Harmonic $SU(3)$- and $G_2$-structures via spinors

In this note, using the spinorial description of $SU(3)$ and $G_2$-structures obtained recently by other authors, we give necessary and sufficient conditions for harmonicity of above mentioned structures. We describe obtained results on appropriate homogeneous spaces. Here, harmonicity means harmonicity of the unique section induced by a $G$-structure in consideration.


Introduction
Let M be a spin manifold equipped with a SU(3) (then n = dim M = 6) or G 2 (then n = dim M = 7) structure. This means, that the special orthonormal frame bundle SO(M) has a reduction of a structure group SO(n) to SU(3) or G 2 , respectively. Recently, the authors of [3] has studied such structures via sponorial approach. A global unit spinor ϕ in the spinor bundle defines, depending on the dimension of a manifold, mentioned above structures. Thus it is natural to study the geometry of a defining spinor ϕ. The crucial observation in [3] is the existence of an endomorphism S : T M → T M and a one form η (which vanishes in the G 2 case by the dimensional reasons) which describe the covariant derivative of ϕ, where j is a certain almost complex structure on the spinor bundle.
On the other hand, C. M. Wood [16,17] and later González-Dávila and Martin Cabrera [8] introduced and studied so called harmonic G-structures. Each Gstructure P ⊂ SO(M) defines a section σ P of the associated bundle SO(M) × G (SO(n)/G) by σ(x) = [p, eG], π SO(M ) (p) = x. In a compact case, if σ is a harmonic section, then we say that the corresponding G-structure is harmonic. This condition is a differential equation i (∇ e i ξ) e i = 0 involving the intrinsic torsion ξ, the main ingredient of all considerations. This condition is treated as a harmonicity condition also in a non-compact case. The intrinsic torsion ξ, shortly speaking, is a (2, 1)-tensor field being the difference of the Levi-Civita connection ∇ and the G-connection ∇ G induced by the gcomponent of the connection form of ∇ (here g is the Lie algebra of G), ξ X Y = ∇ X Y − ∇ G X Y . Thus the intrinsic torsion measures the defect of a G-structure to have holonomy in G.
In this note, the author tries to combine these two approaches for G = SU(3) and G = G 2 . The condition of harmonicity becomes a differential condition on S and η. In some cases, for example η = 0, it takes really simple form. Among others, we conclude that if a SU(3)-structure is in its W 1 or W 3 class from the Gray-Hervella classification of possible intrinsic torsion modules, then it is where m is an orthogonal complement of g in so(n), defines a splitting ω = ω g +ω m . By the fact that (2.1) is ad(G)-invariant, it follows that g-component ω g is a connection form on the G-reduction P ⊂ SO(M) and hence defines a connection ∇ G on M. The difference , defines a tensor ξ ∈ T * M ⊗ m(T M) called the intrinsic torsion. It follows immediately that its alternation is, up to a sign, the torsion T G of ∇ G .
Denote by N the associated bundle SO(M) × SO(n) (SO(n)/G). There is one to one correspondence between G-structures (M, g) and the sections of N. Thus, P ⊂ SO(M) defines the unique section σ P ∈ Γ(N). We say that a G-structure P is harmonic if σ P is a harmonic section [8] (if M is compact). It can be shown [8] that harmonicity is equivalent to vanishing of the following tensor where the sum is taken with respect to any orthonormal basis. We treat this condition as harmonicity condition also in a non-compact case. Since, informally speaking, ∇ G preserves the decomposition (2.1) and by the fact that we see that L ∈ m(T M) [8]. Assume M is equipped with a spin structure. Let ρ : Spin(n) → ∆ n be the real spin representation. We will denote this action, and all induced actions, by a 'dot'. In particular acting on spinors, we have Denote by S(M) the induced spinor bundle, S(M) = Spin(M) × ρ ∆ n , where Spin(M) is a spin structure. Assume G is a stabilizer of some unit spinor ϕ 0 ∈ ∆ n and let ϕ ∈ S be the corresponding unit spinor, which defines a G-structure. Hence, with the usual identification of so(T M) with the space of 2-forms on M the action of m(T M) on ϕ is injective. Therefore, vanishing of L ∈ m(T M) is equivalent to the relation where · denotes the action of skew-forms on spinors. Let us describe above condition with the use of spinorial laplacian. Denote with the same symbol ∇ g the connection on S(M) induced from the Levi-Civita connection ∇ g . Then we put [7] We have [3] (2.5) which follows from the fact that ∇ G ϕ = 0. Differentiating (2.5) we get The second component on the right hand side can be interpreted in the following way. For any tensor T ∈ T * M ⊗ so(T M) define as elements of Clifford bundle acting on spinors. These elements, defined for totally skew tensors, has been already considered and theirs important role have been established [2]. Then (we do not require T to be totally skew-symmetric) The above equation shows that c T acting on spinors contains element of second order, namely σ T , and a scalar |T | 2 . We have proved the following general characterization of harmonic G-structures defined by a spinor.
Proposition 2.1. A G-structure on a spin manifold (M, g, ϕ) defined by a unit spinor field ϕ ∈ S is harmonic if and only if where ξ is the intrinsic torsion.
By the Lichnerowicz formula we have an immediate corollary.
Corollary 2.2. Assume that the defining unit spinor ϕ is harmonic, i.e., ϕ is in the kernel of the Dirac operator. Then, a G-structure is harmonic if and only if σ ξ , equivalently c ξ , acts on ϕ as a scalar. In this in the case, the scalar corresponding to σ ξ is where Scal g is the scalar curvature of g.
Proof. Assume that a given G-structure in harmonic. Then, by above considerations, ∆ϕ = − 1 2 c ξ · ϕ and by (2.6) the second equality holds. Moreover, by the Lichnerowicz formula ∆ϕ = − 1 2 Scal g ϕ, thus the second part follows. Conversely, if c ξ acts as a scalar, say C, then by the Lichnerowicz formula and above considerations Hence, L · ϕ = (Scal g − C)ϕ. Since L · ϕ is either orthogonal to ϕ or 0, it must be 0, so L = 0.
In the following sections we obtain the characterization of harmoniciy of SU(3) and G 2 -structures using the spinorial approach in [3]. We show the relations with the above general approach and state appropriate examples.

SU(3)-structures
Let ρ : Spin(6) → ∆ = ∆ 6 = R 8 be the real spin representation. It can be realized in the following way [3] where E ij is a skew-symmetric matrix such that E ij e j = −e i . Moreover, let j = e 1 · e 2 · . . . · e 6 . Acting on spinors, j : ∆ → ∆ is an almost complex structure anticommuting with the Clifford multiplication by vectors. The crucial observation in [3] is that for a fixed unit spinor ϕ ∈ ∆ we have the following orthogonal decomposition Such a spinor defines a group SU(3) ⊂ SO(6) in a sense that SU(3) is a stabilizer the of ϕ, or equivalently, the Lie algebra that anihilates ϕ is su(3). Moreover, Example 3.1. Let us demonstrate above formulas on the appropriate example. Choose ϕ = (0, 0, 0, 0, 1, 0, 0, 0) = s 5 ∈ ∆. Such a choice is determined by an Examples 5.2 and 5.3 from the last section. Then, simple calculations show that, by a realization of the spin representation (3.1), Let (M, g) be a 6-dimensional spin manifold with a unit spinor (field) ϕ. Since the stabilizer in SO(6) of a unit spinor is SU(3) we get the existence of SU(3)structure on M [3]. This induces the splitting of the real spinor bundle S and implies existence of an emdomorphism S ∈ End(T M) and a one form η such that [3] Hence, comparing both sides with X = Y = e i and taking into account the equality where χ S is a vector field given by We may state and prove the main theorem of this section. Before that, let us say few words about Gray-Hervella classes of possible SU(3)-structures and state some additional simple observations. In general, for any U(n)-structure the intrinsic torsion belongs to the space T * M ⊗ u(n) ⊥ (T M). Under the natural action of U(n) it splits into four modules, so called Gray-Hervella classes [9], W 1 ⊕ W 2 ⊕ W 3 ⊕ W 4 . For SU(n) we have one additional class W 5 , which corresponds to the one form η. The case n = 3 is special. Each module W 1 and W 2 split into two modules W ± i , i = 1, 2. Therefore, we have the following splitting Each class has a nice interpretation in terms of S and η (see [3]): where λ, µ are constants and J ϕ is an almost complex structure induced by ϕ [3], Assume for a while that η = 0. From the definition of J ϕ we immediately get The following lemma shows that in many cases a vector field χ S vanishes.
First proof. Since η = 0, the intrinsic torsion ξ may be described as . Thus ξ X S(X) = 0 for any X ∈ T M. In particular, χ S vanishes.
Second proof. By the assumption η = 0, the intrinsic torsion is in fact the intrinsic torsion of corresponding U(3)-structure. It is well known that in this case Thus, by (3.4) and the definition of J ϕ Hence χ S = 0.
The main theorem of this section reads as follows.
If η = 0, then harmonicity is equivalent to divS = 0. In particular, if the intrinsic torsion ξ belongs to the W + 1 ⊕ W − 1 class or to the pure class W 3 , then a SU(3)structure is harmonic.
Proof. The only thing which is left to prove is harmonicity of SU(3)-structure belonging to the mentioned classes.
W 3 case: Since S is symmetric and traceless, it follows that i e i ·S(e i )·ϕ = 0. Thus Dϕ = 0, where D is the Dirac operator. By Corollary 2.2, it suffices to prove that c ξ acts on ϕ by a scalar. Using (2.3) we obtain  Then [ω, τ ] is a 2-form with the following action on spinors Since the element j corresponds to J ϕ treated as a Kähler form, we have Hence, condition (3.6) reads as Remark 3.5. The fact that in general the U(n)-structure of Gray-Hervella pure classes W 1 or W 3 or W 4 is harmonic was proved, without spinorial approach, in [8]. Our approach is based only on the definitions of J ϕ and S by spinorial approach in [3]. We managed only to show that for W 1 and W 3 we have harmonic structures. In the case W 4 we are only able to establish the correspondence with the classical definition of this class. Let us enlarge on this.
We have that S is skew-symmetric and SJ ϕ = −J ϕ S. In other words, S ∈ u(3) ⊥ (T M). Hence, see (3.2), there is a unique vector field Z ϕ such that Let us first describe Z ϕ . In this case, (3.5) reduces to Moreover, i S(e i ) · e i = 2S as acting on spinors. Applying j to both sides, and using (3.7), we have which by (3.8) implies Therefore, 4Z ϕ is a Lee vector field of a locally conformally Kähler structure [14]. Differentiating the relation (3.7) in the direction of e i and then multiplying by e i we obtain after some tedious calculations On the other hand, by (2.3), 2S(e i ) · ϕ = e i · Z ϕ · ϕ − S · e i · ϕ, which implies Comparing last two relations we get S(Z ϕ ) = 0 and |S| 2 = 4|Z ϕ | 2 .
Thus, by (3.4) we have The operator X → (∇ X J ϕ )(J ϕ Z ϕ ) is skew-symmetric, hence can be considered as a 2-form. Up to a constant factor this form has been considered by Vaisman [15].

G 2 -structures
Analogously as in dimension 6, the spin representation ρ : Spin(7) → ∆ = ∆ 7 = R 8 is real and can be realized identically as in the 6-dimensional case with additional action of e 7 given by Fix a unit spinor ϕ ∈ ∆. By dimensional reasons we have Hence it is g 2 . Moreover, its orthogonal complement m = g ⊥ 2 in so (7) is spanned by Let (M, g) be a 7-dimensional spin manifold with the spinor bundle S and a unit spinor (field) ϕ. Above decomposition induces a splitting of the spinor bundle and implies the existence of an endomorphism S ∈ End(T M) such that Since a stabilizer in SO(7) of a unit spinor is G 2 , (M, g, ϕ) becomes a G 2 -structure [3]. The harmonicity condition, or more generally, the formula for a tensor L becomes, just putting η = 0 in the SU(3) case, Notice, that in G 2 case the vector field χ S vanishes, since the intrinsic torsion may be described as follows g(ξ X Y, Z) = 2 3 ψ ϕ (S(X), Y, Z) with a 3-form ψ ϕ defined in the same way as in the SU(3) case. Thus we have the following observation.
where ψ ϕ is a 3-form defined as π ϕ (X, Y, Z) = X ·Y ·Z ·ϕ, ϕ . The class for which the condition of harmonicity may be explicitly described is W 1 defined by the condition S = λId, where λ is a smooth function. Then divS = grad(λ). Hence, a G 2 -structure in W 1 class is harmonic if and only if λ is constant. Moreover, by the same lines as in the proof of Theorem 3.3, we see that a G 2 -structure belonging to a W 3 class is harmonic. Hence wa may state the following fact.  (1) may be obtained with an alternative approach communicated to the author by I. Agricola. Namely, assuming that a SU(3) or a G 2 -structure admits a characteristic connection ∇ c (see [2, p. 45] for a definition), which holds for considered structures excluding W 2 cases [13,7], the harmonicity condition by [8,Theorem 3.7] is equivalent to δT c = 0, where T c is a torsion of a characteristic connection (characteristic torsion). In particular, if ∇ c T c = 0, then a considered structure is harmonic. It is known that nearly Kähler and nearly parallel G 2structures admit parallel characteristic connection [11,6], thus are harmonic as U(3) and G 2 -structures, respectively.

Examples
We justify described above theory on appropriate examples. We begin with a suitable introduction. We rely on [5]. Consider a homogeneous space M = K/H, where K is a compact, connected Lie group and H its closed subgroup. Denote by k and h the lie algebras of K and H, respectively. Assume we have a decomposition k = h ⊕ n, where n is an orthogonal complement of h with respect to and ad(H)invariant positive bilinear form B on k. Then B induces a Riemannian metric g on M. By a well known theorem by Wang the Levi-Civita connection ∇ g is identified with an invariant linear map Λ : n → so(n).
Denote by λ : H → SO(n) the isotropy representation. Notice, that a tangent bundle of M may be described as T M = K × λ n and hence any tensor bundle T ⊗k M equals K × λ n ⊗k , etc. Consider additionally a G-structure on M. Then we have a splitting so(n) = g ⊕ m. The intrinsic torsion ξ is a section of a bundle K × λ (n * ⊗m). Since, there is a bijection between sections of the associated bundle K × τ V , for a representation τ : H → End(V ) and τ -invariant functions f : K → V , it follows that ξ may be considered as an invariant function f ξ : K → n * ⊗ m, where Λ m is a m-component of Λ.
Following [5], let us introduce a spin structure on M. Assume that there is the liftλ : H → Spin(n), i.e., π •λ = λ, where π : Spin(n) → SO(n) is a double covering. Then M admits a spin structure, namely Spin(M) = K ×λ Spin(n). The connection on the spinor bundle S = K × ρλ ∆, where ρ : Spin(n) → End(∆) is a real spin representation, induced from the Levi-Civita connection is therefore identified with an invariant linear mapΛ : n → spin(n) via the correspondence Assume there is a isotropy invariant unit spinor ϕ 0 . Thus, as a constant function f ϕ (k) = ϕ 0 , it induces a global unit spinor ϕ on the spinor bundle S over M. Then, by the Ikeda result [10], ∇ X ϕ corresponds to Since f ϕ is constant, the first element vanishes. Therefore, the spinorial laplacian of ϕ corresponds to Moreover, the element c ξ by a definition equals Proposition 5.1. Let M = K/H be a reductive homogeneous space with a spin structure induced by the liftλ of the isotropy representation λ. Assume that a Gstructure on M is induced by a spinor, which is a fixed point of λ. If a minimal connection ∇ G is induced by a zero map Λ g : n → g, where k = h⊕n is a reductive decomposition (i.e., is a canonical connection), then a G-structure is harmonic.
Proof. Follows immediately by Proposition 2.1, relations (5.2) and (5.3) and the fact that g-component of Λ : n → so(n) vanishes. Now, we deform B to B t , t > 0, in the following way: we assume that there is a B-orthogonal splitting n = n 0 ⊕ n 1 . Then we put B t = B| n 0 ×n 0 + 2tB| n 1 ×n 1 .
B t induces one parameter family of Riemannian metrics g t on M. Below, we discus the behavior of (M, g t , ϕ) in three cases.
Example 5.2. Consider a complex projective space M = CP 3 , which was studied in detail in [5]. Here, we review all necessary facts and develop these which are indispensable for our purposes. Recall, that CP 3 is a homogeneous space of the form SO(5)/U(2). On the level of Lie algebras, so(5) = u(2) ⊕ n, where Moreover, decompose n into n 0 ⊕ n 1 , where The orthonormal basis of n with respect to B t (here B is (the negative of) the Killing form) can be chosen in the following way orthonormal basis of so(4) with respect to B t (here B is again the negative of the Killing form): With this choice, so(4) = R 6 . The Levi-Civita connection of B t is represented by a map Λ : R 6 → so(6) (compare [5]) The spin structure is the trivial one M × Spin(4) and the spinor bundle is, again, the trivial bundle M × ∆ [5]. Hence, each smooth function f ϕ : M → ∆ defines a global spinor field. Choose a defining spinor being the constant function equal to s 5 ∈ ∆. Then, the equalityΛ(X)s 5 = S(X) · s 5 + η(X)j · s 5 is satisfied by Hence the considered SU(3)-structure is of type W − 2 ⊕ W 3 ⊕ W 4 ⊕ W 5 . Notice that j · s 5 = s 6 (see Example 3.1). Moreover, it is not hard to check that divS = 0, S(η ♯ ) = 0, div(η ♯ ) = 0 and |η ♯ | 2 = 1 2t 3 2 − t 2 . Thus harmonicity condition (Theorem 3.3) has the following form We need to compute ξ η ♯ · s 5 , which corresponds to 3 we see that Thus, harmonicity condition (5.4) simplifies to Since χ S ·s 5 is orthogonal to s 5 and s 6 we have t = 3 2 and, in particular, η vanishes. Thus, by the fact that divS = 0 and Theorem 3.3, the considered SU(3)-structure is harmonic only for t = 3 2 . In this case, it is of type W − 2 ⊕ W 3 ⊕ W 4 . Consider a splitting su(3) = R ⊕ n such that n = n 0 ⊕ n 1 is given by n 0 = span{L, E 12 ,Ẽ 12 }, n 1 = span{E 13 ,Ẽ 13 , E 23 ,Ẽ 23 }, where L = diag(i, i, 0),Ẽ kl = iS kl and S kl is a symmetric matrix with S kl e l = e k . Then, an orthonormal basis of n with respect to B t , induced from the Killing form, can be chosen as follows The Levi-Civita connection of g t gives a map Λ t : n → so(n) [5,4]: The isotropy representation λ : S 1 → SO(n) has a lift to a mapλ : S 1 → Spin(n), thus there is a spin structure on M [5]. Moreover, the spinor ϕ 0 = s 5 is a fixed point of this action, hence as a constant function from SU(3) to ∆ defines a global spinor field ϕ. Consider a G 2 structure induced by ϕ. By Example 4.1 the map Λ t takes values in m if and only if t = 1 8 . In this case, by Proposition 5.1 a G 2 -structure is harmonic. Let us check harmonicity for remaining values of t. It is easy to see that an endomorphism S satisfyingΛ(X)ϕ 0 = S(X) · ϕ 0 equals In particular, considered G 2 -structure is of type W 1 ⊕ W 3 for t = 5 4 and of pure type W 1 for t = 5 4 . The divergence of S, corresponding to i Λ(X i )S(X i ), vanishes. Hence, for any t > 0 considered G 2 -structure is harmonic.