Half-Flat SU(3)-Structures on Lie Groups Project A6: Mathematical Aspects of String Compactifications Fabian Schulte-Hengesbach Department Mathematik, Universit¨ at Hamburg SFB-Meeting, Hamburg-Bergedorf, March 4, 2009
Half-Flat SU(3)-Structures on Lie GroupsProject A6: Mathematical Aspects of String Compactifications
Fabian Schulte-Hengesbach
Department Mathematik, Universitat Hamburg
SFB-Meeting, Hamburg-Bergedorf, March 4, 2009
Project A6: Mathematical Aspects of StringCompactifications
The aim of the project is the study of generalised supersymmetric stringcompactifications. In particular the investigation of manifolds withSU(3)- and SU(3) x SU(3)-structure as possible string backgrounds isproposed. The consistent embedding of D-branes, orientifold-planes andbackground fluxes is also planned.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
String compactifications
(D=10) target space in string theory or supergravity:
N10 = M1,3︸︷︷︸Lorentz/Minkowski space time
× Y 6︸︷︷︸(Compact) Internal Space
Amount of preserved supersymmetries after compactification isrelated to the geometry of Y 6
Standard candidate for Y 6: Compact Calabi-Yau 3-folds
In the presence of background fluxes, also non-integrableSU(3)-structures are considered, e. g. half-flat SU(3)-structures
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
String compactifications
(D=10) target space in string theory or supergravity:
N10 = M1,3︸︷︷︸Lorentz/Minkowski space time
× Y 6︸︷︷︸(Compact) Internal Space
Amount of preserved supersymmetries after compactification isrelated to the geometry of Y 6
Standard candidate for Y 6: Compact Calabi-Yau 3-folds
In the presence of background fluxes, also non-integrableSU(3)-structures are considered, e. g. half-flat SU(3)-structures
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
String compactifications
(D=10) target space in string theory or supergravity:
N10 = M1,3︸︷︷︸Lorentz/Minkowski space time
× Y 6︸︷︷︸(Compact) Internal Space
Amount of preserved supersymmetries after compactification isrelated to the geometry of Y 6
Standard candidate for Y 6: Compact Calabi-Yau 3-folds
In the presence of background fluxes, also non-integrableSU(3)-structures are considered, e. g. half-flat SU(3)-structures
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
String compactifications
(D=10) target space in string theory or supergravity:
N10 = M1,3︸︷︷︸Lorentz/Minkowski space time
× Y 6︸︷︷︸(Compact) Internal Space
Amount of preserved supersymmetries after compactification isrelated to the geometry of Y 6
Standard candidate for Y 6: Compact Calabi-Yau 3-folds
In the presence of background fluxes, also non-integrableSU(3)-structures are considered, e. g. half-flat SU(3)-structures
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Motivation for studying half-flat SU(3)-structures
Half-flat SU(3)-structures appear naturally as mirror duals ofCalabi-Yau 3-folds with certain fluxes, [2003, Gurrieri, Louis, Micu,Waldram, hep-th/0211102]
Fluxes are encoded directly in the geometry of the half-flatSU(3)-structure
Hitchin (2001): The solution of the Hitchin flow on (compact)half-flat SU(3)-structures Y 6 allows the construction of parallelG2-structures on Y 6 × Interval
Physical interpretation: Natural relation to internal space of11-dimensional M-theory
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Motivation for studying half-flat SU(3)-structures
Half-flat SU(3)-structures appear naturally as mirror duals ofCalabi-Yau 3-folds with certain fluxes, [2003, Gurrieri, Louis, Micu,Waldram, hep-th/0211102]
Fluxes are encoded directly in the geometry of the half-flatSU(3)-structure
Hitchin (2001): The solution of the Hitchin flow on (compact)half-flat SU(3)-structures Y 6 allows the construction of parallelG2-structures on Y 6 × Interval
Physical interpretation: Natural relation to internal space of11-dimensional M-theory
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Motivation for studying half-flat SU(3)-structures
Half-flat SU(3)-structures appear naturally as mirror duals ofCalabi-Yau 3-folds with certain fluxes, [2003, Gurrieri, Louis, Micu,Waldram, hep-th/0211102]
Fluxes are encoded directly in the geometry of the half-flatSU(3)-structure
Hitchin (2001): The solution of the Hitchin flow on (compact)half-flat SU(3)-structures Y 6 allows the construction of parallelG2-structures on Y 6 × Interval
Physical interpretation: Natural relation to internal space of11-dimensional M-theory
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Motivation for studying half-flat SU(3)-structures
Half-flat SU(3)-structures appear naturally as mirror duals ofCalabi-Yau 3-folds with certain fluxes, [2003, Gurrieri, Louis, Micu,Waldram, hep-th/0211102]
Fluxes are encoded directly in the geometry of the half-flatSU(3)-structure
Hitchin (2001): The solution of the Hitchin flow on (compact)half-flat SU(3)-structures Y 6 allows the construction of parallelG2-structures on Y 6 × Interval
Physical interpretation: Natural relation to internal space of11-dimensional M-theory
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
SU(3)-structures
Definition
An SU(3)-structure on a six-dimensional manifold M6 consists of
a Riemannian metric g ,
an almost complex structure J, i.e. an endomorphism on everytangent space satisfying J2 = −id ,
which is compatible with g , that is ω := g(J., .) = −g(., J.) is a2-form (called Kahler form or fundamental 2-form)
and a complex volume form, i.e. a complex-valued (3,0)-formΨ = ψ+ + iψ− of constant length.
The tensors (g , J, ω, ψ+ + iψ−) can be reconstructed from a pair(ω, ψ+) ∈ Λ2M × Λ3M of ”stable” forms satisfying ω ∧ ψ+ = 0 anda normalisation condition
The tensors (g , J, ω, ψ+ + iψ−) can also be reconstructed from asingle (globally defined) spinor field η
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
SU(3)-structures
Definition
An SU(3)-structure on a six-dimensional manifold M6 consists of
a Riemannian metric g ,
an almost complex structure J, i.e. an endomorphism on everytangent space satisfying J2 = −id ,
which is compatible with g , that is ω := g(J., .) = −g(., J.) is a2-form (called Kahler form or fundamental 2-form)
and a complex volume form, i.e. a complex-valued (3,0)-formΨ = ψ+ + iψ− of constant length.
The tensors (g , J, ω, ψ+ + iψ−) can be reconstructed from a pair(ω, ψ+) ∈ Λ2M × Λ3M of ”stable” forms satisfying ω ∧ ψ+ = 0 anda normalisation condition
The tensors (g , J, ω, ψ+ + iψ−) can also be reconstructed from asingle (globally defined) spinor field η
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
SU(3)-structures
Definition
An SU(3)-structure on a six-dimensional manifold M6 consists of
a Riemannian metric g ,
an almost complex structure J, i.e. an endomorphism on everytangent space satisfying J2 = −id ,
which is compatible with g , that is ω := g(J., .) = −g(., J.) is a2-form (called Kahler form or fundamental 2-form)
and a complex volume form, i.e. a complex-valued (3,0)-formΨ = ψ+ + iψ− of constant length.
The tensors (g , J, ω, ψ+ + iψ−) can be reconstructed from a pair(ω, ψ+) ∈ Λ2M × Λ3M of ”stable” forms satisfying ω ∧ ψ+ = 0 anda normalisation condition
The tensors (g , J, ω, ψ+ + iψ−) can also be reconstructed from asingle (globally defined) spinor field η
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
SU(3)-structures
Definition
An SU(3)-structure on a six-dimensional manifold M6 consists of
a Riemannian metric g ,
an almost complex structure J, i.e. an endomorphism on everytangent space satisfying J2 = −id ,
which is compatible with g , that is ω := g(J., .) = −g(., J.) is a2-form (called Kahler form or fundamental 2-form)
and a complex volume form, i.e. a complex-valued (3,0)-formΨ = ψ+ + iψ− of constant length.
The tensors (g , J, ω, ψ+ + iψ−) can be reconstructed from a pair(ω, ψ+) ∈ Λ2M × Λ3M of ”stable” forms satisfying ω ∧ ψ+ = 0 anda normalisation condition
The tensors (g , J, ω, ψ+ + iψ−) can also be reconstructed from asingle (globally defined) spinor field η
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
SU(3)-structures
Definition
An SU(3)-structure on a six-dimensional manifold M6 consists of
a Riemannian metric g ,
an almost complex structure J, i.e. an endomorphism on everytangent space satisfying J2 = −id ,
which is compatible with g , that is ω := g(J., .) = −g(., J.) is a2-form (called Kahler form or fundamental 2-form)
and a complex volume form, i.e. a complex-valued (3,0)-formΨ = ψ+ + iψ− of constant length.
The tensors (g , J, ω, ψ+ + iψ−) can be reconstructed from a pair(ω, ψ+) ∈ Λ2M × Λ3M of ”stable” forms satisfying ω ∧ ψ+ = 0 anda normalisation condition
The tensors (g , J, ω, ψ+ + iψ−) can also be reconstructed from asingle (globally defined) spinor field η
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
SU(3)-structures
Definition
An SU(3)-structure on a six-dimensional manifold M6 consists of
a Riemannian metric g ,
an almost complex structure J, i.e. an endomorphism on everytangent space satisfying J2 = −id ,
which is compatible with g , that is ω := g(J., .) = −g(., J.) is a2-form (called Kahler form or fundamental 2-form)
and a complex volume form, i.e. a complex-valued (3,0)-formΨ = ψ+ + iψ− of constant length.
The tensors (g , J, ω, ψ+ + iψ−) can be reconstructed from a pair(ω, ψ+) ∈ Λ2M × Λ3M of ”stable” forms satisfying ω ∧ ψ+ = 0 anda normalisation condition
The tensors (g , J, ω, ψ+ + iψ−) can also be reconstructed from asingle (globally defined) spinor field η
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
SU(3)-structures
Definition
An SU(3)-structure on a six-dimensional manifold M6 consists of
a Riemannian metric g ,
an almost complex structure J, i.e. an endomorphism on everytangent space satisfying J2 = −id ,
which is compatible with g , that is ω := g(J., .) = −g(., J.) is a2-form (called Kahler form or fundamental 2-form)
and a complex volume form, i.e. a complex-valued (3,0)-formΨ = ψ+ + iψ− of constant length.
The tensors (g , J, ω, ψ+ + iψ−) can be reconstructed from a pair(ω, ψ+) ∈ Λ2M × Λ3M of ”stable” forms satisfying ω ∧ ψ+ = 0 anda normalisation condition
The tensors (g , J, ω, ψ+ + iψ−) can also be reconstructed from asingle (globally defined) spinor field η
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
SU(3)-structures II
Basis adapted to SU(3)-structure (g , J, ω,Ψ)
(e1, ...e6) orthonormal with respect to the metric g
Je1 = e2, Je2 = −e1, Je3 = e4, Je4 = −e3, Je5 = e6, Je6 = −e5,
ω = e1 ∧ e2 + e3 ∧ e4 + e5 ∧ e6,
Ψ = ψ+ + iψ− = (e1 + ie2) ∧ (e3 + ie4) ∧ (e5 + ie6).
Stab(g) = O(6) (A Riemannian metric is an ”O(6)-structure”)
Stab(ω) = Sp(6,R)
Stab(J) = GL(3,C)
Stab(Ψ) = SL(3,C)
Stab(g , J, ω,Ψ) = Intersection of all groups = SU(3)
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
SU(3)-structures II
Basis adapted to SU(3)-structure (g , J, ω,Ψ)
(e1, ...e6) orthonormal with respect to the metric g
Je1 = e2, Je2 = −e1, Je3 = e4, Je4 = −e3, Je5 = e6, Je6 = −e5,
ω = e1 ∧ e2 + e3 ∧ e4 + e5 ∧ e6,
Ψ = ψ+ + iψ− = (e1 + ie2) ∧ (e3 + ie4) ∧ (e5 + ie6).
Stab(g) = O(6) (A Riemannian metric is an ”O(6)-structure”)
Stab(ω) = Sp(6,R)
Stab(J) = GL(3,C)
Stab(Ψ) = SL(3,C)
Stab(g , J, ω,Ψ) = Intersection of all groups = SU(3)
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
SU(3)-structures II
Basis adapted to SU(3)-structure (g , J, ω,Ψ)
(e1, ...e6) orthonormal with respect to the metric g
Je1 = e2, Je2 = −e1, Je3 = e4, Je4 = −e3, Je5 = e6, Je6 = −e5,
ω = e1 ∧ e2 + e3 ∧ e4 + e5 ∧ e6,
Ψ = ψ+ + iψ− = (e1 + ie2) ∧ (e3 + ie4) ∧ (e5 + ie6).
Stab(g) = O(6) (A Riemannian metric is an ”O(6)-structure”)
Stab(ω) = Sp(6,R)
Stab(J) = GL(3,C)
Stab(Ψ) = SL(3,C)
Stab(g , J, ω,Ψ) = Intersection of all groups = SU(3)
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
SU(3)-structures II
Basis adapted to SU(3)-structure (g , J, ω,Ψ)
(e1, ...e6) orthonormal with respect to the metric g
Je1 = e2, Je2 = −e1, Je3 = e4, Je4 = −e3, Je5 = e6, Je6 = −e5,
ω = e1 ∧ e2 + e3 ∧ e4 + e5 ∧ e6,
Ψ = ψ+ + iψ− = (e1 + ie2) ∧ (e3 + ie4) ∧ (e5 + ie6).
Stab(g) = O(6) (A Riemannian metric is an ”O(6)-structure”)
Stab(ω) = Sp(6,R)
Stab(J) = GL(3,C)
Stab(Ψ) = SL(3,C)
Stab(g , J, ω,Ψ) = Intersection of all groups = SU(3)
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
SU(3)-structures II
Basis adapted to SU(3)-structure (g , J, ω,Ψ)
(e1, ...e6) orthonormal with respect to the metric g
Je1 = e2, Je2 = −e1, Je3 = e4, Je4 = −e3, Je5 = e6, Je6 = −e5,
ω = e1 ∧ e2 + e3 ∧ e4 + e5 ∧ e6,
Ψ = ψ+ + iψ− = (e1 + ie2) ∧ (e3 + ie4) ∧ (e5 + ie6).
Stab(g) = O(6) (A Riemannian metric is an ”O(6)-structure”)
Stab(ω) = Sp(6,R)
Stab(J) = GL(3,C)
Stab(Ψ) = SL(3,C)
Stab(g , J, ω,Ψ) = Intersection of all groups = SU(3)
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
SU(3)-structures II
Basis adapted to SU(3)-structure (g , J, ω,Ψ)
(e1, ...e6) orthonormal with respect to the metric g
Je1 = e2, Je2 = −e1, Je3 = e4, Je4 = −e3, Je5 = e6, Je6 = −e5,
ω = e1 ∧ e2 + e3 ∧ e4 + e5 ∧ e6,
Ψ = ψ+ + iψ− = (e1 + ie2) ∧ (e3 + ie4) ∧ (e5 + ie6).
Stab(g) = O(6) (A Riemannian metric is an ”O(6)-structure”)
Stab(ω) = Sp(6,R)
Stab(J) = GL(3,C)
Stab(Ψ) = SL(3,C)
Stab(g , J, ω,Ψ) = Intersection of all groups = SU(3)
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Calabi-Yau and Nearly Kahler in dimension 6
The most natural (”geometrical”, ”integrable”) SU(3)-structure is aCalabi-Yau structure (∇LCJ = 0 or equivalently, J is a complexstructure and ω a symplectic structure). Properties:
Usually required to be compact.
Always algebraic.
Metric is Ricci-flat, but cannot be written explicitly.
In real dimension 6 (complex 3-folds), the number of compactmanifolds known to admit a Calabi-Yau structure is huge, but finite.
Space of deformations has been studied in detail by methods ofcomplex and algebraic geometry.
The most interesting non-integrable SU(3)-structure is a Nearly Kahlerstructure ((∇LC
X J)X = 0 for all vector fields X ). Properties:
Automatically compact.
Metric is Einstein ( Ric = λg ).
Admit Killing spinor.
Admit canonical connection with skew-symmetric torsion.
In dimension 6, only 4 examples are known.
Classified in dimension different from 6.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Calabi-Yau and Nearly Kahler in dimension 6
The most natural (”geometrical”, ”integrable”) SU(3)-structure is aCalabi-Yau structure (∇LCJ = 0 or equivalently, J is a complexstructure and ω a symplectic structure). Properties:
Usually required to be compact.
Always algebraic.
Metric is Ricci-flat, but cannot be written explicitly.
In real dimension 6 (complex 3-folds), the number of compactmanifolds known to admit a Calabi-Yau structure is huge, but finite.
Space of deformations has been studied in detail by methods ofcomplex and algebraic geometry.
The most interesting non-integrable SU(3)-structure is a Nearly Kahlerstructure ((∇LC
X J)X = 0 for all vector fields X ). Properties:
Automatically compact.
Metric is Einstein ( Ric = λg ).
Admit Killing spinor.
Admit canonical connection with skew-symmetric torsion.
In dimension 6, only 4 examples are known.
Classified in dimension different from 6.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Calabi-Yau and Nearly Kahler in dimension 6
The most natural (”geometrical”, ”integrable”) SU(3)-structure is aCalabi-Yau structure (∇LCJ = 0 or equivalently, J is a complexstructure and ω a symplectic structure). Properties:
Usually required to be compact.
Always algebraic.
Metric is Ricci-flat, but cannot be written explicitly.
In real dimension 6 (complex 3-folds), the number of compactmanifolds known to admit a Calabi-Yau structure is huge, but finite.
Space of deformations has been studied in detail by methods ofcomplex and algebraic geometry.
The most interesting non-integrable SU(3)-structure is a Nearly Kahlerstructure ((∇LC
X J)X = 0 for all vector fields X ). Properties:
Automatically compact.
Metric is Einstein ( Ric = λg ).
Admit Killing spinor.
Admit canonical connection with skew-symmetric torsion.
In dimension 6, only 4 examples are known.
Classified in dimension different from 6.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Calabi-Yau and Nearly Kahler in dimension 6
The most natural (”geometrical”, ”integrable”) SU(3)-structure is aCalabi-Yau structure (∇LCJ = 0 or equivalently, J is a complexstructure and ω a symplectic structure). Properties:
Usually required to be compact.
Always algebraic.
Metric is Ricci-flat, but cannot be written explicitly.
In real dimension 6 (complex 3-folds), the number of compactmanifolds known to admit a Calabi-Yau structure is huge, but finite.
Space of deformations has been studied in detail by methods ofcomplex and algebraic geometry.
The most interesting non-integrable SU(3)-structure is a Nearly Kahlerstructure ((∇LC
X J)X = 0 for all vector fields X ). Properties:
Automatically compact.
Metric is Einstein ( Ric = λg ).
Admit Killing spinor.
Admit canonical connection with skew-symmetric torsion.
In dimension 6, only 4 examples are known.
Classified in dimension different from 6.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Calabi-Yau and Nearly Kahler in dimension 6
The most natural (”geometrical”, ”integrable”) SU(3)-structure is aCalabi-Yau structure (∇LCJ = 0 or equivalently, J is a complexstructure and ω a symplectic structure). Properties:
Usually required to be compact.
Always algebraic.
Metric is Ricci-flat, but cannot be written explicitly.
In real dimension 6 (complex 3-folds), the number of compactmanifolds known to admit a Calabi-Yau structure is huge, but finite.
Space of deformations has been studied in detail by methods ofcomplex and algebraic geometry.
The most interesting non-integrable SU(3)-structure is a Nearly Kahlerstructure ((∇LC
X J)X = 0 for all vector fields X ). Properties:
Automatically compact.
Metric is Einstein ( Ric = λg ).
Admit Killing spinor.
Admit canonical connection with skew-symmetric torsion.
In dimension 6, only 4 examples are known.
Classified in dimension different from 6.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Calabi-Yau and Nearly Kahler in dimension 6
The most natural (”geometrical”, ”integrable”) SU(3)-structure is aCalabi-Yau structure (∇LCJ = 0 or equivalently, J is a complexstructure and ω a symplectic structure). Properties:
Usually required to be compact.
Always algebraic.
Metric is Ricci-flat, but cannot be written explicitly.
In real dimension 6 (complex 3-folds), the number of compactmanifolds known to admit a Calabi-Yau structure is huge, but finite.
Space of deformations has been studied in detail by methods ofcomplex and algebraic geometry.
The most interesting non-integrable SU(3)-structure is a Nearly Kahlerstructure ((∇LC
X J)X = 0 for all vector fields X ). Properties:
Automatically compact.
Metric is Einstein ( Ric = λg ).
Admit Killing spinor.
Admit canonical connection with skew-symmetric torsion.
In dimension 6, only 4 examples are known.
Classified in dimension different from 6.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Calabi-Yau and Nearly Kahler in dimension 6
The most natural (”geometrical”, ”integrable”) SU(3)-structure is aCalabi-Yau structure (∇LCJ = 0 or equivalently, J is a complexstructure and ω a symplectic structure). Properties:
Usually required to be compact.
Always algebraic.
Metric is Ricci-flat, but cannot be written explicitly.
In real dimension 6 (complex 3-folds), the number of compactmanifolds known to admit a Calabi-Yau structure is huge, but finite.
Space of deformations has been studied in detail by methods ofcomplex and algebraic geometry.
The most interesting non-integrable SU(3)-structure is a Nearly Kahlerstructure ((∇LC
X J)X = 0 for all vector fields X ). Properties:
Automatically compact.
Metric is Einstein ( Ric = λg ).
Admit Killing spinor.
Admit canonical connection with skew-symmetric torsion.
In dimension 6, only 4 examples are known.
Classified in dimension different from 6.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Calabi-Yau and Nearly Kahler in dimension 6
The most natural (”geometrical”, ”integrable”) SU(3)-structure is aCalabi-Yau structure (∇LCJ = 0 or equivalently, J is a complexstructure and ω a symplectic structure). Properties:
Usually required to be compact.
Always algebraic.
Metric is Ricci-flat, but cannot be written explicitly.
In real dimension 6 (complex 3-folds), the number of compactmanifolds known to admit a Calabi-Yau structure is huge, but finite.
Space of deformations has been studied in detail by methods ofcomplex and algebraic geometry.
The most interesting non-integrable SU(3)-structure is a Nearly Kahlerstructure ((∇LC
X J)X = 0 for all vector fields X ). Properties:
Automatically compact.
Metric is Einstein ( Ric = λg ).
Admit Killing spinor.
Admit canonical connection with skew-symmetric torsion.
In dimension 6, only 4 examples are known.
Classified in dimension different from 6.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Calabi-Yau and Nearly Kahler in dimension 6
The most natural (”geometrical”, ”integrable”) SU(3)-structure is aCalabi-Yau structure (∇LCJ = 0 or equivalently, J is a complexstructure and ω a symplectic structure). Properties:
Usually required to be compact.
Always algebraic.
Metric is Ricci-flat, but cannot be written explicitly.
In real dimension 6 (complex 3-folds), the number of compactmanifolds known to admit a Calabi-Yau structure is huge, but finite.
Space of deformations has been studied in detail by methods ofcomplex and algebraic geometry.
The most interesting non-integrable SU(3)-structure is a Nearly Kahlerstructure ((∇LC
X J)X = 0 for all vector fields X ). Properties:
Automatically compact.
Metric is Einstein ( Ric = λg ).
Admit Killing spinor.
Admit canonical connection with skew-symmetric torsion.
In dimension 6, only 4 examples are known.
Classified in dimension different from 6.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Calabi-Yau and Nearly Kahler in dimension 6
The most natural (”geometrical”, ”integrable”) SU(3)-structure is aCalabi-Yau structure (∇LCJ = 0 or equivalently, J is a complexstructure and ω a symplectic structure). Properties:
Usually required to be compact.
Always algebraic.
Metric is Ricci-flat, but cannot be written explicitly.
In real dimension 6 (complex 3-folds), the number of compactmanifolds known to admit a Calabi-Yau structure is huge, but finite.
Space of deformations has been studied in detail by methods ofcomplex and algebraic geometry.
The most interesting non-integrable SU(3)-structure is a Nearly Kahlerstructure ((∇LC
X J)X = 0 for all vector fields X ). Properties:
Automatically compact.
Metric is Einstein ( Ric = λg ).
Admit Killing spinor.
Admit canonical connection with skew-symmetric torsion.
In dimension 6, only 4 examples are known.
Classified in dimension different from 6.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Calabi-Yau and Nearly Kahler in dimension 6
The most natural (”geometrical”, ”integrable”) SU(3)-structure is aCalabi-Yau structure (∇LCJ = 0 or equivalently, J is a complexstructure and ω a symplectic structure). Properties:
Usually required to be compact.
Always algebraic.
Metric is Ricci-flat, but cannot be written explicitly.
In real dimension 6 (complex 3-folds), the number of compactmanifolds known to admit a Calabi-Yau structure is huge, but finite.
Space of deformations has been studied in detail by methods ofcomplex and algebraic geometry.
The most interesting non-integrable SU(3)-structure is a Nearly Kahlerstructure ((∇LC
X J)X = 0 for all vector fields X ). Properties:
Automatically compact.
Metric is Einstein ( Ric = λg ).
Admit Killing spinor.
Admit canonical connection with skew-symmetric torsion.
In dimension 6, only 4 examples are known.
Classified in dimension different from 6.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Calabi-Yau and Nearly Kahler in dimension 6
The most natural (”geometrical”, ”integrable”) SU(3)-structure is aCalabi-Yau structure (∇LCJ = 0 or equivalently, J is a complexstructure and ω a symplectic structure). Properties:
Usually required to be compact.
Always algebraic.
Metric is Ricci-flat, but cannot be written explicitly.
In real dimension 6 (complex 3-folds), the number of compactmanifolds known to admit a Calabi-Yau structure is huge, but finite.
Space of deformations has been studied in detail by methods ofcomplex and algebraic geometry.
The most interesting non-integrable SU(3)-structure is a Nearly Kahlerstructure ((∇LC
X J)X = 0 for all vector fields X ). Properties:
Automatically compact.
Metric is Einstein ( Ric = λg ).
Admit Killing spinor.
Admit canonical connection with skew-symmetric torsion.
In dimension 6, only 4 examples are known.
Classified in dimension different from 6.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Calabi-Yau and Nearly Kahler in dimension 6
The most natural (”geometrical”, ”integrable”) SU(3)-structure is aCalabi-Yau structure (∇LCJ = 0 or equivalently, J is a complexstructure and ω a symplectic structure). Properties:
Usually required to be compact.
Always algebraic.
Metric is Ricci-flat, but cannot be written explicitly.
In real dimension 6 (complex 3-folds), the number of compactmanifolds known to admit a Calabi-Yau structure is huge, but finite.
Space of deformations has been studied in detail by methods ofcomplex and algebraic geometry.
The most interesting non-integrable SU(3)-structure is a Nearly Kahlerstructure ((∇LC
X J)X = 0 for all vector fields X ). Properties:
Automatically compact.
Metric is Einstein ( Ric = λg ).
Admit Killing spinor.
Admit canonical connection with skew-symmetric torsion.
In dimension 6, only 4 examples are known.
Classified in dimension different from 6.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Calabi-Yau and Nearly Kahler in dimension 6
The most natural (”geometrical”, ”integrable”) SU(3)-structure is aCalabi-Yau structure (∇LCJ = 0 or equivalently, J is a complexstructure and ω a symplectic structure). Properties:
Usually required to be compact.
Always algebraic.
Metric is Ricci-flat, but cannot be written explicitly.
In real dimension 6 (complex 3-folds), the number of compactmanifolds known to admit a Calabi-Yau structure is huge, but finite.
Space of deformations has been studied in detail by methods ofcomplex and algebraic geometry.
The most interesting non-integrable SU(3)-structure is a Nearly Kahlerstructure ((∇LC
X J)X = 0 for all vector fields X ). Properties:
Automatically compact.
Metric is Einstein ( Ric = λg ).
Admit Killing spinor.
Admit canonical connection with skew-symmetric torsion.
In dimension 6, only 4 examples are known.
Classified in dimension different from 6.Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Half-flat SU(3)-structures
SU(3)-structures can be classified (Chiossi and Salamon) by the exteriorderivatives of ω, ψ+ and ψ− (Method: Decompose intrinsic torsion intoirreducible components under SU(3)).
Definition
An SU(3)-structure (g , J, ω, ψ+ + iψ−) is
Half-flat if d(ω ∧ ω) = 0, dψ+ = 0.
In particular, Calabi-Yau and Nearly Kahler structures are half-flat since
Calabi-Yau is equivalent to dω = 0, dψ+ = 0, dψ− = 0,
Nearly Kahler is equivalent to dω = 3ψ+, dψ− = −2ω ∧ ω.
Known properties of half-flat SU(3)-structures:
Ricci tensor looks awful
Some invariant examples on nilpotent Lie groups have been studiedand classified under additional assumptions
Can be evolved via Hitchin Flow
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Half-flat SU(3)-structures
SU(3)-structures can be classified (Chiossi and Salamon) by the exteriorderivatives of ω, ψ+ and ψ− (Method: Decompose intrinsic torsion intoirreducible components under SU(3)).
Definition
An SU(3)-structure (g , J, ω, ψ+ + iψ−) is
Half-flat if d(ω ∧ ω) = 0, dψ+ = 0.
In particular, Calabi-Yau and Nearly Kahler structures are half-flat since
Calabi-Yau is equivalent to dω = 0, dψ+ = 0, dψ− = 0,
Nearly Kahler is equivalent to dω = 3ψ+, dψ− = −2ω ∧ ω.
Known properties of half-flat SU(3)-structures:
Ricci tensor looks awful
Some invariant examples on nilpotent Lie groups have been studiedand classified under additional assumptions
Can be evolved via Hitchin Flow
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Half-flat SU(3)-structures
SU(3)-structures can be classified (Chiossi and Salamon) by the exteriorderivatives of ω, ψ+ and ψ− (Method: Decompose intrinsic torsion intoirreducible components under SU(3)).
Definition
An SU(3)-structure (g , J, ω, ψ+ + iψ−) is
Half-flat if d(ω ∧ ω) = 0, dψ+ = 0.
In particular, Calabi-Yau and Nearly Kahler structures are half-flat since
Calabi-Yau is equivalent to dω = 0, dψ+ = 0, dψ− = 0,
Nearly Kahler is equivalent to dω = 3ψ+, dψ− = −2ω ∧ ω.
Known properties of half-flat SU(3)-structures:
Ricci tensor looks awful
Some invariant examples on nilpotent Lie groups have been studiedand classified under additional assumptions
Can be evolved via Hitchin Flow
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Half-flat SU(3)-structures
SU(3)-structures can be classified (Chiossi and Salamon) by the exteriorderivatives of ω, ψ+ and ψ− (Method: Decompose intrinsic torsion intoirreducible components under SU(3)).
Definition
An SU(3)-structure (g , J, ω, ψ+ + iψ−) is
Half-flat if d(ω ∧ ω) = 0, dψ+ = 0.
In particular, Calabi-Yau and Nearly Kahler structures are half-flat since
Calabi-Yau is equivalent to dω = 0, dψ+ = 0, dψ− = 0,
Nearly Kahler is equivalent to dω = 3ψ+, dψ− = −2ω ∧ ω.
Known properties of half-flat SU(3)-structures:
Ricci tensor looks awful
Some invariant examples on nilpotent Lie groups have been studiedand classified under additional assumptions
Can be evolved via Hitchin Flow
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Half-flat SU(3)-structures
SU(3)-structures can be classified (Chiossi and Salamon) by the exteriorderivatives of ω, ψ+ and ψ− (Method: Decompose intrinsic torsion intoirreducible components under SU(3)).
Definition
An SU(3)-structure (g , J, ω, ψ+ + iψ−) is
Half-flat if d(ω ∧ ω) = 0, dψ+ = 0.
In particular, Calabi-Yau and Nearly Kahler structures are half-flat since
Calabi-Yau is equivalent to dω = 0, dψ+ = 0, dψ− = 0,
Nearly Kahler is equivalent to dω = 3ψ+, dψ− = −2ω ∧ ω.
Known properties of half-flat SU(3)-structures:
Ricci tensor looks awful
Some invariant examples on nilpotent Lie groups have been studiedand classified under additional assumptions
Can be evolved via Hitchin Flow
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Half-flat SU(3)-structures
SU(3)-structures can be classified (Chiossi and Salamon) by the exteriorderivatives of ω, ψ+ and ψ− (Method: Decompose intrinsic torsion intoirreducible components under SU(3)).
Definition
An SU(3)-structure (g , J, ω, ψ+ + iψ−) is
Half-flat if d(ω ∧ ω) = 0, dψ+ = 0.
In particular, Calabi-Yau and Nearly Kahler structures are half-flat since
Calabi-Yau is equivalent to dω = 0, dψ+ = 0, dψ− = 0,
Nearly Kahler is equivalent to dω = 3ψ+, dψ− = −2ω ∧ ω.
Known properties of half-flat SU(3)-structures:
Ricci tensor looks awful
Some invariant examples on nilpotent Lie groups have been studiedand classified under additional assumptions
Can be evolved via Hitchin Flow
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Half-flat SU(3)-structures
SU(3)-structures can be classified (Chiossi and Salamon) by the exteriorderivatives of ω, ψ+ and ψ− (Method: Decompose intrinsic torsion intoirreducible components under SU(3)).
Definition
An SU(3)-structure (g , J, ω, ψ+ + iψ−) is
Half-flat if d(ω ∧ ω) = 0, dψ+ = 0.
In particular, Calabi-Yau and Nearly Kahler structures are half-flat since
Calabi-Yau is equivalent to dω = 0, dψ+ = 0, dψ− = 0,
Nearly Kahler is equivalent to dω = 3ψ+, dψ− = −2ω ∧ ω.
Known properties of half-flat SU(3)-structures:
Ricci tensor looks awful
Some invariant examples on nilpotent Lie groups have been studiedand classified under additional assumptions
Can be evolved via Hitchin Flow
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Half-flat SU(3)-structures
SU(3)-structures can be classified (Chiossi and Salamon) by the exteriorderivatives of ω, ψ+ and ψ− (Method: Decompose intrinsic torsion intoirreducible components under SU(3)).
Definition
An SU(3)-structure (g , J, ω, ψ+ + iψ−) is
Half-flat if d(ω ∧ ω) = 0, dψ+ = 0.
In particular, Calabi-Yau and Nearly Kahler structures are half-flat since
Calabi-Yau is equivalent to dω = 0, dψ+ = 0, dψ− = 0,
Nearly Kahler is equivalent to dω = 3ψ+, dψ− = −2ω ∧ ω.
Known properties of half-flat SU(3)-structures:
Ricci tensor looks awful
Some invariant examples on nilpotent Lie groups have been studiedand classified under additional assumptions
Can be evolved via Hitchin Flow
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Half-flat SU(3)-structures
SU(3)-structures can be classified (Chiossi and Salamon) by the exteriorderivatives of ω, ψ+ and ψ− (Method: Decompose intrinsic torsion intoirreducible components under SU(3)).
Definition
An SU(3)-structure (g , J, ω, ψ+ + iψ−) is
Half-flat if d(ω ∧ ω) = 0, dψ+ = 0.
In particular, Calabi-Yau and Nearly Kahler structures are half-flat since
Calabi-Yau is equivalent to dω = 0, dψ+ = 0, dψ− = 0,
Nearly Kahler is equivalent to dω = 3ψ+, dψ− = −2ω ∧ ω.
Known properties of half-flat SU(3)-structures:
Ricci tensor looks awful
Some invariant examples on nilpotent Lie groups have been studiedand classified under additional assumptions
Can be evolved via Hitchin Flow
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Left-Invariant SU(3)-structures on Lie groups
Assume a high degree of symmetry: Let the manifold be asix-dimensional Lie group G and assume all tensors to be invariantunder left multiplication.
Left-invariant tensor fields are in one-to-one correspondence totensors on the Lie algebra g
Exterior differential systems are transformed into algebraic equationssince the exterior derivative restricted to left-invariant forms containsthe same information as the Lie bracket:
[ei , ej ] = ckij ek ⇐⇒ dek = −ck
ij ei ∧ e j
The Jacobi identity is equivalent to d2 = 0.
In particular: Half-flat left-invariant SU(3)-structures on Lie groupscan be described completely by exterior forms (ω, ψ+) on the Liealgebra satisfying algebraic equations (and inequalities).
Hitchin Flow evolution PDEs reduce to ODEs
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Left-Invariant SU(3)-structures on Lie groups
Assume a high degree of symmetry: Let the manifold be asix-dimensional Lie group G and assume all tensors to be invariantunder left multiplication.
Left-invariant tensor fields are in one-to-one correspondence totensors on the Lie algebra g
Exterior differential systems are transformed into algebraic equationssince the exterior derivative restricted to left-invariant forms containsthe same information as the Lie bracket:
[ei , ej ] = ckij ek ⇐⇒ dek = −ck
ij ei ∧ e j
The Jacobi identity is equivalent to d2 = 0.
In particular: Half-flat left-invariant SU(3)-structures on Lie groupscan be described completely by exterior forms (ω, ψ+) on the Liealgebra satisfying algebraic equations (and inequalities).
Hitchin Flow evolution PDEs reduce to ODEs
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Left-Invariant SU(3)-structures on Lie groups
Assume a high degree of symmetry: Let the manifold be asix-dimensional Lie group G and assume all tensors to be invariantunder left multiplication.
Left-invariant tensor fields are in one-to-one correspondence totensors on the Lie algebra g
Exterior differential systems are transformed into algebraic equationssince the exterior derivative restricted to left-invariant forms containsthe same information as the Lie bracket:
[ei , ej ] = ckij ek ⇐⇒ dek = −ck
ij ei ∧ e j
The Jacobi identity is equivalent to d2 = 0.
In particular: Half-flat left-invariant SU(3)-structures on Lie groupscan be described completely by exterior forms (ω, ψ+) on the Liealgebra satisfying algebraic equations (and inequalities).
Hitchin Flow evolution PDEs reduce to ODEs
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Left-Invariant SU(3)-structures on Lie groups
Assume a high degree of symmetry: Let the manifold be asix-dimensional Lie group G and assume all tensors to be invariantunder left multiplication.
Left-invariant tensor fields are in one-to-one correspondence totensors on the Lie algebra g
Exterior differential systems are transformed into algebraic equationssince the exterior derivative restricted to left-invariant forms containsthe same information as the Lie bracket:
[ei , ej ] = ckij ek ⇐⇒ dek = −ck
ij ei ∧ e j
The Jacobi identity is equivalent to d2 = 0.
In particular: Half-flat left-invariant SU(3)-structures on Lie groupscan be described completely by exterior forms (ω, ψ+) on the Liealgebra satisfying algebraic equations (and inequalities).
Hitchin Flow evolution PDEs reduce to ODEs
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Left-Invariant SU(3)-structures on Lie groups
Assume a high degree of symmetry: Let the manifold be asix-dimensional Lie group G and assume all tensors to be invariantunder left multiplication.
Left-invariant tensor fields are in one-to-one correspondence totensors on the Lie algebra g
Exterior differential systems are transformed into algebraic equationssince the exterior derivative restricted to left-invariant forms containsthe same information as the Lie bracket:
[ei , ej ] = ckij ek ⇐⇒ dek = −ck
ij ei ∧ e j
The Jacobi identity is equivalent to d2 = 0.
In particular: Half-flat left-invariant SU(3)-structures on Lie groupscan be described completely by exterior forms (ω, ψ+) on the Liealgebra satisfying algebraic equations (and inequalities).
Hitchin Flow evolution PDEs reduce to ODEs
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Left-Invariant SU(3)-structures on Lie groups
Assume a high degree of symmetry: Let the manifold be asix-dimensional Lie group G and assume all tensors to be invariantunder left multiplication.
Left-invariant tensor fields are in one-to-one correspondence totensors on the Lie algebra g
Exterior differential systems are transformed into algebraic equationssince the exterior derivative restricted to left-invariant forms containsthe same information as the Lie bracket:
[ei , ej ] = ckij ek ⇐⇒ dek = −ck
ij ei ∧ e j
The Jacobi identity is equivalent to d2 = 0.
In particular: Half-flat left-invariant SU(3)-structures on Lie groupscan be described completely by exterior forms (ω, ψ+) on the Liealgebra satisfying algebraic equations (and inequalities).
Hitchin Flow evolution PDEs reduce to ODEs
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Questions asked in PhD thesis
As a starting point, study direct products G × H of two3-dimensional Lie groups
Which admit left-invariant half-flat SU(3)-structures ?
How many ? Parametrisation possible ?
Is it possible to derive general properties or constructions of half-flatmanifolds?
Generalisations to half-flat SU(1, 2)-structures (= almostpseudo-hermitian structures) ?
Generalisations to half-flat SL(3,R)-structures (= almostpara-hermitian structures) ?
( Applications in physics ? )
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Questions asked in PhD thesis
As a starting point, study direct products G × H of two3-dimensional Lie groups
Which admit left-invariant half-flat SU(3)-structures ?
How many ? Parametrisation possible ?
Is it possible to derive general properties or constructions of half-flatmanifolds?
Generalisations to half-flat SU(1, 2)-structures (= almostpseudo-hermitian structures) ?
Generalisations to half-flat SL(3,R)-structures (= almostpara-hermitian structures) ?
( Applications in physics ? )
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Questions asked in PhD thesis
As a starting point, study direct products G × H of two3-dimensional Lie groups
Which admit left-invariant half-flat SU(3)-structures ?
How many ? Parametrisation possible ?
Is it possible to derive general properties or constructions of half-flatmanifolds?
Generalisations to half-flat SU(1, 2)-structures (= almostpseudo-hermitian structures) ?
Generalisations to half-flat SL(3,R)-structures (= almostpara-hermitian structures) ?
( Applications in physics ? )
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Questions asked in PhD thesis
As a starting point, study direct products G × H of two3-dimensional Lie groups
Which admit left-invariant half-flat SU(3)-structures ?
How many ? Parametrisation possible ?
Is it possible to derive general properties or constructions of half-flatmanifolds?
Generalisations to half-flat SU(1, 2)-structures (= almostpseudo-hermitian structures) ?
Generalisations to half-flat SL(3,R)-structures (= almostpara-hermitian structures) ?
( Applications in physics ? )
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Questions asked in PhD thesis
As a starting point, study direct products G × H of two3-dimensional Lie groups
Which admit left-invariant half-flat SU(3)-structures ?
How many ? Parametrisation possible ?
Is it possible to derive general properties or constructions of half-flatmanifolds?
Generalisations to half-flat SU(1, 2)-structures (= almostpseudo-hermitian structures) ?
Generalisations to half-flat SL(3,R)-structures (= almostpara-hermitian structures) ?
( Applications in physics ? )
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Questions asked in PhD thesis
As a starting point, study direct products G × H of two3-dimensional Lie groups
Which admit left-invariant half-flat SU(3)-structures ?
How many ? Parametrisation possible ?
Is it possible to derive general properties or constructions of half-flatmanifolds?
Generalisations to half-flat SU(1, 2)-structures (= almostpseudo-hermitian structures) ?
Generalisations to half-flat SL(3,R)-structures (= almostpara-hermitian structures) ?
( Applications in physics ? )
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Questions asked in PhD thesis
As a starting point, study direct products G × H of two3-dimensional Lie groups
Which admit left-invariant half-flat SU(3)-structures ?
How many ? Parametrisation possible ?
Is it possible to derive general properties or constructions of half-flatmanifolds?
Generalisations to half-flat SU(1, 2)-structures (= almostpseudo-hermitian structures) ?
Generalisations to half-flat SL(3,R)-structures (= almostpara-hermitian structures) ?
( Applications in physics ? )
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Some new existence results
Classification of pairs (G ,H) of two 3-dimensional simplyconnected Lie groups admitting a left-invariant half-flatSU(3)-structure on G × H such that the factors are orthogonalwith respect to the metric g (The list comprises 11 pairs).
Existence result: All pairs of two 3-dimensional unimodular simplyconnected Lie groups admit a left-invariant half-flat SU(3)-structure.
A Lie group is unimodular iff its left-invariant Haar measure is alsoright-invariant. Non-unimodular Lie groups G do NOT admit adiscrete subgroup Γ such that Γ\G is compact.
Some non-unimodular pairs do not admit a half-flat SU(3)-structure.
All examples can be written down explicitly.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Some new existence results
Classification of pairs (G ,H) of two 3-dimensional simplyconnected Lie groups admitting a left-invariant half-flatSU(3)-structure on G × H such that the factors are orthogonalwith respect to the metric g (The list comprises 11 pairs).
Existence result: All pairs of two 3-dimensional unimodular simplyconnected Lie groups admit a left-invariant half-flat SU(3)-structure.
A Lie group is unimodular iff its left-invariant Haar measure is alsoright-invariant. Non-unimodular Lie groups G do NOT admit adiscrete subgroup Γ such that Γ\G is compact.
Some non-unimodular pairs do not admit a half-flat SU(3)-structure.
All examples can be written down explicitly.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Some new existence results
Classification of pairs (G ,H) of two 3-dimensional simplyconnected Lie groups admitting a left-invariant half-flatSU(3)-structure on G × H such that the factors are orthogonalwith respect to the metric g (The list comprises 11 pairs).
Existence result: All pairs of two 3-dimensional unimodular simplyconnected Lie groups admit a left-invariant half-flat SU(3)-structure.
A Lie group is unimodular iff its left-invariant Haar measure is alsoright-invariant. Non-unimodular Lie groups G do NOT admit adiscrete subgroup Γ such that Γ\G is compact.
Some non-unimodular pairs do not admit a half-flat SU(3)-structure.
All examples can be written down explicitly.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Some new existence results
Classification of pairs (G ,H) of two 3-dimensional simplyconnected Lie groups admitting a left-invariant half-flatSU(3)-structure on G × H such that the factors are orthogonalwith respect to the metric g (The list comprises 11 pairs).
Existence result: All pairs of two 3-dimensional unimodular simplyconnected Lie groups admit a left-invariant half-flat SU(3)-structure.
A Lie group is unimodular iff its left-invariant Haar measure is alsoright-invariant. Non-unimodular Lie groups G do NOT admit adiscrete subgroup Γ such that Γ\G is compact.
Some non-unimodular pairs do not admit a half-flat SU(3)-structure.
All examples can be written down explicitly.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Some new existence results
Classification of pairs (G ,H) of two 3-dimensional simplyconnected Lie groups admitting a left-invariant half-flatSU(3)-structure on G × H such that the factors are orthogonalwith respect to the metric g (The list comprises 11 pairs).
Existence result: All pairs of two 3-dimensional unimodular simplyconnected Lie groups admit a left-invariant half-flat SU(3)-structure.
A Lie group is unimodular iff its left-invariant Haar measure is alsoright-invariant. Non-unimodular Lie groups G do NOT admit adiscrete subgroup Γ such that Γ\G is compact.
Some non-unimodular pairs do not admit a half-flat SU(3)-structure.
All examples can be written down explicitly.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Half-flat SU(3)-structures on S3 × S3
Proposition(S-H)
The ”moduli space” of left-invariant half-flat SU(3)-structures moduloLie algebra automorphisms on S3 × S3 is a union M1 ∪M2 ∪M3 of threemanifolds Mi , two of which are six-dimensional and one isfive-dimensional. Their intersection M1 ∩M2 ∩M3 is a four-dimensionalmanifold.
The three families of left-invariant half-flat SU(3)-structures can bedescribed explicitly.In one of the 6-dimensional families, the fundamental two-form ωremains constant.There is a well-known left-invariant Nearly Kahler structure onS3 × S3, which is unique due to a result of Butruille. This structureis contained in the intersection.Therefore, the result can be interpreted such that it describes allleft-invariant deformations of the unique Nearly Kahler structure byhalf-flat SU(3)-structures.This particular moduli space does not seem to have a nice naturalgeometric structure (as in the Calabi-Yau case).
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Half-flat SU(3)-structures on S3 × S3
Proposition(S-H)
The ”moduli space” of left-invariant half-flat SU(3)-structures moduloLie algebra automorphisms on S3 × S3 is a union M1 ∪M2 ∪M3 of threemanifolds Mi , two of which are six-dimensional and one isfive-dimensional. Their intersection M1 ∩M2 ∩M3 is a four-dimensionalmanifold.
The three families of left-invariant half-flat SU(3)-structures can bedescribed explicitly.
In one of the 6-dimensional families, the fundamental two-form ωremains constant.There is a well-known left-invariant Nearly Kahler structure onS3 × S3, which is unique due to a result of Butruille. This structureis contained in the intersection.Therefore, the result can be interpreted such that it describes allleft-invariant deformations of the unique Nearly Kahler structure byhalf-flat SU(3)-structures.This particular moduli space does not seem to have a nice naturalgeometric structure (as in the Calabi-Yau case).
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Half-flat SU(3)-structures on S3 × S3
Proposition(S-H)
The ”moduli space” of left-invariant half-flat SU(3)-structures moduloLie algebra automorphisms on S3 × S3 is a union M1 ∪M2 ∪M3 of threemanifolds Mi , two of which are six-dimensional and one isfive-dimensional. Their intersection M1 ∩M2 ∩M3 is a four-dimensionalmanifold.
The three families of left-invariant half-flat SU(3)-structures can bedescribed explicitly.In one of the 6-dimensional families, the fundamental two-form ωremains constant.
There is a well-known left-invariant Nearly Kahler structure onS3 × S3, which is unique due to a result of Butruille. This structureis contained in the intersection.Therefore, the result can be interpreted such that it describes allleft-invariant deformations of the unique Nearly Kahler structure byhalf-flat SU(3)-structures.This particular moduli space does not seem to have a nice naturalgeometric structure (as in the Calabi-Yau case).
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Half-flat SU(3)-structures on S3 × S3
Proposition(S-H)
The ”moduli space” of left-invariant half-flat SU(3)-structures moduloLie algebra automorphisms on S3 × S3 is a union M1 ∪M2 ∪M3 of threemanifolds Mi , two of which are six-dimensional and one isfive-dimensional. Their intersection M1 ∩M2 ∩M3 is a four-dimensionalmanifold.
The three families of left-invariant half-flat SU(3)-structures can bedescribed explicitly.In one of the 6-dimensional families, the fundamental two-form ωremains constant.There is a well-known left-invariant Nearly Kahler structure onS3 × S3, which is unique due to a result of Butruille. This structureis contained in the intersection.
Therefore, the result can be interpreted such that it describes allleft-invariant deformations of the unique Nearly Kahler structure byhalf-flat SU(3)-structures.This particular moduli space does not seem to have a nice naturalgeometric structure (as in the Calabi-Yau case).
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Half-flat SU(3)-structures on S3 × S3
Proposition(S-H)
The ”moduli space” of left-invariant half-flat SU(3)-structures moduloLie algebra automorphisms on S3 × S3 is a union M1 ∪M2 ∪M3 of threemanifolds Mi , two of which are six-dimensional and one isfive-dimensional. Their intersection M1 ∩M2 ∩M3 is a four-dimensionalmanifold.
The three families of left-invariant half-flat SU(3)-structures can bedescribed explicitly.In one of the 6-dimensional families, the fundamental two-form ωremains constant.There is a well-known left-invariant Nearly Kahler structure onS3 × S3, which is unique due to a result of Butruille. This structureis contained in the intersection.Therefore, the result can be interpreted such that it describes allleft-invariant deformations of the unique Nearly Kahler structure byhalf-flat SU(3)-structures.
This particular moduli space does not seem to have a nice naturalgeometric structure (as in the Calabi-Yau case).
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Half-flat SU(3)-structures on S3 × S3
Proposition(S-H)
The ”moduli space” of left-invariant half-flat SU(3)-structures moduloLie algebra automorphisms on S3 × S3 is a union M1 ∪M2 ∪M3 of threemanifolds Mi , two of which are six-dimensional and one isfive-dimensional. Their intersection M1 ∩M2 ∩M3 is a four-dimensionalmanifold.
The three families of left-invariant half-flat SU(3)-structures can bedescribed explicitly.In one of the 6-dimensional families, the fundamental two-form ωremains constant.There is a well-known left-invariant Nearly Kahler structure onS3 × S3, which is unique due to a result of Butruille. This structureis contained in the intersection.Therefore, the result can be interpreted such that it describes allleft-invariant deformations of the unique Nearly Kahler structure byhalf-flat SU(3)-structures.This particular moduli space does not seem to have a nice naturalgeometric structure (as in the Calabi-Yau case).
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Results in the pseudo- and para-Hermitian setting
Half-flat SU(1, 2)-structures and Nearly pseudo-Kahler structures can bedefined completely analogously by replacing the Riemannian metric by apseudo-Riemannian metric of signature (2, 4).
Proposition (Schafer, S-H)
On SL(2,R)× SL(2,R), there is a unique left-invariant Nearlypseudo-Kahler structure.
The same deformations as on S3 × S3 do exist, however, there may beeven more.Half-flat SL(3,R)-structures and Nearly para-Kahler structures can bedefined completely analogously by replacing the almost complex structureby an almost para-complex structure (J2 = +id). The metric has neutralsignature (3,3) in this case, the ±1-eigenspaces V± are 3-dimensional.
Proposition (S-H)
On a direct product g⊕ h, there exists a half-flat SL(3,R)-structure withg = V + and h = V− iff g and h are unimodular.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Results in the pseudo- and para-Hermitian setting
Half-flat SU(1, 2)-structures and Nearly pseudo-Kahler structures can bedefined completely analogously by replacing the Riemannian metric by apseudo-Riemannian metric of signature (2, 4).
Proposition (Schafer, S-H)
On SL(2,R)× SL(2,R), there is a unique left-invariant Nearlypseudo-Kahler structure.
The same deformations as on S3 × S3 do exist, however, there may beeven more.Half-flat SL(3,R)-structures and Nearly para-Kahler structures can bedefined completely analogously by replacing the almost complex structureby an almost para-complex structure (J2 = +id). The metric has neutralsignature (3,3) in this case, the ±1-eigenspaces V± are 3-dimensional.
Proposition (S-H)
On a direct product g⊕ h, there exists a half-flat SL(3,R)-structure withg = V + and h = V− iff g and h are unimodular.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Results in the pseudo- and para-Hermitian setting
Half-flat SU(1, 2)-structures and Nearly pseudo-Kahler structures can bedefined completely analogously by replacing the Riemannian metric by apseudo-Riemannian metric of signature (2, 4).
Proposition (Schafer, S-H)
On SL(2,R)× SL(2,R), there is a unique left-invariant Nearlypseudo-Kahler structure.
The same deformations as on S3 × S3 do exist, however, there may beeven more.
Half-flat SL(3,R)-structures and Nearly para-Kahler structures can bedefined completely analogously by replacing the almost complex structureby an almost para-complex structure (J2 = +id). The metric has neutralsignature (3,3) in this case, the ±1-eigenspaces V± are 3-dimensional.
Proposition (S-H)
On a direct product g⊕ h, there exists a half-flat SL(3,R)-structure withg = V + and h = V− iff g and h are unimodular.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Results in the pseudo- and para-Hermitian setting
Half-flat SU(1, 2)-structures and Nearly pseudo-Kahler structures can bedefined completely analogously by replacing the Riemannian metric by apseudo-Riemannian metric of signature (2, 4).
Proposition (Schafer, S-H)
On SL(2,R)× SL(2,R), there is a unique left-invariant Nearlypseudo-Kahler structure.
The same deformations as on S3 × S3 do exist, however, there may beeven more.Half-flat SL(3,R)-structures and Nearly para-Kahler structures can bedefined completely analogously by replacing the almost complex structureby an almost para-complex structure (J2 = +id). The metric has neutralsignature (3,3) in this case, the ±1-eigenspaces V± are 3-dimensional.
Proposition (S-H)
On a direct product g⊕ h, there exists a half-flat SL(3,R)-structure withg = V + and h = V− iff g and h are unimodular.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Results in the pseudo- and para-Hermitian setting
Half-flat SU(1, 2)-structures and Nearly pseudo-Kahler structures can bedefined completely analogously by replacing the Riemannian metric by apseudo-Riemannian metric of signature (2, 4).
Proposition (Schafer, S-H)
On SL(2,R)× SL(2,R), there is a unique left-invariant Nearlypseudo-Kahler structure.
The same deformations as on S3 × S3 do exist, however, there may beeven more.Half-flat SL(3,R)-structures and Nearly para-Kahler structures can bedefined completely analogously by replacing the almost complex structureby an almost para-complex structure (J2 = +id). The metric has neutralsignature (3,3) in this case, the ±1-eigenspaces V± are 3-dimensional.
Proposition (S-H)
On a direct product g⊕ h, there exists a half-flat SL(3,R)-structure withg = V + and h = V− iff g and h are unimodular.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Hitchin Flow
A G2-structure on a seven-manifold is a global three-form whichcan be written locally as ϕ =
∑i∈Z7
e i ∧ e i+1 ∧ e i+3.
A family (ω(t), ψ+(t)), t ∈ I , of SU(3)-structures on a 6-manifoldM can always be lifted to a G2-structure
ϕ = ω ∧ dt + ψ+
on the 7-manifold M × I .This G2-structure is parallel (dϕ = 0 = d ∗ϕ) iff the family ishalf-flat for all t and the evolution equations
dω =∂ψ+
∂t, dψ− = −1
2
∂ω2
∂t
are satisfied (solution is called Hitchin flow).
Theorem (Hitchin, 2001)
Let M6 be compact. Given a single half-flat SU(3)-structure as initialvalue, there is always a family of half-flat SU(3)-structures solving theevolution equations and defining therefore a metric with holonomy G2
on M × I .
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Hitchin Flow
A G2-structure on a seven-manifold is a global three-form whichcan be written locally as ϕ =
∑i∈Z7
e i ∧ e i+1 ∧ e i+3.A family (ω(t), ψ+(t)), t ∈ I , of SU(3)-structures on a 6-manifoldM can always be lifted to a G2-structure
ϕ = ω ∧ dt + ψ+
on the 7-manifold M × I .
This G2-structure is parallel (dϕ = 0 = d ∗ϕ) iff the family ishalf-flat for all t and the evolution equations
dω =∂ψ+
∂t, dψ− = −1
2
∂ω2
∂t
are satisfied (solution is called Hitchin flow).
Theorem (Hitchin, 2001)
Let M6 be compact. Given a single half-flat SU(3)-structure as initialvalue, there is always a family of half-flat SU(3)-structures solving theevolution equations and defining therefore a metric with holonomy G2
on M × I .
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Hitchin Flow
A G2-structure on a seven-manifold is a global three-form whichcan be written locally as ϕ =
∑i∈Z7
e i ∧ e i+1 ∧ e i+3.A family (ω(t), ψ+(t)), t ∈ I , of SU(3)-structures on a 6-manifoldM can always be lifted to a G2-structure
ϕ = ω ∧ dt + ψ+
on the 7-manifold M × I .This G2-structure is parallel (dϕ = 0 = d ∗ϕ) iff the family ishalf-flat for all t and the evolution equations
dω =∂ψ+
∂t, dψ− = −1
2
∂ω2
∂t
are satisfied (solution is called Hitchin flow).
Theorem (Hitchin, 2001)
Let M6 be compact. Given a single half-flat SU(3)-structure as initialvalue, there is always a family of half-flat SU(3)-structures solving theevolution equations and defining therefore a metric with holonomy G2
on M × I .
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Hitchin Flow
A G2-structure on a seven-manifold is a global three-form whichcan be written locally as ϕ =
∑i∈Z7
e i ∧ e i+1 ∧ e i+3.A family (ω(t), ψ+(t)), t ∈ I , of SU(3)-structures on a 6-manifoldM can always be lifted to a G2-structure
ϕ = ω ∧ dt + ψ+
on the 7-manifold M × I .This G2-structure is parallel (dϕ = 0 = d ∗ϕ) iff the family ishalf-flat for all t and the evolution equations
dω =∂ψ+
∂t, dψ− = −1
2
∂ω2
∂t
are satisfied (solution is called Hitchin flow).
Theorem (Hitchin, 2001)
Let M6 be compact. Given a single half-flat SU(3)-structure as initialvalue, there is always a family of half-flat SU(3)-structures solving theevolution equations and defining therefore a metric with holonomy G2
on M × I .
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Generalisation of Hitchin Flow
Joint work with Cortes, Leistner, Schafer: Proof of the theorem can besimplified and extended such that it holds
also for non-compact manifolds
also for the pseudo- and para-Hermitian setting (lifting toG∗
2 -structures)
also for nearly half-flat structures (lifting to nearly G(∗)2 -structures)
Explicit examples of half-flat SU(3)-structures can be used to construct
new metrics with full holonomy G(∗)2 . This works particularly well at the
moment for H3 × H3 where H3 denotes the 3-dimensional Heisenberggroup.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Generalisation of Hitchin Flow
Joint work with Cortes, Leistner, Schafer: Proof of the theorem can besimplified and extended such that it holds
also for non-compact manifolds
also for the pseudo- and para-Hermitian setting (lifting toG∗
2 -structures)
also for nearly half-flat structures (lifting to nearly G(∗)2 -structures)
Explicit examples of half-flat SU(3)-structures can be used to construct
new metrics with full holonomy G(∗)2 . This works particularly well at the
moment for H3 × H3 where H3 denotes the 3-dimensional Heisenberggroup.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Generalisation of Hitchin Flow
Joint work with Cortes, Leistner, Schafer: Proof of the theorem can besimplified and extended such that it holds
also for non-compact manifolds
also for the pseudo- and para-Hermitian setting (lifting toG∗
2 -structures)
also for nearly half-flat structures (lifting to nearly G(∗)2 -structures)
Explicit examples of half-flat SU(3)-structures can be used to construct
new metrics with full holonomy G(∗)2 . This works particularly well at the
moment for H3 × H3 where H3 denotes the 3-dimensional Heisenberggroup.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Generalisation of Hitchin Flow
Joint work with Cortes, Leistner, Schafer: Proof of the theorem can besimplified and extended such that it holds
also for non-compact manifolds
also for the pseudo- and para-Hermitian setting (lifting toG∗
2 -structures)
also for nearly half-flat structures (lifting to nearly G(∗)2 -structures)
Explicit examples of half-flat SU(3)-structures can be used to construct
new metrics with full holonomy G(∗)2 . This works particularly well at the
moment for H3 × H3 where H3 denotes the 3-dimensional Heisenberggroup.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Generalisation of Hitchin Flow
Joint work with Cortes, Leistner, Schafer: Proof of the theorem can besimplified and extended such that it holds
also for non-compact manifolds
also for the pseudo- and para-Hermitian setting (lifting toG∗
2 -structures)
also for nearly half-flat structures (lifting to nearly G(∗)2 -structures)
Explicit examples of half-flat SU(3)-structures can be used to construct
new metrics with full holonomy G(∗)2 . This works particularly well at the
moment for H3 × H3 where H3 denotes the 3-dimensional Heisenberggroup.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups
Generalisation of Hitchin Flow
Joint work with Cortes, Leistner, Schafer: Proof of the theorem can besimplified and extended such that it holds
also for non-compact manifolds
also for the pseudo- and para-Hermitian setting (lifting toG∗
2 -structures)
also for nearly half-flat structures (lifting to nearly G(∗)2 -structures)
Explicit examples of half-flat SU(3)-structures can be used to construct
new metrics with full holonomy G(∗)2 . This works particularly well at the
moment for H3 × H3 where H3 denotes the 3-dimensional Heisenberggroup.
Fabian Schulte-Hengesbach Half-Flat SU(3)-Structures on Lie Groups