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String Theory Demysti ed
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The 10-dimensional Gamma matrices obey the anticommutation relation: { , } = 2 As a result, notice that 1 ( + )2 = ( 0 + 9 )( 0 + 9 ) 2 1 = ( 0 0 + 9 0 + 0 9 + 9 9 ) 2 1 = ( 0 0 + 9 9 + { 0 , 9}) 2 1 = ( 0 0 + 9 9 ) 2 1 1 = ( 00 99 ) = (1 1) = 0 2 2 We summarize this result by saying that + and are nilpotent, that is, ( + )2 = ( )2 = 0 (9.20) (9.19)
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To maintain kappa symmetry, a further constraint must be imposed. This is the fact that + annihilates the A : + 1 = + 2 = 0 As is usual in the light-cone gauge, we have X + = x + + p+ (9.22) (9.21)
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It is customary to denote the spinors which contain the remaining eight nonzero components by S Aa. These objects are de ned in the following way: p + 1 S 1a (9.23) p+ 2 S 2 a
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CHAPTER 9 Superstring Theory Continued
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Note that there are dotted spinors in the case of type IIA string theory (see Quantum Field Theory Demysti ed if you are not familiar with this). Making the de nition 1 P = ( h / h ) 2 (9.24)
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we can write down the equations of motion that are derived by adding Eqs. (9.12), (9.15), and (9.16). These are the equations of motion for the GS superstring, which are in general quite complicated: = 1 h h 2 (9.25) (9.26) (9.27)
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P 1 = 0 P+ 2 = 0
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Remarkably, in the light-cone gauge, the equations of motion turn out to be very simple. This is because we can simplify the expression:
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= X i A A
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getting the term A A to drop out in most cases. Using Eq. (9.21), this is immediate when taking = +: A + A = 0 There is only one nonvanishing term, when = . For the cases where = i , we can use the following trick. Consider the fact that 1 + = ( 0 + 9 )( 0 9 ) 2 1 = ( 0 0 + 9 0 0 9 9 9 ) 2 1 = ( 00 + 99 + 9 0 0 9 ) 2 1 = 1 + ( 9 0 0 9 ) 2
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Doing a similar calculation for + one can show that the following provides a representation of the identity operator: 1= + + + 2
(9.28)
So, for = i, one simply inserts Eq. (9.28) into the term A A to make it vanish from the equations of motion. This is only possible in the light-cone gauge, and it can be shown that the equations of motion are 2 2 2 Xi = 0 2 1a S =0 + 2a S =0
(9.29)
(9.30)
(9.31)
Canonical Quantization
Now, we will take a quick look at canonical quantization and the ground state of the type I superstring. First, note that the usual bosonic commutation relations are imposed for the bosonic elds or space-time coordinates X ( , ) and their associated modes. Now we need to extend the theory by de ning quantization conditions for the fermionic elds. Since the supercoordinates are fermionic, we apply equal-time anticommutation relations. These are given by {S Aa ( , ), S Bb ( , )} = ab AB ( ) Open strings in type I theory satisfy boundary conditions given by S 1a S 1a
=0
(9.32)
= S2a
=0 =
2a = = S
(9.33)
CHAPTER 9 Superstring Theory Continued
This helps us determine the modal expansions of the fermion elds, which are given by S 1a = S The modes satisfy