Unified non-metric (1, 0) tensor-Einstein supergravity theories and (4, 0) supergravity in six dimensions

The ultrashort unitary (4, 0) supermultiplet of 6d superconformal algebra OSp(8∗|8) reduces to the CPT-self conjugate supermultiplet of 4d superconformal algebra SU(2, 2|8) that represents the fields of maximal N = 8 supergravity. The graviton in the (4, 0) multiplet is described by a mixed tensor gauge field which can not be identified with the standard metric in 6d. Furthermore the (4, 0) supermultiplet can be obtained as a double copy of (2, 0) conformal supermultiplet whose interacting theories are non-Lagrangian. It had been suggested that an interacting non-metric (4, 0) supergravity theory might describe the strongly coupled phase of 5d maximal supergravity. In this paper we study the implications of the existence of an interacting non-metric (4, 0) supergravity in 6d. The (4, 0) theory can be truncated to non-metric (1, 0) supergravity coupled to 5,8 and 14 self-dual tensor multiplets that reduce to three of the unified magical supergravity theories in d = 5. This implies that the three infinite families of unified N = 2, 5d Maxwell-Einstein supergravity theories (MESGTs) plus two sporadic ones must have uplifts to unified non-metric (1, 0) tensor Einstein supergravity theories (TESGT) in d = 6. These theories have non-compact global symmetry groups under which all the self-dual tensor fields including the gravitensor transform irreducibly. Four of these theories are uplifts of the magical supergravity theories whose scalar manifolds are symmetric spaces. The scalar manifolds of the other unified theories are not homogeneous spaces. We also discuss the exceptional field theoretic formulations of non-metric unified (1, 0) tensor-Einstein supergravity theories and conclude with speculations concerning the existence of higher dimensional non-metric supergravity theories that reduce to the (4, 0) theory in d = 6.


Introduction
Conformal supergravity theories with local Lagrangians based on the conformal superalgebras SU(2, 2|N ) have long been known to exist for N ≤ 4. It was generally believed that one could not go beyond N = 4 without having higher spins (> 2). In [1] it was shown that the fields of maximal N = 8 supergravity of Cremmer and Julia [2] can be fitted into an ultra short CPT-self-conjugate unitary supermultiplet of N = 8 superconformal algebra SU(2, 2|8) referred to as the doubleton supermultiplet. The corresponding ultra short supermultiplet of SU(2, 2|4) is the Yang-Mills supermultiplet in d = 4 [3]. The N = 4 Yang-Mills theory of doubleton supermultiplets of SU(2, 2|4) is conformally invariant both classically and quantum mechanically. This led the authors of [1] to pose the question whether a conformal supergravity theory based on the doubleton supermultiplet of SU(2, 2|8) exists which is closely related to the maximal N = 8 supergravity theory of Cremmer, Julia and Scherk. Since the latter theory is not conformally invariant any superconformal theory based on the doubleton supermultiplet of SU(2, 2|8) must be unconventional or exotic.
The superalgebra SU(2, 2|8) was used to classify the counterterms in maximal supergravity in [4]. Furthermore, it is known that amplitudes of maximal supergravity are SU (8) covariant even though the Lagrangian does not have SU (8) symmetry. This and above mentioned results provided part of the motivation for the work of Chiodaroli, Roiban and the current author [5] who studied the connection between maximal supergravity and superconformal symmetry in all dimensions that admit simple superconformal algebras as classified by Nahm [6]. They showed that the six dimensional counterpart of the doubleton supermultiplet of SU(2, 2|8) is the (4, 0) supermultiplet of the superconformal algebra OSp(8 * |8) with the even subalgebra SO * (8) ⊕ USp (8), where USp(8) is the R-symmetry group, which reduces to the CPT-self-conjugate doubleton supermultiplet of SU(2, 2|8) under dimensional reduction. They also showed that the (4, 0) theory can be obtained as a double copy of the (2, 0) theory based on the CPT-self-conjugate doubleton supermultiplet of OSp(8 * |4). 1 The (2, 0) supermultiplet first appeared in the work of [8] who constructed the entire Kaluza-Klein spectrum of 11-dimensional supergravity over AdS 7 × S 4 by simple tensoring of the (2, 0) doubleton supermultiplet. In the mid 1990s interacting (2, 0) supersymmetric theories in 6d were investigated within the framework of M/Superstring theory [9][10][11][12]. In particular, Seiberg pointed out the existence of four infinite series of new quantum theories with super-Poincare symmetry in six dimensions, which are not local quantum field theories [12]. Later an interacting (2, 0) superconformal theory was proposed by Maldacena as being dual to M-theory on AdS 7 × S 4 [13].
The (4, 0) supermultiplet was studied earlier by Hull using the formalism of double gravitons whose equivalence to the (4, 0) supermultiplet obtained using the twistorial oscillators was shown in [5]. Hull argued that an interacting (4, 0) theory in d = 6 might arise as the effective theory of the strongly coupled phase of five dimensional maximal supergravity when one of the dimensions decompactifies [14][15][16]. On the other hand the interacting (2, 0) JHEP06(2021)081 theory in six dimensions is believed to describe the strong coupling limit of 5d maximal super Yang-Mills theory. Since the maximal supergravity can be obtained as double copy of maximal super Yang-Mills theory in 5d these two proposals are consistent with the result that (4, 0) theory can also be obtained as double copy of (2, 0) theory in 6d [5,7]. More recently, the action for the free (4, 0) theory was written down by Henneaux, Lekeu and Leonard using the formalism of prepotentials in [17] based on their earlier work on (2,2) mixed chiral tensor describing the graviton [18]. The most unorthodox property of the (4, 0) doubleton supermultiplet of OSp(8 * |8) is the fact that the field strength of the graviton does not arise from a metric and hence the corresponding theory in 6d is sometimes referred to as non-metric , exotic or generalized supergravity. However under dimensional reduction it reduces to the standard maximal supergravity in five and four dimensions.
Independently of the work on maximal supergravity, five dimensional N = 2 supergravity theories coupled to vector multiplets (MESGT) were constructed in [19][20][21] and their gaugings were studied in [22][23][24][25][26][27]. Among these MESGTs four are very special in the sense that they are unified theories with symmetric scalar manifolds G/H such that G is a symmetry of the Lagrangian. They were called magical supergravity theories since their symmetry groups in five , four and three dimensions coincide with the symmetry groups of the famous Magic Square of Freudenthal, Rosenfeld and Tits [19]. Later it was shown that there exist three infinite families of unified MESGTs and two isolated ones [28]. Three of the magical supergravities belong to the three infinite families. The scalar manifolds of unified MESGTs outside the magical ones are not homogeneous. One infinite family of unified MESGTs can be gauged to obtain an infinite family of unified Yang-Mills Einstein supergravity theories in d = 5 with the gauge group SU(N, 1) [28].
In this paper we study some of the implications of the existence of an interacting 6d, non-metric (4, 0) supergravity theory. We show that the (4, 0) supergravity can be truncated consistently to non-metric (1, 0) supergravity coupled to 14, 8 and 5 self-dual tensor multiplets such that the resulting non-metric tensor-Einstein supergravity theories are unified theories in the sense that all the tensor fields including the gravitensor transform irreducibly under a simple global symmetry group. This in turn implies that all the three infinite families of unified 5d MESGTs as well as the two sporadic ones must also admit uplifts to non-metric unified (1,0) tensor-Einstein supergravity theories in d = 6. We conclude with some speculations about the possible extensions of the non-metric supergravity theories to higher dimensions with exotic spacetime signatures and the role of generalized superconformal algebras of these spacetimes.
The plan of the paper is as follows. In section 2 we review the 5d , N = 2 Maxwell-Einstein supergravity theories and their gaugings. Section 3 reviews the truncations of 5d, N = 8 supergravity to three of the magical supergravity and the symmetries of octonionic magical supergravity which can not be obtained from maximal supergravity. Section 4 reviews the uplifts of magical supergravity theories to six dimensions as Poincare supergravities. In subsection 5.1 we review the on-shell superfield formulation of (4, 0) supermultiplet and the gauge potentials in the "first order formalism" following [5] and give the gauge potential of the graviton field strength in the "second order formalism". In subsection 5.2 we review the exceptional field theoretic formulation of linearized (4, 0) JHEP06(2021)081 supergravity following the recent work of [29]. Subsection 5.3 is devoted to the question whether interacting conformal supergravity theories with SU(2, 2|8) symmetry in d = 4 and OSp(8 * |8) symmetry in d = 6 exist. In section 6 we give the truncations of (4, 0) supergravity to non-metric (3, 0) supergravity, to non-metric (2, 0) supergravity coupled to (2, 0) tensor multiplets and to non-metric (1, 0) supergravity coupled to (1, 0) tensor multiplets. In section 7 we discuss the metric and non-metric (1, 0) magical supergravity theories. Section 8 is devoted general unified non-metric (1, 0) tensor-Einstein supergravity theories in six dimensions and their exceptional field theoretic formulation. In section 9 we speculate about the possible extensions of non-metric supergravity theories to higher dimensions with non-standard space-time signatures. Appendix A reproduces the CPT-self-conjugate doubleton supermultiplet of SU(2, 2|8) [1].
2 Review of 5D, N = 2 Maxwell-Einstein supergravity theories and their gaugings N = 2 MESGTs in five dimensions describes the coupling of N = 2 supergravity to an arbitrary number, n, of vector multiplets. The supergravity multiplet consists of the fünfbein e m µ , two gravitini Ψ i µ (i = 1, 2) and one vector field A µ (the "bare graviphoton"). On the other hand a N = 2 vector multiplet consists of a vector field A µ , two symplectic Majorana spinor fields λ i and one real scalar field ϕ. The fermions in these theories transform as doublets under the R-symmetry group USp(2) R ∼ = SU(2) R while all the bosonic fields are SU(2) R singlets.
Hence the fields of an N = 2 MESGT can be labelled as where we labelled the bare graviphoton as A 0 µ . The indices a, b, . . . and x, y, . . . correspond to the flat and curved indices on the scalar manifold, M, respectively.
The bosonic part of the Lagrangian is given by [20] where e is the determinant of the fünfbein , R is the scalar curvature and F I µν are the field strengths of Abelian vector fields A I µ . The completely symmetric tensor C IJK , with lower indices is constant and determines the corresponding N = 2 MESGT uniquely [20]. The global symmetries of the Lagrangian JHEP06(2021)081 are the same as symmetries of C IJK . The n dimensional scalar manifold can be identified with a hypersurface in an (n + 1) dimensional ambient space with the metric with real variables h I (I = 0, 1, . . . , n) representing the coordinates of the ambient space. The scalar manifold M is simply the hypersurface V(h) = 1 and the metric , • a IJ (ϕ), of the kinetic energy term of vector fields is simply the restriction a IJ to M: The physical requirements of unitarity and positivity of the MESGT restrict the possible C-tensors. The most general C IJK that satisfy these constraints can be brought to the form where C ijk (i, j, k = 1, 2, . . . , n) are completely arbitrary. This is referred to as the canonical basis. Arbitrariness of C ijk implies that for a given number n of vector multiplets, there exist, in general, MESGTs with different scalar manifolds and different global symmetries.

Unified Maxwell-Einstein supergravity theories
Unified Maxwell-Einstein supergravity theories in d = 5 are those theories with a simple global symmetry group under which all the vector fields A I µ , including the graviphoton, form a single irreducible representation. With a combination of supersymmetry and global noncompact symmetry group any field can be transformed into any other field within this class of theories.
Among MESGTs whose scalar manifolds are homogeneous spaces only four are unified theories. They are defined by the four simple Euclidean Jordan algebras J A 3 of degree three defined by 3 × 3 Hermitian matrices over the four division algebras A, namely the real numbers R, complex numbers C, quaternions H and octonions O. The cubic norm defined by the C-tensor in these theories is identified with the cubic norm of the underlying Jordan algebra. They are referred to as magical supergravity theories because of the deep connection between their geometries and the geometries associated with the "magic square" of Freudenthal, Rosenfeld and Tits [30][31][32].
In N = 2 MESGTs defined by Euclidean Jordan algebras , J , of degree three the scalar manifold is a symmetric space of the form where Str 0 (J) and Aut(J) are the reduced structure and automorphism group of J, respectively. 2 Below we list the corresponding scalar manifolds: We should note that for MESGTs defined by Euclidean Jordan algebras of degree three such as the magical theories the C-tensor is an invariant tensor of the isometry group Str 0 (J) of the scalar manifold and we have where the indices I, J, K, . . . are raised by the inverse • a IJ (ϕ) of the metric of kinetic energy term of vector fields.
In addition to four unified MESGTs defined by four simple Euclidean Jordan algebras of degree three there exist three infinite families of unified theories whose scalar manifolds are not homogeneous as was shown in [28]. These three infinite families are defined by Lorentzian Jordan algebras of arbitrary degree. Now (n × n) Hermitian matrices over various division algebras form Euclidean Jordan algebras with the symmetric Jordan product defined as 1/2 the anticommutator. Their automorphism groups are compact groups. Non-compact analogs of these algebras, denoted as J A (q,n−q) , are generated by matrices over various division algebras , A = R, C, H for n ≥ 3 and over O for n ≤ 3, 3 that are Hermitian with respect to a non-Euclidean "metric" η with signature (q, n − q): It was shown in [28] that the structure constants (d-symbols) of traceless elements T I of noncompact Jordan algebras J A (1,N ) with Lorentzian metric η of signature (1, N ) defined as satisfy the unitarity and positivity requirements and can be identified with the C-tensor of a MESGT: The resulting MESGTs are all unified (for N ≥ 2 ) since all the vector fields including the graviphoton transform in a single irrep of the simple automorphism groups of the underlying Jordan algebras Aut(J A (1,N ) ) which are also the symmetry groups of their Lagrangians.

Unified N = 2 Yang-Mills-Einstein supergravity theories in five dimensions
A unified N = 2 Yang-Mills Einstein supergravity (YMESGT) theory is defined as a theory in which all the vector fields including the graviphoton transform in the adjoint representation of a simple non-Abelian subgroup of the global symmetry group that is gauged. Turning off the gauge coupling constant yields a unified MESGT under whose global symmetry group all the vectors transform irreducibly.

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In [28] the complete list of unified N = 2 YMESGTs in d = 5 was given. They are obtained by gauging the global SU(N, 1) symmetry groups of unified MESGTs defined by complex Lorentzian Jordan algebras J C (1,N ) under which all the vector fields transform in the adjoint representation of SU(N, 1). As stated above the MESGT defined by J C (1,3) is equivalent to the MESGT defined by the Euclidean Jordan algebra J H 3 whose global symmetry is SU * (6). Gauging the SU(3, 1) = SO * (6) subgroup of SU * (6) leads to the unique unified 5d YMESGT whose scalar manifold is a symmetric space [23]. Again in [23] it was shown that the dimensionless ratio g 3 κ involving the non-Abelian gauge coupling constant g and the gravitational constant κ must be quantized in the quantum theory by invariance under large gauge transformations. The same argument extends to all unified YMESGTs since where Π 5 stands for the fifth homotopy group.
Pure YMESGTs in d = 5 without tensor or hypermultiplets do not have a scalar potential. By expanding around the base point where a is some real number fixed by the condition d 000 = 1, one can show that the noncompact gauge fields transforming in N ⊕N of U(N ) become massive by eating scalar fields and around this ground state U(1) × SU(N ) remains unbroken with the U(1) gauge field corresponding to the graviphoton. Spin 1/2 fields transforming in the symplectic N ⊕N also become massive and together with massive gauge fields form short BPS multiplets, with the central charge generated by the U(1) factor.

N = 2 Yang-Mills-Einstein supergravity theories coupled to tensor fields
under the SU(N, 1) subgroup of USp(2N, 2) for N ≥ 2. Therefore in gauging the SU(N, 1) subgroup the N (N + 1) non-adjoint vector fields must be dualized to massive tensor fields satisfying odd dimensional self-duality conditions [25].
As for the family of unified MESGTs defined by the real Jordan algebras J R (1,N ) , the vector fields transform in the symmetric tensor representation of SO(N, 1). For even N = JHEP06(2021)081 2n with N > 3 one can gauge the U(n) subgroup of SO(2n, 1) by dualizing the non-adjoint vector fields transforming in the reducible symplectic representation n(n + 1) 2 ⊕ n(n + 1) 2 of U(n) to tensor fields. For odd N = 2n + 1 (N > 3) in gauging the U(n) subgroup of SO(2n + 1, 1) the remaining vector fields in the reducible representation of U(n) must be dualized to tensor fields.
In the MESGT defined by the octonionic Jordan algebra J O (2,1) with the global symmetry group F 4(−20) one can gauge the SU(2, 1) subgroup with the remaining vector fields transforming in the reducible representation of SU(2, 1) dualized to tensor fields.

Magical supergravity theories and maximal supergravity
The magical Maxwell-Einstein supergravity theories defined by the real, complex and quaternionic Jordan algebras J A 3 ( A = R, C, H ) can all be obtained by a consistent truncation of the maximal supergravity in d = 5, 4 and 3 dimensions [19]. The same is true for their 6 dimensional uplifts as Poincare supergravities [33]. The exceptional supergravity defined by the exceptional Jordan algebra J A 3 on the other hand can not be obtained by a truncation of maximal supergravity. In five dimensions the U-duality group of maximnal supergravity is E 6(6) and that of exceptional supergravity is E 6(−26) . They can both be truncated to the N = 2 MESGT defined by the quaternionic Jordan algebra with the U-duality group SU * (6). Maximal supergravity can be gauged in d = 5 with the gauge group SU(3, 1) and 12 tensor fields which admits an N = 2 supersymmetric vacuum with vanishing cosmological constant [34]. Similarly the exceptional supergravity theory can be gauged with the gauge group SU(3, 1) and 12 tensor fields. The common sector of these two gauged supergravity theories is the unique unified N = 2 YMESGT with the gauge group SU(3, 1) and whose scalar manifold is SU * (6)/USp (6).

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Under the maximal compact subgroup USp(6) the above representations of SU * (6) decompose as Under the maximal compact subgroup USp(8) of E 6(6) we have the decompositions On the other hand maximal compact subgroup of E 6(−26) is F 4 under which we have the decompositions Under the USp(6) × USp(2) subgroup the above representations of F 4 decompose as 26 = (14, 1) ⊕ (6, 2) (3.14) The above decompositions show that restricting to the USp(2) invariant subsector the spectra coincide with that of quaternionic magical theory defined by J H 3 . The global symmetry group SU * (6) of the quaternionic magical theory has the subgroup SL(3, C) × SO(2) where SL(3, C) is the global symmetry group of the complex magical MESGT The U(1) C invariant sector of the quaternionic theory corresponds to the consistent truncation to the complex magical theory. Similarly the global symmetry group of the complex magical theory decomposes as and Z 2 invariant subsector describes the consistent truncation to the real magical supergravity defined by J R 3 .

Magical Poincare supergravity theories in six dimensions
Six dimensional magical supergravity theories coupled to hypermultiplets and their gaugings were studied in [33]. Magical supergravities in six dimensions describe the coupling of (1, 0) Poincare supergravity to n T = 2, 3, 5, 9 tensor fields and vector fields in a definite spinor representation of SO(n T , 1). The coupling between vector fields and tensors involve SO(n T , 1) invariant tensors Γ I AB that are the Dirac Γ-matrices for n T = 2, 3, and the Van der Waerden symbols for n T = 5, 9. They satisfy the identities which are simply the Fierz identities for the existence supersymmetric Yang-Mills theories in 3,4,6 and 10 dimensions. These identities follow from the adjoint identity satisfied by the elements of simple Euclidean Jordan algebras of degree three [35]. We reproduce their field contents in table 2.
Since the vector fields transform in a spinor representation which belong to a unique orbit of the isometry group of the scalar manifold one finds that six dimensional magical supergravity theories admit a unique gauge group which is a centrally extended Abelian nilpotent group. For the octonionic magical theory the unique gauge group is the maximal centrally extended Abelian subgroup of F 4(−20) which is the automorphism group of the Lorentzian octonionic Jordan algebra J O (2,1) . For the quaternionic ( complex) magical theory the unique gauge group is the maximal centrally extended Abelian subgroup of USp(4, 2) ( SU(2, 1) ) which is the automorphism group of the Lorentzian quaternionic ( complex) Jordan algebra J H (2,1) (J C (2,1) ) . They satisfy the inclusions

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These results show that semisimple gaugings of the 5d magical supergravity theories do not admit uplifts to six dimensions as standard Lagrangian Poincare supergravities. On the other hand it is known that the 5d , N = 4 super Yang-Mills theory can be obtained from (1, 1) Poincare supersymmetric Yang-Mills theory in six dimensions by dimensional reduction and it is generally believed that it can also be obtained from an interacting (2,0) superconformal field theory. Similarly the N = 2 super Yang-Mills theory in five dimensions can be obtained from (1, 0) supersymmetric Yang-Mills theory or from a (1, 0) superconformal theory of self-dual tensor multiplets in d = 6. The standard Yang-Mills theories in d = 6 involving vector fields are not conformally invariant. The (2, 0) conformal supermultiplet decomposes as a (1, 0) tensor multiplet plus a conformal hypermultiplet in d = 6 Therefore an interacting (2, 0) theories can be viewed as a special family of interacting (1, 0) tensor multiplets coupled to hypermultiplets. Similarly the N = 4 super Yang-Mills theories that descend from the interacting (2, 0) theories in d = 6 can be viewed as a special class of N = 2 super Yang-Mills theories coupled to N = 2 hypermultiplets in the adjoint representation of the gauge group.

On-shell superfields of 6d , (4, 0) supermultiplet of OSp(8 * |8) in twistorial formulation and first versus second order formalism
The physical degrees of freedom corresponding to the fields of maximal N = 8 supergravity in d = 4 were shown to belong to the CPT-self-conjugate unitary representation (doubleton) of the conformal superalgebra SU(2, 2|8) in [1]. Formulation of this unitary supermultiplet in terms of constrained on-shell superfields was given in [5] which we review in the appendix. Even though the physical degrees of freedom form a unitary supermultiplet of the conformal superalgebra SU(2, 2|8) interactions in maximal supergravity break the conformal symmetry down to Poincare subgroup. Whether a conformal supergravity based on this supermultiplet exists, as contemplated in [1], is still an open problem as discussed below.
In six dimensions the unique superconformal algebra with 64 supersymmetry generators is OSp(8 * |8) with the maximal even subalgebra SO * (8)⊕USp (8). Explicit construction of the CPT self-conjugate doubleton supermultiplet of OSp(8 * |8) in terms of twistorial oscillators was given in [5] and shown to reduce to the doubleton supermultiplet of SU(2, 2|8) under dimensional reduction. In table 3 we reproduce the doubleton supermultiplet of OSp(8 * |8) with R-symmetry group USp(8) given in [5]. 4 This multiplet is referred to as the (4, 0) conformal supermultiplet in d = 6 and was studied earlier by Hull using the JHEP06(2021)081 formalism of double gravitons who argued that an interacting theory based on this supermultiplet may describe a strongly coupled phase of 5d maximal supergravity when one of the dimensions decompactifies [14][15][16].
The fields belonging to the (4, 0) supermultiplet can be fitted into an on-shell superfield satisfying an algebraic and a differential constraint [5]. For this it turns out to be very convenient to represent the coordinates of the six-dimensional extended superspace as antisymmetric tensors in spinorial indices [36,37] where the spinorial indices of the Lorentz group SU * (4) in d = 6 are labelled by hatted Greek indicesα,β . . . and the USp(8) indices by A, B, C . . . . Defining the superspace covariant derivative The symplectic metric satisfies and is used to raise or lower indices, The scalar superfield of the (4, 0) supermultiplet is completely anti-symmetric in its indices and is symplectic traceless i.e.
JHEP06(2021)081 and satisfies the differential constraint The mapping between twistorial formulation of (4, 0) supermultiplet and formulation in terms of vectorial indices M, N, . . . = 0, 1, . . . , 5 of SO(5, 1) as was done by Hull was given in [5]. The field strength Rαβγδ of the non-metric graviton corresponds to the (3, 3) which satisfies self-duality conditions in the first as well as the last 3 indices where * operation is performed with the Levi-Civita tensor in six dimensional Minkowskian spacetime In [5] the gauge potential for the non-metric graviton field strength Rαβγδ transforming in the (4, 0, 0) D representation of SU * (4) was chosen as a tensor field Cα (βγδ) transforming in the (3, 0, 1) D representation such that the field strength involves a single derivative where

It is invariant under the gauge transformations
Cα βγδ → Cα βγδ + ∂ω (β χωα γδ) (5.12) where the gauge parameters χαβ γδ satisfy However, since the standard Riemann tensor involves two derivatives of the metric one can also choose a gauge potential such that the non-metric graviton field strength involves two derivatives of that gauge potential as was done in [15,17]. In terms of spinorial indices such a gauge potential must transform as a mixed tensor Cαβ γδ satisfying the conditions Cαβ γδ = Cβα γδ = Cαβ δγ (5.14) such that the non-metric graviton field strength is given by which is invariant under the gauge transformations where the gauge parameters satisfy which imply that they transform in the 64 dimensional representation of SU * (4) with Dynkin label (1, 1, 1) D . The formulation given in [5] and the formulation in terms of a gauge potential of the form 5.16 given in [15,17] are the analogs of first and second order formalisms in ordinary supergravity [38]. 5 The underlying unitary (4, 0) supermultiplet that describes the physical degrees of freedom is the same for both formulations just as is the case for first and second order formalism of Poincaré supergravity. The gauge potential of the non-metric gravitino field strength is a traceless tensor ψα (βγ) . Under a gauge transformation it transforms as with the gauge parameter χ [αβ] γ such that χαβ β = 0. We should note that in contrast to standart local supersymmetry gauge parameter which involves a single spinor index, the gauge parameter χαβ γ of this local gauge symmetry of the non-metric gravitino field transforms as a spinor vector under SU * (4).
In terms of vectorial indices the non-metric gravitino field can be written as ψ  [39,40] from those of N = 4 super Yang-Mills theory that is conformally invariant [41]. 6 Existence of an interacting 4d Poincare supergravity based on the conformal ultrashort CPT self-conjugate supermultiplet of SU(2, 2|8) suggests that an interacting Poincare supergravity in 6d based on the conformal (4, 0) supermultiplet of OSp(8 * |8) also exists in which the interactions break the superconformal symmetry OSp(8 * |8) down to its Poincare subgroup. On the other hand the existence of conformal supergravity based on the (4, 0) supermultiplet in six dimensions is an open question just like the existence of an interacting N = 8 conformal supergravity in d = 4 as will be discussed below.
In six dimensions the (2, 0) supermultiplet of OSp(8 * |4) is the conformal analog of 4d N = 4 Yang-Mills supermultiplet [8]. It is generally believed that the interacting theories of (2, 0) supermultiplets in six dimensions are not conventional field theories and may only exist as quantum theories [43]. Nonetheless they reduce to conventional field theories in lower dimensions. In [5] it was pointed out that the (4, 0) supermultiplet can be obtained by tensoring the (2, 0) supermultiplets. This raises the possibility that the "amplitudes" or correlation functions of an interacting superconformal (2, 0) theory could yield the amplitudes of an interacting non-metric Poincare supergravity based on the (4, 0) supermultiplet via some generalization of double copy methods of BCJ.
Even though the supermultiplet of fields of (4, 0) supergravity and their free equations were known for a long time the action for linearized non-metric (4, 0) Poincare supergravity in six dimensions was first written down rather recently in [17]. The authors of [17] use the formalism of prepotentials adapted to the self-duality properties of the fields of the (4, 0) supermultiplet. They show that the resulting action is invariant under (4, 0) Poincaré supersymmetry in d = 6 but not manifestly. The reason for loss of manifest Poincaré covariance is due to the fact that to write down the action they split the 6d spacetime coordinates as 5+1 with the singlet coordinate being timelike. In their Lagrangian formulations of the bosonic self-dual tensors which they refer to as chiral two-forms [44] as well as bosonic chiral (2,2)tensor corresponding to the gauge potential of non-metric graviton [18] involve only spatial tensors, and their temporal components are pure gauge. 7 They also present a similar formulation for the non-metric gravitensorino ("chiral spinorial two-form"). The resulting action of free (4, 0) supergravity in terms of prepotentials is fourth order in spatial derivatives. 8 (4, 0) supergravity as well as the (3, 1) supergravity were also studied within the socalled exceptional field theory (ExFT) formalism by the authors of [29] recently. Before giving the exceptional field theoretic formulation of these theories they first present novel JHEP06(2021)081 actions for the bosonic sectors of linearized (4, 0) and (3, 1) using the 5 + 1 split of six dimensional spacetime coordinates such that the singlet coordinate y is space-like. These actions are two-derivative actions that reduce to the bosonic sector of linearized maximal supergravity in five dimensions.
The exceptional field theory formalism is a particular extension of the double field theory formalism and is an outgrowth of the attempts to make the hidden U-duality groups of lower dimensional supergravity theories manifest in higher dimensions from which they can be obtained by toroidal compactification. 9 To achieve this one intoduces an auxiliary "internal" spacetime with coordinates Y M motivated by the U-duality group with which to extend the standard external d-dimensional spacetime with coordinates x µ and imposes a section constraint such that the resulting theory describes the higher dimensional supergravity theory. For 5d maximal supergravity one introduces a 27 dimensional auxiliary internal space-time extending the 5 dimensional external spacetime and imposes a section constraint of the form where C IJK is the E 6(6) invariant symmetric tensor. The above equation is to be interpreted as differentials acting on functions To recover 11-d supergravity corresponding to the 5 + 6 split of the coordinates one decomposes the 27 internal coordinates Y I with respect to SL(6, R) × GL(1) subgroup of E 6(6) 27 = 6 +1 + 15 0 + 6 −1 ⇔ Y I = (y m , y mn = −y nm ,ȳ m ) (5.25) where m, n, . . . = 1, . . . , 6. By restricting the dependence on Y I only to the 6 coordinates y m one obtains a solution to section constraints that leads to the 11 dimensional supergravity [46]. 10 To construct the linearized 6d (4, 0) , (3, 1) and standard (2, 2) supergravity theories as ExFTs in a unified manner describing the maximal N = 8 supergravity in 5d the authors of [29] extend the 27 dimensional internal space-time with an extra singlet coordinate Y • and impose the more general section constraint where ∆ IJ is a constant tensor describing the background spacetime. Setting ∂ • = 0 one has the standart section constraint of the formulation of the maximal N = 8 supergravity as an ExFT whose solutions include the 11d sugra, type IIB supergravity and maximal (2, 2) Poincare supergravity theory in 6d. On the other hand setting ∂ I = 0 the generalized section constraint 5.26 is satisfied trivially and by identifying the extra coordinate Y • with 9 For a review and references on the subject see [45]. 10 Type IIB supergravity corresponds to the decomposition of 27 of E 6(6) with respect to the SL(5, R) × SL(2, R×GL(1) subgroup given by 27 = (5, 1) −4 +(5 , 2) −1 +(10, 1) −2 +(1, 2) − 5 and restricting dependence on internal coordinates to the coordinates y a in (5, 1).

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the space-like coordinate in the (5+1) split of 6d coordinates x µ they show that one obtains the bosonic sector of linearized (4, 0) supergravity. In addition to (2, 2) and linearized (4, 0) supergravity in 6d the generalized section constraint admits a solution corresponding to (3,1) supergravity at the linearized level as well. They leave to future work the supersymmetric extension of the bosonic sector of (4, 0) supergravity. Furthermore the fact that a unified framework exists for ExFT formulations of 6d (2,2), (4,0) and (3,1) supergravity theories and the (2, 2) theory can be extended to the full nonlinear theory is interpreted by the authors of [29] as evidence that the same may be true for the (4, 0) and (3, 1) theories. If the interacting non-metric (4, 0) supergravity exists as a Lagrangian theory it will admit a formulation as an ExFT describing the uplift of maximal Poincare supergravity to six dimensions. However if the interacting (4, 0) theory is non-Lagrangian we shall assume that an appropriate generalization of the ExFT formalism exists with which to uplift the 5d maximal supergravity to 6d as a non-metric (4, 0) Poincare supersymmetric theory.

On the question of existence of interacting N = 8 conformal supergravity theory in d = 4 and (4, 0) conformal supergravity in d = 6
The existence of an interacting conformal supergravity based on the (4, 0) supermultiplet in d = 6 would suggest the existence of an interacting conformal supergravity in d = 4 based on the CPT self-conjugate doubleton supermultiplet of SU(2, 2|8) whose existence was posed as an open problem in [3]. To this date no such supergravity theory has been constructed. Conformal supergravity theories were first studied in the pioneering papers of [47,48]. 11 Recently they have been studied as massless limits of Einstein-Weyl supergravity theories. 12 The standard conformal supergravity theories in d = 4 based on the conformal superalgebras SU(2, 2|N ) exist only for N ≤ 4. All N =4 conformal supergravities in d = 4 have recently been constructed in [51,52]. In addition to a massless graviton they contain a massive spin two ghost field and hence are not unitary. The constraint N ≤ 4 arises from the fact that for N ≥ 4 the conformal supermultiplets containing the massive spin two field must necessarily contain fields of spin greater that two. Furthermore the vector fields associated with the gauge fields of SU(n) and U(1) subgroups inside U(n) ⊂ SU(2, 2|n) have kinetic energy terms that are of opposite sign and hence are not all positive definite. Hence if an interacting N = 8 superconformal theory exists that is unitary its formulation must go beyond the standard local gauging of the underlying conformal superalgebras. It may exist purely at the quantum level without a Lagrangian formulation or its Lagrangian formulation may be non-local. In fact there are non-local formulations of conformal gravity that are both unitary and ultraviolet finite. 13 Their Lagrangians typically involve infinite number of terms that are bilinear in scalar curvature R, Ricci tensor R µν and Riemann tensor R µνρλ with powers of the D'Alembertian sandwiched between them. These results are consistent with the findings of [55] who studied the four 11 For an older review see [49]. 12 For a review see [50] and the references therein. 13 For reviews we refer to [53,54] and references therein.

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derivative action of Stelle [56] of the form where M P l is the Planck mass, ∆ is the cosmological constant and µ indicates the mass scale. This theory has a massive spin two ghost of mass µ whose kinetic energy has opposite sign to that of the massless mode. They show that there exists a ghost free completion of this theory which requires an infinite series of higher derivative terms. The resulting theory is classically equivalent to ghost-free bimetric theory of two symmetric tensor fields studied in [57]. Whether these ultraviolet finite unitary nonlocal (higher derivative) theories admit supersymmetric extensions that go beyond the N = 4 bound and result in a unitary theory whose massless physical spectrum coincides with the CPT-self-conjugate doubleton supermultiplet of SU(2, 2|8) is an open problem. The question about the existence of interacting conformal supergravity theory with OSp(8 * |8) symmetry in d = 6 is more subtle. Such a theory would necessarily involve a nonmetric "graviton" and no interacting non-metric gravity theories have been constructed to date. The metric conformal supergravity theories in d = 6 with superconformal symmetry OSp(8 * |2N ) exist only for N ≤ 2). Whether there exist non-local ( higher derivative) and non-metric conformal supergravity theories with OSp(8 * |8) symmetry and are unitary is an open problem.

Truncations of (4, 0) supergravity
Assuming that there exists an interacting (4, 0) supergravity that reduces to the maximal N = 8 supergravity one can consider its truncations to interacting theories with lower number of supersymmetries.

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The (2, 0) supersymmetric truncation of the (4, 0) conformal supergravity reduces to N = 4 supergravity coupled to 5 vector multiplets in d = 5 with the global symmetry group SO(5, 5) × SO(1, 1) which is also the global symmetry group of the 6 dimensional theory with the moduli space The tensor fields of the six dimensional theory form the (10+1) representation of SO(5, 5)× SO (1, 1). The resulting supergravity can be gauged in d = 5 to obtain Yang-Mills Einstein supergravity theories with various possible gauge groups, in particular SU(2) × U(1). Since it is generally believed that strongly coupled phase of 5d maximal super Yang-Mills is described by an interacting (2, 0) theory in d = 6 this raises the question whether there exist deformations of the non-linear (4, 0) theory that correspond to various gaugings of maximal supergravity in d = 5 or in d = 4. In general not all gaugings of maximal supergravity are expected to have uplifts to higher dimensions since gaugings in general introduce potentials with anti-de Sitter as well as de Sitter vacua.
transforming as symplectic traceless anti-symmetric tensor of rank three which we will denote as 14 .
The (1, 0) truncation of (4, 0) theory that reduces to the quaternionic magical theory in 5d can be further truncated such that the resulting theory describes a unified theory in d = 5. First, by restricting to the U(1) invariant sector in the decomposition of SU * (6) with respect to its subgroup SL(3, C) × U(1), which requires discarding 6 tensor multiplets, one obtains a theory describing the coupling of non-metric (1, 0) graviton supermultiplet to eight tensor multiplets that reduces to the complex magical supergravity theory in d = 5 whose moduli space is SL(3, C)/SU (3). Second, by a further restriction to Z 2 invariant sector under the decomposition SL(3, C) ⊃ SL(3, R)×Z 2 one obtains a 6d theory describing the coupling to 5 tensor multiplets that reduces to the real magical supergravity in 5d with the moduli space SL(3, R)/SO (3).
In all the above truncations of the interacting (4, 0) theory to a (1, 0) supersymmetric theory describing the coupling of generalized graviton multiplet to tensor multiplets all the tensor fields, including the gravitensor, transform in an irrep of the global symmetry group, which are SU * (6), SL(3, C) and SL(3, R), respectively. The quaternionic theory with global SU * (6) symmetry can be extended to the octonionic theory with the global symmetry group E 6(−26) symmetry by coupling additional 12 tensor multiplets and reduces to the octonionic magical supergravity theory with 27 vector fields in d = 5.

(3, 0) supersymmetric truncation of (4, 0) supergravity
Using the labelling of indices as in the previous subsection one can show that by discarding two N = 6 gravitensorino multiplets consisting of the fields one obtains the non-metric (3, 0) graviton supermultiplet consisting of the fields The truncation to the non-metric (3, 0) supergravity theory has the same bosonic field content as the truncation to the maximal (1, 0) tensor Einstein supergravity with 14 selfdual tensor multiplets and the 14 scalars. The corresponding result for the 5d supergravity, namely that N = 6 supergravity has the same bosonic content as the quaternionic magical theory with the symmetric target space SU * (6)/USp (6) as was shown in [19].

ExFT formulation of metric and non-metric (1, 0) magical supergravity theories in six dimensions
The unified ExFT formulation of the bosonic sector of linearized (4, 0) and (2, 2) Poincare supergravities as formulated in [29] can be readily extended to a unified construction of JHEP06(2021)081 (1, 0) metric and non-metric supergravity theories in six dimensions that descend to the four magical supergravity theories in five dimensions. For the octonionic magical supergravity which can not be obtained from maximal supergravity by truncation one imposes the section constraint where the C-tensor C IJK is the one given by the cubic norm of the real exceptional Jordan algebra J O 3 which is Euclidean. It an invariant tensor of E 6(−26) . To obtain the ExFT formulation of the metric (1, 0) supergravity in d = 6 one first decomposes the indices of 27 dimensional representation of E 6(−26) with respect ot its SO(9, 1) subgroup where η ab is the SO(9, 1) invariant metric and (Γ a ) αβ are the gamma matrices of SO(9, 1) with (a, b, . . . = 0, 1, . . . 9) and (α, β, . . . = 1, 2, . . . , 16). Imposing the conditions solves the section constraint and the resulting ExFT describes the metric octonionic magical supergravity in d = 6 with 9 tensor multiplets and 16 vector multiplets and scalar manifold SO(9, 1)/SO(9) [33]. For constructing the non-metric (1, 0) supergravity theory one imposes the condition ∂ I = 0 and identifies the coordinate Y • with the spatial singlet component y in the (5 + 1) split of external 6d spacetime coordinates x µ as was done for the maximal supergravity in [29].
To obtain the unified ExFT formulations of metric and non-metric (1, 0) supergravity theories that reduce to the linearized quaternionic, complex and magical supergravity theories in d = 5 one needs to simply substitute the C-tensors of these theories in the section constraint 5.26 and decompose the indices with respect to subgroups of their 5d U-duality groups listed in the first column of table 2 and the gamma matrices by those listed in column 4 of that table. For the quaternionic magical theory the tensor C IJK becomes the invariant tensor of the internal Lorentz ( reduced structure) group SU * (6) of J H 3 . The resulting metric (1, 0) supergravity describes the coupling of 6 tensor multiplets and eight vector multiplets to metric (1,0) supergravity. The corresponding non-metric (1, 0) supergravity theory describes the coupling of 14 tensor multiplets to non-metric (1, 0) supergravity. In contrast to the metric theory the non-metric supergravity theory describes a unified theory since the 15 tensor multiplets transform irreducibly under the global symmetry group SU * (6). Since the quaternionic magical supergravity can be embedded both in maximal supergravity and octonionic magical supergravity in five dimensions the corresponding ExFts describing the metric and non-metric (1, 0) supergravity theories in 6d can also be obtained by truncation of the unified formulation of ExFTs describing (4, 0) and (2, 2) theories in 6d.
The ExFT defined by J H 3 can be consistently truncated to a (1, 0) unified non-metric tensor-Einstein supergravity described by the Euclidean Jordan algebra J C 3 describing the JHEP06(2021)081 coupling of 8 tensor multiplet to non-metric (1, 0) supergravity. It is an invariant tensor of the Lorentz ( reduced structure ) group SL(3, C) of J C 3 . The latter theory can be further truncated to an ExFT corresponding to the real magical supergravity in d = 5 defined by J R 3 . Its C-tensor is invariant under SL(3, R) and describes the coupling of 5 self-dual tensors to non-metric (1,0) supergravity in six dimensions.
The enlarged symmetry groups SL(3, C), SU * (6) and E 6(−26) are the global symmetry groups of the 5d MESGTs defined by the corresponding Euclidean Jordan algebras. As we discussed above these three 5d unified MESGTs theories can be obtained from unified tensor-Einstein supergravity theories in 6d. For the non-metic tensor-Einstein supergravity theories to be unified theories all the tensor fields including the gravitensor must transform irreducibly under the global symmetry group. Remarkably the corresponding irreducible representations of SL(3, C), SU * (6) and E 6(−26) remain irreducible under the restriction to the manifest symmetry subgroups SO(3, 1), SU(3, 1) and USp(6, 2), respectively. For the other members of the three infinite families of unified MESGTs there is no symmetry enhancement beyond the automorphism groups of the underlying Lorentzian Jordan algebras. Nonetheless we expect them to descend from unified tensor-Einstein supergravity theories in 6d in a similar fashion. In addition to the 3 infinite families there exist a unified MESGT in d = 5 defined by the Lorentzian octonionic Jordan algebra of degree three J O (1,2) . This isolated theory is also expected to descend from a unified tensor-Einstein supergravity in d = 6. In tables 4 and 5 we give the list of unified tensor-Einstein supergravity theories in d = 6, their field content and global symmetry groups, under which all the tensor fields including the gravitensor form an irrep.
As is evident from the tables 4 and 5 in the decomposition of the irreducible representation of the global symmetry group with respect to its maximal compact subgroup there is a unique singlet which is to be identified with the "bare gravitensor". Furthermore all the bosonic fields in tensor Einstein supergravity theories are singlets of the R-symmetry group USp(2).

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One can gauge certain subgroup of the global symmetry groups of the unified MESGTs in d = 5. This naturally leads to the question whether the corresponding six dimensional tensor-Einstein supergravity theories admit interactions among tensor fields that reduce to the non-Abelian gauge interactions in five dimensions. It is generally believed that the interacting superconformal (2, 0) theories do not admit a Lagrangian formulation since they decribe multiple M-5 branes that are strongly coupled with no free parameter for formulating a perturbative Lagrangian theory. Nonetheless a novel method of introducing such non-Abelian couplings in certain (1, 0) superconformal field theories in d = 6 was developed by Samtleben and collaborators [58][59][60]. It is an open problem whether some of these theories can be coupled to non-metric (1, 0) supergravity in 6d that upon dimensional reduction reduce to 5d, N = 2 Yang-Mills-Einstein supergravity theories. (4, 0) and (1, 0) tensor Einstein supergravity theories

Dimensional reduction of non-metric
Dimensional reduction of (4, 0) supergravity multiplet using vectorial indices was performed by Hull who showed that the resulting field content coincides with that of maximal Poincare supergravity in five dimensions. Dimensional reduction of the (4, 0) unitary supermultiplet using the twistorial spinor indices was given in [5]. Twistorial oscillator method yields manifestly unitary supermultiplets which involve only the physical degrees of freedom. Hence the resulting supermultiplets involve only the field strengths and not the gauge fields. For the doubleton supermultiplet of OSp(8 * |8) of the (4, 0) non-metric supergravity in d = 6 the field strengths correspond to symmetric tensors in the spinor indicesα,β, . . . of the Lorentz group SU * (4). Since the spinor representation Sα of 6 dimensional Lorentz group SU * (4) and symmetric tensor representations Sαβδ ,... remain irreducible under the restriction to the five dimensional Lorentz group USp(2, 2), the dimensional reduction to d = 5 in the twistorial formulation as given in [5] is much simpler than in the formulation involving vectorial indices. There is a one-to-one correspondence between the field strengths belonging to the doubleton supermutiplet of OSp(8 * |8) and the field strengths of the fields of maximal supergravity in d = 5. In particular non-metric graviton in 6d with the field strength R (αβδγ) reduces to the five dimensional graviton field strength without an extra vector or scalar field [5,15]. This is to be contrasted with the 6d metric graviton which reduces to a graviton plus a vector and a scalar. Similarly the gravitensor field strength ψ A (αβγ) remains irreducible under restriction to USp(2, 2) subgroup of SU * (4) and becomes the field strength of a gravitino in 5d: where * γ µν = 1 3! µνλ δ γ λ δ . Self-dual tensor fields reduce to vector fields in d = 5 and the symplectic Majorana Weyl spinors go over to symplectic Majorana spinors in d = 5. Therefore non-metric (1, 0) supersymmetric tensor Einstein supergravity theory in d = 6 will reduce to a five dimensional N = 2 supergravity with the same number of 5d vector fields as self-dual tensors in 6d. The bare gravitensor in 6d will reduce to the bare graviphoton in 5d. What distinguishes unified tensor-Einstein supergravity theories from JHEP06(2021)081 others is the fact that gravity sector can not be decoupled from the tensorial matter sector without breaking their global symmetry groups since the gravitensor together with the other tensor fields transform irreducibly under them. Ungauged tensor Einstein theories reduce to Maxwell-Einstein supergravity theories in d = 5.
Unified MESGT theories, in particular the magical supergravity theories , admit gaugings with simple gauge groups with or without tensor fields in five dimensions. It was shown in [33] that Poincare uplifts of magical supergravity theories to six dimensions do not admit gaugings with simple groups. Furthermore Poincare uplifts of magical supergravity theories in 6d are no longer unified since some of the vector fields uplift to selfdual tensors while the others uplift to vector fields in 6d. In addition one finds that magical Poincare supergravity theories in 6d admit a unique gauge group which is a nilpotent Abelian group with (n T −1) translation generators, where n T is the number of tensor multiplets coupled to (1, 0) metric supergravity. These (n t − 1) generators can not lie within the isometry group SO(n T , 1) of the scalar manifold due to appearance of central extensions of the gauge algebra. The gauge algebra with the central extension can be embedded within the isometry group of the corresponding five dimensional magical supergravity. On the other hand in the uplift of the magical supergravity theories to six dimensions as non-metric (1, 0) tensor Einstein supergravity the isometry group of the scalar manifold of the five dimensional theory becomes a global symmetry group in 6d.

On the general ExFT formulation of unified non-metric tensor-Einstein supergravity theories in six dimensions
The 27 dimensional internal space-time that is intoduced in ExFT formulation of the 5d maximal Poincare supergravity can be identified with the generalized spacetime coordinatized by the split exceptional Jordan algebra J Os 3 of 3 × 3 Hermitian matrices over the split octonions O s . This generalized space-time was first intoduced in the early days of spacetime supersymmetry before any supergravity theories was written down [61]. Its automorphism, reduced structure and linear fractional groups were identified with the rotation, Lorentz and conformal groups of this space-time which are F 4(4) , E 6(6) and E 7(7) respectively. The generalized space-times defined by Jordan algebras were later studied further in [62][63][64][65]. For the maximal supergravity in d = 5 the Lorentz group E 6(6) is the invariance of the C-tensor as well as of the Lagrangian.
For the 5d octonionic magical supergravity theory the internal space-time is the generalized spacetime defined by the real exceptional Jordan algebra J O 3 of 3 × 3 Hermitian matrices over the division algebra of octonions whose rotation, Lorentz and conformal groups are F 4 , E 6(−26) and E 7(−25) , respectively. The invariance groups of the C-tensors of magical supergravity theories are given by the Lorentz groups of underlying Euclidean Jordan algebras of degree three. As was discussed above the complex, quaternionic and octonionic magical supergravity theories can equivalently be described by the Lorentzian Jordan algebras of degree four over the reals R , complex numbers C and quaternions H. MESGTs in 5d defined by Lorentzian Jordan algebras J A (1,n) of arbitrary degree over the reals R , complex numbers C and quaternions H.
The maximal (2, 2) supergravity theory in six dimensions can be truncated to (1, 0) supersymmetric Poincare supergravity that reduces to the magical supergravity defined by the Euclidean Jordan algebra J H 3 in five dimensions. Its global symmetry group in six dimensions SU * (4) × USp(2) which is a subgroup of the5d global symmetry group SU * (6).
and describes the coupling of 5 self-dual and 8 vector multiplets to (1, 0) metric supergravity. The global symmetry group of this theory gets enlarged to SU * (6) in five dimensions with the scalars parametrizing the symmetric space SU * (6)/USp (6). This theory can be further truncated to the complex and real magical supergravity theories. The dimensional reduction of unified ExFT formulation of linearized metric and non-metric magical supergravity theories to five dimensions the resulting theories are invariant only under the maximal compact subgroups of their global symmetry groups which are F 4 , USp(6), SU(3) and SO (3).
As summarized in section 2.1 5d MESGTs described by the Euclidean Jordan algebras J H 3 and J C 3 can be equivalently formulated using the Lorentzian Jordan algebras J C (1,3) and J R (1,3) , respectively. Since the section constraint for a unified ExFT formulation of these theories depends only on the C-tensor it extends to the formulation in terms of Lorentzian Jordan algebras. Interestingly the extra singlet coordinate Y • in the ExFT formalism can now be identified with the identity element of the corresponding Lorentzian Jordan algebra. Therefore for the three infinite families as well as the sporadic unified non-metric (1, 0) TESGTs in d = 6 one can give an ExFT formulation of their linearized bosonic sectors using the section constraint where C IJK are now the structure constants of the underlying Lorentzian Jordan algebras which are invariant tensors of their global symmetry groups given by their automorphism groups. In all cases the extra singlet coordinate Y • can be identified with the identity element of the underlying Lorentzian Jordan algebra. Imposing the condition ∂ I = 0 solves JHEP06(2021)081 the section constraint trivially and leads to the bosonic sector of non-metric (1, 0) TESGTs in d = 6. Apart from the magical supergravity theories the higher dimensional origins of the three infinite families of 5d unified MESGTs as Poincare supergravity theories is not known. Whether the unified section constraint admits solutions that lead to three infinite families of (1, 0) supersymmetric metric Poincare supergravities in six dimensions will be left for future investigations.

Conformal path to higher dimensions and non-metric supergravity theories
If an interacting non-metric (4, 0) supergravity exists in d = 6 that reduces to the maximal supergravity in lower dimensions it raises the question as to whether d = 6 is the maximal dimension for the existence of non-metric supergravity theories. Now six is the maximal dimension for the existence of conformal superalgebras that extend the conformal algebra of SO(d, 2) to a simple conformal superalgebra [6]. Here the isomorphism of SO(6, 2) to SO * (8) plays a key role for satisfying spin and statistics constraints. The extensions of the Lie algebra of SO(d, 2) to simple Lie superalgebras for d > 6 do not satisfy the correct spin and statistics connection. Therefore it is believed that conformal metric supergravity theories based on simple superconformal algebras exist only in d ≤ 6.
On the other hand Poincare superalgebras, which are not simple, exist in any dimension. However maximal dimension for Poincare supergravity is d = 11 [6]. The Poincare superalgebra in d = 11 with 32 supercharges can be embedded in the simple Lie superalgebra OSp(1|32, R) with the even subalgebra Sp(32, R) [66][67][68][69]. Its contraction to 11 dimensional Poincare superalgebra involves tensorial central charges Werner Nahm's classification of spacetime superalgebras assumed that the spacetime has Minkowskian signature and gravity is described by a spacetime metric or a corresponding spin connection. In the (4, 0) supermultiplet of OSp(8 * |8) the gauge field of the graviton field strength is not a spacetime metric but rather a mixed tensor both in the first order as well as the second order formalism as reviewed above. As was shown by Hull the gauge symmetries of the mixed tensor reduce to diffeomorphisms in five dimensions and the interacting theory should yield the standard maximal supergravity in five dimensions. This raises the question whether there could be higher dimensional non-metric supergravity theories which reduce to the interacting (4, 0) theory in d = 6 or standard maximal supergravity in five and lower dimensions. At the level of Lie superalgebras the answer appears to be yes. Namely there exist superalgebras of the form OSp(2n * |2m) which extend the
In this context we should point out that M-theory was studied in space-times with exotic signatures by Hull [70]. Later M-theory and superstring theory in exotic signature space-times were studied within the framework of negative branes in [71]. Typically the Lorentz groups of these exotic spacetimes are of the form SO(p, 11−p) or SO(p−10−p) for formulations of M-theory and IIB superstring theory, respectively and the corresponding gravitational theories are of metric type. To our knowledge formulation of M/superstring theory on exotic spacetimes with conformal groups of type SO * (2n) that admit interpretation as non-metric gravity theories has not yet been investigated.
One natural framework for going beyond standard Minkowskian spacetimes with metric gravity is that of generalized spacetimes coordinatized by Euclidean Jordan algebras [61,63,64,72]. The conformal groups of space-times defined by Euclidean Jordan algebras all admit positive energy unitary representations [63] and were shown to be causal spacetimes in [65]. The standart critical Minkowskian space-times that admit supersymmetric Yang-Mills theories can be coordinatized by Euclidean Jordan algebras J A 2 of degree two generated by Hermitian 2 × 2 matrices over the four division algebras A = R, C, H, O. Their conformal groups are Sp(4, R), SU(2, 2), SO * (8) and SO(10, 2), respectively. Except for the octonionic case they all admit extensions to simple Lie superalgebras, namely OSp(n|4, R), SU(2, 2|n) and OSp(8 * |2n) which all satisfy the usual spin and statistics connection. 14 The natural extension of these space-times is to consider those defined by Euclidean Jordan algebras J A 3 of degree 3 over the four division algebras A = R, C, H, O which were studied in [64]. These spacetimes correspond to extensions of the Minkowskian space-times by twistorial coordinates given in the second column of table 2 and an extra singlet coordinate. The C-tensors given by the norm forms of Jordan algebras of degree three satisfy the so-called adjoint identity C IJK C J(M N C P Q)K = δ I (M C N P Q) (9.2) It is a remarkable but little known fact that for simple Jordan algebras of degree three the adjoint identity implies the Fierz identities for the existence of supersymmetric Yang-Mills theories in the critical dimensions [35]. The conformal groups of J A 3 are Sp(6, R), SU(3, 3), SO * (12) and E 7(−25) , respectively. Again except for the octonionic case they admit extensions to simple superalgebras OSp(n|6, R), SU(3, 3|n) and OSp(12 * |2n), respectively. The conformal superalgebras in critical dimensions 3, 4 and 6 are subalgebras of these superalgebras.
We should note that SO * (4n) is the conformal group of a spacetime coordinatized by the Euclidean Jordan algebra J H n of n × n Hermitian matrices over the division algebra of quaternions [63]. Its Lorentz and rotation groups are SU * (2n) and USp(2n), respectively. The quaternionic Jordan algebra J H 2 of degree two describes the standard 6d Minkowskian space-time with the Lorentz group SU * (4) and conformal group SO * (8) , which are isomorphic to Spin(5, 1) and SO(6, 2), respectively. The conformal superalgebras OSp(8 * |2n) JHEP06(2021)081 ( 1 2 , 0)   no conformal supergravity with the same field content as maximal N = 8 supergravity in four dimensions has been constructed. It is clear that such a supergravity would have to have some unusual properties such as non-locality and higher derivaties. On the other hand the superalgebra SU(2, 2|8) and the above supermultiplet play a key role in the construction and classification of potential higher-loop counterterms in maximal supergravity [4]. The physical fields of the CPT-self-conjugate doubleton supermultiplet of SU(2, 2|8) can be represented as a scalar superfield W abcd [5,75,76] in the superspace with coordinates, where α,β = 1, 2 and a = 1, 2, . . . 8. The covariant derivatives in this superspace satisfy The superfield W abcd is completely anti-symmetric and obeys the self-duality condition as well as the differential constraint