No Fermionic Wigs for BPS Attractors in 5 Dimensions

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1 No Fermionic Wigs for BPS Attractors in 5 Dimensions DFPD-1/TH/0 Lorenzo G. C. Gentile 1,, Pietro A. Grassi 1,, Alessio Marrani, Andrea Mezzalira 5, and Wafic A. Sabra 6 arxiv: v1 [hep-th] 0 Mar 01 1 DISIT, Università del Piemonte Orientale, via T. Michel, 11, Alessandria, I-1510, Italy, pgrassi@mfn.unipmn.it Dipartimento di Fisica U01cGalileo GalileiU01d, Università di Padova, via Marzolo 8, I-511 Padova, Italy and INFN, Sezione di Padova, via Marzolo 8, I-511, Padova, Italy, lgentile@pd.infn.it INFN - Gruppo Collegato di Alessandria - Sezione di Torino Instituut voor Theoretische Fysica, KU Leuven, Celestijnenlaan 00D, B-001 Leuven, Belgium, alessio.marrani@fys.kuleuven.be 5 Physique Théorique et Mathématique, Université Libre de Bruxelles, C.P. 1, B-1050 Bruxelles, Belgium, andrea.mezzalira@ulb.ac.be 6 Centre for Advanced Mathematical Sciences and Physics Department, American University of Beirut, Lebanon, ws00@aub.edu.lb Abstract We analyze the fermionic wigging of 1/ BPS electric extremal black hole attractors in N =, D = 5 ungauged Maxwell Einstein supergravity theories, by exploiting anti Killing spinors supersymmetry transformations. Regardless of the specific data of the real special geometry of the manifold defining the scalars of the vector multiplets, and differently from the D = case, we find that there are no corrections for the near horizon attractor value of the scalar fields; an analogous result also holds for 1/ BPS magnetic extremal black string. Thus, the attractor mechanism receives no fermionic corrections in D = 5 at least in the BPS sector.

2 Contents 1 Introduction 1 Ungauged N =, D = 5 MESGT Fermionic Wigging.1 Second Order Evaluation on Purely Bosonic Background 6.1 First Order Second Order Third Order Fourth Order Wigging of BPS Extremal Black Hole First Order Second Order Third Order Fourth Order Conclusion 10 A Notation and Identities 1 B Third Order 1 C Fourth Order 16 1 Introduction The question concerning the presence or absence of hairs of any kind around a black hole is very compelling and, of course, it has been studied from several points of view. Nonetheless, recently some of the authors of the present work re-posed the question by considering possible fermionic hairs first in [1], and then in a series of papers [] for non-extremal, as well as BPS black holes. The first paper on the subject is due to Aichelburg and Embacher []. They considered asymptotically flat black hole solution in N =, D = supergravity without vector multiplets and computed iteratively the supersymmetric variations of the background in terms of the flat-space Killing spinors. In that paper, they were able to compute some of the physical quantities such as the corrections to the angular momentum, while other interesting properties cannot be seen at that order of the expansion. Afterwards, the works [1] and [] applied their technique to some examples of BPS black hole, up to the fourth order in the supersymmetry transformation. In particular, for extremal black hole solutions, the attractor mechanism [5] is a very interesting and important physical property; essentially, it states that the solution at the horizon depends only on the conserved charges of the system, and is independent of the value of the matter fields at infinity. This is related to the no-hair theorem, under which, for example, a BPS black hole solution depends only upon its mass, its angular momentum and other conserved charges. As said, the authors of 1

3 [] addressed the question whether the attractor mechanism has to be modified in the presence of fermions. The conclusion was that, at the level of approximation of their computations, in the case of double-extremal BPS solutions, the mechanism is unchanged. In [1] N =, D = 5 AdS black holes were investigated, and it was found that the solution, as well as its asymptotic charges, get modified at the second order due to fermionic contributions. However, in [1] the attractor mechanism and its possible modifications was not considered. In [6], the fermionic wig for asymptotically flat BPS black holes in N =, D = supergravity coupled to matter was investigated. There, it has been shown that the attractor mechanism gets modified at the fourth order even in the case of double extremal solutions in the simplest example of N = supergravity coupled to a single matter field minimally coupled vector multiplet. The surprising result is that to the lower orders all corrections vanish for the BPS solution, while at the fourth order, despite several cancellations due to special geometry identities, some terms do survive, and thus the attractor gets modified. It has also been noticed that there are situations in which some combinations of charges render the attractor modifications null; this led to the conjecture that, in those D = models admitting an uplift to 5 dimensions, the attractor mechanism is unmodified by the fermionic wig. That motivated us to study in full generality the D = 5 case, by means of the same techniques; we found that there is no modification to the attractor mechanism up to forth order for all the ungauged N =, D = 5 supergravity models coupled to vector multiplets. This is a rather strong result, and it has been obtained for a generic real special geometry of the manifold defined by the scalars of the vector multiplets. The cancellations appear to be due to identities of the special geometry, as well as to the extremal black hole solutions taken into account cfr. Eq We should point out that the wigging is computed by performing a perturbation of the unwigged purely bosonic BPS extremal black hole solution keeping the radius of the event horizon unchanged. The complete analysis, including the study of the fully-backreacted wigged black hole metric, will be presented elsewhere. The plan of the paper is as follows. In Sec. we recall some basics of N =, D = 5 ungauged Maxwell-Einstein supergravity. The fermionic wigging is then presented in Sec., and its evaluation on the purely bosonic background of an extremal BPS black hole is performed in Sec.. The nearhorizon conditions are applied in Sec. 5, obtaining the universal result of vanishing wig corrections to the attractor value of the scalar fields of the vector multiplets in the near-horizon geometry. The universality of this result resides in its independence on the data of the real special geometry endowing the scalar manifold of the supergravity theory. Comments on this result and further remarks and future directions are given in Sec. 6. Three Appendices, specifying notations and containing technical details on the wigging procedure, are presented. Ungauged N =, D = 5 MESGT Following [7] [9], we consider N =, D = 5 ungauged Maxwell Einstein supergravity theory MESGT, in which the N = gravity multiplet { e a µ, ψ i µ, A µ } is coupled to nv Abelian vector

4 multiplets 1 { A µ,λ xi,φ x}, with neither hyper nor tensor multiplets : where δe µ a = 1 ǫγa ψ µ, δψ i µ =D µ ˆωǫ i i 6 h I F I γ µ δ ν µγ ρ ǫ i 1 6 ǫ j λ ix γ µ λ j x 1 1 γ µνǫ j λix γ ν λ j x.1a 1 8 γ µǫ j λix γ λ j x 1 1 γν ǫ j λix γ µν λ j x,.1b δh I = 1 i ǫλ x h I x, 6.1c δφ x = 1 i ǫλx,.1d δa I µ = 1 ǫγ µλ x h I x 6 ihi ǫψ µ,.1e δλ xi = /Dφ i x ǫ i δφ y Γ x yz λzi 1 γ F I h x I ǫi 1 Txyz[ ǫ 6 j λi y λ j z γµ ǫ j λi y γ µ λ j z 1 γµν ǫ j λi y γ µν λ j ] z,.1f F I µν = [µa I ν],.a F µν I =Fµν I ψ [µ γ ν] λ x h I x i 6 ψ µ ψ ν h I,.b T xyz =C IJK h I x hj y hk z,.c Γ w xy =hw I hi x,y T xyzg zw..d From the Vielbein postulate, the N = spin connection reads ˆω µ ab = 1 [ e cµ Ω abc Ω bca Ω cab] K a µ b,. where Ω abc : = e µa e νb µ e c ν νe c µ and K a b µ := 1 ψ [b γ a] ψ µ 1 ψ b γ µ ψ a. The covariant derivatives are defined as D µ φ x = µ φ x 1 i ψ µ λ x,.a D µ h I = µ h I = hi x µφ x = hi x D µφ x,.b D µ λ xi = µ λ xi µ φ y Γ x yzλ zi 1 ω µ ab γ ab λ xi, D µ ψν i = µ 1 ω µ ab γ ab ψν i,.c.d 1 i = 1, of the fundamental of USp SU R symmetry, x = 1,...,n V and I = 0,1,...,n V, where the 0 index pertains to the D = 5 graviphoton. Note that γ µ denote the D = 5 gamma matrices. Moreover, we adopt the convention κ = 1 cfr. e.g. App. C of [9]. When not indicated, spinor indices are contracted using the standard SU metric ε ij see appendix A.

5 and [7]; see also e.g. Eq. C.10 of [9] y h I x = h I g xy T xyz h Iz,.5a y h Ix = h Ig xy T xyz h z I..5b Note that only ω ab µ and not ˆω ab µ occurs in the covariant derivative of the gravitino. Furthermore, it holds that see also e.g. [10, 11, 1] h I x xh I, h Ix a IJ h J x,.6a a IJ = C IJK h K h I h J,.6b C IJK h I h J h K =1, h I h I = 1..6c It is worth pointing out that in D = 5 Lorentzian signature no chirality is allowed, and the smallest spinor representation of the Lorentz group is given by symplectic Majorana spinors; for further details, see App. A. Fermionic Wigging We now proceed to perform the fermionc wigging, by iterating the supersymmetry transformations of the various fields generated by the anti-killing spinor ǫ for a detailed treatment and further details, cfr. e.g. [1, 6]; schematically denoting all wigged fields as Φ and the original bosonic configuration by Φ, the following expansion holds: Φ = e δ Φ = ΦδΦ 1 δ Φ 1! δ Φ 1! δ Φ,.7 where, as in [], the expansion truncates at the fourth order because of the -Grassmannian degrees of freedom that ǫ contains..1 Second Order In order to give an idea on the structure of the iterated supersymmetry transformations on the massless spectrum of the theory under consideration, we present below the second order transformation rules 5 general results on supersymmetry iterations at the third and fourth order are given in Apps. B and In the present treatment, C IJK denotes the C IJK of [1], their difference being just a rescaling factor. In the present paper we will deal with a BPS background so just half of the supersymmetries are preserved. 5 By exploiting Eq..16 of [1], both tt xyz and tγ x yz can be related to the covariant derivative of the Riemann tensor R xyzt; this latter is known to satisfy the the so called real special geometry constraints see e.g. Eq..1 of [1].

6 C, respectively : with δ e a µ = ǫγa 1 δ 1 ψ µ, δ ψµ i = δ 1 D µ ǫ i 1 6 ǫ j λ ix γ µ δ 1 λ j x 1 1 γ µνǫ j λix γ ν δ 1 λ j x 1 8 γ µǫ j λix γ δ 1 λ j x 1 1 γν λix γ µν δ 1 λ j x 1 6 ǫ j λ ix γ a λ j x δ 1 e a µ 1 6 ǫ j δ 1 λix γ µ λ j x 1 1 γ abǫ j λix γ b λ j x δ 1 e a µ 1 1 γ µνǫ j δ 1 λix γ ν λ j x 1 8 γ abcǫ j λix γ bc λ j x δ 1 e a µ 1 8 γ µǫ j δ 1 λix γ λ j x 1 1 γb ǫ j λix γ ab λ j x δ 1 e a µ 1 1 γν ǫ j δ 1 λix γ µν λ j x i 6 h I F [ I δ 1 e a µ e ν b eρ c ea µ δ 1 e ν b e ρ c ea µ eν b δ 1 e ρ c i 1 h Iz δ 1 φ z F I γ µ δµ ν γρ ǫ i i 6 h I δ φ x = ǫ i δ 1 λ x, δ A I µ = 1 ǫγ µ ] γ bc a δ b a γc ǫ i.8 δ 1 FI γ µ δ ν µγ ρ ǫ i,.9 δ 1 λ x h I x 1 ǫhi δ 1 ψ µ δ 1 e a µ ǫγ a λ x h I x δ 1 φ x ǫψ µ.10 i i hi x 1 ǫγ µλ x y h I x δ 1 φ y,.11 δ λ ix = i δ 1 e µ a γ a Dµ φ x ǫ i i γµ δ 1 Dµ φ x ǫ i 1 6 Txyz γ µ ǫ j λi y γ µ δ 1 λ j z 1 Txyz ǫ j λi y δ 1 λ j z Txyz γ µν ǫ j λi y γ µν δ 1 λ j z δ 1 φ y Γ x yz δ 1 λ zi δ φ y Γ x yzλ zi 1 Txyz ǫ j δ 1 λi y λ j z Txyz γ µν ǫ j δ 1 λi y γ µν λ j z 1 6 Txyz γ µ ǫ j δ 1 λi y γ µ λ j z 1 6 tt xyz δ 1 φ t [ ǫj λi y λ j z γ µ ǫ j λi y γ µ λ j z 1 γµν ǫ j λi y γ µν λ j ] z δ 1 φ y t Γ x yz δ 1 φ t λ zi 1 γ F I t h x I δ 1 e µ a e ν b eµ a δ 1 e ν b 1 δ 1 φ t ǫ i 1 γ δ 1 FI h x Iǫ i γ ab FI µν h x I ǫi..1 5

7 δ 1 FI µν = δ 1 Fµν I δ γ 1 ψ[µ ν] λ x h I x ψ [µ γ ν] δ 1 λ x h I x ψ [µ γ ν] λ x y h I x δ 1 φ y i 6 δ 1 ψµ ψ ν h I i 6 ψ µ δ 1 ψ ν h I ψ [µ δ 1 e a ν] γ a λ x h I x i ψ µ ψ ν h I x δ 1 φ x,.1 δ 1 D µ = 1 δ 1 ωµ ab γ ab,.1 δ 1 ωµ ab = 1 [ δ 1 e cµ Ω abc Ω bca Ω cab] 1 [ e cµ δ 1 Ω abc δ 1 Ω bca δ 1 Ω cab] δ 1 K a µ b,.15 δ 1 Ω abc [ = δ 1 e µa e νb e µa δ 1 e νb] µ e c ν νe c µ e µa e νb[ ] µ δ 1 e c ν ν δ 1 e c µ,.16 δ 1 K a µ b = 1 [δ 1 ψρ e ρ[a γ b] ψ µ ψ ρ δ 1 e ρ[a γ b] ψ µ ψ [a γ b] δ 1 ψ µ 1 δ 1 ψρ e ρa γ µ ψ b 1 ψ ρ δ 1 e ρa γ µ ψ b 1 ψ a γ a δ 1 e a µ ψ b 1 ψ a γ µ ψ ρ δ 1 e ρb 1 ψ ] a γ µ δ 1 ψ ρ e ρb,.17 δ 1 Dµ φ x = µ δ 1 φ x i δ 1 ψµ λ x i ψ µ δ 1 λ x..18 Evaluation on Purely Bosonic Background Next, we proceed to evaluate the fermionic wigging on a purely bosonic background characterized by setting ψ = λ = 0identically, anddenoted by bg throughout. Thisresultsin adramaticsimplification of previous formulæ; in particular, all covariant quantities, such as the Ẽ tensor [1], characterizing the real special geometry of the scalar manifold cfr. Apps. B and C, do not occur anymore after evaluation on such a background..1 First Order At the first order, the non-zero supersymmetry variations are: δ 1 ψµ i bg =D µˆωǫ i i 6 h IF I δ 1 λ xi bg = i / φ x ǫ i 1 γ FI h x I ǫi. γ µ δ ν µ γρ ǫ i,.1a.1b Moreover, the supercovariant field strength collapses to the ordinary field strength and the covariant derivative on φ x reduces to an ordinary flat derivative. 6

8 . Second Order δ eµ a bg =1 ǫγa δ 1 bg ψ µ,.a δ φ x bg = ǫ i δ 1 λ x bg,.b δ A I bg µ = i ǫhi δ 1 bg ψ µ ǫγ 1 µ δ 1 λ x bg h I x..c The supercovariant field strength, the covariant derivative on φ x and the variation of the spin connection ωµ ab all collapse to zero.. Third Order At the third order, one obtains the following results : δ ψµ i bg = δ bg D µ ǫ i 1 ǫ j δ 1 λix bg γ µ δ 1 λx j bg 1 6 γ µν δ 1 λix bg γ ν δ 1 λx j bg 1 γ µǫ j δ 1 λix bg γ δ 1 λx j bg 1 6 γ µνǫ j δ 1 λix bg γ ν δ 1 λx j bg i [ 6 h IF I δ eµ a bg eν b eρ c ea µ γa bc δa b γc ǫ i i δ e ν b bg eρ c ea µ eν b δ φ z bg F I γ µ δµ ν γρ ǫ i bg] δ e ρ c 1 h Iz i 6 h I δ bg FI γ µ δµ ν γρ ǫ i,.a δ λ ix bg = i δ e µ bg a γa µ φ x ǫ i i γµ δ Dµ φ x bg ǫi δ φ y bg Γ x yz δ 1 λ zi bg 1 Txyz ǫ j δ 1 λi y bg δ 1 λ j z bg 1 6 Txyz γ µν ǫ j δ 1 λi y bg γ µν δ 1 λ j z bg 1 6 Txyz γ µ ǫ j δ 1 λi bg y γ µ δ 1 λz j bg 1 γ FI t h x I δ φ t bg ǫ i 1 γ δ FI bg h x I ǫi 1 [ γab δ e a µ bg bg] eν b eµ a δ e ν b Fµν I hx I ǫi..b 7

9 For the supercovariant field strength, the covariant derivative on φ x, and the spin connection ω ab µ, it holds that: δ FI µν bg = [µ bg δ 1 ψ[µ bg γ ν] δ 1 λ x bg h I x δ A I ν] i δ 1 ψν bg δ 1 bg ψ µ h I, δ Dµ φ x bg = µ δ φ x bg i 1 δ 1 ψµ bg δ 1 λ x bg, δ ωµ ab bg δ =1 bg e cµ Ω abc Ω bca Ω cab 1 [ δ Ω abc bg δ Ω bca bg δ Ω abc bg =[ δ e µa bg eνb e µa δ e νb bg [ e µa e νb µ δ eν c bg ν δ K a µ b bg = δ 1 ψρ bg e ρ[a γ b] δ 1 bg ψ µ δ 1 bg ψρ γ µ δ 1 bg ψ ν e ρa e νb. ] δ Ω cab bg ] µ e c ν νe c µ δ K a b µ.a.b bg,.c bg] δ e c µ,.d.e. Fourth Order Finally, at the fourth order, one achieves the following expressions : δ eµ a bg =1 ǫγa δ bg ψ µ,.5a δ φ x bg = ǫ i δ λ x bg,.5b δ Aµ I bg = ǫγ 1 µ δ λ x bg h I x i i hi x ǫγ µ ǫhi δ bg ψ µ δ φ x bg ǫ δ 1 bg ψ µ δ e a µ bg ǫγ a δ 1 λ x bg h I x δ 1 λ x bg y h I x δ φ y bg..5c Again, the supercovariant field strength, the covariant derivative on φ x and the spin connection ω ab µ all vanish. 8

10 5 Wigging of BPS Extremal Black Hole Following the treatment of the D = 5 attractor mechanism given in [16, 17] and [18], we consider the 1/ BPS near horizon conditions for extremal electric black hole with near-horizon geometry AdS S : µ h I = 0 = µ φ x = 0, h Ix F I µν = 0, 5.6 and we evaluate the results for purely bosonic background computed in the previous section onto such conditions denoted by BPS, and always understood on the r.h.s. of equations, throughout the following treatment. 5.1 First Order At the first order, the gravitino variation is non zero, while the gaugino variation vanishes : 5. Second Order At the second order, one obtains : 5. Third Order At the third order, it holds that : δ BPS ψ µ = δ BPS D µ ǫ δ 1 ψµ i BPS =D µˆωǫ i i 6 h IF I δ 1 λ xi BPS = 0. γ µ δ ν ργ ρ ǫ i 0, 5.7a 5.7b δ eµ a BPS =1 ǫγa δ 1 BPS ψ µ 0, 5.8a δ φ x BPS = 0, 5.8b δ Aµ I BPS = c [ δ e a µ BPS eν b eρ c ea µ i 6 h IFbc I γa bc δa b γc ǫ i i 6 h I δ e ν b BPS eρ c ea µ eν b BPS] δ e ρ c δ FI BPS γ µ δ ν µ γρ ǫ i, 5.9a and δ λ ix BPS = b 9

11 Concerning the supercovariant field strength, the covariant derivative on φ x and the spin connection ω ab µ, the following expressions hold: δ BPS FI µν = [µ δ A I ν] δ BPS i 1 BPS ψν δ 1 BPS ψ µ h I, 5.10a δ Dµ φ x BPS = µ δ φ x BPS i δ 1 ψµ BPS δ 1 λ x BPS = 0, 5.10b δ ωµ δ ab BPS =1 BPS e cµ Ω abc Ω bca Ω cab 1 [ δ Ω abc BPS δ Ω bca BPS δ Ω cab ] BPS δ K a µ b BPS, 5.10c δ Ω abc [ BPS = δ e µa BPS e νb e µa δ e νb ] µ BPS e c ν ν e c µ e µa e νb[ µ δ eν c BPS BPS] ν δ e c µ, 5.10d δ K a b µ 5. Fourth Order BPS = δ 1 BPS ψρ e ρ[a γ b] δ 1 BPS ψ µ δ 1 ψρ BPS γ µ δ 1 BPS ψ ν e ρa e νb. 1 Finally, at the fourth order, by using the identity [7] one achieves the following results : h I h Ix = 0, 5.10e δ eµ a BPS =1 ǫγa δ BPS ψ µ 0, 5.11a δ φ x BPS = 0, 5.11b δ A I µ BPS = i ǫhi δ BPS ψ µ c Once again, the supercovariant field strength, the covariant derivative on φ x and the spin connection ω ab µ all vanish. 6 Conclusion The general structure of the fermionic wigging.7 along a -component anti-killing spinor, as well as the results reported in Secs. 5. and 5., do imply that the attractor values of the real scalar fields φ x in the near horizon AdS S geometry of the 1/ BPS extremal electric black hole are not corrected by the fermionic wigging itself; an analogous result holds for extremal magnetic black string with a near horizon geometry AdS S cfr. e.g. [18] and [11]. 10

12 Thus, the attractor values of the scalar fields φ x are still fixed purely in terms of the black hole electric charges : φ x BPS = e φ δ BPS = = φ x BPS δ 1 φ x BPS 1 δ φ x BPS! 1 δ φ x BPS 1 δ φ x BPS!! = φ BPS, 6.1 as it holds for the attractor mechanism on the purely bosonic background cfr. e.g. [16, 17, 18]. It should also be stressed that the result 6.1 does not depend on the specific data of the real special geometry of the manifold defined by the scalars of the vector multiplets. We would like to stress once again that we adopted the approximation of computing the fermionic wig by performing a perturbation of the unwigged, purely bosonic BPS extremal black hole solution while keeping the radius of the event horizon unchanged. The complete analysis of the fully-backreacted wigged black hole solution, including the study of its thermodynamical properties and the computation of its Bekenstein-Hawking entropy is left for future work. This study can also be generalized to the non-supersymmetric non-bps case 6. It should also be remarked that in D =, the attractor mechanism receives a priori non-vanishing corrections from bilinear terms in the anti-killing spinor ǫ [6]. Further investigation of such an important difference concerning wig corrections to the attractor mechanism in D = and D = 5 is currently in progress, and results will be reported elsewhere. Here, we confine ourselves to anticipate that the aforementioned non-vanishing wig corrections in D = can be related to the intrinsically dyonic nature of the four-dimensional large charge configurations, namely to the fact that charge configurations giving rise to a non-vanishing area of the horizon, and thus to a well-defined attractor mechanism for scalar dynamics, contain both electric and magnetic charges. As further venues of research, we finally would like to mention that fermionic wigging techniques could also be applied to other asymptotically flat D = 5 solutions, such as black rings [19, 18] and black Saturns [0], as well to extended N > supergravity theories in five dimensions. Acknowledgments We would like to thank Anna Ceresole and Gianguido Dall Agata for useful discussions on black holes in supergravity and on real special geometry. The work of LGCG is partially supported by MIUR grant RBFR10QS5J String Theory and Fundamental Interactions. The work of PAG is partially supported by the MIUR-PRIN contract 009-KHZKRX. The work of A. Marrani is supported in part by the FWO - Vlaanderen, Project No. G , and in part by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy P7/7. 6 Note that in this case the series.7 truncates at the 8 th order 11

13 The work of A. Mezzalira is partially supported by IISN - Belgium conventions and.51.08, by the Communauté Française de Belgique through the ARC program and by the ERC through the SyDuGraM Advanced Grant. A Notation and Identities We follow the notations in [7]. We adopt the Lorentzian D = 5 metric signature,,,, and we consider symplectic Majorana spinors satisfying λ i = λ i γ 0 = λ it C, A.1 where the charge conjugation matrix C fulfills the condition and C T = C = C 1, C = 1, A. Cγ µ C 1 = γ µ T = Cγ µ T C = γ µ, Cγ µν T C = γ µν, A. from which one obtains Cγ µ T = Cγ µ, Cγ µν T = Cγ µν. Notice that C and Cγ µ are antisymmetric matrices, while Cγ µν is a symmetric one. Spinorial indices i = 1, are raised and lowered as follows with V i = ε ij V j, V i = V j ε ji, ε 1 = ε 1 = 1. From these relations, one can derive the following identities : λ i χ i = λ i χ j ε ji = χ i λ i = χ i λ i, λ i γ µ χ i = λ i γ µ χ j ε ji = χ j γ µ λ j = χ i γ µ λ i, λ i γ µν χ i = λ i γ µν χ j ε ji = χ j γ µν λ j = χ i γ µν λ i, A. A.5 A.6 yielding λ i λ i = 0, λ i γ µ λ i = 0, λ i γ µν λ i 0. A.7 A.8 A.9 1

14 B Third Order At third order in N =, D = 5 supersymmetry iterated transformations, one finds 7 δ e a µ = ǫγa 1 δ ψ µ, δ ψµ i = δ D µ ǫ i 1 6 ǫj λ ix γ µ δ λ j x 1 1 γµνǫj λ ix γ ν δ λ j x 1 8 γµǫj λ ix γ δ λ j x 1 1 γν ǫ j λix γ µν δ λ j x 1 δ 1 e a µ ǫ j λix γ a δ 1 λ j x 1 δ ǫj 1 λix γ µ δ 1 λ j x 1 δ 1 e a µ γ ab ǫ 6 j λix γ b δ 1 λ j x 1 6 γµνǫj δ 1 λix γ ν δ 1 λ j x 1 δ 1 e a µ γ abc ǫ j λix γ bc δ 1 λ j x 1 γµǫj δ 1 λix γ δ 1 λ j x 1 δ 1 e a µ γ ab ǫ 6 j λix γ b δ 1 λ j x 1 6 γµνǫj δ 1 λix γ ν δ 1 λ j x 1 6 ǫj λ ix γ aλ j x δ e a µ 1 δ 1 e a µ ǫ j δ 1 λix γ aλ j x 1 δ ǫj λix γ µλ j x 1 1 γ abǫ j λix γ b λ j x δ e a µ 1 6 γ abǫ j δ 1 λix γ b λ j x δ 1 e a µ 1 1 γµνǫj δ λix γ ν λ j x 1 8 γ abcǫ j λix γ bc λ j x δ e a µ 1 γ abcǫ j δ 1 λix γ bc λ j x δ 1 e a µ 1 8 γµǫj δ λix γ λ j x 1 1 γν ǫ j δ λix γ µνλ j x 1 1 γb ǫ j λix γ ab λ j x δ e a µ 1 6 γb ǫ j δ 1 λix γ ab λ j x δ 1 e a µ i 6 hi F [ I δ e a µ ǫ ν be ρ c e a µ δ ǫ ν b e ρ c e a µǫ ν b δ e ρ c ] δ 1 e a µ δ 1 ǫ ν b e ρ c δ 1 e a µ ǫ ν b δ 1 e ρ c e a µ δ 1 ǫ ν b δ 1 e ρ c γa bc δaγ b c ǫ i i 6 hiz δ 1 φ z ] FI [δ 1 e a µ e ν be ρ c e a µ δ 1 e ν b e ρ c e a µe ν b δ 1 e ρ c γa bc δaγ b c ǫ i i 1 hiz δ φ z FI γ µ δ ν µγ ρ ǫ i i 1 yhiz δ 1 φ z δ 1 φ y FI γ i 6 hi δ 1 FI [δ 1 e a µ µ δµγ ν ρ ǫ i e ν be ρ c e a µ δ 1 e ν b e ρ c e a µe ν b ] δ 1 e ρ c γa bc δaγ b c ǫ i i 6 hiz δ 1 φ z γ δ 1 FI µ δµγ ν ρ ǫ i i γ 6 hi δ FI µ δµγ ν ρ ǫ i, B. δ φ x = ǫ i δ λ x, B. δ A I µ = 1 ǫγµ δ λ x h I x 1 i δ e a µ ǫγ aλ x h I x ǫhi δ ψ µ i hi x δ φ x ǫψ µ 7 t yh I x can be elaborated by exploiting Eq..5. Furthermore, w ut xyz = 1Ẽxyz wu, where the rank 5 completely symmetric tensor Ẽxyz wu is the real special geometry analogue [1] of the so called E tensor of special Kähler geometry [15]; by using the last of.a, a similar result holds for u tγ x yz. B.1 1

15 1 ǫγµλx yh I x δ φ y ih I x δ 1 φ x ǫ δ 1 ψ µ i yhi x δ 1 φ y δ 1 φ x ǫψ µ δ 1 e a µ ǫγ a δ 1 λ x h I x δ 1 e a µ ǫγ aλ x yh I x δ 1 φ y ǫγ µ δ 1 λ x yh I x δ 1 φ y 1 ǫγµλx t yh I x δ 1 φ y δ 1 φ t, δ λ ix = δ 1 e µ a γ a δ 1 Dµφ x ǫ i i δ e µ a γ a Dµφ x ǫ i i γµ δ Dµφ x ǫ i 1 6 Txyz γ µ ǫ j λi y γ µ δ λ j z 1 Txyz ǫ j λi y δ λ j z Txyz γ µν ǫ j λi y γ µν δ λ j z δ φ y Γ x yz δ 1 λ zi 1 Txyz ǫ j δ 1 λi y δ 1 λ j z 1 6 Txyz γ µν ǫ j δ 1 λi y γ µν δ 1 λ j z δ 1 φ y Γ x yz 1 1 δ λ zi δ φ y Γ x yzλ zi Txyz ǫ j δ λi y λ j z 1 utxyz δ 1 φ u ǫ j λi y δ 1 λ i z 1 6 utxyz δ 1 φ u γ µ ǫ j λi y γ µ δ 1 λ j z 1 6 utxyz δ 1 φ u γ µ ǫ j δ φ y tγ x yz Txyz γ µ ǫ j δ 1 λi y γ µ δ 1 λ j z 8 6 Txyz γ µν ǫ j δ λi y γ µνλ j z 1 6 utxyz δ 1 φ u γ µν ǫ j δ 1 λi y γ µνλ j z 1 utxyz δ 1 φ u ǫ j δ 1 λi y λ j z δ 1 λi y γ µλ j z δ 1 φ y tγ x yz δ 1 φ t λ zi 1 utxyz δ φ u ǫ j λi y λ j z w utxyz δ 1 φ u δ 1 φ w ǫ j λi y λ j z w utxyz δ 1 φ u δ 1 φ w γ µν ǫ j λi y γ µνλ j z 1 6 w utxyz δ 1 φ u δ 1 φ w γ µ ǫ j λi y γ µλ j z δ 1 φ y u tγ x yz δ 1 φ t δ 1 φ u λ zi 1 γ F I th x I δ φ t ǫ i 1 γ δ 1 FI th x I δ 1 φ t ǫ i 1 γ δ F I h x Iǫ i 1 γ F I u th x I δ 1 φ t δ 1 φ u ǫ i 1 γab[ δ e µ a e ν b δ 1 e µ a δ 1 e ν b e µ a δ eb] ν Fµνh x I γ ab δ 1 e µ a e ν b δ 1 Fµν h x I γ ab δ 1 e µ a e ν F b µν th x I δ 1 φ t δ 1 φ y tγ x yz δ φ t λ zi 1 6 utxyz δ φ u γ µ ǫ j λi y γ µλ j z δ 1 φ t δ 1 λ zi B utxyz δ φ u γ µν ǫ j λi y γ µνλ j z, B.5 1

16 with δ FI µν = δ Fµν I δ 1 ψ[µ γ ν] ψ [µ γ ν] δ 1 λ x yh I x δ 1 φ y δ 1 λ x h I x δ 1 ψ[µ γ ν] λ x yh I x δ ψ[µ γ ν] λ x h I x δ 1 φ y δ 1 φ z δ 1 φ y ψ [µ γ ν] δ λ x h I x ψ [µ γ ν] λ x z yh I x ψ [µ γ ν] λ x yh I x δ φ y i δ ψµ ψ νh I i ψ µ δ ψ ν h I i δ 1 ψµ δ 1 ψ ν h I i ψ µ δ 1 ψ ν h I x δ 1 φ x i ψ µψ νh I x δ φ x i δ 1 ψµ ψ νh I x δ 1 φ x i ψ µψ ν yh I x δ 1 φ x δ 1 φ y δ 1 ψ[µ δ 1 e a ν] γ aλ x h I x ψ [µ δ e a ν] γ aλ x h I x ψ [µ δ 1 e a ν] γ a δ 1 λ x h I x ψ [µ δ 1 e a ν] γ aλ x th I x δ 1 φ t, B.6 δ D µ = 1 δ ωµ ab γ ab, B.7 δ ωµ ab = 1 δ e cµ Ω abc Ω bca Ω cab [ δ 1 e cµ δ 1 Ω abc δ 1 Ω bca δ 1 Ω cab] 1 [δ ecµ Ω abc δ Ω bca δ Ω cab] δ K a µ b, B.8 δ Ω abc [ = δ e µa e νb δ 1 e µa δ 1 e νb e µa δ e νb] µe c ν νe c µ [δ 1 e µa e νb e µa δ 1 e νb][ ] µ δ 1 e c ν ν δ 1 e c µ e µa e νb[ ] µ δ e c ν ν δ e c µ, B.9 δ K a µ b = 1 [δ ψρ e ρ[a γ b] ψ µ δ 1 ψρ δ 1 e ρ[a γ b] ψ µ δ 1 ψρ e ρ[a γ b] δ 1 ψ µ ψ ρ δ e ρ[a γ b] ψ µ ψ ρ δ 1 e ρ[a γ b] δ 1 ψ µ ψ [a γ b] δ ψ µ 1 δ ψρ e ρa γ µψ b δ 1 ψρ γ µ δ 1 ψ ν e ρa e νb δ 1 ψρ γ µψ b δ 1 e ρa δ 1 ψρ γ cψ b e ρa δ 1 e c µ δ 1 ψρ γ µψ νe ρa δ 1 e νb 1 ψ a γ µ δ ψ ν e νb ψ ργ µ δ 1 ψ ν δ 1 e ρa e νb ψ a γ c δ 1 ψ ν δ 1 e c µ e νb ψ a γ µ δ 1 ψ ν δ 1 e νb 1 ψ ργ µψ b δ e ρa ψ ργ cψ b δ 1 e ρa δ 1 e c µ ψ ργ µψ ν δ 1 e ρa δ 1 e νb 1 ψ a γ cψ b δ e c µ ψ a γ cψ ν δ 1 e c µ δ 1 e νb 1 ψ a γ µψ ν δ e νb], B.10 δ Dµφ x = µ δ φ x i δ ψµ λ x i δ 1 ψµ δ 1 λ x i ψ µ δ λ x. B.11 15

17 C Fourth Order Finally, at the fourth order we find 8 δ e a µ = 1 ǫγa δ ψ µ, δ ψµ i = δ D µ ǫ i 1 6 ǫj λ ix γ µ δ λ j x 1 1 γµνǫj λ ix γ ν δ λ j x 1 8 γµǫj λ ix γ δ λ j x 1 1 γν ǫ j λix γ µν δ λ j x 1 6 ǫj λ ix γ aλ j x δ e a µ 1 δ ǫj λix γ µλ j x 1 1 γ abǫ j λix γ b λ j x δ e a µ C.1 1 δ 1 γµνǫj λix γ ν λ j x 1 8 γ abcǫ j λix γ bc λ j x δ e a µ 1 8 γµǫj δ λix γ λ j x 1 1 γν ǫ j δ λix γ µνλ j x 1 1 γb ǫ j λix γ ab λ j x δ e a µ i 6 hi F [ I δ e a µ e ν be ρ c e a µ δ e ν b e ρ c e a µe ν b δ e ρ c e a µ δ e ν b δ 1 e ρ c e a µ δ 1 e ν b δ e ρ c δ e a µ e ν b δ 1 e ρ c δ e a µ δ 1 e ν b e ρ c δ 1 e a µ δ e ν b e ρ c δ 1 e a µ e ν b δ e ρ c ] 6 δ 1 e a µ δ 1 e ν b δ 1 e ρ c γa bc δaγ b c ǫ i i thix F [ I δ 1 e a µ e ν be ρ c e a µ e ρ c δ 1 e ν b ] e a µe ν b δ 1 e ρ c δ 1 φ x δ 1 φ t γa bc δaγ b c ǫ i i hix δ 1 FI [δ 1 e a µ e ν be ρ c e a µ δ 1 e ν b e ρ c ] e a µe ν b δ 1 e ρ c δ 1 φ x γa bc δaγ b c ǫ i i hix F [ I δ e a µ e ν be ρ c e a µ δ e ν b e ρ c e a µe ν b δ e ρ c e a µ δ 1 e ν b δ 1 e ρ c δ 1 e a µ e ν b δ 1 e ρ c ] δ 1 e a µ δ 1 e ν b e ρ c δ 1 φ x γa bc δaγ b c ǫ i i hix F [ I δ 1 e a µ e ν be ρ c e a µ δ 1 e ν b e ρ c e a µe ν b δ 1 e ρ c i hi δ FI [δ 1 e a µ i hi δ 1 FI [δ e a µ e ν be ρ c e a µ δ e ν b e ρ c e a µe ν b δ e ρ c e a µ δ 1 e ν b δ 1 e ρ c δ 1 e a µ e ν b δ 1 e ρ c δ 1 e a µ δ 1 e ν b e ρ c i 1 hiz δ φ z FI γ µ δµγ ν ρ ǫ i i γ 6 hi δ FI µ δµγ ν ρ ǫ i ] δ φ x γ bc a δ b aγ c ǫ i ] e ν be ρ c e a µ δ 1 e ν b e ρ c e a µe ν b δ 1 e ρ c γa bc δaγ b c ǫ i ] γ bc a δ b aγ c ǫ i 8 Note that w t ut xyz = 1 w Ẽ xyz tu [1]; similarly, t z yh I x can be related to Ẽ tensor cfr. footnote 7. 16

18 1 δ 1 e a µ ǫ j λix γ a δ λ j x 1 δ ǫj 1 λix γ µ δ λ j x 1 δ 1 e a µ γ ab ǫ j λix γ b δ λ j x 1 γµνǫj δ 1 λix γ ν δ λ j x 1 δ 1 e a µ γ abc ǫ 16 j λix γ bc δ λ j x 1 16 γµǫj δ 1 λix γ δ λ j x 1 γb ǫ j λix γ ab δ λ j x δ 1 e a µ 1 γν ǫ j δ 1 λix γ µν δ λ j x 1 ǫj λ ix γ a δ 1 λ j x δ e a µ ǫ j δ 1 λix γ a δ 1 λ j x δ 1 e a µ 1 ǫj δ λix γ µ δ 1 λ j x 1 δ e a µ γ ab ǫ j λix γ b δ 1 λ j x 1 δ 1 e a µ γ ab ǫ j δ 1 λix γ b δ 1 λ j x 1 γµνǫj δ λix γ ν δ 1 λ j x 1 δ e a µ γ abc ǫ 16 j λix γ bc δ 1 λ j x 1 δ 1 e a µ γ abc ǫ j δ 1 λix γ bc δ 1 λ j x 8 1 δ 16 γµǫj λix γ δ 1 λ j x 1 γb ǫ j λix γ ab δ 1 λ j x δ e a µ 1 γb ǫ j δ 1 λix γ ab δ 1 λ j x δ 1 e a µ 1 γν ǫ j δ λix γ µν δ 1 λ j x 1 δ e a µ ǫ j δ 1 λix γ aλ j x 1 δ 1 e a µ ǫ j δ λix γ aλ j x 1 δ e a µ γ ab ǫ j δ 1 λix γ b λ j x 1 δ 1 e a µ ǫ jγ ab δ λix γ b λ j x 1 δ e a µ γ abc ǫ j δ 1 λix γ bc λ j x 1 δ 1 e a µ γ abc ǫ j δ λix γ bc λ j x 1 δ e a µ γ b ǫ j δ 1 λix γ ab λ j x 1 δ 1 e a µ γ b ǫ j δ λix γ ab λ j x i [h Iz δ φ z yh Iz δ 1 φ z δ 1 φ y] FI δ 1 e a µ γa δaγ ν ρ ǫ i [ yh Iz δ 1 φ z δ φ y yh Iz δ φ z δ 1 φ y i 1 t yh Iz i δ 1 φ z δ 1 φ y δ 1 φ t] FI γ µ δµγ ν ρ [h Iz δ φ z yh Iz δ 1 φ z δ 1 φ y] γ δ 1 FI i δ hiz 1 φ z δ F I δ φ x = ǫ i δ λ x, δ A I µ = 1 ǫγµ δ λ x h I x 1 i µ δ ν µγ ρ ǫ i γ µ δ ν µγ ρ ǫ i, C. δ e a µ ǫγ aλ x h I x ǫhi δ ψ µ i hi x δ φ x ǫψ µ 1 ǫγµλx yh I x δ φ y i hi x δ 1 φ x ǫ i δ ψ µ i hi x δ φ x ǫ δ 1 ψ µ yhi x δ 1 φ x δ 1 φ y ǫ δ 1 ψ µ i yhi x δ φ y δ 1 φ x ǫψ µ i yh I x δ φ x δ 1 φ y ǫψ µ i z yhi x δ 1 φ x δ 1 φ y δ 1 φ z ǫψ µ δ 1 e a µ ǫγ a δ λ x h I x δ e a µ ǫγ a δ 1 λ x h I x δ ǫγµ λ x yh I x δ 1 φ y δ 1 e a µ ǫγ a δ 1 λ x yh I x δ 1 φ y δ e a µ ǫγ aλ x yh I x δ 1 φ y ǫγµ δ 1 λ x[ yh I x δ φ y C. 17

19 z yh I x δ 1 φ y δ 1 φ z] δ 1 e a µ ǫγ aλ x[ yh I x δ φ y z yh I x δ 1 φ y δ φ z z yh I x 1 ǫγµλx[ z yh I x t z yh I x δ 1 φ y δ 1 φ z δ 1 φ t], δ λ ix = i γµ δ Dµφ x ǫ i i δ 1 φ y δ 1 φ z] δ φ y δ 1 φ z δ 1 Dµφ x ǫ i i γa δ e µ a Dµφ x 1 6 Txyz γ µ ǫ j λi y γ µ δ λ j z γa δ 1 e µ a i γa δ e µ a 1 Txyz ǫ j λi y δ λ j z Txyz γ µν ǫ j λi y γ µν δ λ j z Txyz ǫ j δ 1 λi y δ λ j z 1 8 Txyz γ µν ǫ j δ 1 λi y γ µν δ λ j z 1 Txyz γ µ ǫ j δ 1 λi y γ µ δ λ j z δ φ y Γ x yz δ λ zi δ φ y Γ x yz δ 1 λ zi Txyz ǫ j δ λi y δ 1 λ j z 1 8 Txyz γ µν ǫ j δ λi y γ µν δ 1 λ j z 1 Txyz γ µ ǫ j δ λi y γ µ δ 1 λ j z δ 1 φ y Γ x yz δ λ zi δ φ y Γ x yzλ zi 1 Txyz ǫ j δ λi y λ j z Txyz γ µν ǫ j δ λi y γ µνλ j z 1 6 Txyz γ µ ǫ j δ λi y γ µλ j z ttxyz δ 1 φ t ǫ j δ λi y λ j z 1 8 ttxyz δ 1 φ t γ µν ǫ j λi y γ µν δ λ j z 1 ttxyz δ 1 φ t γ µ ǫ j λi y γ µ δ λ j z ttxyz δ 1 φ t ǫ j δ 1 λi y δ 1 λ j z 1 ttxyz δ 1 φ t γ µν ǫ j δ 1 λi y γ µν δ 1 λ j z 1 ttxyz δ 1 φ t γ µ ǫ j δ 1 λi y γ µ δ 1 λ j z ttxyz δ 1 φ t ǫ j δ λi y λ j z 1 8 ttxyz δ 1 φ t γ µν ǫ j δ λi y γ µνλ j z 1 ttxyz δ 1 φ t γ µ ǫ j δ λi y γ µλ j z 6 δ φ y tγ x yz δ 1 φ t δ 1 λ zi δ φ y tγ x yz δ 1 φ t λ zi δ 1 φ y tγ x yz δ Dµφ x ǫ i δ 1 φ t δ λ zi C. 18

20 [ ǫj λ i y δ 1 λ j z tt xyz δ φ t u tt xyz δ 1 φ t δ 1 φ u] 1 [ 8 γµν ǫ j λi y γ µν δ 1 λ j z tt xyz δ φ t u tt xyz δ 1 φ t δ 1 φ u] 1 [ γµ ǫ j λi y γ µ δ 1 λ j z tt xyz δ φ t u tt xyz δ 1 φ t δ 1 φ u] [ ǫj δ 1 λi y λ j z tt xyz δ φ t u tt xyz δ 1 φ t δ 1 φ u] 1 [ 8 γµν ǫ j δ 1 λi y γ µνλ j z tt xyz δ φ t u tt xyz δ 1 φ t δ 1 φ u] 1 [ γµ ǫ j δ 1 λi y γ µλ j z tt xyz δ φ t u tt xyz δ 1 φ t δ 1 φ u] δ 1 φ y[ tγ x yz δ φ t u tγ x yz δ 1 φ t δ 1 φ u] δ 1 λ zi δ φ y[ tγ x yz δ φ t u tγ x yz δ 1 φ t δ 1 φ u] λ zi 1 [ ut xyz δ φ u t ut xyz δ 1 φ u δ φ t t ut xyz δ φ u δ 1 φ t w t ut xyz δ 1 φ u δ 1 φ t δ 1 φ w] ǫ j λi y λ j z 1 [ 8 6 γµν ǫ j λi y γ µνλ j z ut xyz δ φ u t ut xyz δ 1 φ u δ φ t t ut xyz δ φ u δ 1 φ t w t ut xyz δ 1 φ u δ 1 φ t δ 1 φ w] 1 [ 6 γµ ǫ j λi y γ µλ j z ut xyz δ φ u t ut xyz δ 1 φ u δ φ t t ut xyz δ φ u δ 1 φ t w t ut xyz δ 1 φ u δ 1 φ t δ 1 φ w] δ 1 φ y[ uγ x yz δ φ u u tγ x yz δ 1 φ t δ φ u u tγ x yz δ φ t δ 1 φ u w u tγ x yz δ 1 φ t δ 1 φ u δ 1 φ w] λ zi 1 γ F [ I th x I δ φ t u th x I δ 1 φ t δ φ u u th x I δ φ t δ 1 φ u ǫ i w u th x I δ 1 φ t δ 1 φ u δ 1 φ w] ǫ i [ γ δ 1 FI th x I δ φ t u th x I δ 1 φ t δ 1 φ u] γ δ FI th x I δ 1 φ t 1 γ δ FI h x I 1 γab[ δ e µ a e ν b δ e µ a δ 1 eb] ν F µνh I x I γab[ δ e µ a e ν b δ 1 e µ a δ 1 e ν b ]δ 1 FI µν h x I γab[ δ e µ a e ν b δ 1 e µ a δ 1 eb] ν FI µν th x I δ 1 φ t γab δ 1 e µ a e ν b δ FI µν h x I γ ab δ 1 e µ a e ν b δ 1 FI µν th x I δ 1 φ t γab δ 1 e µ a e ν F b µν I t uh x I δ 1 φ t δ 1 φ u γab δ 1 e µ a e ν F b µν I th x I δ 1 φ t, C.5 with 19

21 δ FI µν = δ Fµν I δ 1 ψ[µ γ ν] δ ψ[µ δ λ x h I x δ ψ[µ γ ν] δ 1 λ x h I x ψ [µ γ ν] δ λ x h I x γ ν] λ x h I x i ψ µ δ ψ ν h I i δ 1 ψµ δ ψ ν h I i δ ψµ δ 1 ψ ν h I i δ ψµ ψ νh I i ψ µ δ ψ ν h I x δ 1 φ x i ψ µ δ 1 ψ ν h I x δ φ x i δ 1 ψµ δ 1 ψ ν h I x δ 1 φ x i ψ µ δ 1 ψ ν yh I x δ 1 φ x δ 1 φ y i ψ µψ νh I x δ φ x i δ 1 ψµ ψ νh I x δ φ x i δ ψµ ψ νh I x δ 1 φ x i ψ µψ ν yh I x δ 1 φ x δ φ y i ψ µψ ν yh I x δ φ x δ 1 φ y i δ 1 ψµ ψ ν yh I x δ 1 φ x δ 1 φ y i ψ µψ ν z yh I x δ 1 φ x δ 1 φ y δ 1 φ z ψ [µ γ ν] δ 1 λ x yh I x δ φ y ψ [µ γ ν] δ λ x yh I x δ 1 φ y 6 δ 1 ψ[µ γ ν] δ 1 λ x yh I x δ 1 φ y ψ [µ γ ν] λ x yh I x δ φ y δ φ y δ 1 φ y δ 1 ψ[µ γ ν] λ x yh I x δ ψ[µ γ ν] λ x yh I x ψ [µ γ ν] δ 1 λ x z yh I x δ 1 φ y δ 1 φ z ψ [µ γ ν] λ x z yh I x δ 1 φ y δ φ z ψ [µ γ ν] λ x z yh I x δ φ y δ 1 φ z δ 1 ψ[µ γ ν] λ x z yh I x δ 1 φ y δ 1 φ z ψ [µ γ ν] λ x w z yh I x δ 1 φ y δ 1 φ z δ 1 φ w δ ψ[µ δ 1 e a ν] γ aλ x h I x δ 1 ψ[µ δ e a ν] γ aλ x h I x 6 δ 1 ψ[µ δ 1 e a ν] γ a δ 1 λ x h I x 6 δ 1 ψ[µ δ 1 e a ν] ψ [µ δ e a ν] γ aλ x h I x ψ [µ δ e a ν] γ a δ 1 λ x h I x ψ [µ δ e a ν] γ aλ x th I x δ 1 φ t ψ [µ δ 1 e a ν] γ a δ λ x h I x 6 ψ [µ δ 1 e a ν] ψ [µ δ 1 e a ν] γ aλ x u th I x δ 1 φ t δ 1 φ u γ aλ x th I x γ a δ 1 λ x th I x δ 1 φ t δ 1 φ t ψ [µ δ 1 e a ν] γ aλ x th I x δ φ t, C.6 δ D µ = 1 δ ωµ ab γ ab, C.7 δ ωµ ab = 1 δ e cµ Ω abc Ω bca Ω cab [ δ e cµ δ 1 Ω abc δ 1 Ω bca δ 1 Ω cab] [ δ 1 e cµ δ Ω abc δ Ω bca δ Ω cab] 1 [δ ecµ Ω abc δ Ω bca δ Ω cab] δ K a µ b, C.8 0

22 δ Ω abc [ = δ K a µ b = δ e µa e νb δ e µa δ 1 e νb δ 1 e µa δ e νb e µa δ e νb] µe c ν νeµ c [δ 1 e µa e νb e µa δ 1 e νb][ ] µ δ e c ν ν δ e c µ [ δ e µa e νb δ 1 e µa δ 1 e νb e µa δ e νb][ ] µ δ 1 e c ν ν δ 1 e c µ e µa e νb[ ] µ δ e c ν ν δ e c µ, C.9 δ 1 ψρ δ 1 e ρa γ c δ 1 e c µ ψ b δ ψρ e ρa γ c δ 1 e c µ ψ b δ ψρ δ 1 e ρa γ µψ b δ 1 ψρ e ρa γ c δ e c µ ψ b ψ ρ δ 1 e ρa γ c δ e c µ ψ b δ 1 ψρ δ e ρa γ µψ b ψ ρ δ e ρa γ c δ 1 e c µ ψ b 1 δ ψρ e ρa γ µψ b 1 ψ a γ c δ e c µ ψ b 1 ψ ρ δ e ρa γ µψ b 1 ψ [a γ b] δ ψ µ 1 ψ a γ µ δ ψ σ e σb δ 1 ψρ e ρ[a γ b] δ ψ µ δ 1 ψρ e ρa γ µ δ ψ σ e σb ψ a γ c δ 1 e c µ δ ψ σ e σb ψ ρ δ 1 e ρ[a γ b] δ ψ µ ψ ρ δ 1 e ρa γ µ δ ψ σ e σb ψ a γ µ δ ψ σ δ 1 e σb δ 1 ψρ e ρa γ c δ 1 e c µ δ 1 ψ σ e σb δ 1 ψρ δ 1 e ρ[a γ b] δ 1 ψ µ δ 1 ψρ δ 1 e ρa γ µ δ 1 ψ σ e σb ψ ρ δ 1 e ρa γ c δ 1 e c µ δ 1 ψ σ e σb δ 1 ψρ e ρa γ µ δ 1 ψ σ δ 1 e σb ψ a γ c δ 1 e c µ δ 1 ψ σ δ 1 e σb ψ ρ δ 1 e ρa γ µ δ 1 ψ σ δ 1 e σb δ ψρ e ρ[a γ b] δ 1 ψ µ δ ψρ e ρa γ µ δ 1 ψ σ e σb ψ a γ c δ e c µ δ 1 ψ σ e σb ψ ρ δ e ρ[a γ b] δ 1 ψ µ ψ ρ δ e ρa γ µ δ 1 ψ σ e σb ψ a γ µ δ 1 ψ σ δ e σb δ 1 ψρ e ρa γ c δ 1 e c µ ψ σ δ 1 e σb δ 1 ψρ δ 1 e ρa γ µψ σ δ 1 e σb ψ ρ δ 1 e ρa γ c δ 1 e c µ ψ σ δ 1 e σb δ ψρ δ 1 e ρ[a γ b] ψ µ δ ψρ e ρa γ µψ σ δ 1 e σb ψ a γ c δ e c µ ψ σ δ 1 e σb δ 1 ψρ δ e ρ[a γ b] ψ µ ψ ρ δ e ρa γ µψ σ δ 1 e σb δ 1 ψρ e ρa γ µψ σ δ e σb ψ a γ c δ 1 e c µ ψ σ δ e σb ψ ρ δ 1 e ρa γ µψ σ δ e σb 1 δ ψρ e ρ[a γ b] ψ µ 1 ψ ρ δ e ρ[a γ b] ψ µ 1 ψ a γ µψ σ δ e σb, C.10 δ Dµφ x = µ δ φ x i δ ψµ λ x i δ ψµ δ 1 λ x i δ 1 ψµ δ λ x i ψ µ δ λ x. C.11 1

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