Component on-shell actions of supersymmetric 3-branes II. 3-brane in D=8
aa r X i v : . [ h e p - t h ] N ov Component on-shell actions of supersymmetric 3-branesII. 3-brane in D=8
S. Bellucci a , N. Kozyrev b , S. Krivonos b and A. Sutulin a,b a INFN-Laboratori Nazionali di Frascati, Via E. Fermi 40, 00044 Frascati, Italy b Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia
Abstract
In the present paper we explicitly construct the on-shell supersymmetric component action for a3-brane moving in D = 8 within the nonlinear realizations framework. Similarly to the previouslyconsidered case of the super 3-brane in D = 6, all ingredients entering the component action followfrom the nonlinear realizations approach. The component action of the 3-brane possesses N =4 , d = 4 supersymmetry partially broken to N = 2 , d = 4 one. The basic Goldstone superfieldis the generalized version of N = 2 , d = 4 hypermultiplet. The action has a structure, such thatall terms of higher orders in the fermions are hidden inside the covariant derivatives and vielbeins.The main part of the component action mimics its bosonic cousin in which the ordinary space-time derivatives and the bosonic worldvolume are replaced by their covariant (with respect to brokensupersymmetry) supersymmetric analogs. The spontaneously broken supersymmetry fixes the Ansatzfor the component action, up to two constant parameters. The role of the unbroken supersymmetryis just to fix these parameters. Introduction
In the famous super p-brane scan [1] there are only two branes with four-dimensional worldvolume -3-brane in D = 6 and in D = 8. While the first 3-brane has been considered in [2] and then furtheranalysed in many other approaches including the superfield ones [3, 4, 5], for the best of our knowledgethe 3-brane in D = 8 has been not considered yet. It is a bit strange, because on the worldvolumethe effective action for such 3-bane will be just the action for N = 2 , d = 4 hypermultiplet [6] - thesuperfield which probably is not less known than the N = 2 , d = 4 vector supermultiplet. Such anaction has to possess hidden, spontaneously broken N = 2 supersymmetry which, together with themanifest N = 2 , d = 4 one, forms a N = 4 , d = 4 supersymmetry algebra with four central charges.Such a situation, probably, can be explained by the on-shell nature of the hypermultiplet constraints inthe standard superspace, so the using of harmonic [7] or/and projective [8] superspaces is unavoidable.Another reason probably is related with the fact that, after dimensional reduction from the known actionsin higher dimensions and fixing the κ -supersymmetry, we will end up with a long tail of fermionic termshaving no geometric meaning. Moreover, in the theories with partially broken supersymmetry there is thepossibility of redefining the fermionic components in many different ways: starting from the fermions ofthe linear realization and finishing by the fermions of the nonlinear realization. Under such redefinitionsthe action changes drastically and a priori it is unknown which basis is preferable.In this second part of our paper we construct and analyse in details the action of supersymmetric3-brane in D = 8 using the nonlinear realization approach [10, 11] properly modified for the constructionof component actions in [12, 13]. In our approach we are paying much attention to the broken super-symmetry, almost ignoring the unbroken one. Thanks to the fact that all physical components appear asthe parameters of the corresponding coset, all of them have Goldstone nature. Keeping in mind that theGoldstone fermions, accompanying the partial breaking of supersymmetry, can enter the component ac-tion either through the covariant derivatives or vielbeins, only, the Ansatz for the action having a properbosonic limit contains just two constants (one constant is related with the possibility to add Wess-Zuminoterm to the action). The fixing of these constants is the role playing by unbroken supersymmetry.The plan of our paper goes as follows. In Section 2 we provide the superspace description of the 3-branein D = 8. The constraints on the basic N = 2 , d = 4 superfield (covariantized hypermultiplet conditions)and the equations of motion - that is all that we can get in this way. Section 3 is the main part of ourpaper. There we derive the bosonic action of our 3-brane, propose the Ansatz for the component action,fixing the parameters entering into such an Ansatz and, finally, demonstrate the relations between spinorderivatives of the fermionic superfields and space-time derivatives of the hypermultiplet. We concludewith a short discussion and further perspectives of our approach. The Appendix contains purely technicalmaterial, concerning the evaluation of the Cartan forms and covariant derivatives. D = 8 The action of the N = 2 supersymmetric three-brane in D = 8 is a minimal action for the Goldstone N = 2 , d = 4 hypermultiplet accompanying the spontaneous breaking of N = 1 , D = 8 supersymmetrydown to N = 2 , d = 4 one, or, in other words, the breaking of N = 4 supersymmetry to N = 2 in fourdimensions [1]. Despite the fact that this information should be enough to construct the superfield actionof the three-brane in terms of the N = 2 , d = 4 hypermultiplet, for the best of our knowledge, such anaction has been not constructed yet. Our goal in this paper is to carry out a less ambitious task - toconstruct the on-shell component action for such a system. In our construction we are going to use thebuilding blocks having clear geometric properties with respect to broken N = 2 supersymmetry. Thebasic tool for this is, similarly to the first part of this paper [14], the method of nonlinear realizations[10, 11] adopted for this task in [12, 13]. Our procedure includes three steps: a) construction of thesuperfield equations of motion, b) deriving the bosonic action, c) construction of the full componentaction. See also [9] and references therein. .1 Superfield equations of motion From the d = 4 standpoint the N = 1 , D = 8 supersymmetry algebra is a four central charges extended N = 4 , d = 4 Poincar´e superalgebra with the following basic relations: (cid:8) Q iα , Q j ˙ α (cid:9) = 2 δ ij (cid:0) σ A (cid:1) α ˙ α P A , (cid:8) S aα , S b ˙ α (cid:9) = 2 δ ab (cid:0) σ A (cid:1) α ˙ α P A , (cid:8) Q iα , S aβ (cid:9) = 2 ǫ αβ Z ia , (cid:8) Q i ˙ α , S a ˙ β (cid:9) = 2 ǫ ˙ α ˙ β Z ia . (2.1)As a reminder about its eight-dimensional nature, the superalgebra (2.1) possesses the so (1 ,
7) automor-phism algebra. Again, from d = 4 perspective, so (1 ,
7) algebra contains the d = 4 Lorentz algebra so (1 , L AB , su (2) ⊕ su (2) subalgebra with generators T ij and R ab , respectively, and the generators K iaA from the coset SO (1 , /SO (1 , × SU (2) × SU (2). The full set of commutation relations can befound in Appendix.Keeping the d = 4 Lorentz and SU (2) × SU (2) symmetries linearly realized, we will choose the cosetelement as g = e i x A P A e θ αi Q iα +¯ θ i ˙ α Q i ˙ α e i q ia Z ia e ψ αa S aα + ¯ ψ a ˙ α S a ˙ α e i Λ Aia K iaA . (2.2)Here, we associated the N = 2 , d = 4 superspace coordinates x A , θ αi , ¯ θ i ˙ α with the generators P A , Q iα , Q i ˙ α of unbroken N = 2 supersymmetry. The remaining coset parameters are Goldstone N = 2 , d = 4superfields.The transformation properties of the coordinates and superfields with respect to all symmetries canbe found by acting from the left on the coset element g (2.2) by the different elements of N = 4 , d = 4central charges extended Poincar´e supergroup. In particular, for the unbroken ( Q, Q ) and broken (
S, S )supersymmetries we have • Unbroken supersymmetry: δ Q x A = i (cid:0) ǫ αi ¯ θ i ˙ α + ¯ ǫ i ˙ α θ αi (cid:1) (cid:0) σ A (cid:1) α ˙ α , δ Q θ αi = ǫ αi , δ Q ¯ θ i ˙ α = ¯ ǫ i ˙ α ; (2.3) • Broken supersymmetry: δ S x A = i (cid:0) ε αa ¯ ψ a ˙ α + ¯ ε a ˙ α ψ αa (cid:1) (cid:0) σ A (cid:1) α ˙ α , δ S ψ αa = ε αa , δ S ¯ ψ a ˙ α = ¯ ε a ˙ α , δ S q ia = 2i (cid:0) ε aα θ iα + ¯ ε a ˙ α ¯ θ i ˙ α (cid:1) . (2.4)The local geometric properties of the system are specified by the Cartan forms. The purely technicalcalculations of these forms, semi-covariant derivatives and their algebra are summarized in Appendix.As we already demonstrated in [12, 13, 14], the covariant superfield equations of motion may beobtained by imposing the following constraints on the Cartan forms: ω Z = ¯ ω Z = 0 ( a ) , ω S | = ¯ ω S | = 0 ( b ) , (2.5)where | means the dθ - and d ¯ θ -projections of the forms. These constraints are similar to superembeddingconditions (see e.g. [15] and references therein).The constraints (2.5a) result in the following equations: ∇ jα q ia + 2i ψ aα δ ji = 0 , ∇ j ˙ α q ia + 2i ¯ ψ a ˙ α ǫ ij = 0 , (2.6) ∇ A q ia = Λ jbA tanh q Λ Bjb Λ iaB q Λ Bjb Λ iaB . (2.7)These equations allow us to express the superfields ψ αa , ¯ ψ a ˙ α and Λ iaA through the covariant derivatives ofsuperfields q ia (this is the so called Inverse Higgs phenomenon [16]): ψ aα = i4 ∇ kα q ka , ¯ ψ a ˙ α = i4 ∇ k ˙ α q ka , (2.8) Λ iaA = ∇ A q ia + . . . , (2.9)where in (2.9) we explicitly write only the leading, linear in ∇ A q ia term. In addition, from (2.6) it followsthat ∇ ( iα q j ) a = 0 , ∇ ˙ α ( i q j ) a = 0 . (2.10)2learly, these are just a covariantized version of the hypermultiplet conditions [6] which put theoryon-shell.The constraints (2.5b) follow from (2.5a), but their explicit form helps to simplify the consideration.Firstly, the d ¯ θ ( dθ ) projection of the form ω S (¯ ω S ) relates the spinor derivative of the superfields ψ αa , ¯ ψ a ˙ α and x − derivative of the superfield q ia ∇ k ˙ γ ψ αb ≡ (cid:0) J α ˙ γ (cid:1) kb = Λ γck ˙ γ (cid:18) tanh √ T √ T (cid:19) cαbγ = Λ αkbγ + . . . , ∇ kγ ¯ ψ b ˙ α ≡ (cid:0) J ˙ αγ (cid:1) kb = Λ kc ˙ γγ (cid:18) tanh √ T √ T (cid:19) b ˙ αc ˙ γ = Λ kb ˙ αγ + . . . . (2.11)At the same time, the dθ ( d ¯ θ ) projection of the form ω S (¯ ω S ) gives the equations ∇ iα ψ βa = 0 , ∇ i ˙ α ¯ ψ a ˙ β = 0 . (2.12)The invariance of the equations (2.6), (2.7), (2.11) and (2.12) with respect to N = 2 , d = 4 superalgebra(A.2), (A.3) is guaranteed by the invariance of the conditions (2.5).Thus, we conclude that the supersymmetric 3-brane in D = 8 is described by the covariantized N = 2 , d = 4 hypermultiplet. Unfortunately, it is not so simple to write down even the bosonic equationsof motion which follow from (2.10). In the next Section we obtain the proper bosonic action starting fromits invariance with respect to the bosonic subalgebra (A.2), and then we will construct the full on-shellcomponent action for our supersymmetric 3-brane. As we already noted, it is not clear how to construct the superfield action within the nonlinear realizationsapproach. It is even technically hard to extract the bosonic equations of motion from the superfield ones(2.10). Therefore, we are going to construct the on-shell supersymmetric component action within thenonlinear realizations approach following the approach developed in [12, 13, 14]. The useful ingredientsfor this construction include the reduced Cartan forms and reduced covariant derivatives, covariant withrespect to broken supersymmetry only. The basic steps of our approach are • construction of the bosonic action • covariantization of the bosonic action with respect to broken supersymmetry • construction of the Wess-Zumino terms • imposing the invariance with respect to the unbroken supersymmetry.Let us perform all these steps for the supersymmetric 3-brane in D = 8. In principle, the bosonic equations of motion can be extracted from the superfield equations (2.10). Butthe calculations are rather involved, because one has to express the superfields Λ iaA in terms of ∇ A q ia (see eq. (2.7)). Instead, one can construct the corresponding action directly, using the fact that suchan action should possess invariance with respect to D = 8 Poincar´e symmetry spontaneously broken to d = 4. One of the key ingredients of such a construction is the bosonic limit of the Cartan forms (A.6)which explicitly reads( ω P ) Abos = dx B cosh q jbB Λ Ajb − dq jb Λ Cjb sinh q jbC Λ Ajb q jbC Λ Ajb , ( ω Z ) iabos = dq jb cosh q Ajb Λ iaA − dx A Λ jbA sinh q Bjb Λ iaB q Bjb Λ iaB . (3.1)3he bosonic part of our constraints (2.5)( ω Z ) bos = (¯ ω Z ) bos = 0 , will result in the bosonic analog of the relations (2.7) ∂ A q ia = Λ jbA tanh q Bjb Λ iaB q Bjb Λ iaB . (3.2)Plugging these expressions in the form ( ω P ) Abos (3.1) one may obtain( ω P ) Abos = dx B e BA = dx B q Bjb Λ jbA . (3.3)Now, the unique invariant which can be constructed from the forms ( ω P ) Abos is a volume form whichexplicitly reads d x det( e ). Thus, the invariant bosonic action is uniquely defined to be S bos = Z d x det( e ) . (3.4)Using the explicit expressions (3.2) and (3.3), one may find the simple expression for the ”metric” g AB in terms of ∂ A q ia g AB ≡ e CA e CB = η AB − ∂ A q ia ∂ B q ia , g = det g AB , (3.5)and, therefore, the bosonic action acquires the form S bos = Z d x √− g . (3.6)This is the static gauge Nambu-Goto action for the 3-brane in D = 8. One may explicitly check that theaction (3.6) is invariant with respect to K iaA transformations (with the parameter A Aia ) from the coset SO (1 , /SO (1 , × SU (2) × SU (2) realized as δ K x A = 2 A Aia q ia , δ K q ia = A iaA x A . (3.7)and therefore, it is invariant with respect to the whole D = 8 Poincar´e group. Working in the component approach, we cannot straightforwardly construct the Ansatz for the actionwhich possesses the unbroken supersymmetry. In contrast, the broken (
S, S ) supersymmetry can bemaintained quite easily due to the transformations δ S θ αi = δ S ¯ θ i ˙ α = 0. Thus, the first task is to modifythe bosonic action (3.6) in such a way as to achieve invariance with respect to broken supersymmetry.Due to the transformation laws (2.4), the coordinates x A and the first components ( q ia , ψ αa , ¯ ψ a ˙ α ) of thesuperfields ( q ia , ψ αa , ¯ ψ a ˙ α ) transform under broken supersymmetry as follows: δ S x A = i (cid:0) ε αa ¯ ψ a ˙ α + ¯ ε a ˙ α ψ αa (cid:1) (cid:0) σ A (cid:1) α ˙ α , δ S ψ αa = ε αa , δ S ¯ ψ a ˙ α = ¯ ε a ˙ α , δ S q ia = 0 . (3.8)Thus, the volume d x and the derivatives ∂ A q ia are not the covariant objects. In order to find the properobjects, let us consider the reduced coset element (2.2) g red = e i x A P A e i q ia Z ia e ψ αa S aα + ¯ ψ a ˙ α S a ˙ α , (3.9)where the fields ( q ia , ψ αa , ¯ ψ a ˙ α ) depend on the coordinates x A only. The corresponding reduced Cartanforms (A.6) read ( ω P ) Ared = E AB dx B , E AB ≡ δ AB − i (cid:0) ψ αa ∂ B ¯ ψ a ˙ α + ¯ ψ a ˙ α ∂ B ψ αa (cid:1) (cid:0) σ A (cid:1) α ˙ α , ( ω Z ) iared = dq ia , ( ω S ) aαred = dψ aα , (¯ ω S ) a ˙ αred = d ¯ ψ a ˙ α . (3.10)4hese forms are invariant with respect to the transformations (3.8). Therefore, the covariant x -derivativewill be D A = (cid:0) E − (cid:1) AB ∂ B , (3.11)while the invariant volume can be constructed from the forms ( ω P ) Ared . Thus, the proper covariantizationof the action (3.6), having the right bosonic limit, will be S = Z d x det( E ) √−G , (3.12)where the covariantized metric tensor G AB , evidently, reads G AB ≡ η AB − D A q ia D B q ia , G = det G AB . (3.13)The action S (3.12) reproduces the fixed kinetic terms for bosons and fermions( S ) lin = − Z d x h i (cid:0) ψ αa ∂ α ˙ α ¯ ψ a ˙ α + ¯ ψ a ˙ α ∂ α ˙ α ψ αa (cid:1) + 2 ∂ A q ia ∂ A q ia i . (3.14)This would be too strong to maintain unbroken supersymmetry. Therefore, we have to introduce onemore, evidently invariant, purely fermionic action S = α Z d x det( E ) , (3.15)which will correct the kinetic terms for the fermions, because( S ) lin = − i α Z d x (cid:0) ψ αa ∂ α ˙ α ¯ ψ a ˙ α + ¯ ψ a ˙ α ∂ α ˙ α ψ αa (cid:1) . (3.16)Thus, our Ansatz for the invariant supersymmetric action of the 3-brane acquires the form S = S + S + S = (1 + α ) Z d x − Z d x det( E ) h α + √−G i , (3.17)where α is a constant parameter that has to be defined, and we have added the trivial invariant action S = R d x to have a proper limit S q,ψ → = 0 . Let us finish this Subsection by some comments on the Ansatz for our action (3.17) • Firstly, the fermions ψ αa , ¯ ψ a ˙ α transform under broken supersymmetry (3.8) as the Goldstinos ofthe Volkov-Akulov model [17]. This means that they may enter an invariant action only throughthe determinant of the super-vielbein det( E ) or the covariant derivatives D A . The action (3.17) hasjust this structure. Moreover, the fermionic limit of the action S ferm = (1 + α ) Z d x h − det( E ) i is just the Volkov-Akulov action for the Goldstino, in full agreement with the results of [18, 19]. • Secondly, the transformation properties of the fields q ia (3.8) show that q ia are just the matterfields with respect to broken supersymmetry. Therefore, it is clear that any action of the form S = Z d x det( E ) F ( D q ) , where F is an arbitrary function, depending on all possible Lorentz and SU (2) × SU (2) invariantcombinations constructed from D A q ia , invariant under unbroken supersymmetry. Thus, the knowl-edge of the proper bosonic limit of the action (3.6) is very important to select the particular systemfrom the quite wide family of the actions invariant with respect to broken supersymmetry.5 Thirdly, the parameter α can be immediately fixed to be α = 1, if we will insist on the invariance ofthe linearized action ( S ) lin + ( S ) lin under linearized transformations of unbroken supersymmetry δ linQ q ia = 2i (cid:0) ǫ iα ψ aα + ¯ ǫ i ˙ α ¯ ψ a ˙ α (cid:1) , δ linQ ψ aα = − ¯ ǫ ˙ αi ∂ α ˙ α q ia , δ linQ ¯ ψ a ˙ α = ǫ αi ∂ α ˙ α q ia . Thus, the suitable Ansatz for our action is S = 2 Z d x − Z d x det( E ) h √−G i . (3.18) • Finally, it seems to be strange to call the action (3.18) by the term Ansatz, because it does notcontain any free parameter. The reason for such a nomenclature is simple - the action (3.18) isnot the most general action possessing the proper bosonic limit (3.6) and invariant under brokensupersymmetry due to existence of Wess-Zumino terms. Thus, the proper Ansatz for the action of3-brane in D = 8 reads S = 2 Z d x − Z d x det( E ) h √−G i + S W Z . (3.19)This additional term S W Z present in the our Ansatz (3.19), will be constructed in the next Sub-section.
The construction of the Wess-Zumino term, which is not strictly invariant, but which is shifted by atotal derivative under broken supersymmetry (3.8), goes in a standard way [20]. First of all, one has todetermine the closed five form Ω , which is invariant under d = 4 Lorentz and broken supersymmetrytransformations (3.8). Moreover, in the present case this form has to disappear in the bosonic limit,because our Ansatz for the action (3.17) already reproduces the proper bosonic action of the 3-brane(3.6). Such a form can be easily constructed in terms of the Cartan forms (3.10) (for the sake of brevity,we have omitted the subscript ”red” below) :Ω = ω Sαa ∧ ¯ ω Sb ˙ α ∧ ω Zia ∧ ω Zib ∧ ω AP ( σ A ) α ˙ α = dψ αa ∧ d ¯ ψ b ˙ α ∧ dq ia ∧ dq ib ∧ (cid:0) dx α ˙ α − (cid:0) ψ cα d ¯ ψ c ˙ α + ¯ ψ c ˙ α dψ cα (cid:1)(cid:1) . (3.20)To see that Ω (3.20) is indeed a closed form, one should take into account that the exterior derivativeof ( ω P ) α ˙ α is given by the expression d ( ω P ) α ˙ α = − dψ cα ∧ d ¯ ψ c ˙ α , (3.21)and, therefore, d Ω = 0, because dψ αa ∧ dψ bα = 12 ǫ ab dψ cα ∧ dψ cα ⇒ d Ω ∼ dψ aα ∧ dψ aα ∧ d ¯ ψ b ˙ α ∧ d ¯ ψ b ˙ α ∧ dq ic ∧ dq ic = 0 . Next, one has to write Ω as the exterior derivative of a 4-form Ω . This step, in contrast with the caseof supersymmetric 3-brane in D = 6 [14], is not completely trivial. Starting with the ”evident” guessΩ (1)4 = 12 (cid:0) ψ αa d ¯ ψ b ˙ α + ¯ ψ b ˙ α dψ αa (cid:1) ∧ dq ia ∧ dq ib ∧ (cid:0) dx α ˙ α − (cid:0) ψ cα d ¯ ψ c ˙ α + ¯ ψ c ˙ α dψ cα (cid:1)(cid:1) , (3.22)we will get d Ω (1)4 = Ω + 2i (cid:0) ψ αa d ¯ ψ b ˙ α + ¯ ψ b ˙ α dψ αa (cid:1) ∧ dψ cα ∧ d ¯ ψ c ˙ α ∧ dq ia ∧ dq ib . (3.23)The last step is to note that the second term in r.h.s of (3.23) may be represented as − i d (cid:8) (cid:2) ( ψ aα dψ bα ) ∧ ( ¯ ψ c ˙ α ∧ d ¯ ψ c ˙ α ) + ( ψ cα ∧ dψ cα ) ∧ ( ¯ ψ a ˙ α d ¯ ψ b ˙ α ) (cid:3) ∧ dq ia ∧ dq ib (cid:9) . (3.24)Thus, the proper form Ω , with the property d Ω = Ω , is given by the expressionΩ = 12 (cid:0) ψ αa d ¯ ψ b ˙ α + ¯ ψ b ˙ α dψ αa (cid:1) ∧ dq ia ∧ dq ib ∧ (cid:0) dx α ˙ α − (cid:0) ψ cα d ¯ ψ c ˙ α + ¯ ψ c ˙ α dψ cα (cid:1)(cid:1) + i (cid:2) ( ψ aα dψ bα ) ∧ ( ¯ ψ c ˙ α ∧ d ¯ ψ c ˙ α ) + ( ψ cα ∧ dψ cα ) ∧ ( ¯ ψ a ˙ α d ¯ ψ b ˙ α ) (cid:3) ∧ dq ia ∧ dq ib . (3.25)6ntegrating this form (3.25) we will get the Wess-Zumino action S W Z = β Z d x det( E ) ǫ ABCD h (cid:0) ψ αa D A ¯ ψ b ˙ α + ¯ ψ b ˙ α D A ψ αa (cid:1) D B q ia D C q ib ( σ D ) α ˙ α − (cid:0) ψ αa D A ψ bα ¯ ψ c ˙ α D B ¯ ψ c ˙ α + ψ cα D A ψ cα ¯ ψ ˙ αa D B ¯ ψ b ˙ α (cid:1) D C q ia D D q ib i . (3.26)By construction, the action (3.26) is invariant with respect to broken supersymmetry transformations(3.8). Thus, the full Ansatz for the component action of 3-brane in D = 8 reads S = 2 Z d x − Z d x det( E ) h √−G i + β Z d x det( E ) ǫ ABCD h (cid:0) ψ αa D A ¯ ψ b ˙ α + ¯ ψ b ˙ α D A ψ αa (cid:1) D B q ia D C q ib ( σ D ) α ˙ α − (cid:0) ψ αa D A ψ bα ¯ ψ c ˙ α D B ¯ ψ c ˙ α + ψ cα D A ψ cα ¯ ψ ˙ αa D B ¯ ψ b ˙ α (cid:1) D C q ia D D q ib i . (3.27)One should note that the Ansatz (3.27) is the unique, minimal, namely containing only the first derivativesof the fields involved, action with the proper bosonic limit, which is invariant with respect to the brokensupersymmetry (3.8). The role of the unbroken supersymmetry is to fix the constant parameter β (wealready used the linearized version of the unbroken supersymmetry to fix the parameter α in (3.17)). The most technically complicated part of our approach is to maintain the unbroken supersymmetry,despite the fact that all we need is to fix one parameter in the action (3.27). In a case which we have now inhands, the situation is even worse than that was in the cases which we considered before [14, 12, 13, 21, 22],because until now we did not present the exact expressions for the quantities ( J A ) ia , ( J A ) ia (2.11) in termsof ∇ A q ia . The corresponding equations, which can be obtained by the action of the anticommutators (cid:8) ∇ iα , ∇ j ˙ α (cid:9) (A.12) on the superfield q mb , have the form δ mj ( J α ˙ α ) bi − δ mi ( J α ˙ α ) bj = ǫ ij ∇ α ˙ α q mb + ( J α ˙ γ ) ai ( J γ ˙ α ) aj ∇ γ ˙ γ q mb . (3.28)Passing to the components does not help, because the θ = ¯ θ = 0 projection of the equations (3.28) readsquite similarly, to be δ mj ( J α ˙ α ) bi − δ mi ( J α ˙ α ) bj = ǫ ij D α ˙ α q mb + ( J α ˙ γ ) ai ( J γ ˙ α ) aj D γ ˙ γ q mb . (3.29)It seems to be a completely hopeless idea to solve the equations (3.29) starting from the most generalAnsatz for ( J A ) ia : ( J A ) ia = f D A q ia + f d BA D B q ia + f d BA d CB D C q ia + f d BA d CB d DC D D q ia + f ǫ ABCD D B q ib D C q jb D D q aj , (3.30)where all functions f are the complex(!) scalar functions depending, in general, on all possible Lorentzand SU (2) × SU (2) invariants constructed from D A q ia , and d AB ≡ D A q ia D B q ia . (3.31)Indeed, it is completely unbelievable, that the system of quadratically nonlinear equations for the fivecomplex functions (if we will succeed in their construction!) can be solved. Nevertheless, the iterativesolution of the equations (3.29) can be straightforwardly found. In the first five orders in D A q ia it reads( J A ) ia = D A q ia + n d BA D B q ia − T r (cid:0) d (cid:1) D A q ia + i3 ǫ ABCD D B q ib D C q jb D D q aj o + n d BA d BC D C q ia − T r (cid:0) d (cid:1) d BA D B q ia − T r (cid:0) d (cid:1) D A q ia + 12 (cid:0) T r (cid:0) d (cid:1)(cid:1) D A q ia + i6 (cid:0) ǫ BCDE D B q jc D C q bj D D q kb D E q ck (cid:1) ∇ A q ia o + n . . . o ≥ . (3.32)7he iterative solution (3.32) suggests another form of the ( J A ) ia :( J A ) ia = N BA D B q ia + i K BA ( X B ) ia , (3.33)where ( X A ) ia = ǫ ABCD D B q ib D C q jb D D q aj (3.34)and the first entries in the real matrices functions N BA and K BA read N BA = h − T r (cid:0) d (cid:1) + 12 (cid:0) T r (cid:0) d (cid:1)(cid:1) − T r (cid:0) d (cid:1) − T r (cid:0) d (cid:1) + 114 T r (cid:0) d (cid:1) T r (cid:0) d (cid:1) − (cid:0) T r (cid:0) d (cid:1)(cid:1) i δ BA + h − T r (cid:0) d (cid:1) + 74 (cid:0) T r (cid:0) d (cid:1)(cid:1) − T r (cid:0) d (cid:1) i d BA + h − T r (cid:0) d (cid:1) i (cid:0) d · d (cid:1) BA + 6 (cid:0) d · d · d (cid:1) BA + . . .K BA = h
13 + 112 (cid:0)
T r (cid:0) d (cid:1)(cid:1) − T r (cid:0) d (cid:1) i δ BA − h − T r (cid:0) d (cid:1) i d BA − (cid:0) d · d (cid:1) BA + . . . . (3.35)As we expected, the solution (3.35) is rather complicated. Fortunately, there is a loophole which helpsus to find the full solution of the equations (3.29) without solving them directly. The idea is to analyzethe variation of our action (3.27) with respect to unbroken supersymmetry keeping ( J A ) ia arbitrary andthen to find and solve the linear equations which guarantee the invariance of our action. β Before performing above mentioned analysis (see the next Subsection), let us firstly fix the parameter β .In order to do this, one has expand the action (3.27) up to terms of the fourth order in ∂ A q ia and thesecond order in the fermions, i.e.det( E ) ⇒ − (cid:0) ψ αa ∂ α ˙ α ¯ ψ a ˙ α + ¯ ψ a ˙ α ∂ α ˙ α ψ αa (cid:1) , √−G ⇒ (cid:16) − T r (cid:0) d (cid:1) + (cid:0) T r (cid:0) d (cid:1)(cid:1) − T r (cid:0) d (cid:1)(cid:17) ,S W Z ⇒ β Z d x ǫ ABCD h (cid:0) ψ αa ∂ A ¯ ψ b ˙ α + ¯ ψ b ˙ α ∂ A ψ αa (cid:1) ∂ B q ia ∂ C q ib ( σ D ) α ˙ α i (3.36)and consider the variation of this reduced action in the first order in the fermions and the third order in ∂ A q ia .Keeping in mind that under the unbroken Q supersymmetry the covariant derivatives ∇ A (A.10) areinvariant, and then for an arbitrary superfield F ( δ Q F ) θ =¯ θ =0 = − (cid:0) ǫ αi D iα F + ¯ ǫ i ˙ α D i ˙ α F (cid:1) θ =¯ θ =0 , ( δ Q ∇ A F ) θ =¯ θ =0 = − (cid:0) ǫ αi D iα ∇ A F + ¯ ǫ i ˙ α D i ˙ α ∇ A F (cid:1) θ =¯ θ =0 , one may find the transformation properties of all ingredients in the action (3.27) (we will explicitly writeonly their ǫ -part) δ Q ψ aα = H B ∂ B ψ aα , δ Q ¯ ψ a ˙ α = H B ∂ B ¯ ψ a ˙ α − ǫ αi ( J A ) ia (cid:0) σ A (cid:1) α ˙ α , (3.37) δ Q D A q ka = H B ∂ B D A q ka + 2 i ǫ αk D A ψ aα − i ǫ αi D A ψ bα ( J B ) ib D B q ka − ǫ αi D A ψ bβ ( J D ) ib (cid:0) σ DB (cid:1) βα D B q ka ,δ Q D A ψ aα = H B ∂ B D A ψ aα − i ǫ βi D A ψ bβ ( J B ) ib D B ψ aα − ǫ γi D A ψ bβ ( J D ) ib (cid:0) σ DB (cid:1) βγ D B ψ aα ,δ Q D A ¯ ψ a ˙ α = H B ∂ B D A ¯ ψ a ˙ α − ǫ αi D A ( J B ) ia (cid:0) σ B (cid:1) α ˙ α − i ǫ αi D A ψ bα ( J B ) ib D B ¯ ψ a ˙ α − i ǫ αi D A ψ bβ ( J D ) ib (cid:0) σ DB (cid:1) βα D B ¯ ψ a ˙ α , where H B = iǫ αi ψ aα ( J B ) ia + ǫ αi ψ aβ ( J A ) ia (cid:0) σ AB (cid:1) βα . (3.38)As a consequence of (3.37) we will have δ Q det( E ) = − det( E ) E AB δ Q (cid:0) E − (cid:1) BA = ∂ A (cid:0) H A det( E ) (cid:1) + 2 i det( E ) ǫ αi h D A ψ aα ( J A ) ia − i D A ψ aβ ( J B ) ia (cid:0) σ BA (cid:1) βα i . (3.39)8hus, we see that all H -dependent terms are converted into full space-time derivatives and, therefore,they can be ignored. For the present analysis, in the above given approximation we will need only thereduced version of these variations δ RedQ ψ aα = 0 , δ RedQ ¯ ψ a ˙ α = − ǫ αi ( J A ) ia (cid:0) σ A (cid:1) α ˙ α , (3.40) δ RedQ D A q ka = 2 i ǫ αk ∂ A ψ aα − i ǫ αi ∂ A ψ bα ( J B ) ib ∂ B q ka − ǫ αi ∂ A ψ bβ ( J D ) ib (cid:0) σ DB (cid:1) βα ∂ B q ka ,δ RedQ D A ψ aα = 0 , δ RedQ D A ¯ ψ a ˙ α = − ǫ αi D A ( J B ) ia (cid:0) σ B (cid:1) α ˙ α ,δ RedQ det( E ) = 2 i ǫ αi h ∂ A ψ aα ( J A ) ia − i ∂ A ψ aβ ( J B ) ia (cid:0) σ BA (cid:1) βα i . Moreover, in these transformations one should insert the object ( J A ) ia , up to the proper order, using thesolution (3.32). Collecting all these together, we will get the following expression for the variation of themain part of the action (3.27): δ RedQ h − det( E ) (cid:16) √−G (cid:17)i ≈ ǫ iα ∂ A ψ αa n i3 ǫ ABCD ∂ B q ib ∂ C q jb ∂ D q aj o +4 ǫ iα ∂ A ψ βa (cid:0) σ AB (cid:1) αβ n (cid:0) − T r (cid:0) d (cid:1)(cid:1) ∂ B q ia + d BC ∂ C q ia + i3 ǫ BCDF ∂ C q ib ∂ D q jb ∂ F q aj o (3.41)+4 ǫ iα ∂ C ψ βa (cid:0) σ AB (cid:1) αβ d AC ∂ B q ia . The term in the first line in r.h.s. of (3.41) cancels as a full divergence. The terms containing both matrixand ǫ -symbol can be simplified using the σ -matrices property σ BC ǫ CF GH = i (cid:0) δ BF σ GH − δ BG σ F H + δ BH σ F G (cid:1) . (3.42)The last term in (3.41) can be also simplified by inserting d CB = ∂ C q ia ∂ B q ia and extracting the termsantisymmetric in { i, j } and { a, b } , respectively. Then the term antisymmetric in { a, b } , also cancels out.Finally, the variation reads δ RedQ h − det( E ) (cid:16) √−G (cid:17)i ≈ ǫ iα ∂ A ψ βa (cid:0) σ AB (cid:1) αβ n (cid:0) − T r (cid:0) d (cid:1)(cid:1) ∂ B q ia + d BC ∂ C q ia o + 4 ǫ iα ∂ C ψ βa (cid:0) σ AB (cid:1) αβ ∂ A q jb ∂ C q ib ∂ B q aj . (3.43)The variation of the reduced Wess-Zumino action (3.36) can be easily found, because in our approximationwe only need to vary ¯ ψ i ˙ α , and take ( J A ) ia in this variation only up to the lowest approximation. Thisvariation involves the product of σ -matrices which can be simplified using (3.42). Finally, we will get δ Q L W Z ≈ − βǫ iα n ∂ C ψ βa (cid:0) σ BC (cid:1) αβ ∂ B q bj ∂ A q ja ∂ A q ib + ∂ C ψ βa (cid:0) σ AB (cid:1) αβ ∂ C q ib ∂ A q ja ∂ B q bj − ∂ C ψ βa (cid:0) σ AC (cid:1) αβ ∂ A q ja ∂ B q bj ∂ B q ib o . (3.44)After a slightly rearranging of the terms, we will find that (3.44) precisely cancels the variation (3.43)if β = 2. Thus, the action (3.27) with β = 2 is invariant under broken and unbroken supersymmetriesin this approximation. One should note that, due to the fact that we do not have at hands any morefreedom to modify the action, and keeping in mind that the terms of higher order in the fermions comeout from the lowest one, due to the invariance under broken supersymmetry, we have to conclude thatthe full component action of 3-brane in D = 8 reads S = 2 Z d x − Z d x det( E ) h √−G i + 2 Z d x det( E ) ǫ ABCD h (cid:0) ψ αa D A ¯ ψ b ˙ α + ¯ ψ b ˙ α D A ψ αa (cid:1) D B q ia D C q ib ( σ D ) α ˙ α − (cid:0) ψ αa D A ψ bα ¯ ψ c ˙ α D B ¯ ψ c ˙ α + ψ cα D A ψ cα ¯ ψ ˙ αa D B ¯ ψ b ˙ α (cid:1) D C q ia D D q ib i . (3.45) ( J A ) ia The last step to complete our analysis of the unbroken supersymmetry is to find a closed expression forthe ( J A ) ia entering the transformation properties (3.37). Our attempts to solve the basic equations (3.29)9esult only in the iterative solution (3.32) which may be prolonged up to any desired order, but whichcannot help us to find the full solution. The idea to find the full expression for ( J A ) ia we shortly discussedabove, is based on the invariance of our action (3.45). If this action is invariant, then the vanishing ofits variation under transformations (3.37) with unspecified ( J A ) ia will result in the linear equations on( J A ) ia which we are going to solve instead of solving the nonlinear equations (3.29). Good news is thatthe bosonic limit of the equations (3.29) simply corresponds to the replacement D A q ia → ∂ A q ia , andtherefore it is enough to consider the variation of the action (3.45) to the first order in the fermions. Ifwe will find the ( J A ) ia which nullify this variation of the action, then the full ( J A ) ia with all fermionicterms can be reconstructed from it by the inverse substitution ∂ A q ia → D A q ia . The linear in the fermionsvariation of the integrand in the action (3.45) has the following form: δ Q L = 4 i √− g (cid:0) g − (cid:1) AB (cid:0) Σ A (cid:1) ia ∂ B q ia − i h δ AB + √− g (cid:0) g − (cid:1) AB i(cid:0) Σ A (cid:1) ia ( J B ) ia (3.46) − h δ AB + √− g (cid:0) g − (cid:1) AB i(cid:0) Σ DBA (cid:1) ia ( J D ) ia − ǫ ABCD ǫ αk ∂ A ψ aα ( J D ) kb ∂ B q ia ∂ C q ib + 4 i ǫ ABCD ǫ αk ∂ A ψ aβ ( J F ) kb (cid:0) σ F D (cid:1) βα ∂ B q ia ∂ C q ib , where we used the following notations: (cid:0) Σ A (cid:1) ia = ǫ αi ∂ A ψ aα , (cid:0) Σ DBA (cid:1) ia = ǫ αi ∂ A ψ aβ (cid:0) σ DB (cid:1) βα . (3.47)Combining together the terms with (cid:0) Σ A (cid:1) ia and (cid:0) Σ DBA (cid:1) ia we will get δ Q S = Z d x (cid:0) Σ A (cid:1) ka n √− g (cid:0) g − (cid:1) AB ∂ B q ka − h δ AB + √− g (cid:0) g − (cid:1) AB i ( J B ) ka − ǫ ABCD ∂ B q ia ∂ C q ib ( J D ) kb o (3.48)+ 2 Z d x (cid:0) Σ A,DB (cid:1) ia nh η AB + √− g (cid:0) g − (cid:1) AB i ( J D ) ka − ǫ ABCF ( J D ) kb ∂ C q ia ∂ F q ib o . (3.49)Both variations, which are proportional to (cid:0) Σ A (cid:1) ia and (cid:0) Σ DBA (cid:1) ia , have to be zero independently. Never-theless, due to a specific structure of (cid:0) Σ A (cid:1) ia and (cid:0) Σ DBA (cid:1) ia (3.47) we cannot conclude that the quantitiesin the curly brackets are equal to zero. Indeed, one may easily check that the terms Z d x (cid:0) Σ A (cid:1) ia ( X A ) ia and Z d x (cid:0) Σ A,DB (cid:1) ia η AB ∂ D q ia (3.50)are equal to zero, being full space-time derivatives. Thus, the expressions in the curly brackets in (3.48)and (3.49) have to vanish up to the integrand in these additional terms. The coefficients before theseterms can be easily fixed from the known lowest orders in the iterative solution (3.32). Thus, we cameto the following equations:4i √− g (cid:0) g − (cid:1) AB ∂ B q ka − M AB ( J B ) ka − β ǫ ABCD ∂ B q ia ∂ C q ib ( J D ) kb = −
83 ( X A (cid:1) ka , (3.51)2 M AB J kaD − β ǫ ABCF ( J D ) kb ∂ C q ia ∂ F q ib = 4 η AB ∂ D q ka , (3.52)where M AB = h δ AB + √− g (cid:0) g − (cid:1) AB i (3.53)and we underline the indices in the equation (3.52) to remind that due to anti-symmetry and self-dualityof σ BD in (3.47) we have only three independent equations over these indices in (3.52).Let us start from the equation (3.51). In order to avoid the appearance of the su (2) indices ( i, a ), wewill convert this equation with ∂ B q ka and substitute the Ansatz (3.33) for ( J A ) ia . Doing so, we finallyget the following matrix equations on the real and imaginary parts of (3.51):12 (cid:0) M · K (cid:1) AB − (cid:0) δ AB δ CD − δ AD δ CB (cid:1) N DC = − δ AB , √− g (cid:0) g − (cid:1) AB d BD − (cid:0) M · N · d (cid:1) AD − (cid:0) δ AD δ CF − δ AF δ CD (cid:1) (cid:0) K · Z (cid:1) FC = 0 , (3.54)10here the matrix Z AB reads Z AB = ( X A ) ia ( X B ) ia . (3.55)This matrix can be expressed through d AB using the definition (3.34)( X A ) ia ( X B ) ia = − / (cid:20)(cid:18) T r ( d ) − T r ( d ) T r ( d ) + 13 ( T r ( d )) (cid:19) η AB + (cid:0) T r ( d ) − ( T r ( d )) (cid:1) d AB +2 T r ( d ) d AC d BC − d AC d DC d BD (cid:3) . (3.56)Besides the rather complicated structure of the matrix Z AB , the equations (3.54) are linear and can beeasily solved. The only problem is to represent the solution in a readable form. We succeeded in thefollowing form of the solution: N AB = 2 F (cid:16) √− g − T r (cid:0) d (cid:1) + 4 (cid:0) T r (cid:0) d (cid:1)(cid:1) − T r (cid:0) d (cid:1) − √− g T r (cid:0) g − (cid:1) (cid:17) δ AB − √− gF (cid:16) √− g T r (cid:0) g − (cid:1) (cid:17)(cid:0) g − (cid:1) AB + 8 F (cid:16) (cid:0) d · d (cid:1) AB + (cid:0) − T r (cid:0) d (cid:1)(cid:1) d AB (cid:17) ,K AB = 43 F h √− g (cid:0) g − (cid:1) AB − g AB − √− g T r (cid:0) g − (cid:1) δ AB i , (3.57)where F = 8 + 8 √− g − T r (cid:0) d (cid:1) + 8 (cid:0) T r (cid:0) d (cid:1)(cid:1) − T r (cid:0) d (cid:1) + g (cid:0) T r (cid:0) g − (cid:1)(cid:1) − √− g T r (cid:0) g − (cid:1) . (3.58)Funny enough, the matrices N and K are related in a quite simple way (cid:0) N · d (cid:1) AB = γK AB + 12 δ AB , (3.59)with γ = − (cid:16) − √− g + √− g T r (cid:0) g − (cid:1)(cid:17) . (3.60)Having at hands the exact solution for ( J A ) ia which guaranteed the invariance of the action (3.45)under unbroken supersymmetry, it is a matter of long and rather complicated calculations to check thatthe equations (3.52) and (3.29) are satisfied. We did not find any simple way, besides the brute forcechecking, to demonstrate this fact. In this second part of our paper we described the partial breaking of N = 1 , D = 8 supersymmetry downto N = 2 , d = 4 one within the nonlinear realization approach. The basic Goldstone superfield associatedwith this breaking is the N = 2 , d = 4 hypermultiplet q ia subjected to a nonlinear generalization of thestandard hypermultiplet constraints (2.10). The dynamical equations which follow from these constraintsare identified with the worldvolume supersymmetric equations of supersymmetric 3-brane in D = 8. Thispart of our paper is quite similar to the previously considered case of spontaneously broken N = 1 , D = 10supersymmetry down to (1 , d = 6 supersymmetry performed in [23]. After this superfield consideration,in the main part of this paper (Section3), we turned to the construction of the component on-shell actionfor this 3-brane. The building blocks for the component action are the Cartan forms for the reduced coset(2.2) which are completely similar to the famous case considered by Volkov and Akulov [17]. Thus, thefirst Ansatz for our action (3.17), possessing the proper bosonic limit, can be constructed immediately.The first parameter α appearing on this stage can be easily fixed to be 1 by the invariance under thesimplest, linear part of the unbroken supersymmetry transformations. The existence of the Wess-Zuminoterm makes the construction slightly more complicated. Fortunately, this additional term with the newparameter β can be constructed from the same Cartan forms in the way discussed in [20]. Thus, our finalAnsatz for the action is defined uniquely (3.27) up to one free parameter β . The fixing of this parameteris a more complicated task. Luckily, it turns out that it is enough to consider the invariance of ourAnsatz to the first nontrivial order in ∂ A q ia . Thus, we came to our main result - the component actionof supersymmetric 3-brane in D = 8 (3.45). 11n contrast with the 3-brane in D = 6 [14], in order to prove the invariance of our new action withrespect to unbroken supersymmetry, firstly one has to solve the nonlinear matrix equation (3.29). Thisequation relates the spinor covariant derivatives of the spinor superfields ∇ iα ¯ ψ a ˙ α , ∇ i ˙ α ψ aα and the space-time derivative of bosonic superfields D α ˙ α q ia . This equation has a nicely defined iterative solution (3.32),but due to a rather complicated most general Ansatz (3.30), it includes five complex functions. In order tosolve this problem we reversed the arguments and wrote the variation of the action (3.45) under unbrokensupersymmetry, still keeping ( J A ) ia unspecified. Then the vanishing of this variation results in the linear equations on ( J A ) ia , which can be immediately solved. Funny enough, the correctness of the approachleads to the fact that this solution of the linear equations solved simultaneously the nonlinear equations.Thus, in these two papers we completed the construction of the component actions for all 3-branesfrom the famous brane scan of the paper [1]. The constructed actions contain only objects with clearmeaning - vielbeins and covariant derivatives. The derivation of the Wess-Zumino term also needs onlythe Cartan forms on the reduced coset. The quite simple form of the final action of the 3-brane in D = 8raised the question of the superfield formulations of such a system within harmonic [7] or projective [8]superspaces. Being constructed, such a description would make it possible to include into the game the N = 2 , d = 4 matter superfields, in such a way as to get N = 4 , d = 4 invariant systems due to the properinteraction with the hypermultiplet. Finally, one should mention that N = 4 , d = 4 → N = 2 , d = 4partial breaking of supersymmetry can be achieved by using the N = 2 , d = 4 vector supermultiplet[24, 25, 26] instead of of the hypermultiplet, which will result in the N = 2 supersymmetric Born-Infeldaction. We are hoping the approach we are using here will help to solve this task. Acknowledgments
We are grateful to Dmitry Sorokin and Igor Bandos for valuable correspondence.This work was partially supported by the ERC Advanced Grant no. 226455 “Supersymmetry, Quan-tum Gravity and Gauge Fields” ( SUPERFIELDS ). The work of N.K. and S.K. was supported by RSCFgrant 14-11-00598. The work of A.S. was partially supported by RFBR grant 13-02-9062 Arm-a.
Appendix: Superalgebra, coset space, transformations and Car-tan forms
In this Appendix we collected some formulas describing the nonlinear realization of N = 1 , D = 8 Poincar´egroup in its coset over its N = 1 , d = 4 subgroup.In d = 4 notation the N = 1 , D = 8 Poincar´e superalgebra is a four central charges extended N = 4super-Poincar´e algebra containing the following set of generators:N=4, d=4 SUSY ∝ (cid:8) P A , Q iα , Q i ˙ α , S aα , S a ˙ α , Z ia , L AB , K iaA , T ij , R ab (cid:9) . (A.1)Here, P A and Z ia are D = 8 translation generators, Q iα , Q i ˙ α and S aα , S a ˙ α are the generators of super-translations, the generators L AB form d = 4 Lorentz algebra so (1 , K iaA belong to thecoset SO (1 , /SO (1 , × SU (2) × SU (2), while the generators T ij and R ab span su (2) × su (2) subalgebra( i, a = 1 , D = 8 Poincar´e algebra in this basis read[ L AB , L CD ] = i ( − η AC L BD + η BC L AD − η BD L AC + η AD L BC ) , [ L AB , P C ] = i ( − η AC P B + η BC P A ) , (cid:2) L AB , K iaC (cid:3) = i (cid:0) − η AC K iaB + η BC K iaA (cid:1) , (cid:2) T ij , T kl (cid:3) = i (cid:0) ǫ ik T jl + ǫ jk T il + ǫ il T jk + ǫ jl T ik (cid:1) , (cid:2) R ab , R cd (cid:3) = i (cid:0) ǫ ac R bd + ǫ bc R ad + ǫ ad R bc + ǫ bd R ac (cid:1) , (cid:2) T ij , K kaA (cid:3) = i (cid:16) ǫ ik K jaA + ǫ jk K iaA (cid:17) , (cid:2) R ab , K icA (cid:3) = i (cid:0) ǫ ac K ibA + ǫ bc K iaA (cid:1) , (cid:2) T ij , Z ka (cid:3) = i (cid:0) ǫ ik Z ja + ǫ jk Z ia (cid:1) , (cid:2) R ab , Z ic (cid:3) = i (cid:0) ǫ ac Z ib + ǫ bc Z ia (cid:1) , (cid:2) P A , K iaB (cid:3) = i η AB Z ia , (cid:2) K iaA , Z jb (cid:3) = − ǫ ij ǫ ab P A , h K iaA , K jbB i = 2i ǫ ij ǫ ab L AB − i η AB (cid:0) ǫ ab T ij + ǫ ij R ab (cid:1) . (A.2)12ere, η AB = diag(1 , − , − , − Q iα , Q i ˙ α , S aα , S a ˙ α obey the following (anti)commutation relations: (cid:8) Q iα , Q ˙ αj (cid:9) = 2 δ ij (cid:0) σ A (cid:1) α ˙ α P A , (cid:8) S aα , S ˙ αb (cid:9) = 2 δ ab (cid:0) σ A (cid:1) α ˙ α P A , (cid:8) Q iα , S aβ (cid:9) = 2 ǫ αβ Z ia , n Q ˙ αi , S ˙ βa o = 2 ǫ ˙ α ˙ β Z ia ; (cid:2) L AB , Q iα (cid:3) = −
12 ( σ AB ) βα Q iβ , (cid:2) L AB , Q ˙ αi (cid:3) = 12 (˜ σ AB ) ˙ β ˙ α Q ˙ βi , [ L AB , S aα ] = −
12 ( σ AB ) βα S aβ , (cid:2) L AB , S ˙ αa (cid:3) = 12 (˜ σ AB ) ˙ β ˙ α S ˙ βa ; (cid:2) K iaA , Q jα (cid:3) = i ( σ a ) α ˙ α ǫ ij S ˙ αa , (cid:2) K iaA , S bα (cid:3) = − i ( σ A ) α ˙ α ǫ ab Q ˙ αi , (cid:2) K iaA , Q ˙ αj (cid:3) = i ( σ A ) α ˙ α δ ij S αa , (cid:2) K iaA , S ˙ αb (cid:3) = − i ( σ a ) α ˙ α δ ab Q αi ; (cid:2) T ij , Q kα (cid:3) = i (cid:0) ǫ ik Q jα + ǫ jk Q iα (cid:1) , (cid:2) T ij , Q ˙ αk (cid:3) = − i (cid:16) δ ik Q j ˙ α + δ jk Q i ˙ α (cid:17) , (cid:2) R ab , S cα (cid:3) = i (cid:0) ǫ ac S bα + ǫ bc S aα (cid:1) , (cid:2) R ab , S ˙ αc (cid:3) = − i (cid:16) δ ac S b ˙ α + δ bc S a ˙ α (cid:17) . (A.3)We define the coset element as follows: g = e i x A P A e θ αi Q iα +¯ θ i ˙ α Q i ˙ α e i q ia Z ia e ψ αa S aα + ¯ ψ a ˙ α S a ˙ α e i Λ Aia K iaA . (A.4)Here, { x A , θ αi , ¯ θ i ˙ α } are N = 2 , d = 4 superspace coordinates, while the remaining coset parameters are N = 2 , d = 4 Goldstone superfields. The local geometric properties of the system are specified by theCartan forms g − dg = i ( ω P ) A P A + i ( ω Z ) ia Z ia + ( ω Q ) αi Q iα + (¯ ω Q ) i ˙ α Q i ˙ α + ( ω S ) αa S aα + (¯ ω S ) a ˙ α S a ˙ α +i ( ω K ) Aia K iaA + i (¯ ω K ) A K A + i ( ω L ) AB L AB . (A.5)In what follows, we will need the explicit expressions of the following forms:( ω P ) A = △ x B cosh q Λ jbB Λ Ajb − △ q jb Λ Cjb sinh q Λ jbC Λ Ajb q Λ jbC Λ Ajb , ( ω Z ) ia = △ q jb cosh q Λ Ajb Λ iaA − △ x A Λ jbA sinh q Λ Bjb Λ iaB q Λ Bjb Λ iaB , ( ω Q ) αi = dθ βj (cid:16) cosh √ W (cid:17) jαiβ + d ¯ ψ c ˙ γ sinh √ T √ T ! b ˙ βc ˙ γ Λ αib ˙ β , ( ω S ) αa = d ψ βb (cid:16) cosh √ T (cid:17) bαaβ − d ¯ θ k ˙ γ Λ γbk ˙ γ sinh √ T √ T ! bαaγ , (A.6)where △ x A = dx A − i (cid:16) θ αi d ¯ θ i ˙ α + ¯ θ i ˙ α dθ αi + ψ αa d ¯ ψ a ˙ α + ¯ ψ a ˙ α d ψ αa (cid:17) (cid:0) σ A (cid:1) α ˙ α , △ q ia = d q ia − (cid:0) ψ aα dθ αi + ¯ ψ a ˙ α d ¯ θ ˙ αi (cid:1) , (A.7)and the matrix-valued functions are defined as follows: W jαiβ = Λ α ˙ αia Λ jaβ ˙ α , W i ˙ αj ˙ β = Λ i ˙ αaα Λ aαj ˙ β , T bαaβ = Λ ibβ ˙ α Λ α ˙ αia , T a ˙ αb ˙ β = Λ iαb ˙ β Λ a ˙ αiα . (A.8)Keeping in mind, that the quantities △ x A , dθ αi and d ¯ θ i ˙ α are invariant with respect to both supersym-metries, one may define the covariant derivatives ∇ A , ∇ iα , ∇ i ˙ α as d F = (cid:18) dx A ∂∂x A + dθ αi ∂∂θ αi + d ¯ θ i ˙ α ∂∂ ¯ θ i ˙ α (cid:19) F = (cid:0) △ x A ∇ A + dθ αi ∇ iα + d ¯ θ i ˙ α ∇ i ˙ α (cid:1) F , (A.9)13nd, therefore, ∇ A = (cid:0) E − (cid:1) BA ∂ B , E BA = δ BA − i (cid:16) ψ αa ∂ A ¯ ψ a ˙ α + ¯ ψ a ˙ α ∂ A ψ αa (cid:17) (cid:0) σ B (cid:1) α ˙ α , (A.10) ∇ iα = D iα − i (cid:16) ψ βa ∇ iα ¯ ψ a ˙ β + ¯ ψ a ˙ β ∇ iα ψ βa (cid:17) (cid:0) σ B (cid:1) β ˙ β ∂ B , ∇ i ˙ α = D i ˙ α − i (cid:16) ψ βa ∇ i ˙ α ¯ ψ a ˙ β + ¯ ψ a ˙ β ∇ i ˙ α ψ βa (cid:17) (cid:0) σ B (cid:1) β ˙ β ∂ B . Here, D iα , D i ˙ α are flat derivatives obeying the relations (cid:8) D iα , D j ˙ α (cid:9) = − δ ij (cid:0) σ A (cid:1) α ˙ α ∂ A , n D iα , D jβ o = n D i ˙ α , D j ˙ β o = 0 . (A.11)The covariant derivatives (A.10) satisfy the following (anti)commutation relations: n ∇ iα , ∇ jβ o = − (cid:16) ∇ iα ψ γa ∇ jβ ¯ ψ a ˙ γ + ∇ iα ¯ ψ a ˙ γ ∇ jβ ψ γa (cid:17) (cid:0) σ A (cid:1) γ ˙ γ ∇ A , (cid:2) ∇ A , ∇ iα (cid:3) = − (cid:16) ∇ A ψ γa ∇ iα ¯ ψ a ˙ γ + ∇ A ¯ ψ a ˙ γ ∇ iα ψ γa (cid:17) (cid:0) σ B (cid:1) γ ˙ γ ∇ B , (cid:8) ∇ iα , ∇ ˙ αj (cid:9) = − δ ij (cid:0) σ A (cid:1) α ˙ α ∇ A − (cid:16) ∇ iα ψ γa ∇ j ˙ α ¯ ψ a ˙ γ + ∇ iα ¯ ψ a ˙ γ ∇ j ˙ α ψ γa (cid:17) (cid:0) σ A (cid:1) γ ˙ γ ∇ A . (A.12) References [1] A. Achucarro, J. M. Evans, P. K. Townsend and D. L. Wiltshire,
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