### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 062, 18 pages      arXiv:1411.7063      https://doi.org/10.3842/SIGMA.2015.062
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

### Topological Monodromy of an Integrable Heisenberg Spin Chain

Jeremy Lane
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4

Received November 27, 2014, in final form July 29, 2015; Published online July 31, 2015

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.

Key words: integrable system; monodromy; Lagrangian fibration; Heisenberg spin chain.

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