AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Locally toric manifolds and singular Bohr-Sommerfeld leaves
About this Title
Mark D. Hamilton, Department of Mathematics, University of Toronto, 40 St. George St., Toronto, Ontario, Canada M5S 2E4
Publication: Memoirs of the American Mathematical Society
Publication Year:
2010; Volume 207, Number 971
ISBNs: 978-0-8218-4714-5 (print); 978-1-4704-0585-4 (online)
DOI: https://doi.org/10.1090/S0065-9266-10-00583-1
Published electronically: March 3, 2010
Keywords: Toric manifold,
geometric quantization,
real polarization,
Bohr-Sommerfeld
MSC: Primary 53D50
Table of Contents
Chapters
- 1. Introduction
- 2. Background
- 3. The cylinder
- 4. The complex plane
- 5. Example: $S^2$
- 6. The multidimensional case
- 7. A better way to calculate cohomology
- 8. Piecing and glueing
- 9. Real and Kähler polarizations compared
Abstract
When geometric quantization is applied to a manifold using a real polarization which is “nice enough”, a result of Śniatycki says that the quantization can be found by counting certain objects, called Bohr-Sommerfeld leaves. Subsequently, several authors have taken this as motivation for counting Bohr-Sommerfeld leaves when studying the quantization of manifolds which are less “nice”.
In this paper, we examine the quantization of compact symplectic manifolds that can locally be modelled by a toric manifold, using a real polarization modelled on fibres of the moment map. We compute the results directly, and obtain a theorem similar to Śniatycki’s, which gives the quantization in terms of counting Bohr-Sommerfeld leaves. However, the count does not include the Bohr-Sommerfeld leaves which are singular. Thus the quantization obtained is different from the quantization obtained using a Kähler polarization.
- Jørgen Ellegaard Andersen, Geometric quantization of symplectic manifolds with respect to reducible non-negative polarizations, Comm. Math. Phys. 183 (1997), no. 2, 401–421. MR 1461965, DOI 10.1007/BF02506413
- Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR 658304
- Ana Cannas da Silva, Symplectic toric manifolds, Symplectic geometry of integrable Hamiltonian systems (Barcelona, 2001) Adv. Courses Math. CRM Barcelona, Birkhäuser, Basel, 2003, pp. 85–173. MR 2000746
- Ana Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, vol. 1764, Springer-Verlag, Berlin, 2001. MR 1853077, DOI 10.1007/978-3-540-45330-7
- Thomas Delzant, Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France 116 (1988), no. 3, 315–339 (French, with English summary). MR 984900
- James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
- L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals, Ph.D. Thesis (1984).
- L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case, Comment. Math. Helv. 65 (1990), no. 1, 4–35. MR 1036125, DOI 10.1007/BF02566590
- Victor Guillemin, Viktor Ginzburg, and Yael Karshon, Moment maps, cobordisms, and Hamiltonian group actions, Mathematical Surveys and Monographs, vol. 98, American Mathematical Society, Providence, RI, 2002. Appendix J by Maxim Braverman. MR 1929136, DOI 10.1090/surv/098
- Roger Godement, Topologie algébrique et théorie des faisceaux, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XIII, Hermann, Paris, 1973 (French). Troisième édition revue et corrigée. MR 0345092
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
- V. Guillemin and S. Sternberg, The Gel′fand-Cetlin system and quantization of the complex flag manifolds, J. Funct. Anal. 52 (1983), no. 1, 106–128. MR 705993, DOI 10.1016/0022-1236(83)90092-7
- Victor Guillemin and Shlomo Sternberg, Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984. MR 770935
- M. D. Hamilton, “The quantization of a toric manifold is given by the integer lattice points in the moment polytope,” preprint www.arxiv.org/abs/0708.2710, to appear in Toric Topology, Conference Proceedings of the International Conference on Toric Topology, Osaka, May 2006
- Birger Iversen, Cohomology of sheaves, Universitext, Springer-Verlag, Berlin, 1986. MR 842190, DOI 10.1007/978-3-642-82783-9
- Lisa C. Jeffrey and Jonathan Weitsman, Bohr-Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula, Comm. Math. Phys. 150 (1992), no. 3, 593–630. MR 1204322
- Mikhail Kogan, On completely integrable systems with local torus actions, Ann. Global Anal. Geom. 15 (1997), no. 6, 543–553. MR 1608655, DOI 10.1023/A:1006592123988
- Bertram Kostant, Quantization and unitary representations. I. Prequantization, Lectures in Modern Analysis and Applications, III, Lecture Notes in Mathematics, Vol. 170, Springer, Berlin, 1970, pp. 87–208. MR 0294568
- Saunders Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR 1344215
- E. Miranda, On symplectic linearization of singular Lagrangian foliations, Ph.D. Thesis, Univ. de Barcelona, 2003
- Rick Miranda, Algebraic curves and Riemann surfaces, Graduate Studies in Mathematics, vol. 5, American Mathematical Society, Providence, RI, 1995. MR 1326604, DOI 10.1090/gsm/005
- Mircea Puta, Hamiltonian mechanical systems and geometric quantization, Mathematics and its Applications, vol. 260, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 1247960, DOI 10.1007/978-94-011-1992-4
- J. Śniatycki, “On Cohomology Groups Appearing in Geometric Quantization”, Differential Geometric Methods in Mathematical Physics I (1975)
- Jędrzej Śniatycki, Geometric quantization and quantum mechanics, Applied Mathematical Sciences, vol. 30, Springer-Verlag, New York-Berlin, 1980. MR 554085
- W. P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), no. 2, 467–468. MR 402764, DOI 10.1090/S0002-9939-1976-0402764-6
- N. M. J. Woodhouse, Geometric quantization, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1992. Oxford Science Publications. MR 1183739
- Nguyen Tien Zung, Symplectic topology of integrable Hamiltonian systems. I. Arnold-Liouville with singularities, Compositio Math. 101 (1996), no. 2, 179–215. MR 1389366