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

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

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

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

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

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

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

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

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

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

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

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

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

Colloque Paul GauduchonPalaiseau, 20/05/05 Intrinsic torsion

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

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

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

Colloque Paul GauduchonPalaiseau, 20/05/05 DIRAC OPERATORS

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

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

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

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

Colloque Paul GauduchonPalaiseau, 20/05/05 Killing spinors

Colloque Paul GauduchonPalaiseau, 20/05/05 MODEL GEOMETRIES

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

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

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

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]

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

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

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

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

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

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

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

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

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

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]

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