Integrable systems and group actions

2021

Anal.Math.Phys.

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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.

Section: Definition 26mentioning

confidence: 75%

Section: Definition 26mentioning

confidence: 84%

Section: Definition 26mentioning

confidence: 99%

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Geometric quantization via cotangent models

Mir

2021

Anal.Math.Phys.

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“…For singular integrable system, there are some subtleties concerning the distribution defined by it (cf. [Mi14,Mi03]). When we say the foliation is defined by a set of first integrals, we mean that the distribution defined by the Hamiltonian vector fields of a set of functions is the same.…”

Section: B-integrable Systems With Singularitiesmentioning

confidence: 99%

“…From a topological point of view, an integrable system on a compact manifold must have singularities. In [E90], [Mi14], [Mi03], [MZ04], in total analogy with the Liouville-Mineur-Arnold theorem, a symplectic Morse-Bott theory is constructed in a neighbourhood of a singular compact orbit.…”

Section: Introductionmentioning

confidence: 99%

Cotangent Models for Integrable Systems

Kiesenhofer

2016

Commun. Math. Phys.

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Abstract. We associate cotangent models to a neighbourhood of a Liouville torus in symplectic and Poisson manifolds focusing on b-Poisson/bsymplectic manifolds. The semilocal equivalence with such models uses the corresponding action-angle theorems in these settings: the theorem of Liouville-Mineur-Arnold [A74] for symplectic manifolds and an action-angle theorem for regular Liouville tori in Poisson manifolds [LMV11]. Our models comprise regular Liouville tori of Poisson manifolds but also consider the Liouville tori on the singular locus of a b-Poisson manifold. For this latter class of Poisson structures we define a twisted cotangent model. The equivalence with this twisted cotangent model is given by an action-angle theorem recently proved in [KMS16]. This viewpoint of cotangent models provides a new machinery to construct examples of integrable systems, which are especially valuable in the b-symplectic case where not many sources of examples are known. At the end of the paper we introduce non-degenerate singularities as lifted cotangent models on b-symplectic manifolds and discuss some generalizations of these models to general Poisson manifolds.

Coupling symmetries with Poisson structures

Laurent-Gengoux

2013

Acta Math Vietnam.

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Abstract. We study local normal forms for completely integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact Lie groups. The existence of Weinstein's splitting theorem for the integrable system is also studied giving some examples in which such a splitting does not exist, i.e. when the integrable system is not, locally, a product of an integrable system on the symplectic leaf and an integrable system on a transversal. The problem of splitting for integrable systems with additional symmetries is also considered.