Search

The purpose of this paper is to present an algebraic theoretical basis for the study of ω-Hamiltonian vector fields defined on a symplectic vector space (V,ω)with respect to coordinates that are not necessarily symplectic. We introduce the concepts of ω-symplectic and ω-semisymplectic groups, and describe some of their properties that may not coincide with the classical context. We show that the Lie algebra of such groups is a useful tool in the recognition and construction of ω-Hamiltonian vector fields.

We construct a full exceptional collection of vector bundles in the bounded derived category of coherent sheaves on the Grassmannian of isotropic 3-dimensional subspaces in a symplectic vector space of dimension 8.

Givental’s Lagrangian cone $${\mathscr {L}}_X$$LXis a Lagrangian submanifold of a symplectic vector space which encodes the genus-zero Gromov–Witten invariants of X. Building on work of Braverman, Coates has obtained the Lagrangian cone as the push-forward of a certain class on the moduli space of stable maps to . This provides a conceptual description for an otherwise mysterious change of variables called the dilaton shift. We recast this construction in its natural context, namely the moduli space of stable maps to relative the divisor . We find that the resulting push-forward is another familiar object, namely the transform of the Lagrangian cone under the action of the fundamental solution matrix. This hints at a generalisation of Givental’s quantisation formalism to the setting of relative invariants. Finally, we use a hidden polynomiality property implied by our construction to obtain a sequence of universal relations for the Gromov–Witten invariants, as well as new proofs of several foundational results concerning both the Lagrangian cone and the fundamental solution matrix.

When the quotient of a symplectic vector space by the action of a finite subgroup of symplectic automorphisms admits as a crepant projective resolution of singularities the Hilbert scheme of regular orbits of Nakamura, then there is a natural isomorphism between the Grothendieck group of this resolution and the representation ring of the group, given by the Bridgeland-King-Reid map. However, this isomorphism is not compatible with the ring structures. For the Hilbert scheme of points on the affine plane, we study the multiplicative behavior of this map.When the quotient of a symplectic vector space by the action of a finite subgroup of symplectic automorphisms admits as a crepant projective resolution of singularities the Hilbert scheme of regular orbits of Nakamura, then there is a natural isomorphism between the Grothendieck group of this resolution and the representation ring of the group, given by the Bridgeland-King-Reid map. However, this isomorphism is not compatible with the ring structures. For the Hilbert scheme of points on the affine plane, we study the multiplicative behavior of this map.

Abstract.: To any finite group Γ⊂Sp(V) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ of the algebra ℂ[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of ℙ r , where r=number of conjugacy classes of symplectic reflections in $Γ$. The algebra Hκ, called a symplectic reflection algebra, is related to the coordinate ring of a Poisson deformation of the quotient singularity V/Γ. This leads to a symplectic analogue of McKay correspondence, which is most complete in case of wreath-products. If Γ is the Weyl group of a root system in a vector space ? and V=?⊕?*, then the algebras Hκ are certain ‘rational’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let $Γ=S n , the Weyl group of ?=?? n . We construct a 1-parameter deformation of the Harish-Chandra homomorphism from ?(?)?, the algebra of invariant polynomial differential operators on ?? n , to the algebra of S n -invariant differential operators with rational coefficients on the space ℂ n of diagonal matrices. The second order Laplacian on ? goes, under the deformed homomorphism, to the Calogero-Moser differential operator on ℂ n , with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: ?(?)? ↠ spherical subalgebra in Hκ, where Hκ is the symplectic reflection algebra associated to the group Γ=S n . This way, the deformed Harish-Chandra homomorphism becomes nothing but a description of the spherical subalgebra in terms of ‘quantum’ Hamiltonian reduction. In the ‘classical’ limit κ→∞, our construction gives an isomorphism between the spherical subalgebra in H∞ and the coordinate ring of the Calogero-Moser space. We prove that all simple H∞-modules have dimension n!, and are parametrised by points of the Calogero-Moser space. The family of these modules forms a distinguished vector bundle on the Calogero-Moser space, whose fibers carry the regular representation of S n . Moreover, we prove that the algebra $H∞ is isomorphic to the endomorphism algebra of that vector bundle.

We study the mod 2 homology of the double loop space of SU(n)/SO(n) using the Serre spectral sequence along with the Eilenberg–Moore spectral sequence. Then we also get the homology of the double loop space of the set of all Lagrangian subspaces of the symplectic vector space R2n.

This paper studies the incidence relation between the points and quadrics in the projective space of a symplectic vector space over a field of even order. The 2-rank of the incidence matrix is determined. This is achieved by viewing the code generated by incidence vectors as a module for the symplectic group and applying the 2-modular representation theory of this group. The radical series of this module is also described.

Abstract: Given a family {L 0(b,L 1(b)} b∈B of pairs of transverse Lagrangian subspaces of a hermitean symplectic vector space we define a family of Dirac operators on the unit interval and consider itsη-formη(L 0,L 1) εΘ*(B). To a family {L 0(b,L 1(b,L 2(b)} b∈B of pairwise transverse Lagrangian subspaces we associate the cocycleη(L 0,L 1)+η(L 1,L 2)+η(L 2,L 1) which is a closed form. We identify its cohomology class with a generalization to families of the triple Maslov index.

A new natural structure on the tangent spaces of a co-tangent bundle is introduced and some of its properties are investigated. This structure is based on a symmetric bilinear form and leads to a geometry that is, in many respects, analogous to the symplectic geometry. The new structure can thus justifiably be called co-symplectic geometry. The null structure of co-symplectic vector spaces is investigated in detail. It is found that the manifold of all maximally isotropic subspaces of a co-symplectic vector space is a homogeneous compact manifold of dimension 1/2n(n-1) consisting of two diffeomorphic components and having fundamental groupZ2?Z2. A representation of the fundamental group of this manifold is explicitly constructed in terms of quadrupoles of co-Lagrangian subspaces.

The mathematical formalism for linear quantum field theory on curved spacetime depends in an essential way on the assumption of global hyperbolicity. Physically, what lie at the foundation of any formalism for quantization in curved spacetime are the canonical commutation relations, imposed on the field operators evaluated at a global Cauchy surface. In the algebraic formulation of linear quantum field theory, the canonical commutation relations are restated in terms of a well-defined symplectic structure on the space of smooth solutions, and the local field algebra is constructed as the Weyl algebra associated to this symplectic vector space. When spacetime is not globally hyperbolic, e.g. when it contains naked singularities or closed time-like curves, a global Cauchy surface does not exist, and there is no obvious way to formulate the canonical commutation relations, hence no obvious way to construct the field algebra. In a paper which appears elsewhere in this journal, we report on a generalization of the algebraic framework for quantum field theory to arbitrary topological spaces which do not necessarily have a spacetime metric defined on them at the outset. Taking this generalization as a starting point, in this paper we give a prescription for constructing the field algebra of a (massless or massive) Klein--Gordon field on an arbitrary background spacetime. When spacetime is globally hyperbolic, the theory defined by our construction coincides with the ordinary Klein--Gordon field theory on a globally hyperbolic background. We explore some basic features of our generalized Klein--Gordon theory on arbitrary spacetimes, and study its specific properties on simple examples of non-globally-hyperbolic backgrounds that contain closed time-like curves or naked singularities.

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 Painlevé transcendentsPV andVI.

A realization of Gromov's nonsqueezing theorem and its applications to uncertainty analysis in Hamiltonian systems are studied in this note. Gromov's nonsqueezing theorem describes a fundamental property of symplectic manifolds, however, this theorem is usually started in terms of topology and its physical meaning is vague. In this note we introduce a physical interpretation of the linear symplectic width, which is the lower bound in the nonsqueezing theorem, in terms of the eigenstructure of a positive-definite, symmetric matrix. Since uncertainty is often represented in terms of a positive definite, symmetric matrix in control theory, our study can be applied to uncertainty analysis by applying the nonsqueezing theorem to the uncertainty ellipsoid. We find a fundamental inequality for the evolving uncertainty in a linear dynamical system and provide some numerical examples. [ABSTRACT FROM AUTHOR]