Lectures on String Theory

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Here c α is called a ghost field, while b αβ is traceless and symmetric, it is called
antighost field. The ghost c α corresponds to infinitezimal reparametrizations while
b aβ corresponds to variations perpendicular to the gauge orbits. Both the ghost and
the antighost fields are real. The last formula can be now written as
∫
|detP | = DbDc e − i R
πα ′ d 2 σ √ h b αβ ∇ α c β .
Thus, the total action is given now by the sum of the Polyakov action and the ghost
action:
S = − T ∫ (
)
d 2 σγ αβ ∂ α X µ ∂ β X µ + 4ib βγ ∇ α c γ
2
There are several subtle issues we have not touched so far
• Conformal anomaly, i.e. possible dependence of the thrown away volume of
the diffeomorphism group on the Weyl (scale) degree of freedom φ.
• Reparametrizations which satisfy P ξ = 0, i.e. conformal Killing vectors. We
see that equations of motion for c α are just conformal Killing equations. Therefore,
in order not to overcount the configurations which are related by a conformal
transformation one has to exclude integration over the zero modes of c α
ghosts.
• So far we assumed that all symmetric traceless deformations of the metric can
be generated by reparametrizations. This is however not the case if P † has zero
modes. These zero modes correspond to zero modes of the b ghosts.
We can define the stress-energy tensor of the ghost fields
∫
δS gh = T d 2 σ √ −h T αβ δh αβ .
Performing the variation one finds
T gh
αβ = i(b αγ∇ β c γ + b βγ ∇ α c γ − c γ ∇ γ b αβ − h αβ b γδ ∇ γ c δ ) .
Here the last term vanishes on shell. In the deriving this expression we also used
the tracelessness of b αβ . One can verify that this tensor is covariantly conserved
∇ α T αβ = 0.
In the world-sheet light-cone coordinates σ ± the non-vanishing components of
the stress-tensor are
T ++ = i ( 2b ++ ∂ + c + + (∂ + b ++ )c +) ,
T −− = i ( 2b −− ∂ − c − + (∂ − b −− )c −) .