Symplectic vector space

About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.

Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds

Abstract: We prove that the group of Hamiltonian automorphisms of a symplectic $4$-manifold $(M,\omega)$, Ham$(M,\omega)$, contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group Symp$(M,\omega)$. We also consider the set of conjugacy classes of\/ $2$-tori in Ham$(M,\omega)$ with respect to Hamiltonian conjugation and show that its finiteness is equivalent to the finiteness of the symplectic mapping class group $\pi_{0}$(Symp$(M,\omega)$). Finally, we extend to rational and ruled manifolds a result of Kedra which asserts that if $(M,\omega)$ is a simply connected symplectic $4$-manifold with $b_{2}\geq 3$, and if $(\widetilde{M},\widetilde{\omega}_{\delta})$ denotes a symplectic blow-up of $(M,\omega)$ of small enough capacity $\delta$, then the rational cohomology algebra of the Hamiltonian group Ham($\widetilde{M},\widetilde{\omega}_{\delta})$ is not finitely generated. Our results are based on the fact that in a symplectic $4$-manifold endowed with any tamed almost structure $J$, exceptional classes of minimal symplectic area are $J$-indecomposable.

Lagrangian isotopy of tori in s 2 × s 2 and cp 2

Abstract: We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space R 4 , the projective plane CP 2 , and the monotone S 2 × S 2 .T he result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for T ∗ T 2 , i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.

Symplectic geometry and the Verlinde Formulas

Abstract: The purpose of this paper is to give a proof of the Verlinde formulas by applying the Riemann-Roch-Kawasaki theorem to the moduli space of flat G-bundles on a Riemann surface Σ with marked points, when G is a connected simply connected compact Lie group G. Conditions are given for the moduli space to be an orbifold, and the strata are described as moduli spaces for semisimple centralizers in G. The contribution of the strata are evaluated using the formulas of Witten for the symplectic volume, methods of symplectic geometry, including formulas of Witten-Jeffrey-Kirwan, and residue formulas. Our paper extends prior work by Szenes and Jeffrey-Kirwan for SU(n) to general groups G.

The Hamilton-Jacobi method and Hamiltonian maps

Abstract: A method for constructing time-step-based symplectic maps for a generic Hamiltonian system subjected to perturbation is developed. Using the Hamilton–Jacobi method and Jacobi’s theorem in finite periodic time intervals, the general form of the symplectic maps is established. The generating function of the map is found by the perturbation method in the finite time intervals. The accuracy of the maps is studied for fully integrable and partially chaotic Hamiltonian systems and compared to that of the symplectic integration method.

On symplectic 4-manifolds with prescribed fundamental group

Abstract: In this article we study the problem of minimizing a? + bs on the class of all symplectic 4-manifolds with prescribed fundamental group G (? is the Euler characteristic, s is the signature, and a,?b ? R), focusing on the important cases ?, ? + s and 2? + 3s. In certain situations we can derive lower bounds for these functions and describe symplectic 4-manifolds which are minimizers. We derive an upper bound for the minimum of ? and ? + s in terms of the presentation of G.