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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 paper, a family of metrics based on a single periodic function defined on the torus spanned by the U(1) isometry directions was constructed and a phase space of geometries in the nearhorizon region of vacuum extremal black holes was defined.
Abstract: We construct the classical phase space of geometries in the near-horizon region of vacuum extremal black holes as announced in [arXiv:1503.07861]. Motivated by the uniqueness theorems for such solutions and for perturbations around them, we build a family of metrics depending upon a single periodic function defined on the torus spanned by the U(1) isometry directions. We show that this set of metrics is equipped with a consistent symplectic structure and hence defines a phase space. The phase space forms a representation of an infinite dimensional algebra of so-called symplectic symmetries. The symmetry algebra is an extension of the Virasoro algebra whose central extension is the black hole entropy. We motivate the choice of diffeomorphisms leading to the phase space and explicitly derive the symplectic structure, the algebra of symplectic symmetries and the corresponding conserved charges. We also discuss a formulation of these charges with a Liouville type stress-tensor on the torus defined by the U(1) isometries and outline possible future directions.
55 citations
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TL;DR: In this paper, the non-linear Schrodinger equation on the complex Grassmannian manifold M is the Hamiltonian equation for the energy functional on C(R,M) with respect to the symplectic form induced from the Kahler form on M.
Abstract: The geometric non-linear Schrodinger equation (GNLS) on the complex Grassmannian manifold M is the Hamiltonian equation for the energy functional on C(R,M) with respect to the symplectic form induced from the Kahler form on M. It has a Lax pair that is gauge equivalent to the Lax pair of the matrix non-linear Schrodinger equation (MNLS). We construct via gauge transformations an isomorphism from C(R,M) to the phase space of the MNLS equation so that the GNLS flow corresponds to the MNLS flow. The existence of global solutions to the Cauchy problem for GNLS and the hierarchy of commuting flows follows from the correspondence. Direct geometric constructions show the flows are given by geometric partial differential equations, and the space of conservation laws has a structure of a non-abelian Poisson group. We also construct a hierarchy of symplectic structures for GNLS. Under pullback, the known order k symplectic structures correspond to the order (k-2) symplectic structures that we find. The shift by two is a surprise, and is due to the fact that the group structures depend on gauge choice.
54 citations
TL;DR: In this article, a Hamiltonian structure for the first variation equation of the Hamiltonian along a given dynamic solution is given, which is different from the well-known and elementary tangent space construction.
Abstract: This paper uses symplectic connections to give a Hamiltonian structure to the first variation equation for a Hamiltonian system along a given dynamic solution. This structure generalises that at an equilibrium solution obtained by restricting the symplectic structure to that point and using the quadratic form associated with the second variation of the Hamiltonian (plus Casimir) as energy. This structure is different from the well-known and elementary tangent space construction. Our results are applied to systems with symmetry and to Lie-Poisson systems in particular.
54 citations
TL;DR: The usual structures of symplectic geometry (symplectic, contact, Poisson) make sense for complex manifolds; they turn out to be quite interesting on projective, or compact Kahler, manifolds.
Abstract: The usual structures of symplectic geometry (symplectic, contact, Poisson) make sense for complex manifolds; they turn out to be quite interesting on projective, or compact Kahler, manifolds. In these notes we review some of the recent results on the subject, with emphasis on the open problems and conjectures.
54 citations
TL;DR: In this article, Donaldson's approximately holomorphic techniques were used to construct a compact isotropic submanifold of a compact symplectic manifold, and the connection with rational convexity results in the Kahler case was discussed.
Abstract: Using Donaldson's approximately holomorphic techniques, we construct symplectic hypersurfaces lying in the complement of any given compact isotropic submanifold of a compact symplectic manifold. We discuss the connection with rational convexity results in the Kahler case and various applications.
54 citations