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

Cotangent bundle quantization: Entangling of metric and magnetic field

For manifolds $\cal M$ 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 $L^2(T^*\cal M)$ and construct an irreducible representation of this algebra in $L^2(\cal M)$. 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 $T^*\cal M$ 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 $\cal M$. The quantization of $\delta$-functions induces a family of symplectic reflections in $T^*\cal M$ and generates a magneto-geodesic connection $\Gamma$ on $T^*\cal M$. This symplectic connection entangles, on the phase space level, the original affine structure on $\cal M$ and the magnetic field. In the classical approximation, the $\hbar^2$-part of the quantum product contains the Ricci curvature of $\Gamma$ and a magneto-geodesic coupling tensor.Comment: Latex, 38 pages, 5 figures, minor correction

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

Quantum Magnetic Algebra and Magnetic Curvature

The symplectic geometry of the phase space associated with a charged particle is determined by the addition of the Faraday 2-form to the standard structure on the Euclidean phase space. In this paper we describe the corresponding algebra of Weyl-symmetrized functions in coordinate and momentum operators satisfying nonlinear commutation relations. The multiplication in this algebra generates an associative product of functions on the phase space. This product is given by an integral kernel whose phase is the symplectic area of a groupoid-consistent membrane. A symplectic phase space connection with non-trivial curvature is extracted from the magnetic reflections associated with the Stratonovich quantizer. Zero and constant curvature cases are considered as examples. The quantization with both static and time dependent electromagnetic fields is obtained. The expansion of the product by the deformation parameter, written in the covariant form, is compared with the known deformation quantization formulas.Comment: 23 page

Selective labeling: identifying representative sub-volumes for interactive segmentation

Automatic segmentation of challenging biomedical volumes with multiple objects is still an open research field. Automatic approaches usually require a large amount of training data to be able to model the complex and often noisy appearance and structure of biological organelles and their boundaries. However, due to the variety of different biological specimens and the large volume sizes of the datasets, training data is costly to produce, error prone and sparsely available. Here, we propose a novel Selective Labeling algorithm to overcome these challenges; an unsupervised sub-volume proposal method that identifies the most representative regions of a volume. This massively-reduced subset of regions are then manually labeled and combined with an active learning procedure to fully segment the volume. Results on a publicly available EM dataset demonstrate the quality of our approach by achieving equivalent segmentation accuracy with only 5 % of the training data

Heisenberg Evolution WKB and Symplectic Area Phases

The Schrodinger and Heisenberg evolution operators are represented in quantum phase space by their Weyl symbols. Their semiclassical approximations are constructed in the short and long time regimes. For both evolution problems, the WKB representation is purely geometrical: the amplitudes are functions of a Poisson bracket and the phase is the symplectic area of a region in phase space bounded by trajectories and chords. A unified approach to the Schrodinger and Heisenberg semiclassical evolutions is developed by introducing an extended phase space. In this setting Maslov's pseudodifferential operator version of WKB analysis applies and represents these two problems via a common higher dimensional Schrodinger evolution, but with different extended Hamiltonians. The evolution of a Lagrangian manifold in the extended phase space, defined by initial data, controls the phase, amplitude and caustic behavior. The symplectic area phases arise as a solution of a boundary condition problem. Various applications and examples are considered.Comment: 32 pages, 7 figure

Discovery of X-ray Pulsations from the HMXB Source AXJ1749.1-2733

We are reporting a discovery of X-ray pulsations from the source AX J1749.1-2733 with the period of ~132 s based on the XMM-Newton data obtained in March 2007. The observed pulse profile has a double-peaked structure with the pulse fraction of about 25-30 % in the 3-10 keV energy band. We have also found that a periodicity with practically the same period has been detected from the source by the IBIS telescope onboard the INTEGRAL observatory during an outburst on Sept. 9, 2003 in the 20-50 keV energy band. Due to the double-peaked pulse profile, there is an additional peak on both periodograms of nearly ~66 s, therefore we have also investigated the possibility that the last value is the true pulse period. The source spectrum obtained by the XMM-Newton observatory in the soft energy band is being heavily absorbed ($N_H\simeq2\times10^{23}$ cm$^{-2}$) due to a strong intrinsic absorption in the binary system that leads to the conclusion that AX J1749.1-2733 is a new transient X-ray pulsar in the high mass X-ray binary system.Comment: MNRAS, 6 pages, 6 figure

Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics

The knowledge of quantum phase flow induced under the Weyl's association rule by the evolution of Heisenberg operators of canonical coordinates and momenta allows to find the evolution of symbols of generic Heisenberg operators. The quantum phase flow curves obey the quantum Hamilton's equations and play the role of characteristics. At any fixed level of accuracy of semiclassical expansion, quantum characteristics can be constructed by solving a coupled system of first-order ordinary differential equations for quantum trajectories and generalized Jacobi fields. Classical and quantum constraint systems are discussed. The phase-space analytic geometry based on the star-product operation can hardly be visualized. The statement "quantum trajectory belongs to a constraint submanifold" can be changed e.g. to the opposite by a unitary transformation. Some of relations among quantum objects in phase space are, however, left invariant by unitary transformations and support partly geometric relations of belonging and intersection. Quantum phase flow satisfies the star-composition law and preserves hamiltonian and constraint star-functions.Comment: 27 pages REVTeX, 6 EPS Figures. New references added. Accepted for publication to JM