# Strings Chan-Paton U(N) gauge symmetry fractional winding number

1. May 15, 2007

### da_willem

I understand why in the presence of a constant vector potential

$$A=-\frac{\theta}{2 \pi R}$$

along a compactified dimension (radius R) the canonical momentum of a -e charged particle changes to P=p-eA. Due to the single valuedness of the wavefunction [itex]\propto e^{iPX}[/tex] P should be K/R with K an integer so the momentum is

$$p=\frac{K}{R} -\frac{e\theta}{2 \pi R}$$

But now a 'N-dimensional gauge field' $$-\frac{1}{2 \pi R} diag(\theta_1,..\theta_N}$$ is introduced. This has probably something to do with the fact that string states have two Chan-Paton labels i,j (both 1..N) at the string endpoints. Now it is said that these states transform with charge +1 under U(1)_i and -1 under U(1)_j...

Can somebody maybe demystify things a bit for me...?!

Ultimately I want to understand why the string wavefunction now picks up a factor $$e^{ip 2\pi R}=e^{-i(\theta _i - \theta _j)}$$

There is some more talk about Wilson lines breaking the U(N) symmetry to the subgroup commuting with the wilson line (or holonomy matrix). This is all a bit above my head, especially given the short (read: no) explanation or claraification in my book (Becker, Becker, Schwarz).

Is there anyone who likes to clarify things for me?!