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8-DIMENSIONAL QUATERNIONIC GEOMETRY
Simon Salamon
Politecnico di Torino
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Contents

• 4-forms and spinors

• Types of Q structures

• Dirac operators

• Model geometries

• Q symplectic manifolds

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4-FORMS AND SPINORS
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4-forms in dimension 8
Possible dimensions include
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A simple example
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A complex variant
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A complex variant
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The quaternionic 4-form
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Symmetric spaces
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Triality for Sp(2)Sp(1)
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Clifford multiplication
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TYPES OF QUATERNIONIC STRUCTURES
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Reduction of structure
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Intrinsic torsion
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Q symplectic manifolds
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Quaternionic manifolds
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Quaternionic manifolds
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DIRAC OPERATORS
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Rigidity principle
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The tautological section
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The tautological section
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Killing spinors
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Killing spinors
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MODEL GEOMETRIES
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Wolf spaces
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1. Projection
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Complex coadjoint orbits
Any nilpotent orbit N has both QK and HK metrics
The hunt for potentials Biquard-Gauduchon,
Swann
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2. The case SL(3,C)
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2. The case SL(3,C)
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QUATERNIONIC SYMPLECTIC MANIFOLDS
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Q contact structures
On hypersurfaces and asymptotic boundaries of QK
manifolds with non-degenerate Levi form
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Q contact structures
Without the integrability condition, extension to
a Q symplectic metric is nonetheless possible
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3. The case SO(5,C)
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3. The case SO(5,C)
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3. The case SO(5,C)
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T2 product examples
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T2 product examples
Compact nilmanifold examples have 3 transverse
simple closed 3-forms, with reduction
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8-DIMENSIONAL QUATERNIONIC GEOMETRY