aa r X i v : . [ h e p - t h ] N ov QMUL-PH-10-17 On AdS × C P T-duality
Ilya Bakhmatov ∗ Queen Mary University of LondonCentre for Research in String TheoryDepartment of PhysicsMile End Road, London, E1 4NS, England
Abstract
We give a supergravity treatment of the set of bosonic and fermionic T-dualities in the
AdS × C P background. We consider T-dualities along three flat AdS directions, threecomplexified isometries of C P , and six fermionic T-dualities. Concentrating on thetransformation of the dilaton, we give description of the singularity that arises in thetransformation. ∗ [email protected] ontents C P
43 Killing vectors 64 Killing spinors 85 Symmetry superalgebra 96 Fermionic T-duality 117 Summary 13A Fermionic T-duality in type IIA 15B Gamma-matrices 16C C P Killing spinors 17
Fermionic T-duality is a tree-level symmetry of type II string theory that can be viewedas extending the idea of ordinary T-duality to the superspace setup [1, 2]. If one has a Green-Schwarz-type sigma-model that describes the embedding of a string worldsheet in type IIsuperspace, then an analog of the classic Buscher procedure [3] can be carried out, resultingin the sigma-model couplings redefinition. The necessary condition is that the backgroundpreserves a supersymmetry, parameterized by some Killing spinors ( ǫ, ˆ ǫ ) (we are consideringan N = 2 theory, hence a couple of supersymmetry parameters).The sigma-model couplings redefinition that results from the fermionic Buscher procedureis quite different from the ordinary T-duality transformation. In fact the entire NS-NS sectoris not modified, except for the dilaton that gets an additive contribution φ ′ = φ + 12 log C, (1.1)where C is determined by the Killing spinors ( ǫ, ˆ ǫ ) that parameterize the fermionic isometries,see appendix A. This transformation law is very similar to the way dilaton changes underordinary T-duality, but the sign of the log term is opposite. This difference will turn out to1e crucial. As for the bosonic fields of the RR sector, their transformation can be writtenconcisely in terms of the bispinor F αβ : e φ ′ F ′ αβ = e φ F αβ + k ǫ α ˆ ǫ β C , (1.2)where the precise value of the numerical coefficient k may be different depending on thesupergravity conventions, and since we will not be using the RR background transformation inthis paper, we do not specify the value of k here. The bispinor F αβ is formed by contracting allthe RR forms of the theory with appropriate antisymmetrized products of gamma-matrices.One can find a more detailed discussion in [1, 4, 5].An important feature of the fermionic T-duality transformation is that it can only be donewith complexified Killing spinors, which means that the resulting target space backgroundwill generically be a solution to complexified supergravity [4]. The paper [5] deals with theextension of fermionic T-duality to a larger class of fermionic symmetries in supergravity, whichalso include some real transformations.A crucial ingredient in the proper theoretical understanding of fermionic T-duality wouldbe to formulate it as a group symmetry [6], in analogy with the O ( d, d ) group representationof the ordinary T-duality. Finally, fermionic T-duality has been recently reformulated as acanonical transformation in phase space [7].In an attempt to further deepen our understanding of the way fermionic T-duality works,in this paper we apply the transformation to an AdS × C P background of type IIA string the-ory. This problem has a rich motivation that comes from the field theory side of the AdS/CFTcorrespondence. The set of problems related to the Yangian invariance [8, 9] and dual super-conformal symmetry [10] of scattering amplitudes in ABJM theory [11] has attracted muchattention recently. Following the N = 4 super-Yang-Mills case where the amplitude/Wilsonloop correspondence [12] and the dual superconformal symmetry [13, 14] have been provento exist there are hopes to find and explain similar structures in ABJM theory. In the SYMcase the amplitude/Wilson loop correspondence has been explained by a combination of 4+8T-dualities on the string theory side of the AdS/CFT correspondence. In particular, four ordi-nary T-dualities along the flat directions of AdS and eight fermionic T-dualities were requiredfor the self-duality of the AdS × S background [1]. Thus, studying the T-duality propertiesof type IIA string theory in the AdS × C P background, which in a certain limit provides thegravity dual to ABJM theory, would in principle do the same to ABJM theory.However, it has been shown that this approach to dual superconformal symmetry cannotbe straightforwardly reproduced in the AdS × C P case [15, 16, 17]. It is clearly impossibleto achieve self-duality in the 3+6 setup (which would be a straightforward mimicking of the2 dS × S case) because three ordinary T-dualities would take us from IIA to IIB theory.There has been a proposal [8] based on the superalgebra arguments that the correct set of T-dualities to perform in this case would be a ‘3+3+6’ set: three flat AdS T-dualities, three C P T-dualities, and six fermionic T-dualities. Furthermore, the authors of [10] have establishedthe existence of dual superconformal symmetry of the tree-level ABJM scattering amplitudesin case when the dual superspace includes three coordinates corresponding to complexifiedisometries of C P . Nevertheless, Adam, Dekel, and Oz have shown [18] that this combinationof T-dualities is singular. The calculation in [18] has been done in the supercoset realization ofthe sigma-model. In the present note we would like to share the complementary point of viewon how does this singularity arise. The derivation here is done in terms of the supergravitycomponent fields.For the sake of simplicity, and also following the conjecture made in [8] that the dilatonshifts coming from the bosonic and the fermionic T-dualities seem to cancel, we confine ourattention to the transformation of the dilaton. This turns out to be sufficient to expose thenature of the singularity involved. The dilaton gets two additive contributions — a negativeone from the bosonic T-dualitites: δ B φ = −
12 log | det g | (1.3)and a positive one from the fermionic dualities: δ F φ = 12 log | det C | . (1.4)Here det g is determinant of the block in the metric tensor that incorporates the directionsthat have been dualized (adapted coordinates have been chosen). An auxiliary function C of(1.1) is promoted to a matrix because we are considering multiple T-dualities here.In what follows we shall consider the transformation of the string coupling e φ , which ac-cording to the above formulae changes as e φ ′ = e φ det C det g . (1.5)The main result will be that not only is this transformation singular, but it is also indetermi-nate, in the sense that both determinants in the above formula vanish. This is to be contrastedwith the AdS × S case [1], where the two detereminants are nonzero and cancel precisely,thus allowing for the self-duality. 3 The background and coordinate systems of C P AdS × C P background has nonzero metric, dilaton, and RR 2- and 4-forms [11]: ds = R k (cid:18) ds AdS + ds CP (cid:19) , (2.1a) e φ = R k , (2.1b) F = 3 R ǫ , (2.1c) F = kJ. (2.1d) ds AdS is a unit radius AdS metric, e.g. in the Poincar´e patch: ds AdS = r (cid:2) − ( dx ) + ( dx ) + ( dx ) (cid:3) + dr r , (2.2)and the corresponding 4-form flux F is proportional to the totally antisymmetric symbol ǫ in 4 dimensions.As regards the C P part of the background, let us introduce several coordinate systemsthat will be useful in what follows. • Fubini-Study coordinates ( z, ¯ z ), where ¯ z α are complex conjugates of z α , α = 1 , ,
3. Lineelement has the well-known form ds C P = dz α d ¯ z α | z | − z α ¯ z β dz β d ¯ z α (1 + | z | ) , (2.3)where | z | = z α ¯ z α . The metric is evidently real, which makes it possible to introduce sixreal coordinates instead. • Starting from the real components of the Fubini-Study coordinates z α = ρ α e iϕ α , we canintroduce six real coordinates ( µ, α, θ, ψ, χ, φ ) as follows [19]: ρ = tan µ sin α sin θ , ϕ = 12 ( ψ − φ + χ ) ,ρ = tan µ cos α, ϕ = 12 χ,ρ = tan µ sin α cos θ , ϕ = 12 ( ψ + φ + χ ) . (2.4)It is convenient to work with the Killing spinors in these coordinates because of the4imple representation of the vielbein forms: e = dµ,e = sin µ dα,e = 12 sin µ sin α (cos ψ dθ + sin θ sin ψ dφ ) ,e = 12 sin µ sin α (sin ψ dθ − sin θ cos ψ dφ ) ,e = 12 sin µ sin α cos α ( dψ + cos θ dφ ) ,e = 12 sin µ cos µ (cid:0) dχ + sin α dψ + sin α cos θ dφ (cid:1) . (2.5)Line element is simply ds C P = δ ab e a e b . We shall use Latin letters for the tangent-spacecomponents. • Finally, we introduce the complexified C P background by means of the following coor-dinate transformation: w α = z α , ¯ w α = ¯ z α | z | . (2.6)Since ¯ w α = ( w α ) ∗ , we can view them as six independent complex coordinates. The lineelement takes the simple form: ds C P = dw α d ¯ w α + ¯ w α ¯ w β dw α dw β . (2.7)The K¨ahler form J in (2.1d) has the simplest representation in the latter coordinates: J = − i dw α ∧ d ¯ w a . (2.8)Transforming it to the real coordinates, we get J = − dµ ∧ ( dψ + dφ cos θ ) sin 2 µ sin α − dµ ∧ dχ sin 2 µ − dα ∧ ( dψ + dφ cos θ ) sin µ sin 2 α + dθ ∧ dφ sin µ sin α sin θ. (2.9)5his looks much simpler in tangent-space components: J ab = e µa e νb J µν = − − − . (2.10) The six isometries that should be T-dualized are the shifts of three flat
AdS directionsand three internal ( C P ) isometries. The contribution of the AdS T-dualities can be triviallyread off from (2.2), and it is nonsingular: δφ = − r. (3.1)Therefore from now on we shall only be concerned with internal isometries.The isometry algebra of C P is su (4), which is 15-dimensional. None of these isometriescommute with any of the supersymmetries, which is the reason for complexifying the Killingvectors. We use the complexified Killing vectors of C P as given in [20]: K α = T α + T βα z β − T z α − T β z β z α ,K α = − T α − T αβ ¯ z β + T ¯ z α + T β ¯ z β ¯ z α , (3.2)for a vector field K = K α ∂∂z α + K α ∂∂ ¯ z α . (3.3)There are precisely 15 independent parameters T AB , A, B = 0 , . . . , T AA = 0.We shall consider the three complex Killing vectors that result from keeping T α in (3.2): K ( α ) = ∂∂z α + ¯ z α ¯ z β ∂∂ ¯ z β , α = 1 , , . (3.4)These three vectors commute with each other and by transforming them to the real coordinates(2.4) one can check that they are of the form a + ib , where a and b are ordinary real Killing6ectors of C P : K (1) = 12 e − i ( ψ − φ + χ ) sin α sin θ ∂∂µ + cot µ cos α sin θ ∂∂α + cot µ cos θ sin α ∂∂θ − i cot µ sin α sin θ ∂∂ψ + i cot µ sin α sin θ ∂∂φ + 2 i tan µ sin α sin θ ∂∂χ ! , (3.5a) K (2) = 12 e − i χ (cid:20) cos α ∂∂µ − cot µ sin α ∂∂α + 2 i cot µ cos α ∂∂ψ − i (cid:18) cot µ cos α − cos α cot µ (cid:19) ∂∂χ (cid:21) , (3.5b) K (3) = 12 e − i ( ψ + φ + χ ) sin α cos θ ∂∂µ + cot µ cos α cos θ ∂∂α − µ sin θ sin α ∂∂θ − i cot µ sin α cos θ ∂∂ψ − i cot µ sin α cos θ ∂∂φ + 2 i tan µ sin α cos θ ∂∂χ ! . (3.5c)Note that alternatively one could also use the three vector fields corresponding to T α ,which are complex conjugates of the vectors (3.4), or those resulting from keeping T αα (nosum). These two groups of complex Killing vectors also commute among themselves.Now we can reveal the reason for the introduction of the ( w, ¯ w ) coordinates in (2.6). Trans-forming the vectors (3.4) to these coordinates one discovers that they are acting as shifts : K ( α ) = ∂∂w α , (3.6)which enables us to calculate det g in (1.5). For this purpose, we read off the metric tensorfrom the expression for the interval in ( w, ¯ w ) coordinates (2.7): g µν = ¯ w ¯ w ¯ w ¯ w ¯ w ¯ w ¯ w ¯ w ¯ w ¯ w ¯ w ¯ w ¯ w ¯ w ¯ w ¯ w ¯ w ¯ w / / . (3.7)The upper-left block here corresponds to the dw dw term in the interval. Rescaling of thestring coupling under the three T-dualities with respect to K (1 , , is given by the determinant This has been pointed out to the author by Carlo Meneghelli.
7f this block, which is identically zero. Now we can rewrite (1.5) as e φ ′ = e φ det C . (3.8)This is clearly a singularity, and now we proceed to showing that the numerator in this formulavanishes as well. In order to get an expression for the matrix C (A.5) we need to know the Killing spinors ǫ, ˆ ǫ . These can be found as solutions to the equations (cid:18) /F − /F Γ (cid:19) E = 0 , (4.1a) ∇ M E = e φ (cid:0) /F Γ M Γ − /F Γ M (cid:1) E , (4.1b)which are conditions that supersymmetry variations of the type IIA fermions vanish. Super-symmetry parameter E is a Majorana spinor, while ǫ and ˆ ǫ are its Majorana-Weyl compo-nents, which can be obtained by applying the projections (1 ± Γ ). We use the notation /F n = n ! F M ...M n Γ M ...M n . Note that the free index M in (4.1b) is a curved index.Original derivation of the Killing spinors of C P can be found in [21], [19], and [22]. Herewe shall briefly overview the derivation for the sake of consistency with our notation andconventions. We decompose the spinor parameter E = κ ⊗ η into the product of the SO (1 , SO (6) spinors κ and η . With the corresponding decomposition of the gamma-matrices(for details see appendix B), the first Killing spinor equation (4.1a) becomes (cid:18) ⊗ F ij β ij (cid:19) ( κ ⊗ η ) = (cid:0) ⊗ β (cid:1) ( κ ⊗ η ) . (4.2)We see that κ is unconstrained, while the equation for η can be rewritten as follows: Q β η = − β η, (4.3)where Q = F ij β ij β . Evaluating this matrix operator using the tangent-space componentsof the 2-form (2.10) shows that indeed there is a − η = (cid:16) − f f f − f f − f − f f (cid:17) T . (4.4)8he exact functional dependence of the parameters f i on spacetime coordinates is fixed by thesecond Killing spinor equation (4.1b).Performing the same decomposition as above we arrive at the following equations for κ and η : (cid:18) ∂ µ + 14 ω µ,ρλ α ρλ (cid:19) κ = α µ α κ, (4.5) (cid:18) ∂ i + 14 ω i,kl β kl (cid:19) η = i β i η − i F i j β j β η, (4.6)where we underline the world indices and our convention for the spin connection is ω A,BC = 12 e DB e EC (Ω ADE − Ω DEA + Ω
EAD ) , (4.7)Ω ABC = ∂ [ A e DB ] e EC η DE . (4.8)The AdS Killing spinor equation is easy to solve and the solution κ is 4-parametric: κ = κ r − / κ r − / r / [ − κ ( x − x ) + κ x + κ ] r / [ κ ( x + x ) − κ x + κ ] . (4.9)Solving the equations for η is more tedious, but it can be done analytically. The solution isvery bulky and is therefore given in the appendix C. The overall result is that the AdS partof the Killing spinor κ is 4-parametric, while the C P part is 6-parametric. Thus there are 24independent Killing spinors in the AdS × C P background. We now need to establish which Killing spinors to use for the T-duality transformation.As long as we have chosen the three isometries generated by (3.4), the choice of the fermionicsymmetries is dictated by the requirement that together they form a commuting subalgebra ofthe symmetry superalgebra. Bosonic generators (3.4) of this subalgebra are commuting; ournext step will be to select the fermionic generators (Killing spinors) that commute with thesethree vectors and finally we shall check the anticommutation of the selected supersymmetriesamong themselves.First of all recall that apart from (3.4) our T-duality setup includes three bosonic dualitiesalong the flat directions of
AdS . Looking at the AdS part of the Killing spinor (4.9) we see9hat we must set κ , = 0 for the product κ ⊗ η to be invariant under the shifts of x , , . Sowhat happens to the C P part of the Killing spinor?From the explicit expressions of the C P spinors (appendix C) it is not easy to tell what aretheir commutation properties with the vectors (3.4). Therefore we calculate the Lie derivativesof our Killing spinor fields with respect to the Killing vectors [23]. Lie derivative of a spinor η with respect to a vector K is given by L K η = K i ∇ i η + 12 ∇ [ i K j ] β ij η, (5.1)where of course the covariant derivatives of a vector and of a spinor are taken correspondinglywith respect to the Christoffel and spin connections.Using the expressions for K (1 , , (3.5) and for η ,..., ((4.4) and appendix C, where thespinor η i results from keeping only the parameter h i = 1 and setting all the rest to zero), onefinds the following algebra: L K (1) η = −
12 ( η − iη + iη − η ) , L K (1) η = − i η − iη + iη − η ) , L K (1) η = 14 ( η + iη ) , L K (1) η = − i η + iη ) , L K (1) η = i η + iη ) , L K (1) η = −
14 ( η + iη ) , (5.2a) L K (2) η = 0 , L K (2) η = 0 , L K (2) η = − i η − iη ) , L K (2) η = i η + iη ) , L K (2) η = 12 ( η − iη ) , L K (2) η = 12 ( η + iη ) , (5.2b)10 K (3) η = − i η + iη + iη + η ) , L K (3) η = 12 ( η + iη + iη + η ) , L K (3) η = i η + iη ) , L K (3) η = −
14 ( η + iη ) , L K (3) η = −
14 ( η + iη ) , L K (3) η = i η + iη ) . (5.2c)It is easy to see that there are three linear combinations of the Killing spinors that are invariantunder the action of all three vectors: η + iη , η + iη , η − iη . (5.3)Tensor multiplying these with the two AdS spinors ( κ , κ = 0 in (4.9)) we get the sixKilling spinors, which is precisely the number needed for the T-duality. Thus the symmetrysuperalgebra constraints unambiguously fix the fermionic directions to be T-dualized.It remains to make sure that the corresponding supersymmetries anticommute. The con-straint on the spinor E = κ ⊗ η is given in the appendix (A.3) and can be checked straightfor-wardly. For multiple supersymmetries one has to generalize this to the matrix constraint¯E i Γ µ E j = 0 , i, j = 1 , . . . , . (5.4) Finally we are in a position to calculate the matrix C ij , i, j = 1 , . . . , ∂ µ C ij = ¯E i Γ µ Γ E j , (6.1)11hich is a generalisation of (A.5) for the case of multiple T-dualities. These equations turnout to be consistent, and the solution is (up to integration constants) C AdS × C P = a b − a c − b − c − a − ba − cb c , (6.2)where a = − r e − i ( ψ + χ ) sin 2 µ sin α (cid:20) cos 12 ( θ + φ ) + i sin 12 ( θ − φ ) (cid:21) , (6.3a) b = 2 r e − i ( ψ + χ ) sin 2 µ sin α (cid:20) i cos 12 ( θ − φ ) + sin 12 ( θ + φ ) (cid:21) , (6.3b) c = − r e − i χ sin 2 µ cos α. (6.3c)The important point to notice here is that the determinant of the matrix (6.2) is identicallyzero, irrespective of the values (6.3). This is the second singularity, which manifests itself inthe numerator of the formula (1.5).The vanishing of det C in the present case is to be contrasted with the AdS × S case[1], where the C -matrix has the same algebraic structure (symmetric matrix with off-diagonalantisymmetric blocks). However since in this setup one does 4 bosonic ( AdS ) dualities and 8fermionic ones, C AdS × S is now an 8 × C AdS × S = a b c − a d e − b − d f − c − e − f − a − b − ca − d − eb d − fc e f , (6.4)12here the entries are given by a = 2 Rr sin y (cid:0) cos y − i sin y cos y (cid:1) , (6.5a) b = 2 Rr (cid:0) i cos y + sin y . . . sin y (cid:1) , (6.5b) c = − Rr sin y sin y sin y (cid:0) cos y − i sin y cos y (cid:1) , (6.5c) d = 2 Rr sin y sin y sin y (cid:0) cos y + i sin y cos y (cid:1) , (6.5d) e = 2 Rr (cid:0) − i cos y + sin y . . . sin y (cid:1) , (6.5e) f = − Rr sin y (cid:0) cos y + i sin y cos y (cid:1) . (6.5f)Here r , as before, is the AdS radial coordinate, R is the AdS radius and the variables { y , . . . , y } are the standard coordinates on S : ds = ( dy ) + sin y (cid:8) ( dy ) + sin y (cid:2) ( dy ) + . . . (cid:3)(cid:9) . (6.6)In the AdS × S case, not only is det C AdS × S nonvanishing, but for these particular valuesof the entries it can be simplified todet C AdS × S = (2 Rr ) . (6.7)This is precisely cancelled by the 4 AdS dualities.
We have shown that under the combination of bosonic and fermionic T-dualities in thedirections given by the three complexified C P isometries and six complexified supersymmetriesthe transformation of the dilaton is indeterminate: e φ ′ = e φ . (7.1)This provides an alternative point of view on T-dualizing AdS × C P background that hasbeen done recently by Adam, Dekel, and Oz [18] in the supercoset formulation of the sigma-model. Perhaps a way to eliminate this ambiguity would be to consider a deformed AdS × C P background, the deformation being parameterized by some λ , such that the dependence on thedeformation parameter e φ ′ = f ( λ ) e φ would have a well-defined limit when the deformationis removed lim λ → f ( λ ).Most likely this deformation would require giving the dilaton some nontrivial coordinate13ependence. The dilaton equation of motion in our conventions is R = 4( ∂φ ) − ∇ φ (7.2)(for a vanishing B -field). If we keep the dilaton constant, then the requirement that the AdS part of the geometry be preserved will only allow for the deformations of the C P part thatpreserve R = 0, which is problematic. One can also consider the Killing spinor equation(4.1a), which in the ABJM background reduces to the eigenspinor condition (4.3). If onewere to deform the RR 2-form, the eigenspinor condition would be broken, and for somesupersymmetry to be preserved one would have to introduce the dilaton into the game. Withnontrivial dilaton the equation (4.3) gets modified to (cid:2) kβ i ∂ i φ − e φ ( Q + 2) (cid:3) β η = 0 , (7.3)where, as before, Q = F ij β ij β , and we have absorbed the numerical factors that depend onthe supergravity conventions into the constant k . An appropriate relative normalization of F and F is also assumed. It is possible that the dilaton field with nontrivial dependence on theinternal manifold could allow for some supersymmetry to be preserved under the deformation.A candidate recipe for the deformation is the TsT transformation [24, 25], which gives thebeta-deformed AdS × C P theory described in [26]. In order for the Killing vectors to bepreserved under the beta-deformation, one may carry out the beta-deformation with respectto these Killing vectors. Therefore if we beta-deform the AdS × C P background using thedirections (3.4), we can then use the same Killing vectors for the T-duality. However the dw dw block in (3.7) is not affected by such a beta-deformation, which means that the correspondingdeterminant is still zero. Thus the use of the TsT transformation for the deformation purposesin our setup is problematic.A general property of fermionic T-duality that has been revealed in the present work is thatthe transformation may be singular. In the case at hand the degeneracy of the matrix (6.2) asopposed to (6.4) is due to their block structure with antisymmetric blocks (an odd-dimensionalantisymmetric matrix has zero determinant). In a different setup the structure of the C -matrixmay be different [4] (as one can observe from the definition (A.5), C is only required to besymmetric). It is yet to be understood what makes singular transformations possible, and inparticular what is the role of complexification of the fermionic symmetries that is obligatoryfor doing fermionic T-duality. 14 cknowledgements The author would like to thank Nathan Berkovits, David Berman, Stefan Hohenegger, andCarlo Meneghelli for valuable discussions.This work is supported by Westfield Trust Scholarship.
A Fermionic T-duality in type IIA
Start with any Majorana-Weyl representation of the gamma-matrices, such thatΓ µ = γ µ ) αβ γ µαβ ! , C = c αβ ¯ c αβ ! , Γ = − ! . (A.1)The indices α, β here take values 1 . . .
16. Different properties of this class of representationsare considered in [27]. We use Majorana conjugation for covariant spinors ¯ ψ = ψ T C .In type IIA it is possible to write the main formulae of fermionic T-duality [1] conciselyby introducing a single Majorana spinor parameter instead of a pair ( ǫ, ˆ ǫ ) of Majorana-Weylspinors. In particular, the abelian constraint for a supersymmetry generated by ( ǫ, ˆ ǫ ) is0 = (cid:16) ¯ ǫ Q + ¯ˆ ǫ ˆ Q (cid:17) = − (cid:0) ¯ ǫ Γ µ ǫ + ¯ˆ ǫ Γ µ ˆ ǫ (cid:1) P µ = ( IIB : (cid:2) ( ǫc ) α γ µαβ ǫ β + (ˆ ǫc ) α γ µαβ ˆ ǫ β (cid:3) P µ , IIA : (cid:2) ( ǫc ) α γ µαβ ǫ β + (ˆ ǫ ¯ c ) α ( γ µ ) αβ ˆ ǫ β (cid:3) P µ . (A.2)The IIA expression can be rewritten in terms of the Majorana spinor E = ǫ + ˆ ǫ :¯E Γ µ E = 0 . (A.3)Another relation of interest is the definition of an auxiliary function C (not to be confusedwith the charge conjugation matrix): ∂ µ C = i (cid:0) ¯ ǫ Γ µ ǫ − ¯ˆ ǫ Γ µ ˆ ǫ (cid:1) = ( IIB : i (cid:2) ( ǫc ) α γ µαβ ǫ β − (ˆ ǫc ) α γ µαβ ˆ ǫ β (cid:3) , IIA : i (cid:2) ( ǫc ) α γ µαβ ǫ β − (ˆ ǫ ¯ c ) α ( γ µ ) αβ ˆ ǫ β (cid:3) . (A.4)Again we can rewrite the IIA expression succinctly as ∂ µ C = ¯E Γ µ Γ E . (A.5)Since the expressions in (A.3) and (A.5) are vectors, these formulae are independent of the15amma-matrix representation. B Gamma-matrices
For the purposes of working with type IIA supergravity, whose spinorial quantities areMajorana spinors of 1 + 9-dimensional spacetime, we need a Majorana representation of thegamma-matrices. This will be built as a product of Majorana representations in 1 + 3 and in6 dimensions.Our spacetime signature convention is ( − + . . . +), hence the following four real anticom-muting matrices α µ furnish a Majorana representation in D = 1 + 3: α = σ ⊗ iσ ,α = σ ⊗ σ ,α = σ ⊗ σ ,α = σ ⊗ . (B.1)Volume element α = α . . . α = iσ ⊗ − D Euclidean space to be β = 1 ⊗ σ ⊗ σ ,β = 1 ⊗ σ ⊗ σ ,β = σ ⊗ ⊗ σ ,β = σ ⊗ ⊗ σ ,β = σ ⊗ σ ⊗ ,β = σ ⊗ σ ⊗ . (B.2)These are imaginary and we define the corresponding volume element to be real: β = − β . . . β = iσ ⊗ iσ ⊗ iσ .Finally, the ten-dimensional real gamma-matrices Γ are the following products (for α =0 , . . . , i = 1 , . . . , µ = α µ ⊗ , Γ i +3 = iα ⊗ β i (B.3)Ten-dimensional chirality operator is Γ = Γ . . . Γ = − α ⊗ β . This representation is clearlynot Weyl. 16 C P Killing spinors
The components of the C P factor (4.4) of the Killing spinor are given by the following: f = 12 n h cos α sin χ h cos α cos χ α (cid:20)(cid:18) h sin φ h cos φ (cid:19) sin 14 (2 θ + χ − ψ ) + (cid:18) h cos φ − h sin φ (cid:19) cos 14 (2 θ − χ + 2 ψ ) − (cid:18) h cos φ − h sin φ (cid:19) cos 14 (2 θ + χ − ψ )+ (cid:18) h sin φ h cos φ (cid:19) sin 14 (2 θ − χ + 2 ψ ) (cid:21)(cid:27) , (C.1) f = 12 n h cos α cos χ − h cos α sin χ α (cid:20)(cid:18) h sin φ h cos φ (cid:19) cos 14 (2 θ + χ − ψ ) + (cid:18) h cos φ − h sin φ (cid:19) sin 14 (2 θ − χ + 2 ψ ) + (cid:18) h cos φ − h sin φ (cid:19) sin 14 (2 θ + χ − ψ ) − (cid:18) h sin φ h cos φ (cid:19) cos 14 (2 θ − χ + 2 ψ ) (cid:21)(cid:27) , (C.2) f = 12 n A (cid:2) (cos α + 1) sin µ − (cos α −
1) cos µ (cid:3) + B (cid:2) (cos α + 1) cos µ − (cos α −
1) sin µ (cid:3) − α (cos µ − sin µ ) (cid:16) h cos χ − h sin χ (cid:17) o , (C.3) f = 12 n A (cid:2) (cos α −
1) sin µ + (cos α + 1) cos µ (cid:3) + B (cid:2) (cos α −
1) cos µ + (cos α + 1) sin µ (cid:3) + 2 sin α (cos µ + sin µ ) (cid:16) h cos χ − h sin χ (cid:17) o , (C.4) f = 12 n A (cid:2) (cos α + 1) sin µ − (cos α −
1) cos µ (cid:3) + B (cid:2) (cos α + 1) cos µ − (cos α −
1) sin µ (cid:3) − α (cos µ − sin µ ) (cid:16) h cos χ − h sin χ (cid:17) o , (C.5) f = 12 n A (cid:2) (cos α −
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