String Theory Demystified

132 String Theory Demystifi ed
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 S we have
F
S
F
T 2 µ α
=− ∫ d σ( −iψ ρ ∂ ψ ) α µ
2
T 2
=− ∫ d σ( −2i)( ψ ⋅∂ψ
+ ψ ⋅∂ ψ )
− + − + − +
2
∫
= iT d σψ ( ⋅∂ ψ + ψ ⋅∂ ψ )
2
− + − + − +
It can be shown that by varying the fermionic action S F , one can obtain the free
fi eld Dirac equations of motion:
µ µ
∂ ψ =∂ ψ = 0
+ − − +
(7.8)
The Majorana fi eld ψ µ
− describes right movers while the Majorana fi eld ψ µ
+ describes
left movers.
SUPERSYMMETRY TRANSFORMATIONS ON THE WORLDSHEET
Now, we introduce a supersymmetry (SUSY for short) transformation parameter
which is denoted by ε. This infi nitesimal object is also a Majorana spinor, which
has real, constant components given by
ε
ε =
ε
⎛ ⎞ −
⎝
⎜
⎠
⎟
Since the components of ε 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
( σ, τ ) . Furthermore, the components of ε are Grassman numbers. Two Grassmann
numbers a, b anticommute such that ab + ba =0.
Now we use ε to defi ne our symmetry. The action which includes the fermionic
fi elds is invariant under the supersymmetry transformations:
δX= εψ
µ µ
µ α µ
δψ =−iρ∂X ε
+
α
(7.9)
Using δψ µ µ α µ α µ
, we also fi nd that δψ =−iρ∂ X ε = iερ∂X . Notice that this takes
α
α
the free boson fi elds into fermionic fi elds, and vice versa. We can relate individual
components as follows. First, we have
δ εψ ε ε ψ
µ ⎛ ⎞
µ µ
−
µ µ
X = = ( ) ⎜ εψ εψ
µ
⎝ψ
⎟ = +
− +
− − + +
(7.10)
⎠
+