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25 25 25 ( pL + pR ) + ( pL25 pR ) + modes 2 2
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(8.7)
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CHAPTER 8 Compacti cation and T-Duality
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The total center of mass momentum of the string is
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25 25 p 25 = pL + pR
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Along the compacti ed dimension, the string acts like a particle moving on a circle. The momentum is quantized according to p 25 = K R (8.9)
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where K is an integer called the Kaluza-Klein excitation number. This is an important result without the compacti ed dimension, the center of mass momentum of the string is continuous. Compactifying a dimension quantizes the center of mass momentum along that dimension. Looking at Eq. (8.7) then, the rst term involving the momenta is the total center of mass momentum of the string. We call this the momentum mode. The second term, however, also involves momentum. In fact this term is the winding mode of the string, which satis es
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25 25 p pR = nR 2 L
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(8.10)
Looking at Eq. (8.3), we see that the winding w can be de ned in terms of the momentum of the left- and right-moving modes as w= 1 nR 25 = p 25 pR 2 L
(8.11)
Modi ed Mass Spectrum
Compactifying a dimension will lead to a modi ed mass spectrum. To obtain the mass spectrum for the state with a compacti ed dimension, let us begin with the Virasoro operators. Recall that L0 =
p R p R + n n 4 n =1
(8.12)
Note the repeated index which is an upper and lower index on the rst term in the right so we have an implied sum. Here the index ranges over the entire
String Theory Demysti ed
space-time, that is, = 0, ..., 2. Now let s write L0 in such a way that we peel of the = 25 term. Then L0 =
25 25 24 pR pR + pR pR + n n 4 4 =0 n =1
(8.13)
Similarly we can write L0 =
25 25 24 pL pL + pL pL + n n 4 4 =0 n =1
(8.14)
With a single compacti ed dimension, the Kaluza-Klein excitations on X 25 are considered to be distinct particles. Hence we can write down the mass operator as a mass term in the 25 noncompacti ed dimensions. That is, m 2 = p p
=0
24
(8.15)
Now, you should recognize the sums n =1 n n and n =1 n n as the number operators N R and N L . Using this fact together with Eq. (8.15) allows us to write the Virasoro operators as
25 25 2 p p m + NR 4 R R 4 25 25 2 L0 = p p m + NL 4 L L 4
L0 =
(8.16) (8.17)
Now we can utilize the mass-shell constraint. This is the condition that L0 1 and L0 1 annihilate physical states : ( L0 1) = 0 ( L0 1) = 0 (8.18) (8.19)
The conditions in Eqs. (8.18) and (8.19) imply that L0 = 1 and L0 = 1. Applying the rst condition to Eq. (8.16) we get L0 = 1
25 25 2 pR pR m + N R 4 4 25 25 m2 = pR pR + 2 N R 2 2 2
(8.20)
CHAPTER 8 Compacti cation and T-Duality
Similarly, using L0 = 1 together with Eq. (8.17) we obtain
2 25 25 m = p p + 2NL 2 2 2 L L
Now using Eq. (8.8) together with Eqs. (8.9) and (8.10) we can write
25 pL =
(8.21)
nR K + R
(8.22)
Which of course allows us to compute
(p )
25 L
nK nR K nR K = + = + +2 R R
(8.23)
25 and similarly pR = ( K / R) (nR / ) so that 2 2 2
(p )
25 R
nK K nR nR K = = + 2 R R
(8.24)
This allows us to obtain the sum and difference formulas:
(p ) +(p )
25 L 2 25 R 25 L 2 25 R
nR 2 K 2 = 2 + R =4 nK
(8.25) (8.26)
(p ) (p )
Using Eqs. (8.25) and (8.26) we can add Eqs. (8.20) and (8.21) to obtain
2 2 nR K m 2 = + + 2( N R + N L ) 4 R
(8.27)
and subtracting Eqs. (8.21) from (8.20) gives N R N L = nK (8.28)
So notice we have extra terms in the formulas for mass [Eq. (8.27)] and the level matching condition [Eq. (8.28)] as compared to the formulas introduced for the
String Theory Demysti ed
bosonic string in Chap. 2. The extra terms are due to two components, the KaluzaKlein excitations and the winding states of the string. The Kaluza-Klein excitations can be regarded as particles and so cannot be thought of as due to strings in a general sense. However the winding excitations can only come from strings, because only strings can wrap around a compacti ed extra dimension. Now let s look at the mass formula in Eq. (8.27) together with the relations for the momenta in Eqs. (8.22) and (8.25). Our task here is to consider limiting behavior. First we consider the case where R . In this limit, the momentum goes to the continuum limit and the Kaluza-Klein excitations disappear. It is simple to show that pL = pR and hence the winding state w= 1 25 25 p pR 0 2 L
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