Lectures on String Theory

– 69 –
These scalar products define natural reparametrization-invariant and Poincaré invariant
measures, however none of them is Weyl invariant.
Let us first assume the simplifying situation when all metric on a world-sheet M
with a given topology are conformally equivalent (i.e. they are related to each other
by diffeomorphism and Weyl rescalings; this is the case when operator P † does not
have zero modes). In this case by using reparametrizations we can bring the metric
to the form
h αβ = e 2φ g αβ ,
where g αβ is a fiducial (reference) metric. Under reparametrizations and the Weyl
rescalings the variation of the metric can be decomposed as
where P is the following operator
δh αβ = (P ξ) αβ + 2˜Λh αβ , ˜Λ = Λ +
1
2 ∇ γξ γ ,
(P ξ) αβ = ∇ α ξ β + ∇ β ξ α − ∇ γ ξ γ h αβ ,
which maps vectors into traceless symmetric tensors. Then the integration measure
can be written as follows
∂(P ξ,
Dh = D(P ξ)D(˜Λ) = D(ξ)D(Λ)
˜Λ)
,
∣ ∂(ξ, Λ) ∣
} {{ }
Jacobian
where in the last formula we changed the variables
(P ξ, ˜Λ) → (ξ, Λ)
for the price of getting a non-trivial Jacobian. Here D(ξ) is the measure which gives
upon integration an infinite volume of the diffeomorphism group and
∂(P ξ, ˜Λ)
∣ ∣ ∣∣∣∣ ∂(P ξ) ∂(P ξ)
∣∣∣∣ ∂(P ξ)
∣ ∂(ξ, Λ) ∣ = ∂ξ ∂Λ
∂ ˜Λ
∣ = 0
∂ξ ∂ ˜Λ
∣ = |detP |
∂ξ
∂ ˜Λ
∂Λ
∂ξ
1
In fact, we have
δ(P ξ) αβ (σ)
δξ γ (σ ′ )
=
(δ γ β ∇ α + δ γ α∇ β − h αβ ∇ γ )
δ(σ − σ ′ )
Thus,
∫
|detP | =
DbDc e −i T 2 2 R )
d 2 σd 2 σ ′√ h b αβ (σ)
(δ γ β ∇ α+δα∇ γ β −h αβ ∇ γ σ δ(σ−σ′ )c γ (σ ′) .