research level - What is the connection between extra dimensions in Kaluza-Klein type theories and those in string theories?

This follows to some extent from a question I asked previously about the flaws of Kaluza-Klein theories.

It appears to me that Kaluza-Klein theories attach additional dimensions to spacetime that are related to the gauge freedoms of field theories. I believe the original model was to attach a $U(1)$ dimension to the usual 4-dimensional spacetime to reproduce electromagnetism. But, as explained in the answers to my previous question, these extra dimensions have all sorts of problems.

String theories also (famously?) require extra dimensions. So, is there a connection between the higher-dimensional descriptions? What do they have in common and how do they differ? For example, the $\rm U(1)\times SU(2)\times SU(3)$ group is 7-dimensional, which, when attached to the 4 dimensions of spacetime, gives 11 dimensions. I hear the same number is bandied about in string theory although there's no obvious reason they should be related at all.

Answer

The group manifold $U(1) \times SU(2)\times SU(3)$ is $1+3+8=12$-dimensional, not 7-dimensional.

You probably meant the dimension of a manifold that may have this group as its isometry group. But one may show that no such low-dimensional manifold can be interpreted as the extra dimensions of string theory to produce a realistic model.

The oldest Kaluza-Klein theory had an extra circular dimension whose isometry is $U(1)$. More generally, one may have more complicated manifolds with the isometry group $G$ (isometry is a map of the manifold onto itself, or a diffeomorphism, that preserves the metric at each point, the true "symmetry" of the manifold). The isometry group always becomes the gauge group in the lower-dimensional description. These facts about the Kaluza-Klein theory are fully reproduced as a low-energy feature of some string compactifications.

But as I have mentioned, realistic models with a large enough gauge group to include the Standard Model which would come purely from the original Kaluza-Klein mechanism don't exist in string theory. That's why realistic stringy vacua have a different origin of the gauge symmetries. For example, a stack of $N$ branes has a $U(N)$ gauge group which may become orthogonal or symplectic at the orientifold planes. M-theory and F-theory admit extra gauge groups from singularities. Heterotic string theory or Hořava-Witten heterotic M-theory contain extra $E_8$ gauge groups, already in the maximum dimension (or codimension one boundary, in the M-theory case) that are simply inherited (and partially broken) in four dimensions.

All these possibilities are related by various dualities (non-obvious but exact equivalences) in string theory. And in some sense, all of them are stringy generalizations of the original Kaluza-Klein theory. For example, the $E_8\times E_8$ or $SO(32)$ gauge group of the heterotic string comes from 16 chiral "purely left-moving" spacetime dimensions in the spacetime where the heterotic string may live. In some stringy sense, the gauge group may still be interpreted as the isometry of the manifold. Well, $U(1)^{16}$ arises as the standard isometry of the torus and the remaining generators of the gauge group have a "stringy origin" which may be interpreted as the "string-generalized geometry".