8-DIMENSIONAL QUATERNIO NIC GEOMETRY

Colloque Paul Gauduchon Palaiseau, 20/05/05

8-DIMENSIONALQUATERNIONIC

GEOMETRY


GEOMETRY

Simon SalamonSimon Salamon

Politecnico di TorinoPolitecnico di Torino


ContentsContents

Dirac operators Dirac operators

Model geometries

Model geometries

4-forms and spinors 4-forms and spinors

Types of Q structures Types of Q structures

Q symplectic manifolds Q symplectic manifolds


4-FORMS AND SPINORS

4-FORMS AND SPINORS


4-forms in dimension 84-forms in dimension 8

Possible dimensions include Possible dimensions include


A simple exampleA simple example


A complex variantA complex variant


The quaternionic 4-formThe quaternionic 4-form


Set of OQS’s Set of OQS’s

Symmetric spacesSymmetric spaces

3-forms

8 = 3 + 5

3-forms

8 = 3 + 5


Triality for Sp(2)Sp(1)Triality for Sp(2)Sp(1)


Clifford multiplicationClifford multiplication

X determines X determines

8 = 3 + 5 8 = 3 + 5


TYPES OFQUATERNIONICSTRUCTURES

TYPES OFQUATERNIONICSTRUCTURES


Reduction of structureReduction 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


Intrinsic torsionIntrinsic torsion


“Q symplectic” manifolds“Q symplectic” manifolds


Quaternionic manifoldsQuaternionic manifolds

“Nijenhuis” = 0“Nijenhuis” = 0


M8 has an integrable “twistor space”M8 has an integrable “twistor space”

I,J,K can be chosen with I complexI,J,K can be chosen with I complex

Quaternionic manifoldsQuaternionic manifolds


DIRAC OPERATORS

DIRAC OPERATORS


Rigidity principleRigidity principle

G acts trivially onG acts trivially on

M Wolf spaceM Wolf space


The tautological sectionThe tautological section

An Sp(2)Sp(1) structure determinesAn Sp(2)Sp(1) structure determines

oror


Proposition [Witt] Proposition [Witt]

The tautological sectionThe tautological section


Killing spinorsKilling spinors

M QK, X an infinitesimal isometryM QK, X an infinitesimal isometry


Killing spinorsKilling spinors


MODEL GEOMETRIES

MODEL GEOMETRIES


M is QK ( )M is QK ( )

M is Einstein ( )M is Einstein ( )

M8 is symmetricM8 is symmetric

Quaternion-Kahler manifoldsQuaternion-Kahler manifolds


Wolf spacesWolf spaces

M8 QK symmetricM8 QK symmetric


1. Projection1. Projection

Links with HK and G2 holonomyLinks with HK and G2 holonomy


Complex coadjoint orbitsComplex coadjoint orbits

Any nilpotent orbit N has both QK and

HK metrics

Any nilpotent orbit N has both QK and

HK metrics

The hunt for potentials: [Biquard-Gauduchon,

Swann]

The hunt for potentials: [Biquard-Gauduchon,

Swann]


2. The case SL(3,C)2. The case SL(3,C)

8 = 3 + 5 8 = 3 + 5


2. The case SL(3,C)2. The case SL(3,C)

M8 parametrizes a subset of OQS’s

M8 parametrizes a subset of OQS’s


QUATERNIONIC SYMPLECTIC MANIFOLDS

QUATERNIONIC SYMPLECTIC MANIFOLDS


Q contact structuresQ contact structures

On hypersurfaces and asymptotic boundaries of QK manifolds with

non-degenerate “Levi form”

On hypersurfaces and asymptotic boundaries of QK manifolds with

non-degenerate “Levi form”


An extra integrability condition is needed for n=1 and allows one to

extend QCS’s on S7 [Duchemin]

An extra integrability condition is needed for n=1 and allows one to

extend QCS’s on S7 [Duchemin]

Without the integrability condition, extension to a Q symplectic metric

is nonetheless possible

Without the integrability condition, extension to a Q symplectic metric

is nonetheless possible

Q contact structuresQ contact structures


3. The case SO(5,C)3. The case SO(5,C)

Fibration based on the reduction Fibration based

on the reduction



Total space is both Kahler and

QK:

Total space is both Kahler and

QK:



X6 has a subspace of 3-forms X6 has a subspace of 3-forms


T2 product examplesT2 product examples

Ingredients:

symplectic with

closed primitive 3-forms

giving closed 4-form

Ingredients:

symplectic with

closed primitive 3-forms

giving closed 4-form


Compact nilmanifold examples have 3 transverse simple closed 3-forms, with reduction

Compact nilmanifold examples have 3 transverse simple closed 3-forms, with reduction

T2 product examplesT2 product examples

Applications to SL/CY geometry [Giovannini,

Matessi]

Applications to SL/CY geometry [Giovannini,

Matessi]



GEOMETRY


GEOMETRY

8-DIMENSIONAL QUATERNIO NIC GEOMETRY

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