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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References

  1. Atiyah M.F., Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15.
  2. Bates L.M., Monodromy in the champagne bottle, Z. Angew. Math. Phys. 42 (1991), 837-847.
  3. Bolsinov A.V., Fomenko A.T., Integrable Hamiltonian systems. Geometry, topology, classification, Chapman & Hall/CRC, Boca Raton, FL, 2004.
  4. Cushman R., Geometry of the energy momentum mapping of the spherical pendulum, CWI Newslett. (1983), 4-18.
  5. Duistermaat J.J., On global action-angle coordinates, Comm. Pure Appl. Math. 33 (1980), 687-706.
  6. Eliasson L.H., Normal forms for Hamiltonian systems with Poisson commuting integrals - elliptic case, Comment. Math. Helv. 65 (1990), 4-35.
  7. Entov M., Polterovich L., Rigid subsets of symplectic manifolds, Compos. Math. 145 (2009), 773-826, arXiv:0704.0105.
  8. Fukaya K., Oh Y.-G., Ohta H., Ono K., Toric degeneration and nondisplaceable Lagrangian tori in $S^2\times S^2$, Int. Math. Res. Not. 2012 (2012), 2942-2993, arXiv:1002.1660.
  9. Grabowski M.P., Mathieu P., Quantum integrals of motion for the Heisenberg spin chain, Modern Phys. Lett. A 9 (1994), 2197-2206, hep-th/9403149.
  10. Hausmann J.-C., Knutson A., Polygon spaces and Grassmannians, Enseign. Math. 43 (1997), 173-198, dg-ga/9602012.
  11. Hausmann J.-C., Knutson A., The cohomology ring of polygon spaces, Ann. Inst. Fourier (Grenoble) 48 (1998), 281-321, dg-ga/9706003.
  12. Izosimov A.M., Classification of almost toric singularities of Lagrangian foliations, Sb. Math. 202 (2011), 1021-1042.
  13. Kapovich M., Millson J.J., The symplectic geometry of polygons in Euclidean space, J. Differential Geom. 44 (1996), 479-513.
  14. Leung N.C., Symington M., Almost toric symplectic four-manifolds, J. Symplectic Geom. 8 (2010), 143-187, math.SG/0312165.
  15. Meinrenken E., Symplectic geometry, course notes, Unpublished lecture notes, 2000, available at http://www.math.toronto.edu/mein/teaching/sympl.pdf.
  16. Oakley J., Lagrangian submanifolds of products of spheres, Ph.D. Thesis, University of Georgia, 2014.
  17. Oakley J., Usher M., On certain Lagrangian submanifolds of $S^2\times S^2$ and $\mathbb{C}P^n$, arXiv:1311.5152.
  18. Pelayo A., Ratiu T.S., Vu Ngoc S., Symplectic bifurcation theory for integrable systems, arXiv:1108.0328.
  19. Williamson J., On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math. 58 (1936), 141-163.
  20. Wu W., On an exotic Lagrangian torus in ${\mathbb C}P^2$, Compos. Math. 151 (2015), 1372-1394, arXiv:1201.2446.
  21. Zung N.T., Symplectic topology of integrable Hamiltonian systems. I. Arnold-Liouville with singularities, Compositio Math. 101 (1996), 179-215, math.DS/0106013.
  22. Zung N.T., A note on focus-focus singularities, Differential Geom. Appl. 7 (1997), 123-130, math.DS/0110147.

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