Locally toric manifolds and singular Bohr-Sommerfeld leaves

Mark D. Hamilton

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arXiv · 0709.4058

Locally toric manifolds and singular Bohr-Sommerfeld leaves

Mark D. Hamilton

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Abstract

When geometric quantization is applied to a manifold using a real polarization which is "nice enough", a result of Sniatycki 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 Sniatycki'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 Kaehler polarization.

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Mark D. Hamilton. 2009-06-24. Locally toric manifolds and singular Bohr-Sommerfeld leaves. https://arxiv.org/abs/0709.4058

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