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8-DIMENSIONAL QUATERNIO NIC GEOMETRY

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.

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8-DIMENSIONAL QUATERNIO NIC GEOMETRY

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  1. 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino

  2. Contents • 4-forms and spinors • Types of Q structures • Dirac operators • Model geometries • Q symplectic manifolds

  3. 4-FORMS AND SPINORS

  4. 4-forms in dimension 8 Possible dimensions include

  5. A simple example

  6. A complex variant

  7. A complex variant

  8. The quaternionic 4-form

  9. 3-forms 8 = 3 + 5 Set of OQS’s Symmetric spaces

  10. Triality for Sp(2)Sp(1)

  11. X determines 8 = 3 + 5 Clifford multiplication

  12. TYPES OF QUATERNIONIC STRUCTURES

  13. The 4-form determines the metric and Levi-Civita connection on the bundle with fibre Reduction of structure

  14. Intrinsic torsion

  15. “Q symplectic” manifolds

  16. “Nijenhuis” = 0 Quaternionic manifolds

  17. M8 has an integrable “twistor space” I,J,K can be chosen with I complex Quaternionic manifolds

  18. DIRAC OPERATORS

  19. G acts trivially on M Wolf space Rigidity principle

  20. An Sp(2)Sp(1) structure determines or The tautological section

  21. Proposition [Witt] The tautological section

  22. M QK, X an infinitesimal isometry Killing spinors

  23. Killing spinors

  24. MODEL GEOMETRIES

  25. Quaternion-Kahler manifolds M is QK ( ) M is Einstein ( ) M8 is symmetric

  26. M8 QK symmetric Wolf spaces

  27. Links with HK and G2 holonomy 1. Projection

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

  29. 8 = 3 + 5 2. The case SL(3,C)

  30. M8 parametrizes a subset of OQS’s 2. The case SL(3,C)

  31. QUATERNIONIC SYMPLECTIC MANIFOLDS

  32. Q contact structures On hypersurfaces and asymptotic boundaries of QK manifolds with non-degenerate “Levi form”

  33. An extra integrability condition is needed for n=1 and allows one to extend QCS’s on S7 [Duchemin] Q contact structures Without the integrability condition, extension to a Q symplectic metric is nonetheless possible

  34. Fibration based on the reduction 3. The case SO(5,C)

  35. Total space is both Kahler and QK: 3. The case SO(5,C)

  36. X6 has a subspace of 3-forms 3. The case SO(5,C)

  37. Ingredients: symplectic with closed primitive 3-forms giving closed 4-form T2product examples

  38. Applications to SL/CY geometry [Giovannini, Matessi] T2product examples Compact nilmanifold examples have 3 transverse simple closed 3-forms, with reduction

  39. 8-DIMENSIONAL QUATERNIONIC GEOMETRY

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