Geometric quantization of $b$-symplectic manifolds

We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose, we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this setting. We proceed by defining a Marsden–Weinstein quotient for our setting and prove a "quantization commutes with reduction" theorem. We explain how our geometric quantization procedure relates to a possible orbit method for Lie groupoids. Our theory encompasses the geometric quantization of symplectic manifolds, Hamiltonian Lie algebra actions, actions of bundles of Lie groups, and foliations, as well as some general constructions from differential geometry.

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On geometric quantization of b-symplectic manifolds

Advances in Mathematics ◽

10.1016/j.aim.2018.04.003 ◽

2018 ◽

Vol 331 ◽

pp. 941-951 ◽

Cited By ~ 5

Author(s):

Victor W. Guillemin ◽

Eva Miranda ◽

Jonathan Weitsman

Keyword(s):

Geometric Quantization ◽

Symplectic Manifolds

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Degenerating Kähler structures and geometric quantization

Reviews in Mathematical Physics ◽

10.1142/s0129055x1430009x ◽

2014 ◽

Vol 26 (09) ◽

pp. 1430009 ◽

Cited By ~ 1

Author(s):

João P. Nunes

Keyword(s):

Moduli Spaces ◽

General Framework ◽

Vector Bundles ◽

Geometric Quantization ◽

Theta Functions ◽

Symplectic Manifolds ◽

Important Class ◽

General Philosophy ◽

Cotangent Bundles ◽

Kähler Structures

We review some recent results on the problem of the choice of polarization in geometric quantization. Specifically, we describe the general philosophy, developed by the author together with his collaborators, of treating real polarizations as limits of degenerating families of holomorphic polarizations. We first review briefly the general framework of geometric quantization, with a particular focus on the problem of the dependence of quantization on the choice of polarization. The problem of quantization in real polarizations is emphasized. We then describe the relation between quantization in real and Kähler polarizations in some families of symplectic manifolds, that can be explicitly quantized and that constitute an important class of examples: cotangent bundles of Lie groups, abelian varieties and toric varieties. Applications to theta functions and moduli spaces of vector bundles on curves are also reviewed.

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Connections on symplectic manifolds and geometric quantization

Lecture Notes in Mathematics - Differential Geometrical Methods in Mathematical Physics ◽

10.1007/bfb0089731 ◽

1980 ◽

pp. 153-166 ◽

Cited By ~ 16

Author(s):

Harald Hess

Keyword(s):

Geometric Quantization ◽

Symplectic Manifolds

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Geometric quantization of symplectic manifolds with respect to reducible non-negative polarizations

Communications in Mathematical Physics ◽

10.1007/bf02506413 ◽

1997 ◽

Vol 183 (2) ◽

pp. 401-421 ◽

Cited By ~ 2

Author(s):

Jørgen Ellegaard Andersen

Keyword(s):

Geometric Quantization ◽

Symplectic Manifolds

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On geometric quantization of $$b^m$$-symplectic manifolds

Mathematische Zeitschrift ◽

10.1007/s00209-020-02590-w ◽

2020 ◽

Author(s):

Victor W. Guillemin ◽

Eva Miranda ◽

Jonathan Weitsman

Keyword(s):

Geometric Quantization ◽

Symplectic Manifolds

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Constructing symplectic manifolds

10.1093/oso/9780198794899.003.0008 ◽

2017 ◽

Author(s):

Dusa McDuff ◽

Dietmar Salamon

Keyword(s):

Final Section ◽

Symplectic Manifolds ◽

Homotopy Equivalence ◽

Codimension Two ◽

Open Manifolds ◽

Connected Sums ◽

Symplectic Forms ◽

Ambient Manifold ◽

Blowing Up ◽

Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

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