Search results for "Symplectic manifold"
showing 10 items of 15 documents
TANGENTIAL DEFORMATIONS ON FIBRED POISSON MANIFOLDS
2005
In a recent article, Cattaneo, Felder and Tomassini explained how the notion of formality can be used to construct flat Fedosov connections on formal vector bundles on a Poisson manifold $M$ and thus a star product on $M$ through the original Fedosov method for symplectic manifolds. In this paper, we suppose that $M$ is a fibre bundle manifold equipped with a Poisson tensor tangential to the fibers. We show that in this case the construction of Cattaneo-Felder- Tomassini gives tangential (to the fibers) star products.
Diffeomorphisms, Noether charges, and the canonical formalism in two-dimensional dilaton gravity
1995
We carry out a parallel study of the covariant phase space and the conservation laws of local symmetries in two-dimensional dilaton gravity. Our analysis is based on the fact that the Lagrangian can be brought to a form that vanishes on-shell giving rise to a well-defined covariant potential for the symplectic current. We explicitly compute the symplectic structure and its potential and show that the requirement to be finite and independent of the Cauchy surface restricts the asymptotic symmetries.
Higher-order polarizations on the Virasoro group and anomalies
1991
In a previous paper the authors showed that the space of (first order) polarized functions on the Virasoro group is not, in general, irreducible. The full reduction was explicitly achieved by taking the orbit of the enveloping algebra through the vacuum. This additional step provided the proper quantization in the “strong-coupling” domain 0<c≦1. In this paper we introduce the concept of “higher order polarization” as a generalization of that of polarization. We prove that the imposing of the additional (higher-order) polarization conditions is equivalent to the taking of the above-mentioned orbit. This demonstrates that the generalized (higher-order) polarization conditions suffice to obtai…
Integrability Conditions: Recent Results in the Theory of Integrable Models
1990
This paper reports various results achieved recently in the theory of integrable models. These are summarised in the Fig.1! At the Chester meeting [1] two of the authors were concerned [1] with the local Riemann-Hilbert problem (double-lined box in the centre of Fig.1), its limit as a non-local Riemann-Hilbert problem used to solve classical integrable models in 2+1 dimensions (two space and one time dimensions) [2,3], and the connection of this Riemann-Hilbert problem with Ueno’s [4] Riemann-Hilbert problem associated with the representation of the algebra gl(∞) in terms of Z⊗Z matrices (Z the integers) and the solution of the K-P equations in 2+1. We were also concerned [1] with the const…
Symplectic Applicability of Lagrangian Surfaces
2009
We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with their integrability equa- tions. The invariant setup is applied to discuss the question of symplectic applicability for elliptic Lagrangian immersions. Explicit examples are considered.
Presymplectic manifolds and conservation laws
2008
In this paper we make use of a new structure called seeded fibre bundle. This allows us to combine the symplectic formalism and general relativity. A theorem of existence is obtained and some examples and properties are studied.
Closedness of Star Products and Cohomologies
1994
We first review the introduction of star products in connection with deformations of Poisson brackets and the various cohomologies that are related to them. Then we concentrate on what we have called “closed star products” and their relations with cyclic cohomology and index theorems. Finally we shall explain how quantum groups, especially in their recent topological form, are in essence examples of star products.
The Nonlinear σ Model
1989
The nonlinear (principal) σ model has been for a long time a theoretical laboratory to test different approaches for quantizing classical field theories. Here we shall discuss as an application of the current algebra representation theory a construction of the quantized σ model.
Star-products, spectral analysis, and hyperfunctions
2000
We study the ⋆-exponential function U(t;X) of any element X in the affine symplectic Lie algebra of the Moyal ⋆-product on the symplectic manifold (ℝ × ℝ;ω). When X is a compact element, a natural specific candidate for U (t;X) to be the exponential function is suggested by the study we make in the non-compact case. U (t;X) has singularities in the t variable. The analytic continuation U(z;X),z = t + iy, defines two boundary values δ+ U (t;X) = limy↓0 U(z;X) and δ-(t;X) = limy↑0 U(z; X). δ+ U (t;X) is a distribution while δ- U (t;X) is a Beurling-type, Gevrey-class s — 2 ultradistribution. We compute the Fourier transforms in t of δ± U (t;X). Both Fourier spectra are discrete but different …
Star calculus on Jacobi manifolds
2002
Abstract We study the Gerstenhaber bracket on differential forms induced by the two main examples of Jacobi manifolds: contact manifolds and l.c.s. manifolds. Moreover, we obtain explicit expressions of the generating operators and the derivations on the algebra of multivector fields. We define star operators for contact manifolds and l.c.s. manifolds and we study some of its properties.