Clifford systems, Clifford structures, and their canonical differential forms
aa r X i v : . [ m a t h . DG ] A ug CLIFFORD SYSTEMS, CLIFFORD STRUCTURES,AND THEIR CANONICAL DIFFERENTIAL FORMS
KAI BRYNNE M. BOYDON AND PAOLO PICCINNI
Abstract.
A comparison among different constructions in H ∼ = R of the quater-nionic 4-form Φ Sp(2)Sp(1) and of the Cayley calibration Φ
Spin(7) shows that one canstart for them from the same collections of ”K¨ahler 2-forms”, entering both in quater-nion K¨ahler and in Spin(7) geometry. This comparison relates with the notions ofeven Clifford structure and of Clifford system. Going to dimension 16, similar con-structions allow to write explicit formulas in R for the canonical 4-forms Φ Spin(8) and Φ
Spin(7)U(1) , associated with Clifford systems related with the subgroups Spin(8)and Spin(7)U(1) of SO(16). We characterize the calibrated 4-planes of the 4-formsΦ
Spin(8) and Φ
Spin(7)U(1) , extending in two different ways the notion of Cayley 4-planeto dimension 16. Introduction
In 1989 R. Bryant and R. Harvey defined the following calibration, of interest inhyperk¨ahler geometry [6]:Φ K = − ω R i − ω R j + 12 ω R k ∈ Λ H n . In this definition, ( ω R i , ω R j , ω R k ) are the K¨ahler 2-forms of the hypercomplex structure( R i , R j , R k ), defined by multiplications on the right by unit quaternions ( i, j, k ) on thespace R n ∼ = H n .When n = 2, the Bryant-Harvey calibration Φ K relates with Spin(7) geometry. Thisis easily recognized by using the map L : H → O , L ( h , h ) = h + ( kh ¯ k ) e ∈ O , from pairs of quaternions to octonions, that yields the identity(1.1) L ∗ Φ Spin(7) = Φ K . Mathematics Subject Classification.
Primary 53C26, 53C27, 53C38.
Key words and phrases.
Octonions, Clifford system, Clifford structure, calibration, canonical form.The first author was supported by University of the Philippines OVPAA Doctoral Fellowship. Partof the present work was done during her visit at Sapienza Universit`a di Roma in the academic year2018-19, and she thanks Sapienza University and Department of Mathematics ”Guido Castelnuovo” forhospitality.The second author was supported by the group GNSAGA of INdAM, by the PRIN Project of MIUR“Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics” and by Sapienza Uni-versit`a di Roma Project “Polynomial identities and combinatorial methods in algebraic and geometricstructures”.
Here Φ
Spin(7) ∈ Λ R is the Spin(7) 4-form, or Cayley calibration , studied since theR. Harvey and H. B. Lawson’s foundational paper [11], and defined through the scalarproduct and the double cross product of R ∼ = O :Φ Spin(7) ( x, y, z, w ) = < x , y × z × w > = < x , y (¯ zw ) >, assuming here orthogonal y, z, w ∈ O .The present paper collects some of the results in the first author Ph.D. thesis [3],inspired from viewing formula (1.1) as a way of constructing the Cayley calibrationΦ Spin(7) through the 2-forms ω R i , ω R j , ω R k . As well known, by summing the squares ofthe latter 2-forms one gets another remarkable calibration, namely the quaternionic right4-form Ω R . Thus ω R i , ω R j , ω R k , somehow building blocks for quaternionic geometry,enter also in Spin(7) geometry.A first result is the following Theorem 1.1, a kind of ”other way around” of formula(1.1). To state it, recall that the Cayley calibration Φ Spin(7) can also be constructedas sum of squares of ”K¨ahler 2-forms” associated with complex structures on R ∼ = O ,defined by the unit octonions. In fact, cf. [18, Prop.10]:(1.2) Φ Spin(7) = −
16 ( φ i + φ j + · · · + φ h ) = 16 ( ϕ ij + φ ik + ϕ ik + · · · + ϕ gh ) . Here φ i , φ j , . . . , φ h are the K¨ahler 2-forms associated with the 7 complex structures R i , R j , . . . , R h on R ∼ = O , the right multiplications by the unit octonions i, j, k, e, f, g, h ,and φ ij , φ ik , . . . , φ gh are the K¨ahler 2-forms associated with the 21 complex structures R ij = R i ◦ R j , R ik = R i ◦ R k , . . . , R gh = R g ◦ R h , compositions of them. Theorem 1.1.
The right quaternionic 4-form Ω R ∈ Λ H can be obtained from the theK¨ahler forms φ i , φ j , . . . , φ h associated with the complex structures R i , R j , . . . , R h as: Ω R = 2[ φ i + φ j + φ k − φ e − φ f − φ g − φ h ] . Moreover, by selecting any five out of the seven ( J = R i , J = R j , . . . , J = R h ) andby looking at the matrix ζ = ( ζ αβ ) ∈ so (5) of K¨ahler 2-forms of their compositions J αβ = J α ◦ J β , one can get the left quaternionic 4-form Ω L as Ω L = − X α<β ζ αβ , up to a permutation or change of signs of some coordinates in R . On the same direction as in Bryant-Harvey’s formula (1.1), a similar result is thefollowing (cf. Section 4 for more details):
Theorem 1.2.
The Cayley calibration Φ Spin(7) ∈ Λ R can be obtained from the K¨ahler2-forms η αβ (1 ≤ α < β ≤ associated to complex structures J Lαβ = I Lα ◦ I Lβ , where I L , . . . I L are anti-commuting self-dual involutions in R . Namely: Φ Spin(7) = 14 (cid:2) η + η + η + η − η − η − η − η − η − η (cid:3) , and on the other hand one can get the right quaternion K¨ahler Ω R as: Ω R = − (cid:2) η + η + η + η + η + η + η + η + η + η (cid:3) . LIFFORD SYSTEMS, CLIFFORD STRUCTURES 3
Moving to dimension 16 and in Section 6, we consider two exterior 4-formsΦ
Spin(8) , Φ Spin(7)U(1) ∈ Λ R , canonically associated with subgroups Spin(8) , Spin(7)U(1) ⊂ SO(16), and we writetheir explicit expressions in the 16 coordinates. We will see in the next two statementswhich 4-planes of R are calibrated by Φ Spin(8) and by Φ
Spin(7)U(1) .The 4-forms Φ
Spin(8) , Φ Spin(7)U(1) and the respective calibrated 4-planes can be com-pared with other calibrations in R , in particular with the previously mentioned Bryant-Harvey 4-form Φ K . It is thus appropriate to remind the main theorem in [6, Theorem2.27], namely that, in any H n ∼ = R n , the Bryant-Harvey 4-form Φ K calibrates the Cayley4-planes that are contained in a quaternionic 2-dimensional vector subspace W H ⊂ H n .Here we prove: Theorem 1.3.
The oriented 4-planes of R ∼ = O calibrated by the 4-form Φ Spin(8) arethe transversal Cayley 4-planes , i.e. the -planes P such that both projections π ( P ) , π ′ ( P ) on the two summands in O = O ⊕ O ′ are two dimensional and both invariant bya same complex structure u ∈ S ⊂ Im O . Also, by recalling that O decomposes in the union of octonionic lines ℓ m = { ( x, mx ) , x ∈ O , m ∈ O ∪ ∞} , meeting pairwise only at (0 , ∈ O , we can state: Theorem 1.4.
The oriented 4-planes of R calibrated by the 4-form Φ Spin(7)U(1) are theones that are invariant under a complex structure u ∈ S ⊂ Im O and that are containedin an octonionic line ℓ m ⊂ O ⊕ O , where only m ∈ R and m = ∞ are allowed. Thusthey are ”Cayley 4-planes”, contained in the oriented 8-planes that are the mentionedoctonionic lines with m ∈ R ∪ ∞ . Preliminaries
The multiplication in the algebra O of octonions can be defined from the one inquaternions H through the Cayley-Dickson process: if x = h + h e, x ′ = h ′ + h ′ e ∈ O ,then xx ′ = ( h h ′ − ¯ h ′ h ) + ( h ¯ h ′ + h ′ h ) e, where product of quaternions is used on the right hand side and ¯ h ′ , ¯ h ′ are the conjugatesof h ′ , h ′ ∈ H . Like for quaternions, the conjugation ¯ x = ¯ h − h e in O relates with thenon-commutativity: xx ′ = ¯ x ′ ¯ x . One has also the associator [ x, x ′ , x ′′ ] = ( xx ′ ) x ′′ − x ( x ′ x ′′ ), that vanishes whenever two among x, x ′ , x ′′ ∈ O are equal or conjugate.The identification x = h + h e ∈ O ↔ ( h , h ) ∈ H , used in the previous formula, isnot an isomorphism of (left or right) quaternionic vector spaces. To get an isomorphismone has instead to go through the following hypercomplex structure ( I, J, K ) on R ∼ = O .For x = h + h e ∈ O , where ( h , h ) ∈ H , define I ( x ) = x · i, J ( x ) = x · j, K ( x ) = ( x · i ) · j or equivalently I ( h , h ) = ( h i, − h i ) , J ( h , h )) = ( h j, − h j ) , K ( x ) = ( h k, h k ) . K. B. BOYDON AND P. PICCINNI
This observation goes likely back to the very discovery of octonions in the mid-1800s.The alternative approach to the same isomorphism used in our Introduction does notseem however to have appeared before 1989, when R. Bryant and R. Harvey [6] lookedat the map L : H → O , L ( h , h ) = h + ( kh ¯ k ) e ∈ O , and observed it satisfies L [( h , h ) i ] = h i +( kh i ¯ k ) e, L [( h , h ) j ] = h j +( kh j ¯ k ) e, L [( h , h ) k ] = h k +( kh ) e. This, in terms of x = h , x = kh ¯ k and of the octonion x = x + x e , can be readexactly as in our previous approach: L [( h , h ) i ] = x · i, L [( h , h ) j ] = x · j, L [( h , h ) k ] = ( x · i ) · j, and as mentioned L ∗ Φ Spin(7) = Φ K .3. The quaternionic -form and the Cayley calibration in R A possible way to produce 4-forms canonically associated with some G -structures isthrough the notion of Clifford system . We recall the definition, originally given in thecontext of isoparametric hypersurfaces, cf. [9].
Definition 3.1. A Clifford system on a Riemannian manifold (
M, g ) is a vector sub-bundle E r ⊂ End
T M locally spanned by self-adjoint anti-commuting involutions I , . . . , I r . Thus I α = Id , I ∗ α = I α , I α ◦ I β = −I β ◦ I α , and the I α are requiredto be related, in the intersections of trivializing sets, by matrices of SO( r ). The rank r of E is said to be the rank of the Clifford system.Possible ranks of irreducible Clifford systems on R N are classified, up to N = 32, asfollows: Table A.
Rank of irreducible Clifford systems in R N dimension N r R can be defined by the classical Paulimatrices: I = (cid:18) (cid:19) , I = (cid:18) − ii (cid:19) , I = (cid:18) − (cid:19) ∈ U(2) ⊂ SO(4) , LIFFORD SYSTEMS, CLIFFORD STRUCTURES 5 and the Clifford system of rank 5 in R by the following similar (right) quaternionicPauli matrices : I = (cid:18) (cid:19) , I = (cid:18) − R i R i (cid:19) , I = (cid:18) − R j R j (cid:19) , I = (cid:18) − R k R k (cid:19) , I = (cid:18) Id 00 − Id (cid:19) ∈ Sp(2) ⊂ SO(8) , (3.1)where as before R i , R j , R k denote the multiplication on the right by i, j, k on H ∼ = R .According to Table, A, there is also a Clifford system with r = 4 in R , explicitlydefined by selecting e.g. I = (cid:18) (cid:19) , I = (cid:18) − R i R i (cid:19) , I = (cid:18) − R j R j (cid:19) , I = (cid:18) − R k R k (cid:19) . Going back to rank r = 5, from the quaternionic Pauli matrices I , I , I , I , I , onegets the 10 complex structures on R I αβ = I α ◦ I β for 1 ≤ α < β ≤ . Their K¨ahler forms θ αβ give rise to a 5 × θ R = ( θ αβ ) , and one can easily see that both the following matrices of K¨ahler 2-forms θ R = ( θ αβ ) ∈ so (5) and ω L = ω L i ω L j − ω L i ω L k − ω L j − ω L k ∈ so (3)allow to write the (left) quaternionic 4-form of H as(3.2) Ω L = − X α<β θ αβ = [ ω L i + ω L j + ω L k ] . On the other hand, as mentioned in the Introduction, the subgroup Spin(7) ⊂ SO(8)(generated by the right translation R u , u ∈ S ⊂ Im O ) gives rise to the Cayley calibra-tion Φ Spin(7) ∈ Λ : Φ Spin(7) = −
16 [ φ i + φ j + . . . φ h ] = 16 X α<β ζ αβ . Here φ i , φ j , φ k , φ e , φ f , φ g , φ h . are the K¨ahler 2-forms associated with the complex struc-tures ( J , J , J , J , J , J , J ) = ( R i , R j , R k , R e , R f , R g , R h ), and ζ = ( ζ αβ ) ∈ so (7) isthe matrix of the K¨ahler 2-forms of compositions J αβ = J α ◦ J β .It is worth to recall that under the action of Sp(2)Sp(1), the space of exterior 2-formsΛ R decomposes as Λ = Λ ⊕ Λ ⊕ Λ , where lower indices denote the dimensions of irreducible components. Here Λ ∼ = sp (2)is generated by the K¨ahler forms θ αβ of the J αβ ( α < β ), compositions of the five quater-nionic Pauli matrices, and Λ ∼ = sp (1) is generated by the K¨ahler forms ω L i , ω L j , ω L k . K. B. BOYDON AND P. PICCINNI
By denoting by τ the second coefficient in the characteristic polynomial of the involvedskew-symmetric matrices, we can rewrite formula (3.2) of Ω L as:Ω L = − τ ( θ R ) = τ ( ω L )where θ R = ( θ αβ ) ∈ so (5), and ω L = ω L i ω L j − ω L i ω L k − ω L j − ω L k ∈ so (3).Similarly, under the Spin(7) action one gets the decomposition:Λ = Λ ⊕ Λ , where Λ is generated by the K¨ahler forms φ α of the J α = R i , R j , . . . , R h and Λ ∼ = spin (7) is generated by the K¨ahler forms ζ αβ of the J α ◦ J β ( α < β ). Thus, in the τ notation: Φ Spin(7) = − X φ α = 16 τ ( ζ ) , ζ = ( ζ αβ ) ∈ so (7) . All the exterior 4-forms θ αβ , φ α and the ζ αβ have been studied systematically as cali-brations in the space R , cf. [8].4. Proof of Theorems 1.1 and 1.2
The matrix η = ( η αβ ) ∈ so (5) in the statement of Theorem 1.2 is defined as follows.Let I Lα ( α = 1 , . . . ,
5) be the left quaternionic Pauli matrices defined as in (3.1) but byusing the left quaternionic multiplications L i , L j , L k by i, j.k . If J Lαβ = I Lα ◦ I Lβ and if η αβ are the K¨ahler 2-forms associated to J Lαβ , a computation shows that2Ω R = η + η + η + η + η + η + η + η + η + η , and note the symmetry with the first identity in formula (3.2).We express now the 2-forms θ αβ and η αβ in the coordinates of R , using the followingabridged notations. Let { dx , . . . , dx } ⊂ Λ R be the standard basis of 1-forms in R .Then αβ (scriptsize) denotes dx α ∧ dx β and αβγδ denotes dx α ∧ dx β ∧ dx γ ∧ dx δ , and ⋆ denotes the Hodge star, so that a + ⋆ = a + ⋆a . One gets:(4.1) θ = − + + − , θ = − − + + , θ = − + + − ,θ = − + − + , θ = + + + + , θ = − + − + , and(4.2) θ = − − − − , θ = − + + − ,θ = − − + + , θ = − + − + , so that, if θ = ( θ αβ ) LIFFORD SYSTEMS, CLIFFORD STRUCTURES 7 τ ( θ ) = θ + θ + · · · + θ == − − − + 4 − − − + ⋆ = − L . (4.3)Next:(4.4) η = − − + + , η = − + + − , η = − − + + ,η = + + + + , η = − + − + , η = + + + + ,η = − − − − , η = − + − + ,η = − + + − , η = − − + + , and, if η = ( η αβ ), τ ( η ) = η + η + · · · + η == 12 − − − + 4 + 4 − + ⋆ = − R . (4.5)Similarly:(4.6) φ i = − + + − , φ j = − − + + , φ k = − + + − ,φ e = − − − − , φ f = − + − + ,φ g = − + + − , φ h = − − + + , and one easily deduce also formulas for the ζ αβ (cf. [18], [3]). Then, by (1.2):(4.7) Φ Spin(7) = + + + − − + + ⋆ . By computing the squares of the 2-forms in (4.6) (4.4) and comparing with Formulas(4.5), (4.3), (4.7), the identities listed in Theorems 1.1 and 1.2 are recognized.5.
Even Clifford structures in dimension
Definition 5.1.
Let (
M, g ) be a Riemannian manifold. An even Clifford structure isthe choice of an oriented Euclidean vector bundle E r of rank r ≥ M , togetherwith a bundle morphism ϕ from the even Clifford algebra bundle ϕ : Cl even E r → End
T M such that Λ E r ֒ → End − T M.r is called the r ank of the even Clifford structure.The even Clifford structure E is said to be parallel if there exists a metric connection ∇ E on E such that ϕ is connection preserving, i.e. ϕ ( ∇ EX σ ) = ∇ gX ϕ ( σ ) , for every tangent vector X ∈ T M and section σ of Cl even E , where ∇ g is the Levi Civitaconnection. K. B. BOYDON AND P. PICCINNI
Rank 2 , , Table B.
Parallel non-flat even Clifford structures of rank ≥ M r M quaternion K¨ahler K¨ahler Spin(7) holonomy Riemannian
A class of examples of even Clifford structures are those coming from Clifford systemsas defined in Section 3. Namely, if the vector sub-bundle E r ⊂ End
T M , locally spannedby self-adjoint anticommuting involutions I , . . . , I r , defines the Clifford system, thenone easily recognizes that through the compositions J αβ = I α ◦I β , the Clifford morphism ϕ : Cl even ( E r ) → End
T M is well defined.An example is given by the first row of the former Table, where the quaternion K¨ahlerstructure is constructed via the local J αβ = I α ◦ I β defined as in Section 3, by usingon the model space H the quaternionic Pauli matrices. The remaining three rows ofthe former Table correspond to essential even Clifford structures, i. e. to even Cliffordstructures that cannot be defined throw a Clifford system, cf. [20] for a discussion onthis notion.The following Table gives a description of the Clifford bundle generators and of thecanonically associated 4-form for each of the four even Clifford structures on R . Table C.
Generators and associated 4-formsr M Clifford bundle generators associated 4-form5 qK I , I , I , I , I Ω L = − τ ( θ αβ ) , Ω R = − τ ( η αβ )6 K¨ahler J , J , J , J , J , J Φ Spin(6) = τ ( ζ αβ ) = − ω J , J , J , J , J , J , J Φ Spin(7) = − P φ α = τ ( ζ αβ )8 Riemannian I, J , J , J , J , J , J , J Φ SO(8) = τ ( ψ αβ ) = 0Here I , I , I , I , I are the (left or right) quaternionic Pauli matrices, ( θ αβ ), ( η αβ )are like in Section 4. Notations ( J , J , J , J , J , J , J ) = ( R i , R j , R k , R e , R f , R g , R h ) LIFFORD SYSTEMS, CLIFFORD STRUCTURES 9 are also used, φ α is the K¨ahler form of J α and ζ αβ is the K¨ahler form of J α ◦ J β . Finally,( ψ αβ ) ∈ so (8), with entries ± φ α in the first line and column and with entries ζ αβ ∈ so (7).It is of course desirable to give examples of Riemannian manifolds ( M , g ) supportingboth a Sp(2) · Sp(1) and a Spin(7) structure. Rarely the metric g can be the same forboth structures, but this is possible of course for parallelizable ( M , g ). On this respect,homogeneous ( M , g ) with an invariant Spin(7) structure have been recently classified[1], by making use of the following topological condition for compact oriented spin M : p ( M ) − p ( M ) + 8 χ ( M ) = 0 . Some of the obtained examples are parallelizable, e. g. diffeomorphic to S × S and S × S . On the latter, S × S , using two natural parallelizations, one can define twoSpin(7) structures, both of general type (in the 1986 M. Fernandez Spin(7) framework),and the hyperhermitian structure associated with one of them corresponds to a familyof Calabi-Eckmann [19].To get examples of 8-dimensional manifolds that admit both a locally conformallyhyperk¨ahler metric g and a locally conformal parallel Spin(7) metric, that is either thesame g as before, or a different metric g ′ , a good point to start with is the class ofcompact 3-Sasakian 7-dimensional manifolds ( S , g ). Many examples of such ( S , g )and with arbitrary second Betti numbers have been given by Ch. Boyer -K. Galicki et al , cf. [5]. In particular, recall that given the 3-Sasakian ( S , g ) one gets a locallyconformally hyperk¨ahler metric g on the product S × S [17]. This can also be expressedby saying that the 3-Sasakian metric g has the property of being nearly parallel G ,and in particular with 3 linearly independent Killing spinors, cf. [4, pages 536-538].Moreover the differentiable manifold S admits, besides the 3-Sasakian metric g , anothermetric g ′ that is also nearly parallel G but proper, i. e. with only one non zeroKilling spinor. This allows to extend the metrics g and g ′ to the product with S andto get both the properties of locally conformally hyperk¨ahler and locally conformallyparallel Spin(7) on ( M , g ) = ( S × S , g ) and of locally conformally parallel Spin(7) on( M , g ′ ) = ( S × S , g ′ ), cf. also [12].Further examples of 8-dimensional differentiable manifolds admitting both a Sp(2) · Sp(1)-structure with respect to a metric g and a Spin(7)-structure with respect to ametric g ′ include the Wolf spaces H P and G / SO(4), cf. [1]. Finally, the non singularsextic Y = { [ z , . . . , z ] ∈ C P , z + · · · + z = 0 } is also an example, where a metric g giving an almost quaternionic structure is insured by a result in [7], and a metric g ′ with holonomy SU(4) ⊂ Spin(7) by Calabi-Yau theorem, cf [13, p. 139].6.
Dimension
16A Clifford system with r = 9 in O ∼ = R is given by the following octonionic Paulimatrices : I = (cid:18) (cid:19) , I = (cid:18) − R i R i (cid:19) , I = (cid:18) − R j R j (cid:19) , I = (cid:18) − R k R k (cid:19) , I = (cid:18) − R e R e (cid:19) , I = (cid:18) − R f R f (cid:19) , I = (cid:18) − R g R g (cid:19) , I = (cid:18) − R h R h (cid:19) , I = (cid:18) Id 00 − Id (cid:19) ∈ SO(16) , and of course now R i , R j , . . . , R h denote the multiplication on the right by the unitoctonions i, j, . . . , h on O ∼ = R .Looking back at Table A, we see that in R there are also irreducible Clifford systemswith r = 8 , ,
6. According to [20], convenient choices are the following: r = 8 : I , . . . , I , r = 7 : I , . . . , I , r = 6 : I , I , I , I , I , I . It is now worth to remind the following parallel situations in complex, quaternionic andoctonionic geometry. The groups U(2) ⊂ SO(4), Sp(2) · Sp(1) ⊂ SO(8), Spin(9) ⊂ SO(16)are the stabilizers of the vector subspaces E ⊂ End + ( R ) , E ⊂ End + ( R ) , E ⊂ End + ( R )spanned respectively by the Pauli, quaternionic Pauli, octonionic Pauli matrices.Moreover, U(2), Sp(2) · Sp(1), Spin(9) are symmetry groups of the Hopf fibrationsrespectively: S S −→ S ∼ = C P , S S −→ S ∼ = H P , S S −→ S ∼ = O P . Finally, U(2), Sp(2) · Sp(1), Spin(9) are stabilizers in Λ C , Λ H , Λ O of the following canonically associated forms , cf. [2]:Φ U(2) = Z C P p ∗ ℓ ν ℓ dℓ ∈ Λ , Φ Sp(2) · Sp(1) = Z H P p ∗ ℓ ν ℓ dℓ ∈ Λ , Φ Spin(9) = Z O P p ∗ ℓ ν ℓ dℓ ∈ Λ , where ν ℓ is the volume form on the line ℓ def = { ( x, mx ) } or ℓ def = { (0 , y ) } in C or H or O , p ℓ : C ∼ = R or H ∼ = R or O ∼ = R −→ ℓ is the projection on the line ℓ ), and note that the integral formula is based on the volumeof distinguished planes. In the three cases one gets in this way the K¨ahler 2-form of C ,the quaternion K¨ahler 4-form of H and the canonical 8-form of O .7. Rank , and Clifford systems on R Look now closer at the nine octonionic Pauli matrices, that define a rank 9 Cliffordsystem in R , and at the choices among them that give rise to ranks r = 8 , , I αβ = I α ◦ I β , α < β , for all choices r = 6 , , , spin (6) ⊂ spin (7) ⊂ spin (8) ⊂ spin (9) ⊂ so (16) . Like in the previous Sections, we can write the matrices of K¨ahler forms ψ αβ associatedto I αβ , and we use for them the following notations: ψ A = ( ψ αβ ) ∈ so (6) , ψ B = ( ψ αβ ) ∈ so (7) , ψ C = ( ψ αβ ) ∈ so (8) , ψ D = ( ψ αβ ) ∈ so (9) . The second coefficients τ of their characteristic polynomial give rise to the followinginvariant 4-forms τ ( ψ A ) , τ ( ψ B ) , τ ( ψ C ) , τ ( ψ D ) ∈ Λ R that can be written in (the differentials of) the coordinates of R = O ⊕ O : , , , , , , , ′ , ′ , ′ , ′ , ′ , ′ , ′ , ′ . We recall in particular that in the Spin(9) situation, the following identity holds: τ ( ψ D ) = 0 , and this gives evidence to the next coefficient τ ( ψ D ) ∈ Λ , proportional to the 8-formΦ Spin(9) , as studied in [18].8.
The -forms Φ Spin(8) and Φ Spin(7)U(1)
Look now only at I , . . . , I and at the matrix ψ C = ( ψ αβ ) ∈ so (8)of K¨ahler forms associated to I αβ = I α ◦ I β . By using coordinate 1-forms , , , , , , , ′ , ′ , ′ , ′ , ′ , ′ , ′ , ′ , an explicit computations based on the explicit formulas for the ψ αβ in [18] yields:Φ Spin(8) = τ ( ψ C ) =
14 8 X α<β ψ αβ = + − + + + − − + + + − + + + X a
Spin(7)U(1) restricts, on any of thetwo summands of R = R ⊕ R , and up to a factor 6, to the usual Cayley calibrationof [11].We are now ready for the proofs of Theorems 1.3 and 1.4. The following notion hasalready implicitly introduced in the statement of Theorem 1.3. Definition 8.1.
Let P be a 4-plane in the real vector space R ∼ = O = O ⊕ O ′ , andlet π : O → O and π ′ : O → O ′ be the orthogonal projections to O and O ′ . P is saidto be a transversal Cayley -plane if both its projections π ( P ), π ′ ( P ) are 2-dimensionaland invariant under a same complex structure u ∈ S ⊂ Im O . Proof of Theorem 1.3 . Recall that Spin(8) can be characterized as the subgroup of thefollowing matrices A ∈ SO(16): A = (cid:18) a + a − (cid:19) where a + , a − ∈ SO(8) are triality companions , i. e. and for any v ∈ O there existsa w ∈ O such that R w = a + R v a t − (cf. [10, p. 278-279]). It follows that Spin(8)contains the diagonal Spin(7) ∆ (characterized by choices a + = a − ) and acts transitivelyon transversal 4-planes of R . On the other hand the 4-form Φ Spin(8) is invariant underthe action of Spin(8). Thus, since Φ
Spin(8) takes value 1 on the 4-plane spanned by thecoordinates ′ ′ , Φ Spin(8) takes value 1 on any tranversal Cayley 4-plane in R .Next, let Q be any 4-plane of R . By looking at the expression of Φ Spin(8) , we seethat the only possibilities for having non zero value on Q are that π ( Q ) and π ′ ( Q ) are2-dimensional. For such 4-planes Q we can use the following canonical form with respectto the complex structure i ∈ S : Q = (cid:2) e ∧ ( R i e cos θ + e sin θ ) (cid:3) ⊕ (cid:2) e ′ ∧ ( R i e ′ cos θ ′ + e ′ sin θ ′ ) (cid:3) , where the pairs e , e and e ′ , e ′ are both orthonormal and respectively in O and in O ′ ,and with angles limited by 0 ≤ θ ≤ π and θ ≤ θ ′ ≤ π − θ . The above canonical formfor Q is a small variation of the canonical forms that are used in a proof of the classicalWirtinger’s inequaliy (cf. [15, p.6]) and in characterizations of Cayley 4-planes in R in the Harvey-Lawson foundational paper (cf. [11, p. 121]). Its proof follows the stepsof proof of the mentioned canonical form, as explained in details in [15]. From thiscanonical form we see that Φ Spin(8) ( Q ) ≤ Q , and that the equalityholds only if θ = θ ′ = 0, i. e. for transversal Cayley 4-planes. Proof of Theorem 1.4 . The leading terms in the expression of Φ
Spin(7)U(1) are those withcoefficient 6, thus terms involving only coordinates among , or only coordinatesamong ′ ′ ′ ′ ′ ′ ′ ′ , or terms aba ′ b ′ . Look first at the first and second types of terms.We already mentioned that the restriction of Φ Spin(7)U(1) to any of the summands in
LIFFORD SYSTEMS, CLIFFORD STRUCTURES 13 O = O ⊕ O ′ is the usual Cayley calibration in R , whose calibrated 4-planes are theCayley planes. Thus, for the first two types of terms, we get as calibrated 4-planesjust the Cayley 4-planes that are contained in the octonionic lines with slope m = 0and m = ∞ . In the remaining case of terms aba ′ b ′ one gets as calibrated 4-planesthe transversal Cayley 4-planes that are contained in the octonionic line ℓ (leadingcoefficient m = 1). Now Spin(7) acts on the individual octonionic lines ℓ , ℓ , ℓ ∞ , andthe only possibility to move planes out of them is through the factor U(1). In fact, thediscussion in [3, Chapter 6, p. 44] shows that the factor U(1) in the group Spin(7)U(1)moves the octonionic lines through the circle, contained in the space S of the octonioniclines, passing through the three points m = 0 , , ∞ . This corresponds to admitting anyreal coefficient: m ∈ R ∪ ∞ as slope of the octonionic lines that are admitted to containthe calibrated 4-planes. Remark . Following the recent work [14] by J. Kotrbat´y, one can use octonionic1-forms, according to the following formal definitions: dx = dα + idβ + jdγ + kdδ + edǫ + f dζ + gdη + hdθ,dx = dα − idβ − jdγ − kdδ − edǫ − f dζ − gdη − hdθ,dx ′ = dα ′ + idβ ′ + jdγ ′ + kdδ ′ + edǫ ′ + f dζ ′ + gdη ′ + hdθ ′ ,dx ′ = dα ′ − idβ ′ − jdγ ′ − kdδ ′ − edǫ ′ − f dζ ′ − gdη ′ − hdθ ′ , referring to pairs of octonions ( x, x ′ ) ∈ O ⊕ O = R . Then, in the same spirit proposedin [14], a straightforward computation yields the following formula, much simpler wayto write the Spin(8) canonical 4-form of R :Φ Spin(8) = 14 ( dx ∧ dx ′ ) ∧ ( dx ′ ∧ dx ) . Similarly, one gets that the Spin(7)U(1) canonical 4-form of R can be written in octo-nionic 1-forms as:Φ Spin(7)U(1) = 14 (cid:2) ( dx ∧ dx ) + ( dx ′ ∧ dx ′ ) (cid:3) − (cid:2) ( dx ∧ dx ′ ) + ( dx ′ ∧ dx ) (cid:3) − (cid:2) ( dx ∧ dx ′ ) ∧ ( dx ′ ∧ dx ) (cid:3) . Details of both computations are in [3].
References [1] D. V. Alekseevsky, I. Chrysikos, A. Fino and A. Raffero Homogeneous 8-manifolds admitting in-variant Spin(7)-structures.
Int. J. Math. , 31, 2020. https://doi.org/10.1142/S0129167X20500603[2] M. Berger. Du cˆot´e de chez Pu.
Ann. Sci. ´Ecole Norm. Sup. (4) , 5:1–44, 1972.[3] K. B. Boydon. Clifford Systems and Clifford Structures with their canonical associated 4-forms indimensions 8 and 16.
Dissertation for the Degree of Doctor of Philosophy in Mathematics , Universityof the Philippines, 2020. [4] Ch. P. Boyer and K. Galicki. 3-Sasakian manifolds.
Surveys in differential geometry vol. VI: essayson Einstein manifolds , pages 123–184. Int. Press, Boston, MA, 1999.[5] Ch. P. Boyer, K. Galicki, B. M. Mann and E. Rees. Compact 3-Sasakian 7-manifolds with arbitrarysecond Betti number.
Invent. Math. , 131:321-344, 1998.[6] R. L. Bryant and R. Harvey. Submanifolds in hyper-K¨ahler Geometry.
J. Am. Math. Soc.
Osaka J. of Math. ,35(1):165–190, 1998.[8] J. Dadok, R. Harvey and F. Morgan. Calibrations on R . Trans. Am. Math. Soc.
Math. Z.
Spinors and calibrations . Academic Press Inc., 1990.[11] R. Harvey and H. B. Lawson Jr. Calibrated Geometries.
Acta Math ,148:47-157, 1982.[12] S. Ivanov, M. Parton, and P. Piccinni. Locally conformal parallel G and Spin(7) manifolds. Math.Res. Lett. , 13(2-3):167–177, 2006.[13] D. D. Joyce.
Compact Manifolds with Special Holonomy . Oxford University Press, 2000.[14] J. Kotrbat´y. Octonion-valued forms and the canonical 8-form on Riemannian manifolds with aSpin(9)-structure. J Geom Anal (2019). https://doi.org/10.1007/s12220-019-00209-z[15] J. D. Lotay. Calibrated Submanifolds. arXiv:1810.08709v1, 2018.[16] A. Moroianu and U. Semmelmann. Clifford structures on Riemannian manifolds
Adv. Math. , 228:940-967, 2011.[17] L. Ornea and P. Piccinni. Locally conformal K¨ahler structures in quaternionic geometry.
Trans.Am. Math. Soc. , 349(2):641–655, 1997.[18] M. Parton and P. Piccinni. Spin(9) and almost complex structures on 16-dimensional manifolds.
Ann. Global Anal. Geom. , 41(3):321–345, 2012.[19] M. Parton and P. Piccinni. Parallelizations on products of spheres and octonionic geometry.
ComplexManifolds, Special Issue on Complex Geometry and Lie Groups
Rend. Sem. Mat.Univ. Pol. Torino , Workshop for Sergio Console, 74:267-288, 2016.
Institute of Mathematics, University of the Philippines Diliman, Philippines
E-mail address : [email protected] Dipartimento di Matematica, Sapienza Universit`a di Roma, Piazzale Aldo Moro 2, I-00185, Roma, Italy
E-mail address ::