Showing papers on "Symplectic vector space published in 1995"

Introduction to symplectic topology

Abstract: Introduction I. FOUNDATIONS 1. From classical to modern 2. Linear symplectic geometry 3. Symplectic manifolds 4. Almost complex structures II. SYMPLECTIC MANIFOLDS 5. Symplectic group actions 6. Symplectic fibrations 7. Constructing symplectic manifolds III. SYMPLECTOMORPHISMS 8. Area-preserving diffeomorphisms 9. Generating functions 10. The group of symplectomorphisms IV. SYMPLECTIC INVARIANTS 11. The Arnold conjecture 12. Symplectic capacities 13. New directions

Algebraic index theorem

Abstract: We prove the Atiyah-Singer index theorem where the algebra of pseudodifferential operators is replaced by an arbitrary deformation quantization of the algebra of functions on a symplectic manifold.

The geometry of symplectic energy

Abstract: "Non-Squeezing Theorem" which says that it is impossible to embed a large ball symplectically into a thin cylinder of the form R2, x B2, where B2 is a 2-disc. This led to Hofer's discovery of symplectic capacities, which give a way of measuring the size of subsets in symplectic manifolds. Recently, Hofer found a way to measure the size (or energy) of symplectic diffeomorphisms by looking at the total variation of their generating Hamiltonians. This gives rise to a bi-invariant (pseudo-)norm on the group Ham(M) of compactly supported Hamiltonian symplectomorphisms of the manifold M. The deep fact is that this pseudo-norm is a norm; in other words, the only symplectomorphism on M with zero energy is the identity map. Up to now, this had been proved only for sufficiently nice symplectic manifolds, and by rather complicated analytic arguments. In this paper we consider a more geometric version of this energy, which was first considered by Eliashberg and Hofer in connection with their study of the extent to which the interior of a region in a symplectic manifold determines its boundary. We prove, by a simple geometric argument, that both versions of energy give rise to genuine norms on all symplectic manifolds. Roughly speaking, we show that if there were a symplectomorphism of M which had "too little" energy, one could embed a large ball into a thin cylinder M x

Holomorphic slices, symplectic reduction and multiplicities of representations

Abstract: I prove the existence of slices for an action of a reductive complex Lie group on a K\"ahler manifold at certain orbits, namely those orbits that intersect the zero level set of a momentum map for the action of a compact real form of the group. I give applications of this result to symplectic reduction and geometric quantization at singular levels of the momentum map. In particular, I obtain a formula for the multiplicities of the irreducible representations occurring in the quantization in terms of symplectic invariants of reduced spaces, generalizing a result of Guillemin and Sternberg.

Symplectic structure on ruled surfaces and a generalized adjunction formula

Abstract: In this section, we state the theorems needed in our paper. Recently, Seiberg and Witten ([SW], [Wi]) have introduced a new set of 4-manifold invariants. These invariants are in similar spirit to Donaldson invariants but much easier to handle. Various longstanding conjectures including the Thom conjecture are proved using Seiberg-Witten invariants. An important ingredient in the proof of the Thom conjecture by Kronheimer and Mrowka is the wall crossing formula for manifolds with b1 = 0. Seiberg-Witten invariants take on a very simple form for Kahler surfaces ([Wi], [B], [FM1]). All the basic classes are explicitly known and in particular, the anticanonical bundle is always a basic class. A large part of this story is generalized to symplectic manifolds by Taubes who ([T1], [T2], [T3], [T4]) proved several remarkable theorems on Seiberg-Witten invariants of symplectic four-manifolds. Recall that every symplectic manifold has a complex line bundle, K (called the canonical bundle), which is canonical up to isomorphism. The first theorem of Taubes is

Symplectic structure of the moduli space of flat connection on a Riemann surface

Abstract: We consider the canonical symplectic structure on the moduli space of flat g-connections on a Riemann surface of genus g with a marked points. For a being a semisimple Lie algebra we obtain an explicit efficient formula for this symplectic form and prove

More constraints on Symplectic forms from Seiberg-Witten invariants

Abstract: Recently, Seiberg and Witten (see [SW1], [SW2], [W]) introduced a remarkable new equation which gives differential-topological invariants for a compact, oriented 4-manifold with a distinguished integral cohomology class which reduces mod(2) to the 2nd Steiffel-Whitney class of the manifold. A brief mathematical description of these new invariants is given in the recent preprint [KM1]. Using the Seiberg-Witten equations, I proved in [T] the following:

On the Arnold conjecture for weakly monotone symplectic manifolds

Abstract: We show the Arnold conjecture concerning symplectic fixed points in the case that the symplectic manifold is weakly-monotone and all the fixed points are non-degenerate. In particular, the conjecture is true in dimension 2, 4, 6 if all the fixed points are non-degenerate.

Star Products on Compact Pre-quantizable Symplectic Manifolds

Abstract: We define in this Letter, a notion of ‘representation’ for a star product (equipped with a star-compatible trace) and show that every compact pre-quantizable symplectic manifold admits a representable star product.

An introduction to Lie groups and symplectic geometry

Abstract: A series of nine lectures on Lie groups and symplectic This is an unofficial version of the notes and was last modified on 19 February 2003. (Mainly to correct some very bad mistakes in Lecture 8 about Kähler and hyperKähler reduction that were pointed out to me by Eugene Lerman.) Please send any comments, corrections or bug reports to the above e-mail address. Introduction These are the lecture notes for a short course entitled " Introduction to Lie groups and symplectic geometry " which I gave at the 1991 Regional Geometry Institute at Park City, Utah starting on 24 June and ending on 11 July. The course really was designed to be an introduction, aimed at an audience of students who were familiar with basic constructions in differential topology and rudimentary differential geometry, who wanted to get a feel for Lie groups and symplectic geometry. My purpose was not to provide an exhaustive treatment of either Lie groups, which would have been impossible even if I had had an entire year, or of symplectic manifolds, which has lately undergone something of a revolution. Instead, I tried to provide an introduction to what I regard as the basic concepts of the two subjects, with an emphasis on examples which drove the development of the theory. I deliberately tried to include a few topics which are not part of the mainstream subject, such as Lie's reduction of order for differential equations and its relation with the notion of a solvable group on the one hand and integration of ODE by quadrature on the other. I also tried, in the later lectures to introduce the reader to some of the global methods which are now becoming so important in symplectic geometry. However, a full treatment of these topics in the space of nine lectures beginning at the elementary level was beyond my abilities. After the lectures were over, I contemplated reworking these notes into a comprehensive introduction to modern symplectic geometry and, after some soul-searching, finally decided against this. Thus, I have contented myself with making only minor modifications and corrections, with the hope that an interested person could read these notes in a few weeks and get some sense of what the subject was about. An essential feature of the course was the exercise sets. Each set begins with elementary material and works up to more involved and delicate problems. My object …

Symplectic and Poisson structures of certain moduli spaces, I

Abstract: Let $G$ be a Lie group with a biinvariant metric, not necessarily positive definite. It is shown that a certain construction carried out in an earlier paper for the fundamental group of a closed surface may be extended to an arbitrary infinite orientation preserving cocompact planar discrete group of euclidean or non-euclidean motions $\\pi$ and yields (i) a symplectic structure on a certain smooth manifold $\\Cal M$ containing the space $\\roman{Hom}(\\pi,G)$ of homomorphisms and, furthermore, (ii) a hamiltonian $G$-action on $\\Cal M$ preserving the symplectic structure together with a momentum mapping in such a way that the reduced space equals the space $\\roman{Rep}(\\pi,G)$ of representations. More generally, the construction also applies to certain spaces of projective representations. For $G$ compact, the resulting spaces of representations inherit structures of {\\it stratified symplectic space\\/} in such a way that the strata have finite symplectic volume . For example, {\\smc Mehta-Seshadri} moduli spaces of semistable holomorphic parabolic bundles with rational weights or spaces closely related to them arise in this way by {\\it symplectic reduction in finite dimensions\\/}.

Four-manifolds without symplectic structures but with nontrivial Seiberg-Witten invariants

Abstract: It was proved in [9] that every closed symplectic four-manifold has a nontrivial Seiberg-Witten invariant. Combining this result with the arguments of [5], we show here that the converse is false. In fact, there are closed oriented four-manifolds with nontrivial Seiberg-Witten (and Donaldson [5]) invariants which do not admit symplectic structures. For some examples the Seiberg-Witten invariants are the same as those of symplectic manifolds, whereas for others they do not satisfy the structure results for the invariants of symplectic manifolds proved in [10]. Our examples are connected sums in which one summand has a negative definite intersection form and a nontrivial fundamental group. Conversely, there are symplectic four-manifolds whose fundamental group splits as a free product [2], but we show that in most cases this splitting is not realized by any decomposition of the manifold as a smooth connected sum. This provides large classes of counterexamples to the four-dimensional Kneser conjecture, which was disproved only recently [7] and only for a few specific groups. We begin with an observation about the possible connected sum decompositions of closed symplectic four-manifolds. This was proved in [5] for the Kähler case.

Equivariant constrained symplectic integration

Abstract: We use recent results on symplectic integration of Hamiltonian systems with constraints to construct symplectic integrators on cotangent bundles of manifolds by embedding the manifold in a linear space. We also prove that these methods are equivariant under cotangent lifts of a symmetry group acting linearly on the ambient space and consequently preserve the corresponding momentum. These results provide an elementary construction of symplectic integrators for Lie-Poisson systems and other Hamiltonian systems with symmetry. The methods are illustrated on the free rigid body, the heavy top, and the double spherical pendulum.

The symplectic structure on moduli space

Abstract: The «moduli space» referred to in the title of this paper is the space ℳ of gauge equivalence classes of flat connections on a principal G bundle over a compact Riemann surface Σ. Atiyah and Bott [1] constructed a symplectic structure ωℳ on ℳ by symplectic reduction from the infinite dimensional symplectic manifold of all connections. Since ℳ is a finite-dimensional object, it seems desirable to have a finite-dimensional construction of its symplectic form. In fact, ℳ can also be realized as the representation space Hom(π, G)/G, where π is the fundamental group of Σ and G acts by conjugation. Using the resulting identification of the tangent spaces of ℳ with cohomology spaces of π with suitable coefficients, Goldman [9] gave a direct construction of ω ℳ as a nondegenerate 2-form, but he was unable to prove that this form is closed without recourse to the infinite-dimensional picture. This gap in Goldman’s approach was filled recently by Karshon [10], who showed that dω ℳ = 0 by methods of group cohomology, without using the space of connections.2)

The Dirac operator on symplectic spinors

Abstract: Symplectic spinors were introduced by B. Kostant in [4] in the context of geometric quantization. This paper presents further considerations concerning symplectic spinors. We define the spinor derivative induced by a symplectic covariant derivative. We compute an explicit formula for this spinor derivative and prove some elementary properties. This makes it possible to define the symplectic Dirac operator in a canonical way. In case of a symplectic and torsion-free covariant derivative it turns out to be formally selfadjoint.

Group-theoretic obstructions to integrability

Abstract: Let V be a four-dimensional complex symplectic vector space. This paper classifies those connected linear algebraic subgroups of the symplectic group Sp(V) that admit two independent rational invariants. As an application we show the non integrability of a three degree of freedom Hamiltonian system.

A symplectic fixed point theorem for toric manifolds

Abstract: In this paper, by a toric manifold we mean a non-singular symplectic quotient M = ℂ n //T k of the standard symplectic space by a linear torus action. Such a toric manifold is in fact a complex Kahler manifold of dimension n - k. We denote p(M) and c(M) the cohomology class of the Kahler symplectic form and the first Chern class of M respectively. They both are effective, that is, Poincare-dual to some holomorphic hypersurfaces. We call a homology class in H 2(M, ℤ) effective if it has non-negative intersection indices with fundamental cycles of all compact holomorphic hypersurfaces in M, and denote ℰ the set of all non-zero effective homology classes. Our main result is the following

Symmetric Symplectic Manifolds and Deformation Quantization

Abstract: In this paper, which we are happy to dedicate to our friend Guy Rideau, we introduce the notion of symplectic symmetric spaces.

Batalin--Vilkovisky Formalism and Odd Symplectic Geometry

Abstract: It is a review of some results in Odd symplectic geometry related to the Batalin-Vilkovisky Formalism

A theorem in complex symplectic geometry

Abstract: We prove that two simple, closed, real-analytic curves in C2n that are polynomially convex are equivalent under the group of symplectic holomorphic automorphisms of C2n if and only if the two curves have the same action integral. Every two simple real-analytic arcs in C2n are so equivalent.

Topology of 2-knots in ℝ4 and symplectic geometry

Abstract: We show in this paper that there exists a deep relationship between the differential topology of S 2-knots in ℝ4 and their symplectic geometry. In particular, we use symplectic tools to define a real-valued topological invariant of a knotted S 2 in ℝ4 (see Section 3.4 below). Here are the main results which motivate this definition.

On the invariance of generating functions for symplectic transformations

Abstract: This paper treats the invariance of generating functions for symplectic transformations under canonical coordinates transformations. A necessary and sufficient condition for the invariance is obtained. A result of Weinstein [9] is recovered as a special case.

Existence of formal integrals of symplectic integrators

Abstract: A recurrent method of solving the formal integrals of symplectic integrators is given. The special examples show that there are no long-term variations in all integrals of the Hamiltonian system in addition to the energy one when symplectic integrators are used in the numerical studies of the system. As an application of the formal integrals, the relation between them and the linear stability of symplectic integrators is discussed.

Casimir operators of semidirect products of semisimple with Heisenberg groups

Abstract: Casimir operators for semidirect products of some semisimple groups with Heisenberg groups are computed. The analysis is carried out using dual representations on Fock space, wherein the action of the semidirect products are related to their dual groups, namely certain unitary, orthogonal, and symplectic groups. The compact symplectic group chain is also investigated; by passing to the complexification, groups `between` the symplectic groups are constructed, which are of the form of semidirect products of symplectic groups with Heisenberg groups.

On the K -theory of the projective symplectic groups

Abstract: In this paper we compute the real ^-groups KO*(PSp(n^) of the projective symplectic groups PSp(n). As for the complex ^-groups, we have in [6, 8] two kinds of methods of computing 7f*(G) in general for a compact connected Lie group G with finite fundamental group of prime order and by using actually those methods its ring structure is explicitly described. Neither of them deals with the projective unitary groups PU(n) except the case when n is prime. In more general, however, the case when n is a power of prime is investigated in more earlier times but not explicitly [18]. The computation in any case is based on the fact [7] that ^*(G0) (where G0 is simply connected) is an exterior algebra generated by elements of degree one arising from the basic irreducible complex representations of G0 . It seems not easy to find a comprehensive method of computing KO* (G) as in the complex case. So we proceed case by case and determined the JfO-groups of SOM, PEQ, PE7, and PSp(2 )in [10, 11, 12, 13, 14, 15]. Then we also use essentially the structure theorem on ^CO*(G0) [17] analogous to that on #*(G0). We compute KO*(PSpW) by applying the modification of the method used for the computation of X"0*(PSp(2")). Making use of the equi variant ^0-theory KOz/2, especially the Thorn isomorphism theorem for KOz/2theory, we reduce the structure of KO*(PSpM) to those of KO*(SpM} and #O*(P*) for a certain integer £ > 0 where P denotes a real projective £ -space. Then we need K*(PSp(n^ with a basis in the form conforming to our method and so we begin by computing this group by a way similar to that used in computing