What are the geometric interpretations of the electric and colour charges in string theory?

Question

(This will be a partially self-answered question)

In string theory, the mass of a particle is determined by the the both the bosonic modes of the string \(\alpha_\mu\), and the fermionic modes \(d_\mu \), through the number operator \(N=\sum\limits_{n=1}^\infty \alpha_{-n}\cdot\alpha_n+\sum\limits_{r/2=1}^\infty d_{-r}\cdot d_r\). On the contrary, the spin number of a particle is intepreted in relation to the fermionic modes of the string \(d_\mu \). This is interesting, because one could say that the mass is related to the vibration of the string, while the spin is related to the rotation, which is very meaningful. For instance, in the bosonic theory, spin is not explained, but the mass can be written in suitable natural units, as \(\sqrt{N-a}\) (the same expression holds with the superstring, but here there are non-zero fermionic modes). 

However, given that mass can be represented in such an elegant, geometric way, there should surely be a similiar geometric interpretation for, e.g., the electric charge, the colour charge, and the weak hypercharge. What are the various geometric interpretations of these, and how are they related (by stringy/QFT dualities)?

Comments

These SM groups usually arise as subgroups of a GUT group. 

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@MitchellPorter I know that, but I was not really asking for the string theoretic origin of the charges, but rather if this had a convenient geometric interpretation like the electric charge, which is linked to the winding number of a closed oriented string about a compactified dimension. This interpretation is linked to the gauge group U(1) of electromagnetism, but the same interpretation does not hold for the colour charge, for instance.

Answer 1

I cannot speak about colour charge, but electric charge sure has a rather interesting geometric interpretation in string theory, in fact, two such interpretations that are equivalent through T-duality.

The first such interpretation is not unique to string theory, and is far from being stringy in nature. It's also present in Kaluza-Klein theory, and also supergravity. Namely, the electric charge is proportional to the momentum of the particle in one of the compactified, periodic dimensions. In other words, the observed electric charge is the number of times the particle oscillates, or "circulates" in this compactified dimension. This is also where the U(1) symmetry of electromagnetism arises. In heterotic string theory, this symmetry is unified into the larger \(\operatorname{Spin}(32)/\mathbb{Z}_2\) or \(E_8\times E_8 \) symmetry, which is a superset of \(U(1)\) symmetry.

The other interpretation is much more stringy in nature. That is, the electric charge is equivalent to, in a suitable choice of natural units, the winding number of the closed string. The sign of the charge is given by the direction of winding; this implies that only theories with closed oriented strings allow for this interpretation.

These two descriptions are T-dual to each other, in other words, they are equivalent descriptions, through T-duality.