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

320 Elementary introduction to pre-big bang cosmologyThis term violates, at one loop, the conformal invariance, unless we restrict to abackground geometry satisfying the conditionR µν = 0, (16.128)which are just the usual Einstein equations in vacuum.A similar procedure can be applied if the string moves in a richer externalbackground (not only pure gravity). Indeed, pure gravity is not enough, as aconsistent quantum theory for closed bosonic strings, for instance, must containat least three massless states (besides the unphysical tachyon, removed bysupersymmetry) in the lowest energy level: the graviton, the scalar dilaton andthe pseudoscalar Kalb–Ramond axion. The σ -model describing the propagationof a string in such a background must thus contain the coupling to the metric, tothe dilaton φ, and to the two-form B µν =−B νµ :S =− 1 ∫4πα ′ d 2 ξ∂ i x µ ∂ j x ν ( √ −γγ ij g µν + ɛ ij B µν )− 1 ∫d 2 ξ √ −γ φ 4π2 R(2) (γ ), (16.129)where ɛ ij is the two-dimensional Levi-Civita tensor density, ɛ 12 =−ɛ 21 = 1, andR (2) (γ ) is the two-dimensional scalar curvature for the world sheet metric γ . Thecondition of conformal invariance, at the one-loop level, leads to the equationsR µν +∇ µ ∇ ν − 1 4 H µαβ H αβ ν = 0, H µνα = ∂ µ B να + ∂ ν B αµ + ∂ α B µν ,R + 2∇ 2 φ − (∇φ) 2 −12 1 H µνα 2 = 0,∇ µ (e −φ H µνα ) = 0, (16.130)which can be obtained by extremizing the effective actionS =− 1 ∫2λ d−1 sd d+1 x √ [|g|e −φ R + (∇φ) 2 − 1 ]12 H µνα2(16.131)(see appendix C).It should be noted that the inclusion of the dilaton in the condition ofconformal invariance cannot be avoided, since the dilaton coupling in the action(16.129) breaks conformal invariance already at the classical level ( √ −γ R (2) isnot invariant under a Weyl rescaling of γ ). However, the dilaton term is of orderα ′ with respect to the other terms of the action (for dimensional reasons), so thatit is correct to sum up the classical dilaton contribution to the quantum, one-loopeffects, as they are all of the same order in α ′ . Without the dilaton, however, theworld sheet curvature density √ −γ R (2) does not contribute to the string equationsof motion, as it is a pure Eulero two-form in two dimensions (just like the Gauss–Bonnet term in four dimensions).