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Strings ChanPaton U(N) gauge symmetry fractional winding number... 
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#1
May1507, 06:51 AM

P: 599

I understand why in the presence of a constant vector potential
[tex]A=\frac{\theta}{2 \pi R}[/tex] along a compactified dimension (radius R) the canonical momentum of a e charged particle changes to P=peA. 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 [tex]p=\frac{K}{R} \frac{e\theta}{2 \pi R}[/tex] But now a 'Ndimensional gauge field' [tex]\frac{1}{2 \pi R} diag(\theta_1,..\theta_N}[/tex] is introduced. This has probably something to do with the fact that string states have two ChanPaton 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 [tex]e^{ip 2\pi R}=e^{i(\theta _i  \theta _j)}[/tex] 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?! 


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