Spin(9), Rosenfeld planes and canonical differential forms in …agricola/rauisch... · 2014. 6....
Transcript of Spin(9), Rosenfeld planes and canonical differential forms in …agricola/rauisch... · 2014. 6....
Spin(9), Rosenfeld planes and canonical differential formsin octonionic geometry
Maurizio Parton - Paolo Piccinni
1 What you should expect in this talk
2 Spin(9)
3 After Spin(9): Rosenfeld projective planes
4 Applications
5 Conclusion
1 What you should expect in this talk
2 Spin(9)
3 After Spin(9): Rosenfeld projective planes
4 Applications
5 Conclusion
MP, Paolo Piccinni.Spin(9) and almost complex structures on 16-dimensional manifolds.Annals of Global Analysis and Geometry, 41 (2012), 321–345.
MP, Paolo Piccinni.Spheres with more than 7 vector fields: all the fault of Spin(9).Linear Algebra and its Applications, 438 (2013), 1113–1131.
Liviu Ornea, MP, Paolo Piccinni, Victor Vuletescu.Spin(9) geometry of the octonionic Hopf fibration.Transformation Groups, 18 (2013), 845–864.
MP, Paolo Piccinni.Canonical Differential Forms, Rosenfeld Planes, and a Matryoshka inOctonionic Geometry.Work in progress.
MP, Paolo Piccinni.Spin(9) and almost complex structures on 16-dimensional manifolds.Annals of Global Analysis and Geometry, 41 (2012), 321–345.
MP, Paolo Piccinni.Spheres with more than 7 vector fields: all the fault of Spin(9).Linear Algebra and its Applications, 438 (2013), 1113–1131.
Liviu Ornea, MP, Paolo Piccinni, Victor Vuletescu.Spin(9) geometry of the octonionic Hopf fibration.Transformation Groups, 18 (2013), 845–864.
MP, Paolo Piccinni.Canonical Differential Forms, Rosenfeld Planes, and a Matryoshka inOctonionic Geometry.Work in progress.
Further references
Robert Bryant.Remarks on Spinors in Low Dimensions.http://www.math.duke.edu/~bryant/Spinors.pdf (1999).
Thomas Friedrich.Weak Spin(9)-Structures on 16-dimensional Riemannian Manifolds.Asian Journal of Mathematics, 5 (2001), 129–160.
Andrei Moroianu, Uwe Semmelmann.Clifford structures on Riemannian manifolds.Advances in Mathematics, 228 (2011), 940–967.
Main aim of the talk
Convince you that Spin(9) is beautiful.
Method
Introduce Spin(9) little brother: Sp(2) · Sp(1)
Show you that Spin(9) is involved in many curious phenomena
Relatives
Construction of relevant differential forms associated with the groups
Spin(9), Spin(10),Spin(12), Spin(16)
appearing as structure and holonomy group in the exceptional symmetricspaces
FII EIII,EVI,EVIII
Cayley plane Rosenfeld planes
1 What you should expect in this talk
2 Spin(9)Why and whatQuaternionic analogyCuriosities about Spin(9)
3 After Spin(9): Rosenfeld projective planes
4 Applications
5 Conclusion
1 What you should expect in this talk
2 Spin(9)Why and whatQuaternionic analogyCuriosities about Spin(9)
3 After Spin(9): Rosenfeld projective planes
4 Applications
5 Conclusion
Berger’s list and Spin(9) refutation
Holonomy of simply connected, irreducible, nonsymmetric?
SO(n)U(n)
SU(n)
Sp(n) · Sp(1)
Sp(n)G2
Spin(7)Spin(9)
Simply connected, complete, holonomy Spin(9)⇔
OP2 = F4Spin(9) (s > 0), R16(flat), OH2 = F4(−20)
Spin(9) (s < 0)[Alekseevsky, Funct. Anal. Prilozhen 1968].
Berger’s list and Spin(9) refutation
Holonomy of simply connected, irreducible, nonsymmetric?
SO(n)U(n)
SU(n)
Sp(n) · Sp(1)
Sp(n)G2
Spin(7)Spin(9)Spin(9)×
Simply connected, complete, holonomy Spin(9)⇔
OP2 = F4Spin(9) (s > 0), R16(flat), OH2 = F4(−20)
Spin(9) (s < 0)[Alekseevsky, Funct. Anal. Prilozhen 1968].
First Spin(9) definition
Definition
Spin(9) is the Lie group which has been excluded from a list.
First Spin(9) definition
Definition
Spin(9) is the Lie group which has been excluded from a list.
What is Spin(9)?
Definition
Spin(9) ⊂ SO(16) is the group of symmetries of the Hopf fibration
O2 ⊃ S15 S7
→ S8 ∼= OP1[Gluck-Warner-Ziller, L’Enseignement Math. 1986].
Λ8(R16)Spin(9)
= Λ81 + . . . [Friedrich, Asian Journ. Math 2001].
Spin(9) is the stabilizer in SO(16) of any element of Λ81
[Brown-Gray, Diff. Geom. in honor of K. Yano 1972].
Definition
Spin(9) is the stabilizer in SO(16) of the 8-form
ΦSpin(9) =
∫OP1
p∗l νl dl Details More details
[Berger, Ann. Ec. Norm. Sup. 1972].
What is Spin(9)?
Definition
Spin(9) ⊂ SO(16) is the group of symmetries of the Hopf fibration
O2 ⊃ S15 S7
→ S8 ∼= OP1[Gluck-Warner-Ziller, L’Enseignement Math. 1986].
Λ8(R16)Spin(9)
= Λ81 + . . . [Friedrich, Asian Journ. Math 2001].
Spin(9) is the stabilizer in SO(16) of any element of Λ81
[Brown-Gray, Diff. Geom. in honor of K. Yano 1972].
Definition
Spin(9) is the stabilizer in SO(16) of the 8-form
ΦSpin(9) =
∫OP1
p∗l νl dl Details More details
[Berger, Ann. Ec. Norm. Sup. 1972].
1 What you should expect in this talk
2 Spin(9)Why and whatQuaternionic analogyCuriosities about Spin(9)
3 After Spin(9): Rosenfeld projective planes
4 Applications
5 Conclusion
The closest relative: the quaternionic group Sp(2) · Sp(1)
Analogy 1
Spin(9) and Sp(2) · Sp(1) are the symmetry groups of the Hopf fibrations:
S15 −→ OP1 S7 −→ HP1
Analogy 2
Spin(9) is the stabilizer in SO(16) of the 8-form
ΦSpin(9) =
∫OP1
p∗l νl dl
Sp(2) · Sp(1) is the stabilizer in SO(8) of the 4-form
Ω =
∫HP1
p∗l νl dl
The closest relative: the quaternionic group Sp(2) · Sp(1)
Analogy 1
Spin(9) and Sp(2) · Sp(1) are the symmetry groups of the Hopf fibrations:
S15 −→ OP1 S7 −→ HP1
Analogy 2
Spin(9) is the stabilizer in SO(16) of the 8-form
ΦSpin(9) =
∫OP1
p∗l νl dl
Sp(2) · Sp(1) is the stabilizer in SO(8) of the 4-form
Ω =
∫HP1
p∗l νl dl
Two alternative constructions for the quaternionic form Ω
Sum of squares: classical
Ω = ω2I + ω2
J + ω2K , where ωI , ωJ , ωK are orthogonal local Kahler forms
Sum of squares: involutions
Ω = τ2(Θ), where
τ2(Θ) =∑
α<β θ2αβ
Θ = (θαβ) matrix of Kahler forms of Jαβ = Iα IβI1, . . . , I5 self-adjoint anti-commuting involutions in R8:
I1, . . . , I5 ∈ SO(8), I∗α = Iα, I2α = Id, Iα Iβ = −Iβ Iα
Explicit Iα
Two alternative constructions for the quaternionic form Ω
Sum of squares: classical
Ω = ω2I + ω2
J + ω2K , where ωI , ωJ , ωK are orthogonal local Kahler forms
Sum of squares: involutions
Ω = τ2(Θ), where
τ2(Θ) =∑
α<β θ2αβ
Θ = (θαβ) matrix of Kahler forms of Jαβ = Iα IβI1, . . . , I5 self-adjoint anti-commuting involutions in R8:
I1, . . . , I5 ∈ SO(8), I∗α = Iα, I2α = Id, Iα Iβ = −Iβ Iα
Explicit Iα
The five involutions of Sp(2) · Sp(1) as 8× 8 matrices
I1 =
(0 Id
Id 0
)I2 =
(0 −RH
i
RHi 0
)
I3 =
(0 −RH
j
RHj 0
)I4 =
(0 −RH
k
RHk 0
)
I5 =
(Id 0
0 − Id
)Go back
Nine involutions for Spin(9)
Analogy 3
ΦSpin(9) = τ4(Θ), where
t9 +τ2(Θ)t7 + τ4(Θ)t5 +τ6(Θ)t3 + τ8(Θ)t characteristic polynomialof Θ
Θ = (θαβ) matrix of Kahler forms of Jαβ = Iα IβI1, . . . , I9 self-adjoint anti-commuting involutions in R16:
I1, . . . , I9 ∈ SO(16), I∗α = Iα, I2α = Id, Iα Iβ = −Iβ Iα
Explicit Iα
[MP-Piccinni, Ann. Glob. Anal. Geom. 2012].
The nine involutions of Spin(9) as 16× 16 matrices
I1 =
(0 Id
Id 0
)I2 =
(0 −Ri
Ri 0
)I3 =
(0 −Rj
Rj 0
)
I4 =
(0 −Rk
Rk 0
)
I5 =
(0 −Re
Re 0
)
I6 =
(0 −Rf
Rf 0
)I7 =
(0 −Rg
Rg 0
)I8 =
(0 −Rh
Rh 0
)I9 =
(Id 0
0 − Id
)Go back
Spin(9) and Sp(2) · Sp(1) as structure groups
Analogy 4
A Spin(9)-structure on M16 is a rank 9 vector subbundle
spanI1, . . . , I9 ⊂ End(M)
A Sp(2) · Sp(1)-structure on M8 is a rank 5 vector subbundle
spanI1, . . . , I5 ⊂ End(M)
Spin(9) and Sp(2) · Sp(1) as structure groups
Analogy 4
A Spin(9)-structure on M16 is a rank 9 vector subbundle
spanI1, . . . , I9 ⊂ End(M)
A Sp(2) · Sp(1)-structure on M8 is a rank 5 vector subbundle
spanI1, . . . , I5 ⊂ End(M)
Due to the dual role Sp-Spin ofSp(1) = Spin(3) and Sp(2) = Spin(5)
1 What you should expect in this talk
2 Spin(9)Why and whatQuaternionic analogyCuriosities about Spin(9)
3 After Spin(9): Rosenfeld projective planes
4 Applications
5 Conclusion
Jαβ = Iα Iβ1≤α<β≤9 generates spin(9)
There are 36 Kahler forms θαβ for 1 ≤ α < β ≤ 9
There are 84 Kahler forms θαβγ for 1 ≤ α < β ≤ 9
Remark
so(16) = Λ2(R16) = Λ236 ⊕ Λ2
84 = spin(9)⊕ Λ284
generated by θαβ generated by θαβγ
Matryoshka-like structure
Nestedness
The family of complex structures Jαβ1≤α<β≤9 is compatible with theinclusion of Lie algebras
spin(7)∆ ⊂ spin(8) ⊂ spin(9)
in the sense that
spin(7)∆ = spanJαβ2≤α<β≤8
⊆ spanJαβ1≤α<β≤8 = spin(8)
⊆ spanJαβ1≤α<β≤9 = spin(9)
Spheres with more than 7 vector fields: Blame Spin(9)!
Spheres Sm−1 ⊂ Rm admit 1, 3 or 7 linearly independent vector fieldsaccording to whether p = 1, 2 or 3 in
m = (2k + 1)2p
In the general case
m = (2k + 1)2p16q with q ≥ 0 and p = 0, 1, 2, 3
the maximum number of vector fields is
σ(m) = 2p − 1 + 8q
C,H,O contribution Spin(9) contribution
No S1-subfibration
Similarly to the quaternionic Hopf fibration, one would expect severalS1-subfibrations for the octonionic Hopf fibration on S15 ⊂ O2.
Theorem
Any global vector field on S15 which is tangent to the fibers of theoctonionic Hopf fibration S15 → S8 has at least one zero.[Ornea-MP-Piccinni-Vuletescu, Transformation Groups, 2013]
[Loo-Verjovsky, Topology, 1992]
1 What you should expect in this talk
2 Spin(9)
3 After Spin(9): Rosenfeld projective planesWhy and whatSpin(10)Spin(12) and Spin(16)
4 Applications
5 Conclusion
1 What you should expect in this talk
2 Spin(9)
3 After Spin(9): Rosenfeld projective planesWhy and whatSpin(10)Spin(12) and Spin(16)
4 Applications
5 Conclusion
Rosenfeld projective planes
What are the Rosenfeld projective planes?
Cayley plane: OP2 =F4
Spin(9)= FII, dimFII = 16
(C⊗O)P2 =E6
Spin(10) ·U(1)=
EIII, dimEIII = 32
(H⊗O)P2 =E7
Spin(12) · Sp(1)=
EVI, dimEVI = 64
(O⊗O)P2 =E8
Spin(16)+=
EVIII, dimEVIII = 128
Rosenfeld projective planes
What are the Rosenfeld projective planes?
Cayley plane: OP2 =F4
Spin(9)= FII, dimFII = 16
(C⊗O)P2 =E6
Spin(10) ·U(1)=
EIII, dimEIII = 32
(H⊗O)P2 =E7
Spin(12) · Sp(1)=
EVI, dimEVI = 64
(O⊗O)P2 =E8
Spin(16)+=
EVIII, dimEVIII = 128
Cayley plane
Rosenfeld projective planes
What are the Rosenfeld projective planes?
Cayley plane: OP2 =F4
Spin(9)= FII, dimFII = 16
(C⊗O)P2 =E6
Spin(10) ·U(1)=
EIII, dimEIII = 32
(H⊗O)P2 =E7
Spin(12) · Sp(1)=
EVI, dimEVI = 64
(O⊗O)P2 =E8
Spin(16)+=
EVIII, dimEVIII = 128
Cayley plane
Rosenfeld projective planes
What are the Rosenfeld projective planes?
Cayley plane: OP2 =F4
Spin(9)= FII, dimFII = 16
(C⊗O)P2 =E6
Spin(10) ·U(1)= EIII, dimEIII = 32
(H⊗O)P2 =E7
Spin(12) · Sp(1)= EVI, dimEVI = 64
(O⊗O)P2 =E8
Spin(16)+= EVIII, dimEVIII = 128
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Cayley plane
Rosenfeld projective planes
Focus on EIII
Cayley plane: OP2 =F4
Spin(9)= FII, dimFII = 16
(C⊗O)P2 =E6
Spin(10) ·U(1)= EIII, dimEIII = 32
(H⊗O)P2 =E7
Spin(12) · Sp(1)=
EVI, dimEVI = 64
(O⊗O)P2 =E8
Spin(16)+=
EVIII, dimEVIII = 128
What is Spin(n)?
Roughly. . .
SO(2): multiplication by a unitary complex number in R2 = CSO(3): conjugation by a unitary quaternion in R3 = ImHSO(4): conjugation by 2 unitary quaternions in R4 = H
. . . we can say that. . .
The action of the above Lie groups SO(n) on Rn, for n = 2, . . . , 4, can bedescribed in terms of multiplication in some algebra embedding Rn as asubspace.
Motivation for Clifford algebras
Spin(n) is the generalization of the above fact to every n. The algebraembedding the group Spin(n) and the vector space Rn is called theClifford algebra. Go to MFD of Spin(n)
What is Spin(n)?
Roughly. . .
SO(2): multiplication by a unitary complex number in R2 = CSO(3): conjugation by a unitary quaternion in R3 = ImHSO(4): conjugation by 2 unitary quaternions in R4 = H
. . . we can say that. . .
The action of the above Lie groups SO(n) on Rn, for n = 2, . . . , 4, can bedescribed in terms of multiplication in some algebra embedding Rn as asubspace.
Motivation for Clifford algebras
Spin(n) is the generalization of the above fact to every n. The algebraembedding the group Spin(n) and the vector space Rn is called theClifford algebra. Go to MFD of Spin(n)
What is Spin(n)?
Roughly. . .
SO(2): multiplication by a unitary complex number in R2 = CSO(3): conjugation by a unitary quaternion in R3 = ImHSO(4): conjugation by 2 unitary quaternions in R4 = H
. . . we can say that. . .
The action of the above Lie groups SO(n) on Rn, for n = 2, . . . , 4, can bedescribed in terms of multiplication in some algebra embedding Rn as asubspace.
Motivation for Clifford algebras
Spin(n) is the generalization of the above fact to every n. The algebraembedding the group Spin(n) and the vector space Rn is called theClifford algebra. Go to MFD of Spin(n)
1 What you should expect in this talk
2 Spin(9)
3 After Spin(9): Rosenfeld projective planesWhy and whatSpin(10)Spin(12) and Spin(16)
4 Applications
5 Conclusion
A basis for spin(10)
spin(10) = liespin(9), u(1) ⊂ su(16), where u(1) is generated by(i Id8 0
0 −i Id8
)
spin(9) is generated by J19, . . . , J89, because Jαβ = [Jβ9, Jα9]/2
=
(i Id8 0
0 i Id8
)·
(Id8 00 − Id8
)
Proposition
spin(10) = spanJαβ = Iα Iβ0≤α<β≤9
A basis for spin(10)
spin(10) = liespin(9), u(1) ⊂ su(16), where u(1) is generated by(i Id8 0
0 −i Id8
)
spin(9) is generated by J19, . . . , J89, because Jαβ = [Jβ9, Jα9]/2
=
(i Id8 0
0 i Id8
)·
(Id8 00 − Id8
)
Proposition
spin(10) = spanJαβ = Iα Iβ0≤α<β≤9
A basis for spin(10)
spin(10) = liespin(9), u(1) ⊂ su(16), where u(1) is generated by(i Id8 0
0 −i Id8
)
spin(9) is generated by J19, . . . , J89, because Jαβ = [Jβ9, Jα9]/2
=
(i Id8 0
0 i Id8
)·
(Id8 00 − Id8
)
Proposition
spin(10) = spanJαβ = Iα Iβ0≤α<β≤9
A basis for spin(10)
spin(10) = liespin(9), u(1) ⊂ su(16), where u(1) is generated by(i Id8 0
0 −i Id8
)
spin(9) is generated by J19, . . . , J89, because Jαβ = [Jβ9, Jα9]/2
=
def= I0
= I9
I0 is a complex structure, not an involution. It acts on the first factor ofC⊗O2 by complex multiplication.
(i Id8 0
0 i Id8
)·
(Id8 00 − Id8
)
Proposition
spin(10) = spanJαβ = Iα Iβ0≤α<β≤9
A basis for spin(10)
spin(10) = liespin(9), u(1) ⊂ su(16), where u(1) is generated by(i Id8 0
0 −i Id8
)
spin(9) is generated by J19, . . . , J89, because Jαβ = [Jβ9, Jα9]/2
=
def= I0
= I9
(i Id8 0
0 i Id8
)·
(Id8 00 − Id8
)
Proposition
spin(10) = spanJαβ = Iα Iβ0≤α<β≤9
Canonical Spin(10)-form
Denote by JN the basis Jαβ = Iα Iβ0≤α<β≤9 of spin(10)
Denote by ΘN its associated skew-symmetric 10× 10 matrix ofKahler forms:
ΘN = (θαβ)0≤α<β≤9
Canonical Spin(10) form: 8-form in R32
The fourth coefficient τ4(ΘN) of the characteristic polynomial of ΘN is acanonical 8-form associated with the representation Spin(10) ⊂ SU(16):
ΦSpin(10) =∑
0≤α1<α2<α3<α4≤9
(θα1α2 ∧θα3α4−θα1α3 ∧θα2α4 +θα1α4 ∧θα2α3)2
ΦSpin(10) generates H8(EIII)
Cohomology of EIII
H∗( (C⊗O)P2 ) =R[d2, d8]
(suitable relations)d2 ∈ H2, d8 ∈ H8
I’m Hermitian symmetric
I’m the Kahler form
What am I?
ΦSpin(10) generates H8(EIII)
Cohomology of EIII
H∗( (C⊗O)P2 ) =R[d2, d8]
(suitable relations)d2 ∈ H2, d8 ∈ H8
I’m Hermitian symmetric
I’m the Kahler form
What am I?
ΦSpin(10) generates H8(EIII)
The d8-Lemma
H∗( (C⊗O)P2 ) =R[d2, d8]
(suitable relations)d2 ∈ H2, d8 ∈ H8
I’m Hermitian symmetric
I’m the Kahler form
What am I?
ΦSpin(10) generates H8(EIII)
0 + 0 2 + 0 4 + 0 6 + 0 8 + 0 10 + 0 12 + 0 14 + 0 16 + 0
0 + 8 2 + 8 4 + 8 6 + 8 8 + 810 + 812 + 814 + 816 + 8
0 + 162 + 164 + 166 + 168 + 1610 + 1612 + 1614 + 1616 + 16
H0 H2 H4 H6 H8 H10 H12 H14 H16
1
2
3
Kahler form
→ generator in H2(EIII)
→ generator in H8(EIII)
ΦSpin(10) generates H8(EIII)
0 + 0 2 + 0 4 + 0 6 + 0 8 + 0 10 + 0 12 + 0 14 + 0 16 + 0
0 + 8 2 + 8 4 + 8 6 + 8 8 + 810 + 812 + 814 + 816 + 8
0 + 162 + 164 + 166 + 168 + 1610 + 1612 + 1614 + 1616 + 16
H0 H2 H4 H6 H8 H10 H12 H14 H16
1
2
3
Kahler form
ΦSpin(10)
→ generator in H2(EIII)
→ generator in H8(EIII)
Time check
How many minutes?
Go on with EVI, EVIII
Skip to applications
1 What you should expect in this talk
2 Spin(9)
3 After Spin(9): Rosenfeld projective planesWhy and whatSpin(10)Spin(12) and Spin(16)
4 Applications
5 Conclusion
EVI: cohomology ring
The cohomology of
EVI = (H⊗O)P2 =E7
Spin(12) · Sp(1)
is given by
H∗( (H⊗O)P2 ) =R[d4, d8, d12]
(suitable relations)d4 , d8 , d12 ∈ H4,H8,H12
I’m quaternion-Kahler
I’m the quaternion-Kahler form
? ?
A basis for spin(12) ⊂ sp(16)
Consider the matrices(i Id8 0
0 −i Id8
),
(j Id8 0
0 −j Id8
),
(k Id8 0
0 −k Id8
)where i , j , k act as quaternionic multiplication on H⊗O2
Denote by I0, I−1, I−2 the i , j , k multiplication
Proposition
spin(12) = spanJαβ = Iα Iβ−2≤α<β≤9
Canonical Spin(12) forms: 8 and 12-form in R64
If ΘO is the matrix of Kahler forms of Jαβ = Iα Iβ, then τ4(ΘO) andτ6(ΘO) are a canonical 8-form and a canonical 12-form on R64 = H⊗O2
associated to Spin(12).
EVI: cohomology ring
0 + 0 + 0 4 + 0 + 0 8 + 0 + 0 12 + 0 + 0 16 + 0 + 0 20 + 0 + 0 24 + 0 + 0 28 + 0 + 0 32 + 0 + 0
0 + 8 + 0 4 + 8 + 0 8 + 8 + 0 12 + 8 + 0 16 + 8 + 0 20 + 8 + 0 24 + 8 + 028 + 8 + 032 + 8 + 0
0 + 16 + 0 4 + 16 + 0 8 + 16 + 0 12 + 16 + 0 16 + 16 + 020 + 16 + 024 + 16 + 028 + 16 + 032 + 16 + 0
0 + 24 + 0 4 + 24 + 0 8 + 24 + 012 + 24 + 016 + 24 + 020 + 24 + 024 + 24 + 028 + 24 + 032 + 24 + 0
0 + 32 + 04 + 32 + 08 + 32 + 012 + 32 + 016 + 32 + 020 + 32 + 024 + 32 + 028 + 32 + 032 + 32 + 0
0 + 0 + 12 4 + 0 + 12 8 + 0 + 12 12 + 0 + 12 16 + 0 + 12 20 + 0 + 1224 + 0 + 1228 + 0 + 1232 + 0 + 12
0 + 8 + 12 4 + 8 + 12 8 + 8 + 12 12 + 8 + 1216 + 8 + 1220 + 8 + 1224 + 8 + 1228 + 8 + 1232 + 8 + 12
0 + 16 + 12 4 + 16 + 128 + 16 + 1212 + 16 + 1216 + 16 + 1220 + 16 + 1224 + 16 + 1228 + 16 + 1232 + 16 + 12
0 + 24 + 124 + 24 + 128 + 24 + 1212 + 24 + 1216 + 24 + 1220 + 24 + 1224 + 24 + 1228 + 24 + 1232 + 24 + 12
0 + 32 + 124 + 32 + 128 + 32 + 1212 + 32 + 1216 + 32 + 1220 + 32 + 1224 + 32 + 1228 + 32 + 1232 + 32 + 12
0 + 0 + 24 4 + 0 + 24 8 + 0 + 2412 + 0 + 2416 + 0 + 2420 + 0 + 2424 + 0 + 2428 + 0 + 2432 + 0 + 24
0 + 8 + 244 + 8 + 248 + 8 + 2412 + 8 + 2416 + 8 + 2420 + 8 + 2424 + 8 + 2428 + 8 + 2432 + 8 + 24
0 + 16 + 244 + 16 + 248 + 16 + 2412 + 16 + 2416 + 16 + 2420 + 16 + 2424 + 16 + 2428 + 16 + 2432 + 16 + 24
0 + 24 + 244 + 24 + 248 + 24 + 2412 + 24 + 2416 + 24 + 2420 + 24 + 2424 + 24 + 2428 + 24 + 2432 + 24 + 24
0 + 32 + 244 + 32 + 248 + 32 + 2412 + 32 + 2416 + 32 + 2420 + 32 + 2424 + 32 + 2428 + 32 + 2432 + 32 + 24
1 1 2 3 4 5 6 6 7
Quaternionic form
τ4(ΘO)
τ6(ΘO)
→ generator in H4(EVI)
→ generator in H8(EVI)
→ generator in H12(EVI)
EVIII: cohomology ring
The cohomology of
EVIII = (O⊗O)P2 =E8
Spin(16)+
is given by
H∗( (O⊗O)P2 ) =R[d8, d12, d16, d20]
(suitable relations)d8 ,d12 ,d16 ,d20∈ H8,H12,H16,H20
?
?
?
?
A basis for spin(16) ⊂ so(128)
Denote by I0, I−1, I−2, I−3, I−4, I−5, I−6 the i , j , k, e, f , g , hmultiplication by the octonion units on the first factor of O⊗O2
Proposition
spin(16) = spanJαβ = Iα Iβ−6≤α<β≤9
Canonical Spin(16) forms: 8, 12, 16 and 20-form in R128
If ΘR is the matrix of Kahler forms of Jαβ = Iα Iβ, then τ4(ΘR),τ6(ΘR), τ8(ΘR) and τ10(ΘR) are canonical forms associated with thestandard Spin(16) structure on R128 = O⊗O2.
EVIII: cohomology ring
The cohomology of
EVIII = (O⊗O)P2 =E8
Spin(16)+
is given by
H∗( (O⊗O)P2 ) =R[d8, d12, d16, d20]
(suitable relations)d8 ,d12 ,d16 ,d20∈ H8,H12,H16,H20
I’m τ4(ΘR)
I’m τ6(ΘR)
I’m τ8(ΘR)
I’m τ10(ΘR)
1 What you should expect in this talk
2 Spin(9)
3 After Spin(9): Rosenfeld projective planes
4 ApplicationsBack to spheresMatryoshkaClifford structures
5 Conclusion
1 What you should expect in this talk
2 Spin(9)
3 After Spin(9): Rosenfeld projective planes
4 ApplicationsBack to spheresMatryoshkaClifford structures
5 Conclusion
More than 7 vector fields on spheres? Spin(9)’s fault. . .
σ(m) = 2p − 1 + 8q
C,H,O contribution Spin(9) contribution
Maximal system of σ(m) = 8, 9, 11, 15 vector fields on S15,S31,S63, S127
Sphere σ(m) Vector fields Structures involved
S15 (p = 0, q = 1) 0 + 8 J19, . . . , J89 Spin(9)
S31 (p = 1, q = 1) 1 + 8 i ·, J19, . . . , J89 C + Spin(9)
S63 (p = 2, q = 1) 3 + 8 i ·, j ·, k ·,J19, . . . , J89 H + Spin(9)
S127 (p = 3, q = 1) 7 + 8 i ·, j ·, k ·, e·, f ·, g ·, h·, J19, . . . , J89 O + Spin(9)
but Spin(10), Spin(12) and Spin(16) are co-conspirators
σ(m) =
0 + 8q Spin(9) contribution
1 + 8q Spin(10) contribution
3 + 8q Spin(12) contribution
7 + 8q Spin(16) contribution
Maximal system of σ(m) = 8, 9, 11, 15 vector fields on S15, S31,S63,S127
Sphere σ(m) Vector fields Structures involved
S15 (p = 0, q = 1) 0 + 8 J19, . . . , J89 Spin(9)
S31 (p = 1, q = 1) 1 + 8 J09,J19, . . . , J89 Spin(10)
S63 (p = 2, q = 1) 3 + 8 J(−2)9, J(−1)9, J09,J19, . . . , J89 Spin(12)
S127 (p = 3, q = 1) 7 + 8 J(−6)9, . . . , J(−1)9, J09,J19, . . . , J89 Spin(16)
1 What you should expect in this talk
2 Spin(9)
3 After Spin(9): Rosenfeld projective planes
4 ApplicationsBack to spheresMatryoshkaClifford structures
5 Conclusion
The dolls. . .
Recall that the Spin(9) family
J I = Jαβ1≤α<β≤9
contains the Spin(8) and Spin(7)∆ subfamilies
JP = Jαβ1≤α<β≤8, JS = Jαβ2≤α<β≤8
Using the first factor multiplication in C⊗O2, H⊗O2, O⊗O2 weobtain the larger families
JN = Jαβ0≤α<β≤9, JO = Jαβ−2≤α<β≤9, J
R = Jαβ−6≤α<β≤9
. . . and their Matryoshka
The Lie group inclusions
Spin(7)∆ ⊂ Spin(8) ⊂ Spin(9) ⊂ Spin(10) ⊂ Spin(12) ⊂ Spin(16)
are preserved by the spinor inclusions
JS ⊂ JP ⊂ J I ⊂ JN ⊂ JO ⊂ JR
= JSPINOR
. . . and their Matryoshka
The Lie group inclusions
Spin(7)∆ ⊂ Spin(8) ⊂ Spin(9) ⊂ Spin(10) ⊂ Spin(12) ⊂ Spin(16)
are preserved by the spinor inclusions
JS ⊂ JP ⊂ J I ⊂ JN ⊂ JO ⊂ JR = JSPINOR
INOR details
so(16) = Λ2(R16) = Λ236 ⊕ Λ2
84 = spin(9)⊕ Λ284
generated by J I = Iα Iβ
Add I0 to obtain JN = Iα Iβ0≤α<β≤9 and
spanJN = spin(10) ⊂ so(32)
Add I−1, I−2 to obtain JO = Iα Iβ−2≤α<β≤9 and
spanJO = spin(12) ⊂ so(64)
Add I−6, . . . , I−3 to obtain JR = Iα Iβ−6≤α<β≤9 and
spanJR = spin(16) ⊂ so(128)
INOR details
so(16) = Λ2(R16) = Λ236 ⊕ Λ2
84 = spin(9)⊕ Λ284
generated by J I = Iα Iβ
Add I0 to obtain JN = Iα Iβ0≤α<β≤9 and
spanJN = spin(10) ⊂ so(32)
Add I−1, I−2 to obtain JO = Iα Iβ−2≤α<β≤9 and
spanJO = spin(12) ⊂ so(64)
Add I−6, . . . , I−3 to obtain JR = Iα Iβ−6≤α<β≤9 and
spanJR = spin(16) ⊂ so(128)
INOR details
so(16) = Λ2(R16) = Λ236 ⊕ Λ2
84 = spin(9)⊕ Λ284
generated by J I = Iα Iβ
Add I0 to obtain JN = Iα Iβ0≤α<β≤9 and
spanJN = spin(10) ⊂ so(32)
Add I−1, I−2 to obtain JO = Iα Iβ−2≤α<β≤9 and
spanJO = spin(12) ⊂ so(64)
Add I−6, . . . , I−3 to obtain JR = Iα Iβ−6≤α<β≤9 and
spanJR = spin(16) ⊂ so(128)
1 What you should expect in this talk
2 Spin(9)
3 After Spin(9): Rosenfeld projective planes
4 ApplicationsBack to spheresMatryoshkaClifford structures
5 Conclusion
Even Clifford structures
A rank r even Clifford structure on a Riemannian manifold Mn is:
a rank r oriented Euclidean vector bundle E over M
an algebra bundle morphism φ : Cl0(E )→ End(TM) such thatφ(Λ2E ) ⊂ End−(TM)
Classification
Rank of E Mn n
. . . . . . . . .
9 OP2 = FII = F4Spin(9) 16
10 (C⊗O)P2 = EIII = E6Spin(10)·U(1) 32
12 (H⊗O)P2 = EVI = E7Spin(12)·Sp(1) 64
16 (O⊗O)P2 = EVIII = E8Spin(16)+ 128
[Moroianu-Semmelmann, Adv. Math. 2011]
Explicit even Clifford structures on Rosenfeld planes
Holonomy representation
Local endomorphisms
Vector bundle
Spin(9)
Local I1, . . . , I9
E = spanI1, . . . , I9
Rank of E E φ Mn n
9 spanI1, . . . , I9 Iα ∧ Iβ 7→ Jαβ = Iα Iβ OP2 16
10 spanI0, . . . , I9 Iα ∧ Iβ 7→ Jαβ = Iα Iβ (C⊗O)P2 32
12 spanI−2, . . . , I9 Iα ∧ Iβ 7→ Jαβ = Iα Iβ (H⊗O)P2 64
16 spanI−6, . . . , I9 Iα ∧ Iβ 7→ Jαβ = Iα Iβ (O⊗O)P2 128
Explicit even Clifford structures on Rosenfeld planes
Holonomy representation
Local endomorphisms
Vector bundle
Spin(9)
Local I1, . . . , I9
E = spanI1, . . . , I9
Rank of E E φ Mn n
9 spanI1, . . . , I9 Iα ∧ Iβ 7→ Jαβ = Iα Iβ OP2 16
10 spanI0, . . . , I9 Iα ∧ Iβ 7→ Jαβ = Iα Iβ (C⊗O)P2 32
12 spanI−2, . . . , I9 Iα ∧ Iβ 7→ Jαβ = Iα Iβ (H⊗O)P2 64
16 spanI−6, . . . , I9 Iα ∧ Iβ 7→ Jαβ = Iα Iβ (O⊗O)P2 128
Explicit even Clifford structures on Rosenfeld planes
Holonomy representation
Local endomorphisms
Vector bundle
Spin(9)
Local I1, . . . , I9
E = spanI1, . . . , I9
Rank of E E φ Mn n
9 spanI1, . . . , I9 Iα ∧ Iβ 7→ Jαβ = Iα Iβ OP2 16
10 spanI0, . . . , I9 Iα ∧ Iβ 7→ Jαβ = Iα Iβ (C⊗O)P2 32
12 spanI−2, . . . , I9 Iα ∧ Iβ 7→ Jαβ = Iα Iβ (H⊗O)P2 64
16 spanI−6, . . . , I9 Iα ∧ Iβ 7→ Jαβ = Iα Iβ (O⊗O)P2 128
1 What you should expect in this talk
2 Spin(9)
3 After Spin(9): Rosenfeld projective planes
4 Applications
5 Conclusion
Wrapping up
Spin(9) is the octonionic version of the quaternionic group Sp(2) · Sp(1).
Spin(9) is the reason why there are more than 7 vector fields on spheres.
The Spin(9) form on R16 = O2 can be extended to C⊗O2, H⊗O2 andO⊗O2, to obtain explicit generators for the cohomology of particularsymmetric spaces called Rosenfeld planes.
Between the Spin(n) groups, Spin(10), Spin(12) and Spin(16) are moreequal than others to Spin(9).
Homology interpretation
Interpretation in terms of the homology of the Rosenfeld planes of thecanonical differential forms living in
H2,H8 H4,H8,H12 H8,H12,H16,H20
EIII EVI EVIII
Extending
Following
Iα1≤α≤9 ∈ End(O2)→ Iα0≤α≤9 ∈ End(C⊗O2)→ . . .
what happens extending the Pauli matrices
I1, I2, I3 ∈ End(C2)→ I0, . . . , I3 ∈ End(C⊗ C2)
and the
I1, . . . , I5 ∈ End(H2)→ I0, . . . , I5 ∈ End(C⊗H2)→ . . .
Subordinated structures
Can we write formulas as in
Ω = ω2I + ω2
J + ω2K
for our Spin(9), . . . ,Spin(16) canonical 8-forms in terms of compatiblequaternonic structures?
Minimal formal definition of Spin(n)
Definition
Cl(n) = Clifford algebra = algebra generated by vectors v ∈ Rn suchthat
v · v = −‖v‖2 · 1
α = canonical involution of Cl(n):
α(v) = −v for vectors v ∈ Rn
Cl0(n) = +1-eigenspace of α.
‖ · ‖ = norm of Cl(n) = extension of ‖ · ‖ to Cl(n).
Spin(n) = x ∈ Cl0(n)|xRnx−1 ⊂ Rnand‖x‖ = 1
Go back
Details for ΦSpin(9) =∫OP1 p∗l νl dl
νl = volume form on the octonionic lines ldef= (x ,mx) or
ldef= (0, y) in O2.
pl : O2 → l = projection on l .
p∗l νl = 8-form in O2 = R16.
The integral over OP1 can be computed over O with polarcoordinates.
The formula arise from distinguished 8-planes in theSpin(9)-geometry → (forthcoming) calibrations.
Go back
Berger and calibrations
Curiosity
Berger appears to know about the fact that ΦSpin(9) is a calibration onOP2 already in 1970 [Berger, L’Enseignement Math. 1970] and more explicitly in 1972[Berger, Ann. Ec. Norm. Sup. 1972, Theorem 6.3], very early with respect to theforthcoming calibration theory.
Time check
ADATTARE E SPOSTAREDo we have at least ... minutes left?
Yes, go ahead as planned No, skip quaternionic analogy
First attempt
Remark
Since Spin(10) ⊂ SU(16), we would like to imitate Spin(9) ⊂ SO(16),and look for I0, . . . , I9 self-adjoint, anti-commuting involutions in C16.
Would-be proposition
Spin(10) ⊂ SU(16) is generated by 10 self-adjoint, anti-commutinginvolutions I0, . . . , I9.
Proposition
C16 with its standard Hermitian scalar product does not admit any familyof 10 self-adjoint, anti-commuting involutions I0, . . . , I9.
First attempt
Remark
Since Spin(10) ⊂ SU(16), we would like to imitate Spin(9) ⊂ SO(16),and look for I0, . . . , I9 self-adjoint, anti-commuting involutions in C16.
Would-be proposition
Spin(10) ⊂ SU(16) is generated by 10 self-adjoint, anti-commutinginvolutions I0, . . . , I9.
Proposition
C16 with its standard Hermitian scalar product does not admit any familyof 10 self-adjoint, anti-commuting involutions I0, . . . , I9.
First attempt
Remark
Since Spin(10) ⊂ SU(16), we would like to imitate Spin(9) ⊂ SO(16),and look for I0, . . . , I9 self-adjoint, anti-commuting involutions in C16.
Would-be proposition
Spin(10) ⊂ SU(16) is generated by 10 self-adjoint, anti-commutinginvolutions I0, . . . , I9.
Proposition
C16 with its standard Hermitian scalar product does not admit any familyof 10 self-adjoint, anti-commuting involutions I0, . . . , I9.
First attempt
Remark
Since Spin(10) ⊂ SU(16), we would like to imitate Spin(9) ⊂ SO(16),and look for I0, . . . , I9 self-adjoint, anti-commuting involutions in C16.
Would-be proposition
Spin(10) ⊂ SU(16) is generated by 10 self-adjoint, anti-commutinginvolutions I0, . . . , I9.
Proposition
C16 with its standard Hermitian scalar product does not admit any familyof 10 self-adjoint, anti-commuting involutions I0, . . . , I9.
351 terms of ΦSpin(9)12345678
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′ 6′ 8
′-1
1358
2′ 3
′ 5′ 7
′1
1358
2′ 3
′ 6′ 8
′1
1358
2′ 4
′ 5′ 8
′1
1358
2′ 4
′ 6′ 7
′1
1367
1′ 3
′ 5′ 8
′-1
1367
1′ 3
′ 6′ 7
′-1
1367
1′ 4
′ 5′ 7
′1
1367
1′ 4
′ 6′ 8
′1
1367
2′ 3
′ 5′ 7
′-1
1367
2′ 3
′ 6′ 8
′-1
1367
2′ 4
′ 5′ 8
′1
1367
2′ 4
′ 6′ 7
′1
1368
1′ 3
′ 6′ 8
′-2
1368
2′ 4
′ 5′ 7
′-2
1378
1′ 2
′ 5′ 7
′1
1378
1′ 2
′ 6′ 8
′1
1378
1′ 3
′ 5′ 6
′-1
1378
1′ 3
′ 7′ 8
′-1
1378
2′ 4
′ 5′ 6
′1
1378
2′ 4
′ 7′ 8
′1
1378
3′ 4
′ 5′ 7
′-1
1378
3′ 4
′ 6′ 8
′-1
13
1′ 2
′ 3′ 4
′ 5′ 7
′2
13
1′ 2
′ 3′ 4
′ 6′ 8
′2
13
1′ 3
′ 5′ 6
′ 7′ 8
′-2
13
2′ 4
′ 5′ 6
′ 7′ 8
′2
145678
1′ 4
′-2
145678
2′ 3
′-2
145678
5′ 8
′2
145678
6′ 7
′-2
1456
1′ 2
′ 5′ 8
′-1
1456
1′ 2
′ 6′ 7
′1
1456
1′ 4
′ 5′ 6
′-1
1456
1′ 4
′ 7′ 8
′-1
1456
2′ 3
′ 5′ 6
′-1
1456
2′ 3
′ 7′ 8
′-1
1456
3′ 4
′ 5′ 8
′1
1456
3′ 4
′ 6′ 7
′-1
1457
1′ 3
′ 5′ 8
′-1
1457
1′ 3
′ 6′ 7
′1
1457
1′ 4
′ 5′ 7
′-1
1457
1′ 4
′ 6′ 8
′1
1457
2′ 3
′ 5′ 7
′-1
1457
2′ 3
′ 6′ 8
′1
1457
2′ 4
′ 5′ 8
′-1
1457
2′ 4
′ 6′ 7
′1
1458
1′ 4
′ 5′ 8
′-2
1458
2′ 3
′ 6′ 7
′-2
1467
1′ 4
′ 6′ 7
′-2
1467
2′ 3
′ 5′ 8
′-2
1468
1′ 3
′ 5′ 8
′-1
1468
1′ 3
′ 6′ 7
′1
1468
1′ 4
′ 5′ 7
′1
1468
1′ 4
′ 6′ 8
′-1
1468
2′ 3
′ 5′ 7
′1
1468
2′ 3
′ 6′ 8
′-1
1468
2′ 4
′ 5′ 8
′-1
1468
2′ 4
′ 6′ 7
′1
1478
1′ 2
′ 5′ 8
′1
1478
1′ 2
′ 6′ 7
′-1
1478
1′ 4
′ 5′ 6
′-1
1478
1′ 4
′ 7′ 8
′-1
1478
2′ 3
′ 5′ 6
′-1
1478
2′ 3
′ 7′ 8
′-1
1478
3′ 4
′ 5′ 8
′-1
1478
3′ 4
′ 6′ 7
′1
14
1′ 2
′ 3′ 4
′ 5′ 8
′2
14
1′ 2
′ 3′ 4
′ 6′ 7
′-2
14
1′ 4
′ 5′ 6
′ 7′ 8
′-2
14
2′ 3
′ 5′ 6
′ 7′ 8
′-2
1567
1′ 2
′ 3′ 5
′-1
1567
1′ 2
′ 4′ 6
′1
1567
1′ 3
′ 4′ 7
′1
1567
1′ 5
′ 6′ 7
′-1
1567
2′ 3
′ 4′ 8
′1
1567
2′ 5
′ 6′ 8
′-1
1567
3′ 5
′ 7′ 8
′-1
1567
4′ 6
′ 7′ 8
′1
1568
1′ 2
′ 3′ 6
′-1
1568
1′ 2
′ 4′ 5
′-1
1568
1′ 3
′ 4′ 8
′1
1568
1′ 5
′ 6′ 8
′-1
1568
2′ 3
′ 4′ 7
′-1
1568
2′ 5
′ 6′ 7
′1
1568
3′ 6
′ 7′ 8
′-1
1568
4′ 5
′ 7′ 8
′-1
1578
1′ 2
′ 3′ 7
′-1
1578
1′ 2
′ 4′ 8
′-1
1578
1′ 3
′ 4′ 5
′-1
1578
1′ 5
′ 7′ 8
′-1
1578
2′ 3
′ 4′ 6
′1
1578
2′ 6
′ 7′ 8
′1
1578
3′ 5
′ 6′ 7
′1
1578
4′ 5
′ 6′ 8
′1
15
1′ 2
′ 3′ 5
′ 6′ 7
′-2
15
1′ 2
′ 4′ 5
′ 6′ 8
′-2
15
1′ 3
′ 4′ 5
′ 7′ 8
′-2
15
2′ 3
′ 4′ 6
′ 7′ 8
′2
1678
1′ 2
′ 3′ 8
′-1
1678
1′ 2
′ 4′ 7
′1
1678
1′ 3
′ 4′ 6
′-1
1678
1′ 6
′ 7′ 8
′-1
1678
2′ 3
′ 4′ 5
′-1
1678
2′ 5
′ 7′ 8
′-1
1678
3′ 5
′ 6′ 8
′1
1678
4′ 5
′ 6′ 7
′-1
16
1′ 2
′ 3′ 5
′ 6′ 8
′-2
16
1′ 2
′ 4′ 5
′ 6′ 7
′2
16
1′ 3
′ 4′ 6
′ 7′ 8
′-2
16
2′ 3
′ 4′ 5
′ 7′ 8
′-2
17
1′ 2
′ 3′ 5
′ 7′ 8
′-2
17
1′ 2
′ 4′ 6
′ 7′ 8
′2
17
1′ 3
′ 4′ 5
′ 6′ 7
′2
17
2′ 3
′ 4′ 5
′ 6′ 8
′2
18
1′ 2
′ 3′ 6
′ 7′ 8
′-2
18
1′ 2
′ 4′ 5
′ 7′ 8
′-2
18
1′ 3
′ 4′ 5
′ 6′ 8
′2
18
2′ 3
′ 4′ 5
′ 6′ 7
′-2
Go
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