On the Six-dimensional Kerr Theorem and Twistor Equation

The Kerr theorem is revisited as part of the twistor program in six dimensions. The relationship between pure spinors and integrable 3-planes is investigated. The real condition for Lorentzian spacetimes is seen to induce a projective property in the space of solutions, reminiscent of the quaternionic structure of the 6-dimensional Lorentz group. The twistor equation (or Killing spinor equations generically) also has an interpretation as integrable null planes and a family of Einstein spacetimes with this property are presented in the Kerr-Schild fashion.


Introduction
The Kerr theorem shows how to generate maximally null integrable submanifolds in flat spacetime. The importance of such characterization was technical at first, with solutions of Einstein's Equations in the Newman-Penrose formalism in mind [1,2]. Many solutions have been found this way, including the Kerr-Newman solution. For complexified spaces, Hughston and Mason [3] showed how to parametrize these null manifolds in terms of pure spinors, and posteriorly related the structure to the integrability of the wave equation [4,5]. The same connection was noted by the mathematicians at around the time [6] to provide a higher-dimensional analogue of "Bateman's Formula", or Penrose transform. These have been recast in the twistor language by Berkovits and Cherkis [7].
From the physical point of view, the existence of null integrable submanifolds is at the heart of the simplification of the metric structure that happens near the horizon of a black hole, even without the presence of supersymmetry, [8,9]. This is a horizon analogue of the "peeling theorem" [10], which originally stated that the corrections to the metric in asymptotic flat spacetimes are ranked in order of algebraic specialty, as defined by Petrov. The Petrov classification of the Weyl tensor sees the latter as a linear application in the space of two vectors. Because of the symmetry of the Weyl tensor, eigenvalues are the outer product of null vectors, dubbed principal null directions. With respect to the peeling theorem, in particular, one can state that the contributions to the metric from charges such as ADM mass and angular momentum allow for the definition of a principal null direction, picking a particular one from the infinite directions of flat spacetime. This fact was used by Penrose to define local conserved charges in Twistor Theory [11,12].
The relationship between null integrable manifolds and spinors have been outlined since its inception [13,14]. In the applications of spinors in General Relativity this connection has always been explicit, however, applications in supergravity and superstrings the algebraic approach has always been preferred. In four dimensions, the work of van Niuwenhuizen and Warner [15] built the bridge between the two formalisms by working the conditions for the existence of Killing spinors in the Newman-Penrose formalism.
This letter will try a similar program in six dimensions. We will revisit the Kerr theorem in six dimensions and define helicity spinors by exploring the symmetry of the solutions. Previous work along those lines was conducted in [16]. We introduce the twistor equation in 6 dimensions as a sufficient condition for the existence of integrable isotropic 3-planes. We close with the relation to the existence of Killing spinors and applications in six-dimensional spaces to a family of examples presented in the Kerr-Schild form.

Six Dimensional Newman-Penrose Formalism
We begin by reviewing some aspects of four dimensional spinors. There, the spinor formalism takes full advantage from the fact that the Lorentz group SO(3, 1) is isomorphic to SU(2) × SU (2). Reality conditions relate both factors, so the decomposition of a vector into SU(2) pieces is done via the van der Waerden symbols to a matricial object A αα . The norm of the vector is written as the determinant of the matrix. When it is zero, it means that both rows are linearly dependent, so that: (2.1) the objects κ α andκα transform as spinors under Lorentz, and parametrize the null vectors. κ andκ will be the complex conjugate of each other if we want the null vector to be real. The correspondence between spinors and null vectors is one up to a multiplicative phase, the "flag" of the spinor, which in itself can be used to encode helicity information, a fact which was much useful for the computation of solutions of arbitrary spin massless fields and their scattering amplitudes. One can go ahead and substitute the tetrad formalism with spinor language in general relativity. Let e 0 , e 1 , e 2 , e 3 be a normalized tetrad. The null vectors can be decomposed with two (complex) spinors: ı α and o α and their complex conjugate: Note that k a and l a are real whereas m a is the complex conjugate ofm a . In six dimensions the decomposition of vectors is similar [17]. The symmetry group is SO (6), which we will see as a complex group SL(4, C). One real form of it is the group of isometries of Euclidian space, SO(6), the compact form of SL(4, C) and is isomorphic to SU(4). The latter has a natural 4-dimensional fundamental 4 representation which can be seen as the spinorial representation of SO (6). Paralleling the 4-dimensional case, a six dimensional vectorial representation can be constructed from the tensor product of the spinorial representation: and then a vector can be written as an antisymmetric tensor of spinor indices: are analogues of the van der Waerden symbols in six dimensions, essentially the positive chirality projection of the Dirac matrices. The inner product of vectors is performed using the completely antisymmetric symbol ǫ ABCD : Now, let us consider a spinorial basis κ A i , normalized so that ǫ ABCD κ A 1 κ B 2 κ C 3 κ D 4 = 1. A generic vector can be expanded in a basis of simple bispinors κ j , but, because the ǫ ABCD is completely antisymmetric, the only chance that this vector will not be null is that the expansion involves all of the κ i . Null vectors will not satisfy this, so they can expanded in a basis involving only three of the κ i : supposing α 2 = 0. Then, paralellizing the result in four dimensions, all null vectors are outer products of spinors, with the converse obviously true, given (2.6). While a vector is associated with the exterior product of two spinors, a spinor κ A is associated with an isotropic 3-plane. This 3-plane is generated by the null vectors for some basis κ A i . It is clear that there are 3 nonzero vectors V AB i and that they generate an isotropic space, since any linear combination of the V a i is a null vector. A tangent space connection ∇ a can be introduced, and from the zero torsion condition and the Leibniz rule it follows that the following operator is linear on spinorial fields: where a and b are coordinate (vector) indices. Since it is antisymmetric under interchange between a ≡ AE and b ≡ F G, one can construct from it the spinorial tensor: which is the spinor analogue of the Riemann tensor. As in four dimensions, one can decompose it into isometry invariant pieces: [CD] with Φ AC AD = 0 being the spinor analogue of the traceless part of the Ricci tensor and Ψ AB CD = Ψ (AB) with Ψ AC AD = 0 being the spinor analogue of the Weyl tensor. The odd form of the first term guarantees that R AB AD = 0, while R AB CA = −15Λδ C B . In terms of SU(4) irreducible representations, the decomposition is 1 + 20 + 84, adding up to 105 independent components of the Riemann tensor in six dimensions.

The Kerr Theorem
The Kerr theorem gives an implicit solution to all analytic, shear-free, geodetic null congruences in flat space. The problem is strongly tied with the existence of integrable null submanifolds. The connection between the two seemingly distinct problems is deep, as will be discussed in the subsection on generic dimensions. The original result, in four dimensions, can be written in terms of an implicit, analytic function in projective twistor space, and actually serves as one geometrical interpretation of a twistor [11]. We will see that most of the geometrical picture translate to six dimensions.
We will follow the discussion in [18], taking a vectorial point of view. This problem has also been considered using spinorial techniques for the general even-dimensional case by [3]. For a more recent consideration see [19]. Let us begin with flat 6dimensional flat space with Lorentzian signature: (3.1) The null vectors: will determine an integrable distribution (submanifold) if their brackets all close, or, in the dual formulation of Frobenius' theorem, if de i can be written as a linear combination of e i ∧ e j tangent to the submanifold. A somewhat tedious computation yields, for de 0 ∧ e 0 ∧ e 1 ∧ e 2 = 0 the following equations: and the equations following from the requisition that de 1 ∧e 0 ∧e 1 ∧e 2 and de 2 ∧e 0 ∧e 1 ∧e 2 vanish:∂ with i = 1, 2. Note that the first two equations imply the third one. Despite the coupling, the equations are simple to solve. After some manipulation, they result in and their complex conjugate. These equations can be solved by the method of characteristics [20]. The solution is given implicitly in terms of a general function F of five complex arguments: This version of the solution of integrable distributions is more economical than the general expression given in [3], but the spinorial interpretation is less clear. In order to recover it, let us introduce the spinors which generate the null vector k a ≡ k [AB] by exterior product: k AB = κ [AκB] . Note thatκ A is related to the complex conjugate of κ A via a conjugation operator: so that the vector k a is real. Lastly, we introduce the "position vector" x AB , an antisymmetric matrix in spinor space: The attribution of the positions of the coordinates is of course arbitrary. There is a SU(2) × SU(2) ≃ SO(4) isometry -the "little group" -in spinor space which keeps the vector invariant. We will discuss its geometrical interpretation. Recasting (3.6) using spinors: where F (κ A , ζ B ) is a holomorphic function from 4×4 to the complex numbers. The pair is a six-dimensional twistor. From the construction above, the function F needs only be defined for κ A ζ A = 0, so one defines in twistor space a natural pairing: With the choice of conjugationZ I = (κ A ,ζ B ),ζ B = (B −1 ) BC (ζ * ) C , this form has signature (6, 2). The crucial fact in six dimensions is that now F has the projective property: These transformations keep the vector k AB = κ [AκB] invariant. This property shows that the relation between twistors and geodesic, shear-free null vector fields is not bijective, as in [3]. There are many isotropic 3-planes which give rise to the same integrable vector field. This contrasts to the situation in 3 + 1 dimensions, where, given a real null vector field, all isotropic planes containing it are related by a phase. It should also be pointed out that without the reality condition enforced by the operator B, the solution F has to be thought of as a projective section on the Grassmanian space Gr (2, 4), which, unlike its 4-dimensional sibling, is not a projective space. Incidentally, in the six dimensional case the antisymmetric operator x AB is not determined uniquely from κ A and ζ A , since the spinor space has more than two dimensions. The notion that the space-time coordinates arise from the "more fundamental" objects, the twistors, cannot work as in four dimensions.

Six Dimensional Helicity Spinors
The results above encode an underlying algebraic structure. The six-dimensional Lorentzian space considered also shows up in the construction of instantons for four dimensional Yang-Mills theories, where the space of solutions can be seen as the twistor space for Euclidian four dimensional spacetime [21]. The construction is relevant for our case. Let us recall that the twistor space in four dimension is defined, e. g. in [11] as the pair of 2-spinors Z a = (ω α , πβ) with a natural linear symmetric inner product of signature (2, 2): The group that keeps this structure invariant is another real form of SL(4, C), SU(2, 2), which is isomorphic to SO(4, 2). For SO(5, 1), the accidental isomorphism is with SL(2, H), or 2 × 2, unit-determinant matrices over the quaternion field H [21,22].
Thus, spinor space in 6 dimensions is essentially another real form of twistor space in four dimensions. Given four complex numbers {z 1 , z 2 , z 3 , z 4 }, one can embed them into a pair of quaternions as follows: (z 1 , z 2 , z 3 , z 4 ) → (z 1 + jz 2 , z 3 + jz 4 ), (3.14) with j = i a basis element of the quaternions. In the quaternion language, the symmetry corresponds to right multiplication by a quaternion q = α + βj [22]. Left and right multiplication from unit-norm quaternions make for a SU(2)×SU(2) ∼ SO(4) symmetry that acts on the spinor κ A but that leaves the null vector κ [AκB] invariant. In terms of the particular spinor representation given in (3.8), the right action is of the form g ⊗iσ 2 , g ∈ SU(2), the generic operator which commutes with B. This symmetry is the "little group" of SO(5, 1), and, among the multiple spinors which represent the same null vector k AB , one line is chosen by representing helicity, a space-like vector orthogonal to k [AB] . The function representing the solution of the Kerr theorem is actually a function on HP 1 , the projective quaternionic line [21].
To each isotropic and simple 3-plane one associates a spinor: given a null basis {e 1 , e 2 , e 3 } of the 3-plane, one constructs a basis of spinors (κ i ) A such that e i = (κ 0 ) [A (κ i ) B] for each i = 1, 2, 3. Following the treatment in [17], given a spinor κ A , there is only one isotropic 3-plane associated to it: the 3-form T abc ≡ T [AB] [CD] [EF ] obtained by inverting the relation: defines a generic 3-plane for symmetricT AB andT AB . If one has either of them simple, and the other zero, likeT AB = 0 andT AB = κ A κ B , the 3-plane is isotropic (and selfdual). This 3-plane contains the null vector k a = κ [AκB] . The vector is actually in the intersection of the 3-planes associated with κ A andκ A . In order to define a helicity of a twistor, take the pair (κ A , ζ B ) and construct from it the 2-plane: Because of the null condition κ A ζ A = 0, this plane is simple and contains the vector . The other vector can be extracted with the following procedure. Define the rest of the spinor basis π andπ such that κ [AκB π CπD] = ǫ ABCD . (3.18) This essentially means that π A is dual to ζ A . We have the helicity as the vector: Likewise, one can associate with the coordinates of HP 1 the null vectors and all helicity vectors associated with the little group. Given this, one can reinterpret formulas given in [7] as holomorphic projective sections in twistor space, or HP 1 . It would be interesting if one can use analytic properties to define n-point functions of 6-d Lorentz invariant massless field theories, as in [23][24][25]. The construction seems similar to that of [16], but with the extra geometric interpretation.

A Digression in Generic (even) Dimensions
In a flat, even dimensional Lorentzian manifold of null coordinates {u, v, z i ,z i }, the analogue of (3.5) is: (3.20) and their complex conjugate. Given the vector fields We can see that (3.20) are equivalent to the vanishing of the "anti-holomorphic" part of the (flat space covariant) derivative of k a : which in turn means that the attribution of "holomorphic" and "anti-holomorphic" parts for the space generated by the m i andm i is mantained under parallel transport by k a .
The solution of (3.20) can also be obtained from the method of characteristics. The result is an implicit function of D − 1 complex arguments: one sees that the number of arguments is much less than the number of components of pure spinors in D dimensions [3]. This should be seen as a function of the Grassmann plane Gr(2, D − 2) in spinorial space, so the generic function of the pure spinor has to be modded out by the subset of the isometries of SO(D −1, 1) which leaves Gr(2, D −2) invariant. These spaces are not projective, so the twistor analysis has to be carried over using other methods. This shows that real space applications of twistor methods in higher dimensions are not trivial, the generic analysis based on (complex) pure spinors is incomplete.

Six Dimensional Twistors
As seen above, an isotropic 3-plane is represented by a simple 3-form T abc = T [abc] = e 1 ∧ e 2 ∧ e 3 , whose translation to the spinor language is the product Frobenius' theorem states that this plane will be integrable if the 3-form T abc is, up to scale, a harmonic (closed and co-closed) 1 . Since an isotropic 3-plane is necessarily self-dual, then one needs only to check for closedness. In terms of spinors, this condition is The six-dimensional twistor equation solves the constraint if π A κ A = 0. The twistor equation is the only plausible simplification for the derivative of a spinor. In general, the derivative of κ A will have the following expansion: with Θ C AB in the 20 of SU(4), being antisymmetric in the lower indices and traceless. Imposing that Θ C AB = 0 is the only Lorentz-invariant condition that leaves the right number of degrees of freedom involved in the Kerr theorem.
The six-dimensional twistor equation (4.3) has the usual integrability condition stemming from the Ricci identity: Contracting with the antisymmetric tensor, one obtains which, using (4.3) and rearranging the indices, can be written as Contracting A and E, and using that R AF GA = −15Λδ G F (2.11), we find that: with Λ related to the scalar curvature. For Einstein manifolds Φ AF EF = 0, Λ is a constant and the antisymmetric part of the left hand side imposes a dual twistor equation for π A : The twistor pair (κ A , π B ) then solves the zero mass half-spin (Weyl) equations. The symmetric part of EG in (4.7) gives an algebraic condition on the Weyl spinor: Because of (4.9), π c obeys a simmilar condition: Ψ AF EG π G = 0. In quaternionic language, the twistor (κ A , π C ) is an eigenvector of the (symmetric) Weyl operator with zero eigenvalue.
If the manifold has a vanishing Riemann tensor, then π C is covariantly constant and therefore the twistor equation (4.3) can be solved as: with ψ A and λ B constant spinors. This is an exact analogue of the four dimensional case. Like its 4-dimensional sibling, it can be defined as the null subspace of the twistor space, where the metric is the usual pairing: With the choice of charge conjugation operator defining a real form in the eightdimensional complex space spanned by the twistors. For generic Einstein manifolds, the pair Z I = (ψ A , χ B ) has the interpretation of a Killing spinor [15]. The twistor equation and the corresponding dual (4.9) are written with the help of gamma matrices as: (4.13) For some constant c, and R being the scalar curvature. One then sees that, as in four dimensions, the algebraic property that κ A is annihilated by the Weyl spinor -a principal spinor -is imperative for the existence of solutions to (4.13). Given that one can associate spinors to integrable harmonic 3-planes, the number of solutions to the twistor equation is a topological invariant, the intersection number (Betti number for half-dimensional submanifolds). Being harmonic, the forms can be seen to saturate the Green's inequality. This property can be generalized to "impure spinors", which are related to lower dimensional isotropic manifolds, and be used to define "calibrations" [26] and are at the heart of the saturation of inequalities that allow for computations in the so-called "attractor mechanism" [27].

A family of Examples
There is a family of spaces allowing for a principal spinor. These are the Kerr-Schild spaces, whose metric is given by: (4.14) whereḡ ab is a maximally symmetric metric (for dS 6 , AdS 6 or R (5,1) ), k a is a geodesic, shear-free null vector field, with respect toḡ ab and S is a function. These metrics have been studied extensively in four dimensions [18,28,29]. The treatment below will follow [17] closely. Solutions of Einstein's equations of this form in four dimensions include all pp-wave backgrounds, as well as the Kerr-Newman family of black holes.
The properties that k a is geodesic and shear free in the metricḡ ab translate to the metric g ab . We will assume that DS = k c ∇ c S = 0. If C c ab is the relative connection between the derivatives ∇ a and∇ a , associated respectively to the metrics g ab andḡ ab , Indices are raised with the maximally symmetric metricḡ ab . These satisfy C c ab k b = 0, so if k a is geodesic with respect to ∇ a , it will also be so with respect to∇ a . Moreover, sinceḡ ab is maximally symmetric, it is conformally flat and the integrability properties of k a which make for the hypothesis of the Kerr theorem remain valid, at least locally. The k a can be thought of as the real part of an integrable 3-plane, as in Section 3.
With a little effort, one can compute the Riemann tensor associated with g ab : withR abc d maximally symmetric. The result is: The Ricci tensor is: where ∆ is the Laplacian in the transverse space, related toḡ ab . We will assume that the space is Einsteinian. Given the form of the Riemann tensor, one can show that the Weyl tensor has the property: Now, we write k a = κ [AκB] , for an integrable κ A satisfying (4.3). We choose an spinorial basis such that ǫ ABCD = κ [AκB (π * ) C (π * ) D] as in Section 3.1. Here the asterisk denotes dual. Given that the dual spinor π B satisfies a similar equation (4.9), we will also assume that it is integrable. We introduce the dual basis {(κ * ) A , (κ * ) A , π A ,π A }, and note that k a = π [AπB] . We find that ( These identities constraint the form of the Weyl spinor so that Ψ AE IJ κ E = 0. The differential equation for S can be solved by usual methods, and the κ A are taken from the general solution in flat space (3.6). As in four dimensions, this family of spacetimes can cover all known families of reduced holonomy manifolds. Some of those, like the cone of Einstein-Sasaki's [30], can be obtained from Wick rotation of Lorentzian manifolds. However, there seems to be more parameters encoded in the κ A as in (3.10), and the constants fixing S than the usual p, q, r involved in the L (p,q,r) and the Reeb vector. Although it should be remembered that in the Euclidian case one would like to impose that the orbits of the vector k a are closed. We hope to address this issue in the future.

Conclusions
In this letter we addressed the relationship between Kerr theorem and pure spinors in six dimensions. We showed that, while the spinorial language is natural to talk about solutions to geodesic, shear-free, null vector fields, the correspondence is not one-to-one, which is reminiscent of the quaternionic structure of supersymmetry in six dimensions [22]. We then turned to the problem of the twistor equation in six dimensions, and showed that, just like the four dimensional case [15], the Killing spinor equation can be cast in the Newman-Penrose language quite naturally. Finally, we presented a generalization of Kerr-Schild metrics which display the algebraic property allowing for solutions of the twistor equation.
A great deal of applications of pure spinors has been put forward in the last ten years, mainly to the covariant superstring [31]. The effort has been basically centered on algebraic aspects of pure spinors. However, most aspects of such applications have had their birth in geometrical aspects of four dimensional general relativity. This work is the result of some effort spent in trying to understand pure spinors from a geometrical point of view in six dimensions. In eight or more dimensions, the geometrical point of view becomes more natural, as the algebraic one becomes more mysterious: for instance, the pure condition becomes a quadratic constraint on spinors, while the relationship to maximal dimension isotropic planes continue to hold. In six dimensions, as we can see here, many of the features translate quite naturally to the new setting. We hope that the fresh point of view will be useful to other people.