Lets continue with compactifications of maximal supergravities. As it is now clear from the previous post the Kaluza-Klein reduction leads to a theory whose vector multiplets are abelian. In other words the gauge group is $U(1)^{n_V}$ with $n_V$ as number of vector multiplets. And none of the matter fields is charged under the gauge group. At the same time both scalars and tensors transform non-trivially under the global duality group. Scalar are elements of the coset space (see the previous post) and vectors transform in some linear representation $\mathcal{R}_{n_V}$. This is what is called ungauged supergravity.
An obvious question arises then: how to get supegravities with non-abelian structure? There are two possible ways of doing this that are represented by this picture
One way is to follow the diagonal arrow that means compactifying not on a torus but on a more complicated manifolds: spheres, Calabi-Yau manifolds etc. Or alternatively compactifying in presence of non-zero p-form fluxes. Mariana Grana wrote a nice review of string compactifications in presence of fluxes.
The other way is represented by the horizontal line on the picture. It suggests to firstly compactify on a torus and get ungauged supergravity. Then deform it in an appropriate way incorporating non-abelian structure into the theory. This procedure is called gauging and the corresponding review was written by Henning Samtleben.
I'll try here to sketch the basic idea of the so-called embedding tensor technique that is the most systematic approach to gaugings of supergravities. It's main feature is that it is a covariant approach.
Recall the transformations of vectors and scalars of the ungauged sugra under the local symmetry group
\begin{equation}
\begin{split}
&\delta \phi_{MN}=0,\\
&\delta A_\mu^M=\partial_\mu\Lambda^M.
\end{split}
\end{equation}
Here I changed the notation for the transformation parameter $\alpha^n$ which is now $\Lambda^M$. The scalar field $\phi_{ab}$ doesn't transform under the local group that is abelian.
The idea is to deform these transformations introducing non-abelian structure constants $X_{MN}^K$
\begin{equation}
\begin{split}
&\delta \phi =g\Lambda^M X_M \phi,\\
&\delta A_\mu^n=\partial_\mu\alpha^n+g X_{PQ}{}^MA_\mu^P\Lambda^Q =D_\mu\Lambda^M.
\end{split}
\end{equation}
Here $X_M$ denote generators of our non-abelian gauge group and $M$ runs from 1 to $\mathcal{R}_{n_V}$.
Obviously this cannot be done in an arbitrary way since one has to respect SUSY, duality symmetries of the theory etc. It appears that one can go through all the restrictions if the gauge group is a subgroup of the global duality group. And the embedding of the gauge group inside the global group is given by the embedding tensor
\begin{equation}
X_{M}=\Theta_M^{\alpha}t_\alpha.
\end{equation}
The embedding tensor $\Theta_M^\alpha$ chooses a subset of generators $X_M$ among generators of the global group $t_\alpha$. The structure constants are just components of the generators in the representation $\mathcal{R}_{n_V}$
\begin{equation}
\label{embed}
X_{MN}{}^K=(X_M) _N{}^K=\Theta_M^{\alpha}(t_\alpha){}_N{}^K.
\end{equation}
Now we set the embedding tensor to be gauge invariant and this leads to the closure of the algebra
\begin{equation}
\label{clos}
[X_M,X_N]=-X_{MN}{}^{K}X_K.
\end{equation}
Very important here is that the structure constants are not necessarily antisymmetric. They contain an symmetric part $Z_{MN}{}^P$ that is non-zero. Moreover, the antisymmetric part satisfy Jacobi identities only up to a term that is proportional to $Z_{MN}^K$. The closure constraint is called quadratic constraint since it is quadratic in $\Theta$.
The other constraint that is called linear constraint appears as a consequence of supersymmetry. In order to provide a supersymmetric theory the embedding tensor should belong to a particular representation. And projection to any other representations should be zero. Schematically this reads
\begin{equation}
\mathbb{P}\Theta=0.
\end{equation}
For example, consider reduction from 11 to 5 dimensions. There are 27 vector fields that means dim$\mathcal{R}_{n_V}=27$. The global duality group is $E_6$ that is 78 dimensional, This means that $\alpha$ runs from 1 to 78. The indices of $\Theta_M^{\alpha}$ implies that it belongs to
\begin{equation}
\mathbf{27\otimes78=27\oplus351\oplus\overline{1728}},
\end{equation}
where the decomposition is done with respect to $E_6$. The linear constraint restrict the embedding tensor to only $\mathbf{351}$ representation. And then one spends some time playing around with all these representations, invariant tensors, projectors and so on. This can be easily found in spires by keywords like "gauged" and "supergravity". There are bunch of papers written by Nicolai, Samtleben, de Wit and others.
The interesting thing happens to the field strength of the vector field. Evidently, it becomes deformed as well
\begin{equation}
F_{\mu\nu}{}^M=\partial_{[\mu}A_{\nu]}^M +gX_{[NP]}{}^{M}A_{\mu}^NA_{\nu}^P.
\end{equation}
The fact that the structure constants are no antisymmetric leads to deformation of the transformation of the field strength.
\begin{equation}
\delta F_{\mu\nu}{}^M=- g\Lambda^PX_{PN}{}^{M}F_{\mu\nu}^N+g Z_{PQ}{}^M\Phi^{PQ}.
\end{equation}It transforms in adjoint only up to further contraction with $X_M$.
To deal with this people introduce 2-form $B_{\mu\nu}^{PQ}$ that contributes to $F_{\mu\nu}$ and defines the full-covariant field strength $H=F+Z B$ that transforms correctly. This leads to a deformation of the transformation of the vector field
\begin{equation}
\delta A_\mu^M=D_\mu\Lambda^M+Z_{PQ}{}^M\Xi_{\mu}^{PQ},
\end{equation}
where $\Xi$ label the gauge transformation of the field $B_{\mu\nu}$. Now, the field strength for $B$ needs in an extra field too and so on. So when it stops?
Obviously, one cannot just add new fields to a supersymmetric theory. These 2-forms and other fields are taken from the set of the fields that already present in the theory. We just deform their transformations in an appropriate way. Hence, the existence of all these papers on the gauged supergravities claims that this procedure can be done in a self-consistent way.
The problem that appears here is a consequence of staying covariant under the global duality group. In most case the dimension of the local gauge group is strictly smaller than $n_V$ and thus not all $X_M$ are linearly independent. Thus the set of the vector fields splits on to part
\begin{equation}
A_{\mu}^M=\left\{
\begin{array}{ll}
A_{\mu}^m & \mbox{transform linearly under gauge group}\\
A_{\mu}^{i} & \mbox{transform in some other representation}
\end{array}
\right.
\end{equation}
For example, in five dimension we have 15 fields of 27 that correspond to the gauge group $SO(6)$ and describe reduction on sphere. The other 12 vector fields are turned into two-forms and become massive. In four dimensional case we have 28 magnetic vector fields that do not participate in gauging and 28 electric vector fields that provide local $SO(8)$.
At the end I would like to mention a gauging with a funny name trombone gauging. This corresponds to the Weyl group $\mathbb{R}^+$ of rescalings of the metric. Thus, starting with the global duality group $\mathbb{R}^+\times G$ one applies embedding tensor technique and finds out that no the embedding tensor is allowed to have $\mathcal{R}_{n_V}$ in is decomposition. For example in five dimension we will have
\begin{equation}
\widehat{\Theta}_M^\alpha \in \mathbb{27\oplus351}.
\end{equation}
Since the Weyl symmetry is a symmetry of the equations of motion and no of the action the trombone gauging does not appear in the action.
One could ask if it is possible to obtain all the gaugings by dimensional reduction. The answer was "no" before bunch of papers on reduction of the extended space formalism in string and M theory by Grana, Roest, Berman, Aldazabal and others. Since then the answer is "may be". The point is that one should dimensionally reduce not the ordinary space-time manifold but some other object called extended space. This leads to all gaugings of the supergravity even to those that cannot be obtained from ordinary dimensional reduction. But nobody knows if it is possible to solve all the equations that give gaugings in terms of reduction matrices in a consistent way.
An obvious question arises then: how to get supegravities with non-abelian structure? There are two possible ways of doing this that are represented by this picture
The other way is represented by the horizontal line on the picture. It suggests to firstly compactify on a torus and get ungauged supergravity. Then deform it in an appropriate way incorporating non-abelian structure into the theory. This procedure is called gauging and the corresponding review was written by Henning Samtleben.
I'll try here to sketch the basic idea of the so-called embedding tensor technique that is the most systematic approach to gaugings of supergravities. It's main feature is that it is a covariant approach.
Recall the transformations of vectors and scalars of the ungauged sugra under the local symmetry group
\begin{equation}
\begin{split}
&\delta \phi_{MN}=0,\\
&\delta A_\mu^M=\partial_\mu\Lambda^M.
\end{split}
\end{equation}
Here I changed the notation for the transformation parameter $\alpha^n$ which is now $\Lambda^M$. The scalar field $\phi_{ab}$ doesn't transform under the local group that is abelian.
The idea is to deform these transformations introducing non-abelian structure constants $X_{MN}^K$
\begin{equation}
\begin{split}
&\delta \phi =g\Lambda^M X_M \phi,\\
&\delta A_\mu^n=\partial_\mu\alpha^n+g X_{PQ}{}^MA_\mu^P\Lambda^Q =D_\mu\Lambda^M.
\end{split}
\end{equation}
Here $X_M$ denote generators of our non-abelian gauge group and $M$ runs from 1 to $\mathcal{R}_{n_V}$.
Obviously this cannot be done in an arbitrary way since one has to respect SUSY, duality symmetries of the theory etc. It appears that one can go through all the restrictions if the gauge group is a subgroup of the global duality group. And the embedding of the gauge group inside the global group is given by the embedding tensor
\begin{equation}
X_{M}=\Theta_M^{\alpha}t_\alpha.
\end{equation}
The embedding tensor $\Theta_M^\alpha$ chooses a subset of generators $X_M$ among generators of the global group $t_\alpha$. The structure constants are just components of the generators in the representation $\mathcal{R}_{n_V}$
\begin{equation}
\label{embed}
X_{MN}{}^K=(X_M) _N{}^K=\Theta_M^{\alpha}(t_\alpha){}_N{}^K.
\end{equation}
Now we set the embedding tensor to be gauge invariant and this leads to the closure of the algebra
\begin{equation}
\label{clos}
[X_M,X_N]=-X_{MN}{}^{K}X_K.
\end{equation}
Very important here is that the structure constants are not necessarily antisymmetric. They contain an symmetric part $Z_{MN}{}^P$ that is non-zero. Moreover, the antisymmetric part satisfy Jacobi identities only up to a term that is proportional to $Z_{MN}^K$. The closure constraint is called quadratic constraint since it is quadratic in $\Theta$.
The other constraint that is called linear constraint appears as a consequence of supersymmetry. In order to provide a supersymmetric theory the embedding tensor should belong to a particular representation. And projection to any other representations should be zero. Schematically this reads
\begin{equation}
\mathbb{P}\Theta=0.
\end{equation}
For example, consider reduction from 11 to 5 dimensions. There are 27 vector fields that means dim$\mathcal{R}_{n_V}=27$. The global duality group is $E_6$ that is 78 dimensional, This means that $\alpha$ runs from 1 to 78. The indices of $\Theta_M^{\alpha}$ implies that it belongs to
\begin{equation}
\mathbf{27\otimes78=27\oplus351\oplus\overline{1728}},
\end{equation}
where the decomposition is done with respect to $E_6$. The linear constraint restrict the embedding tensor to only $\mathbf{351}$ representation. And then one spends some time playing around with all these representations, invariant tensors, projectors and so on. This can be easily found in spires by keywords like "gauged" and "supergravity". There are bunch of papers written by Nicolai, Samtleben, de Wit and others.
The interesting thing happens to the field strength of the vector field. Evidently, it becomes deformed as well
\begin{equation}
F_{\mu\nu}{}^M=\partial_{[\mu}A_{\nu]}^M +gX_{[NP]}{}^{M}A_{\mu}^NA_{\nu}^P.
\end{equation}
The fact that the structure constants are no antisymmetric leads to deformation of the transformation of the field strength.
\begin{equation}
\delta F_{\mu\nu}{}^M=- g\Lambda^PX_{PN}{}^{M}F_{\mu\nu}^N+g Z_{PQ}{}^M\Phi^{PQ}.
\end{equation}It transforms in adjoint only up to further contraction with $X_M$.
To deal with this people introduce 2-form $B_{\mu\nu}^{PQ}$ that contributes to $F_{\mu\nu}$ and defines the full-covariant field strength $H=F+Z B$ that transforms correctly. This leads to a deformation of the transformation of the vector field
\begin{equation}
\delta A_\mu^M=D_\mu\Lambda^M+Z_{PQ}{}^M\Xi_{\mu}^{PQ},
\end{equation}
where $\Xi$ label the gauge transformation of the field $B_{\mu\nu}$. Now, the field strength for $B$ needs in an extra field too and so on. So when it stops?
Obviously, one cannot just add new fields to a supersymmetric theory. These 2-forms and other fields are taken from the set of the fields that already present in the theory. We just deform their transformations in an appropriate way. Hence, the existence of all these papers on the gauged supergravities claims that this procedure can be done in a self-consistent way.
The problem that appears here is a consequence of staying covariant under the global duality group. In most case the dimension of the local gauge group is strictly smaller than $n_V$ and thus not all $X_M$ are linearly independent. Thus the set of the vector fields splits on to part
\begin{equation}
A_{\mu}^M=\left\{
\begin{array}{ll}
A_{\mu}^m & \mbox{transform linearly under gauge group}\\
A_{\mu}^{i} & \mbox{transform in some other representation}
\end{array}
\right.
\end{equation}
For example, in five dimension we have 15 fields of 27 that correspond to the gauge group $SO(6)$ and describe reduction on sphere. The other 12 vector fields are turned into two-forms and become massive. In four dimensional case we have 28 magnetic vector fields that do not participate in gauging and 28 electric vector fields that provide local $SO(8)$.
At the end I would like to mention a gauging with a funny name trombone gauging. This corresponds to the Weyl group $\mathbb{R}^+$ of rescalings of the metric. Thus, starting with the global duality group $\mathbb{R}^+\times G$ one applies embedding tensor technique and finds out that no the embedding tensor is allowed to have $\mathcal{R}_{n_V}$ in is decomposition. For example in five dimension we will have
\begin{equation}
\widehat{\Theta}_M^\alpha \in \mathbb{27\oplus351}.
\end{equation}
Since the Weyl symmetry is a symmetry of the equations of motion and no of the action the trombone gauging does not appear in the action.
One could ask if it is possible to obtain all the gaugings by dimensional reduction. The answer was "no" before bunch of papers on reduction of the extended space formalism in string and M theory by Grana, Roest, Berman, Aldazabal and others. Since then the answer is "may be". The point is that one should dimensionally reduce not the ordinary space-time manifold but some other object called extended space. This leads to all gaugings of the supergravity even to those that cannot be obtained from ordinary dimensional reduction. But nobody knows if it is possible to solve all the equations that give gaugings in terms of reduction matrices in a consistent way.
