- #1

benbenny

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Im trying to learn about string theory in toroidal compactification on an undergraduate level. Im mostly using Zwiebach's "A first course in string Theory" but now im 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 dont 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, travelling 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 travelled 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 travelled 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 travelled [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