Kaluza-Klein in superstring theory

In superstring theory, it says that they wrap 16 dimensions on a torus given by $\mathbb{R}^{16}$ divided by a SO(32) or $E_8 \times E_8$ lattice and this gives a gauge group of the same name.

But in Kaluza-klein theory it is the isometry group of the compact dimensions that gives the gauge group.

Doesn't a 16 dimensional torus have isometry group of $U(1)^n$ since it is the product of $S_1$?

Or is the reduction of the 16 dimensions different to usual Kaluza-Klein compactification?

I'm struggling to see how compactification on an $E_8$ torus would give an $E_8$ gauge group especially as $E_8$ has no 8 dimensional representation? If anything it could only be the isometry of a 248 dimensional surface...

Answer

You are talking about the two ways to construct a consistent 10-dimensional heterotic string theory. The original paper "Heterotic string theory: I. The free heterotic string" by Gross, Harvey, Martinec and Rohm is rather accessible and describes the detailed construction. I'll address your specific question of how the compactification on a torus manages to generate such "large" groups as $\mathrm{SO}(32)$ and $E_8\times E_8$ without giving the details.

You are correct that we usually would expect only a Kaluza-Klein style $\mathrm{U}(1)^{16}$ from compactifying on a 16-dimensional torus. However, this gauge group as a gauge group for a 10d SUGRA theory is forbidden since the gravitational and gauge anomalies don't cancel, so not only does this construction not yield the known heterotic string, it doesn't yield a consistent effective theory at all!

The crucial point is "choosing the right torus", that is, the ratios of the radii of the 16 circles must be specifically chosen to yield a consistent theory. The usual way to encode this choice is to think of the $T^{16}$ as $\mathbb{R}^{16}/\Gamma$, where $\Gamma$ is a discrete sixteen-dimensional lattice. Now one examines the bosonic part of the spectrum of a closed heterotic string on this background. It turns out that the excitations that correspond to the string being "wound" around the compactified dimensions become massless at special choices of $\Gamma$. Those excitations, together with the usual massless Kaluza-Klein modes, now together transform as the adjoint of a larger gauge group $G$ closely related to the lattice $\Gamma$, which turns out to be the root lattice of $G$. Now the dimensions of the geometry and the groups match in the way that the dimension of the lattice/torus corresponds to the rank (not the dimension) of the group, so the 16-dimensional torus has no issue generating the 496 dimensional $E_8\times E_8$.

Further considerations concerning the consistency of the interacting string theory heavily constrain the lattice $\Gamma$ to be integral, self-dual, and even. In 16 dimensions, the only two such lattices that exist are those associated with $E_8\times E_8$ and $\mathrm{SO}(32)$. As an interesting side note, it has recently (2010) been shown by Adams, deWolfe and Taylor in "String universality in ten dimensions" that the other two choices of gauge groups, in particular $\mathrm{U}(1)^{496}$, do not possess a consistent Green-Schwarz mechanism and are anomalous, so these two groups are really the only allowed gauge groups for a 10D SUGRA $\mathcal{N}=1$ gauge theory.