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

250 Infinite-dimensional symmetries in gravityThen, after an integration by parts, we getIn this gauge it is obviously− 1 4 ρ R(2) ˙=− ˜g µν λ −1 ∂ ν λ∂ µ ρ + 1 2 ẽα ν ∂ ν (ẽ αµ ∂ µ ρ). (14.36)˜g µν =ẽ αµẽ να = λ 2 g µν . (14.37)So we have three fields: the dilaton ρ, the λ and the unimodular two-bein ẽ µ α .We can calculate the equations of motion varying the Lagrangian with respect toall these fields. Varying it with respect to λ we get∂ µ ( ˜g µν ∂ ν ρ) ≡ £ρ = 0 (14.38)because in conformal gauge ˜g µν = η µν . The solution of this equation isρ(x) = ρ + (x + ) + ρ − (x − ) (14.39)with x ± = x 0 ± x 1 .The dilaton can be dualized: in two dimensions, the dual of a scalar field isagain a scalar field. We will refer to the dual of the dilaton field as the ‘axion’;itis defined by∂ µ ρ + ɛ µν ∂ ν ˜ρ = 0 (14.40)where ˜ρ is just the axion. In the conformal gauge this field isThe equation obtained by varying ρ is˜ρ(x) = ρ + (x + ) − ρ − (x − ). (14.41)∂ µ ( ˜g µν λ −1 ∂ ν λ) = matter contribution. (14.42)Note that with matter contribution we refer to the fields and B coming out fromdimensional reduction. The terminology matter part will be clear in the followingsection, where we will be able to identify this fields with the fields of a bosonicnonlinear σ -model Lagrangian.Before writing the complete Lagrangian of the two-dimensional reducedgravity we must still consider the equation that is obtained from (14.36) byvariation with respect to the unimodular two-bein. The corresponding equationsmust be interpreted as constraint equations (in standard conformal field theory,they would just correspond to the Virasoro constraints). We have−δẽ αµẽ αν λ −1 ∂ (µ λ∂ ν) ρ + 1 2 δẽ α µ ∂ µ (ẽ αν ∂ ν ρ) − 1 2 ∂ µẽ α µ δẽ αν ∂ ν ρ + matter = 0.(14.43)In conformal gauge, ẽ α µ = δ µ α , this expression becomes− 1 2 δ ˜gµν λ −1 ∂ µ λ∂ ν ρ + 1 4 δ ˜gµν ∂ µ ∂ ν ρ = matter (14.44)