Ivancevic_Applied-Diff-Geom

774 Applied Differential Geometry: A Modern Introduction4.14.11.4 Scaling Limits of CM–SystemsFor the standard elliptic CM–systems corresponding to A N−1 , in the scalinglimit we have [Inozemtsev (1989a); Inozemtsev (1989b)]m = Mq − 12N , q → 0, (4.275)ix i = X i − 2ω 2 ,N1 ≤ i ≤ N, (4.276)where M is kept fixed, the elliptic A N−1 CM Hamiltonian tends to thefollowing Hamiltonian(H Toda = 1 N∑p 2 i − 1 N−1)∑e Xi+1−Xi + e X1−X N. (4.277)2 2i=1i=1The roots e i − e i+1 , 1 ≤ i ≤ N − 1, and e N − e 1 can be recognized as thesimple roots of the affine algebra A (1)N−1. Thus (4.277) can be recognized asthe Hamiltonian of the Toda system defined by A (1)N−1 .Scaling Limits based on the Coxeter NumberThe key feature of the above scaling limit is the collapse of the sum overthe entire root lattice of A N−1 in the CM Hamiltonian to the sum over onlysimple roots in the Toda Hamiltonian for the Kac–Moody algebra A (1)N−1 .Our task is to extend this mechanism to general Lie algebras. For this, weconsider the following generalization of the preceding scaling limitm = Mq − 1 2 δ , x = X − 2ω 2 δρ ∨ , (4.278)Here x = (x i ), X = (X i ) and ρ ∨ are rD vectors. The vector x is thedynamical variable of the CM–system. The parameters δ and ρ ∨ dependon the algebra g and are yet to be chosen. As for M and X, they have thesame interpretation as earlier, namely as respectively the mass parameterand the dynamical variables of the limiting system. Setting ω 1 = −iπ, thecontribution of each root α to the CM potential can be expressed asm 2 ℘(α · x) = 1 2 M 2∞ ∑n=−∞e 2δω2ch(α · x − 2nω 2 ) − 1 . (4.279)It suffices to consider positive roots α. We shall also assume that 0 ≤δ α · ρ ∨ ≤ 1. The contributions of the n = 0 and n = −1 summands in