Lectures on String Theory

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Here D α ɛ = ∂ α ɛ − 1 2 ω α ¯ρɛ and ω α is the connection with torsion:
ω α = ω α (e) + i 4 ¯χ α ¯ρρ β χ β ,
where ω α (e) = − 1 e e αaɛ βγ ∂ β e a γ is the standard composite spin connection (it has
only one-non-trivial component in 2dim).
2. Weyl invariance. Let Λ be a bosonic local parameter Λ = Λ(σ, τ). The Weyl
transformations are
δ Λ X µ = 0 , δ Λ e a α = Λe a α ,
δ Λ ψ µ = − 1 2 Λψµ , δ Λ χ α = 1 2 Λχ α .
3. Super Weyl invariance. Let η be the Majorana spinor. Under the super Weyl
transformations only the gravitino transforms as
δ η χ α = ρ α η .
The invarince of the action easily follows from the identity ρ α ρ β ρ α = 0.
4. Local Lorentz symmetry. Let l be a bosonic local parameter l = l(σ, τ). The
local Lorentz transformations are
δ l X µ = 0 , δ l e a α = lɛ a be b α ,
δ l ψ µ = 1 2 l¯ρψµ , δ l χ α = 1 2 l¯ρχ α .
5. Reparametrizations. Let ξ α be a bosonic vector parameter ξ α = ξ α (σ, τ). The
reparametrizations (diffeomorphisms) are
δ ξ X µ = ξ β ∂ β X µ , δ ξ e a α = ξ β ∂ β e b α + e a β∂ α ξ β ,
δ ξ ψ µ = ξ β ∂ β ψ µ , δ ξ χ α = ξ β ∂ β χ α + χ β ∂ α ξ β .
6.3 Superconformal gauge and supermoduli
The gravitino field is the reducible representation of the Lorentz group. To decompose
it into irreducible representations one can use the following trick:
χ α = δαχ β β =
(δα β − 1 )
2 ρ αρ β χ β + 1 2 ρ αρ β χ β = 1 2 ρβ ρ α χ β + 1
} {{ } 2 ρ αρ β χ β
˜χ α
Here ˜χ α part is called ρ–traceless because, due to the identity ρ α ρ β ρ α = 0 we get
ρ α ˜χ α = ρ a e α a ˜χ α = ρ a ˜χ a = 0. Indeed the gravitino χ a transforms under local Lorentz