Spinorially twisted Spin structures, II: twisted pure spinors, special Riemannian holonomy and Clifford monopoles

We introduce a notion of twisted pure spinor in order to characterize, in a unified way, all the special Riemannian holonomy groups just as a classical pure spinor characterizes the special K\"ahler holonomy. Motivated by certain curvature identities satisfied by manifolds admitting parallel twisted pure spinors, we also introduce the Clifford monopole equations as a natural geometric generalization of the Seiberg-Witten equations, and show that they admit non-trivial solutions on manifolds with special Riemannian holonomy.


Introduction
The purpose of this note is to introduce a suitable notion of pure spinor for spinorially twisted Spin structures [8] in order to give a unified treatment of special Riemannnian holonomy by means of (twisted) spinorial geometry. We begin by noticing that a Spin c structure on a Riemannian ndimensional manifold M consists of the coupling of a (locally defined) Spin(n) structure and an auxiliary (locally defined) U (1) = Spin(2) structure, subject to a topological condition [14,9]. Similarly, a Spin q structure on M consists of the coupling of a (locally defined) Spin(n) structure and an auxiliary (locally defined) Sp(1) = Spin(3) structure, also subject to a topological condition [17]. Here, as in [8], we consider the twisted Spin groups Spin(n) × Z2 Spin(r), r ≥ 3, and recall the definition of spinorially twisted spin structures. We define pure spinors in this context and show that their existence encodes the special Riemannian holonomy groups given in the Berger-Simon's holonomy Theorem [4,20], which states that the holonomy group of an irreducible non-locally symmetric oriented Riemannian manifold is contained in one of the groups in Table 1.

Group
Geometry SO(n) Generic U (n/2) Kähler SU (n/2) Calabi-Yau Sp(n/4)Sp(1) Quaternion-Kähler Sp(n) Hyperkähler Spin (7) Exceptional G 2 Exceptional We refer the reader to [18,12] for extensive accounts on the theory of Riemannian holonomy. The manifolds having holonomies contained in SU (n), Sp(n), Spin(7), G 2 , are known to be Ricci-flat, Spin and to carry parallel spinors for their classical (untwisted) Spin structures [11,21]. However, there are manifolds with holonomies contained in U (n) and Sp(n)Sp(1) which are not even Spin. In [15], A. Moroianu proved that a simply-connected Spin c manifold carrying a parallel spinor field is the Riemannian product of a Ricci-flat Spin manifold and a (not necessarily Spin) Kähler manifold. The relevant notion within the proof was that of pure spinor (with respect to a subbundle of the tangent bundle) in order to identify the Kähler factor. More precisely [14], a (classical) pure spinor φ is a spinor such that for every tangent vector X there exists another tangent vector such that the following equation is fulfilled This condition says that the two spaces T M · φ and i T M · φ not only meet in ∆ n , but actually coincide. This coincidence allows the transfer of the effect of multiplication by the number i = √ −1 in the complex space ∆ n to the tangent space T M . Indeed, manipulation of this equation shows that setting Y = J(X) determines an almost complex structure J on the manifold, while the parallelness of such a spinor implies that the structure J is parallel, i.e. Kähler. From this we draw the conclusion that using a (twisted) Spin c structure allows the holonomy U (n) to be recovered from a spinorial object. Now recall that the tangent spaces of quaternionic Kähler manifolds and 8-manifolds with Spin (7) holonomy are representation spaces of sp(1) ∼ = spin(3) and spin (7) respectively, which are restrictions of representations of even Clifford algebras of rank 3 and 7. Thus, by the arguments above and noticing that U (1) ∼ = Spin(2) ⊂ Cl 0 2 , Sp(1) ∼ = Spin(3) ⊂ Cl 0 3 , Spin(7) ⊂ Cl 0 7 , we were led to speculate that the special Riemannian holonomies must be determined by twisted spinors which, somehow, should induce a transfer of algebraic structure from an even Clifford algebra to the bundle of endomorphisms of the tangent spaces of the manifold (see [19] for our first attempt). By assuming the existence of such a transfer of algebraic structure in the form of a parallel even Clifford structure, Moroianu and Semmelmann [16] verified the relation with special Riemannian holonomies, with the exception of the exceptional Lie group G 2 . Now we will describe briefly our spinorial approach. Let M be a smooth Riemannian manifold and F be an auxiliary Riemannian vector bundle of rank r. Let (e 1 , · · · , e n ) and (f 1 , · · · , f r ) be local orthonormal frames of T M and F respectively, S(T M ) and S(F ) be the locally defined spinor vector bundles of M and F respectively, and suppose m ∈ N is such that the bundle S(T M ) ⊗ S(F ) ⊗m is globally defined. A spinor field φ ∈ Γ(S(T M ) ⊗ S(F ) ⊗m ) determines maps at x ∈ M , for all 1 ≤ k < l ≤ r, where κ m r * is the induced representation of spin(r) on S(F ) ⊗m . These maps are injective at points where φ = 0, since real tangent vectors do not annihilate spinors. Given a pair k < l, we can project X · κ m r * (f k f l ) · φ x , e j · φ x e j · φ x which, in turn, gives the map At this point, we realized that the transfer of algebraic structure of the even Clifford algebra Cl 0 r must be encoded in these maps. Thus, we will define pure spinors in such a way that the local 2-forms and endomorphisms induce a non-trivial representation of Cl 0 r on T x M (cf. Proposition 3.1). Moreover, by assuming the spinor to be parallel, the induced even Clifford structure will also be parallel (cf. Theorem 4.2), and we are able to identify the special holonomies from Berger's list, including G 2 (cf. Corollaries 4.1,4.2,4.3,4.4,4.5 and 4.6). Since all of our considerations hinge on the existence of such special spinors, we give explicit representatives for the ranks r = 3, 7 (cf. Section 3.4). For the benefit of the reader, we provide detailed calculations throughout the paper.
We hope that our notion of pure spinor may arouse the interest of physicists due to its fermionic nature, its relation to physically relevant holonomies such as Spin (7) and G 2 (cf. [3,6,7] and references therein), and the fact that the twisted Spin group Spin(4) × Z2 Spin (6) has been used in the Pati-Salam Grand Unified Theory [2].
The paper is organized as follows. In Section 2, we recall Clifford algebras, twisted spin groups, representations and structures. In Section 3, we define twisted pure spinors, deduce their relevant properties and show explicit representatives. In Section 4, we characterize special Riemannian holonomies by the existence of parallel (twisted) pure spinor fields. denote the complexification of Cl n . The Clifford algebras are isomorphic to matrix algebras and, in particular, The space of spinors is defined as is defined to be either the above mentioned isomorphism for n even, or the isomorphism followed the projection onto the first summand for n odd. In order to make κ explicit, consider the following matrices In terms of the generators e 1 , . . . , e n of the Clifford algebra, κ can be described explicitly as follows, and, if n = 2k + 1,

The vectors
form a unitary basis of C 2 with respect to the standard Hermitian product. Thus, is a unitary basis of ∆ n = C 2 k with respect to the naturally induced Hermitian product.
Remark. We will denote inner and Hermitian products (as well as Riemannian and Hermitian metrics) by the same symbol ·, · trusting that the context will make clear which product is being used.
By means of κ we have Clifford multiplication It is skew-symmetric with respect to the Hermitian product Moreover, µ n can be extended to a map µ n : There exist real or quaternionic structures on the spin representations. A quaternionic structure α on C 2 is given by and a real structure β on C 2 is given by The real and quaternionic structures γ n on ∆ n = (C 2 ) ⊗[n/2] are built as follows if n = 8k + 4, 8k + 5 (quaternionic), γ n = α ⊗ (β ⊗ α) ⊗2k+1 if n = 8k + 6, 8k + 7 (real).

Spin group and representation
The Spin group Spin(n) ⊂ Cl n is the subset endowed with the product of the Clifford algebra. It is a Lie group and its Lie algebra is The restriction of κ to Spin(n) defines the Lie group representation κ n := κ| Spin(n) : Spin(n) −→ GL(∆ n ), which is, in fact, special unitary. We have the corresponding Lie algebra representation κ n * : spin(n) −→ gl(∆ n ).
Both representations can be extended to tensor powers κ m n * : spin(n) −→ End(∆ ⊗m n ), m ∈ N, in the usual way. Recall that the Spin group Spin(n) is the universal double cover of SO(n), n ≥ 3. For n = 2 we consider Spin(2) to be the connected double cover of SO(2). The covering map will be denoted by λ n : Spin(n) → SO(n) ⊂ GL(R n ).
Its differential is given by λ n * (e i e j ) = 2E ij , where E ij = e * i ⊗ e j − e * j ⊗ e i is the standard basis of the skew-symmetric matrices, and e * denotes the metric dual of the vector e. Furthermore, we will abuse the notation and also denote by λ n the induced representation on the exterior algebra * R n . Note that Clifford multiplication µ n is an equivariant map of Spin(n) representations. Now, we summarize some results about real representations of Cl 0 r in the next table (cf. [14]). Here d r denotes the dimension of an irreducible representation of Cl 0 r and v r the number of distinct irreducible representations. Let∆ r denote the irreducible representation of Cl 0 r for r ≡ 0 (mod 4) and∆ ± r denote the irreducible representations for r ≡ 0 (mod 4). Table 2 Note that the representations are complex for r ≡ 2, 6 (mod 8) and quaternionic for r ≡ 3, 4, 5 (mod 8).

Spinorially twisted spin groups and representations
By using the unit complex numbers U (1) or the unit quaternions Sp(1), the Spin group has been "twisted" as follows These give rise to the following short exact sequences respectively, which lead to the notions of Spin c and Spin q structures [9,14,17]. Notice that U (1) = Spin(2) and Sp(1) = Spin(3), so that we are led to define the twisted Spin group Spin r (n) as follows Spin r (n) = (Spin(n) × Spin(r))/{±(1, 1)} = Spin(n) × Z2 Spin(r), where r ∈ N and r ≥ 2. Spin r (n) also fits into a short exact sequence We will call r the rank of the twisting. Note that the groups Spin 2 (n) = Spin c (n) and Spin 3 (n) = Spin q (n). The Lie algebra of Spin r (n) is spin r (n) = spin(n) ⊕ spin(r).
Consider the representations κ m n,r := κ n ⊗ κ m r : where m ∈ N, which are unitary with respect to the Hermitian metric. We will also denote ).
An element φ of ∆ n ⊗ ∆ ⊗m r will be called a twisted spinor, or simply a spinor.
Also consider the map µ n ⊗ µ r : As in the untwisted case, µ n ⊗ µ r is an equivariant homomorphism of Spin r (n) representations. Note that we can also take tensor products with more copies of ∆ r as follows with Clifford multiplication taking place only in the a-th factor. We will also write µ a r (β ⊗ ϕ 1 ⊗ · · · ⊗ ϕ m ) = µ a r (β) · (ϕ 1 ⊗ · · · ⊗ ϕ m ).
Notice that if (f 1 , . . . , f r ) is an orthonormal frame of R r ,

Spin structures on oriented Riemannian vector bundles
Let F be an oriented Riemannian vector bundle over a smooth manifold M , with r = rank(F ) ≥ 3. Let P SO(F ) denote the orthonormal frame bundle of F . A Spin structure on F is a principal Spin(r)bundle P Spin(F ) together with a 2 sheeted covering such that Λ(pg) = Λ(p)λ r (g) for all p ∈ P Spin(F ) , and all g ∈ Spin(r), where λ r : Spin(r) −→ SO(r) denotes the universal covering map. In the case when r = rank(F ) = 2, we set λ 2 : Spin(2) −→ SO (2) to be the connected 2-fold covering of SO(2). When r = 1 a Spin structure is only a 2-fold covering of the base manifold M .
Given a Spin structure P Spin(F ) one can associate a spinor bundle where ∆ r denotes the standard complex representation of Spin(r). In fact, one can also associate spinor bundles whose fibers are tensor powers of ∆ r , where m ∈ N.

Spinorially twisted spin structures on oriented Riemannian manifolds
Definition 2.1 Let M be an oriented n-dimensional Riemannian manifold, P SO(M) be its principal bundle of orthonormal frames and r ∈ N, r ≥ 2. A Spin r structure on M consists of an auxiliary principal SO(r)-bundle P SO(r) and a principal Spin r (n)-bundle P Spin r (n) together with an equivariant 2 : 1 covering map Λ : P Spin r (n) −→ P SO(M)× P SO(r) , where× denotes the fibre-product, such that Λ(pg) = Λ(p)(λ n × λ r )(g) for all p ∈ P Spin r (n) and g ∈ Spin r (n), where λ n × λ r : Spin r (n) −→ SO(n) × SO(r) denotes the canonical 2-fold cover.
A manifold M admitting a Spin r structure will be called a Spin r manifold.

Remark.
A Spin r manifold with trivial P SO(r) auxiliary bundle is a Spin manifold. Conversely, any Spin manifold admits Spin r structures with trivial P SO(r) via the inclusion Spin(n) ⊂ Spin r (n) given by the elements [g, 1].

Remark.
A Spin r manifold has various associated vector bundles such as where the last bundle is globally defined if M and m satisfy certain conditions. Indeed, S(T M ) ⊗ S(F ) ⊗m is defined if one of the following options holds: • M is a non-Spin Spin r manifold and m is odd. The structure group under consideration is Spin r (n).
• Both M and F admit Spin structures, and m ∈ N. The structure group under consideration is Spin(n) × Spin(r), so that we can associate a vector bundle to every representation of the product group.
• M is Spin, F is not Spin, and m must be even. In this case, the representation ∆ ⊗m r must factor through SO(r) in order to get a globally defined bundle. Thus, the structure group we need to consider is Spin(n) × SO(r). Note that although this case falls outside the definition of Spin r structure, we will consider it since one can still work with twisted spinors and twisted Dirac operators.

Example: Homogeneous Spin r structures
Let M be a homogeneous oriented n-dimensional Riemannian manifold and G be its isometry group. Let K be the isotropy subgroup at some point so that M ∼ = G/K. The Lie algebra g of G decomposes where k is the Lie algebra of K and m is the orthogonal complement. Since G can be seen as a principal bundle over M with fiber K, the tangent bundle T M is i.e. the vector bundle associated via the isotropy representation Let F be a homogeneous oriented rank r Riemannian vector bundle over M associated to a representation σ : K −→ SO(r).
A homogeneous Spin r (n) structure on M is given by a homomorphism Ad K × σ : K −→ Spin r (n) that makes the following diagram commute If such a map exists, we can associate the twisted spinor vector bundle Example. Let us consider the real Grassmannians of oriented k-dimensional subspaces of R k+l .

Covariant derivatives on twisted Spin bundles
Let M be a Spin r n-dimensional manifold and F its auxiliary Riemannian vector bundle of rank r. Assume F is endowed with a covariant derivative ∇ F (or equivalently, that P SO(F ) is endowed with a connection 1-form θ) and denote by ∇ the Levi-Civita covariant derivative on M . These two derivatives induce the spinor covariant derivative given locally by . . , e n ) and (f 1 , . . . , f r ) are a local orthonormal frames of T M and F respectively, ω ij and θ kl are the local connection 1-forms for T M (Levi-Civita) and F .
From now on, we shall omit the upper and lower bounds on the indices, by declaring i and j to be the indices for the frame vectors of M , and k and l to be the indices for the frame sections of F . Now, for any tangent vectors X, Y ∈ T x M , where For X, Y vector fields and φ ∈ Γ(S(T M ) ⊗ S(F ) ⊗m ) a spinor field, we also have the compatibility of the covariant derivative with Clifford multiplication,

Special twisted spinors
In this section we define pure spinors and deduce their relevant properties. Throughout this section, let (e 1 , . . . , e n ) and (f 1 , . . . , f r ) be orthonormal frames for R n and R r respectively. A linear basis for Cl 0 r is given by the products . . , r}. In order to simplify notation, we will write f kl := f k f l .
In fact, for any ξ ∈ Remarks.

Lemma 3.2 [8]
Any spinor φ ∈ ∆ n ⊗ ∆ ⊗m r defines two maps (extended by linearity) From now on we shall assume that r ≥ 3.
for all 1 ≤ k < l ≤ r.
Remarks. As mentioned before, the purpose of a pure spinor is to induce the transfer of the algebraic structure of the even Clifford algebra Cl 0 r to End(R n ), i.e. to give us an even Clifford structure [16].
• The first condition ensures that the spinor provides a subalgebra which is part of its own annihilator in spin(n) ⊕ spin(r).
• The second condition ensures that the 2-forms are non-zero and the associated endomorphisms are almost complex structures. In particular, it already tells us that n must be even.
• The combined conditions should produce a copy of spin(r) within End(R n ) by means of span{η φ kl |1 ≤ k < l ≤ r}, as will be shown below.
Proof. For identity (8), the entry in row t and column s of the matrix Analogously, For identity (9), recall that in spin(r) ⊂ Cl 0 r , Thus, on the one hand, By the calculation above We see that the endomorphisms 2η φ kl , if non-zero, satisfy the Lie bracket relations of the Lie algebra so(r). In fact, they satisfy stronger relations as we shall see below.
For identity (11) Compose the last identity on the left and on the right withη φ For (12), we can see thatη Proof. Suppose (f ′ 1 . . . , f ′ r ) is another orthonormal frame of R r so that f ′ k = a k1 f 1 + · · · + a kr f r , for 1 ≤ k ≤ r, and the matrix A = (a kl ) ∈ SO(r). Recall that If we write the left-hand side of the first condition in the definition of pure spinor with respect to the frame (f ′ 1 , . . . , f ′ r ), we have In order to simplify notation, let Now suppose that the second condition of pure spinor is fulfilled for the frame (f 1 , . . . , f r ) With respect to an orthonormal frame (e 1 , . . . , e n ) of R n , (a ks a lt − a kt a ls )J st (X).
Since the bases {f kl |1 ≤ k < l ≤ r} and {f ′ kl |1 ≤ k < l ≤ r} are orthonormal in 2 R r , (a as a bt − a at a bs )(a cu a dv − a cv a du )δ su δ tv = 1≤s<t≤r (a as a bt − a at a bs )(a cs a dt − a ct a ds ).
By rewriting (13) (a ks a lt − a kt a ls )(a ku a lv − a kv a lu )J st J uv .
There are three cases: (i) the indices s, t, u, v are all different; (ii) the pairs (s, t) and (u, v) have one, and only one, common entry; (iii) the pairs (s, t) and (u, v) coincide.
For (i), note that since s < t and u < v, we only have the following six summands with those indices, so that (a ks a lt − a kt a ls )(a ku a lv − a kv a lu )J st J uv +(a ks a lu − a ku a ls )(a kt a lv − a kv a lt )J su J tv +(a ks a lu − a ku a ls )(a kt a lv − a kv a lt )J su J tv +(a ks a lt − a kt a ls )(a ku a lv − a kv a lu )J st J uv +(a ks a lv − a kv a ls )(a kt a lu − a ku a lt )J sv J tu +(a ks a lv − a kv a ls )(a kt a lu − a ku a lt )J sv J tu = (2(a ks a lt − a kt a ls )(a ku a lv − a kv a lu ) −2(a ks a lu − a ku a ls )(a kt a lv − a kv a lt ) +2(a ks a lv − a kv a ls )(a kt a lu − a ku a lt ))J st J uv = 0.
For (ii), suppose s = u but t = v. Now we have two summands (a ks a lt − a kt a ls )(a ks a lv − a kv a ls )J st J sv (a ks a lv − a kv a ls )(a ks a lt − a kt a ls )J sv J st = (a ks a lt − a kt a ls )(a ks a lv − a kv a ls )(J st J sv + J sv J st ) = (a ks a lt − a kt a ls )(a ks a lv − a kv a ls )(J tv + J vt ) = (a ks a lt − a kt a ls )(a ks a lv − a kv a ls )(J tv − J tv ) = 0, and similarly with the other possibilities.
where we have used (14). ✷ Definition 3.3 A linear even-Clifford hermitian structure of rank r on R n , n ∈ N, is a representation such that each bivector e i e j , 1 ≤ i < j ≤ r, is mapped to an antisymmetric endomorphism J ij satisfying is a pure spinor, it induces linear even-Clifford structure of rank r on R n , i.e. there is a morphism of algebras induced by the assignment as a representation of Cl 0 r , for some m, m 1 , m 2 ∈ N, where∆ r denotes the (unique) non-trivial real representation of Cl 0 r if r ≡ 0 (mod 4), and∆ + r and∆ − r denote the two non-trivial real representations of Cl 0 r if r ≡ 0 (mod 4).
Proof. By Lemma 3.4, the map extends to an algebra homomorphism Since the matricesη φ ij square to −Id R n , this representation of Cl 0 r , contains no trivial summands. By [14, Theorem 5.6], we know that the algebra Cl 0 r is isomorphic to a simple matrix algebra and has (up to isomorphism) only one or two non-trivial irreducible representations depending on whether r ≡ 0 (mod 4) or r ≡ 0 (mod 4). ✷ Lemma 3.6 Let φ ∈ ∆ n ⊗∆ ⊗m r be a pure spinor and [g, h] ∈ Spin r (n). Then the spinor κ m n,r ([g, h])(φ) is also a pure spinor.
For the second condition, consider which means that the matrix representing Φ ϕ (f ′ k f ′ l ) with respect to the frame (e ′ 1 , . . . , e ′ n ) has the same coefficients as the matrix representing Φ φ (f k f l ) does with respect to the frame (e 1 , . . . , e n ). Hence, be a pure spinor and [g, h] ∈ Spin r (n). The Stabilizer of φ is given as follows: SO(m) · Spin(r) Table 3 Proof. Let g ∈ Spin r (n) be such that g(φ) = φ, and λ r n (g) = (g 1 , g 2 ) ∈ SO(n) × SO(r). For the sake of convenience we will use the notation of Lemma 3.2. Note that g 1 ∈ N SO(n) (spin(r)), which was computed in [1].
We have where the vertical arrows are isomorphisms and the horizontal arrows correspond to g 2 and g 1 acting via the adjoint representation of SO(r). Since spin(r) is simple, g 1 | span(Φ φ (f k f l )) and g 2 correspond to each other. There exists a frame (f 1 , . . . ,f r ) of R r such that where R ϕ k is a rotation by an angle ϕ k on the plane generated byf 2k−1 andf 2k , 1 ≤ k ≤ [r/2], so that it is the exponential of the element Such an element is mapped to which exponentiates to (cos(ϕ k )Id n×n + sin(ϕ k )Φ φ (f kfl )). Now, there must be an element h 1 ∈ C SO(n) (Spin(r)) such that Thus, where λ n (h 1 ) = h 1 . ✷ r be a pure spinor. Every element in the orbit Spin(r) · φ generates the same orthogonal even Clifford structure on R n , where Spin(r) denotes the canonical copy of the group in Spin r (n) given by the elements [(1, g)].

✷
Remark. The spaceS can be used as the fibre of a twistor space for almost even-Clifford hermitian structures. We shall explore the construction of twistor spaces in a future paper.

Reducing spinors
for all 1 ≤ k < l ≤ r.

Lemma 3.9
The span of the endomorphismsη φ kl associated to a reducing spinor φ ∈ ∆ n ⊗ ∆ ⊗m r , where r ≥ 3 and m ∈ N, form an isomorphic copy of the Lie algebra so(r).
Proof. It follows from calculations similar to those in the proof of Lemma 3.3. Indeed, suppose 1 ≤ i, j, k, l ≤ r are all different. Notice that in spin(r) ⊂ Cl 0 r , and, since κ m r * : spin(r) ⊂ Cl 0 r −→ End(∆ ⊗m r ) is a Lie algebra homomorphism, the entry in row t and column s of the matrix Analogously, Since Proof. Suppose (f ′ 1 . . . , f ′ r ) is another orthonormal frame of R r so that f ′ k = a k1 f 1 + · · · + a kr f r , for 1 ≤ k ≤ r, and the matrix A = (a kl ) ∈ SO(r). Recall that η φ kl = Φ φ (f kl ). If we write the left-hand side of the first condition in the definition of pure spinor with respect to the frame (f ′ 1 , . . . , f ′ r ), we have Note that the right-hand side of be a reducing spinor and [g, h] ∈ Spin r (n). Then, the spinor κ m n,r ([g, h])(φ) is also a reducing spinor.
For the second condition, consider which means that the matrix representing Φ ϕ (f ′ k f ′ l ) with respect to the frame (e ′ 1 , . . . , e ′ n ) has the same coefficients as the matrix representing Φ φ (f k f l ) does with respect to the frame (e 1 , . . . , e n ).
be a reducing spinor. Every element g(φ) ∈ Spin(r) · φ generates the same span of 2-forms where g ∈ Spin(r).

Pure spinors: r = 2
We have left out the case r = 2 due to the following two reasons: 1. The prototypical pure Spin c spinor is ϕ = u 1,...,1 ∈ ∆ 2n . It satisfies the equation for 1 ≤ j ≤ n. This means that the complex structure determined by ϕ is the standard complex structure on R 2n Thus, i.e. the associated 2-form η ϕ and the spinor ϕ satisfy which contains the coefficient n instead of 2. (2) is very different from all other spin groups Spin(r), r ≥ 3, since it is abelian, non-simple and non-simply-connected. All of these differences are somehow reflected by the fact that there are, in fact, no pure Spin 2 (2n)-spinors according to our Definition 3.2. Instead, there are spinors satisfying the equations (η φ 12 + nκ 1 2 (f 12 )) · φ = 0, (η φ 12 ) 2 = −Id R 2n , just as in the Spin c description above.

Special Riemannian holonomy
In this section, we present the geometrical consequences of the existence of parallel pure spinors on manifolds with spinorially twisted spin structures. In particular, we establish a correspondence between special Riemannnian holonomies and parallel pure spinors. Let M be an oriented Riemannian manifold admitting a Spin r structure, and F the auxiliary Riemannian vector bundle.

Definition 4.1
• A rank r almost even-Clifford hermitian structure, r ≥ 2, on a Riemannian manifold M is a smoothly varying choice of linear even-Clifford hermitian structure on each tangent space of M . Let Q ⊂ End − (T M ) denote the subbundle with fiber spin(r).
• A Riemannian manifold carrying such a structure will be called an almost even-Clifford hermitian manifold.
• An almost even-Clifford hermitian structure on a Riemannian manifold M is called a parallel even Clifford structure if the bundle Q is parallel with respect to the Levi-Civita connection on M .
Our terminology differs from that of [16]. We have added the words "almost" and "hermitian" since, in principle, there is no integrability condition on the structure and the compatibility with a Riemannian metric is an extra condition. We shall explore integrability conditions in the style of Gray [10] in a future paper.

Generic holonomy SO(n)
Proposition 4.1 [8] Every oriented Riemannian manifold admits a spinorially twisted spin structure such that an associated spinor bundle admits a parallel spinor field. ✷ Indeed, there exists a lift of the diagonal map given in the horizontal row of the following diagram .
Let β be the unitary basis of ∆ n described in Section 2 and γ n be the corresponding real or quaternionic structure of ∆ n . The twisted spinor φ 0 ∈ ∆ n ⊗ ∆ n ,  Note that φ 0 is not pure. However, it satisfies the equations e p e q · φ 0 + κ 1 n * (f p f q ) · φ 0 = 0, for 1 ≤ p < q ≤ n, i.e. it is a reducing spinor.  The Kähler and hyperkähler cases have been treated spinorially by various authors [11,13,14,15,21]. For the sake of completeness, we collect and use some of their ideas to prove the following two corollaries. Proof. Let us assume M is a 2m-dimensional Kähler manifold, J its complex structure, p,q denote the vector bundle of exterior differential forms of type (p, q) and κ M = m,0 = det( 1,0 ).
By [11], the locally defined Spin bundle decomposes as follows so that the anti-canonical Spin c bundle contains a trivial summand. Thus, the manifold M admits a parallel spinor field ψ ∈ Γ( 0,0 ) such that [9] (X + i J(X)) · ψ = 0 for all X ∈ Γ(T M ).
Conversely, suppose M admits a Spin c structure carrying a parallel pure spinor field ψ ∈ Γ(S c (T M )). If X ∈ Γ(T M ), there exists Y ∈ Γ(T M ) such that By defining Y = J(X), we see that J is an orthogonal complex structure, and by differentiating Note that the vector ∇ Z X satisfies so that (∇ Z (J(X)) − J(∇ Z X)) · ψ = 0.
Conversely, suppose M admits a Spin c structure carrying two parallel pure spinor fields such that at each x ∈ M , τ x = γ(σ x ), where γ denotes the corresponding real or quaternionic structure, which is complex-conjugate linear and either commutes or anticommutes with Clifford multiplication. If X ∈ Γ(T M ), there exists Y ∈ Γ(T M ) such that X · σ = iY · σ.
Although we obtain the same complex structure, the common stabilizer of σ and τ is SU (m) ⊂ U (m). Proof. Let us assume M is quaternion-Kähler so that its orthonormal frame bundle has a parallel reduction to a principal bundle with fiber Sp(m)Sp(1). We have the following diagram Spin 3 (4m) ր ↓ Sp(m)Sp(1) −→ SO(4m) × SO (3) so that the manifold admits a Spin 3 structure with an induced connection. We can associate a twisted spinor bundle with fibre ∆ 4m ⊗ ∆ m 3 which contains a trivial Sp(m)Sp(1) summand generated by a pure spinor, such as the one in Subsection 3.4.1.
Conversely, if M admits a Spin 3 structure with a connection and carrying a parallel pure spinor, by Theorem 4.2, we have a parallel quaternion-Kähler structure. ✷ Corollary 4.4 A Riemannian manifold is hyperkähler if and only if it admits a Spin 3 structure endowed with a connection and a twisted spinor bundle carrying two parallel pure spinor fields σ and τ such that where g ∈ Spin(r) ⊂ Spin r (n).
Proof. Let us assume M is hyperkähler. Its structure group reduces further to Sp(m) so that the auxiliary SO(3) bundle is trivial, and we can take the flat connection on it. The associated Spin 3 bundle ∆ 4m ⊗ ∆ m 3 contains the Spin(3) orbit of the pure spinor in Subsection 3.4.1, which consists of pure spinors inducing the same quaternionic structure (cf. Lemma 3.12) and fixed by Sp(m).
Conversely, suppose M admits a Spin 3 structure with a connection and carrying two parallel pure spinors σ and τ such that σ x = g · τ x for all x ∈ M , where g ∈ Spin(3) ⊂ Spin 3 (4m). By Theorem 4.2, M admits a quaternion-Kähler structure. On the other hand, the common stabilizer of σ x and τ x is Sp(m)U (1), so that the holonomy of the manifold is contained in Sp(m) (cf. [5]). ✷ (7) and G 2 Corollary 4.5 A Riemannian 8-dimensional manifold has holonomy contained in Spin (7) if and only if it admits a Spin 7 structure endowed with a connection and carrying a parallel pure spinor field.

Exceptional holonomies Spin
Proof. Let us assume M is an 8-dimensional Riemannian manifold with holonomy contained in Spin (7). Its orthonormal frame bundle has a parallel reduction to a principal bundle with fiber Spin (7). We have the following diagram Spin 7 (8) ր ↓ Spin(7) −→ SO(8) × SO (7) so that the manifold admits a Spin 7 structure with an induced connection. We can associate a twisted spinor bundle with fibre ∆ 8 ⊗∆ 7 which contains a trivial Spin(7) summand generated by a pure spinor, such as the one in Subsection 3.4.2.
Conversely, if M admits a Spin 7 structure with a connection and carrying a parallel pure spinor, by Theorem 4.2, it admits a parallel rank 7 even Clifford structure. ✷ Corollary 4.6 The Riemannian product M = N × S of a Riemannian 7-manifold N with holonomy contained in G 2 and a flat line or circle S admits a Spin 7 structure endowed with a connection and carrying a parallel pure spinor field and a parallel reducing spinor field.
Conversely, an 8-dimensional Riemannian manifold admitting a Spin 7 structure endowed with a connection, carrying a parallel pure spinor field and a parallel reducing spinor field factors as a Riemannian product of a 7-manifold with holonomy contained in G 2 and a flat line or circle.
Proof. Let us assume the 7-dimensional manifold N is a G 2 -manifold. Since G 2 ⊂ Spin (7), we can use the pure and reducing spinors of Subsection 3.4.2 in conjunction with the previous corollary.
Conversely, the holonomy group of an 8-dimensional Riemannian manifold admitting a Spin 7 structure endowed with a connection, carrying a parallel pure spinor field and a parallel reducing spinor field must be contained in the common stabilizer of such spinors which, by Subsection 3.4.2, is a copy of G 2 . ✷