How do compact dimensions determine the particle content of string theory?

Question

In string theory, 10 spatial dimensions are required for mathematical consistency. One way to model our 3-dimensional universe is by compactifying the extra dimension on a Calabi-Yau manifold. They are 'flat enough' to be vacuum solutions to gravity.

Different Calabi-Yau manifolds give different particle content. I would like to understand a bit (really anything) about how this works. I know on a cylinder, motional states around the circumference give a tower of states that look like a spectrum of progressively more massive particles. But somehow Calabi-Yau manifolds determine spin, mass, and charge under various gauge fields etc.. Is it possible to give any relatively simple geometric intuition for how different types of particles are 'represented' as motional states of strings on a particularly-shaped Calabi-Yau manifold?

Answer 1

I would like to answer this question for the case of E8xE8 heterotic string theory, since that was the basis of many models. But it doesn't really work the way you're thinking. For example, in 10 completely uncompactified space-time dimensions, the massless states of this string include E8xE8 gauge bosons, their gaugino superpartners, the graviton and the gravitino. All these states are created by particular combinations of left-moving and right-moving excitations of the string. In particular, this accounts for the spin.

So the uncompactified E8xE8 string approximately behaves like an E8xE8 gauge theory with N=1 supersymmetry, coupled to a supergravity field. Then when you compactify on a Calabi-Yau, E8xE8 reduces to E6xE8. Then, if this manages to reduce to something like the standard model, it is because of various forms of Higgs mechanism that break the gauge group (thus producing the charges) and that couple the chiral fermions (thus producing the masses).

There are a few geometric comments that can be made about a number of steps in this process - where E8xE8 comes from, how it breaks to E6xE8, how you get three generations. I may or may not get the time and clarity needed to describe them. But meanwhile I would emphasize that, for mass and charge at least, the way they are obtained in these models are by starting with an N=1 E6xE8 gauge theory, and then breaking the E6 and coupling the chiral fermions with a Higgs field, just as happens in a grand unified field theory. All of that happens at a scale where the string can be approximated as a point particle.