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

TL;DR

This paper constructs twisted Calogero-Moser systems with spins from Higgs bundles over elliptic curves, linking automorphisms of Lie algebras to integrable systems, r-matrices, and KZB equations.

Contribution

It introduces a novel spin generalization of twisted integrable systems derived from Higgs bundles, incorporating automorphisms of Lie algebras.

Findings

Constructed twisted Calogero-Moser systems with spins from Higgs bundles.

Derived twisted classical dynamical r-matrices and KZB equations.

Extended known integrable systems to include automorphism-based twists.

Abstract

We construct twisted Calogero-Moser (CM) systems with spins as the Hitchin systems derived from the Higgs bundles over elliptic curves, where transitions operators are defined by an arbitrary finite order automorphisms of the underlying Lie algebras. In this way we obtain the spin generalization of the twisted D'Hoker- Phong and Bordner-Corrigan-Sasaki-Takasaki systems. As by product, we construct the corresponding twisted classical dynamical r-matrices and Knizhnik-Zamolodchikov-Bernard equations related to the automorphisms of the Lie algebras.