Lectures on String Theory

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By using reparametrizations and local Lorentz transformations the zweibein can
be brought to the form e a α = e φ δ a α which is locally always possible. The gauge
e a α = e φ δ a α ,
χ α = ρ α λ
is called superconformal gauge. In classical theory the Weyl symmetry and super
Weyl symmetry can be used to eliminate the remaining gravitational degrees of
freedom φ and λ. In quantum theory it will be possible in critical dimension only.
6.4 Action in the superconformal gauge
In the superconformal gauge the action becomes rather simple
S = − 1 ∫
d 2 σ
(∂ α X µ ∂ α X µ + 2i
8π
¯ψ
)
µ ρ α ∂ α ψ µ . (6.3)
The world-sheet indices are now raised and lowered with the help of the flat worldsheet
metric η aβ and ρ α = δ α a ρ a . This action is invariant w.r.t. local reprametrizations
and supersymmetry transformations which satisfy the requirement
P ξ = 0 , Πɛ = 0 .
We would like to check directly that the action (6.3) is invariant under the supersymmetry
transformations
δ ɛ X µ = i¯ɛψ µ ,
δ ɛ ψ µ = 1 2 ρα ∂ α X µ ɛ
δ ɛ ¯ψµ = − 1 2¯ɛρα ∂ α X µ
provided the parameter ɛ satisfies the following equation
ρ β ρ α ∂ β ɛ = 0 . (6.4)
To check the invariance we perform the variation
δ ɛ S = − 1 ∫
d 2 σ
(2∂ α X µ ∂ α (i¯ɛψ µ ) + i
8π
¯ψ
)
µ ρ α ∂ α (ρ β ∂ β X µ ɛ) − ¯ɛρ α ∂ α X µ ρ β ∂ β ψ µ .
Now we integrate by parts the first term and write out the second term more explicitly
δ ɛ S = − 1 ∫ (
d 2 σ − 2i□X µ ¯ɛψ µ + i
8π
¯ψ µ ρ α ρ β ∂ α ∂ β X µ ɛ
}{{}
η αβ
+ i ¯ψ µ ρ α ρ β ∂ β X µ ∂ α ɛ − ¯ɛρ α ∂ α X µ ρ β ∂ β ψ µ
)
.