Lectures on String Theory

– 17 –
Thus, eom for h αβ is
Performing a variation we obtain
T αβ = 0 .
T αβ = 1 2 ∂ αX µ ∂ β X µ − 1 4 h αβh γδ ∂ γ X µ ∂ δ X µ . (3.12)
From here we immediately notice that the stress-energy tensor is traceless
T α α = T αβ h αβ = 0
This is a direct consequence of the Weyl symmetry.
Due to the equations of motion for the scalar fields X µ this tensor is covariantly
conserved:
∇ α T αβ = 0 .
This can be easily derived as follows. Consider a variation of the action:
∫ ( δL
δS p = d 2 σ
δh αβ δhαβ + δL )
δX µ δXµ
We see that on the equations of motion for X µ , i.e. on δL = 0, we have
δX
∫
µ
δS p = −T d 2 σ √ ∫
−h T αβ δh αβ = −2T d 2 σ √ −h T αβ ∇ α ξ β ,
where on the r.h.s. we specified the variation to be a diffeomorphism transformation.
The last expression can be integrated by parts and, die to δS p = 0, we conclude that
∇ α T αβ = 0. This derivation is similar to the derivation of the charge conservation
law in electrodynamics, the latter being a consequence of the gradient invariance.
Let us show directly that ∇ α T αβ = 0. We have
2∇ α T αβ = ∇ α ∂ α X µ ∂ β X µ + ∂ α X µ ∇ α ∂ β X µ − 1 2 ∇ β
h γδ ∂ γ X µ ∂ δ X µ
(3.13)
Since ∇ α ∂ αX µ = 0 is eom for X µ we find
2∇ α T αβ
= ∂ α X µ ∇ α ∂ β X µ − 1 2 ∂ β
h γδ ∂ γ X µ ∂ δ X µ
=
= ∂ αX µ h αδ ∇ δ ∂ β X µ − 1 2 ∂ β h γδ ∂ γ X µ ∂ δ X µ − h γδ ∂ γ X µ ∂ δ ∂ β X µ =
= ∂ αX µ h αδ ∂ δ ∂ β X µ − ∂ αX µ ∂ sX µh αδ Γ s δβ − 1 2 ∂ β h γδ ∂ γ X µ ∂ δ X µ − h γδ ∂ γ X µ ∂ δ ∂ β X µ .
Here the first term cancels with the last one and we find
2∇ α T αβ = −∂ α X µ ∂ s X µ h αδ Γ s δβ − 1 2 ∂ β h γδ ∂ γ X µ ∂ δ X µ
Only symmetric part of h αδ Γ s δβ = 1 2 hαδ h sp ∂ δ h pβ + ∂ β h pδ − ∂ p h βδ
in the indices α, s matters! This translates into symmetry of
β, δ. Finally,
Equations
2∇ α T αβ = − 1 2 hαδ ∂ β h pδ h sp ∂ αX µ ∂ sX µ − 1 2 ∂ β h γδ ∂ γ X µ ∂ δ X µ = 0 . (3.14)
T α α = 0 , ∇ β T αβ = 0
are consequences of the Weyl and diffeomorphism symmetry, respectively. These
two are gauge symmetries, and the relations above can be understood as the second
Noether theorem.