N-dimensional torus compactification and string winding

[benbenny]

Hello,

Im trying to learn about string theory in toroidal compactification on an undergraduate level. I am mostly using Zwiebach's "A first course in string Theory" but now I am trying to do something that doesn't seem to be covered in the book or any other literature explicitly. Perhaps because I am misunderstanding the issue. The periodicity condition for a closed string with winding number m, in a 1-D compact space is

[tex] X(\tau, \sigma) = X(\tau, \sigma +2\pi) +2 \pi R m \label{1} [/tex]

I can understand this if I imagine one extended dimension and one compactified dimension. Such a space looks like a cylinder and a closed string can be wrapped around the cylinder m number of times. To return to the same point on the string thus we have to travel a distance of 2\pi R m.

My confusion is when you generalize to arbitrary number of compactified dimensions:

[tex]X^{j}(\tau,\sigma)=X^{j}(\tau,\sigma+2\pi)+2\pi R_{j}m^{j} \label{2} [/tex]

I can understand this if I think about a particle in compact space and disregard the first term on the right hand side which pertains to a string: The distance a particle will travel to return to the same point is the sum over the lengths of all the compact dimensions in the space.

What I don't understand is why this equation holds for a string.

I have tried to think about the 2 compact dimensional case. In this case the compact space can be said to look like a 2-torus. Say you have a 2-D ring-torus with R the radius of the torus, and r the radius of the tube.

For a point on a string which is wrapped around the 2-torus, traveling around the r circle will bring us back to the same point after a distance of 2\pi r. But if the string travels along the torus, around the R circle it will have traveled a distance of 2\pi R only if r=0. Otherwise, as far as I can see, if the point is on the inwards side of the torus, it will have traveled a distance of [tex] 2 \pi (R-r) [/tex], and if the point on the string is on the outwards side of the torus it will have traveled [tex] 2 \pi (R+r) [/tex].

So as far as I can see the periodicity condition for a 2-D torus should be

[tex] X = X(\sigma+2\pi) +2\pi r m +2\pi (R+r\cos \theta) \label{3} [/tex]
where \theta is the parametrization angle of the r circle.

Then I have no idea how to proceed for a d-torus.

I would appreciate any help on this.

B

 

[Dick]

Maybe the word 'torus' is confusing you. The d dimensional case is just a cross product of d copies of 1-D compact space with periodicity condition pertaining in each of them. That's what the formula is trying to tell you. In the 2-d case that's not what you usually think of as the usual torus inheriting the 3-d metric. Sometimes this is called the 'flat torus'. You can't embed that in 3-d space.

 

[benbenny]

Thanks for the reply.

So I shouldn't try to make the analogy with the usual 2-torus in 3d space? is there another way to try to understand the concept of a string wrapping around a d-torus?

Thanks again.
B

 

[Dick]

Thanks for the reply.

So I shouldn't try to make the analogy with the usual 2-torus in 3d space? is there another way to try to understand the concept of a string wrapping around a d-torus?

Thanks again.
B

You can bend a flat piece of paper into a cylinder without distorting any of the point-to-point distances within the sheet of paper, right? That means a cylinder is still 'flat' in the metric sense. Now to make the torus you have to imagine you can do the same thing and join the two circular ends of the cylinder without distorting internal distances. In 3d you can't do this, the paper will wrinkle and tear. But don't let that get in the way of imagining it. Since you aren't letting it stretch all of the distances within the torus remain the same as within the original sheet of paper.

 

[benbenny]

Ok. So does it make sense then to imagine the 2-torus in a 4d space so that it remains flat?
I'm trying to convince myself that the periodicity condition for a torus is really just the sum of the lengths of the different circles(compact dimension) times the winding term.

For a 2-torus, which is the only kind if torus I can imagine, this doesn't seem to be the case.

 

[Dick]

Ok. So does it make sense then to imagine the 2-torus in a 4d space so that it remains flat?
I'm trying to convince myself that the periodicity condition for a torus is really just the sum of the lengths of the different circles(compact dimension) times the winding term.

For a 2-torus, which is the only kind if torus I can imagine, this doesn't seem to be the case.

I don't think you are supposed to read the periodicity condition as a sum over dimensions. It's a different condition for each dimension.

 

[benbenny]

I don't think you are supposed to read the periodicity condition as a sum over dimensions. It's a different condition for each dimension.

ah ok.

But the periodicity equation above seems like it would hold for any any type of compactification, isn't that so?

What is then the meaning of toroidal compactifcation? How is that different from other types of compactifications, in terms of the string's equations of motion, and possibly its effect on T-duality?

Im sorry if these questions seem general. I was told to look into multi-dimensional toroidal compactification, and I am trying to figure out what is meant by that.
I haven't been able to figure this out from Zwiebach, and all other texts that I can find are highly advanced. In those texts I have mostly seen toroidal copactification in the context of Conformal Field Theory, and although I have some background of complex analysis, it doesn't seem likely that this is what I am supposed to be studying at this point.

Its plausible that the answer to my confusion is something trivial.

Again, I appreciate the help.

Ben

 

[Dick]

Why would it hold for a general compactification? Toroidal is simple. More general ones aren't. They don't have to be periodic in each coordinate.

 

[benbenny]

i see