-structures in dimension five

Research Group ‘Global Analysis’

* * * * * *

SO(3)ir-geometries in dimension five and seven – results and open problems

Thomas Friedrich (Humboldt-Universit¨at zu Berlin) Srn´ı Winter School, January 2008

Introduction

Fix a subgroup G ⊂ SO(n) and consider a Riemannian manifold (Mⁿ, g, R) equipped with a G-structure R.

Examples:

G = U(n) ⊂ SO(2n) −→ almost hermitian geometry.

G = U(n) ⊂ SO(2n + 1) −→ contact geometry G = G₂ ⊂ SO(7) −→ G₂-geometry in dimension 7

First Question: Does there exist a metric connection ∇^c preserving the structure R such that the torsion

T^c(X, Y, Z) := g(∇^cXY − ∇^cY X − [X, Y ], Z) is totally skew-symmetric ? → characteristic connection.

Second Question: Study the curvature and the spin geometry of the new connection ∇^c.

In particular, we look for solutions of type II string equations involving a spinor field (super-symmetry) Ψ and a non-trivial ‘B-field’ T^c:

Ric^c = λ · g , δ(T^c) = 0 ,

∇^cΨ = 0, T^c · Ψ = µ · Ψ.

Approach: We construct in a systematic way solutions in type II superstring theory starting from non-integrable geometric structures.

Results: In dimension 5 ≤ n ≤ 8 for contact, almost hermitian, G₂- and Spin(7)-geometries (Agricola, Friedrich, Ivanov, . . . ).

Reference: the ’Srni lectures’ of Ilka Agricola, 2006.

In dimension n = 5 one usually considers contact geometries; they are related to the subgroup U(2) ⊂ SO(5) (Fr/Ivanov, Crelle J., 2004).

New Geometries: Consider the subgroup SO(3)_ir ⊂ SO(5). Study the geometry of 5-manifolds with such a structure. Apply the described method to them.

First Results:

• M. Bobienski, P. Nurowski: started this program during their stay in Berlin (Crelle J., 2006)

• S. Chiossi, A. Fino: SO(3)ir-structures on Lie groups (J. Lie Th., 2007)

Aim of this lecture: discuss some topological and geometric problems for SO(3)_ir-structures in dimensions n = 5,7.

SO(3)

-structures in dimension five

The group SO(5) contains two subgroups isomorphic to SO(3),

SO(3)_st ⊂ SO(5), SO(3)_ir ⊂ SO(5) .

The subgroup SO(3)_ir is the SO(3)-action on S₀²(R³) = R⁵. The generators of the Lie algebra so(3)_ir ⊂ so(5) are

X₁ = e₁₃ + √

3e₂₃ + e₄₅, X₂ = 2e₁₄ + e₃₅ X₃ = −e₁₅ + √

3e₂₅ − e₃₄ .

Question: Under which conditions a compact oriented 5-manifold M⁵ admits an SO(3)_st- or an SO(3)_ir-structure ?

First case: SO(3)_st-structures

In order to formulate the condition, we need some invariants.

Definition: The semi-characteristics (Kervaire) are defined by

k(M⁵) :=

2

X

i=0

dim^R H²ⁱ(M⁵;R)

mod 2 ,

ˆ

χ₂(M⁵) :=

2

X

i=0

dim^Z₂ H_i(M⁵;Z₂)

mod 2 .

Theorem:(Lusztig-Milnor-Peterson 1969)

k(M⁵) − χˆ₂(M⁵) = w₂(M⁵) ∪ w₃(M⁵) . In particular, if M⁵ is spin, then k(M⁵) = ˆχ₂(M⁵).

Theorem: A compact, oriented 5-manifold admits an SO(3)_st-structure (i.e. two vector fields) if and only if

w₄(M⁵) = 0 , k(M⁵) = 0 .

Proof: E. Thomas in 1967 for spin manifolds (w₄(M⁵) = 0 = ˆχ₂(M⁵)), M.F. Atiyah in 1969 for the general case.

Second case: SO(3)_ir-structures

Example 1: M⁵ = SU(3)/SO(3) has an SO(3)_ir-structure.

Some topological properties of this space:

• M⁵ is simply connected and a rational homology sphere.

• M⁵ does not admit any Spin- or Spin^C-structure.

• k(M⁵) − χˆ₂(M⁵) = w₂(M⁵) ∪ w₃(M⁵) = 1

• k(M⁵) = 1 and χˆ₂(M⁵) = 0

In particular, M⁵ = SU(3)/SO(3) does not admit any SO(3)_st-structure!

Example 2: M⁵ = S⁵ has no SO(3)_st- or SO(3)_ir-structure.

• M⁵ admits a Spin-structure

• k(M⁵) − χˆ₂(M⁵) = w₂(M⁵) ∪ w₃(M⁵) = 0

• k(M⁵) = 1 and χˆ₂(M⁵) = 1

Example 3: Consider the subgroup H = {(A, A²), A ∈ SO(2)} ⊂ SO(3) × SO(3) as well as the homogeneous space M⁵ = (SO(3) × SO(3))/H. Then M⁵ has an SO(3)_ir-structure and the following topological data:

• H₁(M⁵;Z) = Z₂ , H₂(M⁵;Z) = Z.

• k(M⁵) = 0 and χˆ₂(M⁵) = 0.

More examples: Bobienski/Nurowski (Crelle Journal 2006): there is a 2-parameter family G⁶(s, t) of 6-dimensional Lie groups containing SO(2) such that the isotropy representation of M⁵ = G⁶(s, t)/SO(2) is the maximal torus T_max = {(A, A²,1), A ∈ SO(2)} ⊂ SO(3)_ir ⊂ SO(5).

The groups are, for example,

G⁶(s, t) = SO(3) × SO(3), SO(3) × SO(1,2),

R¹ × (SO(2) on R⁴), (SO(2) on R²) × SO(3) .

The obstructions for SO(3)_ir-structures:

The relevant space is X⁷ := SO(5)/SO(3)_ir. Let us list some of its homotopy groups:

π₁(X⁷) = 0 , π₂(X⁷) = 0 , π₃(X⁷) = Z₁₀ , π₄(X⁷) = Z₂ .

Consequence: The obstructions for the existence of an SO(3)_ir- structure on a compact 5-manifold M⁵ are in H⁴(M⁵;Z₁₀) = H⁴(M⁵;Z₅) ⊕ H⁴(M⁵;Z₂) and in H⁵(M⁵;Z₂) .

Problem: Compute the topological conditions for the existence in general.

The criterion given in Bobenski, math.dg/0601066 is wrong (w₄ = 0, k = 0, p₁/5 ∈ Z). The space M⁵ = SU(3)/SO(3) does not satisfy it

!

Proposition 1: M⁵ admits an SO(3)_ir-structure if and only if there exists a 3-dimensional bundle E³ such that T(M⁵) = S₀²(E³).

Proposition 2: Suppose that T(M⁵) = S₀²(E³). Then

• p₁(M⁵) = 5 · p₁(E³).

• w₁(M⁵) = 0 and w₄(M⁵) = 0.

• w₂(M⁵) = w₂(E³) and w₃(M⁵) = w₃(E³).

Corollary: If M⁵ admits an SO(3)_ir-structure, then

• w₄(M⁵) = 0 in H⁴(M⁵;Z₂) ;

• p₁(M⁵)/5 ∈ H⁴(M⁵;Z) is integral.

Conjecture: M⁵ admits an SO(3)_ir-structure if and only if p (M⁵)

SO(3)

- and G

-structures in dimension seven

The real, irreducible 7-dimensional representation S₀³(R³) = R⁷ of SO(3) yield an embedding SO(3)_ir ⊂ G₂ ⊂ SO(7).

The sub-algebra so(3)_ir ⊂ g₂ ⊂ so(7) is given by

X₁ =

r1 5

e₁₂ + 2e₃₄ − 3e₅₆

X₂ = − r6

5 e₂₇ −

r1

2 e₁₄ +

r1

2 e₂₃ +

r 3

10 e₃₅ −

r 3

10 e₄₆

X₃ = − r6

5 e₁₇ −

r1

2 e₁₃ −

r1

2 e₂₄ +

r 3

10 e₃₆ +

r 3

10 e₄₅ .

The group G₂ preserves the 3-form

ω³ = e₅₆₇ + e₃₄₇ + e₁₂₇ − e₁₄₆ + e₁₃₅ − e₂₄₅ − e₂₃₆ .

The following formula proves the inclusion SO(3)_ir ⊂ G₂ :

∗ X₁ ∧ X₁ + X₂ ∧ X₂ + X₃ ∧ X₃

= − 6

5 ω³ .

Theorem: SO(3)_ir ⊂ SO(7) is the stabilizer of two symmetric tensors in S⁴(R⁷). The first polynomial is (x²₁ + . . . + x²₇)² and the second polynomial is given by the formula

4√

15x⁴₁ + 4√

15x⁴₂ + 120x³₂x₅ − 120x³₁x₆ + 60x1`

10x3x₄x₅ + (6x²₂ + 5x²₃ − 5x²₄)x6 − (4x2x₄ + 2√

15x4x₅ + 2√

15x3x₆)x7´

− 60x₂`

5x²₃x₅ − 5x²₄x₅ − 10x₃x₄x₆ + 2√

15x₃x₅x₇ − 2√

15x₄x₆x₇´ +

√15`

25x⁴₃ + 25x⁴₄ − 30x²₄x²₇ + 10x²₃(5x²₄ − 3x²₇) + 9x²₇(10x²₅ + 10x²₆ + x²₇)´ + 2x²₁`

4√

15x²₂ + 10√

15x²₃ − 180x₂x₅ − 60x₃x₇ + √

15(10x²₄ + 30x²₅ + 30x²₆ + 9x²₇)´ +2x²₂`

10√

15x²₃ + 60x₃x₇ + √

15(10x²₄ + 30x²₅ + 30x²₆ + 9x²₇)´

Some consequences:

• An SO(3)_ir-structure on a Riemannian 7-manifolds is defined by a special symmetric tensor (polynomial) of degree 4.

• Any SO(3)_ir-structure on a Riemannian 7-manifold (M⁷, g) induces a unique G₂-structure ω³.

• There are 4 basic classes W¹,W²,W³,W⁴ of G₂-structures on a 7- dimensional Riemannian manifold (Fernandez/Gray 1982). They are given by the G₂-components of the representation R⁷ ⊗ (so(7)/g₂).

• A G₂-manifold admits a characteristic connection if and only if it is of type W¹ ⊕ W³ ⊕ W⁴ (d ∗ ω³ = θ ∧ ∗ω³ – Fr/Ivanov 2002).

• Denote by Vk the real, (2k + 1)-dimensional, irreducible representation of SO(3). The SO(3)ir-representations decompose into

W¹ = V₀ , W² = V₂ ⊕ V₄ , W³ = V₂ ⊕ V₄ ⊕ V₆ , W⁴ = V₃ .

Theorem: There is a bijection between

• SO(3)_ir-structures admitting a characteristic connection;

• G₂-structures of type W¹⊕W³⊕W⁴ such that the holonomy hol(∇^c) ⊂ so(3)_ir of their characteristic connection is contained in so(3)_ir.

Problem: Construct 7-dimensional Riemannian manifolds with SO(3)_ir- structure such that the underlying G₂-structure is of type

W¹ ⊕ W³ ⊕ W⁴ = V₀ ⊕ V₂ ⊕ V₃ ⊕ V₄ ⊕ V₆ .

Can any V_α-type be realized ?

An equivalent formulation: Construct G₂-structures on Riemannian 7- manifolds with characteristic connection such that hol(∇^c) ⊂ so(3)_ir and of a fixed V_α-type.

Remark: A parallel G -manifold (i.e. ∇^c = ∇^g,T^c = 0) cannot have a

Theorem:

A compact, 7-dimensional SO(3)_ir-manifold of type W⁴ = V₃ and hol(∇^c) ⊂ so(3)_ir , T^c 6= 0 does not exist. Equivalently, a compact G₂-manifold of type W⁴ and hol(∇^c) ⊂ so(3)_ir , T^c 6= 0 does not exist.

Sketch of the proof:

Up to a conformal change of the metric, the universal covering splits into Y ⁶ × R¹, where Y ⁶ is a nearly K¨ahler manifold (see Agricola/Friedrich, J. Geom. Phys. 2006). Then we conclude that

hol(∇^c) ⊂ su(3) ∩ so(3)_ir = so(2) .

In particular, Y ⁶ is nearly K¨ahler with a reduced, 1-dimensional characteristic holonomy, a contradiction (Belgun/Moroianu, Ann. Glob.

Anal. Geom. 2002).

The basic example: X⁷ = SO(5)/SO(3)_ir . Geometric properties of the basic example:

• X⁷ admits an SO(3)_ir-structure. It is not symmetric.

• The underlying G₂-structure is of type W¹ (nearly parallel). In particular, it realizes the type V₀. Moreover, X⁷ admits one real Killing spinor. This spinor field is ∇^c-parallel.

Theorem: (TF 2006)

X⁷ is the unique G₂-manifold of type W¹ ⊕ W³ ⊕ W⁴ such that

hol(∇^c) ⊂ so(3)_ir , ∇^cT^c = 0 , T^c 6= 0 .

Topological properties of the basic example:

• X⁷ is simply connected and a rational homology sphere,

H₁(M⁷;Z) = 0 , H₂(M⁷;Z) = 0 , H₃(M⁷;Z) = Z₁₀ , H₄(M⁷;Z) = 0 , H₅(M⁷;Z) = 0 , H₆(M⁷;Z) = 0 .

• k(X⁷) = 1 and χˆ₂(X⁷) = 0 .

• All Stiefel-Whitney classes w_i(X⁷) and the Pontrjagin class p₁(X⁷) are trivial.

• X⁷ is not parallelizable, but admits 4 vector fields (Goette/Kitchloo/Shankar 2002). In particular, there is no further reduction of the frame bundle to a subgroup of SO(3)_ir.

Existence of SO(3)

-structures in dimension seven

Problem: Study the conditions for the existence of a topological SO(3)_ir- structure on a compact 7-manifold. The relevant space is

Z¹⁸ = SO(7)/SO(3)_ir .

The homotopy groups of Z¹⁸:

π₁(Z¹⁸) = Z₂ , π₂(Z¹⁸) = Z₂ , π₃(Z¹⁸) = Z₂₈ , π₄(Z¹⁸) = 0 , π₅(Z¹⁸) = Z₂ , π₆(Z¹⁸) = Z₂ .

Consequence: Let M⁷ be a compact, oriented 7-manifold. Then the obstructions for the existence of an SO(3)ir-structure are in

H²(M⁷; ) , H³(M⁷; ) , H⁴(M⁷; ) , H⁶(M⁷; ) , H⁷(M⁷; ) .

Proposition: M⁷ admits an SO(3)_ir-structure if and only if there exists a 3-dimensional bundle E³ such that T(M⁷) = S₀³(E³).

Now we compute again characteristic classes.

Theorem: Let F⁷ be a real, oriented, 7-dimensional vector bundle over some space Y and suppose that there exists a 3-dimensional bundle E³ such that F⁷ = S₀³(E³). Then we have:

• w₂(F⁷) = w₃(F⁷) = w₅(F⁷) = w₇(F⁷) = 0 ;

• w₄(F⁷) = w₂²(E³) and w₆(F⁷) = w₃²(E³) ;

• p₁(F⁷)/14 = p₁(E³) is an integral cohomology class and p₁(F⁷)/14 = w₄(F⁷) mod 2.

If F⁷ = T(M⁷) is the tangent bundle of some 7-manifold, then w₁(M⁷) = w₂(M⁷) = w₃(M⁷) = 0 implies the vanishing of w₄(M⁷) and w₆(M⁷) (use the Wu formulas !).

Corollary: Let M⁷ be an oriented, compact 7-manifold. If it admits an SO(3)_ir-structure, then

• all Stiefel-Whitney classes are trivial.

• The Pontrjagin class p₁(M⁷) is divisible by 28.

Remark: The criterion is only necessary, but not sufficient. Again the highest obstruction in H⁷(M⁷;Z₂) is missing.

Remark: This obstruction cannot be χˆ₂(M⁷). Indeed, the sphere S⁷ is parallelizable and χˆ₂(S⁷) = 1. On the other hand, SO(5)/SO(3)_ir admits an SO(3)ir-structure, but χˆ₂(SO(5)/SO(3)ir) = 0. However, both manifolds have k = 1.

Bundles over twistor spaces

Let X⁴ be a compact, oriented 4-manifold such that

w₃(X⁴) = 0, 6σ(X⁴) − 2χ(X⁴) ≡ 0 mod 28

holds. Denote by Z⁶ its twistor space and consider a principal S¹-bundle M⁷ → Z⁶. Then all Stiefel-Whitney classes of M⁷ vanish and the Pontrjagin class p₁(M⁷) is divisible by 28.

Remark: The condition 0 = w₃(X⁴) ∈ H³(X⁴;Z₂) = H₁(X⁴;Z₂) is satisfied for example if

• X⁴ is simply connected.

• X⁴ is a spin manifold (apply Wu’s formula !).

Example: Consider X⁴ = A · (S¹ × S³) #B ·(S² × S²). X⁴ is spin and σ(X⁴) = 0, χ(X⁴) = 2(B + 1 − A).

Example: Consider X⁴ = CP² #k · (−CP²) = blow up of CP² in k points. X⁴ is simply connected and

σ(X⁴) = 1 − k, χ(X⁴) = 3 + k, 6σ(X⁴) − 2χ(X⁴) = −8k .

In particular, del Pezzo surfaces (k = 7) satisfy the necessary conditions.

These surfaces admit K¨ahler-Einstein metrics with positive scalar curvature.

More example: X⁴ = A · (S² × S²) # B · (K3) , X⁴ = A · (S¹ × S³) # B · (K3) , . . .

Construction of T

⊂ SO(3)

-structures

The algebraic fact

T_max = diag(A, A², A⁻³,1) = SO(3)_ir ∩ SU(3) ⊂ G₂ , A ∈ SO(2)

yields the following

Theorem: Let M⁵ be a 5-manifold with an T_max = {(A, A²,1), A ∈ SO(2)} ⊂ SO(3)_ir ⊂ SO(5)-structure R and denote by ρ the 2- dimensional representation of T_max given by ρ(A) = A⁻³ , A ∈ SO(2).

Then M⁷ := R ×^ρ R² admits an T_max ⊂ SO(3)_ir ⊂ SO(7)-structure.

M⁷ is a complex vector bundle over M⁵.

Remark: In a similar way we can consider 5 manifolds with an {(A, A⁻³,1), A ∈ SO(2)} ⊂ SO(5)- or an {(A², A⁻³,1), A ∈ SO(2)} ⊂ SO(5)-structure, Then a vector bundle M⁷ over M⁵ again admits an T_max ⊂ SO(3)_ir ⊂ SO(7)-structure.

Construction of SO(3)

-structures via twistor theory

The algebraic fact

T_max = diag(z, z², z⁻³) = SO(3)_ir ∩ SU(3) ⊂ G₂

yields the following

Proposition: Let Y ⁶ be a 6-manifold such that its tangent bundle splits T Y ⁶ = (E ⊕ E² ⊕ E⁻³)^R, where E is a complex, 1-dimensional bundle.

Then any S¹-bundle M⁷ → Y ⁶ admits a topological T_max ⊂ SO(3)_ir- structure.

Consider a compact spin 4-manifold X⁴. Then the twistor space Z⁶ = P(S⁻) is the projective spin bundle and there exists the tautological bundle H → Z⁶. The tangent bundle is given by

T(Z⁶) = T^v ⊕ T^h, T^v = H⁻² .

Ansatz: T^h = H⁻¹ ⊕ H³.

Then any S¹-bundle over Z⁶ admits a T_max ⊂ SO(3)_ir-structure.

Theorem: If T^h = H⁻¹ ⊕ H³, then 9σ(X⁴) = 10χ(X⁴).

Conversely, if 9 σ(X⁴) = 10χ(X⁴), then

• c₁(T^h) = c₁(H⁻¹ ⊕ H³), c₂(T^h) = c₂(H⁻¹ ⊕ H³).

Moreover, the following conditions are equivalent:

• T^h splits into H⁻¹ ⊕ H³.

• T^h splits into the sum of two line bundles.

Remark: If 9 σ(X⁴) = 10χ(X⁴) = 0, then Z⁶ is parallelizable.

Consequently, the interesting case is 9 σ(X⁴) = 10χ(X⁴) 6= 0

Problem to handle: Suppose that 9 σ(X⁴) = 10χ(X⁴) 6= 0. Then we have to decide whether or not the complex 2-dimensional bundle T^h over Z⁶ splits. These bundles are basically given by the homotopy classes

[Z⁶ , P(H)^∞] = [Z⁶ , S⁴] Theorem: (Steenrod classification theorem)

Let Z⁶ be a compact 6-dimensional manifold and consider two SU(2)- principal fiber bundles with the same Chern class, c₂(P₁) = c₂(P₂) ∈ H⁴(Z⁶ ; Z). Then there exist a cohomology class

δ(P₁, P₂) ∈ H⁵(Z⁶ ; Z₂)/Sq² H³(Z⁶ ; Z₂)

such that δ(P₁, P₂) = 0 if and only if P₁ and P₂ are isomorphic over Z⁶ − {point}. In this case, the last obstruction to P₁ = P₂ over Z⁶ is in H⁶(Z⁶ ; π₅(SU(2)) = H⁶(Z⁶ ; Z₂) = Z₂.

Corollary: Let X⁴ be a compact, oriented 4-manifold such that

• X⁴ is spin.

• 9 σ(X⁴) = 10 χ(X⁴) 6= 0.

Then the twistor space Z⁶ of X⁴ is not parallelizable. The tangent bundle splits into

T(Z⁶) = H⁻¹ ⊕ H⁻² ⊕ H³ . over the 4-skeleton of Z⁶

Examples:

The spaces

X⁴ = 20 · (S¹ × S³) # 5 · (K3) and

X⁴ = (1 + A) · (S¹ × S³) # A · (S² × S²) satisfy the condition 9 σ(X⁴) = 10 χ(X⁴).