Reddit mentions of String Theory and Particle Physics: An Introduction to String Phenomenology

Sure, there is a lot of research on this, but sadly it's not even remotely as simple as this:

>Separately, I am wondering if there is any connection beyond coincidence whereby the 6 compactified dimensions are related to the dimensionality of the three gauge groups at play in the standard model: SU(3) x SU(2) x U(1). Such as theorems relating certain families of (Lie?) groups to certain classes of compactifications? Or put another way, do you know of any connection between group representation theory and the compactification topology/geometry?

There is a very complex interplay between compactification geometry+topology, brane configurations and fluxes, and low-energy matter, gauge groups and couplings. If you are interested I strongly suggest this book.

Actually I would say this is rather well-understood, in the sense that if you give a knowledgeable string phenomenologist a certain "mild enough" compactification he can work out the exact 4D effective theory, as complex as it may be it is essentially a "solved" problem.

Modern research is concentrated more on the problem of why a specific compactification should happen, so the problem of moduli stabilization and dynamical selection. Thus bringing down those 10^500 candidates to a reasonable number by dynamical arguments.

Let me just make a small technical point about gauge group, just to paint a sketch of how things tend to work out: it turns out it's hard to make the SM gauge group pop out in the Kaluza-Klein way as the isometries of the compact dimensions. It is much more fruitful to use brane stacks. N coincident identical D-branes have a natural SU(N) symmetry (just like N particles) and that will be gauged on their worldvolume. That is D-brane stacks naturally host SU(N) gauge theories - so the simplest string pheno models use a stack of 3 "colour" branes.

Or, you could start from the large gauge groups of the heterotic strings that they already automatically have and try to see what can be pulled out from them under compactifications.