Manifolds of Special Holonomy

2.2.1 Hyperk¨ahler structure

Definition2.40(Hyperk¨ahler structure). A4-dimensional oriented Riemannian mani-foldMis said to have ahyperk¨ahler structureif its holonomy group reduces toSpp1q Ă SOp4q. The structure is called an integrable hyperk¨ahler structureif the Levi-Civita con-nection reduces too.

Proposition2.41(Description by local forms). LetMbe a4-dimensional oriented Rieman-nian manifold and

F“

pω₁,ω₂,ω₃q P Λ²T_x^_M^‘3

| DpPFr_SOpMqx:p^˚ω_i “ω¹_i

(2.56) be the admissible bundle of ω¹₁,ω¹₂ andω¹₃. Then a hyperk¨ahler structure is equivalent to a section

pω1,ω2,ω3q PΓpFq, (2.57) and it is integrable if and only if the sections are parallel.

Proof. This follows immediately from the holonomy theorem (see e.g. [Bau09, Satz

5.3]) and Proposition2.28.

Lemma 2.42 (Hitchin). LetM be an oriented Riemannian manifold with hyperk¨ahler struc-turepω₁,ω₂,ω₃q. Then the structure is integrable if and only if

dω1 “dω2“dω3“0. (2.58)

Notation 2.43. In the following Λⁱ_j will denote a j-dimensional subspace of the i-th exterior product ofT^_M.

Proposition 2.44 (Λ

¨

of a hyperk¨ahler manifold). Let Mbe a4-dimensional hyperk¨ahler manifold. Then

1 “Rare the trivial representations,Λ¹₄ “C²_Ris the realified tautological representa-tion andΛ²₃“^sup²qis the adjoint representation.

Proof. Λ¹T^_M“Λ¹₄“C²_Rfollows immediately by restrictingSOp4qñR⁴toSUp2qãÑ SOp4q. Furthermore

Λ²T^_M“^Λ2R⁴“^Λ2_{`</sub>‘<sup>Λ2</sup><sub>´</sub> (2.59) as SOp4q representations. It is then a tedious but simple calculation to show that Λ<sup>2</sup><sub>`} – 3R and Λ²_´ – sup2q as SUp2q representations. First note that SUp2q is simply connected, so that we can equivalently work with the its Lie Algebrasup²q.

We use the usual orthogonal basis ofΛ²_˘

f¹_˘“e¹²˘e³⁴, f²_˘“e¹³¯e²⁴, f³_˘“e¹⁴˘e²³. (2.60) Heree^ij“eⁱ^e^j ande¹,. . .,e⁴ is the dual of the standard basise1,. . .,e4 ofR⁴. If we identifyC²_R withR⁴, we have the identification

e1Ø ¹ Using the standard basis ofsup2qgiven by

u1 “ ^{i 0} matrices. A quick inspection shows that the dual representation X ÞÑ ´ρpXq^T has exactly the same matrices (with respect to the dual basis) and hence the induced rep-resentationΛ²ρ^_acts by

which shows that the representation onΛ²_`is trivial and the isomorphism induced by f¹_´ ÞÑ u1, f²_´ ÞÑ ´u2 and f³_´ ÞÑ u3 gives an sup2q-module isomorphisms between sup2qandΛ²_´.

Proposition 2.45 (Description of the various representations). We have the following description of the various representations of proposition (2.44)

Λ²₃“ Proof. The second assertion follows immediately from the proof of proposition (2.44).

To see the first, note that wedging withωi gives aSUp2q-map3Λ²₁‘Λ²₃ ÑΛ⁴₁, and a quick check shows thatωi^ωi ‰0, so the kernel of the map is of dimension5. This

shows the first assertion.

2.2.2 Hypo structure

Definition 2.46 (Hypo structure). A5-dimensional oriented Riemannian manifoldM possesses a hypo structure, if the structure group of the frame bundle reduces to ι2: SUp2q ãÑ SOp5q. If, in addition, the Levi-Civita connection reduces, we say that Mpossesses anintegrable hypo structure.

Proposition2.47(Descritpion by local forms). LetMbe a5-dimensional oriented Rieman-nian manifold and let be the admissible subbundle to the hypo structure forms. Then a hypo structure onMis given by a sections

pη,ω1,ω2,ω3q PΓpFq (2.66) and it is integrable if and only if all sections are parallel.

Proof. This follows from the holonomy theorem (see e.g. [Bau09, Satz5.3]) and

Propo-sition2.28.

Proposition 2.48 (Λ

¨

of a hypo manifold). Let M be a manifold with a hypo structure.

Then the exterior powers of the cotangent bundle decompose as follows Space Decomposition Isomorphic to by‹ Λ⁰T^_M Λ⁰₁ Λ⁵T^_M Λ¹T^_M Λ¹₁‘Λ¹₄ Λ⁴T^_M Λ²T^_M Λ²₄‘^3Λ2₁‘^Λ2_´ ^Λ3T_M

(2.67) whereΛ

¨

1 “Rare the trivial representations,Λ

¨

4 “C²_Rare the realified tautological represen-tations andΛ²_´“sup2qis the adjoint representation.

Proposition 2.49 (Description of the various representations). We have the following description of the representations of proposition2.48

Λ¹₁“hηi

Λ¹₄“kerp^ι_η⁷|Λ¹q Λ²₄“impFp^ηq|Λ¹q

3Λ²₁“hω₁i\hω₂i\hω₃i

Λ²_´“kerpFpη2q|_Λ²q XkerpFpη2q|_Λ²q XkerpFpη3q|_Λ²q XkerpFpηq|_Λ²q

Proof. This is can be checked by calculating with the hypo structure forms onR⁵ and

realizing that all maps above areSUp2q-equivariant.

2.2.3 SUp3qstructure

Definition2.50. A6-dimensional oriented Riemannian manifoldMpossesses aSUp3 q-structure if the q-structure group of the frame bundle reduces toι₃:SUp³qãÑSOp⁶q. If, in addition, the Levi-Civita connection reduces, we say thatMpossesses anintegrable SUp3q-structure.

Proposition2.51. LetMbe a6-dimensional oriented Riemannian manifold and let Fx“

pω,Ωq PΛ²T_x^_M‘Λ³T_x^_M

DpPFr_SOpMq: p^˚ω“^ω1, p^˚Ω“Ω¹

(2.68) be the corresponding admissible bundle. ASUp3qstructure onMis a pair of sections

pω,Ωq PΓpFq, (2.69) and it is integrable if and only if the sections are parallel.

Proof. This follows from the holonomy theorem (see e.g. [Bau09, Satz5.3]) and

Propo-sition2.28.

Proposition 2.52 (Λ

¨

of a SUp3q manifold). Let M be a6-dimensional SUp3q-manifold.

Then

Space Decomposition Isomorphic to by‹ Λ⁰T^_M Λ⁰₁ Λ⁶T^_M Λ¹T^_M Λ¹₆ Λ⁵T^_M Λ²T^_M Λ²₁‘Λ²₆‘Λ²₈ Λ⁴T^_M Λ³T^_M 2Λ³₁‘^Λ3₆‘^Λ3₁₂

(2.70) where the representations of the decomposition are uniquely determined by their dimensions (compare to Table2.1).

Proof. See e.g. [Xu08, section 2.1.1].

2.2.4 G₂structure

Definition 2.53. A 7-dimensional oriented Riemannian manifold M possesses a G2 -structure if the -structure group of the frame bundle reduced to ι4: G₂ ãÑ SOp7q. If, in addition, the Levi-Civita connection reduces, we say thatMpossesses anintegrable G2-structure.

Proposition2.54. LetMbe a7-dimensional oriented Riemannian manifold and let Fx “

ϕPΛ³T_x^_M

DpPFr_SOpMq:p^˚ϕ“ϕ¹

(2.71) be the corresponding admissible bundle. AG₂structure onMis section

ϕPΓpFq, (2.72)

and it is integrable if and only if the section is parallel.

Proof. This follows from the holonomy theorem (see e.g. [Bau09, Satz5.3]) and

Propo-sition2.28.

Lemma2.55. A sectionϕPΓpFqis parallel if and only ifϕis closed and co-closed.

Proposition2.56(Λ

¨

of aG2 manifold). LetMbe a7-dimensionalG2-manifold. Then Space Decomposition Isomorphic to by‹

Λ⁰T^_M Λ⁰₁ Λ⁷T^_M Λ¹T^_M Λ¹₇ Λ⁶T^_M Λ²T^_M Λ²₇‘Λ²₁₄ Λ⁵T^_M Λ³T^_M Λ³₁‘Λ³₇‘Λ³₂₇ Λ⁴T^_M

where the representations of the decomposition are uniquely determined by their dimensions (compare to Table2.2).

Proof. See e.g. [Bry87].

Proposition2.57. LetMbe a7-dimensional G₂-manifold. Then

Λ²₇ “impGpψq|Λ¹q “Eig^(Gpϕq|Λ²;²⁾ (2.73) Λ²₁₄ “kerpFp^ψq|Λ²q “Eig(Gp^ϕq|Λ²;´¹⁾ (2.74)

Λ³₁ “hϕi (2.75)

Λ³₇ “impGpϕq|_Λ¹q (2.76) Λ³₂₇ “kerpFpϕq|_Λ³q XkerpFpψq|_Λ³q (2.77) where the representations of the decomposition are uniquely determined by their dimensions (compare to Table2.2).

Proposition2.58. There is aG₂-structure onΛ²_`Xof a4-dimensional Riemannian manfiold.

IfX is an anti self-dual Einstein manifold, then an integrableG2-structure can be defined on Λ²_`X. These structures reduce algebraically toSOp4q.

Proof. See [Sal89, Theorem11.10].

2.2.5 Spinp7qstructure

Definition2.59. An8-dimensional oriented Riemannian manifoldMis said to posses a Spinp7q-structure if the structure group of the frame bundle reduced toι₅: Spinp7qãÑ SOp8q. If, in addition, the Levi-Civita connection reduces, we say thatMpossesses an integrableSpinp7q-structure.

Proposition2.60. LetMbe a8-dimensional oriented Riemannian manifold and let Fx “

ΘPΛ⁴T_x^_M

DpPFr_SOpMq:p^˚Θ“Θ¹

(2.78) be the corresponding admissible bundle. ASpinp7qstructure onMis section

ΘPΓpFq, (2.79)

and it is integrable if and only if the section is parallel.

Proof. This follows from the holonomy theorem (see e.g. [Bau09, Satz5.3]) and

Propo-sition2.28.

Lemma2.61. A sectionΘPΓpFqis parallel if and only ifΘis closed.

Proposition2.62(Λ

¨

of aSpinp7qmanifold). LetMbe a7-dimensionalSpinp7q-manifold.

Then

Space Decomposition Isomorphic to by‹ Λ⁰T^_M Λ⁰₁ Λ⁸T^_M Λ¹T^_M Λ¹₈ Λ⁷T^_M Λ²T^_M Λ²₇‘Λ²₂₁ Λ⁶T^_M Λ³T^_M Λ³₈‘Λ³₄₈ Λ⁵T^_M Λ⁴T^_M Λ⁴₁‘^Λ4₇‘^Λ4₂₇‘^Λ4₃₅ —

Proof. See e.g. [Bry87].

Proposition2.63. LetMbe a8-dimensional Spinp7q-manifold. Then

Λ²₇ “Eig(GpΘq|_Λ2;3) (2.80) Λ²₂₁ “Eig(GpΘq|_Λ²;´1) (2.81) Λ³₈ “impGpΘq|Λ¹q (2.82) Λ³₄₈ “kerpFpΘq|Λ³q (2.83)

Λ⁴₁ “hΘi (2.84)

Λ⁴₃₅ “Eig(‹8|_Λ4;´1) (2.85) Proof. See [Bry87] for a justification and a description of the remaining spaces (which

we are not interested in here).

2.2.6 Connection of Holonomies

Corollary2.64. If we apply the above to the holonomy groups of manifolds, we get the

Im Dokument Generalized Seiberg-Witten and the Nahm Transform (Seite 23-29)