Lectures on String Theory

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Consider first the commutator of two supersymmetry variations applied to a bosonic
field X µ :
[δ ɛ1 , δ ɛ2 ]X µ = i¯ɛ 1 δ ɛ2 ψ µ − i¯ɛ 1 δ ɛ2 ψ µ = i 2(¯ɛ1 ρ α ɛ 2 − ¯ɛ 2 ρ α ɛ 1
)
∂α X µ = i¯ɛ 1 ρ α ɛ 2 ∂ α X µ ,
where the last formula stems from the spin-flip property. We see that the commutator
of two super-symmetry transformations generates a diffeomorphism transformation
[δ ɛ1 , δ ɛ2 ]X µ = ξ α ∂ α X µ
with the parameter ξ α = i¯ɛ 1 ρ α ɛ 2 . In fact, this is not an arbitrary diffeomorohism,
rather it is a conformal transformation, because ξ α is nothing else as a conformal
Killing vector! Thus, bilinear combinations made of conformal spinors
Consider now the commutator of two supersymmetry variations applied to a
fermion ψ µ :
[δ ɛ1 , δ ɛ2 ]ψ µ = 1 2 ∂ α(δ ɛ1 X µ )ρ α ɛ 2 − 1 2 ∂ α(δ ɛ2 X µ )ρ α ɛ 1 = i 2 ∂ α(¯ɛ 1 ψ µ )ρ α ɛ 2 − i 2 ∂ α(¯ɛ 2 ψ µ )ρ α ɛ 1
Therefore,
[δ ɛ1 , δ ɛ2 ]ψ µ = i 2 (∂ α¯ɛ 1 ψ µ )ρ α ɛ 2 − i 2 (∂ α¯ɛ 2 ψ µ )ρ α ɛ 1 +
+ i 2 (¯ɛ 1∂ α ψ µ )ρ α ɛ 2 − i 2 (¯ɛ 2∂ α ψ µ )ρ α ɛ 1 (6.5)
Constraints
In the original (gauge-unfixed) theory we have a world-sheet metric (vielbein) and
the gravitino field. They are removed upon imposition of the superconformal gauge.
However, before we fix the gauge, the metric and the gravitino have their equations
of motion which become the constraints on the other fields of the theory after fixing
the gauge. The stress tensor in now defined as
T αβ = − 2π e
δS
e αa .
We can also define the supercurrent as response of the action for variation of the
gravitino field
G α = −i 2π δS
e δ ¯χ . α
Analogously to what was in the bosonic case the stress tensor T αβ will generate
conformal transformations, while the new object G α appears to be a generator of the
supersymmetries. Equations of motion
δe β a
T αβ = 0 = G α