Vlasov equation on a symplectic leaf

The infinite dimensional phase space of the Vlasov equation is foliated by symplectic manifolds (leaves) which are invariant under the dynamics. By adopting a Lie transform representation, exp{W, }, for near-identity canonical transformations we obtain a local coordinate system on a leaf. The evolution equation defined by restricting the Vlasov equation to the leaf is approximately represented by the evolution of W. We derive the equation for ∂_tW and show that it is hamiltonian relative to the nondegenerate Kirillov-Kostant-Souriau symplectic structure.     

Symplectic algebraic dynamics algorithm

Based on the algebraic dynamics solution of ordinary differential equations andintegration of  ,the symplectic algebraic dynamics algorithm sn is designed,which preserves the local symplectic geometric structure of a Hamiltonian systemand possesses the same precision of the na ve algebraic dynamics algorithm n.Computer experiments for the 4th order algorithms are made for five test modelsand the numerical results are compared with the conventional symplectic geometric algorithm,indicating that sn has higher precision,the algorithm-inducedphase shift of the conventional symplectic geometric algorithm can be reduced,and the dynamical fidelity can be improved by one order of magnitude.     

Infinitesimal Deformations of a Formal Symplectic Groupoid

Given a formal symplectic groupoid G over a Poisson manifold (M, π ₀), we define a new object, an infinitesimal deformation of G, which can be thought of as a formal symplectic groupoid over the manifold M equipped with an infinitesimal deformation \({\pi_0 + \varepsilon \pi_1}\) of the Poisson bivector field π ₀. To any pair of natural star products \({(\ast,\tilde\ast)}\) having the same formal symplectic groupoid G we relate an infinitesimal deformation of G. We call it the deformation groupoid of the pair \({(\ast,\tilde\ast)}\) . To each star product with separation of variables \({\ast}\) on a Kähler–Poisson manifold M we relate another star product with separation of variables \({\hat\ast}\) on M. We build an algorithm for calculating the principal symbols of the components of the logarithm of the formal Berezin transform of a star product with separation of variables \({\ast}\) . This algorithm is based upon the deformation groupoid of the pair \({(\ast,\hat\ast)}\) .     

A Module Structure on the Symplectic Floer Cohomology

The aim of this paper is to offer an affirmative answer to the Floer conjectures in [2, p. 589] which states that there is a module structure on the Z _{2 N} -graded symplectic Floer cohomology for monotone symplectic manifolds. By constructing a Z-graded symplectic Floer cohomology as an integral lift of the Z _{2 N} -graded symplectic Floer cohomology, we gain control of the holomorphic bubbling spheres. This makes a module structure on the Z-graded Floer cohomology. There is a spectral sequence with E ¹ _*,* given by the Z-graded symplectic Floer cohomology. Such a spectral sequence converges to the Z _{2 N} -graded symplectic Floer cohomology. Hence we induce a module structure for the Z _{2 N} -graded symplectic Floer cohomology by the spectral sequence and algebraic topology methods. Received: 2 August 1999 / Accepted: 25 October 1999     

Lie Algebras of Derivations and Resolvent Algebras

This paper analyzes the action δ of a Lie algebra X by derivations on a C*–algebra ${\mathcal{A}}$ . This action satisfies an “almost inner” property which ensures affiliation of the generators of the derivations δ with ${\mathcal{A}}$ , and is expressed in terms of corresponding pseudo–resolvents. In particular, for an abelian Lie algebra X acting on a primitive C*–algebra ${\mathcal{A}}$ , it is shown that there is a central extension of X which determines algebraic relations of the underlying pseudo–resolvents. If the Lie action δ is ergodic, i.e. the only elements of ${\mathcal{A}}$ on which all the derivations in δ _X vanish are multiples of the identity, then this extension is given by a (non–degenerate) symplectic form σ on X. Moreover, the algebra generated by the pseudo–resolvents coincides with the resolvent algebra based on the symplectic space (X, σ). Thus the resolvent algebra of the canonical commutation relations, which was recently introduced in physically motivated analyses of quantum systems, appears also naturally in the representation theory of Lie algebras of derivations acting on C*–algebras.     

Normal forms generalizing action-angle coordinates for Hamiltonian actions of Lie groups

Let (M, Ω) be a symplectic manifold on which a Lie group G acts by a Hamiltonian action. Under some restrictive assumptions, we show that there exists a symplectic diffeomorphism ψ of a G-invariant open neighbourhood U of a given G-orbit in M, onto an open subset ψ(U) of a vector bundle F ^*, with base space G. Explicit expressions are given for the symplectic 2-form, for the momentum map and for a Hamiltonian vector field whose Hamiltonian function is G-invariant, on the model symplectic manifold ψ(U).     

Generalized Cohomologies and Supersymmetry

We show that the complex cohomologies of Bott, Chern, and Aeppli and the symplectic cohomologies of Tseng and Yau arise in the context of type II string theory. Specifically, they can be used to count a subset of scalar moduli fields in Minkowski compactification with RR fluxes in the presence of either O5/D5 or O6/D6 brane sources, respectively. Further, we introduce a new set of cohomologies within the generalized complex geometry framework which interpolate between these known complex and symplectic cohomologies. The generalized complex cohomologies play the analogous role for counting massless fields for a general supersymmetric Minkowski type II compactification with Ramond–Ramond flux.     

Lie 2-Bialgebras

In this paper, we study Lie 2-bialgebras, paying special attention to coboundary ones, with the help of the cohomology theory of L _∞-algebras with coefficients in L _∞-modules. We construct examples of strict Lie 2-bialgebras from left-symmetric algebras (also known as pre-Lie algebras) and symplectic Lie algebras (also called quasi-Frobenius Lie algebras).     

A variational principle for symplectic connections

We introduce a variational principle for symplectic connections and study the corresponding field equations. For two-dimensional compact symplectic manifolds we determine all solutions of the field equations. For two-dimensional non-compact simply connected symplectic manifolds we give an essentially exhaustive list of solutions of the field equations. Finally we indicate how to construct from solutions of the field equations on (M, ω) solutions of the field equations on the cotangent bundle to M with its standard symplectic structure.     

q-Deformation of symplectic dynamical symmetries in algebraic models of nuclear structure

With a view toward further nuclear structure applications of approaches based on quantum-deformed (or q-deformed) algebras, introduced to the authors by Yu.F. Smirnov, we construct a q analog of a boson realization of the symplectic noncompact sp(4, R) algebra together with a q analog of a fermion realization of the symplectic compact sp(4) algebra. The first study, on the q-deformed Sp(4,R) symmetry, is applied to the development of a q analog of the two-dimensional Interacting Boson Model with q-deformed SU(3) the underpinning dynamical symmetry group. An explicit realization in terms of q-tensor operators with respect to the standard su _q (2) algebra is given. The group-subgroup structure of this framework yields the physical interpretation of the generators of the groups under consideration. The second symplectic algebra, the q-deformed sp(4), is applied to studying isovector pairing correlations in atomic nuclei. A specific q deformation of the sp(4) algebra is realized in terms of q deformed fermion creation and annihilation operators of the shell model. The generators of the algebra close on four distinct realizations of the u _q (2) subalgebra. These reductions, which correspond to different types of pairing interactions, yield a complete classification of the basis states. An analysis of the role of the q deformation is based on a comparison of the results for energies of the lowest isovector-paired 0⁺ states in the deformed and nondeformed cases.     

Poisson–Lie Generalization of the Kazhdan–Kostant–Sternberg Reduction

The trigonometric Ruijsenaars–Schneider model is derived by symplectic reduction of Poisson–Lie symmetric free motion on the group U(n). The commuting flows of the model are effortlessly obtained by reducing canonical free flows on the Heisenberg double of U(n). The free flows are associated with a very simple Lax matrix, which is shown to yield the Ruijsenaars–Schneider Lax matrix upon reduction.     

A new family of four-dimensional symplectic and integrable mappings

We investigate the generalisations of the Quispel, Roberts and Thompson (QRT) family of mappings in the plane leaving a rational quadratic expression invariant to the case of four variables. We assume invariance of the rational expression under a cyclic permutation of variables and we impose a symplectic structure with Poisson brackets of the Weyl type. All mappings satisfying these conditions are shown to be integrable either as four-dimensional mappings with two explicit integrals which are in involution with respect to the symplectic structure and which can also be inferred from the periodic reductions of the double-discrete versions of the modified Korteweg–deVries (ΔΔMKdV) and sine-Gordon (ΔΔsG) equations or by reduction to two-dimensional mappings with one integral of the symmetric QRT family.     

The Maslov Gerbe

Let Lag(E) be the Grassmannian of Lagrangian subspaces of a complex symplectic vector space E. We construct a Maslov class which generates the second integral cohomology of Lag(E), and we show that its mod 2 reduction is the characteristic class of a flat gerbe with structure group Z ₂. We explain the relation of this gerbe to the well-known flat Maslov line bundle with structure group Z ₄ over the real Lagrangian Grassmannian, whose characteristic class is the mod 4 reduction of the real Maslov class.     

Quantum mechanics as a matrix symplectic geometry

The main purpose of this work is to describe the quantum analog of the usual classical symplectic geometry and then to formulate quantum mechanics as a noncommutative symplectic geometry. First, we describe a discrete Weyl-Schwinger realization of the Heisenberg group and we develop a discrete version of the Weyl-Wigner-Moyal formalism. We also study the continuous limit and the case of higher degrees of freedom. In analogy with the classical case, we present the noncommutative (quantum) symplectic geometry associated with the matrix algebraM _N (C) generated by the Schwinger matrices.     

Black holes and groups of type E 7

We report some results on the relation between extremal black holes in locally supersymmetric theories of gravity and groups of type E ₇, appearing as generalized electric-magnetic duality symmetries in such theories. Some basics on the covariant approach to the stratification of the relevant symplectic representation are reviewed, along with a connection between special K?hler geometry and a ??generalization?? of groups of type E ₇.     

Structures symplectiques, structures complexes, spineurs symplectiques et transformation de Fourier

Planck's constant is very useful in the development of the theory of symplectic Clifford algebras introduced by the author in 1977 [1,a], and to solve many connected problems for example the Poisson Lie algebra deformations [1,c]. In this paper we give a precise link between a complex structure J and the Fourier transform which is nothing but the natural left action of the covering J? of J in a symplectic convenient spinor space (modulo a constant factor).Thus Fourier transform becomes a geometric transformation separated from integration technics, good peculiarity for global problems. We explain nice algebraic properties of the Fourier transform taking them in the symplectic context with adapted metric in any signature. Some applications are given: Hermite's functions, Plancherel-Parseval's theorem, covariance problemes … . Our approach is particularly convenient for explain results in Maslov's theory [1,b] and the difficulties in defining a global Fourier transform over a symplectic manifold.     

Symplectic three-algebra and \mathcal{N}=6, Sp(2N)×U(1) superconformal Chern–Simons-matter theory

We introduce an antisymmetric metric into a 3-algebra and call it a symplectic 3-algebra. The $\mathcal{N}=6$ , Sp(2N)×U(1) superconformal Chern–Simons-matter theory with SU(4) R-symmetry in three dimensions is constructed by specifying the 3-brackets in a symplectic 3-algebra. We also demonstrate that the $\mathcal{N}=6$ , U(M)×U(N) theory can be recast into this symplectic 3-algebraic framework.