Symplectic Operators Research Articles

Unitary Howe dualities in fermionic and bosonic algebras and related Dirac operators

In this paper we use the canonical complex structure \mathbb{J}𝕁 on \mathbb{R}^{2n}ℝ2n to introduce a twist of the symplectic Dirac operator. This can be interpreted as the bosonic analogue of the Dirac operators on a Hermitian manifold. Moreover, we prove that the algebra of these Dirac operators is isomorphic to the Lie algebra \mathfrak{su}(1,2)𝔰𝔲(1,2) which leads to the Howe dual pair (U(n),\mathfrak{su}(1,2))(U(n),𝔰𝔲(1,2)).

Novel finite difference approach to discretize the symplectic dirac operator

Symplectic Dirac operator is an intertwining differential operator. Discretising symplectic Dirac operator gives a new direction to study the quantum space. The construction of discrete symplectic Dirac operator requires the theory of discrete symplectic Clifford analysis or the concept of discrete symplectic connections, which are not explained in literature. In this work, a discretization approach for symplectic Dirac operator is suggested by considering the forward and backward basis vectors on symplectic Clifford algebra. The suggested discrete symplectic Dirac operator is Ds=Ds++Ds- where the Ds+ and Ds- are the forward and backward discrete symplectic Dirac operators, respectively. The new discrete symplectic Dirac operator gives the factorization of discrete Laplacian on symplectic spaces. Further, we establish commutation relations involving forward and backward discrete symplectic Dirac operators in the representation theory.

Unchaining surgery and topology of symplectic 4-manifolds

We study a symplectic surgery operation we call unchaining, which effectively reduces the second Betti number and the symplectic Kodaira dimension at the same time. Using unchaining, we give novel constructions of symplectic Calabi–Yau surfaces from complex surfaces of general type and completely resolve a conjecture of Stipsicz on the existence of exceptional sections in Lefschetz fibrations. Combining the unchaining surgery with others, which all correspond to certain monodromy substitutions for Lefschetz pencils, we provide further applications, such as new constructions of exotic symplectic 4-manifolds, and inequivalent pencils of the same genera and the same number of base points on families of symplectic 4-manifolds. Meanwhile, we present a handy criterion for determining from the monodromy of a pencil whether its total space is spin or not.

A Whittaker category for the symplectic Lie algebra and differential operators

A Whittaker category for the symplectic Lie algebra and differential operators

SYMPLECTIC DIRAC OPERATORS FOR LIE ALGEBRAS AND GRADED HECKE ALGEBRAS

The aim of this paper is to define a pair of symplectic Dirac operators (D+, D–) in an algebraic setting motivated by the analogy with the algebraic orthogonal Dirac operators in representation theory. We work in the settings of ℤ/2-graded quadratic Lie algebras 𝔤 = 𝔨 + 𝔭 and of graded affine Hecke algebras ℍ. In these contexts, we show analogues of the Parthasarathy’s formula for [D+, D–] and certain generalisations of the Casimir inequality.

Transversely symplectic Dirac operators on transversely symplectic foliations

We study transversely metaplectic structures and transversely symplectic Dirac operators on transversely symplectic foliations. And we give the Weitzenböck type formula for transversely symplectic Dirac operators. Moreover, we estimate the lower bound of the eigenvalues of the transversely symplectic Dirac operator defined by the transverse Levi-Civita connection on transverse Kähler foliations.

The symplectic fermion ribbon quasi-Hopf algebra and the [formula omitted]-action on its centre

We introduce a family of factorisable ribbon quasi-Hopf algebras Q(N) for N a positive integer: as an algebra, Q(N) is the semidirect product of CZ2 with the direct sum of a Graßmann and a Clifford algebra in 2N generators. We show that RepQ(N) is ribbon equivalent to the symplectic fermion category SF(N) that was computed in [54] from conformal blocks of the corresponding logarithmic conformal field theory. The latter category in turn is conjecturally ribbon equivalent to representations of Vev, the even part of the symplectic fermion vertex operator super algebra.Using the formalism developed in [20] we compute the projective SL(2,Z)-action on the centre of Q(N) as obtained from Lyubashenko's general theory of mapping class group actions for factorisable finite ribbon categories. This allows us to test a conjectural non-semisimple version of the modular Verlinde formula: we verify that the SL(2,Z)-action computed from Q(N) agrees projectively with that on pseudo trace functions of Vev.

A study on reflection and transmission characteristics of magnetized plasma based on S-MRTD scheme

Employing symplectic algorithm in electromagnetic simulation enables the advantages of energy conservation, high stability, low dispersion and high accuracy to be maintained. A new numerical scheme for simulating magnetized plasma, which is symplectic multi-resolution time-domain (S-MRTD) scheme is presented. First, the discrete iterative process of S-MRTD scheme for magnetized plasma is constructed based on the symplectic operator technique. Then, the correctness and effectiveness of the S-MRTD scheme framework are verified by the numerical simulation of left-handed circular polarized wave (LCP) and right-handed circular polarized wave (RCP). Finally, the influence of electron cyclotron frequency(wb) on reflection and transmission coefficients is analyzed based on S-MRTD scheme.

Examples of Four- or Six-Dimensional Symplectic-Haantjes Manifolds and About a Relationship with Recursion Operators

Certain ways of characterizing integrable systems with $(1,1)$-tensor field have been investigated, so far. For example, recursion operators and Haantjes operators are known. We show that geometrical examples of four- or six-dimensional symplectic Haantjes manifolds and recursion operators for several Hamiltonian systems. Through these examples, we consider the relation between recursion operators and Haantjes operators.

Generalized chain surgeries and applications

We describe the Stein handlebody diagrams of Milnor fibers of Brieskorn singularities $x^p + y^q + z^r = 0$. We also study the natural symplectic operation by exchanging two Stein fillings of the canonical contact structure on the links in the case $p = q = r$, where one of the fillings comes from the minimal resolution and the other is the Milnor fiber. We give two different interpretations of this operation, one as a symplectic sum and the other as a monodromy substitution in a Lefschetz fibration.

The Symplectic Fueter–Sce Theorem

In this paper we present a symplectic analogue of the Fueter theorem. This allows the construction of special (polynomial) solutions for the symplectic Dirac operator $D_s$, which is defined as the first-order $\mathfrak{sp}(2n)$-invariant differential operator acting on functions on ${\mathbb R}^{2n}$ taking values in the metaplectic spinor representation.

Higher symmetries of symplectic Dirac operator

We construct in projective differential geometry of the real dimension 2 higher symmetry algebra of the symplectic Dirac operator acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overdetermined operators of arbitrarily high order acting on symmetric tensors. The higher symmetry algebra structure corresponds to a completely prime primitive ideal having as its associated variety the minimal nilpotent orbit of $${\mathfrak {sl}}(3,{\mathbb {R}})$$ .

Linear Differential Operators Invertible in the Integro-differential Sense

The paper studies linear differential operators in derivatives with respect to one variable. Such operators include, in particular, operators defined on infinite prolongations of evolutionary systems of differential equations with one spatial variable. In this case, differential operators in total derivatives with respect to the spatial variable are considered. In parallel, linear differential operators with one independent variable are investigated. The known algorithms for reducing the matrix to a stepwise or diagonal form are generalized to the operator matrices of both types. These generalizations are useful at points, where the functions, into which the matrix components are divided when applying the algorithm, are nonzero.In addition, the integral operator is defined as a multi-valued operator that is the right inverse of the total derivative. Linear operators that involve both the total derivatives and the integral operator are called integro-differential. An invertible operator in the integro-differential sense is an operator for which there exists a two-sided inverse integro-differential operator. A description of scalar differential operators that are invertible in this sense is obtained. An algorithm for checking the invertibility in the integro-differential sense of a differential operator and for constructing the inverse integro-differential operator is formulated.The results of the work can be used to solve linear equations for matrix differential operators arising in the theory of evolutionary systems with one spatial variable. Such operator equations arise when describing systems that are integrable by the inverse scattering method, when calculating recursion operators, higher symmetries, conservation laws and symplectic operators, and also when solving some other problems. The proposed method for solving operator equations is based on reducing the matrices defining the operator equation to a stepwise or diagonal form and solving the resulting scalar operator equations.

Symplectic MRTD for solving three‐dimensional Maxwell equations based on optimised operators

A new set of symplectic operators for the symplectic multi-resolution time-domain (S-MRTD) method is obtained by using the time reversible constraint and the growth factor. Then, the perfectly matched layer absorbing boundary condition based on splitting field technique is introduced into the S-MRTD method and its iterative formula is derived. The stability and numerical dispersion of S-MRTD method with optimised symplectic operators are discussed in detail. Finally, the optimal S-MRTD method is applied to the calculation of electromagnetic radiation and scattering. Numerical calculation and simulation results show that the S-MRTD is more accurate and efficient than the traditional FDTD method.

Acquiring the Symplectic Operator Based on Pure Mathematical Derivation then Verifying It in the Intrinsic Problem of Nanodevices

The symplectic algorithm can maintain the symplectic structure and intrinsic properties of the system, its cumulative error is small and suitable for multi-step calculation. At present, the widely accepted symplectic operators are obtained by solving the Hamilton equation based on artificial definitions and assumptions in advance. There are inevitable dispersion errors. We solve the equation by pure mathematical derivation without any artificial limitations and assumptions. The way to accurately obtain high-precision symplectic operators greatly reduces the dispersion error from the beginning. The numerical solution of the one-dimensional Schrödinger equation for describing the intrinsic problem of nanodevices is used as an application environment to compare the total energy distribution of the particle wave function in the box, thus verifying the properties of the Symplectic Operator based on Pure Mathematical Derivation by comparing with Finite-Difference Time-Domain (FDTD) and the widely accepted symplectic operator.

Variational Operators, Symplectic Operators, and the Cohomology of Scalar Evolution Equations

For a scalar evolution equation ut = K(t, x, u, ux,...,u2m+1) with m ≥ 1, the cohomology space H1,2(ℜ∞) is shown to be isomorphic to the space of variational operators and an explicit isomorphism is given. The space of symplectic operators for ut = K for which the equation is Hamiltonian is also shown to be isomorphic to the space H1, 2(ℜ∞) and subsequently can be naturally identified with the space of variational operators. Third order scalar evolution equations admitting a first order symplectic (or variational) operator are characterized. The variational operator (or symplectic) nature of the potential form of a bi-Hamiltonian evolution equation is also presented in order to generate examples of interest.

Simulating Electron Dynamics of Complex Molecules with Time-Dependent Complete Active Space Configuration Interaction.

Time-dependent electronic structure methods are growing in popularity as tools for modeling ultrafast and/or nonlinear processes, for computing spectra, and as the electronic structure component of mean-field molecular dynamics simulations. Time-dependent configuration interaction (TD-CI) offers several advantages over the widely used real-time time-dependent density functional theory: namely, that it correctly models Rabi oscillations; it offers a spin-pure description of open-shell systems; and a hierarchy of TD-CI methods can be defined that systematically approach the exact solution of the time-dependent Schrodinger equation (TDSE). In this work, we present a novel TD-CI approach that extends TD-CI to large complete active-space configuration expansions. Such extension is enabled by use of a direct configuration interaction approach that eliminates the need to explicitly build, store, or diagonalize the Hamiltonian matrix. Graphics processing unit (GPU) acceleration enables fast solution of the TDSE even for large active spaces-up to 12 electrons in 12 orbitals (853776 determinants) in this work. A symplectic split operator propagator yields long-time norm conservation. We demonstrate the applicability of our approach by computing the response of a large molecule with a strongly correlated ground state, decacene (C42H24), to various pulses (δ-function, transform limited, chirped). Our simulations predict that chirped pulses can be used to induce dipole-forbidden transitions. Simulations of decacene using the 6-31G(d) basis set and a 12 electrons/12 orbitals active space took 20.1 h to propagate for 100 fs with a 1 attosecond time step on a single NVIDIA K40 GPU. Convergence with respect to time step is found to depend on the property being computed and the chosen active space.

Discrete chaotic maps obtained by symmetric integration

Chaotic return maps are widely used to model various dynamical systems such as charged particle movement, laser beam dynamic, celestial body orbiting and many others. Return maps are commonly obtained by discretization of continuous equations using the Euler–Cromer operator with the only motivation that it is the simplest symplectic operator. Recent progress in geometric integration raised considerable interest to symmetric operators due to their ability to preserve some geometric properties of continuous flows this way yielding better agreement between discrete and continuous dynamical systems. Here we compare symmetric and asymmetric discretization approaches applied to several examples of Hamiltonian systems. In particular, we suggest symmetric modifications of Chirikov and Hénon maps and show explicitly that the implied symmetric integration procedure yields reflectional symmetry in the phase space. For verification, we show that a smooth even perturbation function used instead of a discontinuous delta pulse provides asymptotically similar results. Numerical experiments using several statistical methods show that symmetric and asymmetric maps, while yielding similar asymptotic behavior, often exhibit considerably different statistical properties for intermediate regimes providing smoother transitions that are more reminiscent to those observed in various natural phenomena. We believe that the proposed approach may be useful for modeling empirical systems by preserving their keynote physical properties.

Explicit symplectic geometric algorithms for quaternion kinematical differential equation

Solving quaternion kinematical differential equations U+0028 QKDE U+0029 is one of the most significant problems in the automation, navigation, aerospace and aeronautics literatures. Most existing approaches for this problem neither preserve the norm of quaternions nor avoid errors accumulated in the sense of long term time. We present explicit symplectic geometric algorithms to deal with the quaternion kinematical differential equation by modelling its time-invariant and time-varying versions with Hamiltonian systems and adopting a three-step strategy. Firstly, a generalized Euler U+02BC s formula and Cayley-Euler formula are proved and used to construct symplectic single-step transition operators via the centered implicit Euler scheme for autonomous Hamiltonian system. Secondly, the symplecticity, orthogonality and invertibility of the symplectic transition operators are proved rigorously. Finally, the explicit symplectic geometric algorithm for the time-varying quaternion kinematical differential equation, i.e., a non-autonomous and non-linear Hamiltonian system essentially, is designed with the theorems proved. Our novel algorithms have simple structures, linear time complexity and constant space complexity of computation. The correctness and efficiencies of the proposed algorithms are verified and validated via numerical simulations.

Hamiltonian structures of a coupled Ramani equation

With the aid of Hamiltonian structures for the constrained modified KP flow, we show that a coupled Ramani equation is a bi-Hamiltonian system with a local Hamiltonian operator P¯2 and a symplectic operator. After a reduction, we recover the bi-Hamiltonian formalism of the Ramani equation. Furthermore, we obtain the explicit bi-Hamiltonian structure for the Drinfeld–Sokolov hierarchies of C3(1) using the local Hamiltonian operator P¯2.