Calogero-Moser Lax Pairs with Spectral Parameter for General Lie Algebras

Abstract

We construct a Lax pair with spectral parameter for the elliptic Calogero-Moser Hamiltonian systems associated with each of the finite dimensional Lie algebras, of the classical and of the exceptional type. When the spectral parameter equals one of the three half periods of the elliptic curve, our result for the classical Lie algebras reduces to one of the Lax pairs without spectral parameter that were known previously. These Calogero-Moser systems are invariant under the Weyl group of the associated untwisted affine Lie algebra. For non-simply laced Lie algebras, we introduce new integrable systems, naturally associated with twisted affine Lie algebras, and construct their Lax operators with spectral parameter (except in the case of $G_2$).

84 pages, Plain TeX, 1 figure; minor typos corrected, 2 refs added

References in corpus (12)

• Solutions Of Four-Dimensional Field Theories Via M Theory
• N=2 Supersymmetric Gauge Theories, Branes and Orientifolds
• Calogero-Moser Systems in SU(N) Seiberg-Witten Theory
• M Theory And Seiberg-Witten Curves: Orthogonal and Symplectic Groups
• Multiscale N=2 SUSY field theories, integrable systems and their stringy/brane origin -- I
• Singular Monopoles and Supersymmetric Gauge Theories in Three Dimensions
• More Evidence for the WDVV Equations in N=2 SUSY Yang-Mills Theories
• Spectral Curves for Super-Yang-Mills with Adjoint Hypermultiplet for General Lie Algebras
• Strong Coupling Expansions of SU(N) Seiberg-Witten Theory
• Calogero-Moser and Toda Systems for Twisted and Untwisted Affine Lie Algebras
• On Integrable Systems and Supersymmetric Gauge Theories
• Prepotentials and Riemann surfaces

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