Kac-Moody algebras in 5 dimensional Kaluza-Klein theory I am trying to make sense to the issue of how does the Kac-Moody algebra encode the symmetries of the non-truncated theory.
Let's contextualize a little bit. Ok, so in the 5 dimensional Kaluza-Klein we start with pure (5 dimensional) gravity. Then we assume a $M_4×S^1$ ground state where we have Minkowski space with a circle. The topology of this ground state breaks the 5D diffeomorphism invariance because now only periodic transformations are allowed. So considering a coordinate change
$$x\to{}x'=x+\xi,$$
we can Fourier expand
$$\xi(x,\theta)=\sum{}\xi(x)e^{in\theta}.$$
It is known that if we truncate at $n=0$ we get a theory of gravitation in 4 dimensional space-time, electromagnetism and a scalar field (sometimes called the Dilaton).
So, the remnants of the initial 5 dimensional diffeo invariance are 4 dimensinal diffeo invariance,  $U(1)$ gauge and a scale invariance for the dilaton.
In the non truncated theory we will keep all the modes in the previous Fourier expansion and we will have more symmetries. The 4d diffeo and $U(1)$ gauge do survive in the non truncated version but not the scale invariance of the dilaton.
So far so good.
But, how to know the symmetries of the full theory? well,  with Kac-Moody generalization of Poincaré  algebras. For example in http://arxiv.org/abs/hep-th/9410046 says this. But HOW exactly? I would like a clear explanation of how this Kac-Moody algebra encodes the remnant symmetries of the original diffeo 5 and also how the zero modes of this algebra correspond to diffeo4 and $U(1)$
EDIT:: this is the algebra I am talking about
$[P_{\mu}^{(n)},P_{\nu}^{(m)}]=0$
$[M_{\mu\nu}^{(m)},P_{\lambda}^{(n)}]=i(\eta_{\lambda\nu}P_{\mu}^{(m+n)}-\eta_{\lambda\mu}P_{\nu}^{(m+n)})$
$[M_{\mu\nu}^{(n)},M_{\rho\sigma}^{(m)}]=i(\eta_{\nu\rho}M_{\mu\sigma}^{(m+n)}+\eta_{\mu\sigma}M_{\nu\rho}^{(m+n)}-\eta_{\mu\rho}M_{\nu\sigma}^{(m+n)}-\eta_{\nu\sigma}M_{\mu\rho}^{(m+n)})$
$[Q^{(n)},Q^{(m)}]=(n-m)Q^{(n+m)}$
$[Q^{(n)},P^{(m)}_{\mu}]=-mP^{(n+m)}_{\mu}$
$[Q^{(n)},M^{(m)}_{\mu\nu}]=-mM^{(n+m)}_{\mu\nu}$
 A: As the very formulation of your question makes clear, we know what the actual algebra of local symmetries is. It is the five-dimensional diffeomorphism invariance assuming the $M^4\times S^1$ topology of the five-dimensional spacetime.
The term "Kač-Moody generalization of an algebra" is nothing else than an alternative name for this algebra, especially for the form of the algebra that is obtained by considering not generators as continuous functions of the fifth coordinate $\theta$, but using the discrete Fourier modes in this direction labeled by $n$.
The original, Kač-Moody-non-generalized algebra has commutators of the generators like
$$ [G_i,G_j] = f_{ijk} G_k $$
with the appropriately raised indices. In your case, all $G_i$ are linear combinations of generators of diffeomorphisms, i.e. integrals of the stress-energy tensor with some tensor-valued coefficients as functions of the 4D spacetime.
The Kač-Moody generalization arises when all the generators are allowed to depend on the extra coordinate $\theta$ or, equivalently, to depend on the Fourier mode integer index $n$. Then the generalization replaces $G_i$ by $G^n_i$ and the commutator becomes something like
$$ [G_i^m,G_j^n] = (\pm m \pm n)^{\text{0 or 1}} f_{ijk} G_k^{m+n} $$
There may also be $n^3$ terms weighted by $\delta_{m,-n}$ etc. but I don't want to present all possible subtleties and generalizations of Kač-Moody algebras here.
The appearance of $m+n$ as the superscript on the right hand side is guaranteed by the $\theta\to \theta+c$ translational symmetry, the $U(1)$ gauge subalgebra you mentioned. Otherwise it's not suprising that we must get a similar  algebra to the original 4D one, just with some extra indices $m,n,m+n$ moderately  inserted and with some moderate coefficients.
To check the equivalence of the two descriptions, one only needs to know some basics of integrals and derivatives or the Fourier transform etc.
