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.
Manifolds with boundary, with corners, b-manifolds and foliations model configuration
spaces for particles moving under constraints and can be described as E-manifolds. E-manifolds
were introduced in [NT01] and investigated in depth in [MS21]. In this article we explore their
physical facets by extending gauge theories to the E-category. Singularities in the configuration
space of a classical particle can be described in several new scenarios unveiling their Hamiltonian
aspects on an E-symplectic manifold. Following the scheme inaugurated in [Wei78], we show
the existence of a universal model for a particle interacting with an E-gauge field. In addition,
we generalize the description of phase spaces in Yang-Mills theory as Poisson manifolds and their
minimal coupling procedure, as shown in [Mon86], for base manifolds endowed with an E-structure.
In particular, the reduction at coadjoint orbits and the shifting trick are extended to this framework.
We show that Wong’s equations, which describe the interaction of a particle with a Yang-Mills field,
become Hamiltonian in the E-setting. We formulate the electromagnetic gauge in a Minkowski space
relating it to the proper time foliation and we see that our main theorem describes the minimal
coupling in physical models such as the compactified black hole.
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