The torsion of spinor connections and related structure

In this text we introduce the torsion of spinor connections. In terms of the torsion we give conditions on a spinor connection to produce Killing vector fields. We relate the Bianchi type identities for the torsion of spinor connections with Jacobi identities for vector fields on supermanifolds. Fur...

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Дата:	2006
Автор:	Klinker, F.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2006
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	The torsion of spinor connections and related structure / F. Klinker // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 32 назв. — англ.

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spelling	irk-123456789-1460902019-02-08T01:23:06Z The torsion of spinor connections and related structure Klinker, F. In this text we introduce the torsion of spinor connections. In terms of the torsion we give conditions on a spinor connection to produce Killing vector fields. We relate the Bianchi type identities for the torsion of spinor connections with Jacobi identities for vector fields on supermanifolds. Furthermore, we discuss applications of this notion of torsion. 2006 Article The torsion of spinor connections and related structure / F. Klinker // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 32 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 17B66; 53C27; 53B20 http://dspace.nbuv.gov.ua/handle/123456789/146090 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In this text we introduce the torsion of spinor connections. In terms of the torsion we give conditions on a spinor connection to produce Killing vector fields. We relate the Bianchi type identities for the torsion of spinor connections with Jacobi identities for vector fields on supermanifolds. Furthermore, we discuss applications of this notion of torsion.
format	Article
author	Klinker, F.
spellingShingle	Klinker, F. The torsion of spinor connections and related structure Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Klinker, F.
author_sort	Klinker, F.
title	The torsion of spinor connections and related structure
title_short	The torsion of spinor connections and related structure
title_full	The torsion of spinor connections and related structure
title_fullStr	The torsion of spinor connections and related structure
title_full_unstemmed	The torsion of spinor connections and related structure
title_sort	torsion of spinor connections and related structure
publisher	Інститут математики НАН України
publishDate	2006
url	http://dspace.nbuv.gov.ua/handle/123456789/146090
citation_txt	The torsion of spinor connections and related structure / F. Klinker // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 32 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT klinkerf thetorsionofspinorconnectionsandrelatedstructure AT klinkerf torsionofspinorconnectionsandrelatedstructure
first_indexed	2025-07-10T23:08:13Z
last_indexed	2025-07-10T23:08:13Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 077, 28 pages The Torsion of Spinor Connections and Related Structures Frank KLINKER University of Dortmund, 44221 Dortmund, Germany E-mail: frank.klinker@math.uni-dortmund.de Received August 25, 2006, in final form November 03, 2006; Published online November 09, 2006 Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper077/ Abstract. In this text we introduce the torsion of spinor connections. In terms of the tor- sion we give conditions on a spinor connection to produce Killing vector fields. We relate the Bianchi type identities for the torsion of spinor connections with Jacobi identities for vector fields on supermanifolds. Furthermore, we discuss applications of this notion of torsion. Key words: spinor connection; torsion; Killing vector; supermanifold 2000 Mathematics Subject Classification: 17B66; 53C27; 53B20 1 Introduction In this article we introduce the torsion of arbitrary spinor connections. Although the construc- tion depends on additional data on the spinor bundle, namely a choice of charge conjugation, the notion of torsion of a spinor connection is a natural extension of what is usually known as the torsion of a connection on a manifold. In Section 3 we give the relevant definitions and discuss certain properties. In particular, in Proposition 4 we list Bianchi-type identities which connect the torsion and the curvature of the given spinor connection. The spinor connections for which parallel spinors leads to infinitesimal transformations of the underlying manifold are discussed in Section 4. This turns out to be a symmetry condition on the torsion and lead to the definition of admissibility. In the case of metric connections, admissibility recovers the connections with totally skew symmetric torsion. The latter have been discussed in detail during the last years, e.g. [18] and references therein. Beside these metric connections there are a lot of examples coming from supergravity models and we em- phasize on them, e.g. [15] for the basic one. In view of the Fierz relation we formulate the admissibility condition in terms of forms. In Theorem 2 and its extension 3 we give a list of all admissible connections, i.e. connections such that the supersymmetry bracket of paral- lel spinor fields – when identified with the projection from the endomorphisms of the spinor bundle to the one-forms – closes into the space of Killing vector fields without further as- sumptions. Such connections are always used when we consider supergravity theories and exa- mine the variations of the odd fields. Moreover admissible pairs are one of the basic objects in our current work on natural realizations of supersymmetry on non-flat manifolds. In Sec- tion 4.3 we draw a connection to the geometry of a special class of supermanifold. We show that torsion enters naturally into higher order commutators of canonically defined super vec- tor fields. This yields a connection between the graded Jacobi identity on the superalgebra of vector fields and the Bianchi identities derived in Section 3. The motivation for the intro- duction of supermanifolds and the consideration of the canonical vector field is taken from the constructions in [24] and [25]. The canonical vector field we consider has also been discussed in [30] from another point of view: One of the vector fields is considered as first order opera- mailto:frank.klinker@math.uni-dortmund.de http://www.emis.de/journals/SIGMA/2006/Paper077/ 2 F. Klinker tor on the bundle of exterior powers of the spin bundle and it is asked when this operator is a differential. Section 5 is devoted to examples and applications. We introduce three notions of torsion freeness which are motivated by the discussion so far, and we shortly discuss torsion freeness in the case of flat space. Some properties of spinor connections on flat space have recently been discussed in [11]. In Section 5.3 we discuss brane metrics admitting torsion free admissible subsets. 2 Preliminaries We consider the graded manifold M̂ = (M,ΓΛS). where M denotes a (pseudo) Riemannian spin manifold and ΛS the exterior bundle of the spinor bundle S. The splitting ΛS = Λ0S⊕Λ1S into even and odd forms define the even and odd functions on M̂ . An inclusion of vector fields on the base manifold M and sections of the spinor bundle S into the vector fields on M̂ via  : X(M)⊕ ΓS ↪→ X(M̂) yields a splitting1 X(M̂) = ΓΛS ⊗ X(M)⊕ ΓΛS ⊗ ΓS, compare [26] or [28]. The even and odd parts of the vector fields are given by X(M̂)α = ΓΛαmod2S ⊗ X(M)⊕ ΓΛα+1 mod2S ⊗ ΓS, α = 0, 1. The v- and s-like fields are defined by Xv(M̂) := ΓΛS ⊗ X(M), Xs(M̂) := ΓΛS ⊗ ΓS. We call a vector field X of order (k, 1) or (k, 0) if X ∈ ΓΛkS ⊗ ΓS or X ∈ ΓΛkS ⊗ X(M), respectively. The graded manifold M̂ = (M,ΓΛS) is equipped with a bilinear form g + C where g is the metric on M and C a charge conjugation on S. The latter is a spin-invariant bilinear form on S. Another important map is the Clifford multiplication γ : X(M)⊗ ΓS → ΓS, γ(X ⊗ η) = γ(X)η = Xη with XY + Y X = −2g(X,Y ). As the notation indicates, we often consider the induced map γ : X(M) → Γ End(S). We call the images of a local frame {eµ} on M γ-matrices and write γ(eµ) = γµ. We always use the abbreviation γµ1···µk = γ[µ1 · · · γµk], e.g. γµν = 1 2 ( γµγν − γµγν ) . The charge conjugation and the Clifford multiplication give rise to the well known morphism ΓS ⊗ ΓS ↪→ ΛX(M), compare [25, 30, 1]. We denote the projection ΓS ⊗ ΓS → ΛkX(M) by Ck and its symmetry by ∆k ∈ {±1}. The projection is explicitly given by the k-form (Ck(φ⊗ ψ))µ1...µk = C(φ, γµ1···µk ψ). (2.1) The symmetry of the morphisms obeys ∆k = −∆k−2 and so may be written as2 ∆k = (−) k(k−1) 2 ∆k+1 0 ∆k 1. (2.2) 1We often use the identifications Γ(E ⊕ F ) = γE ⊕ ΓF , Γ(E ⊗ F ) = ΓE ⊗C∞(M) ΓF , Γ(Hom(E, F )) = HomC∞(M)(ΓE, ΓF ) etc, for sections of vector bundles over the manifold M . 2This can be made more explicit by evaluating ∆k for k = 0, 1, compare [25]. The Torsion of Spinor Connections and Related Structures 3 The charge conjugations as well as the Clifford multiplication γ : X(M)⊗ΓS → ΓS are parallel with respect to the Levi-Civita connection and so are all maps Ck. The map Γ End(S) ↪→ ΛX(M) is called Fierz relation and an isomorphism onto the image is explicitly given by Ω 7→ 2−[D 2 ] 〈D〉∑ n=0 (−) n(n−1) 2 1 n! tr(γ(n)Ω)γ(n), with 〈dimM〉 := dimM if dimM is even and 〈D〉 := 1 2(dimM−1) if D is odd, compare [32, 23]. If we take into account the charge conjugation to identify S and S∗ and use (2.2), the Fierz identity is written as φ⊗ ψ = 1 dimS ∑ n ∆0(∆0∆1)n n! C(φ, γ(n)ψ)(Cγ(n)). (2.3) We will often use the notations{ ϕ,ψ } := 2C1(ϕ⊗ ψ), 〈φ, ψ〉 := C(φ, ψ). Charge conjugations with ∆1 = 1 are of special interest, because { ·, · } may be seen as a super- symmetry bracket in this case. In particular, this choice is possible for Lorentzian space-times, i.e. spin manifold of signature (−1, 1, . . . , 1), compare [25]. Furthermore, we draw the attention to [3] for a classification of bilinear forms also for the case of extended supersymmetry algebras. Remark 1. Even in the case of ∆1 = −1 we may construct a graded manifold with super- symmetry bracket by taking the direct sum of the spinor bundle with itself and provide it with a modified charge conjugation C ⊗ τ2. Although there is a choice of charge conjugation with the appropriate symmetry, we are sometimes forced to use the “wrong” one. For example when we want to deal with real spinors. We will discuss such a construction in section 4 so that we will omit it here. Special vector fields on M̂ . The charge conjugation C yields an identification S∗ ' S. Using this identification a natural inclusion  : ΓS → X(M̂)1 is given by the interior multiplication of forms and its image is a vector field of degree −1. Explicitly we have (φ) : ΓS → ΓΛS, (φ)(η) = 〈φ, η〉 with the extension as derivation of degree −1. Let us consider a connection D on the spinor bundle S. For every vector field X ∈ X(M) the action of DX on ΛS is of degree zero. This connection gives rise to an inclusion D : X(M) → X(M̂)0 given by D(X) : ΓΛS → ΓΛS, D(X)(η) = DXη. (2.4) These two inclusions give the natural splitting X(M̂) = ΓΛS ⊗ X(M)⊕ ΓΛS ⊗ ΓS. The endomorphisms of S are vector fields of degree zero on M̂ in the natural way. Suppose Φ ∈ Γ End(S) ⊂ X(M̂)0 ∩ Xv(M̂), then the action is given by ΓΛS ⊃ ΓS 3 η 7−→ Φ(η) ∈ ΓS ⊂ ΓΛS. With respect to a local frame {θk} of S the endomorphism Φ has the components Φi j and the associated vector field is given by Φ = Φj iC ikθj ⊗ (θk), where CijCjk = δik and Cij = C(θi, θj). 4 F. Klinker For X,Y ∈ X(M), ϕ,ψ ∈ ΓS and Φ ∈ Γ End(ΓS) the following fundamental commutation relations hold:[ D(X), D(Y ) ] = R(X,Y ) + D([X,Y ]),[ (ϕ), (ψ) ] = 0, [ D(X), (ϕ) ] = (DC Xφ),[ Φ, (ϕ) ] = (−ΦCϕ), [ D(X),Φ ] = DXΦ. Consider the space X(M)⊗ ΓS of vector-spinors. The decomposition into irreducible repre- sentation spaces yields X(M)⊗ΓS = ΓS⊕ΓS 3 2 . Using the identification X(M) ' Ω1(M) via g, the inclusion of ΓS ↪→ X(M) ⊗ ΓS is given by the Clifford multiplication ξ(Y ) = Y ξ. In this way the spin-3 2 fields are given by the kernel of the Clifford multiplication. Given a frame {eµ} on M with associated γ-matrices γµ the inclusion is given by ΓS ↪→ ΓS ⊗ X(M), φ 7→ (dimM)−1 γµφ⊗ eµ. This identification of the spinors in the vector-spinors is used to define a v-like vector field of degree one on the graded manifold. For φ ∈ ΓS we denote this vector field by ıD(φ) and it is defined by the above formula up to the dimension dependent factor together with (2.4): ıD(φ) = γµφ⊗ D(eµ). In [24, 25] we used this map with D = ∇ the Levi-Civita connection on M and S. In [30] this object is considered to construct a (spinor dependent) differential on ΛS. The action of the differential corresponds to the action of the vector field ıD(φ) on the (super)functions ΓΛS of M̂ , i.e. ΓΛS ⊃ ΓS 3 η 7−→ ı(φ)η = γµφ ∧Dµη ∈ ΓΛ2S ⊂ ΓΛS. This vector field will be considered in section 4.3. 3 The torsion of spinor connections Given a connection D on S we associate to D the field A := D−∇ ∈ Ω1(M)⊗Γ End(S) where as before ∇ denotes the Levi-Civita connection on M . Furthermore, if we denote by A the pro- jection of A onto the sub algebra which is locally given by span { γµν } ⊂ {Φ ∈ Γ End(S)\|[Φ, γµ] ⊂ span{γν} for all µ}, then the connection ∇D = ∇+A is a metric connection on M . As noted in the last section, the charge conjugation C : S → S∗ as well as the Clifford multiplication γ : TM ⊗ S → S are parallel with respect to the Levi-Civita connection. More precisely we have the following well known result: Proposition 1. The Clifford multiplication is parallel with respect to the connection D on S and ∇̃ on M if and only if D = ∇̃ is a metric connection. The charge conjugation is parallel with respect to the connection D on S if and only if A takes its values in span { γµ1···µk ;∆k∆0 = −1 } = span { γµ1···µ4k+2 , γµ1···µ4k−∆0∆1 } . In particular, C is parallel with respect to every metric connection. Example 1. In 11-dimensional space-time, i.e. t = 1, s = 10 we have ∆1 = −∆0 = 1 so that the map Φ 7→ ΦC , with C(ΦCη, ξ) := C(η,Φξ) has (−1)-eigenspace span { γµ1···µ4k+2 , γµ1···µ4k+1 } The Torsion of Spinor Connections and Related Structures 5 and (+1)-eigenspace span { γµ1···µ4k+3 , γµ1···µ4k } . In particular, the Clifford multiplication is skew symmetric. For example, consider the super- covariant derivation which come from the supergravity variation of the gravitino and for which AX has a three-form and a five-form part, compare [15, 14, 31, 17]. This connection does not make the charge conjugation parallel. Due to this example, parallelism of the charge conjugation is not the appropriate notion to be related to supersymmetry in general. To the connection D on S we will associate another connection DC . To construct this we consider the connection D⊗1+1⊗DC on S⊗S and the induced connection on S∗⊗S∗. Then D ⊗ 1 + 1 ⊗DC shall make the charge conjugation parallel, i.e. (D ⊗ 1 + 1 ⊗DC)C = 0. For D = ∇+A this implies DC = ∇−AC . The next remark is obtained immediately. Remark 2. The curvature R of the connection D and the curvature RC of the connection DC are related by (R(X,Y ))C = −RC(X,Y ). We endow the bundle of End(S)-valued tensors on M with a connection induced by D, DC and ∇. Definition 1. Let Φ ∈ X(M)⊗k ⊗ Ω1(M)⊗` ⊗ Γ End(S). The connection D̂ is defined by (D̂ZΦ)(X)ξ := DZ(Φ(X)ξ)− Φ(∇ZX)ξ − Φ(X)DC Z ξ for all vector fields Z,X ∈ Ω1(M)⊗k ⊗ X(M)⊗`, and ξ ∈ ΓS. We consider the following ad-type representation of End(S) on itself. Definition 2. Let Ω ∈ End(S). We define adCΩ : End(S) → End(S) by adCΩΦ := ΩΦ + ΦΩC . This is indeed a representation, because adC[Ω1,Ω2]Φ = [ adCΩ1 , adCΩ2 ] Φ. For Ω = Ω+ + Ω−, i.e. ΩC = Ω+ − Ω− we have adCΩΦ = [ Ω−,Φ ] + { Ω+,Φ } . Furthermore we have (adCΩΦ)C = (ΩΦ + ΦΩC)C = ΦCΩC + ΩΦC = adCΩΦC (3.1) which yields Proposition 2. adCΩ preserves the (±1)-eigenspaces of the linear map Φ 7→ ΦC for all Ω ∈ End(S). Proposition 3. Let D̂ be the connection associated to D cf. Definition 1. Then D and the charge adjoint are compatible in the way that D̂(adCΩΨ) = adCDΩΨ + adCΩD̂Ψ (3.2) for all Ω,Ψ ∈ Γ End(S). 6 F. Klinker Definition 3. Let D be a connection on the spinor bundle S over the (pseudo) Riemannian manifold M and denote by ∇ the Levi-Civita connection on M . The torsion T ∈ Ω2(M) ⊗ Γ End(S) of D is the defined by two times the skew symmetrization of D̂γ : X(M) ⊗ X(M) → Γ End(S). Remark 3. 1. We have (D̂Xγ)(Y ) = D̂X(γ(Y )) − γ(∇XY ). Using this and ∇XY −∇YX = [X,Y ] and omitting the map γ we may also write T (X,Y ) = D̂XY − D̂YX − [X,Y ]. 2. In terms of the difference A = D−∇ ∈ Ω1(M)⊗ Γ End(S) the torsion may be written as T (X,Y ) = adCA(X)Y − adCA(Y )X. 3. The last point and (3.1) yield that the torsion has symmetry ∆1, i.e. for all η, ξ we have C(η, Tµνξ) = ∆1C(ξ, Tµνη) . 4. For a metric connection D on S the torsion is exactly the torsion which is defined by the connection D on the manifold M . The torsion obeys some Bianchi-type identities. Proposition 4. Let D be a connection on the spinor bundle S over the (pseudo) Riemannian manifold M . The torsion T and the curvature R of D obey D̂[κTµν] = adC(R[κµ)γν], (3.3) D̂[κ(adCRγ)µνρ] = adC(R[κµ)Tνρ]. (3.4) In this context we add the following identity for the curvature R of D3: D[κRµν] = 0. (3.5) Proof. With Definition 1 the left hand side of (3.3) is given by (D̂XT )(Y, Z)ξ = DX(T (X,Y )ξ)− T (∇XY, Z)ξ − T (Y,∇XZ)ξ − T (Y, Z)DC Xξ. We use the definition of the torsion and get (D̂XT )(Y, Z)ξ = DX(T (Y, Z)ξ)− T (∇XY, Z)ξ − T (Y,∇XZ)ξ − T (Y, Z)DC Xξ = DXDY (Zξ)−DX(∇Y Zξ)−DX(ZDC Y ξ)−DXDZ(Y ξ) +DX(∇ZY ξ) +DX(Y DC Z ξ)−D∇XY (Zξ) +∇∇XY Zξ + ZDC ∇XY ξ +DZ(∇XY ξ)−∇Z∇XY ξ −∇XY D C Z ξ −DY (∇XZξ) +∇Y∇XZξ +∇XZD C Y ξ +D∇XZ(Y ξ) −∇∇XZY ξ − Y DC ∇XZ ξ −DY (ZDC Xξ) +∇Y ZD C Xξ + ZDC YD C Xξ +DZ(Y DC Xξ)−∇ZY D C Xξ − Y DC ZD C Xξ. 3This identity holds for any connection D on a vector bundle over M , if we endow all tensor bundles with the connection induced by D and the Levi-Civita connection on M . The Torsion of Spinor Connections and Related Structures 7 The underlined terms are symmetric with respect to X, Z or X, Y . So they vanish when we skew symmetrize the above expression with respect to X, Y , Z. So we are left with (D̂XT )(Y, Z)ξ + (D̂ZT )(X,Y )ξ + (D̂Y T )(Z,X)ξ = R(X,Y )(Zξ) +R(Z,X)(Y ξ) +R(Y, Z)(Xξ) + ZRC(Y,X)ξ + Y RC(X,Z)ξ +XRC(Y, Z)ξ + (R0(Y,X)Z +R0(Z, Y )X +R0(X,Z)Y )︸︷︷︸ =0 ξ. With Remark 2, i.e. RC(Y,X) = R(X,Y )C , we may rewrite this as D̂[µTνκ] = R[µνγκ] + γ[κ(Rµν]) C = adC(R[µν)γκ]. The proof of (3.5) is done by similar calculations. (3.4) follows from (3.2) and (3.5) after skew symmetrization of (D̂κadCRγ)µνρ = adCDκRµν γρ + adCRµν D̂κγρ. This completes the proof. � Example 2. We consider a manifold which admits geometric Killing spinors. These are spinors which fulfill the equation ∇Xφ = −aXφ for a constant a 6= 0, the Killing number. This equation has been extensively examined in the literature [8, 29, 21] and in particular [9]. Moreover we would like to stress on [7] where the author draws a remarkable connection between geometric Killing spinors on a manifold and parallel spinors on the cone over the manifold, at least in the Riemannian case. From the above equation we read that the connection D on the spinor bundle for which the geometric Killing spinors are parallel is given by D = ∇+ a · γ. Suppose ∆1∆0 = −1, i.e. the Clifford multiplication is skew symmetric. This yields a condition on the connection which will be important in the next section: adC(A{µ)γν} = aγ{µγν} + aγ{νγ C µ} = −agµν + ∆0∆1aγ{νγµ} = −a(1 + ∆1∆0)gµν = 0. The torsion and the curvature of this connection are given by Tµν = 4aγµν and Rµν = R0 µν + 2a2γµν and obey D̂κTµν = −16agκ[µγν] and adCRµν γκ = R0 µνκλγ λ + 8a2gκ[µγν]. such that both sides of (3.3) vanish. 4 Admissible spinor connections 4.1 Killing equations and admissible connections We examine the conditions on the connection D = ∇+A, such that the vector field {ϕ,ψ} built up by the Killing spinors DCϕ = DCψ = 0 is a Killing vector field, i.e. L{ϕ,ψ}g = 0. We have L{ϕ,ψ}g(eµ, eν) = g(∇µ{ϕ,ψ}, eν) + g(∇ν{ϕ,ψ}, eµ) 8 F. Klinker = g({∇µϕ,ψ}, eν) + g({ϕ,∇µψ}, eν) + {µ↔ ν} = g({ACµϕ,ψ}, eν) + g({ϕ,ACµψ}, eν) + {µ↔ ν} = 2〈ACµϕ, γνψ〉+ 2〈ϕ, γνACµψ〉+ {µ↔ ν} = 2〈ϕ, adCAµ γνψ〉+ {µ↔ ν}. This yields Theorem 1. Let D be a connection on the spinor bundle S over M . Suppose φ, ψ ∈ S are parallel with respect to the associated connection DC . Then the vector field {φ, ψ} = 2C1(φ⊗ψ) is a Killing vector field if the symmetric part of D̂γ : X(M)⊗X(M) → End(S) acts trivially on the parallel spinors. In this case we have ∇µ{η, ξ}ν = C(η, Tµνξ). This motivates the next definition. Definition 4. Let D be a connection on the spinor bundle S over M and K ⊂ ΓS be a subset. 1. We call (K, D) admissible if the symmetric part of D̂γ acts trivially on K. If D is fixed we call K admissible. 2. We call D admissible if D̂γ is skew symmetric. In this case is T = 2D̂γ. Remark 4. Due to Theorem 1 the admissible subsets of DC-parallel spinors are of particular interest. Example 3. Consider the supergravity connection D = ∇+A with A = F 3 + F 5 given by F 3(X) = − 1 36 XµFµνρσγ νρσ and F 5(X) = 1 288 XµFνρστγ µνρστ for a 4-form F on M . This connection obeys F 5 µ = −(F 5 µ)C and F 3 µ = (F 3 µ)C due to Example 1. Furthermore we have adC(Aµ)γν = [ F 5 µ , γν ] + { F 3 µ , γν } = 1 144 F κρστγµνκρστ + 1 9 Fµνκργ κρ which is indeed skew symmetric with respect to µ and ν, i.e. the supergravity connection is admissible. This example can be generalized. Theorem 2. Let D be a connection on the spinor bundle S of M and A := D−∇ ∈ Ω1(M)⊗ Γ End(S). Suppose AX is homogeneous with respect to Γ End(S) ' ⊕ k Ωk(M). Consider the decomposition4 Ω1(M)⊗ Ω`(M) = Ω`+1(M)⊕ Ω`−1(M)⊕ Ω(`,1). A(X) may be written as AX = XcF `+1 + X ∧ G`−1 + A0(X) with an (` + 1)-form F , an (`− 1)-form G and A0 ∈ Ω(`,1). Then D is admissible if and only if A0 = 0 and ∆1∆deg = −1 or equivalently ∆0∆deg−1 = (−)deg, i.e. deg ≡ 3 mod 4, or 1 + ∆0∆1 mod4. Here deg denotes the degree of the forms F and G respectively. 4Ω(`,1) denotes the irreducible representation space with highest weight e1 + e`. The Torsion of Spinor Connections and Related Structures 9 Proof. Consider A either to be of the form Aµ = Fµκ1...κ` γκ1...κ` or Aµ = Gκ1...κ`−1γµκ1...κ`−1 with F ∈ Ω`+1 ⊕ Ω(`,1) and G ∈ Ω`−1. In the first case we have Tµν = Fµ κ1...κ` ( γκ1...κ` γν + γνγ C κ1...κ` ) = Fµ κ1...κ` ( γκ1...κ` γν + ∆0∆`γνγκ1...κ` ) = Fµ κ1...κ` ( γκ1...κ`ν + ∆0∆`γνκ1...κ` ) − `Fµ κ1...κ` ( (−)`−1 + ∆0∆` ) gν[κ1 γκ2...κ`] = Fµ κ1...κ` ( 1 + (−)`∆0∆` ) γκ1...κ`ν − `Fµν κ2...κ` ( (−)`−1 + ∆0∆` ) γκ2...κ` . This expression is skew symmetric if and only if F is totally skew symmetric, i.e. A0 = 0, and ∆0∆` = (−)`−1. With deg = ` + 1 this is exactly the condition stated. The second case is treated in almost the same way. Tµν = Gκ1...κ`−1 ( γµκ1...κ`−1 γν + γνγ C µκ1...κ`−1 ) = Gκ1...κ`−1 ( γµκ1...κ`−1 γν + ∆0∆`γνγµκ1...κ`−1 ) = Gκ1...κ`−1 ( γµκ1...κ`−1ν + ∆0∆`γνµκ1...κ`−1 ) − `Gκ1...κ`−1 ( (−)`−1 + ∆0∆` ) gν[µγκ1...κ`−1] = Gκ1...κ`−1 ( (−)`−1 + ∆0∆` ) γµνκ1...κ` − `Gκ1...κ`−1 ( (−)`−1 + ∆0∆` ) gν[µγκ1...κ`−1]. This is skew symmetric if and only if ∆0∆` = (−)` or ∆0∆`−2 = (−)`−1 which with deg = `− 1 finishes the proof. � If A is of the form AX = α̂X ∧ F + β̂XcF for an `-form F we may rewrite it as AX = αX · F + βF · X where (·) denotes Clifford multiplication and α, β are linear combinations of α̂, β̂. Therefore, we will restrict ourself often to the two cases F ·X and X · F . Remark 5. • To be admissible is a property which has to be checked for every degree of adCAγ. This yields that the connection D on S is admissible if and only if every homogeneous summand is. Furthermore D is admissible iff DC is admissible, because this fact does only depend on the degree of Aµ in Ω(M) which is independent of the charge conjugation. • For AX = X ∧ F +XcG admissible the torsion is given by T (X,Y ) = ±X ∧ Y ∧ F ±XcY cG. Example 4. Let A(X) be of the form X∧F (`) or XcF (`). In eleven dimensional space time this leads to an admissible connection for ` = 0, 3, 4, 7, 8, 11. In Example 3 we have F 3 X ∼ XcF (4) and F 5 X ∼ X ∧ F (4). Theorems 1 and 2 have an important consequence for metric connections on the spinor bundle. Corollary 1. Let D be a metric connection on S. D is admissible if and only if AX is of the form XcF (3). We write Aµ = 1 4Aµνκγ νκ. The torsion tensor in this case is totally skew symmetric and given by Tµνκ = 2A[µν]κ = 2Aµνκ. In other words D is admissible if and only if its torsion is totally skew symmetric. Metric connections with skew symmetric torsion play an important role in string theory as well as supergravity theories. A lot of literature on this topic has been published during the past few years, see for example [19] or [18] and references therein. 10 F. Klinker 4.2 Admissible connections on twisted spinor bundles Sometimes it is necessary to introduce `-form fields which have degree different from those which are allowed by Theorem 2. This is possible in two different ways. The first way is, in particular, interesting if M is of even dimension 2n. Suppose n is even. In this case the `-forms with ` ≡ 1 or 1 + ∆0∆1 mod4 contribute to an admissible connection by F ν1...ν`γν1...ν` γµγ ∗. (4.1) This is due to ∆(γ(`)γ∗) = ∆2n−` (compare (A.3) in Appendix A) and 2n− 1 ≡ 3 mod 4, 2n− 3 ≡ 1 mod 4, 2n− (1±∆1∆0) ≡ 1±∆1∆0 mod4 for n even as well as Theorem 2. If n is odd we have 2n− 1 ≡ 1 mod 4, 2n− 3 ≡ 3 mod 4, 2n− (1±∆1∆0) ≡ 1∓∆1∆0 mod4. In this case the `-form with ` ≡ 3 or 1−∆0∆1 mod4 contributes cf. (4.1), for the same reason. The introduction of γ∗ is a bit artificial, because we may express for example F ν1...ν`γν1...ν` · Xγ∗ as ±(∗F )ν1...ν2n−`γν1...ν2n−` X. Nevertheless, we will see in Section 5.2 that this is a useful description. Corollary 2. We consider the projections Π± : S = S+⊕S− → S±. An `-form contributes to an admissible connection by F(`)γ (`)γµΠ± if and only if ` ≡ 3 mod 4 for n odd, or ` ≡ 1+∆0∆1 mod4 for n even. The second way uses the forms without considering duality, i.e. without adding γ∗. This bypasses the last remark. We replace the spinor bundle S by the direct sum S ⊕ S. This space is equipped with a charge conjugation which is given by the charge conjugation C on S twisted by a modified Pauli-matrix, i.e. C ⊗ τi. For τ0 we get the direct sum of C and we denote this usually by C, too. The connection D for an `-form F may be written as Dµ = ∇µ + 1 `!Fi1...i`γ i1...i`γµ ⊗ τj = ∇µ + Fγµτj with a matrix τj . We have C ⊗ τi(Fγµτjη, γνξ) = C(Fγµτjη, γντiξ) = ∆`∆1εjC(η, γµFγντjτiξ) = ∆`∆1εjεijC ⊗ τi(η, γµFγντjξ). This yields Theorem 3. For the twisted spinor bundle S ⊕ S with charge conjugation C ⊗ τi the `-form F contributes to an admissible connection in the form 1 `!Fi1...i`γ i1...i`γµ ⊗ τj if and only if ∆`∆1εjεij = −1. All possible values for (`, i, j) are listed in Table 1. If we fix ` we see that the possible values of j depend on the choice of τi in the charge conjugation and on ∆0∆1 (at least for even `). For i = j = 0 the two components decouple and we recover the result from Theorem 2. The Torsion of Spinor Connections and Related Structures 11 Table 1. Possible choices for τj so that the `-form contributes to an admissible connection if the charge conjugation is given by C ⊗ τi. i `mod4 j 0 1 2 3 0, 1, 3 1 1 3 3 0, 1, 2 2 1 1, 2, 3 3 0 3 1 1 3 0, 2, 3 i `mod4 j 0 1−∆0∆1 2 1 + ∆0∆1 0, 1, 3 1 1−∆0∆1 3 1 + ∆0∆1 0, 1, 2 2 1−∆0∆1 1, 2, 3 1 + ∆0∆1 0 3 1−∆0∆1 1 1 + ∆0∆1 0, 2, 3 Remark 6. We draw the attention to the fact that we change the symmetry of C1 if we use τ2 to modify the charge conjugation. Example 5. • In [4] the authors discuss pp-wave solutions of type IIA supergravity. The starting point is a Killing equation for the spinors constructed by a 3-form and a 4-form in the following way D = ∇+ F 3γ ⊗ τ3 + F 4γ ⊗ τ1. In ten dimensional space time we have two natural ways to choose the charge conjugation (∆0 = +1 or −1) and in both cases we have ∆1 = 1. The above connection is admissible for the choice ∆0 = −1 and i = 0 or 3 as we read from Table 1. • In [10] the type IIB supergravity and the variation of its fields are discussed. The vanishing of the gravitino variation leads to a Killing equation where AC contains all odd `-forms, F ` which are twisted by τ2 if the degree is ` ≡ 1 mod 4 and by τ1 if the degree is ` ≡ 3 mod 4 and furthermore a second three-form, H3, twisted by τ3. This is possible only for i = 0 independent of ∆0. Moreover the four Z2-symmetries which are given by multiplying the fermion doublets by τ1 or τ3 may be seen as a change of the charge conjugation from C ⊗ τ0 to C ⊗ τ1 or C ⊗ τ3. Now it is evident from Theorem 3 that not all off fields are allowed if we want to keep the connection admissible. In particular, these are F ` = 0 for all ` if j = 1 and F 1 = F 5 = F 9 = H3 = 0 if j = 3. These are exactly the truncations which are made in [10]. We carry on considering the supergravity connection cf. [10] which is given by DC = ∇−AC with ACµ = Hµκλγ κλ + F 1 κγ κγµ ⊗ τ2 + 1 3!F 3 κ1κ2κ3 γκ1κ2κ3γµ ⊗ τ1 + 1 5!F 5 κ1...κ5 γκ1...κ5γµ ⊗ τ2 + 1 7!F 7 κ1...κ7 γκ1...κ7γµ ⊗ τ1 + 1 9!F 9 κ1...κ9 γκ1...κ9γµ ⊗ τ2, (4.2) where H is a torsion three form and the `-forms are connected by ∗F 1 = F 9, ∗F 3 = −F 7, and ∗F 5 = F 5. As we mentioned in Example 5 this connection is admissible for the charge conjugation C ⊗ τ0. 12 F. Klinker Due to the nature of the gravity theories the parallel spinors have a fixed chirality property. More precisely the chirality of the two components of η and a relation between both entries, are fixed for all supersymmetry parameters. This may be described by an operator 1⊗ τi ± γ∗ ⊗ τj . (4.3) In the last part of this section we describe admissible connections which are compatible with such chirality property. In contrast to admissibility it is essential to distinguish between D and DC , as we will see. We consider a manifold of even dimension 2n with twisted spinor bundle S ⊕ S and charge conjugation5 C ⊕ C. We suppose that the connection D has an admissible contribution of the form 1 `!F ` κ1...κ` γκ1...κ`γµ ⊗ τi + 1 (2n−`)!F 2n−` κ1...κ2n−` γκ1...κ2n−`γµ ⊗ τj , (4.4) where the two forms are connected by ∗F ` = w`F 2n−`. We insert this as well as (A.2) into the connection and get 1 `!F ` κ1...κ` γκ1...κ`γµ ( 1⊗ τi − w`(−)n(−) `(`−1) 2 γ∗ ⊗ τj ) . We define Πij,w := 1 2(1⊗ τi + wγ∗ ⊗ τj) which has the following properties: Lemma 1. 1. Πij,w has eigenvalue zero if (i, j) is none of the pairs (0, 2), (2, 0), (1, 3), or (3, 1) and in the latter cases we have Π2 02,w = 1 2wγ ∗ ⊗ τ2 and Π2 13,w = 1 21. 2. The dimension of the zero eigenspace is dim(kerΠij,w) = dimS. 3. For the operators with eigenvalue zero we have Πij,wΠij,−w = 0 if i = j or ij = 0 and Π12,wΠ12,−w = Π03,w, Π23,wΠ23,−w = Π01,w but in all cases ker Πij,± = im Πij,∓. Proof. The proof is done by taking a look at Π2 ij,w = εi + εj 4 1+ 1 + εij 4 wγ∗ ⊗ τiτj , Πij,wΠij,−w = εi − εj 4 1+ εij − 1 4 wγ∗ ⊗ τiτj for the different cases. The kernel of Πij,w which match with the image of Πij,−w is listed in Table 2. � Due to this lemma we may take Πij,w as a kind of projection which defines the chirality properties of the spinors η ∈ S ⊕ S. (4.4) with ∗F ` = w`F 2n−` contributes non trivially to an admissible connection in case of a chirality property of the form Πkl,wη = 0 if and only if ker Πkl,w ∩ ker Πij,α 6= 0 for α = w`(−)n(−) `(`−1) 2 . This is the case in (4.2) where all projections have the same image imΠ11,− = imΠ22,− = S+ ⊕ S+ ⊂ S ⊕ S. We now ask in what way this chirality operator is transferred to the torsion. The observation which is summarized in the next proposition will, in particular, be used in Section 5.1. 5We restrict ourself to the case C ⊗ τ0 = C ⊕ C. The other possibilities are treated in the same way. The Torsion of Spinor Connections and Related Structures 13 Table 2. The kernels of Πij,w as subsets of S ⊕ S. Π00,w Π01,w Π03,w Π11,w S−w ⊕ S−w { (η,−wγ∗η)\|η ∈ S } S−w ⊕ Sw S−w ⊕ S−w Π12,w Π22,w Π23,w Π33,w S−w ⊕ Sw S−w ⊕ S−w { (η, wγ∗η)\|η ∈ S } S−w ⊕ S−w Proposition 5. Consider a connection D which has a contribution proportional to a projection cf. (4.3). Then the associated part of the connection DC as well as the associated part of the torsion of D are proportional to the opposite projection. Proof. We restrict to the the case i = j = 0 where the connection has a contribution of the form Aµ = F(`)γ (`)γµΠ± with Π± = 1± γ∗. The associated part of the connection DC is given by ACµ = F(`)(γ (`)γµΠ±)C = F(`) ( ∆1∆`γµγ (`) ∓ (γ(`)γ∗γµ)C ) = F(`) ( ∆1∆`γµγ (`) ∓∆1∆2n−`γµγ (`)γ∗ ) = −F(`)γµγ (`)Π∓, where the last equality is due to the admissibility of the connection. Furthermore we have Tµν = Aµγν + γνACµ = F(`) ( γ(`)γµΠ±γν − γνγµγ (`)Π∓) = F(`) ( γ(`)γµγν − γνγµγ (`) ) Π∓ = F(`) ( γ(`)γµν + γµνγ (`) ) Π∓. The proof for Aµ = F(`)γµγ (`)Π± or (i, j) 6= (0, 0) is almost the same. � 4.3 Jacobi versus Bianchi In this section we consider a graded manifold of the form M̂ = (M,ΛΓS) and calculate com- mutators of the vector fields ı(φ) which have been defined in the preliminaries. The (graded) Jacobi identity on the (super) Lie algebra of vector fields will be seen to be related to the Bianchi identities. We recall the inclusions of the vector fields on M , the spinors, and of the endomorphisms of S into the vector fields on M̂ as given in Section 2. Due to the fact that we will fix a connection D on S, we will drop the index and will write  : ΓS ⊕ X(M) → X(M̂) for the inclusions. Proposition 6. We consider the graded manifold (M,ΛΓS) and a connection D on S which defines the inclusion  and the map ı : ΓS → X(M̂). Furthermore we consider a linear subspace K ⊂ { φ ∈ ΓS \|DCη = 0 } such that (K, D) is admissible. Then the following holds for all ϕ,ψ ∈ K[ ı(ϕ), ı(ψ) ] = B(R;ϕ,ψ) + 1 2D(T ;ϕ,ψ), (4.5) where we use the short notations B(R;ϕ,ψ) = γµϕ ∧ γνψ ?Rµν , (4.6) D(T ;ϕ,ψ) = ( γµϕ ∧ Tµνψ + γµψ ∧ Tµνϕ ) ⊗Dν .. (4.7) 14 F. Klinker Proof. For all φ, ψ ∈ ΓS we have[ ı(φ), ı(ψ) ] = γµφ? [ (eµ), γνψ ⊗ (eν) ] + γνψ ∧Dν(γµφ)⊗ (eν) = γµφ ∧Dµ(γνψ)⊗ (eν) + γµφ ∧ γνψ ? [ (eν), (eν) ] = B(R;φ, ψ) + γµφ ∧ D̂µγν ψ ⊗Dν + γµψ ∧ D̂µγν φ⊗Dν + γµφ ∧ γνDC µ ψ ⊗Dν + γµψ ∧ γνDC µ φ⊗Dν . In particular, these relations reduce to (4.5) if we restrict to K. � Corollary 3. Consider an admissible metric connection on S, i.e. with skew symmetric torsion Tµνκ = 2Aµνκ. In this case (4.5) is given by[ ı(ϕ), (ψ) ] = 1 4Rµνκργ µϕ ∧ γνψ ? γκρ + 1 2Tµνκγ µϕ ∧ γνψ ⊗Dκ. For the following calculations we restrict to the case that the spinors belong to an admissible subspace K ⊆ { η ∈ ΓS \|DCη = 0 } [ ı(ϕ),B(R; η, ξ) ] = 1 2γκϕ ∧ T κµη ∧ γνξ ?Rµν + 1 2γκϕ ∧ γ µη ∧ T κνξ ?Rµν − γµη ∧ γνξ ∧ adCRµν γκϕ⊗Dκ + γκϕ ∧ γµη ∧ γνξ ? (DκR)µν , (4.8)[ ı(ϕ),D(T ; η, ξ) ] = 1 2γ κϕ ∧ Tκµη ∧ T µνξ ⊗Dν + 1 2γ κϕ ∧ Tκµξ ∧ T µνη ⊗Dν + 1 2T νκϕ ∧ γµη ∧ Tµκξ ⊗Dν + 1 2T νκϕ ∧ γµξ ∧ Tµκη ⊗Dν + γκϕ ∧ γµη ∧ D̂κTµνξ ⊗Dν + γκϕ ∧ γµξ ∧ D̂κTµνη ⊗Dν + γκϕ ∧ γµη ∧ T µνξ ?Rκν + γκϕ ∧ γµξ ∧ T µνη ?Rκν . (4.9) From (4.8) and (4.9) we read of the terms of different order in [ ı(ϕ), [ ı(η), ı(ξ) ]] : [ ı(ϕ), [ ı(η), ı(ξ) ]](3,0) = 1 4 ( γκϕ ∧ Tκµη ∧ T µνξ + γκϕ ∧ Tκµξ ∧ T µνη + T νκϕ ∧ γµη ∧ Tµκξ + T νκϕ ∧ γµξ ∧ Tµκη ) ⊗Dν + ( 1 2γ κϕ ∧ γµη ∧ D̂κTµνξ + 1 2γ κϕ ∧ γµξ ∧ D̂κTµνη − γκη ∧ γµξ ∧ adCRκµ γνϕ ) ⊗Dν , (4.10)[ ı(ϕ), [ ı(η), ı(ξ) ]](4,1) = 1 2 ( γµϕ ∧ γκη ∧ T κνξ + γµϕ ∧ γκξ ∧ T κνη + γκϕ ∧ γµξ ∧ T κνη + γκϕ ∧ γµη ∧ T κνξ ) ?Rµν + γκϕ ∧ γµη ∧ γνξ ? (DκR)µν . (4.11) The Jacobi identity, i.e. the vanishing of ∑ ϕ,η,ξ [ ı(ϕ), [ ı(η), ı(ξ) ]] holds independently for the terms of different degree – here ∑ denotes the graded cyclic sum. More precisely: • The cyclic sums of the first summands in (4.10) and (4.11) vanish due to the symmetry of the involved objects. • The vanishing of the cyclic sum of the second summand in (4.11) is equivalent to the Bianchi identity (3.5). • The cyclic sum of the second summand in (4.10) vanishes due to the algebraic Bianchi- identity of the curvature of the Levi-Civita connection. This is due to the following sup- plement to Proposition 4. The Torsion of Spinor Connections and Related Structures 15 Lemma 2. Let K ⊂ { η ∈ ΓS \|DCη = 0 } such that (D,K) is admissible. Let T be the torsion of D. Then (3.3) in Proposition 4 reduces to( D̂[κTµ]ν − adCRκµ γν −R0 κµνλγ λ ) η = 0 for all η ∈ K. This yields Corollary 4. Let K, D and T as before. For all spinors ϕ, η, ξ ∈ K the following holds∑ ϕ,η,ξ {( D̂[κTµ]ν − adCRκµ γν ) ξ ∧ γκϕ ∧ γµη } = 0. Remark 7. As we saw above, the action of D̂[κTµ]ν − adCRκµ γν on K coincides with the action of the curvature of the Levi-Civita connection R0 on K. If D is admissible this yields a way to express R0 in terms of R and T . Let furthermore D be metric, i.e. a connection with totally skew symmetric torsion. Then the above expression may be written as R0 κλµν = Rκλµν −DT [κTλ]µν − 1 4TκλρTµν ρ − σTκλµν with σTκλµν = 3Tρ[κλTµ]ν ρ which is indeed a 4-form. This is due to [16] or [20]. Here DT denotes the connection which differs from D by (DX −DT X)T (Y, Z) = 1 2T (T (X,Y ), Z) + 1 2T (Y, T (X,Z)), i.e. (Dµ −DT µ )Tκλν = Tρν[λTκ]µ ρ. 5 Applications and examples 5.1 Torsion freeness We consider a connection D on the spinor bundle S and K ⊆ ΓS such that (D,K) is admissible. In (4.5) we defined the map D : S2(ΓS) → Λ2ΓS ⊗ X(M) which motivates the following definition. Definition 5. Let D be a connection on S with torsion T and K ⊆ ΓS such that (D,K) is admissible. 1. We call (D,K) torsion free if D(T ; η, ξ) = 0 for all η, ξ ∈ K. 2. We call (D,K) strongly torsion free if Tµνη = 0 for all η ∈ K. And in view of (4.5) 3. We call (D,K) flat if B(R;ϕ,ψ) = D(T ;ϕ,ψ) = 0 for all ϕ,ψ ∈ K. There are two natural problems: firstly fix D and restrict K such that one of the properties are obtained, secondly look for conditions on the connection – or the torsion – such that an admissible set K is “as large as needed”. Of course, admissible subsets K ⊆ { η ∈ ΓS \|DCη = 0 } will be of particular interest. Due to Theorem 1 the Killing vector fields which we obtain by {K,K} are parallel with respect to the Levi-Civita connection if K is strongly torsion free. Therefore, to get non parallel Killing vector fields by C1, it is necessary for the connection D on S to admit a part which contribution to the torsion acts non trivially on K. 16 F. Klinker 5.1.1 On strongly torsion freeness in Rn We consider flat Rn with spinor bundle S and connection DC Xψ = X(ψ)−ACµψ. where the potential ~A = (AC1 , . . . ,ACn ) is constructed from forms on Rn with constant coeffi- cients. Example 6. Consider R2n with connection DC µ = dµ − ACµ on its spinor bundle. Let AC be determined by a three-form F , moreover F shall be a one-form with values in su(n). Then hol ⊂ su(n) and there exist two parallel pure spinors η, η̄ which are associated via charge conjugation. These spinors obey B(R; η, η) = B(R, η̄, η̄) = 0. We use the decomposition C2n = n ⊕ n̄ where the complex structure obeys nη̄ = n̄η = 0. If F ∈ Λ3C2n ∩ (n̄ ⊗ su(n)) the torsion acts trivially on η. In this case the subspace spanned by this sole odd generator is strongly torsion free, in particular, it would have vanishing center. If F ∈ Λ3C2n ∩ (n⊗ su(n)) the same holds for η̄. We emphasize that in both cases the three-form is not real and that for a real three-form a trivial action on one of the spinors is only possible in case of vanishing torsion. Example 7. Suppose A is obtained by a constant form and AX ∝ Π+ (Π−) for a projection Π± cf. (4.3). Due to Proposition 5 DC and T are proportional to the opposite projection Π− (Π+). So K spanned by the constant positive (negative) spinors is strongly torsion free. The last example can be generalized to Remark 8. Strongly torsion freeness can not be achieved by pure chirality considerations due to Proposition 5, when we want to deal with spinors which are not Levi-Civita parallel. In this case strongly torsion freeness leads to new algebraic constraints on the fields. We will discuss torsion free structures which are not strongly torsion free in Section 5.2 (generalizing Example 6) and 5.3. 5.1.2 A comment on differentials As we mentioned in the introduction and as performed in [30] we may take the vector field ıD(η) = γµη ⊗ D(eµ) as degree-one operator on ΛS and look for conditions such that this operator is a differential. We immediately get Proposition 7. Let D be a connection on a spinor bundle S over the (pseudo) Riemannian manifold M . Consider the vector field ı(η) on the graded manifold (M,ΓΛS). Let DCη = 0, then ı(η) is a dif ferential on ΓΛS if and only if (D, {η}) is flat. When we consider admissible subspaces K of order one we have to take the collection of all elements in { B(R; η, ξ)\|DCη = DCξ = 0 } and { D(T ; η, ξ)\|DCη = DCξ = 0 } and discuss whether or not these terms vanish. In particular, if the dimension of K is large the conditions on the torsion are very restrictive. When we consider the differential point of view we only have to discuss the terms B(R; η, η) and D(T ; η, η) for one fixed spinorial entry. In [30] and [25] the condition on B is discussed for the untwisted case. The twisted case is touched when the authors discuss real spinors. The main emphasis is on metric connections D of holonomy g ⊂ so(n) ⊂ sl(2[n 2 ]) with g = su(n2 ), sp(n4 ), spin(7) if n = 8, or g2 if n = 7. The discussion in [25] is restricted to the torsion free Levi-Civita connection. If we want to cover non-torsion free metric connections – or general spinorial connections – we have to take into account the D-contribution which yields further restrictions and we recall Example 6 and the examples below. The Torsion of Spinor Connections and Related Structures 17 5.2 Parallel pure spinors We consider a Riemannian manifold M of even dimension 2n ≥ 4. Consider a pure spinor η ∈ ΓS. We will discuss conditions on a connection D such that B(R; η, η) or D(T ; η, η) vanish. As before, the case of a DC-parallel pure spinor is of particular interest due to Theorem 1, Section 4.3, and Proposition 7. Although we deal with forms of arbitrary degree, we always specialize to the metric case. A pure spinor is characterized by the following two equivalent conditions (compare [13, 22]). (1) The space {X ∈ TM \| Xη = 0} has maximal dimension, namely n. (2) Ck(η, η) = 0 for all k 6= n. Furthermore a pure spinor is either of positive or of negative chirality and the vector field {η, η} vanishes. The symmetry ∆k and the chirality of Ck are given by 2nmod8 0 2 4 6 ∆2m (−)m ±(−)m −(−)m ∓(−)m ∆2m+1 ±(−)m (−)m ∓(−)m −(−)m chirality non chiral chiral non chiral chiral The different signs belong to the choice of charge conjugation. Chiral means C : S±⊗S∓ → C and non-chiral (nc) means C : S±⊗S± → C. Examining this table yields that the second part 2 in the characterization may be relaxed as follows (2′) The chiral (or anti-chiral) spinor η is pure if Ck(η, η) = 0 for all k − n ≡ 0 mod 4, k 6= n. In particular Cn has symmetry ∆n = 1 in all cases. We take a closer look at B(R; η, η) = γµη ∧ γνη ?Rµν . We use the Fierz identity (2.3) to to rewrite this expression. γ[µη ∧ γν]η = 1 dimS ∑ ∆k=−1 ∆0(∆0∆1)k k! C(γ[µϕ, γ(k)γ ν]ψ)γ(k) = 1 dimS ∑ ∆k=−1 (−∆0∆1)k+1 ( 1 k! C(γµν(k)ψ,ϕ)γ(k) + 1 (k − 2)! C(γ(k−2)ψ,ϕ)γµν(k−2) ) = (−∆0∆1)n+1 dimS ( 1 (n− 2)! C(γµν(n−2)η, η)γ(n−2) + 1 n! C(γ(n)η, η)γ µν(n) ) . The second last equality holds because of (A.1) and the last due to the fact that only the summands with k = n− 2 and k = n+ 2 survive. Furthermore we needed 1 = ∆n = −∆n−2 = −∆n+2. Using the duality relation (A.2) to manipulate the first or second summand, we get the following two equivalent expressions γ[µη ∧ γν]η = (−∆0∆1)n+1 n! dimS C(γ(n)η, η)γ µν(n)(1− (−)nwηγ∗) and γ[µη ∧ γν]η = (∆0∆1)n+1 (n− 2)! dimS C(γµν(n−2)η, η)γ(n−2)(1− (−)nwηγ∗), (5.1) where wη is defined by γ∗η = wηη. 18 F. Klinker Suppose dimM = 4. Then (5.1) is self dual if η is of negative chirality and anti-self dual if η is of positive chirality in the sense that 1 2ερσµνγ [µη ∧ γν]η = −wηγ[ρη ∧ γσ]η. This yields Proposition 8. Let M be of dimension four and the pure spinor η be of negative (positive) chirality. Then B(R; η, η) vanishes if the curvature R of D is self dual (resp. anti-self dual). The last proposition is an extension of the result we obtained in [25] where we examined the four dimensional case with D = ∇ and holonomy su(2) which implies self-duality of the curvature tensor R0. Moreover in dimension four there is a further symmetry which yields B(R0; η, ηC) = 0 for the parallel pure spinors η and its parallel pure charge conjugated ηC . Self duality of the curvature tensor as a necessary condition for the vanishing of B(R; η, η) is too restrictive. Suppose η is positive so that γ[µη ∧ γν]η is anti-self dual. This is half of the game. More precisely we find γ[µη ∧ γν]η in the Λ2,0 part of anti-self dual forms Λ2 − ⊗ C. Here Λ2,0 is defined by the complex structure given by η (compare [27]). If we use complex matrices {γa, γā}1≤a,ā≤2 associated to this complex structure, i.e. γāη = 0, and write R in this frame as Rab, Rab̄, Rāb̄ the necessary condition for the vanishing of B(R; η, η) is R12 = 0. If the connection D, and so the curvature R, is in a real representation the vanishing of the Λ2,0-part of the curvature is equivalent to two of the three self duality equations. Furthermore we have6 Rij = AR∗ ı̄̄A −1 and the Λ0,2 part R1̄2̄ vanishes, too. So the condition for the vanishing of B reduces to R ∈ Λ1,1. The part which prevent the curvature from being self dual is the trace of the Λ1,1-part. This is due to the isomorphism Λ2 + = Λ1,1 0 , cf. [6]. Similar considerations as in the four dimensional case can be made for arbitrary even dimen- sion. For this we introduce complex coordinates associated to the null space of η, {γa, γā}1≤a,ā≤n with γāη = 0 . The only surviving part of the form which is associated to η via the Fierz identity is C(γ1...nη, η)γ1...n with only unbarred indices. So (5.1) reads as γµη ∧ γνη ?Rµν = η(n)εa1...anγa1...an−2(1− (−)nwηγ∗) ?Ran−1an with η(n) := (−∆0∆1)n+1 (n−2)! dimS C(γ1...nη, η) and εa1...an the totally skew-symmetric symbol of unbarred indices. This yields Proposition 9. Let η be a pure spinor on the even dimensional manifold M . Then B(R; η, η) vanishes if and only if εa1...anγ a1...an−2(1− (−)nwηγ∗) ?Ran−1an = 0. (5.2) Here the sum is over the unbarred indices with respect to the complex structure given by the pure spinor η. A class of connections for which the above is applicable is given in the following corollary. The proof needs the decomposition of Λ2 which can be taken from the discussion of the four dimensional case. Corollary 5. Let D be a metric connection on M , and suppose it is of holonomy su(n). Then condition (5.2) holds for the two parallel pure spinors. 6A denotes the matrix which defines the charge conjugation ϕC := Aϕ∗ compare [25]. The Torsion of Spinor Connections and Related Structures 19 Using the complex coordinates which have been introduced above, the condition Rab = 0 as a necessary condition for B(R; η, η) = 0 could be seen directly from (4.6). Nevertheless, we used the Fierz identity here to draw a connection to the forms defined by the spinor η and to make the condition more precise. We turn to the torsion dependent term D(T ; η, η) and distinguish the two cases Tµν = 1 (`−2)!Fµν(`−2)γ (`−2) and Tµν = 1 `!F (`)γµν(`). In both cases we use the Fierz identity as well as (A.1) and condition (2′) above and get after some careful calculations 1 (`− 2)! γµη ∧ Fµν(`−2)γ (`−2)η = 1 dimS ∑ ∆k=−1 (∆0∆1)k+1 (−) m(m−2k−1) 2 (−)k−1(n− k) m!(k −m)!(`−m− 1)! × × Fν(m) (`−1−m)C ( γ(k−m)(`−1−m)η, η ) γ(m)(k−m) (5.3) and 1 `! γµη ∧ F (`)γµν(`)η = 1 dimS ∑ ∆k=−1 (∆0∆1)k+1 (−) m(m−2k−1) 2 (−)m+1(n− k) (m+ 1)!(k −m− 1)!(`−m)! × × ( (m+ 1)F (m) (`−m)C ( γ(k−m−1)(`−m)η, η ) γν(m)(k−m−1) + (−)`(`−m)F(m+1) (`−m−1)C ( γ(k−m−1)(`−m−1)νη, η ) γ(m+1)(k−m−1) ) (5.4) with m = 1 2(k+`−n−1). This may be used to get conditions on the forms and their contribution to the connection D to let D(T ; η, η) vanish. We will not explicitly use this formulas in the next example, but we will see that this would have been possible. Example 8. We turn again to the case of dimension four. In the case ` = 3, i.e. the case of metric connection Dµ −∇0 µ = DC µ −∇0 µ = Aµ = Tµνκγ νκ the term γµη ∧ Tµνη = Tµνκγ µη ∧ γκη vanishes in the case of self duality. We recall the decomposition Λ2 ⊗ Λ1 = Λ1 ⊕ Λ3 ⊕ Λ(2,1). If we denote the projections on Λ1 ' Λ3 and Λ3 by π1 and π3 respectively, we have T ∈ Λ2 ± ⊗ Λ1 ⇐⇒ ∗π1(T ) = ±π3(T ). (5.5) This example fits into the discussion of admissible connections, in particular, when we added “non-allowed” forms to the connection in the artificial way (4.1). Moreover if we would have taken an arbitrary one-form V κγµκγ ∗ and three-form Tµνκγ νκ as contributions to A = D −∇, equations (5.3) and (5.4) would have yield exactly the right hand side of (5.5). As before we may generalize the result to dimensions greater than four. When we consider three-form potentials we see that the D- and the B-term have similar shape. So we get Proposition 10. Let η be a pure spinor and D be constructed by a 3-form. Then D(T ; η, η) = 0 if F ian−1anεa1...anγ a1...an−2(1− (−)nwηγ∗)⊗ ei = 0. (5.6) Here the sum over the a∗ is over the unbarred indices with respect to the complex structure given by the pure spinor η, and the sum over i is over the complete set of indices. 20 F. Klinker Remark 9. • (5.6) is solved by F ∈ (n ⊕ n̄) ⊗ (su(n) ⊕ Λ0,2). Of course, the strongly torsion free Example 6 fits into this discussion. • Propositions 9 and 10 give the conditions on the connection such that the parallel pure spinor yields a differential. We will make a short comment on the twisted case. Consider a doubled spinor bundle. Suppose there are two pure spinors ξ, ξ̂ ∈ ΓS, and let Ξ = (ξ, ξ̂) be one parallel spinor of the twisted bundle. Furthermore, suppose that the two null-spaces defined by ξ and ξ̂ intersect transversally7. The necessary condition for B(R; Ξ,Ξ) to vanish is R = 0. Now suppose that the null spaces of the two spinors have non empty intersection N and the tangent space splits orthogonally into T = N⊕N⊥, i.e. Λ2T = Λ2N⊕N⊗N⊥⊕Λ2N⊥. Then the necessary condition reduces and only the part of curvature which acts on Λ2N⊥ has to vanish. 5.3 Torsion freeness from brane metrics We consider a Lorentzian manifold M = (RD, g) such that the coordinates are orthogonal with respect to the metric g. Furthermore we consider a spinor connection DC which is determined by a single q-form F . This q-form is Hodge-dual to a vector field X, where the Hodge-duality is with respect to only one part of the whole space. Furthermore the metric g shall depend on this vector field in such way that the Christoffel symbols obey ΓABC ∝ XAgBC . We take X to be the gradient of a function f and use the following ansatz for the metric on RD: g = f2 µ(x, y) (dxµ)2 + f2 i (x, y) (dyi)2, (5.7) where ( xµ, ym ) 0≤µ≤p,1≤m≤d is a partition of coordinates into a (p + 1)-dimensional space-time determined by ( xµ ) and a d-dimensional space determined by ( ym ) 1≤m≤d We discuss two choices for the q-form F . Either q = p+ 2 with Fµ1...µp+1m = εµ1...µp+1∂mf(y) or q = d− 1 with Fm1...md−1 = εm1...md−1mδ mn∂nf(y), (5.8) where the function f depends on {ym} only. We call F electric or magnetic field strength in the first or second case, respectively. This notation is due to the fact that the two forms are connected via F (p+2) ∝ ∗DF (d−1). Which values for p are possible to yield an admissible connection in one of the two cases may be checked using Theorem 2 and its extension Theorem 3. Remark 10. This metric together with the q-form for low dimensions is considered in the discussion of p-brane solutions of supergravity. E.g. in dimension D = 11 we have a 5-brane with magnetic four-form or a 2-brane with electric four-form. More general p-branes may be obtained by using a non-flat metric in the space-time part (pp-waves or AdS) or in the space part (see for example [5] or [12] and references therein). 7This is true for the parallel pure spinor and its charge conjugated counterpart in the case of Levi-Civita connection of holonomy su(n). In this case Ξ = (ξ, ξC) is real and B(R; Ξ, Ξ) does not vanish. This has been used in [25] to show that the real supersymmetric Killing structure is not finite in the case of quaternionic spin representation where a twist of the spinor bundle is necessary to yield a real structure. Nevertheless, it has been shown that in this case there exist two isomorphic finite sub-structures. The Torsion of Spinor Connections and Related Structures 21 We specialize our discussion to the case where the metric is determined by two functions which depend on {yi} only: g = f2 1 (y)dx2 + f2 2 (y)dy2 . (5.9) We refer to the coordinate frame by unchecked indices and to the orthonormal frame by checked indices. The two frames are connected by eµ̌ = f1(y)−1∂µ, em̌ = f2(y)−1∂m and eµ̌ = f1(y)dxµ, em̌ = f2(y)dxm. The Levi-Civita connection of (5.9) is determined by the Christoffel symbols ΓABC = ΓCBA = 1 2 ( ∂AgBC + ∂CgBA − ∂BgAC ) , Γµνi = −Γµiν = ∂i(ln f1)gµν , Γijk = ∂i(ln f2)gjk + ∂k(ln f2)gij − ∂j(ln f2)gki, Γµκν = Γµij = Γiµj = 0, and given by ∇A = ∂A + 1 4ΓABCγBC with ∇µ = ∂µ + 1 2Γµνiγνi = ∂µ + 1 2∂i(ln f1)f1f −1 2 γµ̌ ı̌, ∇i = ∂i + 1 4Γiµνγµν + 1 4Γijkγjk = ∂i + 1 2∂j(ln f2)γı̌ ̌. The additional part −AC of the spinor connection DC = ∇−AC is determined by the q-form F and given by a linear combination of FµA1...Aq−1γ A1...Aq−1 and FA1...AqγµA1...Aq . The magnetic case. We consider the (d− 1)-form F cf. (5.8) and calculate −ACµ = αFµA1...Ad−2 γA1...Ad−2 + βFA1...Ad−1 γµ A1...Ad−1 = βεi1...id−1jδ jk(∂kf)γµγi1...id−1 = (−)dβ(d− 1)!f2 2 (det gd)− 1 2 gjk(∂jf)γµkγ[d] = (−)dβ(d− 1)!(∂jf)f1f 2 2 f −1 2 f−d2 γµ̌ ̌γ[d] = (−)dβ(d− 1)!(∂jf)f1f 1−d 2 γµ̌ ̌γ[d] as well as −ACi = αFiA1...Ad−2 γA1...Ad−2 + βFA1...Ad−1γiA1...Ad−1 = αεij1...jd−2kδ kj(∂jf)γj1...jd−2 + βεj1...jd−1kδ jk(∂jf)γij1...jd−1 = (−)d−1 ( α(d− 2)!gjkf2 2 (det gd)− 1 2 (∂jf)γik + βgjkgii′f 2 2 (det gd) 1 2 (∂jf)εj1...jd−1kε i′j1...jd−1 ) γ[d] = (−)d−1 ( α(d− 2)!(∂jf)f2 2 (det gd)− 1 2γı̌ ̌ + β(d− 1)!(∂if)f2 2 (det gd)− 1 2 ) γ[d] = (−)d−1 ( α(d− 2)!(∂jf)f2−d 2 γı̌ ̌ + β(d− 1)!(∂if)f2−d 2 γ[d] ) . From now on we suppose that at least one of the two factors in the brane ansatz is even dimensional. The matrix γ[d] = 1 d!εi1...idγ i1...id is connected to the volume element of the space factor in M and obeys γ[d]γj = (−)d+1γjγ [d] and γ[d]γµ = (−)dγµγ[d]. We choose ε ∈ {1, i} such that (εγ[d])2 = 1. Then Π± = 1 2 ( 1 ± εγ[d] ) is the projection on one half of the spinor bundle. When we denote the spinor bundle of M , of its d-dimensional factor, and of its (p+ 1)-dimensional factor by SD, Sd and Sp+1, respectively, we have S±D = Π±(SD) = S ± d ⊗ Sp+1 if d even, Sd ⊗ S±p+1 if d odd. 22 F. Klinker Furthermore we suppose that f1, f2, f and α, β obey Xi := ∂i(ln f1)f1f −1 2 = (−)dδ1 2 ε β(d− 1)!(∂if)f1f 1−d 2 , Yi := ∂i(ln f2) = −(−)dδ2 2 ε α(d− 2)!(∂if)f2−d 2 (5.10) for some choice of signs δ1, δ2 ∈ {±1}. Remark 11. (5.10) can be obtained by the ansatz f`(y) = eα`u(y) which yields the following system for the constants α`: α1 = (−)dδ1 2 ε β(d− 1)!α3, α2 = −(−)dδ2 2 ε α(d− 2)!α3, α3 = (d− 2)α2. For d = 5 we deal with a four-form F which leads to an admissible connection when we have ∆0∆1 = −1. Then a possible solution for δ1 = −δ2 = −1 and ε = i is β = − i 288 , α = 8i 288 , α1 = −1 6 , α2 = 1 3 , and α3 = 1. In dimension eleven this is the supergravity M5-brane solution. With (5.10) the connection DC is given by DC µ = ∂µ +Xiγµ̌ ı̌Π±, DC i = ∂i + Yjγı̌ ̌Π± + δ2ε (d− 1)β 2α Yiγ [d]. (5.11) The signs δ∗ in (5.10) determine which projection is present. Nevertheless, the projections should be the same in both terms. Proposition 11. 1. The holonomy of the connection (5.11) is given by hol = so(d) n  (p+ 1) · 2 d−1 2 · Sd if d odd, (i.e. (p+ 1) even), (p+ 1) · 2 d 2 −1 · S̃d if d ≡ 0 mod 4, (p+ 1) · 2 d 2 · S̃d if d ≡ 2 mod 4, Here so(d) ⊂ sl(S±D) and for d even S̃d denotes the 2 d 2 −1-dimensional (not specified) half spinor representation S±d . 2. The torsion of the connection D – the charge conjugated of (5.11) – is given by Tµν = δ1εf1f2X · γµ̌ν̌γ[d], Tµi = −δ1εf2Xiγµ̌γ [d] = δ2ε(d− 1)βα−1f1Yiγµ̌γ [d], (5.12) Tij = δ2εf2Ykγ ǩ ı̌̌γ [d]. Proof. The bracket [Dµ, Dν ] vanishes due to Π±γµ̌ı̌ = γµ̌ı̌Π∓ and Π∓Π± = 0 whereas [Dµ, Di] is given by[ ∂µ +Xjγµ̌ ̌Π±, ∂i + Yjγı̌ ̌Π± + δ2ε (d− 1)β 2α Yiγ [d] ] = [ Xjγµ̌ ̌Π±, Ykγı̌ ǩΠ±] + [ Xjγµ̌ ̌Π±, δ2ε (d− 1)β 2α (d− 1)Yiγ[d] ] − ∂iXjγµ̌ ̌Π± = XjYk [ γµ̌ ̌Π±, γı̌ ǩΠ±] + δ2 (d− 1)β 2α YiXj [ γµ̌ ̌Π±, εγ[d] ] − ∂iXjγµ̌ ̌Π± = XjYk ( γµ̌ ̌Π±γı̌ ǩΠ± − γı̌ ǩΠ±γµ̌ ̌Π±) + δ2 (d− 1)β 2α YiXj ( γµ̌ ̌Π±εγ[d] − εγ[d]γµ̌ ̌Π±) − ∂iXjγµ̌ ̌Π± The Torsion of Spinor Connections and Related Structures 23 = XjYkγµ̌ ̌γı̌ ǩΠ± ± δ2 (d− 1)β 2α YiXjγµ̌ ̌Π± − ∂iXjγµ̌ ̌Π± = XjYkγµ̌(−δ̌ı̌γ ǩ + gǰǩγı̌)Π± ± δ2 (d− 1)β 2α YiXjγµ̌ ̌Π± − ∂iXjγµ̌ ̌Π± = −XiYjγµ̌ ̌Π± +XkY kf2 2γµ̌ı̌Π ± ± δ2 (d− 1)β 2α YiXjγµ̌ ̌Π± − ∂iXjγµ̌ ̌Π± = ((±δ2(d− 1)β − 2α 2α XiYj − ∂iXj ) γµ̌ ̌ +XkY kf2 2γµ̌ı̌ ) Π±. Here we used γµ̌ı̌γ[d] = −γ[d]γµ̌ı̌ and XjYk = XkYj . When we calculate [Dµ, Di] we furthermore use γı̌̌γ[d] = γ[d]γı̌̌. This yields[ ∂i + Ykγı̌ ǩΠ± + δ2ε (d− 1)β 2α Yiγ [d], ∂j + Y`γ̌ ˇ̀Π± + δ2ε (d− 1)β 2α Yjγ [d] ] = (∂iY`)γ̌ ˇ̀Π± − (∂jY`)γı̌ ˇ̀Π± + δ2ε (d− 1)β 2α (∂iYj − ∂jYi)γ[d] + YkY` [ γı̌ ǩ, γ̌ ˇ̀]Π± = 2f2 2∂[iY `γ̌]ˇ̀Π ± + 2YkY` ( gı̌̌γ ǩ ˇ̀− δ ˇ̀ ı̌γ ǩ ̌ + gǩ ˇ̀ γı̌̌ − δǩ̌ γı̌ ˇ̀)Π± = 2 ( f2 2∂[iY kγ̌]ǩ + f−2 2 YkY kγı̌̌ + 2YkY[iγ̌] ǩ ) Π±. We have two different families of generators for the holonomy algebra: first { γı̌̌Π±} and second{ γµ̌̌Π±} . The first one generates a so(d) sub algebra of sl(S±D) ⊂ sl(SD). Suppose d is odd. The action of so(d) on the second family generates the commuting set span { γµ̌ı̌1...̌ırΠ ± ∣∣ r odd } ' C`odd d . (5.13) The action of so(d) on this set is given by right multiplication [γı̌̌Π±, γµ̌ı̌1...̌ırΠ ±] = −γµ̌ı̌1...̌ırγı̌̌Π±. As a spin module via right (or left) multiplication the Clifford algebra is isomorphic to a direct sum of copies of the minimal spinor representation Sd and so is the 2d−1-dimensional odd part due to Spin-invariance. The minimal representation Sd is of dimension 2 d−1 2 . Therefore, the commuting set is iso- morphic to (p+ 1) · 2 d−1 2 Sd as representation space. Suppose d is even. Consider once more the set generated by the action of so(d) on the second family. If (γ[d])2 = 1 (d ≡ 0 mod 4) we have γoddΠ± ∝ γoddγ[d]Π± = γoddΠ± in the other case (d ≡ 2 mod 4 ) there is an extra i-factor in the proportionality. Therefore, the commuting set is (5.13) of dimension 2d−1 if d ≡ 2 mod 4, and only one half of this if d ≡ 0 mod 4 due to the duality above. The minimal representation S±d is of dimension 2 d 2 −1. Therefore, as representation space the commuting set is isomorphic to (p+1) ·2 d 2 −1S̃d if d ≡ 0 mod 4 and to (p+1) ·2 d 2 S̃d if d ≡ 2 mod 4. The torsion of the admissible connection D = ∇+A is given by TAB = adCAA γB = AAγB + γBACA. We have −ACµ = δ1ε 2 Xiγµ̌ ı̌γ[d], −ACi = δ2ε 2 Yjγı̌ ̌γ[d] + δ2ε(d− 1)β 2α Yiγ [d]. Due to Theorem 2 we have ∆d−1∆1 = −1 or equivalently ∆d∆0 = (−)d which yields −Aµ = δ1ε 2 Xi ( γµ̌ ı̌γ[d] )C = (−)d δ1ε 2 Xiγµ̌ ı̌γ[d], 24 F. Klinker −Ai = δ2ε 2 Yj ( γı̌ ̌γ[d] )C + δ2ε(d− 1)β 2α Yi(γ[d])C = −(−)d δ2ε 2 Yjγı̌ ̌γ[d] + (−)d δ2ε(d− 1)β 2α Yiγ [d]. This is used to calculate the torsion of the brane connection: Tµν = Aµγν + γνACµ = −δ1ε 2 Xi ( (−)dγµ̌ı̌γ[d]γν + γνγµ̌ ı̌γ[d] ) = −δ1ε 2 f1Xi ( − γµ̌γν̌γ ı̌γ[d] + γν̌γµ̌γ ı̌γ[d] ) = δ1εf1f2X · γµ̌ν̌γ[d] as well as Tµi = Aµγi + γiACµ = −δ1ε 2 Xj ( (−)dγµ̌̌γ[d]γi + γiγµ̌ ̌γ[d] ) = δ1ε 2 f2Xj ( γ ̌γı̌ + γı̌γ ̌ ) γµ̌γ [d] = −δ1εf2Xiγµ̌γ [d]. Last but not least we have Tij = Aiγj + γjACi = δ2ε 2 Yk ( (−)dγı̌ǩγ[d]γj − γjγı̌ ǩγ[d] ) − δ2ε(d− 1)β 2α Yi ( (−)dγ[d]γj + γjγ [d] ) = −δ2ε 2 f2Yk ( γı̌ ǩγ̌ + γ̌γı̌ ǩ ) γ[d] = δ2εf2Ykγ ǩ ı̌̌γ [d]. � Corollary 6. The spinors which are parallel with respect to the connection (5.11) form a sub- space of the kernel of Π±. Explicitly we have η(y) = f(y)η0 with constant η0 ∈ S∓D and f obeys ∂if = ±δ2 (d−1)β 2α Yif . The electric case. Due to the fact that the electric (p + 2)-form is dual to the magnetic (d − 1)-form we will only give a rough sketch of what is used to get a similar result. We will assume that at least one of the factors is even dimensional. Then we have the duality relation induced by γ[d]γ[D] ∝ γ[p+1]. For a suitable choice of X and Y we get DC µ = ∂µ +Xiγµ̌ ı̌Π̂±, DC i = ∂µ + Yjγµ̌ ̌Π̂± + αYiγ [p+1] which is of the same type as in the magnetic case. The projections are given by Π̂± : SD = Sd ⊗ Sp+1 → S±D = Sd ⊗ S±p+1 if (p+ 1) is even, S±d ⊗ Sp+1 if (p+ 1) is odd. The expressions for the holonomy and the torsion can be taken directly from Proposition 11. In the remaining part of this section we analyze in what way we have to restrict the set of parallel spinors to yield a torsion free subset K in the sense of Definition 5. I.e. we look for solutions of D(T , η, ξ) = 0 or equivalently Tiµη ∧ γµξ + Tijη ∧ γjξ + (η ↔ ξ) = 0, Tµiη ∧ γiξ + Tµνη ∧ γνξ + (η ↔ ξ) = 0 (5.14) with TAB given by (5.12). We discuss the four summands separately and get: Tiµη ∧ γµξ = −δ1εf2Xiγµ̌γ [d]η ∧ γµξ = δ1εf2f −1 1 Xiγµ̌ξ ∧ γµ̌γ[d]η The Torsion of Spinor Connections and Related Structures 25 = δ1εf2Xiγµ̌γ [d]ξ ∧ γµη = −Tiµξ ∧ γµη, where the second last equality holds because the spinors are of the same chirality with respect to γ[d]. From this result we see that the first summand in (5.14) vanishes after symmetrization over the spinorial entries. With Zk = δ2εf2Yk we keep on calculating Tijη ∧ γjξ = Zkγ ǩ ı̌̌γ [d]η ∧ γjξ = Zǩγı̌̌γ ǩγ[d]η ∧ γ ̌ξ + Zǩγ[̌ıδ ǩ ̌]γ [d]η ∧ γ ̌ξ = γı̌̌ Z · γ[d]η ∧ γ ̌ξ + γı̌γ [d]η ∧ Zξ − Zı̌ γ̌γ [d]η ∧ γ ̌ξ. The last summand vanishes when we symmetrize with respect to η and ξ. Furthermore we have Tµiη ∧ γiξ = −δ1εf2γµ̌γ [d]η ∧ (Xξ) = −(−)dδ1εf2γ [d]γµ̌η ∧ (Xξ) and Tµνη ∧ γνξ = δ1εf2f1X γµ̌ν̌γ [d]η ∧ γνξ = (−)d+1δ1εf2γµ̌ν̌γ [d](Xη) ∧ γν̌ξ. If we put all this together and use X ∝ Z then equations (5.14) reduce to γı̌̌γ [d](Xη) ∧ γ ̌ξ + γ[d]γı̌η ∧ (Xξ) + (η ↔ ξ) = 0, γµ̌ν̌γ [d](Xη) ∧ γν̌ξ + γ[d]γµ̌η ∧ (Xξ) + (η ↔ ξ) = 0. We collect the brane-example in the following theorem. Theorem 4. Consider the manifold M which is Rp+1×Rd equipped with the p-brane metric (5.7) and denote its spinor bundle by S. Let F be a magnetic (d−1)-form on M , i.e. it is ∗d-dual to a gradient field X(y) on the transversal space Rd. The form F and the metric are compatible such that they define an admissible connection D on the spinor bundle cf. (5.11). Then the space K given by K = { η ∈ ΓS \|DCη = 0, Xη = 0 } is admissible and torsion free. 6 Outlook As stated in the introduction admissible spinorial connections, i.e. connections with further sym- metry condition on its torsion c.f. Definition 3, are basic objects when we look for infinitesimal automorphisms of the underlying manifold constructed from parallel spinors, compare Theo- rem 1. This condition may be relaxed by considering admissible pairs cf. Definition 4. The notion of torsion enters naturally, when we look at commutators of vector fields on supermani- folds constructed from the spinor bundle. This will be one tool in constructing a purely geometric representation of the supersymmetry algebra extending the work of [2] or [25]. Work on this construction is in progress. A Useful identities and symmetries for Clifford multiplication and charge conjugation In this appendix we collect some identities concerning gamma matrices as well as some properties of the symmetry of the morphisms (2.1)8. 8We note that most of the formulas are valid without additional (det g)-factors only if the indices belong to an orthonormal frame (compare the calculations in Section 5.3). 26 F. Klinker For the Clifford multiplication we use the convention γ{µγν} = −gµν which yields γµ1...µk γν1...ν` = min{k,`}∑ m=0 (−) m(m−2k−1) 2 k!`! m!(k −m)!(`−m)! δ [ν1...νm [µ1...µm γµm+1...µk] νm+1...ν`]. (A.1) We have γµ1...µk = 1 (D − k)! (−) k(k+1) 2 (−) D(D+1) 2 εµ1...µDγ µk+1...µDγ[D] (A.2) with γ[D] := γ1 · · · γD = 1 D! εµ1...µDγ µ1 · · · γµD . This matrix obeys (γ[D])2 = (−) D(D+1) 2 +t where t denotes the amount of time-like directions in the metric. For D odd γ[D] is proportional to 1. For D = 2n even we define the modified matrix γ∗ = { γ[2n] σ̃ ≡ 0 mod 4, i γ[2n] σ̃ ≡ 2 mod 4, where σ̃ denotes the signature of the metric g. It obeys γ∗γ(k) = (−)kγ(k)γ∗ and (γ∗)2 = 1 and yields a splitting of the spinors in the two eigenspaces S = S+ ⊕ S−. The symmetry property (2.2) implies ∆k = −1 ⇔ k ∈ {4m−∆1, 4m+ 1 + ∆0}, ∆0∆k = −1 ⇔ k ∈ {4m+ 2, 4m−∆0∆1}, ∆1∆k = −1 ⇔ ∆0∆k−1 = (−)k ⇔ k ∈ {4m+ 3, 4m+ 1 + ∆0∆1}. The symmetries ∆k and ∆D−k are connected via ∆D−k = (−) D(D−1) 2 (−)Dk(−)k(∆0∆1)D∆k. This yields ∆k = (−)n+k∆D−k = ∆(γ(D−k)γ∗) if D = 2n even, (A.3) ∆k = ∆D−k if D = 2n+ 1 odd. Introducing the complex coordinates γa = γa + iγa+n and γā = γa − iγa+n, for a, ā = 1, . . . , n, yields γ{aγb} = γ{āγ b̄} = 0, γ{aγ b̄} = −2gab̄, (−) σ(σ−1) 2 γ∗ = γ11̄ · · · γnn̄ = (1+ γ1γ1̄) · · · (1+ γnγn̄), γ1...nγ∗ = γ1...n, γ1̄...n̄γ∗ = (−)nγ1̄...n̄. 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[31] Papadopoulos G., Tsimpis D., The holonomy of the supercovariant connection and Killing spinors, JHEP, 2003, V.7, 018, 28 pages, hep-th/0306117. [32] van Proeyen A., Tools for supersymmetry, hep-th/9910030. http://arxiv.org/abs/math.DG/0410494 http://arxiv.org/abs/hep-th/0306117 http://arxiv.org/abs/hep-th/9910030 1 Introduction 2 Preliminaries 3 The torsion of spinor connections 4 Admissible spinor connections 4.1 Killing equations and admissible connections 4.2 Admissible connections on twisted spinor bundles 4.3 Jacobi versus Bianchi 5 Applications and examples 5.1 Torsion freeness 5.1.1 On strongly torsion freeness in Rn 5.1.2 A comment on differentials 5.2 Parallel pure spinors 5.3 Torsion freeness from brane metrics 6 Outlook A Useful identities and symmetries for Clifford multiplication and charge conjugation

The torsion of spinor connections and related structure

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