 8 = 3 + 5 3-forms 8 = 3 + 5

10. Colloque Paul GauduchonPalaiseau, 20/05/05 Triality for Sp(2)Sp(1)

11. Colloque Paul GauduchonPalaiseau, 20/05/05 Clifford multiplication X determines 8 = 3 + 5

12. Colloque Paul GauduchonPalaiseau, 20/05/05 TYPES OF QUATERNIONIC STRUCTURES TYPES OF QUATERNIONIC STRUCTURES

13. Colloque Paul GauduchonPalaiseau, 20/05/05 Reduction 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

14. Colloque Paul GauduchonPalaiseau, 20/05/05 Intrinsic torsion

15. Colloque Paul GauduchonPalaiseau, 20/05/05 Q symplectic manifolds

16. Colloque Paul GauduchonPalaiseau, 20/05/05 Quaternionic manifolds Nijenhuis = 0

17. Colloque Paul GauduchonPalaiseau, 20/05/05 M 8 has an integrable twistor space I,J,K can be chosen with I complex Quaternionic manifolds

18. Colloque Paul GauduchonPalaiseau, 20/05/05 DIRAC OPERATORS

19. Colloque Paul GauduchonPalaiseau, 20/05/05 Rigidity principle G acts trivially on M Wolf space

20. Colloque Paul GauduchonPalaiseau, 20/05/05 The tautological section An Sp(2)Sp(1) structure determines or

21. Colloque Paul GauduchonPalaiseau, 20/05/05 Proposition [Witt] The tautological section

22. Colloque Paul GauduchonPalaiseau, 20/05/05 Killing spinors M QK, X an infinitesimal isometry

23. Colloque Paul GauduchonPalaiseau, 20/05/05 Killing spinors

24. Colloque Paul GauduchonPalaiseau, 20/05/05 MODEL GEOMETRIES

25. Colloque Paul GauduchonPalaiseau, 20/05/05 M is QK ( ) M is Einstein ( ) M 8 is symmetric Quaternion-Kahler manifolds

26. Colloque Paul GauduchonPalaiseau, 20/05/05 Wolf spaces M 8 QK symmetric

27. Colloque Paul GauduchonPalaiseau, 20/05/05 1. Projection Links with HK and G 2 holonomy

28. Colloque Paul GauduchonPalaiseau, 20/05/05 Complex coadjoint orbits Any nilpotent orbit N has both QK and HK metrics The hunt for potentials: [Biquard-Gauduchon, Swann]

29. Colloque Paul GauduchonPalaiseau, 20/05/05 2. The case SL(3,C) 8 = 3 + 5

30. Colloque Paul GauduchonPalaiseau, 20/05/05 2. The case SL(3,C) M 8 parametrizes a subset of OQSs

31. Colloque Paul GauduchonPalaiseau, 20/05/05 QUATERNIONIC SYMPLECTIC MANIFOLDS

32. Colloque Paul GauduchonPalaiseau, 20/05/05 Q contact structures On hypersurfaces and asymptotic boundaries of QK manifolds with non-degenerate Levi form

33. Colloque Paul GauduchonPalaiseau, 20/05/05 An extra integrability condition is needed for n=1 and allows one to extend QCSs on S 7 [Duchemin] Without the integrability condition, extension to a Q symplectic metric is nonetheless possible Q contact structures

34. Colloque Paul GauduchonPalaiseau, 20/05/05 3. The case SO(5,C) Fibration based on the reduction

35. Colloque Paul GauduchonPalaiseau, 20/05/05 3. The case SO(5,C) Total space is both Kahler and QK:

36. Colloque Paul GauduchonPalaiseau, 20/05/05 3. The case SO(5,C) X 6 has a subspace of 3-forms

37. Colloque Paul GauduchonPalaiseau, 20/05/05 T 2 product examples Ingredients: symplectic with closed primitive 3-forms giving closed 4-form Ingredients: symplectic with closed primitive 3-forms giving closed 4-form

38. Colloque Paul GauduchonPalaiseau, 20/05/05 Compact nilmanifold examples have 3 transverse simple closed 3-forms, with reduction T 2 product examples Applications to SL/CY geometry [Giovannini, Matessi]

39. Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY