What is the role of twistors in supergeometry?

V. Cortés^{1}

Mathematisches Institut der Universität Bonn, Beringstr. 1, 53115 Bonn, Germany

Abstract

The present is an extended version of the talk given by the author on the occasion of the “3rd International Workshop on Differential Geometry and its Applications and 1st German-Romanian Seminar on Geometry” (Sibiu 1997). It is based on joint work with D.V. Alekseevsky, C. Devchand and U. Semmelmann, see [A-C] and [A-C-D-S].

A supermanifold M is canonically associated to any pseudo Riemannian spin manifold (M_{0}, g_{0}). Its structure sheaf is the sheaf of local sections of the exterior algebra ∧S over the spinor bundle S → M_{0}. M carries a G-structure F_{G} for a certain linear supergroup G whose even part is the spinor group. G is constructed using the coadjoint representation of a super extended Poincaré group. For any spinor field s on M_{0} there is a corresponding odd vector field X_{s} on M. The twistor equation has the following supergeometric interpretation: X_{s} is an infinitesimal automorphism of the G-structure F_{G} iff S is a solution of the twistor equation.

1. Spin geometry Let (M, g) be an oriented pseudo Riemannian manifold of signature (k, l) and denote by P_{SO} → M the SO(k, l)-principal bundle of positively oriented orthonormal frames. (M, g) is called a spin manifold if there is a Spin(k, l)-equivariant lift of P_{SO} to a Spin(k, l)-principal bundle P_{Spin} → M. The spinor bundle is the vector bundle S = P_{Spin} ×_{ρ} Σ over M associated to the spinor representation ρ : Spin(k, l) → GL(Σ). The Levi-Civita connection on M induces a connection on P_{Spin} and hence on all associated vector bundles; in particular, S carries a covariant derivative ∇. Finally, we have the Clifford multiplication μ : T M ⊗ S → S or equivalently the embedding ι_{μ} : S → T^{∗}M ⊗ S, which turns each fibre S_{p} into a module over the Clifford algebra Cℓ(T_{p}M), p ∈ M. We have the canonical decomposition

T^{∗}M ⊗ S = ker μ ⊕ ι_{μ}(S) ;

1Supported by SFB 256 (Bonn University). e-mail: vicente@math.uni-bonn.de

here we have identified T M and T^{∗}M via g. Correspondingly there are two fundamental differential operators: the Dirac operator

D : Γ(S) → Γ(S)

and the twistor operator

D : Γ(S) → Γ(ker μ)

defined by the equation

∇ = D − \frac{1}{m}ι_{μ} ∘ D ,

where m = k + l = dim M. A spinor field s ∈ Γ(S) is called a twistor if it is a solution of the twistor equation Ds = 0. From the definition of the twistor operator D it follows that s is a twistor iff there exists a spinor field s̃ such that ∇s = ι_{μ}(s̃), i.e. if

∇_{X}s = X · s̃:= μ(X ⊗ s̃)

for all vector fields X on M.

2. Supergeometry We consider now pairs M = (M_{0}, A), where M_{0} is a C^{∞}-manifold and A = A_{0} + A_{1} a sheaf of ℤ_{2}-graded ℝ-algebras on M_{0}; dim M_{0} = m. M is called a supermanifold of dimension m|n if it is locally isomorphic to the following model: Let E → M_{0} be a vector bundle of rank n and E the corresponding sheaf of local sections. Then the exterior sheaf ∧E = ∧^{ev}E + ∧^{odd}E is a sheaf of ℤ_{2}-graded ℝ-algebras on M_{0} and M(E) := (M_{0}, ∧E) is the standard supermanifold associated to the vector bundle E. It is known that any supermanifold is globally (although not canonically) isomorphic to a standard supermanifold.

Let A be a ℤ_{2}-graded ℝ-algebra. A derivation of degree α ∈ ℤ_{2} = {0, 1} is an ℝ-linear map X : A → A satisfying the following Leibniz rule:

X(fg) = X(f)g + (−1)^{αf̃}fX(g)

for all homogeneous f, g ∈ A, where f̃ is the ℤ_{2}-degree of f. Of course, derivations of degree 0 (respectively 1) are called even (respectively odd). A derivation of A is by definition a sum of an even and an odd derivation. The ℤ_{2}-graded vector space of all derivations of A is denoted by Der A = (Der A)_{0} + (Der A)_{1}.

Let M = (M_{0}, A) be a supermanifold of dimension m|n. Te correspondence U ↦ Der A(U), U ⊂ M_{0} open, defines a locally free sheaf of (ℤ_{2}-graded) A-modules on M_{0}.

Definition 1 The sheaf T_{M} := Der A is called tangent sheaf of M. (Local) sections of T_{M} are called (local) vector fields on M. A frame field over U ⊂ M_{0} is a basis (X_{1}, …, X_{m}, Y_{1}, …, Y_{n}) of the A(U)-module T_{M}(U); the X_{i} are even and the Y_{j} are odd local vector fields.

3. G-Structures on supermanifolds Recall that a ℤ_{2}-graded algebra A is said to be supercommutative if the relation ab = (−1)^{ãb̃}ba holds for all homogeneous elements a, b ∈ A of degree ã, b̃. In the following A will always denote a ℤ_{2}-graded associative and supercommutative ℝalgebra; short a superalgebra. The ℤ_{2}-graded vector space of rank m|n is denoted by ℝ^{m|n} and, more generally, the free A-module of rank m|n by A^{m|n} = ℝ^{m|n} ⊗ A. Finally, GL_{m|n}(A) stands for the group of even invertible elements of End_{A}(A^{m|n}).

Definition 2 A supergroup G is a contravariant functor M ↦ G(M) from the category of supermanifolds into that of groups.

Example: The general linear supergroup M ↦ GL_{m|n}(M) = GL_{m|n}(A(M_{0})) is the composition of two functors:

supermanifolds (contravariant)

⟶ superalgebras

\underrightarrow{(covariant)}

groups

M = (M_{0}, A) ↦ A = A(M_{0}) ↦ GL_{m|n}(A) .

A supergroup G is called linear if G ⊂ GL_{m|n}.

Definition 3 A super Lie algebra is a ℤ_{2}-graded vector space 𝔤 = 𝔤_{0} + 𝔤_{1} together with a bilinear map [·, ·] : 𝔤 × 𝔤 → 𝔤 (super Lie bracket) such that:

i) [𝔤_{α}, 𝔤_{β}] ⊂ 𝔤_{α+β}, α, β ∈ ℤ_{2},

ii) [x, y] = −(−1)^{x̃ỹ}[y, x] (skew symmetry) and

iii) [x, [y, z]] = [[x, y], z] + (−1)^{x̃ỹ}[y, [x, z]] (Jacobi identity) for all homogeneous elements x, y, z ∈ 𝔤.

Example: The supercommutator [a, b] := ab − (−1)^{ãb̃}ba defines on the (naturally) ℤ_{2}-graded algebra End ℝ^{m|n} the structure of super Lie algebra, which is denoted by 𝔤𝔩_{m|n}. Subalgebras 𝔤 ⊂ 𝔤𝔩_{m|n} are called linear super Lie algebras.

For any linear super Lie algebra 𝔤 ⊂ 𝔤𝔩_{𝔪|𝔫} there is a linear supergroup denoted by exp 𝔤. In fact, if M = (M_{0}, A) is a supermanifold we define the group

(exp 𝔤)(M) := ⟨(𝔤 ⊗ A(M_{0}))_{0}⟩ ⊂ GL_{m|n}(A(M_{0})) .

Note that the even part (𝔤 ⊗ A(M_{0}))_{0} of the super Lie algebra 𝔤 ⊗ A(M_{0}) is a matrix Lie algebra for which the usual exponential series is defined and converges in the standard Fréchet topology of A(M_{0}) ≅ Γ(∧E) (here E is a vector bundle such that M ≅ M(E)).

Let now M = (M_{0}, A) be a supermanifold, dim M = m|n, F the sheaf of local frame fields and G ⊂ GL_{m|n} a linear supergroup. The group GL_{m|n}(A(U)) acts naturally from the right on the set F(U), U ⊂ M_{0} open.

Definition 4 A subsheaf F_{G} ⊂ F is called G-structure if all p ∈ M_{0} have an open neighborhood U such that G(M|_{U}) ⊂ GL_{m|n}(A(U)) acts simply transitively on F_{G}(U). Here M|_{U} stands for the supermanifold (U, A|_{U}).

Example: F is always a GL_{m|n}-structure.

We have a natural left-action of vector fields X on (m + n)-tuples of vector fields F = (X_{1}, … X_{m+n}):

F ↦ L_{X}F := ([X, X_{1}], … [X, X_{m+n}]) ,

where [X, Y] := XY − (−1)^{X̃Ỹ} Y X is the supercommutator of vector fields.

Definition 5 X is called an infinitesimal automorphism of the G-structure F_{G} if

F ∈ F_{G}(U) ⟹ L_{X}F ∈ F · (𝔤 ⊗ A(U)) .

(The dot stands for the natural right action.)

4. Spinors as odd vector fields Let (M_{0}, g_{0}) be a pseudo Riemannian spin manifold with spinor bundle S → M_{0}. From now on we will consider the supermanifold M = M(S) = (M_{0}, A). Functions on M are by definition sections of the exterior algebra ∧S (A(U) = Γ(U, ∧S) for any open subset U ⊂ M_{0}). Let s^{∗} ∈ Γ(S^{∗}) be a section of the dual spinor bundle. Interior multiplication by s^{∗} is an odd derivation ι(s^{∗}) : Γ(∧S) → Γ(∧S), and hence an odd vector field on M. Choosing a nondegenerate bilinear form g_{1} ∈ Γ(S^{∗} ⊗ S^{∗}) ≅ Γ(Hom(S, S^{∗})) we can associate to any spinor field s ∈ Γ(S) the vector field X_{s} := ι(g_{1}s). Now we can ask: Does there exist a G-structure F_{G} on M such that a spinor field s is a twistor iff X_{s} is an infinitesimal automorphism of F_{G}?

In the following I will present the G-structure F_{G} which provides a positive answer to this question. The first step is the construction of the linear supergroup G = exp 𝔤 or equivalently of the linear super Lie algebra 𝔤 5. Construction of the super Lie algebra 𝔤

Definition 6 A (super) extended Poincaré algebra of signature (k, l) is a (super) Lie algebra 𝔭 with ℤ_{2}-grading 𝔭 = 𝔭_{0} + 𝔭_{1} ([𝔭_{α}, 𝔭_{β}] ⊂ 𝔭_{α+β}) such that

i) 𝔭_{0} = Lie Isom ℝ^{k,l} = 𝔰𝔬(k, l)+ℝ^{k,l} (Poincaré algebra of signature (k, l)), where ℝ^{k,l} stands for the pseudo Euclidean vector space with scalar product ⟨·, ·⟩ of signature (k, l).

ii) 𝔭_{1} = Σ = spinor module of 𝔰𝔬(k, l) trivially extended to a 𝔭_{0}-module and [𝔭_{1}, 𝔭_{1}] ⊂ ℝ^{k,l}.

Definition 7 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 s ↦ vs, s ∈ Σ, is either symmetric or skew symmetric for all v ∈ ℝ^{k,l}. Accordingly, we define the type τ of β to be τ(β) = ±1.

Let β be admissible. Then the equation

(⋆) ⟨[s, t], v⟩ := β(vs, t) , s, t ∈ Σ, v ∈ ℝ^{k,l} ,

defines the bracket [·, ·] : Σ × Σ → ℝ^{k,l} of an extended (respectively super extended) Poincaré algebra if σ(β)τ(β) = −1 (respectively σ(β)τ(β) = +1).

Theorem 1 [A-C] Let [·, ·] : Σ × Σ → ℝ^{k,l} be the Lie bracket (respectively the super Lie bracket) of an extended (respectively super extended) Poincaré algebra. Then [·, ·] satisfies equation (⋆) for some bilinear form β which is a linear combination of admissible forms β_{i} such that σ(β_{i})τ(β_{i}) = −1 (respectively σ(β_{i})τ(β_{i}) = +1).

Put L^{−} (respectively L^{+}) := dimension of the vector space of extended (respectively super extended) Poincaré algebras of signature (k, l); m := k + l, s := k − l. The following remarkable mirror symmetry holds:

Theorem 2 [A-C] L^{+} and L^{−} depend only on m and s modulo 8 and

L^{+}(−m, s) = L^{−}(m, s) .

Now we choose a nondegenerate bilinear form β on Σ as in Thm. 1 defining a super extended Poincaré algebra 𝔭. Given a super Lie algebra 𝔩 the coadjoint representation ad^{∗} : 𝔩 → 𝔤𝔩(𝔩^{∗}), x ↦ ad_{x}^{∗}, is defined by the equation

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

for homogeneous elements x ∈ 𝔩 and y ∈ 𝔩^{∗}. The coadjoint representation ad^{∗} : 𝔭 → 𝔤𝔩(𝔭^{∗}) of the super extended Poincaré algebra 𝔭 preserves the annihilator (ℝ^{k,l} + Σ)^{⊥} and hence induces a representation ρ of 𝔭 on 𝔭^{∗}/(ℝ^{k,l} + Σ)^{⊥} ≅ (ℝ^{k,l} + Σ)^{∗} ≅ (via ⟨·, ·⟩ and β) ℝ^{k,l} + Σ ≅ ℝ^{m|n}, m = k + l, n = dim Σ. Its image 𝔤 := ρ(𝔭) ⊂ 𝔤𝔩_{m|n}(ℝ) is a linear super Lie algebra with even part 𝔤_{0} = 𝔰𝔭𝔦𝔫(k, l).

Now we construct a G-structure on the supermanifold M = M(S) for the linear supergroup G = exp 𝔤.

6. The G-structure on M Consider the ℤ_{2}-graded vector bundle V := T M_{0} + S of rank m|n. The Levi-Civita connection of the pseudo Riemannian manifold (M_{0}, g_{0}) induces a canonical covariant derivative ∇ on V. Using its holonomy we can reduce the structure group from GL_{m} × GL_{n} to Hol(∇) ⊂ Spin(k, l). Without restriction of generality we assume that the holonomy group Hol (∇) ⊂ Spin_{0}(k, l) (otherwise it is sufficient to replace G by the linear supergroup generated by G and Spin(k, l)). The bilinear form β on the spinor module Σ is Spin(k, l)-invariant and hence defines an extension of g_{0} to a parallel field g = g_{0} + g_{1} of bilinear forms on V. The data (V, ∇, g_{1}) immediately provide us with the following fundamental correspondence.

Proposition 1 Any (local) trivialization (e_{i}, s_{j}), i = 1, …, m, j = 1, …, n, of the ℤ_{2}-graded vector bundle V defines a (local) frame field (∇_{e_{i}}, X_{s_{j}}) on M, X_{s} = ι(g_{1}s).

Let (e_{i}, s_{j}) be a trivilization of V |_{U} which belongs to the holonomy reduction, U ⊂ M_{0} open. Denote by F = (∇_{e_{i}}, X_{s_{j}}) ∈ F(U) the corresponding local frame field on M. Now the linear supergroup G = exp 𝔤 associated to the the linear super Lie algebra 𝔤 constructed in the previous section comes into play. The orbit

F_{G}(U) = F · G(M|_{U})

is independent of the trivialization of V |_{U} (as long as the trivialization belongs to the holonomy reduction) since 𝔤 ⊃ 𝔰𝔭𝔦𝔫(k, l) = 𝔤_{0}. Moreover F_{G} is a G-structure on M.

Theorem 3 [A-C-D-S] s ∈ Γ(S) is a twistor iff X_{s} is an infinitesimal automorphism of F_{G}.

Proof: We compute:

L_{X_{s}}F = (−X_{∇_{e_{i}}s}, 0) = F · ( \begin{matrix}0 & 0 \\ C̃(s) & 0 \end{matrix} ) , C̃(s) = −(s^{j}(∇_{e_{i}}s))_{ij} ,

where (s^{j}) is the basis dual to (s_{j}). This shows that X_{s} is an infinitesimal automor-

phism iff ( \begin{matrix}0 & 0 \\ C̃(s) & 0 \end{matrix} ) ∈ 𝔤_{1} ⊗ C^{∞}(U).

On the other hand, the explicit description of the representation ρ defining 𝔤 = ρ(𝔭) gives:

𝔤_{1} ⊗ C^{∞}(U) = \{( \begin{matrix}0 & 0 \\ C(s̃) & 0 \end{matrix} ) |C(s̃) = −(s^{j}(e_{i}s̃))_{ij} , s̃ ∈ Γ(S)\} ,

Therefore we obtain the equation C̃(s) = C(s̃) or, more explicitely, ∇_{e_{i}}s = e_{i}s̃, i = 1, …, m. This is equivalent to the twistor equation. □

References

[A-C] D.V. Alekseevsky, V. Cortés: Classification of N-(super)-extended Poincaré algebras and bilinear invariants of the spinor representation of Spin(p, q), Commun. Math. Phys. 183 (1997), 477-510.

[A-C-D-S] D.V. Alekseevsky, V. Cortés, C. Devchand, U. Semmelmann: Killing spinors are Killing vector fields in Riemannian supergeometry, Journal of Geometry and Physics 25 (3) (1997), 14p.