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I'm taking a course in String Theory, a first course, and I wish to perform a calculation. Related to that I have a question, I request someone to please help me with it:

There is a closed string in 26 dimensions and one of the dimensions is compactified into a circle. The winding number along this circle is $w$ and the momentum excitation number corresponding to this compactified dimension is $n$.

Now this string goes through a one-loop scattering. There are thus two branches into which the string has split and then it rejoins to make a single closed string and goes away to infinity.

My question is that throughout this process, is it possible to keep $w$ fixed while $n$ changes? That is, in each of the branches (the propagators), the momenta become loop momenta and are thus variable. However we wish to keep $w$ fixed. Can we perform an experiment tuned to these specific conditions ? The answer helps in the following way:

Once we make one component of the momentum of a string discrete and then make the string go through a loop, we would need to divide the integer into two integer parts (or more parts depending on the number of loops). This limits the number of solutions possible for the momenta running through the loops due to the discrete property of that momentum component. This is because a positive integer can be divided into n smaller parts only in finitely many ways. This problem of limited solutions exists only if the momentum takes positive integer values.

Thus if we take the modulus of the winding number to be constant and greater than the ultraviolet momentum cut-off, the discrete component of the momentum which goes as (n-w) [n=momentum excitation number, w= winding number] will be positive, although variable, in the external leg as well as in the loop. Then this component of the external momentum, which is positive needs to get segregated into two or more positive integer parts which has only limited solutions/possibilities.

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  • $\begingroup$ Well there is T-duality exchaning $n$ and $w$ so i don't think you can keep $w$ fixed in a duality-preserving way. $\endgroup$
    – ɪdɪət strəʊlə
    Commented Nov 15 at 9:32
  • $\begingroup$ Yes that's what my professor also told me... But I was thinking that in the dual space it would mean that the string is going through a loop whose dynamical variable is winding number, not it's excitation of momentum (which we now hold fixed in this dual space). This is because a interaction process can take place with some variables changing in the loop and some remaining constant... We just have to reconsider what kind of process we want. $\endgroup$
    – SX849
    Commented Nov 15 at 11:36
  • $\begingroup$ There's no dual space. Say you do your computation at the critical radius $R=\sqrt{\alpha'}$. The string doesn't know which variable is winding and which one is momentum. $\endgroup$
    – ɪdɪət strəʊlə
    Commented Nov 15 at 13:33
  • $\begingroup$ Okay I understood... Thanks a lot for your answer, it clears my long standing doubt :) $\endgroup$
    – SX849
    Commented Nov 15 at 15:05

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