code 128 font vb.net Conformal Field Theory Part I in Java

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CHAPTER 5 Conformal Field Theory Part I
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Closed String Conformal Field Theory
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Now let s move forward so we can see what the developments laid out thus far in the chapter really mean. We will see that the energy-momentum tensor can be expanded in a Laurent series, and that the Virasoro operators turn out to be the coef cients of the expansion. In other words, they describe the modes of the energymomentum tensor. In particular, the operator L0 is proportional to the energy operator or hamiltonian. As a speci c example, consider a closed string with worldsheet coordinates ( , ). The spatial dimension is compacti ed, that it is periodic with
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= + 2
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(5.27)
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The time coordinate satis es < < . We can describe the worldsheet of the closed string, which is an in nite cylinder, using conformal eld theory in the following way. We begin by making the following conformal transformation: z = e +i z = e i (5.28)
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The effect of this transformation is to map the cylinder to the complex plane. The radial coordinate plays the role of time, with the in nite past at the origin. With increasing radius, we move forward in time. Spatial integrals on the worldsheet are translated into contour integrals about the origin in the complex plane as the result of the conformal transformation [Eq. (5.28)]. A slice through the cylinder, which corresponds to a slice at constant time i , is transformed into a circle of radius ri in the complex plane. That is, radius in the z plane is a measure of Euclidean worldsheet time as R = z = e So, at time 1 a closed string is a circle of radius R1 = z = e 1 in the z plane, with the angular coordinate given by = . This is illustrated in Fig. 5.1. As time increases, from say 1 2 , 2 > 1 , the radius of the circle in the z plane increases from R1 to R2 > R1 . Let s recall the left-moving and right-moving modes described in Eq. (2.55) and (2.56). Given Eq. (5.28), it is clear that + ln z , and ln z . So we have
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XL (z ) =
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2 x i s p ln z + i s n z n 2 2 2 n 0 n 2 x i s p ln z + i s n z n 2 2 2 n 0 n
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(5.29) (5.30)
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X R (z) =
String Theory Demysti ed
z=et+is R
Worldsheet of closed string
z plane
Figure 5.1. The worldsheet of a closed string is mapped to the z plane. A slice through the cylinder, at a constant time, is mapped to a circle of a xed radius in the z plane. The radius of the circle in the z plane corresponds to (Euclidean) time on the worldsheet.
The energy-momentum tensor of the worldsheet is a conserved quantity, meaning that T = 0 (5.31)
The energy-momentum tensor is also traceless. This means that in the coordinates ( z, z ), the components Tzz = 0. The conservation condition [Eq. (5.31)] implies that zTzz = z Tzz = 0 (5.32)
That is, the energy-momentum tensor is composed of a holomorphic and antiholomorphic functions given by Tzz and Tzz , respectively. A holomorphic function has a Laurent series expansion, which we write as Tzz ( z ) =
m =
(5.33)
We have written this expression in a way anticipating that the Laurent coef cients are the Viarasoro generators. The antiholomorphic component also has a Laurent expansion: Tzz ( z ) =
m =
(5.34)
CHAPTER 5 Conformal Field Theory Part I
Using standard complex analysis, this tells us that we can invert these formulas in the following way: 1 2 i
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