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.

Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds

Abstract: In their work on symplectic manifolds, Donaldson and Auroux use analogues of holomorphic sections of an ample line bundle L over a symplectic manifold M to create symplectically embedded zero sections and almost holomorphic maps to various spaces. Their analogues were termed ‘asymptotically holomorphic’ sequences {sN} of sections of L . We study another analogue H J (M, L) of holomorphic sections, which we call ‘almostholomorphic’ sections, following a method introduced earlier by Boutet de Monvel Guillemin [BoGu] in a general setting of symplectic cones. By definition, sections in H J (M, L ) lie in the range of a Szegö projector ΠN . Starting almost from scratch, and only using almostcomplex geometry, we construct a simple parametrix for ΠN of precisely the same type as the Boutet de Monvel-Sjöstrand parametrix in the holomorphic case [BoSj]. We then show that ΠN (x, y) has precisely the same scaling asymptotics as does the holomorphic Szegö kernel as analyzed in [BSZ1]. The scaling asymptotics imply more or less immediately a number of analogues of well-known results in the holomorphic case, e.g. a Kodaira embedding theorem and a Tian almost-isometry theorem. We also explain how to modify Donaldson’s constructions to prove existence of quantitatively transverse sections in H J (M, L ).

More constraints on Symplectic forms from Seiberg-Witten invariants

Abstract: Recently, Seiberg and Witten (see [SW1], [SW2], [W]) introduced a remarkable new equation which gives differential-topological invariants for a compact, oriented 4-manifold with a distinguished integral cohomology class which reduces mod(2) to the 2nd Steiffel-Whitney class of the manifold. A brief mathematical description of these new invariants is given in the recent preprint [KM1]. Using the Seiberg-Witten equations, I proved in [T] the following:

Dual isomonodromic deformations and moment maps to loop algebras

Abstract: The Hamiltonian structure of the monodromy preserving deformation equations of Jimboet al [JMMS] is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on Poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps. This construction is shown to lead to “dual” pairs of matrix differential operators whose monodromy is preserved under the same family of deformations. As illustrative examples, involving discrete and continuous reductions, a higher rank generalization of the Hamiltonian equations governing the correlation functions for an impenetrable Bose gas is obtained, as well as dual pairs of isomonodromy representations for the equations of the Painleve transcendentsPV andVI.

Symplectic geometry of the convariant phase space

Abstract: Develops a formalism for a covariant treatment of the phase space of field theories. Within that formalism, an expression for the symplectic form of a general Lagrangian field theory is presented. As examples, the author derives the symplectic forms for general relativity and for the Green-Schwarz covariant superstring action. Finally, an extension of the covariant phase space formalism to theories in superspace is given.