Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 777Λ ⊗ Λ ∗ associated to the root u I − u J of GL(N, C). The Lax pairs for bothuntwisted and twisted CM–systems will be of the formL = P + X, M = D + X,where the matrices P, X, D, and Y are given byX = ∑ I≠JC IJ Φ IJ (α IJ , z)E IJ , Y = ∑ I≠jC IJ Φ ′ IJ(α IJ , z)E IJand by P = p · h, D = d · (h ⊕ ˜h) + ∆.Here h is in a Cartan subalgebra H g for g, ˜h is in the Cartan–Killingorthogonal complement of H g inside a Cartan subalgebra H for GL(N, C),and ∆ is in the centralizer of H g in GL(N, C). The functions Φ IJ (x, z)and the coefficients C IJ are yet to be determined. We begin by statingthe necessary and sufficient conditions for the pair L(z), M(z) of (4.275)to be a Lax pair for the (twisted or untwisted) CM–systems. For this, it isconvenient to introduce the following notationΦ IJ = Φ IJ (α IJ · x),℘ ′ IJ = Φ IJ (α IJ · x, z)Φ ′ JI(−α IJ · x, z) − Φ IJ (−α IJ · x, z)Φ ′ JI(α IJ · x, z).Then the Lax equation ˙L(z) = [L(z), M(z)] implies the CM–system ifand only if the following three identities are satisfied (K ≠ I ≠ J)∑C IJ C JI ℘ ′ IJα IJ = s 2 m 2 |α| ℘ ν(α)(α · x), C IJ C JI ℘ ′ IJ(v I − v J ) = 0,α∈R(g)C IK C KJ (Φ IK Φ ′ KJ − Φ ′ IKΦ KJ ) =sC IJ Φ IJ d · (v I − v J ) + ∆ IJ C KJ Φ KJ − C IK Φ IK ∆ KJWe have the following Theorem [D’Hoker and Phong (1998b)]:A representation Λ, functions Φ IJ , and coefficients C IJ with a spectralparameter z satisfying (4.280–4.281) can be found for all twisted and untwistedelliptic CM–systems associated with a simple Lie algebra g, exceptpossibly in the case of twisted G 2 . In the case of E 8 , we have to assumethe existence of a ±1 cocycle.Lax Pairs for Untwisted CM SystemsHere are some important features of the Lax pairs obtained in this manner[D’Hoker and Phong (1998a); D’Hoker and Phong (1998b); D’Hoker andPhong (1998c)]: