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Register a new userGeometric quantization of integrable systems with hyperbolic singularities
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We construct the geometric quantization of a compact surface using a singular real polarization coming from an integrable system. Such a polarization always has singularities, which we assume to be of nondegenerate type. In particular, we compute the effect of hyperbolic singularities, which make an infinite-dimensional contribution to the quantization, thus showing that this quantization depends strongly on polarization.
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We define prequantization for Dirac manifolds to generalize known procedures for Poisson and (pre) symplectic manifolds by using characteristic distributions obtained from 2-cocycles associated to Dirac structures. Given a Dirac manifold $(M,D)$, we construct Poisson structure on the space of admissible functions on $(M,D)$ and a representation of the Poisson algebra to establish the prequantization condition of $(M,D)$ in terms of a Lie algebroid cohomology. Additional to this, we introduce a polarization for a Dirac manifold $M$ and discuss procedures for quantization in two cases where $M$ is compact and where $M$ is not compact.
We introduce a pre-symplectic structure on the space of connections in a G-principal bundle over a four-manifold and a Hamiltonian action on it of the group of gauge transformations that are trivial on the boundary. The moment map is given by the square of curvature so that the 0-level set is the space of flat connections. Thus the moduli space of flat connections is endowed with a pre-symplectic structure. In case when the four-manifold is null-cobordant we shall construct, on the moduli space of connections, as well as on that of flat connections, a hermitian line bundle with connection whose curvature is given by the pre-symplectic form. This is the Chern-Simons pre-quantum line bundle. The group of gauge transformations on the boundary of the base manifold acts on the moduli space of flat connections by an infinitesimally symplectic way. When the base manifold is a 4-dimensional disc we show that this action is lifted to the pre-quantum line bundle by its abelian extension. The geometric description of the latter is related to the 4-dimensional Wess-Zumino-Witten model. The previous version of this arxiv text had several incoincidence with the published article in the Differential Geometry and its Applications vol.29, so the author corrected them.
We introduce a method of geometric quantization for compact $b$-symplectic manifolds in terms of the index of an Atiyah-Patodi-Singer (APS) boundary value problem. We show further that b-symplectic manifolds have canonical Spin-c structures in the usual sense, and that the APS index above coincides with the index of the Spin-c Dirac operator. We show that if the manifold is endowed with a Hamiltonian action of a compact connected Lie group with non-zero modular weights, then this method satisfies the Guillemin-Sternberg ``quantization commutes with reduction property. In particular our quantization coincides with the formal quantization defined by Guillemin, Miranda and Weitsman, providing a positive answer to a question posed in their paper.
In this paper we outline the construction of semiclassical eigenfunctions of integrable models in terms of the semiclassical path integral for the Poisson sigma model with the target space being the phase space of the integrable system. The semiclassical path integral is defined as a formal power series with coefficients being Feynman diagrams. We also argue that in a similar way one can obtain irreducible semiclassical representations of Kontsevichs star product.
About 6 years ago, semitoric systems were classified by Pelayo & Vu Ngoc by means of five invariants. Standard examples are the coupled spin oscillator on $mathbb{S}^2 times mathbb{R}^2$ and coupled angular momenta on $mathbb{S}^2 times mathbb{S}^2$, both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a 6-parameter family of integrable systems on $mathbb{S}^2 times mathbb{S}^2$ and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focus-focus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focus-focus points, this paper provides systems for the future study of the twisting index.
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