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The center of mass momentum Eq. (8.8) returns to the nonquantized, continuous momentum of the noncompacti ed case. Now let s think about the opposite limit where R 0. You might also expect that this is like returning to the noncompacti ed case. After all, taking the limit R 0 is like making the extra dimension go away. In quantum eld theory we might expect the elds to completely decouple from that unseen extra dimension. However, things don t quite work this way in string theory. As R 0, we nd that the Kaluza-Klein modes become in nitely massive and decouple from the theory. Since these can be regarded as particle states, maybe this isn t so surprising. What s left behind for the center of mass momentum are the winding states. First note that as R 0 we obtain p R = pL Hence p 25 0 but the winding term behaves in the following way: w= 1 25 25 25 pL pR pL 2
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25 (or to pR if you like). Now the winding states, rather than the momentum states, form a continuum of states. This should not be so surprising, as R 0 the circle gets smaller and smaller. So it gets easier and easier to wrap a string around it that is, it costs less energy. When the circle is very small it doesn t require a lot of energy to wrap the string around it. So you see as the radius gets very large or very small there is a trade-off between winding states and momentum. This trade-off leads us to a discussion of T-duality, the topic of the next section.
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CHAPTER 8 Compacti cation and T-Duality
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T-duality is a symmetry which exists between different string theories. This symmetry relates small distances in one theory to large distances in another, seemingly different theory and shows that the two theories are in fact the same theory expressed from different viewpoints. This is an important recognition; before T-duality was discovered it was believed that there were ve different string theories, when in fact they were all different versions of the same theory that could be related to one another by transformations or dualities. One can transform between small and large distances when considering the compacti ed dimension in one theory, and arrive at another dual theory. This is the essence of T-duality. We will see later that other dualities exist in string theory as well. T-duality relates type IIA and type IIB string theories, as well as the heterotic string theories. It applies to the type of compacti cation that we have been studying in this chapter, namely the compacti cation of a spatial dimension to a circle of radius R. The transformation that is used in T-duality is to transform the radius to a new large radius R which is de ned by the exchange R
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(8.29)
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The T-duality transformation also exchanges winding states characterized by a winding number n with high-momentum states in the other theory (Kaluza-Klein excitations). That is, n K (8.30)
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The symmetry of T-duality, described by these exchanges, makes its appearance in the mass formula [Eq. (8.27)], which we reproduce here:
2 2 nR K m 2 = + + 2( N R + N L ) 4 R
Now exchange R / R and n K , then: K n nR = R R
(8.31)
We also have: K nR R = K R
(8.32)
String Theory Demysti ed
So we see that the mass formula Eq. (8.27) is invariant under the exchange R / R and n K . It assumes the form nR K + + 2( N R + N L ) 4 m = R
2 2 2
That is, it keeps the same form but now with the new radius R . That s the math of the transformation. The physics is that if we started with a theory with a small compacti ed dimension R, we have transformed to a dual theory with a large extra dimension R . What this means for the string is that a string in a type IIA theory (with small compacti ed dimension), which winds around the small compact dimension (with winding states) is dual to a string in type IIB theory (with the dimension transformed to a large dimension of radius R ), which has momentum along that dimension. Each time the string in type IIA theory winds around the compact dimension, this corresponds to increasing the momentum in type IIB theory by one unit. 25 25 Now let s examine how pL and pR , and by extension 0 and 0, transform under this symmetry. Recall Eq. (8.22) that states
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