Clifford systems, Clifford structures, and their canonical differential forms

A comparison among different constructions of the quaternionic $4$-form $\Phi_{Sp(2)Sp(1)}$ and of the Cayley calibration $\Phi_{Spin(7)}$ shows that one can start for them from the same collections of"K\"ahler 2-forms", entering in dimension 8 both in quaternion K\"ahler and in $Spin(7)$ geometry. This comparison relates with the notions of even Clifford structure and of Clifford system. Going to dimension $16$, similar constructions allow to write explicit formulas for the canonical $4$-forms $\Phi_{Spin(8)}$ and $\Phi_{Spin(7)U(1)}$, 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 $4$-forms $\Phi_{Spin(8)}$ and $\Phi_{Spin(7)U(1)}$, extending in two different ways the notion of Cayley $4$-plane to dimension $16$.


Introduction
In 1989 R. Bryant and R. Harvey defined the following calibration, of interest in hyperkähler geometry [6]: In this definition, (ω R i , ω R j , ω R k ) are the Kähler 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 the space R 4n ∼ = H n . When n = 2, the Bryant-Harvey calibration Φ K relates with Spin(7) geometry. This is easily recognized by using the map from pairs of quaternions to octonions, that yields the identity (1.1) L * Φ Spin(7) = Φ K .
2010 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. Part of the present work was done during her visit at Sapienza Università di Roma in the academic year 2018-19, and she thanks Sapienza University and Department of Mathematics "Guido Castelnuovo" for hospitality.
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 Università di Roma Project "Polynomial identities and combinatorial methods in algebraic and geometric structures".
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 calibration Φ Spin (7) through the 2-forms ω R i , ω R j , ω R k . As well known, by summing the squares of the latter 2-forms one gets another remarkable calibration, namely the quaternionic right 4-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 constructed as sum of squares of "Kähler 2-forms" associated with complex structures on R 8 ∼ = O, defined by the unit octonions. In fact, cf. [18,Prop.10]: . . , φ h are the Kähler 2-forms associated with the 7 complex structures R i , R j , . . . , R h on R 8 ∼ = O, the right multiplications by the unit octonions i, j, k, e, f, g, h, and φ ij , φ ik , . . . , φ gh are the Kähler 2-forms associated with the 21 complex structures Theorem 1.1. The right quaternionic 4-form Ω R ∈ Λ 4 H 2 can be obtained from the the Kähler forms φ i , φ j , . . . , φ h associated with the complex structures R i , R j , . . . , R h as: . Moreover, by selecting any five out of the seven (J 1 = R i , J 2 = R j , . . . , J 7 = R h ) and by looking at the matrix ζ = (ζ αβ ) ∈ so(5) of Kähler 2-forms of their compositions J αβ = J α • J β , one can get the left quaternionic 4-form Ω L as up to a permutation or change of signs of some coordinates in R 8 .
On the same direction as in Bryant-Harvey's formula (1.1), a similar result is the following (cf. Section 4 for more details): are anti-commuting self-dual involutions in R 8 . Namely: , and on the other hand one can get the right quaternion Kähler Ω R as: Moving to dimension 16 and in Section 6, we consider two exterior 4-forms canonically associated with subgroups Spin (8), Spin(7)U(1) ⊂ SO (16), and we write their explicit expressions in the 16 coordinates. We will see in the next two statements which 4-planes of R 16 are calibrated by Φ Spin (8) and by Φ Spin (7)U (1) . The 4-forms Φ Spin (8) , Φ Spin (7)  Also, by recalling that O 2 decomposes in the union of octonionic lines

Preliminaries
The multiplication in the algebra O of octonions can be defined from the one in quaternions H through the Cayley-Dickson process: if where product of quaternions is used on the right hand side andh ′ 1 ,h ′ 2 are the conjugates of h ′ 1 , h ′ 2 ∈ H. Like for quaternions, the conjugationx =h 1 − h 2 e in O relates with the non-commutativity: xx ′ =x ′x . One has also the associator [x, , that vanishes whenever two among x, x ′ , x ′′ ∈ O are equal or conjugate.
The identification x = h 1 + h 2 e ∈ O ↔ (h 1 , h 2 ) ∈ H 2 , used in the previous formula, is not an isomorphism of (left or right) quaternionic vector spaces. To get an isomorphism one has instead to go through the following hypercomplex structure (I, J, K) on 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 not seem however to have appeared before 1989, when R. Bryant and R. Harvey [6] looked at the map L : and observed it satisfies This, in terms of x 1 = h 1 , x 2 = kh 2k and of the octonion x = x 1 + x 2 e, can be read exactly as in our previous approach: and as mentioned L * Φ Spin(7) = Φ K .

The quaternionic 4-form and the Cayley calibration in R 8
A possible way to produce 4-forms canonically associated with some G-structures is through the notion of Clifford system. We recall the definition, originally given in the context of isoparametric hypersurfaces, cf. [9].
and the I α are required to 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, as follows:  In particular, the Clifford system of rank 3 in R 4 can be defined by the classical Pauli matrices: , and the Clifford system of rank 5 in R 8 by the following similar (right) quaternionic Pauli matrices: where as before R i , R j , R k denote the multiplication on the right by i, j, k on H 2 ∼ = R 8 . According to Table, A, there is also a Clifford system with r = 4 in R 8 , explicitly defined by selecting e.g.
Going back to rank r = 5, from the quaternionic Pauli matrices I 1 , I 2 , I 3 , I 4 , I 5 , one gets the 10 complex structures on R 8 Their Kähler forms θ αβ give rise to a 5 × 5 skew-symmetric matrix and one can easily see that both the following matrices of Kähler 2-forms (5) and allow to write the (left) quaternionic 4-form of H 2 as On the other hand, as mentioned in the Introduction, the subgroup Spin(7) ⊂ SO(8) (generated by the right translation R u , u ∈ S 6 ⊂ Im O) gives rise to the Cayley calibration Φ Spin(7) ∈ Λ 4 : Here It is worth to recall that under the action of Sp (2)Sp (1), the space of exterior 2-forms Λ 2 R 8 decomposes as is generated by the Kähler forms θ αβ of the J αβ (α < β), compositions of the five quaternionic Pauli matrices, and Λ 2 By denoting by τ 2 the second coefficient in the characteristic polynomial of the involved skew-symmetric matrices, we can rewrite formula (3.2) of Ω L as: Similarly, under the Spin (7) action one gets the decomposition: (7) is generated by the Kähler forms ζ αβ of the J α • J β (α < β). Thus, in the τ 2 notation: (7).
By computing the squares of the 2-forms in (4.6) (4.4) and comparing with Formulas

Even Clifford structures in dimension 8
We recall first the following notion, proposed in 2001 by A. Moroianu and U. Semmelmann, [16].
Definition 5.1. Let (M, g) be a Riemannian manifold. An even Clifford structure is the choice of an oriented Euclidean vector bundle E r of rank r ≥ 2 over M , together with a bundle morphism ϕ from the even Clifford algebra bundle 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(∇ E X σ) = ∇ g X ϕ(σ), for every tangent vector X ∈ T M and section σ of Cl even E, where ∇ g is the Levi Civita connection. Rank 2, 3, 4 parallel even Clifford structures are equivalent to complex Kähler, quaternion Kähler, product of two quaternion Kähler. Besides them, higher rank parallel non-flat even Clifford structures in dimension 8 are listed in the following Table, cf. [16]  A class of examples of even Clifford structures are those coming from Clifford systems as defined in Section 3. Namely, if the vector sub-bundle E r ⊂ End T M , locally spanned by self-adjoint anticommuting involutions I 1 , . . . , I r , defines the Clifford system, then one 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 Kähler structure is constructed via the local J αβ = I α • I β defined as in Section 3, by using on the model space H 2 the quaternionic Pauli matrices. The remaining three rows of the former Table correspond to essential even Clifford structures, i. e. to even Clifford structures that cannot be defined throw a Clifford system, cf. [20] for a discussion on this notion.
The following Table gives a description of the Clifford bundle generators and of the canonically associated 4-form for each of the four even Clifford structures on R 8 .
It is of course desirable to give examples of Riemannian manifolds (M 8 , g) supporting both a Sp(2) · Sp(1) and a Spin (7) structure. Rarely the metric g can be the same for both structures, but this is possible of course for parallelizable (M 8 , g). On this respect, homogeneous (M 8 , 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 8 : Some of the obtained examples are parallelizable, e. g. diffeomorphic to S 7 × S 1 and S 5 × S 3 . On the latter, S 5 × S 3 , using two natural parallelizations, one can define two Spin(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 family of Calabi-Eckmann [19].
To get examples of 8-dimensional manifolds that admit both a locally conformally hyperkähler metric g and a locally conformal parallel Spin(7) metric, that is either the same g as before, or a different metric g ′ , a good point to start with is the class of compact 3-Sasakian 7-dimensional manifolds (S 7 , g). Many examples of such (S 7 , 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 7 , g) one gets a locally conformally hyperkähler metric g on the product S 7 ×S 1 [17]. This can also be expressed by saying that the 3-Sasakian metric g has the property of being nearly parallel G 2 , and in particular with 3 linearly independent Killing spinors, cf. [4, pages 536-538]. Moreover the differentiable manifold S 7 admits, besides the 3-Sasakian metric g, another metric g ′ that is also nearly parallel G 2 but proper, i. e. with only one non zero Killing spinor. This allows to extend the metrics g and g ′ to the product with S 1 and to get both the properties of locally conformally hyperkähler and locally conformally parallel Spin(7) on (M 8 , g) = (S 7 × S 1 , g) and of locally conformally parallel Spin(7) on (M 8 , g ′ ) = (S 7 × S 1 , g ′ ), cf. also [12].

Dimension 16
A Clifford system with r = 9 in O 2 ∼ = R 16 is given by the following octonionic Pauli matrices: and of course now R i , R j , . . . , R h denote the multiplication on the right by the unit octonions i, j, . . . , h on O 2 ∼ = R 16 . Looking back at Table A, we see that in R 16 there are also irreducible Clifford systems with r = 8, 7, 6. According to [20], convenient choices are the following: r = 8 : I 1 , . . . , I 8 , r = 7 : I 2 , . . . , I 8 , r = 6 : I 1 , I 2 , I 3 , I 4 , I 5 , I 9 .
We are now ready for the proofs of Theorems 1.3 and 1.4. The following notion has already implicitly introduced in the statement of Theorem 1.3. Next, let Q be any 4-plane of R 16 . By looking at the expression of Φ Spin (8) , we see that the only possibilities for having non zero value on Q are that π(Q) and π ′ (Q) are 2-dimensional. For such 4-planes Q we can use the following canonical form with respect to the complex structure i ∈ S 6 : Q = e 1 ∧ (R i e 1 cos θ + e 2 sin θ) ⊕ e ′ 1 ∧ (R i e ′ 1 cos θ ′ + e ′ 2 sin θ ′ ) , where the pairs e 1 , e 2 and e ′ 1 , e ′ 2 are both orthonormal and respectively in O and in O ′ , and with angles limited by 0 ≤ θ ≤ π 2 and θ ≤ θ ′ ≤ π − θ. The above canonical form for Q is a small variation of the canonical forms that are used in a proof of the classical Wirtinger's inequaliy (cf. [15, p.6]) and in characterizations of Cayley 4-planes in R 8 in the Harvey-Lawson foundational paper (cf. [11, p. 121]). Its proof follows the steps of proof of the mentioned canonical form, as explained in details in [15]. From this canonical form we see that Φ Spin(8) (Q) ≤ 1 for any 4-plane Q, and that the equality holds 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 with coefficient 6, thus terms involving only coordinates among 12345678, or only coordinates among 1 ′ 2 ′ 3 ′ 4 ′ 5 ′ 6 ′ 7 ′ 8 ′ , 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 O 2 = O ⊕ O ′ is the usual Cayley calibration in R 8 , whose calibrated 4-planes are the Cayley planes. Thus, for the first two types of terms, we get as calibrated 4-planes just the Cayley 4-planes that are contained in the octonionic lines with slope m = 0 and m = ∞. In the remaining case of terms aba ′ b ′ one gets as calibrated 4-planes the transversal Cayley 4-planes that are contained in the octonionic line ℓ 1 (leading coefficient m = 1). Now Spin (7) acts on the individual octonionic lines ℓ 0 , ℓ 1 , ℓ ∞ , and the only possibility to move planes out of them is through the factor U(1). In fact, the discussion 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 8 of the octonionic lines, passing through the three points m = 0, 1, ∞. This corresponds to admitting any real coefficient: m ∈ R ∪ ∞ as slope of the octonionic lines that are admitted to contain the calibrated 4-planes.
Remark 8.2. Following the recent work [14] by J. Kotrbatý, one can use octonionic 1-forms, according to the following formal definitions: referring to pairs of octonions (x, x ′ ) ∈ O ⊕ O = R 16 . Then, in the same spirit proposed in [14], a straightforward computation yields the following formula, much simpler way to write the Spin(8) canonical 4-form of R 16 : Similarly, one gets that the Spin(7)U(1) canonical 4-form of R 16 can be written in octonionic 1-forms as: Details of both computations are in [3].