Symplectic Reduction Research Articles

Symplectic model reduction methods for the Vlasov equation

AbstractParticle‐based simulations of the Vlasov equation typically require a large number of particles, which leads to a high‐dimensional system of ordinary differential equations. Solving such systems is computationally very expensive, especially when simulations for many different values of input parameters are desired. In this work, we compare several model reduction techniques and demonstrate their applicability to numerical simulations of the Vlasov equation. The necessity of symplectic model reduction algorithms is illustrated with a simple numerical experiment.

Symplectic Reduction of Classical Mechanics on Shape Space

One of the foremost goals of research in physics is to find the most basic and universal theories that describe our universe. Many theories assume the presence of absolute space and time in which the physical objects are located and physical processes take place. However, it is more fundamental to understand time as relative to the motion of another object, e.g., the number of swings of a pendulum, and the position of an object primarily relative to other objects. This paper aims to explain how using the principle of relationalism (to be introduced below), classical mechanics can be formulated on a most elementary space, which is freed from absolute entities: shape space. In shape space, only the relative orientation and length of subsystems are taken into account. A sufficient requirement for the validity of the principle of relationalism is that by changing a system’s scale variable, all the theory’s parameters that depend on the length, get changed accordingly. In particular, a direct implementation of the principle of relationalism introduces a proper transformation of the coupling constants of the interaction potentials in classical physics. This change leads consequently to a transformation in Planck’s measuring units, which enables us to derive a metric on shape space in a unique way. In order to find out the classical equations of motion on shape space, the method of “symplectic reduction of Hamiltonian systems” is extended to include scale transformations. In particular, we will give the derivation of the reduced Hamiltonian and symplectic form on shape space, and in this way, the reduction of a classical system with respect to the entire similarity group is achieved.

Symplectic reduction along a submanifold

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden–Weinstein–Meyer reduction, Mikami–Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg–Kazhdan construction of Moore–Tachikawa varieties in topological quantum field theory. A key feature of our construction is a concrete and systematic association of a Hamiltonian $G$-space $\mathfrak {M}_{G, S}$ to each pair $(G,S)$, where $G$ is any Lie group and $S\subseteq \mathrm {Lie}(G)^{*}$ is any submanifold satisfying certain non-degeneracy conditions. The spaces $\mathfrak {M}_{G, S}$ satisfy a universal property for symplectic reduction which generalizes that of the universal imploded cross-section. Although these Hamiltonian $G$-spaces are explicit and natural from a Lie-theoretic perspective, some of them appear to be new.

Double framed moduli spaces of quiver representations

Motivated by problems in the neural networks setting, we study moduli spaces of double framed quiver representations and give both a linear algebra description and a representation theoretic description of these moduli spaces. We define a network category whose isomorphism classes of objects correspond to the orbits of quiver representations, in which neural networks map input data. We then prove that the output of a neural network depends only on the corresponding point in the moduli space. Finally, we present a different perspective on mapping neural networks with a specific activation function, called ReLU, to a moduli space using the symplectic reduction approach to quiver moduli.

Explicit constructions of quilts with seam condition coming from symplectic reduction

Associated to a symplectic quotient $M/\!/G$ is a Lagrangian correspondence $\Lambda_G$ from $M/\!/G$ to $M$. In this note, we construct in two examples quilts with seam condition on such a correspondence, in the case of $S^1$ acting on $\mathbb{CP}^2$ with symplectic quotient $\mathbb{CP}^2/\!/ S^1 = \mathbb{CP}^1$. First, we study the quilted strips that would, if not for figure eight bubbling, identify the Floer chain groups $CF(\gamma,S_{\text{Cl}}^1)$ and $CF(\mathbb{RP}^2,T_{\text{Cl}}^2)$, where $\gamma$ is the connected double-cover of $\mathbb{RP}^1$. Second, we answer a question due to Akveld-Cannas da Silva-Wehrheim by explicitly producing a figure eight bubble which obstructs an isomorphism between two Floer chain groups. The figure eight bubbles we construct in this paper are the first concrete examples of this phenomenon.

A dual pair for the contact group

AbstractGeneralizing the canonical symplectization of contact manifolds, we construct an infinite dimensional non-linear Stiefel manifold of weighted embeddings into a contact manifold. This space carries a symplectic structure such that the contact group and the group of reparametrizations act in a Hamiltonian fashion with equivariant moment maps, respectively, giving rise to a dual pair, called the EPContact dual pair. Via symplectic reduction, this dual pair provides a conceptual identification of non-linear Grassmannians of weighted submanifolds with certain coadjoint orbits of the contact group. Moreover, the EPContact dual pair gives rise to singular solutions for the geodesic equation on the group of contact diffeomorphisms. For the projectivized cotangent bundle, the EPContact dual pair is closely related to the EPDiff dual pair due to Holm and Marsden.

Romans Massive QP Manifolds

We introduce QP manifolds that capture the generalised geometry of type IIA string backgrounds with Ramond–Ramond fluxes and Romans mass. Each of these is associated with a BPS brane in type IIA: a D2, D4, or NS5-brane. We explain how these probe branes are related to their associated QP-manifolds via the AKSZ topological field theory construction and the recent brane phase space construction. M-theory/type IIA duality is realised on the QP-manifold side as symplectic reduction along the M-theory circle (for branes that do not wrap it); this always produces IIA QP-manifolds with vanishing Romans mass.

Locally conformally symplectic reduction of the cotangent bundle

In Stanciu (Ann Global Anal Geom 56:245-275, 2019), we introduced a reduction procedure for locally conformally symplectic manifolds at any regular value of the natural momentum mapping. We use this construction to prove an analogue of a well-known theorem in the symplectic setting about the reduction of cotangent bundles.

Symplectic and Cosymplectic Reduction for Simple Hybrid Forced Mechanical Systems with Symmetries

Symplectic and Cosymplectic Reduction for Simple Hybrid Forced Mechanical Systems with Symmetries

Optimal Bases for Symplectic Model Order Reduction of Canonizable Linear Hamiltonian Systems

Symplectic model order reduction is a structure-preserving reduction technique for Hamiltonian systems. Apart from theoretical results like the preservation of stability, it has been demonstrated to give improved numerical results compared to classical MOR techniques. A key element in this procedure is the choice of a good symplectic reduced order basis (ROB). In our work, we introduce so-called canonizable Hamiltonian systems in energy coordinates. For such systems with the assumption of a periodic solution, we derive a globally optimal symplectic ROB in the sense of the proper symplectic decomposition (PSD). To this end, we show that the proper orthogonal decomposition (POD) of a canonizable Hamiltonian system is automatically symplectic from which we deduce optimality of the PSD. To verify our findings numerically, we consider a reproduction experiment for the linear wave equation.

Shifted symplectic reduction of derived critical loci

Shifted symplectic reduction of derived critical loci

Stratification of singular hyperkähler quotients

Abstract Hyperkähler quotients by non-free actions are typically singular, but are nevertheless partitioned into smooth hyperkähler manifolds. We show that these partitions are topological stratifications, in a strong sense. We also endow the quotients with global Poisson structures which recover the hyperkähler structures on the strata. Finally, we give a local model which shows that these quotients are locally isomorphic to linear complex-symplectic reductions in the GIT sense. These results can be thought of as the hyperkähler analogues of Sjamaar–Lerman’s theorems for singular symplectic reduction. They are based on a local normal form for the underlying complex-Hamiltonian manifold, which may be of independent interest.

A monotone Lagrangian casebook

We present an array of new calculations in Lagrangian Floer theory which demonstrate observations relating to symplectic reduction, grading periodicity, and the closed-open map. We also illustrate Perutz's symplectic Gysin sequence and the quilt theory of Wehrheim and Woodward.

Geometric hydrodynamics and infinite-dimensional Newton’s equations

We revisit the geodesic approach to ideal hydrodynamics and present a related geometric framework for Newton’s equations on groups of diffeomorphisms and spaces of probability densities. The latter setting is sufficiently general to include equations of compressible and incompressible fluid dynamics, magnetohydrodynamics, shallow water systems and equations of relativistic fluids. We illustrate this with a survey of selected examples, as well as with new results, using the tools of infinite-dimensional information geometry, optimal transport, the Madelung transform, and the formalism of symplectic and Poisson reduction.

Symplectic reduction of Yang-Mills theory with boundaries: from superselection sectors to edge modes, and back

I develop a theory of symplectic reduction that applies to bounded regions in electromagnetism and Yang--Mills theories. In this theory gauge-covariant superselection sectors for the electric flux through the boundary of the region play a central role: within such sectors, there exists a natural, canonically defined, symplectic structure for the reduced Yang--Mills theory. This symplectic structure does not require the inclusion of any new degrees of freedom. In the non-Abelian case, it also supports a family of Hamiltonian vector fields, which I call ``flux rotations,'' generated by smeared, Poisson-non-commutative, electric fluxes. Since the action of flux rotations affects the total energy of the system, I argue that flux rotations fail to be dynamical symmetries of Yang--Mills theory restricted to a region. I also consider the possibility of defining a symplectic structure on the union of all superselection sectors. This in turn requires including additional boundary degrees of freedom aka ``edge modes.'' However, I argue that a commonly used phase space extension by edge modes is inherently ambiguous and gauge-breaking.

Reduction of multisymplectic manifolds

We extend the Marsden-Weinstein-Meyer symplectic reduction theorem to the setting of multisymplectic manifolds. In this context, we investigate the dependence of the reduced space on the reduction parameters. With respect to a distinguished class of multisymplectic moment maps, an exact stationary phase approximation and nonabelian localization theorem are also obtained.

Reduction by Symmetries of Simple Hybrid Mechanical Systems

Abstract This paper investigates reduction by symmetries in simple hybrid mechanical systems, in particular, symplectic and Poisson reduction for simple hybrid Hamiltonian and Lagrangian systems. We give general conditions for whether it is possible to perform a symplectic reduction for simple hybrid Lagrangian system under a Lie group of symmetries and we also provide sufficient conditions for performing the reduction by symmetries of hybrid Poisson manifolds for simple hybrid Hamiltonian systems.

余切群胚的辛约化

余切群胚的辛约化

Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey

Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. Integrability results for few point-vortices on various domains is a vivid topic, with many results and techniques scattered in the literature. Here, we give a unified framework for proving integrability results for N=2, 3, or 4 point-vortices (and also more general Hamiltonian systems), based on symplectic reduction theory. The approach works on any two-dimensional manifold with a symmetry group; we illustrate it on the sphere, the plane, the hyperbolic plane, and the flat torus. A systematic study of integrability is prompted by advances in two-dimensional turbulence, bridging the long-time behaviour of 2D Euler equations with questions of point-vortex integrability. A gallery of solutions is given in the appendix.

Quantization of Dynamical Symplectic Reduction

A long-standing problem in quantum gravity and cosmology is the quantization of systems in which time evolution is generated by a constraint that must vanish on solutions. Here, an algebraic formulation of this problem is presented, together with new structures and results, which show that specific conditions need to be satisfied in order for well-defined evolution to be possible.