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With the enlargement of the space-time coordinates to include the supercoordinates A, and the action in Eq. (9.9), which is invariant under supersymmetry transformations, we now see that we have the super-Poincar group. In Example 9.1, we illustrate an interesting result. We compute the commutator of two in nitesimal SUSY transformations applied to a space-time coordinate and show that the result is a space-time translation.
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CHAPTER 9 Superstring Theory Continued
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EXAMPLE 9.1 Let 1 and 2 be two in nitesimal supersymmetry transformations on x . Compute [ 1 , 2 ]x . SOLUTION This is actually rather easy. The commutator is [ 1 , 2 ]x = 1 2 x 2 1 x For the rst term, we have
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1 2 x = 1 (i 2A A )
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= i 2A 1 A = i 2A 1A The second term is
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A [ 1 , 2 ]x = 1 2 x 2 1 x = i 2A 1A i 1A 2
It s a simple exercise to rewrite one term like the other, which gives
A [ 1 , 2 ]x = i 2 1A 2
This is just a number, so we can write [ 1 , 2 ]x = a . Carrying on, we de ne a momentum term:
= x i A A
(9.10)
where = 0, 1, ..., p for a general p-brane. In the case of the point particle, which is a 0-brane, only = 0 applies and so
0 = x i A A
(9.11)
In fact, we have already seen that Eq. (9.11) is invariant under a SUSY transformation.
String Theory Demysti ed
Space-Time Supersymmetry and Strings
We have introduced some of the basic ideas of space-time supersymmetry by considering the point particle (known as the D0-brane). Now we move on to consider the supersymmetric generalization of bosonic string theory. Recall that the action for a bosonic string can be written as SB = 1 d 2 h h X X 2 (9.12)
Following the procedure used to write down the D0-brane action which was invariant under SUSY transformations, we introduce a new eld:
= X i A A
(9.13)
This approach is different than the RNS formalism discussed in Chap. 7. The are actual fermion elds on space-time. In Chap. 7, we had spinors but the were space-time vectors and not genuine fermion elds. It turns out that in string theory the number of supersymmetries is restricted to N 2. If we consider the most general case allowed which has N = 2, then there are two fermionic coordinates:
(9.14)
To get the full action we need to extend Eq. (9.12) in two steps. The rst step is to simply add a corresponding piece containing the fermion elds de ned in Eq. (9.13). It has basically the same form: S1 = 1 d 2 h h 2 (9.15)
Now things get hairy for technical reasons. In supersymmetry, there is a local fermionic symmetry called kappa symmetry. To avoid getting weighted down with mathematical details in a Demysti ed series book, we are going to leave it to you to read about kappa symmetry in more advanced treatments. Here we simply take it as a given that we need to preserve this kappa symmetry and that we can only do so by adding the following unwieldy piece to the action: S2 = 1 2 d i X ( 1 1 2 2 ) + 1 1 2 2 (9.16)
CHAPTER 9 Superstring Theory Continued
Light-Cone Gauge
As we found in Chap. 7, the quantum theory will force us to take the number of space-time dimensions to be D = 10. Since a general Dirac spinor has components 1, ..., 2 D / 2, in 10 space-time dimensions a general Dirac spinor is going to have 32 components. I am sure the reader found dealing with 4 components in quantum eld theory enough of a headache, what are we going to do with 32 components Luckily certain restrictions will cut this down dramatically. The rst thing to note is that the complete action, which is given by adding up Eqs. (9.12), (9.15), and (9.16) S = S1 + S2 which is invariant under SUSY transformations and the mysterious local Kappa symmetry only under very speci c conditions that restrict the number of space-time dimensions and the type of spinors in the theory. These conditions are given as follows: D = 3 with Majorana fermions. D = 4 with Majorana or Weyl fermions. D = 6 with Weyl fermions. D = 10 with Majorana-Weyl fermions. It is clear that we don t live in atland, so that rules out the rst case. The quantum theory forces us to take D = 10, which is no surprise since this was explored in Chap. 7. Therefore the spinors that are relevant to our discussion are MajoranaWeyl fermions. This helps us in two ways: The Majorana condition makes the spinor components real. The Weyl condition eliminates half of the components. This leaves us with a 16-component spinor. Once again the Kappa symmetry reveals its hand by cutting the number of components by half. So we are left with an eight component Majorana-Weyl spinor. With this in mind we will proceed with some aspects of light-cone quantization. This procedure imposes several conditions. First let s begin by de ning light-cone components of the Dirac matrices. This is done by singling out the = 9 component to make the following de nitions: + = = 0 + 9 2 0 9 2 (9.17) (9.18)