Notophs of N= 8 Supergravity

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1 Introduction 2

2 N =8 supergravity superspace 5

2.1 Geometry of N = 8 superspace and Cartan forms of E7(+7) 5 2.2 N =8,D = 4 superspace constraints and their consequences 6

2.3 E7(+7)/SU(8) Cartan forms in N = 8 supergravity superspace 7

3 1-form gauge potentials in N =8 supergravity superspace 7

4 2-form gauge potentials in N =8 supergravity superspace 8 4.1 Strategy 9 4.2 Constraints and generalized Bianchi identities for 3-form field strengths 9 4.3 Superfield duality equations 11 4.4 Identities for identities and the proof of the consistency of the constraints 11 4.5 Scalar (super)field equation of motion and duality equation 13

5 Conclusion and outlook 13

A 4D Weyl spinors and sigma matrices 15

B More on diﬀerential forms in curved N =8,D =4 superspace 15

1 Introduction

The action of the maximal N = 8,D = 4 supergravity was obtained in [1] by dimensional reduction of the D = 11 supergravity [2] followed by dualization of 7 antisym- I metric tensor gauge potentials Bµν originating in the 11-dimensional 3-form, called “notophs” in [3],1 to scalars. Then the complete set of 70(=28+35+7) scalars of the N = 8,D = 4 supergravity multiplet was found to parametrize the coset space

E7(+7)/SU(8) [1]. The natural question is whether this duality can be performed in an opposite direction, introducing a dual “notoph” for each scalar of the theory. In this paper we

1“Notoph” is “photon” read from the right to the left. Other, more popular names are Kalb- Ramond ﬁeld [4], 2-form, and even “B-ﬁeld” [5].

–2– study this problem in the N =8,D = 4 superspace formulation of supergravity. To be more precise, we search for a “duality symmetric” formulation of the theory, containing both the scalar fields and the notophs rather than trying to replace everywhere the former by the latter (which is not possible beyond the linear approximation in fields). The motivation for such a study is two-fold. On one hand, we hope that our results will contribute to a deeper understanding of the U-duality group of the N =8,D =4 supergravity, the exceptional Lie group E7(+7). The interest in this symmetry has remained high during the almost 36 years passed since its discovery in [1], and, recently, a relation with the exceptional convergence properties of its loop amplitudes has been proposed (see Refs. [6] and references therein.) On the other hand, the knowledge on existence of (p + 1)-form gauge potentials in a supergravity superspace might indicate the existence of supersymmetric extended ob- jects, p-branes, coupled electrically to these potentials. In this sense our results imply the possible existence of a family of supersymmetric strings in an N =8,D = 4 supergravity superspace2. The search for possible worldvolume actions of such hypothetical superstrings is one of the natural applications of our results. A first result showed by our study is that, to be consistent, one has to introduce G G/H H a 2-form potential for each of the generators of G = E7(+7) group, B2 = (B2 , B2 ), and not just for the generators of the coset G/H. This result can be generalized to other theories with scalars parametrizing a symmetric space [8]. An early example of how the dualization of scalars requires the introduction of a (d − 2)-form potential for each generator of the isometry group, even though their numbers do not match, is the dualization of the dilaton and RR 0-form of N = 2B,D = 10 supergravity in [9] (see also [10]): the two real scalars parametrize an SL(2, R)/SO(2) coset space and they are dualized into a triplet of 8-forms transforming in the adjoint. The existence of this triplet of 8-forms is required by the symmetry algebra E11 [11] and has clear implications in the classification of the possible 7-branes of the theory [9, 12–14]. In the context of the embedding tensor formalism for 4-dimensional gauged supergravities [15– 18] (bosonic, spacetime) 2-form potentials in the adjoint representation of the duality group have to be introduced for different technical reasons, unrelated to the dualization of scalar fields, and for the specific case of N = 8,D = 4 supergravity this was done some years ago in Refs. [18, 19]. The general duality rule between scalars and (d − 2)- forms was established in Refs. [16, 20, 21] using the embedding tensor formalism, but the results remain valid in the ungauged limit. The study of supersymmetrization of these and other higher-rank gauge potentials

2The BPS branes of the maximal supergravity theories were studied originally in Refs. [7], but their worldvolume actions are, in general, unknown.

–3– has received much less attention3 and in this paper we are going to start filling this gap for the case of the notophs of N = 8,D = 4 supergravity using the superspace formalism. The knowledge of the gauge and supersymmetry transformations of these fields is a key ingredient in the construction of κ-symmetric worldvolume actions for possible associated supersymmetric string (p-brane) models. In superspace formalism the problem of duality symmetric formulation, including the scalars and 2-form potentials dual to them on the mass shell, can be posed as G G searching for a set of constraints for 3-forms H3 = dB2 +... which are generalized field G strengths of the corresponding 2-form potentials B2 defined on superspace. Below we E G 7(+7) present such superspace constraints for the E7(+7) algebra valued H3 = H3 on the curved N = 8 superspace of maximal D = 4 supergravity, study their self-consistency G conditions: the generalized Bianchi identities (gBIs) dH3 = .... The explicit form of these gBIs are part of the definition of the tensor hierarchy of the Cremmer–Julia (CJ) N = 8 supergravity.4 They reflect the group theoretical structure associated to the E7(+7) symmetry of N = 8 supergravity in the dual language. We are going to recover this piece of the tensor hierarchy starting from the natural G candidate for superspace constraints for H3 and requiring that the algebraic part of the suitable gBIs, concentrated in their lower dimensional components, should be satisfied identically when the candidate constraints are taken into account. At this stage we find, G in particular, that the standard Bianchi identities dH3 = 0, if imposed, would lead to inconsistency and also that one cannot formulate a consistent set of constraints for the 3- G/H forms corresponding to the coset generators, H3 , without introducing simultaneously H the 3-forms H3 corresponding to the generators of the stability subgroup H = SU(8) of the coset. In this sense one of the message of this paper is that the superspace approach can be used in search for a consistent tensorial hierarchies of supergravity (as well as also of the theories invariant under rigid supersymmetry). After this is done, we further study their higher dimensional components and show G/H that the duality relations between the field strengths of the notophs, Hµνρ , and of G/H the scalar fields of N = 8 supergravity (generalized Cartan forms Pµ ) are the consequences of our superspace constraints. The field strengths of the stability subgroup H generators, Hµνρ, are found to be dual to fermionic bilinear; this reflects the auxiliary

3Some partial results on the supersymmetrization of the 2-forms dual to scalars in 4-dimensional N = 2, 1 theories can be found in [22, 23]. Supersymmetry has, nevertheless, been one of the main tools to find higher-rank potentials that can be added to the 10-dimensional maximal supergravities [11, 13, 24], in particular for (d − 1)- and d-form potentials. 4The tensor hierarchy arises naturally in the democratic gauging of theories using the embedding- tensor formalism [15–18], but the fields still make sense when the embedding tensor and any other deformation parameters are switched off, in the ungauged, undeformed theory.

–4– H character of the corresponding notophs Bµν .

2 N =8 supergravity superspace

2.1 Geometry of N =8 superspace and Cartan forms of E7(+7) Let us denote the bosonic and fermionic supervielbein forms of N = 8,D = 4 superspace Σ(4|32) by

A a α a α ¯αi˙ M a E ≡ (E , E )=(E , Ei , E )= dZ EM (Z) . (2.1)

Here ZM =(xµ, θαˇ) are local bosonic and fermionic coordinates of Σ(4|32), a =0, 1, 2, 3 is Lorentz group vector index, α =1, 2 andα ˙ =1, 2 are Weyl spinor indices of different chirality (see Appendix A), i =1,..., 8 is the index of the fundamental representation of the SU(8) R-symmetry group, and α is the 32-valued cumulative index of SL(2, C)⊗ SU(8). In the case of world indices, only the counterpart of this cumulative index seems to make sense (until the Wess–Zumino gauge is fixed); it is carried by the fermionic (Grassmann-odd) coordinate θαˇ. Finally, µ =0, 1, 2, 3 is the world vector index carried by bosonic (Grassmann-even) coordinate xµ. The curved superspace of N = 8,D = 4 supergravity is endowed with a spin ab ba M ab connection ω = −ω = dZ ωM (Z) and with the composite connection of the SU(8) j i ∗ M j i R-symmetry group, Ωi = −(Ωj ) = dZ ΩMi (Z), Ωi = 0; these are used to define the SL(2, C) ⊗ SU(8) covariant derivative D. The exterior covariant derivatives of the supervielbein forms are called bosonic and fermionic torsion 2-forms,

a a a b a 1 B C a T := DE = dE − E ∧ wb = 2 E ∧ E TCB , (2.2) α α α β α j α 1 B C α Ti := DEi = dEi − Ei ∧ wβ − Ωi ∧ Ej = 2 E ∧ E TCBi , (2.3) αi˙ ¯αi˙ ¯αi˙ ¯βi˙ α˙ ¯αj˙ i 1 B C αi˙ T := DE = dE − E ∧ wβ˙ − E ∧ Ωj = 2 E ∧ E TCB . (2.4)

Here ∧ denotes the exterior product of diﬀerential forms with the basic properties

Ea ∧ Eb = −Eb ∧ Ea , Eα ∧ Eβ = Eβ ∧ Eα , Ea ∧ Eα = −Eα ∧ Ea , and d is exterior derivative which acts from the right (see Appendix B). By construction, the torsion 2-forms obey the Bianchi identities

a a b a I3 := DT + E ∧ Rb =0 , (2.5) α α β α j α I3i := DTi + Ei ∧ Rβ − Ri ∧ Ej =0 , (2.6) αi˙ αi˙ ¯βi˙ α˙ ¯αj˙ i I3 := DT + E ∧ Rβ˙ + E ∧ Rj =0 , (2.7)

–5– β 1 ab β β˙ ∗ which involve the curvature of the spin connection (ωα = 4 ω σab α =(ωα˙ ) ),

ab ab ba 1 C D ab R = (dω − ω ∧ ω) = −R = 2 E ∧ E RDC , (2.8) β 1 ab β β 1 B A β Rα = 4 R σab α =(dω − ω ∧ ω)α = 2 E ∧ E RABα , (2.9) β˙ β ∗ 1 ab β˙ β˙ 1 B A β˙ Rα˙ = (Rα ) = − 4 R σ˜ab α˙ =(dω − ω ∧ ω)α˙ = 2 E ∧ E RABα˙ , (2.10)

j j k j and also the curvature of the induced SU(8) connection, Ri := dΩi − Ωi ∧ Ωk . The j compositeness of Ωi is reﬂected by the fact that its curvature is expressed as [25]

j i ∗ 1 P P¯jklp Ri = −(Rj ) = 3 iklp ∧ , (2.11)

ijkl where Pijkl is the covariant Cartan form of the E7(+7)/SU(8) coset and P¯ is its complex conjugate, which is also its SU(8) dual up to an arbitrary constant phase β:

P¯ijkl P ∗ 1 −iβ ijklpqrsP =( ijkl) = 4! e ε pqrs . (2.12)

The Cartan forms are covariantly closed,

p ijkl DPijkl := dPijkl − 4Ω[i| ∧ Pp|jkl] =0 ,DP¯ =0 . (2.13)

Some further properties obeyed by these forms can be found in Appendix B.

Eqs. (2.13), and(2.11) with Pijkl obeying (2.12) are structure equations of the E7(+7) Lie group. These can be solved providing the expressions for the covariant Cartan forms j Pijkl and SU(8) connection Ωi in terms of scalar superﬁelds of the N = 8 supergravity, the explicit form of which is not needed for our discussion below.

2.2 N =8,D =4 superspace constraints and their consequences The constraints of N =8,D = 4 supergravity [25, 26] can be collected in the following expressions for the bosonic and fermionic torsion 2-forms

a α ¯βi˙ a T = −iEi ∧ E σαβ˙ , (2.14) α 1 ¯βj˙ ¯γk˙ α c β j α c ¯βj˙ α 1 c b α Ti = 2 E ∧ E ǫβ˙γ˙ χijk + E ∧ Ej Tβ c i + E ∧ E Tβj˙ c i + 2 E ∧ E Tbc i , (2.15) αi˙ 1 β γ αijk˙ c β j αi˙ c ¯βj˙ αi˙ 1 c b αi˙ T = − 2 Ej ∧ Ekǫβγ χ¯ + E ∧ Ej Tβ c + E ∧ E Tβj˙ c + 2 E ∧ E Tbc . (2.16)

α αijk˙ ∗ Here χijk =(¯χ ) is the main fermionic superﬁeld of N =8,D = 4 supergravity and the dimension 1 fermionic torsion components have the expressions

T j α = 1 χ (¯χjklσ˜ )α , T αi˙ = 1 χ¯ikl(˜σ χ )α˙ , β b i 4 iklβ b βj˙ b 4 β˙ b jkl ˙ (2.17) α i αβ i αα˙ ¯ j αi˙ i α˙ β ij i αα˙ ij Tβj˙ b i = − 2 σbββ˙ Mij − 2 σ˜b Nα˙ βij˙ , Tβ b = − 2 σbββ˙ M − 2 σ˜b Nαβ ,

–6– in terms of the fermionic bilinears5

−iβ iβ ′ ij e ij[3][3′] e [3] [3 ] N = ε χ χ ′ , N¯ ˙ = − ε ′ χ¯ χ¯ , (2.18) αβ 6 · 4! α[3] β[3 ] α˙ βij 6 · 4! ij[3][3 ] α˙ β˙ and the bosonic superﬁelds M = M = (M¯ ij )∗. These appear as irreducible ijαβ [ij](αβ) α˙ β˙ parts of the fermionic covariant derivatives of the main fermionic superﬁeld,

Di χ = −3δi M , D¯ χ¯ jkl = −3δ[jM¯ kl] , (2.19) (α β)jkl [j kl]αβ i(˙α β˙) i α˙ β˙ The other irreducible components of these covariant derivatives of the main superﬁeld are expressed through their bilinears:

eiβ e−iβ Dαiχ = − ε χ¯i[2]χ¯α˙ [3] , D¯ α˙ χ¯jkl = − εjkl[2][3]χα χ , (2.20) αjkl 12 jkl[2][3] α˙ i α˙ 12 i[2] α[3]

2.3 E7(+7)/SU(8) Cartan forms in N =8 supergravity superspace The covariantly constant Cartan 1-forms obey the constraints

eiβ P = 2Eαχ − 2 E¯αp˙ ε χ¯[3] + EaP , (2.21) ijkl [i jkl]α 4! ijklp[3] α˙ a ijkl e−iβ P¯ijkl = 2 Eαεijklp[3]χ − 2E¯α˙ [iχ¯jkl] + EaP¯ijkl . (2.22) 4! p α[3] α˙ a These coincide with those in Refs. [25, 26] up to the constant phase parameter β. With the constraints (2.21), (2.22) the “Bianchi identities” (2.13) imply

¯ a P i jkl a P¯ijkl Dαi˙ χαjkl =2iσαα˙ aijkl ,Dαχ¯α˙ = −2iσαα˙ a , (2.23)

αi The results of Eq. (2.13) are also of help to ﬁnd the expression Eq. (2.20) for D χαjkl, P P¯ijkl and the duality relation between the vector aijkl and its conjugate a eiβ P = ε P¯pqrs . (2.24) aijkl 4! ijklpqrs a Just after this stage the superspace 1-forms in Eq. (2.21) and (2.22) become related by Eq. (2.12).

3 1-form gauge potentials in N =8 supergravity superspace

Although the supervielbein forms restricted by the torsion constraints already con- tain all the fields of supergravity multiplets, including the vector fields and their field

′ 5 [3] [3 ] klm npq ε ′ χ¯ χ¯ ≡ ε χ¯ χ¯ ij[3][3 ] α˙ β˙ ijklmnpq α˙ β˙

–7– strength, it is possible and also convenient to introduce the corresponding 1-form gauge potentials in superspace. As it was found already in Ref. [25], to preserve manifest SU(8) R-symmetry, one should introduce the super-1-forms corresponding to both the “electric” gauge ﬁelds of the supergravity multiplet and to their magnetic duals, packed M in the complex 1-form Aij = A[ij] = dZ AM ij(Z) in the 28 representation of SU(8), ¯ij ¯[ij] M ¯ij ∗ and its complex conjugate A = A = dZ AM (Z)=(Aij) in its 28 representation. Their 2-form ﬁeld strengths, which obey the generalized Bianchi identities (gBIs)

kl ij ijkl DFij = Pijkl ∧ F¯ ,DF¯ = P¯ ∧ Fkl , (3.1) are restricted by the constraints

α β 1 a ¯γk˙ γ 1 c b Fij = −iEi ∧ Ej ǫαβ − 2 E ∧ E σaγγ˙ χijk + 2 E ∧ E Fbc ij , (3.2) ¯ij αi˙ βj˙ 1 a γ γijk˙ 1 c b ¯ij F = −iE ∧ E ǫα˙ β˙ + 2 E ∧ Ek σaγγ˙ χ¯ + 2 E ∧ E Fbc , (3.3)

The antisymmetric tensor superﬁeld can be decomposed in the two irreducible parts6

a b σαα˙ σββ˙ Fab ij =2ǫαβFα˙ β˙ ij − 2ǫα˙ β˙ Fαβ ij . (3.4)

The Bianchi identities, including (3.1), imply, in particular,

iβ ′ i i e [3] [3 ] F = M , F ˙ = N¯ ˙ = −i ε ′ χ¯ χ¯ . (3.5) αβ ij 2 αβ ij α˙ β ij 2 α˙ β ij 12 · 4! ij[3][3 ] α˙ β˙

4 2-form gauge potentials in N =8 supergravity superspace

Σ˜ Now we are ready to turn to the main subject of this paper: 2-form gauge potentials B2 (“notophs”) in the complete supersymmetric description of N =8,D = 4 supergravity. As we have discussed in the introduction, although the appearance of seven 2-form potentials after dimensional reduction from D = 11 down to D = 4 is manifest and was noticed already in [1], these were immediately dualized to scalars. Only then the global E7(+7) duality becomes manifest. The inverse transformations relating all the scalars of N = 8 supergravity, parametrizing E7(+7)/SU(8), to 2-form potential have not been studied, at least in a complete form and especially in superspace, and this is our goal here. As we are going to see, in addition to the 2-forms associated to the coset G/H generators, B2 , which were expected as dual to the physical scalars parametrizing G/H = E7(+7)/SU(8) (basically because there are 70 of them), it is necessary to in- su H troduce (8) valued 2-form B2 . These are auxiliary and do not correspond to any 6Notice that F =+ 1 F σ˜ab = −(F¯ ij )∗. α˙ β˙ ij 4 ab ij α˙ β˙ αβ

–8– dynamical degrees of freedom of N =8,D = 4 supergravity. The general situation will be discussed in the companion paper [8]. Here we adopt a more technical superspace- based approach to establishing the content and the structure of the tensorial hierarchy of N =8,D = 4 supergravity.

4.1 Strategy Our strategy to search for higher form potentials in maximal supergravity is essen- tially superspace based: we begin by searching for an ansatz for possible superspace Σ˜ Σ˜ constraints for 3-form field strengths H3 = dB2 + ... suggested by the indices carried by the potentials. Checking their consistency, we can find whether more forms have to be introduced and what kind of ”free differential algebra” (FDA) they have to gen- A erate. This is described by a set of generalized Bianchi identities (gBIs) DF4 = ... The further study of the gBIs (FDA relations) for the constrained field strength should result (provided the constraints are consistent and the potentials are dynamical fields) in equations of motion which, in the case of the 2-form potentials, should have the form of duality of their field strength to the covariant derivatives of the scalar fields. Since, in our case, these are (the bosonic leading components of) the covariant E7(7)/SU(8) ′ ′ ′ ′ 1 ′ ′ ′ ′ ¯i j k l Cartan forms, i.e. the complex self-dual 1-forms Pijkl = 4! εijkli j k l P in the 70 of −iβ/2P 1 iβ/2P¯pqrs SU(8) (e aijkl = 4! εijklpqrse a in terms of bosonic component of superforms), the “physical” 2-form potentials are expected to be B2 ijkl and its complex conjugate ¯ijkl and dual B2 . However, as discussed in the introduction, experience suggests that, when the scalars parametrize a coset space G/H, it is not sufficient to introduce only the dual (D − 2)-form potentials with indices of the generators of the coset: the (D − 2)-forms associated to the generators of the subgroup H must be included as well (see [8] for a general discussion). In our case, these correspond to the hermitian traceless matrix of j j ∗ j j 2-forms B2i =(B2i ) with the generalized field strength H3 i = dB2 i + ....

4.2 Constraints and generalized Bianchi identities for 3-form ﬁeld strengths The natural candidate for the superspace constraints are

iβ e ′ ˙ ′ ′ ′ α (2) β ′ ′ ′ ′ ¯αi˙ (2) β j k l H3 ijkl = E[i ∧ σ α χjkl]β − εijkli j k l E ∧ σ˜ α˙ χ¯β˙ + 4! (4.1) 1 c b a + 3! E ∧ E ∧ E Habc ijkl ,

(2) β 1 b a β (2) β˙ ∗ where σ α = 2 E ∧ E σabα = −(˜σ α˙ ) , and

j a α αj˙ i j a α αk˙ 1 c b a j H3 i = iE ∧ Ei ∧ E σaαα˙ − 8 δi E ∧ Ek ∧ E σaαα˙ + 3! E ∧ E ∧ E Habc i . (4.2)

–9– Clearly, the leading term in the expression for H3 ijkl should be dB2 ijkl. But the question to be answered is whether other terms are also present, and the answer is affirmative. Indeed, if we assume H3 ijkl = dB2 ijkl (or, keeping the SU(8) invariance, H3 ijkl = DB2 ijkl), the generalized field strength should obey the simplest Bianchi p identities dH3 ijkl = 0 (or DH3 ijkl =4R[i| ∧ H3 |jkl]p), and the constraints (4.1) are not consistent if consistency is expressed by such a simple Bianchi identity. Similarly one can check that no consistent FDA can be formulated without intro- j ducing also the su(8) valued field strength H3i . It might also look tempting to omit i the tracelessness condition H3 i = 0 and thus to consider the u(8) rather than su(8) valued 3-form field strength obeying simpler constraints given by (4.2) without the second term in the r.h.s. However, as we have checked, this is also inconsistent with the superspace constraints of N = 8 supergravity. Thus the structure of the tensor hierarchy of N = 8 supergravity is quite rigid. To make a long story short, we have found that the constraints (4.1) and (4.2) are consistent with the FDA relations (generalized Bianchi identities) j j ¯jk 1 j ¯kl 1 P¯jkpq 1 ¯ jkpq P I4 i := DH3 i +2Fik∧F − 4 δi Fkl∧F + 3 H3ikpq∧ + 3 H3 ∧ ikpq =0 , (4.3) and iβ j′ 3e i′j′ k′l′ I := DH − 4H ∧ P ′ − 3F ∧ F + ε ′ ′ ′ ′ F¯ ∧ F¯ =0 . (4.4) 4 ijkl 3 ijkl 3[i jkl]j [ij kl] 4! ijkli j k l Let us stress that

−iβ ′ ′ ′ ′ ˜ [i P¯jkl]p e ijkli j k l ˜ ′ p P ′ ′ ′ 1. As long as H3 p ∧ = − 4! ε H3 i ∧ j k l p, the identity (4.4) and the ijkl ∗ complex conjugate identity for H¯3 =(H3 ijkl) are consistent with the duality relation (cf. with (2.12); notice the sign)

−iβ ijkl e ijkli′j′k′l′ H¯ = − ε H ′ ′ ′ ′ , (4.5) 3 4! 3 i j k l 2. When this property is taken into account, the traces of last two terms in the r.h.s. of (4.3) cancel one another.

3. The terms quadratic in 2-form ﬁeld strengths are those that occur in the E7(+7) Noether-Gaillard-Zumino current [35]. This current, whose components are all

conserved, even for the E7(+7) transformations which are not symmetries of the action, may play an important role in the UV ﬁniteness of the theory [6]. To check the consistency of our ansatz for the generalized Bianchi identities (gBIs) G G (4.3) and (4.4) one has to study the ”identities for identities” I5 = DI4 = 0:

j j I5 i := DI4 i =0 , I5ijkl := DI4ijkl =0 , (4.6)

– 10 – taking into account the Ricci identities. In application to our 3-form the latter read

j p j p j p DDH3 i = Ri ∧ H3 p − H3 i ∧ Rp , DDH3ijkl =4R[i| ∧ H3p|jkl] , (4.7) and can be further speciﬁed substituting the explicit expression (2.11) for the curvature of induced SU(8) connection. In such a way, after some algebra, one can prove that the proposed gBIs (4.3) and (4.4) are consistent provided the following identity holds

[3]q p[3] p[3] P[3][i| ∧ P¯ ∧ H3 |jkl]q − Pp[ijk| ∧ P¯ ∧ H3 |l][3] − Pp[ijk ∧ Pl][3] ∧ H¯3 =0 . (4.8)

This equation is proven in Appendix B using only the complex self-duality and anti- self-duality of Pijkl and H3 ijkl, respectively.

4.3 Superfield duality equations Substituting Eqs. (4.2) and (4.1) and using the superspace supergravity constraints, we have checked that the dim 2 and 5/2 components of the gBIs (4.4) and (4.3) are b a α β satisfied. As far as dim 3 components are concerned, the ∝ E ∧E ∧Ep ∧Eq component of Eq. (4.4) is satisfied identically (due to the basic constraints and properties of main b a α ¯βq˙ superfields, like (2.19) with (3.5)), while its ∝ E ∧ E ∧ Ep ∧ E component shows Pd that Habc ijkl is dual to the generalized Cartan form ijkl,

i Pd Habc ijkl = 2 ǫabcd ijkl . (4.9)

b a α ¯βq˙ j The ∝ E ∧ E ∧ Ep ∧ E component of (4.4) shows that Habci is dual to a bilinear of fermionic superﬁelds,

H j ∝ ǫ χ σdχ¯j[2] − 1 δjχ σdχ¯[3] . (4.10) abci abcd i[2] 8 i [3] This reﬂects the auxiliary character of the su(8) “(pseudo-)notophs”.

4.4 Identities for identities and the proof of the consistency of the constraints Instead of studying the higher-dimensional components of the gBIs, we simplify our study by proving that they are dependent and cannot produce independent consequences; this implies that our constraints are consistent and all the dynamical equations are contained as higher components in the superﬁeld duality equations (4.9) and (4.10). G j G To this end we solve the identities for identities (4.6),0= I5 =(I5 ijkl,I5 i )= DI4 , G with respect to the (l.h.s. of the) gBIs, IABCD, in the same manner as we solve Bianchi

– 11 – identities for the torsion and curvature tensors (and also gBIs for the 3-forms above) expressing them in terms of the main superfields (see Ref. [27]). As we have already said, the lower dimensional, dim 2 and 5/, components of the 4-form gBIs are satisfied algebraically, without any involvement of superfields. Set- G ting these to zero, IαβγA = 0, we obtain a counterpart of the torsion constraints of supergravity. Substituting

G 1 b a α β G 1 c b a α G I4 = 4 E ∧ E ∧ E ∧ E Iβαab + 3! E ∧ E ∧ E ∧ E Iαabc (4.11) 1 d c b a G + 4! E ∧ E ∧ E ∧ E Iabcd , into Eq. (4.6) and using the torsion constraints of N =8,D = 4 supergravity, Eqs. (2.14), (2.15) and (2.16), we ﬁnd

G i b α αq˙ β γ p a G b a 0= I5 = − 2 E ∧ Ep ∧ E ∧ E ∧ E δq σαα˙ Iβ γ ab+ ∝ E ∧ E (4.12)

Thus, the lowest dimensional (dim 3) nontrivial components of the identities for identities imply the following algebraic equations for the l.h.s. of the dim 3 gBIs:

p a Gk k a Gp 0 = δq σαα˙ I β γl˙ ab + δq σβα˙ I α γl˙ ab + (αq ˙ 7→ γl˙ ) , (4.13)

p a Gk l k a Glp l a Gpk 0 = δq σαα˙ I β γ ab + δq σβα˙ I γ α ab + δq σγα˙ I α β ab , (4.14) plus the complex conjugate of Eq. (4.14). It is not difficult to find that the latter as well Gk l as Eq. (4.14) have only trivial solutions I β γ ab = 0. In contrast, the general solution Gi i a ˜G ˜G ˜G of Eq. (4.13) reads I ααjbc˙ = δj σαα˙ Iabc with an arbitrary antisymmetric Iabc = I[abc]. This implies that the only independent consequences for the superfields can be obtained Gj αα˙ from I ααj˙ [bcσã] = 0. This is exactly what we have observed in the explicit calculations of the dimension

3 Bianchi identities for H3 ijkl (see Sec. 4.3). Namely, we have found that

0=(I )p ≡ −iδp σc H − i ǫ Pd , (4.15) 4 ijkl ααq˙ ab q αα˙ abc ijkl 2 abcd ijkl which implies the superﬁeld duality equation (4.9). The above general statement allows one to escape the exhausting algebraic calculations necessary to check explicitly the cancellation of diﬀerent terms in the equation G I αβab = 0. Furthermore, the higher-dimensional components of identities for identities Eq. (4.12) G G show the dependence of higher-dimensional Bianchi identities Iαabc = 0 and Iabcd = 0. This implies that their results can be obtained by applying covariant derivatives to the

– 12 – results of the dimension 3 gBIs, this is to say to the superfield duality equations (4.9) and (4.10), with the use of the superspace constraints for torsion, Cartan forms and 2-form field strength of the 1-form gauge fields and of their consequences. The latter include the equations of motion of N =8,D = 4 CJ supergravity.

4.5 Scalar (super)ﬁeld equation of motion and duality equation To illustrate this statement let us consider the dimension 4 Bianchi identity corresponding to E7(+7)/SU(8) generators:

p 0= Iijkl abcd = 4D[aHbcd] ijkl + 16H[abc|[i Pjkl]p |d] − 18F[ij|[abFcd]|kl]

iβ ′ ′ 3e i j ′k′l′ α + ε ′ ′ ′ ′ F¯ F¯ +6T (σ χ ) 4 ijkli j k l [ab cd] [ab|[i| |cd] |jkl] α iβ e αi˙ ′ j′k′l′ − ε ′ ′ ′ ′ T (¯χ σ˜ ) . (4.16) 4 ijkli j k l [ab| |cd] α˙ Using (4.9) we can equivalently write this as

iβ ′ ′ ′ ′ a 4i abcd p 3i ab cd ie abcd i j k l D P = − ε H P − ε F F + ε ′ ′ ′ ′ ε F¯ F¯ aijkl 3 abc[i jkl]p d 2 abcd [ij kl] 16 ijkli j k l ab cd iβ α ab e αi˙ ′ j′k′l′ ab +T (σ χ ) + ε ′ ′ ′ ′ T (¯χ σ˜ ) . (4.17) ab[i jkl] α 4! ijkli j k l ab α˙ After using Eq. (4.10), this expression acquires the usual form of the scalar (super)ﬁeld equation of N =8,D = 4 supergravity,

iβ ′ ′ ′ ′ a 3i ab cd ie abcd i j ′k l D P = − ε F F + ε ′ ′ ′ ′ ε F¯ F¯ + ..., (4.18) aijkl 2 abcd [ij kl] 16 ijkli j k l ab cd where the dots stand for the terms bilinear in fermions. To reflect the dependence of the higher-dimensional Bianchi identities proved in the previous Sec. 4.4, the above line should be read in the opposite direction: the results of the dimension 4 Bianchi identity Eq. (4.16) can be obtained by taking the bosonic covariant derivative of the duality equation (4.9) and using the scalar (super)field equation (as obtained from the torsion constraints of [25, 26]) and Eq. (4.10). Thus, the results of Sec. 4.3 and the arguments of Sec. 4.4 allow us to conclude that our constraints for the 3-form field strength are consistent and describe a set of “notophs” dual to the scalar fields of N =8,D = 4 supergravity.

5 Conclusion and outlook

In this paper we have provided the complete supersymmetric description of the “notophs” (2-form gauge potentials) of the Cremmer-Julia N =8,D = 4 supergravity [1].

– 13 – More specifically, we have presented the set of superspace constraints for the 3-form field strengths of the 2-form gauge potentials defined on N = 8,D = 4 supergravity superspace [25] and we have shown that these are consistent and produce the duality relation between the field strengths of the “physical notophs” and the scalar fields of the N = 8,D = 4 CJ supergravity parametrizing the G/H = E(7(+7)/SU(8) coset. We have found that the consistency, expressed by the generalized Bianchi identities, requires to introduce also the auxiliary 2-form potentials corresponding to the generators of the stability subgroup H = SU(8) of the coset. In the companion paper [8] we will discuss the reasons for this in detail. Here we have adopted a purely superspace approach and arrived at this conclusion starting from the natural candidate for the superspace constraints and searching for their consistency. The generalized Bianchi identities for the 3-form field strengths of the notophs, which define the tensorial hierarchy (or free differential algebra) of the N = 8,D = 4 CJ supergravity, have been also obtained in this manner. The list of natural directions of development of our approach includes the stud- ies of the superfield description of the notophs of gauged N = 8,D = 4 supergravity [15, 28, 29] using the torsion constraints of [30] and of the supersymmetric aspects of the generalized “notophs” of the exceptional field theories [31–33] in N = 8,D = 4 superspace enlarged by 56 bosonic “central charge” coordinates (see [34]). Another obvious extension of this work is the search for worldvolume actions of possible super- string models carrying the “electric” charges with respect to the antisymmetric tensor gauge fields. Probably the correct posing of this problem may also require to work in the Howe-Linmdstöm enlarged N =8,D = 4 superspace.

Notice added

After this paper appeared on the net, the superspace description of higher form gauge ﬁelds in D-dimensional maximal and half maximal supergravities have been discussed in [36], where the cases of D=11 and D=10 are elaborated explicitly. For 3 ≤ D < 10 cases the representations carried by higher forms in maximal and half maximal superspaces and their generalized Bianchi identities have been tabulated in Appendix A of [36]. Higher forms in maximal and half-maximal D=3 dimensional superspaces were studied in [37].

Acknowledgments

This work has been supported in part by the Spanish MINECO grants partially ﬁ- nanced with FEDER funds: No FPA2012-35043-C02-01, the Centro de Excelencia

– 14 – Severo Ochoa Program grant SEV-2012-0249 and the Spanish Consolider-Ingenio 2010 program CPAN CSD2007-00042, as well as by the Basque Government research group grant ITT559-10 and the Basque Country University program UFI 11/55. We are thankful to the Theoretical Department of CERN for hospitality and support of our visits in 2013, when the present project has been initiated. T.O. wishes to thank M.M. Fern´andez for her permanent support.

A 4D Weyl spinors and sigma matrices

a aβα˙ We use the relativistic Pauli matrices σβα˙ = ǫβαǫα˙ β˙ σ˜ which obey a b ab i abcd a b ab i abcd σ σ˜ = η + 2 ǫ σcσ˜d , σ˜ σ = η − 2 ǫ σ˜cσd , (A.1) where ηab = diag(1, −1, −1, −1) is the Minkowski metric and ǫabcd = ǫ[abcd] is the 0123 antisymmetric tensor with ǫ =1= −ǫ0123. αβ βα The spinorial (SL(2, C)) indices are raised and lowered by ǫ = −ǫ = iτ2 = 0 1 βγ γ β α αβ ( −1 0 )= −ǫαβ obeying ǫαβǫ = δα: θα = ǫαβθ and θ = ǫ θβ. The antisymmetrized ab α [a b] 1 a b b a abα˙ [a b] products σ β = σ σ˜ := 2 (σ σ˜ − σ σ˜ ) andσ ˜ β˙ =σ ˜ σ are self-dual and anti- ab i abcd ab i abcd self-dual, σ = 2 ǫ σcd,σ ˜ = − 2 ǫ σ˜cd.

B More on diﬀerential forms in curved N = 8, D = 4 superspace

Exterior derivative The exterior derivative d acts on a q-form

1 Mq M1 1 Aq A1 Ωq = q! dZ ∧ ... ∧ dZ ΩM1...Mq (Z)= q! E ∧ ... ∧ E ΩA1...Aq (Z) as

1 Mq M1 dΩq = q! dZ ∧ ... ∧ dZ ∧ dΩM1...Mq (Z) (B.1) 1 Mq+1 M2 M1 = (q+1)! dZ ∧ ... ∧ dZ ∧ dZ (q + 1)∂[M1 ΩM2...Mq+1}(Z) .

In action on the product of diﬀerential forms, e.g. the q-form Ωq and the p-form Ωp, it obeys the Leibnitz rule p d(Ωq ∧ Ωp) = Ωq ∧ dΩp +(−) dΩq ∧ Ωp . (B.2) The mixed brackets [...} denote the graded antisymmetrization of the enclosed indices with the weight unity, so that (q + 1)!∂[M1 ΩM2...Mq+1}(Z) = ∂M1 ΩM2...Mq+1 (Z) − ε(M1)ε(M2) M (−) ∂M2 ΩM1M3...Mq+1 (Z)+ ..., where ε(M) := ε(Z ) is the Grassmann parity (fermionic number), ε(µ) := ε(xµ) = 0, ε(α)= ε(θα) = 1.

– 15 – On E7(+7) Cartan forms

Using the complex self-duality of Pijkl Eq. (2.12) and the antisymmetry of the exterior product of P one ﬁnds [4] P[4] ∧ P¯ =0 . (B.3)

kl[2] Then, using Eq. (2.12) and this last property Eq. (B.3) in Pij[2] ∧ P¯ one ﬁnds

P P¯klpq 2 [kP P¯l][3] ijpq ∧ = 3 δ[i j][3] ∧ . (B.4)

The Ricci identity

P p P 4 P P¯[3]p P DD ijkl = −4R[i ∧ jkl]p = − 3 [3][i ∧ ∧ jkl]p = 0 (B.5) is satisﬁed because, by virtue of Eq. (B.4), the r.h.s. is equivalent to

¯pqrs Ppq[ij ∧ P ∧ Pkl]rs , (B.6) which vanishes automatically on account of the antisymmetry of the wedge product and the symmetry under the interchange of pairs of the SU(8) indices. From Eq. (B.4) it follows that the ﬁrst term in Eq. (4.8) can be reexpressed as

′ P P¯[3]q 3 P P¯[2][2 ] ′ [3][i| ∧ ∧ H3 |jkl]q = − 2 [2][ij| ∧ ∧ H3 |kl][2 ] . (B.7)

Using again the complex self-duality of Pijkl and the complex anti-self-duality of H3 ijkl, the third term in Eq. (4.8) can be reexpressed as

′ P P ¯ p[3] 1 P P [4] 3 P P¯[2][2 ] ′ p[ijk ∧ l][3] ∧ H3 = − 8 ijkl ∧ [4] ∧ H3 − 4 [2][ij| ∧ ∧ H3 |kl][2 ] . (B.8)

The same properties and this last identity allow us to rewrite the second term in Eq. (4.8) as

′ P P¯p[3] 1 P P [4] 3 P P¯[2][2 ] ′ p[ijk| ∧ ∧ H3 |l][3] = 8 ijkl ∧ [4] ∧ H3 − 4 [2][ij| ∧ ∧ H3 |kl][2 ] . (B.9)

After rewriting the three terms of Eq. (4.8) using the above identities, we ﬁnd that Eq. (4.8) is identically satisﬁed.

On E7(+7) Cartan forms in N =8 supergravity superspace Eqs. (2.20) can be derived also from (2.13) with (2.21). To this end it is useful to notice the trivial identity αpq˙ [i jkl] 5 αp˙ [qi jkl] 3 αp˙ [ij kl]q χ¯ χ¯α˙ = 2 χ¯ χ¯α˙ − 2 χ¯ χ¯α˙ . (B.10)

– 16 – Its l.h.s. is antisymmetric, while the second term in its r.h.s is symmetric. Hence

αp˙ [ij kl]q 5 αp˙ [qi jkl] αq˙ [pi jkl] χ¯ χ¯α˙ = 6 (¯χ χ¯α˙ +χ ¯ χ¯α˙ ) , (B.11) and αpq˙ [i jkl] 5 αp˙ [qi jkl] αq˙ [pi jkl] χ¯ χ¯α˙ = 4 (¯χ χ¯α˙ − χ¯ χ¯α˙ ) . (B.12) As a consequence

ijkli′j′k′l′ α˙ [p jkl[2][3] α˙ ′ ′ ′ ′ ε χ¯pq[i χα˙ [j k l ] = −2δ[i ε χ¯q][2]χα˙ [3] . (B.13) Curvature 2 forms of N =8 superspace The study of the Bianchi identities results in the following expressions for the curvature of the spin connection (the “Riemann” curvature 2-form) (see [25], [26])

a ββ˙ b β β˙ β˙ β σαα˙ σ˜b Ra = 2δα Rα˙ +2δα˙ Rα , (B.14)

αβ 1 ab αβ 1 γ δ αβij α β ij 1 βi˙ γj˙ αβ R = R σ = E ∧ E ǫ N +2δ δ S + E¯ ∧ E¯ ǫ ˙M 4 ab 2 i j γδ (γ δ) 2 γ˙ δ ij

γ ¯γj˙ i αβ c β αβ 1 c b αβ +Ei ∧ E Rγ γj˙ + E ∧ E Rβ c + 2 E ∧ E Rbc , (B.15)

˙ ˙ Rα˙ β˙ = − 1 Rabσ˜α˙ β = − 1 Eγ ∧ Eδǫ M¯ α˙ β˙ ij − 1 E¯βi˙ ∧ E¯γj˙ ǫ N¯ γ˙ δ +2δ α˙ δ β˙ S¯ 4 ab 2 i j γδ 2 γδ ij (˙γ δ˙) ij

γ ¯γj˙ i α˙ β˙ c β α˙ β˙ 1 c b α˙ β˙ +Ei ∧ E Rγ γj˙ + E ∧ E Rβ c + 2 E ∧ E Rbc . (B.16) Eqs. (3.4) and (3.5) can be combined as

iβ ′ 1 αβ 1 α˙ β˙ i αβ ie [3] [3 ] F = σ F + σ˜ F ˙ = σ M + ε ′ χ¯ σ˜ χ¯ . (B.17) ab ij 2 ab αβ ij 2 ab α˙ β ij 4 ab αβ ij 12 · 4! ij[3][3 ] ab

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