Symplectic Framework Research Articles

The role of complex structures in w-symmetry

In a symplectic framework, the infinitesimal action of symplectomorphisms together with suitable reparametrizations of the two-dimensional complex base space generate some type of w-algebras. It turns out that complex structures parametrized by Beltrami differentials play an important role in this context. The construction parallels very closely two-dimensional Lagrangian conformal models where Beltrami differentials are fundamental.

Semisimple Symplectic Symmetric Spaces

A symplectic symmetric space is a connected affine symmetric manifold M endowed with a symplectic structure ω which is invariant under the geodesic symmetries. When the transvection group G0 of such a symmetric space M is semisimple, its action on (M,ω) is strongly Hamiltonian; a classical theorem due to Kostant implies that the moment map associated to this action realises a G0-equivariant symplectic covering of a coadjoint orbit O in the dual of the Lie algebra $$\mathcal{G}^0 $$ of G0. We show that this orbit itself admits a structure of symplectic symmetric space whose transvection algebra is $$\mathcal{G}^0 $$ . The main result of this paper is the classification of symmetric orbits for any semisimple Lie group. The classification is given in terms of root systems of transvection algebras and therefore provides, in a symplectic framework, a theorem analogous to the Borel–de Siebenthal theorem for Riemannian symmetric spaces. When its dimension is greater than 2, such a symmetric orbit is not regular and, in general, neither Hermitian nor pseudo-Hermitian.

Poisson Lie Groups and Non-Linear Convexity Theorems

In this paper we describe how one can derive nonlinear convexity theorems which are similar to Kostant's nonlinear convexity theorem for the Iwasawa decomposition from Hamiltonian actions on Poisson Lie groups. Our results generalize those of LU and Ratiu ([24]) and aim at a unified symplectic framework for the convexity theorem in [32] and the linear convexity theorems for coadjoint orbits of convex type in [15].

Orthosymplectic integration of linear Hamiltonian systems

The authors describe a continuous, orthogonal and symplectic factorization procedure for integrating unstable linear Hamiltonian systems. The method relies on the development of an orthogonal, symplectic change of variables to block triangular Hamiltonian form. Integration is thus carried out within the class of linear Hamiltonian systems. Use of an appropriate timestepping strategy ensures that the symplectic pairing of eigenvalues is automatically preserved. For long-term integrations, as are needed in the calculation of Lyapunov exponents, the favorable qualitative properties of such a symplectic framework can be expected to yield improved estimates. The method is illustrated and compared with other techniques in numerical experiments on the Henon-Heiles and spatially discretized Sine-Gordon equations.

On Lie algebra extensions in a symplectic framework

It is shown that the construction carried out by Cariñena and Ibort [J. Math. Phys. 29, 541–545 (1988)] involving nonsymplectic actions of Lie groups gives rise to “true” noncentral extensions of the corresponding Lie algebras.

Orthogonal lie algebras with cone potential

We say that a Lie algebra g is orthogonal if it admits a non-degenerate invariant symmetric bilinear form. In this paper we give a complete decription of all orthogonal Lie algebras with cone potential, a property defined in terms of the root decomposition with respect to a compactly embedded Cartan subalgebra. The class of these Lie algebras arises naturally in the study of convexity properties of Hamiltonian torus actions which are related to decompositions of groups similar to Iwasawa decompositions G = KAN of real semisimple Lie groups. In [HiNe94] we have generalized Kostant’s nonlinear convexity theorem for the Iwasawa decomposition for a complex Lie group to the class of complex groups whose Lie algebras have a real form which is an orthogonal Lie algebra with cone potential. We even obtained a generalization to the class of real semisimple Lie algebras having a solvable minimal parabolic subalgebra. This work generalized results of Lu and Ratiu ([LR91]) and aims at a unified symplectic framework for the convexity theorem in [Ne94] and the linear convexity theorems for coadjoint orbits of convexity type in [HNP93]. The motivation for this paper is to clarify the scope of the convexity theorems proved in [HiNe94]. Our description gives a rather clear picture of the situation. Every orthognal Lie algebra with cone potential is the ideal direct sum of a solvable Lie algebra and a semisimple Lie algebra of the same type. A semisimple Lie algebras has cone potential if and only if has a compactly embedded Cartan subalgebra. The orthogonal solvable Lie algebras with cone potential also decompose into certain indecomposable pieces which can be described by a root decomposition.

Electromagnetic duality transformations in supersymmetric field theories

We explore the properties of theories which are invariant under electromagnetic duality.N=2 supergravity Yang-Mills systems are discussed in more details. In particular, a symplectic and coordinate covariant framework is established which allows one to discuss ‘classical and quantum’ duality symmetries (T andS dualities), as they emerge in superstrings in four dimensions withN=2 spacetime supersymmetry.

Duality transformations in supersymmetric Yang-Mills theories coupled to supergravity

We consider duality transformations in N = 2, d = 4 Yang-Mills theory coupled to N = 2 supergravity. A symplectic and coordinate covariant framework is established, which allows one to discuss stringy “classical and quantum duality symmetries” (monodromies), incorporating T- and S-dualities. In particular, we shall be able to study theories (like N = 2 heterotic strings) which are formulated in symplectic basis where a “holomorphic prepotential” F does not exist, and yet give general expressions for all relevant physical quantities. Duality transformations and symmetries for the N = 1 matter coupled Yang-Mills supergravity system are also exhibited. The implications of duality symmetry on all N > 2 extended supergravities are briefly mentioned. We finally give the general form of the central charge and the N = 2 semiclassical spectrum of the dyonic BPS saturated states (as it comes by truncation of the N = 4 spectrum).

Classification of N-electron states in a quantum dot.

The physical symmetries of an N-electron gas in a d-dimensional quantum dot are studied. The symplectic group framework is used to discuss the dynamical modes of the system and to classify the N-electron states including spin. We focus on the results for a few-electron (N\ensuremath{\le}12) dot in d=2 dimensions. For N4 electrons the types of configurations appearing are strongly N dependent. The allowed configurations for N4 are also highly restricted compared to those for N\ensuremath{\ge}4. These findings suggest that results of few-electron (N4) cluster calculations may have limited applicability to large-N systems.

Author index

Solar radiation pressure affects the evolution of high area-to-mass geostationary space debris. In this work, we extend the stability study of Valk et al. (2009) by considering the influence of Earth’s shadows on the short- and long-term time evolutions of space debris. To assess the orbits stability, we use the Mean Exponential Growth factor of Nearby Orbits (MEGNO), which is an efficient numerical tool to distinguish between regular and chaotic behaviors. To reliably compute long-term space debris motion, we resort to the Global Symplectic Integrator (GSI) of Libert et al. (2011) which consists in the symplectic integration of both Hamiltonian equations of motion and variational equations. We show how to efficiently compute the MEGNO indicator in a complete symplectic framework, and we also discuss the choice of a symplectic integrator, since propagators adapted to the structure of the Hamiltonian equations of motion are not necessarily suited for the associated variational equations. The performances of our method are illustrated and validated through the study of the Arnold diffusion problem. We then analyze the effects of Earth’s shadows, using the adapted conical and cylindrical Earth’s shadowing models introduced by Hubaux et al. (2012) as the smooth shadow function deriving from these models can be easily included into the variational equations. Our stability study shows that Earth’s shadows greatly affect the global behaviour of space debris orbits by increasing the size of chaotic regions around the geostationary altitude. We also emphasize the differences in the results given by conical or cylindrical Earth’s shadowing models. Finally, such results are compared with a non-symplectic integration scheme.

A symplectic framework of linear field theories

A framework for linear field theories based on the geometry of Lagrangian subspaces of symplectic spaces is formulated. Fields defined on finite compact domains and infinitesimal limits are treated in terms of de Rham currents. Field sources are included in the formulation. No distinction between elliptic and hyperbolic field equations is made.