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Locally Toric Manifolds and Singular Bohr-Sommerfeld Leaves
Mark D. Hamilton, University of Toronto, ON, Canada
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Memoirs of the American Mathematical Society
2010; 60 pp; softcover
Volume: 207
ISBN-10: 0-8218-4714-7
ISBN-13: 978-0-8218-4714-5
List Price: US$58
Individual Members: US$34.80
Institutional Members: US$46.40
Order Code: MEMO/207/971
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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, the author examines 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. The author computes the results directly and obtains 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.

Table of Contents

  • Introduction
  • Background
  • The cylinder
  • The complex plane
  • Example: \(S^2\)
  • The multidimensional case
  • A better way to calculate cohomology
  • Piecing and glueing
  • Real and Kähler polarizations compared
  • Bibliography
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