aa r X i v : . [ h e p - t h ] J u l July 2020HU-EP-20/08
Toward Exotic 6D Supergravities
Yannick Bertrand a , Stefan Hohenegger a , Olaf Hohm b , Henning Samtleben ca Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3,IP2I Lyon, UMR 5822, F-69622, Villeurbanne, France [email protected], [email protected] b Institute for Physics, Humboldt University Berlin,Zum Großen Windkanal 6, D-12489 Berlin, Germany [email protected] c Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS,Laboratoire de Physique, F-69342 Lyon, France [email protected]
Abstract
We investigate exotic supergravity theories in 6D with maximal p , q and p , q supersymmetry, which were conjectured by C. Hull to exist and to describe strongcoupling limits of N “ ` p , q and p , q theory, respectively, includingthe (self-)duality relations. Evidence is presented for a master exceptional fieldtheory formulation with an extended section constraint that, depending on the so-lution, produces the p , q , p , q or the conventional p , q theory. We comment onthe possible construction of a fully non-linear master exceptional field theory. ontents N “ p , q Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 The N “ p , q Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 The N “ p , q Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 ` Split 7 N “ p , q Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 The N “ p , q Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 The N “ p , q Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 N “ p , q Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Action for the N “ p , q Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 N “ p , q model . . . . . . . . . . . . . 175.2 Beyond standard ExFT: embedding of the N “ p , q and p , q couplings . . . . 195.3 The spin-2 sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 A.1 Henneaux-Teitelboim Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . 24A.2 ExFT type Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
B 6D field equations from the new Lagrangians 26
B.1 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26B.2 Going back to the original equations . . . . . . . . . . . . . . . . . . . . . . . . . 261
Introduction
Among the surprising features of string/M-theory is the possible existence of exotic supercon-formal field and gravity theories in six dimensions, which display generalizations of electric-magnetic duality. Specifically, the supermultiplets of these theories are such that the cor-responding fields must be subject to (self-)duality constraints, some of which involve exoticYoung tableaux representations. In this paper our focus will be on a conjecture by Hull [1, 2]according to which there are strong coupling limits of N “ N “ p , q and N “ p , q supersymmetry,respectively. Such theories must be exotic or non-geometric since they feature mixed symmetrytensors of Young tableaux type and , respectively, instead of a conventional graviton,hence suggesting the need for a generalized notion of spacetime and diffeomorphism invariance.They are set to play a distinguished role among the maximally supersymmetric theories [3–5]A possible window into these somewhat mysterious structures is offered by a Kaluza-Kleinperspective from five dimensions. The supermultiplets of five-dimensional (5D) theories withmaximal supersymmetry (32 real supercharges) were classified by Strathdee [6] and furtherclarified by Hull in [7]. The 5D superalgebra reads t Q aα , Q bβ u “ Ω ab p Γ µ C q αβ P µ ` C αβ p Z ab ` Ω ab K q , (1.1)where α, β, . . . “ , . . . , a, b, . . . “ , . . . , p q R -symmetry indices. This superalgebra features 27 central charges Z ab , satisfying Z ab “ ´ Z ba ,Ω ab Z ab “
0, and a singlet central charge K . The BPS multiplets of this superalgebra describethe possible Kaluza-Klein towers that any six-dimensional (6D) theory with maximal supersym-metry displays when compactified on a circle. For the conventional maximal 6D supergravity,which features N “ p , q supersymmetry, the massive Kaluza-Klein states do not carry thesinglet central charge K . Instead, they carry a particular central charge Z ab transforming as asinglet under the six-dimensional R -symmetry group USp p qˆ USp p q . In contrast, the massivemultiplets of the exotic theories carry non-vanishing singlet charge K (together with nonvan-ishing Z ab , singlet under USp p q ˆ USp p q in the case of the N “ p , q multiplets) [7]. Thispoints to a unifying framework in the spirit of exceptional field theories [8–11] which we willelaborate on in this paper.Exceptional field theory (ExFT) provides in particular a formulation of 11-dimensional(11D) and type IIB supergravity in a form that is covariant under the global symmetry groupE p q of 5D maximal supergravity, thanks to extended coordinates in the representation ofthis group, which are added to the five coordinates of 5D supergravity. The resulting theory isthus based on a p ` q -dimensional spacetime split, in which the 27 coordinates are subjectto an E p q covariant ‘section constraint’ restricting them to a suitable physical subspace, fromwhich the complete (untruncated) 11D supergravity can be reconstructed, albeit in a Kaluza-Klein type formulation with a 5 ` p q exceptional field theory [10,11] by adding one more ‘exotic’coordinate to the 27, as suggested by the singlet central charge in (1.1). As one of the mostenticing outcomes of our investigation we find evidence for a master exceptional field theoryformulation in which the conventional N “ p , q theory as well as the N “ p , q and N “ p , q theory are obtained through different solutions of an extended section constraint ofthe form d MNK B N b B K ´ ?
10 ∆ MN pB N b B ‚ ` B ‚ b B N q “ , (1.2)where M, N “ , . . . , , are fundamental E p q indices, d MNK denotes the E p q invariant fullysymmetric tensor, and B ‚ is the derivative dual to the exotic coordinate. Moreover, ∆ MN denotes the (constant) background part of the generalized metric M MN encoding all scalarfields. The first term in equation (1.2) defines the section constraint of E p q exceptional fieldtheory, whose solutions restrict to the standard D “
11 and IIB sections. The second termencodes the extension of the constraint allowing for two more solutions corresponding to N “p , q and N “ p , q , respectively. More precisely, we recover the N “ p , q exotic theoryby dropping all dependence on the 27 standard coordinates, and keeping only the dependenceon the exotic coordinate. The N “ p , q model in turn is recovered by superposing thiscoordinate with the F p q singlet under 27 Ñ ` N “ p , q andthe N “ p , q model that are based on a 5+1 split of the six-dimensional space-time, sacrificingmanifest 6D Poincar´e invariance. In the spirit of ExFT, these are two-derivative actions whichupon dimensional reduction to five dimensions reduce to the same action of linearized maximal5D supergravity. All dual fields, in particular the entire dual graviton sector, only appear underderivative along the sixth dimension. The full field equations obtained by variation combine thesecond order Fierz-Pauli equations with first-order duality equations defining the dual gravitonsector. Actions for selfdual fields based on a 5+1 split of spacetime date back to [13] with thedescription of selfdual 6D tensor fields. More recently, actions for the N “ p , q and N “ p , q models have been constructed in [14–16], based on the prepotential formalism developed in [17]in the context of linearized gravity. Introduction of prepotentials for the gauge fields adaptedto their self-duality properties allows for the construction of an action of fourth order in spatialderivatives. Our construction is closer in spirit to the original construction of [13], albeit dualin a sense discussed in more detail in appendix A. It provides a novel mechanism for describingself-dual exotic tensor fields.The rest of this paper is organized as follows. In sec. 2 we review the bosonic sector of the N “ p , q , N “ p , q and N “ p , q theories at the level of the equations motion, whichare manifestly 6D Lorentz invariant. In order to find actions for these theories we abandonmanifest Lorentz invariance by performing a 5 ` Note added:
While finalizing the present paper the preprint [18] appeared, which also in-vestigates exotic theories in 6D.
We study six-dimensional field theories in Minkowski space with flat metric η ˆ µ ˆ ν “ diag t´ , , , , , u , ˆ µ, ˆ ν “ , . . . , . (2.1)The Poincar´e algebra in six dimensions admits chiral p N ` , N ´ q supersymmetric extensionswhere N ˘ count the cumber of right- and left-handed supercharges, respectively [6]. In particu-lar, maximal supersymmetry (32 real supercharges) allows for the three possibilities p N ` , N ´ q “p , q , p N ` , N ´ q “ p , q , and p N ` , N ´ q “ p , q . The corresponding lowest-dimensional mass-less supermultiplets have the field content [6] p , q : p ,
3; 1 , q ` p ,
2; 4 , q ` p ,
3; 5 , q ` p ,
1; 1 , q ` p ,
1; 5 , q` p ,
3; 4 , q ` p ,
2; 1 , q ` p ,
1; 4 , q ` p ,
2; 5 , q , (2.2a) p , q : p ,
2; 1 , q ` p ,
2; 14 , q ` p ,
1; 6 , q ` p ,
1; 14 , q` p ,
1; 1 , q ` p ,
2; 6 , q ` p ,
1; 14 , q ` p ,
2; 14 , q , (2.2b) p , q : p ,
1; 1 q ` p ,
1; 27 q ` p ,
1; 42 q ` p ,
1; 8 q ` p ,
1; 48 q , (2.2c)organized into representations of the little groupG “ SU p q ˆ SU p q ˆ USp p N ` q ˆ USp p N ´ q . (2.3)In this section, we briefly review the six-dimensional free theories associated to these multiplets. N “ p , q Model
Let us start from the N “ p , q multiplet corresponding to maximal supergravity in sixdimensions. Its bosonic field content comprises a metric, 25 scalar fields, 16 vectors, and5 two-forms. The full non-linear theory has been constructed in [19] with the scalar fieldsparametrizing an SO p , q{ p SO p q ˆ SO p qq coset space. For the purpose of this paper, we willonly consider the linearized (free) theory with no couplings among the different types of matter.The linearized spin-2 sector carries the symmetric Pauli-Fierz field h ˆ µ ˆ ν . With the linearizedRiemann tensor given by R ˆ µ ˆ ν, ˆ ρ ˆ σ “ ´B ˆ µ B r ˆ ρ h ˆ σ s ˆ ν ` B ˆ ν B r ˆ ρ h ˆ σ s ˆ µ , (2.4)linearization of the Einstein-Hilbert Lagrangian gives rise to the massless Fierz-Pauli Lagrangian L h “ ´ B ˆ µ h ˆ µ ˆ ν B ˆ ν h ˆ ρ ˆ ρ ` B ˆ µ h ˆ ρ ˆ σ B ˆ ρ h ˆ σ ˆ µ ´ B ˆ µ h ˆ ρ ˆ σ B ˆ µ h ˆ ρ ˆ σ ` B ˆ µ h ˆ ν ˆ ν B ˆ µ h ˆ ρ ˆ ρ “ ´
14 Ω ˆ µ ˆ ν ˆ ρ Ω ˆ µ ˆ ν ˆ ρ `
12 Ω ˆ µ ˆ ν ˆ ρ Ω ˆ ν ˆ ρ ˆ µ ` Ω ˆ µ Ω ˆ µ , (2.5)4ith Ω ˆ µ ˆ ν ˆ ρ ” B r ˆ µ h ˆ ν s ˆ ρ . The vector fields A ˆ µi couple with a standard Maxwell term L A “ ´ F ˆ µ ˆ νi F ˆ µ ˆ ν i , i “ , . . . , , (2.6)for F ˆ µ ˆ νi “ B r ˆ µ A ˆ ν s i , while scalar couplings take the form L φ “ ´ B ˆ µ φ α B ˆ µ φ α , α “ , . . . , . (2.7)The couplings (2.6) and (2.7) break the global SO p , q symmetry of the non-linear theory downto its compact part SO p q ˆ SO p q , as expected for the free theory. Finally, the two-forms B ˆ µ ˆ νp couple with a standard kinetic term L B “ ´ H ˆ µ ˆ ν ˆ ρq H ˆ µ ˆ ν ˆ ρ q , q “ , . . . , , (2.8)for H ˆ µ ˆ ν ˆ ρq “ B r ˆ µ B ˆ ν ˆ ρ s q . For the following it will be convenient to combine these fields togetherwith their magnetic duals into a set of 10 two-forms B ˆ µ ˆ νa , satisfying first order (anti-)selfdualityfield equations δ ab H ˆ µ ˆ ν ˆ ρb “ ε ˆ µ ˆ ν ˆ ρ ˆ σ ˆ κ ˆ λ η ab H ˆ σ ˆ κ ˆ λ b , a “ , . . . , , (2.9)with the SO p , q invariant constant tensor η ab . Equations (2.9) amount to a description ofthese degrees of freedom in terms of 5 selfdual and 5 anti-selfdual two forms. N “ p , q Model
Let us now turn to the free field equations associated with the N “ p , q multiplet (2.2b).This multiplet does not carry a standard graviton field, but an exotic three-index tensor fieldof mixed-symmetry type [20] : C ˆ µ ˆ ν, ˆ ρ “ ´ C ˆ ν ˆ µ, ˆ ρ , C r ˆ µ ˆ ν, ˆ ρ s “ . (2.10)Its field equation is given by a selfduality equation [1] S ˆ µ ˆ ν ˆ ρ, ˆ σ ˆ τ “ ε ˆ µ ˆ ν ˆ ρ ˆ η ˆ κ ˆ λ S ˆ η ˆ κ ˆ λ ˆ σ ˆ τ , (2.11)in terms of its second order curvature S ˆ µ ˆ ν ˆ ρ, ˆ σ ˆ τ “ B ˆ σ B r ˆ µ C ˆ ν ˆ ρ s , ˆ τ ´ B ˆ τ B r ˆ µ C ˆ ν ˆ ρ s , ˆ σ . (2.12)Counting reveals that the field equation (2.11) captures the 8 degrees of freedom as countedin the multiplet (2.2b). Moreover, curvature and field equation are invariant under the gaugesymmetries δC ˆ µ ˆ ν, ˆ ρ “ B r ˆ µ α ˆ ν s ˆ ρ ` B ˆ ρ β ˆ µ ˆ ν ´ B r ˆ ρ β ˆ µ ˆ ν s , (2.13)with parameters α ˆ µ ˆ ν “ α p ˆ µ ˆ ν q and β ˆ µ ˆ ν “ β r ˆ µ ˆ ν s . An action principle for the field equations(2.11) has been constructed in [16] based on the prepotential formalism introduced in [17] inthe context of linearized gravity. 5n addition to the exotic tensor field, the bosonic field content of the N “ p , q multiplet(2.2b) contains 14 vectors, 12 selfdual 2-forms and 28 scalar fields. The dynamics of vector andscalar fields can be captured by standard Lagrangians (2.6) and (2.7) (with different range ofinternal indices). The selfdual 2-forms B ˆ µ ˆ ν a obey a selfduality equation similar to (2.9) H ˆ µ ˆ ν ˆ ρa “ ε ˆ µ ˆ ν ˆ ρ ˆ σ ˆ κ ˆ λ H ˆ σ ˆ κ ˆ λ a , a “ , . . . , , (2.14)contrary to (2.9), no indefinite tensor η ab appears in this equation, all forms are selfdual. Asa consequence there is no standard action principle for these field equations, they can howeverbe derived from an action with non-manifest Lorentz invariance [13] or upon coupling to theauxiliary PST scalar [21]. The free N “ p , q theory is invariant under the R -symmetry group USp p q ˆ USp p q .The (yet elusive) interacting theory is conjectured to exhibit a global F p q symmetry with inparticular the 28 scalars parametrizing the coset space F p q { p USp p q ˆ USp p qq [1]. N “ p , q Model
The N “ p , q multiplet carries an exotic four-index tensor field with the symmetries of theRiemann tensor : T ˆ µ ˆ ν, ˆ ρ ˆ σ “ T ˆ ρ ˆ σ, ˆ µ ˆ ν “ ´ T ˆ ν ˆ µ, ˆ ρ ˆ σ , T r ˆ µ ˆ ν, ˆ ρ s ˆ σ “ . (2.15)Its field equation is given by a selfduality equation [1] G ˆ µ ˆ ν ˆ λ, ˆ ρ ˆ σ ˆ τ “ ε ˆ µ ˆ ν ˆ λ ˆ α ˆ β ˆ γ G ˆ α ˆ β ˆ γ ˆ ρ ˆ σ ˆ τ , (2.16)in terms of its second order curvature G ˆ µ ˆ ν ˆ λ, ˆ ρ ˆ σ ˆ τ “ B ˆ ρ B r ˆ µ T ˆ ν ˆ λ s , ˆ σ ˆ τ ` B ˆ σ B r ˆ µ T ˆ ν ˆ λ s , ˆ τ ˆ ρ ` B ˆ τ B r ˆ µ T ˆ ν ˆ λ s , ˆ ρ ˆ σ . (2.17)Counting confirms that this field equation describes the 5 degrees of freedom as counted inthe multiplet (2.2c). Moreover, curvature and field equation are invariant under the gaugesymmetries δT ˆ µ ˆ ν, ˆ ρ ˆ σ “ B r ˆ µ λ ˆ ν s , ˆ ρ ˆ σ ` B r ˆ ρ λ ˆ σ s , ˆ µ ˆ ν , (2.18)with the (2,1) gauge parameter λ ˆ µ, ˆ ρ ˆ σ “ λ ˆ µ, r ˆ ρ ˆ σ s , λ r ˆ µ, ˆ ρ ˆ σ s “ N “ p , q multiplet (2.2c) combines the exotic tensor field T ˆ µ ˆ ν, ˆ ρ ˆ σ with 42 scalars and27 selfdual 2-forms. Their dynamics is described by a free Lagrangian (2.7) and selfdualityequations (2.14), respectively.The free N “ p , q theory is invariant under the R -symmetry group USp p q . The (yetelusive) interacting theory is conjectured to exhibit a global E p q symmetry with in particularthe 42 scalars parametrizing the coset space E p q USp p q [1]. For uniformity, we use the same indices a, b, to label two-forms in all three models, despite the fact thatthe range of these indices differs among the different models according to the number of two-form fields. Thisshould not be a source of confusion. For more recent constructions, see also [22], [23]. ` Split
Upon dimensional reduction to D “ D “ N “ p , q and the N “ p , q model after dimensional reduction carry the D “ D “ D “ D “ ` ` x ˆ µ ( ÝÑ t x µ , y u , µ “ , . . . , , (3.1)by singling out one of the spatial coordinates. Of course, an analogous construction can beperformed with a split along the time-like coordinate which may be of interest for example ina Hamiltonian context. N “ p , q Model
With the coordinate split (3.1), we parametrize the graviton of the N “ p , q theory as h ˆ µ ˆ ν “ ˜ h µν ´ η µν φ A µ A µ φ ¸ , (3.2)which is the linearized form of the standard Kaluza-Klein reduction ansatz. Recall that allfields still depend on 6 coordinates. Working out the Lagrangian (2.5) in this parametrizationgives rise to its expression L h ÝÑ L “ ´ B µ h µν B ν h ρρ ` B µ h ρσ B ρ h σµ ´ B µ h νρ B µ h νρ ` B µ h ν ν B µ h ρρ ´ B y h µν B y h µν ` B y h µν B µ A ν ´ B y h σσ B ρ A ρ ` B y h σσ B y h ρρ ´ B y h σσ B y φ ´ F µν F µν ´ B µ φ B µ φ ` B y A µ B µ φ ` B y φ B y φ , (3.3)7p to total derivatives. As an illustration of the above discussion let us note the explicit formof the equations for the five-dimensional spin-2 field G µν “ ´ B y B y h µν ` η µν B y B y h ρρ ´ η µν B y B y φ , (3.4)in terms of the linearized Einstein tensor G µν “ ´B ρ D p µ h ν q ρ ` B ρ D ρ h µν ` B p µ D ν q h ρρ ` η µν B ρ D σ h ρσ ´ η µν B ρ D ρ h σσ , with covariant derivatives D µ h νρ ” B µ h νρ ´ B y A µ η νρ . (3.5)The form of (3.4) shows that upon dimensional reduction to D “ p q ExFT [10, 11] upon proper identification of the coordinate y among the 27 internal coordinateson which this ExFT is based.Let us also note, that the Lagrangian (3.3) can be put to the more compact form L “ ´
14 Ω µνρ Ω µνρ `
12 Ω µνρ Ω νρµ ` Ω µ Ω µ ´ pB µ φ ´ B y A µ qpB µ φ ´ B y A µ q´ F µν F µν ` B y φ B y φ ´ B y h σσ B y φ ` B y h σσ B y h ρρ ´ B y h µν B y h µν , (3.6)with the linearized (and covariantized) anholonomity objectsΩ µνρ ” B r µ h ν s ρ ´ B y A r µ η ν s ρ , Ω µ ” Ω µνν . (3.7)The remaining part of the six-dimensional degrees of freedom described by (3.6) are capturedby a (modified) five-dimensional Maxwell and Klein-Gordon equation for A µ and φ , respectively,obtained by varying (3.6). It is useful to note the symmetries of the Lagrangian (3.6) descendingfrom six-dimensional spin-2 gauge transformations δh µν “ B p µ ξ ν q ` η µν B y λ ,δA µ “ B µ λ ` B y ξ µ ,δφ “ B y λ , (3.8)upon decomposition of the six-dimensional gauge parameter as t ξ ˆ µ u “ t ξ µ , λ u .In a similar way, the six-dimensional Maxwell and Klein-Gordon Lagrangians (2.6) and (2.7)take the form L A “ ´ F µν i F µν i ´ ` B µ φ i ´ B y A µ i ˘ ` B µ φ i ´ B y A µi ˘ , L φ “ ´ B µ φ α B µ φ α ´ B y φ α B y φ α , (3.9)8espectively, after splitting t A ˆ µi u “ t A µi , φ i u , and with abelian F µν i “ B r µ A ν s i , giving riseto modified Maxwell and Klein-Gordon equations for their components. The rewriting of thetensor field sector is slightly less straightforward: rather than evaluating the Lagrangian (2.8),we choose to evaluate the first-order field equations (2.9) after splitting the 6D tensor fieldsinto t B ˆ µ ˆ νa u “ t B µν a , B µ a ” A µa u η ab H µνρb “ ε µνρκλ δ ab p F κλ b ` B y B κλ b q , (3.10)where we use conventions ε µνρκλ “ ε µνρκλ , and abelian field strengths H µνρa “ B r µ B νρ s a , and F µν a “ B r µ A ν s a , respectively. These equations can be integrated to a Lagrangian L “ ´ p F µν a ` B y B µνa q p F µν a ` B y B µν a q ´ ε µνρστ η ab B y B µν a H ρστ b . (3.11)Again, this Lagrangian can be deduced from the linearized version of exceptional field theory.We discuss this mechanism in more detail in appendix A.2. As we will see in the following, thisform of the Lagrangian allows for the most uniform treatment of the different six-dimensionalmodels. After dimensional reduction to D “ B µν a induces equations(3.10) (under B y derivative) as duality equations relating vector and tensor fields.In summary, the D “ N “ p , q , model can be equivalently reformulated in terms of aLagrangian given by the sum of (3.6), (3.9), and (3.11). Upon dimensional reduction to fivedimensions, i.e. setting B y Ñ
0, and rescaling of the scalar fields, this Lagrangian reduces to L D “ ´
14 ˚Ω µνρ ˚Ω µνρ `
12 ˚Ω µνρ ˚Ω νρµ ` ˚Ω µ ˚Ω µ ´ B µ φ A B µ φ A ´ F µν M F µν M ´ B y h µν B y h µν ,M “ , . . . , , A “ , . . . , , (3.12)with ˚Ω µνρ ” B r µ h ν s ρ , and where we have combined the various vector and scalar fields into jointobjects A µM ( , M “ , . . . , , φ A ( , A “ , . . . , . (3.13)The Lagrangian (3.12) is the free limit of D “ p q . N “ p , q Model
We now turn to the N “ p , q model. Its most characteristic element is the mixed-symmetrytensor field C ˆ µ ˆ ν, ˆ ρ whose field equation (2.11) cannot be derived from a standard action principle.We thus perform the Kaluza-Klein reorganization of the model on the level of the field equations.To this end, we again split coordinates as (3.1) and parametrize the mixed-symmetry tensor as t C ˆ µ ˆ ν, ˆ ρ u “ C µν,ρ ´ A r µ η ν s ρ ; C µ ,ν “ h µν ` B µν ; C µ , “ A µ ( , (3.14)with symmetric h µν “ h νµ , antisymmetric B µν “ ´ B νµ , and a (2,1) tensor C µν,ρ . Afterdimensional reduction to five dimensions, the fields h µν and A µ satisfy the linearized Einstein9nd Maxwell equations while the fields C µν,ρ and B µν describe their on-shell duals, togetheraccounting for the 8 degrees of freedom of the six-dimensional tensor field. Explicitly, in theparametrization (3.14), the six-dimensional selfduality equations (2.11) split into two equations B ρ ´ F µν ` ε µνλστ H λστ ¯ “ B y B r µ h ν s ρ ` ε µνκλτ B y B κ C λτ ρ ` B y B y C µν,ρ ´ B y B y A r µ η ν s ρ ´ ε ρµνστ B y F στ ` B y B r µ B ν s ρ ´ B y B ρ B µν , (3.15)and R µν,ρσ “ B ρ ´ H µνσ ´ ε µνσκλ F κλ ¯ ´ B σ ´ H µνρ ´ ε µνρκλ F κλ ¯ ` ε µνκλτ B κ B r ρ C λτ σ s ` B y B ρ C µν,σ ´ B y B σ C µν,ρ ´ B y B ρ A r µ η ν s σ ` B y B σ A r µ η ν s ρ , (3.16)with abelian field strengths F µν “ B r µ A ν s , H µνρ “ B r µ B νρ s , and the linearized Riemanntensor R µν,ρσ defined as in (2.4) however for the field h µν . Contraction of (3.16) gives rise toan equation G µν “ ´ B y B ρ C ρ p µ,ν q ´ B y B p µ C ν q ρρ ` η µν B y B ρ C ρσσ , (3.17)where G µν denotes the linearized Einstein tensor defined as in (3.5), however with covariantderivatives now given by D µ h νρ ” B µ h νρ ´ B y A µ η νρ , (3.18)i.e. with a different value of the coupling constant (which could be absorbed into rescaling thevector field). Equation (3.17) confirms that upon reduction to five dimensions ( B y Ñ h µν satisfies the linearized Einstein equations. As in the N “ p , q model, the coordinatedependence along the sixth coordinate induces a nontrivial gauge structure (3.18) togetherwith non-vanishing source terms in (3.17) — which differ from those of (3.4) illustrating theinequivalence of the N “ p , q and the N “ p , q model before dimensional reduction.The full field equation (3.16) takes the form of a vanishing curl (in r ρσ s ) and can locally beintegrated into the first order equation B r µ h ν s ρ ` ε µνκλτ B κ C λτ ρ ` ´ H µνρ ´ ε µνρκλ F κλ ¯ ` B y C µν,ρ ´ B y A r µ η ν s ρ “ B ρ u µν , (3.19)with an antisymmetric tensor u µν “ ´ u νµ . Combining this equation with the field equation(3.15) implies that B ρ ˆ F µν ` ε µνκλτ H κλτ ` B y B µν ´ B y u µν ˙ “ , (3.20)which can be further integrated into another first order duality equation F µν ` ε µνκλτ H κλτ ` B y B µν ´ B y u µν “ , (3.21) Here, and in the following we work locally and ignore potential subtleties that may arise from a non-trivialtopology. We refer to [27] for a discussion of such issues in the context of chiral p -forms.
10p to a function f µν p y q that can be absorbed into u µν . Eventually, we can use (3.21) to bring(3.19) into the form B r µ h ν s ρ ` ε µνκλτ B κ C λτ ρ ´ B ρ u µν “ ε µνρκλ B y ´ u κλ ´ B κλ ¯ ´ B y C µν,ρ ` B y A r µ η ν s ρ . (3.22)To sum up, we have cast the original second order field equations (2.11) of the six-dimensionalmixed-symmetry tensor field into the form of two first-order duality equations (3.21) and (3.22),upon parametrizing the six-dimensional fields in terms of its components (3.14) and introduc-tion of an additional field u µν . Upon reduction to five dimensions, these equations constitutethe duality equations relating the vector-tensor fields, and the graviton-dual graviton fields,respectively.It is instructive to work out the gauge symmetries of these equations which originate fromthe D “ α ˆ µ ˆ ν “ ˜ α µν ´ η µν λ p ξ µ ` µ q p ξ µ ` µ q λ ¸ , β ˆ µ ˆ ν “ ˜ β µν p ξ µ ´ Λ µ q p Λ µ ´ ξ µ q ¸ , (3.23)their action on the various components of (3.14) is derived as δA µ “ B µ λ ` B y p ξ µ ´ µ q ,δB µν “ B r µ Λ ν s ` B y β µν ,δh µν “ B p µ ξ ν q ` η µν B y λ ´ B y α µν ,δC µν,ρ “ B r µ α ν s ρ ` B ρ β µν ´ B r ρ β µν s ` B y ` ξ r µ η ν s ρ ´ r µ η ν s ρ ˘ . (3.24)With the field u µν defined by equation (3.19), its gauge variation is found by integrating up thevariation of (3.19) and takes the form δu µν “ B r µ ξ ν s ` ε µνρστ B ρ β στ ` B y β µν . (3.25)For later use, let us note that contraction of (3.22) with the fully antisymmetric ε -tensoryields 16 B ρ C µν,ρ ` B r µ C ν s ρρ ` ε µνρστ B ρ u στ “ B y ´ u µν ´ B µν ¯ , (3.26)while contraction gives rise to B µ h νµ ´ B ν h µµ ` B y C µν µ ` B y A ν “ B µ u µν . (3.27)This gives rise to an equivalent rewriting of (3.22) as B r µ h ν s ρ ` B σ h r µσ η ν s ρ ` η ρ r µ B ν s h σσ “ ´ ε µνκλτ B κ ´ C λτ ρ ` ε λταβσ u αβ η ρσ ¯ ´ B y A r µ η ν s ρ ` ε µνρκλ B y ´ u κλ ´ B κλ ¯ ´ B y C µν,ρ ´ B y C σ r µσ η ν s ρ . (3.28)11et us further note that taking the divergence of (3.21) yields the Maxwell type equation B µ F µν “ B y B µ h νµ ´ B y B ν h µµ ` B y B y C µν µ ´ B y B µ B µν ` B y B y A ν , (3.29)where we have used (3.27) in order to eliminate the divergence of u µν .For the remaining fields of the N “ p , q model, the 5+1 Kaluza-Klein split is achievedjust as for the N “ p , q model discussed above. The six-dimensional field equations of the14 vector fields and 28 scalar fields take the form obtained from variation of Lagrangians of theform (3.9), respectively. The field equations of the 12 selfdual forms take the form H µνρa “ ε µνρκλ p F κλ a ` B y B κλ a q , (3.30)after splitting the two-forms according to t B ˆ µ ˆ νa u “ t B µν a , B µ a ” A µa u . The equations maybe integrated up to an action in precise analogy with (3.11), c.f. the discussion in appendix A.2. N “ p , q Model
In this model, the exotic graviton is given by the rank four tensor (2.15) whose dynamics isdefined by the selfduality equations (2.16) for its second-order curvature. According to the splitof coordinates (3.1), we parametrize the various components of this field as t T ˆ µ ˆ ν, ˆ ρ ˆ σ u “ t T µν,ρσ ; T µν,ρ “ C µν,ρ ; T µ ,ν “ h µν u . (3.31)After dimensional reduction to five dimensions, these fields describe the graviton, dual gravitonand double dual graviton, respectively. Explicitly, in this parametrization the six-dimensionalfield equations (2.16) split into two equations R µν,ρσ “ B y B µ C ρσ,ν ´ B y B ν C ρσ,µ ` B y B ρ C µν,σ ´ B y B σ C µν,ρ ` ε µνκλτ B r ρ B κ C λτ σ s ` ε µνκλτ B y B κ T λτ ρσ ` B y B y T µν,ρσ , (3.32) ε µναβγ B α B r ρ T στ s βγ “ ´ B µ B r ρ C στ s ,ν ` B ν B r ρ C στ s ,µ ´ B y B r ρ T στ s ,µν , (3.33)with the linearized Riemann tensor R µν,ρσ defined as in (2.4) for the field h µν . The secondequation (3.33) has the form of a curl in r ρστ s and can be integrated up into12 ε µναβγ B α T στ βγ ` B µ C στ,ν ´ B ν C στ,µ ` B y T στ,µν “ B r σ v τ s ,µν , (3.34)up to a tensor v τ,µν “ ´ v τ,νµ , determined by this equation up to the gauge freedom δv τ,µν “B τ ζ µν . Combining (3.34) with the first field equation (3.32), we find R µν,ρσ “ B y B ρ C µν,σ ´ B y B σ C µν,ρ ` ε µνκλτ B r ρ B κ C λτ σ s ` B y B r ρ v σ s ,µν , (3.35)which in turn is a curl in r ρσ s and can be integrated up into B r µ h ν s ρ ` ε µνλστ B λ C στ ρ ` B y C µν,ρ ` B y v ρ,µν “ B ρ u µν , (3.36)12p to an antisymmetric field u µν “ ´ u νµ . As for the N “ p , q model, we have obtainedan equivalent reformulation of the dynamics in terms of two first-order equations (3.34) and(3.36) from which the original second-order field equations (3.32), (3.33), can be obtainedby derivation. After reduction to five dimensions, equations (3.34) and (3.36) describe theduality relations between graviton and dual graviton and between dual graviton and doubledual graviton, respectively. In particular, equation (3.36) differs from equation (3.22) in the N “ p , q model only if fields depend on the sixth coordinate.It is instructive to work out the gauge symmetries of these equations which originate fromthe D “ t λ ˆ ρ, ˆ µ ˆ ν u “ " λ ρ,µν ; λ µ,ν “ α µν ´ β µν ; λ ,µ “ ξ µ * , (3.37)with symmetric α µν , and antisymmetric β µν , their action on the various components of (3.31)is derived as δh µν “ B p µ ξ ν q ´ B y α µν ,δC µν,ρ “ B r µ α ν s ρ ` B ρ β µν ´ B r ρ β µν s ´ B y λ ρ,µν ,δT µν,ρσ “ B r µ λ ν s ,ρσ ` B r ρ λ σ s ,µν . (3.38)Gauge variations of the two new fields v ρ,µν and u µν are obtained by integrating up the variationof (3.34) and (3.36), respectively, giving rise to δu µν “ B r µ ξ ν s ` ε µνλστ B λ β στ ` B y β µν ` B y ζ µν ,δv ρ,µν “ ε µνκλσ B κ λ ρλσ ` B r µ α ν s ρ ` B r µ β ν s ρ ` B ρ ζ µν ` B y λ ρ,µν , (3.39)where the antisymmetric gauge parameter ζ µν “ ´ ζ νµ has been introduced after (3.34).Let us finally note that from (3.32) and (3.35), we may obtain the modified Einstein equa-tions ˚ G µν “ ´ B y B ρ C ρ p µ,ν q ´ B y B p µ C ν q ρρ ` B y B ρ v p µ,ν q ρ ´ B y B p µ v ρν q ρ ` η µν B y B ρ C ρσσ ´ η µν B y B ρ v σσρ , (3.40)with the linearized Einstein tensor ˚ G µν defined as˚ G µν “ ´B ρ B p µ h ν q ρ ` B ρ B ρ h µν ` B µ B ν h ρρ ` η µν B ρ B σ h ρσ ´ η µν B ρ B ρ h σσ , (3.41)which differs from the previous models by the absence of covariant derivatives, c.f. (3.5).For the remaining fields of the N “ p , q model, the 5+1 Kaluza-Klein split is achievedjust as for the previous models discussed above. The field equations of the 42 scalar fields areobtained from variation of a Lagrangian of the form L φ in (3.9). The field equations of the 27selfdual forms take the form of (3.30) above, again after splitting the two-forms according to t B ˆ µ ˆ νa u “ t B µν a , B µ a ” A µa u . 13 Actions for (free) exotic graviton fields
In the above, we have reformulated the dynamics of the six-dimensional exotic tensor fieldsin terms of first order differential equations upon breaking six-dimensional Poincar´e invarianceaccording to the split (3.1), and introducing some additional tensor fields. As a key propertyof the resulting equations, we have put the dynamics of the different models into a form whichreduces to the same equations after dimensional reduction B y Ñ
0. E.g. all three models featurelinearized Einstein equations for the field h µν , given by (3.4), (3.17), and (3.40), respectively.The three equations only differ by terms carrying explicit derivatives along the sixth dimension.We will use this as a guiding principle to construct uniform Lagrangians for the N “ p , q and the N “ p , q model which after setting B y Ñ D “ D “ L “ ´ p F µν ` B y B µν q p F µν ` B y B µν q ´ ε µνρστ B y B µν H ρστ , (4.1)after a Kaluza-Klein (5 `
1) decomposition t B ˆ µ ˆ ν u “ t B µν , B µ ” A µ u of the six-dimensionaltensor field. After dimensional reduction to five dimensions, the 3 degrees of freedom of theselfdual tensor field are described as a massless vector with the standard Maxwell Lagrangian towhich (4.1) reduces at B y Ñ
0. In presence of the sixth dimension, variation of the Lagrangian(4.1) w.r.t. the vector field gives rise to modified Maxwell equations while variation w.r.t. thetensor field yields the duality equation relating A µ and B µν , which is of first order in thederivatives B µ and appears under a global B y derivative. Combining these two equations onemay infer the full six-dimensional selfduality equation. Details are spelled out in appendix A.2.The Lagrangians for exotic gravitons are constructed in analogy to (4.1) with the role of A µ and B µν now taken by the graviton h µν and its duals, respectively. N “ p , q Model
The main result of this subsection is the following: the first order field equations (3.21) and(3.22), which describe the dynamics of the six-dimensional exotic graviton field C ˆ µ ˆ ν, ˆ ρ in the N “ p , q model, can be derived from the Lagrangian L “ ´ p Ω µνρ p Ω µνρ ` p Ω µνρ p Ω νρµ ` p Ω µ p Ω µ ´ ε µνρστ B y p C µν,λ B ρ p C στ,λ ´ F µν F µν ´ ε µνρστ B y B µν B ρ B στ ´ ε µνρστ B y B µν B y p C ρσ,τ , (4.2)with p Ω µνρ ” B r µ h ν s ρ ´ B y A r µ η ν s ρ ` B y p C µν,ρ , p C µν,ρ ” C µν,ρ ` ε µνρστ u στ , F µν ” B r µ A ν s ` B y B µν . (4.3)The Lagrangian (4.2) is invariant under the gauge transformations (3.24), (3.25). After reduc-tion to five dimensions, i.e. at B y Ñ
0, this Lagrangian reduces to the Fierz-Pauli Lagrangian14or h µν together with a free Maxwell Lagrangian for A µ ; the dual fields p C µν,ρ and B µν dropout in this limit. In presence of the sixth dimension, variation of the Lagrangian (4.2) w.r.t.to the dual fields yields the first-order duality equations (3.21) and (3.22), however under anoverall derivative B y . Together, one recovers the full six-dimensional dynamics. Details of theequivalence are presented in appendix B.The bosonic Lagrangian for the full N “ p , q model is then given by combining (4.2)with the Lagrangians of the type (3.9) and (4.1) for the remaining matter fields of the theory.Putting everything together, we obtain L p , q “ L ´ ` B µ φ i ´ B y A µ i ˘ ` B µ φ i ´ B y A µi ˘ ´ B µ φ α B µ φ α ´ B y φ α B y φ α (4.4) ´ F µν i F µν i ´ p F µν a ` B y B µν a q p F µν a ` B y B µν a q ´ ε µνρστ B y B µν a H ρστ a , with indices ranging along i “ , . . . , , α “ , . . . , , a “ , . . . , . (4.5)After dimensional reduction to five dimensions (and rescaling of the vector field A µ ), thisLagrangian coincides with the Lagrangian (3.12) of linearized maximal supergravity. The La-grangian (4.4) describes the full six-dimensional theory, with the field content of five-dimensionalmaximal supergravity enhanced by the field p C µν,ρ . D “ N “ p , q Model
The main result of this subsection is the following: the first order field equations (3.34) and(3.36), which describe the dynamics of the six-dimensional exotic graviton field T ˆ µ ˆ ν, ˆ ρ ˆ σ in the N “ p , q model, can be derived from the Lagrangian L “ ´ p Ω µνρ p Ω µνρ ` p Ω µνρ p Ω νρµ ` p Ω µ p Ω µ ´ ε µνσκλ B µ p C νσρ B y p C κλ,ρ ` ε µνσκλ B µ C νσρ B y C κλ,ρ ´ B y C στ,ν B µ T µν,στ ` B y C κλ,τ B κ T λσ,τ σ ` B ν C σµµ B y T στ,ντ ´ B y C σµµ B σ T τν τν ´ ε µναβγ B α T στ βγ B y T µν,στ ´ B y T στ,µν B y T µν,στ ` B y T σµ,ν µ B y T στ,ντ ´ B y T µν µν B y T στ στ , (4.6)with p Ω µνρ “ B r µ h ν s ρ ` B y p C µν,ρ ´ B y C µν,ρ , p C µν,ρ “ C µν,ρ ` ε µνρστ u στ , C µν,ρ “ C µν,ρ ´ v ρ,µν ` v r ρ,µν s ` ε µνρστ u στ . (4.7)After reduction to five dimensions, i.e. at B y Ñ
0, this Lagrangian reduces to the Fierz-PauliLagrangian for h µν ; the dual fields p C µν,ρ , C µν,ρ , and T µν,ρσ drop out in this limit. In presence ofthe sixth dimension, variation of the Lagrangian (4.6) w.r.t. to the dual fields yields the first-order duality equations (3.34) and (3.36), however under an overall derivative B y . Together,15ne recovers the full six-dimensional dynamics. The computation works in close analogy withthe derivation for the N “ p , q model, c.f. appendix B.Let us spell out the gauge transformations (3.38), (3.39) in terms of the fields (4.7) δ p Ω µνρ “ B ρ B r µ ξ ν s ´ B y B r µ β ν s ρ ´ B y B r µ ζ ν s ρ ` B y B κ λ r µστ ε ν s κστρ ,δ p C µν,ρ “ B r µ α ν s ρ ´ B r µ β ν s ρ ` ε µνρστ B σ ξ τ ` ε µνρστ B y ´ β στ ` ζ στ ¯ ´ B y λ ρ,µν ,δ C µν,ρ “ ε µνρστ B σ ξ τ ´ B r µ β ν s ρ ` B r µ ζ ν s ρ ` ε µνρστ B y ´ β στ ` ζ στ ¯ ` ε κστρ r µ B κ λ ν s στ ´ B y λ ρ,µν , (4.8)which allows to confirm gauge invariance of the Lagrangian (4.6).The bosonic Lagrangian for the full N “ p , q model is finally given by combining (4.6)with the Lagrangians of the type (3.9) and (4.1) for the remaining matter fields of the theory.Putting everything together, we obtain L p , q “ L ´ ` F µν M ` B y B µνM ˘ ` F µν M ` B y B µν M ˘ ´ ε µνρστ B y B µν M H ρστ M ´ B µ φ A B µ φ A ´ B y φ A B y φ A , (4.9)with indices ranging along M “ , . . . , , A “ , . . . , . (4.10)After dimensional reduction to five dimensions, this Lagrangian coincides with the Lagrangian(3.12) of linearized maximal supergravity. The Lagrangian (4.9) describes the full six-dimensionaltheory, with the field content of five-dimensional maximal supergravity enhanced by the fields p C µν,ρ , C µν,ρ , and T µν,ρσ . D “ In the previous sections, we have constructed Lagrangians (3.6), (4.4), and (4.9), for the threesix-dimensional models which share a number of universal features and structures. In particular,after dimensional reduction to five dimensions they all reduce to the same Lagrangian (3.12)corresponding to linearized maximal supergravity in five dimensions. The three distinct six-dimensional theories are then described as different extensions of this Lagrangian by termscarrying derivatives along the sixth dimension. In the various matter sectors, these terms ensurecovariantization under non-trivial gauge structures and provide sources to the field equationsof five-dimensional supergravity.This reformulation within a common framework is very much in the spirit of exceptional fieldtheories. In that framework, higher-dimensional supergravity theories are reformulated in termsof the field content of a lower-dimensional supergravity keeping the dependence on all coordi-nates. More precisely, their formulation is based on a split of coordinates into D external and16 internal coordinates of which the latter are formally embedded into a fundamental represen-tation R v of the global symmetry group E ´ D, p ´ D q of D -dimensional maximal supergravity.Different embeddings of the internal coordinates into R v then correspond to different higher-dimensional origins. Here, we will discuss a similar uniform description of the six-dimensionalmodels based on D “ D “ N “ p , q model The theory relevant for our discussion is E p q exceptional field theory (ExFT) [10, 11]. Itsbosonic field content is given by a graviton g µν together with 27 vector fields A µM and their dualtensors B µν M , together with 42 scalars parametrizing the internal metric M MN “ p V V T q MN with V a representative of the coset space E p q { USp p q . Fields depend on 5 external and27 internal coordinates with the latter transforming in the fundamental of E p q and withinternal coordinate dependence of the fields restricted by the section constraint [9] d KMN B M b B N “ , (5.1)with the two differential operators acting on any couple of fields and gauge parameters of thetheory. The tensor d KMN denotes the cubic totally symmetric E p q invariant tensor, which wenormalize as d MNP d MNQ “ δ QP . The section condition (5.1) admits two inequivalent solutions[10] which reduce the internal coordinate dependence of all fields to the 6 internal coordinatesfrom D “
11 supergravity, or 5 internal coordinates from IIB supergravity, respectively. Fordetails of the ExFT Lagrangian we refer to [10, 11]. Here, we spell out its ‘free’ limit, obtainedby linearizing the full theory according to g µν “ η µν ` h µν , M MN “ ∆ MN ` φ MN , (5.2)around the constant background given by the Minkowski metric η µν and the identity matrix∆ MN . The scalar fluctuations φ MN are further constrained by the coset properties of M MN .To quadratic order in the fluctuations, the ExFT Lagrangian then yields L ExFT , free “ ´
14 Ω µνρ Ω µνρ `
12 Ω µνρ Ω νρµ ` Ω µ Ω µ ´ F µν M F µν N ∆ MN ´ ? ε µνρστ d MNK B µ B νρ M B N B στ K ´ D µ φ MN D µ φ MN ` L pot , (5.3)17ith indices M, N raised and lowered by ∆ MN and its inverse, and with the various elementsof (5.3) given byΩ µνρ “ B r µ h ν s ρ ´ B M A r µM η ν s ρ , Ω µ ” Ω µν ν , F µν M “ B r µ A ν s M ` d MNK B N B µν K ,D µ φ MN “ B µ φ MN ` B K A µ p M ∆ N q K ` B K A µK ∆ MN ´ B K A µL d P LR d RK p M ∆ N q P , L pot “ ´
124 ∆ MN B M φ KL B N φ KL `
12 ∆ MN B M φ KL B L φ NK ´ B M h ν ν B N φ MN `
14 ∆ MN B M h µµ B N h ν ν ´
14 ∆ MN B M h µν B N h µν . (5.4)The Lagrangian we have presented above for the six-dimensional N “ p , q model naturallyfits into this framework. This does not come as a surprise since the six-dimensional model isnothing but linearized maximal supergravity known to be described by E p q ExFT upon properselection of the sixth coordinate among the internal B M . This choice is uniquely fixed by therequirement that the resulting theory exhibits the global SO p , q symmetry group of maximalsix-dimensional supergravity, thus breakingE p q ÝÑ SO p , q , ÝÑ ‘ ‘ , tB M u ÝÑ tB , B i , B a u , (5.5)and keeping only coordinate-dependence along the SO p , q singlet. In this split, the E p q invariant symmetric tensor d MNK has the non-vanishing components d ab “ ? η ab , d aij “ ? p Γ a q ij , (5.6)in terms of SO p , q Γ-matrices and its invariant tensor η ab of signature p , q , showing thatthe section constraint (5.1) is trivially satisfied is B i “ “ B a . Putting this together with thelinearized ExFT Lagrangian (5.3), and splitting fields as t A µM u “ t A µ , A µi , A µa u , etc. , (5.7)we arrive at L p , q “ ´
14 Ω µνρ Ω µνρ `
12 Ω µνρ Ω νρµ ` Ω µ Ω µ ´ F µν F µν ´ F µν i F µν i ´ p F µν a ` B y B µνa q p F µν a ` B y B µν a q ´ ε µνρστ η ab B y B µν a H ρστ b ´ B µ φ α B µ φ α ´ pB µ φ ´ c B y A µ qpB µ φ ´ c B y A µ q ´ ` B µ φ i ´ B y A µ i ˘ ` B µ φ i ´ B y A µi ˘ , ´ B y φ α B y φ α ` B y φ B y φ ´ B y h σσ B y φ ` B y h σσ B y h ρρ ´ B y h µν B y h µν , (5.8)which precisely produces the sum of Lagrangians (3.6), (3.9), (3.11), after proper rescaling ofthe singlet scalar field φ . The non-trivial checks of this coincidence include all the coefficientsin the various connection terms, as well as in the St¨uckelberg-type couplings between vector18nd tensor fields, and the coefficients in front of the various B y φ B y φ terms in the last line.Again, this is not a surprise but a consequence of the proven equivalence of ExFT with higher-dimensional maximal supergravity. Note that although the free theory only exhibits a compactUSp p q ˆ USp p q global symmetry, the couplings exhibited in (5.8) are far more constrainedthan allowed by this symmetry and witness the underlying E p q structure broken to SO p , q according to (5.5), (5.6).The ExFT Lagrangian is to a large extent determined by invariance under generalizedinternal diffeomorphisms acting with a gauge parameter Λ M in the . After linearization (5.2)these diffeomorphisms act as δφ MN “ K p M B N q Λ K ` B K Λ K ∆ MN ´ d P KR d RL p M ∆ N q P B K Λ L ,δ A µM “ B µ Λ M , δh µν “ B M Λ M η µν , (5.9)and one can show invariance of the linearized Lagrangian (5.8), provided the section constraint(5.1) is satisfied. N “ p , q and p , q couplings As we have discussed in the introduction, the charges carried by the massive BPS multipletsin the reduction of the N “ p , q and the N “ p , q model, respectively, suggest that aninclusion of these models into the framework of ExFT necessitates an extension of the space of27 internal coordinates by an additional exotic coordinate corresponding to the singlet centralcharge [7]. Denoting derivatives along this coordinate by B ‚ , this would amount to a relaxationof the standard section constraint (5.1) to a constraint of the form d KMN B M b B N ´ ?
10 ∆ KM pB M b B ‚ ` B ‚ b B M q “ , (5.10)which at the present stage only makes sense in the linearized theory where ∆ KM is a constantbackground tensor. Apart from the standard ExFT solutions d KMN B M b B N “ , B ‚ “ , (5.11)of this constraint, which allow the embedding of the N “ p , q model as described above, theextended section constraint also allows for two exotic solutions p , q : B p , q y “ ? B “ ´ B ‚ , with the F p q singlet B Ă B M , p , q : B p , q y “ ´ B ‚ , B M “ , (5.12)corresponding to the two exotic six-dimensional models in precise correspondence with thecentral charges carried by the corresponding BPS multiplets [7]. While the (4,0) solutiontrivially solves the constraint (5.10), the N “ p , q solution is based on the decompositionE p q ÝÑ F p q , ÝÑ ‘ , tB M u ÝÑ tB , B A u , (5.13)19nder which the symmetric d -tensor decomposes into d “ ´ ? , d AB “ ? η AB , d ABC , (5.14)with the F p q invariant symmetric tensor η AB of signature p , q , and the symmetric invarianttensor d ABC satisfying d ABC η BC “ , d ABC d ABD “ δ C D . (5.15)This shows explicitly how the (3,1) assignment of (5.12) also provides a solution to the extendedsection constraint (5.10).It is intriguing to study the fate of diffeomorphism invariance of the ExFT Lagrangian (5.3)if the original section constraint is relaxed to (5.10). Except for the last term in (5.3), theLagrangian remains manifestly invariant without any use of the section constraint. Explicitvariation of the potential term L pot under linearized diffeomorphisms (5.9) on the other handyields (up to total derivatives) δ Λ L pot “ ` LS d LMN d KP Q ´ MN ∆ KL d LSR d RP Q ˘ Λ S B P B Q B M φ NK ´ h µµ ∆ MK d KLR d RP Q B M B P B Q Λ L , (5.16)which consistently vanishes modulo the standard section constraint (5.1). For the weaker con-straint (5.10), this variation no longer vanishes and may be recast in the following form δ Λ L pot “ ∆ KM Λ N B ‚ B ‚ B M φ NK ´ h ν ν B ‚ B ‚ B N Λ N , (5.17)after repeated use of (5.10) and further manipulation of the expressions. In order to com-pensate for this variation let us first note that there is no possible covariant extension of thetransformation rules (5.10) by terms carrying B ‚ Λ M , such that invariance can only be restoredby extending the potential. A possible such extension is given by L pot , ‚ “ L pot ´ B ‚ φ MN B ‚ φ MN ´ B ‚ h σσ B ‚ h ρρ ` B ‚ h µν B ‚ h µν , (5.18)and it is straightforward to verify that the variation of the additional terms in (5.18) preciselycancels the contributions in (5.17), such that δ Λ L pot , ‚ “ . (5.19)For the exotic solutions of the section constraint, the B ‚ φ MN B ‚ φ MN terms in (5.18) give riseto additional contributions of the type B y φ B y φ in the Lagrangian. Collecting all such terms in(5.18) for the two exotic solutions (5.12) yields p , q ÝÑ ´ B y φ α B y φ α , α “ , . . . , , p , q ÝÑ ´ B y φ A B y φ A , A “ , . . . , . (5.20)These are precisely the terms found in our explicit construction of actions (4.4) and (4.9)above! In other words, the relaxation (5.10) of the section constraint together with generalizeddiffeomorphism invariance precisely implies the correct scalar couplings in the Lagrangians of20he exotic models. In addition, the B ‚ h B ‚ h terms in (5.18) cancel the corresponding terms in L pot (5.4) upon selecting the (3,1) solution of the section constraint (5.12), just as required inorder to reproduce the correct Lagrangian of the N “ p , q model (4.2). We may continue the symmetry analysis for the tensor gauge transformations given by agauge parameter Λ µ M in standard ExFT. For these transformations there is a natural extensionof the standard ExFT transformation rules in presence of the exotic coordinate and exotic fieldsas δ Λ µ A µM “ ´ d MNK B N Λ µ K ´ ?
10 ∆ MK B ‚ Λ µ K ,δ Λ µ B µν M “ B r µ Λ ν s M . (5.21)Computing the action of these transformations on the connection featuring in the covariantscalar derivatives D µ φ MN in (5.4), we obtain after some manipulation δ Λ µ D µ φ MN “ ˆ
13 ∆ MN δ P Q ` ∆ Q p M δ P N q ´
10 ∆ S p M d N q QR d RSP ˙ d P KL B K B L Λ µ Q ´ ? ˆ
13 ∆ MN δ P Q ` δ P p M ∆ N q Q ´
10 ∆ L p M d N q QR d RP L ˙ ∆ P K B K B ‚ Λ µ Q . (5.22)The resulting expression precisely vanishes with the modified section constraint (5.10). Thisshows the necessity of the B ‚ Λ µ M terms in (5.21) in order to maintain gauge invariance of thekinetic term D µ φ MN D µ φ MN in presence of the relaxed section constraint. It is straightforwardto verify that these additional terms in the transformation induce a modification of the gaugeinvariant vector field strengths to F µν M ” B r µ A ν s M ` d MNK B N B µν K ` ?
10 ∆ MK B ‚ B µν K , (5.23)as well an extension of the topological term, such that the combined vector-tensor couplingstake the form L vt , ‚ “ ´
14 ∆ MN F µν M F µν N ´ ε µνρστ B µ B νρ M ´ ? d MNK B N B στ K ` ∆ MK B ‚ B στ K ¯ , (5.24)and are invariant under these gauge transformations. Let us work out the effect of thesemodifications for the exotic solutions of the section constraint. With the kinetic scalar termunchanged, the resulting couplings are directly inferred from evaluating the covariant derivatives(5.4) for the d -symbol (5.14), giving rise to p , q ÝÑ ´ ` B µ φ i ´ B y A µ i ˘ ` B µ φ i ´ B y A µi ˘ ´ B µ φ α B µ φ α , i “ , . . . , , α “ , . . . , , p , q ÝÑ ´ B µ φ A B µ φ A , A “ , . . . , , (5.25) In contrast, these terms appear in conflict with embedding the spin-2 sector of the N “ p , q model asthey survive under the (4,0) solution in (5.12) but should be absent in the final Lagrangian (4.6). We come backto this in section 5.3. A useful identity for this computation is given by d PLQ d PSR d KMR B K B L “ δ KS d LQM B K B L ` δ MS d QKL B K B L ` δ QS d MKL B K B L ´ d QMR d RSP d PKL B K B L , generalizing equations (2.12), (2.13) of [11]. p , q ÝÑ ´ p F µν ` ? B y B µν qp F µν ` ? B y B µν q ´ p F µν a ` B y B µνa q p F µν a ` B y B µν a q´ F µν i F µν i ´ ε µνρστ B y B µν H ρστ ´ ε µνρστ B y B µνa H ρστ a , p , q ÝÑ ´ ` F µν M ` B y B µν M ˘ ` F µν M ` B y B µν M ˘ ´ ε µνρστ B y B µν M H ρστ M , (5.26)with indices in range i “ , . . . , a “ , . . . , M “ , . . . ,
27, as above. Again, this preciselyreproduces the couplings found above (after proper rescaling of the vector field A µ )!To summarize, in the scalar, vector and tensor sector, we have constructed an extension ofthe ExFT Lagrangian (at the linearized level), given by L “ ´ D µ φ MN D µ φ MN ` L vt , ‚ ` L pot , ‚ , (5.27)which is invariant under the gauge transformations (5.9), (5.21) modulo the relaxed sectionconstraint (5.10). The weaker section constraint necessitates a numer of additional contributionsto the Lagrangian (and transformation rules) which precisely reproduce the explicit couplingsfound in the Lagrangians of the exotic models (4.4), (4.9) constructed above. It is remarkablethat this match confirms the couplings that have been determined from an underlying non-compact E p q and F p q structure, respectively, despite the fact that the free theory only exhibitsinvariance under the compact R -symmetry subgroup USp p N ` q ˆ USp p N ´ q which might inprinciple allow for much more general couplings. We take this as evidence for the conjecturedE p q and F p q invariance of the putative interacting theories [1]. The above findings have revealed a very intriguing common structure of the couplings in thescalar, vector and tensor sectors of the different models which can be consistently embedded intoan extension of (linearized) exceptional field theory. For the spin-2 sector carrying the Pauli-Fierz field and its duals on the other hand the picture appears not yet complete. Extrapolationof the Lagrangian of the N “ p , q model (4.6) suggests an extension of the standard ExFTLagrangian by couplings carrying B ‚ derivatives and the dual graviton fields as L “ ´ p Ω µνρ p Ω µνρ ` p Ω µνρ p Ω νρµ ` p Ω µ p Ω µ ` ε µνσκλ B µ p C νσρ B ‚ p C κλ,ρ ´ ε µνσκλ B µ C νσρ B ‚ C κλ,ρ ` B ‚ C στ,ν B µ T µν,στ ´ B ‚ C κλ,τ B κ T λσ,τ σ ´ B ν C σµµ B ‚ T στ,ντ ` B ‚ C σµµ B σ T τν τν ` ε µναβγ B α T στ βγ B ‚ T µν,στ ´ B ‚ T στ,µν B ‚ T µν,στ ` B ‚ T σµ,ν µ B ‚ T στ,ντ ´ B ‚ T µν µν B ‚ T στ στ ` ε µνρστ d KMN B K B µν M B N p C ρσ,τ , (5.28)22ith p Ω µνρ “ B r µ h ν s ρ ´ B M A r µM η ν s ρ ´ B ‚ p C µν,ρ ` B ‚ C µν,ρ . (5.29)By construction, this reproduces the N “ p , q and the N “ p , q models upon choosingthe corresponding solutions of the section constraint. It remains unclear however, how thespin-2 sector of the N “ p , q model can find its place in this construction. In particular,the appearance of the extra fields C µν,ρ and T µν,ρσ appearing in (5.28), whose couplings remainpresent upon selecting the (3,1) solution (5.12) of the section constraint, poses a challengefor recovering the Lagrangian (4.4) of the N “ p , q model. The structure of the gaugetransformations of C as extrapolated from (4.8) appears to suggest a gauge fixing of the ζ µν and λ ρ,µν gauge symmetries — absent in the N “ p , q model — in order to remove this field.Another apparent problem in the spin-2 sector is the lacking reconciliation between the B ‚ h B ‚ h terms from (5.18) and the B ‚ T B ‚ T terms of (5.28) which mutually violate the correct limitsto the exotic models. Resolution of this problem may require to implement algebraic relationsbetween the Pauli-Fierz h µν field and the double dual graviton [2] (see also [28]). In this paper we have taken the first step in constructing action principles for exotic supergravitytheories in 6D by giving such actions for the free bosonic part. These actions show alreadyintriguing new features such as the simultaneous appearance of (linearized) diffeomorphismsand dual diffeomorphisms, which are realized on exotic Young tableaux fields as well as moreconventional gravity fields. Our formulation abandons manifest 6D Lorentz invariance, asexpected to be necessary on general grounds, by being based on a 5 ` N “ p , q , as wellas the exotic N “ p , q and N “ p , q models all emerge through different solutions of anextended section constraint, but clearly much more needs to be done. We close with a briefdiscussion of possible future developments.First, it remains to exhibit the (maximal) supersymmetries in these non-standard formula-tions, even just at the free level. We have no doubt that this can be achieved as in exceptionalfield theory where different supersymmetries (such as type IIB versus type IIA) are realizedwithin a single master formulation. Second, it would be interesting to study possible embed-dings into exceptional field theories of higher rank, such as for U-duality groups E p q and E p q ,which may illuminate some issues and which can also be done already at linearized level. Fi-nally, the most important outstanding problem is clearly the question whether our formulationcan be extended to the non-linear interacting theory. We would like to emphasize that thepresent formulations seem quite promising in this regard since they feature not only the exoticfields but also the more conventional gravity fields, which come with an action that allows anatural embedding into the full non-linear Einstein-Hilbert action. In turn this suggests thatall these fields might become part of a tensor hierarchy that extends to the gravity sector. Ifso this could quite naturally lend itself to a formulation of non-linear dynamics in terms of ahierarchy of duality relations as in [29]. 23 cknowledgements We wish to thank X. Bekaert, C. Hull and V. Lekeu for enlighteningdiscussions. The work of O.H. is supported by the ERC Consolidator Grant “Symmetries &Cosmology”.
AppendixA Actions for selfdual tensor fields
It is well known that the first-order field equations for D “ H ˆ µ ˆ ν ˆ ρ “ ε ˆ µ ˆ ν ˆ ρ ˆ σ ˆ κ ˆ λ H ˆ σ ˆ κ ˆ λ , H ˆ µ ˆ ν ˆ ρ “ B r ˆ µ B ˆ ν ˆ ρ s , (A.1)do not integrate to a standard action principle, yet various mechanisms with different charac-teristics have been devised such as to provide a Lagrangian description of these equations [13,21–23]. In this appendix, we briefly review the construction of Henneaux and Teitelboim [13]which is somewhat closest in spirit to the construction employed in this paper, together with itsdual formulation that is naturally embedded within exceptional field theory. Both formulationsare based on a coordinate split (3.1) t x ˆ µ u ÝÑ t x µ , y u , (A.2)and sacrifice manifest D “ With the corresponding split t B ˆ µ ˆ ν u “t B µν , B µ ” A µ u of the six-dimensional tensor field, the selfduality equations (A.1) take theform F µν ` ε µνρστ H ρστ “ , for F µν ” F µν ` B y B µν . (A.3)In particular, the divergence and curl of this equation give rise to B µ F µν “ ,ε µνλστ B y H λστ ´ B λ H λµν “ , (A.4)respectively. A.1 Henneaux-Teitelboim Lagrangian
The Lagrangian proposed by Henneaux and Teitelboim [13] for the description of the self-dualtensors takes the form L “ ε µνρστ F µν H ρστ ´ H µνρ H µνρ , (A.5)when applied to equations (A.3), i.e. evaluated for space-like split and flat background. As afirst observation, this Lagrangian depends on the vector field A µ only via total derivatives, suchthat it does not show up in the field equations ε µνρστ B y H ρστ “ B ρ H µνρ , (A.6) The original construction of [13] defines the 5 ` A µ via the equation2 B r µ A ν s “ ´B y B µν ´ ε µνρστ H ρστ . (A.7)Indeed, the curl of the r.h.s. vanishes by virtue of (A.6). Defining the vector field A µ by (A.7),we precisely recover the equations of motion (A.3). A.2 ExFT type Lagrangian
Exceptional field theory (ExFT) typically yields formulations of higher-dimensional supergrav-ity theories based on the field content of lower-dimensional theories. In particular, it offersactions for theories that do not admit actions in terms of their original variables, such as IIBsupergravity, c.f. [12]. In the context of (anti-)selfdual tensor fields appearing in six dimensions,an exceptional field theory formulation based on a split (A.2) gives rise to an action L “ ´ F µν F µν ´ ε µνρστ B y B µν H ρστ , (A.8)carrying the fields of equation (A.3). The field equations are now given by0 “ B ν F νµ “ B ν F νµ ` B y B ν B νµ , (A.9)0 “ B y ´ F µν ` ε µνρστ H ρστ ¯ . (A.10)In particular, equation (A.10) implies the original field equations (A.3) up to some functionthat does not depend on y : F µν ` ε µνρστ H ρστ “ χ µν , B y χ µν “ . (A.11)Comparing the divergence of this equation to (A.9), we find that locally the field χ µν can beintegrated to B µ χ µν “ ùñ χ µν “ ε µνρστ B ρ b στ , (A.12)in terms of a function b µν , such that the field equations (A.11) can be rewritten as ´ F µν ` B y ˜ B µν ¯ ` ε µνρστ B ρ ˜ B στ “ , (A.13)with the modified two-form ˜ B µν ” B µν ´ b µν . (A.14)In terms of the fields A µ , ˜ B µν , we thus recover the desired original field equations (A.3). Notefinally, that the Lagrangian (A.8) precisely comes with a gauge freedom of the type (A.14)which allows to absorb b µν into B µν .We thus arrive at two complementary Lagrangians (A.5), (A.8), which both describe thesix-dimensional selfdual tensor field upon sacrificing manifest D “ D “ B y Ñ
0, the Lagrangian (A.8) describes the 3 degrees of freedom in terms of afree Maxwell field whereas (A.5) describes them in terms of the dual massless tensor field B µν .Similarly, the two Lagrangians (A.5) and (A.8) can be dualized into each other in presence ofthe sixth dimension. 25
6D field equations from the new Lagrangians
In this appendix, we present in detail how the second-order field equations obtained by vari-ation of the Lagrangian (4.2) for the N “ p , q model can be integrated to the first-orderfield equations (3.21) and (3.22) which in turn imply the original 6D second-order selfdualityequations (2.11). For the N “ p , q model (4.6), the discussion goes along the same lines. B.1 Field equations
Here, we spell out the field equations obtained from variation of the Lagrangian (4.2).Variation w.r.t. A µ : B µ F µν ` B y B µ B µν ´ B y ´ B µ h µν ´ B ν h µµ ´ B y p C νµµ ` B y A ν ¯ “ , (B.1)which is exactly (3.29).Variation w.r.t. B µν : B y ˆ F µν ` B y B µν ` ε µνρστ B ρ B στ ` B y ε µνρστ p C ρστ ˙ “ , (B.2)which is the B y derivative of equation (3.21).Variation w.r.t. h µν : G µν ` B y ´ B ρ p C ρ p µ,ν q ` B p µ p C ν q ρρ ´ η µν B ρ p C ρσσ ¯ “ , (B.3)with the linearized Einstein tensor as it appears in (3.17), this variation thus exactly reproducesthe Einstein equation (3.17).Variation w.r.t p C µν,ρ : B y ˆ B r µ h ν s ρ ` B σ h σ r µ η ν s ρ ´ η ρ r ν B µ s h σσ ` ε µνλστ B λ p C στ ρ ` B y p p C µν,ρ ´ p C νρ,µ ´ p C ρµ,ν q´ η ρ r ν B y p C µ s σσ ` B y A r µ η ν s ρ ` ε µνρστ B στ ˙ “ , (B.4)which we can further project onto its (2,1) part and totally antisymmetric part B y ˆ B r µ h ν s ρ ` B σ h σ r µ η ν s ρ ´ η ρ r ν B µ s h σσ ` ε µνλστ B λ p C στ ρ ´ ε λστ r µν B λ p C στ ρ s ` B y p p C µν,ρ ´ p C νρ,µ ´ p C ρµ,ν ` p C r µν,ρ s q ´ η ρ r ν B y p C µ s σσ ` B y A r µ η ν s ρ ˙ “ , (B.5) B y ˆ ε λστ r µν B λ p C στ ρ s ´ B y p C r µν,ρ s ` B y ε µνρστ B στ ˙ “ . (B.6) B.2 Going back to the original equations
The goal of this section is to recover the full 6D system (3.19) and (3.21) from the equationsderived in the section B.1. Let us first rewrite them in terms of the original fields of the (3,1)26odel and integrate all the equations under B y by introducing three functions χ µν p x µ q , ψ µνρ p x µ q and ϕ µν,ρ p x µ q which are respectively antisymmetric, antisymmetric and of p , q type, and donot depend on the sixth coordinate. B µ F µν ` B y B µ B µν ´ B y pB µ h µν ´ B ν h µµ ´ B y C νµµ ` B y A ν q “ , (B.7) F µν ` B y B µν ´ B y u µν ` ε µνρστ B ρ B στ “ χ µν , (B.8) ε λστ r µν B λ C στ ρ s ´ B r ρ u µν s ´ B y ε µνρστ ˆ u στ ´ B στ ˙ “ ψ µνρ , (B.9) G µν ` B y ` B ρ C ρ p µ,ν q ` B p µ C ν q ρρ ´ η µν B ρ C ρσσ ˘ “ , (B.10) B r µ h ν s ρ ` B σ h σ r µ η ν s ρ ´ η ρ r ν B µ s h σσ ` ε µνλστ B λ C στ ρ ´ ε λστ r µν B λ C στ ρ s ` B y C µν,ρ ´ η ρ r ν B y C µ s σσ ` B y A r µ η ν s ρ ´ B ρ u µν ` B r ρ u µν s ´ B σ u σ r µ η ν s ρ “ ϕ µν,ρ . (B.11) A-B duality
Combining (B.7) and (B.8) gives B y B µ u µν “ B y pB µ h µν ´ B ν h µµ ´ B y C νµµ ` B y A ν q ´ B µ χ µν , (B.12)while the trace of (B.11) in p µρ q gives B µ u µν “ pB µ h µν ´ B ν h µµ ´ B y C νµµ ` B y A ν q ` ϕ µν µ . (B.13)Together, these two equations imply that locally, we can define a 2-form b such that χ µν “ ε µνρστ B ρ b στ p x µ q . (B.14)This 2-form can be absorbed in B (following exactly the same process as in section A.2) suchthat equations (B.8) reproduces (3.21). h-C duality Contracting (B.11) with B µ , we can extract both symmetric and antisymmetricparts: p νρ q : G νρ ` B y ` B µ C µ p ν,ρ q ´ η νρ B µ C µσσ ` B p ρ C ν q σσ ˘ “ B µ ϕ µ p ν,ρ q , (B.15) r νρ s : ´ ε λστνρ B µ B λ C στ µ ` B y ` B µ C µ r ν,ρ s ` B r ρ C ν s σσ ´ B r ρ A ν s ˘ ` B µ B r µ u νρ s “ B µ ϕ µ r ν,ρ s . (B.16)Using (B.10) we can conclude that B µ ϕ µ p ν,ρ q “ . (B.17)The divergence of (B.9) reads ´ ε µναβγ B ρ B α C αβρ ` B ρ B r ρ u µν s ` B y ε µνραβ B ρ ˆ u αβ ´ B αβ ˙ “ ´ B ρ ψ µνρ , (B.18)27nd combining it with (B.16) and (B.8), we eventually get2 B µ ϕ µ r ν,ρ s “ ´ B ρ ψ µνρ . (B.19)Together with (B.17), one has B µ p ϕ µν,ρ ` B ρ ψ µνρ q “ , (B.20)such that locally there exist 5D tensors c µν,ρ and a µνρ , where c is of p , q type and a iscompletely antisymmetric, such that2 ϕ µν,ρ ` B ρ ψ µνρ “ ε µναβγ B α p c βγ ρ ` a βγ ρ q . (B.21)Consequently ϕ µν µ “ ε µναβγ B α a βγµ , (B.22) ϕ µν,ρ “ ε µναβγ B α ´ c βγρ ` a βγρ ¯ ´ ε αβγ r µν B α ´ c βγ ρ s ` a βγ ρ s ¯ , (B.23) ψ µνρ “ ε αβγ r µν B α ´ c βγ ρ s ` a βγρ s ¯ . (B.24)Plugging the expression for ϕ and its trace back into (B.11), one has2 B r µ h ν s ρ ` ε µνλστ B λ C στ ρ ´ ε λστ r µν B λ C στ ρ s ` B y C µν,ρ ´ B y A r µ η ν s ρ ´ B ρ u µν ` B r ρ u µν s “ ε µναβγ B α p c βγ ρ ` a βγρ q ´ ε αβγ r µν B α p c βγ ρ s ` a βγ ρ s q ` ε αβγσ r µ B α a βγσ η ν s ρ . (B.25)Then, using the following two Schouten identities ε r µν αβγ B α a βγρ s “ “ ´ ε αβγ r µν B ρ s a αβγ ` B α ε αβγ r µν a ρ s βγ , (B.26) ε r σµαβγ B α a βγσ η ν s ρ “ “ ε αβγσ r µ B α a βγσ η ν s ρ ´ B ρ ε βγσµν a βγσ ` ε αβγµν B α a βγρ , (B.27)one obtains2 B r µ h ν s ρ ` ε µνλστ B λ p C στ ρ ´ c στ ρ q ` p H µνρ ´ ε µνραβ F αβ q ` B y p C µν,ρ ´ A r µ η ν s ρ q´ B ρ ˆ u µν ` ε µναβγ a αβγ ˙ “ . (B.28)We recover the 6D equation (3.19) after the following redefinitions u µν Ñ u µν ` ε µναβγ a αβγ ,B µν Ñ B µν ´ b µν , C µν,ρ Ñ C µν,ρ ´ c µν,ρ . (B.29)One can check that these redefinitions are consistent with the expression of ψ in (B.9). Finally,derivative of (3.19) and (3.21) give rise to the original 6D equations of motion (2.11) as discussedin section 3.2 above. 28 eferences [1] C. Hull, “Strongly coupled gravity and duality,” Nucl.Phys.
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