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CHAPTER 9 Superstring Theory Continued
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To write down the SUSY transformations, we simply compare with * and look for corresponding terms in the expansion. Doing this we nd
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X = = i X + B B = i
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We can write the action in terms of super elds. Since the super elds are functions of the Grassman variables, we will have to utilize a new kind of integration, called Grassman integration. We take a brief detour to describe this now.
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Grassman Integration
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Another important tool in the supersymmetry toolkit is the technique of Grassman integration. It turns out that the integration of anticommuting Grassman variables is quite a bit different (and actually a lot simpler, although less intuitive) than the ordinary integration of a function of real variables. Even so, we develop the notion of Grassman integration by our wish to preserve one important property of integration. If you integrate a function of a real variable over the entire number line, that integral is translation invariant. That is, let a be some real constant then it must be true that
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f ( x ) dx = f ( x + a ) dx
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Now let ( ) be a function of the Grassman variable . We also want integration here to be translation invariant:
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d ( ) = d ( + c)
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where c is a constant (a Grassman number in this case). To deduce the properties of Grassman integration while preserving translation invariance, we expand the function ( ) in Taylor:
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( ) = a + b
Then,
d ( ) = d (a + b )
Now, let + c and we obtain
String Theory Demysti ed
d [a + b( + c)] = d (a + b + bc) = (a + bc) d + b d
In order for this integral to be translation invariant, it cannot depend on c, and so we conclude that
d = 0
when is a Grassman variable. By convention, the Grassman integral is normalized to one in the following way:
d = 1
So that altogether we have the rule
d (a + b ) = b
For double integration over two Grassman coordinates, there is only one rule to remember
d = 2i
With these rules in hand, you are on your way to becoming a supersymmetry expert.
A Manifestly Supersymmetric Action
The action written in Chap. 7 [Eq. (7.2)] includes fermionic elds but supersymmetry is not manifest. This situation can be remedied by writing down an action in terms of super elds. The action we use is given by S= i d 2 d 2 DY DY 8
CHAPTER 9 Superstring Theory Continued
where, i DY = + B i X + 2 i DY = + B + i X 2
Performing the Grassman integration in the action, we can obtain the component form, which is S= 1 d 2 ( X X i B B ) 4
The equation of motion for B , as mentioned earlier, is B = 0, which allows us to discard the auxiliary eld, and we arrive back at the theory described in Chap. 7. To see how to arrive at this, you can just apply the rules of Grassman integration, considering and as separate variables and using d 2 = d d . We illustrate by computing a couple of terms. For example,
d B
= d d B
d = 1 and so, d d B
= d B = 0
On the other hand,
d ( B )( B ) = d ( ) B
B = 2iB B
Using these types of computations, one can transform the manifestly supersymmetric action into the coordinate form to recover the theory of the RNS superstring.
The Green-Schwarz Action
In this section, we use the idea of supersymmetry applied to the space-time coordinates. For worldsheet supersymmetry, we extended the coordinates = ( , ) of the worldsheet by introducing fermionic super-worldsheet coordinates. Now, we are going to utilize this same idea but apply it to the actual space-time coordinates,
String Theory Demysti ed
which are the bosonic elds X ( , ). This can be done by adding new elds, typically denoted by a ( , ), which map the worldsheet to fermionic coordinates. Taking the X ( , ) together with the a ( , ) will enable us to map the worldsheet to superspace. This approach to superstring theory is known as the Green-Schwarz (GS) formalism. To summarize, when applying worldsheet supersymmetry We extend the coordinates ( , ) by introducing fermionic coordinates 1 and 2. This gives us super-worldsheet coordinates. In this case: We are developing an extension of space-time itself, creating a superspace described by the pair X ( , ) and a ( , ). An N = m supersymmetric theory will have a = 1, ..., m , or m fermionic coordinates.
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