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Locally Toric Manifolds and Singular Bohr-Sommerfeld Leaves
 
Mark D. Hamilton University of Toronto, Toronto, ON, Canada
Locally Toric Manifolds and Singular Bohr-Sommerfeld Leaves
eBook ISBN:  978-1-4704-0585-4
Product Code:  MEMO/207/971.E
List Price: $61.00
MAA Member Price: $54.90
AMS Member Price: $36.60
Locally Toric Manifolds and Singular Bohr-Sommerfeld Leaves
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Locally Toric Manifolds and Singular Bohr-Sommerfeld Leaves
Mark D. Hamilton University of Toronto, Toronto, ON, Canada
eBook ISBN:  978-1-4704-0585-4
Product Code:  MEMO/207/971.E
List Price: $61.00
MAA Member Price: $54.90
AMS Member Price: $36.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2072010; 60 pp
    MSC: Primary 53;

    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
     
     
    • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2072010; 60 pp
MSC: Primary 53;

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.

  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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