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".

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