Showing papers on "Symplectic vector space published in 1998"
TL;DR: In this article, the Hamiltonian reduction of a finite-dimensional symplectic space where the mapping class group acts by symplectic rational transformations is realized as a Hamiltonian representation of the Teichmuller space of punctured surfaces.
Abstract: The Teichmuller space of punctured surfaces with the Weil–Petersson symplectic structure and the action of the mapping class group is realized as the Hamiltonian reduction of a finite-dimensional symplectic space where the mapping class group acts by symplectic rational transformations. Upon quantization, the corresponding (projective) representation of the mapping class group is generated by the quantum dilogarithms.
236 citations
TL;DR: In this paper, the authors give new proofs of the convexity and connectedness properties of the moment map using the technique of symplectic cutting and extend these results to the case of orbifolds.
175 citations
TL;DR: In this article, a geometric formulation which describes extended supergravities in any dimension in the presence of electric and magnetic sources is presented, and the underlying duality symmetries of the theories are manifest.
Abstract: A geometric formulation which describes extended supergravities in any dimension in the presence of electric and magnetic sources is presented. In this framework, the underlying duality symmetries of the theories are manifest. Particular emphasis is given to the construction of central and matter charges and to the symplectic structure of all D=4, N-extended theories. The latter may be traced back to the existence, for N>2, of a flat symplectic bundle which is the N>2 generalization of N=2 Special Geometry.
171 citations
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19 Oct 1998TL;DR: In this paper, fundamental algebraic and geometric concepts applied to the theory of self-adjoint boundary value problems were applied to quasi-differential operators for real and real GKN-theorem for real operators.
Abstract: Introduction: Fundamental algebraic and geometric concepts applied to the theory of self-adjoint boundary value problems Maximal and minimal operators for quasi-differential expressions, and GKN-theory Symplectic geometry and boundary value problems Regular boundary value problems Singular boundary value problems Appendix A. Constructions for quasi-differential operators Appendix B. Complexification of real symplectic spaces, and the real GKN-theorem for real operators References.
117 citations
TL;DR: In this article, it was shown that rational blowdown on smooth 4-manifolds admits a closed non-degenerate 2-form, which is known as a symplectic manifold.
Abstract: We prove that the rational blowdown, a surgery on smooth 4-manifolds introduced by Fintushel and Stern, can be performed in the symplectic category. As a consequence, interesting families of smooth 4-manifolds, including the exotic K3 surfaces of Gompf and Mrowka, admit symplectic structures. A basic problem in symplectic topology is to understand what smooth manifolds admit a symplectic structure (a closed non-degenerate 2-form). In this paper we focus on this question in dimension 4. Currently, the primary methods for constructing smooth (irreducible) 4-manifolds in such a way that one can distinguish them by Donaldson or Seiberg-Witten invariants are surgery constructions that use complex manifolds as building blocks. These surgery methods are (smooth) logarithmic transforms, rational blowdowns, and connect sums along surfaces. It is interesting to see when these surgeries can be performed in the symplectic category. In this paper we prove that performing a rational blowdown of a symplectic manifold along symplectic surfaces yields a symplectic manifold. This result establishes that certain exotic 4-manifolds, including the exotic K3 surfaces of Gompf and Mrowka [9], are symplectic. In any even dimension, two symplectic manifolds can be summed along codimension 2 symplectic submanifolds to yield a symplectic manifold. We refer to this symplectic operation, which was proposed by Gromov [11], as the symplectic sum. Gompf [8] used the symplectic sum to construct a plethora of interesting symplectic manifolds, including the first examples of simply connected symplectic 4-manifolds that are not homotopic to any complex surface and some exotic K3 surfaces. More recently, Fintushel and Stern [6] have used the connect sum along smoothly embedded tori to produce a rich class of exotic 4-manifolds, some homeomorphic to a K3 surface, many of which cannot admit a symplectic structure. � The author is grateful for the support of an NSF post-doctoral fellowship, DSM9627749.
103 citations
TL;DR: In this article, a Hamiltonian theory for 2D soliton equations is developed, in which the spaces of doubly periodic operators on which a hierarchy of commuting flows can be introduced are identified, and these flows are Hamiltonian with respect to a universal symplectic form.
Abstract: We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with respect to a universal symplectic form $\omega={1\over 2}\r_{\infty} \d k$. We also construct other higher order symplectic forms and compare our formalism with the case of 1D solitons. Restricted to spaces of finite-gap solitons, the universal symplectic form agrees with the symplectic forms which have recently appeared in non-linear WKB theory, topological field theory, and Seiberg-Witten theories. We take the opportunity to survey some developments in these areas where symplectic forms have played a major role.
93 citations
TL;DR: In this article, various forms of convexity in symplectic geometry are surveyed, with particular attention paid to applications in low-dimensional topology, and the authors present a survey of the most common convexities in low dimensional topology.
74 citations
TL;DR: In this article, a lower bound for the number of fixed points of Hamiltonian automorphisms of closed symplectic manifolds (M,ω) is established, and an analogon to the cup-length based on the quantum cohomology ring of M,ω is defined.
Abstract: A new lower bound for the number of fixed points of Hamiltonian automorphisms of closed symplectic manifolds (M,ω) is established. The new estimate extends the previously known estimates to the class of weakly monotone symplectic manifolds. We prove for arbitrary closed symplectic manifolds with rational symplectic class that the cup-length estimate holds true if the Hofer energy of the Hamiltonian automorphism is sufficiently small. For arbitrary energy and on weakly monotone symplectic manifolds we define an analogon to the cup-length based on the quantum cohomology ring of (M,ω) providing a quantum cup-length estimate.
69 citations
65 citations
45 citations
TL;DR: In this paper, the Atiyah-Guillemin-Sternberg convexity theorem was used to prove that a non-Kahler manifold with large non-trivial symmetries does not admit an invariant Kahler structure.
Abstract: An important question with a rich history is the extent to which the symplectic category is larger than the Kahler category. Many interesting examples of non-Kahler symplectic manifolds have been constructed [T] [M] [G]. However, sufficiently large symmetries can force a symplectic manifolds to be Kahler [D] [Kn]. In this paper, we solve several outstanding problems by constructing the first symplectic manifold with large non-trivial symmetries which does not admit an invariant Kahler structure. The proof that it is not Kahler is based on the Atiyah-Guillemin-Sternberg convexity theorem [At] [GS]. Using the ideas of this paper, C. Woodward shows that even the symplectic analogue of spherical varieties need not be Kahler [W].
TL;DR: In this article, the authors discuss some aspects of the recently proposed symplectic butterfly form which is a condensed form for symplectic matrices, and present a Lanczos-like algorithm for reducing a symplectic matrix to butterfly form.
TL;DR: In this article, it was shown that the first eigenvalue of a Riemannian metric compatible with a symplectic fibration admits a universal upper bound in terms of the size.
Abstract: There are two themes in the present paper. The first one is spelled out in the title, and is inspired by an attempt to find an analogue of Hersch-Yang-Yau estimate for $lambda_1$ of surfaces in symplectic category. In particular we prove that every split symplectic manifold $T^4 times M$ admits a compatible Riemannian metric whose first eigenvalue is arbitrary large. On the other hand for Kahler metrics compatible with a given integral symplectic form an upper bound for $lambda_1$ does exist. The second theme is the study of Hamiltonian symplectic fibrations over the 2-sphere. We construct a numerical invariant called the size of a fibration which arises as the solution of certain variational problems closely related to Hofer's geometry, K-area and coupling. In some examples it can be computed with the use of Gromov-Witten invariants. The link between these two themes is given by an observation that the first eigenvalue of a Riemannian metric compatible with a symplectic fibration admits a universal upper bound in terms of the size.
TL;DR: In this article, the authors obtained a real canonical form for real symplectic pencils, which is restricted to the case where the elementary divisors of eigenvalues on the unit circle have an even degree.
TL;DR: In this article, a survey of some selected topics in the field of low-dimensional symplectic and contact topology is presented, including applications of generating functions, Donaldson's theory of approximately complex manifolds, and some other recent developments.
Abstract: This is a survey of some selected topics in symplectic topology. In particular, we discuss lowdimensional symplectic and contact topology, applications of generating functions, Donaldson's theory of approximately complex manifolds and some other recent developments in the field.
TL;DR: In this article, the authors studied certain natural differential forms and their equivariant extensions on the space of connections and explained their relationship with the Einstein metric and the stability of vector bundles.
Abstract: We study certain natural differential forms \(\) and their \(\) equivariant extensions on the space of connections. These forms are defined using the family local index theorem. When the base manifold is symplectic, they define a family of symplectic forms on the space of connections. We will explain their relationships with the Einstein metric and the stability of vector bundles. These forms also determine primary and secondary characteristic forms (and their higher level generalizations).
TL;DR: In this paper, the authors show that for any semisimple Lie group, a symmetric orbit is not regular and neither Hermitian nor pseudo-Hermitian.
Abstract: 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.
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TL;DR: In this paper, the weak homotopy type of the group S of those symplectic automorphisms of M which are trivial at infinity was determined by adapting an argument of Gromov.
Abstract: Let M be the cotangent bundle of S^2, with the standard symplectic structure. By adapting an argument of Gromov we determine the weak homotopy type of the group S of those symplectic automorphisms of M which are trivial at infinity. It turns out that S is weakly homotopy equivalent to \Z. \pi_0(S) is generated by the class of the standard "generalized Dehn twist". As a consequence, we show that there are different connected components of S which lie in the same connected component of the corresponding group of diffeomorphisms.
TL;DR: In this article, the authors calculate linear stability boundaries for natural symplectic maps, which are symplectic mappings derived from Lagrangian generating functions having positive definite kinetic energy, and apply them to a four-dimensional pair of coupled standard maps.
TL;DR: In this article, a Lagrangian approach for constructing a symplectic structure for singular systems is presented, which gives a simple and unified framework for understanding the origin of pathologies that appear in the Dirac-Bergmann formalism.
Abstract: We develop a Lagrangian approach for constructing a symplectic structure for singular systems. It gives a simple and unified framework for understanding the origin of the pathologies that appear in the Dirac-Bergmann formalism, and offers a more general approach for a symplectic formalism, even when there is no Hamiltonian in a canonical sense. We can thus overcome the usual limitations of the canonical quantization, and perform an algebraically consistent quantization for a more general set of Lagrangian systems.
TL;DR: In this paper, the integrability of non-commutatively integrable, possibly degenerate, Hamiltonian systems is characterized from the point of view of symplectic geometry.
TL;DR: In this paper, the generalized symplectic sum is defined along positively intersecting symplectic surfaces in 4-manifolds and proved an existence theorem for the special case of a 3-fold sum.
TL;DR: In this paper, the Hamiltonian canonical equation for the statics and dynamics analysis of Reissner plates is derived from the mixed variational principle, by selection of the variables and its dual variables, so it leads the problem to the mathematical frame of symplectic geometry and its algorithms, and exact solution for the arbitary boundary conditions is also derived by the adjoint ortho-normalized symplectic expansion method.
TL;DR: Bouchet as discussed by the authors showed that every Eulerian multimatroid is representable with a symplectic vector space over GF(2) and adapted the construction to symplectic matroids.
TL;DR: In this paper, a class of affine symmetric spaces with a non-Abelian solvable transvection group is defined, where the underlying manifold M of each element (M, ▿) belonging to this class is diffeomorphic to Rn.
Abstract: A symplectic is a symmetric space endowed with a symplectic structure which is invariant by the symmetries. We give here a classification of four-dimensional symplectic which are simply connected. This classification reveals a remarkable class of affine symmetric spaces with a non-Abelian solvable transvection group. The underlying manifold M of each element (M, ▿) belonging to this class is diffeomorphic to Rnwith the property that every tensor field on M invariant by the transvection group is constant; in particular, ▿ is not a metric connection. This classification also provides examples of nonflat affine symmetric connections on Rnwhich are invariant under the translations. By considering quotient spaces, one finds examples of locally affine symmetric tori which are not globally symmetric.
TL;DR: In this paper, a universal construction of a Lefschetz pencil with infinite fundamental groups is given for the case of Riemann surfaces with positive self-intersection and a small image of the fundamental group inside the ambient symplectic fourfold.
TL;DR: The hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces were studied in this paper for flat connections in the context of flat connection networks.
Abstract: The hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces
01 Jan 1998
TL;DR: In this article, a generalization of the Marsden-Weinstein reduction procedure in symplectic geometry is presented, which is shown how this procedure is quantized by Rieffel induction, a technique in operator algebra theory.
Abstract: This is an introduction to the author's recent work on constrained systems. Firstly, a generalization of the Marsden-Weinstein reduction procedure in symplectic geometry is presented - this is a reformulation of ideas of Mikami-Weinstein and Xu. Secondly, it is shown how this procedure is quantized by Rieffel induction, a technique in operator algebra theory. The essential point is that a symplectic space with generalized moment map is quantized by a pre-(Hilbert) C^*-module. The connection with Dirac's constrained quantization method is explained. Three examples with a single constraint are discussed in some detail: the reduced space is either singular, or defined by a constraint with incomplete flow, or unproblematic but still interesting. In all cases, our quantization procedure may be carried out. Finally, we re-interpret and generalize Mackey's quantization on homogeneous spaces. This provides a double illustration of the connection between C^*-modules and the moment map.
TL;DR: In this article, Mass-weighted symplectic forms provide a unified framework for the treatment of both finite and vanishingly small masses in the N-body problem, and their properties are discussed.
Abstract: Mass-weighted symplectic forms provide a unified framework for the treatment of both finite and vanishingly small masses in the N-body problem. These forms are introduced, compared to previous approaches, and their properties are discussed. Applications to symplectic mappings, the definition of action-angle variables for the Kepler problem, and Hamiltonian perturbation theory are outlined