Showing papers on "Symplectic representation published in 2023"

Multi-symplectic Method for an Infinite-Dimensional Hamiltonian System

Abstract: AbstractBased on the multi-symplectic theory proposed by Bridges and Marsden, the mathematical formulations for the multi-symplectic method are reviewed and several applications of which are presented in this chapter. Generalizing the concept of reversibility to high-dimensional systems, the bisymplectic structure, named as the multi-symplectic structure of the infinite-dimensional Hamiltonian system with several conservation laws are presented. For the multi-symplectic form, the explicit midpoint discretization method and the Euler box discretization method are discussed in detail. The multi-symplectic structure as well as the discrete scheme for some evolution equations, including the membrane free vibration equation, the generalized fifth-order KdV equation, the generalized (2+1)-dimensional KdV–mKdV equation, the Landau–Ginzburg–Higgs equation, the Boussinesq-type equation, the quasi-Degasperis–Procesi equation and the logarithmic–KdV equation, are given.

Coulomb branch algebras via symplectic cohomology

Abstract: Let $(\bar{M}, \omega)$ be a compact symplectic manifold with convex boundary and $c_1(T\bar{M})=0$. Suppose that $(\bar{M}, \omega)$ is equipped with a convex Hamiltonian $G$-action for some connected, compact Lie group $G$. We construct an action of the pure Coulomb branch of $G$ on the $G$-equivariant symplectic cohomology of $\bar{M}.$ Building on work of Teleman, we use this construction to characterize the Coulomb branches of Braverman-Finkelberg-Nakajima in terms of equivariant symplectic cohomology.

Explicit K-Symplectic and Symplectic-like Methods for Charged Particle System in General Magnetic Field

Abstract: We propose explicit K-symplectic and explicit symplectic-like methods for the charged particle system in a general strong magnetic field. The K-symplectic methods are also symmetric. The charged particle system can be expressed both in a canonical and a non-canonical Hamiltonian system. If the three components of the magnetic field can be integrated in closed forms, we construct explicit K-symplectic methods for the non-canonical charged particle system; otherwise, explicit symplectic-like methods can be constructed for the canonical charged particle system. The symplectic-like methods are constructed by extending the original phase space and obtaining the augmented separable Hamiltonian, and then by using the splitting method and the midpoint permutation. The numerical experiments have shown that compared with the higher order implicit Runge-Kutta method, the explicit K-symplectic and explicit symplectic-like methods have obvious advantages in long-term energy conservation and higher computational efficiency. It is also shown that the influence of the parameter ε in the general strong magnetic field on the Runge-Kutta method is bigger than the two kinds of symplectic methods.

"Equation missing" Hamiltonian Symplectic Structures: An Introduction

Abstract: The final chapter is devoted to formulating Hamiltonian Mechanics on symplectic manifolds and on bundles over symplectic manifolds. We will take the chance to discuss in detail the phase-space action of the Galilean and Poincaré groups in terms of canonical transformations.

Unchaining surgery and topology of symplectic 4-manifolds

Abstract: 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.

The momentum map of the affine real symplectic group

Abstract: In this paper we explain how the cocycle of the momentum map of the action of the affine symplectic group on $\mathbb{R}^{2n}$ gives rise to a coadjoint orbit of the odd real symplectic group with a modulus.

Symplectic Methods for a Finite-Dimensional System

Abstract: In this chapter, symplectic approaches for the finite-dimensional Hamiltonian systems are discussed. Firstly, the foundations of the symplectic method are reviewed, which include the symplectic map, the symplectic matrix, the symplectic structure, and so on. Then, two typical symplectic discretization methods are presented. One is the symplectic Runge–Kutta method and another is the splitting/composition method. With these foundations, some research progresses on the applications of the symplectic approach, including the symplectic precise integration of folding and unfolding processes of undercarriage and the symplectic Runge–Kutta method for aerospace dynamics problems are given.

On $T$-invariant subvarieties of symplectic Grassmannians and representability of rank $2$ symplectic matroids over ${\mathbb C}$

Abstract: For the symplectic Grassmannian $\text{SpG}(2,2n)$ of $2$-dimensional isotropic subspaces in a $2n$-dimensional vector space over an algebraically closed field of characteristic zero endowed with a symplectic form and with the natural action of an $n$-dimensional torus $T$ on it, we characterize its irreducible $T$-invariant subvarieties. This characterization is in terms of symplectic Coxeter matroids, and we use this result to give a complete characterization of the symplectic matroids of rank $2$ which are representable over $\mathbb{C}$.

Discrete symplectic fermions on double dimers and their Virasoro representation

Abstract: A discrete version of the Conformal Field Theory of symplectic fermions is introduced and discussed. Specifically, discrete symplectic fermions are realised as holomorphic observables in the double-dimer model. Using techniques of discrete complex analysis, the space of local fields of discrete symplectic fermions on the square lattice is proven to carry a representation of the Virasoro algebra with central charge $-2$.

Branching symplectic monogenics using a Mickelsson--Zhelobenko algebra

Abstract: In this paper we consider (polynomial) solution spaces for the symplectic Dirac operator (with a focus on $1$-homogeneous solutions). This space forms an infinite-dimensional representation space for the symplectic Lie algebra $\mathfrak{sp}(2m)$. Because $\mathfrak{so}(m)\subset \mathfrak{sp}(2m)$, this leads to a branching problem which generalises the classical Fischer decomposition in harmonic analysis. Due to the infinite nature of the solution spaces for the symplectic Dirac operators, this is a non-trivial question: both the summands appearing in the decomposition and their explicit embedding factors will be determined in terms of a suitable Mickelsson-Zhelobenko algebra.

Evaluation of Hamiltonians from Complex Symplectic Matrices

Abstract: Gaussian unitaries play a fundamental role in the field of continuous variables. In the general n mode, they may formulated by a second-order polynomial in the bosonic operators. Another important role related to Gaussian unitaries is played by the symplectic transformations in the phase space. The paper investigates the links between the two representations: the link from Hamiltonian to symplectic, governed by an exponential, and the link from symplectic to Hamiltonian, governed by a logarithm. Thus, an answer is given to the non-trivial question: which Hamiltonian produces a given symplectic representation? The complex instead of the traditional real symplectic representation is considered, with the advantage of getting compact and elegant relations. The application to the single, two, and three modes illustrates the theory.

On parabolic subgroups of symplectic reflection groups

Abstract: Abstract Using Cohen’s classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group, are themselves symplectic reflection groups. This is the symplectic analog of Steinberg’s Theorem for complex reflection groups. Using computational results required in the proof, we show the nonexistence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open. Another immediate consequence of our result is that the singular locus of the symplectic quotient singularity associated to a symplectic reflection group is pure of codimension two.

No $C^1$-recurrence of iterations of symplectomorphisms

Abstract: In this article, we study the behavior of iterations of symplectomorphisms and Hamiltonian diffeomorphisms on symplectic manifolds. We prove that symplectomorphisms and Hamiltonian diffeomorphisms do not have $C^1$-recurrence on negatively monotone symplectic manifolds. This is a generalization of the results of the study of Polterovich, Ono, Atallah-Shelukhin. Hamiltonian group actions play very important roles in symplectic geometry. We see that negatively monotone symplectic manifolds are far from being Hamiltonian $G$-manifolds.