Symplectic Operators Research Articles

Basic Aspects of Symplectic Clifford Analysis for the Symplectic Dirac Operator

In the present article we study basic aspects of the symplectic version of Clifford analysis associated to the symplectic Dirac operator. Focusing mostly on the symplectic vector space of real dimension 2, this involves the analysis of first order symmetry operators, symplectic Clifford-Fourier transform, reproducing kernel for the symplectic Fischer product and the construction of bases of symplectic monogenics for the symplectic Dirac operator.

Projective structure, $\widetilde{\operatorname{SL}}(3,\mathbb{R})$ and the symplectic Dirac operator

Inspired by the results on symmetries of the symplectic Dirac operator, we realize symplectic spinor fields and the symplectic Dirac operator in the framework of (the double cover of) homogeneous projective structure in two real dimensions. The symmetry group of the homogeneous model of the double cover of projective geometry in two real dimensions is ${\widetilde{}}(3,)$.

The Hilbert–Schmidt Lagrangian Grassmannian in infinite dimension

In this paper we study the action of the symplectic operators which are a perturbation of the identity by a Hilbert-Schmidt operator in the Lagrangian Grassmannian manifold.

Riemannian metrics on an infinite dimensional symplectic group

The aim of this paper is the geometric study of the symplectic operators which are a perturbation of the identity by a Hilbert–Schmidt operator. This subgroup of the symplectic group was introduced in Pierre de la Harpe's classical book of Banach–Lie groups. Throughout this paper we will endow the tangent spaces with different Riemannian metrics. We will use the minimal curves of the unitary group and the positive invertible operators to compare the length of the geodesic curves in each case. Moreover we will study the completeness of the symplectic group with the geodesic distance.

Nonclassical Correlation Dynamics in a System of Mesoscopic Josephson Junction Coupled to Single-mode Optical Cavity

We investigate the time evolutions of the continuous-variable entanglement and Gaussian quantum discord in a system consisting of a mesoscopic Josephson junction coupled to a single-mode optical cavity field. We can obtain the time-dependent covariance matrix using known symplectic operation and local canonical transformations. We compare the dynamics of Gaussian quantum discord with that of entanglement. It is shown that the entanglement dynamics of two-mode squeezed thermal state is richer and undergoes three different features: periodical oscillation, sudden death and revival, and no-creation of entanglement, conditioned on the average number of thermal photons in each mode, whereas the Gaussian quantum discord can only exhibit a periodical oscillation behavior during the evolution.

On Mpc-structures and symplectic Dirac operators

We prove that the kernels of the restrictions of symplectic Dirac or symplectic Dirac-Dolbeault operators on natural subspaces of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute those kernels for the complex projective spaces. We construct injections of subgroups of the symplectic group (the pseudo-unitary group and the stabilizer of a Lagrangian subspace) in the group Mpc and classify G-invariant Mpc-structures on symplectic spaces with a G-action. We prove a variant of Parthasarathy's formula for the commutator of two symplectic Dirac-type operators on a symmetric symplectic space.

Symplectic Calabi–Yau 6-manifolds

In this article, we construct simply connected symplectic Calabi–Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along T4. Using our method, we also construct symplectic non-Kähler Calabi–Yau 6-manifolds with fundamental group Z. This paper also produces the first examples of simply connected and non-simply connected symplectic Calabi–Yau 6-manifolds with fundamental groups Zp×Zq, and Z×Zq for any p≥1 and q≥2via co-isotropic Luttinger surgery.

The metaplectic Howe duality and polynomial solutions for the symplectic Dirac operator

We study various aspects of the metaplectic Howe duality realized by the Fischer decomposition for the metaplectic representation space of polynomials on R2n valued in the Segal–Shale–Weil representation. As a consequence, we determine symplectic monogenics, i.e. the space of polynomial solutions of the symplectic Dirac operator Ds.

Erratum to: Symplectic Dolbeault operators on Kähler manifolds

Such a structure on G/T turns it into what is called a “Kahler with torsion (KT)” manifold. The general results and constructions established in the previous sections carry over unchanged to KT manifolds except for Proposition 3 which states that the symplectic Dirac operators D and D are formally self-adjoint. For connections with torsion and parallel complex structure, a sufficient condition for D and D to be self-adjoint is the vanishing of the torsion vector field, defined by

Lie algebras and Hamiltonian structures of multi-component Ablowitz–Kaup–Newell–Segur hierarchy

Isospectral and non-isospectral hierarchies of multi-component Ablowitz–Kaup–Newell–Segur (AKNS) are obtained from a matrix spectral problem, then by means of the zero curvature representations of the isospectral flows {Km} and non-isospectral flows {σn}, we construct the symmetries and their algebraic structures for isospectral multi-component AKNS hierarchies, demonstrate the recursive operator L is a strong and hereditary symmetry for the isospectral hierarchy. We also derive that there are implectic operator θ and symplectic operator J such that L = θJ, and discuss the multi-Hamiltonian structures and the Liouville integrability of the isospectral hierarchies.

Symplectic Twistor Operator on $$\mathbb R ^{2n}$$ and the Segal–Shale–Weil Representation

The aim of our article is the study of solution space of the symplectic twistor operator $$T_s$$ in symplectic spin geometry on standard symplectic space $$(\mathbb R ^{2n},\omega )$$ , which is the symplectic analogue of the twistor operator in (pseudo) Riemannian spin geometry. In particular, we observe a substantial difference between the case $$n=1$$ of real dimension $$2$$ and the case of $$\mathbb R ^{2n}, n>1$$ . For $$n>1$$ , the solution space of $$T_s$$ is isomorphic to the Segal–Shale–Weil representation.

Singular Casimir Elements of the Euler Equation and Equilibrium Points

The problem of the nonequivalence of the sets of equilibrium points and energy-Casimir extremal points, which occurs in the noncanonical Hamiltonian formulation of equations describing ideal fluid and plasma dynamics, is addressed in the context of the Euler equation for an incompressible inviscid fluid. The problem is traced to a Casimir deficit, where Casimir elements constitute the center of the Lie-Poisson algebra underlying the Hamiltonian formulation, and this leads to a study of the symplectic operator defining the Poisson bracket. The kernel of the symplectic operator, for this typical example of an infinite-dimensional Hamiltonian system for media in terms of Eulerian variables, is analyzed. For two-dimensional flows, a rigorously solvable system is formulated. The nonlinearity of the Euler equation makes the symplectic operator inhomogeneous on phase space (the function space of the state variable), and it is seen that this creates a singularity where the nullity of the symplectic operator (the "dimension" of the center) changes. Singular Casimir elements stemming from this singularity are unearthed using a generalization of the functional derivative that occurs in the Poisson bracket.

Invertible Coupled KdV and Coupled Harry Dym Hierarchies

In this paper, we discuss the conditions under which the coupled KdV and coupled Harry Dym hierarchies possess inverse (negative) parts. We further investigate the structure of nonlocal parts of tensor invariants of these hierarchies, in particular, the nonlocal terms of vector fields, conserved one‐forms, recursion operators, Poisson and symplectic operators. We show that the invertible coupled KdV hierarchies possess Poisson structures that are at most weakly nonlocal while coupled Harry Dym hierarchies have Poisson structures with nonlocalities of the third order.

Symplectic Dolbeault operators on Kähler manifolds

For a K\"ahler Manifold $M$, the "symplectic Dolbeault operators" are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $\bar\partial$ and $\bar\partial^*$, arise from Dirac operators on the canonical complex spinors on $M$. We give special attention to two special classes of K\"ahler manifolds: Riemann surfaces and flag manifolds ($G/T$ for $G$ a simply-connected compact semisimple Lie group and $T$ a maximal torus). In the case of flag manifolds, we work with the Hermitian structure induced by the Killing form and a choice of positive roots (this is actually not a K\"ahler structure but is a K\"ahler with torsion (KT) structure). For Riemann surfaces the symplectic Dolbeault operators are elliptic and we compute their indices. In the case of flag manifolds, we will see that the representation theory of $G$ plays a role and that these operators can be used to distinguish (as Hermitian manifolds) between the flag manifolds corresponding to the Lie algebras $B_n$ and $C_n$. We give a thorough analysis of these operators on $\C P^1$ (the intersection of these classes of spaces), where the symplectic Dolbeault operators have an especially interesting structure.

Symplectic twistor operator and its solution space on ${\mathbb{R}}^2$

We introduce the symplectic twistor operator $T_s$ in symplectic spin geometry of real dimension two, as a symplectic analogue of the Dolbeault operator in complex spin geometry of complex dimension 1. Based on the techniques of the metaplectic Howe duality and algebraic Weyl algebra, we compute the space of its solutions on ${\mathbb{R}}^2$.

Intertwining operators and fusion rules for vertex operator algebras arising from symplectic fermions

We determine fusion rules (dimensions of the spaces of intertwining operators) among simple modules for the vertex operator algebra obtained as an even part of the symplectic fermionic vertex operator superalgebra. By using these fusion rules we show that the fusion algebra of this vertex operator algebra is isomorphic to the group algebra of the Klein four group over Z.

Helical bifurcation and tearing mode in a plasma—a description based on Casimir foliation

The relation between the helical bifurcation of a Taylor relaxed state (a Beltrami equilibrium) and a tearing mode is analyzed in a Hamiltonian framework. Invoking an Eulerian representation of the Hamiltonian, the symplectic operator (defining a Poisson bracket) becomes non-canonical, i.e. the symplectic operator has a nontrivial cokernel (dual to its nullspace), foliating the phase space into level sets of Casimir invariants. A Taylor relaxed state is an equilibrium point on a Casimir (helicity) leaf. Changing the helicity, equilibrium points may bifurcate to produce helical relaxed states; a necessary and sufficient condition for bifurcation is derived. Tearing yields a helical perturbation on an unstable equilibrium, producing a helical structure approximately similar to a helical relaxed state. A slight discrepancy found between the helically bifurcated relaxed state and the linear tearing mode viewed as a perturbed, singular equilibrium state is attributed to a Casimir element (named ‘helical flux’) pertinent to a ‘resonance singularity’ of the non-canonical symplectic operator. While the helical bifurcation can occur at discrete eigenvalues of the Beltrami parameter, the tearing mode, being a singular eigenfunction, exists for an arbitrary Beltrami parameter. Bifurcated Beltrami equilibria appearing on the same helicity leaf are isolated by the helical-flux Casimir foliation. The obstacle preventing the tearing mode to develop in the ideal limit turns out to be the shielding current sheet on the resonant surface, preventing the release of the ‘potential energy’. When this current is dissipated by resistivity, reconnection is allowed and tearing instability occurs. The Δ′ criterion for linear tearing instability of Beltrami equilibria is shown to be directly related to the spectrum of the curl operator.

A Unified Approach to Computation of Integrable Structures

We expose a unified computational approach to integrable structures (including recursion, Hamiltonian, and symplectic operators) based on geometrical theory of partial differential equations.

Symplectic Dirac operators and $${Mp^{\rm c}}$$ -structures

Given a symplectic manifold $(M,\omega)$ admitting a metaplectic structure, and choosing a positive $\omega$-compatible almost complex structure $J$ and a linear connection $\nabla$ preserving $\omega$ and $J$, Katharina and Lutz Habermann have constructed two Dirac operators $D$ and ${\wt{D}}$ acting on sections of a bundle of symplectic spinors. They have shown that the commutator $[ D, {\wt{D}}]$ is an elliptic operator preserving an infinite number of finite dimensional subbundles. We extend the construction of symplectic Dirac operators to any symplectic manifold, through the use of $\Mpc$ structures. These exist on any symplectic manifold and equivalence classes are parametrized by elements in $H^2(M,\Z)$. For any $\Mpc$ structure, choosing $J$ and a linear connection $\nabla$ as before, there are two natural Dirac operators, acting on the sections of a spinor bundle, whose commutator $\mathcal{P}$ is elliptic. Using the Fock description of the spinor space allows the definition of a notion of degree and the construction of a dense family of finite dimensional subbundles; the operator $\mathcal{P}$ stabilizes the sections of each of those.

Cosymmetries and Nijenhuis recursion operators for difference equations

In this paper we discuss the concept of cosymmetries and co-recursion operators for difference equations and present a co-recursion operator for the Viallet equation. We also discover a new type of factorization for the recursion operators of difference equations. This factorization enables us to give an elegant proof that the pseudo-difference operator presented in Mikhailov et al 2011 Theor. Math. Phys. 167 421–43 is a recursion operator for the Viallet equation. Moreover, we show that the operator is Nijenhuis and thus generates infinitely many commuting local symmetries. The recursion operator and its factorization into Hamiltonian and symplectic operators have natural applications to Yamilov's discretization of the Krichever–Novikov equation.