scispace - formally typeset
Search or ask a question
Topic

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.


Papers
More filters
Journal ArticleDOI
TL;DR: In this article, the Calabi-Yau structures are not rigid in the class of symplectic half-flat structures and the real part of the complex volume form is d-exact.
Abstract: We construct examples of symplectic half-flat manifolds on compact quotients of solvable Lie groups We prove that the Calabi-Yau structures are not rigid in the class of symplectic half-flat structures Moreover, we provide an example of a compact 6-dimensional symplectic half-flat manifold whose real part of the complex volume form is d-exact Finally we discuss the 4-dimensional case

18 citations

Journal ArticleDOI
TL;DR: The Hitchin-Kobayashi correspondence for vortices on the complex affine line with Kahler targets was shown in this article, which generalizes a result of Taubes for the case of a line target.
Abstract: We prove a Hitchin–Kobayashi correspondence for vortices on the complex affine line with Kahler target, which generalizes a result of Taubes for the case of a line target. More precisely, suppose that K is a compact Lie group and that the target X is either a compact Kahler K-Hamiltonian manifold or X is a symplectic vector space with linear K-action and a proper moment map. Suppose that the action of the complexified Lie group G satisfies stable = semistable. Then, for some sufficiently divisible integer n, there is a bijection between gauge equivalence classes of K-vortices with target X and isomorphism classes of maps from the weighted projective line P(1,n) to X/G that map the stacky point at infinity P(n) to the semistable locus of X. The results allow the construction and partial computation of the quantum Kirwan map from Woodward and play a role in the conjectures of Dimofte, Gukov, and Hollands relating vortex counts to knot invariants.

18 citations

Journal ArticleDOI
TL;DR: In this paper, the authors review classical and recent results on Hamiltonian and non-Hamiltonian symplectic group actions roughly starting from the results of these authors and also serve as a quick introduction to the basics of symplectic geometry.
Abstract: Classical mechanical systems are modeled by a symplectic manifold $(M,\omega)$, and their symmetries, encoded in the action of a Lie group $G$ on $M$ by diffeomorphisms that preserves $\omega$ These actions, which are called "symplectic", have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on symplectic actions of compact abelian Lie groups that are, in addition, of "Hamiltonian" type, ie they also satisfy Hamilton's equations Since then a number of connections with combinatorics, finite dimensional integrable Hamiltonian systems, more general symplectic actions, and topology, have flourished In this paper we review classical and recent results on Hamiltonian and non Hamiltonian symplectic group actions roughly starting from the results of these authors The paper also serves as a quick introduction to the basics of symplectic geometry

18 citations

Journal ArticleDOI
TL;DR: In this paper, a construction of 81 symplectic resolutions of a 4-dimensional quotient singularity obtained by an action of a group of order 32 is presented. But the existence of such resolutions is known by a result of Bellamy and Schedler.
Abstract: We provide a construction of 81 symplectic resolutions of a 4-dimensional quotient singularity obtained by an action of a group of order 32. The existence of such resolutions is known by a result of Bellamy and Schedler. Our explicit construction is obtained via GIT quotient of the spectrum of a ring graded in the Picard group generated by the divisors associated to the conjugacy classes of symplectic reflections of the group in question. As the result we infer the geometric structure of these resolutions and their flops. Moreover, we represent the group in question as a group of automorphisms of an abelian 4-fold so that the resulting quotient has singularities with symplectic resolutions. This yields a new Kummer-type symplectic 4-fold.

18 citations

Journal ArticleDOI
TL;DR: In this article, a set of operators can be used for defining a basis in a finite dimensional irreducible representation space of the symplectic group Sp(2n) in which each basis vector is an eigenvector of all invariant operators of the canonical subgroups U(1)⊂ ⊂Sp(2 n−2), U(n−1), Sp( 2n−2, Sp(1), U

18 citations

Network Information
Related Topics (5)
Symplectic geometry
18.2K papers, 363K citations
91% related
Manifold
18.7K papers, 362.8K citations
87% related
Lie group
18.3K papers, 381K citations
86% related
Lie algebra
20.7K papers, 347.3K citations
85% related
Cohomology
21.5K papers, 389.8K citations
85% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202310
202221
202113
20208
201910
201818