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(7.62)
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From Eq. (7.61), we deduce that the ground state in the R sector is a massless Dirac spinor in 10 dimensions.
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In the previous section, we saw that the theory still has a major problem it admits an imaginary mass or tachyon state. This indicates that the vacuum is unstable. We can rid the theory of the tachyon state, however, giving superstring theory a major advantage over bosonic string theory (aside from bringing fermions into the picture). This is done using GSO projection. GSO projection reduces the number of states in the theory, and rids it of unwanted problems like the tachyon state. In the NS sector, we keep states with an odd number of fermion excitations and reject states with an even number of fermion excitations. This is done by de ning a fermion number operator: F = b r br
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Then we de ne a parity operator given by 1 PNS = [1 ( 1) F ] 2 (7.64)
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CHAPTER 7 RNS Superstrings
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The parity operator determines the states that we can have in the theory. Notice that if F = 0, PNS = 0 . Only half integer values of the number operator N = n n + rb r br rb r br are allowed, giving a mass spectrum for the NS sector:
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n =1 r =1/ 2 r =1/ 2
m 2 = 0,
1 2 , ,
(7.65)
This means that the spin-0 ground state of the NS sector is now massless. The tachyon state has been removed from the theory. In the R sector, we de ne the Klein operator which is given by ( 1) F = 11 Here, 11 = 0 1 9 (7.67) (7.66)
is a 10-dimensional chirality operator. It acts on spinors according to 11 = (7.68)
That is, states have positive or negative chirality. Weyl spinors are states with de nite chirality, and states can be projected into spinors with opposite space-time chirality using the operator 1 P = (1 11 ) 2 (7.69)
Critical Dimension
We will not pursue light-cone quantization in this chapter, but if that procedure is used the number of space-time dimensions is easily extracted. One obtains a relation for the Lorentz generators M i : [M i , M j ] = 1 ( p + )2
(
n =1
i n
i nj j n n )( n n )
(7.70)
where
String Theory Demysti ed
D 2 D 2 1 n = n + n 2 aNS 8 8
(7.71)
In order to maintain Lorentz invariance, we must have [ M i , M j ] = 0 . This can only be true if the rst term on the right-hand side of Eq. (7.71) is n and the second term vanishes. This implies that D 2 =1 8 D = 10 So, we see that Lorentz invariance requires us to take the critical space-time dimension to be 10 (9 space and 1 time dimension) in superstring theory. Using Eq. (7.72), we can deduce the value of the normal-ordering constant: 2 aNS D 2 =0 8 D = 10 aNS = 1 2
(7.72)
Summary
In this chapter, we made the rst attempt to introduce fermions to string theory. This was done by adding supersymmetry as a global symmetry on the worldsheet. The conserved current and supercurrent was derived. Next, we wrote down the superVirasoro algebra and determined how physical states behave in the theory, and the spectrum of the open string was described including the two sectors, the NS and R sectors which give rise to bosonic and fermionic states, respectively. Using GSO projection, one can remove unwanted states like the Tachyon from the theory. Finally, we showed how Lorentz invariance forces us to take the critical dimension to be 10.
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