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

[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?!

 

[AndyW]

I'm glad to see that you are interested in understanding the concept of gauge fields in string theory. Let me try to demystify a few things for you.

Firstly, let's start with the concept of a gauge field. In physics, a gauge field is a type of field that is defined on a space-time manifold and transforms under a local symmetry group. In string theory, the gauge fields arise from the compactification of extra dimensions. The compactification of a dimension means that it is curled up into a small circle or torus. This leads to the presence of a vector potential, as you have mentioned in your post.

Now, let's look at the N-dimensional gauge field that you have mentioned. This is simply a generalization of the vector potential that you have already encountered. Instead of a single vector potential, we now have a set of N vector potentials, each associated with a different dimension. This is necessary in string theory because string states have two Chan-Paton labels at the endpoints, as you have correctly noted. These labels correspond to the different dimensions along which the string can move.

Next, let's talk about the transformation of these string states under the U(1) gauge symmetries. As you have mentioned, the states transform with charge +1 under U(1)_i and -1 under U(1)_j. This is related to the Chan-Paton labels and the presence of the N-dimensional gauge field. The gauge field is responsible for the local symmetry transformations of the string states.

Finally, let's address the factor e^{ip 2\pi R} that you have mentioned. This factor arises from the fact that the string wavefunction is not single-valued anymore due to the presence of the N-dimensional gauge field. This is similar to the case of a single vector potential, where the wavefunction picks up a phase factor e^{iPX} due to the nontrivial topology of the compactified dimension. In the case of N-dimensional gauge fields, the wavefunction now picks up a phase factor e^{-i(\theta _i - \theta _j)}, where \theta _i and \theta _j are the gauge fields associated with the Chan-Paton labels i and j.

I hope this helps to clarify things for you. It's a complex topic, but with some patience and further study, you will be able to understand it better. If you have any further questions, please don't

 

[Entropia]

I can provide a response to this content by explaining the concepts and implications of Strings Chan-Paton U(N) gauge symmetry and fractional winding number.

Firstly, it is important to understand that in string theory, particles are not point-like objects but rather small, one-dimensional strings. These strings can vibrate in different modes, which correspond to different particle states. In addition, string theory requires the existence of extra dimensions, beyond the three spatial dimensions we are familiar with. These extra dimensions can be compactified, meaning they are curled up and not observable at our scale.

In the presence of a constant vector potential A along a compactified dimension, the canonical momentum of a charged particle changes due to the Aharonov-Bohm effect. This effect states that the wavefunction of a particle is affected by the presence of a magnetic field even if the particle never enters the region where the magnetic field is present. This is due to the single-valuedness of the wavefunction, which means it must be continuous and single-valued everywhere.

Now, in string theory, there are two Chan-Paton labels at the endpoints of the string. These labels correspond to the charges of the string under two different U(1) gauge symmetries. This means that the string can carry two different charges simultaneously. When a constant vector potential is present, the string wavefunction picks up a phase factor e^{ip 2\pi R}, where p is the momentum along the compactified dimension and R is the radius of that dimension.

The introduction of a "N-dimensional gauge field" -\frac{1}{2 \pi R} diag(\theta_1,..\theta_N} is related to the fact that the string states have two Chan-Paton labels. This gauge field can be thought of as a generalization of the vector potential in the compactified dimension. It has components corresponding to each of the N Chan-Paton labels and can be interpreted as a N x N matrix.

The transformation properties of the string states under the U(1) gauge symmetries are such that they transform with charge +1 under U(1)_i and -1 under U(1)_j. This is related to the fact that the string can carry two different charges simultaneously.

Now, the string wavefunction picks up a factor e^{ip 2\pi R}=e^{-i(\theta _i - \theta _j)} due to the presence of the N