String Theory Demystified

138 String Theory Demystifi ed
β
The divergence term ∂ ε is not going to contribute anything, so we drop it. The
β
fi rst and third terms cancel, leaving us with
β ⎛ i µ α ⎞
δL=∂ ε − ψ ρ ∂ψ
F α
β µ
⎝
⎜
2 ⎠
⎟
β
This is what we want, because terms that multiply ∂ ε are going to be terms that
α
make up the energy-momentum tensor T . This isn’t quite right, because we want
αβ
it to be symmetric. So, we take
β ⎛ i µ α i µ β ⎞
δL=∂ ε − ψ ρ ∂ψ − ψ ρ ∂ ψ
F α
β µ
α µ
⎝
⎜
4 4 ⎠
⎟
(7.16)
In the Chapter Quiz, you will derive an expression for the bosonic part of the
energy-momentum tensor. When all is said and done
µ i µ i µ
T =∂ X ∂ X + ψ ρ ∂ ψ + ψ ρ ∂ ψ −(
Trace )
αβ α β µ
α β µ
β α µ
4 4
(7.17)
The “Trace” is explicitly removed to ensure that T αβ remains traceless as required
for scale invariance.
The energy-momentum tensor and supercurrent can be written compactly using
worldsheet light-cone coordinates. The energy-momentum tensor has two nonzero
components given by
µ i µ
µ i µ
T =∂ X ∂ X + ψ ∂ ψ T =∂ X ∂ X + ψ ∂
µ
µ µ
2 2
ψψ ++ + + + + + −− − − − − µ
The components of the supercurrent are
µ
µ
J = ψ ∂ X J = ψ ∂ X
+ + + µ − − −
The equations of motion for the fermion fi elds are
µ
− (7.18)
(7.19)
µ µ
∂ ψ =∂ ψ = 0 (7.20)
+ − − +
Together with the equations of motion of the boson fi elds
µ
∂∂ X =
+ −
0
(7.21)