Abstract
We develop an invariant approach to SU(2)–structures on spin 5–manifolds. We characterize (via spinor approach) the subspaces in the spinor bundle which induce the same group isomorphic to SU(2). Moreover, we show how to induce quaternionic structure on the contact distribution of the considered SU(2)–structure. We show the invariance of certain components of the covariant derivative \(\nabla \varphi \), where \(\varphi \) is any spinor field defining SU(2)–structure. This shows, as expected, that (at least some of) the intrinsic torsion modules can be derived invariantly with the spinorial approach. We conclude with the explicit description of the intrinsic torsion and the characteristic connection.
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1 Introduction
Holonomy plays an important role in Riemannian geometry. It measures the behavior of parallel displacement with respect to the Levi-Civita connection. The celebrated theorem by Berger states that the list of possible (restricted) holonomy groups is limited to a few cases. In the list there are geometries, called exceptional, which appear only in certain dimensions:
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\(G_2\) in dimension 7.
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\(\textrm{Spin}(7)\) in dimension 8,
There are two additional geometries, SU(3) in dimension 6 and SU(2) in dimension 5, which are also called exceptional (despite the fact that they fall into general category of SU(n)–structures in dimension n or \(n+1\) as a codimension one distribution). This is due to the fact that each of these geometries in dimension k induces appropriate geometry in dimension \(k+1\).
In very interesting articles [4, 6, 15] the authors study these exceptional geometries from the spinorial point of view. In fact, it can be shown that the unit spinor field induces an exceptional geometry. However, in the SU(2) case the choice of the unit spinor is not unique.
In this article, we study SU(2)–geometry from the perspective of spinors, focusing on the invariant approach, i.e., independent on the choice of the defining spinor. We concentrate on the action of vectors on spinors. Let us be more precise. Consider the spinor representation \(\rho :\textrm{Spin}(5)\rightarrow \textrm{End}(\Delta )\), where \(\Delta =\mathbb {C}^4\). Let \(\varphi _0\in \Delta \) be a unit spinor (defining a group SU(2)). Then an SU(2)–structure on a spin 5–dimensional manifold M is a subbundle P in the bundle \(\textrm{Spin}(M)\) with the structure group SU(2) or, equivalently, a choice of the unit spinor field \(\varphi \) and a subbundle P of all frames u such that \(\varphi =[u,\varphi _0]\).
We show that there are different choices of spinors in \(\Delta \) defining the same SU(2) [3]. In fact, we show that the subspace \(V\subset \Delta \) of spinors defining the same group SU(2) is of real dimension four. We characterize these spaces from two perspective: complex and quaternionic. The quaternionic approach seems to be well known to the experts in the field, whereas the complex approach is most likely new.
The other case is concerned with a characterization of the intrinsic torsion modules. It is well known [9] that the intrinsic torsion is characterized by the covariant derivative \(\nabla \varphi \), where \(\varphi \) is a defining spinor. We study the invariance of this approach. We show that for another spinor \(\psi \) defining the same SU(2)–structure a certain decomposition of \(\nabla \psi \) induces the same components. To derive the invariance, we slightly modify the quaternionic structure on \(\Delta \) and use the action of two–forms on spinors.
We begin, in the first section, with an algebraic approach to spinors defining SU(2), or equivalently, its Lie algebra \(\mathfrak {su}(2)\). Majority of the results in this section is well known, however it is hard to find appropriate citations. We give a characterization of the subspaces of spinors defining a given Lie algebra \(\mathfrak {su}(2)\) in terms of complex and quaternionic structures. Moreover, we study a correspondence between the complex structures on the spinor space \(\Delta \) and an associated four dimensional space D of vectors acting on spinors. We show how to obtain, in a canonical way, a quaternionic structure on D by the invariant spinorial approach. It is interesting, that the map which assigns a complex structure (from the quaternionic structure) to a unit spinor is, in fact, the Hopf fibration. Moreover, we show nonexistence of a complex structure on D induced from the complex structure on V.
In the second section, we show how algebraic approach developed in the first section induces a SU(2)–structure on a 5–dimensional spin manifold. Moreover, we show relations with the approaches from [6] and [9].
In the final—third—section, we show that with a slight modification the spinorial approach developed in [6] is invariant, i.e., independent on the choice of a defining spinor. Moreover, we derive explicit formula for the intrinsic torsion as well as for the characteristic connection.
2 Decomposition of the Space of Spinors
2.1 Spin Representation
Consider a real Clifford algebra \(\textrm{Cl}_5\). Then the irreducible representation \(\Delta \) of \(\textrm{Cl}_5\) is complex, \(\Delta =\mathbb {C}^4\). It can be given by the following action of the vectors \(e_i\in \mathbb {R}^5\subset \textrm{Cl}_5\) [1]:
There is a Hermitian product \((\cdot ,\cdot )\) on \(\Delta \) such that the spinor representation is unitary and the Clifford product by vectors is skew–symmetric. Denote by \(\langle \cdot ,\cdot \rangle \) the inner product, which is the real part of \((\cdot ,\cdot )\),
Fix a (unit) spinor \(\varphi \in \Delta \) and define
The action \(\mathbb {R}^5\ni x\mapsto x\cdot \varphi \in W_{\varphi }\) is an isomorphism (see, for example, the proof of Lemma 1.1 below), i.e., \(\dim W_{\varphi }=5\).
We begin with the first well–known easy observation (see for example Lemma 6.2 in [10]).
Lemma 1.1
There is a unique vector \(y=y_{\varphi }\) such that \(y\cdot \varphi =i\varphi \) and a unique complex 2–dimensional subspace \(V_{\varphi }\) of \(W_{\varphi }\). They satisfy
Moreover, there is a real 4–dimensional subspace \(D_{\varphi }\subset \mathbb {R}^5\) such that \(V_{\varphi }=D_{\varphi }\cdot \varphi \) and
where for any \(\psi \in V_{\varphi }^{\bot }\) we have
Proof
Writing \(\varphi =(\varphi ^1,\varphi ^2,\varphi ^3,\varphi ^4)\in \mathbb {C}^4\) the action \(R_{\varphi }:\mathbb {R}^5\rightarrow \mathbb {C}^4\), \(R_{\varphi }(x)=x\cdot \varphi \) is represented by the matrix
It can be checked that the rank of \(R_{\varphi }\) is 5 as a real \(5\times 8\) matrix and is equal to the rank of extended block matrix \((R_{\varphi }\, i\varphi ^{\top })\). Moreover, if x is a solution to \(R_{\varphi }(x)=i\varphi \), then
It implies \(\Vert x\Vert =1\). Thus, there is only one solution to the equation \(R_{\varphi }(x)=i\varphi \) and this solution has unit norm. Denote it by \(y=y_{\varphi }\) and let the action of a vector \(y=\sum _i y^ie_i\in \mathbb {R}^5\) on \(\Delta \) be denoted by \(L_y\). Then \(L_y\) is represented by a matrix
It is not hard to check that there exists \(\tilde{\varphi }\) orthogonal and \(\mathbb {C}\)–linearly independent with \(\varphi \) such that \(y\cdot \tilde{\varphi }=i\tilde{\varphi }\).
We show that \(V_{\varphi }\) equals \(\langle \varphi ,\tilde{\varphi }\rangle _{\mathbb {C}}^{\bot }\). For x orthogonal to y, we have
This implies \((x\cdot \varphi ,\tilde{\varphi })=0\) and \((x\cdot \varphi ,\varphi )=0\) for any x orthogonal to y. Put
By the above \(V_{\varphi }\) is orthogonal with respect to \((\cdot ,\cdot )\) to the complex space spanned by \(\varphi \) and \(\tilde{\varphi }\). Thus, \(V_{\varphi }\) is complex. By dimensional reasons \(V_{\varphi }\) is the unique complex 2–dimensional subspace in \(W_{\varphi }\). Moreover, \(D_{\varphi }=\{x\in \mathbb {R}^5\mid \langle x,y\rangle =0\}\). \(\square \)
We define an action of skew–forms on \(\Delta \) as usual
Denote by \(\mathfrak {su}(2)_{\varphi }\) the anihilator of this action on two–forms (for a given \(\varphi \)). It is well known that \(\mathfrak {su}(2)_{\varphi }\) is isomorphic to the Lie algebra \(\mathfrak {su}(2)\). Moreover let \(\mathbb {R}^4_{\varphi }\) be the subspace of \(\mathfrak {so}(5)\) of 2–forms \(\omega \) such that \(\omega \wedge y^{\flat }=0\). In other words \(\omega \in \mathbb {R}^4_{\varphi }\) if \(\omega =\alpha \wedge y^{\flat }\) for some 1–form on \(D_{\varphi }\). The action of such forms equals
Since \(\dim \mathfrak {su}(2)_{\varphi }=3\), it follows that the subspace \(\mathfrak {so}(5)\cdot \varphi \subset \Delta \) is 7–dimensional and, clearly, orthogonal to \(\varphi \). Hence, \(\mathfrak {so}(5)\cdot \varphi =\langle \varphi \rangle ^{\bot }\).
2.2 Complex Approach
Definition
We say that a complex 2–dimensional subspace V in \(\Delta \) is admissible if for any \(\varphi \in V^{\bot }\) we have \(V\subset W_{\varphi }\).
Lemma 1.2
Fix a (unit) spinor \(\varphi \in \Delta \). Then \(V_{\varphi }\) is admissible.
Proof
By Lemma 1.1\(V_{\varphi }=V_{\tilde{\varphi }}\), where \(\tilde{\varphi }\) is as in the proof of Lemma 1.1. Take any linear combination \(\psi \) of \(\varphi \) and \(\tilde{\varphi }\). Then, \(y_{\psi }=y_{\varphi }\), hence \(D_{\psi }=D_{\varphi }\). Moreover, for x orthogonal to \(y_{\psi }\) we have
for some \(a,b\in \mathbb {C}\). Thus \(V_{\psi }=V_{\varphi }\). \(\square \)
We are now going to show that the construction of \(y_{\varphi }\), as well as \(D_{\varphi }\), is independent of a unit spinor \(\varphi \) in the orthogonal complement \(V^{\bot }\) of an admissible space V.
Lemma 1.3
Let V be an admissible subspace. Then
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1.
\(y_{\varphi }\) coincide for all \(\varphi \in V^{\bot }\),
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2.
\(D_{\varphi }\) coincide for all \(\varphi \in V^{\bot }\),
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3.
\(V=V_{\varphi }\) for any \(\varphi \in V^{\bot }\).
Proof
Fix \(\varphi \in V^{\bot }\). By Lemma 1.1 there is a unique y such that \(y\cdot \varphi =i\varphi \) and \(W_{\varphi }=V_{\varphi }\oplus \langle i\varphi \rangle \), where \(V_{\varphi }=\{x\cdot \varphi \mid \langle x,y\rangle =0\}\). Thus \(D_{\varphi }=\langle y\rangle ^{\bot }\). Moreover, there is \(\tilde{\varphi }\) which is \(\mathbb {C}\)–linearly independent with \(\varphi \) and such that \(y\cdot \tilde{\varphi }=i\tilde{\varphi }\) and \(\Delta =V_{\varphi }\oplus \langle \varphi ,\tilde{\varphi }\rangle _{\mathbb {C}}\). Since V is maximal complex in \(W_{\varphi }\), by admissibility we have \(V_{\varphi }=V\). This proves the third condition.
Now, take any \(\psi \in V^{\bot }\). Then, \(\psi =a\varphi +b\tilde{\varphi }\) for some \(a,b\in \mathbb {C}\). Thus \(y\cdot \psi =i\psi \), which implies \(D_{\psi }=D_{\varphi }\), what proves the first and the second condition. \(\square \)
Theorem 1.4
Assume V is admissible. Then \(\mathfrak {su}(2)_{\varphi }\) coincide for all \(\varphi \in V^{\bot }\). Conversely, for a maximal space U such that all \(\mathfrak {su}(2)_{\varphi }\) coincide for \(\varphi \in U\) the orthogonal complement \(U^{\bot }\) is admissible.
Proof
Assume V is admissible. Take an orthonormal \(\mathbb {C}\)–basis \(\{\varphi , \tilde{\varphi }\}\) of \(V^{\bot }\). By Lemma 1.3\(D_{\varphi }=D_{\tilde{\varphi }}\). We will write just D. Choose an orthonormal basis \((e_j)\) of D and let \((u_j)\) be a basis of D such that \(e_j\cdot \varphi =u_j\cdot \tilde{\varphi }\). Then
Hence \((u_j)\) is also orthonormal. Moreover,
Fixing \(e_1\) and the corresponding \(u_1\), we may take \(e_2=u_1\). Hence, \(u_1\cdot \varphi =u_2\cdot \tilde{\varphi }\). We have
Since both \(e_1\) and \(u_2\) are unit, it follows that \(u_2=-e_1\). In particular, \(\textrm{span}\{e_1,e_2\}=\textrm{span}\{u_1,u_2\}\). Analogously, we set \(e_4=u_3\) and we obtain \(u_4=-e_3\). Consider the following 2–forms
It is not hard to check that \(\omega _j\cdot \varphi =\omega _j\cdot \tilde{\varphi }=0\) and \(\omega _j\) are linearly independent. In particular \(\omega _j\cdot \psi =0\) for any \(\mathbb {C}\)–linear combination of \(\varphi \) and \(\tilde{\varphi }\). Hence, all \(\mathfrak {su}(2)_{\varphi }\) for \(\varphi \in V^{\bot }\) coincide.
Conversely, let \(\omega _0,\omega _1,\omega _2\) be 2–forms in \(\mathbb {R}^5\) defining a Lie algebra \(\mathfrak {g}\) isomorphic to \(\mathfrak {su}(2)\) and let U be the subspace of these \(\varphi \in \Delta \) such that \(\omega _j\cdot \varphi \,=0\). Clearly, U is complex.
Fix \(\varphi \in U\) and denote by V the orthogonal complement of U, i.e., \(V=U^{\bot }\). By Lemma 1.1, \(V_{\varphi }\) is complex 2–dimensional and orthogonal to \(\varphi \). By Lemma 1.3, \(V_{\varphi }\) is admissible. Since \(\tilde{\varphi }\in V_{\varphi }^{\bot }\), where \(\tilde{\varphi }\) is as above, by the first part \(\mathfrak {su}(2)_{\tilde{\varphi }}=\mathfrak {su}(2)_{\varphi }=\mathfrak {g}\), i.e., \(\omega _j\cdot \tilde{\varphi }=0\). Thus \(\tilde{\varphi }\in U\). We have shown that \(V_{\varphi }^{\bot }\subset U\). In other words, \(V\subset V_{\varphi }\subset W_{\varphi }\). It suffices to show that \(\dim _{\mathbb {C}}V=\dim _{\mathbb {C}}V^{\bot }=2\). Suppose \(\dim _{\mathbb {C}}V=1\) and let u be a unit vector in \(\mathbb {R}^5\) orthogonal to \(y_{\varphi }\) and such that \(\psi =u\cdot \varphi \) is orthogonal to V. Then \(\psi \in V^{\bot }\), hence \(\mathfrak {su}(2)_{\psi }=\mathfrak {g}\). Therefore
For \(x\in D_{\varphi }\), by the fact that \(V_{\varphi }\) is a complex subspace, we have
This implies \(\mathbb {R}^4_{\varphi }\cdot \varphi =V_{\varphi }\). Since \(\mathfrak {so}(5)\cdot \varphi =\langle \varphi \rangle ^{\bot }\), it follows that \(\mathfrak {g}\subset \mathfrak {so}(D_{\varphi })\). By (3), we see that in fact \(\mathfrak {g}\subset \mathfrak {so}(3)\), where \(\mathfrak {so}(3)\) is taken with respect to the 3–dimensional subspace of \(D_{\varphi }\) orthogonal to \(u\in D_{\varphi }\). Thus \(\mathfrak {g}=\mathfrak {so}(3)\). This is impossible, since \(\mathfrak {so}(3)\) contains pure elements \(\omega =\alpha \wedge \beta \) and \(\omega \cdot \varphi \) cannot vanish. Finally, \(\dim _{\mathbb {C}}V=2\). \(\square \)
Notice that by Theorem 1.4 we may write \(\mathfrak {su}(2)_V\) for the Lie algebra induced by any spinor \(\varphi \in V^{\bot }\), where V is an admissible space i.e., \(\mathfrak {su}(2)_V=\mathfrak {su}(2)_{\varphi }\) for any \(\varphi \in V^{\bot }\). Moreover, the subspace \(\mathbb {R}^4_{\varphi }\subset \mathfrak {so}(5)\) is also independent of the choice of \(\varphi \) in the orthogonal complement of an admissible space V. Hence we may denote it by \(\mathbb {R}^4_V\).
Let us now describe the decomposition of \(\mathfrak {so}(5)\) into irreducible \(\mathfrak {su}(2)\)–modules. Fix an admissible space V. The orthogonal complement \(V^{\bot }\) is again an admissible space (as will be seen below). Therefore we introduce the ±–notation: \(V^-=V\) and \(V^+=V^{\bot }\). Denote by \(\mathfrak {su}(2)_-\) the Lie algebra corresponding to \(V^-\), which is isomorphic to \(\mathfrak {su}(2)\). Analogously, we write \(\mathbb {R}^4_-\) instead of \(\mathbb {R}^4_{V^-}\).
Lemma 1.5
The following holds:
-
1.
The orthogonal complement of an admissible space is again admissible.
-
2.
Let \(V^-\) be an admissible space. Denoting by \(\mathfrak {su}(2)_+\) the Lie algebra corresponding to \(V^+\) (isomorphic to \(\mathfrak {su}(2)\)) we have
$$\begin{aligned} \Delta =V^-\oplus V^+,\quad \mathfrak {so}(5)=\mathfrak {su}(2)_-\oplus \mathfrak {su}(2)_+\oplus \mathbb {R}^4_-. \end{aligned}$$In particular, \(\mathbb {R}^4_-=\mathbb {R}^4_+\).
Proof
Let \(V^-\) be admissible and choose \(\psi \in V^-\). Choose any \(\varphi \in V^+\). By Lemma 1.3\(V^-=V_{\varphi }\), hence, there is \(x_0\in \mathbb {R}^5\) such that \(\psi =x_0\cdot \varphi \). Therefore, \(x_0\cdot \psi =-\Vert x_0\Vert ^2\varphi \), i.e., \(\varphi \in W_{\psi }\). We have shown that \(V^+\subset W_{\psi }\), thus \(V^+\) is admissible.
By Lemma 1.1, \(x_0\) is orthogonal to y, where \(y\cdot \varphi =i\varphi \). Thus
Hence \(y_{\psi }=-y\). In particular, \(D_{\psi }=D_{\varphi }\) for any \(\varphi \in V^+\). By the proof of Theorem 1.4 we have \(\mathfrak {su}(2)_{\psi }\subset \mathfrak {so}(D_{\psi })\). For \(\omega \in \mathfrak {su}(2)_{\varphi }\) we have
The right hand side vanishes only if \(x_0\lrcorner \omega =0\). Choose a basis \((e_j)\) in \(D_{\varphi }\) such that \(e_1=\frac{1}{\Vert x_0\Vert }x_0\) and a basis \((u_j)\) as in the proof of Theorem 1.4. Then \(\omega \) is a linear combination of \(\omega _0,\omega _1,\omega _2\) given by (2), say \(\omega =\sum _j a_j\omega _j\). If \(\omega \ne 0\), then
Thus \(\omega \cdot \varphi \ne 0\). Therefore, \(\mathfrak {su}(2)_{\psi }\) is transversal to \(\mathfrak {su}(2)_{\varphi }\). This completes the proof. \(\square \)
Corollary 1.6
Let \(V^-\) be an admissible space. Then the following actions are surjective
Proof
Follows from the fact that for a given spinor \(\varphi \in V^+\) we have \(\mathfrak {su}(2)_+\cdot \varphi =V^+\cap \langle \varphi \rangle ^{\bot }\). \(\square \)
Remark 1.7
From the considerations above we have a useful observation concerning the Clifford action. Namely, let \(V^-\) be admissible and let \(\varphi \in (V^-)^{\bot }=V^+\) be unit. Then the right multiplication \(R_{\varphi }:\mathfrak {so}(5)\rightarrow \Delta \) by \(\varphi \) satisfies the following restrictions
which are isomorphisms.
Example 1.8
(The fundamental example). Denote by \(s_1,\ldots ,s_4\) the canonical \(\mathbb {C}\)–basis in \(\Delta =\mathbb {C}^4\). Fix a spinor \(\varphi =s_1\). Then
Hence \(V^-=\langle s_3,s_4\rangle _{\mathbb {C}}\) and \(V^+=\langle s_1,s_2\rangle _{\mathbb {C}}\). Indeed,
Now it suffices to apply Lemma 1.1. Moreover,
Here and further, \(e_{jk}\) denotes the two–form \(e_j\wedge e_k\). Now, it is easy to see that the equation \(\omega \cdot \varphi =0\) is satisfied by the following 2–forms
which define \(\mathfrak {su}(2)_-\). Hence, \((\mathfrak {su}(2)_-)^{\bot }\subset \mathfrak {so}(5)\) is spanned by \(\mathfrak {su}(2)_+\) generated by the elements
and \(\mathbb {R}^4_-\) (see the definition after proof of Theorem 1.4) is generated by the elements
Notice that for the spinor \(s_2\) we have
which confirms that \(V^-\) and \(V^+\) are exactly as stated above. Moreover, we have the following relations
2.3 Complex Structures
Firstly, we will show nonexistence of complex structures on \(\Delta \) satisfying certain relations.
Assuming \(j:\Delta \rightarrow \Delta \) is a complex structure on a real vector space \(\Delta \), there is a complex structure \(I_{\varphi }\) on \(D_{\varphi }\) for fixed \(\varphi \) in the orthogonal complement of the admissible space \(V_{\varphi }\) (compare [4, 6]). Namely,
The definition of \(I_{\varphi }\) depends on \(\varphi \). Moreover, in the definition (5) we only need the values of j on \(V_{\varphi }\), since for any \(x\in D_{\varphi }\) we have \(x:V_{\varphi }^{\bot }\rightarrow V_{\varphi }\).
We want to find all complex structures j on V such that the induced complex structure \(I_{\varphi }\) is independent of the choice of \(\varphi \in V^{\bot }\). We will prove below that there is no such complex structure. Even more, let \(t:V\rightarrow V\) be an \(\mathbb {R}\)–linear map and define the induced linear map \(T^{\varphi }:D_{\varphi }\rightarrow D_{\varphi }\), for \(\varphi \in V^{\bot }\), analogously as in (5):
Theorem 1.9
The only linear map \(t:V\rightarrow V\) such that the induced linear map \(T_{\varphi }\) is independent of the choice of \(\varphi \in V^{\bot }\) is a scalar multiple of the identity map.
Proof
Without loss of generality we may take V to be \(\langle s_3,s_4\rangle _{\mathbb {C}}\). Assume t induces a map T, which does not depend on the choice of the spinor in \(V^{\bot }=\langle s_1,s_2\rangle _{\mathbb {C}}\). Then \(D=\textrm{span}_{\mathbb {R}}\{e_1,e_2,e_3,e_4\}\). Let \(T(e_1)=(a,b,c,d)\). Then
This implies that the matrix of \(t:V\rightarrow V\) with respect to the basis \(s_3,is_3,s_4,is_4\) is of the form
From this we conclude that T is represented by the same matrix with respect to the basis \(e_1,e_2,e_3,e_4\).
Let us study the independence from \(\varphi \). For \(x,y\in D_{\varphi }\) and \(\varphi ,\psi \in V^{\bot }\) such that \(x\cdot \varphi =y\cdot \psi \) it must hold
Substituting relations (4) we obtain \(b=d=0\). Moreover, for \(\varphi \in V^{\bot }\) and \(x\in D_{\varphi }\) there is y (depending on \(\varphi \)) such that \(y\cdot \varphi =x\cdot i\varphi \). Considering this condition we conclude, as above, that \(c=0\). Hence, \(t=a\cdot \textrm{id}_V\). Finally, it is clear that such t satisfies the assumptions of the theorem. \(\square \)
Corollary 1.10
There is no complex structure I on \(D_{\varphi }\) induced from the complex structure j on V by the formula (5), which does not dependent on the choice of \(\varphi \in V^{\bot }\).
Secondly, we give a natural procedure on how to define an associated quaternionic structure on an SU(2)–structure via spinorial approach. We have already shown that an approach similar to the one considered in [4] (compare [6]) is not valid. Nevertheless, these is a nice description of the complex structures on \(D_{\varphi }\) with the spinorial (invariant) approach. The intuition has been already used in the proof of Theorem 1.9.
Let V be an admissible space. We know that all \(D_{\varphi }\) coincide for \(\varphi \in V^{\bot }\). Denote this space by D. Fix \(\varphi \in V^{\bot }\) and \(x\in D\). Then there exists a unique element \(J^{\varphi }(x)\in D\) such that
The second equality follows from the fact that the action of \(x\in D\) on \(\Delta \) is \(\mathbb {C}\)–linear.
Notice that the map \(J^{\varphi }:D\rightarrow D\) is a complex structure,
Hence \((J^{\varphi })^2=-\textrm{id}_D\).
Moreover, we have \(J^{\lambda \varphi }=J^{\varphi }\) for any complex number \(\lambda \ne 0\). Thus we have the correspondence
Proposition 1.11
The correspondence (7) is in fact the Hopf fibration \(\mathbb {S}^1\rightarrow \mathbb {S}^3\rightarrow \mathbb {S}^2\). In particular, the image \(\{J^{\varphi }:\varphi \in D\}\) is a 2–sphere which defines a quaternionic structure on D.
Proof
Without loss of generality we may take \(V=\langle s_3,s_4\rangle _{\mathbb {C}}\). Then \(V^{\bot }=\langle s_1,s_2\rangle _{\mathbb {C}}\). Any spinor \(\varphi \in V^{\bot }\) equals \(\varphi =as_1+bis_1+cs_2+dis_2\) for some \(a,b,c,d\in \mathbb {R}\). Moreover, let \(J^{\varphi }(e_1)=\sum _j x_je_j\). Substituting \(e_1\) for x in the definition of \(J^{\varphi }\) we get
This implies \(x_1=0\), \(x_2=-(a^2+b^2-c^2-d^2)\), \(x_3=2(ad-bc)\), \(x_4=2(ac+db)\). We proceed in a similar way taking \(x=e_2,e_3,e_4\), respectively. Finally, denoting
we obtain
Notice that \(\alpha ^2+\beta ^2+\gamma ^2=1\) and that the map \((a,b,c,d)\mapsto (\alpha ,\beta ,\gamma )\) is the Hopf fibration of \(\mathbb {S}^3\) onto \(\mathbb {S}^2\).
Let us end by showing that the 2–sphere of complex structures defines a quaternionic structure on D. Denote by \((\alpha ,\beta ,\gamma )\in \mathbb {S}^2\) a point on the 2–sphere inducing the complex structure \(J=J(\alpha ,\beta ,\gamma )\) from the image of the map (7). Then,
if and only if the vectors \((\alpha ,\beta ,\gamma )\) and \((\tilde{\alpha },\tilde{\beta },\tilde{\gamma })\) are orthogonal (with respect to the standard inner product in \(\mathbb {R}^3\)). Now, it suffices to take two orthogonal vectors \(u,v\in \mathbb {S}^3\) and define
It is easy to see that the triple \((J_1,J_2,J_3)\) is a quaternionic structure. \(\square \)
2.4 Quaternionic Approach
Let us rewrite some of the results from subsection 1.1 with the quaternionic approach.
Let us begin with a choice of a quaternionic structure on \(\Delta \). The choice is not unique. We begin with the one considered in [6]. It can be shown [11] that in \(\Delta \) there is a quaternionic structure \(i_2:\Delta \rightarrow \Delta \) which anticommutes with multiplication by vectors. Recall that a quaternionic structure j may be seen as an antilinear map such that \(j^2=-\textrm{id}\). Let \(i_1\) be the complex structure on \(\Delta =\mathbb {C}^4\) induced by the volume element \(\textrm{vol}=e_1\cdot e_2\cdot e_3\cdot e_4\cdot e_5\). It is easy to see that \(\textrm{vol}\) induces the standard complex structure given by the multiplication by i, which clearly commutes with the multiplication by vectors. Define \(i_3=i_1\circ i_2\). \(i_3\) anticommutes with multiplication. Then we have a triple \((i_1,i_2,i_3)\) of complex structures on \(\Delta \). Each \(i_k\) is an isometry [6].
We will need the following useful fact.
Lemma 1.12
([6]). Fix a unit spinor \(\varphi \in \Delta \). Then the subspace U generated by \(\varphi ,i_1\varphi ,i_2\varphi ,i_3\varphi \) and its orthogonal complement \(V=U^{\bot }\) are \(i_k\)–invariant, \(k=1,2,3\). Moreover, the subspace \(D\subset \mathbb {R}^5\) such that \(D\cdot \varphi =V\) inherits a quaternionic structure induced by the complex structures \(I_k\), \(k=1,2,3\), defined by \(I_k(x)\cdot \varphi =i_k(x\cdot \varphi )\). In particular, each \(i_k\) leaves the decomposition \(\Delta =V\oplus V^{\bot }\) invariant.
This allows us to prove the main result of this subsection.
Theorem 1.13
A real 4–dimensional subspace \(V\subset \Delta \) is admissible if and only if it is a quaternionic subspace with respect to \((i_1,i_2,i_3)\).
Proof
Firstly, assume V is admissible. Choose any \(\varphi \in V^{\bot }\). Then \(V=V_{\varphi }\). By Lemma 1.12, it suffices to show that \(V^{\bot }=\langle \varphi ,i_1\varphi ,i_2\varphi ,i_3\varphi \rangle \). Since \(i_1\) is the multiplication by i and \(i_3\varphi =i i_2\varphi \) we need to show that \(i_2\varphi \) is orthogonal to V. Any element in V is of the form \(x\cdot \varphi \). We may assume x is unit. Hence
Thus \(\langle i_2\varphi ,x\cdot \varphi \rangle =0\).
Conversely, assume V is quaternionic. Thus its orthogonal complement \(V^{\bot }\) is also quaternonic. Take \(\varphi \in V^{\bot }\). By Lemma 1.3 it suffices to show that \(V=V_{\varphi }\).
We have \(V^{\bot }=\langle \varphi ,i_1\varphi ,i_2\varphi ,i_3\varphi \rangle \). By Lemma 1.12 there is a subspace \(D\subset \mathbb {R}^5\) such that \(D\cdot \varphi =V\). Since V is complex, by Lemma 1.1, \(V=V_{\varphi }\). \(\square \)
For our approach, to make the description at least partially invariant, we need to modify the quaternionic structure \((i_1,i_2,i_3)\) a little bit. The modification depends on the choice of an admissible space V. Let \(j_1=i_1\) and we define \(j_2\) as follows
Finally, let \(j_3=j_1 j_2\). Then \((j_1,j_2,j_3)\) equals \((i_1,i_2,i_3)\) on V and \((i_1,-i_2,-i_3)\) on \(V^{\bot }\). The triple \((j_1,j_2,j_3)\) is in fact a quaternionic structure (since V and \(V^{\bot }\) are invariant with respect to \((i_1,i_2,i_3)\)). Moreover,
Hence, all complex structures \(j_1, j_2\) and \(j_3\) commute with multiplication by vectors.
Now, we move to a description of a quaternionic structure on D. As discussed in the previous subsection take
Then \(J_3=J_1J_2\) and \(J_kJ_l=-J_lJ_k\) for distinct k, l. There are unit spinors (not unique) \(\varphi _1, \varphi _2, \varphi _3\in V^{\bot }\) such that
Moreover, define three 2–forms \(\omega _k\) by
Then
which implies
This relation shows that \(\omega _k\) belongs to the Lie algebra dual to \(\mathfrak {su}(2)\).
Remark 1.14
If \(V=\langle s_3,s_4\rangle _{\mathbb {C}}\), then we may take, for example,
Moreover, the 2–forms \(\omega _k\) are
and the corresponding complex structures \(J_1,J_2,J_3\) are given by the following matrices
We would like to add that a construction of a quaternionic structure from the given data was studied in [7] where the authors use the approach from [9]. Obtained (almost) complex structures agree with our approach.
2.5 Conjugacy Classes
In this subsection we find a condition for admissible spaces, to induce conjugate Lie algebras (equivalently, groups) isomorphic to \(\mathfrak {su}(2)\). More precisely, we deal with the problem when \(\mathfrak {su}(2)_V\) and \(\mathfrak {su}(2)_{V'}\) are conjugate for \(V,V'\) admissible. Recall that \(\mathfrak {su}(2)_V\) is the Lie algebra of all \(\omega \) such that \(\omega \cdot \varphi =0\) for any \(\varphi \in V^{\bot }\).
Lemma 1.15
Assume \(V\subset \Delta \) is an admissible space and let \(g\in \textrm{Spin}(5)\). Then gV is also admissible.
Proof
Since g acts as a complex linear map, it follows that the space gV is complex. Moreover, let \(\psi \in (gV)^{\bot }\) and let \(\varphi \in V\). By the invariance of the Hermitian product, we see that \(g^{-1}\psi \in V{^\bot }\). Since V is admissible, it follows that \(V\subset W_{g^{-1}\psi }\). Thus \(\varphi =x\cdot (g^{-1}\psi )\) for some \(x\in \mathbb {R}^5\). Hence
Since \(\textrm{Ad}(g)x\) is a vector in \(\mathbb {R}^5\), we have \(g\varphi \in W_{\psi }\). This proves admissibility of gV. \(\square \)
Lemma 1.16
The isotropy group of a fixed element of the action of \(\textrm{Spin}(5)\) on admissible subspaces is isomorphic to \(\textrm{Spin}(4)\).
Proof
It suffices to take \(\varphi =s_1\). Then \(V=\langle s_3,s_4\rangle _{\mathbb {C}}\). It is easy to see that \(gV=V\) if and only if \(g\in \textrm{Spin}(4)\) where we consider the spin group with respect to the first component in the decomposition \(\mathbb {R}^5=\mathbb {R}^4\oplus \mathbb {R}\). \(\square \)
Lemmas 1.15 and 1.16 imply the following interpretation of \(\textrm{Spin}(5)\) in terms of Grassmanians.
Corollary 1.17
\(\textrm{Spin}(5)\) acts transitively on the space of admissible subspaces.
Proof
By Theorem 1.13 any admissible space is of the form \(V=\langle \psi ,i_1\psi ,i_2\psi ,i_3\psi \rangle \) for some spinor \(\psi \). Choose another admissible space \(V'\) induced in this way, by a spinor \(\varphi \). Since the action of \(\textrm{Spin}(5)\) on \(\Delta \) is transitive, there is \(g\in \textrm{Spin}(5)\) such that \(g\cdot \varphi =\psi \). We know that \(i_1\) acts by multiplication, \(i_2\) commutes with the action of g, whereas \(i_3\) is the composition of \(i_1\) and \(i_3\). Therefore
Hence \(gV=V'\). \(\square \)
To justify, in a sense, the above fact let us count appropriate dimensions. We have \(\dim \textrm{Spin}(5)=10\) and \(\dim \textrm{Spin}(4)=6\). Since the space of all possible admissible subspaces is, by Theorem 1.13, the quaternionic Grassmanian \(\textrm{Gr}_1(\mathbb {H}^2)\), its dimension is \(\dim \textrm{Gr}_1(\mathbb {H}^2)=4\).
Proposition 1.18
Let \(g\in \textrm{Spin}(5)\). We have
Proof
Let \(\omega \in \mathfrak {su}(2)_V\), i.e., \(\omega \cdot \varphi =0\) for all \(\varphi \in V^{\bot }\). Thus \(0=(\textrm{Ad}(g)\omega )\cdot (g\varphi )\), which implies \(\textrm{Ad}(g)\mathfrak {su}(2)_V=\mathfrak {su}(2)_{g\varphi }\). Since \(g\varphi \in (gV)^{\bot }\), by admissibility of gV, proposition follows. \(\square \)
Corollary 1.19
Let \(g\in \textrm{Spin}(5)\). Then \(\textrm{Ad}(g)\mathfrak {su}(2)_V=\mathfrak {su}(2)_V\), for V admissible, if and only if \(gV=V\). In particular, the stabilizer of \(\mathfrak {su}(2)_V\) with respect to the adjoint action of \(\textrm{Spin}(5)\) is isomorphic to \(\textrm{Spin}(4)\).
Let us compare the above considerations with a quaternionic approach. Choose a quaternionic structure \((j_1,j_2,j_3)\) on \(\Delta \) in the way such that each \(j_k\) commutes or anticommutes with the multiplication by vectors (see subsection 1.7). Consider the natural action of \(\mathbb {H}\) on \(\Delta \): for \(a=a_0+a_1i+a_2j+a_3k\in \mathbb {H}\) and \(\varphi \in \Delta \) let
This action commutes with the action of \(\textrm{Spin}(5)\). Moreover, by Theorem 1.13 each admissible space is of the form \(V_{\varphi }=\{a\cdot \varphi \mid a\in \mathbb {H}\}\) for each \(\varphi \in \Delta \). These arguments give another proof of Lemma 1.16.
The quotient space \(\Delta /\mathbb {H}\) of this action, which is isomorphic to \(\mathbb {R}^4\), is the space of all admissible spaces. In addition, for fixed \(\varphi \) and any \(\psi \in V_{\varphi }^{\bot }\) the action of \(\mathbb {H}\) on \(\varphi \) and \(\psi \) spans \(\Delta \).
3 Invariant Description of SU(2)–Structures
Let (M, g) be a spin 5–manifold with the corresponding Riemannian structure g. Denote by \(\textrm{Spin}(M)\) the spinor structure (with the structure group \(\textrm{Spin}(5)\)) and let \(\textbf{S}\) be the associated spinor bundle, \(\textbf{S}=\textrm{Spin}(M)\times _{\textrm{Spin}(5)}\Delta \), where \(\Delta =\mathbb {C}^4\) is as in the first section. An SU(2)–structure on M is a reduction P of the frame bundle SO(M) to the structure group \(SU(2)\subset SO(5)\). We can extend P to \(P_{\textrm{Spin}(5)}=P\times _{SU(2)}\textrm{Spin}(5)\). Alternatively, as shown by Conti and Salamon [9], an SU(2)–structure is given by a quadruplet \((\alpha ,\omega _1,\omega _2,\omega _3)\) consisting of a 1–form \(\alpha \) and 2–forms \(\omega _1, \omega _2, \omega _3\) such that
for some 4–form v satisfying \(\alpha \wedge v \ne 0\). The third approach is the following [9]. Fix a unit spinor field \(\varphi \). Then we define a subundle P as the set of all frames u such that \(\varphi (x)=[u,\varphi _0]\), where u is a frame over \(x\in M\) and \(\varphi _0\in \Delta \) is a fixed unit spinor. Then P is an SU(2) structure with \(SU(2)=\textrm{Stab}(\varphi _0)\).
Motivated by the final approach and the discussions in the first section we may consider the following approach to SU(2)–structures: Fix an admissible space \(V\subset \Delta \), denote the corresponding Lie algebra by \(\mathfrak {su}(2)_V\) and its Lie group by \(SU(2)_V\). In other words, \(\mathfrak {su}(2)_V=\mathfrak {su}(2)_{\varphi _0}\) for any \(\varphi _0\in V^{\bot }\) (see Theorem 1.4). If P is an \(SU(2)_V\)–structure we may define the space (of real dimension 4) of certain spinor fields
Since spinors from \(V^{\bot }\) are the fixed points of the action of \(SU(2)_V\), it follows that, in fact, \(\textbf{S}\) is a real 4–dimensional subspace. Directly from the definition we see that \(\textbf{S}_P\) is isomorphic to \(V^{\bot }\) (and V). Any spinor field in \(\textbf{S}_P\) is said to induce given SU(2)–structure P.
There is a natural subbundle in the spinor bundle \(\textbf{S}\) over the SU(2)–structure.
Definition
Assume P is an SU(2)–structure on a spin manifold M. We say that a subbundle \(\textbf{V}\) in the spinor bundle \(\textbf{S}\) is adapted to P if it is of the form \(\textbf{V}=P\times _{SU(2)}V\), where V is an admissible space in \(\Delta \) such that \(SU(2)=SU(2)_V\).
Consider the almost complex structure on \(\textbf{S}\) induced by \(j_1\) on \(\Delta \) (see also discussion on a quaternionic structure below). From the definition it follows that the adapted subbundle is complex 2–dimensional.
Notice that two spinor fields \(\varphi \) and \(\psi \), which are sections of \(\textbf{V}^{\bot }\), i.e., an orthogonal complement of the adapted subbundle, do not in general induce the same SU(2)–structure. Indeed, assume \(\varphi \) defines the underlying SU(2)–structure P, i.e., \(\varphi =[u,\varphi _0]\), \(u\in P\), where \(SU(2)=\textrm{Stab}(\varphi _0)\). Then \(\psi \) defines the same structure if and only if \(\psi =[u,\psi _0]\), \(u\in P\), for some \(\psi _0\) in the admissible space V. In other words, \(\varphi \) and \(\psi \) must lie in the space \(\textbf{S}_P\) for some P (which they induce).
We may consider the quaternionic structure \(i_2\) on \(\textbf{S}\) induced from the quaternionic structure \(i_2\) on \(\Delta \). Therefore, \(i_3=i_1\circ i_2\), where \(i_1=j_1\), defines an additional almost complex structure on \(\textbf{S}\) and \((i_1,i_2,i_3)\) forms a triple of almost complex structures [6]. By Theorem 1.13 we have the following result.
Corollary 2.1
An admissible subbundle \(\textbf{V}\) is quaternionic with respect to the quaternionic structure \((i_1,i_2,i_3)\).
If an admissible subbundle \(\textbf{V}\) is fixed, we have, additionally, a quaternionic structure \((j_1,j_2,j_3)\) induced from \((i_1,i_2,i_3)\) on \(\Delta \). These two structures differ only by a sign (for \(i_2\) and \(i_3\) on the orthogonal complement \(\textbf{V}^{\bot }\) of the admissible distribution).
Corollary 2.2
An admissible distribution is quaternionic with respect to a quaternionic structure \((j_1,j_2,j_3)\).
It is important to notice that the spinor fields \(\varphi \), \(j_1\varphi \), \(j_2\varphi \) and \(j_3\varphi \) (equivalently, \(\varphi \), \(i_1\varphi ,i_2\varphi ,i_3\varphi \)) induce the same SU(2)–structure. In other words, if \(\varphi \in \textbf{S}_P\), then \(j_k\varphi \in \textbf{S}_P\) for any \(k=1,2,3\). Indeed, each complex structure \(j_k\) on \(\Delta \) commutes with the action of \(\textrm{Spin}(5)\), i.e., \(j_k(gs)=g(j_ks)\), \(g\in \textrm{Spin}(5)\), \(s\in \Delta \). Therefore, if \(\varphi \) is induced by a spinor \(\varphi _0\in \Delta \), then \(j_k\varphi \) is induced by \(j_k\varphi _0\). Since the quaternionic structure \((j_1,j_2,j_3)\) leaves the admissible space V and its orthogonal complement \(V^{\bot }\) invariant, it follows that \(j_k\varphi \) induces the same SU(2)–structure as \(\varphi \). Therefore if \(\varphi \) is fixed, any spinor field \(\psi \) defining the same SU(2)–structure is given by
where \(j_0\) denotes the identity. Form these considerations, we also see that
where \(\nabla \) is a connection in the spinor bundle induced from the Levi-Civita connection on M. This follows from the fact that each \(j_k\) commutes with the action on \(\textrm{Spin}(5)\). Let us gather these observations in the proposition below.
Proposition 2.3
For any two spinor fields \(\varphi ,\psi \) defining the same SU(2)–structure, i.e., \(\varphi ,\psi \in \textbf{S}_P\), there is a quaternion \((a_0,a_1,a_2,a_3)\) such that \(\psi =\sum _{k=0}^3 a_kj_k\varphi \).
We may describe the above arguments on the level of principal bundles. Namely, fix a group \(SU(2)\subset \textrm{Spin}(5)\). We may assume that SU(2) is induced by a unit spinor \(\varphi _0\in V^{\bot }\) for some admissible space V. There is a bijection between SU(2) structures on M and sections \(\sigma \) of the associated homogeneous bundle \(N=\textrm{Spin}(M)\times _{\textrm{Spin}(5)}(\textrm{Spin}(5)/SU(2))\) – reduction \(P\subset \textrm{Spin}(M)\) to a subgroup SU(2) defines \(\sigma _P\in \Gamma (N)\) as follows
where e is the coset induced by the neutral element. We wish to show that there is a bijection between \(\Gamma (N)\) and \(\Gamma (\textbf{V}^{\bot }_1)/\mathbb {H}_1\), where \(\textbf{V}^{\bot }_1\) denotes the bundle of unit spinors in \(\textbf{V}^{\bot }\) and the right action of unit quaternions on sections \(\varphi \) of this bundle is given by conjugation
The correspondence is the following: Let \(\sigma \in \Gamma (N)\). Since \(\textrm{Spin}(5)/SU(2)\) is isomorphic to \(\mathbb {S}^7\subset \Delta \), the isomorphism depending on \(\varphi _0\), section \(\sigma \) induces a spinor field \(\varphi _{\sigma }\in \Gamma (S)\). It suffices to show that \(\varphi _{\sigma }\in \textbf{V}^{\bot }_1\) and changing \(\varphi _0\) by \(\varphi _0a\), \(a\in \mathbb {H}_1\), leads to \(\varphi _{\sigma }a\). Indeed, \(\sigma \) may be treated as an equivariant function \(\sigma :\textrm{Spin}(M)\rightarrow \textrm{Spin}(5)/SU(2)\). Hence, the induced spinor at \(x\in M\) is given by \(\varphi (x)\sigma (u)\cdot \varphi _0\), \(\pi (u)=x\). Thus \(\varphi (x)\) belongs to the same admissible space as \(\varphi _0\). Secondly, substituting \(\varphi _0\) by \(\varphi _0a\), \(a\in \mathbb {H}_1\), we get
Let us relate how to derive the quadruplet \((\alpha ,\omega _i)\) defining an SU(2)–structure in the sense of [9] from the admissible distribution \(\textbf{V}\). A choice of an SU(2)–structure P gives existence of a codimension one distribution \(\textbf{D}\), defined as \(\textbf{D}=P\times _{SU(2)}D\), where the existence of D follows from the first section, and a unit orthogonal vector field \(\zeta \) called Reeb. \(\zeta \) is induced by a vector \(y\in \mathbb {R}^5\) in Lemma 1.1. Fix an admissible distribution \(\textbf{V}\) and consider the induced SU(2)–structure P. Denote by \(\mathfrak {su}(2)_+\) the Lie algebra dual to \(\mathfrak {su}(2)\) (compare the first section for details). Since the adjoint representation of \(\textrm{SU}(2)\) on \(\mathfrak {su}(2)_+\) is trivial, the bundle
of 2–forms is trivial. Hence, there are global linearly independent three 2–forms \(\omega _1,\omega _2,\omega _3\). Consider, moreover, the quaternionic structure \((J_1,J_2,J_3)\) on the distribution \(\textbf{D}\) described as follows (see subsection 1.7):
where \(\varphi _k\) are three \(\mathbb {R}\)–linearly independent spinor fields in \(\textbf{V}^{\bot }\) (the representation of SU(2) on \(\textbf{V}^{\bot }\) is trivial).
Proposition 2.4
The forms \(\alpha \) and \(\omega _k\) defining an SU(2)–structure in the sense of [9] may be given by the following relations
Proof
We may choose a local section of the orthonormal frame \((e_1,e_2,e_3,e_4,e_5)\) such that the quadruplet \((\alpha ,\omega _k)\) defining an SU(2)–structure in the sense of [9] is given by (see [9])
Then locally \(\textbf{V}^{\bot }=\langle s_1,s_2\rangle _{\mathbb {C}}\) (compare the fundamental example in subsect. 1.4). It suffices to choose \(\varphi _1,\varphi _2,\varphi _3\) as in Remark 1.14. \(\square \)
Remark 2.5
The relations contained in Proposition 2.4, adapted to the considered setting, have been already obtained in [6].
4 Characterization of the Intrinsic Torsion and its Modules
In this section we want to derive a decomposition of the module of all possible intrinsic torsions via spinorial approach.
Let M be a spin 5–manifold with the corresponding Riemannian structure g. Let SO(M) be a frame bundle of oriented orthonornal frames and \(\textrm{Spin}(M)\supset SO(M)\) the induced spin structure with the structure group \(\textrm{Spin}(5)\). The Levi–Civita connection \(\nabla \) on M induces a connection form \(\omega \) on SO(M) and \(\tilde{\omega }\) on \(\textrm{Spin}(M)\). Let \(P\subset SO(M)\) be an SU(2)–structure. Identifying 2–forms with skew–symmetric endomorphism, i.e., considering the isomorpism \(\Lambda ^2((\mathbb {R}^5)^{*})\equiv \mathfrak {so}(5)\) and taking into account the decomposition \(\mathfrak {so}(5)=\mathfrak {su}(2)\oplus \mathfrak {su}(2)^{\bot }=\mathfrak {su}(2)\oplus \mathfrak {su}(2)_+\oplus \mathbb {R}^4\) described in the previous sections, we have the following subbundles of 2–forms
Since the adjoint action on \(\mathfrak {su}(2)_+\) is trivial, the bundle \(\mathfrak {su}_+(M)\) admits, as mentioned earlier, three global linearly independent 2–forms.
An \(\mathfrak {su}(2)\)–component of \(\omega \) induces a Riemannian connection \(\nabla ^P\) on M. The intrinsic torsion of the considered SU(2)–structure is a (1, 2)–tensor field \(\xi \) of the form
From the definition of \(\xi \) is follows that \(\xi \in T^{*}M\otimes \mathfrak {su}^{\bot }(M)\).
Moreover, \(\tilde{\omega }\) and its \(\mathfrak {su}(2)\)–component induce connections on the spinor bundle \(\textbf{S}=\textrm{Spin}(M)\times _{\textrm{Spin}(5)}\Delta =P\times _{SU(2)}\Delta \). Denote them by \(\nabla \) and \(\nabla ^P\) (as on M), respectively. If \(\varphi \in \textbf{S}\) is a spinor field defining P, it follows that \(\nabla ^P\varphi =0\).
Let \(\textbf{V}\) be a subbundle adapted to P. Let V be the corresponding admissible space, \(V^{\bot }\) is its orthogonal complement in \(\Delta \). It is clear from the previous considerations that with respect to the map \(\omega \mapsto \omega \cdot \varphi _0\) for a fixed spinor \(\varphi _0\in V^{\bot }\), we have an isomorphism of \(\mathfrak {su}(2)^{\bot }\) onto \(\langle \varphi _0\rangle ^{\bot }\). It can be shown [9] that with respect to this isomorphism
where we consider \(\xi _X\) as an element of \(\mathfrak {su}(2)^{\bot }\). More precisely, \(\xi _X\) is treated as an invariant function from P to \(\mathfrak {su}(2)^{\bot }\).
Denote by \(\mathcal {T}\) the space \(T^{*}(M)\otimes \mathfrak {su}^{\bot }(M)\) of all possible intrinsic torsions. This space splits into irreducible modules under the action of the group SU(2). In [6], the authors, applying the intuition developed in [4], show how to rewrite \(\nabla \varphi \) for a fixed unit spinor field \(\varphi \) inducing the considered SU(2)–structure into components lying in each irreducible component of \(\mathcal {T}\). Let us recall this approach. Since for a unit spinor field \(\varphi \), \(\nabla _X\varphi \) is orthogonal to \(\varphi \) is follows that there is a linear map \(S^{\varphi }:TM\rightarrow \textbf{D}\) and three one–forms \(\beta ^{\varphi }_k\) on M such that
Here, \((i_1,i_2,i_3)\) is a quaternionic structure on \(\textbf{S}\), \(i_2\) anticommutes with the multiplication by vectors, \(i_1\) is induced by multiplication by the volume element [6]. Let \(S^{\varphi }(\zeta )=V^{\varphi }\), where \(\zeta \) is the Reeb field. Thus, we may write,
where \(\alpha \) is a one–form dual to \(\zeta \) and \(S_\textbf{D}\) is an endomorphism of \(\textbf{D}\). Analogously, we may “decompose” each \(\beta _k\) with respect to the splitting \(\textbf{D}\oplus \langle \zeta \rangle \) as
for some function \(f^{\varphi }_k\). Finally, \(S^{\varphi }_\textbf{D}\) splits as (we skip writing indices \(\varphi \) and \(\textbf{D}\) to make the formula more readable) [6]
where \(S_0\in \mathfrak {su}(M)=P\times _{SU(2)}\mathfrak {su}(2)\) and \(\sigma _k\) is traceless, symmetric and such that \(J_l\sigma _k=(-1)^{\delta _{kl}+1}\sigma _k J_l\), \(l=1,2,3\). Thus elements \(\lambda _0,\lambda _k,f_k,S_0,\sigma _k,\beta ^{\varphi ,\textbf{D}}_k,V^{\varphi }\) (\(k=1,2,3\)) are the components with respect to the splitting of \(\mathcal {T}\) into irreducible modules [6, 9]
4.1 Partial Invariance
The aim is to make the above decomposition independent of the choice of \(\varphi \in \textbf{S}_P\). Let \(\textbf{V}\) be a subbundle in \(\textbf{S}\) adapted to P and let \(V\subset \Delta \) be corresponding admissible subspace. We introduce a slight modification. Instead of considering the quaternionic structure \((i_1,i_2,i_3)\) and elements \(\varphi ,i_1\varphi ,i_2\varphi ,i_3\varphi \) spanning \(\textbf{V}^{\bot }\) we consider the quaternionic structure \((j_1,j_2,j_3)\). For a unit spinor field \(\varphi \) we may write
The first main result of this section gives partial invariance.
Proposition 3.1
\(S^{\varphi }\) is independent on the choice of \(\varphi \in \textbf{S}_P\). Moreover, the one–forms \(\beta ^{\varphi }_k\) change with respect to the following formula: if \(\psi =\sum _{k=0}^3 a_kj_k\varphi \), where \(j_0\) is the identity, then
Proof
For \(\varphi _0\in V\) denote by \((\varphi _0)_{\mathbb {H}}\) the following quadruplet of elements spanning V:
We have a natural action of \(a=(a_0,a_1,a_2,a_3)\in \mathbb {H}\) on V (compare subsection 1.8), namely
where \(j_0\) is the identity. We have
Notice that for \(a\ne 0\), \(\rho (a)^{-1}=\frac{1}{\Vert a\Vert ^2}\rho (\bar{a})\) (\(\rho \) is one of possible inclusions of \(\mathbb {H}\) into \(\mathfrak {gl}(4,\mathbb {R})\)).
Take two unit spinor fields \(\varphi ,\psi \) defining the same SU(2)–structure. Let \(S^{\varphi },S^{\psi }\) and \(\beta ^{\varphi }_k,\beta ^{\psi }_k\) be the corresponding elements with respect to the decomposition (8). Firstly, we will show that \(S^{\varphi }=S^{\psi }\). We have
Since, by the definition of \(j_k\), \(j_k(S^{\varphi }(X)\cdot \varphi )=S^{\varphi }(X)\cdot j_k\varphi \), we get
Hence, \(S^{\varphi }=S^{\psi }\). Moreover, by the considerations at the beginning of the proof
where \(a=(a_0,a_1,a_2,a_3)\). Substituting this relation we get the desired formula for the change of \(\beta ^{\varphi }_k\). \(\square \)
Remark 3.2
Notice that the components of each \(\beta ^{\psi }_k\) constitute the Hopf fibration (compare Proposition 1.11 and its proof). Moreover, treating \((\beta ^{\varphi }_1,\beta ^{\varphi }_2,\beta ^{\varphi }_3)\) as a trivialization of \(\mathfrak {su}(2)_+\) isomorphic to \(\mathfrak {su}(2)\), transformation for the one-forms \(\beta ^{\varphi }_k\) restricted to unit quaternions is just the adjoint action of SU(2) on \(\mathfrak {su}(2)\).
4.2 Full Invariance
Let us now show how to decompose \(\nabla _X\varphi \) to obtain all “components” independent of \(\varphi \). Recall that the multiplication of two–forms in \(\mathfrak {so}(5)\) by \(\varphi _0\in \Delta \) is a surjective map onto \(\langle \varphi _0\rangle ^{\bot }\) with the kernel \(\mathfrak {su}(2)\). Moreover, restricted to the \(\mathfrak {su}(2)_+\) (the dual to \(\mathfrak {su}(2)\)) it is an isomorphism onto \(V^{\bot }\cap \langle \varphi _0\rangle ^{\bot }\), where V is admissible space such that \(\varphi _0\in V^{\bot }\). Hence, the component \(\sum _{k=0}^3 \beta ^{\varphi }_k(X)j_k\varphi \) in (8) can be written as
where \(\omega ^{\varphi }_X\in \mathfrak {su}_+(M)\) is a 2–form. Therefore, for a unit spinor field \(\varphi \) we have
The advantage of the use of \(\omega ^{\varphi }\) is that we do not specify the “coordinates” \((j_1\varphi ,j_2\varphi ,j_3\varphi )\) on the space \(\textbf{V}^{\bot }\cap \langle \varphi \rangle ^{\bot }\). Moreover, \(S^{\varphi }\) and \(\omega ^{\varphi }\) do not depend on the choice of \(\varphi \) in \(\textbf{S}_P\).
Proposition 3.3
There exists an endomorphism \(S:TM\rightarrow \textbf{D}\) and an element \(\omega \in T^{*}(M)\otimes \mathfrak {su}_+(M)\), such that
for any unit spinor field \(\varphi \in \textbf{S}_P\).
Proof
Fix a spinor field \(\varphi \) and let \(\psi =\sum _{k=0}^3 a_kj_k\varphi \in \textbf{S}_P\). Then (9) holds. We will show that \(S^{\psi }=S^{\varphi }\) and \(\omega ^{\varphi }_X=\omega ^{\psi }_X\). We have
We used the fact that for any k, \(j_k(Z\cdot \varphi )=Z\cdot j_k\varphi \), where \(X\in TM\), and \(j_k(\eta \cdot \varphi )=\eta \cdot j_k\varphi \), where \(\eta \) is a two–form. \(\square \)
Notice that \(\omega _X\) splits as
4.3 Intrinsic Torsion Explicitly
Let us now provide an explicit formula for the intrinsic torsion. We proceed analogously as in [4]. Fix an SU(2)–structure on M. Take any \(\varphi \in \textbf{S}_P\) and define a 3–form \(\Psi \) as follows
An easy calculation shows that \(\Psi \) does not depend on the choice of \(\varphi \in S_P\).
Lemma 3.4
For any \(\varphi \in S_P\) the following relation holds
where \(p:TM\rightarrow TM\) is given by \(p(X)=X-\frac{1}{2}\alpha (X)\zeta \). In particular, if \(X_\textbf{D}\in \textbf{D}\), then
Proof
We may assume that \(\varphi =s_1\). Then it can be verified that
Let \(X=\sum _{j=1}^5 x_je_j\). Then
which implies
\(\square \)
Now we may state an explicit formula for the intrinsic torsion.
Proposition 3.5
The intrinsic torsion \(\xi \) of an SU(2)–structure P is given by
where S and \(\omega _X\) are given by the formula (10).
Proof
Firstly notice that \(\omega _X\in \mathfrak {su}(2)_+(M)\subset \mathfrak {su}(2)^{\bot }(M)\). Secondly, by the proof of Lemma 3.4 we see that \(p(X)\lrcorner \Psi \in \mathfrak {su}(2)^{\bot }(M)\). Moreover, by Lemma 3.4 we have
for any \(\varphi \in \textbf{S}_P\). This ends the proof. \(\square \)
Considering the decomposition \(S=S_\textbf{D}+\alpha \otimes V\), where \(S_\textbf{D}\in \textrm{End}(\textbf{D})\) is a restriction of S and \(V=S(\zeta )\in \textbf{D}\), we have
In particular, for \(X=\zeta \),
4.4 Characteristic Connection
We end this section by providing a formula and conditions for existence of the characteristic connection \(\nabla ^c\) [2]. Recall, that \(\nabla ^c\) is a metric connection which has totally skew–symmetric torsion \(T^c\) and which is also an SU(2)–connection, i.e., \(\nabla ^c\varphi =0\) for any unit spinor \(\varphi \in \textbf{S}_{P}\) defining the underlying SU(2)–structure P.
In order to do so, we need some additional definitions. We rely on the approach developed in [4]. We will derive a formula for \(\nabla ^c\) using the formula for characteristic connection for \(U(2)\subset SO(5)\) structures [12]. An U(2)–structure on M, i.e., an almost contact metric structure, is defined by a 1–form \(\alpha \) and an almost complex structure J on \(\textbf{D}=\textrm{ker}\alpha \) (compatible with the metric). Fixing an SU(2)–structure on M the choice of an U(2)–structure is not unique, in fact, it can be any J taken from a quaternionic structure (we extend J to the Reeb field trivially). We fix any such J and denote by \(F\in \mathfrak {su}_+(M)\) the induced 2–form. There is a natural choice of a spinor \(\varphi \) associated with J. Namely, there is unique, up to multiplication by complex numbers, unit \(\varphi \in V^{\bot }\) such that \(F\cdot \varphi =2j\cdot \varphi =2i\varphi \), where j is an almost complex structure on \(\textbf{S}\) (compare Lemma 6.2 in [10]). Moreover, j, J and \(\varphi \) are also related by the formula (compare subsection 1.3).
In fact, since j commutes with multiplication by vectors,
This implies
Moreover, let \(\Psi ^J\) be a 3–form defined by
and \(\Phi \) a 3–tensor in \(T^{*}M\otimes \Lambda ^2(T^{*}M)\) given by
Lemma 3.6
The following formula holds
where S is given by the formula (9).
Proof
Since j is \(\nabla \)-parallel, we have
Using the formula \(\nabla _X\varphi =S(X)\cdot \varphi +\omega _X\cdot \varphi \) we obtain
\(\square \)
Consider now the Nijenhuis tensor of an almost contact metric structure \((M,g,\alpha ,J)\)
and treat it as a 3-tensor contracting it with the metric.
Lemma 3.7
The Nijenhuis tensor may be described as follows
Proof
Firstly notice that
Now, the formula follows by (11) and Lemma 3.6. \(\square \)
We are almost ready to describe the characteristic connection \(\nabla ^c\) for an SU(2)–structure. Since an SU(2)–structure is a U(2)–structure, it follows that \(\nabla ^c\) is also the characteristic connection for a U(2)–structure. By the results in [12] it follows that such a connection is unique and its torsion \(T^c\) equals
where \(d^JF(X,Y,Z)=-(dF)(JX,JY,JZ)\). The existence of \(\nabla ^c\) implies that N is a 3–form and the Reeb field \(\zeta \) is Killing (with respect to the Levi–Civita connection) [12].
Theorem 3.8
An SU(2)–structure on a spin 5–manifold M admits a characteristic connection if and only if the Nijenhuis tensor is totally skewsymmetric, the Reeb field is Killing and \(4F(\omega _X)=\delta F\).
Proof
The proof follows the same lines as the proof of Theorem 3.18 in [4]. Assume \(\nabla ^c\) is a characteristic connection for an U(2)–structure on M. Then, in particular, \(\nabla ^cF=0\). Let \(\varphi \) be a unit spinor considered above, i.e., such that \(j(X\cdot \varphi )=J(X)\cdot \varphi \). By the formula for F and the fact that j is \(\nabla ^c\)–parallel we obtain
Thus \(\langle \nabla ^c_X\varphi ,YZ\cdot j\varphi \rangle =0\) for Y orthogonal to Z. This implies that \(\nabla ^c_X\varphi \) is in the span of \(j\varphi \). Concluding, a U(2) characteristic connection \(\nabla ^c\) is in fact an SU(2) connection, i.e. \(\nabla ^c\varphi =0\), if and only if \(\langle \nabla ^c_X\varphi ,j\varphi \rangle =0\) for all X. Since (see for example [4])
we have
Thus \(4F(\omega _X)=-\delta F\). \(\square \)
Theorem 3.9
Consider an U(2)–structure \((M,g,F,\zeta ,\alpha )\) on a 5–dimensional spin manifold induced from an SU(2)–structure. Assume that the Nijenhuis tensor N is skew–symmetric, the Reeb field \(\zeta \) is Killing and \(4F(\omega _X)=-\delta F(X)\), where \(\omega _X\) is given by (9). Then the torsion \(T^c\) of the characteristic connection \(\nabla ^c\) is given by the formula
Proof
Follows by Lemma 3.7, formula (12) and the facts that \(\nabla _{\zeta }\zeta =0\) and \((\zeta \lrcorner N)(Y,Z)=-d\alpha (Y,Z)+d\alpha (JY,JZ)\) [12]. \(\square \)
Notice that some components in the formula (13) as well as in the formula for the Nijenhuis tensor in Lemma 3.7 may be derived and rewritten in a different way. Firstly, we state and prove a formula analogous to the one contained in Lemma 3.4. Namely, we have the following simple observation.
Lemma 3.10
For a fixed unit spinor \(\varphi \), a vector \(X\in TM\) and the corresponding almost complex structure J the following formula holds
where \(\alpha \wedge (JX)^{\flat }\) lies in the \(\mathbb {R}^4\)–component of \(\mathfrak {so}(5)\).
Proof
By the definition of J, noticing that \(\zeta \cdot \varphi =j\varphi \), we have
\(\square \)
By the lemma above we have
For an element \(\eta \in \Lambda ^2((\mathbb {R}^5)^{*})\equiv \mathfrak {so}(5)\) denote its components with respect to the decomposition \(\mathfrak {so}(5)=\mathfrak {su}(2)\oplus \mathfrak {su}(2)_+\oplus \mathbb {R}^4\) by \(\eta _-\), \(\eta _+\) and \(\eta _4\), respectively. Then by the above
Applying the formula (12) and noticing that the projection to the \(\mathbb {R}^4\) factor equals \(\eta \mapsto \alpha \wedge (\zeta \lrcorner \eta )\), we get
where \(V=S(\zeta )\) and \(X_\textbf{D}\in \textbf{D}\). Here, we used the facts that \(\zeta \lrcorner T=d\alpha \) and \(\zeta \lrcorner d\alpha =0\) [12]. This gives the partial correspondence already established in [6] between the approach by differential forms [9] and the spinorial approach [6] in the case of the existence of the characteristic connection.
Remark 3.11
The formulas for the Nijenhuis tensor N and the torsion \(T^c\) of the characteristic connection contained in Lemma 3.7 and Theorem 3.9 depend on the almost complex structure J. Since J was chosen arbitrary from a corresponding quaternionic structure associated with an SU(2)–structure, these formulas are valid for any such choice. In the approach in [9] and [6] the considered almost complex structure is induced by a 2–form \(\omega _1\) or by the almost complex structure j on \(\textbf{S}\) induced from the volume form and a fixed unit spinor defining SU(2)–structure.
4.5 Intrinsic Torsion of a U(2)–Structure
Any SU(2)–structure in dimension 5 is a U(2)–structure, hence an almost contact structure. Let us compare the intrinsic torsion of these two structures. For the formula for the intrinsic torsion of a U(2)–structure, or more generally, an U(n)–structure on a \((2n+1)\)–dimensional Riemannian manifold we refer to [13].
Let M be a spin 5 dimensional manifold with the corresponding Riemannian structure g. Consider an SU(2)–structure on M. Then, as we have seen in the previous subsection, we may choose a U(2)–structure on M corresponding to the choice of an almost complex structure J on the distribution \(\textbf{D}=\textrm{ker}\alpha \), where \(\alpha \) is a 1–form, whose existence follows from the choice of an SU(2)–structure on M. Then J defines an endomorphism (denoted by the same letter) \(J:TM\rightarrow TM\) which is J in \(\textbf{D}\) and zero on \(\zeta =\alpha ^{\sharp }\). Moreover, let F be the 2–form associated with J, \(F(X,Y)=g(X,JY)\). The intrinsic torsion of the U(2)–structure is given by [13]
Let us rewrite \(\xi ^{U(2)}\), with the use of the objects defined in the previous section, which are induced by the spinors defining the SU(2)–structure. In fact, we have to deal with the first component. By Lemma 3.6 we have
Moreover, by the fact that \((\nabla _X\alpha )Y=(\nabla _XF)(\zeta ,JY)\) [8], we get
Finally,
Recall that the intrinsic torsion of an SU(2)–structure is given by
The difference of \(\xi \) and \(\xi ^{U(2)}\) should be of the form \(\gamma (X)F(Y,Z)\) for some 1–form \(\gamma \). We believe this is the case and leave the details to the reader. This follows from the fact that we have the splitting \(\mathfrak {u}(2)=\mathfrak {su}(2)\oplus \mathbb {R}\), where the second component is induced by the element J.
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Acknowledgements
I would like to thank Ilka Agricola for pointing that there is no uniqueness of choice of the spinor defining a subgroup \(SU(2)\subset \textrm{Spin}(5)\) [3]. This was the starting point of this article. Moreover, I wish to thank anonymous referee for careful reading of the manuscript and valuable suggestions that led to the improvement of the article.
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Niedziałomski, K. A Note on Invariant Description of SU(2)–Structures in Dimension 5. Results Math 79, 15 (2024). https://doi.org/10.1007/s00025-023-02042-x
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DOI: https://doi.org/10.1007/s00025-023-02042-x