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1. Translational invariance along leads to the condition ( L0 L0 ) = 0 for 25 25 physical states . Use this to nd a relation between pL , N L, and pR , N R.
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Superstring Theory Continued
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In Chap. 7, we took our rst look at superstring theory by considering supersymmetry on the worldsheet. The result is the RNS superstring. We can learn a lot from this method but space-time supersymmetry is not manifest with this theory. In this chapter, we have a somewhat random collection of material on supersymmetry and superstrings that will give you a general overview of what these topics are about so that you can pursue more advanced treatments if desired. In short, we are going to do two things. First, we will deepen and extend our discussion of supersymmetry and superstrings, and then we will introduce a space-time supersymmetry approach. These are more advanced topics so we aren t going to go into great detail, and leave out a lot of important information. But our purpose here is to provide the reader with an introductory overview that exposes you to some of the basic ideas of superstring theory. A detailed study can be undertaken by reading any of the references listed at the end of the book. We hope this rst exposure will make going through material in other textbooks a bit easier. The
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material in this chapter will not be necessary to understand D-branes or black hole physics as discussed in this book, so if you d rather avoid it for now you can do so without much harm.
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Adding space-time supersymmetry is going to involve a couple of things. Speci cally: We will extend the coordinates to add a supersymmetric partner to the space-time coordinate x . The result will be superspace de ned by coordinates x and A . We will introduce a super eld which is a function of the superspace coordinates. The super eld will be added to the action to generate a supersymmetric theory. With these points in mind let s rst move ahead by describing the concept known as superspace. As noted above, the idea here is to add to the usual space-time coordinate x 0 , x1 , ..., x d by adding fermionic or Grassman coordinates A. The index A used on the superspace or Grassman coordinates corresponds to the spinor index used on the spinors A . Taking the case of worldsheet supersymmetry that we have discussed already, we had two component spinors, and so A = 1, 2 . Fermionic coordinates A are also called Grassman coordinates because they satisfy an anticommution relation. That is,
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A B + B A = 0
(9.1)
2 Notice that this relation implies that A A = A = 0. In the case of the worldsheet, the A are super-worldsheet coordinates that are two component spinors:
A = + To characterize superspace, we also need to understand how the fermionic coordinates behave with respect to normal space-time coordinates this is encapsulated in commutation and anticommutation relations. Sticking to the worldsheet as an a example, we denote the coordinates of the worldsheet by = ( , ). Since these are ordinary coordinates, they commute with themselves:
a b b a = 0
(9.2)
CHAPTER 9 Superstring Theory Continued
They also commute with the fermionic coordinates:
a A A a = 0
(9.3)
So, what we ve seen here is that supersymmetry doesn t just pair up bosons and fermions, it also enlarges the notion of space-time to pair up ordinary coordinates with fermionic coordinates, with superspace being characterized by the relations given by Eqs. (9.1) through (9.3). Now let s consider the notion of a super eld. It is possible to de ne functions on superspace, meaning that we can introduce elds Y that are functions of space-time coordinates and fermionic coordinates. We can indicate this by writing Y Y ( , ) for a given eld Y. We call a eld that is a function of superspace a super eld. A super eld can be introduced into the action to construct a supersymmetric theory. Next, we introduce the supercharge which can also be called the supersymmetry generator. In the case of worldsheet supersymmetry, this is given by QA = i ( ) A A
We call the supercharge the supersymmetry generator because it generates supersymmetry transformations on superspace. That is, it acts on the coordinates as follows. Using the worldsheet example, the role of the space-time coordinates are a played by = ( , ). The supersymmetry generator acts on them as follows: Q = i = i
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