String Theory Demystified

178 String Theory Demystifi ed
Now of course we can ignore the 1 e term when varying the action since the
SUSY transformation is Eq. (9.8). Proceeding
Ok, now we have
Using Eq. (9.7) then, we have
1 1 µ A µ A 2
δS = δ ∫ dτ ( x−iθ Γ θ)
2 e
1 1 µ A µ
= ∫ dτ δ( x−iθ Γ θ
A 2
)
2 e
1 µ A µ µ µ
τ θ Γ θA A
δ θ Γ θA
= d ( x −i ) ( x −i
)
e
∫
1 µ A µ A µ A µ A
= ∫ dτ ( x−iθ Γ θ) ⎡⎣ δx−iδ( θ Γ θ)
⎤
e
⎦
A µ A A µ A A µ A
δθ ( Γ θ ) = ( δθ) Γ θ + θ Γ ( δθ
)
A µ A
= ( δθ ) Γ θ
= ε θ
µ A A
Γ
µ A µ µ µ
δ δ θ θA A A A A
x − i ( Γ ) = iε Γ θ − iε
Γ θ = 0
Therefore, δS = 0 and the action is invariant under a SUSY transformation.
Since we are dealing with an enlargement of space-time coordinates, take a step
back and recall that the actions described in Chap. 2
• Are invariant under space-time translations a µ .
• Are invariant under Lorentz transformations ω µ ν
x .
We combine these two results in the Poincaré group and note that the action in
Eq. (9.4) is invariant under
µ µ µ
δx = a + ω x
With the enlargement of the space-time coordinates to include the supercoordinates
θ A , and the action in Eq. (9.9), which is invariant under supersymmetry transformations,
we now see that we have the super-Poincaré group.
In Example 9.1, we illustrate an interesting result. We compute the commutator
of two infi nitesimal SUSY transformations applied to a space-time coordinate and
show that the result is a space-time translation.
ν
ν
ν