\section{Introduction}
Conformal supergravity theories with local Lagrangians based on the conformal superalgebras $SU(2,2|N)$ have long been known to exist for $N\leq 4$. It was generally believed that one could not go beyond $N=4$ without having higher spins ($>2)$.
In \cite{Gunaydin:1984vz} it was shown that the fields of maximal $N=8$ supergravity of Cremmer, Julia and Scherk 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 $d=4$ \cite{Gunaydin:1984fk}. 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  \cite{Gunaydin:1984vz} 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 \cite{Beisert:2010jx}. 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 \cite{Chiodaroli:2011pp} who studied the connection between maximal supergravity  and superconformal symmetry in all dimensions that admit simple superconformal algebras as classified by Nahm \cite{Nahm:1977tg}. 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 $SO^*(8) \oplus USp(8)$, 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 $OSp(8^*|4)$\footnote{Double copy construction of the $(4,0)$ theory in terms of the $(2,0)$ theory was later reformulated by  showing  that it can be obtained as the square of  $(2,0)$ Abelian theory by using an involutive field-theoretic product at the linearized level\cite{Borsten:2017jpt}.}. The latter supermultiplet first appeared in the work of  \cite{Gunaydin:1984wc} who constructed the entire Kaluza-Klein spectrum of 11-dimensional supergravity over $AdS_7\times S^4$ by simple tensoring of the $(2,0)$ doubleton supermultiplet.  Witten argued that there must exist interacting superconformal field theories based on the (2,0) supermultiplet that are non-Lagrangian \cite{Witten:1995zh}.  Later an interacting $(2,0)$ superconformal theory was proposed by  Maldacena as being dual to  M-theory on  $AdS_7\times S^4$ \cite{Maldacena:1997re}.

 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 \cite{Chiodaroli:2011pp}.  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\cite{Hull:2000zn,Hull:2000rr,Hull:2001iu}.  On the other hand the interacting $(2,0)$ 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 $6d$  \cite{Chiodaroli:2011pp,Borsten:2017jpt}. More recently, the  action for the free $(4,0)$ theory was written down by Henneaux, Lekeu and Leonard using the formalism of prepotentials in \cite{Henneaux:2017xsb} based on their earlier work on  (2,2) mixed chiral tensor describing the graviton \cite{Henneaux:2016opm}.
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 \cite{Gunaydin:1983rk,Gunaydin:1983bi,Gunaydin:1986fg} and their gaugings were studied in \cite{Gunaydin:1984ak,Gunaydin:1984nt,Gunaydin:1984pf,Gunaydin:1999zx,Gunaydin:2000xk,Gunaydin:2000ph}. 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 \cite{Gunaydin:1983rk}. Later it was shown that there exist three infinite families of unified MESGTs and two isolated ones \cite{Gunaydin:2003yx}. 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 $SU(N,1)$ \cite{Gunaydin:2003yx}.

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 also argue that the infinite family of unified Yang-Mills Einstein supergravity theories must admit uplifts to an infinite family of   unified tensor Einstein supergravity theories with non-Abelian interactions in $d=6$. We conclude with a discussion of possible extensions of the  non-metric supergravity theories to higher dimensions.

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 section 5 we review the on-shell superfield formulation of  $(4,0)$ supermultiplet and the gauge potentials  in the "first order formalism"  following \cite{Chiodaroli:2011pp} and give  the gauge potential of the graviton field strength in the "second order formalism". In section 6 we give  the truncations of $(4,0)$ supergravity to $(3,0)$ 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 give the list of unified non-metric tensor-Einstein supergravity theories in six dimenions and discuss the corresponding unified Maxwell-Einstein and Yang-Mills Einstein supergravity theories in five dimensions in section 8. In section 9 we discuss possible extensions of non-metric supergravity theories to higher dimensions. Appendix A reproduces the CPT-self-conjugate doubleton supermultiplet of $SU(2,2|8)$ \cite{Gunaydin:1984vz}.









\section{Review of $5D$, $\mathcal{N}=2$ Maxwell-Einstein supergravity theories \label{unifiedMESGT} and their gaugings }
\setcounter{equation}{0}



 $\mathcal{N}=2$ MESGTs in five dimensions describes
the coupling of  $\mathcal{N}=2$
 supergravity to an arbitrary number, $\ensuremath{n}$, of vector
multiplets.
The  supergravity multiplet consists of the
f\"{u}nfbein $e_{\mu}^{m}$, two gravitini $\Psi_{\mu}^{i}$
($i=1,2$) and one vector field $A_{\mu}$ (the "bare graviphoton").
On the other hand a $\mathcal{N} =2$ vector multiplet consists of  a
vector field $A_{\mu}$, two symplectic Majorana spinor fields $\lambda^{i}$ and
one real scalar field $\varphi$. The fermions in these theories  transform as doublets under the R-symmetry group $USp(2)_{R}\cong
SU(2)_{R}$ while all the bosonic  fields are $SU(2)_{R}$ singlets.

Hence the  fields
of an $\mathcal{N}=2$ MESGT can be labelled as
\begin{equation}
\{ e_{\mu}^{m}, \Psi_{\mu}^{i}, A_{\mu}^{\ensuremath{I}}, \lambda^{i\ensuremath{a}}, \varphi^{\ensuremath{x}}\}
\end{equation}
with
\begin{eqnarray*}
\ensuremath{I}&=& 0,1,\ldots, \ensuremath{n}\\
\ensuremath{a}&=& 1,\ldots, \ensuremath{n}\\
\ensuremath{x}&=& 1,\ldots, \ensuremath{n}.
\end{eqnarray*}
where we labelled the bare graviphoton as $A_{\mu}^{0}$.
The indices $\ensuremath{a}, \ensuremath{b}, \ldots$ and $\ensuremath{x}, \ensuremath{y},
\ldots$ correspond to  the
flat and curved indices on  the
 scalar manifold, $\mathcal{M}$, respectively.

The bosonic part of the Lagrangian is
given by\cite{Gunaydin:1983bi}
\begin{eqnarray}\label{Lagrange}
e^{-1}\mathcal{L}_{\rm bosonic}&=& -\frac{1}{2}R
-\frac{1}{4}{\stackrel{\circ}{a}}_{\ensuremath{I}\ensuremath{J}}F_{\mu\nu}^{\ensuremath{I}}
F^{\ensuremath{J}\mu\nu}-\frac{1}{2}g_{\ensuremath{x}\ensuremath{y}}(\partial_{\mu}\varphi^{\ensuremath{x}})
(\partial^{\mu}
\varphi^{\ensuremath{y}})+\nonumber \\ &&+
 \frac{e^{-1}}{6\sqrt{6}}C_{\ensuremath{I}\ensuremath{J}\ensuremath{K}}\varepsilon^{\mu\nu\rho\sigma\lambda}
 F_{\mu\nu}^{\ensuremath{I}}F_{\rho\sigma}^{\ensuremath{J}}A_{\lambda}^{\ensuremath{K}},
\end{eqnarray}
where  $e$ is the determinant of  the f\"{u}nfbein
, $R$ is the scalar curvature and
$F_{\mu\nu}^{\ensuremath{I}}$ are the field strengths of Abelian vector
fields $A_{\mu}^{\ensuremath{I}}$.

The completely symmetric
tensor $C_{\ensuremath{I}\ensuremath{J}\ensuremath{K}}$, with lower indices is constant and determines the corresponding  $\mathcal{N}=2$
MESGT uniquely\cite{Gunaydin:1983bi}. The global symmetries of the Lagrangian are the same as symmetries of $C_{\ensuremath{I}\ensuremath{J}\ensuremath{K}}$. The $n$ dimensional scalar manifold can be identified with a hypersurface   in an $(n+1)$ dimensional ambient space with the metric
\begin{equation}\label{aij}
a_{\ensuremath{I}\ensuremath{J}}(h):=-\frac{1}{3}\frac{\partial}{\partial h^{\ensuremath{I}}}
\frac{\partial}{\partial h^{\ensuremath{J}}} \ln \mathcal{V}(h)\ .
\end{equation}
where
\begin{equation}
\mathcal{V}(h):=C_{\ensuremath{I}\ensuremath{J}\ensuremath{K}}h^{\ensuremath{I}}h^{\ensuremath{J}}h^{\ensuremath{K}}\ .
\end{equation}
with real variables $h^{\ensuremath{I}}$ $(\ensuremath{I}=0,1,\ldots,\ensuremath{n})$ representing the coordinates of the ambient space.

The  scalar manifold  $\mathcal{M}$ is simply  the hypersurface $
{\cal V} (h)=1 $ and
the metric , ${\stackrel{\circ}{a}}_{\ensuremath{I}\ensuremath{J}}(\varphi)$,  of the kinetic energy term of vector fields  is simply the  restriction
$a_{\ensuremath{I}\ensuremath{J}}$ to $\mathcal{M}$:
\[
{\stackrel{\circ}{a}}_{\ensuremath{I}\ensuremath{J}}(\varphi)=a_{\ensuremath{I}\ensuremath{J}}|_{{\cal V}=1} \ .
\]

The physical requirements of unitarity  and positivity of the MESGT restrict the possible C-tensors.
The most general  $C_{\ensuremath{I}\ensuremath{J}\ensuremath{K}}$ that
satisfy these constraints can be brought to the form
\begin{equation}\label{canbasis}
C_{000}=1,\quad C_{0ij}=-\frac{1}{2}\delta_{ij},\quad  C_{00i}=0,
\end{equation}
where  $C_{ijk}$
 ($i,j,k=1,2,\ldots , \ensuremath{n}$) are completely arbitrary.
This is referred to as  the canonical basis. Arbitrariness  of  $C_{ijk}$ implies that
 for a given
number $\ensuremath{n}$ of vector multiplets, there exist, in general,  MESGTs with different
scalar manifolds and different global symmetries.



\subsection{ Unified Maxwell-Einstein Supergravity Theories}
Unified
 Maxwell-Einstein supergravity
theories in $d=5 $ are those theories with a simple global symmetry group under which
\emph{all}
the vector fields $A_{\mu}^{\ensuremath{I}}$, 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 $J_3^{\mathbb{A}}$ of degree three defined by $3 \times 3$ Hermitian matrices over the four division algebras $\mathbb{A}$, namely the real numbers $\mathbb{R}$, complex numbers $\mathbb{C}$, quaternions $\mathbb{H}$ and octonions $\mathbb{O}$. 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 \cite{MR0146231,MR0170974,MR0077076}.

In $N=2$ MESGTs defined by Euclidean Jordan algebras , $J$ , of degree three the scalar manifold is a symmetric space of the form
\begin{equation}
\mathcal{M}(J)= Str_0(J) / Aut(J) \end{equation}
where $Str_0(J)$ and $Aut(J)$ are the reduced structure and automorphism group of $J$, respectively.\footnote{Reduced structure group is the invariance group of the norm form of underlying Jordan algebra.}
Below we list the corresponding scalar manifolds:
\begin{eqnarray}
\mathcal{M}(J_3^{\mathbb{R}})&=& SL(3,\mathbb{R})/
SO(3)\qquad
(\ensuremath{n}=5)\nonumber\cr
\mathcal{M}(J_3^{\mathbb{C}})&=& SL(3,\mathbb{C})/
SU(3)\qquad
(\ensuremath{n}=8)\nonumber\cr
\mathcal{M}(J_3^{\mathbb{H}})&=& SU^{*}(6)/
USp(6)\qquad
(\ensuremath{n}=14)\nonumber\cr
\mathcal{M}(J_3^{\mathbb{O}})    &=& E_{6(-26)}/
F_{4}\qquad \qquad
(\ensuremath{n}=26)\nonumber
\end{eqnarray}

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
\begin{equation}
C_{IJK}=C^{IJK}
\end{equation}
where the indices $I,J,K,..$ are raised by the inverse ${\stackrel{\circ}{a}}^{\ensuremath{I}\ensuremath{J}}(\varphi)$ 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 \cite{Gunaydin:2003yx}. These three infinite families are defined by Lorentzian  Jordan algebras of arbitrary degree.








Now  $(n\times n)$ 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_{(q,n-q)}^{\mathbb{A}} $,  are generated by  matrices over
various  division
algebras , $\mathbb{A}= \mathbb{R},\mathbb{C}, \mathbb{H} $
for $n \geq 3$ and over  $ \mathbb{O}$ for $n\leq 3$,  \footnote{The Hermitian
$(n\times n)$ matrices
over the octonions do not form Jordan algebras for $n>3$.}
 that are Hermitian with respect to a non-Euclidean
``metric'' $\eta $ with signature $(q,n-q)$:
\begin{equation}\label{eta}
(\eta X)^\dag = \eta X  \hspace{1cm} \forall \, \, X\in J_{(q,n-q)}^{\mathbb{A}}
\ .
\end{equation}


It was shown in \cite{Gunaydin:2003yx} that  the structure constants (d-symbols) of traceless elements $T_{\ensuremath{I}}$ of noncompact Jordan algebras $J_{(1,N)}^{\mathbb{A}}$  with Lorentzian metric $\eta$ of signature $(1,N)$  defined as
\begin{equation}\label{dsymbols}
d_{\ensuremath{I}\ensuremath{J}\ensuremath{K}}\equiv  \frac{1}{2}
\textrm{tr}(T_{\ensuremath{I}}\{ T_{\ensuremath{J}}, T_{\ensuremath{K}}\})=
\textrm{tr} (T_{\ensuremath{I}}\circ (T_{\ensuremath{J}}\circ
T_{\ensuremath{K}}))
\end{equation}
satisfy the unitarity and positivity  requirements and can be identified
 with the C-tensor of
 a MESGT:
 \begin{equation} C_{\ensuremath{I}\ensuremath{J}\ensuremath{K}}=d_{\ensuremath{I}\ensuremath{J}\ensuremath{K}} \end{equation}
The resulting MESGTs are all unified (for $N
\geq 2 $ ) since all the vector fields including the graviphoton transform in a single irrep of the simple automorphism groups of  the underlying Jordan algebras
$\textrm{Aut}(J_{(1,N)}^{\mathbb{A}})$ which are also the symmetry groups of their Lagrangians.










 \begin{table}[ht]

\label{LorentzianJAs}
\begin{center}
\begin{displaymath}
\begin{array}{|c|c|c|c|c|}
\hline
~&~&~&~&~\\
J & D&  \textrm{Aut}(J)
& \textrm{No. of vector fields} & \textrm{No.  of scalars}  \\
\hline
~&~&~&~&~\\
J_{(1,N)}^{\mathbb R}& \frac{1}{2} (N+1)(N+2) & SO(N,1) &
\frac{1}{2}N(N+3) &\frac{1}{2}N(N+3)-1 \\
~&~&~&~&~\\
J_{(1,N)}^{\mathbb C} & (N+1)^2 & SU(N,1) & N(N+2) & N(N+2)-1 \\
~&~&~&~&~\\
J_{(1,N)}^{\mathbb H} &(N+1)(2N+1) & USp(2N,2) & N(2N+3) & N(2N+3)-1 \\
~&~&~&~&~\\
J_{(1,2)}^{\mathbb O} & 27& F_{4(-20)} & 26 & 25 \\
~&~&~&~&~ \\
\hline
\end{array}
\end{displaymath}
\end{center}

\caption{Simple Lorentzian Jordan algebras
$J_{(1,N)}^{\mathbb{A}}$. Second and third columns list
  their  dimensions $D$ and
automorphism groups $\textrm{Aut}
(J_{(1,N)}^{\mathbb{A}})$. Third and fourth columns list
the number of vector fields $(D-1)$
and the number of scalars $(D-2)$ in the $N=2$ MESGTs defined by them.}
\end{table}
In Table \ref{LorentzianJAs}  we list
 all the simple Lorentzian Jordan algebras of
type $J_{(1,N)}^{\mathbb{A}} $,
their  automorphism  groups and the number of vector and scalar fields in the MESGTs defined by them.

Remarkably the structure constants of the Lorentzian Jordan algebras of degree four
$J_{(1,3)}^{\mathbb R}$, $J_{(1,3)}^{\mathbb C}$ and $J_{(1,3)}^{\mathbb H}$
coincide with the C-tensors of  the magical MESGTs defined by  the Euclidean Jordan algebras
$J_{3}^{\mathbb C}$, $J_{3}^{\mathbb H}$ and $J_{3}^{\mathbb O}$, respectively\cite{Gunaydin:2003yx}.
Hence the magical
MESGTs based
on Euclidean Jordan algebras $J_{3}^{\mathbb C}$, $J_{3}^{\mathbb H}$ and $J_{3}^{\mathbb O}$
\cite{Gunaydin:1983rk}
are \emph{equivalent}
to the MESGTs defined by
the Minkowskian algebras
$J_{(1,3)}^{\mathbb R}$, $J_{(1,3)}^{\mathbb C}$ and $J_{(1,3)}^{\mathbb H}$,
respectively.
As a consequence the global symmetries of these theories get extended to the reduced structure groups $SL(3,\mathbb{C}), SU^*(6)$ and $E_{6(-26)}$  of the Euclidean Jordan algebras $J_{3}^{\mathbb C}$, $J_{3}^{\mathbb H}$ and $J_{3}^{\mathbb O}$, respectively. They   have the automorphism groups $SO(3,1), SU(3,1)$ and $USp(6,2)$ of $J_{(1,3)}^{\mathbb R}$, $J_{(1,3)}^{\mathbb C}$ and $J_{(1,3)}^{\mathbb H}$ as subgroups,
respectively. Furthermore their scalar manifolds are symmetric spaces as reviewed above, while the scalar manifolds of all the other MESGTs defined by Lorentzian Jordan algebras are not even homogeneous. The smallest magical  MESGT defined by the Euclidean Jordan algebra $J_{3}^{\mathbb R}$, does not belong to the three infinite families. In addition the octonionic Lorentzian Jordan algebra $J_{(1,2)}^{\mathbb O}$ defines a unified MESGT which does not belong to an infinite family. It has the global symmetry group $F_{4(-20)}$ with the maximal compact subgroup $SO(9)$.








\subsection{Unified $\ensuremath{\mathcal{N}}=2$  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.





In \cite{Gunaydin:2003yx} the {\it 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 $J_{(1,N)}^{\mathbb{C}}$ under which all the vector fields transform in the adjoint representation of $SU(N,1)$. As stated above the MESGT defined by $J_{(1,3)}^{\mathbb{C}}$ is equivalent to the MESGT defined by the Euclidean Jordan algebra $J_3^{\mathbb{H}}$ 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 \cite{Gunaydin:1984nt}. Again in \cite{Gunaydin:1984nt} it was shown that the dimensionless
 ratio $\frac{g^3}{\kappa}$  involving the
 non-Abelian gauge
coupling constant $g$ and the gravitational constant $\kappa$
must be quantized in the quantum theory by invariance under large gauge transformations.
The same argument extends to all unified YMESGTs since
\[ \Pi_5 (SU(N,1))=\Pi_5 (U(N))=\Pi_5(SU(N)) = \mathbb{Z} \ ,  \]
where $\Pi_5$ 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
\begin{equation}
T_{0}=\left(
\begin{array}{cc}
a & 0 \\
0 & -\frac{a}{N}\mathbf{1}_{(N)}%
\end{array}
\right) ,
\end{equation}
where $a$ is some real number
fixed by the condition  $d_{000}=1$, one can show that the non-compact gauge fields transforming in $ N \oplus \bar{N}$  of $U(N)$ become massive by eating scalar fields and around this ground state $U(1)\times SU(N)$ remains unbroken with the $U(1)$ gauge field corresponding to the graviphoton. Spin 1/2 fields transforming in the symplectic $N \oplus \bar{N}$  also become massive and together with massive gauge fields form short BPS multiplets, with the central charge generated by the $U(1)$ factor.
\subsection{ ${\cal N} =2 $  Yang-Mills-Einstein
  Supergravity Theories coupled to tensor fields}
  \setcounter{equation}{0}
  Unified YMESGTs  are obtained by gauging the $SU(N,1)$ global symmetry groups of  unified MESGTs defined by the Jordan algebras $J_{(1,N)}^{\mathbb{C}}$. Since the Jordan algebras $J_{(1,N)}^{\mathbb{C}}$ are subalgebras of the quaternionic Jordan algebras$J_{(1,N)}^{\mathbb{H}}$  one can also gauge the $SU(N,1)$ subgroups of the global symmetry groups $USp(2N,2)$ of the MESGTs defined by  $J_{(1,N)}^{\mathbb{H}}$.
Under the automorphism group $USp(2N,2)$ of
$J_{(1,N)}^{\mathbb{H}}$,
the vector fields  transform in the $(2N^2+3N) $ dimensional
anti-symmetric symplectic traceless tensor representation. They  decompose as
\[ [(N+1)^2 -1 ] \oplus \frac{N(N+1)}{2} \oplus
\overline{\frac{N(N+1)}{2}} \] under the  $SU(N,1)$ subgroup of $USp(2N,2)$  for $N \geq 2$.
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\cite{Gunaydin:1999zx}.

As for the family of unified MESGTs defined by the real Jordan algebras  $J_{(1,N)}^{\mathbb{R}}$,
 the vector fields transform in the symmetric
tensor representation of  $SO(N,1)$.
For even $N=2n$  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
\[ \frac{n(n+1)}{2} \oplus \overline{\frac{n(n+1)}{2}} \]
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
\[ ( n \oplus \bar{n} ) \oplus (\frac{n(n+1)}{2} \oplus
\overline{\frac{n(n+1)}{2}}) \oplus ( 1 \oplus
\bar{1})   \] of $U(n)$ must be dualized to tensor fields.

In the  MESGT defined by the octonionic Jordan algebra
$J_{(2,1)}^{\mathbb{O}}$ 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
\[ (3 \oplus \bar{3} ) \oplus (3 \oplus \bar{3} ) \oplus
(3 \oplus \bar{3} ) \ .  \]
of $SU(2,1)$ dualized to tensor  fields.

\section{Magical supergravity theories and maximal supergravity}
\setcounter{equation}{0}

The magical Maxwell-Einstein supergravity theories defined by the real, complex and quaternionic Jordan algebras $J_3^{\mathbb{A}}$ ( $\mathbb{A}= \mathbb{R}, \mathbb{C},\mathbb{H}$ )  can all be obtained by a  consistent truncation of the maximal supergravity in $d=5,4$ and $3$ dimensions\cite{Gunaydin:1983rk}. The same is true  for their 6 dimensional uplifts as Poincare supergravities\cite{Gunaydin:2010fi}. The exceptional supergravity defined by the exceptional Jordan algebra  $J_3^{\mathbb{A}}$ 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 \cite{Gunaydin:1985tb}. 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 $SU^*(6)/USp(6)$.

Both real forms $E_{6(6)}$ and $E_{6(-26)}$ have $SU^*(6)\times SU(2)$ as subgroups under which they decompose as
\begin{eqnarray}
27 = (15,1) \oplus (\bar{6},2) \\
78 = (35,1) \oplus (1,3) \oplus (20,2)
\end{eqnarray}
Under the maximal compact subgroup $USp(6)$ the above representations of  $SU^*(6)$ decompose as
\begin{eqnarray}
6=6 \\
15= 14 \oplus 1 \\
20 = 6 \oplus 14' \\
35= 21 \oplus 14
\end{eqnarray}

Under the maximal compact subgroup $USp(8)$ of $E_{6(6)}$ we have the decompositions
\begin{eqnarray}
27=27 \\
78 =36 \oplus 42
\end{eqnarray}
which further decompose under the $USp(6)\times USp(2)$ as
\begin{eqnarray}
27 = (15,1) \oplus (6,2) \\
36 = (21,1) \oplus (1,3) \oplus (6,2) \\
42= (14,1) \oplus (14',2)
\end{eqnarray}
On the other hand maximal compact  subgroup of $E_{6(-26)}$  is  $F_4$ under which we have the decompositions
\begin{eqnarray}
27=26\oplus 1 \\
78 = 52 \oplus 26
\end{eqnarray}
Under the $USp(6)\times USp(2)$ subgroup the above representations of $F_4$ decompose as
\begin{eqnarray}
26 = (14,1) \oplus (6,2) \\
52 = (21,1) \oplus (1,3) \oplus (14',2)
\end{eqnarray}

The above decompositions show that restricting to the $USp(2)$ invariant subsector the spectra coincide with that of quaternionic magical theory defined by $J_3^{\mathbb{H}}$.  The global symmetry group $SU^*(6)$ of the quaternionic magical theory has the subgroup $SL(3,\mathbb{C})\times SO(2)$ where $ SL(3,\mathbb{C}) $ is the global symmetry group of the complex magical MESGT
\begin{equation}
SU^*(6) \supset SL(3,\mathbb{C})\times U(1)_C
\end{equation}
 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
 \begin{equation}
SL(3,\mathbb{C}) \supset SL(3,\mathbb{R}) \times \mathbb{Z}_2
\end{equation}
and $ \mathbb{Z}_2$ invariant subsector describes the consistent truncation to the real magical supergravity defined by $J_3^{\mathbb{R}}$.
\section{Magical   Poincare supergravity theories in six dimensions}
\setcounter{equation}{0}
Six dimensional magical supergravity theories coupled  to hypermultiplets and  their gaugings were studied in \cite{Gunaydin:2010fi}.
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
$\Gamma^I_{AB}$ that are the Dirac $\Gamma$-matrices for $n_T=2,3 $, and the Van der Waerden symbols for $n_T=5,9$. They satisfy the identities
\begin{eqnarray}
\Gamma^{\vphantom{I}}_{I\,(AB}\Gamma^{I}_{C)D} &=& 0\;.
\label{magic}
\end{eqnarray}
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\cite{Sierra:1986dx}.
We reproduce their field contents  in Table \ref{tab:reality}.

\begin{table}
\begin{center}
\begin{tabular}{c||c|c|c|c}
$G_T$ & ${\cal R}_{\rm v}$ & $A_\mu^A$ & $\Gamma^I_{AB}$ & ${\cal R}_{\rm ten}$ \\[1ex] \hline\hline
&&&&\\[-2ex]
${SO}(9,1)$ &
${\bf 16}_c$
& MW & $\Gamma^I_{AB}$
& ${\bf 10}$
\\[1ex]\hline&&&&\\[-2ex]
${SO}(5,1)\times {USp}(2)$ &
${\bf (4}_c,{\bf 2)}$
& SMW,\, $A=(\alpha r)$ &
$\Gamma^I_{\alpha r,\beta s} = \Gamma^I_{\alpha\beta} \epsilon_{rs}$
& ${\bf (6,1)}$
\\[1ex]\hline&&&&\\[-2ex]
${SO}(3,1)\times {U}(1)$ &
${\bf (2,1)}_+ + {\bf (1,2)}_-$
&W,\, $A=\{\alpha,\dot{\beta}\}$
&
$\left(
\begin{array}{cc}
0 & \Gamma^I_{\alpha\dot{\beta}}\\
\bar\Gamma^I_{\dot\alpha{\beta}} & 0
\end{array}
\right)$
& ${\bf (2,2)_0}$
\\[3ex]\hline&&&&\\[-2ex]
${SO}(2,1)$  &
${\bf 2}$&
M & $\Gamma^I_{AB}$
& ${\bf 3}$
\\[1ex]\hline
\end{tabular}
\caption{
{\small Above we reproduce the field content and symmetries of magical supergravity theories in six dimensions.
The first column lists their global symmetry groups $G_T$
and the second column lists the representations ${\cal R}_{\rm v}$ of the vector fields $A_\mu^A$ under $G_T$. The reality properties of these representations are given in the third column: Majorana (M), Weyl (W), Majorana-Weyl (MW), symplectic Majorana-Weyl (SMW).
The last column lists  the representations ${\cal R}_{\rm ten}$ of the tensor fields under $G_T$. $\Gamma^I$ are the gamma matrices in the respective dimensions
 }}
\label{tab:reality}
\end{center}
\end{table}

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_{(2,1)}^{\mathbb{O}} $.
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_{(2,1)}^{\mathbb{H}} $ ($J_{(2,1)}^{\mathbb{C}} $) .

They satisfy the inclusions

\begin{equation}
F_{4(-20)} \supset USp(4,2) \times USp(2) \supset SU(2,1) \times U(1)
\end{equation}

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 $d=6$
\begin{equation}
(2,0) = (1,0) \,  \textrm{tensor multiplet} \oplus (1,0) \,  \textrm{hypermultiplet}
\end{equation}
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.
\section{ Superconformal symmetry and non-metric  (4,0)  supergravity in six dimensions}
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 \cite{Gunaydin:1984vz}. Formulation of this unitary supermultiplet in terms of constrained on-shell superfields was given in \cite{Chiodaroli:2011pp} 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 \cite{Gunaydin:1984vz}, is still an open problem.

In six dimensions the unique superconformal algebra with 64 supersymmetry generators is $OSp(8^*|8)$ with the maximal even subalgebra $SO^*(8) \oplus USp(8)$. Explicit construction of the CPT self-conjugate doubleton supermultiplet of $OSp(8^*|8)$ in terms of twistorial oscillators was given in \cite{Chiodaroli:2011pp} and shown to reduce to the doubleton supermultiplet of $SU(2,2|8)$ under dimensional reduction.
In Table \ref{USp(8)_doubleton} we  reproduce the doubleton supermultiplet of $OSp(8^*|8)$ with
$R$-symmetry group $USp(8)$ given in \cite{Chiodaroli:2011pp}\footnote{We should note that in the manifestly unitary construction using twistorial oscillators the field strengths corresponding to physical degrees of freedom and not the corresponding gauge fields  form the unitary supermultiplets. With that caveat we will use the terms fields and field strengths interchangeably.}. This multiplet is referred to as the $(4,0)$ conformal supermultiplet in $d=6$ and  was studied earlier  by Hull using the  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 \cite{Hull:2000zn,Hull:2000rr,Hull:2001iu}.


\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
 Field Strengths &  ${SU^{*}(4)}_{D}$ & $USp(8)$
\\ \hline

$\phi^{[ABCD]|}(x)$  & (0,0,0)       & 42     \\ \hline
$\lambda_{\hat{\alpha}}^{[ABC]|} (x) $ & (1,0,0)
& 48\\ \hline

$h_{(\hat{\alpha}\hat{\beta})}^{[AB]|}(x)$ & (2,0,0)       & 27     \\ \hline
$\psi_{(\hat{\alpha}\hat{\beta}\hat{\gamma})}^A (x) $& (3,0,0) & 8 \\ \hline
$R_{(\hat{\alpha}\hat{\beta} \hat{\gamma}\hat{\delta})}(x) $& (4,0,0) & 1 \\ \hline
\end{tabular}
\end{center}
\caption {\label{USp(8)_doubleton} \small
(4,0) doubleton supermultiplet of $OSp(8^*|8)$. First column lists the field strengths corresponding to actual physical degrees of freedom. Second column gives their Dynkin labels under the Lorentz group  $SU^*(4)$. Third column gives the  dimensions
of their $USp(8)$  representations.  The field strengths are labelled by chiral
$SU^*(4)$ spinor indices $\hat{\alpha}, \hat{\beta},\dots$ and $USp(8)$ indices $A,B,\dots$.}
\end{table}






The fields belonging to the $(4,0)$ supermultiplet  can be fitted into an on-shell
superfield satisfying an algebraic and a differential constraint \cite{Chiodaroli:2011pp}.
For this it turns out to be very convenient to represent the coordinates of the  six-dimensional
extended superspace  as anti-symmetric tensors in spinorial indices \cite{Howe:1983fr,Koller:1982cs}
\begin{equation}
\big( x^{{\hat \alpha} {\hat \beta}}= -x^{{\hat \beta} {\hat \alpha}}, \theta^{\hat \alpha}_A \big) \qquad {\hat \alpha}, {\hat \beta} = 1, \dots, 4 \ ;
\quad A=1, \dots, 8 \ ;
\end{equation}
where the spinorial indices of the Lorentz group $SU^*(4)$ in $d=6$
 are labelled by hatted Greek
indices ${\hat \alpha}, {\hat \beta}  \dots$ and the $USp(8)$  indices
by $A,B,C \dots$.
Defining the  superspace covariant derivative
\begin{equation}
D^A_{\hat \alpha} = \partial^A_{\hat \alpha} +
i \Omega^{AB} \theta^{\hat \beta}_B \partial_{\hat \alpha  {\hat \beta}} \ ,
\end{equation}
where $\partial^A_{\hat \alpha} \theta^{\hat \beta}_B = \delta^A_B \delta^{\hat \beta}_{\hat \alpha}$ and  $\Omega_{AB}=-\Omega_{BA}$
 one finds
\begin{equation} \{ D^A_{\hat \alpha}, D^B_{\hat \beta} \} = 2 i \Omega^{AB} \partial_{{\hat \alpha} {\hat \beta}} \ ,
\qquad \{ D_{{\hat \alpha} A}, D^B_{{\hat \beta}} \} = 2 i \delta^B_A \partial_{{\hat \alpha} {\hat \beta}} \ . \end{equation}
The symplectic metric satisfies
\begin{equation}
\Omega_{AB} \Omega^{BC} = \delta^C_A \ .
\end{equation}
and is  used to raise or lower indices,
\begin{equation}
\theta^{A{\hat \alpha}} = \Omega^{AB} \theta^{\hat \alpha}_B, \qquad D_{A{\hat \alpha}}= \Omega_{AB} D^B_{\hat \alpha} \ .
\end{equation}


The scalar  superfield  of the $(4,0)$ supermultiplet
\begin{equation}
\Phi^{ABCD}\big( x^{{\hat \alpha} {\hat \beta}}, \theta^{\hat \alpha}_A \big)
\end{equation}
is completely anti-symmetric in its indices and is symplectic traceless i.e.
\begin{equation}
\Phi^{ABCD} \Omega_{CD}= 0  ~~ ,
\label{algebraicconstraint}
\end{equation}
and satisfies the
differential constraint
\begin{equation}
D^A_{\hat \alpha} \Phi^{BCDE} + {1 \over 21} D_{{\hat \alpha} F} \Big\{ \Omega^{A[B} \Phi^{CDE]F}
+ {3 \over 4} \Omega^{[BC} \Phi^{DE]AF} \Big\}=0~~.
\label{diffconstraint}
 \end{equation}



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 \cite{Chiodaroli:2011pp}.
The field strength $R_{{\hat \alpha} {\hat \beta} {\hat \gamma} {\hat \delta}}$ of the non-metric graviton corresponds to the $(3,3)$ tensor $R_{[MNO][PQR]}= R_{[PQR][MNO]}$ which satisfies self-duality conditions in the first as well as the last 3 indices
\begin{equation}
*R=R* =R  ~~  ,
\end{equation}
where $*$ operation is performed with the Levi-Civita tensor in six dimensional Minkowskian spacetime
\[ (*R)_{[MNO][PQR]}= \frac{1}{6} \epsilon_{MNOSTU}R^{STU}_{~~~~ PQR} ~~. \]

In \cite{Chiodaroli:2011pp} the gauge potential for the non-metric graviton field strength  $R_{{\hat \alpha} {\hat \beta} {\hat \gamma} {\hat \delta}}$ transforming in the $(4,0,0)_D$ representation of $SU^*(4)$ was chosen as a tensor field $C^{{\hat \alpha}}_{({\hat \beta} {\hat \gamma} {\hat \delta})}$ transforming in the $(3,0,1)_D$ representation such that the field strength involves a single derivative
\begin{equation}
R_{({\hat \alpha} {\hat \beta} {\hat \gamma} {\hat \delta})}=\partial^{\vphantom{ \hat \omega \hat \lambda}}_{ \hat \lambda ( {\hat \alpha}}  C^{\hat \lambda }_{{\hat \beta} {\hat \gamma} {\hat \delta})}
\end{equation}
where
\begin{equation}  C^{\hat \lambda }_{({\hat \beta} {\hat \gamma} {\hat \delta}) } = C^{\hat \lambda }_{{\hat \beta} {\hat \gamma} {\hat \delta} } \, \quad .
 \end{equation}

It is invariant under the gauge transformations
\begin{equation}  C^{{\hat \alpha}}_{{\hat \beta} {\hat \gamma} {\hat \delta}} \rightarrow C^{{\hat \alpha}}_{{\hat \beta} {\hat \gamma} {\hat \delta}} +
\partial_{ \hat \omega ( {\hat \beta} }^{\vphantom{{\hat \alpha}}} \chi^{\hat \omega {\hat \alpha}}_{{\hat \gamma} {\hat \delta})} \end{equation}
where the gauge parameters $\chi^{{\hat \alpha} {\hat \beta}}_{{\hat \gamma} {\hat \delta}}$ satisfy
\begin{equation}
\chi^{{\hat \alpha} {\hat \beta}}_{{\hat \gamma} {\hat \delta}} = - \chi^{{\hat \beta} {\hat \alpha}}_{{\hat \gamma} {\hat \delta}} = \chi^{{\hat \alpha} {\hat \beta}}_{{\hat \delta} {\hat \gamma}} \, \quad .
\end{equation}

 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 \cite{Hull:2000rr,Henneaux:2017xsb}.
 In terms of spinorial indices such a gauge potential  must transform  as a mixed tensor $ C^{ {\hat \alpha} {\hat \beta} }_{ {\hat \gamma} {\hat \delta}} $
  satisfying the conditions
  \begin{eqnarray}
  C^{ {\hat \alpha} {\hat \beta} }_{ {\hat \gamma} {\hat \delta}}=C^{ {\hat \beta} {\hat \alpha} }_{ {\hat \gamma} {\hat \delta}}=C^{ {\hat \alpha} {\hat \beta} }_{ {\hat \delta} {\hat \gamma}} \\
  C^{ {\hat \alpha} {\hat \beta} }_{ {\hat \gamma} {\hat \beta}}=0
  \end{eqnarray}
 such that  the non-metric graviton field strength is given by \begin{equation}
R_{({\hat \alpha} {\hat \beta} {\hat \gamma} {\hat \delta})}=\partial^{\vphantom{ \hat \omega \hat \lambda}}_{ \hat \lambda ( {\hat \alpha}}  \partial^{\vphantom{ \hat \omega \hat \lambda}}_{ \hat \epsilon  {\hat \beta}} C^{\hat \lambda \hat \epsilon}_{{\hat \gamma} {\hat \delta} )}  \label{secondorder}
\end{equation}
which is invariant under the gauge transformations
\begin{equation}
 C^{{\hat \alpha} {\hat \beta}}_{{\hat \gamma} {\hat \delta} } \longrightarrow C^{{\hat \alpha} {\hat \beta}}_{{\hat \gamma} {\hat \delta} } + \partial_{\hat \lambda ( {\hat \gamma} } \chi^{{\hat \alpha} {\hat \beta} \hat \lambda}_{{\hat \delta} )}
 \end{equation}
 where the gauge parameters satisfy
 \begin{eqnarray}
  \chi^{{\hat \alpha} {\hat \beta} \hat \lambda}_{{\hat \delta} }=- \chi^{{\hat \alpha} \hat{\lambda}  {\hat \beta}}_{{\hat \delta} } \\
   \chi^{{\hat \alpha} {\hat \beta} {\hat \gamma} }_{{\hat \gamma} } =0
   \end{eqnarray}
 which imply that they transform in the 64 dimensional  representation of $SU^*(4)$ with Dynkin label  $(1,1,1)_D$. The formulation given in \cite{Chiodaroli:2011pp} and the formulation in terms of a gauge potential of the form  \ref{secondorder} given in \cite{Hull:2000rr,Henneaux:2017xsb} are the analogs of first and second order formalisms in ordinary supergravity \cite{Freedman:2012zz}.\footnote{ I would like to thank Marc Henneaux for posing the question about the difference between the formulations in  \cite{Chiodaroli:2011pp} and in \cite{Hull:2000rr,Henneaux:2017xsb} that triggered the investigations that led to this paper. } 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 Poincar\'e supergravity.

   The gauge potential of the non-metric gravitino field strength is a traceless tensor
$\psi^{{\hat \alpha}}_{({\hat \beta} {\hat \gamma})}$.
 Under a gauge transformation it transforms as
\begin{equation}
\psi^{{\hat \alpha} }_{{\hat \beta} {\hat \gamma}}  \rightarrow \psi^{{\hat \alpha}}_{{\hat \beta} {\hat \gamma}} +
\partial_{\hat \omega  ( {\hat \beta} } \chi^{\hat \omega {\hat \alpha} }_{{\hat \gamma})} \ ;
\end{equation}
with the gauge parameter $\chi^{[{\hat \alpha} {\hat \beta}]}_{\hat \gamma}$ such that  $\chi^{{\hat \alpha} {\hat \beta}}_{\hat \beta} =0$.
We should note that  in contrast to standart local supersymmetry gauge parameter which involves a single spinor index, the gauge parameter $\chi^{{\hat \alpha} {\hat \beta}}_{\hat \gamma}$ 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  $\psi_{[MN]{\hat \alpha}}$ such that its  field strength $\psi_{[MNO]{\hat \alpha}} $ satisfies the self-duality condition
\begin{equation}
* \psi = \psi
\end{equation}
Similarly the tensor field with field strength $h_{({\hat \alpha} {\hat \beta})}$ is $ b^{\hat \alpha}_{\hat \beta} $ so that
\begin{equation}
 h_{({\hat \alpha} {\hat \beta})}= \partial^{\vphantom{{\hat \gamma}}}_{{\hat \gamma} ({\hat \alpha}} b^{{\hat \gamma}}_{{\hat \beta})} \, , \quad b^{\hat \alpha}_{\hat \alpha}=0 \end{equation}
 It undergoes gauge transformations with parameters $\chi^{{\hat \alpha}{\hat \beta}}=-\chi^{{\hat \beta}{\hat \alpha}}$.
 In terms of vectorial indices it is described by an anti-symmetric tensor field $b_{[MN]}$ whose field strength $h_{[MNP]}$ satisfies the selfduality condition
 \[ *h =h \]

\section{ Truncations of $(4,0)$ supergravity}
\setcounter{equation}{0}
Assuming that there exists an interacting superconformal  $(4,0)$ supergravity that reduces to the maximal $N=8$ supergravity one can consider its
 truncations to  interacting theories with lower number of supersymmetries.
\subsection{ $(2,0)$ supersymmetric truncations}
The decomposition of the $(4,0)$ supermultiplet of $OSp(8^*|8)$ into supermultiplets of $N=4$  superconformal algebra $OSp(8^*|4)$ in $d=6$ and some of its implications were studied in \cite{Chiodaroli:2011pp}. One finds that $(4,0)$ multiplet decomposes into  one $(2,0)$ non-metric graviton supermultiplet\cite{Chiodaroli:2011pp}
\begin{equation}
R_{(\hat{\alpha}\hat{\beta} \hat{\gamma}\hat{\delta})}(x),  \quad  \psi_{(\hat{\alpha}\hat{\beta}\hat{\gamma})}^a (x),  \quad  h_{(\hat{\alpha}\hat{\beta})}^{[ab]|}(x) \oplus h_{(\hat{\alpha}\hat{\beta})}(x) , \quad \lambda_{\hat{\alpha}}^{[abc]|} (x) , \quad \phi(x)
\label{confgrav6d}\end{equation}
four $(2,0)$  non-metric gravitino supermultiplets
\begin{equation}
 \psi_{(\hat{\alpha}\hat{\beta}\hat{\gamma})} (x),  \quad  h_{(\hat{\alpha}\hat{\beta})}^a(x) , \quad \lambda_{\hat{\alpha}}^{[ab]|} (x) \oplus \lambda_{\hat{\alpha}} (x) , \quad \phi^{[abc]|}(x)
\end{equation}
and five  $(2,0)$ self-dual tensor multiplets ( doubletons )
\begin{equation}
  h_{(\hat{\alpha}\hat{\beta})}(x) , \quad \lambda_{\hat{\alpha}}^{a} (x)  , \quad \phi^{[ab]|}(x)
\end{equation}
where   $a,b,c=1,2,3,4$ are the $USp(4)$ indices.
Therefore the interacting $(4,0)$ theory must admit a consistent truncation that  describes the coupling of the non-metric $(2,0)$ graviton supermultiplet to five
$(2,0)$ doubleton supermultiplets.


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 $SO(5,5)\times SO(1,1)$ which is also the global symmetry group of the 6 dimensional theory with the moduli space
\[ SO(5,5)\times SO(1,1)/ SO(5)\times SO(5)
\]
The tensor fields of the six dimensional theory form the  (10+1) representation of $SO(5,5) \times SO(1,1)$.  The resulting supergravity can be gauged in $d=5$ to obtain  Yang-Mills Einstein supergravity theories with various possible gauge groups, in particular $SU(2)\times U(1)$. Since it is generally believed that $5d$ super Yang-Mills describes an interacting $(2,0)$ theory in $d=6$ this implies that fully non-linear $(4,0)$ theory must also admit novel interactions corresponding to various gaugings of maximal supergravity in $d=5$ or in $d=4$. Not all gaugings of maximal supergravity are expected to have uplifts to "gauged" (4,0) theories since gaugings in general introduce potentials and anti-de Sitter  as well as de Sitter vacua.
\subsection{ $(1,0)$ supersymmetric truncations of $(4,0)$ conformal supergravity }
\setcounter{equation}{0}
The $(4,0)$ supermultiplet of $OSp(8^*|8)$ can be decomposed into $(1,0)$ supermultiplets. Splitting the $USp(8)$ indices $A,B,..$ as $A=(a,i), B=(b,j),..$ where $i,j,..=1,2$ and $a,b,..=3,4,..,8$ we find
\begin{itemize}
\item non-metric graviton supermultiplet
\[  R_{(\hat{\alpha}\hat{\beta} \hat{\gamma}\hat{\delta})}(x),  \quad  \psi_{(\hat{\alpha}\hat{\beta}\hat{\gamma})}^i (x),  \quad  h_{(\hat{\alpha}\hat{\beta})}^{[ij]}(x) \]

\item 6 non-metric gravitino ( gravitensorino) supermultiplets \[
\psi^a_{(\hat{\alpha}\hat{\beta}\hat{\gamma})} (x),  \quad  h^{ai}_{(\hat{\alpha}\hat{\beta})}(x) , \quad \lambda_{\hat{\alpha}}^{a[ij]} (x)  \]
\item
14 (1,0) self-dual tensor multiplets
\[
  h_{(\hat{\alpha}\hat{\beta})}^{[ab]|} (x) , \quad \lambda_{\hat{\alpha}}^{[ab]|i} (x)  , \quad \phi^{[ab]|ij}(x)
\] transforming in the antisymmetric symplectic traceless representation of $USp(6)$.
\item
14 (1,0) hypermultiplets
\[  \lambda_{\hat{\alpha}}^{[abc]|} (x)  , \quad \phi^{[abc]|i}(x) \]
transforming as symplectic traceless anti-symmetric tensor of rank three which we will denote as $14'$.
\end{itemize}

In parallel to full $5d$, $N=8$ supergravity we expect the interacting $(4,0)$ superconformal theory to admit  two maximal $(1,0)$ supersymmetric truncations. By discarding the 6 generalized gravitino multiplets and 14 hyper multiplets one  obtains an interacting theory describing the coupling of  non-metric $(1,0)$  graviton supermultiplet to 14 tensor multiplets with the global symmetry group $SU^*(6)$ under which 15 tensor fields, including the gravitensor transform irreducibly. 14 scalar fields belong to the coset $SU^*(6)/USp(6)$ as they do in the quaternionic magical theory in $d=5$. On the other hand by discarding the 6 non-metric gravitino multiplets together with the 14 tensor multiplets one obtains a consistent truncation to a $6d$ theory describing the coupling of  non-metric $(1,0)$ graviton supermultiplet to 14 hypermultiplets whose scalars sit in the coset $F_{4(4)}/USp(6)\times SU(2)$ which reduces in five dimensions to $N=2$ supergravity coupled to 14 hypermultiplets\cite{Gunaydin:1983rk}.

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 $SL(3,\mathbb{C})\times U(1)$, 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 $SL(3,\mathbb{C})/SU(3)$.  Second, by a further restriction to $\mathbb{Z}_2$ invariant sector under the decomposition  $SL(3,\mathbb{C})\supset SL(3,\mathbb{R}) \times \mathbb{Z}_2$ 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 $SL(3,\mathbb{R})/SO(3)$.

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 $SU^*(6), SL(3,\mathbb{C})$ and $SL(3,\mathbb{R})$, 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 $d=5$.
\subsection{ $(3,0)$ supersymmetric truncation of $(4,0)$  supergravity }
\setcounter{equation}{0}
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
\[ \psi^i_{(\hat{\alpha}\hat{\beta}\hat{\gamma})} (x),  \quad  h^{ia}_{(\hat{\alpha}\hat{\beta})}(x) , \quad \lambda_{\hat{\alpha}}^{i[ab]|} (x) , \quad
\phi^{i[abc]|}(x) \]
one obtains the non-metric $(3,0)$ graviton supermultiplet consisting of the fields
\[  R_{(\hat{\alpha}\hat{\beta} \hat{\gamma}\hat{\delta})}(x),  \quad  \psi_{(\hat{\alpha}\hat{\beta}\hat{\gamma})}^a (x),  \quad  h_{(\hat{\alpha}\hat{\beta})}^{[ab]|}(x)+ h^{[ij]}_{({\hat \alpha}{\hat \beta})}(x) \, , \quad \lambda_{\hat \alpha}^{[abc]|}(x) + \lambda^{[ij]a}_{\hat \alpha} (x) \, , \quad \phi^{[ij][ab]|}(x)  \]
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 self-dual tensor multiplets and the 14 scalars in both theories parameterize the symmetric space $SU^*(6)/USp(6)$. 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 $SU^*(6)/USp(6)$  was shown in \cite{Gunaydin:1983rk}.

\section{  Unified non-metric Tensor-Einstein Supergravity theories in $d=6$  }
As we reviewed in section \ref{unifiedMESGT}  there exist three infinite families of unified $N=2$ MESGTs
in $d=5$. Their C-tensors are given by the structure constants $d_{IJK}$  of the traceless elements of Lorentzian Jordan algebras $J_{(1,N)}^{\mathbb{A}} $  of degree three over the associative division algebras $\mathbb{A}=\mathbb{R},\mathbb{C},\mathbb{H}$ which are invariant under their automorphism groups of $J_{(1,N)}^{\mathbb{A}} $.
Since the  structure constants $d_{IJK}$ of three Lorentzian Jordan algebras of degree four, namely
$J_{(1,3)}^{\mathbb R}$, $J_{(1,3)}^{\mathbb C}$ and $J_{(1,3)}^{\mathbb H}$,
coincide with the C-tensors defined by the norms of the Euclidean Jordan algebras of degree three,
$J_{3}^{\mathbb C}$, $J_{3}^{\mathbb H}$ and $J_{3}^{\mathbb O}$ respectively\cite{Gunaydin:2003yx}, they have accidental enlarged hidden symmetries:
\begin{eqnarray}
Aut [J_{(1,3)}^{\mathbb R}]= SO(3,1) \quad \Longrightarrow \quad Str_0(J_3^{\mathbb{C}})= SL(3,\mathbb{C}) \\
Aut [J_{(1,3)}^{\mathbb C}]= SU(3,1) \quad \Longrightarrow \quad Str_0(J_3^{\mathbb{H}})= SU^*(6) \\
Aut [J_{(1,3)}^{\mathbb H}]= USp(6,2) \quad \Longrightarrow \quad Str_0(J_3^{\mathbb{O}})=E_{6(-26)}
\end{eqnarray}
The enlarged symmetry groups  $SL(3,\mathbb{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,\mathbb{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. 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_{(1,2)}^{\mathbb{O}} $.   This isolated theory is also expected to descend  from a unified tensor-Einstein supergravity in $d=6$.  In Tables \ref{unifiedTESGT} and \ref{unifiedMESGT} 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.

\begin{table}[ht]
\scriptsize
\begin{center}
\begin{displaymath}
\begin{array}{|c|c|c|c|}
\hline
~&~&~&~\\
J &  \textrm{ Aut}(J)\supset K
& \textrm{Decompositon of self-dual tensor fields under $K$} & \textrm{Number  of scalars}  \\
\hline
~&~&~&~\\
J_{(1,N)}^{\mathbb R} & SO(N,1)\supset SO(N) &
\frac{1}{2}N(N+3)=[\frac{1}{2}N(N+1)-1]\oplus N \oplus 1 &\frac{1}{2}N(N+3)-1 \\
~&~&~&~\\
J_{(1,N)}^{\mathbb C}  & SU(N,1)\supset SU(N)\times U(1) & N(N+2)= [N^2-1]^0 \oplus N^{+1} \oplus N^{-1} \oplus 1^0 & N(N+2)-1 \\
~&~&~&~\\
J_{(1,N)}^{\mathbb H}  & USp(2N,2)\supset USp(2N)\times USp(2) & N(2N+3)=[(N(2N-1)-1,1)]\oplus (2N,2) +(1,1)  & N(2N+3)-1 \\
~&~&~&~\\
J_{(1,2)}^{\mathbb O} & F_{4(-20)}\supset SO(9) & 26=9 \oplus 16 \oplus 1 & 25 \\
~&~&~&~ \\
\hline
\end{array}
\end{displaymath}
\end{center}

\caption{\label{unifiedTESGT} Unified tensor-Einstein supergravity theories in $d=6$ that reduce to unified MESGTs in $d=5$ defined by simple Lorentzian Jordan algebras
$J_{(1,N)}^{\mathbb{A}}$ whose scalar manifolds are not homogeneous spaces for $N\neq 3$. Second  column lists
  their
global symmetry  groups $\textrm{Aut}
(J_{(1,N)}^{\mathbb{A}})$ and their maximal compact subgroups $K$. Third  column lists
dimension of the irrep of tensor fields under $\textrm{Aut}
(J_{(1,N)}^{\mathbb{A}})$   and their decomposition with respect to the maximal compact subgroup $K$. The last column lists the number of scalars. The theories with $N=3$ have hidden larger symmetries and correspond to three of the magical supergravity theories. }
\end{table}
\begin{table}[ht]
\scriptsize
\begin{center}
\begin{displaymath}
\begin{array}{|c|c|c|c|}
\hline
~&~&~&~\\
J &  \textrm{Global symmetry group}
& \textrm{Number of self-dual tensor fields} & \textrm{Scalar Manifold}  \\
\hline
~&~&~&~\\
J_{3}^{\mathbb R} & SL(3,\mathbb{R})\supset SO(3) &
6= 5 \oplus 1 & SL(3,\mathbb{R})/SO(3) \\
~&~&~&~\\
J_{3}^{\mathbb C}  & SL(3,\mathbb{C})\supset SU(3) & 9= 8  \oplus 1  & SL(3,\mathbb{C})/SU(3) \\
~&~&~&~\\
J_{3}^{\mathbb H}  & SU^*(6)\supset USp(6) & 15=14 \oplus 1  & SU^*(6)/USp(6) \\
~&~&~&~\\
J_{(3)}^{\mathbb O} & E_{6(-26)} \supset F_4 & 27=26 \oplus 1 & E_{6(-26)}/F_4 \\
~&~&~&~ \\
\hline
\end{array}
\end{displaymath}
\end{center}

\caption{\label{unifiedMESGT} Unified tensor-Einstein supergravity theories in $d=6$ that reduce to magical supergravity theories in $d=5$ defined by simple Euclidean Jordan algebras $J_{3}^{\mathbb{A}}$ of degree three where $\mathbb{A}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$.
 Second  column lists
  their
global symmetry  groups  and their maximal compact subgroups. Third  column lists
the number of tensor fields  and their decomposition with respect to the maximal compact subgroup. The last column lists their scalar manifolds which are symmetric spaces. }
\end{table}

As is evident from the Tables \ref{unifiedTESGT} and \ref{unifiedMESGT} 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 $USp(2)$.

Since one can gauge a subgroup of the global symmetry group of the unified MESGTs in $d=5$ the corresponding six dimensional tensor-Einstein supergravity theories must 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 \cite{Samtleben:2011fj,Samtleben:2012fb,Samtleben:2012mi}.  It is an open problem whether some of these theories can be coupled to non-metric
$(1,0)$ graviton supermultiplet to obtain Lagrangian formulation of some of the $(1,0)$ supersymmetric gauged tensor-Einstein supergravity theories.


\section{Dimensional reduction of non-metric $(4,0)$ and $(1,0)$ tensor Einstein supergravity theories}

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 \cite{Chiodaroli:2011pp}. 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 indices ${\hat \alpha},{\hat \beta}, ..$ of the Lorentz group $SU^*(4)$.
Since the spinor representation $S_{\hat \alpha}$  of 6 dimensional Lorentz group $SU^*(4)$ and symmetric tensor representations $ S_{{\hat \alpha}{\hat \beta}{\hat \delta},...}$ 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 \cite{Chiodaroli:2011pp} 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 $R_{({\hat \alpha}{\hat \beta}{\hat \delta}{\hat \gamma})}$ reduces to the five dimensional graviton field strength without an extra vector or scalar field\cite{Hull:2000rr,Chiodaroli:2011pp}. 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 $\psi^A_{({\hat \alpha}{\hat \beta}{\hat \gamma})}$ remains irreducible under restriction to $USp(2,2)$ subgroup of $SU^*(4)$ and becomes the field strength of a gravitino in $5d$:
\[ \psi^A_{(\alpha\beta\gamma)}= \partial_{[\mu} \psi^A_{\nu] (\alpha} *\gamma^{\mu\nu}_{\beta\gamma)} \]
where $*\gamma^{\mu\nu}=\frac{1}{3!}\epsilon^{\mu\nu\lambda\epsilon\delta} \gamma_{\lambda\epsilon\delta}$.
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 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 \cite{Gunaydin:2010fi} 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 $6d$. Furthermore  we expect the tensor Einstein supergravity theories admit interactions that reduce to Yang-Mills Einstein supergravity theories in five dimensions with simple and semisimple gauge groups.
\section{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 \cite{Nahm:1977tg}. 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 based on simple superconformal algebras exist only in $d\leq 6$.

 On the other hand Poincare superalgebras, which are not simple, exist in any dimension. However maximal dimension for Poincare supergravity is $d=11$ \cite{Nahm:1977tg}. The Poincare superalgebra in $d=11$ with 32 supercharges can be embedded in the simple Lie superalgebra $OSp(1|32,\mathbb{R})$ with the even subalgebra $Sp(32,\mathbb{R})$ \cite{vanHolten:1982mx,Townsend:1997wg,Gunaydin:1998bt,Bergshoeff:2000qu}.
Its contraction to 11 dimensional Poincare superalgebra involves tensorial central charges
\begin{equation}
\{ Q_{A}, Q_{B} \} = (C\Gamma^M)_{AB} P_M + \frac{1}{2}
(C\Gamma^{MN})_{AB} Z_{MN} + \frac{1}{5!} (C\Gamma^{M_1..M_5})_{AB}
Y_{M_1 ..M_5}  \label{eq:malgebra}
\end{equation}
where $\Gamma^M  (M,N=0,1,..,10)$ are the  Dirac gamma matrices and $C$ is the charge conjugation matrix.
The embedding is such that the fundamental representation of $Sp(32,\mathbb{R})$ is identified with the Majorana spinor of the Lorentz group $SO(10,1)$. The $Z_{MN}$ and $Y_{M_1,...,M_5}$ are the tensorial central charges  with $P_M$ representing translations.


 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 Lie superalgebra $OSp(8^*|8)$ to higher dimensions. For 64 supercharges the possibilities are $OSp(16^*|4)$ and $OSp(32^*|2)$. In this context we should note that $SO^*(4n)$ is the conformal group of a spacetime coordinatized by the Euclidean Jordan algebra $J_n^{\mathbb{H}}$ of $ n \times n $ Hermitian matrices over the division algebra of quaternions \cite{Gunaydin:1992zh}. The conformal groups of space-times defined by Euclidean Jordan algebras all admit positive energy unitary representations \cite{Gunaydin:1992zh} and were shown to be causal space-times in \cite{Mack:2004pv}. The quaternionic Jordan algebra  $J_2^{\mathbb{H}}$ describes the standard $6d$ Minkowskian space-time with the Lorentz group $SU^*(4)$.  The Lorentz group of the spacetime defined by $J_n^{\mathbb{H}}$ is $SU^*(2n)$. Using the oscillator method \cite{Gunaydin:1984wc,Gunaydin:1990ag} one can construct the positive energy unitary supermultiplets of  $OSp(16^*|4)$ and $OSp(32^*|2)$ , in particular the analogs of the $(4,0)$ and $(2,0)$ supermultiplets readily. The twistorial oscillators transform in the fundamental representation of $SU^*(2n)$ and its conjugate.  Whether one can construct interacting  generalized superconformal theories based these supermultiplets and how they may be  related to M/Superstring  theory is an interesting open problem and will be the subject of a separate study.

 {\bf Note added: } After the completion of this paper two papers on $(4,0)$ supergravity in $6d$ appeared \cite{Bertrand:2020nob,Minasian:2020vxn}. In \cite{Bertrand:2020nob} the free $(4,0)$ theory is studied from the point of view of exceptional field theory. The authors of  \cite{Minasian:2020vxn} study  the anomalies in $(4,0)$ supergravity and its truncations  and argue that only their dimensional reduction to three dimensions make contact with standard metric supergravity theories.


 {\bf Acknowledgements:} The results reported above were obtained  at the  Stanford Institute of Theoretical Physics during my visit there  during the Summer/Fall of 2018. I would like to thank SITP for its kind hospitality. I would like to thank Marc Henneaux for posing a question that triggered this invesigation and  Henning Samtleben for a discussion on an earlier draft of this paper. Thanks  also to Marco Chiodaroli and  Radu Roiban for their helpful comments on the final draft of this paper.

 