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.

An explicit description of the symplectic structure of moduli spaces of flat connections

Abstract: The moduli space of flat connections on a principal G‐bundle over a compact oriented surface of genus g≥1 is considered herein. Using the holonomies around noncontractible loops, the moduli space is described as a quotient of a submanifold of G2g. An explicit expression is obtained for the symplectic form on the smooth part of moduli space, and several properties of this form are established.

A Hofer-like metric on the group of symplectic diffeomorphisms

Abstract: Using a "Hodge decomposition" of symplectic isotopies on a compact symplectic manifold $(M,\omega)$, we construct a norm on the identity component in the group of all symplectic diffeomorphisms of $(M,\omega)$ whose restriction to the group $Ham(M,\omega)$ of hamiltonian diffeomorphisms is bounded from above by the Hofer norm. Moreover, $Ham(M,\omega)$ is closed in $Symp(M,\omega)$ equipped with the topology induced by the extended norm. We give an application to the $C^0$ symplectic topology. We also discuss extensions of Oh's spectral distance.

The moduli space of complex Lagrangian submanifolds

Abstract: Following an earlier paper on the differential-geometric structure of the moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold, we follow an analogous approach for compact complex Lagrangian submanifolds of a (K\"ahlerian) complex symplectic manifold. The natural geometric structure on the moduli space is a special K\"ahler metric, but we offer a different point of view on the local differential geometry of these, based on the structure of a submanifold of $V\times V$ (where $V$ is a symplectic vector space) which is Lagrangian with respect to two constant symplectic forms. As an application, we show using this point of view how the hyperk\"ahler metric of Cecotti, Ferrara and Girardello associated to a special K\"ahler structure fits into the Legendre transform construction of Lindstr\"om and Ro\v cek.

Multi-interval linear ordinary boundary value problems and complex symplectic algebra

Abstract: Introduction: Goals, organization Some definitions for multi-interval systems Complex symplectic spaces Single interval quasi-differential systems Multi-interval quasi-differential systems Boundary symplectic spaces for multi-interval systems Finite multi-interval systems Examples of complete Lagrangians Bibliography.

An example of moduli for singular symplectic forms

Abstract: In [3] Mar t ine t shows that there are four generic types of s ingulari t ies for germs of closed C ~ 2-forms on 4-manifolds and then defines a no t ion of s tabil i ty for these germs. The stabi l i ty of the first s ingular i ty type is just the classical D a r b o u x theorem for symplect ic forms. Mar t ine t proved the s tabi l i ty of the second type; while, more recently, Roussar ie [6] has shown the s tabi l i ty of the third. In this paper we shall show that forms exhibi t ing this last type of s ingular i ty are unfor tunate ly not stable. In fact, we show that near any generic X2.2.1 singular i ty there is, at least, a one pa rame te r family of moduli . In w we briefly descr ibe the var ious singularit ies. In w we will show how to reduce the p rob lem of s tabi l i ty to one involving a contact s t ructure on IR 3 at 0. Section 3 conta ins the p r o o f of instabi l i ty . Note : we assume that all functions, forms, vector fields, etc. are C ~.