At the moment we have 5 different string theories that are connected to each other by duality transformations.
As it is shown on this picture low energy limits of IIA and Heterotic $E_8\times E_8$ string theories can be obtained by compactification of the 11D SUGRA on a circle $S^1$ and a line $I$ respectively.
What we know is that the $11$ dimensional supergravity is a low energy limit of the mysterious M-theory. Actually, this is more or less the only thing we know about M-theory. This means that it is worth understanding supergravity.
So what supergravity actually is? Obviously, it is supersymmetric version of the ordinary gravity -- General Relativity. The most transparently it is constructed in 11 dimensions. This is because we are forced to use $\mathcal{N}=1$ SUSY in order to have only spins $\leq2$.
The field content of the 11 dimensional SUGRA contains graviton field $g_{MN}$ (44 components), gravitino $\psi_{M}$ (128 components) and 3-form field $C_{MNK}$ (84 components).
But wait, where are vector fields that we have around us? And moreover we do not live in 11 dimensions. That's the point. To get supergravities in lower dimensions (e.g. $D=4$) it is easier to use dimensional reduction from 11 dimensions rather that construct them in lower dimensions directly.
Let's go say to 4 dimensions $\mu=1..4$ and $m=1..7$ with $x^\mu$ to be space-time coordinates and $y^m$ that denote inner coordinates. We obtain the following fields (bosonic part)
\begin{equation}
\begin{split}
g_{MN} &\rightarrow g_{\mu \nu}, A_{\mu, m}, \phi_{a b};\\
C_{MNK} &\rightarrow C_{\mu\nu\rho}, B_{\mu\nu,m}, \tilde{A}_{\mu,a b}, \tilde{\phi}_{abc}.
\end{split}
\end{equation}
Thus we have 1 graviton, $28=7+21$ vector fields, one 3-form, seven 2-forms and $63 = 28 + 35$ scalar fields.
Here we note, that the field strength for the 2-form is a 3-form $F=dB$, its Hodge dual is 1-form. And a 1-form field strength corresponds to a scalar $*F=d\rho$. That is to say that a 2-form in 4 dimensions is dual to a 0-form and we have 7 additional scalars. Finally
\begin{equation}
\begin{split}
1 & graviton\\
7+21=28 & vectors\\
28+35+7=70 & scalars.\\
\end{split}
\end{equation}
Three form represents fluxes since its field strength is a 4-form and has no dynamics in 4 dimensions. For a while lets forget about it.
The field content we obtained looks like $\mathcal{N}=8$ supergravity multiplet in 4 dimensions (its bosonic part). The only problem is that we need 56 vector fields but have only half of it. The solution is in the self-duality of 2 form field strength in 4 dimensions. These additional 28 vector fields are magnetic duals of the ones we have. They do not show up in the Lagrangian.
For the details of all this I send you to the great lectures by Prof. Samtleben that are very clear and readable.
From these lectures we find out that the diffeomorphisms in the 11 dimensional space
\begin{equation}
\delta g_{MN}=\left(\mathcal{L}_{\xi^A}g\right)_{MN}=\xi^A\partial_Ag_{MN}+g_{AN}\partial_{M}\xi^A+g_{AM}\partial_{N}\xi^A
\end{equation}
induce diffeomorphisms in 4 dimensions, gauge transformations and some global $SL(n)$ transformations.
\begin{equation}
\begin{split}
&\delta g_{\mu\nu} = \left(\mathcal{L}_{\xi^\alpha}g\right)_{\mu\nu},\\
&\delta A^{n}_{\mu}=\left(\mathcal{L}_{\xi^\alpha}A^n\right)_{\mu}+\partial_\mu\alpha^n-G^n_mA^m_\mu,\\
&\delta\phi_{ab}=\mathcal{L}_{\xi^\alpha}\phi_{ab}+G_{a}^{n}\phi_{nb}+G_{b}^{n}\phi_{an}.
\end{split}
\end{equation}
Here the 11-dimensional parameter was splitted as $\xi^A(x,y)=(\xi^\alpha(x),\alpha^{m}(x)+G^m_ny^n)$. This choice is allowed by the demand that fields do not depend on the inner coordinates $y^m$.
Thus we see, that except of obvious diffeos we have a gauge group $U(1)^7$ acting on the gauge fields and a global group $SL(7)$ represented by the matrices $G^m_n$. I would like to emphasise that this compactification leads to abelian gauge fields.
More carefull analysis shows that the scalar fields $\phi_{ab}$ live in the coset space $SL(7)/SO(7)$. Here the orthogonal group $SO(7)$ appears as a group of local Lorentz transformations that act on flat indices of the vielbein. Indeed the dimension of this coset space is 27, that matches the previous counting.
Actually, the target space of the scalar sector of the theory in question is bigger. It is $E_7/SU(8)$. Indeed, we have 56 vector fields that (appear to) transform in the $\mathbf{56}$ representation of $E_7$. And 70 scalar fields exactly match the dimension of the coset. Obviously, these are not the only reasons for this structure.
Moreover this is not the end of the story. Return to the 3-form that we forgot for a while previously. Its equations of motion are trivial and imply that the field strength is a constant giving one deformation parameter. This parameter appears to be one of the components of the $\mathbf{912}$ representation of $E_7$. This leads to non-abelian fields. But this topic is for some of the next posts.
What we know is that the $11$ dimensional supergravity is a low energy limit of the mysterious M-theory. Actually, this is more or less the only thing we know about M-theory. This means that it is worth understanding supergravity.
So what supergravity actually is? Obviously, it is supersymmetric version of the ordinary gravity -- General Relativity. The most transparently it is constructed in 11 dimensions. This is because we are forced to use $\mathcal{N}=1$ SUSY in order to have only spins $\leq2$.
The field content of the 11 dimensional SUGRA contains graviton field $g_{MN}$ (44 components), gravitino $\psi_{M}$ (128 components) and 3-form field $C_{MNK}$ (84 components).
But wait, where are vector fields that we have around us? And moreover we do not live in 11 dimensions. That's the point. To get supergravities in lower dimensions (e.g. $D=4$) it is easier to use dimensional reduction from 11 dimensions rather that construct them in lower dimensions directly.
Let's go say to 4 dimensions $\mu=1..4$ and $m=1..7$ with $x^\mu$ to be space-time coordinates and $y^m$ that denote inner coordinates. We obtain the following fields (bosonic part)
\begin{equation}
\begin{split}
g_{MN} &\rightarrow g_{\mu \nu}, A_{\mu, m}, \phi_{a b};\\
C_{MNK} &\rightarrow C_{\mu\nu\rho}, B_{\mu\nu,m}, \tilde{A}_{\mu,a b}, \tilde{\phi}_{abc}.
\end{split}
\end{equation}
Thus we have 1 graviton, $28=7+21$ vector fields, one 3-form, seven 2-forms and $63 = 28 + 35$ scalar fields.
Here we note, that the field strength for the 2-form is a 3-form $F=dB$, its Hodge dual is 1-form. And a 1-form field strength corresponds to a scalar $*F=d\rho$. That is to say that a 2-form in 4 dimensions is dual to a 0-form and we have 7 additional scalars. Finally
\begin{equation}
\begin{split}
1 & graviton\\
7+21=28 & vectors\\
28+35+7=70 & scalars.\\
\end{split}
\end{equation}
Three form represents fluxes since its field strength is a 4-form and has no dynamics in 4 dimensions. For a while lets forget about it.
The field content we obtained looks like $\mathcal{N}=8$ supergravity multiplet in 4 dimensions (its bosonic part). The only problem is that we need 56 vector fields but have only half of it. The solution is in the self-duality of 2 form field strength in 4 dimensions. These additional 28 vector fields are magnetic duals of the ones we have. They do not show up in the Lagrangian.
For the details of all this I send you to the great lectures by Prof. Samtleben that are very clear and readable.
From these lectures we find out that the diffeomorphisms in the 11 dimensional space
\begin{equation}
\delta g_{MN}=\left(\mathcal{L}_{\xi^A}g\right)_{MN}=\xi^A\partial_Ag_{MN}+g_{AN}\partial_{M}\xi^A+g_{AM}\partial_{N}\xi^A
\end{equation}
induce diffeomorphisms in 4 dimensions, gauge transformations and some global $SL(n)$ transformations.
\begin{equation}
\begin{split}
&\delta g_{\mu\nu} = \left(\mathcal{L}_{\xi^\alpha}g\right)_{\mu\nu},\\
&\delta A^{n}_{\mu}=\left(\mathcal{L}_{\xi^\alpha}A^n\right)_{\mu}+\partial_\mu\alpha^n-G^n_mA^m_\mu,\\
&\delta\phi_{ab}=\mathcal{L}_{\xi^\alpha}\phi_{ab}+G_{a}^{n}\phi_{nb}+G_{b}^{n}\phi_{an}.
\end{split}
\end{equation}
Here the 11-dimensional parameter was splitted as $\xi^A(x,y)=(\xi^\alpha(x),\alpha^{m}(x)+G^m_ny^n)$. This choice is allowed by the demand that fields do not depend on the inner coordinates $y^m$.
Thus we see, that except of obvious diffeos we have a gauge group $U(1)^7$ acting on the gauge fields and a global group $SL(7)$ represented by the matrices $G^m_n$. I would like to emphasise that this compactification leads to abelian gauge fields.
More carefull analysis shows that the scalar fields $\phi_{ab}$ live in the coset space $SL(7)/SO(7)$. Here the orthogonal group $SO(7)$ appears as a group of local Lorentz transformations that act on flat indices of the vielbein. Indeed the dimension of this coset space is 27, that matches the previous counting.
Actually, the target space of the scalar sector of the theory in question is bigger. It is $E_7/SU(8)$. Indeed, we have 56 vector fields that (appear to) transform in the $\mathbf{56}$ representation of $E_7$. And 70 scalar fields exactly match the dimension of the coset. Obviously, these are not the only reasons for this structure.
Moreover this is not the end of the story. Return to the 3-form that we forgot for a while previously. Its equations of motion are trivial and imply that the field strength is a constant giving one deformation parameter. This parameter appears to be one of the components of the $\mathbf{912}$ representation of $E_7$. This leads to non-abelian fields. But this topic is for some of the next posts.
