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MAJORANA SPINORS
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The elds introduced in the action, = ( , ), are two-component Majorana spinors on the worldsheet. Given that they have two components, they are sometimes written with two indices A , where = 0, 1, , D 1 is the space-time index and A = is the spinor index. We can write A as a column vector in the following way (suppressing the space-time index): = + Under Lorentz transformations, these elds transform as vectors in space-time [recall that a contravariant vector eld V ( x ) is one that transforms as V ( x ) V ( x ) = V ( x ) under x x = x where is a Lorentz transformation]. Following the convention used with Dirac spinors in quantum eld theory, we have the de nition:
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= ( ) 0
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Note that the de nitions used here depend on the basis used to write down the Dirac matrices Eq. (7.3), and that other conventions are possible. We can also introduce a third Dirac matrix analogous to the 5 matrix you re familiar with from studies of the Dirac equation, which in this context we denote by 3: 1 0 3 = 0 1 = 0 1 It will be of interest to make left movers and right movers manifest. This can be done by recalling the following de nitions:
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= 1 = ( ) 2 = + + = +
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(7.5) (7.6) (7.7)
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EXAMPLE 7.2 Show that = 2( + + + + ).
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CHAPTER 7 RNS Superstrings
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SOLUTION We can rewrite in a more enlightening way by expanding out the sum explicitly:
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= ( 0 0 + 1 1 )
Now 0 i 0 0 = i 0 0 i 1 1 = i 0 So, the summation is 0 i 0 i 0 0 + 1 1 = + i 0 i 0 i( ) 0 0 = = 2i 0 i ( + ) + Hence, 2i 0
= ( 0 0 + 1 1 )
0 = 2i + 2i 0
Now, we write out the components of the spinors. For simplicity, we suppress the space-time index for a moment. First, note that 0 2i 2i 0 = 2i 0 + 2i + = 2i 0 + +
Using = ( ) 0 , we have 0 2i + 2i i + 0 = + i 0 2i 0 +
2 + = + = 2 + + 2 + + 2 + = 2( + + + + )
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The result obtained in Example 7.2 allows us to write the fermionic part of the action in a relatively simple way. Denoting the fermionic action by SF we have SF = T 2 d ( i ) 2 T 2 = d ( 2i )( + + + + ) 2 = iT d 2 ( + + + + )
It can be shown that by varying the fermionic action SF , one can obtain the free eld Dirac equations of motion:
+ = + = 0
(7.8)
The Majorana eld describes right movers while the Majorana eld + describes left movers.
SUPERSYMMETRY TRANSFORMATIONS ON THE WORLDSHEET
Now, we introduce a supersymmetry (SUSY for short) transformation parameter which is denoted by . This in nitesimal object is also a Majorana spinor, which has real, constant components given by = + Since the components of are taken to be constant, this represents a global symmetry of the worldsheet. If it were a local symmetry, it would depend on the coordinates ( , ). Furthermore, the components of are Grassman numbers. Two Grassmann numbers a, b anticommute such that ab + ba = 0. Now we use to de ne our symmetry. The action which includes the fermionic elds is invariant under the supersymmetry transformations:
X = = i X
(7.9)
Using , we also nd that = i X = i X . Notice that this takes the free boson elds into fermionic elds, and vice versa. We can relate individual components as follows. First, we have
X = = (
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