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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, the Schlafli identity is examined from a symplectic and semiclassical standpoint in the special case of flat, 3-dimensional space, and a proof is given, based on symplectic geometry.
Abstract: The Schlafli identity, which is important in Regge calculus and loop quantum gravity, is examined from a symplectic and semiclassical standpoint in the special case of flat, 3-dimensional space. In this case a proof is given, based on symplectic geometry. A series of symplectic and Lagrangian manifolds related to the Schlafli identity, including several versions of a Lagrangian manifold of tetrahedra, are discussed. Semiclassical interpretations of the various steps are provided. Possible generalizations to 3-dimensional spaces of constant (nonzero) curvature, involving Poisson-Lie groups and q-deformed spin networks, are discussed.
19 citations
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TL;DR: A review of some results in odd symplectic geometry related to the Batalin-Vilkovisky Formalism can be found in this article, where the same authors also present a review of the results of odd manifold geometry.
Abstract: It is a review of some results in Odd symplectic geometry related to the Batalin-Vilkovisky Formalism
19 citations
TL;DR: In this article, a 10-dimensional moduli space of torsion sheaves on the cubic 4-fold in the 5-dimensional projective space is constructed, which is closely related to the quasi-symplectic structure induced by the Yoneda pairing.
Abstract: We construct a 10-dimensional symplectic moduli space of torsion sheaves on the cubic 4-fold in the 5-dimensional projective space. It parametrizes the stable rank 2 vector bundles on hyperplane sections of the cubic 4-fold which are obtained by the Serre construction from normal elliptic quintics. The natural projection of this moduli space onto the dual projective 5-space is a Lagrangian fibration. The symplectic structure is closely related (and conjecturally equal) to the quasi-symplectic structure induced by the Yoneda pairing on the moduli space.
19 citations
TL;DR: In this article, it was shown that a closed symplectic toric manifold with infinite dimensional spaces of symplectic quasi-state and quasi-morphisms can be constructed under a superheavy level set of a Hamiltonian torus action.
Abstract: We prove that symplectic quasi-states and quasi-morphisms on a symplectic manifold descend under symplectic reduction on a superheavy level set of a Hamiltonian torus action. Using a construction due to Abreu and Macarini, in each dimension at least four we produce a closed symplectic toric manifold with infinite dimensional spaces of symplectic quasi-states and quasi-morphisms, and a one-parameter family of non-displaceable Lagrangian tori. By using McDuff's method of probes, we also show how Ostrover and Tyomkin's method for finding distinct spectral quasi-states in symplectic toric Fano manifolds can also be used to find different superheavy toric fibers.
19 citations
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TL;DR: In this paper, Gelfand, Kalinin and Fuks showed that the transverse symplectic class can be decomposed as a product of a certain leaf cohomology class of degree 5.
Abstract: In the early 1970's, Gelfand, Kalinin and Fuks found an exotic characteristic class of degree 7 in the Gelfand-Fuks cohomology of the Lie algebra of formal Hamiltonian vector fields on the plane. We prove that this cohomology class can be decomposed as a product of a certain leaf cohomology class of degree 5 and the transverse symplectic class. This is similar to the well known factorization of the Godbillon-Vey class for codimension n foliations. We also interpret the characteristic classes of transversely symplectic foliations introduced by Kontsevich in terms of the known classes and prove non-triviality for some of them.
19 citations