Geometry of Higgs bundles over elliptic curves related to automorphisms of simple Lie algebras, Calogero–Moser systems, and KZB equations

A. M. Levin^ab, M. A. Olshanetsky^c, A. V. Zotov<sup>ade

a</sup> Institute for Theoretical and Experimental Physics, Moscow, Russia
^b Department of Mathematics, National Research University "Higher School of Economics", Moscow, Russia
^c Kharkevich Institute for Information Transmission Problems, RAS, Moscow, Russia
^d Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Oblast, Russia
^e Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: We construct twisted Calogero–Moser systems with spins as Hitchin systems derived from the Higgs bundles over elliptic curves, where the transition operators are defined by arbitrary finite-order automorphisms of the underlying Lie algebras. We thus obtain a spin generalization of the twisted D'Hoker–Phong and Bordner–Corrigan–Sasaki–Takasaki systems. In addition, we construct the corresponding twisted classical dynamical $r$-matrices and the Knizhnik–Zamolodchikov–Bernard equations related to the automorphisms of Lie algebras.

Keywords: elliptic integrable system, finite-order Lie algebra automorphism, Higgs bundle, Knizhnik–Zamolodchikov–Bernard equation.

Received: 14.07.2015

DOI: 10.4213/tmf9005

 English version: Theoretical and Mathematical Physics, 2016, 188:2, 1121–1154

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ArXiv: 1507.04265