Lax formalism for Gelfand-Tsetlin integrable systems

TL;DR

This paper develops a Lax pair formalism for Gelfand-Tsetlin integrable systems on (co)adjoint orbits of compact Lie groups, linking algebraic curves, topology, and action coordinates.

Contribution

It introduces a novel Lax pair approach for Gelfand-Tsetlin systems on (co)adjoint orbits, connecting algebraic geometry and symplectic topology.

Findings

Constructed algebraic curves encoding integrable systems

Linked topology of singular fibers with algebraic curves and vanishing cycles

Reformulated action coordinates using hyperelliptic integrals

Abstract

In the present work, we study Hamiltonian systems on (co)adjoint orbits and propose a Lax pair formalism for Gelfand-Tsetlin integrable systems defined on (co)adjoint orbits of the compact Lie groups ${\rm{U}}(n)$ and ${\rm{SO}}(n)$. In the particular setting of (co)adjoint orbits of ${\rm{U}}(n)$, by means of the associated Lax matrix we construct a family of algebraic curves which encodes the Gelfand-Tsetlin integrable systems as branch points. This family of algebraic curves enables us to explore some new insights into the relationship between the topology of singular Gelfand-Tsetlin fibers, singular algebraic curves and vanishing cycles. Further, we provide a new description for Guillemin and Sternberg's action coordinates in terms of hyperelliptic integrals.