No Arabic abstract
This paper is devoted to the study of Morita equivalence for twisted Poisson manifolds. We review some Morita invariants and prove that integrable twisted Poisson manifolds which are gauge equivalent are Morita equivalent. Moreover, we introduce the notion of weak Morita equivalence and show that if two twisted Poisson manifolds are weak Morita equivalent, there exists a one-to-one correspondence between their twisted symplectic leaves.
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 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.
Logicians and philosophers of science have proposed various formal criteria for theoretical equivalence. In this paper, we examine two such proposals: definitional equivalence and categorical equivalence. In order to show precisely how these two well-known criteria are related to one another, we investigate an intermediate criterion called Morita equivalence.
This is an exposition of the Donaldson geometric flow on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. The original work appeared in [1].
These notes give an introduction to the quantization procedure called geometric quantization. It gives a definition of the mathematical background for its understanding and introductions to classical and quantum mechanics, to differentiable manifolds, symplectic manifolds and the geometry of line bundles and connections. Moreover, these notes are endowed with several exercises and examples.