half-flat su(3)-structures on lie groupshalf-flat su(3)-structures on lie groups project a6:...

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

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

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

SU(3)-structures

Definition

SU(3)-structures

Definition

SU(3)-structures

Definition

SU(3)-structures

Definition

SU(3)-structures

Definition

SU(3)-structures

Definition

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)

SU(3)-structures II

ω = e1 ∧ e2 + e3 ∧ e4 + e5 ∧ e6,

Ψ = ψ+ + iψ− = (e1 + ie2) ∧ (e3 + ie4) ∧ (e5 + ie6).

Stab(ω) = Sp(6,R)

Stab(J) = GL(3,C)

Stab(Ψ) = SL(3,C)

SU(3)-structures II

ω = e1 ∧ e2 + e3 ∧ e4 + e5 ∧ e6,

Ψ = ψ+ + iψ− = (e1 + ie2) ∧ (e3 + ie4) ∧ (e5 + ie6).

Stab(ω) = Sp(6,R)

Stab(J) = GL(3,C)

Stab(Ψ) = SL(3,C)

SU(3)-structures II

ω = e1 ∧ e2 + e3 ∧ e4 + e5 ∧ e6,

Ψ = ψ+ + iψ− = (e1 + ie2) ∧ (e3 + ie4) ∧ (e5 + ie6).

Stab(ω) = Sp(6,R)

Stab(J) = GL(3,C)

Stab(Ψ) = SL(3,C)

SU(3)-structures II

ω = e1 ∧ e2 + e3 ∧ e4 + e5 ∧ e6,

Ψ = ψ+ + iψ− = (e1 + ie2) ∧ (e3 + ie4) ∧ (e5 + ie6).

Stab(ω) = Sp(6,R)

Stab(J) = GL(3,C)

Stab(Ψ) = SL(3,C)

SU(3)-structures II

ω = e1 ∧ e2 + e3 ∧ e4 + e5 ∧ e6,

Ψ = ψ+ + iψ− = (e1 + ie2) ∧ (e3 + ie4) ∧ (e5 + ie6).

Stab(ω) = Sp(6,R)

Stab(J) = GL(3,C)

Stab(Ψ) = SL(3,C)

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.

Always algebraic.

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

Definition

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

ij ei ∧ e j

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 ? )

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.

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).

Proposition(S-H)

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).

Proposition(S-H)

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).

Proposition(S-H)

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).

Proposition(S-H)

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).

Proposition(S-H)

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).

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.

Proposition (S-H)

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)

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

∂ω2

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 .

Hitchin Flow

∑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

∂ω2

on M × I .

Hitchin Flow

∑i∈Z7

ϕ = ω ∧ dt + ψ+

dω =∂ψ+

∂t, dψ− = −1

∂ω2

on M × I .

Hitchin Flow

∑i∈Z7

ϕ = ω ∧ dt + ψ+

dω =∂ψ+

∂t, dψ− = −1

∂ω2

on M × I .

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.

2 -structures)

half-flat su(3)-structures on lie groupshalf-flat su(3)-structures on lie groups project a6:...

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