Lectures on String Theory

– 103 –
Now we apply the spin-flip identity in the second and the third term and get
δ ɛ S = − 1 ∫ (
d 2 σ − 2i□X µ ¯ɛψ µ + i¯ɛψ µ □X µ
8π
)
+ i∂ α¯ɛρ β ρ α ψ µ ∂ β X µ − ¯ɛρ β ρ α ∂ β X µ ∂ α ψ µ .
Finally, we integrate the last term by parts and get
δ ɛ S = − 1 ∫
d 2 σ 2i∂ α¯ɛρ β ρ α ψ µ ∂ β X µ .
8π
This vanishes as the consequence of eq.(6.4).
We can write equation (6.4) more explicitly
) ( )
)
ρ β ρ 0 ∂ β ɛ =
(ρ 0 ρ 0 ∂ 0 + ρ 1 ρ 0 ∂ 1 ɛ = − ρ 0 ρ 0 ∂ 0 + ρ 0 ρ 1 ∂ 1 ɛ =
(∂ 0 + ¯ρ∂ 1 ɛ = 0 ,
)
)
)
ρ β ρ 1 ∂ β ɛ =
(ρ 0 ρ 1 ∂ 0 + ρ 1 ρ 1 ∂ 1 ɛ =
(ρ 0 ρ 1 ∂ 0 + ρ 1 ρ 1 ∂ 1 ɛ =
(¯ρ∂ 0 + ∂ 1 ɛ = 0 .
Since ¯ρ 2 = 1 the second equation is obtained from the first by multiplying with ¯ρ
and by this reason it is redundant. To analyze the first equation it is convenient to
denote the components of any spinor as follows
ψ =
(
ψ+
ψ −
)
and ɛ =
We thus see that the first equation reduces to
(
ɛ+
)
.
ɛ −
(∂ 0 + ∂ 1 )ɛ + = ∂ + ɛ + = 0 , (∂ 0 − ∂ 1 )ɛ − = ∂ − ɛ − = 0
One cab define the spinors with upper indices by using the following convention
ψ − = ψ + , ψ + = −ψ − .
With this convention we obtain that components of the Majorana spinor which is a
parameter of the supersymmetry transformations satisfy the equations
∂ + ɛ − = ∂ − ɛ + = 0 =⇒ ɛ ± ≡ ɛ ± (σ ± ) .
This equations should be contracted with the equations defining the conformal Killing
vectors, i.e. reparametrizations which do not destroy the conformal gauge choice:
∂ + ξ − = ∂ − ξ + = 0 =⇒ ξ ± ≡ ξ ± (σ ± ) .
On-shell closer of the supersymmetry algebra