327 research outputs found
Poisson sigma models and symplectic groupoids
We consider the Poisson sigma model associated to a Poisson manifold. The
perturbative quantization of this model yields the Kontsevich star product
formula. We study here the classical model in the Hamiltonian formalism. The
phase space is the space of leaves of a Hamiltonian foliation and has a natural
groupoid structure. If it is a manifold then it is a symplectic groupoid for
the given Poisson manifold. We study various families of examples. In
particular, a global symplectic groupoid for a general class of two-dimensional
Poisson domains is constructed.Comment: 34 page
Analogues of the central point theorem for families with -intersection property in
In this paper we consider families of compact convex sets in
such that any subfamily of size at most has a nonempty intersection. We
prove some analogues of the central point theorem and Tverberg's theorem for
such families
Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model
General boundary conditions ("branes") for the Poisson sigma model are
studied. They turn out to be labeled by coisotropic submanifolds of the given
Poisson manifold. The role played by these boundary conditions both at the
classical and at the perturbative quantum level is discussed. It turns out to
be related at the classical level to the category of Poisson manifolds with
dual pairs as morphisms and at the perturbative quantum level to the category
of associative algebras (deforming algebras of functions on Poisson manifolds)
with bimodules as morphisms. Possibly singular Poisson manifolds arising from
reduction enter naturally into the picture and, in particular, the construction
yields (under certain assumptions) their deformation quantization.Comment: 21 pages, 2 figures; minor corrections, references updated; final
versio
Graphene as a quantum surface with curvature-strain preserving dynamics
We discuss how the curvature and the strain density of the atomic lattice
generate the quantization of graphene sheets as well as the dynamics of
geometric quasiparticles propagating along the constant curvature/strain
levels. The internal kinetic momentum of Riemannian oriented surface (a vector
field preserving the Gaussian curvature and the area) is determined.Comment: 13p, minor correction
Tverberg-type theorems for intersecting by rays
In this paper we consider some results on intersection between rays and a
given family of convex, compact sets. These results are similar to the center
point theorem, and Tverberg's theorem on partitions of a point set
Cotangent bundle quantization: Entangling of metric and magnetic field
For manifolds of noncompact type endowed with an affine connection
(for example, the Levi-Civita connection) and a closed 2-form (magnetic field)
we define a Hilbert algebra structure in the space and
construct an irreducible representation of this algebra in . This
algebra is automatically extended to polynomial in momenta functions and
distributions. Under some natural conditions this algebra is unique. The
non-commutative product over is given by an explicit integral
formula. This product is exact (not formal) and is expressed in invariant
geometrical terms. Our analysis reveals this product has a front, which is
described in terms of geodesic triangles in . The quantization of
-functions induces a family of symplectic reflections in
and generates a magneto-geodesic connection on . This
symplectic connection entangles, on the phase space level, the original affine
structure on and the magnetic field. In the classical approximation,
the -part of the quantum product contains the Ricci curvature of
and a magneto-geodesic coupling tensor.Comment: Latex, 38 pages, 5 figures, minor correction
Approximations and selections of multivalued mappings of finite-dimensional spaces
Abstract. We prove extension-dimensional versions of finite dimensional selection and approximation theorems. As applications, we obtain several results on extension dimension
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