Killing spinors are Killing vector fields in Riemannian Supergeometry

D.V. Alekseevsky^{a}^{1}, V. Cortés^{c}^{2}, C. Devchand^{d}^{3}, U. Semmelmann^{s}^{4}

a Sophus Lie Center, Gen. Antonova 2, kv. 99, 117279 Moscow, Russian Federation c Mathematisches Institut der Universität Bonn, Beringstr. 1, 53115 Bonn, Germany d Max-Planck-Institut für Gravitationsphysik, Schlaatzweg 1, 14473 Potsdam s Mathematisches Institut der Universität Bonn, Beringstr. 1, 53115 Bonn, Germany.

Abstract

A supermanifold M is canonically associated to any pseudo Riemannian spin manifold (M_{0}, g_{0}). Extending the metric g_{0} to a field g of bilinear forms g(p) on T_{p}M, p ∈ M_{0}, the pseudo Riemannian supergeometry of (M, g) is formulated as G-structure on M, where G is a supergroup with even part G_{0} ≅ Spin(k, l); (k, l) the signature of (M_{0}, g_{0}). Killing vector fields on (M, g) are, by definition, infinitesimal automorphisms of this G-structure. For every spinor field s there exists a corresponding odd vector field X_{s} on M. Our main result is that X_{s} is a Killing vector field on (M, g) if and only if s is a twistor spinor. In particular, any Killing spinor s defines a Killing vector field X_{s}.

1 Introduction to supergeometry

First we introduce the supergeometric language which is needed to formulate the main result of the paper. Standard references on supergeometry are [M], [L] and [K].

1.1 Supermanifold. We consider pairs (M_{0}, A), where M_{0} is a C^{∞}-manifold and A = A_{0} + A_{1} is a sheaf of ℤ_{2}-graded ℝ-algebras; dim M_{0} = m.

Example 1: We denote by C_{M_{0}}^{∞} the sheaf of (smooth) functions of M_{0}. It associates to an open set U ⊂ M_{0} the algebra C_{M_{0}}^{∞}(U) = C^{∞}(U) of smooth functions on U. Let E be a (smooth) vector bundle over M_{0} and E the corresponding locally free sheaf of C_{M_{0}}^{∞}-modules: E associates to an open set U ⊂ M_{0} the C^{∞}(U)-module E(U) = Γ(U, E) of sections of E over U. Conversely, any locally free sheaf E of C_{M_{0}}^{∞}-modules defines a vector bundle E → M_{0}. The exterior sheaf ∧E = ∧^{ev}E +∧^{odd}E is a sheaf of ℤ_{2}-graded ℝ-algebras on M_{0}.

1Supported by Max-Planck-Institut für Mathematik (Bonn). e-mail: aba@math.genebee.msu.su 2Supported by AvH Foundation, MSRI (Berkeley) and SFB 256 (Bonn University). e-mail:

vicente@math.uni-bonn.de

3e-mail: devchand@ictp.trieste.it 4Supported by Graduiertenkolleg Bonn. e-mail: uwe@rhein.iam.uni-bonn.de

Definition 1 The pair M = (M_{0}, A) is called a (differentiable) supermanifold of dimension m|n over M_{0} if for all p ∈ M_{0} there exists an open neighborhood U ∋ p and a rank n free sheaf E_{U} of C_{U}^{∞}-modules over U such that A|_{U} ≅ ∧E_{U} (as sheaves of ℤ_{2}-graded ℝ-algebras). The (local) sections of A are called (local) functions on M.

From Def. 1 it follows that there exists a canonical epimorphism ϵ : A → C_{M_{0}}^{∞}, which is called the evaluation map. Its kernel is the ideal J generated by A_{1}: ker ϵ = J = ⟨A_{1}⟩ = A_{1} +A_{1}^{2}. By the construction of Example 1 to any vector bundle E → M_{0} we have associated a supermanifold M(E) = (M_{0}, A = ∧E). In this case the exact sequence

0 → J = ⟨E⟩ → A = ∧E →^{ϵ} C_{M_{0}}^{∞} → 0

of sheafs of ℤ_{2}-graded ℝ-algebras has a canonical splitting C_{M_{0}}^{∞} ↪ ∧E = C_{M_{0}}^{∞} + ⟨E⟩.

Let (x^{1}, …, x^{m}) be local coordinates for M_{0} defined on an open set U ⊂ M_{0} such that A|_{U} ≅ ∧E_{U}, where E_{U} is a rank n free sheaf of C_{U}^{∞}-modules, cf. Def. 1. Let θ_{1}, …, θ_{n} be sections of E_{U} trivializing the vector bundle E_{U} associated to the sheaf E_{U}. Note that x^{1}, …, x^{m}, θ_{1}, …, θ_{n} can be considered as local functions on the supermanifold M. Moreover, any local function f ∈ A(U) is of the form

f = ∑_{α∈ℤ_{2}^{n}} f_{α}(x^{1}, …, x^{m})θ^{α} , f_{α}(x^{1}, …, x^{m}) ∈ C^{∞}(U) = C_{M_{0}}^{∞}(U) ,  (1)

where θ^{α} := θ_{1}^{α_{1}} ∧ … ∧ θ_{n}^{α_{n}}, α = (α_{1}, …, α_{n}).

Definition 2 The tupel (x^{i}, θ_{j}) = (x^{1}, …, x^{m}, θ_{1}, …, θ_{n}) is called a local coordinate system for M over U.

The evaluation map applied to a (local) function f = f(x^{1}, …, x^{m}, θ_{1}, …, θ_{n}) with expansion (1) is given by:

ϵ(f) = f(x^{1}, …, x^{m}, 0, …, 0) = f_{(0,...,0)}(x^{1}, …, x^{m}) .

Let M = (M_{0}, A) and N = (N_{0}, B) be supermanifolds.

Definition 3 A morphism Φ : M → N is a pair Φ = (φ, ϕ), where φ : M_{0} → N_{0} is a differentiable map and ϕ : B → φ_{∗}A is a morphism of sheaves of ℤ_{2}-graded ℝ-algebras. Φ is called an isomorphism if φ is a diffeomorphism and ϕ is an isomorphism. An isomorphism Φ : M → M is called automorphism of M.

In local coordinate systems (x^{i}, θ_{j}) for M and (x̃^{k}, θ̃_{l}) for N a morphism Φ is expressed by p even functions x̃^{k}(x^{1}, …, x^{m}, θ_{1}, …, θ_{n}), k = 1, …, p, and q odd functions θ̃_{l}(x^{1}, …, x^{m}, θ_{1}, …, θ_{n}), l = 1, …, q; where (p, q) = dim N.

1.2 Tangent vector/vector field. Let M = (M_{0}, A) be a supermanifold. For any point p ∈ M_{0} the evaluation map ϵ : A → C_{M_{0}}^{∞} induces an epimorphism ϵ_{p} : A_{p} → ℝ, ϵ_{p}(f) := ϵ(f)(p), where A_{p} denotes the stalk of A at p. For α ∈ ℤ_{2} = {0, 1} we define

(T_{p}M)_{α} := {v : A_{p} → ℝ ℝ-linear|v(fg) = v(f)ϵ_{p}(g) + (−1)^{αf̃}ϵ_{p}(f)v(g)} ,

where the equation is required for all f, g ∈ A_{p} of pure degree and f̃ ∈ {0, 1} denotes the degree of f.

Definition 4 The tangent space of M at p ∈ M_{0} is the ℤ_{2}-graded vector space T_{p}M = (T_{p}M)_{0} + (T_{p}M)_{1}. The elements of T_{p}M are called tangent vectors. Any morphism Φ = (φ, ϕ) : M = (M_{0}, A) → N = (N_{0}, B) induces linear maps dΦ(p) : T_{p}M → T_{φ(p)}N, defined by (dΦ(p)v)(f) := v(ϕ_{p}(f)), p ∈ M_{0}, v ∈ T_{p}M, f ∈ B_{φ(p)}, where ϕ_{p} : B_{φ(p)} → A_{p} is the morphism of stalks associated to ϕ : B → φ_{∗}A. The map dΦ(p) is called the differential at p of Φ.

The sheaf DerA of derivations of A over ℝ is a sheaf of ℤ_{2}-graded A-modules: DerA = (DerA)_{0} + (DerA)_{1}, where

(DerA)_{α} = {X : A → A ℝ-linear|X(fg) = X(f)g + (−1)^{αf̃}fX(g)} ,

where the equation is required for all f, g ∈ A of pure degree.

Definition 5 The sheaf T_{M} = DerA is called the tangent sheaf of M = (M_{0}, A). The sections of T_{M} are called vector fields.

Any local coordinate system (x^{i}, θ_{j}) over U gives rise to even vector fields \frac{∂}{∂x^{i}} and odd vector fields \frac{∂}{∂θ_{j}} over U. The action of the vector fields \frac{∂}{∂x^{i}} , \frac{∂}{∂θ_{j}} on a function f with expansion (1) is given by:

\frac{∂f}{∂x^{i}} = ∑_{α} \frac{∂f_{α}(x^{1}, …, x^{m})}{∂x^{i}}θ^{α} ,

\frac{∂f}{∂θ_{j}} = ∑_{α} α_{j}(−1)^{α_{1}+···+α_{j−1}}f_{α}(x^{1}, …, x^{m})θ_{1}^{α_{1}} ∧ … ∧ θ_{j}^{α_{j}−1} ∧ … ∧ θ_{n}^{α_{n}} .

Any vector field X on M over U can be written as

X = ∑_{i=1}^{m} X^{i}(x^{1}, …, x^{m}, θ_{1}, …, θ_{n})\frac{∂}{∂x^{i}} + ∑_{j=1}^{n} Y^{j}(x^{1}, …, x^{m}, θ_{1}, …, θ_{n})\frac{∂}{∂θ_{j}} ,

where X^{i}, Y^{j} ∈ A(U).

If Φ = (φ, ϕ) : M = (M_{0}, A) → N = (N_{0}, B) is an isomorphism then φ^{−1} and ϕ^{−1} : φ_{∗}A → B exist and give rise to an isomorphism A → φ_{∗}^{−1}B. The induced isomorphism between the corresponding sheaves of derivations is denoted by

dΦ : T_{M} → φ_{∗}^{−1}T_{N}

and is called the differential of Φ. For any open U ⊂ M_{0} the differential dΦ is expressed by an A(U)-linear map dΦ_{U} : T_{M}(U) → T_{N}(φ(U)), where the action of A(U) on T_{N}(φ(U)) is defined using the isomorphism A(U) →^{∼} B(φ(U)) induced by ϕ^{−1}.

Let X be a vector field defined on some open set U ⊂ M_{0} and p ∈ U. Then we can define the value X(p) ∈ T_{p}M of X at p:

X(p)(f) := ϵ_{p}(X(f)) , f ∈ A_{p} .

However, unless dim M = m|n = m|0, a vector field is not determined by its values.

Finally, we relate the tangent spaces and tangent sheaves of M and M_{0}. Any even tangent vector v ∈ (T_{p}M)_{0} annihilates the ideal J = ker ϵ in the exact sequence

0 → J → A →^{ϵ} C_{M_{0}}^{∞} → 0  (2)

and hence defines a tangent vector to M_{0}. More explicitly, we define a map ϵ : T_{p}M → T_{p}M_{0} by the equation

ϵ(v)(ϵ(f)) = v_{0}(f) ,

where v = v_{0} + v_{1} ∈ (T_{p}M)_{0} + (T_{p}M)_{1}, f ∈ A_{p} and f ↦ ϵ(f) is the evaluation map of stalks ϵ : A_{p} → (C_{M_{0}}^{∞})_{p}.

Proposition 1 There is a canonical exact sequence of ℤ_{2}-graded vector spaces:

0 → (T_{p}M)_{1} → T_{p}M →^{ϵ} T_{p}M_{0} → 0 .

In particular, ϵ induces a canonical isomorphism (T_{p}M)_{0} →^{∼} T_{p}M_{0}.

Similarly, on the level of tangent sheaves we define ϵ : T_{M} → T_{M_{0}} by the equation

ϵ(X)(ϵ(f)) = ϵ(X_{0}(f)) ,

where X = X_{0} + X_{1} ∈ (T_{M}(U))_{0} + (T_{M}(U))_{1}, f ∈ A(U) and U ⊂ M_{0} open.

Proposition 2 There is a canonical exact sequence of sheaves of A-modules

0 → ker ϵ → T_{M} →^{ϵ} T_{M_{0}} → 0 ,  (3)

where ker ϵ = (T_{M})_{1} + J T_{M}. In particular, there is the following exact sequence of A-modules:

0 → (J T_{M})_{0} → (T_{M})_{0} → T_{M_{0}} → 0 .

1.3 Frame/frame field/local coordinates.

Definition 6 Let V = V_{0}+V_{1} be a ℤ_{2}-graded vector space of rank m|n, i.e. dim V_{0} = m and dim V_{1} = n. A basis of V is a tupel (b_{1}, …, b_{m+n}) such that (b_{1}, …, b_{m}) is a basis of V_{0} and (b_{m+1}, …, b_{m+n}) is a basis of V_{1}. Let M = (M_{0}, A) be a supermanifold and p ∈ M_{0}. A frame at p is a basis of T_{p}M. A tupel (X_{1}, … X_{m+n}) of vector fields defined on an open subset U ⊂ M_{0} is called a frame field if (X_{1}(p), … X_{m+n}(p)) is a frame at all points p ∈ U. We denote by F(U) the set of all frame fields over U. The sheaf of sets U ↦ F(U) is called the sheaf of frame fields.

Any local coordinate system (x^{i}, θ_{j}) over U gives rise to the frame field (\frac{∂}{∂x^{i}} , \frac{∂}{∂θ_{j}}) over U.

1.4 Supergroup. Let A = A_{0} + A_{1} be an associative ℤ_{2}-graded ℝ-algebra with unit. We will always assume that A is supercommutative, i.e. ab = (−1)^{ãb̃}ba for all a, b ∈ A_{0}∪A_{1}. Under this assumption any left-A-module carries a canonical right-A-module structure and vice versa; so we will simply speak of A-modules. For any supermanifold M = (M_{0}, A) the algebra of functions A(M_{0}) is supercommutative, associative and has a unit.

For any set Σ and non-negative integers r, s we denote by Mat(r, s, Σ) the set of r × s-matrices with entries in Σ and put Mat(r, Σ) := Mat(r, r, Σ). Any partition (r = m + n, s = k + l) defines a ℤ_{2}-grading on the A-module V = Mat(r, s,A):

V_{0} = \{( \begin{matrix}A & B \\ C & D \end{matrix} ) |A ∈ Mat(m, k, A_{0}), D ∈ Mat(n, l, , A_{0}),
B ∈ Mat(m, l, A_{1}), C ∈ Mat(n, k, A_{1})} ,
V_{1} = \{( \begin{matrix}A & B \\ C & D \end{matrix} ) |A ∈ Mat(m, k, A_{1}), D ∈ Mat(n, l, , A_{1}),
B ∈ Mat(m, l, A_{0}), C ∈ Mat(n, k, A_{0})} .

The ℤ_{2}-graded A-module V = V_{0} + V_{1} is denoted by Mat(m|n, k|l, A). Matrix multiplication turns Mat(m|n, A) := Mat(m|n, m|n, A) into an associative ℤ_{2}-graded algebra with unit.

Definition 7 A super Lie bracket on a ℤ_{2}-graded vector space V = V_{0} + V_{1} is a bilinear map [·, ·] : V × V → V such that for all x, y, z ∈ V_{0} ∪ V_{1} we have:

i) ˜[x, y] = x̃ + ỹ,

ii) [x, y] = −(−1)^{x̃ỹ}[y, x] and

iii) [x, [y, z]] = [[x, y], z] + (−1)^{x̃ỹ}[y, [x, z]].

The pair (V, [·, ·]) is called a super Lie algebra.

The supercommutator

[X, Y] = XY − (−1)^{X̃Ỹ} Y X , X, Y ∈ Mat(m|n, A)_{0} ∪ Mat(m|n, A)_{1}

defines a super Lie bracket on the ℤ_{2}-graded vector space Mat(m|n, A). The super Lie algebra (Mat(m|n, A), [·, ·]) is denoted by 𝔤𝔩_{m|n}(A). We put

GL_{m|n}(A) := {g ∈ Mat(m|n, A)_{0}|g is invertible} .

Similarly, if V is a ℤ_{2}-graded A-module End_{A}(V) carries a canonical super Lie algebra structure, which is denoted by 𝔤𝔩_{A}(V). By definition GL_{A}(V) is the group of invertible elements of End_{A}(V). Finally, we will use the convention 𝔤𝔩_{m|n} := 𝔤𝔩_{m|n}(ℝ), 𝔤𝔩(V) := 𝔤𝔩ℝ(V), GL(V) := GLℝ(V).

Definition 8 A supergroup G is a contravariant functor M ↦ G(M) from the category of supermanifolds into the category of groups. Let H, G be supergroups. We say that H is a super subgroup of G and write H ⊂ G if H(M) ⊂ G(M) is a subgroup and H(Φ) = G(Φ)|H(N) for all supermanifolds M, N and morphisms Φ : M → N.

Example 2: The general linear supergroup GL_{m|n} is the supergroup M → GL_{m|n}(M) obtained as composition of the following two functors:

i) the contravariant functor M = (M_{0}, A) → A(M_{0}) from the category of supermanifolds into that of asssociative, supercommutative algebras with unit,

ii) the covariant functor A → GL_{m|n}(A) from the category of associative, supercommutative algebras with unit into that that of groups.

Definition 9 A linear super Lie algebra 𝔤 is a super Lie subalgebra 𝔤 ⊂ 𝔤𝔩_{m|n} (for some m|n). A linear supergroup is a super subgroup G ⊂ GL_{m|n} (for some m|n).

Example 3: Let 𝔤 ⊂ 𝔤𝔩_{m|n} be a linear super Lie algebra. For any associative, supercommutative algebra with unit A we can consider the super Lie algebra 𝔤⊗A ⊂ 𝔤𝔩_{m|n}(A). Its even part 𝔤(A) := (𝔤 ⊗ A)_{0} is a Lie algebra. If A = A(M_{0}) is the algebra of functions of a supermanifold M = (M_{0}, A) then it is easy to see that the exponential series

∑_{n=0}^{∞} \frac{1}{n!}X^{n} , X ∈ Mat(m|n, A) ,

converges (locally uniformly) to an element exp X ∈ GL_{m|n}(A). Now let G(A) be the subgroup of GL_{m|n}(A) generated by exp 𝔤(A). then the functor M = (M_{0}, A) ↦ G(M) := G(A(M_{0})) is a linear supergroup, which we denote by exp 𝔤.

1.5 G-structure. Let M = (M_{0}, A) be a super manifold of dim M = m|n. For any open subset U ⊂ M_{0} we consider the supermanifold M|_{U} := (U, A|_{U}). The general linear supergroup GL_{m|n} induces a sheaf GL_{M} of groups over M_{0}: GL_{M}(U) := GL_{m|n}(M|_{U}) = GL_{m|n}(A(U)), U ⊂ M_{0} open. The group GL_{M}(U) acts naturally (from the right) on the set F(U) of frame fields over U. This action turns F into a sheaf of GL_{M}-sets. Now let G ⊂ GL_{m|n} be a linear supergroup and G the corresponding sheaf of groups, i.e. G(U) = G(M|_{U}) for all open U ⊂ M_{0}. Since G is a sheaf of subgroups G ⊂ GL_{M} the sheaf F of frame fields of M is, in particular, a sheaf of G-sets.

Definition 10 Let M = (M_{0}, A), dim M = m|n, be a supermanifold and G ⊂ GL_{m|n} a linear supergroup. A G-structure on M is a sheaf F_{G} of G-subsets F_{G} ⊂ F such that for all p ∈ M_{0} there exists an open neighborhood U ∋ p for which G(U) acts simply transitively on F_{G}(U).

Example 4: For any supermanifold M, dim M = m|n, the sheaf of frame fields F is a GL_{m|n}-structure.

1.6 Automorphism of G-structure. We denote by Aut(M) the group of all automorphisms of the supermanifold M, see Def. 3. The differential dΦ : T_{M} → φ_{∗}^{−1}T_{M} of any Φ = (φ, ϕ) ∈ Aut(M) induces an isomorphism F → φ_{∗}^{−1}F, again denoted by dΦ. Now let F_{G} ⊂ F be a G-structure on M, for some linear supergroup G ⊂ GL_{m|n}. For simplicity we can assume that G = exp 𝔤 as in Example 3.

Definition 11 Φ = (φ, ϕ) ∈ Aut(M) is called an automorphism of the G-structure F_{G} if dΦF_{G} ⊂ φ_{∗}^{−1}F_{G}.

We recall that any p ∈ M_{0} has an open neighborhood U such that G(U) acts simply transitively on F_{G}(U). Such open sets U ⊂ M_{0} will be called small. If U ⊂ M_{0} is small then F_{G}(U) = EG(U) for any frame field E ∈ F_{G}(U). Here the right-action of the group G(U) on F_{G}(U) is simply denoted by juxtaposition.

Proposition 3 Φ ∈ Aut(M) is an automorphism of the G-structure F_{G} iff

dΦ_{U′}E|_{U′} ∈ E|_{φ(U′)}G(φ(U′))

for all small U ⊂ M_{0}, E ∈ F_{G}(U) and open U′ ⊂ U such that φ(U′) ⊂ U.

For any open set U ⊂ M_{0} the vector space T_{M}(U)^{m+n} of (m+n)-tupels of vector fields is naturally a right-module of the associative, ℤ_{2}-graded algebra Mat(m|n, A(U)). In particular, it is a right-module of the super Lie algebra 𝔤 ⊗ A(U) ⊂ 𝔤_{m|n}(A(U)). On the other hand, T_{M}(U) (and hence T_{M}(U)^{m+n}) is naturally a left-module for the super Lie algebra T_{M}(U) of local vector fields. The action on T_{M}(U) is given by the adjoint representation, i.e. by the supercommutator ad_{X}Y = X ∘ Y − (−1)^{X̃Ỹ} Y ∘ X, X, Y ∈ T_{M}(U) of pure degree. The corresponding action on T_{M}(U)^{m+n} is denoted by L_{X} (“Lie derivative”):

L_{X}E := ([X, X_{1}], …, [X, X_{m+n}]) , E = (X_{1}, …, X_{m+n}) ∈ T_{M}(U)^{m+n} .

Proposition 3 motivates the following definition.

Definition 12 A vector field X on M is an infinitesimal automorphism of the G-structure F_{G} if

L_{X|_{U}} E ∈ E(𝔤 ⊗ A(U))

for all small U ⊂ M_{0}, E ∈ F_{G}(U).

2 Supergeometry associated to the spinor bundle

2.1 The supermanifold M(S). Let (M_{0}, g_{0}) be a (smooth) pseudo Riemannian spinmanifold with spinor bundle S → M_{0}. The corresponding locally free sheaf of C_{M_{0}}^{∞}-modules will be denoted by S; S(U) = Γ(U, S), U ⊂ M_{0} open. To the vector bundle S → M_{0} we associate the supermanifold M : M(S) = (M_{0}, A = ∧S).

Consider the ℤ_{2}-graded vector bundle T M_{0} + S^{∗} → M_{0} with even part T M_{0} and odd part S^{∗}.

Proposition 4 For any p ∈ M_{0} there is a canonical isomorphism of ℤ_{2}-graded vector

spaces ι_{p} : T_{p}M_{0} + S_{p}^{∗} →^{∼} T_{p}M.

Proof: We define ι_{p}^{−1}|(T_{p}M)_{0} := ϵ|(T_{p}M)_{0}, see Prop. 1. Now it is sufficient to construct a canonical isomorphism S^{∗} →^{∼} (T_{p}M)_{1}. For any section s ∈ Γ(U, S^{∗}) interior multiplication ι(s) by s defines an odd derivation of the ℤ_{2}-graded algebra A(U) = Γ(U, ∧S), i.e. a vector field X_{s} := ι(s) ∈ T_{M}(U)_{1}. The value X_{s}(p) ∈ (T_{p}M)_{1} depends only on s(p) ∈ S_{p}^{∗} and we can define ι_{p}(s(p)) := X_{s}(p). □

Using the embedding C_{M_{0}}^{∞} ↪ ∧S, we can consider T_{M} as a sheaf of C_{M_{0}}^{∞}-modules. Interior multiplication s ↦ ι(s) = X_{s} defines a monomorphism S^{∗} ↪ (T_{M})_{1} of sheaves of C_{M_{0}}^{∞}-modules. We want to extend this map to ι : T_{M_{0}} + S^{∗} → T_{M}. For a local vector field X ∈ T_{M_{0}}(U) on M_{0} we put

ι(X) := ∇_{X} ∈ T_{M}(U)_{0} ,

where ∇ is the canonical connection on ∧S, i.e. the one induced by the Levi-Civita-connection on (M_{0}, g_{0}).

Proposition 5 The map ι : T_{M_{0}} + S^{∗} → T_{M} is a monomorphism of sheaves of ℤ_{2}-graded C_{M_{0}}^{∞}-modules. Moreover, ι|T_{M_{0}} defines a splitting of the sequence (3), i.e. ϵ ∘ ι|T_{M_{0}} = id.

Note that given any vector bundle E and connection D on E we can canonically define ι_{E,D} : T_{M_{0}} + E^{∗} ↪ T_{M}, where M = M(E) and E is the sheaf of local sections of E. In Prop. 5 we have ι = ι_{S,∇}.

2.2 The coadjoint representation of the Poincaré super Lie algebras. Let (V_{0}, ⟨·, ·⟩) be a pseudo Euclidean vector space of signature (k, l), k + l = m, and V_{1} the spinor module of the group Spin(V_{0}), n := dim V_{1}. Put V := V_{0} + V_{1}. The vector space 𝔭(V) := 𝔰𝔭𝔦𝔫(V_{0}) + V carries the structure of 𝔰𝔭𝔦𝔫(V_{0})-module. We want to extend this structure to a super Lie bracket [·, ·] on 𝔭(V) which satisfies [V_{0}, V] = 0 and [V_{1}, V_{1}] ⊂ V_{0}. Such an extension is precisely given by a Spin(V_{0})-equivariant map π : ∨^{2}V_{1} → V_{0}; here ∨^{2} denotes the symmetric square.

Definition 13 The structure of super Lie algebra defined on 𝔭(V) by the map π is called a Poincaré super Lie algebra.

We denote by ρ : V_{0} → End(V_{1}) the (standard) Clifford multiplication.

Definition 14 A bilinear form β on the spinor module is called admissible if

1) β is symmetric or skew symmetric. We define the symmetry σ of β to be σ(β) = +1 in the first case and σ(β) = −1 in the second.

2) Clifford multiplication ρ(v), v ∈ V_{0}, is either symmetric or skew symmetric. Accordingly, we define the type τ of β to be τ(β) = ±1.

An admissible form β is called suitable if σ(β)τ(β) = +1.

Given a suitable bilinear form β on V_{1} we define π = π_{ρ,β} : ∨^{2}V_{1} → V_{0} by

⟨π(s_{1} ∨ s_{2}), v⟩ = β(ρ(v)s_{1}, s_{2}), s_{1}, s_{2} ∈ V_{1}, v ∈ V_{0} .  (4)

The map π is Spin(V_{0})-equivariant. Hence it defines on the vector space 𝔭(V) the structure of Poincaré super Lie algebra. The following theorem was proved in [A-C].

Theorem 1 Any Spin(V_{0})-equivariant map ∨^{2}V_{1} → V_{0} is a linear combination of maps π_{ρ,β_{i}}, β_{i} suitable.

All admissible bilinear forms on the spinor module were explicitly determined in [A-C]. The spinor module carries a non-degenerate suitable bilinear form β for all values of m = k + l and s = k − l except for (m, s) = (5, 7), (6, 0), (6, 6) and (7, 7) (mod 8, mod 8). Now we assume that a non-degenerate suitable bilinear form β on V_{1} is given. The map π = π_{ρ,β} defines on 𝔭(V) the structure of Poincaré super Lie algebra such that [V_{1}, V_{1}] = V_{0}.

Given a super Lie algebra 𝔤 the coadjoint representation ad^{∗} : 𝔤 → 𝔤𝔩(𝔤^{∗}), x ↦ ad_{x}^{∗}, is defined by the equation

ad_{x}^{∗}(y^{∗}) = −(−1)^{x̃ỹ^{∗}}y^{∗} ∘ ad_{x} ,

for x ∈ 𝔤 and y^{∗} ∈ 𝔤^{∗} of pure degree.

Proposition 6 The coadjoint representation of 𝔭(V) preserves the subspace V^{⊥} = {x^{∗} ∈ 𝔭(V)^{∗}|x^{∗}(V) = 0} ⊂ 𝔭(V)^{∗} and hence induces a representation α : 𝔭(V) → 𝔤𝔩(V^{∗}) on V^{∗} ≅ 𝔭(V)^{∗}/V^{⊥}. It has kernel ker α = V_{0} and therefore induces a faithful representation of the super Lie algebra 𝔭(V)/V_{0} on V^{∗}.

Once we choose a basis b = (b_{1}, …, b_{m+n}) of V^{∗}, we can identify α(𝔭(V)) ⊂ 𝔤𝔩(V^{∗}) with a subalgebra α(𝔭(V))^{b} ⊂ 𝔤𝔩_{m|n}, where A ↦ A^{b} denotes the isomorphism 𝔤𝔩(V^{∗}) → 𝔤𝔩_{m|n} defined by b. If moreover (b_{1}, …, b_{m}) is an orthonormal basis of V_{1}^{⊥} ≅ V_{0}^{∗} then the even part α(𝔭(V))_{0}^{b} ≅ 𝔰𝔭𝔦𝔫(k, l) is a canonically embedded spinor Lie algebra, i.e.

α(𝔭(V))_{0}^{b} = 𝔰𝔭𝔦𝔫_{σ} := \{( \begin{matrix}A & 0 \\ 0 & σ(A) \end{matrix} ) |A ∈ 𝔰𝔬(k, l) ⊂ 𝔤𝔩_{m}\} ,

where σ : 𝔰𝔬(k, l) → 𝔤𝔩_{n} is equivalent to the spinor representation.

The linear group Spin_{σ} ⊂ GL_{m|n}(ℝ) generated by the Lie algebra 𝔰𝔭𝔦𝔫_{σ} ⊂ (𝔤𝔩_{m|n})_{0} ≅ 𝔤𝔩_{m} ⊕ 𝔤𝔩_{n} acts on the set of bases of V^{∗} from the right.

Proposition 7 Assume that α(𝔭(V))_{0}^{b} = 𝔰𝔭𝔦𝔫_{σ} and b′ = bg for some g ∈ Spin_{σ}. Then α(𝔭(V))^{b} = α(𝔭(V))^{b′}.

Proof: This follows from the fact that α(𝔭(V))_{0}^{b} = 𝔰𝔭𝔦𝔫_{σ} and α(𝔭(V))_{1}^{b} = α(V_{1})^{b} are invariant under 𝔰𝔭𝔦𝔫_{σ} = α(𝔰𝔭𝔦𝔫(V_{0}))^{b}. □

Now let (e_{1}, …, e_{m}) be an orthonormal basis of V_{0} and (θ^{1}, …, θ^{n}) a basis of V_{1}. The dual bases of V_{0}^{∗} and V_{1}^{∗} will be denoted by (e^{i}) and (θ_{j}).

Proposition 8 With respect to the basis b = (e^{1}, …, e^{m}, θ_{1}, …, θ_{n}) of V^{∗} ≅ V_{0}^{∗} +V_{1}^{∗} the super Lie algebra α(𝔭(V)) ⊂ 𝔤𝔩(V^{∗}) is identified with

α(𝔭(V))^{b} = \{( \begin{matrix}A & 0 \\ C & σ(A) \end{matrix} ) |A ∈ 𝔰𝔬(k, l), C^{ji} = e^{i}(π(s ∨ θ^{j})), s ∈ V_{1}\} ,

where C = (C^{ji}), j = 1, …, n, i = 1, …, m, and σ : 𝔰𝔬(k, l) → 𝔤𝔩_{n} is equivalent to the spinor representation.

2.3 The (pseudo) Riemannian supergeometry associated to the spinor bundle. Now we carry over the construction of 2.2 to the ℤ_{2}-graded vector bundle V := T M_{0} + S over M_{0}. We assume that M_{0} is simply connected. The vector bundle V carries the canonical connection induced by the Levi-Civita connection of the pseudo Riemannian manifold (M_{0}, g_{0}). The holonomy algebra of V at p ∈ M_{0} is a subalgebra of 𝔰𝔭𝔦𝔫(T_{p}M_{0}) ⊂ 𝔤𝔩(V_{p})_{0}. This implies, in particular, that the bundle of Spin(T M_{0})-invariant bilinear forms on S is flat. Let g_{1} be a parallel non-degenerate suitable bilinear form on S, see Def. 14 and the remarks following Thm. 1.

The Spin(T M_{0})-invariant bilinear form g = g_{0} + g_{1} on V should be thought of as a pseudo Riemannian metric for the supermanifold M = M(S). Note that, due to Prop. 4, g(p) induces a non-degenerate bilinear form on T_{p}M. However, recall that g_{1} is symmetric or skew-symmetric. The map π = π_{ρ,g_{1}} : ∨^{2}S → T M_{0} defines on 𝔭(V) = 𝔰𝔭𝔦𝔫(T M_{0})+S ⊂ 𝔤𝔩(V) the structure of bundle of Poincaré super Lie algebras. 𝔭(V) is a parallel bundle. Now let α : 𝔭(V) → 𝔤𝔩(V^{∗}) be the field of representations induced by the coadjoint representation, cf. Prop. 6. The image α(𝔭(V)) ⊂ 𝔤𝔩(V^{∗}) is a parallel bundle of super Lie algebras.

Proposition 9 The frame bundle of V^{∗} → M has a subbundle P_{Spin_{σ}} with structure group Spin_{σ} ⊂ GL_{m|n}(ℝ), Spin_{σ} ≅ Spin(k, l), such that for all b = (e^{i}, θ_{j}) ∈ (P_{Spin_{σ}})_{p}:

1) (e^{i}) is an orthonormal basis of T_{p}^{∗}M_{0} and

2 ) α(𝔭(V_{p})) is identified via b with the subalgebra 𝔤 = α(𝔭(V_{p}))^{b} ⊂ 𝔤𝔩_{m|n}(ℝ), where

𝔤_{0} = 𝔰𝔭𝔦𝔫_{σ} = \{( \begin{matrix}A & 0 \\ 0 & σ(A) \end{matrix} ) |A ∈ 𝔰𝔬(k, l)\} and

𝔤_{1} = \{( \begin{matrix}0 & 0 \\ C & 0 \end{matrix} ) |C = (C^{ji}), C^{ji} = e^{i}(π(s ∨ θ^{j})), s ∈ S_{p}\}

are independent of b and p. Here (θ^{j}) is the basis of S_{p} dual to (θ_{j}).

Proof: This follows from the holonomy reduction and Propositions 7 and 8. □

We denote by V the sheaf of local sections of V. Identifying T M_{0} and T^{∗}M_{0} via g_{0}, the map ι of Prop. 5 corresponds to a monomorphism ι : V = T_{M_{0}}^{∗} + S^{∗} ↪ T_{M}. This induces a map

ι : Γ(U, P_{Spin_{σ}}) → F(U) ,

where F(U) is the set of frame fields of M over the open set U ⊂ M_{0}. The image of ι generates a Spin_{σ}-structure on M, where Spin_{σ} is now considered as (purely even) linear supergroup Spin_{σ} ⊂ GL_{m|n}. More precisely, recall that Spin_{σ}(A(U)) is the group generated by exp 𝔰𝔭𝔦𝔫_{σ}(A(U)) ⊂ GL_{m|n}(A(U)). It acts on F(U) from the right. Put

F_{Spin_{σ}}(U) := ι(Γ(U, P_{Spin_{σ}}))Spin_{σ}(A(U)) .

Proposition 10 F_{Spin_{σ}} is a Spin_{σ}-structure on M.

Denote by G the linear supergroup defined by the linear super Lie algebra 𝔤, see Example 3. Since 𝔰𝔭𝔦𝔫_{σ} ⊂ 𝔤 ⊂ 𝔤𝔩_{m|n}(ℝ), we have the following inclusions of linear supergroups:

Spin_{σ} ⊂ G ⊂ GL_{m|n} .  (5)

Put F_{G}(U) := F_{Spin_{σ}}(U)G(A(U)) for all open U ⊂ M_{0}.

Proposition 11 F_{G} is a G-structure on M.

Definition 15 A Killing vector field on (M, g) is an infinitesimal automorphism of the G-structure F_{G}, see Def. 12.

2.4 Twistor spinors as Killing vector fields.

Definition 16 A section s of the spinor bundle S → M_{0} is called a twistor spinor if there exists a section s̃ of S such that

∇_{X}s = ρ(X)s̃  (6)

for all vector fields X on M_{0}. Here ρ(X) : S → S is Clifford multiplication. A twistor spinor s is called a Killing spinor if s̃ = λs for some constant λ ∈ ℝ Remark: From (6) it follows that s̃ = −\frac{1}{m}Ds, where D is the Dirac operator.

The non-degenerate bilinear form g_{1} on S induces the isomorphism

S →^{∼} S^{∗}, s ↦ s^{∗} := g_{1}(s, ·) .

Recall that ι|S^{∗} : S^{∗} ↪ T_{M} is simply given by interior multiplication, s. 2.1. To any spinor field S we associate the odd vector field X_{s} := ι(s^{∗}) on M. Now we can state the main result of this paper.

Theorem 2 Let (M_{0}, g_{0}) be a pseudo Riemannian spin manifold with spinor bundle (S, g_{1}); g_{1} a parallel non-degenerate suitable bilinear form on S, see Def. 14 and 2.3. Consider the supermanifold M = M(S) with the bilinear form g = g_{0} + g_{1} and let s be a section of S. The vector field X_{s} is a Killing vector field on (M, g) iff s is a twistor spinor, see Def. 15 and 16.

Corollary 1 A Killing vector field X_{s} for an extension g of g_{0} is a Killing vector field for any other extension; the extensions beeing as in 2.3.

Lemma 1 For all sections s^{∗}, t^{∗} of S^{∗} and X of T M_{0} we have:

i) [ι(s^{∗}), ι(t^{∗})] = 0,

ii) [ι(s^{∗}), ι(X)] = [ι(s^{∗}), ∇_{X}] = −ι((∇_{X})^{∗}).

Proof: i) By definition of the supercommutator [·, ·] on T_{M}, we have [ι(s^{∗}), ι(t^{∗})] = ι(s^{∗}) ∘ ι(t^{∗}) + ι(t^{∗}) ∘ ι(s^{∗}) = 0. ii) Recall that s^{∗} = g_{1}(s, ·). If t is a section of S we have [ι(s^{∗}), ι(X)](t) = s^{∗}(∇_{X}t) − ∇_{X}s^{∗}(t) = g_{1}(s, ∇_{X}t) − ∇_{X}g_{1}(s, t) = −g_{1}(∇_{X}s, t) = −(∇_{X}s)^{∗}(t). □

Proposition 12 Let s be a twistor spinor. For all vector fields X and spinor fields t on M_{0} we have:

i) [ι(s^{∗}), ι(X)] = −ι((ρ(X)s̃)^{∗}) = −τ(g_{1})ι(ρ(X)^{∗}s̃^{∗}), where τ(g_{1}) ∈ {±1} is the type of g_{1}, see Def. 14.

ii) [ι(s^{∗}), ι(X)](t) = −g_{1}(ρ(X)s̃, t) = −g_{0}(π(s̃ ∨ t), X).

Proof: The first equation of i) follows from Lemma 1 ii), since ∇_{X}s = ρ(X)s̃. Now the second equation of i) and the first equation of ii) follow from the definition of the type τ: (ρ(X)s̃)^{∗}(t) = g_{1}(ρ(X)s̃, t) = τ(g_{1})g_{1}(s̃, ρ(X)t). The last equation of ii) is simply the definition of π = π_{ρ,g_{1}}, cf. (4). □

Proof (of Theorem 2): Let (e^{i}, θ_{j}) ∈ Γ(U, P_{Spin_{σ}}), U ⊂ M_{0} open, and (e_{i}, θ^{j}) the dual local frame for V = T M_{0} + S. Put

E := (ι(e^{i}), ι(θ_{j})) ∈ Γ(U, F_{Spin_{σ}}) ⊂ Γ(U, F_{G}) .

Since (e_{i}) is orthonormal, i.e. g_{0}(e_{i}, e_{j}) = ε_{i}δ_{ij}, ε_{i} ∈ {±1}, we have e^{i} = ε_{i}g_{0}(e_{i}, ·). Hence, by definition of ι on T_{M_{0}}^{∗}, we have ι(e^{i}) = ε_{i}ι(e_{i}). Therefore by Lemma 1 for any s ∈ Γ(U, S) we have

L_{X_{s}}E = ([X_{s}, ι(e^{i})], [X_{s}, ι(θ_{j})]) = (−ε_{i}ι((∇_{e_{i}}s)^{∗}), 0) ,  (7)
(∇_{e_{i}}s)^{∗}(θ^{j}) = g_{1}(∇_{e_{i}}s, θ^{j}) .  (8)

From this computation it follows that L_{X_{s}}E ∈ E(𝔤 ⊗ A(U)) iff there exists a t ∈ Γ(U, S) such that

L_{X_{s}}E = EC_{t} , where  (9)
C_{t} = ( \begin{matrix}0 & 0 \\ (C_{t}^{ji}) & 0 \end{matrix} ) ∈ 𝔤 ⊗ A(U) , C_{t}^{ji} = e^{i}(π(t ∨ θ^{j})) ,  (10)

see Prop. 8. By (7), (8) and (10) equation (9) is equivalent to

g_{1}(∇_{e_{i}}s, θ^{j}) = −ε_{i}e^{i}(π(t ∨ θ^{j})) , i = 1, …, m, j = 1, …, n.  (11)

The right-hand-side is

−ε_{i}e^{i}(π(t ∨ θ^{j})) = −g_{0}(π(t ∨ θ^{j}), e_{i}) = −g_{1}(ρ(e_{i})t, θ^{j}) ,  (12)

hence (11) is equivalent to the twistor equation (6) with s̃ = −t. □

Acknowledgments. D.V.A. is grateful to Max-Planck-Institut für Mathematik and V.C. thanks the Mathematical Sciences Research Institute, S.-S. Chern and R. Osserman, for hospitality and support. V.C. would also like to thank W. Kramer, E. Poletaeva and V. Serganova for discussions related to the subject of this paper. Three of us (D.V.A., C.D., U.S.) should like to thank the Mathematisches Forschungsinstitut Oberwolfach, where this work was begun, for hospitality in the framework of the Research-in-Pairs programme supported by the Volkswagen Stiftung.

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