Locally toric manifolds and singular Bohr-Sommerfeld leaves

Abstract

When geometric quantization is applied to a manifold using a real polarization which is "nice enough", a result ofŚniatycki 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 toŚniatycki'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. * Supported by a PIMS Postdoctoral Fellowship Let V be a vector bundle over a manifold M , Γ(V ) be the space of smooth 1 sections of V , and Ω k (M ) the space of (smooth) differential k-forms on M .Definition. Formally, a connection on a vector bundle V is a map ∇ : Γ(V ) → Ω 1 (M ) ⊗ Γ(V ) which satisfies the following properties:1. ∇(σ 1 + σ 2 ) = ∇σ 1 + ∇σ 2

Geometric quantization of integrable systems with hyperbolic singularities

“…From [10], we have a Mayer-Vietoris principle for this cohomology (see Propositions 3.4.2 and 6.3.1). Putting together the results of this section with the results from [18] and [10] (which give the regular and elliptic cases, respectively), and patching together with Mayer-Vietoris, we obtain the following: …”

“…Although Čech cohomology is defined as the direct limit over the set of all coverings, in ( [10], §3) we saw that the interesting features of the cohomology appeared already in the computation using the simplest covering, and so this is what we use here. In §8 we will show that we have computed the actual sheaf cohomology.…”

“…The case of neighbourhoods of regular leaves is covered by the theorems in [18] and [10], so we concentrate on a neighbourhood of a singular leaf, where we compute the cohomology groups using a Čech approach.…”

“…As described in [10] Chapter 5, this system has k − 1 non-singular BohrSommerfeld leaves, corresponding to the circles with integer height. A picture of the integrable system with the Bohr-Sommerfeld leaves marked on it for k = 4 is shown in Figure 7.1.…”

“…In [10], the first author computed the quantization of systems with only elliptic singularities. The result obtained was similar to Śniatycki's: all cohomology groups are zero except in degree n, and H n has dimension equal (1) Some authors, particularly those who take an index theory approach to quantization (e.g.…”