Showing papers on "Symplectic vector space published in 1977"

Lectures on Symplectic Manifolds

Abstract: Introduction Symplectic manifolds and lagrangian submanifolds, examples Lagrangian splittings, real and complex polarizations, Kahler manifolds Reduction, the calculus of canonical relations, intermediate polarizations Hamiltonian systems and group actions on symplectic manifolds Normal forms Lagrangian submanifolds and families of functions Intersection theory of lagrangian submanifolds Quantization on cotangent bundles Quantization and polarizations Quantizing lagrangian submanifolds and subspaces, construction of the Maslov bundle References.

Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field.

Abstract: This note is to show how to use symplectic geometry to write equations of motion of a “classical particle” in the presence of a Yang-Mills field, for any gauge group, G, and any differentiable manifold, M. In the case that M is Minkowski space and G = U(1), the equations reduce to the Lorentz equations for a charged particle in an electromagnetic field. Our procedure in the general case uses the connection form as defined on the principle bundle to introduce a symplectic structure on certain associated bundles and is automatically gauge invariant.

Closed 2-forms and an embedding theorem for symplectic manifolds

Abstract: The existence of universal connections was shown by Narasimhan and Ramanan [5], and Kostant [3] showed that any integral closed 2-form is the curvature form of a connection on some circle bundle. These results can be combined to show the existence of a universal closed 2-form with integral periods. In this paper we will use the symplectic structure of a complex projective space to give an elementary proof of this result the precise statement is given in Theorem A. The result of Kostant is in fact a corollary of the existence of a universal closed 2-form, as is indicated below. Another immediate corollary of Theorem A is the result of Gromov [3] that closed symplectic manifolds can be symplecticalΓy immersed in CP, for large enough n see Theorem B. First we indicate why the proof which we are going to give here is a simple and natural generalization of an elementary fact about exact 2-forms. Consider the standard symplectic form Ω = Σιl=i dXidyt on R . Any exact 2-form on a manifold M can be induced from Ω by a mapping to R for some n, since any exact 2-form on M can be written in the form 2*=i dft A dgt, where /*, gi are real valued functions on M. CP has a symplectic structure Ωo which is locally given by Ωo = 2?=i dxi A dyt. Furthermore, CP n is the 2π-skeleton of an Eilenberg-MacLane space of type K(Z, 2). It is thus natural to expect that any closed 2-form with integral periods can be induced from Ωo by a map to CP, because there is some map to CP, for large n, which pulls back Ωo to within an exact 2-form of the given closed 2-form. The only complication that is met in CP to adjusting the map to account for the exact 2-form is that, unlike in R, the symplectic charts on CP have finite radius, so the fi9 g/s utilized would have to be bounded. The proof we give of Theorem A depends only on estimating the bounds on fi9 gt as n becomes large. A closed &-form on a manifold M will be said to be integral if its de Rham cohomology class is in the image of the canonical coefficient map H(M Z) -*H(M;R). Complex projective space CP has a Kahlerian structure, and we will denote its Kahler form by flj. The 2-form Ω% can be chosen to represent a generator in the image of H\\CP Z) -> H\\CP R), and we can assume that /*(βJ0 = Ωl where i is the standard inclusion of CP in CP,

Eisenstein series on the symplectic group

Abstract: Analytic continuation is proved for certain Eisenstein series on the symplectic group which are associated with nonparabolic forms. In the case of the full modular group an explicit functional equation is obtained, and the singularities of the series are completely described.

Spin in kink-type field theories

Abstract: This paper studies some classical three-dimensional field theories for which the ranges of the field variables are a 3-sphere, a 2-sphere, the symplectic group,Sp(n), the special orthogonal group,SO(3), and theS4,1 space of general relativistic metrics. The main result is the proof that these theories admit half-odd-integer spin, so that the 1-kink states are classical analogs of fermion states.

The manifold of the lagrangean subspaces of a symplectic vector space

Abstract: Let (E, a) be a real 2n-dimensional symplectic vector space with symplectic form a, i.e., a is a nondegenerate skew-symmetric bilinear form on E. Then an n-dimensional subspace λ of E will be called a Lagrangean subspace if alλ = 0 holds. The set Λ(E) of all Lagrangean subspaces of (E, a) has a structure of n(n + l)-dimensional compact connected regular algebraic variety. If we put A\λ): = [μ 6 Λ(E) I dim (λ Π μ) = k} for λ £ Λ(E), then Λ°U) is a cell (i.e., diίϊeomorphic to i r ( n + 1 ) / 2 ) for any λ e Λ(E). Moreover Σ tf): = |J*2>i Λ*W) is an algebraic subvariety of Λ(E), and defines an oriented cycle of codimension one, whose Poincare dual is a generator of H(Λ(E), Z) = Z and defines the Maslov-Arnold index [1], [3], [4]. This index plays an important role in the proof of Morse index theorem in the calculus of variations [4]. In the present note, we shall give a differential geometric characterization of 2 U)> i ^ ? by introducing an appropriate riemannian metric on Λ{E) we shall show that 2 (λ) is the cut locus of some μ e Λ(E) and Λ°(X) is the interior set of μ. In fact, take a basis {eu fά] (1 , and ao((p, q), (p\ q')): = (q, p'y —