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.

This post imported from StackExchange Physics at 2014-06-27 15:38 (UCT), posted by SE-user Luboš Motl