String Theory Demystified

CHAPTER 9 Superstring Theory Continued 169
They also commute with the fermionic coordinates:
a
a
σθ − θσ =0 (9.3)
A A
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 superfi eld.
It is possible to defi ne functions on superspace, meaning that we can introduce
fi elds Y that are functions of space-time coordinates and fermionic coordinates. We
can indicate this by writing
Y ≡ Y(
σ, θ)
for a given fi eld Y. We call a fi eld that is a function of superspace a superfi eld. A
superfi 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
Q = i
A
A
∂
− ρθ ∂
∂θ
α
( )
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 σ = ( τ, σ).
The supersymmetry generator acts on them as follows:
α ⎛ ∂ β ⎞ α
εQσ = ε −iρθ∂ σ
⎝
⎜
β
∂θ ⎠
⎟
α ⎛ ∂σ
⎞
β α
= ε −iρθ∂ σ
⎝
⎜
β
∂θ
⎠
⎟
β α α
=− iερ θδ =−iερ
θ
β
α
(the σ coordinate is not a function
of t
α
he fermion coordinate θ)
Hence we conclude that under a supersymmetry transformation
α α α
σ →σ −iερ
θ