(ed.). Gravitational waves (IOP, 2001)(422s).

Appendix A. The string effective action 319Let us formally assume a deformation of the number of world sheet dimensions,from 2 to 2 + ɛ, and perform the conformal transformation: η ij → η ij exp(ρ).Expanding we get, for small ɛ,− 14πα ′ ∫= − 14πα ′ ∫d 2+ɛ ξ e ρɛ/2 ∂ i x µ ∂ i x ν g µνd 2+ɛ ξ∂ i x µ ∂ i x ν g µν(1 + ɛ 2 ρ +···). (16.123)For ɛ → 0 the ρ dependence disappears and the classical action isconformally invariant. In order to preserve this invariance also for the quantumtheory, at the one-loop level, let us treat the σ -model as a quantum field theoryfor x µ (σ, τ), and let us consider the quantum fluctuations ˆx µ around a givenexpectation value x µ 0. For the general reparametrization invariance of the theorywe can always choose for x 0 a locally inertial frame, such that g µν (x 0 ) = η µν .By expanding the metric around x 0 , the leading corrections are of second orderin the fluctuations, because in a locally inertial frame the first derivatives of themetric (and then the Cristoffel connection) can always be set to zero (but not thecurvature). With an appropriate choice of coordinates, called Riemann normalcoordinates, the metric can thus be expanded as:g µν (x) = η µν − 1 3 R µναβ(x 0 ) ˆx α ˆx β +··· (16.124)and the action for the quantum fluctuations becomes, to lowest order in thecurvature,S =− 1 ∫4πα ′ d 2+ɛ ξ[∂ i ˆx µ ∂ i ˆx µ(1 + ɛ )2 ρ− 1 ( 3 ∂ i ˆx µ ∂ i ˆx ν R µναβ (x 0 ) ˆx α ˆx β 1 + ɛ ) ]2 ρ +··· . (16.125)It must be noted that, at the quantum level, the dependence of ρ does notdisappear in general from the action in the limit ɛ → 0, since there are oneloopterms that diverge like ɛ −1 , just to cancel the ɛ dependence and to give acontribution proportional to ρ to the effective action. By evaluating, for instance,the two-point function for the quantum operator ˆx α ˆx β , in the coincidence limitσ → σ ′ (the tadpole graph), one obtains [83]〈ˆx α (σ ) ˆx β (σ ′ )〉 σ →σ ′ ∼ η αβ∫limσ →σ ′d 2+ɛ k eik·(σ−σ ′ )k 2 ∼ η αβ ɛ −1 , (16.126)which gives the one-loop contribution to the action∫S ∼ d 2+ɛ ξ∂ i ˆx µ ∂ i ˆx ν R µν ρ. (16.127)