References of "Bonechi, Francesco"
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See detailQuantization of Poisson manifolds from the integrability of the modular function
Bonechi, Francesco; Tarlini, Marco; Ciccoli, Nicola et al

in Communications in Mathematical Physics (2014), 331(2), 851885

We discuss a framework for quantizing a Poisson manifold via the quantization of its symplectic groupoid, that combines the tools of geometric quantization with the results of Renault's theory of groupoid ... [more ▼]

We discuss a framework for quantizing a Poisson manifold via the quantization of its symplectic groupoid, that combines the tools of geometric quantization with the results of Renault's theory of groupoid C*-algebras. This setting allows very singular polarizations. In particular we consider the case when the modular function is "multiplicatively integrable", i.e. when the space of leaves of the polarization inherits a groupoid structure. If suitable regularity conditions are satisfied, then one can define the quantum algebra as the convolution algebra of the subgroupoid of leaves satisfying the Bohr-Sommerfeld conditions. We apply this procedure to the case of a family of Poisson structures on CP_n, seen as Poisson homogeneous spaces of the standard Poisson-Lie group SU(n+1). We show that a bihamiltoniam system on CP_n defines a multiplicative integrable model on the symplectic groupoid; we compute the Bohr-Sommerfeld groupoid and show that it satisfies the needed properties for applying Renault theory. We recover and extend Sheu's description of quantum homogeneous spaces as groupoid C*-algebras. [less ▲]

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See detailWilson Lines from Representations of NQ-Manifolds
Bonechi, Francesco; Zabzine, Maxim; Qiu, Jian UL

in International Mathematics Research Notices (2013), 2013(24),

An NQ-manifold is a nonnegatively graded supermanifold with a degree 1 homological vector field. The focus of this paper is to define the Wilson loops/lines in the context of NQ-manifolds and to study ... [more ▼]

An NQ-manifold is a nonnegatively graded supermanifold with a degree 1 homological vector field. The focus of this paper is to define the Wilson loops/lines in the context of NQ-manifolds and to study their properties. The Wilson loops/lines, which give the holonomy or parallel transport, are familiar objects in usual differential geometry, we analyze the subtleties in the generalization to the NQ-setting and we also sketch some possible applications of our construction. [less ▲]

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