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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.
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TL;DR: In this article, the authors extend the techniques of Weinstein [W] to give a construction in group cohomology of symplectic forms on twisted moduli spaces of representations of the fundamental group of a 2-manifold.
Abstract: Let $G$ be a compact connected semisimple Lie group. We extend the techniques of Weinstein [W] to give a construction in group cohomology of symplectic forms $\omega$ on \lq twisted' moduli spaces of representations of the fundamental group $\pi$ of a 2-manifold $\Sigma$ (the smooth analogues of ${\rm Hom} (\pi_1(\Sigma), G)/G$) and on relative character varieties of fundamental groups of 2-manifolds. We extend this construction to exhibit a symplectic form on the extended moduli space [J1] (a Hamiltonian $G$-space from which these moduli spaces may be obtained by symplectic reduction), and compute the moment map for the action of $G$ on the extended moduli space.
29 citations
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TL;DR: In this paper, it was shown that a compact log symplectic manifold has a class in the second cohomology group whose powers, except for the top, are nontrivial, which gives cohomological obstructions for the existence of b-log structures similar to those in symplectic geometry.
Abstract: We prove that a compact log symplectic manifold has a class in the second cohomology group whose powers, except maybe for the top, are nontrivial This result gives cohomological obstructions for the existence of b-log symplectic structures similar to those in symplectic geometry
29 citations
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TL;DR: In this paper, the authors considered compact symplectic manifolds, which arise as the symplectic quotients of a symplectic manifold by a compact torus, and showed that the characteristic classes of these bundles can be determined explicitly, and another result uses this to give formulae for integrals of cohomology classes over the manifold.
Abstract: This paper gives methods for understanding invariants of symplectic quotients. The symplectic quotients considered here are compact symplectic manifolds (or more generally orbifolds), which arise as the symplectic quotients of a symplectic manifold by a compact torus. (A companion paper examines symplectic quotients by a nonabelian group, showing how to reduce to the maximal torus.)
Let X be a symplectic manifold, with a Hamiltonian action of a compact torus T. The main topological result of this paper describes an explicit cobordism that exists between a symplectic quotient of X by T, and a collection of iterated projective bundles over components of the set of T-fixed-points.
The characteristic classes of these bundles can be determined explicitly, and another result uses this to give formulae for integrals of cohomology classes over the symplectic quotient, in terms of data localized at the T-fixed points of X.
29 citations
TL;DR: In this article, the authors study orthogonal and symplectic matrix models with polynomial potentials and multi-interval supports of the equilibrium measure and find the bounds on the convergence rate of linear eigenvalue statistics and variance of the variance.
Abstract: We study orthogonal and symplectic matrix models with polynomial potentials and multi interval supports of the equilibrium measure. For these models we find the bounds (similar to the case of hermitian matrix models) for the rate of convergence of linear eigenvalue statistics and for the variance of linear eigenvalue statistics and find the logarithms of partition functions up to the order O(1). We prove also universality of local eigenvalue statistics in the bulk.
29 citations
TL;DR: In this article, the authors prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian in the presence of rational or ruled manifolds, via a symplectic cutting construction.
Abstract: In this paper, we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, $$S^{2}$$
or $$\mathbb{RP }^{2}$$
, in symplectic manifolds that are rational or ruled. Via a symplectic cutting construction, this is a natural extension of McDuff’s connectedness of ball packings in other settings and this result has applications to several different questions: smooth knotting and unknottedness results for spherical Lagrangians, the transitivity of the action of the symplectic Torelli group, classifying Lagrangian isotopy classes in the presence of knotting, and detecting Floer-theoretically essential Lagrangian tori in the del Pezzo surfaces.
29 citations