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

264 Infinite-dimensional symmetries in gravityThe sum now reads + = 1 + t1 − t D + P − − 1 − t1 + t D − P ++ 2t(1 − t) 2 t−1 ∂ + tP − + 2t(1 + t) 2 t−1 ∂ − tP + (14.141)where the integrability relation seen in section 14.2 has been used. Now let uspostulatet −1 ∂ ± t = 1 ∓ t1 ± t ρ−1 ∂ ± ρ (14.142)so it follows + = 1 + t21 − t 2 (D + P − − D − P + ) + 2t1 − t 2 (D + P − + D − P + )+ 2t1 − t 2 (ρ−1 ∂ + ρ P − + ρ −1 ∂ − ρ P + ). (14.143)Now the first term is null for the integrability relation D + P − = D − P + andthe second for the equation (14.126). Therefore, the integrability condition ischecked.Let us now focus on equation (14.142): it is integrable once one has asolution of £ρ = 0. This can be explicitly verified, as it follows. First, let usmultiply (14.142) by (1 − t 2 ); after a little algebra this equation reduces to(∂ ±[ρ t + 1 ) ]− 2 ˜ρ = 0 (14.144)twhere the axion ˜ρ has been introduced. So one must have12 ρ (t + 1 t)−˜ρ = w (14.145)where w is an integration constant. When we substitute in this relation the explicitexpression of the dilaton and the axion as functions of incoming and outgoingfields, we get√w + ρ+ (xt(x; w) =+ ) − √ w − ρ − (x − )√w + ρ+ (x + ) + √ w − ρ − (x − ) . (14.146)For fixed x, the function t(x; w) lives on a two-sheeted Riemann surface over thecomplex w-plane, with an x-dependent cut extending from ρ − (x − ) to ρ + (x + ).The integration constant w can be regarded as an alternative spectral parameter.The inverse of the spectral parameter is also important⎧⎪2ty ≡ 1 w = 2t ⎨ρ(1 + t 2 ) − 2t ˜ρ = ρ +···, t ∼ 0⎪2(14.147)⎩ρt +···, t ∼∞