On rigid supersymmetry and notions of holomorphy in five dimensions

We study the equations governing rigid N=1 supersymmetry in five dimensions. If the supersymmetry spinor satisfies a reality condition, these are foliations admitting families of almost complex structures on the leaves. In other words, all these manifolds have families of almost Cauchy-Riemann (CR) structures. After deriving integrability conditions under which circumstances the almost CR structure defines a CR manifold or a transversally holomorphic foliation (THF), we discuss implications on localization. We also discuss potential global obstructions to the existence of solutions.


I. INTRODUCTION
The work of Festuccia and Seiberg [1] provides an easy to implement recipe to construct gauge theories with rigid supersymmetry on curved manifolds. One picks an (off-shell) supergravity theory coupled to the vector and matter multiplets and solves the Killing spinor equations arising from the gravitino and dilatino variations. These in turn impose constraints on the auxiliary fields that then appear as parameters in the lagrangian of the field theory once one has decoupled gravity using a suitable scaling limit. Using supergravity techniques, the approach allows for a classification of admissible spacetime manifolds [2][3][4][5][6][7][8].
In five dimensions, initial progress came not so much from the rigid limit of supergravity, as from the direct construction of supersymmetric field theories on increasingly intricate manifolds [9][10][11]. Here, a key observation was that the theory on the five-sphere can be readily generalized to generic Sasaki-Einstein manifolds and even to manifolds that support only a K-contact structure. Contact with the formalism of the rigid limit was made by Imamura and Matsuno [12] who solved the Killing spinor equations of the N = 1 supergravity of Kugo, Ohashi and Zucker [13][14][15] locally, as well as in [16] where the equations arising from the gravitino were discussed in the context of the same supergravity theory. For maximal supergravity in d = 5 consider [17].
Of course, the interest in rigid supersymmetry goes hand in hand with the idea of localization following [18]. In this context, the Sasakian case has proven to be interesting. Sasakian manifolds, being the odd dimensional cousin of Kähler ones, support something akin to an integrable complex structure on the space transverse to the Reeb vector R. That is, the tangent bundle admits the decomposition This is integrable in a sense that we will discuss shortly and it follows that one can introduce differentials ∂ b and∂ b that correspond to the Dolbeault operators ∂ and∂ that are familiar from complex geometry. In [19,20] this was used in order to simplify the calculation of the perturbative part of the supersymmetric partition function and to solve the BPS equations on the Higgs branch. This note focusses on the question to what extend this notion and use of holomorphy can be extended to general five-dimensional backgrounds admitting rigid N = 1 supersymmetry. Our analysis is based on the gravitino and dilatino equations of [13,14] which in our conventions and in Euclidean signature are and 4 D m t I J Γ m + t I J (F + 2V) mn Γ mn ξ J + 4∇ m V mn Γ n + F mn F kl Γ mnkl + C ξ I = 0 Here, I = 1, 2 are indices for the fundamental representation of SU (2) R . F = dA is a U (1) field strength and V an antisymmetric tensor. The triplet t J I is valued in the adjoint representation of SU (2) R . The covariant derivatives are D m ξ I = ∇ m ξ I − A J mI ξ J and D m t J I = ∇ m t J I − [A m , t] J I . For later convenience, note that (2) can also be rewritten as In the Lorentzian theory, the spinors ξ I satisfy a symplectic Majorana condition (A1). Transitioning to the Euclidean theory one usually drops such reality conditions and effectively doubles the degrees of freedom of all fields involved. In general, the spinor ξ I defines a possibly complex vector R. Imposing the reality condition (A1) for ξ I it follows that R is real and non-vanishing and that the tangent space decomposes as in (1), which one refers to as an almost Cauchy-Riemann (CR) structure (of hypersurface type) [21].
In opposite to the familiar case of almost complex structures, the integrability condition for (1) is not unique. Indeed, there are two possibilities. To begin, there is the case of a integrable CR structure, that defines a CR manifold. CR manifolds have previously appeared in the context of the rigid limit of new minimal supergravity with Lorentzian signature in [6]; there, the authors found fibrations of the real line over three dimensional CR manifolds. Alternatively, there is the condition which defines a transversally holomorphic foliation (THF). 1 The work of [2] relates rigid supersymmetry in three dimensions with the existence of a THF. Note that Sasakian manifolds fulfill both (5) and (6) as here [R Sasakian , T 1,0 ] ⊆ T 1,0 . Naturally, the question whether solutions to the Killing spinor equations (2) and (3) admit integrable CR structures of THFs is closely related to the question whether a given five dimensional manifold M admits any solution in the first place. As we alluded above, this question was already addressed in [12] and [16], but not exhaustively answered. As we will see, existence of a solution to the Killing spinor equations that satisfies the symplectic Majorana condition implies the existence of a globally non-vanishing Killing vector field parallel to R. We will show that the existence of such a Killing vector field is not only necessary, but also sufficient. While we will do so by directly constructing a single solution and arguing that there are no topological obstructions, one can already give a short argument why one should be able to expect this result. The existence of a non-vanishing vector field implies that M admits an SO(4) structure. Since the theory has an SU (2) R symmetry, one can perform an operation akin to a Witten twist in four dimensions and identify the the SU (2) R with an SU (2) factor inside the structure group.
The structure of this note is as follows: The relation between the supersymmetry spinor ξ I , almost CR-structures and almost contact structures is the topic of section II. Then, we will discuss the integrability of the Killing spinor equations, possible obstructions and general differential properties of (2) and (3) in section III. Section IV is concerned with the implications for localization. We will argue that the results of [16,19] can be generalized to CR-manifolds and THFs. Subsequently we discuss the existence of globally well-defined solutions (section V) before concluding with some examples from the literature in section VI. Various appendices complement the discussion.
During the final stages of this project [24] appeared, which has some overlap with our work. There, the authors study rigid supersymmetry on Riemannian five-manifolds using a holographic approach.

II. ALGEBRAIC PROPERTIES
In this section we will discuss the algebraic structures arrising from the existence of the spinors ξ I .

A. The Almost Contact Structure
The bulk of our analysis is based on the following set of bi-spinors that can be defined for any given ξ I : Let us emphasize the presence of the minus sign as well as the normalizing factor s −1 in the definition of R where we tacitly assume that s = 0. If one imposes the symplectic Majorana condition (A1) one finds that s and R are a real function and a real vector field respectively. A further consequence of (A1) is that s ≥ 0 with equality if and only if ξ I = 0. It follows that s > 0 everywhere on M since the gravitino equation is linear and of first order. Finally, the two forms Θ IJ lie in the adjoint representation of SU (2) R . Using Fierz-identities, one can show the following identities involving the bispinors: Here, * is the usual five-dimensional Hodge dual and ι R denotes interior multiplication. The first of the above equations tells us that M carries an SO(4) structure. This allows us to introduce a lot of structure that is familiar from four-dimensional geometry. As is usual, we will refer to vectors and forms parallel to R and κ respectively as vertical and their orthogonal complement as horizontal. I.e. forms can be decomposed as ω = ω H + ω V . Then the Hodge dual defines the notion of self-dual and anti self-dual forms on the horizontal subspace. See equations (A4) and (A5) for full definitions. Since the Θ IJ are both horizontal and self-dual, Θ IJ = (Θ IJ ) + , they define an isomorphism between su(2) R and the su(2) + factor in the typical so(4) ∼ = su(2) + × su(2) − decomposition of the Lie algebra of the structure group. One can also verify some more involved identities involving Θ IJ : Here Π mn = g mn − κ m κ n and thus the latter of these is a projection to horizontal, self-dual two-forms. Suppose now that m IJ is an SU (2) R triplet. Later we will show that m IJ = t IJ emerges naturally when imposing integrability and we will refer to this as the canonical choice. Yet for now, we continue with a generic m IJ and define 2 det m ≡ −1/2 IJ m I J m J I . Once we impose the reality condition (16) for m IJ , det m will be positive semi-definite. For now we proceed with the milder assumption det m = 0 and define the following tensor As follows from (9), Φ satisfies the following condition: Mathematicians refer to a multiplet (κ, R, Φ) as an almost contact structure if As we have shown, the quantities defined using ξ I and a suitable m IJ satisfy these relations, and therefore define an almost contact structure. Note that Φ is invariant under m IJ → f m IJ for any non-zero function f .

B. The Almost CR Structure
Equations (11) and (12) indicate that for each m, Φ[m] defines an almost CR structure. Indeed, each Φ[m] induces a decomposition of the complexified tangent bundle as in (1) via The decomposition holds also for the exterior algebra and all horizontal n-forms ω = ω H can be decomposed into (p, q)-forms via 2 Note that In this context Φ mn is a horizontal (1, 1)-form. Similar to the case of four-dimensional Kähler manifolds, self-dual and anti-self-dual 2-forms have a simple (p, q)-decomposition, with ω 1,1 primitive and thus annihilated by contraction with Φ. We continue by discussing the integrability of the almost CR structure (1). While this can be done using a direct analysis of the Niejenhuis tensor, we prefer to do a spinorial analysis in the spirit of [1]. 3 This is computationally more straight forward, yet requires us to impose the reality condition for the triplet which we alluded to previously. The bar denotes complex conjugation. Let us emphasize that we are also using the symplectic Majorana condition since we assume R to be real.
In appendix B we show that one can characterize elements of T 1,0 in terms of a spinorial equation: where Similarly, one can also characterize the tangent vectors in T 1,0 ⊕ RR by the spinorial equation Recall that Π m n = δ m n − R m κ n is a projection that maps a generic tangent vector to its horizontal component.

III. DIFFERENTIAL PROPERTIES
We finally turn to the integrability conditions for the decomposition (1). To do so, we will first establish some useful identities involving the bispinors (7) and the gravitino (2) and dilatino (3) variations. Subsequently we consider the case of CR structures as a warm-up before studying the integrability conditions for THFs.

A. Supersymmetry variations and bispinors
Studying the gravitino variation (2), one finds that the scalar s satisfies ∇ n s = 2sR m F mn , from which it follows that L R F = L R s = 0. Similarly, the non-normalized vector field sR is Killing: One also finds that There is a more involved relation involving the two-form t IJ Θ IJ : Similarly we are interested in the consequences of the dilatino equation (3) for bispinors and background fields. By contraction with t IJ ξ I one finds that R m ∇ m (t IJ t IJ ) = 2L R det t = 0. Contraction with ξ I on the other hand fixes the value of the scalar, We can extract additional information from the dilatino equation and start by projecting it onto its "chiral" components. Recalling the last identity in (8) we consider the projector 1 2 (1 − R m Γ m ). Acting on (3) and using (22), one finds A related identity can be obtained by contracting (3) with ξ I Γ mn and projecting onto the horizontal subspace: where (η × ω) mn = η p m ω pn − ω p m η pn . In passing, one needs to use the simple identity (A6). As a point of consistency note that one can obtain the same result by contracting (23) with ξ I Γ mn and again projecting onto the horizontal part.

Cauchy-Riemann structures
Having established the existence of the almost CR structure (1), it is natural to ask if it satisfies any integrability condition. As a warm-up to the integrability condition of a THF (6), we consider the slightly simpler case of a CR structure (5).
Thus we study the condition (17) for the commutator [X, Y ] for arbitrary X, Y ∈ T 1,0 . I.e. by acting with Y n D n on (17) and antisymmetrizing in X, Y , one finds that This reduces quickly to Per usual, (26) can be mapped to two equations by suitable contractions.
To begin, we contract (26)  Finally, we contract with ξ J Γ k : By symmetrizing and antisymmetrizing over I and J, it is clear that both terms in parantheses have to vanish independently. It follows that D X H J I = 0. In summary, the almost CR-structure is integrable and the manifold is CR if and only if Note that due to our reality condition for t J I , the last statement is actually equivalent to where T M H = T 1,0 ⊕ T 0,1 .

Transversally holomorphic foliations
Having discussed integrable CR structures, we now turn to the integrability condition for transversal holomorphic foliations (6). Using identical arguments to those from the previous section, we note that the integrability condition is To begin, consider (30) for X, Y ∈ T 1,0 . Direct substitution gives Now, since X, Y ∈ T 1,0 , the only contributions to the last term arise from the components of ⋆V that lie in Ω 2,1 ⊕ Ω 2,0 ∧ R. However, since (Γ k + R k )ξ I = Π k l Γ l ξ I the latter of these is annihilated by the projection while the former vanishes due to holomorphy -i.e. for any ω ∈ Ω 0,1 , H J I ω k Γ k ξ J = 0. Thus we are left with Once again, contraction with ξ I gives the first necessary condition, ([H, t] IJ Θ IJ ) 2,0 = 0, from which it follows once again that m J I = f t J I . Just as in the CR case the second condition is D X H J I = 0, ∀X ∈ T 1,0 . We continue our analysis of (30) by considering X ∈ T 1,0 and Y = R. Using the results from the previous paragraph, one finds that the necessary and sufficient condition is the vanishing of By inspection one finds that the only contributing terms including F or V lie in Ω 2,0 -(Ω 1,0 ⊕ Ω 0,1 ) ∧ R as well as Ω 0,2 components are projected to zero while those in Ω 1,1 vanish due to holomorphy. The components in Ω 2,0 are of course self-dual under ι R ⋆ so the above can be rewritten in terms of F + 2V instead of F + 2ι R ⋆ V.
To further simplify this, we consider the chiral projection of the dilatino equation (23). Acting with X m H J I Γ m on (23) one finds that Now, we first note that As before we argue that only the Ω 2,0 and Ω 0,2 terms contribute. Thus we find that (33) vanishes without any further conditions. In the end, the integrability conditions are As in the case of the CR structure the reality condition for t J I implies that the last condition holds for all horizontal sections of the tangent bundle. By comparison with equation (28) it is clear that any solution defining a THF also defines an integrable CR structure while the converse is not the case.
From d = ∂ b +∂ b + d v and d 2 = 0 it follows directly that ∂ 2 b =∂ 2 b = 0 and one can define cohomology groups H p,q ∂ b via the exact sequence

B. Mode counting and partition functions
As mentioned in the introduction, partition functions for supersymmetric gauge theories calculated in the context of topological field theories or localization simplify significantly on Kähler and Sasakian manifolds. The argument relies not only on the existence of the differential∂ b (∂ in the Kähler case). Indeed, one also requires the compatibility of the decomposition (1) with the action of the Lie derivative £ sR . In this section we will go over this argument of [19,25] in some detail and discuss under what circumstances it applies to the manifolds in question.
Consider a vector multiplet with Lie algebra g. The bosonic modes lie in Ω 1 (g)⊕H 0 (g)⊕H 0 (g), where H 0 (g) denotes harmonic Lie algebra valued functions. Fermionic modes on the other hand can be mapped to Ω + (g) ⊕ Ω 0 (g) ⊕ Ω 0 (g). The one-loop contribution to the perturbative partition function is given by 4 If £ sR Φ = £ sR κ = 0 we can calculate the determinants using the decomposition (1). Clearly £ sR κ = 0, so we need to evaluate £ sR Φ = ι sR dΦ. Direct calculation using (21) yields Thus where we used (24). In conclusion we can rewrite (41) as where we used the notation det Ω p,q £ sR = det £sR Ω p,q and dropped the various appearances of g for readability. As [£ sR ,∂ b ] = 0 it follows that the above simplifies to It is interesting to note that the above argument does not require a property akin to Lefschetz decomposition on Kähler manifolds. Recall that the Lefschetz theorem relates cohomology groups of the Dolbeault operator as H 0,0 ∂,ω , where the subscript ω denotes forms parallel to the symplectic form ω. Such a decomposition, while true for e.g. Sasaki-Einstein manifolds does not hold in general for the operator Φ. That is, for α ∈ Ω 1,1 Φ one can write α = aΦ for some scalar function a, yet∂ b α = 0 is not in one-to-one correspondence with∂ b a = 0 since∂ b Φ does not vanish in the general case.

C. BPS equations on the Higgs branch
The nilponency of∂ b has also immediate implications on the Higgs branch BPS equations of N = 1 theories. In [20] these were studied for supersymmetric backgrounds that are K-contact. Defining∂ a ≡∂ b − ıa 0,1 for a U (1) connection a with field strength F a , some of the relevant equations arē Here, α is a 0-form and β is a (0, 2)-form; both are related to the scalar in the hypermultiplet. The superscript dκ denotes the component along dκ. The BPS equations and the nilpotence then imply that∂ a∂ * a β = −∂ a∂a α = ıF 0,2 a α = −2|α| 2 β. Thus 2 |α| 2 |β| 2 + |∂ * a β| 2 = 0, and it follows that In other words, similar to our discussion in the previous section we see that results for Sasaki (-Einstein) manifolds can be extended to geometries that are either THF or CR.

V. A KARLHEDE-ROCEK-WITTEN TWIST IN FIVE DIMENSIONS
As discussed above in section III A as well as in [12] a necessary condition for the existence of a solution of the background supergravity variations for supersymmetry spinors satisfying the symplectic Majorana condition is the existence of a Killing vector. Recall that the symplectic Majorana condition (A1) implies that s > 0 from which it follows that v has no zeroes. In other words, the Killing vector is globally non-vanishing. 5 In this section we will show that the existence of a globally non-vanishing Killing vector is also sufficient for the manifold M to admit supersymmetry spinors that solve (2) and (3). 6 At the heart of the argument is the idea that the existence of the vector implies that the manifold supports an SO(4) structure. This in turn allows us to do a standard Witten twist [26,27]. Our strategy is to work in a patch using methods familiar from Kaluza-Klein reduction, yet show that we can write the overall result in terms of globally well-defined objects. In principal one should be able to make the same argument using the general, local solution of [12].
Given a manifold M with a Killing vector v = ∂ τ we can write the vielbein as 7 I.e. the metric takes the form ds 2 = g mn dx m dx n + k 2 (dτ + a) 2 , where ∂ τ = v. The spin connection iŝ Here, f = da. Keeping in mind (8), we demand the spinor ξ I to be anti-chiral. That is, Γ 5 ξ I = −ξ I which is why we write ξ I ≡ ξ − I .

A. Gravitino Equation
One can then decompose the gravitino variation (2) into components along a = 1, . . . 4, components along a = 5 as well as chiral and anti-chiral parts: 5 If one does not impose the symplectic Majorana condition, the situation is more complicated. I.e. both s and R are generally complex; it is also clear that the vector vanishes if the spinors are parallel. Moreover, note that R does not even vanish at a single point. Assume ∃p ∈ M such that R|p = 0. It follows that s(p) = 0 and thus ξ I |p = 0. From the gravitino equation it follows immediately that ξ I vanishes identically on M. 6 We would like to thank Diego Rodriguez-Gomez for many discussions and collaboration that lead to the approach used in this section. 7 In this section, greek indices run from one to five while roman ones only run from one to four.
The last of these, (53), is solved by A = − 1 2 k −1 v. It follows that Equation (50) is solved by setting A J τ I = 0, ξ − I = √ kχ I , where χ I χ I = 1, and -more importantly -D a χ I = ∇ a χ I − A J aI χ J = 0. The possibility of finding a χ such that ∇ a χ I = A J aI χ J is of course at the heart of this argument. As long as Γ 5 χ I = −χ I , it is possible to find such a spinor; explicit calculations can be done using 't Hooft matrices for example [28]. With all our previous assumptions and observations (51) becomes Substituting this into (52) we find that t IJ = 0 since In summary, the gravitino equation is fully solved by By now it is clear that the spinor bilinears s, v coincide with the scalar and vector defined by the background, k, v, i.e. s = k, v = v, so we drop the distinction.

B. Dilatino Equation
Performing a similar decomposition of the Dilatino equation, one finds Imposing the solution to the gravitino equations (57), this simplifies of course considerably. Also, note that Then, (59) is solved by C = − 1 4 k 2 f ab f cd ǫ abcd5 . Since 4s∇ b V ba = − 1 2 sǫ abcd5 f bc ∂ d k, one finds that (58) is solved trivially.

C. Topological Issues
To conclude, we discuss whether the solution (57) is globally well-defined. Since F is globally exact we only have to worry about the SU (2) R field strength. Our strategy is to rewrite this in terms of the Riemann tensor. Thus we use the integrability condition for the spinor χ I , This implies F IJ ab = −2s −1 R abκλ Θ IJκλ from which it follows that we can express the SU (2) R connection in terms of a projection of the Riemann tensor. In summary, the two connections are where R H µν = Π σ κ Π τ λ R στ µν dx κ ⊗ dx λ denotes the horizontal part of the curvature two-form. In both cases, all objects appearing on the right hand side are globally well defined. We proceed to consider characteristic classes defined by F IJ . Using (9) one finds that The expression is completely horizontal and since v is Killing, 0 = £ v (F J I ∧ F I J ) = ι v d(F J I ∧ F I J ) from which it follows that (63) is closed and defines thus an element of the de Rham cohomolgy group H 4 (M) as it should. Usually the next question would be whether this element is trivial and whether it might be an obstruction to the existence of the solution given by (57) and C. However, equation (63) clearly show that this class has a representative that is independent of our specific solution since it can be expressed in terms of v and the Riemann tensor. Thus, in the case that the class is non-trivial, it is clear that the corresponding cycle in homology exists and vice versa.
One might worry about the f dependence of V. In general, the manifolds are not bundles yet only foliations and one cannot necessarily think of f as the curvature of a connection. Yet as we saw above, f is a projection of F onto the horizontal space -f = −2k −1 F H . While one might not consider f globally as the curvature of a connection, it is well-defined as a two-form. Since it doesn't enter the solution directly yet only via V, this is good enough and we conclude that any manifold M admits a solution to (2) and (3) with symplectic Majorana spinor if and only if there is a non-vanishing Killing vector v.

VI. EXAMPLES
It follows from the previous section that any direct product R × M 4 or S 1 × M 4 admits a solution to the Killing spinor equations and thus rigid supersymmetry. Similarly, it is clear that such manifolds do at least not trivially 8 admit an integrable CR-structure or a THF if M 4 does not admit a complex structure -the example coming to mind here being R × S 4 . See however the discussion in [24].

A. Sasakian manifolds
Sasakian manifolds are the odd-dimensional analogs of Kähler manifolds. They are either characterized by having Kähler metric cones, or by the existence of a Killing spinor satisfying Here, A is the connection one-form associated to the Ricci-form on the metric cone. The equation and its complec conjugate corresponds to the special case of (2) with Since both t and A have only components along σ 3 one finds that ∇ m t IJ = 0. The dilatino equation is solved by Hence, N = 1 supersymmetry can be defined on any 5-dimensional Sasakian structure as was first observed without resorting to supergravity [11]. Sasakian structures are examples of both Cauchy-Riemann or transversal-holomorphic structures, as follows from the fact that ∇ m t IJ = F = V = 0.
Squashed five-spheres have appeared in various literatures. In particular, [29,30] discussed a class of squashed S 5 , defined by metric Our discussion follows that of [30] closely. The real constant b is the squashing parameter, which gives a round sphere when b = 1, h is a 1-form defined as where ω can be viewed as the Kahler form on CP 2 , satisfying dω = 0. The metric is written in a form adapted to the smooth U (1)-fibration over CP 2 , where b −2 (dτ + h) 2 is the metric in the U (1)-fiber direction, and b is there to squash the radius. In this way it is easy to see the metric has U (1) × SU (3) one finds where we have introduced the left-invariant one forms This class of squashed sphere admits solutions to the Killing spinor equations where Q is a real parameter. And of course one can define bilinears as in (7). In terms of (A8), the quarter BPS solution with Q = −3 is given by The symplectic Majorana condition (A1) corresponds to (c − ) * = c + . For more involved 3/4 BPS solutions refer to [30]. Comparing (2) with (72) one identifies (74) Note that κ = −e 5 and one finds that ω is horizontal and self-dual since ⋆ω = κ ∧ ω. Furthermore dκ = −2b −1 ω and ∇ m ω mn = 4b −1 κ n . Moreover ω mn ω kl Γ mnkl ξ I = ω mn ω kl ǫ mnkl r Γ r ξ I = 2ω mn ( * ω) mn r Γ r ξ I = 2ω mn ω mn κ r Γ r ξ I = −8ξ I .
Finally, substituting everything into the dilatino equation (3), one finds From (22) it follows that so the above simplifies to which vanishes identically for the above solution. Now compare the "algebraic equation" of [30]. Rewritten in our conventions, it is where we used (8). Contracting with ξ I one finds (σ 3 ) J I ω mn Θ I mnJ = 2ıs(1 + Q). Substituting this into (78) yields (79), which tells us that the Dilatino equation and the "algebraic equation" are equivalent in the case of squashed S 5 .
Comparing (74) with (28) and (36) it is clear that the squashing does not change the fact that S 5 admits both a CR-structure and a THF. In principle this is already clear from the form of the metric (67) since changes in the parameter b do not affect the CP 2 base of the bundle. 12 (A7) where the first and the second equality in the second line come from applying on the first line the self-duality on ω + and the anti-self-duality on ω − .
The charge conjugation matrix is C = Γ 24 .
2. £sRΦ and [T 1,0 , R] In section IV we showed gravitino and dilatino equations imply that for the canonical almost CR structure £ sR Φ = 0. For any X ∈ T M it follows that In other words, we have confirmed that the canonical almost CR structure defines a THF as long as the triplet t J I is covariantly constant; i.e. equation (36).