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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 simple construction yielding homology classes in (non-simply-connected) symplectic four-manifolds which admit infinitely many pairwise non-isotopic symplectic representatives was given.
Abstract: We give a simple construction yielding homology classes in (non-simply-connected) symplectic four-manifolds which admit infinitely many pairwise non-isotopic symplectic representatives. Examples are constructed in which the symplectic curves can have arbitrarily large genus. The examples are built from surface bundles over surfaces and involve only elementary techniques. As a corollary we see that a blow-up of any simply-connected complex projective surface contains a connected symplectic surface not isotopic to any complex curve.
24 citations
TL;DR: In this paper, the authors studied the symplectic geometry of the moduli spaces of closed n-gons with fixed side-lengths in hyperbolic three-space and proved that these moduli space have almost canonical symplectic structures.
Abstract: We study the symplectic geometry of the moduli spaces Mr = Mr(H 3 ) of closed n-gons with fixed side-lengths in hyperbolic three-space. We prove that these moduli spaces have almost canonical symplectic structures. They are the symplectic quotients of B n by the dressing action of SU(2) (here B is the subgroup of the Borel subgroup of SL2(C) defined below). We show that the hyperbolic Gauss map sets up a real analytic isomorphism between the spaces Mr and the weighted quotients of (S 2 ) n by P SL2(C) studied by Deligne and Mostow. We construct an integrable Hamiltonian system on Mr by bending polygons along nonintersecting diagonals. We describe angle variables and the momentum polyhedron for this system. The results of this paper are the analogues for hyperbolic space of the results of (KM2) for Mr(E 3 ), the space of n-gons with fixed side-lengths in E 3 . We prove Mr(H 3 ) and Mr(E 3 ) are symplectomorphic.
24 citations
24 citations
TL;DR: In this paper, the authors study the Berezin-Toeplitz quantization using as quantum space the space of eigenstates of the renormalized Bochner Laplacian corresponding to eigenvalues localized near the origin on a symplectic manifold.
Abstract: We study the Berezin–Toeplitz quantization using as quantum space the space of eigenstates of the renormalized Bochner Laplacian corresponding to eigenvalues localized near the origin on a symplectic manifold. We show that this quantization has the correct semiclassical behavior and construct the corresponding star-product.
24 citations
TL;DR: The most general action, quadratic in the B fields as well as in the curvature F, having SO(3, 1) or SO(4) as the internal gauge group for a four-dimensional BF theory is presented and its symplectic geometry is displayed in this paper.
Abstract: The most general action, quadratic in the B fields as well as in the curvature F, having SO(3, 1) or SO(4) as the internal gauge group for a four-dimensional BF theory is presented and its symplectic geometry is displayed. It is shown that the space of solutions to the equations of motion for the BF theory can be endowed with symplectic structures alternative to the usual one. The analysis also includes topological terms and cosmological constant. The implications of this fact for gravity are briefly discussed.
24 citations