String Theory Demystified

CHAPTER 9 Superstring Theory Continued 171
Now, use the fact that the components are anticommuting to write
εθ= εθ − − + εθ + + =−θε − − − θε + + =−θε
0α
α α
This means that 2η ( θε + εθ)
= 0,
and so we are able to write ερ θ =− θρ ε .
Returning to the main thrust of our discussion, in general, space-time coordinates
will transform as
µ µ µ
x → x −iεγ
θ
under a supersymmetry transformation, where γ µ are the usual Dirac matrices.
Now let’s consider the action of the supersymmetry generator on the fermionic or
super-worldsheet coordinates. We simply state the result which you can work out in
the Chapter Quiz:
δθ A = ε A
Hence, the supercoordinates transform as
θA → θA + εA
A tool we will use to write down the action is the supercovariant derivative. This is
given by
D = i
A A A
∂
+ ρθ ∂
∂θ
α
( )
A key property of the supercovariant derivative is that under a supersymmetry
transformation, the supercovariant derivative of a superfi eld F, DF transforms the
same way as F does.
α
Superfi eld for Worldsheet Supersymmetry
We will use the case of worldsheet supersymmetry to illustrate how to write down an
action where supersymmetry is manifest. To do this, we start with the superfi eld:
µ α µ α µ α µ α
Y ( σ , θ) = X ( σ ) + θψ ( σ ) + θθB ( σ )
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