String Theory and M-Theory

4.3 Constraint equations and conformal invariance 119
momentum tensor of the RNS string is
Tαβ = ∂αX µ ∂βXµ + 1
4 ¯ ψ µ ρα∂βψµ + 1
4 ¯ ψ µ ρβ∂αψµ − (trace). (4.36)
The conserved current associated with the global world-sheet supersymmetry
of the RNS string is the world-sheet supercurrent. It can be constructed
using the Noether method. Specifically, taking the supersymmetry parameter
ε to be nonconstant, one finds that up to a total derivative the variation
of the action (4.2) takes the form
δS ∼ d 2 σ(∂α¯ε)J α , (4.37)
where
This current satisfies
J α A = − 1
2 (ρβ ρ α ψµ)A∂βX µ . (4.38)
(ρα)ABJ α B = 0 (4.39)
as a consequence of the identity ραρ β ρ α = 0. This is the analog of the
tracelessness of the Tαβ. In fact, it can be traced back to local super-Weyl
invariance in the formalism with local world-sheet supersymmetry. As a
result, J α A has only two independent components, which can be denoted J+
and J−.
Written in terms of world-sheet light-cone coordinates, the nonzero components
of the energy–momentum tensor in Eq. (4.36) are
T++ = ∂+Xµ∂+X µ + i
2 ψµ + ∂+ψ+µ , (4.40)
T−− = ∂−Xµ∂−X µ + i
2 ψµ −∂−ψ−µ . (4.41)
Similarly, the nonzero components of the supercurrent in Eq. (4.38) are
J+ = ψ µ
+ ∂+Xµ and J− = ψ µ
− ∂−Xµ. (4.42)
The supercurrent (4.38) is conserved, ∂αJ α A
equations of motion, which leads to
= 0, as a consequence of the
∂−J+ = ∂+J− = 0. (4.43)
The energy–momentum tensor satisfies analogous relations
∂−T++ = ∂+T−− = 0. (4.44)
These relations follow immediately from the equations of motion ∂+∂−X µ =