aa r X i v : . [ m a t h . DG ] J un The curvature of almost Robinson manifolds
Arman Taghavi-Chabert
Masaryk University, Faculty of Science, Department of Mathematics and Statistics,Kotl´aˇrsk´a 2, 611 37 Brno, Czech Republic
Abstract
An almost Robinson structure on an n -dimensional Lorentzian manifold ( M , g ), where n = 2 m + ǫ , ǫ ∈ { , } , is a complex m -plane distribution N that is totally null with respect to the complexified metric,and intersects its complex conjugate in a real null line distribution K , say. When N and its orthogonalcomplement N ⊥ are in involution, the line distribution K is tangent to a congruence of null geodesics,and the quotient of M by this flow acquires the structure of a CR manifold. In four dimensions, such acongruence is shearfree.We give classifications of the tracefree Ricci tensor, the Cotton-York tensor and the Weyl tensor,invariant under i) the stabiliser of a null line, and ii) the stabiliser of an almost Robinson structure. Forthe Weyl tensor, these are generalisations of the Petrov classification to higher dimensions. Since analmost Robinson structure is equivalent to a projective pure spinor field of real index 1, the present workcan also be viewed as spinorial classifications of curvature tensors.We illustrate these algebraic classifications by a number of examples of higher-dimensional generalrelativity that admit integrable almost Robinson structures, emphasising the degeneracy type of the Weyltensor in each case. Let ( M , g ) be a Lorentzian manifold of dimension n = 2 m + ǫ , ǫ ∈ { , } . There are two natural geometricstructures we can endow ( M , g ) with:1. either a preferred real null line distribution, whereby the structure group of the frame bundle is reducedto Sim( n − ⊂ SO( n − , complex distribution of rank m intersecting its complex conjugate in a real nullline distribution, whereby the structure group is reduced to Sim( m − , C ) ⊂ Sim( n − ⊂ SO( n − , almost Robinson structure [Tra02a, NT02]. It has also beenreferred to as an optical structure in other places e.g. [Nur96, TC11]. Clearly, structure 2 implies structure1, and the additional geometric datum here can be seen to be a Hermitian structure on the fibers of thescreenspace bundle of the real null line distribution.The main aims of the article are • to give Sim( n − • to give Sim( m − , C )-invariant classifications of the Weyl tensors and other curvature tensors; • to apply the classifications to known solutions of Einstein’s equations. The motivation for the study of such geometrical structures comes from four-dimensional general relativity,where they are equivalent and intrinsically related to Petrov’s classification of the Weyl tensor [Pet00].In four-dimensional general relativity, there is a fundamental relation between the existence of shearfreecongruences of null geodesics (SCNG) on a spacetime ( M , g ) and algebraically degenerate or special solutions1o certain field equations. For instance, a 2-form F ab that is a solution of the vacuum Maxwell equations ∇ [ a F bc ] = 0, and ∇ b F ab = 0 is algebraically special if and only if ( M , g ) admits a SCNG – this is thecontent of the Robinson theorem [Rob61]. Another important example comes from the study of Einstein’sfield equations: the Goldberg-Sachs theorem [GS09] tells us that the Weyl tensor C abcd of an Einsteinspacetime ( M , g ) is algebraically special, i.e. of Petrov type II or more degenerate [Pet00] if and only if( M , g ) admits a SCNG. In both cases, the algebraic degeneracy of F ab and C abcd can be characterised bythe existence of a repeated principal null direction (PND) , i.e. a null vector k a such that F ab k a = 0 and k d k e C ade [ b k c ] = 0, respectively, and such a vector field generates the SCNG by virtue of the field equations.A further development in the theory came from the introduction of spinors, which simplified much ofthe formalism of general relativity, and stems essentially from the remark that in four dimensions, a nullvector field is equivalent to a chiral spinor field (up to scale): such a spinor defines a complex null 2-planedistribution, which intersects its complex conjugate in a line bundle spanned by this real null vector field.Quite naturally, to the notion of principal null direction corresponds a notion of principal spinor , and thePetrov classification can then be expressed more transparently in this context [Wit59, Pen60]. Similarly, thegeometric properties of a PND can be translated into this formalism: a SCNG is equivalent to its corre-sponding principal spinor being foliating in the sense that its associated complex null 2-plane distributionis integrable. Many of the classical results of general relativity such as the Robinson and Goldberg-Sachstheorems can then be viewed from this spinorial angle, as explained in the monographs [PR84, PR86].How does all this generalise to higher dimensions? Unless n = 4, the correspondence between nulllines and totally null complex m -planes is not one-to-one: each null line lifts to a compact subset of thegrassmannian of totally null complex m -planes of dimension ( m − m − − ǫ )). This implies that, forspacetimes of dimensions higher than four, one can define two distinct, yet related, algebraic classificationsof the Weyl tensor, generalising the Petrov classification: • one, put forward in [CMPP04, PPCM04], is based on the notion of principal null directions, and is nowknown as the null alignment formalism . It has given rise to a large amount of literature, generalising,or at least partially, results from four to higher general relativity - for a survey see [OPP13a]. Ourclassification essentially emphasises the Sim( n − • the other, already advocated in [Hug95,Jef95,TC12b,TC13], is based on the notion of principal spinors,and provides a convenient setting in which generalisations of the Robinson and Kerr theorems [HM88],and to some extent, the Goldberg-Sachs theorem [TC11, TC12a] can be formulated. Since an almostRobinson structure is also equivalent to the line spanned by a pure spinor field of real index
1, theSim( m − , C )-invariant classification of the Weyl tensor is essentially a spinorial classification.Furthermore, we can view the Sim( m − , C )-invariant classification of the Weyl tensor as a refinement ofthe Sim( n − m − integrable almost Robinson structures occur in important higher-dimensional solutions to Einstein’s field equations such as the Kerr black hole [MP86, GLPP05, HHT99,CLP06] or the black ring [ER02]. This is unlike shearfree congruences of null geodesics, which have not beenas ubiquitous as in dimension four. The structure of the paper is as follows. Section 2 lays the algebraic foundation of the paper. We first reviewthe properties of the stabiliser sim ( n −
2) of a (real) null line in n -dimensional Minkowski space. We broadlyfollow the theory of parabolic Lie algebras of [ ˇCS09] by making use of invariant filtrations and gradings onvector spaces. We then show how a Robinson structure, i.e. a conjugate pair of totally null complex m -planesintersecting in a real null line, induces a Hermitian structure on the orthogonal complement of this null line.Drawing from standard results of Hermitian geometry, e.g. [Sal89], we describe the stabiliser sim ( m − , C )in the Lie algebra so ( n − ,
1) of a Robinson structure and its irreducible representations. In addition,Proposition 2.8 describes the space of all Robinson structures on Minkowski space, while Propositions 2.10the space of all Robinson structures intersecting a given real null line.This calculus is then extended in section 3 to the classifications of curvature tensors. We first focus onthe Sim( n − n − n −
2) or Sim( m − , C ).Section 4 is concerned with the geometric applications of the classifications of section 3. After a briefreview of the geometry associated to a null line distribution, we shift our attention to Robinson manifolds, i.e.Lorentzian manifolds equipped with an integrable almost Robinson structure. We distinguish various degreesof integrability in odd dimensions in Definition 4.7. We recast some of the previous results of the author[TC11, TC12a, TC12b, TC13] within the framework of the present Sim( m − , C )-invariant classification ofthe Weyl tensor. Proposition 4.12 gives necessary Weyl curvature conditions for the existence of a Robinsonstructure. This yields a natural definition of an almost Robinson structure aligned with the Weyl tensor -Definition 4.13. We exhibit aligned Robinson structures in the Petrov type G static KK bubble – Example4.15. Proposition 4.16 and Corollary 4.17 establish a relation between a Sim( n − repeated aligned almost Robinson structures is given in Definition 4.18. The algebraically special conditionis then reformulated in the present context in Proposition 4.19. Under additional curvature assumptions, thisforms the basis of a higher-dimensional version of the Goldberg-Sachs theorem given in the same references,and restated as Theorem 4.20. Proposition 4.22 establishes a relation between a Sim( n − n − n − m − , C )-invariant classifications of section 3. The short appendix C highlights some of the specialfeatures occuring in low dimensions, where a number of simplifications can be made. Null lines
Let ( V , g ) be oriented n -dimensional Minkowski space, i.e. a vector space V equipped witha non-degenerate symmetric bilinear form g ab of signature ( n − ,
1) (i.e. (+ , . . . , + , − )). We adopt theabstract index convention of [PR84], elements of V and V ∗ will carry upstairs and downstairs lower caseRoman indices respectively. Indices will be lowered and raised freely be means of g ab and its inverse g ab respectively. Symmetrisation and skew-symmetrisation will be denoted by round and square brackets aroundgroups of indices respectively, e.g. g ab = g ( ab ) ∈ ⊙ V ∗ and α abc = α [ abc ] ∈ ∧ V ∗ .3et K be a null line in V , i.e. for any V a ∈ K , V a V a = 0. For convenience, we fix a generator k a in K .Since K is null, it is contained in its orthogonal complement K ⊥ = (cid:8) V a ∈ V : g ab k a V b = 0 (cid:9) . Thus, a nullline in V induces a filtration { } =: V ⊂ V ⊂ V ⊂ V − := V . (2.1)of vector subspaces on V , where V := K and V := K ⊥ . By the metric isomorphism V ∗ ∼ = V , it is clearthat K also defines a filtration { ( V ∗ ) i } on the dual vector space V ∗ where ( V ∗ ) i ∼ = V i for each i .One can fix a vector subspace V − dual to V , so that V = V ⊕ V ⊕ V − , (2.2)where V ∼ = V , V ⊕ V ∼ = V . In this case, V is the orthogonal complement of both V and V − . Fix agenerator ℓ a of V − satisfying the normalisation ℓ a k a = 1. The metric tensor g ab induces a non-degeneratepositive definite symmetric bilinear form h ab = g ab − S ab , where S ab := 2 k ( a ℓ b ) , on V . The Lie subalgebra sim ( n −
2) The filtration (2.1) and grading (2.1) on V induce filtrations and gradingson any tensor product of V . In particular, the Lie algebra g := so ( n − , ∧ V , admits the filtration { } =: g ⊂ g ⊂ g ⊂ g − := g , (2.3)of vector subspaces g i , and a grading g = g − ⊕ g ⊕ g , (2.4)where g i = g i ⊕ g i +1 for i = − , ,
1, and g ± ∼ = V ± ⊗ V and g ∼ = ∧ V ⊕ ( V ⊗ V − ). It is straightforwardto see that the vector subspace g is the stabiliser of V , i.e. k c φ [ ac k b ] = 0 for any φ ab ∈ g . Such a Liesubalgebra is known as a parabolic Lie subalgebra of g [FH91,BE89, ˇCS09]. It is also known as the Lie algebra sim ( n −
2) of the group Sim( n −
2) of similarities of R n − . As for any parabolic Lie subalgebra, it consistsof a nilpotent part g ∼ = R n − , and a reductive part g = co ( n −
2) := so ( n − ⊕ R . For convenience,we shall denote the center of g and its complement in g by z and so respectively. The center containsa unique grading element , which, given our choice of k a and ℓ a , can be expressed as E ab := − k [ a ℓ b ] , andhas eigenvalues i on V i , i.e. V b E ab = ± V a if V a ∈ V ± , and V b E ab = 0 if V a ∈ V , and so on for tensorproducts of V .At this stage, we note that g + z and g + so are sim ( n − of g . Their quotientsby g are linear isomorphic to the irreducible g -modules z and so . In particular, they are irreducible sim ( n − g ) associated to the filtration (2.3), i.e.gr( g ) = gr − ( g ) ⊕ gr ( g ) ⊕ gr ( g ) , where gr i ( g ) := g i / g i +1 , Each summand gr i ( g ) is a totally reducible sim ( n − co ( n − g i that depends on the splitting (2.4) chosen. When n = 6, each irreducible sim ( n − g ji , say,of gr i ( g ) is linearly isomorphic to a co ( n − g ji , say, of g i – here, ˘ g ± := g ± , ˘ g := z and˘ g := so ( n − n = 6, the sim (4)-module g splits further into a self-dual part g , +0 and an anti-self-dual part g , +0 , i.e. g ∼ = g , +0 ⊕ g , − . In both cases, the nilpotent part g of sim ( n −
2) acts trivially on each In fact, their invariance can be made more explicit by noting that they can be defined in terms of kernels of the maps gK Π ji defined in appendix B.1: g = { φ ab ∈ g : gK Π − ( φ ) = 0 } , g + z := { φ ab ∈ g : gK Π ( φ ) = 0 } , g + so := { φ ab ∈ g : gK Π ( φ ) = 0 } . ji . The splitting of gr( g ) into irreducibles together with the effect of the action of g on the corresponding co ( n − g ji can be conveniently represented in terms of the graph g ( ( PPPPPP g ' ' ❖❖❖❖❖❖ ♦♦♦♦♦♦ g − g ♥♥♥♥♥♥ (2.5)where we draw an arrow from g ji to g ki − whenever ˘ g ji ⊂ g · ˘ g ki − – here, the · denotes the action of g on a g -module, and in this case, corresponds with the Lie bracket. Lie group level and the null grassmanian of null lines
The stabiliser of an (unoriented) null line K ⊂ V in the orthogonal group O( V , g ) ∼ = O( n − ,
1) is isomorphic to the group Sim( n − ∼ = ( R ∗ × O( n − ⋉R n − of similarities of R n − , with Lie algebra sim ( n −
2) as described above. We can identify the spaceGr ( V , g ) of all unoriented null lines in V with the homogeneous spaceGr ( V , g ) = O( n − , / Sim( n − , which is simply diffeomorphic to the ( n − S n − .We note in passing that Sim( n −
2) contains the Lie subgroup Sim + ( n − ∼ = ( R + × O( n − ⋉ R n − stabilising an oriented null line. Thus, the space of all oriented null lines is given by the homogeneous spaceO( n − , / Sim + ( n −
2) and splits into two connected components, the space of all future-pointing null linesand the space of all past-pointing null lines, each being isomorphic to the ( n − S n − . As before, ( V , g ) will denote oriented n -dimensional Minkowski space. Denote by ( C V , g ) the complexificationof ( V , g ), where it is now understood that g becomes complex-valued. Let ¯: C V → C V : V a ¯ V a be thereal structure, or complex conjugation, on C V preserving V , i.e. ¯ V a = V a if and only if V a ∈ V .Set n = 2 m + ǫ where ǫ ∈ { , } . Let N be a totally null complex m -plane, i.e. an m -dimensional vectorsubspace of C V , on which the complexified bilinear form g ab is totally degenerate. Denote by ¯ N the complexconjugate of N , i.e. ¯ N := { V a ∈ C V : ¯ V a ∈ N } . Definition 2.1
We call the dimension of the intersection of N and ¯ N the real index of N . Lemma 2.2 ( [KT92, Kop97] ) When ǫ = 0 , the real index of N is . When ǫ = 1 , the real index of N iseither or . Definition 2.3 A Robinson structure on ( V , g ) is a totally null complex m -plane of ( V , g ) of real index 1.Since a Robinson structure N determines a null line K in V , we can henceforth assume the setting of theprevious section 2.1 with V := K and V := K ⊥ together with the filtration (2.1) and grading (2.2) on V .Clearly both N and ¯ N are contained in the complexification of V . For clarity, we first deal with the even-and odd-dimensional cases separately. ǫ = 0 ) In this case, it is a standard result [PR86] that the totally null complex m -plane N of the Robinson structurecan be either self-dual or anti-self-dual according to a chosen orientation on V , i.e. ∗ ω = ± ( − i) ( m − m − ω for ω ∈ ∧ m N , where ∗ : ∧ k V → ∧ n − k V is the Hodge duality operator. The orientation of the complexconjugate ¯ N relative to that of N depends on the dimension of V . The following lemma is standard (see eg.[Car81, KT92]). In the extant literature, Sim( n −
2) may be taken to stabilise an oriented null line. emma 2.4 When m is even, a Robinson structure is self-dual, respectively anti-self-dual, if and only if itscomplex conjugate is anti-self-dual, respectively self-dual.When m is odd, a Robinson structure is self-dual, respectively anti-self-dual, if and only if its complexconjugate is self-dual, respectively anti-self-dual. Referring to the grading (2.2) adapted to the filtration (2.1), we can define the ( m − V (1 , := C V ∩ N , V (0 , := C V ∩ ¯ N . (2.6)of C V := C ⊗ R V , so that C V = V (1 , ⊕ V (0 , . (2.7)Being subspaces of N and ¯ N respectively, each of V (1 , and V (0 , are totally null with respect to thecomplexification of g ab , and in fact, with respect to the complexification of h ab since they are also orthogonalto both V ± . By a standard argument of linear algebra [NT02], they can therefore be identified with the +iand − i-eigenspaces of a complex structure J ba on V compatible with H ab := h ab , i.e. J ca J bc = − H ba . Thereason for the relabelling of h ab as H ab is for notational consistency with the odd-dimensional case. We takethe convention that V a J ba = i V a for any V a ∈ V (1 , . In fact, this complex structure has the associatedhermitian 2-form ω ab := J ca H cb on V .To make the description independent of the splitting (2.2), we extend the vector subspaces V (1 , and V (0 , of C V to vector subspacesgr (1 , ( V ) := (cid:16) V (1 , + C V (cid:17) / C V , gr (0 , ( V ) := (cid:16) V (0 , + C V (cid:17) / C V , (2.8)respectively, of the complexification of gr ( V ) = V / V . These can be identified with the +i and − ieigenspaces of a complex structure on the screenspace gr ( V ), that is an equivalence class of complex struc-tures [ J ba ] on gr ( V ) where J ba , ˆ J ba ∈ [ J ba ] ⇔ ˆ J ba = J ba + α a k b − k a α b , for some α a ∈ V .Similarly, one can also define a Hermitian 2-form on gr ( V ) as an equivalence class of Hermitian 2-forms[ ω ab ] on gr ( V ). ǫ = 1 ) The odd-dimensional description of the Robinson structure N is almost identical to the even-dimensionalcase, except that now the totally null complex m -plane N of the Robinson structure is strictly contained inits ( m + 1)-dimensional orthogonal complement N ⊥ := { V a ∈ C V : V a W a = 0 , for all W a ∈ N } , and N ⊥ / N is one-dimensional. In the splitting (2.2), we can then distinguish a one-dimensional subspace V (0 , := C V ∩ N ⊥ ∩ ¯ N ⊥ of C V , which can be seen to be the complexification of a real vector subspace U of V , on which g ab (andthus h ab ) is non-degenerate. We thus have a splitting of the complexification of V C V = V (1 , ⊕ V (0 , ⊕ V (0 , , (2.9)where V (1 , and V (0 , are defined by (2.6) as in the even-dimensional case. Choosing a unit vector u a in U , we can define a positive definite symmetric bilinear form H ab := h ab − u a u b , V ∩ U ⊥ . Then each of V (1 , and V (0 , are totally null ( m − − i-eigenspaces of a H ab -compatible complex structure J ba on V ∩ U ⊥ , with corresponding hermitian 2-form ω ab := J ca H cb . The vector space U is the kernel of J ba . Thus, the odd-dimensional case differs from theeven-dimensional case only by an additional one-dimensional vector subspace. Remark 2.5
In odd dimensions, the endomorphism J ba on V could be called an h ab -compatible CRstructure on V , but it is usually referred to as a contact Riemannian structure . In this article, we shallreferred to J ba as a Hermitian structure regardless of the dimension with the understanding that it appliesto V ∩ U ⊥ .Again, this complex structure defines an equivalence class of complex structures on gr ( V ) with ± i-eigenspaces gr (1 , ( V ) and gr (1 , ( V ) as defined by (2.8), and kernelgr (0 , ( V ) := (cid:16) V (0 , + C V (cid:17) / C V . The sim ( n − ( V ) is equipped with an equivalence class of unit vectors [ u a ] where u a , ˆ u a ∈ [ u a ] ⇔ ˆ u a = u a + λk a , for some λ ∈ R . Notation 2.6
For future use, we define the real vector spaces˘ V , := [[ V (1 , ]] , V , := (cid:16) ˘ V , + V (cid:17) / V , ˘ V , := [ V (0 , ] , V , := (cid:16) ˘ V , + V (cid:17) / V , where the ‘bracket’ notation of [Sal89] denotes the real span of the bracketed complex vector space, i.e. V (1 , = C ⊗ R [[ V (1 , ]] , V (0 , = C ⊗ R [ V (0 , ] . Here, we have been careful to distinguish ˘ V , and V , as vector subspaces of V and V / V respectively. Assume as before n = 2 m + ǫ where ǫ ∈ { , } . Henceforth, we shall combine the even- and odd-dimensionalcases, the distinction between which will be made by the use of ǫ ∈ { , } . Thus, the splittings (2.7) and(2.9) will be merged by writing C V = V (1 , ⊕ V (0 , ⊕ ǫ V (0 , . The stabiliser of N in so ( n − ,
1) also stabilises its complex conjugate and thus their intersection. Hence, itmust be contained in sim ( n − K . Then,using our previous notation, the complexifications C g i of g i admit direct sum decompositions C g ± = g (1 , ± ⊕ g (0 , ± ⊕ ǫ g (0 , ± , C g = C z ⊕ g (2 , ⊕ g (0 , ⊕ g (1 , ⊕ ǫ (cid:16) g (1 , ⊕ g (0 , (cid:17) , where g (0 , ± ∼ = V ± ⊗ V (0 , , g (1 , ± ∼ = V ± ⊗ V (1 , , g (2 , ∼ = ∧ V (1 , , g (0 , ∼ = ∧ V (0 , , g (1 , ∼ = V (1 , ⊗ V (0 , , and in addition, when ǫ = 1, g (0 , ± ∼ = V ± ⊗ V (0 , , g (1 , ∼ = V (1 , ⊗ V (0 , . Here, we have made use of the standard identifications so ( n, C ) ∼ = ∧ ( C V ) and so ( n − , ∼ = ∧ V .Further, the complex structure J ba on V is contained in g (1 , and spans a 1-dimensional invariantsubspace, which we shall denote g ω . The complement of g ω in g (1 , will be denoted by g (1 , ◦ , i.e. g (1 , ∼ = g ω ⊕ g (1 , ◦ . eal representations Since we shall essentially be interested in real representations, we again draw fromthe notation of [Sal89] and define C ⊗ R [[ g ( j,k ) i ]] := g ( j,k ) i k = j , C ⊗ R [ g ( i,i )0 ] := g ( i,i )0 . Putting things together, we see that the Lie algebra so ( n − ,
1) admits the direct sum decomposition g ± = [[ g (1 , ± ]] ⊕ ǫ [ g (0 , ± ] , g = z ⊕ (cid:16) [ g ω ] ⊕ [ g (1 , ◦ ] ⊕ [[ g (2 , ]] ⊕ ǫ [[ g (1 , ]] (cid:17) . By standard results of Hermitian geometry, we have [ g (1 , ] ∼ = u ( m − g ω ] ∼ = R and [ g (1 , ◦ ] ∼ = su ( m − u ( m −
1) and su ( m −
1) are the Lie algebras of the unitary group U( m −
1) and the special unitarygroup SU( m −
1) respectively. Clearly, ( z ⊕ [ g (1 , ]) ⊕ g preserves V + V (1 , and V ⊕ V (0 , . Thus, Lemma 2.7
The stabilizer of a Robinson structure in so ( n − , is the Lie subalgebra sim ( m − , C ) := cu ( m − ⊕ R n − , where cu ( m −
1) := R ⊕ u ( m − , (2.10) of the Lie algebra sim ( n − . Associated graded module
The associated graded sim ( n − g ) of the filtration (2.3) is also a sim ( m − , C )-module, and each of the irreducible sim ( n − g ji decomposes further into irreducible sim ( m − , C )-modules, i.e. for each i, j , gr ji ( g ) = M k g j,ki , where each g j,ki is linearly isomorphic to a cu ( m − g j,ki given by˘ g , ± := [[ g (1 , ± ]] , ˘ g , ± := [ g (0 , ± ] , ˘ g , ± := [[ g (1 , ± ]] , ˘ g , := [ g ω ] , ˘ g , := [ g (1 , ◦ ] , ˘ g , := [[ g (2 , ]] . Since the modules ˘ g j,ki are not invariant under g , we can encode the action of the nilpotent part g of g on ˘ g j,ki by means of a sim ( m − , C )-invariant graph, which refines the graph (2.5) g , (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ g , ( ( ◗◗◗◗◗◗ g , (cid:24) (cid:24) ✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ (cid:28) (cid:28) ✽✽✽✽✽✽✽✽✽✽✽ ( ( ◗◗◗◗◗◗ ♠♠♠♠♠♠ ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ g , − g , ♠♠♠♠♠♠ g , (cid:31) (cid:31) ( ( g , − g , B B ✝✝✝✝✝✝✝✝✝✝ g , F F ✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍ ? ? (2.11)where an arrow from g j,ki to g m,ni − for some i, j, k, m, n means that ˘ g j,ki ⊂ g · ˘ g m,ni − . Here, the modules occuringonly in odd dimensions are in grey, and dotted arrows apply only to odd dimensions. Alternatively, the irreducible sim ( m − , C )-modules can be invariantly defined by means of the projection maps g Π j,ki tobe found in appendix B, i.e. g j,k := { φ ∈ g : g Π m,ni ( φ ) = 0 , m = j , n = k } / g i +1 , and so on. .2.4 The space of Robinson structures Quite naturally, the stabiliser of N and N ⊥ in the orthogonal group O( n − ,
1) is the group Sim( m − , C )of similarities of an ( m − R ∗ × U( m − ⋉ R n − , when n is even ( ǫ = 0),( R ∗ × U( m − × Z ) ⋉ R n − , when n is odd ( ǫ = 1),with Lie algebras sim ( m − , C ) as described above, and have two, respectively four, connected componentswhen n is even, respectively odd. Both O( n − ,
1) and SO( n − ,
1) act transitively on the space of all totallynull complex m -planes of C V of real index 1. In particular, Proposition 2.8
The space Gr m ( C V , g ) of all totally null complex m -planes of real index in C V is iso-morphic to the ( m ( m −
1) + ǫ (2 m − -dimensional homogeneous space O( n − , / Sim( m − , C ) . In evendimensions ( ǫ = 0) , this space splits into two connected components, the space Gr + , m ( C V , g ) of all self-dualRobinson structures in C V , and the space Gr − , m ( C V , g ) of all anti-self-dual Robinson structures in C V , eachisomorphic to SO( n − , / Sim( m − , C ) . Remark 2.9
When n = 2 m and m is even, points in Gr + , m ( C V , g ) and points in Gr − , m ( C V , g ) can becanonically identified by sending a self-dual Robinson structure to its complex conjugate, which is necessarilyanti-self-dual.Clearly, the group Sim( m − , C ) is a subgroup of Sim( n − In four dimensions, the correspondence between null lines and Robinson structures is one-to-one. In higherdimensions, this is no longer case. Instead, while a fixed Robinson structure N determines a real null line K as the real span of its intersection N ∩ ¯ N , a given null line K yields a family of totally null complex m -planes,those that intersect K .More specifically, there is a projection π : Gr m ( C V , g ) → Gr ( V , g ), which sends a totally null complex m -plane N of real index 1 to the real span of N ∩ ¯ N . The inverse image of a point K is then π − ( K ) = { N ∈ Gr m ( V , g ) : dim( N ∩ K ) > } , which must be isomorphic to the quotient Sim( n − / Sim( m − , C ), orequivalently O( n − / U( m − m − / U( m − m − m −
2) and m ( m −
1) in even and odd dimensions respectively. In summary,
Proposition 2.10
Let ( V , g ) be n -dimensional Minkowski space with n = 2 m + ǫ , ǫ ∈ { , } . Then every nullline in V corresponds to a family of Robinson structures in C V parametrised by points of the homogeneousspace O( n − / U( m − of dimensions ( m − m + 2( ǫ − . Further, when n is even, this family splitsinto two disjoint connected families of self-dual and anti-self-dual Robinson structures, each isomorphic to SO(2 m − / U( m − . The algebraic setting of section 2 extends naturally to tensor products of V , the standard representationof so ( n − , This can also be seen at the Lie algebra level by considering the quotient sim ( n −
2) by sim ( m − , C ), or equivalently thecomplement of cu ( m −
1) (see (2.10)) in co ( n − g (2 , ]] ⊕ ǫ [[ g (1 , ]]. so ( n − , F := { Φ abc ∈ ⊗ V : Φ ab = Φ ( ab ) , Φ aa = 0 } , (3.1) A := { A abc ∈ ⊗ V : A abc = A a [ bc ] , A [ abc ] = 0 , A aab = 0 } , (3.2) C := { C abcd ∈ ⊗ V : C abcd = C [ ab ][ cd ] , C [ abc ] d = 0 , C babc = 0 } , (3.3)of tracefree Ricci tensors, Cotton-York tensors, and of Weyl tensors respectively. We recall the dimensionsof these modules in the following table V m m + 1 g ∼ = ∧ V (2 m − m (2 m + 1) m F (2 m − m + 1) m (2 m + 3) A m ( m + 1)( m − (2 m − m + 1)(2 m + 3) C m ( m + 1)(2 m + 1)(2 m − ( m − m + 1)(2 m + 1)(2 m + 3)where, for convenience, we distinguish the even- and odd-dimensional cases by shading the latter in gray.The sim ( n − sim ( m − , C )-modules. Needless to say that these classifications are also invariant under the Lie groups Sim( n −
2) andSim( m − , C ). In the tables, we have abbreviated so ( n − , sim ( n − co ( n − sim ( m − , C )-modules and cu ( m − so ( n − , sim -mod, co -mod, sim C -modand cu -mod respectively. Again, we treat both the even- and odd-dimensional cases at once: the rows of theadditional modules occurring in odd dimensions only are shaded in gray. The symbol ⊚ denotes the Cartanproduct, which corresponds to the tracefree symmetric product for co ( n − cu ( m − sim ( n − -invariant classifications As in section 2.1, we single out a null line K on n -dimensional Minkowski space ( V , g ), so that V admitsa filtration (2.1) of sim ( n − sim ( n − V . This applies in particular to F , A and C . Each of the graded sim ( n − sim ( n − sim ( n −
2) on any associated co ( n − co ( n − F , A and C can all be expressed as the Cartan product of the co ( n − V and g , which wepresently recall: sim -mod co -mod Dimension ( n = 2 m ) Dimension ( n = 2 m + 1) V ± V ± V V m − m − sim -mod co -mod Dimension ( n = 2 m ) Dimension ( n = 2 m + 1) g ± V ± m − m − g z g so ( m − m −
3) ( m − m − im -mod co -mod Dimension ( n = 6) g ± V ± sim -mod co -mod Dimension ( n = 6) g z g ± so ± The filtration (2.1) on V induces a filtration { } =: F ⊂ F ⊂ F ⊂ F ⊂ F − ⊂ F − := F , (3.4) of sim ( n − -modules on the space F defined by (3.1) .Further, the associated graded sim ( n − -module gr( F ) = L i = − gr i ( F ) where gr i ( F ) := F i / F i +1 splitsinto a direct sum gr ± ( F ) = F ± , gr ± ( F ) = F ± , gr ( F ) = F ⊕ F , of irreducible sim ( n − -modules, where sim -mod co -mod Dimension ( n = 2 m ) Dimension ( n = 2 m + 1 ) F ± V ± ⊚ V ± F ± V ± ⊚ V m − m − F V − ⊚ V F V ⊚ V m (2 m −
3) ( m − m + 1) Finally, the sim ( n − -module gr( F ) can be expressed by means of a sim ( n − -invariant graph F ( ( ❘❘❘❘❘❘ F / / F ( ( ◗◗◗◗◗◗ ♠♠♠♠♠♠ F − / / F − F ❧❧❧❧❧❧ where an arrow from F ji to F ki − for some i, j, k implies that ˘ F ji ⊂ g · ˘ F ki − for any choice of irreducible co ( n − -modules ˘ F ji and ˘ F ki − linearly isomorphic to F ji and F ki − respectively. The filtration (2.1) on V induces a filtration { } =: A ⊂ A ⊂ A ⊂ A ⊂ A − ⊂ A − := A , (3.5) of sim ( n − -modules on the space A defined by (3.2) Further, when n = 6 , the associated graded p -module gr( A ) = L i = − gr i ( A ) where gr i ( A ) := A i / A i +1 splits into a direct sum gr ± ( A ) = A ± , gr ± ( A ) = A ± ⊕ A ± ⊕ A ± , gr ± ( A ) = A ⊕ A ⊕ A . of irreducible sim ( n − -modules, where im -mod co -mod Dimension ( n = 2 m > ) Dimension ( n = 2 m + 1 ) A ± V ± ⊚ g ± m − m − A ± V ± ⊚ z A ± V ± ⊚ so ( m − m −
3) ( m − m − A ± V ⊚ g ± m (2 m −
3) ( m − m + 1) A V ⊚ z m − m − A V ∓ ⊚ g ± m − m − A V ⊚ so m ( m − m − (2 m − m − m + 1) with the proviso that when n ≤ , A ± do not occur.Finally, when n = 6 , the sim ( n − -module gr( A ) can be expressed by means of a sim ( n − -invariantgraph A ⊕ A / / ! ! ❉❉❉❉❉❉❉❉❉❉❉❉❉ A / / A − ⊕ A − ( ( ◗◗◗◗◗◗◗ A ♣♣♣♣♣♣♣ ' ' ◆◆◆◆◆◆◆◆◆ A − A / / A ⊕ A / / ; ; ①①①①①①①①①①①①① A − ♠♠♠♠♠♠♠♠♠ where an arrow from A ji to A ki − for some i, j, k implies that ˘ A ji ⊂ g · ˘ A ki − for any choice of irreducible co ( n − -modules ˘ A ji and ˘ A ki − isomorphic to A ji and A ki − respectively. Proposition 3.3
Assume n = 6 . The filtration (2.1) on V induces a filtration (3.5) of sim (4) -modules onthe space A defined by (3.2) .Further, the associated graded p -module gr( A ) = L i = − gr i ( A ) where gr i ( A ) := A i / A i +1 splits into adirect sum gr ± ( A ) = A ± , gr ± ( A ) = A ± ⊕ A , + ± ⊕ A , −± ⊕ A ± , gr ± ( A ) = A ⊕ A ⊕ A , +0 ⊕ A , − . of irreducible sim (4) -modules, where sim -mod co -mod Dimension A ± V ± ⊚ g ± A ± V ± ⊚ z A , + ± V ± ⊚ so +0 A , −± V ± ⊚ so − A ± V ⊚ g ± sim -mod co -mod Dimension A V ⊚ z A V ∓ ⊚ g ± A , +0 V ⊚ so − A , − V ⊚ so +0 inally, the sim (4) -module gr( A ) can be expressed by means of a sim (4) -invariant graph A A A , − A , +1 A A , − A , +0 A ⊕ A A − A , −− A , + − A − A − ttttttttttttttt : : ❞❞❞❞❞❞❞❞❞❞❞❞❞ ❩❩❩❩❩❩❩❩❩❩❩❩❩ , , ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ $ $ ❩❩❩❩❩❩❩❩❩❩❩ , , ❥❥❥❥❥❥❥❥❥❥❥ ❚❚❚❚❚❚❚❚❚❚❚❚❚ * * ❚❚❚❚❚❚❚❚❚❚❚❚❚ * * ttttttttttttttt : : ☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ A A ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ ❞❞❞❞❞❞❞❞❞❞❞❞❞ ❞❞❞❞❞❞❞❞❞❞❞❞ ❚❚❚❚❚❚❚❚❚❚❚ * * ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ $ $ ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ (cid:29) (cid:29) ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ' ' ❥❥❥❥❥❥❥❥❥❥❥❥❥ ❩❩❩❩❩❩❩❩❩❩❩❩❩ , , ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ $ $ ❩❩❩❩❩❩❩❩❩❩❩❩❩ , , ❥❥❥❥❥❥❥❥❥❥❥❥❥ ❞❞❞❞❞❞❞❞❞❞❞❞❞ ttttttttttttttttt : : where an arrow from A ji to A ki − for some i, j, k implies that ˘ A ji ⊂ g · ˘ A ki − for any choice of irreducible co ( n − -modules ˘ A ji and ˘ A ki − isomorphic to A ji and A ki − respectively. The filtration (2.1) on V induces a filtration { } =: C ⊂ C ⊂ C ⊂ C ⊂ C − ⊂ C − := C . (3.6) of sim ( n − -modules on the space C defined by (3.3) .Further, when n = 6 , the associated graded sim ( n − -module gr( C ) = L i = − gr i ( C ) where gr i ( C ) := C i / C i +1 splits into a direct sum gr ± ( C ) = C ± , gr ± ( C ) = C ± ⊕ C ± , gr ± ( C ) = C ⊕ C ⊕ C ⊕ C . of irreducible sim ( n − -modules, where sim -mod co -mod Dimension ( n = 2 m = 6 ) Dimension ( n = 2 m + 1 ) C ± g ± ⊚ g ± m (2 m −
3) ( m − m + 1) C ± g ± ⊚ z m − m − C ± g ± ⊚ so m ( m − m − (2 m − m + 1)(2 m − C z ⊚ z C z ⊚ so ( m − m −
3) ( m − m − C g ⊚ g − m (2 m −
3) ( m − m + 1) C so ⊚ so m ( m − m − m − m ( m − m − m + 1) with the proviso that when n = 4 , C does not occur.Finally, when n = 6 , the sim ( n − -module gr( A ) can be expressed by means of a sim ( n − -invariant raph C (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ C @ @ ✁✁✁✁✁✁✁✁ / / (cid:27) (cid:27) ✻✻✻✻✻✻✻✻✻✻ C / / (cid:28) (cid:28) ✽✽✽✽✽✽✽✽✽✽ C − & & ▲▲▲▲▲▲ C : : ✉✉✉✉✉✉ $ $ ■■■■■■ C − C (cid:30) (cid:30) ❂❂❂❂❂❂❂❂ / / C C ✟✟✟✟✟✟✟✟✟✟ C B B ✝✝✝✝✝✝✝✝✝✝ / / C − rrrrrr C ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ where an arrow from C ji to C ki − for some i, j, k implies that ˘ C ji ⊂ g · ˘ C ki − for any choice of irreducible co ( n − -modules ˘ C ji and ˘ C ki − isomorphic to C ji and C ki − respectively. Proposition 3.5
Assume n = 6 . The filtration (2.1) on V induces a filtration (3.6) of sim (4) -modules onthe space C defined by (3.3) .Further, the associated graded sim (4) -module gr( C ) = L i = − gr i ( C ) where gr i ( C ) := C i / C i +1 splits into adirect sum gr ± ( C ) = C ± , gr ± ( C ) = C ± ⊕ C , + ± ⊕ C , −± , gr ± ( C ) = C ⊕ C , +0 ⊕ C , − ⊕ C ⊕ C , +0 ⊕ C , − . of irreducible sim (4) -modules, where sim -mod g -mod Dimension C ± g ± ⊚ g ± C ± g ± ⊚ z C , + ± g ± ⊚ so +0 C , −± g ± ⊚ so − sim -mod g -mod Dimension C z ⊚ z C , ± z ⊚ so ± C g ⊚ g − C , ± so ± ⊚ so ± Finally, the sim (4) -module gr( A ) can be expressed by means of a sim (4) -invariant graph C C , +1 C , − C C , +0 C , − C C , +0 C , − C C , + − C , −− C − C − ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ ❚❚❚❚❚❚❚❚❚❚❚❚❚ * * ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ $ $ ❩❩❩❩❩❩❩❩❩❩❩❩❩ , , ❞❞❞❞❞❞❞❞❞❞❞❞❞ tttttttttttttttt : : ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ " " ❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ' ' ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ? ? / / ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ? ? ❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ' ' ❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ $ $ ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ (cid:31) (cid:31) ③③③③③③③③③③③③③③③③③③ < < / / ❚❚❚❚❚❚❚❚❚❚❚❚❚❚ * * ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ ❩❩❩❩❩❩❩❩❩❩❩❩❩❩ , , ❞❞❞❞❞❞❞❞❞❞❞❞❞❞ ③③③③③③③③③③③③③③③③③③ < < ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ' ' ❥❥❥❥❥❥❥❥❥❥❥❥❥❥ tttttttttttttttt : : here an arrow from C ji to C ki − for some i, j, k implies that ˘ C ji ⊂ g · ˘ C ki − for any choice of irreducible co ( n − -modules ˘ C ji and ˘ C ki − linearly isomorphic to C ji and C ki − respectively. Let D denote any of F , A or C (or any irreducible so ( n − , { D i } the sim ( n − K , and { gr i ( D ) } its associated graded module withirreducibles D ji . Given a choice of splitting, irreducible co ( n − D ji as before.In explicit computations, it is often easier to characterise elements of D that do not lie in a particular sim ( n − D Π i : D → D / D i +1 , D ˘Π ji : D → ˘ D ji . (3.7)Clearly, the kernel of D Π i is the sim ( n − D i +1 . However, the kernel of D ˘Π ji is not sim ( n − D i and ˘ D ji since it consists of D i +1 , D i − and ˘ D ki for all k = j . Since we are essentially concerned with sim ( n − D ∈ D , we shall write D Π ji ( D ) = 0 , ⇐⇒ D ˘Π ji ( D ) = 0 , for any splitting (2.2). (3.8) Remark 3.6
One way of verifying the RHS of (3.8) is to trace down the arrows in the sim ( n − D ji , in effect computing the action of g − on a given ˘ D ji , therebyproduces a subgraph. If, for a homogeneity k, i , say, the co ( n − { ˘ D ℓk } lying in this subgraph areof distinct dimensions, then D ˘Π ℓk ( D ) = 0. However, for isotypic co ( n − { ˘ D ℓk } (i.e. of the samedimensions), one cannot in general expect D ˘Π ℓk ( D ) = 0, but algebraic conditions among { D ˘Π ℓk ( D ) } . Remark 3.7
The maps D Π ji occurring in (3.8) are expressed explicitly in appendix B.1, where they areinterpreted as generalisations of the Bel-Debever criteria of [Ort09]. To make contact with reference [CMPP04], we recast a number of their definitions in our language: • Let K be a null line in ( V , g ), i.e. an element of Gr ( V , g ) so that K induces a so ( n − , K -invariantfiltration (3.6) on C where so ( n − , K ∼ = sim ( n −
2) is the stabiliser of K . Then K is a Weyl alignednull direction (WAND) of a Weyl tensor C abcd if C abcd ∈ C − . • A Weyl tensor C abcd ∈ C is said to be (at least) of Petrov type I, II, III, or N, respectively, if thereexists a WAND K ∈ Gr ( V , g ) with induced sim ( n − C , such that C abcd ∈ C − , C , C , or C , respectively.In the null alignment formalism, one generally insists on choosing a null line with respect to which theWeyl tensor degenerates ‘most’. Thus, once such a null line has been fixed, the null alignment classificationcoincides with the present sim ( n − We refer the reader tothe literature for details. We summarise the comparison between the terminology and notation of [CMPP04],and ours in the following table: This subgraph is however not sim ( n − g − ⊕ g . Our approach emphasises the sim ( n − i , D and III i defined in [CMPP04]. Such subtypes break the sim ( n − C ), in which case the sim ( n − co ( n − n −
2) condition Petrov subtypes Sim( n −
2) conditionG – – –I C Π − ( C ) = 0 I(a) C Π − ( C ) = 0I(b) C Π − ( C ) = 0II C Π − ( C ) = 0 II(a) C Π ( C ) = 0II(d) C Π ( C ) = 0II(b) C Π ( C ) = 0II(c) C Π ( C ) = 0III C Π ( C ) = 0 III(a) C Π ( C ) = 0III(b) C Π ( C ) = 0N C Π ( C ) = 0 – –O C Π ( C ) = 0 – –Here the maps C Π ji are defined in section 3.1.4. We must emphasise that the above subtypes are only thebuildling blocks of other types given in [CMPP04,Ort09]. Thus, for instance, the Petrov subtype type II(abc)is characerised by the Weyl tensor belonging to the module C and satisfying C Π ( C ) = C Π ( C ) = C Π ( C ) =0, and so on. We finally note that the sim ( n − C Π ( C ) = 0 (since it implies C Π i ( C ) = 0 for i = 0 , , C Π − ( C ) = 0) corresponds to the Petrov type denoted II’(abd) in [OPP13a].All together, when n >
5, we count 28 distinct Sim( n − C . In terms of the Petrov types,these define • each of types G, N and O; • •
17 of type II; • n = 5, we count 19 distinct Sim( n − C , 8 of which being subtypes of type II. sim ( m − , C ) -invariant classifications We now introduce a Robinson structure N on ( V , g ) with associated null line K , i.e. N ∩ ¯ N = C K . Assume n = 2 m + ǫ with ǫ ∈ { , } . The null line K gives rise to the sim ( n − F , A and C of theprevious section, which we shall henceforth assume. There is a further splitting of the associated graded sim ( n − F ), gr( A ) and gr( C ) into irreducible sim ( m − , C )-modules, where sim ( m − , C )is the stabiliser of N . In effect, this splitting is the result of the branching rule arising from the inclusion u ( m − ⊂ so ( m −
1) of the semi-simple part g of sim ( n − sim ( m − , C )-invariant graphs, reflecting the nilpotent action of sim ( m − , C ). Unlike in the sim ( n −
2) case, one doesnot need to distinguish between the cases n = 6 and n = 6 – see appendix C. Notation 3.8
The even-dimensional and odd-dimensional cases differ only by the existence of additionalirreducible sim ( m − , C )-modules in the latter. The distinction between these two cases will be made bythe appropriate value of ǫ and the use of gray fonts and dotted arrows in the sim ( m − , C )-invariant graphs.As in the sim ( n −
2) case, the irreducible cu ( n − F , A and C can all be expressed as the Cartan product of the cu ( n − V and g , which we recall below: sim C -mod cu -mod Dimension V ± V ± sim C -mod cu -mod Dimension V , [[ V (1 , ]] 2 m − V , [ V (1 , ] 116 im C -mod cu -mod Dimension g , ± [[ g (1 , ± ]] 2 m − g , ± [ g (0 , ± ] 1 sim C -mod cu -mod Dimension g z g , [ g ω ] 1 g , [[ g (2 , ]] ( m − m − g , [ g (1 , ◦ ] m ( m − g , [[ g (1 , ]] 2 m − The associated graded sim ( n − -module gr( F ) of the filtration (3.4) on the space F definedby (3.1) splits into a direct sum F ± = F , ± , F ± = F , ± ⊕ ǫ F , ± , F = F , , F = (cid:16) F , ⊕ F , (cid:17) ⊕ ǫ (cid:16) F , ⊕ F , (cid:17) . of irreducible sim ( m − , C ) -modules, where sim C -mod cu -mod Dimension F , ± V ± ⊚ V ± F , ± V ± ⊚ [[ V (1 , ]] 2 m − F , ± V ± ⊚ [[ V (0 , ]] 1 sim C -mod cu -mod Dimension F V − ⊚ V F , [ V (1 , ⊚ V (0 , ] m ( m − F , [[ V (1 , ⊚ V (1 , ]] m ( m − F , [ V (0 , ⊚ V (0 , ] 1 F , [[ V (0 , ⊚ V (1 , ]] 2 m − Further, the sim ( m − , C ) -module gr( F ) can be expressed in terms of the sim ( m − , C ) -invariant graph F , (cid:25) (cid:25) ✹✹✹✹✹✹✹✹✹✹✹✹✹✹ F , % % ▲▲▲▲▲▲ F , (cid:21) (cid:21) ✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯✯ % % ❑❑❑❑❑❑ ssssss E E ✡✡✡✡✡✡✡✡✡✡✡✡✡✡ F , − % % ❑❑❑❑❑❑ F , ssssss % % F , ssssss % % F , − F , (cid:25) (cid:25) % % F , − F , F , E E I I ✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔ where an arrow from F j,ki to F p,qi − for some i, j, k, p, q implies that ˘ F j,ki ⊂ g · ˘ F p,qi − for any choice of irreducible cu ( m − -modules ˘ F j,ki and ˘ F p,qi − linearly isomorphic to F j,ki and F p,qi − respectively. .2.2 The Cotton-York tensorProposition 3.10 The associated graded sim ( n − -module gr( A ) of the filtration (3.5) on the space A defined by (3.2) splits into a direct sum A ± = A , ± ⊕ ǫ A , ± , , A ± = A , ± , A ± = (cid:16) A , ± ⊕ A , ± ⊕ A , ± (cid:17) ⊕ ǫ A , ± , A ± = (cid:16) A , ± ⊕ A , ± (cid:17) ⊕ ǫ (cid:16) A , ± ⊕ A , ± (cid:17) , A = A , ⊕ ǫ A , , A = A , ⊕ ǫ A , , A = (cid:16) A , ⊕ A , ⊕ A , ⊕ A , (cid:17) ⊕ ǫ (cid:16) A , ⊕ A , ⊕ A , ⊕ A , ⊕ A , ⊕ A , (cid:17) . of irreducible sim ( m − , C ) -modules, where sim C - cu -mod Dimension A , ± [[ V ± ⊚ g (1 , ± ]] 2 m − A , ± [[ V ± ⊚ g (0 , ± ]] 1 A ± V ± ⊚ z A , ± [[ V ± ⊚ g (2 , ]] ( m − m − A , ± [ V ± ⊚ g (1 , ◦ ] m ( m − A , ± [ V ± ⊚ g ω ] 1 A , ± [[ V ± ⊚ g (1 , ]] 2 m − A , ± [[ V (1 , ⊚ g (1 , ± ]] m ( m − A , ± [[ V (1 , ⊚ g (0 , ± ]] m ( m − A , ± [[ V (0 , ⊚ g (0 , ± ]] 1 A , ± [[ V (0 , ⊚ g (1 , ± ]] 2 m − sim C - cu -mod Dimension A , [[ V (1 , ]] ⊚ z m − A , V (0 , ⊚ z A , V ∓ ⊚ [[ g (1 , ± ]] 2 m − A , V ∓ ⊚ g (0 , ± A , [[ V (1 , ⊚ g ω ]] 2 m − A , [[ V (1 , ⊚ g (2 , ]] m ( m − m − A , [[ V (0 , ⊚ g (2 , ]] m ( m − m − A , [[ V (1 , ⊚ g (1 , ◦ ]] ( m + 1)( m − m − A , [[ V (0 , ⊚ g ω ]] 1 A , [[ V (0 , ⊚ g (1 , ]] 2 m − A , [[ V (0 , ⊚ g (2 , ]] ( m − m − A , [[ V (0 , ⊚ g (1 , ◦ ]] m ( m − A , [[ V (0 , ⊚ g (1 , ]] m ( m − A , [[ V (1 , ⊚ g (1 , ]] m ( m − Further, the sim ( m − , C ) -module gr( A ) can be expressed in terms of the sim ( m − , C ) -invariant graph1 where an arrow from A j,ki to A p,qi − for some i, j, k, p, q implies that ˘ A j,ki ⊂ g · ˘ A p,qi − for any choice ofirreducible cu ( m − -modules ˘ A j,ki and ˘ A p,qi − linearly isomorphic to A j,ki and A p,qi − respectively. Arrow heads have been omitted for clarity. , A , A , A , A , A , A , A , A , A , A , A , A , A , A , A , A , A , A , A , A , A , ⊕ A , A , ⊕ A , A , − A , − A , − A , − A , − A , − A , − A , − A , − A , − A , − ✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎ ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ttttttttttttttt ☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞ ❚❚❚❚❚❚❚❚❚❚❚❚❚ ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏qqqqqqqqqqqqqqqqqqqqqqqqqqq ✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇ ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ ✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾ ✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹ ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄ ✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷ ✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞ ✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎ ✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁ ❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂ ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀ ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴ ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ ❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂ ✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾ ✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆ ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ qqqqqqqqqqqqqqqqqqqqqqqqqqqq ☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞ ✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁ ③③③③③③③③③③③③③③③③③③③③③③③③③③③③③③③③✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴ ✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷ ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ ❥❥❥❥❥❥❥❥❥❥❥❥❥ ttttttttttttttt Table 1: sim ( m − , C )-invariant graph for A .2.3 The Weyl tensorProposition 3.11 The associated graded sim ( n − -module gr( C ) of the filtration (3.6) on the space C defined by (3.3) splits into a direct sum C ± = (cid:16) C , ± ⊕ C , ± (cid:17) ⊕ ǫ (cid:16) C , ± ⊕ C , ± (cid:17) , C ± = C , ± ⊕ ǫ C , ± , C ± = (cid:16) C , ± ⊕ C , ± ⊕ C , ± ⊕ C , ± (cid:17) ⊕ ǫ (cid:16) C , ± ⊕ C , ± ⊕ C , ± ⊕ C , ± ⊕ C , ± ⊕ C , ± (cid:17) , C = C , C = (cid:16) C , ⊕ C , ⊕ C , (cid:17) ⊕ ǫ C , , C = (cid:16) C , ⊕ C , (cid:17) ⊕ ǫ (cid:16) C , ⊕ C , (cid:17) , C = (cid:16) C , ⊕ C , ⊕ C , ⊕ C , ⊕ C , ⊕ C , ⊕ C , ⊕ C , (cid:17) ⊕ ǫ (cid:16) C , ⊕ C , ⊕ C , ⊕ C , ⊕ C , (cid:17) . of irreducible sim ( m − , C ) -modules, where sim C - cu -mod Dimension C , ± [ g (1 , ± ⊚ g (0 , ± ] m ( m − C , ± [[ g (1 , ± ⊚ g (1 , ± ]] m ( m − C , ± [ g (0 , ± ⊚ g (0 , ± ] 1 C , ± [[ g (1 , ± ⊚ g (0 , ± ]] 2 m − C , ± [[ g (1 , ± ]] ⊚ z m − C , ± [ g (0 , ± ] ⊚ z C , ± [[ g (1 , ± ⊚ g ω ]] 2 m − C , ± [[ g (1 , ± ⊚ g (2 , ]] m ( m − m − C , ± [[ g (1 , ± ⊚ g (0 , ]] m ( m − m − C , ± [[ g (1 , ± ⊚ g (1 , ◦ ]] ( m +1)( m − m − C , ± [ g (0 , ± ⊚ g ω ] 1 C , ± [[ g (0 , ± ⊚ g (1 , ]] 2 m − C , ± [[ g (0 , ± ⊚ g (2 , ]] ( m − m − C , ± [[ g (0 , ± ⊚ g (1 , ◦ ]] m ( m − C , ± [[ g (1 , ± ⊚ g (0 , ]] m ( m − C , ± [[ g (1 , ± ⊚ g (1 , ]] m ( m − sim C - cu -mod Dimension C , z ⊚ z C , z ⊚ [ g ω ] 1 C , z ⊚ [[ g (2 , ]] ( m − m − C , z ⊚ [ g (1 , ◦ ] m ( m − C , z ⊚ [[ g (1 , ]] 2 m − C , [ g (1 , ⊚ g (0 , − ] m ( m − C , [[ g (1 , ⊚ g (1 , − ]] m ( m − C , [ g (0 , ⊚ g (0 , − ] 1 C , [[ g (1 , ± ⊚ g (0 , ∓ ]] 2 m − C , [ g ω ⊚ g ω ] 1 C , [[ g (2 , ⊚ g ω ]] ( m − m − C , [ g (1 , ◦ ⊚ g ω ] m ( m − C , [[ g (2 , ⊚ g (2 , ]] m ( m − ( m − C , [ g (2 , ⊚ g (0 , ] m ( m − ( m − C , [ g (1 , ◦ ⊚ g (1 , ◦ ] ( m +2)( m − ( m − C , [[ g (2 , ⊚ g (1 , ◦ ]] ( m +1)( m − ( m − C , [[ g (1 , ⊚ g ω ]] 2 m − C , [ g (1 , ⊚ g (0 , ] m ( m − C , [[ g (1 , ⊚ g (1 , ]] m ( m − C , [[ g (2 , ⊚ g (1 , ]] m ( m − m − C , [[ g (2 , ⊚ g (0 , ]] m ( m − m − C , [[ g (1 , ⊚ g (1 , ◦ ]] ( m +1)( m − m − with the proviso that when n ≤ , C , ± do not occur, and when n ≤ , C , does not occur. urther, the sim ( m − , C ) -module gr( C ) can be expressed in terms of the sim ( m − , C ) -invariant graphs2 & 3 where an arrow from C j,ki to C p,qi − for some i, j, k, p, q implies that ˘ C j,ki ⊂ g · ˘ C p,qi − for any choice ofirreducible cu ( m − -modules ˘ C j,ki and ˘ C p,qi − linearly isomorphic to C j,ki and C p,qi − respectively. Continuing on from section 3.1.4 with D = F , A or C , we denote by D j,ki the irreducible sim ( m − , C )-modulesof gr( D ), and by ˘ D j,ki the corresponding cu ( m − D ˘Π j,ki : D → ˘ D j,ki ⊂ ˘ D ji , for each i, j, k . This time, the kernel of D ˘Π j,ki is not sim ( m − , C )-invariant, as it depends on a choice of asplitting ˘ D j,ki . Since we are essentially concerned with sim ( m − , C )-invariant conditions, for any D ∈ D ,we shall write D Π j,ki ( D ) = 0 , ⇐⇒ D ˘Π j,ki ( D ) = 0 , for any splitting (2.2). (3.9)To verify the RHS of (3.9), one can go through the same reasoning as Remark 3.6. Regarding Remark3.7, it must be said that explicit formulae for the maps D Π j,ki occurring in (3.8) are far more involved thanin the sim ( n −
2) case – see appendix B.1. Arrow heads have been omitted for clarity. , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫❫ ✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾ ✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻ ❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛ ✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ ❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛ ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅ ❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀ ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ ❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽ ✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸ ❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋ ❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ ✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆ ④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④ Table 2: sim ( m − , C )-invariant graph for C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , − C , − C , − C , − C , − C , − C , − C , − C , − C , − C , − C , − C , − C , − C , − C , − ✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝ ①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①① ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅ ✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾ ☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛ ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄ ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ ❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ ❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵ ❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪ ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP ❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪ ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP ✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆ ✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈ ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ Table 3: sim ( m − , C )-invariant graph for C (continued)23 Geometric applications
We shall presently apply the algebraic machinery of sections 2 and 3 to Lorentzian geometry. Throughoutthis section,( M , g ) will denote an oriented Lorentzian manifold of dimension n . The Levi-Civita will bedenoted by ∇ a with Riemann curvature tensor given by R cabd V d := 2 ∇ [ a ∇ b ] V c , for any vector field V a . The Riemann tensor decomposes into SO( n − , n >
3, thecase we shall mostly be concerned with, this decomposition is given by R abcd = C abcd + 4 n − [ c | [ a g b ] | d ] + 2 n ( n − Rg a [ c g d ] b , (4.1)where C abcd is the Weyl tensor, Φ ab the tracefree part of the Ricci tensor R ab := R cacb , and R := R aa theRicci scalar. There is no Weyl tensor when n ≤ M of interest for us will be the bundles of irreducible curvature tensors. From arepresentational point of view, these can be defined D := FM ×
SO( n − , D , where FM is the frame bundle,and D is an SO( n − , D = F , A or C as defined by (3.1), (3.2)and (3.3) respectively.We first briefly review spacetimes endowed with a distinguished null line distribution. Many geometricproperties of such manifolds have been investigated, especially within the framework of the null alignmentformalism, see [OPP13a] for a recent survey. We then examine the bundle generalisation of Robinsonstructures. Let
K ⊂ T M be a null line distribution, so that the structure group of the frame bundle FM is reduced toSim( n − n − sim ( n − D i = FM ×
Sim( n − D i , etc... in the obviousway (here D = F , A or C ). We shall therefore draw from the notation of the previous sections and theappendices. In particular, we shall characterise the curvature tensors as elements of sim ( n − D Π ji defined by (3.7) and (3.8) of section 3.1.4.Further, since many of the examples we shall consider are equipped with more than one distinguishednull line distributions, it will be convenient to consider the bundle Gr (T M , g ) of all unoriented null linedistributions on ( M , g ). This is the S n − -bundle whose fiber over a point p of M is the null GrassmannianGr (T p M , g ). We shall refer to a null line distribution K specifically in the maps D Π ji by writing D K Π ji .A natural arena for the study of the geometric properties of K and its orthogonal complement K ⊥ isprovided by the intrinsic torsion of the Sim( n − K . In broad terms, the intrinsictorsion associated to K splits into classes, which can be identified with sim ( n − ∇ a k b (mod α a k b ). In general relativity, the resulting classification gives rise to the well-known character-isation of the congruence generated by K in terms of geometric optics: geodesy, shear, twist, dilation andparallelism. The latter is the strongest Sim( n − K that one can impose.Equivalently, any generator k a of K is recurrent with respect to ∇ a , i.e. (cid:0) ∇ a k [ b (cid:1) k c ] = 0. The holonomy of ∇ a is then contained in a subgroup of Sim( n − Remark 4.1
In the following statements and the rest of the paper, special features of low dimensions canbe obtained by excluding those sim ( n − roposition 4.2 Suppose that ( M , g ) admits a parallel null line distribution K , and let k a be any sectionof K so that ( ∇ a k [ b ) k c ] = 0 . Then R eab [ c k d ] k e = 0 , C K Π ( C ) = 0 , F K Π − (Φ) = 0 , i.e. Φ e [ a k b ] k e = 0 . Further, F K Π (Φ) = 0 ⇐⇒ C K Π ( C ) = 0 , and if any two of the following conditions C K Π ( C ) = 0 , F K Π (Φ) = 0 , R = 0 , hold, the remaining one holds too.Proof. [Sketch] It is a matter of differentiating the recurrent vector field twice and commuting the covariantderivatives. Using the decomposition (4.1) then leads to C K Π ( C ) = 0, and in addition, C K Π ( C ) ab = n − n − k ( a Φ b ) c k c + n − n ( n − Rk a k b , C K Π ( C ) = − n − F K Π (Φ) , which completes the proof. (cid:3) More restrictive yet are the Lorentzian manifolds, known as pp-waves , that admit a parallel null vectorfield . Note that this is no longer a Sim( n − Proposition 4.3
Let k a be a parallel null vector field on ( M , g ) , i.e. ∇ a k b = 0 . Then R dabc k d = 0 , Φ ab k b = − n Rk a C K Π ( C ) = 0 . Further, F K Π (Φ) = 0 ⇐⇒ R = 0 ⇐⇒ C K Π ( C ) = 0 , F K Π (Φ) = 0 ⇐⇒ C K Π ( C ) = 0 , F K Π (Φ) = 0 ⇐⇒ C K Π ( C ) = 0 . Proof. [Sketch] Clearly, a parallel null vector field is a special type of recurrent null vector field. We canrecycle the proof of Proposition 4.2, and we find in addition C K Π ( C ) ab = 1( n − n − Rk a k b , C K Π ( C ) abc = 2 n − F K Π (Φ) abc − n ( n − n − Rk [ a g b ] c , and the result follows. (cid:3) We shall presently translate the algebraic setting of section 2.2 into the language of bundles. As we shalloccasionally consider pseudo-Riemannian manifolds of arbitrary signature, we introduce the following defi-nition.
Definition 4.4
Let ( M , g ) be a pseudo-Riemannian manifold of dimension 2 m + ǫ where ǫ ∈ { . } . An almost null structure is a totally null complex m -plane distribution on M .25enceforth, unless otherwise stated, ( M , g ) will denote an n -dimensional smooth Lorentzian manifold,with n = 2 m + ǫ , ǫ ∈ { , } . Definition 4.5 An almost Robinson structure on ( M , g ) is an almost null structure of real index 1. We saythat ( M , g ) is an almost Robinson manifold if it is equipped with an almost Robinson structure.In other words, an almost Robinson structure is a smooth assignment of a Robinson structure N p on thetangent space T p M at every point p of M . It defines a real null line distribution K on M , i.e. C K p := N p ∩ ¯ N p for all p in M – here, C K p denotes the complexification of K p . Each fiber of the screenbundle K ⊥ / K isequipped with a Hermitian structure. An almost Robinson manifold is characterised by a reduction of thestructure group of the frame bundle to Sim( m − , C ) stabilising N , the subgroup of the stabiliser Sim( n − K . We can then apply the sim ( m − , C )-invariant decompositions of the various tensor representationsof so ( n − ,
1) given in sections 2 and 3 to the curved setting, and in particular, in Propositions 3.9, 3.10and 3.11. Again, to characterise curvature tensors as elements of some irreducible sim ( m − , C )-invariantsubspace, we shall refer to sections 3.1.4 and 3.2.4 and make use of the maps D K Π i , D K Π ji and D N Π j,ki definedby (3.7), (3.8) and (3.9) respectively. Here, we have specified which almost Robinson structure N and itsassociated null line distribution K the maps refer to.To make the description somewhat more tractable, it is more convenient to choose a splitting of C T M adapted to N , and work with representations of the reductive part cu ( m −
1) of sim ( m − , C ). To this end,we introduce a null line distribution L dual to K , so that the almost Robinson structure gives rise to analmost Hermitian structure J ba on the distribution orthogonal to both K and L , and C T M splits as C T M = C K ⊕ T (1 , ⊕ T (0 , ⊕ ǫ T (0 , ⊕ C L , N = C K ⊕ T (1 , , N ⊥ = C K ⊕ T (1 , ⊕ ǫ T (0 , , (4.2)where T (1 , and T (0 , are the i- and − i-eigenbundle of J ba respectively, and when n is odd, T (0 , is itsone-dimensional kernel. This splitting is the curved version of (4.2), and evidently carries through to thebundles of curvature tensors. Remark 4.6
Unless otherwise specified, we shall not in general distinguish the even- and odd-dimensionalcases. When applying any of the following statements to even dimensions, the reader should exclude those sim ( m − , C )-modules occuring in odd dimensions only. The same applies to low dimensions, where a numberof sim ( m − , C )-modules do not occur. See section 3 and appendix C for details. The bundle of almost Robinson structures
For convenience, we can define the bundle Gr m ( C T M , g )of all almost Robinson structures on ( M , g ): the fiber over a point p of M is the null GrassmannianGr m ( C T p M , g ), which has dimension m ( m −
1) + ǫ (2 m − n = 2 m , the bundle Gr m ( C T M , g )consists of the disjoint union of the bundle Gr + , m ( C T M , g ) of all almost self-dual Robinson structures, andthe bundle Gr − , m ( C T M , g ) of all almost anti-self-dual Robinson structures. As usual, when m is even,sections of Gr + , m ( C T M , g ) can be identified with those of Gr − , m ( C T M , g ) by complex conjugation. Robinson structures
As in the Sim( n −
2) case above, almost Hermitian geometry [GH80], and complexRiemannian geometry [TC12b,TC13], the natural framework to study the differential properties of an almostRobinson structure is by classifying its intrinsic torsion. This is beyond the scope of the present article,and will be treated elsewhere. In this article, we shall nonetheless be concerned with integrable Robinsonstructures, which feature in a number of solutions of Einstein’s field equations.
Definition 4.7
An almost null structure N on a pseudo-Riemannian manifold ( M , g ) is said to be • integrable if [Γ( N ) , Γ( N )] ⊂ Γ( N ), • totally geodetic if for all X a , Y b ∈ Γ( N ), X a ∇ a Y b ∈ Γ( N ), • co-integrable if [Γ( N ⊥ ) , Γ( N ⊥ )] ⊂ Γ( N ⊥ ), 26 totally co-geodetic if for all X a , Y b ∈ Γ( N ⊥ ), X a ∇ a Y b ∈ Γ( N ⊥ ).When ( M , g ) is Lorentzian, the same definitions apply to almost Robinson structures. A (locally) integrablealmost Robinson structure is called a (local) Robinson structure , and a Lorentzian manifold equipped witha Robinson structure is called a Robinson manifold . Remark 4.8
The definition of integrable almost Robinson structure was first introduced in [NT02] in evendimensions, and then extended to the odd-dimensional case in [TC11], where it is also referred to as an optical structure . The above definitions differ from the one given in [TC11], which did not reflect thesubtleties involved in odd dimensions. In particular, what is referred to as an integrable almost Robinsonstructure or integrable almost optical structure in [TC11] is an almost Robinson structure that is bothintegrable and co-integrable according to Definition 4.7.
Remark 4.9
It is shown in [TC13] that, for any almost null structure N , and thus for any almost Robinsonstructure, • if N is totally geodetic, then it is integrable; • if N is both integrable and co-integrable, it is also totally geodetic; • if N is totally co-geodetic, then it is both integrable and co-integrable.In even dimensions, since N = N ⊥ , all definitions reduce to N being integrable.A shearfree congruence of null geodesics is equivalent to a Robinson structure in four dimensions, but thisis not so in higher dimensions [Tra02b]. In higher dimensions, we still wish to retain the geodesy property ofthe congruence of null curves generated by K associated to a Robinson structure N . By the remark above,in even dimensions, this will be automatically satisfied by virtue of the integrability of N . However, thisproperty must be added in odd dimensions, and quite naturally follows from a totally geodetic Robinsonstructure. In fact, as we shall see in section 4.2.2, it appears that co-integrable Robinson structures play abigger rˆole in higher odd dimensions. Let us recall the properties enjoyed by a Robinson manifold.
Proposition 4.10
Let N be a Robinson structure on ( M , g ) with associated real null line distribution K ,i.e. N ∩ ¯ N = C K . Then K is tangent to a congruence of null curves whose leaf space is a CR manifold.Further, if N is totally geodetic, then this congruence is geodetic.Proof. [Sketch] For the almost Robinson structure N to descend to an almost CR-structure on the leaf space of K , one must require that [ k , V ] ∈ Γ( N ) for any k ∈ Γ( K ), and all V ∈ Γ( N ). This is clearly satisfied when N is integrable. The integrability of the almost CR-structure then follows from that of N . In even dimensions,this is proved in [NT02] where the geodesy property of K follows automatically from the integrability of N .In odd dimensions, this geodesy property must be an additional requirement. (cid:3) Since in dimensions two and three, an almost Robinson structure is equivalent to a real null line distri-bution, one obtains the following
Lemma 4.11
Suppose ( M , g ) is a two- or three-dimensional Lorentzian manifold. Then an integrable andco-integrable almost Robinson structure on ( M , g ) is equivalent to the existence a congruence of null geodesics.Further, there always exists a local Robinson structure, and when M is two-dimensional, there is a uniquealmost Robinson structure, which is always integrable. Using the results of [TC11, TC12a, TC12b, TC13] on almost null structures, the curvature condition forthe existence of a totally geodetic Robinson structure N is given by C abcd X a Y b Z c W d = 0 , for all X a , Y a , Z a ∈ Γ( N ), W a ∈ Γ( N ⊥ ). (4.3)This also applies to co-integrable almost Robinson structure [TC11, TC12a]. Taking the real span of (4.3)yields the following proposition. 27 roposition 4.12 Let N be a totally geodetic or co-integrable Robinson structure on ( M , g ) . Then theWeyl tensor (locally) satisfies C N Π ,i − ( C ) = 0 , for i = 1 , C N Π ,i − ( C ) = 0 , for i = 1 , , C N Π , ( C ) = 0 , for i = 3 ,
10 (4.4c)
Proof.
We fix a splitting (4.2) adapted to N . Then with reference to appendix B.2, we have C abcd k a X b k c W d = 0 ⇔ C N Π , − ( C ) = 0 ,C abcd k a X b k c u d = 0 ⇔ C N Π , − ( C ) = 0 ,C abcd k a X b Y c Z d = 0 ⇔ C N Π , − ( C ) = 0 ,C abcd k a X b Y c u d = 0 ⇔ C N Π , − ( C ) = C N Π , − ( C ) = 0 ,C abcd X a Y b Z c W d = 0 ⇔ C N Π , ( C ) = 0 ,C abcd X a Y b Z c u d = 0 ⇔ C N Π , ( C ) = 0 , for all local sections k a of K , X a , Y a , Z a , W a of T (1 , and u a of T (0 , , hence the result. (cid:3) Definition 4.13
We shall say that an almost Robinson structure N on a Lorentzian manifold is aligned with the Weyl tensor if the Weyl tensor tensor satisfies (4.3), or equivalently, (4.4) with respect to N . Remark 4.14
We note that except for n = 4, the integrability conditions (4.4) for the existence of alocal Robinson structure does not imply the Petrov type I condition with respect to its associated null linedistribution K (or any section k a thereof): C K Π − ( C ) = 0 , i.e. k [ a C b ] ef [ c k d ] k e k f = 0 , (4.5)which is also the defining property for a null direction to be a Weyl aligned null direction (WAND) .As shown in [TC11], an example of a vacuum spacetime of Petrov type I that admits co-integrableRobinson structures, for each of which condition (4.4) is satisfied, is provided by the black ring solution of[ER02]. The following example is an explicit example of a Lorentzian manifold that does not admit anyWAND (i.e. it is of Petrov type G), but that nevertheless admits almost Robinson structures aligned withthe Weyl tensor.
Example 4.15 (The five-dimensional static Kaluza-Klein bubble)
The static Kaluza-Klein bubbleis the direct product of a Euclidean Schwarzschild black hole and a one-dimensional timelike manifold. Itsmetric has thus the form g = − d t + (cid:18) − Mr (cid:19) − d r + (cid:18) − Mr (cid:19) d z + r dΩ , where dΩ is the 2-sphere metric. Both the self-dual and anti-self-dual parts of the Weyl tensor of the four-dimensional Euclidean Schwarzschild metric are algebraically special. Thus, by the Goldberg-Sachs theorem,it locally admits two distinct Hermitian structures of opposite orientations.Define the 1-forms κ := − d t + (cid:18) − Mr (cid:19) − d r , λ := d t + (cid:18) − Mr (cid:19) − d r , ν := (cid:18) − Mr (cid:19) d z , and let µ be a (1 , S . Then, it is straightforward to check that thetwo sets of 1-forms { κ , µ , ν } and { λ , µ , ν } define two distinct almost Robinson structures that are bothintegrable and co-integrable. 28urther, since the almost Robinson structures are integrable and co-integrable, the Weyl tensor satisfiesthe condition (4.4) (or equivalently (4.3)). However, it is not algebraically ‘special’ in the sense of (4.9). Forotherwise, it would be of Petrov type II (or D) in the null alignment formalism of [CMPP04]. But it is shownin [GR09] that this spacetime is algebraically general, i.e. of Petrov type G, i.e. there is no null directionwith respect to which the Weyl tensor satisfies the weakest algebraic condition (4.5) in this scheme.In addition, the two sets of 1-forms { κ ′ , µ ′ , ν ′ } and { λ ′ , µ ′ , ν ′ } where κ ′ := − d t + (cid:18) − Mr (cid:19) d z , λ ′ := d t + (cid:18) − Mr (cid:19) d z , ν ′ := (cid:18) − Mr (cid:19) − d r , define two distinct almost Robinson structures. Since d κ ′ = M r (cid:0) − Mr (cid:1) − ν ′ ∧ ( κ ′ + λ ′ ), these are clearlynon-integrable. Computing the Weyl tensor shows that (4.3) is nonetheless satisfied with respect to any ofthese. The subbundle of almost Robinson structures incident on a null line distribution
Recall that( M , g ) is naturally equipped with the bundles Gr (T M , g ) and Gr m ( C T M , g ) of unoriented null line dis-tributions and almost Robinson structures respectively, the latter splitting into a self-dual componentGr + , m ( C T M , g ) and an anti-self-dual component Gr − , m ( C T M , g ) when n = 2 m .On an even-dimensional pseudo-Riemannian manifold of any signature, it is a standard result (see [Tra02a]and references therein), which follows directly from the integrability condition (4.3), that if at a point p , eachof the self-dual totally null m -planes is tangent to an integrable self-dual totally null m -plane distribution,then the Weyl tensor must vanish at p when m >
2, and its self-dual part when m = 2. Imposing specificreality conditions on these distributions leads to different interpretation. For instance, in Euclidean signature,a hyper-K¨ahler manifold admits a two-sphere of global self-dual Hermitian structures, and must thereforebe conformally half-flat in four dimensions. On the other hand, a four-dimensional Lorentzian manifold thatadmits a two-sphere of local Robinson structures must be locally conformally flat, and similarly in higherdimensions.Instead, one can pick out a set of co-integrable, or more simply totally geodetic, Robinson structures.A natural way to go about this is to fix a real null line distribution K , i.e. a section of Gr (T M , g ). Thenat every point p of M , the inverse image of the projection π : Gr m ( C T p M , g ) → Gr (T p M , g ) is simply π − ( K p ) = {N p ∈ Gr m ( C T p M , g ) : N p ∩ ¯ N p = C K p } , i.e. K p determines a family of Robinson structures inGr m ( C T p M , g ), precisely, those intersecting K p . By Proposition 2.10, this family is parametrised by pointsin the homogeneous space O( n − / U( m −
1) of dimension ( m − m + 2( ǫ − n = 2 m , this familysplits into a self-dual family and an anti-self-dual family parametrised by the points of SO( n − / U( m − π − ( K p ) over all points p forms the bundle of all almost Robinson structuresincident on K , and which we shall denote Gr m ( C T M , g ) K . In even dimensions, we shall have an additionaltwo subbundles, the bundle Gr + , m ( C T M , g ) K of all almost self-dual Robinson structures incident on K , andthe bundle Gr − , m ( C T M , g ) K of all almost anti-self-dual Robinson structures incident on K . Proposition 4.16
Let ( M , g ) be an n -dimensional Lorentzian manifold equipped with a null line distribution K with n = 2 m + ǫ , ǫ ∈ { , } . Then the Weyl tensor C abcd satisfies C K Π − ( C ) = 0 , when n ≥ , C K Π ( C ) = 0 , when n > ,at a point p of M if and only if every element of Gr m ( C T p M , g ) K is aligned with the Weyl tensor at p , i.e. C abcd satisfies (4.3) (or equivalently (4.4) ).When n = 2 m , the Weyl tensor C abcd satisfies C K Π ( C ) = 0 , when m > , C K Π , +0 ( C ) = 0 , when m = 3 ,at a point p of M if and only if if every element of Gr + , m ( C T p M , g ) K is aligned with the Weyl tensor at p i.e. C abcd satisfies (4.3) (or equivalently (4.4) ). roof. Let N be an almost Robinson structure incident on K such that the Weyl tensor satisfies (4.3). Wethen have C abcd k a Y b k c W d = 0 , (4.6) C abcd k a Y b Z c W d = 0 , (4.7) C abcd X a Y b Z c W d = 0 , (4.8)for all k a ∈ Γ( K ), X a , Y a ∈ Γ( N ⊥ ), Z a , W a ∈ Γ( N ), with X a , Y a , Z a , W a / ∈ Γ( K ). We now let vary N whilekeeping K fixed, we then see that (4.6) = ⇒ C K Π − ( C ) = 0, (4.7) = ⇒ C K Π − ( C ) = 0 and (4.8) = ⇒ C K Π ( C ) = 0– this last condition is trivial in dimension five.In the case, when n = 2 m with m > m = 3, however, conditions (4.7) and (4.8) splitinto self-dual and anti-self-dual parts, so that restriction to almost self-dual Robinson structures yields therequired condition C K Π , +0 ( C ) = 0 (which implies in particular C K Π , + − ( C ) = 0 and C K Π − ( C ) = 0). (cid:3) Now applying Proposition 4.12 proves
Corollary 4.17
Let ( M , g ) be an n -dimensional Lorentzian manifold equipped with a null line distribution K with n = 2 m + ǫ , ǫ ∈ { , } . Suppose that at every point p of a region U of M and for any N p ∈ Gr m ( C T p M , g ) K , there exists a totally geodetic integrable almost Robinson structure tangent to N p . Then atevery point of U , the Weyl tensor C abcd satisfies C K Π − ( C ) = 0 , when n ≥ , C K Π ( C ) = 0 , when n > .When n = 2 m , suppose that at every point p of a region U of M and any N p ∈ Gr + , m ( C T p M , g ) K , thereexists an integrable self-dual almost Robinson structure tangent to N p . Then the Weyl tensor C abcd satisfies C K Π ( C ) = 0 , when m > , C K Π , +0 ( C ) = 0 , when m = 3 ,at every point of U . According to one interpretation, the Goldberg-Sachs theorem in four dimensions [GS09] gives a relationbetween (local) Robinson structures and solutions to Einstein’s field equations, whose Weyl tensor is alge-braically special , i.e. of Petrov type II or more degenerate. Among several equivalent approaches, this notionessentially rests on the classification of the multiplicites of the roots of the ‘Weyl spinor’ [Wit59, Pen60].The various notions of algebraic special Weyl tensors, equivalent in four dimensions, are no longer so inhigher dimensions. In the case at hand, since we are interested in the geometric properties of an almostRobinson structure N , a ‘special’ algebraic degeneracy condition for the Weyl tensor should be defined withrespect to N . In [TC11, TC12a], a simple and natural generalisation of the Petrov type II condition fromfour to higher dimensions was put forward:
Definition 4.18
A Weyl tensor C abcd on a (2 m + ǫ )-dimensional pseudo-Riemannian manifold ( M , g ), with ǫ ∈ { , } , is said to be (locally) algebraically special with respect to an almost null structure N if it satisfies C abcd X a Y b Z c = 0 , for all X a ∈ Γ( N ⊥ ) and all Y a , Z a ∈ Γ( N ), (4.9)and we say that N is a repeated aligned almost null structure of the Weyl tensor .The same definitions apply to almost Robinson structures when ( M , g ) is Lorentzian.What (4.9) means for the present classification of the Weyl tensor is given by Whether this can be made independent of such a choice is a different matter. roposition 4.19 Let ( M , g ) be a Lorentzian manifold equipped with an almost Robinson structure N .Then condition (4.9) is equivalent to C K Π − ( C ) = 0 , (4.10a) C N Π ,i ( C ) = C N Π ,i ( C ) = 0 , for i = 1 , C N Π ,i ( C ) = 0 , for i = 1 , , , , , , ,
12 (4.10c) C N Π ,i ( C ) = 0 , for i = 1 , , . (4.10d) Proof.
For clarity, we work in a splitting (4.2) adapted to N . By imposing reality conditions, condition(4.9) implies that C abcd k a X b Y c Z d = 0 for all k a ∈ Γ( K ) and X a , Y a , Z a ∈ Γ( K ⊥ ), which is equivalentlyto (4.10a). The remaining conditions of (4.10) can be obtained from Proposition (4.12) and the additionalconditions C abcd X a Y b Z c ¯ W d = 0 ⇔ C N Π , ( C ) = C N Π , ( C ) = 0 ,C abcd u a X b Y c ¯ W d = 0 ⇔ C N Π , ( C ) = C N Π , ( C ) = C N Π , ( C ) = 0 ,C abcd X a u b Y c u d = 0 ⇔ C N Π , ( C ) = 0 ,C abcd k a X b Y c ℓ d = 0 ⇔ C N Π , ( C ) = C N Π , ( C ) = 0 ,C abcd k a u b X c ℓ d = C abcd k a ℓ b u c X d = 0 ⇔ C N Π , ( C ) = C N Π , ( C ) = 0 ,C abcd X a Y b Z c ℓ d = 0 ⇔ C N Π , ( C ) = 0 ,C abcd u a X b Y c ℓ d = 0 ⇔ C N Π , ( C ) = C N Π , ( C ) = 0 , for all local sections k a ∈ Γ( K ), ℓ a ∈ Γ( L ), X a , Y a , Z a of T (1 , , ¯ W a of T (0 , and u a of T (0 , . (cid:3) We can now restate the generalisation of the Goldberg-Sachs theorem given in [TC12a].
Theorem 4.20 ( [TC11, TC12a] ) Let ( M , g ) be an Einstein Lorentzian manifold equipped with an almostRobinson structure N . Suppose that the Weyl tensor is (locally) algebraically special along N . Assumefurther that the Weyl tensor is otherwise generic. Then N is integrable and co-integrable. In [TC11,TC12a], further degeneracy conditions on the Weyl tensors are also considered, and the Einsteincondition is replaced by weaker conditions on the Cotton-York tensor, as in the version of [KT62, RS63],which reflect the conformal invariance of the theorem.
Remark 4.21
In four dimensions, shearfree congruences of null geodesics (SCNG) are equivalent to Robin-son structures. Generalising the former to higher dimensions provides an alternative natural definition of‘algebraically special’ in line with the null alignment formalism of [CMPP04]: the Weyl tensor is said to bealgebraically special if it admits a null direction k a with respect to which it satisfies C K Π − ( C ) = 0 , i.e. C ade [ b k c ] k d k e = k [ a C b c ] f [ d k e ] k f = 0 , (4.10a)i.e. it is of Petrov type II (or more degenerate) – see section 3.1.5. In particular, by Proposition 4.19 tells usthat (4.10) implies that C abcd satisfies (4.10a), i.e. it is algebraically special in the null alignment formalism.But except in dimension four, the converse is clearly not true.Also, the algebraic condition (4.10a) leads to an ‘optical matrix’ treatment of the Goldberg-Sachs theoremin higher dimensions [DR09, OPPR12, OPP13b], but does not appear to be a sufficient condition for theexistence of a SCNG.We now give an analogue of Proposition 4.16 in the context of algebraically special spacetimes. Proposition 4.22
Let ( M , g ) be an n -dimensional Lorentzian manifold equipped with a null line distribution K , with n = 2 m + ǫ , ǫ ∈ { , } . The Weyl tensor C abcd satisfies C K Π ( C ) = 0 , t a point p of M if and only if the Weyl tensor is algebraically special along all elements of Gr m ( C T p M , g ) K at p , i.e. it satisfies (4.9) (or equivalently (4.10) ) with respect to any element of Gr m ( C T p M , g ) K .When n = 2 m , the Weyl tensor satisfies C K Π ( C ) = 0 , when m > , C K Π , +1 ( C ) = 0 , when m = 3 ,at a point p of M if and only if the Weyl tensor is algebraically special along all elements of Gr + , m ( C T p M , g ) K at p , i.e.Proof. This is analogous to the proof of Proposition 4.16. Let N be an almost Robinson structure incidenton K . The algebraically special condition (4.9) can then be expressed as C abcd k a Y b k c = 0 , (4.11) C abcd k a Y b Z c = 0 , (4.12) C abcd X a Y b Z c = 0 , (4.13)for all k a ∈ Γ( K ), X a , Y a ∈ Γ( N ⊥ ), Z a ∈ Γ( N ), with X a , Y a , Z a / ∈ Γ( K ). We now let vary N whilekeeping K fixed, we obtain (4.11) = ⇒ C K Π − ( C ) = C K Π − ( C ) = 0, (4.12) = ⇒ C K Π ( C ) = C K Π ( C ) = 0, and(4.13) = ⇒ C K Π ( C ) = C K Π ( C ) = 0.Finally, this result remains unchanged if we restrict ourselves to almost self-dual Robinson structureswhen n = 2 m >
6. When n = 6, however, conditions (4.11) and (4.12) split into self-dual and anti-self-dualparts, so that restriction to almost self-dual Robinson structures yields the required condition. (cid:3) The existence of (local) integrable and co-integrable almost Robinson structures on higher-dimensional so-lutions to Einstein’s field equations and their connection to the special algebraic condition (4.9) was firsthighlighted in [MT10, TC11]. Further instances of Robinson structures in higher dimensions were given in[OPPR12, OPP13a], but their relation to the degeneracy of the Weyl tensor was not investigated there.We shall review these examples paying more attention to their curvature properties. We also examine theintegrability conditions for the existence of a parallel Robinson structure and for the existence of a parallelpure spinor field of real index 1.
Conformal Killing-Yano -formsTheorem 4.23 ( [MT10] ) Let ( M , g ) be a (2 m + ǫ ) -dimensional pseudo-Riemannian manifold, where ǫ ∈{ , } , equipped with a conformal Killing-Yano -form φ ab , i.e. a -form φ ab that satisfies ∇ a φ bc = τ abc + 2 g a [ b K c ] , where τ abc = ∇ [ a φ bc ] and -form K a = (2 m + ǫ − ∇ c φ ca . Assume that φ ba has distinct eigenvalues. Let N be the totally null complex m -plane distribution associated to some (pure) eigenspinor of φ ab , and let N ⊥ denote its orthogonal complement. Then the Weyl tensor satisfies (4.9) with respect to N .Suppose further that τ abc X a Y b Z b = 0 , (4.14) for all X a , Y b , Z b ∈ Γ( N ⊥ ) . Then N and N ⊥ are locally both integrable. As stated, the theorem applies to any pseudo-Riemannian manifolds, and while this article deals primarilywith Lorentzian geometry, the following Riemannian example is worth mentioning as it sheds light on therˆole played by both the genericity of the eigenvalues of a CKY 2-form and the additional condition (4.14).32 xample 4.24 (The Iwasawa manifold)
The Iwasawa manifold is the quotient of the three-dimensionalcomplex Heisenberg group by a discrete subgroup. The set of all invariant Hermitian structures on this six-dimensional real manifold is known to consist of the union of a point (its bi-invariant Hermitian structure)and a 2-sphere ([KS04] and references therein). It was also shown in [BDS12] that it admits a
Killing-Yano -form , i.e. a co-closed conformal Killing-Yano 2-form, which is not closed. We will presently make theconnection between these two geometric entities more explicit.To describe the Iwasawa manifold, we introduce complex coordinates { z , z , z } together with complexvalued 1-forms θ := d z , θ := d z and θ := − d z + z d z , so that the metric and the Killing-Yano 2-formtake the form g = 2 θ ⊙ ¯ θ + 2 θ ⊙ ¯ θ + 2 θ ⊙ ¯ θ , φ = i (cid:0) θ ∧ ¯ θ + θ ∧ ¯ θ + 3 θ ∧ ¯ θ (cid:1) , respectively. As a spinor endomorphism, φ has four complex conjugate pairs of eigenvalues, and each pairof projective pure eigenspinors defines a conjugate pair of totally null complex 3-plane distributions, i.e. analmost Hermitian structure. To be precise, φ admits • an eigenvalue of multiplicity 1 with associated distribution N annihilated by θ ∧ θ ∧ θ , • an eigenvalue i4 of multiplicity 1 with associated distribution N annihilated by ¯ θ ∧ ¯ θ ∧ θ , • an eigenvalue 3i of multiplicity 2 with an associated 2-sphere of distributions N [ a,b ] annihilated by (cid:0) a θ + b θ (cid:1) ∧ (cid:0) b ¯ θ − a ¯ θ (cid:1) ∧ θ where [ a, b ] ∈ CP ,and similarly for their respective complex conjugates.Since d θ = 0, d θ = 0 and d θ = θ ∧ θ , we see at once that N and N [ a,b ] for any [ a, b ] ∈ CP are integrable. On the other hand, N is not integrable, but defines a quasi-K¨ahler structure [Gra76].Computing τ := d φ = 3i (cid:0) θ ∧ θ ∧ ¯ θ − ¯ θ ∧ ¯ θ ∧ θ (cid:1) , we see that τ is degenerate on N and N [ a,b ] for any [ a, b ] ∈ CP , but not on N . Thus, as expected fromthe additional condition (4.14) of Theorem 4.23, τ is the obstruction to the integrability of N .As for the curvature, we first define µ = 2i (cid:0) θ ∧ ¯ θ + θ ∧ ¯ θ (cid:1) , ν = 2 θ ⊙ ¯ θ + 2 θ ⊙ ¯ θ , α = 2i θ ∧ ¯ θ , β = 2 θ ⊙ ¯ θ . Then, in abstract index notation, the Weyl tensor, the tracefree Ricc tensor and the Ricci scalar are givenby C abcd = − (cid:0) µ ab µ cd − µ a [ c µ d ] b (cid:1) + 12 (cid:0) µ ab α cd + α ab µ cd − µ [ a | [ c α d ] | b ] (cid:1) + 710 ν a [ c ν d ] b + 35 α ab α cd − ν [ a | [ c β d ] | b ] , Φ ab = 23 ν ab − β ab ,R = 2 . In particular, the Weyl tensor is algebraically special with respect to any of N , N and N [ a,b ] for any[ a, b ] ∈ CP . The Iwasawa manifold is not Einstein, but the Cotton-York tensor is given by A = 23 (cid:0) ¯ θ ⊙ (cid:0) θ ∧ θ (cid:1) + θ ⊙ (cid:0) ¯ θ ∧ ¯ θ (cid:1)(cid:1) , i.e. it is degenerate with respect to of N and N [ a,b ] for any [ a, b ] ∈ CP , but not with respect to N . TheCotton-York tensor is therefore the obstruction to the integrability of N corroborating [TC11]. In fact, on inspection of their intrinsic torsion, they define
Hermitian semi-K¨ahler (also known as special Hermitian )structures [Gra76]. xample 4.25 (The Kerr-NUT-AdS metric) The higher-dimensional Euclidean Kerr-NUT-AdS met-ric [CLP06] is a (partial) generalisation of the Plebanski-Demianski metric [PD76], and was shown to admita conformal Killing-Yano tensor that satisfies the requirement of Theorem 4.23 (in fact, with τ abc = 0)[KF07, KK07, KKPF07, KFK08]. In even dimensions, it then follows that all 2 m projective eigenspinors ofthe CKY 2-forms define Hermitian structures [MT10]. These metrics also come in Lorentzian flavour, inwhich case they locally admit a discrete set of 2 m Robinson structures. Similar considerations apply in odddimensions.Other metrics admitting a special CKY 2-form as in Theorem 4.23 include those obtained by ‘switchingoff’ the mass or NUT parameters, or cosmological constants of the Kerr-NUT-(A)dS metric, e.g. the Myers-Perry black hole [MP86, GLPP05]. While one cannot apply Theorem 4.23 when the eigenvalues of the CKY2-form become degenerate, one can still deduce the existence of Robinson structures as a limiting case. Thisis precisely the case when all the rotation coefficients of the Kerr-NUT-(A)dS metric are set to zero, and inparticular, for the Schwarzschild metric. which is discussed in Example 4.28.
Kerr-Schild metrics
A remarkable property of the Kerr metric [GLPP05] is that they can be put in
Kerr-Schild form , i.e. exact first order perturbations of Minkowski space , with a preferred null line distribution K , as given by (4.15). This makes the study of the differential properties of K particularly tractable, asalready discussed in [OPP09]. There are further interesting curvature properties that these metrics enjoywhen considering Robinson structures incident on K . Proposition 4.26
Let ( M , g ) be a Lorentzian manifold equipped with a co-integrable Robinson structure N .Let K be the associated null line distribution of N , i.e. N ∩ ¯ N = C K . Suppose that the metric on M is ofthe Kerr-Schild form g ab = η ab + Hk a k b , (4.15) where η ab is a flat metric, H is a smooth function, and k a is a section of K . Then the Weyl tensor isalgebraically special with respect to N . In addition, the Ricci tensor and its tracefree part satisfy R ab X a Y b =0 and Φ ab X a Y b = 0 for all X a ∈ Γ( N ) and Y a ∈ Γ( N ⊥ ) , i.e. F N Π , (Φ) = F N Π , (Φ) = 0 .Proof. In [TC15], it is shown that for a complex Riemannian manifold equipped with an almost null structure N and a metric of the form g ab = η ab + H ab , where η ab is the flat complex Euclidean metric and H ab is asection of ⊙ N , if both N and N ⊥ are integrable then the Weyl tensor satisfies (4.9). This result can alsobe applied to (complexified) Lorentzian metric, in which case the additional structure forces H ab to be ofthe form Hk a k b for some function H and real null vector field k a . The condition on the Ricci tensor is givenin the same reference. (cid:3) Example 4.27 (Myers-Perry black holes in Kerr-Schild form)
We introduce the standard flat co-ordinates { x a } = { t, x i , y i , ǫz } on Minkowski space. The higher-dimensional Kerr-Schild ansatz [MP86,GLPP05] describing a rotating black hole in higher dimensions admits a Kerr-Schild form (4.15) with k a d x a := d t + m − ǫ X i =1 r ( x i d x i + y i d y i ) + a i ( x i d y i − y i d x i ) r + a i + (1 − ǫ ) z d zr ,U := 1 r ǫ +1 − m − X i =1 a i ( x i + y i )( r + a i ) ! n − Y j =1 ( r + a j ) , H := 2 MU .
Here, the constants a i are the rotation coefficients and M the mass of the black hole, and r satisfies theconstraint m − X i =1 x i + y i r + a i + ( ǫ − z r = 1 . Generalisation of this ansatz can also be obtained by considering non-flat background metrics. φ ba for the metric. Solutions r to this equation are theeigenvalues of φ ba . In fact, this metric admits a second Kerr-Schild ansatz with a null direction ℓ a , dual to k a , obtained by sending r to − r . The vector fields k a and ℓ a are two distinct real eigenvectors for the realeigenvalues ± r of φ ab on ( M , g ). The remaining eigenvalues are purely imaginary with complex eigenvectors.Thus, by Theorem 4.23 one can associate a discrete set of 2 m − local Robinson structures to each of k a and ℓ a [MT10]. Finally, being a special case of the Kerr-NUT-(A)dS metric, the Weyl tensor is algebraicallyspecial with respect to any of the 2 m Robinson structures of the Kerr-Myers-Perry metric.
Example 4.28 (Schwarzschild metric)
When all rotation coefficients a i are zero, the Kerr-Schild/Myers-Perry black hole of Example 4.27 degenerates to the Kerr-Schild form of the Schwarzschild black hole. Boththe vector fields k a and ℓ a become hypersurface-orthogonal, with k a d x a := d t + d r , ℓ a d x a := d t − d r , H := 2 Mr n − ǫ . We recast the flat metric of (4.15) into the form η ab = 2 k ( a ℓ b ) + h ab , (4.16)where h ab is the round metric on the ( n − S n − (up to a factor r ) – the hypersurfacesorthogonal to both k a and ℓ a are ( n − S n − .The (co-integrable) Robinson structures of the Kerr metric persist as the parameters a i are set to zero inthe Scharzschild metric [OPPR12]. In fact, the rank of the CKY 2-form of the Kerr-NUT-(A)dS metric willdrop to 2, which implies in particular that as a spinor endomorphism, the CKY 2-form will have only a pairof eigenvalues ± r , each of multiplicity 2 m − . This is not unlike Example 4.24, which admits infinitely manyHermitian structures. An eigenspinor for φ need not be pure, but a pure eigenspinor will determine a (notnecessarily integrable) almost Robinson structure incident on either K or L . We shall presently, describethese pure eigenspinors in terms of their associated almost Robinson structures.For our purpose, however, we can use the more standard form of the Schwarschild metric (4.16). Forspecificity, we assume n = 2 m . An arbitrary almost Robinson structure incident on K := h k a i must bespanned by k a and m − { ζ A } , say, which define a local almost Hermitianstructure on S n − . Since k a commutes with any of { ζ A } , the question then boils down to seek integrable almost Hermitian structures on S n − .Except on S there are no global Hermitian structures on S n − , but this is not so locally. We introducethe standard complex coordinates { z A , ¯ z A } on an affine subset R n − of S n − with flat metric 2 d z A ⊙ d¯ z A .Then an arbitrary totally null complex ( m − ζ A = ∂∂ ¯ z A + φ AB ∂∂z B for some ( m − m −
2) functions φ AB = φ [ AB ] on M . For { ζ A } to be integrable, thesefunctions must satisfy ζ C φ AB = 0. The general analytic solution to this system of PDEs can be obtainedfrom the prescription of a set of ( m − m −
2) holomorphic functions in m ( m −
1) complex variables: this is the content of the Kerr theorem [CF76, PR86, HM88]. Thus, there exist infinitely many local analyticHermitian structures on S n − . Consequently, at every point p of the Schwarzschild black hole ( M , g ), andfor every element N p of Gr m ( C T p M , g ) K and Gr m ( C T p M , g ) L , we can find an integrable almost Robinsonstructure tangent to N p .A similar argument applies in odd dimensions too: locally, on S m − , there are infinitely many analyticmetric-compatible CR structures whose orthogonal complements are integrable too. Their description isgiven in [TC15] where an odd-dimensional version of the Kerr theorem is also presented. In turn, these giverise to infinitely many Robinson structures on the Schwarschild spacetime.All these (integrable or not) almost Robinson structures will subsist when considering the full Kerr-Schildmetric. By Proposition 4.26, locally the Weyl tensor of the Schwarzschild metric is algebraically special alongany local integrable and co-integrable section of Gr m ( C T M , g ) K and Gr m ( C T M , g ) L . In particular, we cannow apply Proposition 4.22 to conclude that C K Π ( C ) = 0 , C L Π ( C ) = 0 , (4.17) This defines an ( m − twistor space [HM88]. M (1) × S n − where M (1) is a two-dimensional Lorentzian manifold.What is more, from Proposition 4.22, we discover that any local, not necessarily integrable , section ofGr m ( C T M , g ) K and Gr m ( C T M , g ) L will satisfy (4.9). Thus the Schwarzschild metric admits infinitely manyRobinson structures, and its Weyl tensor is algebraically special to infinitely many (not necessarily integrable)almost Robinson structures.In sum, the above example shows Proposition 4.29
On an Einstein Lorentzian manifold, not every almost Robinson structure with respectto which the Weyl tensor satisfies the algebraically special conditions (4.9) (i.e. (4.10) ) is integrable andco-integrable.
Remark 4.30
An analogous statement in the null alignment formalism is that not every WAND is geodetic[GR09]. In this case, any of the almost Robinson structures associated to a non-geodetic WAND is necessarilynon-integrable.Further examples of non-integrable totally null distributions on complex Riemannian or split signaturemanifolds, which satisfy (4.9), were already pointed out in [TC12b, TC13] and references therein.
Robinson-Trautman
The Schwarzschild metric belongs to the class of Robinson-Trautman metrics [RT61]that have been generalised to higher dimensions in [PO06, OPZ08]. These are spacetimes admiting an ex-panding, twistfree and shearfree congruence K of null geodesics. In other words, the orthogonal complement K ⊥ of K is integrable, i.e. [Γ( K ⊥ ) , Γ( K ⊥ )] ⊂ Γ( K ⊥ ), so that M is foliated by null hypersurfaces, and theconformal Riemannian metric on the screenspace K ⊥ / K is preserved along the flow of K .The local description of these metrics is given in [PO06]. There exist local coordinates { u, r, x i } , where u parametrises the field of hyperplanes tangent to K ⊥ , and r is an affine parameter along a null geodesic of K , so that k := ∂∂r spans K , and { x i } are coordinates on the ( n − H u tangentto K ⊥ and complementary to K . Each of H u is equipped with a metric h ij = h ij ( u ; x ). For simplicity, weassume that ( M , g ) is Ricci-flat. In this case, the Robinson-Trautman metric takes the local form g = 2 d u ⊙ d r + H d u ⊗ d u + 2 r h ij d x i ⊙ d x j , (4.18)where H is a smooth function on M , and h ij = h ij ( u ; x ) is a family of Einstein metrics on H u . Denote by L the null line distribution dual to K , spanned by ℓ := ∂∂u − H ∂∂r . It turns out [PO06] that when n >
4, themetric (4.18) can only be of Petrov type D, with Weyl aligned null directions K and L , or conformally flat.More specifically, in our notation, the Weyl curvature is given by C K Π ( C ) = C L Π ( C ) = µ ( u ) , C K Π ( C ) = C K Π ( C ) = C L Π ( C ) = C L Π ( C ) = 0 , C K Π ( C ) = C L Π ( C ) = ( , n = 5 ,r C ijkℓ ( u ; x ) , n > , (4.19)where µ ( u ) is some function, and for each u , C ijkℓ ( u ; x ) is the Weyl tensor of the Einstein metric h ij ( u ; x )on H u . For the Schwarzschild metric, conditions (4.19) reduce further to (4.17) since C ijkℓ ( u ; x ) = 0 for any u , and since this condition is always satisfied when n = 5, the vacuum Robinson-Trautman metric coincideswith the Schwarzschild metric then.From conditions (4.19), it is clear that any further properties of the conformal curvature of ( M , g ) willdepend exclusively on the family of screenspace metrics h ij ( u ; x ), and thus on any additional structure com-patible with h ij ( u ; x ). A natural candidate for such a structure is evidently an almost Hermitian structure. It is noteworthy to mention [OPZ08] that the screenspace metrics of the non-vacuum Robinson-Trautman solutions mustbe almost K¨ahler .
36n even dimensions, each C ijkℓ ( u ; x ) can be characterised in terms of the classification given in [TV81,FFS94].In odd dimensions, such a classification can be derived with no difficulty from the article [TC13].The case when ( M , g ) is a six-dimensional Robinson-Trautman spacetime is of particular interest: eachmanifold H u on M is a four-dimensional Riemannian manifold, which means that one can apply the Petrov-Penrose classification of the Weyl tensor of h ij ( u ; x ). The Weyl tensor associated to the screenspace Rieman-nian metric h ij ( u ; x ) splits into a self-dual part Ψ + ijkℓ and anti-self-dual part Ψ − ijkℓ , each of a particular Petrovtype: Ψ + ijkℓ is said to be of Petrov type D or algebraically special if there is an almost self-dual Hermitianstructure J ji with respect to which Ψ + ijkℓ X i Y j Z k = 0 for any vector fields X i , Y j , Z k in the i-eigenbundleof J ji , and similarly for Ψ − ijkℓ . Algebraically general and vanishing (anti)-self-dual Weyl tensors are calledPetrov type G and O respectively – see [GHN10] and references therein for details.We can adopt our orientation convention to be such that C K Π , ± ( C ) = C L Π , ± ( C ) = r Ψ ± ijkℓ ( u ; x ) . Now, suppose that N is an almost self-dual Robinson structure incident on K . The following observationsare easy to make: • if C N Π ,i ( C ) = 0 for i = 1 , + is of Petrov type D; • if C N Π ,i ( C ) = 0 for i = 0 , , + is of Petrov type O; • if C N Π , ( C ) = 0 then Ψ − is of Petrov type O.For the converse, we note that generically Ψ + is of type G, which singles out a field self-dual 2-plane, i.e.a principal almost Hermitian structure. This almost Hermitian structure can naturally be paired with anyof the null line distributions K and L . In particular, there always exists a pair of almost self-dual Robinsonstructures on M , and the Weyl tensor must satisfy C N Π , ( C ) = 0. If Ψ + is of type D, respectively of typeO, there always exists an almost Robinson structure N for which C N Π ,i ( C ) = 0 for i = 1 ,
3, respectively, C N Π ,i ( C ) = 0 for i = 0 , ,
3. Finally, if Ψ − is of Petrov type O, we can find an almost Robinson structuresuch that C N Π , ( C ) = 0.Since h ij ( u ; x ) is Einstein, we can go a step further by applying the four-dimensional RiemannianGoldberg-Sachs theorem [PB83, Nur93, AG97, GHN10]. If h ij ( u ; x ) is algebraically special, then H u admitsa Hermitian structure, which lifts to a Robinson structure on M . Conversely, suppose that N is a (locally)integrable almost Robinson structure on M incident on K . Then N restricts to a Hermitian structure oneach H u . In particular, by the Goldberg-Sachs theorem, the Weyl tensor is algebraically degenerate.Summing up these observations, we have Proposition 4.31
Let ( M , g ) be a Ricci-flat six-dimensional Robinson-Trautman spacetime, i.e. a Ricci-flat Lorentzian manifold equipped with a null line distribution K that generates a twistfree, shearfree andexpanding congruence of null geodesics. Let N be an almost Robinson structure incident on K , i.e. N is a(local) section of Gr m ( C T M , g ) K . Then locally the Weyl tensor is algebraically special with respect to N ifand only if N is integrable. In a similar vein, one could extend this analysis to Kundt spacetimes and warped product of manifolds,which share some of the properties of the Robinson-Trautman metrics. There is however too little space forthis purpose here.
Higher-dimensional Taub-NUT-(A)dS metrics
It was pointed out in [OPP13b] that in six dimensionsthese metrics admit local Robinson structures. Here, we extend this to higher dimensions and examine thealgebraic properties of their Weyl tensor.
Example 4.32
These metrics generalise the four-dimensional Taub-NUT-(A)dS of [NTU63,Tau04,HHP99],and are constructed as metrics on a spacetime whose boundary at infinity is a U(1)-fibration over a (2 m − B , H, J ). In fact, they admit an odd-dimensional version by con-structing a U(1)-fibration over the first factor of the direct product B × Y where Y is an odd-dimensional37iemannian manifold. Throughout this example, we suspend the Einstein summation convention. We shallalso specialise to the case where Y is simply R . To describe the construction, we introduce a (local) unitarybasis { µ A , ¯ µ A } A =2 ,...,m on B adapted to the K¨ahler structure, i.e.d µ A = m X B =2 µ B ∧ α AB , where α AB is the connection 1-form for the K¨ahler metric on B . In odd dimensions, we take { µ } to spanT ∗ Y with d µ = 0. The Taub-NUT-(A)dS metric of [AC02, MS04, Awa06] is then given by g = F − d r − F (d t + A ) + 2 m X A =2 ( r + N A ) µ A ⊙ ¯ µ A + ǫr µ ⊗ µ , (4.20)where t is the coordinate on the U(1)-fiber, the constants N A are the NUT parameters, F = F ( r ) is a smoothfunction of r . The 1-form A can be thought of as a K¨ahler potential in the sense that it satisfiesd A = 2 i m X C =2 N C µ C ∧ ¯ µ C . In what follows, we shall make either of the following two assumptions: • either we take B = B × . . . × B m where each B i is a 2-dimensional Riemannian manifold of constantcurvature, i.e. spheres, tori, or hyperboloids; • or, B is a more general K¨ahler manifold, but we impose N A = N B for all A, B .Define the 1-forms θ = κ := 2 − (cid:16) F − d r + F (d t + A ) (cid:17) , ˜ θ = λ := 2 − (cid:16) F − d r − F (d t + A ) (cid:17) , θ A := ( r + N A ) µ A , θ := r µ . Then the connection 1-form for the metric (4.20) takes the form Γ = 2 − F − F ′ ( κ − λ ) , Γ A = 2 − − r + i N A r + N A F ¯ θ A , Γ A = − − r + i N A r + N A F ¯ θ A , Γ = 2 − F r θ = Γ Γ BA = α BA − − i N A r + N A F ( κ − λ ) δ BA , from which we immediately see that ( M , g ) is endowed with 2 m local co-integrable Robinson structuresannihilated by the set of ( m + ǫ ) 1-forms { κ , θ A , ¯ θ B , ǫ θ } A = B , { λ , θ A , ¯ θ B , ǫ θ } A = B , Computing the curvature 2-form yields R = − − F ′′ κ ∧ λ + i X C N C ( F ′ ( r + N C ) − rF ) ( r + N C ) θ C ∧ ¯ θ C , R A = − (cid:0) F ′ r ( r + N A ) + 2 F N A (cid:1) + i N A (cid:0) F ′ ( r + N A ) − F r (cid:1) r + N A ) κ ∧ ¯ θ A , R = F ′ r λ ∧ θ , R BA = R BA + i N A ( F ′ ( r + N A ) − rF )( r + N A ) κ ∧ λ + 2 N A Fr + N A X C N C r + N C θ C ∧ ¯ θ C ! δ BA + F ( N A N B − r )( r + N A )( r + N B ) θ B ∧ ¯ θ A , R A = Fr + N A ¯ θ A ∧ θ . F = F ( r ) as given in [MS04, AC02, Awa06],in which case it is immediate that the Weyl tensor satisfies (4.9). Parallel Robinson structures
The holonomy of the Levi-Civita of a spacetime admitting a parallel nullline distribution is contained in Sim( n − m − , C ), the manifold admits a parallel Robinson structure, i.e.( ∇ a X b ) Y b = 0 , for all X a ∈ Γ( N ) , Y a ∈ Γ( N ⊥ ) (4.21)Equivalently, the manifold admits a recurrent pure spinor of real index 1. This is one of the cases consideredin [Gal13].We now examine the integrability condition for the existence of such a parallel Robinson structure.
Proposition 4.33
Let ( M , g ) be a Lorentzian manifold equipped with a parallel Robinson structure N withassociated null line distribution K . Then the Weyl tensor satisfies conditions (4.10) together with the addi-tional conditions C K Π ( C ) = 0 , (4.22a) C N Π , ( C ) = 0 , (4.22b) C N Π , ( C ) = 0 . (4.22c) Further, the tracefree Ricci tensor satisfies F N Π ,i (Φ) = 0 , for i = 1 , . (4.23) Finally:1. If any two of the following conditions C N Π , ( C ) = 0 , C N Π , ( C ) = 0 , F N Π , (Φ) = 0 , R = 0 , hold, then the remaining two hold too. In addition, in odd dimensions, C N Π , ( C ) = 0 ⇐⇒ C N Π , ( C ) = 0 ⇐⇒ F N Π , (Φ) = 0 .
2. We have C N Π , ( C ) = 0 ⇐⇒ C N Π , ( C ) = 0 ⇐⇒ F N Π , (Φ) = 0 , and in addition, in odd dimensions, C N Π , ( C ) = 0 ⇐⇒ F N Π , (Φ) = 0 .
3. If any two of the following conditions C N Π , ( C ) = 0 , C N Π , ( C ) = 0 , F N Π , (Φ) = 0 , hold, then the remaining one holds too. In addition, in odd dimensions, C N Π , ( C ) = 0 ⇐⇒ C N Π , ( C ) = 0 . As explained in appendix A, a pure spinor of real index 1 is equivalent to the existence of sim ( m − , C )-invariant 3-form ̺ abc and 2-form µ ab . Recurrence of the spinor is equivalent to recurrence of these forms. roof. Taking another covariant derivative of (4.21) to compute the Riemann curvature tensor, we find0 = R abcd X c Y d = C abcd X c Y d + 2 n − (cid:16) X c Φ c [ a Y b ] − Y c Φ c [ a X b ] (cid:17) + 2 n ( n − RX [ a Y b ] , (4.24)for all X a ∈ Γ( N ), Y a ∈ Γ( N ⊥ ). By the results of [TC12b, TC13], this implies C abcd X c Y d Z b = 0 andΦ ab X a Y b = 0 for all X a , Z b ∈ Γ( N ), Y a ∈ Γ( N ⊥ ). The first of these conditions yields the ‘complexGoldberg-Sachs’ conditions of Proposition 4.19, while the second of these gives the additional algebraicconstraints (4.23). With reference to the proof of Proposition 4.2, we reobtain C N Π ,i ( C ) = 0 for i = 1 ,
3, asfollows from Proposition 4.19.On the other hand, since the associated real null distribution is also parallel, Proposition 4.2 applies, i.e.(4.22a) holds.Introducing a splitting (4.2) of the tangent bundle adapted to N , we now consider the remaining condi-tions implied by (4.24), more precisely, R abcd X a Y b ¯ V c ¯ W d = R abcd k a Y b ℓ c ¯ V d = R abcd X a Y b ℓ c ¯ V d = 0 R abcd X a u b ¯ V c u d = R abcd X a u b ¯ W c u d = R abcd k a u b ℓ c u d = 0 , ( ǫ = 1 only), (4.25)for all k a ∈ Γ( K ), X a , Y a , V a , W a ∈ Γ(T (1 , ), u a ∈ Γ(T (0 , ) and ℓ a ∈ Γ( L ). These conditions imposealgebraic relations between irreducible components of the Weyl tensor and the tracefree Ricci tensor, andthe Ricci scalar, which lie in isotypic irreducible cu ( m − C , and ˘ C , of respective dimensions m ( m − ( m −
4) and m ( m − m −
3) are not isotypic toany other, we can safely conclude (4.22b) and (4.22c). Now, with reference to equations (4.25) and appendixB.2.2, we prove the additional conditions 1, 2 and 3 by noting:1. first, 0 = 6 Ψ , + n − n − , + n − n ( n − R , n − n − , − m m − ǫ ) − , + nn − , , and additionally, in odd dimensions,2(2 m − m − n −
4) Φ , = − m − n − n − m −
2) Ψ , = 2 m m − , , where Ψ , , Ψ , , Ψ , , Φ , and Φ , are the components for the 1-dimensional modules ˘ C , , ˘ C , ,˘ C , , ˘ F , , ˘ F , respectively;2. further,0 = (Ψ , ) AB + 2 n − , ) AB , − (cid:18) n − n − (cid:19) (cid:18) m − m − (cid:19) (Ψ , ) AB + 2 m − m − , ) AB , and additionally, in odd dimensions,0 = 2 n − , ) AB + (Ψ , ) AB , where (Ψ , ) AB , (Ψ , ) AB , (Ψ , ) AB and (Φ , ) AB are the components for m ( m − C , , ˘ C , , ˘ C , and ˘ F , respectively;3. finally, 0 = 1 n − , ) A + 2 m − m −
4) (Ψ , ) A − n − , ) A , and additionally, in odd dimensions0 = 2 m − m −
4) (Ψ , ) A − (Ψ , ) A , where (Ψ , ) A , (Ψ , ) A , (Ψ , ) A and (Φ , ) A are the components for (2 m − C , , ˘ C , , ˘ C , and ˘ F , respectively.This ends the proof of Proposition 4.33. (cid:3) arallel pure spinor fields of real index N . When such a spinor is parallel, one obtains a counterpart ofProposition 4.3 in the context of almost Robinson geometry. For an account of metrics admitting a parallelpure spinor, see for instance [Bry00, Lei02]. Proposition 4.34
Let ( M , g ) be a Lorentzian manifold admitting a parallel pure spinor field ξ of real index . Let N be the almost Robinson structure annihilating ξ with associated null line distribution K . Then C K Π ( C ) = 0 , i.e. C abcd k d = 0 , (4.26a) C N Π i ( C ) = 0 , for i = 0 , , , (4.26b) C N Π ,i ( C ) = 0 , for all i = 5 , (4.26c) C N Π ,i ( C ) = 0 , for all i = 3 , , , (4.26d) F K Π (Φ) = 0 , i.e. Φ a [ b k c ] = 0 , (4.26e) R = 0 . (4.26f) Proof.
For simplicity, we treat the case n = 2 m + 1 only, the even-dimensional case being similar. Wefirst note that since the pure spinor field is parallel, so must be its conjugate and its associated null vector k a . In particular, Proposition 4.3 holds. Using the notation of appendix A, we know (see [TC12b, TC13]and references therein) that the existence of a parallel pure spinor ξ A implies R = 0, i.e. (4.26f), andΦ ab γ a BA ξ A = 0. The latter condition leads to (4.26e), so that by Proposition 4.3, we have that (4.26a).Moreover, since N is parallel, Proposition 4.33 holds with our additional constraints (4.26a), (4.26e), and(4.26e), i.e. C abcd X c Y d = 0 for all X a ∈ Γ( N ) and Y a ∈ Γ( N ). The extra constraints can be found to be C N Π ,i ( C ) = 0 for i = 0 , , C N Π ,i ( C ) = 0 for i = 0 , C abcd γ cd BA ξ A = 0 can be re-expressed as C abcd ̺ cde = 0 and C abcd µ cd = 0 , where ̺ abc := 3 k [ a ω bc ] and µ ab = 2 k [ a u b ] , ω ab is any Hermitian structure and u a a unit vectoron the screenspace of K . Thus, those components of C abcd which have non-vanishing trace with respect toany choice ω ab will vanish. The only additional condition on the Weyl tensor is then C N Π , ( C ) = 0, whichcan happen only in odd dimensions. This completes the proof. (cid:3) Acknowledgements
I would like to thank Thomas Leistner for interesting discussions and for his hos-pitality during my stay at the University of Adelaide in November 2012. I am also grateful to DimitriAlekseevsky and Anton Galaev for helpful discussions. Finally, I extend my thanks to Lionel Mason, whoseideas came to influence the content of this article.This work was funded by a SoMoPro (South Moravian Programme) Fellowship: it has received a financialcontribution from the European Union within the Seventh Framework Programme (FP/2007-2013) underGrant Agreement No. 229603, and is also co-financed by the South Moravian Region.The author has also benefited from an Eduard ˇCech Institute postdoctoral fellowship GPB201/12/G028,and a GA ˇCR (Czech Science Foundation) postdoctoral grant GP14-27885P.
A Pure spinors and Robinson structures
The aim of this appendix is to give an explicit description of a Robinson structure building on the spinorcalculus of references [TC12b,TC13]. Spinor representations for a real inner product space ( V , g ) are complexvector spaces equipped with additional reality or quaternionic structures depending on the dimension of V ,the signature g , and the convention adopted. There is a wealth of literature on the details and subtletiesinvolved in their description [Car81, PR86, BT89, KT92, Kop97, LM89].Our treatment will however be overwhelmingly dimension-independent, and, more in the spirit of Cartan’stheory of spinors [Car81], we shall exploit the geometric properties of pure spinors to construct our calculus.In fact, many of the algebraic properties of Clifford and spinor representations can be derived from suchgeometric considerations. In this way, our spinor calculus will merely boil down to a tensor calculus adaptedto a given Robinson structure. Alternative spinorial approaches to five-dimensional spacetimes can be foundin [DS05, God10] and [GPGLMG09]. 41hroughout, ( V , g ) will denote n -dimensional Minkowski space, and as before, the signature of g will betaken to be ( n − , C V , g ) of ( V , g ), in which case the metrictensor will be assumed to be complex-valued. The spinor representations for ( V , g ) will then be obtained byadditional algebraic restrictions. We first treat the even- and odd-dimensional cases separately for clarity.Our setup and notation will allow us to dispense with such a distinction later. A.1 Background algebra
A.1.1 Even dimensions
Assume n = 2 m . The spinor representation S of ( C V , g ) splits into a direct sum of two 2 m − -dimensionalirreducible chiral spin spaces S + and S − , the spaces of positive and negative spinors respectively. Elementsof S + and S − will be adorned with upstairs primed and unprimed upper case Roman indices respectively,e.g. α A ′ ∈ S + and β A ∈ S − , and similarly for the dual spinor spaces ( S ± ) ∗ with downstairs indices.The relation between ( C V , g ) and S is given by the Clifford algebra C ℓ ( C V , g ) of ( V , g ), which establishes anisomorphism between the exterior algebra ∧ • V and the space End( S ) of endomorphisms of S . We introducethe van der Waerden γ -matrices γ B ′ aA and γ BaA ′ which satisfy γ C ( aA ′ γ B ′ b ) C = g ab δ B ′ A ′ , γ C ′ ( aA γ Bb ) C ′ = g ab δ BA . We emphasise that we shall use the van der Waerden γ -matrices in a purely abstract way , i.e. in the sensethat we shall think of γ BaA ′ and γ B ′ aA as injective maps from V to the space of homomorphisms from S ∓ to S ± , or as a projection from ( S ∓ ) ∗ ⊗ S ± to V .The real structure¯on C V preserving V induces a real structure, when m = 1 , m = 0 , S . Thus, the spinor representation S of ( V , g ) will be thatof ( C V , g ) together with this additional structure. Further, this complex conjugation interchanges thechirality of spinors when m is even, and preserve them when m is odd. One can then use the isomor-phisms S ± ∼ = ( S ± ) ∗ when m is even, and S ± ∼ = ( S ± ) ∗ when m is odd, to define a complex conjugation¯: S ± → ( S ∓ ) ∗ , i.e. ¯: ξ A ′ ¯ ξ A , and¯: η A ¯ η A ′ . The pure spinors associated to a Robinson structure
By definition, a Robinson structure on ( V , g )is a totally null complex m -plane N in ( V , g ), which, for definiteness, we shall assume to be self-dual. Then N is the annihilator of a pure positive spinor ξ A ′ in the sense that N = ker ξ Aa : C V → S − , where wehave written ξ Aa := ξ B ′ γ AaB ′ , and ξ A ′ satisfies the algebraic condition ξ aA ξ Ba = 0. Similarly, the conjugate m -plane ¯ N annihilates the conjugate pure spinor ¯ ξ A . We note that the chirality of ¯ ξ A is consistent with theorientation of ¯ N , i.e. a β -plane when m is even, and an α -plane when m is odd.Any element V a of N is of the form V a = ξ aA v A for some v A ∈ ( S − ) ∗ / { ker ξ Aa : C V ← ( S − ) ∗ } . Theconjugate spinor ¯ ξ A is the distinguished element of ( S − ) ∗ for which the real null vector k a = ξ aB ¯ ξ B , (A.1)spans K , the real span of the intersection of N and ¯ N . A dual Robinson structure and its pure spinors
We now introduce a Robinson structure N ∗ dual to N . Here, the totally null m -plane N ∗ , its complex conjugate ¯ N ∗ and their real intersection L are dual to N ,¯ N and K respectively. The m -plane N ∗ annihilates a pure spinor ¯ η A ′ dual to ξ A ′ , i.e. N ∗ = ker ¯ η aA : C V → ( S − ) ∗ , where ¯ η aA := γ B ′ aA ¯ η B ′ . Similarly, to ¯ N ∗ corresponds a pure spinor η A dual to ¯ ξ A and conjugate of¯ η A ′ . We can choose to normalise these spinors as ξ A ′ ¯ η A ′ = and ¯ ξ A η A = − . Any generator of L is a realmultiple of ℓ a := 2 η A ¯ η aA . (A.2)We can now set V = K , V − = L and V = K ⊥ ∩ L ⊥ , and refer to the splitting (2.2) of the filtration(2.1). The spinor I AB := ξ aA ¯ η aB : S − → S − is the identity map on im ξ aA , or dually on im ¯ η aB . Now, since Our convention differs from [TC12b] by a sign. (1 , = N ∩ ¯ N ∗ , we have ξ aA η B ′ a = 2 ξ B ′ η A so that ¯ ξ B I BA = ¯ ξ A and η B I AB = η A . It follows that im ξ aA andim ¯ η aA contain h ¯ ξ A i and h η A i respectively. Thus, the map defined by ι AB := I AB − ξ B η A can be identifiedwith the identity map on the ( m − S (0 , := { im ξ Aa : C V → S − } ∩ { ker ¯ ξ A : C ← S − } , S (1 , − := { im ¯ η aA : C V → ( S − ) ∗ } ∩ { ker η A : C ← ( S − ) ∗ } . At this stage, we set S := h ξ A ′ i and ¯ S := h ¯ ξ A i . This notation is consistent with the fact that V := K injects into the tensor product S ⊗ ¯ S . In fact, both ξ A ′ and ¯ ξ A are eigenspinors of the grading elementof the Lie algebra (2.4) for the eigenvalue . Putting things together, we have the identifications N = C V ⊕ V (1 , ∼ = S ⊗ (cid:16) ¯ S ⊕ S (1 , − (cid:17) , N ∗ = C V − ⊕ V (0 , ∼ = ¯ S − ⊗ (cid:16) S − ⊕ S (0 , (cid:17) , (A.3)so that any elements of V (1 , and V (0 , admit spinorial representatives in S (1 , and S (0 , − respectively.The complex conjugation on C V interchanges V (1 , and V (0 , . It must therefore extends to an invo-lution on S which interchanges S (1 , and S (0 , − . This new complex conjugation, which we shall denote ˇ,must depend on the complex conjugation¯on S and a choice of splitting. A.1.2 Odd dimensions
Now assume n = 2 m + 1. The irreducible spinor representation S of ( V , g ) is a 2 m -dimensional complexvector space. Elements of S will be adorned with upstairs upper case Roman indices, e.g. α A ∈ S , andsimilarly for the dual spinor space S ∗ with downstairs indices. The relation between ( C V , g ) and S is givenby the Clifford algebra C ℓ ( C V , g ) of ( V , g ), which establishes an isomorphism between the exterior algebra ∧ • V and the space End( S ) of endomorphisms of S . We introduce the van der Waerden γ -matrices γ BaA which satisfy γ C ( aA γ Bb ) C = g ab δ BA , As in the even-dimensional case, these γ -matrices will be thought of as abstract maps.The real structure¯on C V preserving V induces a real structure, when m = 0 , m = 2 , S . Using the isomorphism S ∼ = S ∗ , we can define a complexconjugation, denoted by¯, that sends elements of S to elements of S ∗ , i.e. ¯: ξ A ¯ ξ A . The pure spinors associated to a Robinson structure
We associate to a Robinson structure N apure spinors ξ A , which is annihilated by the complex null m -plane N , i.e. N = ker ξ Aa : C V → S , where ξ Aa := ξ B ′ γ AaB ′ . Here, the pure spinor ξ A satisfies the algebraic constraint ξ aA ξ Ba = ξ A ξ B . Similarly, ¯ N annihilates a pure spinor ¯ ξ A conjugate to ξ A . Moreover, the real index of N imposes an additional algebraicconstraints on these pure spinors, i.e. ξ A ¯ ξ A = 0 , which is the algebraic condition for N and ¯ N to intersect in C K . The real span K of this intersection is infact generator by the element k a = ξ aB ¯ ξ B , (A.4)as in even dimensions.More generally, any element V a of N ⊥ is of the form V a = ξ aA v A for some v A ∈ ( S − ) ∗ / { ker ξ Aa : C V ← ( S − ) ∗ } . We then have that V a belongs to N if and only if v A ξ A = 0. Recall that V is equipped with a positive definite metric tensor h ab , and we can identify S (1 , and S (0 , − as subspacesof a chiral spinor representation for ( V , h ) and its dual respectively. The complex conjugationˆcan then be identified with thecanonical complex conjugation on the spinor representation of ( V , h ) Again, our convention differs from [TC13] by a sign. dual Robinson structure and its pure spinor Again, we choose a Robinson structure N ∗ , dual to N , to which we associate a pure spinor ¯ η A dual to ξ A , i.e. N ∗ = ker ¯ η aA : C V → S ∗ , where ¯ η aA := γ BaA ¯ η B .Similarly, ¯ N ∗ annilates the complex conjugate pure spinor η A dual to ¯ ξ A . The spinors ¯ η A and η A are of realindex 1, and as such satisfy ¯ η A η A = 0, and the real intersection of N ∗ and ¯ N ∗ is spanned by the real element ℓ a = 2 η B ¯ η aB . (A.5)We choose the normalisation ξ A ¯ η A = − and ¯ ξ A η A = . Referring to [TC13], we know that since N and ¯ N ∗ intersect in a totally null ( m − ξ A and η B must satisfy ξ aA η Ba = − ξ A η B + 2 η A ξ B . We also knowthat I AB := ξ aA ¯ η aB + ¯ η B ξ A : S → S is the identity map on im ξ Aa ∩ ker ¯ η A , or dually, im ¯ η aA ∩ ker ξ A . Wetherefore have ¯ ξ B I BA = ¯ ξ A and ¯ ξ B ¯ I BA = 0, i.e. im ξ aA and im ¯ η aA contain h ¯ ξ A i and h η A i respectively. Hencethe map ι AB := I AB − ξ B η A is the identity on the ( m − S (0 , := { im ξ Aa : C V → S } ∩ { ker ¯ η A : C ← S } ∩ { ker ¯ ξ A : C ← S } , S (1 , − := { im ¯ η aA : C V → S ∗ } ∩ { ker ξ A : C ← S ∗ } ∩ { ker η A : C ← S ∗ } . As in the even-dimensional, one obtains isomorphisms (A.3). In addition, we have N ⊥ = C V ⊕ V (1 , ⊕ V (0 , ∼ = S ⊗ (cid:16) ¯ S ⊕ S (1 , − ⊕ S − (cid:17) , ( N ⊥ ) ∗ = C V − ⊕ V (0 , ⊕ V (0 , ∼ = ¯ S − ⊗ (cid:16) S − ⊕ S (0 , ⊕ S (cid:17) . (A.6)where S := h ξ A i and ¯ S := h ¯ ξ A i . In particular, the vector space [ V (0 , ] is spanned by the real unit vector u a := ξ aB ¯ η B + η aB ¯ ξ B . (A.7)The complex conjugation on C V interchanges V (1 , and V (0 , , and must therefore induce an involu-tion that interchanges S (1 , and S (0 , − , we shall denote this involution byˇ. A.1.3 Robinson -forms and Robinson -forms The pairing of the conjugate pair of pure spinors ξ A ′ (or ξ A ) and ¯ ξ A associated to a Robinson structure N and its complex conjugate ¯ N yields sim ( m − , C )-invariant k -forms as is shown in [KT92]. In other words,the tensor product S ⊗ ¯ S projects into ∧ k V for some k . When ǫ = 0, k has to be odd, but is otherwiseunrestricted. As we have already seen with (A.1) and (A.4), when k = 1, this pairing is simply the vector k a spanning the intersection K of N and ¯ N . When k = 2 ,
3, we obtain the real 3-form and 2-form ̺ abc := i ξ Babc ¯ ξ B , when ǫ = 0 , µ ab := ξ Bab ¯ ξ B , when ǫ = 1 only.Here ξ Babc = ξ A ′ γ BabcA ′ when ǫ = 0 and ξ Babc = ξ A γ BabcA when ǫ = 1. Using the purity condition, it isstraightforward to check that these forms satisfy ̺ abe ̺ ecd = 4 k [ a k [ c g b ] d ] − ǫµ ab µ cd , when ǫ = 0 , µ ac µ cb = k a k b , when ǫ = 1 only.These conditions are equivalent to writing ̺ abc = 3 k [ a ω bc ] , when ǫ = 0 , µ ab = 2 k [ a u b ] , when ǫ = 1 only,for some Hermitian 2-form ω ab on V , and when ǫ = 1, some unit vector u a orthogonal to ω ab . Since ω ab := ℓ c ̺ abc and u a := ℓ b µ ba for some choice of ℓ a dual to k a with k a ℓ a = 1, both ω ab and u a depend onthe choice of splitting of K ⊥ / K . On the other hand, both ̺ abc and µ ab are independent of such a choice. As in the even-dimensional case, we note that V is equipped with a positive definite metric tensor h ab , and we can identify S (1 , and S (0 , − as being subspaces of a chiral spinor representation for ( V , h ) and its dual respectively. emark A.1 In four dimensions, one has k a = ε bcda ̺ bcd , while in five dimensions, µ ab = ε cdeab ̺ cde ,which result in a number of simplifications in these dimensions – see appendix C. A.2 Tensor-spinor translation
As we have set up our notation and convention, primed spinor indices will not appear in even dimensions,so we can effectively combine the even- and odd-dimensional cases. This is due to the fact that our primaryobjects here are ξ aA and ¯ η aA , which we shall use to convert spinorial quantities into cu ( m − k a , and ℓ a , and in odd dimensions, u a as defined by (A.1), (A.4), (A.2), (A.5) and (A.7). Theremaining isomorphisms V (1 , ∼ = S ⊗ S (1 , − and V (0 , ∼ = S − ⊗ S (0 , allow us to realise the embeddingof u ( m − so ( n − S (1 , of u ( m −
1) and its dual S (0 , − . In general, if φ a ...a p b ...b q is in the cu ( m − V ( q,p )0 := (cid:16)N p V (0 , (cid:17) ⊗ (cid:16)N q V (1 , (cid:17) , then we can write φ a ...a p b ...b q = ξ a A . . . ξ a p A p ¯ η b B . . . ¯ η b q B q φ B ...B q A ...A p , for some φ B ...B q A ...A p in S ( q,p )0 := (cid:16)N p S (0 , − (cid:17) ⊗ (cid:16)N q S (1 , (cid:17) .The complex conjugate of φ a ...a p b ...b q is then simply given by¯ φ a ...a p b ...b q = ¯ η a A . . . ¯ η a p A p ξ a B . . . ξ a q B q ˇ φ A ...A p B ...B q , where ˇ φ A ...A p B ...B q belongs to S ( q,p )0 .An element of some irreducible u ( m − V ( q,p )0 will lie in the (invariant) kernel of someprojections from V ( q,p )0 to this irreducible submodule. Of particular interest is the identity element ι BA on S (1 , which has images ω ab = i (cid:16) ξ A [ a ¯ η b ] A − η A [ a ¯ ξ b ] A (cid:17) , H ab = ξ A ( a ¯ η b ) A − η A ( a ¯ ξ b ) A − k ( a ℓ b ) − ǫu a u b , in ∧ V and ⊙ V respectively. It is then not too difficult to show that a tensor lying in [ V (0 , ⊗ V (1 , ]is tracefree with respect to either ω ab and H ab if and only if its spinor representative in S (0 , − ⊗ S (1 , istracefree with respect to ι BA , i.e. φ BA ι AB = 0. Remark A.2
An alternative description of the above spinorial approach of a Robinson structure is toidentify the spinor module S with the vector space V • N . Then using the decomposition N = C V ⊕ V (1 , ,we have S ∼ = C V ⊗ m − M k =0 ∧ k V (1 , ! ⊕ m − M k =0 ∧ k V (1 , ! , and the spinors ξ A ′ and ¯ ξ A can be identified with generators of C V ⊗ (cid:16) ∧ m − V (1 , (cid:17) and C V respectively.Similarly, having fixed a dual N ∗ , we can identify the spinors ¯ η A ′ and η A ′ with C V − ⊗ (cid:16) ∧ m − V (0 , (cid:17) and C V − respectively, so that S (1 , ∼ = C V ⊗ (cid:16) ∧ m − V (1 , (cid:17) , S (0 , ∼ = V (1 , . This applies to both even and odd dimensions. 45
Tensorial and spinorial descriptions of irreducible sim ( n − -modules and sim ( m − , C ) -modules This appendix complements sections 2 and 3 by providing a number of explicit formulae to describe thevarious irreducible sim ( n − co ( n − sim ( m − , C )-and cu ( m − B.1 Projection maps
B.1.1 sim ( n − -invariant projection maps Here, we fix a null line K of an n -dimensional Minkowski space ( V , g ), so that V admits the sim ( n − k a of K . The sim ( n − g , F , A and C can then be described in terms of the kernels of the maps defined below. These project into themodules g ji , F ji , A ji and C ji on restriction to the associated graded sim ( n − g ), gr( F ), gr( A )and gr( C ) respectively. In the case of the Weyl tensor, these maps can be used to extend the Bel-Debevercriterion of [Ort09]. The Lie algebra so ( n − ,
1) For φ ab ∈ g , we define g Π − ( φ ) := k c φ c [ a k b ] , g Π ( φ ) := φ ba k a , g Π ( φ ) := φ [ a b k c ] . and when n = 6, g Π , ± ( φ ) := k [ d φ ef ] ± k a φ bc ε abcdef , where ε abcdef is a normalised volume 6-form. The tracefree Ricci tensor
For Φ ab ∈ F , we define F Π − (Φ) := k a k b Φ ab , F Π − (Φ) := k c Φ c [ a k b ] , F Π (Φ) := k [ a Φ b ][ c k d ] + 1 n − (cid:16) k [ a g b ][ c Φ d ] e k e + k [ c g d ][ a Φ b ] e k e (cid:17) , F Π (Φ) := k b Φ ab , F Π (Φ) := k [ a Φ b ] c . The Cotton-York tensor
For A abc ∈ A , we define A Π − ( A ) := k c k d A cd [ a k b ] , A Π − ( A ) := k b k c A bca , A Π − ( A ) := k [ a A b ] e [ c k d ] k e − k [ c A d ] e [ a k b ] k e + 1 n − (cid:16) k [ a g b ][ c | k e k f A ef | d ] − k [ c g d ][ a | k e k f A ef | b ] (cid:17) , A Π − ( A ) := k [ a A b ] e [ c k d ] k e + k [ c A d ] e [ a k b ] k e + 1 n − (cid:16) k [ a g b ][ c | k e k f A ef | d ] + k [ c g d ][ a | k e k f A ef | b ] (cid:17) , A Π ( A ) := 2 k d A ad [ b k c ] − k a k d A dbc , A Π ( A ) := 2 k d A ad [ b k c ] + k a k d A dbc , A Π ( A ) := k [ a A b ][ cd k e ] − n − g [ c | [ a (cid:16) k f A b ] f | d k e ] − k b ] A f | de ] k f (cid:17) − n − n − g [ c | [ a g b ] | d k f k g A fg | e ] , A Π ( A ) := A abc k c , A Π ( A ) := k [ a A b ] cd − k [ c A d ] ab + 2 n − (cid:16) g [ a | [ c A d ] | b ] e k e − g [ c | [ a A b ] | d ] e k e (cid:17) , A Π ( A ) := k [ a A b ] cd + k [ c A d ] ab + 2 n − (cid:16) g [ a | [ c A d ] | b ] e k e + g [ c | [ a A b ] | d ] e k e (cid:17) , he Weyl tensor For C abcd ∈ C , we define C Π − ( C ) := k [ a C b ] ef [ c k d ] k e k f , C Π − ( C ) := C ade [ b k c ] k d k e , C Π − ( C ) := k [ a C b c ] f [ d k e ] k f + 2 n − g [ a | [ d C e ] fg | b k c ] k f k g , C Π ( C ) := C acdb k c k d , C Π ( C ) := k [ a C b c ] de k e , C Π ( C ) := C abe [ c k d ] k e + C cde [ a k b ] k e − n − g [ c | [ a C b ] ef | d ] k e k f , C Π ( C ) := k [ a C b c ][ d e k f ] − n − (cid:16) g [ a | [ d C e f ] | g | b k c ] k g + g [ d | [ a C b c ] | g | e k f ] k g (cid:17) + 49( n − n − (cid:16) g d [ a C b ] gh [ e g f ] c + g d [ b C c ] gh [ e g f ] a + g d [ c C a ] gh [ e g f ] b + g e [ a C b ] gh [ f g d ] c + g e [ b C c ] gh [ f g d ] a + g e [ c C a ] gh [ f g d ] b + g f [ a C b ] gh [ d g e ] c + g f [ b C c ] gh [ d g e ] a + g f [ c C a ] gh [ d g e ] b (cid:17) k g k h , C Π ( C ) := C abcd k d , C Π ( C ) := k [ a C b c ] de + 2 n − g [ a | [ d C e ] | f | b c ] k f , C Π ( C ) := C abcd . B.1.2 sim ( m − , C ) -invariant projection maps We now fix a Robinson structure N on ( V , g ) with associated real null line K . In principle, given an irreducible so ( n − , R , say, one could define its sim ( m − , C )-submodules with respect to N as kernels ofprojection maps by means of certain algebraic operations with the sim ( m − , C )-invariant tensors k a , ̺ abc and µ ab defined in appendix A.1.3. If these maps are ‘saturated with symmetries’, they will correspond tosome irreducible sim ( m − , C )-modules of an associated graded module. In practice, the fact that we aredealing with a 3-form ̺ abc makes this approach rather cumbersome. To illustrate the point, we consider the sim ( m − , C )-invariant graph (2.11) of the Lie algebra g . We define, for φ ab ∈ g , g Π , ( φ ) := ̺ fab φ f [ c k d ] k e + ̺ ecd φ e [ a k b ] k e + 2 n − (cid:16) k [ c g d ][ a ̺ fgb ] φ fg + k [ a g b ][ c ̺ fgd ] φ fg (cid:17) k e + ǫ (cid:16) ̺ fab φ fg µ g [ c µ d ] e + ̺ fcd φ fg µ g [ a µ b ] e (cid:17) , g Π , ( φ ) := ̺ eab φ ef ̺ fcd + 4 k [ a φ b ][ c k d ] − ǫ (cid:16) µ ab µ e [ c φ d ] e − µ cd µ e [ a φ b ] e (cid:17) , and in addition, in odd dimensions, g Π , − ( φ ) := 4 k c φ c [ a k b ] + µ ab φ cd µ cd , g Π , − ( φ ) := µ ab φ ab , g Π , ( φ ) := k [ a φ b ] d µ dc , g Π , ( φ ) := φ ac µ cb , g Π , ( φ ) := φ [ ab µ c ] d . To give another example, one can give explicit expressions for the maps C Π j,ki of Proposition (4.12), byusing the spinorial expression for the integrability condition given in [HM88, TC12b, TC13]. When ǫ = 0, wecan recast the integrability condition (4.3) as [TC12b] C abcd ξ aA ξ bB ξ cC ξ dD = 0 , (B.1)where ξ A ′ is the spinor field annihilating the Robinson structure – we refer the reader to appendix A for thenotation. Contracting each free spinor index with ¯ ξ abA in equation (B.1), and using the identity ξ Ca ¯ ξ bcC = − i ̺ abc − g a [ b k c ] ,
47e find C Π , ( C ) := ρ iab ρ jcd ρ kef ρ lgh C ijkl + 4 ρ iab ρ jcd C ij [ e | [ g k h ] k | f ] + 4 k [ a | k [ c C d ] | b ] kl ρ kef ρ lgh + 16 k [ a | k [ c C d ] | b ][ e | [ g k h ] k | f ] − ρ jab k [ c C d ] jk [ e k f ] ρ kgh + 4 ρ jcd k [ a C b ] jk [ e k f ] ρ kgh + 4 ρ jab k [ c C d ] jk [ g k h ] ρ kef − ρ jcd k [ a C b ] jk [ g k h ] ρ kef = 0 . (B.2)The other cases are similar. B.2 Representatives
B.2.1 co ( n − -representatives Assume the existence of a null line K on ( V , g ), with generator k a , together with the choice of a vector ℓ a dual to k a with k a ℓ a = 1. In other words, V is equipped with a sim ( n − co ( n − k a ∈ V and ℓ a ∈ V − . The irreducible sim ( n − g ji , F ji , A ji and C ji are linearly isomorphic to irreducible co ( n − g ji , ˘ F ji , ˘ A ji and ˘ C ji respectively. To describeelements of these modules, we introduce the co ( n − E ab := − k [ a ℓ b ] , S ab := 2 k ( a ℓ b ) , and h ab := g ab − S ab . The Lie algebra so ( n − ,
1) Let φ ab ∈ g . Then • φ ab ∈ ˘ g if and only if φ ab = 2 k [ a φ b ] for some φ a such that φ a k a = φ a ℓ a = 0; • φ ab ∈ ˘ g if and only if φ ab = φ E ab for some real φ ; • φ ab ∈ ˘ g if and only if φ ab k a = φ ab ℓ a = 0.Elements of ˘ g − can be obtained from ˘ g by interchanging k a and ℓ a . The tracefree Ricci tensor
Let Φ ab ∈ F . Then • Φ ab ∈ ˘ F if and only if Φ ab = Φ k a k b for some real Φ; • Φ ab ∈ ˘ F if and only if Φ ab = 2 k ( a Φ b ) for some Φ a ∈ V ; • Φ ab ∈ ˘ F if and only if Φ ab = Φ (cid:16) S ab − n − h ab (cid:17) for some real Φ; • Φ ab ∈ ˘ F if and only if Φ ab k a = Φ ab ℓ a = 0;Elements of ˘ F j − i can be obtained from those of ˘ F ji by interchanging k a and ℓ a . The Cotton-York tensor
Let A abc ∈ A . Then • A abc ∈ ˘ A if and only if A abc = 2 k a k [ b A c ] for some A c ∈ V ; • A abc ∈ ˘ A if and only if A abc = a (cid:16) k a E bc − n − h a [ b k c ] (cid:17) for some real a ; • A abc ∈ ˘ A if and only if A abc = k a A bc − A a [ b k c ] for some A ab = A [ ab ] ∈ ˘ g ; • A abc ∈ ˘ A if and only if A abc = 2 A a [ b k c ] for some A ab = A ( ab ) ∈ ˘ F ; Equation (B.2) can be compared with its Hermitian analogue given in [Gra76, TV81, FFS94]. If J ba is a metric-compatiblecomplex structure, then its integrability condition is given by J ae J bf J cg J dh C abcd − J ae J bf C abgh − C efcd J cg J dh + C efgh − J a [ e C f ] ac [ g J ch ] = 0 , which can also be derived from the spinorial expression (B.1) with suitable reality conditions. A abc ∈ ˘ A if and only if A abc = A a E bc − E a [ b A c ] for some A c ∈ V ; • A abc ∈ ˘ A if and only if A abc = S a [ b A c ] − n − h a [ b A c ] for some A c ∈ V ; • A abc ∈ ˘ A if and only if A abc k a = A abc ℓ a = 0;Elements of ˘ A j − i can be obtained from those of ˘ A ji by interchanging k a and ℓ a . The Weyl tensor
Let C abcd ∈ C . Then • C abcd ∈ ˘ C if and only if C abcd = Ψ (cid:18) E ab E cd − E [ a | [ c E d ] | b ] + 6 n − (cid:16) h [ a | [ c S d ] | b ] (cid:17) − n − n − (cid:16) h a [ c h d ] b (cid:17)(cid:19) , for some real Ψ; • C abcd ∈ ˘ C if and only if C abcd = 2 E ab Ψ cd + 2 E cd Ψ ab − E [ a | [ c Ψ d ] | b ] for some Ψ ab = Ψ [ ab ] ∈ ˘ g ; • C abcd ∈ ˘ C if and only if C abcd = 2 S [ a | [ c Ψ d ] | b ] − n − (cid:16) h [ a | [ c Ψ d ] | b ] (cid:17) for some Ψ ab = Ψ ( ab ) ∈ ˘ F ; • C abcd ∈ ˘ C if and only if C abcd k a = C abcd ℓ a = 0; • C abcd ∈ ˘ C if and only if C abcd = 2 k [ a Ψ b ] E cd + 2 k [ c Ψ d ] E ab − n − (cid:16) h [ a | [ c Ψ d ] k | b ] + h [ c | [ a Ψ b ] k | d ] (cid:17) for some Ψ a ∈ V satisfying Ψ a k a = Ψ a ℓ a = 0; • C abcd ∈ ˘ C if and only if C abcd = k [ a Ψ b ] cd + k [ c Ψ d ] ab for some Ψ abc = Ψ a [ bc ] ∈ ˘ A ; • C abcd ∈ ˘ C if and only if C abcd = k [ a Ψ b ][ c k d ] for some Ψ ab = Ψ ( ab ) ∈ ˘ F .Elements of ˘ C j − i can be obtained from those of ˘ C ji by interchanging k a and ℓ a . B.2.2 cu ( m − -representatives Here, we assume the existence of a Robinson structure N with associated real null line K on ( V , g ) and adual Robinson structure N ∗ with associated real null line L dual to K , as described in appendix A. Thisfixes linear isomorphisms from the irreducible sim ( m − , C )-modules g j,ki , F j,ki , A j,ki , and C j,ki to irreducible cu ( m − g j,ki , ˘ F j,ki , ˘ A j,ki , and ˘ C j,ki . To describe elements of these cu ( m − co ( n − g , ˘ F , ˘ A and ˘ C , which may be seenas the buildling blocks of the other irreducible cu ( m − g ,i ± ∼ = V ± ⊗ ˘ V ,i , ˘ F ,i ± ∼ = V ± ⊗ ˘ V ,i , ˘ A ,i ± ∼ = V ± ⊗ V ± ⊗ ˘ V ,i , ˘ A ,i ± ∼ = V ± ⊗ ˘ g ,i , ˘ A ,i ± ∼ = V ± ⊗ ˘ F ,i , ˘ C ,i ± ∼ = V ± ⊗ ˘ V ,i , ˘ C ,i ± ∼ = V ± ⊗ ˘ A ,i , ˘ C ,i ∼ = ˘ g ,i , ˘ C ,i ∼ = ˘ F ,i . For instance, from Proposition 3.10, we know that the cu ( m − A , is isomorphic to V ⊗ ˘ g , .Thus, an element A abc of ˘ A , is simply given by the representative A abc = k a A bc − A a [ b k c ] , since A abc belongs to ˘ A , where A ab = A [ ab ] belongs to the irreducible cu ( m − g , .49e shall use the spinor notation introduced in appendix A to which the reader is referred. In particular,upstairs and downstairs upper case Roman indices will always refer to the ( m − S (0 , and S (1 , − of the spinor space respectively. We shall also use the short-hand c.c. to refer to the complexconjugate of a (complex) tensor in the usual sense – what this means ‘spinorially’ is explained in appendixA. We shall use the u ( m − J ba , H ab , ω ab = J ca H cb and u a that were defined in section2: J ca J bc = − H ba , H ab k a = H ab ℓ a = H ab u a = 0, u a u a = 1. They can also be constructed from the choice ofa pure spinor and its dual. Spinorial representatives for ˘ g Let φ ab ∈ ˘ g . Then • φ ab ∈ ˘ g , if and only if φ ab = φ ω ab for some real φ ; • φ ab ∈ ˘ g , if and only if φ ab = 2i ξ Ba ξ Cb φ BC + c.c. for some φ BC = φ [ BC ] ; • φ ab ∈ ˘ g , if and only if φ ab = 2i ξ B [ a ¯ η b ] D φ DB + c.c. for some tracefree φ DB ; • φ ab ∈ ˘ g , if and only if φ ab = 2 u [ a φ b ] for some φ a ∈ V , ; Spinorial representatives for ˘ F Let Φ ab ∈ ˘ F . Then • Φ ab ∈ ˘ F , if and only if Φ ab = 2 ξ A ( a ¯ η b ) B Φ BA + c.c. for some tracefree Φ BA ; • Φ ab ∈ ˘ F , if and only if Φ ab = ξ Aa ξ Bb Φ AB + c.c. for some Φ AB = Φ ( AB ) ; • Φ ab ∈ ˘ F , if and only if Φ ab = Φ (cid:16) u a u b − m − H ab (cid:17) for some real Φ; • Φ ab ∈ ˘ F , if and only if Φ ab = 2 u ( a Φ b ) for some Φ a ∈ V , . Spinorial representatives for ˘ A Let A abc ∈ ˘ A . Then • A abc ∈ ˘ A , if and only if A abc = A a ω bc − A [ b ω c ] a + m − H a [ b J dc ] A d for some A a ∈ V , ; • A abc ∈ ˘ A , if and only if A abc = ξ Aa ξ Bb ξ Cb A ABC + c.c. for some A ABC = A A [ BC ] such that A [ ABC ] = 0; • A abc ∈ ˘ A , if and only if A abc = ¯ η aA ξ Bb ξ Cc A ABC − ¯ η [ bA ξ Bc ] ξ Ca A ABC + c.c. for some tracefree A ABC = A A [ BC ] ; • A abc ∈ ˘ A , if and only if A abc = 2 ξ Aa ξ B [ b ¯ η c ] C A CAB + c.c. for some tracefree A CAB = A C ( AB ) ; • A abc ∈ ˘ A , if and only if A abc = a (cid:0) u a ω bc − u [ b ω c ] a (cid:1) for some real a ; • A abc ∈ ˘ A , if and only if A abc = 2 u a u [ b A c ] − m − H a [ b A c ] for some A a ∈ V , ; • A abc ∈ ˘ A , , ˘ A , if and only if A abc = u a A bc − u [ b A c ] a for some A ab ∈ ˘ g , , ˘ g , respectively; • A abc ∈ ˘ A , , ˘ A , if and only if A abc = 2 A a [ b u c ] for some A ab ∈ ˘ F , , ˘ F , respectively;50 pinorial representatives for ˘ C Let C abcd ∈ ˘ C . Then • C abcd ∈ ˘ C , if and only if C abcd = Ψ (cid:16) ω ab ω cd − ω a [ c ω d ] b − m − H a [ c H d ] b (cid:17) for some real Ψ; • C abcd ∈ ˘ C , , ˘ C , if and only if C abcd = ω ab Ψ cd + Ψ ab ω cd − ω [ a | [ c Ψ d ] | b ] − m − (cid:16) H [ a | [ c J ed ] Ψ | b ] e + H [ c | [ a J eb ] Ψ | d ] e (cid:17) for some Ψ cd = Ψ [ cd ] ∈ ˘ g , , ˘ g , respectively; • C abcd ∈ ˘ C , if and only if C abcd = ξ Aa ξ Bb ξ Cc ξ Dd Ψ ABCD + c.c. for some Ψ ABCD = Ψ [ AB ][ CD ] satisfyingΨ [ ABC ] D = 0; • C abcd ∈ ˘ C , if and only if C abcd = ξ Aa ξ Bb ¯ η cC ¯ η dD Ψ CDAB + ξ Ac ξ Bd ¯ η aC ¯ η bD Ψ CDAB − ξ A [ a | ξ C [ c ¯ η d ] | D ¯ η b ] B Ψ DBAC + c.c. for some tracefree Ψ DBAC = Ψ [ DB ][ AC ] ; • C abcd ∈ ˘ C , if and only if C abcd = ξ A [ a | ξ C [ c ¯ η d ] | D ¯ η b ] B Ψ DBAC + c.c. for some tracefree Ψ DBAC =Ψ ( DB )( AC ) ; • C abcd ∈ ˘ C , if and only if C abcd = ξ Aa ξ Bb ξ C [ c ¯ η d ] D Ψ DABC + ξ Ac ξ Bd ξ C [ a ¯ η b ] D Ψ DABC + c.c. for some tracefreeΨ DABC = Ψ D [ AB ] C satisfying Ψ D [ ABC ] = 0; • C abcd ∈ ˘ C , if and only if C abcd = ω ab Ψ [ c u d ] + ω cd Ψ [ a u b ] − ω [ a | [ c Ψ d ] u | b ] − ω [ c | [ a Ψ b ] u | d ] + 32 m − (cid:16) H [ a | [ c u d ] J e | b ] Ψ e + H [ c | [ a u b ] J e | d ] Ψ e (cid:17) for some Ψ a ∈ ˘ V , ; • C abcd ∈ ˘ C , , ˘ C , if and only if C abcd = u [ a Ψ b ][ c u d ] + m − H [ a | [ c Ψ d ] | b ] for some Ψ cd = Ψ ( cd ) ∈ ˘ F , , ˘ F , respectively; • C abcd ∈ ˘ C , , ˘ C , , ˘ C , if and only if C abcd = u [ a Ψ b ] cd + u [ c Ψ d ] ab for some Ψ abc = Ψ a [ bc ] ∈ ˘ A , , ˘ A , , ˘ A , respectively. C Low dimensions
There are a number of simplifications that can be made in the classification of curvature tensors in lowdimensions, notably up to dimension six. In the following, we give a brief description of the special featuresof dimensions four, five and six.
C.1 Four dimensions
In four dimensions, a Robinson structure is equivalent to a choice of a null line K , which is a reflection of theisomorphisms of Lie algebras sim (2) ∼ = sim (1 , C ) and so (2) ∼ = u (1). As a consequence, the sim (1 , C )-invariantclassification of the Weyl tensor reduces to its sim (2)-classification. The corresponding sim ( m − , C )-invariantgraph then reads C ( ( ❘❘❘❘❘❘ C / / C ( ( ◗◗◗◗◗◗ ♠♠♠♠♠♠ C − / / C − . C ❧❧❧❧❧❧ C.2 Five dimensions
In five dimensions, V is three-dimensional, so we can identify the Hermitian 2-form on V with the unitvector orthogonal to it. Thus, a Robinson structure can be determined by a choice of a null line k a andan equivalence class of spacelike subspaces of V / V . For clarity, the diagram below is the five-dimensionalspecialisation of diagram 2: C , C , C , C , C , C , C , C , C , C , C , C , C , C , C , − C , − C , − C , − C , − C , − C , − C , − ✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞ C C ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ $ $ ttttttttttttttt : : ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ (cid:27) (cid:27) ttttttttttttttt : : ✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞ C C / / ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ $ $ ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ (cid:27) (cid:27) rrrrrrrrrrrrrrrrrrrrrrrrrrrr ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻ (cid:26) (cid:26) ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ , , ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ (cid:31) (cid:31) ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ % % ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ (cid:31) (cid:31) ✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻ (cid:26) (cid:26) rrrrrrrrrrrrrrrrrrrrrrrrrrrr ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ % % rrrrrrrrrrrrrrrrrrrrrrrrrrrr ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ , , ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ? ? ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ , , ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ % % ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ , , ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ % % ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ rrrrrrrrrrrrrrrrrrrrrrrrrrrr ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ % % ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ (cid:31) (cid:31) ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ D D ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ? ? ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ , , ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ? ? ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ , , ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ D D rrrrrrrrrrrrrrrrrrrrrrrrrrrr ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ (cid:27) (cid:27) ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ $ $ ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ $ $ ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ (cid:27) (cid:27) / / ✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞ C C ttttttttttttttt : : ttttttttttttttt : : ✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞ C C C.3 Six dimensions
In six dimensions, the semi-simple part so (4 , R ) in the Levi decomposition of sim (4) splits further as so (4 , R ) ∼ = su + (2) ⊕ su − (2), where su ± (2) are two copies of the simple Lie algebra su (2), which can beidentified with the self-dual part and anti-self-dual part of so (4 , R ). Here, the Hodge duality operator on so (4 , R ) is given by ε efabcd E ef for any choice of grading element E ab .Such a splitting of sim (4) is automatically taken care in the sim (2 , C ) decomposition of g . To be precise,we can make the identifications su + (2) = [[ g (2 , ]] ⊕ g ω and su − (2) = [ g (1 , ◦ ]. We can thus ‘regroup’ some ofthe sim (2 , C )-modules into self-dual and anti-self-dual sim (4)-modules occuring in the classifications of theCotton-York tensor and of the Weyl tensor as follows: A , + ± ∼ = A , ± ⊕ A , ± , A , −± ∼ = A , ± , A , +0 ∼ = A , ⊕ A , , A , − ∼ = A , , C , + ± ∼ = C , ± ⊕ C , ± , C , −± ∼ = C , ± , C , +0 ∼ = C , ⊕ C , , C , − ∼ = C , , C , +0 ∼ = C , ⊕ C , ⊕ C , , C , − ∼ = C , . eferences [AC02] A. M. Awad and A. Chamblin, A bestiary of higher-dimensional Taub-NUT-AdS spacetimes , Classical QuantumGravity (2002), no. 8, 2051–2061. MR1901703 (2003g:53133)[AG97] V. Apostolov and P. Gauduchon, The Riemannian Goldberg-Sachs theorem , Internat. J. Math. (1997), no. 4,421–439.[Awa06] A. M. Awad, Higher dimensional Taub-NUTs and Taub-bolts in Einstein-Maxwell gravity , Classical QuantumGravity (2006), no. 9, 2849–2859. MR2220862 (2007j:83036)[BDS12] M. L. Barberis, I. G. Dotti, and O. Santill´an, The Killing-Yano equation on Lie groups , Classical QuantumGravity (2012), no. 6, 065004, 10. MR2902941[BE89] R. J. Baston and M. G. Eastwood, The Penrose transform , Oxford Mathematical Monographs, The ClarendonPress Oxford University Press, New York, 1989. Its interaction with representation theory, Oxford SciencePublications.[Bry00] R. L. Bryant,
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