zbMATH — the first resource for mathematics

Geometric quantization of integrable systems with hyperbolic singularities. (English) Zbl 1191.53058
The main result of this very interesting paper is Theorem 6.1: Let \((M,\omega,F)\) be a two dimensional, compact, completely integrable system whose moment map has only nondegenerate singularities. Suppose \(M\) has a prequantum line bundle \(\mathbb L\) and let \({\mathcal J}\) be the sheaf of sections of \(\mathbb L\) flat along the leaves. The cohomology \(H^1(M, {\mathcal J})\) has two contributions of the form \(\mathbb C^{\mathbb N} \) for each hyperbolic singularity, each one corresponding to a space of Taylor series in one complex variable. It also has one \(\mathbb C\) term for each non-singular Bohr-Sommerfeld leaf. The cohomology in other degrees is zero.

53D50 Geometric quantization
Full Text: DOI EuDML arXiv
[1] Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N., Singularities of Differentiable Maps, 1, 2, (1988), Birkhäuser · Zbl 0659.58002
[2] Bolsinov, A. V.; Fomenko, A. T., Integrable Hamiltonian systems: geometry, topology, classification, (2004), Chapman & Hall/CRC · Zbl 1056.37075
[3] Colin de Verdière, Y.; Vey, J., Le lemme de Morse isochore, Topology, 18, 4, 283-293, (1979) · Zbl 0441.58003
[4] Cushman, R. H.; Bates, L. M., Global aspects of classical integrable systems, (1997), Birkhäuser Verlag, Basel · Zbl 0882.58023
[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., 318, 10, 949-952, (1994) · Zbl 0808.58025
[6] Eliasson, L. H., Normal forms for Hamiltonian systems with Poisson commuting integrals, (1984)
[7] Eliasson, L. H., Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case, Comment. Math. Helv., 65, 1, 4-35, (1990) · Zbl 0702.58024
[8] Ginzburg, V.; Guillemin, V.; Karshon, Y., Moment maps, cobordisms, and Hamiltonian group actions, (2004), AMS Monographs
[9] Guillemin, V.; Sternberg, S., The gel’fand-cetlin system and quantization of the complex flag manifolds, J. Funct. Anal., 52, 1, 106-128, (1983) · Zbl 0522.58021
[10] Hamilton, M., Locally toric manifolds and singular Bohr-Sommerfeld leaves, to appear in Mem. AMS, http://arxiv.org/abs/0709.4058 · Zbl 1201.53088
[11] Kostant, B., On the definition of quantization, Géométrie Symplectique et Physique Mathématique, Coll. CNRS, No. 237, Paris, 187-210, (1975) · Zbl 0326.53047
[12] Marsden, J.; Ratiu, T., Introduction to mechanics and symmetry: A basic exposition of classical mechanical systems, 17, (1999), Springer-Verlag, New York · Zbl 0933.70003
[13] Milnor, J., Morse theory, (1963), Princeton University · Zbl 0108.10401
[14] Miranda, E., On symplectic linearization of singular Lagrangian foliations, (2003)
[15] Miranda, E.; San Vu Ngoc, A singular Poincaré lemma, IMRN, 1, 27-46, (2005) · Zbl 1078.58007
[16] Petzsche, H.-J., On E. borel’s theorem, Math. Ann., 282, 299-313, (1988) · Zbl 0633.46033
[17] Rawnsley, J., On the cohomology groups of a polarization and diagonal quantization, Transaction of the American Mathematical Society, 230, 235-255, (1977) · Zbl 0313.58016
[18] Śniatycki, J., On Cohomology Groups Appearing in Geometric Quantization, (1975), Differential Geometric Methods in Mathematical Physics · Zbl 0353.53019
[19] Śniatycki, J., Geometric quantization and quantum mechanics, 30, (1980), Springer-Verlag, New York-Berlin · Zbl 0429.58007
[20] Tougeron, J. C., Idéaux de fonctions différentiables, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71, vii+219 pp., (1972) · Zbl 0251.58001
[21] Williamson, J., On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math., 58:1, 141-163, (1936) · JFM 63.1290.01
[22] Woodhouse, N. M. J., Geometric quantization, (1992), Oxford Science Publications, The Clarendon Press, Oxford University Press, New York · Zbl 0747.58004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.