String Theory Demystified

136 String Theory Demystifi ed
The leftover term multiplying the infi nitesimal ∂ α
a is our conserved current.
µ
Being that we started with a translation of space-time coordinates, we identify this
as the momentum:
µ
P = T∂ X
α
With Example 7.3 in mind, we can easily fi nd the conserved supercurrent, which
is the conserved current associated with the supersymmetry transformation. Let’s
µ α
µ α
just grind it out. Starting with L =−T/ 2( ∂ X ∂ X −iψ ρ ∂ ψ ) , we have
α
µ
α µ
T
µ α
µ α
µ α
δL =− ⎡2
∂ ( δX ) ∂ X −i( δψ ) ρ ∂ ψ −iψ ρ ∂ ( δψ ) ⎤
α
µ
α µ
α µ
2 ⎣
⎦
T
=− ∂ ( ) ∂ X + ( ∂ X ) ∂ −
2 2 µ α
β µ α
µ α β µ
⎡
⎣
εψ ερ ρ ψ ψ ρ ∂ ( ρ ∂ X ε)
⎤
α
µ
β
α µ
α β ⎦
T
µ α
β µ α
µ α β
=− ⎡2
∂ ( εψ ) ∂ X −∂ ( ερ ∂ X ) ρψ ψ ρ ( ρ ε)
α
µ α β
µ
2 ⎣
− ∂ ∂ X ⎤
α β µ ⎦
µ α
β µ α
=−T⎡ ⎣
∂ ( εψ ) ∂ X −∂ ( ερ ∂ X ) ρ ψ ⎤
α
µ α β
µ ⎦
µ α
β µ α
β α
µ
=−T⎡ ⎣
∂ ( εψ ) ∂ X −∂ ε ( ρ ∂ X ) ρψ −ερρ( ∂ ∂ X ) ψ ⎤
α
µ α β
µ
α β µ ⎦
µ α
µµ α
β α µ
α
=−T⎡ ⎣
∂ ( εψ ∂ X ) −εψ
∂∂X −∂ε( ρ ρ ψ ∂ X ) + ε( ∂∂X
) ψ
α
µ
α µ α
β µ α µ
µ α
β α µ
=−T⎡
⎣
∂ ( εψ ∂ X ) −∂ ε ( ρ ρ ψ ∂ X ) ⎤
α
µ α
β µ ⎦
The fi rst term is a total derivative, so it does not contribute to the variation of the
action. So we identify the conserved current with the second term. It is taken to be
α
µ 1 β µ
J = ρρψ∂
X
α
α
2
The Energy-Momentum Tensor
µ
β µ
µ
⎤
⎦
(7.13)
The next item of interest in our description of strings with worldsheet supersymmetry
is the derivation of the energy-momentum tensor. The energy-momentum tensor is
associated with translation symmetry on the worldsheet. Consider an infi nitesimal
translation ε α which is used to vary the worldsheet coordinates as
α α α
σ → σ +
ε