String Theory Demystified

172 String Theory Demystifi ed
This expression is a general expression, it’s a Taylor expansion of the superfi eld.
Due to the anticommuting properties of Grassman variables θθ is the highest-order
term in the expansion. It can be shown that the equation of motion for B µ is given
by B µ = 0, so this is an auxiliary fi eld that plays no role in the physics. The superfi eld
transforms as
δY = [ εQ, Y ] = εQY
µ µ µ
Now recall Eq. (7.9), which gave the SUSY transformations of the boson and
µ µ
fermion fi elds X and ψ :
δX= εψ
µ µ
µ α µ
δψ =−iρ∂X ε
We can derive these transformations by calculating δ µ
Y explicitly. Using Q = (/ ∂∂θ) −
A A
( ρθ)
α
∂ , we have
i A
α
µ α µ
δY ( σ , θ) = εQ
Y A
⎡ ∂ α ⎤ ⎡ µ α
⎤
= ε ⎢ −i( ρθ) ∂ X ( σ )
A α
⎣∂θ
⎥ + +
A
⎦ ⎣
⎢
⎦
⎥
∂
= +
∂
∂
µ α 1 µ α
θψ ( σ ) θθ ( σ )
2
µ µ
ε
θ ψ
θ ∂θ
B
B α B B µ α
X
( σ ) + θ θ B ( σ )
B
θ
A A
A
α
µ
ε i( ρ θ)
X A α
∂
⎡
1
⎤
⎢
⎣
∂ 2
⎥
⎦
⎡
α
µ α 1 µ α ⎤
− ∂ + i( ρθ) ∂ θψ + i( ρθ) ∂ θθB ( σ )
A α B A α
⎣
⎢
2 ⎦
⎥
µµ α µ α
= ε ψ ( σ ) + θ θ ( σ )
θ
α
µ
ε ( ρ θ)
α
∂
⎡
1 B B ⎤
⎢
B
⎣ ∂ 2
⎥
A
⎦
α
µ
− ⎡⎣ i ∂ X + i( ρθ) ∂ θψ ⎤ A
A α B ⎦
To get the last step, we dropped the third-order term which is 0 due to the
anticommuting nature of Grassman variables, and we dropped the fi rst term since
X µ
does not depend on the supercoordinates. Using the Fierz transformation
We can fi nally write this as
α
θθ A B =− δABθθ C C
1
2
µ µ α µ µ α µ
δY = εψ + θ( iερ ∂ X + εB ) + θθ( iερ
∂ ψ )
α
α