Geometric quantization via cotangent models

In this article we give a universal model for geometric quantization associated to a real polarization given by an integrable system with non-degenerate singularities. This universal model goes one step further than the cotangent models in [13] by both considering singular orbits and adding to the c...

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Detalles Bibliográficos
Autores: Mir Garcia, Pau|||0000-0002-6761-2445, Miranda Galcerán, Eva|||0000-0001-9518-5279
Tipo de recurso: artículo
Fecha de publicación:2021
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/348245
Acceso en línea:https://hdl.handle.net/2117/348245
https://dx.doi.org/10.1007/s13324-021-00559-4
Access Level:acceso abierto
Palabra clave:Geometric quantization
Singularities (Mathematics)
Symplectic geometry
Real polarization
Singularities
Cotangent models
Quantització geomètrica
Singularitats (Matemàtica)
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra
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spelling Geometric quantization via cotangent modelsMir Garcia, Pau|||0000-0002-6761-2445Miranda Galcerán, Eva|||0000-0001-9518-5279Geometric quantizationSingularities (Mathematics)Geometric quantizationSymplectic geometryReal polarizationSingularitiesCotangent modelsQuantització geomètricaSingularitats (Matemàtica)Àrees temàtiques de la UPC::Matemàtiques i estadística::ÀlgebraIn this article we give a universal model for geometric quantization associated to a real polarization given by an integrable system with non-degenerate singularities. This universal model goes one step further than the cotangent models in [13] by both considering singular orbits and adding to the cotangent models a model for the prequantum line bundle. These singularities are generic in the sense that are given by Morse-type functions and include elliptic, hyperbolic and focus-focus singularities. Examples of systems admitting such singularities are toric, semitoric and almost toric manifolds, as well as physical systems such as the coupling of harmonic oscillators, the spherical pendulum or the reduction of the Euler’s equations of the rigid body on T *(SO(3)) to a sphere. Our geometric quantization formulation coincides with the models given in [11] and [21] away from the singularities and corrects former models for hyperbolic and focus-focus singularities cancelling out the infinite dimensional contributions obtained by former approaches. The geometric quantization models provided here match the classical physical methods for mechanical systems such as the spherical pendulum as presented in [4]. Our cotangent models obey a local-to-global principle and can be glued to determine the geometric quantization of the global systems even if the global symplectic classification of the systems is not known in general.Peer Reviewed20212021-09-0120212021-07-01journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfapplication/pdfhttps://hdl.handle.net/2117/348245https://dx.doi.org/10.1007/s13324-021-00559-4reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution 3.0 Spainhttp://creativecommons.org/licenses/by/3.0/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/3482452026-05-27T15:37:01Z
dc.title.none.fl_str_mv Geometric quantization via cotangent models
title Geometric quantization via cotangent models
spellingShingle Geometric quantization via cotangent models
Mir Garcia, Pau|||0000-0002-6761-2445
Geometric quantization
Singularities (Mathematics)
Geometric quantization
Symplectic geometry
Real polarization
Singularities
Cotangent models
Quantització geomètrica
Singularitats (Matemàtica)
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra
title_short Geometric quantization via cotangent models
title_full Geometric quantization via cotangent models
title_fullStr Geometric quantization via cotangent models
title_full_unstemmed Geometric quantization via cotangent models
title_sort Geometric quantization via cotangent models
dc.creator.none.fl_str_mv Mir Garcia, Pau|||0000-0002-6761-2445
Miranda Galcerán, Eva|||0000-0001-9518-5279
author Mir Garcia, Pau|||0000-0002-6761-2445
author_facet Mir Garcia, Pau|||0000-0002-6761-2445
Miranda Galcerán, Eva|||0000-0001-9518-5279
author_role author
author2 Miranda Galcerán, Eva|||0000-0001-9518-5279
author2_role author
dc.subject.none.fl_str_mv Geometric quantization
Singularities (Mathematics)
Geometric quantization
Symplectic geometry
Real polarization
Singularities
Cotangent models
Quantització geomètrica
Singularitats (Matemàtica)
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra
topic Geometric quantization
Singularities (Mathematics)
Geometric quantization
Symplectic geometry
Real polarization
Singularities
Cotangent models
Quantització geomètrica
Singularitats (Matemàtica)
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra
description In this article we give a universal model for geometric quantization associated to a real polarization given by an integrable system with non-degenerate singularities. This universal model goes one step further than the cotangent models in [13] by both considering singular orbits and adding to the cotangent models a model for the prequantum line bundle. These singularities are generic in the sense that are given by Morse-type functions and include elliptic, hyperbolic and focus-focus singularities. Examples of systems admitting such singularities are toric, semitoric and almost toric manifolds, as well as physical systems such as the coupling of harmonic oscillators, the spherical pendulum or the reduction of the Euler’s equations of the rigid body on T *(SO(3)) to a sphere. Our geometric quantization formulation coincides with the models given in [11] and [21] away from the singularities and corrects former models for hyperbolic and focus-focus singularities cancelling out the infinite dimensional contributions obtained by former approaches. The geometric quantization models provided here match the classical physical methods for mechanical systems such as the spherical pendulum as presented in [4]. Our cotangent models obey a local-to-global principle and can be glued to determine the geometric quantization of the global systems even if the global symplectic classification of the systems is not known in general.
publishDate 2021
dc.date.none.fl_str_mv 2021
2021-09-01
2021
2021-07-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/348245
https://dx.doi.org/10.1007/s13324-021-00559-4
url https://hdl.handle.net/2117/348245
https://dx.doi.org/10.1007/s13324-021-00559-4
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution 3.0 Spain
http://creativecommons.org/licenses/by/3.0/es/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution 3.0 Spain
http://creativecommons.org/licenses/by/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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