String Theory Demystified

CHAPTER 9 Superstring Theory Continued 177
where θ Aa is anticommuting spinor coordinate. In the case we are studying here, for
Aa Aa
a point particle, these are functions of τ , that is, θ = θ ( τ).
The index A ranges
over the number of supersymmetries in the theory. If there are N of them, then
A= 1, ..., N
Hence, if we have an N = 2 supersymmetry, then we have the two fermionic
1a 2a
coordinates θ and θ . You may be a little confused by the notation. We actually
have a second index here. The second index is the spinor index. Consider a general
Dirac spinor. In D dimensions it has 2 2 D/
components. So,
D
a = 1 2 2 /
, ...,
For Majorana spinors, this number is cut in half. Now, we are actually going to
proceed in a manner which is not too different from what you learned for worldsheet
supersymmetry. Once again, we consider a constant Majorana spinor that we denote
by ε A (suppressing the spinor index) to emphasize that it is infi nitesimal. Now we
introduce the following SUSY transformations:
µ µ
δx = iε Γ θ
A A
δθ = ε
A A
δθ = ε
A A
(9.7)
In addition, we have to worry about the auxiliary fi eld. We suppose that the SUSY
transformation in this case is
δe = 0 (9.8)
The simplest supersymmetric action that can be conceived of is an extension of the
action in Eq. (9.5) written as follows:
1 1 µ A µ A 2
S = ∫ dτ ( x−iθ Γ θ)
(9.9)
2 e
Now, since ε A is a constant, it does not depend on τ and hence ε A = 0. Given that
plus Eq. (9.7), it’s very easy to see that Eq. (9.9) is invariant under a SUSY
transformation. First note that
δθ δ
τ θ
τ δθ
τ ε
A d A d A d A
=
d d d
⎛ ⎞
⎝
⎜
⎠
⎟ = ( ) = = 0