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

SIGMA 14 (2018), 016, 43 pages      arXiv:1502.07698      https://doi.org/10.3842/SIGMA.2018.016

Classifying Toric and Semitoric Fans by Lifting Equations from ${\rm SL}_2({\mathbb Z})$

Daniel M. Kane, Joseph Palmer and Álvaro Pelayo
University of California, San Diego, Department of Mathematics, 9500 Gilman Drive #0112, La Jolla, CA 92093-0112, USA

Received April 17, 2017, in final form February 13, 2018; Published online February 22, 2018

Abstract
We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group ${\rm SL}_2({\mathbb Z})$ to its preimage in the universal cover of ${\rm SL}_2({\mathbb R})$. With this method we recover the classification of two-dimensional toric fans, and obtain a description of their semitoric analogue. As an application to symplectic geometry of Hamiltonian systems, we give a concise proof of the connectivity of the moduli space of toric integrable systems in dimension four, recovering a known result, and extend it to the case of semitoric integrable systems with a fixed number of focus-focus points and which are in the same twisting index class. In particular, we show that any semitoric system with precisely one focus-focus singular point can be continuously deformed into a system in the same isomorphism class as the Jaynes-Cummings model from optics.

Key words: symplectic geometry; integrable system; semitoric integrable systems; toric integrable systems; focus-focus singularities; ${\rm SL}_2({\mathbb Z})$.

pdf (738 kb)   tex (122 kb)

References

  1. Abraham R., Marsden J.E., Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.
  2. Atiyah M.F., Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15.
  3. Babelon O., Douçot B., Classical Bethe Ansatz and normal forms in an integrable version of the Dicke model, Phys. D 241 (2012), 2095-2108.
  4. Babelon O., Douçot B., Higher index focus-focus singularities in the Jaynes-Cummings-Gaudin model: symplectic invariants and monodromy, J. Geom. Phys. 87 (2015), 3-29, arXiv:1312.6087.
  5. Borman M.S., Li T.-J., Wu W., Spherical Lagrangians via ball packings and symplectic cutting, Selecta Math. (N.S.) 20 (2014), 261-283, arXiv:1211.5952.
  6. Cannas da Silva A., Lectures on symplectic geometry, Lecture Notes in Math., Vol. 1764, Springer-Verlag, Berlin, 2001.
  7. Charles L., Pelayo Á., Vũ Ngoc S., Isospectrality for quantum toric integrable systems, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), 815-849, arXiv:1111.5985.
  8. Colin de Verdière Y., Spectre conjoint d'opérateurs pseudo-différentiels qui commutent. I. Le cas non intégrable, Duke Math. J. 46 (1979), 169-182.
  9. Colin de Verdière Y., Spectre conjoint d'opérateurs pseudo-différentiels qui commutent. II. Le cas intégrable, Math. Z. 171 (1980), 51-73.
  10. Cox D., What is a toric variety?, in Topics in Algebraic Geometry and Geometric Modeling, Contemp. Math., Vol. 334, Amer. Math. Soc., Providence, RI, 2003, 203-223.
  11. Cox D.A., Little J.B., Schenck H.K., Toric varieties, Graduate Studies in Mathematics, Vol. 124, Amer. Math. Soc., Providence, RI, 2011.
  12. Cummings F.W., Stimulated emission of radiation in a single mode, Phys. Rev. 140 (1965), A1051-A1056.
  13. Danilov V.I., The geometry of toric varieties, Russian Math. Surveys 33 (1978), no. 2, 97-154.
  14. Delzant T., Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988), 315-339.
  15. Duistermaat J.J., Pelayo Á., Reduced phase space and toric variety coordinatizations of Delzant spaces, Math. Proc. Cambridge Philos. Soc. 146 (2009), 695-718, arXiv:0704.0430.
  16. Dullin H.R., Semi-global symplectic invariants of the spherical pendulum, J. Differential Equations 254 (2013), 2942-2963, arXiv:1108.4962.
  17. Eliashberg Y., Polterovich L., Symplectic quasi-states on the quadric surface and Lagrangian submanifolds, arXiv:1006.2501.
  18. Fulton W., Introduction to toric varieties, Annals of Mathematics Studies, Vol. 131, Princeton University Press, Princeton, NJ, 1993.
  19. Gross M., Siebert B., Mirror symmetry via logarithmic degeneration data. I, J. Differential Geom. 72 (2006), 169-338, math.AG/0309070.
  20. Gross M., Siebert B., Mirror symmetry via logarithmic degeneration data, II, J. Algebraic Geom. 19 (2010), 679-780, arXiv:0709.2290.
  21. Gross M., Siebert B., From real affine geometry to complex geometry, Ann. of Math. 174 (2011), 1301-1428, math.AG/0703822.
  22. Guillemin V., Kaehler structures on toric varieties, J. Differential Geom. 40 (1994), 285-309.
  23. Guillemin V., Moment maps and combinatorial invariants of Hamiltonian $T^n$-spaces, Progr. Math., Vol. 122, Birkhäuser Boston, Inc., Boston, MA, 1994.
  24. Guillemin V., Sternberg S., Convexity properties of the moment mapping, Invent. Math. 67 (1982), 491-513.
  25. Hohloch S., Palmer J., A family of compact semitoric systems with two focus-focus singularities, arXiv:1710.05746.
  26. Jaynes E.T., Cummings F.W., Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proc. IEEE 51 (1963), 89-109.
  27. Kane D.M., Palmer J., Pelayo Á., Minimal models of compact symplectic semitoric manifolds, J. Geom. Phys. 125 (2018), 49-74, arXiv:1610.05423.
  28. Karshon Y., Kessler L., Circle and torus actions on equal symplectic blow-ups of $\mathbb C{\rm P}^2$, Math. Res. Lett. 14 (2007), 807-823, math.SG/0501011.
  29. Kassel C., Turaev V., Braid groups, Graduate Texts in Mathematics, Vol. 247, Springer, New York, 2008.
  30. Kontsevich M., Soibelman Y., Affine structures and non-Archimedean analytic spaces, in The Unity of Mathematics, Progr. Math., Vol. 244, Birkhäuser Boston, Boston, MA, 2006, 321-385, math.AG/0406564.
  31. Kostant B., On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. (4) 6 (1973), 413-455.
  32. Leung N.C., Symington M., Almost toric symplectic four-manifolds, J. Symplectic Geom. 8 (2010), 143-187, math.SG/0312165.
  33. Miller E., What is $\ldots$ a toric variety?, Notices Amer. Math. Soc. 55 (2008), 586-587.
  34. Oda T., Convex bodies and algebraic geometry. An introduction to the theory of toric varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 15, Springer-Verlag, Berlin, 1988.
  35. Palmer J., Moduli spaces of semitoric systems, J. Geom. Phys. 115 (2017), 191-217, arXiv:1502.07296.
  36. Pelayo Á., Pires A.R., Ratiu T.S., Sabatini S., Moduli spaces of toric manifolds, Geom. Dedicata 169 (2014), 323-341, arXiv:1207.0092.
  37. Pelayo Á., Vũ Ngoc S., Semitoric integrable systems on symplectic 4-manifolds, Invent. Math. 177 (2009), 571-597, arXiv:0806.1946.
  38. Pelayo Á., Vũ Ngoc S., Constructing integrable systems of semitoric type, Acta Math. 206 (2011), 93-125, arXiv:0903.3376.
  39. Pelayo Á., Vũ Ngoc S., Symplectic theory of completely integrable Hamiltonian systems, Bull. Amer. Math. Soc. (N.S.) 48 (2011), 409-455, arXiv:1306.0115.
  40. Pelayo Á., Vũ Ngoc S., First steps in symplectic and spectral theory of integrable systems, Discrete Contin. Dyn. Syst. 32 (2012), 3325-3377, arXiv:1306.0124.
  41. Pelayo Á., Vũ Ngoc S., Hamiltonian dynamics and spectral theory for spin-oscillators, Comm. Math. Phys. 309 (2012), 123-154, arXiv:1005.0439.
  42. Poonen B., Rodriguez-Villegas F., Lattice polygons and the number 12, Amer. Math. Monthly 107 (2000), 238-250.
  43. Sepe D., Vũ Ngoc S., Integrable systems, symmetries and quantization, arXiv:1704.06686.
  44. Symington M., Four dimensions from two in symplectic topology, in Topology and Geometry of Manifolds (Athens, GA, 2001), Proc. Sympos. Pure Math., Vol. 71, Amer. Math. Soc., Providence, RI, 2003, 153-208, math.SG/0210033.
  45. Vũ Ngoc S., On semi-global invariants for focus-focus singularities, Topology 42 (2003), 365-380, math.SG/0208245.
  46. Vũ Ngoc S., Moment polytopes for symplectic manifolds with monodromy, Adv. Math. 208 (2007), 909-934, math.SG/0504165.
  47. Vianna R., On exotic Lagrangian tori in $\mathbb{CP}^2$, Geom. Topol. 18 (2014), 2419-2476, arXiv:1305.7512.
  48. Zelditch S., The inverse spectral problem for surfaces of revolution, J. Differential Geom. 49 (1998), 207-264, math-ph/0002012.

Previous article  Next article   Contents of Volume 14 (2018)