837 research outputs found
Weak-Hamiltonian dynamical systems
A big-isotropic structure is an isotropic subbundle of ,
endowed with the metric defined by pairing. The structure is said to be
integrable if the Courant bracket ,
. Then, necessarily, one also has
, \cite{V-iso}. A weak-Hamiltonian dynamical system is a vector field
such that . We obtain the
explicit expression of and of the integrability conditions of under
the regularity condition We show that the
port-controlled, Hamiltonian systems (in particular, constrained mechanics)
\cite{{BR},{DS}} may be interpreted as weak-Hamiltonian systems. Finally, we
give reduction theorems for weak-Hamiltonian systems and a corresponding
corollary for constrained mechanical systems.Comment: 19 pages, minor improvement
On the geometry of double field theory
Double field theory was developed by theoretical physicists as a way to
encompass -duality. In this paper, we express the basic notions of the
theory in differential-geometric invariant terms, in the framework of
para-Kaehler manifolds. We define metric algebroids, which are vector bundles
with a bracket of cross sections that has the same metric compatibility
property as a Courant bracket. We show that a double field gives rise to two
canonical connections, whose scalar curvatures can be integrated to obtain
actions. Finally, in analogy with Dirac structures, we define and study
para-Dirac structures on double manifolds.Comment: The paper will appear in J. Math. Phys., 201
Reduction and construction of Poisson quasi-Nijenhuis manifolds with background
We extend the Falceto-Zambon version of Marsden-Ratiu Poisson reduction to
Poisson quasi-Nijenhuis structures with background on manifolds. We define
gauge transformations of Poisson quasi-Nijenhuis structures with background,
study some of their properties and show that they are compatible with reduction
procedure. We use gauge transformations to construct Poisson quasi-Nijenhuis
structures with background.Comment: to appear in IJGMM
Tri-hamiltonian vector fields, spectral curves and separation coordinates
We show that for a class of dynamical systems, Hamiltonian with respect to
three distinct Poisson brackets (P_0, P_1, P_2), separation coordinates are
provided by the common roots of a set of bivariate polynomials. These
polynomials, which generalise those considered by E. Sklyanin in his
algebro-geometric approach, are obtained from the knowledge of: (i) a common
Casimir function for the two Poisson pencils (P_1 - \lambda P_0) and (P_2 - \mu
P_0); (ii) a suitable set of vector fields, preserving P_0 but transversal to
its symplectic leaves. The frameworks is applied to Lax equations with spectral
parameter, for which not only it unifies the separation techniques of Sklyanin
and of Magri, but also provides a more efficient ``inverse'' procedure not
involving the extraction of roots.Comment: 49 pages Section on reduction revisite
On the geometric quantization of twisted Poisson manifolds
We study the geometric quantization process for twisted Poisson manifolds.
First, we introduce the notion of Lichnerowicz-twisted Poisson cohomology for
twisted Poisson manifolds and we use it in order to characterize their
prequantization bundles and to establish their prequantization condition. Next,
we introduce a polarization and we discuss the quantization problem. In each
step, several examples are presented
Gauge field theories with covariant star-product
A noncommutative gauge theory is developed using a covariant star-product
between differential forms defined on a symplectic manifold, considered as the
space-time. It is proven that the field strength two-form is gauge covariant
and satisfies a deformed Bianchi identity. The noncommutative Yang-Mills action
is defined using a gauge covariant metric on the space-time and its gauge
invariance is proven up to the second order in the noncommutativity parameter.Comment: Dedicated to Ioan Gottlieb on the occasion of his 80th birthday
anniversary. 12 page
A class of Poisson-Nijenhuis structures on a tangent bundle
Equipping the tangent bundle TQ of a manifold with a symplectic form coming
from a regular Lagrangian L, we explore how to obtain a Poisson-Nijenhuis
structure from a given type (1,1) tensor field J on Q. It is argued that the
complete lift of J is not the natural candidate for a Nijenhuis tensor on TQ,
but plays a crucial role in the construction of a different tensor R, which
appears to be the pullback under the Legendre transform of the lift of J to
co-tangent manifold of Q. We show how this tangent bundle view brings new
insights and is capable also of producing all important results which are known
from previous studies on the cotangent bundle, in the case that Q is equipped
with a Riemannian metric. The present approach further paves the way for future
generalizations.Comment: 22 page
Twistor theory on a finite graph
We show how the description of a shear-free ray congruence in Minkowski space
as an evolving family of semi-conformal mappings can naturally be formulated on
a finite graph. For this, we introduce the notion of holomorphic function on a
graph. On a regular coloured graph of degree three, we recover the space-time
picture. In the spirit of twistor theory, where a light ray is the more
fundamental object from which space-time points should be derived, the line
graph, whose points are the edges of the original graph, should be considered
as the basic object. The Penrose twistor correspondence is discussed in this
context
A sigma model field theoretic realization of Hitchin's generalized complex geometry
We present a sigma model field theoretic realization of Hitchin's generalized
complex geometry, which recently has been shown to be relevant in
compactifications of superstring theory with fluxes. Hitchin sigma model is
closely related to the well known Poisson sigma model, of which it has the same
field content. The construction shows a remarkable correspondence between the
(twisted) integrability conditions of generalized almost complex structures and
the restrictions on target space geometry implied by the Batalin--Vilkovisky
classical master equation. Further, the (twisted) classical Batalin--Vilkovisky
cohomology is related non trivially to a generalized Dolbeault cohomology.Comment: 28 pages, Plain TeX, no figures, requires AMS font files AMSSYM.DEF
and amssym.tex. Typos in eq. 6.19 and some spelling correcte
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