Geometric quantization of integrable systems with hyperbolic singularities

Hamilton, Mark D.; Miranda, Eva

Annales de l'Institut Fourier, Tome 60 (2010) no. 1, pp. 51-85.

On construit la quantification géométrique d’une surface compacte en utilisant une polarisation singulière donnée par un système intégrable. Cette polarisation présente toujours des singularités qu’on suppose de type non-dégénéré. En particulier, on calcule l’effet des singularités hyperboliques qui donnent une contribution de dimension infinie à la quantification, en démontrant que cette quantification dépend fortement de la polarisation choisie.

We construct the geometric quantization of a compact surface using a singular real polarization coming from an integrable system. Such a polarization always has singularities, which we assume to be of nondegenerate type. In particular, we compute the effect of hyperbolic singularities, which make an infinite-dimensional contribution to the quantization, thus showing that this quantization depends strongly on polarization.

Classification : 53D50
Keywords: Geometric quantization, integrable system, non-degenerate singularity
Mot clés : quantification géométrique, système intégrable, singularité non-dégénérée

[1] Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. Singularities of Differentiable Maps, Monographs in Mathematics, 1, 2, Birkhäuser, 1988 | MR | Zbl

[2] Bolsinov, A. V.; Fomenko, A. T. Integrable Hamiltonian systems: geometry, topology, classification, Chapman & Hall/CRC, 2004 | MR | Zbl

[3] Colin de Verdière, Y.; Vey, J. Le lemme de Morse isochore, Topology, Volume 18 (1979) no. 4, pp. 283-293 | DOI | MR | Zbl

[4] Cushman, R. H.; Bates, L. M. Global aspects of classical integrable systems, Birkhäuser Verlag, Basel, 1997 | MR | Zbl

[5] Dufour, J. P.; Molino, P.; Toulet, A. Classification des systèmes intégrables en dimension $2$ et invariants des modèles de Fomenko, C. R. Acad. Sci. Paris Sér. I Math., Volume 318 (1994) no. 10, pp. 949-952 | MR | Zbl

[6] Eliasson, L. H. Normal forms for Hamiltonian systems with Poisson commuting integrals, Stockholm University (1984) (Ph. D. Thesis)

[7] Eliasson, L. H. Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case, Comment. Math. Helv., Volume 65 (1990) no. 1, pp. 4-35 | DOI | MR | Zbl

[8] Ginzburg, V.; Guillemin, V.; Karshon, Y. Moment maps, cobordisms, and Hamiltonian group actions, AMS Monographs, 2004

[9] Guillemin, V.; Sternberg, S. The Gel’fand-Cetlin system and quantization of the complex flag manifolds, J. Funct. Anal., Volume 52 (1983) no. 1, pp. 106-128 | DOI | MR | Zbl

[10] Hamilton, M. Locally toric manifolds and singular Bohr-Sommerfeld leaves, to appear in Mem. AMS, http://arxiv.org/abs/0709.4058

[11] Kostant, B. On the Definition of Quantization, Géométrie Symplectique et Physique Mathématique, Coll. CNRS, No. 237, Paris (1975), pp. 187-210 | MR | Zbl

[12] Marsden, J.; Ratiu, T. Introduction to mechanics and symmetry: A basic exposition of classical mechanical systems, Second edition, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999 | MR | Zbl

[13] Milnor, J. Morse theory, Princeton University, 1963 | MR | Zbl

[14] Miranda, E. On symplectic linearization of singular Lagrangian foliations, Univ. de Barcelona (2003) (Ph. D. Thesis)

[15] Miranda, E.; San Vu Ngoc A singular Poincaré lemma, IMRN, Volume 1 (2005), pp. 27-46 | DOI | MR | Zbl

[16] Petzsche, H.-J. On E. Borel’s Theorem, Math. Ann., Volume 282 (1988), pp. 299-313 | DOI | Zbl

[17] Rawnsley, J. On the Cohomology Groups of a Polarization and Diagonal quantization, Transaction of the American Mathematical Society, Volume 230 (1977), pp. 235-255 | DOI | MR | Zbl

[18] Śniatycki, J. On Cohomology Groups Appearing in Geometric Quantization, Differential Geometric Methods in Mathematical Physics, 1975 | Zbl

[19] Śniatycki, J. Geometric quantization and quantum mechanics, Applied Mathematical Sciences, 30, Springer-Verlag, New York-Berlin, 1980 | MR | Zbl

[20] Tougeron, J. C. Idéaux de fonctions différentiables, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71 (1972), pp. vii+219 | MR | Zbl

[21] Williamson, J. On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math., Volume 58:1 (1936), pp. 141-163 | DOI | MR

[22] Woodhouse, N. M. J. Geometric quantization, Second edition. Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992 | MR | Zbl