Calogero-Moser and Toda Systems for Twisted and Untwisted Affine Lie Algebras

Abstract

The elliptic Calogero-Moser Hamiltonian and Lax pair associated with a general simple Lie algebra $\G$ are shown to scale to the (affine) Toda Hamiltonian and Lax pair. The limit consists in taking the elliptic modulus $\tau$ and the Calogero-Moser couplings $m$ to infinity, while keeping fixed the combination $M=m{e}^{i\pi \delta \tau }$ for some exponent $\delta$. Critical scaling limits arise when $1/\delta$ equals the Coxeter number or the dual Coxeter number for the untwisted and twisted Calogero-Moser systems respectively; the limit consists then of the Toda system for the affine Lie algebras $\G^{(1)}$ and $(\G ^{(1)})^\vee$. The limits of the untwisted or twisted Calogero-Moser system, for $\delta$ less than these critical values, but non-zero, consists of the ordinary Toda system, while for $\delta =0$, it consists of the trigonometric Calogero-Moser systems for the algebras $\G$ and $\G^\vee$ respectively.

34 pages, Plain TeX, minor typos corrected

References in corpus (4)

• Calogero-Moser Systems in SU(N) Seiberg-Witten Theory
• Calogero-Moser Lax Pairs with Spectral Parameter for General Lie Algebras
• Spectral Curves for Super-Yang-Mills with Adjoint Hypermultiplet for General Lie Algebras
• Strong Coupling Expansions of SU(N) Seiberg-Witten Theory

Cited by in corpus (22)

• Physics and Mathematics of Calogero particles
• Calogero-Moser Lax Pairs with Spectral Parameter for General Lie Algebras
• Unoriented Strings, Loop Equations, and N=1 Superpotentials from Matrix Models
• Spectral Curves for Super-Yang-Mills with Adjoint Hypermultiplet for General Lie Algebras
• Calogero-Moser Models II: Symmetries and Foldings
• Calogero-Moser Models III: Elliptic Potentials and Twisting
• Calogero-Moser Models IV: Limits to Toda theory
• Elliptic Calogero-Moser Systems and Isomonodromic Deformations
• Euler-Calogero-Moser system from SU(2) Yang-Mills theory
• Geometric construction of elliptic integrable systems and N=1^* superpotentials
• Seiberg-Witten Theory and Integrable Systems
• The N=1* Theories on R^{1+2} X S^1 with Twisted Boundary Conditions
• Lax Pairs and Spectral Curves for Calogero-Moser and Spin Calogero-Moser Systems
• sl(M+1) Construction of Quasi-solvable Quantum M-body Systems
• Seiberg-Witten Theory and Calogero-Moser Systems
• On the N=1* Gauge Theory on a Circle and Elliptic Integrable Systems
• The Lax pairs for elliptic C_n and BC_n Ruijsenaars-Schneider models and their spectral curves
• Affine Toda-Sutherland Systems
• Instanton Correction of Prepotential in Ruijsenaars Model Associated with N=2 SU(2) Seiberg-Witten Theory
• A Family of Quasi-solvable Quantum Many-body Systems
• Solution of quantum integrable systems from quiver gauge theories
• Nonperturbative Recursion Relations in N=2 Supersymmetric Gauge Theories