Toward Exotic 6D Supergravities

We investigate exotic supergravity theories in 6D with maximal (4,0) and (3,1) supersymmetry, which were conjectured by C. Hull to exist and to describe strong coupling limits of ${\cal N}=8$ theories in 5D. These theories involve exotic gauge fields with non-standard Young tableaux representations, subject to (self-)duality constraints. We give novel actions in a 5+1 split of coordinates whose field equations reproduce those of the free bosonic (4,0) and (3,1) theory, respectively, including the (self-)duality relations. Evidence is presented for a master exceptional field theory formulation with an extended section constraint that, depending on the solution, produces the (4,0), (3,1) or the conventional (2,2) theory. We comment on the possible construction of a fully non-linear master exceptional field theory.


Introduction
Among the surprising features of string/M-theory is the possible existence of exotic superconformal field and gravity theories in six dimensions, which display generalizations of electricmagnetic duality. Specifically, the supermultiplets of these theories are such that the corresponding fields must be subject to (self-)duality constraints, some of which involve exotic Young 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 " 8 theories in five dimensions that are given by six-dimensional theories with chiral N " p4, 0q and N " p3, 1q supersymmetry, respectively. Such theories must be exotic or non-geometric since they feature mixed symmetry tensors 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][4][5] A possible window into these somewhat mysterious structures is offered by a Kaluza-Klein perspective from five dimensions. The supermultiplets of five-dimensional (5D) theories with maximal supersymmetry (32 real supercharges) were classified by Strathdee [6] and further clarified by Hull in [7]. The 5D superalgebra reads tQ a α , Q b β u " Ω ab pΓ µ Cq αβ P µ`Cαβ pZ ab`Ωab Kq , (1.1) where α, β, . . . " 1, . . . , 4 are the space-time spinor indices and a, b, . . . " 1, . . . , 8 are USpp8q 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 describe the possible Kaluza-Klein towers that any six-dimensional (6D) theory with maximal supersymmetry displays when compactified on a circle. For the conventional maximal 6D supergravity, which features N " p2, 2q supersymmetry, the massive Kaluza-Klein states do not carry the singlet central charge K. Instead, they carry a particular central charge Z ab transforming as a singlet under the six-dimensional R-symmetry group USpp4qˆUSpp4q . In contrast, the massive multiplets of the exotic theories carry non-vanishing singlet charge K (together with nonvanishing Z ab , singlet under USpp4qˆUSpp4q in the case of the N " p3, 1q multiplets) [7]. This points to a unifying framework in the spirit of exceptional field theories [8][9][10][11] which we will elaborate 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 group E 6p6q of 5D maximal supergravity, thanks to extended coordinates in the 27 representation of this group, which are added to the five coordinates of 5D supergravity. The resulting theory is thus based on a p5`27q-dimensional spacetime split, in which the 27 coordinates are subject to an E 6p6q covariant 'section constraint' restricting them to a suitable physical subspace, from which the complete (untruncated) 11D supergravity can be reconstructed, albeit in a Kaluza-Klein type formulation with a 5`6 split of coordinates. (Equivalently, one may think of this as coupling the infinite towers of massive Kaluza-Klein multiplets to 5D supergravity, which reconstructs the complete 11D supergravity.) A different physical section reproduces the IIB theory [10,12]. From a higher-dimensional perspective, the extra coordinates can be thought of as accounting for the possible brane windings, which in turn are related to the 27 central charges of the supersymmetry algebra (1.1). The structure of the exotic supermultiplets then suggests an inclusion of their couplings within a suitable extension of this framework.
In this paper we will present actions for (the bosonic sectors of) the free exotic 6D theories that generalize the (linearized) E 6p6q 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 most enticing outcomes of our investigation we find evidence for a master exceptional field theory formulation in which the conventional N " p2, 2q theory as well as the N " p4, 0q and N " p3, 1q theory are obtained through different solutions of an extended section constraint of the form where M, N " 1, . . . , 27, are fundamental E 6p6q indices, d M N K denotes the E 6p6q invariant fully symmetric tensor, and B ‚ is the derivative dual to the exotic coordinate. Moreover, ∆ M N denotes the (constant) background part of the generalized metric M M N encoding all scalar fields. The first term in equation (1.2) defines the section constraint of E 6p6q exceptional field theory, whose solutions restrict to the standard D " 11 and IIB sections. The second term encodes the extension of the constraint allowing for two more solutions corresponding to N " p4, 0q and N " p3, 1q, respectively. More precisely, we recover the N " p4, 0q exotic theory by dropping all dependence on the 27 standard coordinates, and keeping only the dependence on the exotic coordinate. The N " p3, 1q model in turn is recovered by superposing this coordinate with the F 4p4q singlet under 27 Ñ 26`1 among the 27 coordinates of ExFT. While we give several independent pieces of evidence for the existence of this master formulation (some of which entail highly non-trivial numerical agreement), we also point out some gaps of the master formulation as understood so far. This implies that the complete non-linear theory requires new ingredients, not the least of which is a section constraint that makes sense for the non-linear theory and that reduces to (1.2) in the appropriate limit.
As a technical result, we present novel actions for the bosonic sectors of the N " p3, 1q and the N " p4, 0q model that are based on a 5+1 split of the six-dimensional space-time, sacrificing manifest 6D Poincaré invariance. In the spirit of ExFT, these are two-derivative actions which upon dimensional reduction to five dimensions reduce to the same action of linearized maximal 5D supergravity. All dual fields, in particular the entire dual graviton sector, only appear under derivative along the sixth dimension. The full field equations obtained by variation combine the second order Fierz-Pauli equations with first-order duality equations defining the dual graviton sector. Actions for selfdual fields based on a 5+1 split of spacetime date back to [13] with the description of selfdual 6D tensor fields. More recently, actions for the N " p3, 1q and N " p4, 0q models have been constructed in [14][15][16], based on the prepotential formalism developed in [17] in the context of linearized gravity. Introduction of prepotentials for the gauge fields adapted to their self-duality properties allows for the construction of an action of fourth order in spatial derivatives. Our construction is closer in spirit to the original construction of [13], albeit dual in a sense discussed in more detail in appendix A. It provides a novel mechanism for describing self-dual exotic tensor fields.
The rest of this paper is organized as follows. In sec. 2 we review the bosonic sector of the N " p2, 2q, N " p3, 1q and N " p4, 0q theories at the level of the equations motion, which are manifestly 6D Lorentz invariant. In order to find actions for these theories we abandon manifest Lorentz invariance by performing a 5`1 split of coordinates in sec. 3 for each of these models. In sec. 4 we then present, as one of our main technical results, actions whose secondorder Euler-Lagrange equations can be integrated in order to reproduce the correct dynamics of the three theories. In sec. 5 we present the master formulation as currently understood, highlight its successes, which strike us as significant, but also discuss the structural problems that remain. We close with a brief outlook.
Note added: While finalizing the present paper the preprint [18] appeared, which also investigates exotic theories in 6D.

The N " p2, 2q Model
Let us start from the N " p2, 2q multiplet corresponding to maximal supergravity in six dimensions. Its bosonic field content comprises a metric, 25 scalar fields, 16 vectors, and 5 two-forms. The full non-linear theory has been constructed in [19] with the scalar fields parametrizing an SOp5, 5q{ pSOp5qˆSOp5qq coset space. For the purpose of this paper, we will only 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 hμν. With the linearized Riemann tensor given by for Fμν i " 2 B rμ Aν s i , while scalar couplings take the form The couplings (2.6) and (2.7) break the global SOp5, 5q symmetry of the non-linear theory down to its compact part SOp5qˆSOp5q, as expected for the free theory. Finally, the two-forms Bμν p couple with a standard kinetic term Hμνρ q Hμνρ q , q " 1, . . . , 5 , (2.8) for Hμνρ q " 3 B rμ Bνρ s q . For the following it will be convenient to combine these fields together with their magnetic duals into a set of 10 two-forms Bμν a , satisfying first order (anti-)selfduality field equations with the SOp5, 5q invariant constant tensor η ab . Equations (2.9) amount to a description of these degrees of freedom in terms of 5 selfdual and 5 anti-selfdual two forms.
In addition to the exotic tensor field, the bosonic field content of the N " p3, 1q multiplet (2.2b) contains 14 vectors, 12 selfdual 2-forms and 28 scalar fields. The dynamics of vector and scalar fields can be captured by standard Lagrangians (2.6) and (2.7) (with different range of internal indices). The selfdual 2-forms Bμν a obey a selfduality equation similar to (2.9) Hμνρ a " 1 6 εμνρσκλ Hσκλ a , a " 1, . . . , 12 , (2.14) contrary to (2.9), no indefinite tensor η ab appears in this equation, all forms are selfdual. 1 As a consequence there is no standard action principle for these field equations, they can however be derived from an action with non-manifest Lorentz invariance [13] or upon coupling to the auxiliary PST scalar [21]. 2 The free N " p3, 1q theory is invariant under the R-symmetry group USpp6qˆUSpp2q. The (yet elusive) interacting theory is conjectured to exhibit a global F 4p4q symmetry with in particular the 28 scalars parametrizing the coset space F 4p4q { pUSpp6qˆUSpp2qq [1].
The free N " p4, 0q theory is invariant under the R-symmetry group USpp8q. The (yet elusive) interacting theory is conjectured to exhibit a global E 6p6q symmetry with in particular the 42 scalars parametrizing the coset space E 6p6q USpp8q [1].

5`1 Split
Upon dimensional reduction to D " 5 dimensions, the three models discussed in the previous section all reduce to the same theory: the free limit of maximal D " 5 supergravity [1,2]. The bosonic sector of this theory carries a spin-2 field and 27 vector fields together with 42 scalar fields. In particular, the exotic tensor fields of the N " p3, 1q and the N " p4, 0q model after dimensional reduction carry the D " 5 dual graviton and double dual graviton, respectively. Within the free theory, these fields can be dualized into the standard Pauli-Fierz field [1,24,25], and do not represent independent degrees of freedom. In order to make the equivalence explicit, the fields of D " 5 supergravity (together with their on-shell duals) have to be properly identified among the various components of the D " 6 fields.
In this section, we discuss for every of the three models the reorganization of the D " 6 fields which allows their identification after reduction to five dimensions. However, throughout this section (and this paper) we keep the full dependence of all fields on six space-time coordinates. More precisely, we break 6-dimensional Poincaré invariance down to 5`1 and perform a standard Kaluza-Klein decomposition on the six-dimensional fields without dropping the dependence on the 6th coordinate. We then rearrange the equations such that they take the form of the five-dimensional (free) supergravity equations however sourced by derivatives of matter fields along the sixth direction. The resulting reformulation of the six-dimensional models casts their dynamics into a common framework -which ultimately allows us to construct uniform actions for the three models.
For the purpose of this paper, we choose the 5`1 coordinate split by singling out one of the spatial coordinates. Of course, an analogous construction can be performed with a split along the time-like coordinate which may be of interest for example in a Hamiltonian context.

The N " p2, 2q Model
With the coordinate split (3.1), we parametrize the graviton of the N " p2, 2q theory as which is the linearized form of the standard Kaluza-Klein reduction ansatz. Recall that all fields still depend on 6 coordinates. Working out the Lagrangian (2.5) in this parametrization gives rise to its expression 3) up to total derivatives. As an illustration of the above discussion let us note the explicit form of the equations for the five-dimensional spin-2 field in terms of the linearized Einstein tensor The form of (3.4) shows that upon dimensional reduction to D " 5 dimensions, these equations reproduce the (linearized) five-dimensional Einstein field equations. In contrast, the coordinate dependence along the sixth coordinate induces a non-trivial gauge structure via covariant derivatives (3.5) and non-vanishing source terms in (3.4). This is very much in the spirit of the reformulation of higher-dimensional supergravities as exceptional field theories (ExFTs).

Indeed, equation (3.4) can be equivalently obtained upon linearizing the corresponding E 6p6q
ExFT [10,11] upon proper identification of the coordinate y among the 27 internal coordinates on which this ExFT is based.
Let us also note, that the Lagrangian (3.3) can be put to the more compact form with the linearized (and covariantized) anholonomity objects The remaining part of the six-dimensional degrees of freedom described by (3.6) are captured by 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) descending from six-dimensional spin-2 gauge transformations upon decomposition of the six-dimensional gauge parameter as tξμu " tξ µ , λu .
In a similar way, the six-dimensional Maxwell and Klein-Gordon Lagrangians (2.6) and (2.7) take the form respectively, after splitting tAμ i u " tA µ i , φ i u, and with abelian F µν i " 2 B rµ A νs i , giving rise to modified Maxwell and Klein-Gordon equations for their components. The rewriting of the tensor 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 fields into tBμν a u " tB µν a , B µ5 a " A µ a u where we use conventions ε µνρκλ5 " ε µνρκλ , and abelian field strengths H µνρ a " 3 B rµ B νρs a , and F µν a " 2 B rµ A νs a , respectively. These equations can be integrated to a Lagrangian 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, this form of the Lagrangian allows for the most uniform treatment of the different six-dimensional models. After dimensional reduction to D " 5 dimensions, it simply reduces to a collection of Maxwell terms, such that all degrees of freedom of (2.8) are described as massless vector fields in five dimensions. In presence of the sixth dimension, the Lagrangian (3.11) gives rise to modified Maxwell equations while variation w.r.t. the tensor fields B µν a induces equations (3.10) (under B y derivative) as duality equations relating vector and tensor fields.
In summary, the D " 6 N " p2, 2q, model can be equivalently reformulated in terms of a Lagrangian given by the sum of (3.6), (3.9), and (3.11). Upon dimensional reduction to five dimensions, i.e. setting B y Ñ 0, and rescaling of the scalar fields, this Lagrangian reduces to withΩ µνρ " B rµ h νsρ , and where we have combined the various vector and scalar fields into joint objects The Lagrangian (3.12) is the free limit of D " 5 maximal supergravity [26]. In the interacting theory, the fields (3.13) transform in the fundamental and a non-linear representation of its global symmetry group E 6p6q .

The N " p3, 1q Model
We now turn to the N " p3, 1q model. Its most characteristic element is the mixed-symmetry tensor field Cμν ,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 with symmetric h µν " h νµ , antisymmetric B µν "´B νµ , and a (2,1) tensor C µν,ρ . After dimensional reduction to five dimensions, the fields h µν and A µ satisfy the linearized Einstein and Maxwell equations while the fields C µν,ρ and B µν describe their on-shell duals, together accounting for the 8 degrees of freedom of the six-dimensional tensor field. Explicitly, in the parametrization (3.14), the six-dimensional selfduality equations (2.11) split into two equations and with abelian field strengths F µν " 2 B rµ A νs , H µνρ " 3 B rµ B νρs , and the linearized Riemann tensor R µν,ρσ defined as in (2.4) however for the field h µν . Contraction of (3.16) gives rise to an equation where G µν denotes the linearized Einstein tensor defined as in (3.5), however with covariant derivatives now given by i.e. with a different value of the coupling constant (which could be absorbed into rescaling the vector field). Equation (3.17) confirms that upon reduction to five dimensions (B y Ñ 0), the field h µν satisfies the linearized Einstein equations. As in the N " p2, 2q model, the coordinate dependence along the sixth coordinate induces a nontrivial gauge structure (3.18) together with non-vanishing source terms in (3.17) -which differ from those of (3.4) illustrating the inequivalence of the N " p2, 2q and the N " p3, 1q model before dimensional reduction.
The full field equation (3.16) takes the form of a vanishing curl (in rρσs) and can locally be integrated into the first order equation 3 with an antisymmetric tensor u µν "´u νµ . Combining this equation with the field equation (3.15) implies that which can be further integrated into another first order duality equation up to a function f µν pyq that can be absorbed into u µν . Eventually, we can use (3.21) to bring (3.19) into the form To sum up, we have cast the original second order field equations (2.11) of the six-dimensional mixed-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 introduction of an additional field u µν . Upon reduction to five dimensions, these equations constitute the 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 from the D " 6 gauge transformations (2.13). Parametrizing the six-dimensional gauge parameters as their action on the various components of (3.14) is derived as With the field u µν defined by equation (3.19), its gauge variation is found by integrating up the variation of (3.19) and takes the form For later use, let us note that contraction of (3.22) with the fully antisymmetric ε-tensor yields while contraction gives rise to This gives rise to an equivalent rewriting of (3.22) as (3.28) Let us further note that taking the divergence of (3.21) yields the Maxwell type equation where we have used (3.27) in order to eliminate the divergence of u µν .
For the remaining fields of the N " p3, 1q model, the 5+1 Kaluza-Klein split is achieved just as for the N " p2, 2q model discussed above. The six-dimensional field equations of the 14 vector fields and 28 scalar fields take the form obtained from variation of Lagrangians of the form (3.9), respectively. The field equations of the 12 selfdual forms take the form after splitting the two-forms according to tBμν a u " tB µν a , B µ5 a " A µ a u . The equations may be integrated up to an action in precise analogy with (3.11), c.f. the discussion in appendix A.2.

The N " p4, 0q Model
In this model, the exotic graviton is given by the rank four tensor (2.15) whose dynamics is defined by the selfduality equations (2.16) for its second-order curvature. According to the split of coordinates (3.1), we parametrize the various components of this field as tTμν ,ρσ u " tT µν,ρσ ; T µν,ρ5 " C µν,ρ ; T µ5,ν5 " h µν u . (3.31) After dimensional reduction to five dimensions, these fields describe the graviton, dual graviton and double dual graviton, respectively. Explicitly, in this parametrization the six-dimensional field equations (2.16) split into two equations with the linearized Riemann tensor R µν,ρσ defined as in (2.4) for the field h µν . The second equation (3.33) has the form of a curl in rρστ s and can be integrated up into 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 which in turn is a curl in rρσs and can be integrated up into up to an antisymmetric field u µν "´u νµ . As for the N " p3, 1q model, we have obtained an 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 obtained by derivation. After reduction to five dimensions, equations (3.34) and (3.36) describe the duality relations between graviton and dual graviton and between dual graviton and double dual graviton, respectively. In particular, equation (3.36) differs from equation (3.22) in the N " p3, 1q model only if fields depend on the sixth coordinate.
Let us finally note that from (3.32) and (3.35), we may obtain the modified Einstein equa-tionsG with the linearized Einstein tensorG µν defined as which differs from the previous models by the absence of covariant derivatives, c.f. (3.5).
For the remaining fields of the N " p4, 0q model, the 5+1 Kaluza-Klein split is achieved just as for the previous models discussed above. The field equations of the 42 scalar fields are obtained from variation of a Lagrangian of the form L φ in (3.9). The field equations of the 27 selfdual forms take the form of (3.30) above, again after splitting the two-forms according to tBμν a u " tB µν a , B µ5 a " A µ a u .
4 Actions for (free) exotic graviton fields In the above, we have reformulated the dynamics of the six-dimensional exotic tensor fields in terms of first order differential equations upon breaking six-dimensional Poincaré invariance according to the split (3.1), and introducing some additional tensor fields. As a key property of the resulting equations, we have put the dynamics of the different models into a form which reduces to the same equations after dimensional reduction B y Ñ 0. E.g. all three models feature linearized 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 " p3, 1q and the N " p4, 0q model which after setting B y Ñ 0 both reduce to the Lagrangian (3.12) of linearized D " 5 maximal supergravity.
This construction follows the toy model of D " 6 selfdual tensor fields whose dynamics can be described by a Lagrangian (3.11) after a Kaluza-Klein (5`1) decomposition tBμνu " tB µν , B µ5 " A µ u of the six-dimensional tensor field. After dimensional reduction to five dimensions, the 3 degrees of freedom of the selfdual tensor field are described as a massless vector with the standard Maxwell Lagrangian to which (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. the tensor field yields the duality equation relating A µ and B µν , which is of first order in the derivatives B µ and appears under a global B y derivative. Combining these two equations one may 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.

Action for the N " p3, 1q 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 Cμν ,ρ in the N " p3, 1q model, can be derived from the Lagrangian with p Ω µνρ " B rµ h νsρ´By A rµ η νsρ`1 2 B y p C µν,ρ , p C µν,ρ " C µν,ρ`εµνρστ u στ , for h µν together with a free Maxwell Lagrangian for A µ ; the dual fields p C µν,ρ and B µν drop out 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 an overall derivative B y . Together, one recovers the full six-dimensional dynamics. Details of the equivalence are presented in appendix B.
The bosonic Lagrangian for the full N " p3, 1q 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 with indices ranging along i " 1, . . . , 14 , α " 1, . . . , 28 , a " 1, . . . , 12 . (4.5) After dimensional reduction to five dimensions (and rescaling of the vector field A µ ), this Lagrangian coincides with the Lagrangian (3.12) of linearized maximal supergravity. The Lagrangian (4.4) describes the full six-dimensional theory, with the field content of five-dimensional maximal supergravity enhanced by the field p C µν,ρ . D " 6 Poincaré invariance is no longer manifest although it can still be realized on the equations of motion.

Action for the N " p4, 0q 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 Tμν ,ρσ in the N " p4, 0q model, can be derived from the Lagrangian with p Ω µνρ " B rµ h νsρ`By p C µν,ρ´1 2 B y C µν,ρ , p C µν,ρ " C µν,ρ`εµνρστ u στ , C µν,ρ " C µν,ρ´vρ,µν`2 v rρ,µνs`2 ε µνρστ u στ . (4.7) After reduction to five dimensions, i.e. at B y Ñ 0, this Lagrangian reduces to the Fierz-Pauli Lagrangian for h µν ; the dual fields p C µν,ρ , C µν,ρ , and T µν,ρσ drop out in this limit. In presence of the sixth dimension, variation of the Lagrangian (4.6) w.r.t. to the dual fields yields the firstorder duality equations (3.34) and (3.36), however under an overall derivative B y . Together, one recovers the full six-dimensional dynamics. The computation works in close analogy with the derivation for the N " p3, 1q model, c.f. appendix B.
Let us spell out the gauge transformations (3.38), (3.39) in terms of the fields (4.7) which allows to confirm gauge invariance of the Lagrangian (4.6).
The bosonic Lagrangian for the full N " p4, 0q 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 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-dimensional theory, 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 " 6 Poincaré invariance is no longer manifest although it can still be realized on the equations of motion.

Progress toward an exceptional master action
In the previous sections, we have constructed Lagrangians (3.6), (4.4), and (4.9), for the three six-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 sixdimensional theories are then described as different extensions of this Lagrangian by terms carrying derivatives along the sixth dimension. In the various matter sectors, these terms ensure covariantization under non-trivial gauge structures and provide sources to the field equations of five-dimensional supergravity.
This reformulation within a common framework is very much in the spirit of exceptional field theories. In that framework, higher-dimensional supergravity theories are reformulated in terms of the field content of a lower-dimensional supergravity keeping the dependence on all coordinates. More precisely, their formulation is based on a split of coordinates into D external and n internal coordinates of which the latter are formally embedded into a fundamental representation R v of the global symmetry group E 11´D,p11´Dq of D-dimensional maximal supergravity. Different embeddings of the internal coordinates into R v then correspond to different higherdimensional origins. Here, we will discuss a similar uniform description of the six-dimensional models based on D " 5 external dimensions which encompasses the three different models upon proper identification of the sixth coordinate within the internal coordinates. As discussed in the introduction this will require an enhancement of the internal coordinates of exceptional field theory by an additional exotic coordinate related to the singlet central charge in the D " 5 supersymmetry algebra.

Linearized ExFT and embedding of the N " p2, 2q model
The theory relevant for our discussion is E 6p6q exceptional field theory (ExFT) [10,11]. Its bosonic field content is given by a graviton g µν together with 27 vector fields A µ M and their dual tensors B µν M , together with 42 scalars parametrizing the internal metric M M N " pV V T q M N with V a representative of the coset space E 6p6q {USpp8q. Fields depend on 5 external and 27 internal coordinates with the latter transforming in the fundamental 27 of E 6p6q and with internal coordinate dependence of the fields restricted by the section constraint [9] with the two differential operators acting on any couple of fields and gauge parameters of the theory. The tensor d KM N denotes the cubic totally symmetric E 6p6q invariant tensor, which we normalize as d M N P d M N Q " δ Q P . The section condition (5.1) admits two inequivalent solutions [10] which reduce the internal coordinate dependence of all fields to the 6 internal coordinates from D " 11 supergravity, or 5 internal coordinates from IIB supergravity, respectively. For details of the ExFT Lagrangian we refer to [10,11]. Here, we spell out its 'free' limit, obtained by linearizing the full theory according to with indices M, N raised and lowered by ∆ M N and its inverse, and with the various elements of (5.3) given by The Lagrangian we have presented above for the six-dimensional N " p2, 2q model naturally fits into this framework. This does not come as a surprise since the six-dimensional model is nothing but linearized maximal supergravity known to be described by E 6p6q ExFT upon proper selection of the sixth coordinate among the internal B M . This choice is uniquely fixed by the requirement that the resulting theory exhibits the global SOp5, 5q symmetry group of maximal six-dimensional supergravity, thus breaking in terms of SOp5, 5q Γ-matrices and its invariant tensor η ab of signature p5, 5q, showing that the section constraint (5.1) is trivially satisfied is B i " 0 " B a . Putting this together with the linearized ExFT Lagrangian (5.3), and splitting fields as we arrive at which precisely produces the sum of Lagrangians (3.6), (3.9), (3.11), after proper rescaling of the singlet scalar field φ . The non-trivial checks of this coincidence include all the coefficients in the various connection terms, as well as in the Stückelberg-type couplings between vector and tensor fields, and the coefficients in front of the various B y φB y φ terms in the last line. Again, this is not a surprise but a consequence of the proven equivalence of ExFT with higherdimensional maximal supergravity. Note that although the free theory only exhibits a compact USpp4qˆUSpp4q global symmetry, the couplings exhibited in (5.8) are far more constrained than allowed by this symmetry and witness the underlying E 6p6q structure broken to SOp5, 5q according to (5.5), (5.6).
The ExFT Lagrangian is to a large extent determined by invariance under generalized internal diffeomorphisms acting with a gauge parameter Λ M in the 27. After linearization (5.2) these diffeomorphisms act as and one can show invariance of the linearized Lagrangian (5.8), provided the section constraint (5.1) is satisfied.

Beyond standard
ExFT: embedding of the N " p3, 1q and p4, 0q couplings As we have discussed in the introduction, the charges carried by the massive BPS multiplets in the reduction of the N " p3, 1q and the N " p4, 0q model, respectively, suggest that an inclusion of these models into the framework of ExFT necessitates an extension of the space of 27 internal coordinates by an additional exotic coordinate corresponding to the singlet central charge [7]. Denoting derivatives along this coordinate by B ‚ , this would amount to a relaxation of the standard section constraint (5.1) to a constraint of the form which at the present stage only makes sense in the linearized theory where ∆ KM is a constant background tensor. Apart from the standard ExFT solutions of this constraint, which allow the embedding of the N " p2, 2q model as described above, the extended section constraint also allows for two exotic solutions p3, 1q : B p3,1q corresponding to the two exotic six-dimensional models in precise correspondence with the central charges carried by the corresponding BPS multiplets [7]. While the (4,0) solution trivially solves the constraint (5.10), the N " p3, 1q solution is based on the decomposition with the F 4p4q invariant symmetric tensor η AB of signature p14, 12q, and the symmetric invariant tensor d ABC satisfying This shows explicitly how the (3,1) assignment of (5.12) also provides a solution to the extended section 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), the Lagrangian remains manifestly invariant without any use of the section constraint. Explicit variation of the potential term L pot under linearized diffeomorphisms (5.9) on the other hand yields (up to total derivatives) which consistently vanishes modulo the standard section constraint (5.1). For the weaker constraint (5.10), this variation no longer vanishes and may be recast in the following form after repeated use of (5.10) and further manipulation of the expressions. In order to compensate for this variation let us first note that there is no possible covariant extension of the transformation rules (5.10) by terms carrying B ‚ Λ M , such that invariance can only be restored by extending the potential. A possible such extension is given by and it is straightforward to verify that the variation of the additional terms in (5.18) precisely cancels the contributions in (5.17), such that For the exotic solutions of the section constraint, the B ‚ φ M N B ‚ φ M N terms in (5.18) give rise to additional contributions of the type B y φB y φ in the Lagrangian. Collecting all such terms in (5.18) for the two exotic solutions (5.12) yields 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 generalized diffeomorphism invariance precisely implies the correct scalar couplings in the Lagrangians of the exotic models. In addition, the B ‚ hB ‚ 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 in order to reproduce the correct Lagrangian of the N " p3, 1q model (4.2). 4 We may continue the symmetry analysis for the tensor gauge transformations given by a gauge parameter Λ µ M in standard ExFT. For these transformations there is a natural extension of the standard ExFT transformation rules in presence of the exotic coordinate and exotic fields as Computing the action of these transformations on the connection featuring in the covariant scalar derivatives D µ φ M N in (5.4), we obtain after some manipulation 5 The resulting expression precisely vanishes with the modified section constraint (5.10). This shows the necessity of the B ‚ Λ µ M terms in (5.21) in order to maintain gauge invariance of the kinetic term D µ φ M N D µ φ M N in presence of the relaxed section constraint. It is straightforward to verify that these additional terms in the transformation induce a modification of the gauge invariant vector field strengths to as well an extension of the topological term, such that the combined vector-tensor couplings take the form 24) and are invariant under these gauge transformations. Let us work out the effect of these modifications for the exotic solutions of the section constraint. With the kinetic scalar term unchanged, the resulting couplings are directly inferred from evaluating the covariant derivatives (5.4) for the d-symbol (5.14), giving rise to In contrast, these terms appear in conflict with embedding the spin-2 sector of the N " p4, 0q model as they survive under the (4,0) solution in (5.12) but should be absent in the final Lagrangian (4.6). We come back to this in section 5.3. 5 A useful identity for this computation is given by generalizing equations (2.12), (2.13) of [11].
To summarize, in the scalar, vector and tensor sector, we have constructed an extension of the ExFT Lagrangian (at the linearized level), given by 27) which is invariant under the gauge transformations (5.9), (5.21) modulo the relaxed section constraint (5.10). The weaker section constraint necessitates a numer of additional contributions to the Lagrangian (and transformation rules) which precisely reproduce the explicit couplings found in the Lagrangians of the exotic models (4.4), (4.9) constructed above. It is remarkable that this match confirms the couplings that have been determined from an underlying noncompact E 6p6q and F 4p4q structure, respectively, despite the fact that the free theory only exhibits invariance under the compact R-symmetry subgroup USpp2N`qˆUSpp2N´q which might in principle allow for much more general couplings. We take this as evidence for the conjectured E 6p6q and F 4p4q invariance of the putative interacting theories [1].

The spin-2 sector
The above findings have revealed a very intriguing common structure of the couplings in the scalar, vector and tensor sectors of the different models which can be consistently embedded into an 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. Extrapolation of the Lagrangian of the N " p4, 0q model (4.6) suggests an extension of the standard ExFT Lagrangian by couplings carrying B ‚ derivatives and the dual graviton fields as By construction, this reproduces the N " p2, 2q and the N " p4, 0q models upon choosing the corresponding solutions of the section constraint. It remains unclear however, how the spin-2 sector of the N " p3, 1q 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 remain present upon selecting the (3,1) solution (5.12) of the section constraint, poses a challenge for recovering the Lagrangian (4.4) of the N " p3, 1q model. The structure of the gauge transformations 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 " p3, 1q model -in order to remove this field. Another apparent problem in the spin-2 sector is the lacking reconciliation between the B ‚ hB ‚ h terms from (5.18) and the B ‚ T B ‚ T terms of (5.28) which mutually violate the correct limits to the exotic models. Resolution of this problem may require to implement algebraic relations between the Pauli-Fierz h µν field and the double dual graviton [2] (see also [28]).

Conclusions and Outlook
In this paper we have taken the first step in constructing action principles for exotic supergravity theories in 6D by giving such actions for the free bosonic part. These actions show already intriguing new features such as the simultaneous appearance of (linearized) diffeomorphisms and dual diffeomorphisms, which are realized on exotic Young tableaux fields as well as more conventional gravity fields. Our formulation abandons manifest 6D Lorentz invariance, as expected to be necessary on general grounds, by being based on a 5`1 split of coordinates.
Remarkably, the field equations implied by our actions can be integrated to reconstruct the correct dynamics of these exotic supergravites. Moreover, we have seen the first glimpses of an exceptional field theory master formulation, in which the conventional N " p2, 2q, as well as the exotic N " p3, 1q and N " p4, 0q models all emerge through different solutions of an extended section constraint, but clearly much more needs to be done. We close with a brief discussion of possible future developments.
First, it remains to exhibit the (maximal) supersymmetries in these non-standard formulations, even just at the free level. We have no doubt that this can be achieved as in exceptional field theory where different supersymmetries (such as type IIB versus type IIA) are realized within a single master formulation. Second, it would be interesting to study possible embeddings into exceptional field theories of higher rank, such as for U-duality groups E 7p7q and E 8p8q , which may illuminate some issues and which can also be done already at linearized level. Finally, the most important outstanding problem is clearly the question whether our formulation can be extended to the non-linear interacting theory. We would like to emphasize that the present formulations seem quite promising in this regard since they feature not only the exotic fields but also the more conventional gravity fields, which come with an action that allows a natural embedding into the full non-linear Einstein-Hilbert action. In turn this suggests that all these fields might become part of a tensor hierarchy that extends to the gravity sector. If so this could quite naturally lend itself to a formulation of non-linear dynamics in terms of a hierarchy of duality relations as in [29].
reproducing the second equation of (A.4). This equation now serves an an integrability equation in order to locally define the vector field A µ via the equation 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 supergravity theories based on the field content of lower-dimensional theories. In particular, it offers actions for theories that do not admit actions in terms of their original variables, such as IIB supergravity, 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 carrying the fields of equation (A.3). The field equations are now given by In particular, equation (A.10) implies the original field equations (A.3) up to some function that does not depend on y: Comparing the divergence of this equation to (A.9), we find that locally the field χ µν can be integrated to in terms of a function b µν , such that the field equations (A.11) can be rewritten aś with the modified two-formB In terms of the fields A µ ,B µν , we thus recover the desired original field equations (A.3). Note finally, 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 .

B 6D field equations from the new Lagrangians
In this appendix, we present in detail how the second-order field equations obtained by variation of the Lagrangian (4.2) for the N " p3, 1q model can be integrated to the first-order field equations (3.21) and (3.22) which in turn imply the original 6D second-order selfduality equations (2.11). For the N " p4, 0q 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).

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 equations derived in the section B.1. Let us first rewrite them in terms of the original fields of the (3,1) model and integrate all the equations under B y by introducing three functions χ µν px µ q, ψ µνρ px µ q and ϕ µν,ρ px µ q which are respectively antisymmetric, antisymmetric and of p2, 1q type, and do not depend on the sixth coordinate. Together, these two equations imply that locally, we can define a 2-form b such that χ µν " 1 2 ε µνρστ B ρ b στ px µ q. (B.14) This 2-form can be absorbed in B (following exactly the same process as in section A.2) such that equations (B.8) reproduces (3.21).
h-C duality Contracting (B.11) with B µ , we can extract both symmetric and antisymmetric parts: