Harmonic SU(3)- and $G_2$-Structures via Spinors

Abstract

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 the above mentioned structures. We describe the results on appropriate homogeneous spaces. Here, harmonicity means harmonicity of a certain unique section induced by the G-structure in consideration.

1 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 the structure group SO(n) to SU(3) or $G_2$, respectively. Recently, the authors of [2] have studied such structures via sponorial approach. A global unit spinor $\varphi$ in the spinor bundle defines, depending on the dimension of a manifold, the structures mentioned above. Thus it is natural to study the geometry of a defining spinor $\varphi$. The crucial observation in [2] is the existence of an endomorphism $S{:}TM\rightarrow TM$ and a 1-form $\eta$ (which vanishes in the $G_2$ case for dimensional reasons) which describe the covariant derivative of $\varphi$,

$$\begin{aligned} \nabla _X\varphi =S(X)\cdot \varphi +\eta (X)j\cdot \varphi , \end{aligned}$$

where j is a certain almost complex structure on the spinor bundle.

On the other hand, C. M. Wood [14, 15] and later J. C. González-Dávila and F. Martin Cabrera [7] introduced and studied the so called harmonic G-structures. Each G-structure $P\subset SO(M)$ defines a section $\sigma _P$ of the associated bundle $SO(M)\times _G (SO(n)/G)$ as follows

$$\begin{aligned} \sigma (x)=[p,eG],\quad \pi _{SO(M)}(p)=x. \end{aligned}$$

In the compact case, if $\sigma$ is a harmonic section, then we say that the corresponding G-structure is harmonic. This condition is a differential equation $\sum _i (\nabla _{e_i}\xi )_{e_i}=0$ involving the intrinsic torsion $\xi$, the main ingredient of all considerations. We treat it as a harmonicity condition also in the non-compact case. The intrinsic torsion $\xi$, shortly speaking, is a (2, 1)-tensor field which is the difference of the Levi-Civita connection $\nabla$ and the G-connection $\nabla ^G$ induced by the $\mathfrak {g}$–component of the connection form of $\nabla$ (here $\mathfrak {g}$ is the Lie algebra of G), $\xi _XY=\nabla _XY-\nabla ^G_XY$. Thus the intrinsic torsion measures the defect of the 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 for harmonicity becomes a differential condition on S and $\eta$. In some cases, for example when $\eta =0$, it takes a really simple form. Among others, we conclude that if a SU(3)-structure is in its ${\mathcal {W}}_1$ or ${\mathcal {W}}_3$ class from the Gray–Hervella classification of possible intrinsic torsion modules, then it is harmonic. Analogously, if a $G_2$-structure is in ${\mathcal {W}}_1$ class with the defining function $\lambda$ being constant or in ${\mathcal {W}}_3$ class, then it is harmonic.

Finally, we present some examples on homogeneous spaces. We begin with general considerations, which lead to the following conclusion—if a unit spinor defining a G-structure is induced by a fixed point of an isotropy representation and the minimal G-connection is induced by the zero map $\Lambda _{\mathfrak {g}}$ (see the last section for details), then the structure is harmonic. We justify these results by presenting appropriate examples. Although, these examples have been already considered in the literature, they have not been studied from this point of view. In other words, we find “new” examples of harmonic SU(3) and $G_2$-structures.

2 The Intrinsic Torsion and Spinors—A General Approach

Let (M, g) be an oriented Riemannian manifold equipped with a G-structure, $G\subset SO(n)$, where $n=\dim M$. Let $\nabla ^g$ be the Levi–Civita connection of g and $\omega$ be its connection form on SO(M). The splitting on the level of Lie algebras

$$\begin{aligned} \mathfrak {so}(n)=\mathfrak {g}\oplus \mathfrak {m}, \end{aligned}$$

(1)

where $\mathfrak {m}$ is the orthogonal complement of $\mathfrak {g}$ in $\mathfrak {so}(n)$, defines the splitting $\omega =\omega _{\mathfrak {g}}+\omega _{\mathfrak {m}}$. By the fact that (1) is $\mathrm{ad}(G)$-invariant, it follows that the $\mathfrak {g}$-component $\omega _{\mathfrak {g}}$ is a connection form on the G-reduction $P\subset SO(M)$ and hence defines a connection $\nabla ^G$ on M. The difference

$$\begin{aligned} \xi _XY=\nabla ^G_XY-\nabla ^g_XY,\quad X,Y\in \Gamma (TM), \end{aligned}$$

defines a tensor $\xi \in T^{*}M\otimes \mathfrak {m}(TM)$ called the intrinsic torsion. It follows immediately that its alternation is, up to a sign, the torsion $T^G$ of $\nabla ^G$.

Denote by N the associated bundle $SO(M)\times _{SO(n)}(SO(n)/G)$. There is a one to one correspondence between the G-structures and the sections of N. Thus, $P\subset SO(M)$ defines a unique section $\sigma _P\in \Gamma (N)$. We say that the G-structure P is harmonic if $\sigma _P$ is a harmonic section [7] (if M is compact). It can be shown [7] that harmonicity is equivalent to vanishing of the following tensor

$$\begin{aligned} L=\sum _i (\nabla ^g_{e_i}\xi )_{e_i}, \end{aligned}$$

(2)

where the summation is taken with respect to any orthonormal basis. We treat this condition as the harmonicity condition also in the non-compact case. Since, informally speaking, $\nabla ^G$ preserves the decomposition (1) and

$$\begin{aligned} L=\sum _i ((\nabla ^G_{e_i}\xi )_{e_i}+\xi _{\xi _{e_i}e_i}), \end{aligned}$$

we see that $L\in \mathfrak {m}(TM)$ [7].

Assume M is equipped with a spin structure. Let $\rho :\mathrm{Spin}(n)\rightarrow \varDelta _n$ be a spin representation. We will denote the Clifford multiplication, and all induced actions, by a “dot”. In particular acting on spinors, we have

$$\begin{aligned} X^{\flat }\cdot \omega -\omega \cdot X^{\flat }=2 X\lrcorner \,\omega . \end{aligned}$$

(3)

Denote by S(M) the induced spinor bundle, $S(M)=\mathrm{Spin}(M)\times _{\rho }\varDelta _n$, where $\mathrm{Spin}(M)$ is a spin structure. Assume G is the stabilizer of some unit spinor $\varphi _0\in \varDelta _n$ and let $\varphi \in S$ be the corresponding unit spinor, which defines a G-structure. Hence, with the usual identification of $\mathfrak {so}(TM)$ with the space of 2-forms on M, the action of $\mathfrak {m}(TM)$ on $\varphi$ is injective. Therefore, vanishing of $L\in \mathfrak {m}(TM)$ is equivalent to the relation

$$\begin{aligned} L\cdot \varphi =0, \end{aligned}$$

(4)

where $\cdot$ denotes the action of skew-forms on spinors.

Let us describe the above condition with the use of spinorial laplacian. Denote by the same symbol $\nabla ^g$ the connection on S(M) induced from the Levi-Civita connection $\nabla ^g$. Then we put [6]

$$\begin{aligned} \varDelta \psi =-\sum _i (\nabla ^g_{e_i}\nabla ^g_{e_i}\psi -\nabla ^g_{\nabla ^g_{e_i}e_i}\psi ). \end{aligned}$$

We have [2]

$$\begin{aligned} \nabla ^g_X\varphi =\frac{1}{2}\xi _X\cdot \varphi \end{aligned}$$

(5)

which follows from the fact that $\nabla ^G\varphi =0$. Differentiating (5) we get

$$\begin{aligned} \varDelta \varphi =-\frac{1}{2}L\cdot \varphi -\frac{1}{4}\sum _i\xi _{e_i}\cdot \xi _{e_i}\cdot \varphi . \end{aligned}$$

The second component on the right hand side can be interpreted in the following way. For any tensor $T\in T^{*}M\otimes \mathfrak {so}(TM)$ define

$$\begin{aligned} c_T=\frac{1}{2}\sum _i T_{e_i}\cdot T_{e_i}\quad \text {and}\quad \sigma _T=\frac{1}{2}\sum _i T_{e_i}\wedge T_{e_i} \end{aligned}$$

as elements of the Clifford bundle acting on spinors. These elements, defined for totally skew tensors, have been already considered and their important role has been established [1]. Then (we do not require T to be totally skew–symmetric)

$$\begin{aligned} c_T=\sigma _T-\frac{3}{2}|T|^2. \end{aligned}$$

(6)

The above equation shows that $c_T$ acting on spinors contains element of second order, namely $\sigma _T$, and the scalar $|T|^2$. We have proven the following general characterization of harmonic G-structures defined by a spinor.

Proposition 1

A G-structure on a spin manifold $(M,g,\varphi )$ defined by a unit spinor field $\varphi \in S$ is harmonic if and only if

$$\begin{aligned} \varDelta \varphi =-\frac{1}{2}c_{\xi }\cdot \varphi =-\frac{1}{2}\sigma _{\xi }\cdot \varphi +\frac{3}{4}|\xi |^2\varphi , \end{aligned}$$

where $\xi$ is the intrinsic torsion.

By the Lichnerowicz formula we have an immediate corollary.

Corollary 1

Assume that the defining unit spinor $\varphi$ is harmonic, i.e., $\varphi$ is in the kernel of the Dirac operator. Then, a G–structure is harmonic if and only if $\sigma _{\xi }$, equivalently $c_{\xi }$, acts on $\varphi$ as a scalar. In this in the case, the scalar corresponding to $\sigma _{\xi }$ is

$$\begin{aligned} -\frac{1}{2}\mathrm{Scal}^g+\frac{3}{2}|\xi |^2, \end{aligned}$$

where $\mathrm{Scal}^g$ is the scalar curvature of g.

Proof

Assume that a given G-structure in harmonic. Then, by the discussion above, $\varDelta \varphi =-\frac{1}{2}c_{\xi }\cdot \varphi$ and by (6) the second equality holds. Moreover, by the Lichnerowicz formula, $\varDelta \varphi =-\frac{1}{2}\mathrm{Scal}^g\varphi$. Thus the second part follows.

Conversely, if $c_{\xi }$ acts as a scalar, say C, then by the Lichnerowicz formula and the above considerations

$$\begin{aligned} -\frac{1}{2}\mathrm{Scal}^g\varphi =\varDelta \varphi =-\frac{1}{2}L\cdot \varphi -\frac{1}{2}C\varphi . \end{aligned}$$

Hence, $L\cdot \varphi =(\mathrm{Scal}^g-C)\varphi$. Since $L\cdot \varphi$ is either orthogonal to $\varphi$ or 0, it must be 0, so $L=0$. $\square$

Let us provide a useful sufficient condition for harmonicity in the case of a homogeneous reductive space. We rely on [4]. Consider a homogeneous space $M=K/H$, where K is a compact, connected Lie group and H its closed subgroup. Denote by $\mathfrak {k}$ and $\mathfrak {h}$ the Lie algebras of K and H, respectively. Assume we have a decomposition $\mathfrak {k}=\mathfrak {h}\oplus \mathfrak {n}$, where $\mathfrak {n}$ is the orthogonal complement of $\mathfrak {h}$ with respect to and $\mathrm{ad}(H)$–invariant positive bilinear form $\mathbf{B}$ on $\mathfrak {k}$. Then $\mathbf{B}$ induces a Riemannian metric g on M. By a well known theorem by Wang the Levi-Civita connection $\nabla ^g$ is identified with an invariant linear map $\Lambda :\mathfrak {n}\rightarrow \mathfrak {so}(\mathfrak {n})$.

Denote by $\lambda :H\rightarrow \mathrm{SO}(\mathfrak {n})$ the isotropy representation. Notice, that the tangent bundle of M may be described as $TM=K\times _{\lambda }\mathfrak {n}$ and hence any tensor bundle $T^{\otimes k}M$ equals $K\times _{\lambda }\mathfrak {n}^{\otimes k}$, etc. Consider additionally a G-structure on M. Then we have a splitting $\mathfrak {so}(\mathfrak {n})=\mathfrak {g}\oplus \mathfrak {m}$. The intrinsic torsion $\xi$ is a section of the bundle $K\times _{\lambda }(\mathfrak {n}^{*}\otimes \mathfrak {m})$. Since, there is a bijection between sections of the associated bundle $K\times _{\tau }V$, for a representation $\tau :H\rightarrow \mathrm{End}(V)$, and $\tau$-invariant functions $f:K\rightarrow V$, it follows that $\xi$ may be considered as an invariant function $f_{\xi }:K\rightarrow \mathfrak {n}^{*}\otimes \mathfrak {m}$,

$$\begin{aligned} f_{\xi }(g)=\mathrm{ad}(g^{-1})\Lambda _{\mathfrak {m}}, \end{aligned}$$

(7)

where $\Lambda _{\mathfrak {m}}$ is the $\mathfrak {m}$-component of $\Lambda$.

Following [4], let us introduce a spin structure on M. Assume that there is a lift $\tilde{\lambda }:H\rightarrow \mathrm{Spin}(\mathfrak {n})$, i.e., $\pi \circ \tilde{\lambda }=\lambda$, where $\pi :\mathrm{Spin}(\mathfrak {n})\rightarrow SO(\mathfrak {n})$ is the double covering. Then M admits a spin structure, namely $\mathrm{Spin}(M)=K\times _{\tilde{\lambda }}\mathrm{Spin}(\mathfrak {n})$. The connection on the spinor bundle $S=K\times _{\rho \tilde{\lambda }}\varDelta$, where $\rho :\mathrm{Spin}(\mathfrak {n})\rightarrow \mathrm{End}(\varDelta )$ is a spin representation, induced from the Levi-Civita connection is therefore identified with an invariant linear map $\tilde{\Lambda }:\mathfrak {n}\rightarrow \mathfrak {spin}(\mathfrak {n})$ via the correspondence

$$\begin{aligned} \pi _{*}\circ \tilde{\Lambda }=\Lambda . \end{aligned}$$

Assume there is an isotropy invariant unit spinor $\varphi _0$. Thus, as a constant function $f_{\varphi }(k)=\varphi _0$, $k\in K$, it induces a global unit spinor $\varphi$ on the spinor bundle S over M. Then, by the result of Ikeda [9], $\nabla _X\varphi$ corresponds to

$$\begin{aligned} f_X(f_{\varphi })+\rho _{*}(\tilde{\Lambda }(f_X))f_{\varphi }. \end{aligned}$$

Since $f_{\varphi }$ is constant, the first element vanishes. Therefore, the spinorial laplacian of $\varphi$ corresponds to

$$\begin{aligned} -\sum _i \rho _{*}(\tilde{\Lambda }(E_i))\rho _{*}(\tilde{\Lambda }(E_i))\varphi _0, \end{aligned}$$

(8)

where $(E_i)$ is an orthonormal basis of $\mathfrak {n}$. Moreover, the element $c_{\xi }$ acting on $\varphi$ by the definition equals

$$\begin{aligned} \frac{1}{2}\sum _i\rho _{*}(\tilde{\Lambda }_{\mathfrak {m}}(E_i))\rho _{*} (\tilde{\Lambda }_{\mathfrak {m}}(E_i))\varphi _0. \end{aligned}$$

(9)

Proposition 2

Let $M=K/H$ be a reductive homogeneous space with a spin structure induced by a lift $\tilde{\lambda }$ of the isotropy representation $\lambda$. Assume that M has a G-structure is induced by a spinor $\varphi _0$, which is a fixed point of $\lambda$. Then, the G–structure is harmonic if and only if

$$\begin{aligned}&\sum _i (\rho _{*}(\tilde{\Lambda }_{\mathfrak {g}}(E_i))\rho _{*}(\tilde{\Lambda }_{\mathfrak {m}}(E_i))\varphi _0+\rho _{*}(\tilde{\Lambda }_{\mathfrak {m}}(E_i))\rho _{*}(\tilde{\Lambda }_{\mathfrak {g}}(E_i))\varphi _0 \nonumber \\&\quad \quad +\rho _{*}(\tilde{\Lambda }_{\mathfrak {g}}(E_i))\rho _{*}(\tilde{\Lambda }_{\mathfrak {g}}(E_i))\varphi _0)=0. \end{aligned}$$

(10)

Proof

Follows immediately by Proposition 1, relations (8) and (9). $\square$

As an immediate consequence we have the following useful fact.

Corollary 2

If the minimal connection $\nabla ^G$ is induced by the zero map $\Lambda _{\mathfrak {g}}:\mathfrak {n}\rightarrow \mathfrak {g}$, where $\mathfrak {k}=\mathfrak {h}\oplus \mathfrak {n}$ is a reductive decomposition (i.e., $\nabla ^G$ is the canonical connection), then the G–structure is harmonic.

The converse implication is not true as can be seen by appropriate examples (see the last section).

In the following sections we obtain a characterization of harmoniciy of SU(3) and $G_2$-structures using the spinorial approach in [2]. We show relations with the above general approach and state appropriate examples.

3 SU(3)-Structures

Let $\rho :\mathrm {Spin}(6)\rightarrow \mathrm {End}(\varDelta ), \varDelta =\varDelta _{6}=\mathbb {R}^{8}$, be a spin representation. It can be realized via the Clifford multiplication defined in the following way [2]

$$\begin{aligned}&e_1=+E_{18}+E_{27}-E_{36}-E_{45}, \quad \quad e_2=-E_{17}+E_{28}+E_{35}-E_{46}, \nonumber \\&e_3=-E_{16}+E_{25}-E_{38}+E_{47}, \quad \quad e_4=-E_{15}-E_{26}-E_{37}-E_{48},\\&e_5=-E_{13}-E_{24}+E_{57}+E_{68}, \quad \quad e_6=+E_{14}-E_{23}-E_{58}+E_{67},\nonumber \end{aligned}$$

(11)

where $E_{ij}$ is a skew–symmetric matrix such that $E_{ij}e_j=-e_i$. Moreover, let $j=e_1\cdot e_2\cdot \ldots \cdot e_6$. Acting on spinors, $j:\varDelta \rightarrow \varDelta$ is a complex structure anti–commuting with the Clifford multiplication by vectors. The crucial observation in [2] is that for a fixed unit spinor $\varphi \in \varDelta$ we have the following orthogonal decomposition

$$\begin{aligned} \varDelta =\mathrm{span}\{\varphi \}\oplus \mathrm{span}\{j\cdot \varphi \}\oplus \{X\cdot \varphi \mid X\in \mathbb {R}^6\}. \end{aligned}$$

Such a spinor defines the group $SU(3)\subset SO(6)$ in a sense that SU(3) is the stabilizer of $\varphi$, or equivalently, the Lie algebra that anihilates $\varphi$ is $\mathfrak {su}(3)$. Moreover,

$$\begin{aligned} \begin{aligned} \mathfrak {su}(3)^{\bot }\cdot \varphi&=\mathrm{span}\{\varphi \}^{\bot },\\ \mathfrak {u}(3)^{\bot }\cdot \varphi&=\mathrm{span}\{\varphi ,j\cdot \varphi \}^{\bot }=\{X\cdot \varphi \mid X\in \mathbb {R}^6\}. \end{aligned} \end{aligned}$$

(12)

Example 1

Let us demonstrate the above formulas by an appropriate example. Choose $\varphi =(0,0,0,0,1,0,0,0)=s_5\in \varDelta$. Such a choice is determined by Examples 3 and 4 from the last section. Simple calculations show, by the realization (11), that

$$\begin{aligned} \{X\cdot \varphi \mid X\in \mathbb {R}^6\}=\mathrm{span}\{s_1,s_2,s_3,s_4,s_7,s_8\},\quad j\cdot \varphi =s_6 \end{aligned}$$

and the Lie algebra $\mathfrak {su}(3)$ of the anihilator of the spinor $\varphi$ is generated by

$$\begin{aligned} e_{13}-e_{24},\, e_{14}+e_{23},\, e_{15}+e_{26},\, e_{16}-e_{25},\, e_{35}-e_{46},\, e_{36}+e_{45},\, e_{12}+e_{34},\, e_{34}+e_{56}. \end{aligned}$$

Hence, $\mathfrak {m}=\mathfrak {su}(3)^{\bot }$ is the span of

$$\begin{aligned} e_{35}+e_{46},\, e_{36}-e_{45},\,e_{15}-e_{26},\, e_{16}+e_{25},\, e_{13}+e_{24},\, e_{14}-e_{23},\, e_{12}-e_{34}+e_{56}. \end{aligned}$$

Let (M, g) be a 6-dimensional spin manifold with a unit spinor (field) $\varphi$. Since the stabilizer in SO(6) of a unit spinor is SU(3) we get the existence of SU(3)-structure on M [2]. This induces the splitting of the real spinor bundle S and implies the existence of an endomorphism $S\in \mathrm{End}(TM)$ and a 1–form $\eta$ such that [2]

$$\begin{aligned} \nabla ^g_X\varphi =S(X)\cdot \varphi +\eta (X)j\cdot \varphi . \end{aligned}$$

(13)

By (13) and (5) we have

$$\begin{aligned} \frac{1}{2}\nabla ^g_Y(\xi _X\cdot \varphi )&=\frac{1}{2}\left( (\nabla ^g_Y\xi _X)\cdot \varphi +\xi _X\cdot \nabla ^g_Y\varphi \right) \\&=\frac{1}{2}\left( (\nabla ^g_Y\xi _X)\cdot \varphi +\xi _X\cdot S(Y)\cdot \varphi +\eta (Y)\xi _X\cdot j\cdot \varphi \right) \end{aligned}$$

and, on the other hand, since j is $\nabla ^g$–parallel,

$$\begin{aligned} \frac{1}{2}\nabla ^g_Y(\xi _X\cdot \varphi )&=(\nabla ^g_Y S)(X)\cdot \varphi +S(\nabla ^g_YX)\cdot \varphi +S(X)\cdot S(Y)\cdot \varphi \\&\quad \quad +\eta (Y)S(X)\cdot j\cdot \varphi +(\nabla ^g_Y\eta )(X)j\cdot \varphi +\eta (\nabla ^g_YX)j\cdot \varphi \\&\quad \quad +\eta (X)j\cdot S(Y)\cdot \varphi -\eta (X)\eta (Y)\varphi . \end{aligned}$$

Hence, comparing both sides with $X=Y=e_i$ and taking into account the equality

$$\begin{aligned} \frac{1}{2}\xi _{\nabla ^g_{e_i}e_i}\cdot \varphi =S(\nabla ^g_{e_i}e_i)\cdot \varphi +\eta (\nabla ^g_{e_i}e_i)\varphi . \end{aligned}$$

we get

$$\begin{aligned} \frac{1}{2}L\cdot \varphi&=-\frac{1}{2}\sum _i \xi _{e_i}\cdot S(e_i)\cdot \varphi -\frac{1}{2}\xi _{\eta ^{\sharp }}\cdot j\cdot \varphi +(\mathrm{div}S)\cdot \varphi -|S|^2\varphi +S(\eta ^{\sharp })\cdot j\cdot \varphi \\&\quad \quad +\mathrm{div}(\eta ^{\sharp })j\cdot \varphi +j\cdot S(\eta ^{\sharp })\cdot \varphi -|\eta |^2\varphi . \end{aligned}$$

Applying (3) we obtain

$$\begin{aligned} \frac{1}{2}L\cdot \varphi =\chi ^S\cdot \varphi -\frac{1}{2}\xi _{\eta ^{\sharp }}\cdot j\cdot \varphi +(\mathrm{div}S)\cdot \varphi +\mathrm{div}(\eta ^{\sharp })j\cdot \varphi +j\cdot S(\eta ^{\sharp })\cdot \varphi -|\eta |^2\varphi , \end{aligned}$$

where $\chi ^S$ is a vector field given by

$$\begin{aligned} \chi ^S=\sum _i \xi _{e_i}S(e_i). \end{aligned}$$

We may state and prove the main theorem of this section. Before we do that, let us say a few words about the 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 $T^{*}M\otimes \mathfrak {u}(n)^{\bot }(TM)$. Under the natural action of U(n) it splits into four modules, so called Gray-Hervella classes [8], ${\mathcal {W}}_1\oplus {\mathcal {W}}_2\oplus {\mathcal {W}}_3\oplus {\mathcal {W}}_4$. For SU(n) we have one additional class ${\mathcal {W}}_5$, which corresponds to the 1–form $\eta$. The case $n=3$ is special. Each module ${\mathcal {W}}_1$ and ${\mathcal {W}}_2$ splits into two modules ${\mathcal {W}}^{\pm }_i$, $i=1,2$. Therefore, we have the following splitting

$$\begin{aligned} T^{*}M\otimes \mathfrak {su}(3)^{\bot }(TM) ={\mathcal {W}}^+_1\oplus {\mathcal {W}}^-_1\oplus {\mathcal {W}}^+_2\oplus {\mathcal {W}}^-_2\oplus {\mathcal {W}}_3\oplus {\mathcal {W}}_4\oplus {\mathcal {W}}_5. \end{aligned}$$

Each class has a nice interpretation in terms of S and $\eta$ (see [2]):

$$\begin{aligned} {\mathcal {W}}^+_1&:\quad S=\lambda \, J_{\varphi },\quad \eta =0,\\ {\mathcal {W}}^-_1&:\quad S=\mu \, \mathrm{Id},\quad \eta =0,\\ {\mathcal {W}}^+_2&:\quad S\in \mathfrak {su}(3),\quad \eta =0,\\ {\mathcal {W}}^-_2&:\quad S\in \mathrm{Sym}^2_0(T^{*}M), SJ_{\varphi }=J_{\varphi }S,\quad \eta =0,\\ {\mathcal {W}}_3&:\quad S\in \mathrm{Sym}^2_0(T^{*}M), SJ_{\varphi }=-J_{\varphi }S,\quad \eta =0,\\ {\mathcal {W}}_4&:\quad S\in \Lambda ^2(T^{*}M), SJ_{\varphi }=-J_{\varphi }S,\quad \eta =0,\\ {\mathcal {W}}_5&:\quad S=0, \end{aligned}$$

where $\lambda ,\mu$ are constants and $J_{\varphi }$ is the almost complex structure induced by $\varphi$ [2],

$$\begin{aligned} J_{\varphi }(X)\cdot \varphi =j\cdot X\cdot \varphi ,\quad X\in TM. \end{aligned}$$

Assume for a while that $\eta =0$. From the definition of $J_{\varphi }$ we immediately get

$$\begin{aligned} (\nabla ^g_YJ_{\varphi })(X)\cdot \varphi =2S(Y)\cdot J_{\varphi }(X)\cdot \varphi +2g(J(X),S(Y))\varphi -2g(X,S(Y))j\cdot \varphi , \end{aligned}$$

(14)

which implies

$$\begin{aligned} (\mathrm{div}J_{\varphi })\cdot \varphi =2\sum _i S(e_i)\cdot J_{\varphi }(e_i)\cdot \varphi -2(\mathrm{tr}S) j\cdot \varphi -2\mathrm{tr}(J_{\varphi }S)\varphi . \end{aligned}$$

(15)

The following lemma shows that in many cases the vector field $\chi ^S$ vanishes.

Lemma 1

If $\eta =0$, then $\chi ^S=0$.

Proof (First proof)

Since $\eta =0$, the intrinsic torsion $\xi$ may be described as follows $g(\xi _XY,Z)=\psi _{\varphi }(S(X),Y,Z)$, where $\psi _{\varphi }$ is a 3–form induced by $\varphi$, $\psi _{\varphi }(X,Y,Z)=-\langle X\cdot Y\cdot Z\cdot \varphi ,\varphi \rangle$ [2]. Thus $\xi _X S(X)=0$ for any $X\in TM$. In particular, $\chi ^S$ vanishes. $\square$

Proof (Second proof)

By the assumption $\eta =0$, the intrinsic torsion is in fact the intrinsic torsion of the corresponding U(3)-structure. It is well known that in this case

$$\begin{aligned} \xi _XY=-\frac{1}{2}J_{\varphi }(\nabla ^g_XJ_{\varphi })Y. \end{aligned}$$

Thus, by (14) and the definition of $J_{\varphi }$

$$\begin{aligned} 2J_{\varphi }(\chi ^S)&=\sum _i (\nabla ^g_{e_i}J_{\varphi })S(e_i)\\&=2\sum _i S(e_i)\cdot J_{\varphi }(S(e_i))-2|S|^2j\cdot \varphi \\&=2\sum _i S(e_i)\cdot j\cdot S(e_i)\cdot \varphi -2|S|^2j\cdot \varphi \\&=0. \end{aligned}$$

Hence $\chi ^S=0$. $\square$

The main theorem of this section reads as follows.

Theorem 1

An SU(3)-structure on a 6-dimensional spin manifold induced by a unit spinor $\varphi$ is harmonic if and only if the following condition holds

$$\begin{aligned} \chi ^S\cdot \varphi -\frac{1}{2}\xi _{\eta ^{\sharp }}\cdot j\cdot \varphi +(\mathrm{div}S)\cdot \varphi +\mathrm{div}(\eta ^{\sharp })j\cdot \varphi +j\cdot S(\eta ^{\sharp })\cdot \varphi -|\eta |^2\varphi =0. \end{aligned}$$

(16)

If $\eta =0$, then harmonicity is equivalent to $\mathrm{div}S=0$. In particular, if the intrinsic torsion $\xi$ belongs to ${\mathcal {W}}^+_1\oplus {\mathcal {W}}^-_1$ class or to the pure class ${\mathcal {W}}_3$ or defines a locally conformally Kähler structure (contained in ${\mathcal {W}}_4$), then the SU(3)-structure is harmonic.

Proof

The only thing which is left to prove is harmonicity of the SU(3)–structure belonging to the mentioned classes.

$\mathbf{{\mathcal {W}}^+_1\oplus {\mathcal {W}}^-_1 case:}$ In this case, $S=\lambda J_{\varphi }+\mu \mathrm{Id}$ for two constants $\lambda ,\mu$. Then, $\mathrm{div}(S)=\lambda \mathrm{div} J_{\varphi }$, thus it suffices to show that the divergence of $J_{\varphi }$ vanishes, but this follows immediately by (15).

$\mathbf{{\mathcal {W}}_3 case:}$ Since S is symmetric and traceless, it follows that $\sum _i e_i\cdot S(e_i)\cdot \varphi =0$. Thus $D\varphi =0$, where D is the Dirac operator. By Corollary 1, it suffices to prove that $c_{\xi }$ acts on $\varphi$ by a scalar. Using (3) we obtain

$$\begin{aligned} 2c_{\xi }\cdot \varphi&=\sum _i \xi _{e_i}\cdot \xi _{e_i}\cdot \varphi =2\sum _i \xi _{e_i}\cdot S(e_i)\cdot \varphi \\&=2\sum _i (S(e_i)\cdot \xi _{e_i}\cdot \varphi +2\chi ^S\cdot \varphi ) =-4|S|^2\varphi , \end{aligned}$$

since by Lemma 1, $\chi ^S$ vanishes.

Locally conformally Kähler case: This means that $\xi$ is in ${\mathcal {W}}_4$ module and the Lee form $\theta$ is closed [13]. We have that S is skew-symmetric and $SJ_{\varphi }=-J_{\varphi }S$. In other words, $S\in \mathrm{u}(3)^{\bot }(TM)$. Hence, see (12), there is a unique vector field $Z_{\varphi }$ such that

$$\begin{aligned} S\cdot \varphi =Z_{\varphi }\cdot \varphi . \end{aligned}$$

(17)

Let us first describe $Z_{\varphi }$. In this case, (15) reduces to

$$\begin{aligned} (\mathrm{div}J_{\varphi })\cdot \varphi =2\sum _i S(e_i)\cdot J_{\varphi }(e_i)\cdot \varphi . \end{aligned}$$

(18)

Moreover, $\sum _i S(e_i)\cdot e_i=2S$ as acting on spinors. Applying j to both sides and using (17), we have

$$\begin{aligned} \sum _i S(e_i)\cdot J_{\varphi }(e_i)\cdot \varphi =-2 j\cdot S\cdot \varphi =-2 J_{\varphi }(Z_{\varphi })\cdot \varphi , \end{aligned}$$

which by (18) implies

$$\begin{aligned} J_{\varphi }(\mathrm{div} J_{\varphi })=4Z_{\varphi }. \end{aligned}$$

Therefore, $4Z_{\varphi }$ is dual to the Lee form $\theta$ of the locally conformally Kähler structure.

Now we wish to determine S in terms of $Z_{\varphi }$. Choose a local coordinate system such that $\varphi =s_5$, as in Example 1. Writing $Z_{\varphi }=\sum _i z_ie_i$, we see that S is represented by the following matrix

$$\begin{aligned} S=\frac{1}{2}\left( \begin{array}{cccccc} 0 &{} 0 &{} -z_5 &{} -z_6 &{} z_3 &{} z_4 \\ 0 &{} 0 &{} z_6 &{} -z_5 &{} z_4 &{} -z_3 \\ z_5 &{} -z_6 &{} 0 &{} 0 &{} -z_1 &{} z_2 \\ z_6 &{} z_5 &{} 0 &{} 0 &{} -z_2 &{} -z_1 \\ -z_3 &{} -z_4 &{} z_1 &{} z_2 &{} 0 &{} 0 \\ -z_4 &{} z_3 &{} -z_2 &{} z_1 &{} 0 &{} 0 \end{array}\right) . \end{aligned}$$

Computing the divergence of S we get

$$\begin{aligned} 2\mathrm{div}S=\sum _{\text {some}} i,j,k dZ_{\varphi }^{\flat }(e_i,e_j)e_k, \end{aligned}$$

where $\sum _{\text {some}} {i,j,k}$ denotes the summation over certain (twelve) permutations of pairwise different indices $i,j,k=1,\ldots ,6$. Since the Lee form is closed, it follows that $\mathrm{div}S=0$. $\square$

Remark 1

The harmonicity in Theorem 1 can be stated, maybe in more elegant but less applicable way for our further considerations, as follows. Introduce an operation (commutator) $[\omega ,\tau ]$ defined for any 2-forms $\omega$ and $\tau$ by (compare [11] and [1])

$$\begin{aligned}{}[\omega ,\tau ]=\sum _i (e_i\lrcorner \omega )\wedge (e_i\lrcorner \tau ). \end{aligned}$$

Then $[\omega ,\tau ]$ is a 2–form with the following action on spinors

$$\begin{aligned} \omega \cdot \tau -\tau \cdot \omega =2[\omega ,\tau ]. \end{aligned}$$

Since the element j corresponds to $J_{\varphi }$ treated as a Kähler form, we have

$$\begin{aligned} 2[\xi _{\eta ^{\sharp }},J_{\varphi }]\cdot \varphi =\xi _{\eta ^{\sharp }}\cdot j\cdot \varphi -2j\cdot S(\eta ^{\sharp })\cdot \varphi +2|\eta |^2\varphi . \end{aligned}$$

Hence, condition (16) reads as

$$\begin{aligned} \chi ^S\cdot \varphi -[\xi _{\eta ^{\sharp }}, J_{\varphi }]\cdot \varphi +(\mathrm{div}S)\cdot \varphi +\mathrm{div}(\eta ^{\sharp })j\cdot \varphi =0. \end{aligned}$$

Remark 2

The fact that, in general, the U(n)-structure of Gray-Hervella pure classes ${\mathcal {W}}_1$ or ${\mathcal {W}}_3$ or locally conformally Kähler structure is harmonic was proved, without spinorial approach, in [7]. Our approach is based only on the definitions of $J_{\varphi }$ and S by spinorial approach in [2].

4 $G_2$-Structures

Analogously as in dimension 6, a spin representation $\rho :\mathrm {Spin}(7)\rightarrow \mathrm {End}(\varDelta ), \varDelta =\varDelta _7=\mathbb {R}^8,$ is realized via Clifford multiplication defined identically as in the 6-dimensional case with additional action of $e_7$ given by

$$\begin{aligned} e_7=+E_{12}-E_{34}-E_{56}+E_{78}. \end{aligned}$$

Fix a unit spinor $\varphi \in \varDelta$. By dimensional reasons we have

$$\begin{aligned} \varDelta =\mathrm{span}\{\varphi \}\oplus \{X\cdot \varphi \mid X\in \mathbb {R}^7\}. \end{aligned}$$

Example 2

Analogously, as in SU(3) case, let us perform some calculations on a concrete example. Choose a unit spinor $\varphi =s_5\in \varDelta$. Then, the Lie algebra, which anihilates $\varphi$ via Clifford multiplication on 2-forms is spanned by the elements

$$\begin{aligned}&e_{16}+e_{37},\quad e_{16}-e_{25},\quad e_{15}+e_{26},\quad e_{26}+e_{47},\quad e_{17}-e_{36},\quad e_{17}+e_{45},\quad e_{27}-e_{35},\\&e_{27}-e_{46},\quad e_{12}+e_{34},\quad e_{12}-e_{56},\quad e_{13}-e_{24},\quad e_{13}-e_{67},\quad e_{14}+e_{23},\quad e_{14}+e_{57}. \end{aligned}$$

Hence it is $\mathfrak {g}_2$. Moreover, its orthogonal complement $\mathfrak {m}=\mathfrak {g}_2^{\bot }$ in $\mathfrak {so}(7)$ is spanned by

$$\begin{aligned}&e_{16}-e_{37}+e_{25},\quad e_{15}-e_{26}+e_{47},\quad e_{17}+e_{36}-e_{45},\quad e_{27}+e_{35}+e_{46},\\&e_{12}-e_{34}+e_{56},\quad e_{13}+e_{24}+e_{67},\quad e_{14}-e_{23}-e_{57}. \end{aligned}$$

Let (M, g) be a 7-dimensional spin manifold with the spinor bundle S and a unit spinor (field) $\varphi$. The above decomposition induces a splitting of the spinor bundle and implies existence of an endomorphism $S\in \mathrm{End}(TM)$ such that

$$\begin{aligned} \nabla _X\varphi =S(X)\cdot \varphi . \end{aligned}$$

(19)

Since the stabilizer in SO(7) of a unit spinor is $G_2$, $(M,g,\varphi )$ becomes a $G_2$-structure [2]. The harmonicity condition, or more generally, the formula for the tensor L becomes, just putting $\eta =0$ in the SU(3) case,

$$\begin{aligned} \frac{1}{2} L\cdot \varphi =(\mathrm{div}S)\cdot \varphi . \end{aligned}$$

Notice, that in the $G_2$ case the vector field $\chi ^S$ vanishes, since the intrinsic torsion may be described as follows $g(\xi _XY,Z)=\frac{2}{3}\psi _{\varphi }(S(X),Y,Z)$ with a 3-form $\psi _{\varphi }$ defined (up to a sign) in the same way as in the SU(3) case, i.e., $\psi _{\varphi }(X,Y,Z)=\langle X\cdot Y\cdot Z\cdot \varphi ,\varphi \rangle$. Thus we have the following observation.

Theorem 2

A $G_2$-structure on a spin 7-dimensional manifold is harmonic if and only if $\mathrm{div}S=0$.

For a $G_2$-structure the space of all possible intrinsic torsions $T^{*}M\otimes \mathfrak {g}_2^{\bot }(TM)$ splits into four irreducible modules ${\mathcal {W}}_1,\ldots ,{\mathcal {W}}_4$ [2]:

$$\begin{aligned} {\mathcal {W}}_1&:\quad S=\lambda \mathrm{Id},\\ {\mathcal {W}}_2&:\quad S\in \mathfrak {g}_2,\\ {\mathcal {W}}_3&:\quad S\in \mathrm{Sym}^2_0(T^{*}M),\\ {\mathcal {W}}_4&:\quad S=V\lrcorner \psi _{\varphi },\quad V\in TM. \end{aligned}$$

The class for which the condition of harmonicity may be explicitly described is ${\mathcal {W}}_1$ defined by the condition $S=\lambda \mathrm{Id}$, where $\lambda$ is a smooth function. Then $\mathrm{div}S=\mathrm{grad}(\lambda )$. Hence, a $G_2$-structure in ${\mathcal {W}}_1$ class is harmonic if and only if $\lambda$ is constant. Moreover, by the same argument as in the proof of Theorem 1, we see that a $G_2$-structure belonging to the ${\mathcal {W}}_3$ class is harmonic. Hence we may state the following fact.

Corollary 3

A $G_2$-structure belonging to the pure class

• 1. ${\mathcal {W}}_1$ is harmonic if and only if $\lambda$ is a constant, i.e., a $G_2$-structure is nearly parallel,

• 2. ${\mathcal {W}}_3$ is harmonic.

Remark 3

Some of the results, especially the second part of Theorem 1 and Corollary 3(1), may be obtained with an alternative approach communicated to the author by I. Agricola. Namely, assuming that an SU(3) or a $G_2$-structure admits a characteristic connection $\nabla ^c$ (see [1, p. 45] for a definition), which holds for the considered structures excluding ${\mathcal {W}}_2$ cases [6, 12], harmonicity condition by [7, Theorem 3.7] is equivalent to $\delta T^c=0$, where $T^c$ is a torsion of a characteristic connection (characteristic torsion). In particular, if $\nabla ^c T^c=0$, then the considered structure is harmonic. It is known that nearly Kähler and nearly parallel $G_2$–structures admit parallel characteristic connections [5, 10], thus are harmonic as U(3)- and $G_2$-structures, respectively.

5 Examples

We present examples that justify our results. We begin with a suitable introduction.

Consider a homogeneous space $M=K/H$, where K is a compact, connected Lie group and H its closed subgroup, as in the second section, i.e., assume that on the level of Lie algebras we have a decomposition $\mathfrak {k}=\mathfrak {h}\oplus \mathfrak {n}$, where $\mathfrak {n}$ is the orthogonal complement of $\mathfrak {h}$ with respect to an $\mathrm{ad}(H)$-invariant positive bilinear form $\mathbf{B}$ on $\mathfrak {k}$. Now, we deform $\mathbf{B}$ to $\mathbf{B}_t$, $t>0$, in the following way (see [4]): we assume that there is a $\mathbf{B}$-orthogonal splitting $\mathfrak {n}=\mathfrak {n}_0\oplus \mathfrak {n}_1$. Then we put

$$\begin{aligned} \mathbf{B}_t=\mathbf{B}|_{\mathfrak {n}_0\times \mathfrak {n}_0}+2t\mathbf{B}|_{\mathfrak {n}_1\times \mathfrak {n}_1}. \end{aligned}$$

$\mathbf{B}_t$ induces a one parameter family of Riemannian metrics $g_t$ on M. The Levi-Civita connection of $g_t$ is identified with the invariant linear map $\Lambda _t:\mathfrak {n}\rightarrow \mathfrak {so}(\mathfrak {n})$.

Below, we discuss a behavior of $(M,g_t,\varphi )$ in three cases.

Example 3

Consider the complex projective space $M=\mathbb {CP}^3$, which was studied in detail in [4]. Here, we review all necessary facts and develop these which are indispensable for our purposes. Recall, that $\mathbb {CP}^3$ is a homogeneous space of the form SO(5)/U(2). On the level of Lie algebras, $\mathfrak {so}(5)=\mathfrak {u}(2)\oplus \mathfrak {n}$, where

$$\begin{aligned} \mathfrak {u}(2)&=\mathrm{span}\{E_{12}, E_{34}, E_{13}-E_{24}, E_{14}+E_{23}\},\\ \mathfrak {n}&=\mathrm{span}\{E_{15}, E_{25}, E_{35}, E_{45}, E_{13}+E_{24}, E_{14}-E_{23}\}. \end{aligned}$$

Moreover, decompose $\mathfrak {n}$ into $\mathfrak {n}_0\oplus \mathfrak {n}_1$, where

$$\begin{aligned} \mathfrak {n}_0=\mathrm{span}\{E_{15}, E_{25}, E_{35}, E_{45}\},\quad \mathfrak {n}_1=\mathrm{span}\{E_{13}+E_{24}, E_{14}-E_{23}\}. \end{aligned}$$

An orthonormal basis of $\mathfrak {n}$ with respect to $\mathbf{B}_t$ (here $\mathbf{B}$ is (the negative of) the Killing form) can be chosen in the following way

$$\begin{aligned} X_1&=E_{15}, \quad \quad X_2=E_{25}, \quad \quad X_3=E_{35},\\ X_4&=E_{45}, \quad \quad X_5=\frac{1}{2\sqrt{t}}(E_{13}+E_{24}), \quad \quad X_6=\frac{1}{2\sqrt{t}}(E_{14}-E_{23}). \end{aligned}$$

Thus we have an identification $\mathfrak {n}=\mathbb {R}^6$. The Levi-Civita connection $\Lambda _t$ takes the form

$$\begin{aligned} \Lambda _t(X_1)&=\frac{\sqrt{t}}{2}(E_{35}+E_{46}), \quad \quad \Lambda _t(X_2)=\frac{\sqrt{t}}{2}(E_{45}-E_{36}),\\ \Lambda _t(X_3)&=\frac{\sqrt{t}}{2}(E_{26}-E_{15}), \quad \quad \Lambda _t(X_4)=\frac{\sqrt{t}}{2}(-E_{16}-E_{25}),\\ \Lambda _t(X_5)&=\frac{1-t}{2\sqrt{t}}(E_{13}+E_{24}),\quad \quad \Lambda _t(X_6)=\frac{1-t}{2\sqrt{t}}(E_{14}-E_{23}). \end{aligned}$$

The (differential of the) isotropy representation $\lambda _{*}:\mathfrak {u}(2)\rightarrow \mathfrak {so}(6)$ is of the form

$$\begin{aligned} \lambda _{*}(E_{12})&=E_{12}-E_{56},\quad \quad \lambda _{*}(E_{34})=E_{34}+E_{56},\\ \lambda _{*}(E_{13}-E_{24})&=E_{13}-E_{24},\quad \quad \lambda _{*}(E_{14}+E_{23})=E_{14}+E_{23}. \end{aligned}$$

It can be shown [4] that $\lambda$ has a lift to a map $\tilde{\lambda }:U(2)\rightarrow \mathrm{Spin}(6)$, which implies existence of a spin structure on M. Via the Clifford representation (11) we see that $\tilde{\lambda }_{*}$ has the following form

$$\begin{aligned} \tilde{\lambda }_{*}(E_{12})&=E_{34}+E_{78}, \quad \quad \tilde{\lambda }_{*}(E_{34})=E_{12}-E_{78},\\ \tilde{\lambda }(E_{13}-E_{24})&=-E_{13}+E_{24}, \quad \quad \tilde{\lambda }_{*}(E_{12}+E_{23})=-E_{14}-E_{23}. \end{aligned}$$

Thus, spinors $\varphi =s_5$ and $\varphi =s_6$ are anihilated by the above isotropy representation. Thus, each spinor in the span of $s_5$ and $s_6$ defines a global spinor field. Fix the spinor $\varphi =s_5$ defining an SU(3)-structure on M.

From Example 1 we see that $\Lambda _t$ has values in $\mathfrak {m}$, hence corresponds to the intrinsic torsion, whereas, the SU(3)–connection corresponds to the zero map, $\Lambda _{\mathfrak {su}(3)}\equiv 0$. Thus, by Corollary 2, the structure is harmonic for all $t>0$.

Let us consider a different approach. By the definition of $\tilde{\Lambda }$, which corresponds to the induced connection on the spinor bundle, we have $\tilde{\Lambda }(X)\varphi =S(X)\cdot \varphi$, where S is diagonal of the form (compare [2])

$$\begin{aligned} S=-\mathrm{diag}\left( \frac{\sqrt{t}}{2},\frac{\sqrt{t}}{2}, \frac{\sqrt{t}}{2}, \frac{\sqrt{t}}{2}\frac{1-t}{2\sqrt{t}}, \frac{1-t}{2\sqrt{t}}\right) , \end{aligned}$$

Hence the 1-form $\eta$ vanishes. Since

$$\begin{aligned} S=-\frac{t+1}{6\sqrt{t}}\mathrm{Id}-\frac{2t-1}{6\sqrt{t}}\mathrm{diag}(1,1,1,1,-2,-2), \end{aligned}$$

the considered SU(3)-structure is of class ${\mathcal {W}}_1^-\oplus {\mathcal {W}}_2^-$ for $t\ne \frac{1}{2}$ and ${\mathcal {W}}_1^-$ for $t=\frac{1}{2}$. In both cases, $\mathrm{div}S$, which corresponds to $\sum _i (\Lambda (X_i)S)X_i$, vanishes, hence the considered SU(3)–structures are harmonic.

Example 4

Consider the Lie group $M=\mathrm{Spin}(4)\equiv \mathbb {S}^3\times \mathbb {S}^3$. Then its Lie algebra is isomorphic to $\mathfrak {so}(4)$. Consider the following decomposition $\mathfrak {so}(4)=\mathfrak {n}_0\oplus \mathfrak {n}_1$, where $\mathfrak {n}_0=\mathrm{span}\{E_{14},E_{24},E_{34}\}$ and $\mathfrak {n}_1=\mathrm{span}\{E_{12},E_{13},E_{23}\}$ [4]. Computing the Lie brackets of generators, we see that $\mathfrak {n}_1$ is a Lie subalgebra. Choose the following orthonormal basis of $\mathfrak {so}(4)$ with respect to $\mathbf{B}_t$ (here $\mathbf{B}$ is again the negative of the Killing form):

$$\begin{aligned} X_1&=E_{14},\quad X_2=E_{24},\quad X_3=E_{34},\\ X_4&=\frac{1}{\sqrt{2t}}E_{12},\quad X_5=\frac{1}{\sqrt{2t}}E_{13},\quad X_6=\frac{1}{\sqrt{2t}}E_{23}. \end{aligned}$$

With this choice, $\mathfrak {so}(4)=\mathbb {R}^6$. The Levi-Civita connection of $\mathbf{B}_t$ is represented by the map $\Lambda _t:\mathbb {R}^6\rightarrow \mathfrak {so}(6)$ (compare [4])

$$\begin{aligned} \Lambda _t(X_1)&=\frac{1}{2}\sqrt{2t}E_{24}+\frac{1}{2}\sqrt{2t}E_{35}, \quad \quad \Lambda _t(X_2)=-\frac{1}{2}\sqrt{2t}E_{14}+\frac{1}{2}\sqrt{2t}E_{36},\\ \Lambda _t(X_3)&=-\frac{1}{2}\sqrt{2t}E_{15}-\frac{1}{2}\sqrt{2t}E_{26}, \quad \quad \Lambda _t(X_4)=\frac{1-t}{\sqrt{2t}}E_{12}+\frac{1}{2\sqrt{2t}}E_{56},\\ \Lambda _t(X_5)&=\frac{1-t}{\sqrt{2t}}E_{13}-\frac{1}{2\sqrt{2t}}E_{46}, \quad \quad \Lambda _t(X_6)=\frac{1-t}{\sqrt{2t}}E_{23}+\frac{1}{2\sqrt{2t}}E_{45} \end{aligned}$$

The spin structure is the trivial one $M\times \mathrm{Spin}(4)$ and the spinor bundle is, again, the trivial bundle $M\times \varDelta$ [4]. Hence, each smooth function $f_{\varphi }:M\rightarrow \varDelta$ defines a global spinor field. Choose the defining spinor to be the constant function equal to $s_5\in \varDelta$. Then, the equality $\tilde{\Lambda }(X)s_5=S(X)\cdot s_5+\eta (X)j\cdot s_5$ is satisfied by

$$\begin{aligned} S=\left( \begin{array}{cccccc} -\frac{1}{2}\sqrt{2t} &{} 0 &{} 0 &{} 0 &{} \frac{1}{2\sqrt{2t}} &{} 0 \\ 0 &{} \frac{1}{2}\sqrt{2t} &{} 0 &{} 0 &{} 0 &{} -\frac{1}{2\sqrt{2t}} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ -\frac{1}{2}\sqrt{2t} &{} 0 &{} 0 &{} &{} -\frac{1-t}{\sqrt{2t}} &{} 0 \\ 0 &{} \frac{1}{2}\sqrt{2t} &{} 0 &{} 0 &{} 0 &{} \frac{1-t}{\sqrt{2t}} \end{array}\right) ,\quad \eta =\left( \frac{3}{2}-t\right) \frac{1}{\sqrt{2t}}X_4^{\flat }. \end{aligned}$$

Hence the considered SU(3)–structure is of type ${\mathcal {W}}_{2}^-\oplus {\mathcal {W}}_{3}\oplus {\mathcal {W}}_4\oplus {\mathcal {W}}_5$. Notice that $j\cdot s_5=s_6$ (see Example 1). Moreover, it is not hard to check that $\mathrm{div}S=0$, $S(\eta ^{\sharp })=0$, $\mathrm{div}(\eta ^{\sharp })=0$ and $|\eta ^{\sharp }|^2=\frac{1}{2t}\left( \frac{3}{2}-t\right) ^2$. Thus the harmonicity condition (Theorem 1) has the following form

$$\begin{aligned} \chi ^S\cdot s_5-\frac{1}{2}\xi _{\eta ^{\sharp }}\cdot s_6-\frac{1}{2t}\left( \frac{3}{2}-t\right) ^2s_5=0. \end{aligned}$$

(20)

We need to compute $\xi _{\eta ^{\sharp }}\cdot s_5$, which corresponds to $\left( \frac{3}{2}-t\right) \frac{1}{\sqrt{2t}}\Lambda _{\mathfrak {su(3)}^{\bot }}(X_4)$. Since

$$\begin{aligned} \Lambda _{\mathfrak {su}(3)^{\bot }}(X_4)=\frac{\sqrt{t}}{2}(-E_{16}-E_{25})_{\mathfrak {su}(3)^{\bot }}=\frac{3-2t}{6\sqrt{2t}}(E_{12}-E_{34}+E_{56}). \end{aligned}$$

we see that

$$\begin{aligned} \xi _{\eta ^{\sharp }}\cdot s_5=\left( \frac{3}{2}-t\right) ^2\frac{1}{6t}s_6. \end{aligned}$$

Thus, the harmonicity condition (20) simplifies to

$$\begin{aligned} \chi ^S\cdot s_5-\frac{1}{12t}\left( \frac{3}{2}-t\right) ^2s_6-\frac{1}{2t} \left( \frac{3}{2}-t\right) ^2s_5=0. \end{aligned}$$

Since $\chi ^S\cdot s_5$ is orthogonal to $s_5$ and $s_6$ we have $t=\frac{3}{2}$ and, in particular, $\eta$ vanishes. Thus, by the fact that $\mathrm{div}S=0$ and by Theorem 1, the considered SU(3)-structure is harmonic only for $t=\frac{3}{2}$. In this case, it is of type ${\mathcal {W}}_{2}^-\oplus {\mathcal {W}}_3\oplus {\mathcal {W}}_4$.

Example 5

Let M be the Aloff-Wallach space $N(1,1)=SU(3)/\mathbb {S}^1$, where the action $\mathbb {S}^1\rightarrow SU(3)$ is by the diagonal matrices

$$\begin{aligned} \theta \mapsto \mathrm{diag}(e^{\theta i},e^{\theta i},e^{-2\theta i}). \end{aligned}$$

Consider the splitting $\mathfrak {su}(3)=\mathbb {R}\oplus \mathfrak {n}$ such that $\mathfrak {n}=\mathfrak {n}_0\oplus \mathfrak {n}_1$ is given by

$$\begin{aligned} \mathfrak {n}_0=\mathrm{span}\{L,E_{12},\tilde{E}_{12}\},\quad \mathfrak {n}_1=\mathrm{span}\{ E_{13},\tilde{E}_{13},E_{23},\tilde{E}_{23}\}, \end{aligned}$$

where

$$\begin{aligned} L=\mathrm{diag}(i,i,0),\quad \tilde{E}_{kl}=iS_{kl} \end{aligned}$$

and $S_{kl}$ is a symmetric matrix with $S_{kl}e_l=e_k$. Then, an orthonormal basis of $\mathfrak {n}$ with respect to $\mathbf{B}_t$, induced from the Killing form, can be chosen as follows

$$\begin{aligned}&X_1=E_{12},\quad X_2=\tilde{E}_{12},\quad X_3=\frac{1}{\sqrt{2t}}E_{13},\quad X_4=\frac{1}{\sqrt{2t}}\tilde{E}_{13},\\&X_5=\frac{1}{\sqrt{2t}}E_{23},\quad X_6=\frac{1}{\sqrt{2t}}\tilde{E}_{23},\quad X_7=L. \end{aligned}$$

The Levi-Civita connection of $g_t$ defines the map $\Lambda _t:\mathfrak {n}\rightarrow \mathfrak {so}(\mathfrak {n})$ [3, 4]:

$$\begin{aligned} \Lambda _t(X_1)&=E_{27}-\left( 1-\frac{1}{4t}\right) (E_{35}+E_{46}), \quad \quad \Lambda _t(X_2)=-E_{17}-\left( 1-\frac{1}{4t}\right) (E_{45}-E_{36}),\\ \Lambda _t(X_3)&=\frac{1}{4t}E_{47}-\frac{1}{4t}(E_{26}-E_{15}), \quad \quad \Lambda _t(X_4)=-\frac{1}{4t}E_{37}+\frac{1}{4t}(E_{16}+E_{25}),\\ \Lambda _t(X_5)&=-\frac{1}{4t}E_{67}-\frac{1}{4t}(E_{13}+E_{24}), \quad \quad \Lambda _t(X_6)=\frac{1}{4t}E_{57}-\frac{1}{4t}(E_{14}-E_{23}),\\ \Lambda _t(X_7)&=E_{12}+\left( 1-\frac{1}{4t}\right) (E_{34}-E_{56}). \end{aligned}$$

The isotropy representation $\lambda :\mathbb {S}^1\rightarrow \mathrm{SO}(\mathfrak {n})$ has a lift to a map $\tilde{\lambda }:\mathbb {S}^1\rightarrow \mathrm{Spin}(\mathfrak {n})$, thus there is a spin structure on M [4]. Moreover, the spinor $\varphi _0=s_5$ is a fixed point of this action, hence as a constant function from $\mathrm{SU}(3)$ to $\varDelta$ defines a global spinor field $\varphi$. Consider the $G_2$ structure induced by $\varphi$. By Example 2 the map $\Lambda _t$ takes values in $\mathfrak {m}$ if and only if $t=\frac{1}{8}$. In this case, by Corollary 2, the $G_2$-structure is harmonic. Let us check harmonicity for remaining values of t. It is easy to see that an endomorphism S satisfying $\tilde{\Lambda }(X)\varphi _0=S(X)\cdot \varphi _0$ equals

$$\begin{aligned} S&=\frac{1}{2}\mathrm{diag}\left( \left( \frac{1}{2t}-1\right) ,\left( \frac{1}{2t}-1\right) ,-\frac{3}{4t},-\frac{3}{4t},-\frac{3}{4t},-\frac{3}{4t},\left( \frac{1}{2t}-1\right) \right) . \end{aligned}$$

In particular, the considered $G_2$-structure is of type ${\mathcal {W}}_1\oplus {\mathcal {W}}_3$ for $t\ne \frac{5}{4}$ and of pure type ${\mathcal {W}}_1$ for $t=\frac{5}{4}$. The divergence of S, corresponding to $\sum _i \Lambda (X_i)S(X_i)$, vanishes. Hence, for any $t>0$ the considered $G_2$-structure is harmonic.