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148 10 Generation techniques10.7 Riemann–Hilbert problemsWe again seek a new solution of (10.58) by the ansatz (10.69), viz.dΦ(λ) =H(λ)Φ(λ) , Φ(λ) =P (λ)Φ 0 (λ) , (10.79)and assume without loss of generality det (Φ) ̸= 0 and Φ (0) = 1. Theparameter λ is extended into the complex plane. The requirement thatΦ(λ) has the same analyticity properties as Φ 0 (λ) implies that P (λ) hasto be a holomorphic function in a neighbourhood of λ = 0. However,P (λ) cannot be holomorphic in the whole complex plane without beingconstant. To get a non-trivial transformation one has to use differentmatrices in different neighbourhoods, say of the origin and λ = ∞, andmatch them on the intersection of those neighbourhoods.Let L be a contour enclosing the origin in the complex λ-plane, L +its interior and L − its exterior. The Riemann–Hilbert problem consistsof finding matrices P + (λ) and P − (λ) such that P + (λ) is holomorphic inL + and continuous with non-vanishing determinant in L + ∪L and P − (λ)is holomorphic in L − and continuous with non-vanishing determinant inL − ∪L. On the contour L we have the condition[dP + (λ)+P + (λ) H 0 (λ)]P + (λ) −1=[dP − (λ)+P − (λ) H 0 (λ)]P − (λ) −1 = H (λ) ∀λ ∈L. (10.80)This is equivalent toP − (λ) =P + (λ)Φ 0 (λ) u (λ)Φ 0 (λ) −1 ∀λ ∈L, (10.81)where u (λ) is a constant matrix depending on λ only.The solution of this problem leads to integral equations of the Cauchytype which are discussed in e.g. Muskhelishvili (1953) and Ablowitz andFokas (1997), where also methods for solving them are given. For therelations of Riemann–Hilbert problems to Kac–Moody algebras cp. Chauand Ge (1989) and Liand Hou (1989).10.8 Harmonic mapsLet M be a m-dimensional Riemannian space with coordinates x a andmetric γ ab (x) and let N be an n-dimensional space with coordinates ϕ Aand metric G AB (ϕ); M is called the background space and N is thepotential space. A map M→Nis called harmonic if it is such thatϕ A (x a ) satisfies the Euler–Lagrange equations of a variational principleδL/δϕ A = 0 (10.82)