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Kühlungsborn 16/03/2006
SU(3) STRUCTURES AND SPINORS by Simon Salamon
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A sequel to the papers of
Abbena-Garbiero-Salamon Chiossi-Salamon
with material common to
Goldstein-Prokushin Cardoso-Curio-DellAgata-Lust
-... Fidanza-Minasian-Tomasiello and many others
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CONTENTS
Definitions and a general scheme for calculation
and classification
1. Spinors and complex structures
2. Intrinsic torsion from scratch
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CONTENTS
Applications to the Iwasawa manifold
3. Spinors and covariant derivatives
4. Classification of halfflat structures
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1. SPINORS AND COMPLEX STRUCTURES
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Projective spaces
Unitary and special unitary reductions in 6
dimensions are parametrized by
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Representations of SU(4)
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Representations of SU(4)
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1-forms as 2-spinors
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1-forms as 2-spinors
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Standard complex structure
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Klein correspondence
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Conclusion
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Conclusion
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Exterior powers
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Conclusion
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Example
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2. INTRINSIC TORSION FROM SCRATCH
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Second fundamental form
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Second fundamental form
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Torsion as a 2-tensor
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Gray-Hervella classes
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Gray-Hervella classes
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Halfflat or hypo structures
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The Dirac operator
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The Dirac operator
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The Dirac operator
Take a spinor of unit norm

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Lemma
The Dirac operator maps a defining spinor to
components W1 , bW4cW5 of the intrinsic torsion
for SU(3)
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3. SPINORS AND COVARIANT DERIVATIVES
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The Heisenberg Lie algebra
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The Iwasawa Lie algebra
is the underlying real Lie algebra obtained by
substituting

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The Iwasawa Lie algebra
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Parallelizations
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The Levi-Civita Connection
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Corollary
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4. CLASSIFICATION OF HALFFLAT STRUCTURES
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Invariant spinors
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The invariant Dirac operator
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The language of 3-forms
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The language of 3-forms
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Toric picture
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Related theorems
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Almost Kähler structures
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The halfflat condition
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Lemma
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Final theorem
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Halfflat structures
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Mary Poppins bag
(0,0,0,0,1342,1423)
The Iwasawa manifold has a.everything in it!
Symplectic non-Kähler structures Halfflat
SU(3) structures ...for
constructing metrics with holonomy G2
Disconnected families of complex
structures ...connected by gen complex
structures Obstructed Kodaira deformation
classes Strong KT structures
......