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

318 Elementary introduction to pre-big bang cosmologyIt is important to note, at this point, that for a classical string it is alwayspossible to impose the so-called ‘conformal gauge’ in which the world sheetmetric is flat, γ ij = η ij . In fact, in an appropriate basis, the two-dimensionalmetric tensor can always be set in a diagonal form, γ ij = diag(a, b), and then,by using reparametrization invariance on the world sheet, b 2 dσ 2 = a 2 dσ ′2 , themetric can be set in a conformally flat form, γ ij = a 2 η ij . Since the action (16.113)is invariant under the conformal (or Weyl) transformation γ ij → 2 (ξ k )γ ij ,√ −γγij→√ 4 −2√ −γγ ij , (16.119)we can always eliminate the conformal factor a 2 in front of the Minkowski metric,by choosing = a −1 . In the conformal gauge the equations of motion (16.117)reduce toẍ µ − x ′′µ + Ɣ αβ µ (ẋ α + x ′α )(ẋ β − x ′β ) = 0, (16.120)where ẋ = dx/dτ, x ′ = dx/dσ , and the constraints (16.118) becomeg µν (ẋ µ ẋ ν + x ′µ x ′ν ) = 0, g µν ẋ µ x ′ν = 0. (16.121)We now come to the crucial observation which leads to the effective actiongoverning the motion of the background fields. The conformal transformation(16.119) is an invariance of classical theory. Let us require that there are no‘anomalies’, i.e. no quantum violations of this classical symmetry. By imposingsuch a constraint, we will obtain a set of differential equations to be satisfiedby the background fields coupled to the string. Thus, unlike a point particlewhich does not impose any constraint on the external geometry in which it ismoving, the consistent quantization of a string gives constraints for the externalfields. The background geometry cannot be chosen arbitrarily, but must satisfy theset of equations (also called β-function equations) which guarantee the absenceof conformal anomalies. The string effective action used in this paper is theaction which reproduces such a set of equations for the background fields, andin particular for the metric.The derivation of the background equations of motion and of the effectiveaction, from the σ -model action (16.113), can be performed order by order byusing a perturbative expansion in powers of α ′ (indeed, in the limit α ′ → 0the action becomes very large in natural units, so that the quantum correctionsare expected to become smaller and smaller). Such a procedure, however, is ingeneral long and complicated, even to lowest order, and a detailed derivation ofthe background equations is outside the scope of these lectures. Let us sketch herethe procedure for the simplest case in which the only external field coupled to thestring is the metric tensor g µν .In the conformal gauge, the action (16.113) becomes:S =− 1 ∫4πα ′ d 2 ξ∂ i x µ ∂ i x ν g µν . (16.122)