Topological Monodromy of an Integrable Heisenberg Spin Chain

TL;DR

This paper studies the topological properties of an integrable Heisenberg spin chain system on a product of spheres, revealing non-trivial monodromy and its geometric implications.

Contribution

It demonstrates the presence of non-trivial topological monodromy in a specific integrable system related to a classical Heisenberg spin chain.

Findings

The system has a Lagrangian fiber diffeomorphic to RP^3.

The system exhibits non-trivial topological monodromy.

Monodromy is related to the geometric interpretation of integrals.

Abstract

We investigate topological properties of a completely integrable system on $S^2\times S^2 \times S^2$ which was recently shown to have a Lagrangian fiber diffeomorphic to $\mathbb{R} P^3$ not displaceable by a Hamiltonian isotopy [Oakley J., Ph.D. Thesis, University of Georgia, 2014]. This system can be viewed as integrating the determinant, or alternatively, as integrating a classical Heisenberg spin chain. We show that the system has non-trivial topological monodromy and relate this to the geometric interpretation of its integrals.